An Alternative Gyroscope Calibration Methodology

Transcription

1 An Alternative Gyroscope Calibration Methodology by Jan Abraham Francois du Plessis A thesis submitted in partial fulfilment for the degree of DOCTOR INGENERIAE in ELECTRICAL AND ELECTRONIC ENGINEERING SCIENCE FACULTY OF ENGINEERING AND THE BUILT ENVIRONMENT UNIVERSITY OF JOHANNESBURG Supervisor: Prof A Nel 1th August 213

2 Declaration TO WHOM IT MAY CONCERN This serves to confirm that I, Jan Abraham Francois du Plessis, ID number , Student number , enrolled for the Qualification D.Ing Electrical and Electronic Engineering Faculty: Engineerinng and the Built Environment. I hereby declare that the thesis submitted for the Doctor in Engineering degree at the University of Johannesburg, apart from the help recognized, is my own work and has not previously been submitted to another university or institution of higher education for a degree. Signed at Auckland Park, Johannesburg on this 1 th day of August 213. Signiture: Name: Jan Abraham Francois du Plessis 2

3 Abstract The objective of the research performed in this thesis is to address the calibration process of Fiber-Optic Gyroscopes (FOGs) a class of gyroscopes that make use of the Sagnac effect to determine rotational information from laser-light traveling in an optical fiber. The calibration process has traditionally been a time-consuming and therefore an expensive one due to the various environmental parameters that can influence the sensor under operation. Calibration is not a step that can be neglected as it is the process whereby the residual manufacturing errors in the sensor are characterized. If these measurement errors are not eliminated, the sensors would result in the host vehicle s assumed position rapidly diverging from its true position. Once the errors are characterized, they can be removed from the sensor output to improve the accuracy of the complete navigation system. The class of the sensor is determined by the amount of residual errors and the smaller the residual errors, the more expensive the sensor. The specific focus of the study is to determine whether it is possible to reduce the calibration cost of the Fiber-Optic Gyroscopes through the use of innovative calibration strategies. The use of neural networks are investigated as an alternative to the traditional calibration strategies which consists of the estimation of the constant error parameters through stochastic estimation strategies such as Kalman filters. The whole calibration problem is recast into the well-defined Systems Identification (SID) domain where the whole calibration problem is considered in terms of the systems identification design steps. The main contributions presented in this study are that the traditional calibration strategy is reviewed by casting the calibration problem into the Systems Identification domain; that a unified FOG error model is developed that combines a number of seemingly contradictory error models available in the technical literature; that computational intelligence techniques are used to perform gyro calibration; that a novel, non-linear gyro calibration strategy is developed; and i

4 ii that the sensors are calibrated under the simultaneous dynamic excitation of the full range of multi-dimensional environmental conditions. In the process of the development of this new calibration strategy the need for a problemspecific Criterion of Fit was observed. Such a Criterion of Fit was therefore developed and it acted as the core criterium whereby the accuracy of the new calibration strategy was assessed. One of the most important results obtained from the research presented in this thesis is that the new strategy significantly outperforms the traditional strategies and that, with the availability of high-performance embedded computational platforms, it has potential to be used within an operational environment as the gyro compensation strategy of choice.

5 Acknowledgements I want to acknowledge the following people and parties for their involvement in this research project. Firstly, Prof. André Nel, my supervisor, for his persistence and support. If it was not for everything that was chasing me I would have been finished a long time ago. Thank you for guiding me through the process of learning what a doctorate is and not dropping me along the way. To Friedl Swanepoel at Incomar in Centurion for allowing me to use the data from their sensors to perform this study. A significant amount of the supporting work was also performed during the time that I spent on their projects. I also learned almost everything that I know about navigation from you. Thank you. To André Steenekamp and Carl Havenga at Denel Dynamics for their guidance, interest and assistance through the project. You helped me to not get lost in this difficult field. To Prof. Jan du Plessis, who supported me through my early post-graduate studies when I did not believe that I could do it. My grandfather, who will never read this, who believed in me and saw something that I did not know about. Before he died, he made sure that I could study for as long as I wanted to. I wish I could say thank you. To my wife and my children who were there throughout this lengthy process and through the storms that accompanied it. I know it was not easy, but without you I would never have made it. We will reap benefits from this and it was worth it. iii

6 Contents 1 Introduction Background Calibration Motivation for IMU Calibration FOG Error Modelling Problem Statement Research Objectives Hypothesis Postulate Contributions Scope Research Methodology Document Layout Literature Survey Introduction Traditional Calibration Strategies Calibration Parameter Non-linearities Temperature Stabilization Other Environmental Conditions A Fundamental Perspective on Calibration Alternative Calibration Strategies Deterministic and Estimation Based Approaches Computational Intelligence Based Approaches What has not been done? Calibration as a System Identification Problem Calibration Experiment iv

7 CONTENTS v Model Selection Criterion of Fit Parameter identification Conclusion Unified FOG Sensor Model Introduction Unified FOG Sensor Model FOG Model Equation Standard Terminology Definition of Parameters Axes Definition Development of a Unified FOG Error Model Equation Development of a Gyro Simulation Environment Extending the Error Model An Empirical Simulation Model Misalignment Computations Axes Systems General Misalignment Components Gyro Misalignment Matrix Practical Insight Conclusion A New Calibration Method Introduction Calibration Strategy Model Selection Network Architecture Network Training Strategy Experimental Design Training Trajectory Test Trajectory Criterion of Fit Definition Background Argument for a New Criterion of Fit Practical Aspects of the New Criterion of Fit

8 CONTENTS vi 4.6 The Identification Strategy The New Calibration Methodology Conclusion Implementation and Analysis Introduction Aspects of Implementation Neural Network Implementation Experimental Data-sets Calibration Accuracy Interpretation Criterion of Fit Implementation Compensation Accuracy Analysis Noise-free Modeling Evaluation Accuracy Analysis Criterion of Fit Evaluation of Noise-free Network Conclusion and Discussion: Noise-free Training Modeling Evaluation in the Presence of Noise Basic Investigation into Noise-handling Ability Monte Carlo Analysis Investigation into the Training with Other Types of Noise Conclusion and Discussion: Training with Noise Conclusion Results and Discussions Introduction Specific Results Literature Study FOG Model A New Calibration Method Implementation and Analysis Results Summary Research Hypothesis Publications Contributions and Future Work Contributions

9 CONTENTS vii 7.2 Future Research Extensive Testing Recursive Real-time Calibration Low-cost Inertial Sensors A Rogers Rotation Matrix Computation Method 22 B Definitions of Terminology 24 C Test Trajectories 25 C.1 Temperature Model C.2 aps TurnAround48g C.3 aps vl C.4 aps UAVref data D INS Simulation 225 D.1 INS Model Overview D.2 INS Model Implementation E Test Definition 23 E.1 Introduction E.2 Equipment E.3 Measurement Strategy E.3.1 Temperature-Angular Rate Relationship E.3.2 Allan Variance Computation

10 List of Figures 1.1 Local geographic navigation frame mechanization of the strapdown INS equations as presented in Titterton and Weston [115] Current gyro applications as a function of the sensor performance as presented by Schmidt [94] Three axis rate and positioning table. This table is commonly referred to as a motion simulator FOG measurement error due to scale factor non-linearity [75] Gyro scale factor nonlinearities. Taken from Titterton and Weston [115] Graphical representation of the classical temperature calibration sequence The system identification process [65] Complementary filter as used in INS/GPS sensor data fusion for hybrid inertial navigation Drift rate definition breakdown presented in the IEEE Std [55] Gyro axes definition with misalignment angles Polynomial expression for temperature dependent scale factor error Polynomial expression for angular rate dependent scale factor error D surface description of the raw scale factor measurement data as a function of both the temperature and the angular rate D surface description of the raw scale factor measurement data as a function of both the temperature and the angular rate Temperature dependent drift data Theoretical contribution of the Shupe effect to changes in the FOG bias Raw gyro output for the computation of the bias and the standard deviation of the noise Allan variance plot of raw gyro output data Histogram of raw gyro output for bias computation viii

11 LIST OF FIGURES ix 3.12 Error surface used for the empirical error model IMU assembly diagram with proposed body axes overlay Overview of the components of the misalignment matrices Misalignment angle definitions between the CA and the body-axis Gyro casing with the axes for the misalignment computation between the input and the reference axes IMU assembly with the axes for the misalignment computation Radial basis function neural networks Demonstration of the modeling ability of an RBFN for a single dimensional problem axis rotation table (motion simulator) with environmental chamber mounted on the inner axis An example of a single-axis rate table and an indication of how it is mounted within an environmental chamber Single temperature scan training trajectory Sample locations for the single scan training trajectory Temperature and angular rate data for the single scan training trajectory Dual temperature scan training trajectory Sample locations for the dual scan training trajectory Temperature and angular rate data for the dual scan training trajectory Sample location for the linear up and down trajectory with 1 sample points and a NN scale of Sample location for the linear up and down trajectory with 1111 sample points and a NN scale of D view of the test trajectory consisting of 4 temperature cycles with 4 rate cycles per temperature cycle Test trajectory training data consisting of 4 temperature cycles with 4 rate cycles per temperature cycle Coverage of the functional space for the test trajectory consisting of 4 temperature cycles with 4 rate cycles per temperature cycle D view of the test trajectory consisting of 4 temperature cycles with 16 rate cycles per temperature cycle RBFN training and input data architecture RBFN training and input data architecture for the case where the network does not estimate the error, but the corrected rate

12 LIST OF FIGURES x 4.19 RBFN operational data architecture Functional space coverage by the various validation trajectories Functional space coverage by the various validation trajectories after the angular rates were clipped to fit the calibration space INS performance test strategy Reference error surface and polynomial-based error surface comparisons Training error histogram for the linear up and down trajectory with 1 sample points and a NN scale of Comparison between the estimated and reference error surfaces Network node placement for the linear up and down trajectory with 1 sample points and a NN scale of D error surface for the linear up and down trajectory with 1 sample points and a NN scale of Side view of the 3D error surface for the linear up and down trajectory with 1 sample points and a NN scale of Histogram of the error distribution onto the 3D error surface for the linear up and down trajectory with 1 sample points and a NN scale of Impact of parameter adjustments on the estimation accuracy Impact of parameter adjustments on the estimation accuracy for the network generated with the high-density training set High accuracy error surfaces used for the interpretation of the ridge-estimation anomaly Combined high accuracy error surfaces used for the interpretation of the ridge-estimation anomaly Criterion of Fit evaluation as a function of the network node width and the training trajectory for the short evaluation trajectory Criterion of Fit evaluation as a function of the network node width and the training trajectory for the medium evaluation trajectory Criterion of Fit evaluation as a function of the network node width and the training trajectory for the long evaluation trajectory Description of the Monte Carlo simulation strategy used Breakdown of the end-of-trajectory error into along-track and cross-track components Monte Carlo mean error analysis Monte Carlo error standard deviation analysis

13 LIST OF FIGURES xi 5.22 Monte Carlo mean error deviation expressed as a percentage deviation of the noiseless case Impact of the noise amplitude on the Criterion of Fit performance standard deviation Impact of the noise amplitude on the Criterion of Fit performance mean Impact of the noise amplitude on the Criterion of Fit performance mean as a function of the network node width Impact of the noise amplitude on the Criterion of Fit performance standard deviation as a function of the network node width Impact of the noise amplitude on the Criterion of Fit performance - normal plots Estimation error as a function of different scaling parameters when a randomwalk noise source with a correlation time of 1 seconds is included in the training data Estimation error as a function of different scaling parameters when a randomwalk noise source with a correlation time of 1 seconds is included in the training data Estimation error as a function of the inclusion of regularization when a random-walk noise source with a correlation time of 1 seconds is included in the training data Proposed recursive calibration procedure A.1 Rogers rotation matrix method C.1 Test trajectory aps TurnAround48g: angular rates C.2 Test trajectory aps TurnAround48g: acceleration C.3 Test trajectory aps TurnAround48g: Euler angles C.4 Test trajectory aps TurnAround48g: velocity C.5 Test trajectory aps TurnAround48g: position C.6 Test trajectory aps TurnAround48g: horizontal displacement C.7 Test trajectory aps vl668 : angular rates C.8 Test trajectory aps vl668 : acceleration C.9 Test trajectory aps vl668 : Euler angles C.1 Test trajectory aps vl668 : velocity C.11 Test trajectory aps vl668 : position

14 LIST OF FIGURES xii C.12 Test trajectory aps vl668 : horizontal displacement C.13 Test trajectory aps UAVref data: angular rates C.14 Test trajectory aps UAVref data: acceleration C.15 Test trajectory aps UAVref data: Euler angles C.16 Test trajectory aps UAVref data: velocity C.17 Test trajectory aps UAVref data: position C.18 Test trajectory aps UAVref data: horizontal displacement D.1 Local geographic navigation frame mechanization of the strapdown INS equations as presented in Titterton and Weston [115] D.2 Simulink implementation of the simulated INS D.3 Simulink implementation of the simulated INS E.1 Equipment setup for the gyro calibration

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16 LIST OF FIGURES xiv List of Acronyms 6-DOF AV CI CoF DCM DoE EKF FOG GPS IMU INS IEEE LMN LOLIMOT MC MEMS MFFN MLFNN NN OLS PPM RBF RBFN RBFNN RLG RW SID SLAM SVM UAV Six degrees of freedom Allan Variance Computational Intelligence Criterion of Fit Direction Cosine Matrix Design of Experiments Extended Kalman Filter Fiber Optic Gyroscope Global Positioning System Inertial Measurement Unit Inertial Navigation System Institute of Electrical and Electronics Engineers Local Modeling Networks Local Linear Modeling Trees Monte Carlo Micro-Electro-Mechanical Systems Multi-layer Feed-forward (Neural) Network Multi-layer Feed-forward Neural Network Neural Networks Orthogonal Least Squares Parts Per Million Radial Basis Functions Radial Basis Function (Neural) Network Radial Basis Function Neural Network Ring Laser Gyro Random Walk System Identification Simultaneous Localization and Mapping Support Vector Machines Unmanned Aerial Vehicle

17 Chapter 1 Introduction 1.1 Background The field of inertial navigation has become one of the most important enabling technologies in the aviation, aerospace and recently the robotics industries since the end of World War II. The concept of inertial navigation consists of the measurement of the motion of a moving vehicle using a set of inertial sensors and then processing these measurements to determine the position and orientation of the vehicle along its path of motion. The primary motion parameters that need to be measured are acceleration (which is measured using accelerometers) and the rate of angular rotation (which is measured with gyroscopes or commonly referred to as gyros). These sensors are usually combined into a single sensor pack known as an Inertial Measurement Unit (IMU) consisting of a combination of three orthogonal gyros and three orthogonal accelerometers. In recent years traditional mechanical rotating gyros have been widely replaced by Fiber Optic Gyroscopes (FOGs) and Vibrating Structure Gyros (VSGs commonly referred to as Micro-Electro-Mechanical System (MEMS) gyros) 1. These sensors are used in the place of their mechanical counterparts as there are no mechanical rotating or moving parts, they can achieve longer life, have higher reliability and have reduced cost of manufacturing. However, the problem with the new generation of sensors is that they exhibit measurement errors which are not as well-defined and deterministic as those that are present in the traditional mechanical instruments. Inertial sensors are classified in terms of their measurement accuracy. For gyroscopes, 1 Refer to Jekeli [57], Aggarwal [4] and Titterton and Weston [115] for a comprehensive discussion on the operation of and history of inertial sensors. 1

18 Chapter 1 Introduction 2 the primary classification parameter is the sensor drift, which refers to the positional error that will be experienced as a result of the sensor measurement errors. The drift rate of a gyro is defined in terms of the equivalent positional error along the equator after one hour of navigation (Titterton and Weston [115]). One degree of arc on the equator is equal to 111 km or roughly 6 nautical miles. A gyro that is specified as having a measurement drift of 1 degree per hour will therefore contain a residual measurement error that will result in a positional error of 111 km if used as part of a dead-reckoning Inertial Navigation System (INS). A dead-reckoning Inertial Navigation System (or a stand-alone INS) consists of the integration of the IMU outputs to compute velocity and then position without the aid of any additional sensors. The basic INS architecture is presented in figure 1.1. For an in-depth discussion of the implementation of an INS the reader is referred to Titterton and Weston [115]. In figure 1.1 the red block (the IMU) represents the triad of gyros and accelerometers and the compensation of the raw measurements. Both sets of measurements are performed in the body axis 2. The angular rate measurements from the gyros are processed in the Attitude Computer block to determine the attitude angles of the vehicle and these are used to transform the accelerometer measurements to the navigation axis. Following this transformation, the acceleration measurements are integrated twice to determine the velocity and position in the navigation axis. The various feedback loops in figure 1.1 represent the various perturbation forces that have an impact on the navigation computation. Based on the resulting positional drift, gyros are therefore classified as indicated in figure 1.2 (Schmidt [94]). It can be seen that the higher end performance of FOGs (listed as IFOG (Interferometric Fiber Optic Gyro) in the figure) is around.4 deg/hour with the lower end performance of the FOGs being in the region of more than 1 deg/hour. Scale factor performance for these sensors range from around 2 parts per million (PPM) for the better sensors to around 1 PPM for the less accurate sensors. These figures of merit refer to the post-calibration accuracy of the sensors. One of the key steps in the manufacturing of any sensor is the calibration thereof. In the case of gyroscopes the calibration process is so critical to the final sensor performance, that the cost of the laboratory calibration can sometimes be around 25% of the final sensor cost. In addition to the laboratory calibration during manufacturing, most high quality navigation sensors are subjected to a second round of calibration during operation. The 2 From there the name strapdown navigation, since the sensors are strapped onto the body of the vehicle.

19 Chapter 1 Introduction 3 Local gravity vector Gravity computer Position information IMU Coriolis correction Navigation computer Body mounted accelerometers f b Raw accelerometer compensation f b Resolution of specific force measurements f n Σ Σ x n x n x n Position and velocity estimates Body mounted gyroscopes ω b ib Raw gyro compensation ω b ib C n b Attitude computer Initial position and velocity estimates Transport rate ω n ie + ω n en Initial attitude estimates Complete INS Figure 1.1: Local geographic navigation frame mechanization of the strapdown INS equations as presented in Titterton and Weston [115]. in-operation calibration is usually aimed at addressing the time varying error parameters in the system 3 while the laboratory calibration is aimed at the characterization of the deterministic components of the measurement errors. The general method that is followed in the calibration of high quality gyros is a tedious and therefore expensive process. It consists of the mounting of the sensors (as part of a complete IMU) on a three-axis rotation table (also known as a motion simulator) as shown in figure 1.3, and performing a number of predetermined rotations under a number of fixed environmental operating temperature points. Very high accuracy (and therefore very expensive) rotation tables are required for the calibration of the tactical and navigation grade sensors and the environmental (usually temperature) testing is usually performed within an environmental test chamber that is mounted within the gimbals of the rotation table. 3 Unless specifically indicated otherwise, the term calibration in this document will always refer to the laboratory calibration of the sensor.

20 Chapter 1 Introduction 4 Figure 1.2: Current gyro applications as a function of the sensor performance as presented by Schmidt [94]. 1.2 Calibration Motivation for IMU Calibration As listed by Titterton and Weston [115], typical instrument electronics blocks contained within the support electronics block of an IMU are instrument power supplies; re-balance loop electronics (for some types of gyroscopes); temperature monitoring and control electronics; instrument compensation processing; analogue-to-digital conversion electronics; output interface conditioning; and built-in-test (BIT) facility.

21 Chapter 1 Introduction 5 Figure 1.3: Three axis rate and positioning table. This table is commonly referred to as a motion simulator. This study will focus on the instrument compensation processing block that is contained within the IMU. This block contains a micro-processor that... enables some form of on-line compensation of the instrument outputs to be performed based on instrument characterization data obtained during laboratory or production testing. Since such computing tasks are very specific to the type of instrument used, they may well be implemented here rather than as part of the subsequent attitude and navigation processing (Titterton and Weston [115]). It is the objective of this study to investigate the methods used to obtain the instrument characterization data that is referred to here. A thorough discussion of what exactly this entails and how it is determined will follow.

22 Chapter 1 Introduction FOG Error Modelling Ring laser and fiber optic gyroscopes started to emerge as viable replacements for traditional mechanical gyros around 197. The field of inertial navigation sensors in the period from approximately 192 to 197, when these sensors appeared on the scene, was dominated by mechanical gyroscopes. Mechanical gyros are relatively low bandwidth sensors that can be described by a deterministic transfer function. Unlike the mechanical gyros, the ring-laser gyros and FOGs are known for their noisy behaviour, thereby significantly deteriorating the measurement accuracy of these devices. The stochastic nature of these sensors can be attributed to their high sensing bandwidth, to unstable effects in the sensor electronics and to the impact that environmental changes have on the sensor electronics. Similar to the mechanical gyros, it is generally accepted that there are three deterministic parameters that need to be determined to properly characterize the errors in the measured output of a FOG. These terms are the bias, the misalignment matrix and the scale factor matrix. These parameters will be referred to as the primary calibration parameters. As in indicated by Gebre-Egziabher[36] the FOG measurement equation is generally described by the expression where ω m = (1 + S + M) ω t + B(t) (1.1)

23 Chapter 1 Introduction 7 S is the gyro scale-factor error matrix S x = S y S z M g is the misalignment error matrix M g12 M g13 = M g21 M g23 M g31 M g32 ω t is the input angular rate vector describing the physical angular rate = ω xt ω yt, and ω zt ω m is the noise-corrupted angular rate vector that is measured by the sensor = ω xm ω ym ω zm The misalignment matrix describes the coupling of motion in the other two orthogonal system axis into the particular measurement axis and is presented as a matrix that describes the angular misalignment between the gyro sensor axis and the sensor measurement axis. Small angle assumptions are made for the determination of this matrix. These assumptions can be considered valid as the mechanical alignment of the measurement axis and the sensor axis are usually performed to within a couple of milli-radians. The scale factor matrix describes the measurement error due to erroneous scale factor terms (converting the measurement signal to angular rate) used in the sensor. For mechanical gyros, the calibration process consists of the determination of these components in a rather straight-forward manner. However, as noted before, the difference between mechanical gyros and FOGs is that the modern sensors exhibit significant non-deterministic error components as part of the measurements. These error components need to be modeled and characterized during calibration in an attempt to improve the FOG measurement accuracy during operation. As noted by Rogers [86], the misalignment and scale factor error terms are often

24 Chapter 1 Introduction 8 combined into a single matrix where the scale factor terms appear on the diagonal and the misalignment terms appears on the off-diagonal positions of the matrix. In its simplest form, the bias is a constant measurement offset that is present in the angular measurement of the gyro. The term B(t) in equation (1.1) presents the measurement bias, which can be further expanded into the expression where B(t) = B + B tt (t) + B noise (t) (1.2) B B tt (t) B noise (t) is the constant, non-time-varying measurement bias; is the time-varying turn-on-to-turn-on bias, that is constant for the duration of the sensor being powered, but has a different value following the next power-up of the sensor predominantly due to thermal effects; and is a time-varying noise component of the bias. The B term is a constant value (simply called the bias) that is normally used to refer to the constant measurement error in the gyro. This term is usually determined from the average value of the measurement error made by the gyro when no rotational rate is applied to the sensor. The term B noise (t) (sometimes simply called the drift), refers to the time-varying component of the bias. The sensor measurement noise can consist of various noise components such as quantization noise, angle random walk (modeled as a first order Gauss-Markov process), flicker noise and rate random walk (IEEE Std [54]). As explained in Borenstein et al. [75], this term is often used as the governing criteria when selecting a gyro as it describes the rate at which the sensor will degrade in units of either degrees per second or equivalently in degrees per hour. Some aspects of B tt (t) and B noise (t) is usually estimated during operation of the gyro using Kalman filtering, so it is usually not considered to be part of the calibration process due to the non-deterministic nature of these terms. Some confusion can be avoided by noting that the process of the online estimation of this variation is often also referred to as calibration. Equation (1.2) could in fact be extrapolated to describe additional error components in the scale factor and misalignment components as well, but such time-varying effects in

25 Chapter 1 Introduction 9 these terms are usually regarded as being insignificant in comparison to the bias errors and the environmental error components that are to be discussed subsequently. Environmental Conditions Apart from the stochastic error components discussed above, the environmental conditions within which the sensor operates have a significant impact on its measurement accuracy. The impact of the applied angular rate, environmental temperature, environmental temperature gradient (known as the Shupe effect), vibration and applied magnetic field strength need to be considered to improve the sensor accuracy. Changes in the applied angular rate generally do not reflect as a linear relationship in the measured angular rate. Generally referred to as parameter non-linearity, this effect is mostly observed in the bias and the scale factor components. Chung et al. [21, 75] present the curve in figure 1.4 which indicates the angular measurement error that is made by the FOG due to scale factor non-linearity 4. In this figure ω g is the angular measurement measured by the gyro and ɛ is the error between the measured value and the actual rate of the gyro as mounted on the rotation table. The range of input values that were applied to the gyro for the generation of this graph was 1 /s to +1 /s. Due to the importance of this parameter, it is explicitly introduced as part of the traditional calibration method. The approach followed was to fit a straight line through the data of the scale factor error curve as presented in figure 1.4, thereby ignoring the non-linear aspects of this error. If the non-linearity error does not exhibit a skew-symmetric nature as portrayed in figure 1.4, a separate value for positive and negative rotations is sometimes defined. 4 This curve is indicative of the non-linearity in the parameters. It differs from the curve that will be used later on in this document to model the rate dependent scale factor component as this curve typically represents a closed-loop FOG while the model for an open-loop FOG will be used to model this parameter in this document.

26 Chapter 1 Introduction 1 Figure 1.4: FOG measurement error due to scale factor non-linearity [75]. All of the primary calibration parameters are sensitive to temperature variations, usually exhibiting a non-linear relationship between the change in temperature and the variation in the calibration parameter. As mechanical gyros were extremely sensitive to thermal variations a calibration approach was developed where the complete set of calibration parameters for all three axis were recomputed for a number of stabilized temperature points within the valid operational thermal range of the sensor. This practice was continued when mechanical gyros were replaced with FOGs. The validity of this assumption needs to be revisited with respect to the Shupe effect [13] as it potentially has a dominant thermal effect on the system. The Shupe effect is observed when a temperature gradient is applied across the coil of the FOG. The error can be compensated for during manufacturing by following different techniques of winding the gyro coil, however it is usually not completely eliminated from the system response and this effect is not characterized during the calibration of the sensor using the classical calibration technique. Vibration impacts a FOG by deforming the coil, thereby impacting the scale factor and the bias of the sensor [115]. The impact of this potential source of error is usually addressed on a mechanical level by ensuring that the sensor is packaged as a mechanically stiff structure. Most gyros are subjected to vibration tests as part of the sensor qualification process, but to add active vibration sensing to the gyro for calibrating the vibratory response as part of the complete system environmental sensitivity is not always practical.

27 Chapter 1 Introduction 11 In principle one should be able to use the accelerometers in the IMU to obtain vibration information. Stray magnetic fields can impact the sensor measurement by interacting with the nonoptical components of the sensor. It can also affect the state of polarization of the light inside the sensor through the Faraday effect [115], thereby influencing the bias. This error is usually compensated for during sensor design by adding magnetic shielding to the system. It should be noted that all temperature, temperature gradient, vibration and magnetic influences on the sensor can be calibrated and compensated for during sensor operation if the environmental effect is measured on board the sensor. 1.3 Problem Statement As mentioned in section 1.1, the cost involved with the current methods of calibration is excessive due to the duration of the traditional calibration process. The duration of the calibration process is determined by the time that is needed to subject the sensor under calibration to the specified range of environmental conditions that is considered to be representative of the operational environment of such a sensor. As the FOGs are sensitive to changes in a number of environmental conditions, the sensor needs to be exposed to these environmental variations during the calibration process. Only a single environmental parameter is usually stimulated at a time, thereby significantly extending the time that is required for the system to be on the rotation table and in the environmental chamber. The actual extraction of the calibration compensation coefficients from the measured data is usually not a time-consuming process, but the time needed to properly expose the sensors to these environmental conditions is usually the dominant factor in the total calibration time. In addition to this, it must also be noted that the documented FOG calibration process has not been reviewed and published in recent years. Such a review of the calibration process is required to determine whether it is acceptable to calibrate FOGs with the same techniques as mechanical gyros. The problem to be addressed during this study can therefore be summarized in terms of the following points. A calibration methodology is required that will lead to the reduction of calibration time and cost and thereby lead to the reducing the overall sensor cost;

28 Chapter 1 Introduction 12 It is to be determined whether a calibration strategy can be defined that will result in a performance enhancement of the sensor; and It is desirable that any new calibration strategy should produce a dataset that can be interpreted by a domain expert and implemented on an embedded navigation processor as a sensor compensation algorithm. 1.4 Research Objectives Hypothesis The hypotheses that will be tested in this thesis are that it is possible to derive an alternative gyro calibration strategy; while retaining the inherent accuracy related to present calibration techniques; this strategy will address time constraints in present calibration strategies; and the new calibration strategy will still result in a dataset that can be interpreted by a domain expert Postulate It is postulated that the hypothesis for this thesis can be proven through the following argument: 1. The calibration task has traditionally been approached through a very deterministic (repeatable) algorithmic approach to the problem or using Kalman filters. Both these approaches was marked by the expert knowledge from a domain expert being used to develop a customized calibration procedure for a particular sensor. 2. It is observed that the calibration process can be considered to be a subset of the well-defined and structured system identification (SID) domain. (a) If the calibration problem is considered to be an SID problem, a wide range of predefined SID tools can be used to solve the calibration problem; (b) A number of constraints that form part of the traditional calibration approach can be reviewed or completely eliminated in the much broader perspective of addressing calibration as an SID problem;

29 Chapter 1 Introduction 13 (c) The research methodology for the development of a new calibration strategy can be aligned with the well-defined SID approach to problem solving; (d) In particular, it can be observed that calibration is actually a non-linear problem. This leads one to the fairly new field of non-linear SID where a number of new approaches are being developed to perform the characterization of nonlinear systems; and (e) One of the most significant tools in the non-linear SID field is the use of computational intelligence 5 methods such as neural networks, fuzzy logic and genetic algorithms to solve the problem. It is therefore postulated that, by using computational intelligence techniques from the field of non-linear SID, the gyro calibration problem can be solved in an innovative way Contributions The contributions in this thesis are as follows: 1. A review of the classical calibration approach through the casting of the calibration problem as an SID problem. 2. The development of a unified FOG error model and the development of an empirical FOG simulation model. The focus of the Unified FOG model is to bring together three different gyro error models and combine them into a single expression that is representative of all three. All assumptions were either mathematically justified or were referenced as being part of the field of navigation. This section presents one of the key contributions in the thesis since it unifies the field consisting of scattered gyro error presentations. 3. The misalignment computations section aims to present a single strategy for the computation of the misalignment error. It is presented in a way that has not been published anywhere else and therefore contains very little references since it is derived from first principles. Since the misalignment angles are usually determined 5 Throughout this research work the term Computational Intelligence is preferred over the traditional term of Artificial Intelligence as the focus is on the addition of intelligence to a system via computational means. There is not really anything artificial to this process it is the application of a well-defined, but complex, computational paradigm to a particular domain.

30 Chapter 1 Introduction 14 as a single value, this section aims to give the user more insight into the actual composition of these angles. 4. The use of computational intelligence techniques to perform the calibration of gyros. 5. The development of a novel non-linear gyro calibration strategy. 6. The calibration of the sensors under simultaneous dynamic excitation of the full range of multi-dimensional environmental conditions. 1.5 Scope Calibration of any instrument can be defined as the characterization of the repeatable (deterministic) measurement error within the instrument in such way that the knowledge obtained from the calibration process can be used to increase the measurement accuracy of the instrument. The focus of this study will be to determine the extent of the research that has been done on the calibration of FOGs. Throughout this study it will be seen that some researchers have looked into the inclusion of stochastic components into the calibration process, but since the essence of stochastic error components is their non-predictability, these components have to be excluded from the calibration process. The scope of the research presented here will therefore be on the improvement of the characterization process of the deterministic errors in FOGs in such a way that it can be used to improve the sensor measurement accuracy. The work presented in this study is based on a very specific fiber optic gyro under development at Incomar, a South African company that specializes in fiber optic and inertial sensors. The gyro is an open-loop configuration, which necessarily means that it will portray more pronounced system errors before compensation. The fact that the specific sensor is in an open-loop configuration should not prohibit the method develop in this thesis from being used to calibrate a closed-loop gyro or even an accelerometer. Although the error models of MEMS gyros portray similar characteristics to FOGs, only FOGs will be analyzed under the research presented here. It should be apparent how the developed calibration methodology could be extended to MEMS sensors, but a detailed development thereof does not fall within the scope of this project. The focus of this project is on the calibration of FOGs only and full IMUs or accelerometers will not be characterized as part of this research. Similar to MEMS gyros, it should be easy to extend the theory developed for the FOG calibration to be used for the calibration of accelerometers and complete IMUs.

31 Chapter 1 Introduction 15 Neural network or hybrid neural network architectures will be used to solve the problem presented. When using the neural networks, the focus will be on the selection of various architectures and training methodologies that can be applied to and that complement these network architectures. The focus will therefore be on using the existing methods as a set of tools to solve the calibration problem and not to develop new neural network architectures or training methodologies. The work presented in this document is aimed at graduate engineers with a background in electronic, mechatronic or aeronautical engineering. A basic knowledge of inertial navigation systems and statistical signal processing is assumed. It should be noted that the terms FOG, gyro and sensor will be used throughout this thesis as terms that can be interchangeably used to refer to a Fiber Optic Gyro. Unless explicitly stated, none of the work that is presented here will address any other type of gyroscopes such as MEMS, ring-laser or mechanical gyroscopes. 1.6 Research Methodology In general the process of calibration consists of both a theoretical and an experimental component. The first step in the calibration consists of the development of a mathematical description of the error present in the sensor measurement. In the second step the coefficients of the measurement error model are characterized by means of an experimental validation process. The focus of this project is not to advance the theory of FOG measurement error modeling, but rather on the development of a new experimental measurement technique. Similar to the calibration process which focuses on the measurements instead of the theory, the application of the neural networks to the calibration process will not focus on the development of theoretical extensions in neural network theory. The aim will be on the application of existing neural network theory to the calibration process and on determining the optimal (or at least a suitable) combination of neural network configuration and training strategies that can be used to solve the calibration problem. It should therefore be apparent that a research methodology focusing on experimental measurements should be followed during the execution of the research presented in this document. In addition to this, it is also required to use or develop criteria that can be used to determine the applicability of a particular solution to a problem and to distinguish between different solution architectures to find the one optimal to the particular solution. The field of system identification presents a well-defined and structured approach to

32 Chapter 1 Introduction 16 both the theoretical and experimental aspects of research. It could therefore be stated that an experimental research methodology will be followed that is strongly aligned with the system identification approach to problem solving. Refer to chapter 2 for a more in-depth discussion on the system identification method as applicable to the calibration problem. 1.7 Document Layout The structure of this thesis is as follows: Chapter 2 presents a literature survey that will address the current stance of research using the following steps. The classical calibration process is presented; The systems identification (SID) process is proposed as an alternative strategy for the development of a new calibration strategy; and The stance of research in the area of alternative strategies to the classical calibration process is determined and the research is classified in terms of its compliance to the SID process. A unified FOG error model and the development of an empirical FOG simulation model is presented in chapter 3. A new calibration methodology for FOGs is presented in chapter 4. Chapter 5 presents the implementation of the new calibration methodology and discusses the results of this implementation. A summary of the research results is presented in chapter 6 and the results are used to test the research hypothesis. A conclusion to this thesis is presented in chapter 7 where the research contributions are highlighted and recommendations are made for future research. The appendices are used to augment the work presented in the main text without distracting from the flow of the document. Appendix A presents a method for the computation of coordinate system transformation matrices in an easy and accurate way. The method presented in this

33 Chapter 1 Introduction 17 appendix is used in chapter 3 during the development of the unified gyro error model. Some definitions of terminology used throughout the thesis are presented in appendix B. The aircraft and missile trajectories used in the validation of the calibration process are presented in appendix C as a set of graphs presenting the various components of each trajectory. Appendix D presents the inertial navigation system (INS) simulation that was used to evaluate the new calibration strategy.

34 Chapter 2 Literature Survey 2.1 Introduction The hypothesis for this thesis is presented in section 1.4. The essence of the hypothesis is whether it is possible to develop a calibration method for inertial sensors that will fill the gap left by the current calibration strategies. This chapter will attempt to address this question through a literature survey. The field of calibration of inertial sensors is very wide and the term calibration is used for a number of different scenarios within the usage profile of inertial sensors. The first place where the term calibration is used is in the laboratory calibration of the static and dynamic system parameters of the inertial sensors. Calibration can also refer to the determination of these parameters while the sensors are exposed to operational conditions. In these cases the calibration procedure is called operational calibration. During operational calibration it is usually not only the sensor constants that are determined, but also the time-varying components of the system and some components of the system noise. The focus of this literature study will be on the laboratory calibration, but one could argue that there does not need to be a distinction drawn between laboratory and operational calibration as the laboratory calibration is simply a special case of the more general case of operational calibration. Even the operational calibration can be seen as a special case of the state estimation problem where the calibration parameters are simply defined as additional states of the dynamic system. The structure of this chapter will be as follows: The first step will be to look at the current calibration strategies and determine what the traditional approach to calibration entails. This will be followed by a survey of variations on the traditional method. 18

35 Chapter 2 Literature Survey 19 From these results the focus will shift to look at calibration from a fundamental perspective and ask whether it is possible to abstract the calibration problem. The question asked will be: What is calibration essentially? If we consider the fact that calibration consists of the identification of a suitable system model, the excitation of the system by a suitable excitation signal and the characterization of the model parameters, it becomes possible to consider calibration as an SID problem. The rest of the chapter will then look at calibration from the perspective of SID and will categorize previous work in terms of the SID steps. As a summary to the literature study the areas for research in the area of gyro calibration will be identified as an outflow of the literature survey. 2.2 Traditional Calibration Strategies A number of traditional gyro calibration methods are used with all of these methods portraying some slight variation from the other methods in use. Titterton and Weston [115], Rogers [86], Aggarwal et al. [4], Lawrence [62] and Chatfield [14] all present slight variations of the traditional gyro calibration approach. In all of these methods the primary calibration parameters as defined in section are determined as a function of static temperature and applied rotational rate. Thompson [112] presents an overview of Chatfield s work which includes a number of worked examples on the parameter estimation process of calibration. The traditional calibration method uses a 3-axis rotation table (such as the Acutronic [2] one shown in figure 1.3) or the rotation of the Earth to determine the calibration terms. For low dynamic applications the Earth s rotation can be used, but if the sensors are to be used in high dynamic rate applications, it is necessary to calibrate the sensors using the 3-axis rotation table to expand the dynamic range wherein the calibration results are applicable. If only a gyro cluster needs to be calibrated, a sequence of six orientations is needed to determine the misalignment/scale factor and the bias terms. If a complete IMU needs to be calibrated (gyros and accelerometers), a sequence of 18 orientations needs to be performed. The sequence of orientations for the respective scenarios is presented in Rogers [86], Chatfield [14], Aggarwal [4] and Lawrence [62]. There are two approaches that are generally followed to determine the calibration parameters from the data collected during the sequence of rotations performed on the rotation table. The method presented by Rogers [86] consists of a very deterministic

36 Chapter 2 Literature Survey 2 calculation of these parameters from the data through a series of sequential additions and subtractions of the measurements. The sensor is placed in a particular orientation under a fixed rotation rate and sampled for a couple of seconds. The measurements are sampled for some time (3 seconds per position is usually considered to be adequate) at a typical rate of 5Hz and averaged to remove the noise. Once the average of these measurements has been calculated, the various calibration parameters can be determined by following the specified sequence of calculations. Chatfield [14] presents a slightly different approach to calibration. It starts out from the same basis as the deterministic method followed by Rogers with the difference that the calibration parameters are determined using least squares fitting of the data instead of deterministic mathematical methods. This approach is often extended to include the use of Kalman filters to aid in the processing of the data and the estimation of the calibration parameters. Bekkeng [8] and Fountain [34] present Kalman filter formulations for use in inertial sensor calibration. Table 2.1 provides the orientations of the rotation table if the FOG is calibrated using rotations applied by the rotation table instead of the Earth s rate of rotation. If the rotation occurs for example at 1 /s (36 /h) we can ignore Earth rate coupling parameters as these parameters represent only.42% of the total range of the input rotation applied to the sensor 1. The abbreviations X, Y, Z in table 2.1 refers to the orientation of the body axis system. The abbreviations N, E, D refers to the North, East, Down navigation axis system. The Earth gravitational acceleration is defined by the parameter g. The Earth s rotation rate about the North axis is defined by Ω N, about the East axis by Ω E and about the Down axis by Ω D. Ω r refers to the rate of rotation of the rotation table on which the gyro is mounted for calibration. As can be seen from the accompanying diagrams, the table assumes a single axis rate table to rotate in a positive direction about the positive down axis Calibration Parameter Non-linearities Under the traditional calibration approach, the non-linear aspects of the scale factor and bias are usually included as part of the sensor characterization. As indicated by 1 The Earth s rate of rotation (Ω) is /h or.42 /s on the equator. At a particular location on the Earth the North component of the Earth s rate of rotation is Ω cos(lat), the South component Ω cos(lat), the Up component Ω sin(lat) and the Down component Ω sin(lat).

37 Chapter 2 Literature Survey 21 1 N X D Z Ω r E Y No. Position IMU Orientation X Y Z = N E D Gravity input f x f y f z = g Rotation Rate Input ω x ω y ω z = Ω N Ω E Ω D + Ω r Z X 2 Y N E X Y = N E f x f y = ω x ω y = Ω N Ω E D Ω r Z D f z g ω z Ω D Ω r X N 3 Z Ω r E X Y = N D f x f y = g ω x ω y = Ω N Ω D + Ω r D Z E f z ω z Ω E Y Y X 4 N E Z X Y = N D f x f y = g ω x ω y = Ω N Ω D Ω r D Ω r Z E f z ω z Ω E X Z 5 N E Y X Y = D E f x f y = g ω x ω y = Ω D Ω r Ω E D Ω r Z N f z ω z Ω N N 6 Ω r Z E Y X Y = D E f x f y = g ω x ω y = Ω D + Ω r Ω E D X Z N f z ω z Ω N Table 2.1: IMU reference orientations for FOG calibration on a rotation table.

38 Chapter 2 Literature Survey 22 Lawrence [62], Grewal et al. [4] and Titterton and Weston [115], the nonlinearities in the system errors include measurement non-linearity, asymmetry, hysteresis and dead-band. Dead-band is a common phenomenon with mechanical gyros due to the stiction of the bearings inside the gyro, but with FOGs one should not expect any form of dead-band due to the inherent difference in the operation of FOGs from mechanical gyros. Figure 2.1 (taken from Titterton and Weston [115]) indicates the nonlinearities present in a typical gyro. A similar but more formal description of the nonlinearities in a gyro is presented in the IEEE Standard for Inertial Sensor Terminology [55]. Figure 2.1: Gyro scale factor nonlinearities. Taken from Titterton and Weston [115]. These parameters are determined by sweeping the angular rate of the table through quite a large range of rates (the parameter Ω r in table 2.1) and temperatures while the orientation of the table is kept in a particular position as defined in table 2.1. The system is left to stabilize at a particular rate while data are sampled before the applied rate is changed. When the complete range under which the sensor needs to be calibrated for has been covered, the calibration parameters at each one of the measurement points are determined and the relationship between the calibration parameter and the rotational rate or temperature is described in terms of a polynomial function determined from fitting a curve to the measured data. This approach is described in a number of references such as Titterton and Weston [115] and Chung et al. [75] and was followed to determine the data presented in figure 1.4. It should be apparent that one would need multiple polynomials

39 Chapter 2 Literature Survey 23 to accurately characterize all the nonlinear aspects of a particular sensor. Although identified as part of the sensor characterization, hysteresis is not usually included as part of the compensation algorithm, but Gulmammadov [41] presents a method for the compensation of temperature hysteresis effects in MEMS gyros. Lai and Crassidis [61] describe a way to compensate for the asymmetric components of the scale factor by making use of the direction of rotation Temperature Stabilization As mentioned in section 1.2.2, an important aspect of the calibration process is the temperature dependency of the gyro calibration parameters, specifically the bias and scale factor. Traditional mechanical gyros were extremely sensitive to thermal variations. The traditional approach consists of the gyros being calibrated at the stabilized temperatures in the range [ 6 C to +8 C] in increments of 2 C. This approach is often referred to as a temperature soak test (Titterton and Weston [115], Aggarwal et al. [4]) and is the most frequently used strategy for gyro thermal calibration. In some cases a thermal ramp test is also used to perform the temperature calibration (Titterton and Weston [115], Aggarwal et al. [4], Stave [16]). Aggarwal et al. [4] indicates that care needs to be taken when a ramp test is performed as the temperature in the environmental chamber is not necessarily the same as that inside the gyro casing. Stave [16] recorded good results from following a temperature ramp test compared to the soak test with the most important improvement the reduction in the overall calibration time. To perform a thermal soak test, it is required that an environmental chamber be fixed within the gimbals of the rotation table and that the temperature stabilize to within.5 C of the final value before measurements would commence. Once the temperature has stabilized, the rotation table would be taken through the complete set of rotations as presented in table 2.1. A typical temperature cycle for temperature calibration is graphically presented in figure 2.2. The rotational sequence is executed at the horizontal (constant temperature) sections of the temperature cycle. The following example should help to put this procedure in context. If the temperature takes 3 minute to stabilize following an adjustment in the reference value, the whole temperature stabilization sequence will take 3 hours to complete. This time must be added to the time needed to complete the 6 rotations (18 for a full IMU) per temperature measurement point, thereby highlighting the fact that the calibration of a single IMU is a rather time-consuming exercise. From the set of measurements taken at the various temperature measurement points

40 Chapter 2 Literature Survey 24 Temperature Increasing temperature Decreasing temperature Wait for temperature to settle before rotational sequence commences Time Figure 2.2: Graphical representation of the classical temperature calibration sequence. used (6 in this case), 6 sets of temperature sensitive calibration parameters are determined. Polynomial approximations that are based on the measured data are usually used during operation of the sensor to dynamically determine the specific value of the calibration parameter needed at that time. Aspects of temperature calibration that are not often addressed are the impact of temperature rate and time-dependent temperature rate on the gyro bias. The IEEE FOG Standard [54] suggests that these parameters be characterized, but they are seldom included as part of the online sensor compensation algorithm as the exact impact may be difficult to quantify or the sensors needed to detect the environmental conditions are not built into the gyro assembly Other Environmental Conditions The IEEE FOG Standard [54] also indicates that apart from temperature and angular rate, one needs to characterize the sensors for the impact of voltage changes in the supply electronics, vibration, magnetism and radiation. The impact of magnetism through the Faraday effect is one of the key differences between mechanical and optical gyros. As mentioned in the previous paragraph, one can characterize the impact of some environmental condition on the sensor, but it is not always feasible to correct for the particular impact as the sensors needed to measure the particular environmental condition cannot always be installed as part of the gyro assembly. Supply voltage can be measured directly and vibration can be detected using the accelerometers in an IMU configuration. Magnetism is usually shielded using some form of magnetic shielding, but a magnetic sensor can be

41 Chapter 2 Literature Survey 25 installed inside the gyro casing. Radiation plays an important role for space applications and one could characterize and sense for this condition as well. Apart from the possibility to measure and compensate for all these environmental conditions, compensation usually excludes these components due to tradition, the insignificant contribution towards the overall measurement error or the lack of sufficient computational power. 2.3 A Fundamental Perspective on Calibration The objective of this section is to revisit the calibration problem and critically evaluate the problem against the available scientific approaches. By taking a very broad and fundamental look at the calibration problem, it is seen that it can be considered to be a very specific application of system identification (SID) and that a new range of options opens up when calibration is viewed to be a (potentially non-linear) SID problem. Lennart Ljung opens his authoritative text on System Identification [65] with the following statement: Inferring models from observations and studying their properties is really what science is about. The models ( hypotheses, laws of nature, paradigms, etc.) may be of more or less formal character, but they have the basic feature that they attempt to link observations together into some pattern. System identification deals with the problem of building mathematical models of dynamical systems based on observed data from the system. The subject is thus part of basic scientific methodology, and since dynamical systems are abundant in our environment, the techniques of system identification have a wide application area. From this statement one could argue that the field of SID to a large extent encapsulates the engineering fields of systems modeling, characterization and calibration. An overview of the SID field can be obtained when one considers figure 2.3 which presents a description of the SID process as presented by Ljung [65]. The SID steps are defined as: Step 1: From the inspection of figure 2.3, it can be seen that the first step is to design an experiment that will generate data that will be maximally informative (Ljung [65]), meaning that the data has a dynamic richness such that the system can be characterized from only the measurements.

42 Chapter 2 Literature Survey 26 Prior knowledge Experiment design Gather data Choose Model Set Choose Criterion of Fit Calculate Model Validate model Not OK: Revise OK: Use it! Figure 2.3: The system identification process [65]. Step 2: The second step is usually to define a set of models that could potentially describe the system. The types of models can be described in three categories being black box models, gray box models and white box models (Jategaonkar [56]). In the case of black box models, the model parameters do not have any physical significance and are only considered to be adjustment parameters in an attempt to obtain a good model fit to the data. In the gray box modeling approach, the adjustable parameters do have physical meaning and the identified model can be evaluated

43 Chapter 2 Literature Survey 27 through inspection of the parameters that was identified as part of the SID process. The classical calibration approach is a white box modeling approach where the system description is determined from a fundamental physical analysis and the characterization steps are used to determine the parameters of the physical system. Step 3: In the third step the criterion that will be used to define the goodness of fit of any particular model to the data is defined. Step 4: The fourth step is the gathering of the data from which the system description will be obtained. Step 5: Step 5 consists of the process where the best model is computed from the measured data and is referred to as the actual system identification step. Step 6: Once the model is computed from the data, the evaluation criterion is used to determine whether the various models are suitable system descriptions. If the model appears to be a good system description, it can be used for the purpose for which the model was required, if not, the SID process needs to go through another iteration by adjusting some of the parameters in the process. Based on this diagram, one can see that the use of prior knowledge plays an important role in SID. Such knowledge is used to define the experimental design phase, the type of model that will be used to describe the system, the evaluation criterion and the method that will be used to compute the system model. In chapter 2 of Ruano [88] the following statement is made: The main goal of system identification is to determine models from experimental data. This quote in conjunction with the earlier quote from Ljung describes the core idea on which this thesis is built, being that calibration is a subset of the larger and much more comprehensive SID problem. If one compares this discussion of the SID process with the discussion of the classical approach to the calibration problem that was presented in chapter 1.2, it can be seen that the classical approach to calibration can be considered to be a simplification of the SID process. Calibration consists of the following steps: 1. Obtain a model that will adequately describe the errors in the sensor output measurements.

44 Chapter 2 Literature Survey Design an experiment that can be used to stimulate the sensor in such a way that it will generate output data over a suitable dynamic range. 3. Execute this experiment to generate the sensor data. 4. Analyze the experimental data to determine the coefficients of the sensor model. These steps can be mapped onto the SID process defined in figure 2.3 to support the argument that calibration is an SID subset. During the development of the classical calibration approach prior system expert knowledge was used extensively to define the calibration experiment, the model set that was used to describe the sensor measurement errors and the criterion of fit. The criterion of fit is usually not explicitly defined for the classical approach, but is usually considered to be implicit to the calibration process as a kind of as accurate as possible criterion. One needs to possibly question the applicability of these expert assumptions to modern gyros and in particular regarding FOGs as the classical approach was developed with a different type of sensor and different operational environment in focus. Especially the criterion of fit needs to be revisited as it was not explicitly defined for the classical approach that was based on the calibration of low-noise mechanical gyros. If the SID process is used to develop a new calibration strategy, a range of tools and techniques are available that could be used to aid in the development of an improved calibration strategy for FOGs. The components from the SID field that could be used in the development of a new FOG (or any inertial sensor) calibration algorithm are Efficient experimental design using guidelines such as the ones proposed by Ljung [65] in chapter 13 of his book. The use of non-linear SID strategies. Higher complexity calibration models that could capture multi-dimensional system characteristics. Gray-box strategies that will allow for the inclusion of prior knowledge into the SID process and the characterization of the systems in term of interpretable system parameters. These topics will be discussed throughout the rest of this literature survey using the blocks from the SID process presented in figure 2.3 as these are the components that are considered to constitute a comprehensive view on calibration. The steps to be investigated are the

45 Chapter 2 Literature Survey 29 design of the calibration experiment, which included the gathering of the data; model selection; criterion of fit; and parameter identification strategies. The questions that need to be answered during the development of a calibration strategy are therefore: What approaches (identification algorithms) can be followed to determine the model parameters? How should the excitation signal look? How should the model look? Can the current models handle or represent the high complexity required from an improved calibration strategy? How can the calibration equipment be used in an efficient way through an alternative calibration strategy? As a final comment on this topic, it should be noted that inertial sensor laboratory calibration has traditionally been seen as the process whereby constants are determined within a static system. This idea is slightly modified for the operational calibration where the system parameters are estimated as part of the state estimation operation. The SID process usually deals with the estimation of the parameters of a dynamic system. If the inertial sensors are seen as being part of a dynamic system (which is sometimes static, such as during the laboratory calibration) the more comprehensive viewpoint of calibration as a state estimation problem results in the same tools being usable for laboratory and operational calibration. As mentioned before, one could even take it further and consider calibration as the more generic sensor fusion problem. This is true as calibration makes use of different sensor measurements to determine the system errors with data fusion combining the measurement form different sensors to improve the overall system measurement. From this perspective one could then see that the same algorithmic structures could be used for laboratory calibration, operational calibration, alignment and sensor fusion and in all of these the principles of SID could be used to optimize the performance of the particular algorithmic implementation.

46 Chapter 2 Literature Survey Alternative Calibration Strategies It is important to determine what variations have been proposed on the classical calibration method in the literature. The standard method and Kalman filter implementations discussed earlier will be taken as the norm and variations on these methods will be discussed. A number of alternative modeling/calibration approaches have been published in the technical literature. The areas that have received the most attention from the research fraternity are improvements to the gyro error modeling, the use of alternative techniques (including neural networks) to estimate the gyro errors and different methods that could be used to reduce the random noise present in the measured gyro output data. Due to the integrated nature in which these topics appear within the following publications, it will not necessarily be possible to group the various publications together by topical groups. The literature survey will therefore be performed and a topical analysis will be performed thereafter. If the Allan variance plots (as presented by Gebre-Egziabher [36]) of the FOGs and MEMS based gyroscopes are analyzed, it can be seen that there is a close similarity between the noise characteristics of these two types of sensors. Although the two types of sensors are based on completely different physical processes, the similarity between the two types of gyros can be exploited on the level of the system calibration. The last decade has seen a significant number of publications on the calibration of MEMS gyros and the integration of MEMS-based IMUs with other navigation sensors such as Global Positioning System (GPS). During the rest of this literature survey attention will be given to the publications on MEMS sensors as well in the hope that some of the innovative approaches published in recent times could be adapted to the calibration of FOGs. To have a proper review of the laboratory calibration, it is worth the effort to look at the operational calibration strategies as well. The objective being that these strategies should perhaps provide a broader picture from which new ideas could be extracted. The operational strategies considered will therefore include aircraft and spacecraft applications Deterministic and Estimation Based Approaches McConley[66] addressed gyro calibration from the perspective of Systems Identification, but by making use of the maximum likelihood systems identification (MLSI) strategy presented by Maybeck [92] he effectively increased the complexity of the solution by having

47 Chapter 2 Literature Survey 31 to compute the partial derivative matrices used for the implementation of the extended Kalman filter. Although McConley makes use of a system identification algorithm he does not employ the full system identification methodology to improve the calibration strategy. His use of the system identification concept is limited to the estimation of the constant components of a non-linear model equation. As part of his research McConley indicates the importance of the use of adequate motion profiles to ensure the observability of all the non-linear parameters, but he does not present a detailed discussion of these profiles. Thompson [113] also makes use of a systems identification approach to calibration by making use of the Full Information Maximum Likelihood Optimal Filtering (FIMLOF) approach to perform IMU noise parameter calibration. He started to apply system identification theory to the calibration process, but predominantly focussed on the estimation algorithm. No attention was paid to the design of the excitation signal using systems identification principles. Musoff [68] presents an argument for a review of the classical calibration procedure of IMUs to reduce the calibration time and thereby reduce the overall system cost. He focussed on mechanical gyros and used a Kalman filter based calibration strategy. Although his ideas are older than 3 years, some of it was never fully absorbed into the main stream of navigation research and some of his other ideas are still the prevalent techniques used even today. One of his main contributions was the development of a calibration technique which did not require the use of expensive, high accuracy test equipment. This is accomplished by using the measured and known values of the gravity and Earth rates at the calibration location to solve a set of simultaneous equations. Another key aspect to his approach was to do a very basic calibration (more a functional performance test) of the sensors before integration into an IMU and then spending more time calibrating on the level of the IMU. Lintereur [64] looked into the generation of optimal trajectories for the calibration of inertial sensors on rotational tables that will maximize the observability of the inertial instrument errors. Whereas Musoff [68] referred very positively towards the decoupling of the various parameters through a decoupling trajectory, Lintereur proposes trajectories which will make more than one parameter at a time observable. He also discusses the concept of static and dynamic errors for gyros and accelerometers. Static errors are those that can be determined without dynamic motion or that are corrupted by dynamic motion and dynamic errors are the ones which only become observable once the sensor is experiencing dynamic motion on a rotational table. He followed a linear approach to the calibration of the dynamic error and proposed the application of a control signal to

48 Chapter 2 Literature Survey 32 the gimbals of the test platform to minimize the calibration cost function and therefore optimize observability of the calibration parameters. Grewal et al. [39] also discuss the development of optimal excitation trajectories to calibrate an IMU. Although they do not provide any specifics regarding the trajectory, a set of 6 references is provided for the computation of the optimal signal. Interestingly enough, all these references are from the field of system identification. They generated parameter excitation trajectories that maximize the information matrix and made use of a state estimation instead of a parameter estimation strategy. Their calibration strategy was based on the use of a dual extended Kalman filter (EKF). Only the standard, constant calibration parameters was determined as part of the calibration. Guo et al. [42] simultaneously used temperature and measurement voltage information to estimate the temperature dependent bias and scale factor dependent errors. They used a statistical software package to estimate a multi-factor linear regression model that would fit the data. Good results were obtained from their calibration strategy. Bekkeng [8] presents a calibration strategy for MEMS sensors that uses a single-axis rate table and that uses a ramp profile to perform the scale factor characterization. He makes use of a Kalman filter to perform the calibration. An interesting observation that is reported is the reduction of the noise in the system when the temperature compensation is implemented. No explanation for this observation is provided. A very innovative approach to inertial sensor calibration is presented by Hwangbo and Kanade [53]. They present a factorization-based calibration method from the field of computer vision to perform the calibration. The results from this method appear promising, but not really conclusive since the presented results were conceptual in nature without a true analysis of the impact of the method on the resulting navigation performance of the system. Gulmammadov [41] presents a method for the compensation of temperature hysteresis effects in MEMS gyros. His approach essentially consists of the identification of two different bias-temperature relationships: one for increasing temperature and one for decreasing temperature. His method is used to effectively eliminate the impact of temperature variations on the bias of the sensor. In addition to the alternative approaches to calibration presented up to this point, a number of publications presenting modifications to mostly MEMS sensor calibration that focus on the adjustment of the data measurement profile, modifications to the estimation algorithms or the elimination of the need for expensive test equipment from the calibration process have been published. These publications include Bo and Feng [1], Buschmann

49 Chapter 2 Literature Survey 33 et al. [13], Fong et al. [33], Han et al. [44], Jiancheng [58], Jurman et al. [59], Li et al. [63], Li et al. [18], Reddy and Herr [84], Shcheglov [99], Shin [11], Skog and Händel [14], Syed et al.[11, 3], Ojeda et al.[75], Zhang et al. [126] and Zhu [128]. Although some significant breakthroughs in the area of calibration was published by these authors, the focus of the work is on small modifications to the classical deterministic or estimation based calibration methods. The impact of these publications on the current research is therefore not considered relevant enough to justify lengthy discussions on the content of these publications within this dissertation Computational Intelligence Based Approaches A number of implementations of neural networks for the estimation of the FOG errors have been published. These implementations range from the pure estimation of the gyro errors without any real application of the technique, to partial calibration implementations to the integration of Global Positioning System (GPS) data with the Inertial Navigation System (INS) data. Zhu et al. [127] indicated that a Radial Basis Function Neural Network [47, 16] (RBFN) could be used to estimate the bias drift component due to temperature variation if the noise in the signal can be reduced. They effectively used a series-single-layer network to act as a low pass filter that was situated in front of the RBFN. Hongwei et al. [5] followed a similar approach using a Projection Pursuit Learning Network for modeling the temperature drift of the FOG. They also indicated the importance of smoothing the training data supplied to the network. Arguably the best smoothing results were obtained by El-Sheimy et al. [28] and by Sharaf and Noureldin [98] where they introduced the Wavelet Multi-Resolution Analysis (WMRA) approach to remove the high-frequency noise from the gyro measurements. Sharaf and Noureldin [98] combined the WMRA method with an RBFN in a complementary filter architecture to estimate the random walk noise in a non-moving FOG mounted in a fixed position. From the results presented, their noise reduction strategy proved to be extremely effective. Nassar et al. [72, 71, 7] used the autocorrelation sequence (ACS), the numeric equivalent of the autocorrelation function (ACF) to indicate that the practice of representing the INS sensor errors by a first order Gauss-Markov model is not always accurate. Although they indicate in Nassar et al. [71] that the ACS is not completely accurate when applied to the measured output data of the gyro, the actual stochastic part of the INS modeling errors still needs to be presented by a higher order Gauss-Markov model. They therefore used a combination of wavelet de-noising (such as that presented by Sharaf and Nourel-

50 Chapter 2 Literature Survey 34 din [98]) and Autoregressive (AR) modeling to estimate the INS position error drift. The best results was obtained when 3 rd order AR models were applied to lower accuracy gyros (1 deg/h) rather than to the higher accuracy systems. Accuracy improvements of up to 36% were obtained. It was also found that an increase in the order of the AR model beyond 3 rd order usually resulted in reduced system performance, most probably due to divergence in the Kalman filter that was used for the parameter estimation. Fan et al. [29] used the RBFN in a novel way to model the temperature drift in a FOG. They integrated the grey Accumulated Generating Operation (AGO) into the RBFN processing of the data. The main advantage of the grey AGO is that it reduces the randomness in the training data, thereby increasing the modeling accuracy of the RBFN. During training and implementation the grey AGO is applied to the raw data before it is supplied to the RBFN and the output of the RBFN is processed using the grey Inverse AGO (IAGO). They call their system the Grey RBFN (GRBFN). It is found that the resulting GRBFN has a much shorter training time than the normal RBFN and that it exhibits significant improvements in modeling accuracy over the normal RBFN. The RBFN structure used by these authors is different from that presented by Sharaf and Noureldin [98] in that they make use of an on-line training strategy where they present the last 5 temperature drift measurements and the last 5 environmental temperature measurements as input to the system with the required output being the predicted temperature drift value at the next time step. RBFNs and traditional multi-layer perceptrons trained by back-propagation have been used extensively to perform sensor fusion and sensor error estimation in the area of aided inertial navigation. Sharaf et al. [97], Sharaf and Noureldin [96], Noureldin et al. [74] and Semeniuk and Noureldin [95] all present applications where neural networks were used to estimate the navigation errors in INS/GPS integration schemes. In all these applications the neural network is trained to replace the navigation error state estimation Kalman filter in a typical complementary filter as presented by Bar-Shalom et al. [7] and presented in figure 2.4. In most of these applications a Wavelet Multi-Resolution Analysis (WMRA) de-noising strategy was used to perform data conditioning before using the data to train the neural network. El-Gizawy et al. [26] used a slightly different approach to the estimation of the navigation errors by making use of a Recursive Least Square Lattice (RLSL) filter to remove the high frequency noise from the positional and velocity data in the INS and then feeding the filtered data into an Adaptive Neuro-Fuzzy Inference System (ANFIS) to model the underlying vehicle dynamics. The RLSL filter proved to be very effective in removing the high frequency noise and was well-suited to the real-time

51 Chapter 2 Literature Survey 35 implementation of the system. Very good results were obtained with this strategy. The importance of the data fusion work presented by these authors is that the complementary filter structure that was used to train and implement the neural networks could be adapted to be used during sensor calibration. INS Uncorrected INS outputs + Corrected INS outputs Aiding sensor + Kalman filter INS corrections Figure 2.4: Complementary filter as used in INS/GPS sensor data fusion for hybrid inertial navigation. Tekinalp and Ozemre [111] used a MLFFN to perform that transfer alignment of an inertial navigation system. Transfer alignment is usually performed on an aircraft mounted missile where a high-accuracy reference INS is available on the aircraft and a lower accuracy INS is installed on the missile. During transfer alignment, the initial conditions for the missile INS are obtained from the aircraft INS and the INS error parameters is determined from the differences in measurements between the reference and missile INS. In their application Tekinalp and Ozemre therefore used the reference INS to determine the measurement errors of the missile INS and used these errors as the output training data to the network. They reported a ten times improvement in the positional navigation accuracy due to the neural network implementation of the IMU error parameter estimation. Similar results for a neural network based transfer alignment of a missile were also reported by Wang et al. [123] and by Wang and Guo [122] with the last publication indicating that the real-time calibration of the sensor errors is feasible. Hao and Tian [46] used grey theory to filter the random noise sources from the temperature calibration data of a MEMS gyro. Following the filtering, the bias-temperature relationship was modeled using an RBFN. No real analysis of the estimation ability of the network is presented. The grey filtering appears to be effective in filtering the noise from the training data.

52 Chapter 2 Literature Survey 36 Kim and Hong [6] used a fuzzy logic approach to perform the thermal bias compensation of an alternative type of mechanical gyroscope called the resonant rate sensor. A number of piecewise linear models were developed to model the bias-temperature relationship and fuzzy logic is used to combine the piecewise models into a smooth description of the systems response. Their results indicate that the method is quite robust and that it presents an improvement in thermal bias compensation over the traditional polynomial based compensation strategy. There appeared a number of publications more recently that partially addressed the objectives of this dissertation. Wang et al. [121], Wang and Wang [119] and Wang et al. [12] presented an investigation into the use of MLFNN with a fuzzy training strategy to perform the thermal calibration of MEMS gyros and accelerometers. The biastemperature relationship was modeled using the neural network and the random walk component of the bias was estimated using a Kalman filter. Some reference is made towards the use of both angular rates and temperature to perform the calibration and some graphs are presented to indicate that both parameters were used to train the network, but no conclusive results or discussions of the results are presented. The inclusion of scale factor into the calibration process was completely neglected and no analysis of the true estimation ability of the network was performed. The excitation signal that was used to gather the calibration data was based on the classical calibration profile. The impact of noise in the training data on the neural network training ability was not discussed either as it appears that the network was being overtrained. Shiau et al. [1] and Xia et al. [124] presented similar approaches as Wang et al. [121] to perform the modeling of the bias-temperature relationship using a MLFNN, but once again they did not look at the inclusion of other parameters nor did they really look into the effectiveness of the calibration strategy. Completely outside the field of inertial sensor calibration one finds the combination of RBFNs and calibration procedures in the work of Yang and Zhao [125] and Zong et al. [129]. Yang and Zhao [125] used self-organizing RBFNs to estimate the non-linear parametric model of a ship s motion and Zong et al. [129] used the RBFN to perform camera calibration. In both cases they looked wider than a single input to a single output configuration, but included a multi-dimensional input-output approach to the calibration of the systems under investigation. The work presented in this dissertation has a close resemblance to the camera calibration performed by Zong et al. [129] while the data fusion architecture of Sharaf and Noureldin [98] will be used.

53 Chapter 2 Literature Survey What has not been done? All the neural network based calibration methods presented here used a blind approach to the implementation of the network. No insight into the estimation action of the network was presented. The working of the networks was not analyzed to determine the actual calibration accuracy and the ability of the networks to estimate the deterministic error parameters in the system was not analyzed. In some cases there was not even a distinction made between the estimation of the deterministic and stochastic components of the sensor errors. The essence of laboratory calibration of an inertial sensor is the characterization of the repeatable error components in the sensor. The saying that if we know how it looks, we can subtract it holds true. Nothing is gained by including stochastic error components into the laboratory calibration, since these errors will not look the same in the next measurement. Any laboratory calibration using neural networks (or in fact any method) should therefore focus on the deterministic errors with the estimation of stochastic error components being reserved for an online adaptive type of implementation. 2.5 Calibration as a System Identification Problem The calibration techniques that have been presented from the literature in the first part of this chapter highlight the obvious gaps in the approach to the calibration of gyros. When the complete calibration process is considered from the perspective of SID, ons can identify what has already been done and where the areas for future research in this area lie. This section will place the previous research listed in section 2.4 within the SID framework for calibration presented in section 2.3. From the SID framework presented in section 2.3 the contribution of the most important references listed in the literature study on the four areas of the SID process needs to be investigated. Table 2.2 contains a matrix of these references and lists the contributions from the various publications to the following criteria: Is the experimental design process addressed? Is the process of the collection of the calibration data modified in any way to improve the accuracy or applicability of the resulting calibration model? Is the selection of an appropriate or alternative model structure for the calibration model considered? Is an appropriate criterion of fit developed to match the modeling structure or to

54 Chapter 2 Literature Survey 38 accurately describe the system performance? Is the parameter identification strategy addressed in a new or alternative way? The identification strategy is usually closely related to the structure of the calibration model. Table 2.2: Analysis of the ways in which the various references address the aspects of the SID process. Author Experimental Model selec- Criterion of Parameter design fit tion identification McConley[66] x x Thompson [113] x Musoff [68] x Lintereur [64] x x Grewal et al. [39] x x Guo et al. [42] x? x Bekkeng [8] x Hwangbo and Kanade [53] x x x Gulmammadov [41] x Zhu et al. [127] x x Hongwei et al. [5] x x Sharaf and Noureldin [98] x? x Nassar et al. [72, 71, 7] x? x Fan et al. [29] x x Noureldin et al. [97, 96, 74, 95] x? x El-Gizawy et al. [26] x? x Tekinalp and Ozemre [111] x? x Hao and Tian [46] x x Kim and Hong [6] x x Wang et al. [121, 119, 12] x? x Shiau et al. [1] x x Xia et al. [124] x x Yang and Zhao [125] x? x Zong et al. [129] x x

55 Chapter 2 Literature Survey Calibration Experiment From table 2.2 it can be seen that it was only when an estimation based approach or modifications to the traditional approach to calibration was used that modifications to the calibration experiment through the use of an alternative calibration profile was considered. One would have expected the researchers who used neural network models to look into the use of alternative experimental strategies due to the sensitivity of neural networks to the training data. The development of experimental procedures that explicitly address the requirements and constraints of computational intelligence techniques is therefore clearly an area that is open to further research Model Selection Apart from Guo et al. [42] and to a lesser extent Hwangbo and Kanade [53], none of the research into estimation based calibration looked into alternative ways to model the sensor errors. A significant amount of work has been done to use computational intelligence strategies to model the gyro measurement errors, but most of the research published was of an exploratory nature. Very few in-depth investigations into the ability of the new modeling approaches to perform the fundamental calibration have been performed. In some cases no distinction was even made between the characterization of the deterministic and the stochastic errors as part of the new approach to modeling. The gyro/inertial sensor calibration strategies that focused on computational intelligence strategies almost exclusively focussed on single input single output (SISO) strategies. Wang et al. [119] referred to the fact that more than one input parameter was used to perform the calibration, but provided no analysis to support it. Multi-input calibration configurations were only performed on applications outside the inertial sensor calibration sphere by Yang and Zhao [125] and Zong et al. [129]. From the work considered up to this point one can see that a large field of modeling options have emerged in recent year with the calibration community not really taking notice of these developments. This could partly be described to the fact that calibration has been seen as an established area of navigation where little needs to be adjusted. This could have been true for the large governmental organization with the backing of significant budgets, but the emergence of and need for lower cost navigation sensors in the areas of unmanned aerial vehicles (UAVs) and robotic systems have necessitated a review of the calibration strategies. The disciplines of Machine Learning and of SID and in particular non-linear SID have started to act as the guardians of novel modeling tech-

56 Chapter 2 Literature Survey 4 niques, with the SID discipline being more focussed towards the engineering applications of the theory. Nelles [73], Jategaonkar [56], Ljung [65], Ruano [88], Shin and Xu [12], Haykin [48] and Wang and Hill [118] are just some of the references that list the possible modeling techniques for dynamic systems that have emerged over the last 2 to 3 years. The possibilities available for new and innovative ways to model inertial sensors are vast and present an open research area. The modeling strategy will be dependent on the resources available, the type of nonlinearity that needs to be modeled and the preference of the systems expert. It could also be closely linked to the available calibration equipment and the types of excitation signals that are going to be used as part of the calibration process. One cannot decouple the model and the identification strategy from one another. It is possible, but perhaps not optimal, to use a neural network for parametric systems identification and to use a parametric-focussed identification technique for training a neural network. This being said, it is also true that the boundaries between the application of the various training and optimization algorithms are disappearing (refer to Nelles [73], Haykin [48] and Ljung [65]) with algorithms that were traditionally applied to SID problems being used for training of computational intelligence systems and vice versa. The next two sub-sections will address the potential approaches to modeling of inertial sensors with the objective of using an SID approach to calibration. Non-parametric Models Methods for developing non-parametric sensor models can be taken from the traditional fields of neural networks (NN), evolutionary algorithms (which included genetic algorithms) and fuzzy logic, although fuzzy logic could be considered as a semi-parametric approach as well. In addition to these approaches, techniques such as decision trees could be taken from the discipline of machine learning to describe the systems. Nelles [73] presents a significant number of the traditional non-parametric approaches to systems modeling from a nonlinear systems identification perspective. Russell and Norvig [9] presents a much broader angle on the CI field and even moves towards the application of some of the techniques to the field of robotics. As already discussed in this chapter, some success has been achieved with the use of a number of NN in the modeling of specifically gyros. These techniques have mostly focussed on Multi-layer Feed-forward Neural Networks (MLFNN) and on Radial Basis Function Neural Networks (RBFN). As discussed by Haykin [47] and by Girosi and Poggio [38], general MLFNN are universal approximators (can approximate any type of function), but

57 Chapter 2 Literature Survey 41 the RBFN can be considered as the best function approximators. For this reason it is suggested that the RBFN be investigated for the development of non-parametric models. The use of RBFNs within gyro calibration has been presented by du Plessis et al. [25] in an exploratory study as part of this research project. The complexity of the network has been highlighted as a challenge. Semi-parametric Models When one considers the elements of the research problem statement in chapter 1.3, the last point indicates that 1. the calibration dataset needs to be interpretable by a domain expert and 2. that it must be possible to implement the compensation algorithm (or computational structure) that is obtained from the calibration exercise with relative ease on an embedded navigation processor. Both these requirements point to the fact that a general neural network architecture such as the RBFN or a MLFNN is probably not the best type of architecture for the development of a calibration algorithm that needs to be interpreted by a domain expert and that needs to be implemented on an embedded processor. For the RBFN architectures a large number of neurons are often included in the network, resulting in a large number of computations and significant system memory required to implement the network. This is not necessarily the case for the MLFNN, but for both systems it is almost impossible to gain insight into the inner workings of the network and for a domain expert to evaluate the network for consistency and correctness. Although the need for low computational requirements could possibly be relaxed, it is of great importance for the domain expert to be able to gain insight into the system description generated by the calibration process. This requirement leads one to an investigation of computational intelligence strategies to the SID problem that has a high degree of interpretability. Semi-parametric or grey-box modeling strategies consist of the approaches to system modeling where the classical CI techniques are used in such a way that information can be extracted from the complex modeling structures. The strategies that could potentially be used to satisfy this requirement are Local Modeling Networks (Nelles [73], Murray-Smith [67]); Fuzzy models (Shin and Xu [12]);

58 Chapter 2 Literature Survey 42 Neuro-fuzzy systems (Nelles [73], Shin and Xu [12]); and the Delta Method and Modified Delta-Method that is used to extract information from neural networks (Ghosh [37] and Jategaonkar [56]). An alternative perspective on the interpretability requirement can found if the interpretation of the model is linked to the criterion of fit presented below. Traditionally the domain expert would make use of knowledge of the error model and the system performance to judge whether the calibration has been accurately performed. It should be possible to make use of a non-parametric model (which does not really provide any interpretability) to model the system, but to link the interpretability requirement to the criteria of fit. In this way the model does not have to be transparent, but it must be clear what the impact of the model is on the overall navigation performance of a navigation system that will make use of the model Criterion of Fit No explicit criterion of fit was defined in the references that were studied. As indicated by the question marks in table 2.2 some of the publications did look at the impact of the alternative sensor error model on the system performance, but the more formal definition of an explicit criterion of fit was not addressed. It is assumed that the studied research looked into this as an explicit way of determining the accuracy of the new model that has been defined as part of the identification/training process. Most publications use some kind of analysis to show that their method results in a performance increase, but almost no-one explicitly defines a criteria of fit that can be used to determine whether the new method improves the system accuracy. The definition of an explicit criteria of fit was probably not studied as part of the methods that focused on the determination of calibration constants as these methods were based on the implicit accuracy of the estimation process. This meant that the calibration constants were automatically suitable to the systems as it was part of the systems model. The criterion of fit is more important for the alternative approaches as the model that is derived has no obvious interpretation as being suitable to model the errors in an inertial sensor. In the absence of a method that can be used to gain physical insight into the model defined during the calibration process, a more formalized criterion of fit should be used to judge the applicability of the trained network to the sensor that was characterized. This area is open to further investigation.

59 Chapter 2 Literature Survey Parameter identification The parameter identification steps can be argued to be an integral part of neural network training and does not really present any innovation in terms of the calibration of inertial sensors. One aspect which has not really been investigated is real-time or recursive training strategies that are specifically focussed on the calibration problem. Batch calibration of gyros could greatly benefit from a recursive real-time calibration strategy to reduce the overall calibration time. The common measurement error present in a batch of sensors could be determined only on the first sensor with subsequent sensors only having to be incrementally calibrated for the deviation in the sensor errors from the nominal sensor. 2.6 Conclusion The aim of this chapter was to identify areas where the current calibration strategies for inertial sensors can be improved on. The first step was to provide an overview of the field of inertial navigation sensor calibration by defining the classical approach to calibration and listing the main references that describe this approach. From there a literature survey was performed into modifications of the classical approach and this led to a fundamental perspective of calibration where it was postulated that the field of calibration could be seen as a specific application of the broad field of Systems Identification (SID). The key components of Systems Identification was presented and the SID philosophy was projected onto the field of inertial sensor calibration. From this step the identified references in the area of gyro and accelerometer calibration was mapped onto the SID-calibration framework and the possible improvements in the traditional approach to inertial sensor calibration was identified. Some of the identified open research areas of the SID-based gyro calibration strategy will form the foundation for the work presented in the rest of this dissertation.

60 Chapter 3 Unified FOG Sensor Model 3.1 Introduction The focus of this chapter will be to present a comprehensive model of a fiber optic gyroscope. Such a comprehensive model should in the first place describe the various measurement aspects of the sensor, but it should also discuss the common measurement errors encountered by the sensor and it should be usable to simulate the sensor in cases where actual sensor measurements are not available. The unified FOG error model presented here does not aim to simplify the IEEE model by the omission of the products of errors. The model was developed to show the equivalence of the different models presented and the equivalence argument was not based on the omission of the products of errors, but rather on the development of a systematic argument to show the equivalence. From this perspective the chapter has been divided into three parts. The first part consists of the development of a unified gyro measurement and error model that brings together a number of available gyro modeling equations. The second part will consist of the extension of the unified error model into a practical simulation model through the addition of an empirical sensor model. Since the unified gyro model is focussed on a single axis of operation and gyros are usually used in a tri-axes configuration, it is important to define how the gyro model fits into the tri-axes configuration. The last part of this chapter will therefore define the role of the misalignment matrix and axes transformations in the overall modeling of the gyro. 44

61 Chapter 3 Unified FOG Sensor Model Unified FOG Sensor Model In this section the terminology used throughout this document for describing gyros will be defined and a unified FOG error equation will be developed. The error equation will be used in the simulation of FOG measurements and following a slight adjustment, the same equation would also be useful in the compensation of the raw FOG measurements inside an IMU. Such a unified error equation is needed as most FOG design specifications and development documentation define the system performance parameters in the format of the IEEE standard specification format guide and test procedure for single-axis interferometric fiber optic gyros (IEEE Std ) [54], but the correspondence between this format and the more widely published gyro error models is not obvious. This chapter will attempt to bridge this gap through the derivation of a unified FOG sensor model. The main references for this section are the IEEE standard for inertial sensor terminology (IEEE Std ) [55] and the IEEE standard specification format guide and test procedure for single-axis interferometric fiber optic gyros (IEEE Std ) [54]. Unless stated otherwise, quotes of definitions are taken from the IEEE Std [55]. The unified model presented in this section has already been accepted for publication in du Plessis [24] FOG Model Equation Any discussion of FOG error modeling should start with the standard FOG model equation. As defined in IEEE Std [54], the model equation for a single-axis FOG expresses the relationship between the input rotation rate and the gyro output in terms of parameters whose coefficients are necessary to specify the performance of the gyro. The FOG model equation presented in reference [54], is defined as [ ] N S = Ω + E + D (3.1) t ɛ K where S N/ t Ω E D ɛ K is the nominal scale factor in units of degrees per second ( /s), is the output pulse rate in units of pulses per second, is the inertial input terms in unit of ( /h), is the environmentally sensitive drift terms in unit of ( /h), is the drift excluding the environmentally sensitive components in units of ( /h), is the scale factor error terms in unit of parts per million (ppm).

62 Chapter 3 Unified FOG Sensor Model 46 Note that the IEEE Std [54] uses the notation I for the inertial input terms, but Ω is rather used here as I is usually used to indicate the identity matrix. Equation (3.1) will be used as the baseline for our development of a unified FOG error equation Standard Terminology Before the terms in equation (3.1) are explained in more detail, it is important to define some of the standard terminology that is used for describing inertial sensors in accordance with IEEE Std [55]. The concept of drift is probably the most confusing term amongst the terminology used for describing the performance of inertial sensors. For this reason it is important to define this concept in terms of the standardized terminology. The IEEE Std defines drift rate as Drift rate: The component of gyro output that is functionally independent of input rotation. Figure 3.1 expands the concept of drift rate into its components. In this figure it is indicated that the broad term of drift rate can be divided into two terms called the Systematic Drift Rate and the Random Drift Rate, which are defined as follows: Systematic Drift Rate: The component of drift rate comprising bias, environmentally sensitive drift rates and elastic-restraint drift rates. Random Drift Rate: The random time-varying component of drift rate. As indicated in the formal definition, Systematic Drift Rate consists of a number of terms, the Bias, Environmentally Sensitive Drift Rate and Elastic Restraint Drift Rate. The Elastic Restraint Drift Rate is only defined for mechanical gyros and will therefore be omitted from this discussion. These terms are defined as follows: Bias: The average over a specified time of gyro output measured at specified operating conditions that has no correlation with input rotation or acceleration. Bias is typically measured in degrees per hour. Environmentally Sensitive Drift Rate: The component of Systematic Drift Rate that includes acceleration-sensitive, acceleration squared-sensitive and acceleration-insensitive drift rates.

63 Chapter 3 Unified FOG Sensor Model 47 Figure 3.1: Drift rate definition breakdown presented in the IEEE Std [55]. The three components of the Environmentally Sensitive Drift Rate are defined as follows: Acceleration-Sensitive Drift Rate: The component of Systematic Drift Rate correlated with the first power of a linear acceleration component, typically expressed in units of ( /h)/g. Acceleration-Squared-Sensitive Drift Rate: The component of Systematic Drift Rate correlated with the second power of a linear acceleration component or the product of two linear acceleration components, typically expressed in units of ( /h)/g 2. Acceleration-Insensitive Drift Rate: The component of Systematic Drift Rate not correlated with acceleration. Note: This term includes the effects of temperature, magnetic and other influences Definition of Parameters Some of the parameters presented in equation (3.1) need to be expanded further to define the various error parameters that are associated with FOG calibration. This expansion will be presented in the following paragraphs.

64 Chapter 3 Unified FOG Sensor Model 48 Scale factor error: The scale factor error coefficient ɛ K is defined in the IEEE Std [54] to consist of the following terms ɛ K = ɛ T. T + ɛ(ω) (3.2) where ɛ T. T ɛ(ω) is the scale factor error attributable to a change in temperature, T, where ɛ T is the scale factor temperature sensitivity coefficient and is the scale factor error dependent on input rate. This error is indicated by the term f(i) in the IEEE Std [54]. Environmentally Sensitive Terms: The environmentally sensitive error terms E is defined in the IEEE Std [54] to represent the following sources of measurement error: E = D T. T + D δt dδt dt + D T. d T dt (3.3) where D T. T D δt dδt dt D T. d T dt is the drift rate attributable to a change in temperature, T, where D T is the drift rate temperature sensitivity coefficient, is the drift rate attributable to a temperature gradient, dδt dt, where D δt is the temperature-rate dependent drift-rate sensitivity coefficient and is the drift rate attributable to a time-varying temperature gradient, d T dt, where D T is the coefficient of the time-varying temperature-gradient dependent drift-rate sensitivity. Drift Terms: The drift error terms D is defined in the IEEE Std [54] to represent the following sources of measurement error: D = D F + D Q + D R (3.4) where

65 Chapter 3 Unified FOG Sensor Model 49 D F is the bias term as defined in section 3.2.2, D Q D R is the equivalent random drift rate attributable to angle quantization, where Q is the quantization coefficient and is the random drift rate that will be defined in the next equation. The random drift rate consists of a number of components that are defined as follows D R = D RN + D RB + D RK + D RR (3.5) where D RN D RB D RK D RR is the random drift rate attributable to the angle random walk, where N is the angle random walk coefficient, is the random drift rate attributable to the bias instability, where B is the bias instability coefficient, is the random drift rate attributable to the rate random walk, where K is the rate random walk coefficient and is the random drift rate attributable to the rate ramp, where R is the rate ramp coefficient. The formal definition for these terms are as follows Random Walk: A zero-mean Gaussian stochastic process with stationary independent increments and with standard deviation that grows as the square root of time. Angle Random Walk: The angular error build-up with time that is due to white noise in angular rate. This error is typically expressed in degrees per square root of hour [ / h]. Bias Instability: The random variation in bias as computed over specified finite sample time and averaging time intervals. This non-stationary (evolutionary) process is characterized by a 1/f power spectral density. It is typically expressed in degrees per hour ( /h) or [m/s 2, g], respectively. Rate Random Walk: The drift rate error build-up with time that is due to white noise in angular acceleration. This error is typically expressed in degrees per hour per square root of hour [( /h)/ h]. Rate Ramp: A gyro behaviour characterized by quadratic growth with averaging time of the rate Allan variance. NOTE: When estimated by a linear least-squares fit to the data time series, this behaviour is usually called trend.

66 Chapter 3 Unified FOG Sensor Model 5 IAY θ x IRY θ x θ y IAX IRA IA IA θ y IRX Figure 3.2: Gyro axes definition with misalignment angles Axes Definition The axes definition for a FOG is presented in figure 3.2. The primary sensing axis of a single axis FOG is defined as the Input Axis (IA). This axis is defined as the axis about which a rotation of the case causes a maximum output. This axis is perpendicular to the plane of the gyro coil while passing through the centre of the coil. In terms of the axes definition presented in figure 3.2, IA represent the z-axis of a orthogonal triad of axis defined in terms of the right hand rule. The x-axis (defined as IAX) and y-axis (defined as IAY ) are perpendicular to IA passing through the rim of the coil. IAX is not linked to a particular reference point, so a rotation of the axes around IA does not affect the axes definition. The single-axis gyro is usually mounted as part of a triad of orthogonal gyros within the IMU casing. The Input Reference Axis (IRA) is defined as the direction of an axis (nominally parallel to an input axis) as defined by the case mounting surfaces, or external case markings, or both. The IRA is therefore the mounting axis of the gyro. The IRA axis is combined with the IRX and IRY axis to complete the orthogonal set of the Reference Axis. It is assumed that the single-axis gyro (which measures in the IA) is misaligned with respect to the IMU axis (the RA) by the misalignment angles θ x and θ y as indicated in figure 3.2. To resolve the actual measurement in the RA, the gyro measurement needs to be rotated through the misalignment angles to transform the gyro measurement from the IA to the RA. This transformation consists of the following

67 Chapter 3 Unified FOG Sensor Model 51 rotations: 1. Perform a negative rotation through θ y around the IAY axis to change the IAX axis to the IRX axis and the IA axis to the IA axis. 2. Perform a negative rotation through θ x around the IRX axis to change the IA axis to the IRA axis and the IAY axis to the IRY axis. If the measurement in the IA is defined as ω 1 and the measurement as resolved in the IA axis is defined as ω, the relationship between these two terms is defined by the first rotation mentioned above as 2 ω = cθ y sθ y 1 sθ y cθ y ω (3.6) where the rotation matrix computation method of Rogers as defined in appendix A was used. Defining the rotation as resolved in the RA as ω and using Rogers method once again, the second rotation defined above equates to ω = 1 cθ x sθ x sθ x cθ x ω (3.7) The full sequence of rotations that is necessary to transform a measurement from the IA to the RA is therefore defined through a combination of equations (3.6) and (3.7) with the full transformation being defined as ω = T RA IA ω (3.8) where T RA IA = cθ y sθ y sθ x sθ y cθ x sθ x cθ y cθ x sθ y sθ x cθ x cθ y (3.9) 1 The angular rate applied to the sensor and that measured by the sensor will be designated by the symbol ω from here on until the relationship between the IEEE defined term Ω and the angular rates used in this derivation has been established. 2 Note that a shorthand notation is used during the development of the rotation matrices where cosθ is abbreviated as cθ and sinθ is abbreviated as sθ.

68 Chapter 3 Unified FOG Sensor Model 52 When a gyro measurement ω m of a particular rotation ω, where the axis of rotation coincides with IA, is resolved in terms of IA, it is presented by a vector only having a z-component, or ω m = ω (3.1) If this rotation were to be resolved in the IMU axis (which is the RA), the resulting vector would be obtained through the rotation ω RA = ω XRA ω Y RA ω IRA (3.11) = T RA IA ω m (3.12) = ωsθ y ωsθ x cθ y ωcθ x cθ y (3.13) These terms (ω XRA, ω Y RA and ω IRA ) are the standard terms that are used to describe the rotation measured by the gyro in terms of the IMU axis. As these become the system parameters, it is sometimes required to use the actual gyro measurement in a computation. The rotation is then resolved as with T IA RA RA = (T IA ) 1 = (T RA IA )T, or T IA RA = ω IA = T IA RAω RA (3.14) cθ y sθ x sθ y cθ x sθ y cθ x sθ x sθ y sθ x cθ y cθ x cθ y (3.15) If equation (3.11) is combined with equation (3.15) and substituted into equation (3.14), the following expression is obtained: ω IA = ω XRA cθ y + ω Y RA sθ x sθ y ω IRA cθ x sθ y ω Y RA cθ x + ω IRA sθ x ω XRA sθ y ω Y RA sθ x cθ y + ω IRA cθ x cθ y (3.16)

69 Chapter 3 Unified FOG Sensor Model 53 As it is known that the gyro only has a measurement component along its z-axis (the IA axis), equation (3.16) can be simplified into the following expression: ω IA = ω XRA sθ y ω Y RA sθ x cθ y + ω IRA cθ x cθ y (3.17) As noted on page 41 of Titterton and Weston [115], for small angle rotations, the assumptions can be made that sinθ x θ x, sinθ y θ y, cosθ x 1 and cosθ y 1. Using this assumption, equation (3.17) simplifies to ω IA = ω XRA θ y ω Y RA θ x + ω IRA (3.18) or to ω IA = ω XRA sθ y ω Y RA sθ x + ω IRA (3.19) if only the cosine angle simplifications are implemented. Equation (3.19) is used as part of the IEEE Std [54] as presented in the next section. It should be noted that the magnitude of the original measurement could also be determined from the magnitude computation of the rotation components as resolved in the RA. These components are presented in equation (3.11). Taking the square root of the sum of the squares of the respective components results in the magnitude of the original rotation measured in the IA Development of a Unified FOG Error Model Equation As equation (3.1) is regarded as the standard expression for modeling a FOG, it is important to understand this expression and to equate it to the modeling equations that are generally found in literature. The format of equation (3.1) is quite different to the gyro modeling expressions presented in some of the major texts on inertial navigation such as Rogers [86], Farrell [3] and Chatfield [14] and at first sight it is not clear how the errors defined in these equations equate to those presented in equation (3.1). In the following derivation, equation (3.1) will be written in a more general format and will then be equated to one of the more well-known expressions for the gyro measurement equation in an attempt to develop a standard expression for the FOG error equation. The approach that will be taken is to use the gyro error equation presented on page 411 of Farrell [3] as being representative of the generally published gyro model (referred to hereafter simply as the general model) and develop it into a form where it can be equated to the gyro model equation of Savage [93]. Savage s equation is more comprehensive than the general model and its error components more closely resemble those of the IEEE

70 Chapter 3 Unified FOG Sensor Model 54 Standard FOG model equation in equation (3.1). Following these steps, the general model and Savage s model will be equated to equation (3.1) to present a unified FOG error model equation. General FOG Error Equation To develop a standard expression for modeling the gyro errors, the approach presented on page 411 of Farrell [3] will be used. Using this approach, we can define the gyro measurements in the RA as where ˆω RA T IA RA ω IA is the computed angular rate in the RA, ˆω RA = T IA RA ω IA (3.2) is the rotation through the misalignment angles from the IA to the RA and is the computed angular rate in the IA. On the level of notation it should be noted that, throughout the rest of this chapter, the barred notation (e.g. ω IA ) will be used to refer to the true rate or rotation applied to the sensor, whereas the tilde notation (e.g. ω IA ) will be used to indicate the measured or computed rotational rate. T IA RA in equation (3.2) can be computed directly from equation (3.9), or the abovementioned simplifications from Titterton and Weston [115] can be used to present T IA RA as T RA IA = Equation (3.21) can be expressed as where g is defined as g = 1 θ y 1 θ x θ y θ x 1 (3.21) T RA IA = I g (3.22) θ y θ x θ y θ x (3.23) Using the formulation of Farrell [3] for the computed angular rate in the IA, we can express ω IA as where ω IA = (I δsf g )( ω IA δb g µ g ) (3.24)

71 Chapter 3 Unified FOG Sensor Model 55 I δsf g ω IA is the identity matrix, is a diagonal matrix representing the gyro scale factor error, is the actual rate of rotation that is applied to the gyro, δb g is the combined gyro drift terms and µ g is the combined gyro measurement noise. The term drift is used here in a very general sense as referring to the combination of both the environmentally sensitive and environmentally insensitive measurement errors as presented in section A formal definition of these error components in terms of the IEEE Std [55] was presented in section If we simplify equation (3.24) and we neglect the terms that are products of errors, we obtain the following expression which defines the general model for the gyro measurement. ω IA = ω IA δsf g. ω IA δb g µ g (3.25) By combining equations (3.2), (3.22) and (3.25), we can express the computed angular rate in the RA as ˆω RA = T IA RA ω IA (3.26) = (I g ) ω IA = (I g )( ω IA δsf g. ω IA δb g µ g ) = ω IA g. ω IA δsf g. ω IA δb g µ g (3.27) where products of error terms were once again omitted from the final expression. From equation (3.27) we can define the gyro measurement error ɛ g RA as resolved in the RA as ɛ g RA = ˆω RA ω IA = g. ω IA δsf g. ω IA δb g µ g (3.28) The signs in this expression can be incorporated into the various error terms, so it is in order to neglect all the negative signs and define the general gyro measurement error equation as ɛ g RA = g. ω IA + δsf g. ω IA + δb g + µ g (3.29) Equation (3.29) resembles the format of the derivations presented in the major texts on navigation error modeling. In attempting to standardize the error equation, it is necessary to determine the relationship between equation (3.29) and equation (3.1). This will be pursued in section

72 Chapter 3 Unified FOG Sensor Model 56 Savage s Gyro Error Equation Savage [93] uses a slightly different and more comprehensive formulation of the gyro error equation presented in equation (1.1). He defines the gyro error model as where ω IB puls = 1 Ω ωt (I + F scal ) (F Algn ω ω ω + δω Bias + δω Quant + δω Rand ) (3.3) ω IB puls Ω ωt I F scal F Algn ω ω ω IB δω Bias δω Quant δω Rand is the raw gyro output in units of pulses per second, is the pulse weight in units of rad/s per pulses, is an identity matrix, is the scale factor correction matrix, is the DCM for transformation from IMU body axes to gyro measurement axes, is the actual rotational rate applied to the gyro, is the gyro bias vector, is the gyro triad pulse quantization vector associated with the output only being provided when the cumulative input equals the pulse weight per axis, and is the gyro random error output vector. If we take note of the fact that the term F Algn effectively transforms the applied rotational rate from the IMU body axes to the gyro measurement axes 3, (the Input Axes), we can see that which means that F Algn = T IA B (3.31) F Algn ω ω ω = ω ω ω IA (3.32) This results in the whole of equation (3.3) being defined in the gyro measurement axes. Taking this into consideration and using the notation defined up to this point in this derivation, we can rewrite equation (3.3) as ω ω ω IA = (I + F scal ) ( ω ω ω IA + δω Bias + δω Quant + δω Rand ) (3.33) 3 These axes are defined in section 3.4.

73 Chapter 3 Unified FOG Sensor Model 57 where we defined the measured angular rate in the input axis ( ω ω ω IA ) as ω ω ω IA = ω IB puls Ω ωt (3.34) If we multiply the terms in brackets and ignore the products of errors, we can rewrite equation (3.33) as ω ω ω IA = ω ω ω IA + F scal ω ω ω IA + δω Bias + δω Quant + δω Rand (3.35) which is essentially identical to equation (3.25) apart from the additional error terms. Unified Gyro Error Equation A unified gyro error equation can be obtained if we manage to determine the relationship between equations (3.25), (3.33) and (3.1). This can be achieved by first rewriting equation (3.1) in the following form: [ ] N S t + S [ N t or equivalently, if we rearrange some terms, in the form ] 1 6 ɛ K = Ω + E + D (3.36) [ ] [ ] N N Ω = S + S 1 6 ɛ K E D (3.37) t t We will now show that each term in equation (3.37) can effectively be related to the terms in equations (3.25) and (3.33). Before we continue with the derivation of a unified FOG modeling equation, it is important to highlight a potential notational discrepancy in the derivation. Equation (3.36) presents the gyro measurements in the sensor input axis, which is effectively scalar measurements. The IEEE Std [54] defines the Ω term as Ω = ω XRA sθ y ω Y RA sθ x + ω IRA (3.38) which is exactly the equation for the angular rate resolved in the IA that was presented in equation (3.19). Realizing that equation (3.36) describes the RA rotation resolved in the IA, we can modify equation (3.26) into the following form ω IA = T RA IA ˆω RA (3.39) which is the expression that was used to derive equation (3.19). This therefore points to the fact that the transformation matrix T RA IA converts the scalar Input Axis measurements

74 Chapter 3 Unified FOG Sensor Model 58 to a vectorized set of measurements in the Reference Axes. As equation (3.36) is in scalar form and we are trying to equate it to equations (3.25) and (3.33), which is in vectorized format, it should be a potential conflict of notation will surface if we try to equate the scalars to the vectors. Equation (3.25) was presented in vectorized form to be general enough to accommodate both the Reference Axes expression of the measurements and the combination of measurements from a triad of sensors (for which this expression is generally used). Equation (3.33) was presented in vectorized form to be general enough to accommodate both the IA and the RA representations. The conversion from the IA to the RA therefore changes the expression from a scalar to a vector form. For the rest of this derivation, we will therefore assume a slightly conflicting notation where we will maintain all the vectorized forms that we have been using up to now in both the IA and the RA. In the RA the vector notation will be absolutely correct, but in the IA, the vectorized expressions will effectively simplify to scalar terms. The interpretation of the terminology should be apparent from the context within which it is being used. A consequence of this notational strategy is that, when vectorized expressions are used to represent an entity in the IA, it is possible to equate these vectorized terms to terms which have not been expressed in vectorized notation. If we take equation (3.24) and we multiply it out, we get ω IA + δsf g ω IA = ω IA δb g µ g (3.4) where the products of error terms have been neglected. We can also write this expression as ω IA = (1 + δsf g ) ( ω ) 1 IA δb g µ g or as follows if we multiply it out and neglect the products of errors (3.41) ω IA = (1 + δsf g ) 1 ω IA δb g µ g (3.42) By direct comparison of equations (3.37) and (3.42), it should be noted that Ω and ω IA both define the true rate applied to the sensor. By taking this into consideration we can equate the following terms δb g + µ g = E + D as both sides represent the cumulative drift terms, [ ] N ω IA = S t

75 Chapter 3 Unified FOG Sensor Model 59 as both terms represent the inertial rate of rotation measured by the system and ɛ K = (1 + δsf g ) 1 (3.43) as both terms represent the measurement error due to the scale factor error. This last term may appear odd, but this apparent discrepancy can be clarified by expressing it as a set of scalar parameters (which it is in this case) as ɛ K = δsf g If the equality is solved, the following expression for δsf g is obtained: δsf g = 1 6 ɛ K ɛ K This expression can be used to resolve the apparent scale factor ambiguity through the substitution of some numbers. If, as an example, the scale factor error on the specification of the FOG is defined in IEEE standard units as 1 PPM (parts per million) and this value is substituted into this expression as ɛ K, then it results in δsf g = The negative sign reflects a convention where the scale factor error is always subtracted rather than added to the true measurement. From this numerical investigation it can be seen that the two methods of scale factor error formulation is directly equivalent. From these expressions of equivalence, we can now write a unified FOG calibration equation as ω IA = ( ɛ K ) ω IA E D (3.44) This equation describes the way in which the measurements obtained from the sensor are corrected to obtain the true rotational rate that was applied to the sensor. An additional formulation that is required is the model equation that can be used to generate the perturbed gyro measurement if knowledge of the true rotational rate and the sensor errors are known. Since the equivalence between the two scale factor descriptions have been proven in equation (3.43), equation (3.44) can be expressed as ω IA = (1 + δsf g ) 1 ω IA E D and if the products of error terms are neglected, this expression can be rewritten as ω IA = ω IA + δsf g ω IA + E + D (3.45) This expression can be used to simulate FOG measurements since it defines a unified FOG model equation. It can be used to develop a simulation of the gyro measurements,

76 Chapter 3 Unified FOG Sensor Model 6 but can also be used to describe the measurements made by the sensor when needed for different calibration or estimation strategies such as those presented by Bar-Itzhack and Harman [6]. The significance of this particular expression is that it relates the standard IEEE specified way in which the system performance is specified to the more generally used gyro modeling equations found in most of the mainstream navigation references. We can therefore conclude that the general gyro error equation presented in equation (3.29) can indeed be standardized in terms of the IEEE standard FOG model equation presented in equation (3.1). Products of Errors Omission The omission of the products of errors can be considered to be standard practice as presented by Savage [93] (page 8-9), furthermore we have F scal = δsf g 1 3, δ bias 1 4, δ quant approx1 5 and δ bias 1 5. This would result in the following omissions: On page 57, in the computation of equation(3.35), the omitted term are F scal.δ bias 1 7, F scal.δ quant 1 8 and F scal.δ bias 1 8, all which can indeed be considered to be negligible. On page 58 in equation (3.4) the omitted error terms are δsf g.δb g 1 7, and δsf g.δ g 1 7. On page 58 in equations (3.42) the wording can be changed to also read as follows: (1 + SF g ) 1.δB g δb g and (1 + SF g ) 1.µ g µ g. The same comment as the last one mentioned above applies to equation (3.45) as well. 3.3 Development of a Gyro Simulation Environment Extending the Error Model The unified error model developed in the first part of this chapter can be extended to be used for the simulation of the gyro measurement errors by extending the error model from equation (3.45). For the model equation the true rate of rotation is taken as the input to the system and the perturbed angular rate is provided as the output. One can therefore rewrite equation (3.45) as ω ω ω IA = ω ω ω IA + ω ω ω IA (3.46)

77 Chapter 3 Unified FOG Sensor Model 61 where ω ω ω IA = δsf g ω ω ω IA + E + D (3.47) but, since the scale factor error is not necessarily linear and usually a function of the angular rate and environmental temperature, one could define this expression as ω ω ω IA = δsf + E + D (3.48) where δsf δsf g ω ω ω IA (3.49) The terms on the right hand side of equation (3.48) are taken from the formulation of the IEEE FOG model equation in section on page 47. If the various coefficients presented in these expressions were available, one would therefore be able to simulate the measurements for the particular gyro. The determination of these coefficients are the topic of the following section An Empirical Simulation Model The objective of this section is to describe the development of an empirical error model for use in the development of the alternative calibration algorithm. It should be stated at the start of this section that the measurements used for this derivation were made during the development and extensive characterization of a particular FOG that was used as the basis for this research. These measurements were crude at times and one could question the integrity of some of them. With that kept in mind, the data was used as presented and assumptions were made in some cases to obtain a feasible description of the particular error parameter. Irrespective of the quality of the data, the objective of this chapter is to present a method to develop an empirical model of the system. The process presented here can therefore be repeated to present a more realistic system description if a high-integrity data-set is available. The empirical model was derived from a set of readings that had as its objective the elimination of any stochastic effects from the data and purely focussing on the deterministic measurement errors. It was therefore not a single measurement that was used to derive the model, but rather a single set of measurements.

78 Chapter 3 Unified FOG Sensor Model 62 Scale Factor Error If we redefine ɛ K = δsf in equation (3.2), the expression for the scale factor errors can be rewritten as δsf = ɛ T. T + ɛ(ω) where the symbol Ω was retained from equation (3.2) to define the applied angular rate instead of ω, since the equivalence of these two parameters was determined in section If it is recognized that both the temperature and rate dependent scale factor parameters are nonlinear functions of the environmental conditions, then this expression can be further generalized into the following expression: δsf = ɛ T (T ) + ɛ(ω) (3.5) The parameter δsf is defined as the deviation from a 1:1 ratio between the angular rate measurement and the applied rate, or δsf = Ω measured Ω applied 1 From the format of equation (3.5) one can see that the scale factor error is actually a function of two parameters, or δsf = f(t, Ω). There are two approaches that were considered to represent this relationship. The first was a pure polynomial expression and the second was a two-dimensional mapping approach. The polynomial approximation will be discussed first. Polynomial approximations The first approach that was followed for the description of the scale factor error was to make use of polynomial approximations. As mentioned by Titterton and Weston [115] and Aggarwal et al [4], it is common practice in the calibration of inertial sensors to fit the datapoints from the environmentally sensitive parameters with polynomial functions. For the temperature dependence, the average deviation in the linear scale factor as a function of the temperature was determined and a second order polynomial expression was used to fit the data. Two different sets of data were generated for positive and negative rates of rotation. The data with the polynomial approximations overlaid on them are presented in figure 3.3. From the differences in the curves for positive and negative rotations, it can be seen that the angular rate has an impact on the temperature dependence. One could therefore use two different expressions for the temperature dependent scale factor based on whether

79 Chapter 3 Unified FOG Sensor Model x Pos rotations Neg rotations Pos fit Neg fit Temperature dependent SF error 6 δ SF Temperature Figure 3.3: Polynomial expression for temperature dependent scale factor error. the system is experiencing positive of negative angular rotations, but the curves presented in figure 3.3 are descriptions of the average values over all the angular rates. The curves could therefore be used, but it would be a much simplified expression of the temperature dependence. The angular rate dependence of the scale factor can also be described by a polynomial expression. An example of a 6 th order polynomial fit to the scale factor error as a function of the angular rate is presented in figure 3.4. The data presented here was collected at a temperature of 2 C, but this curve hides the fact that the relationship is highly dependent on temperature variations. One could therefore also use this curve to define the impact of the angular rate on the scale factor error, but it would probably be an oversimplification of the true relationship. 2D Mapping Given the indicated limitations for the polynomial descriptions of the scale factor error, the question presents itself as to the possibility of using a two-dimensional (2D) curve fitting to describe the relationships. To investigate this option, the grid of mea-

80 Chapter 3 Unified FOG Sensor Model 64.8 Angular rate dependent scale factor δ SF Angular rate Figure 3.4: Polynomial expression for angular rate dependent scale factor error. surements of the scale factor error as various combinations of angular rate and temperature was presented as a 3D surface plot. The result is presented in figure 3.5. Temperature measurements were made at -2,, 2, 25, 4 and 6 C. Angular rate measurements were made at rates of approximately -1, -8, -6, -4, -2, -1, 1, 2, 4, 6, 8 and 1 degrees per second. Although the surface presented in figure 3.5 is not very smooth, from visual inspection one can see that there are aspects of the 2D scale factor error function that were not captured by the simplified polynomial expressions. In an attempt to capture the 2D characteristics of the scale factor error, a 2D interpolation (using Matlab s interp2 function) was used to approximate the data from the raw data points using a bicubic spline interpolation. The objective with the use of the bicubic spline was to obtain an error surface that is intuitively realistic and smooth taken the limited number of data-points available. The 2D interpolated scale factor error surface is presented in figure 3.6. This interpolated 2D function will be used to model the scale factor error as part of the empirical error model.

81 Chapter 3 Unified FOG Sensor Model 65 Scale factor error surface.1.5 Scale factor error Applied angular rate Temperature 1 Figure 3.5: 2D surface description of the raw scale factor measurement data as a function of both the temperature and the angular rate. Environmentally Sensitive Terms Equation (3.3) presents the environmentally sensitive terms. One can express these errors in a similar way to the scale factor errors in a functional format as where D T ( T ) ( D dδt ) δt dt D T. ( ) d T dt E = D T ( T ) + D δt ( dδt dt ) ( ) d T + D T. dt indicates the drift rate as a function of the change in temperature, T, is the drift rate as a function of the temperature gradient, dδt, and dt is the drift rate as a function of the time-varying temperature gradient, d T dt. (3.51) This more general expression for the environmentally sensitive error components lends itself to a more complex representation of the errors. As indicated by Handrich et al. [45], the temperature dependant drift components are often modeled using polynomial expres-

82 Chapter 3 Unified FOG Sensor Model 66 Figure 3.6: 2D surface description of the raw scale factor measurement data as a function of both the temperature and the angular rate. sions. The starting point for the functional models presented here will therefore be polynomial models that can be modified in a similar way to the scale factor errors presented previously when deemed necessary. Temperature Dependent Drift As indicated by Titterton and Weston [115], changes in ambient temperature have an effect on the drift (effectively the bias). To determine the relationship between the bias and the temperature, bias measurement experiments were conducted at a number of static temperatures. The results for 3 different sets of experiments conducted for the same gyro is presented in figure 3.7. The green curve represents the 3 rd order polynomial approximation to the three sets of data. This polynomial will be used to model the sensor. The variation observed in this relationship between different experimental sets indicates that the sensor is still under development and that it is yet thermally stable not. Temperature Gradient Dependent Drift The Shupe effect (Lawrence [62], Titterton and Weston [115]) is a drift component observed in a FOG when a temperature gra-

83 Chapter 3 Unified FOG Sensor Model A axis bias vs temperature repeatability Set 1 Set 2 Set 3 Polyfit Bias ( /h) Temperature ( C) Figure 3.7: Temperature dependent drift data. dient across the gyro coil results in a time-dependent temperature gradient along the coil. The temperature gradient distorts the coil in a way that changes its optical properties and thereby results in a change to the gyro bias. To determine the impact of the temperature gradient and time-varying temperature gradient on the bias, it is necessary to measure these gradients across the gyro coil during the calibration. During the characterization of the current system no sensors were in place to determine these gradients. An indication of the temperature gradient to bias relationship that is typical of a FOG can be obtained from Ruffin et al. [89]. This relationship, based on the physical parameters of the gyro under investigation, is presented in figure 3.8. It can be seen that it portrays a linear relationship between the temperature gradient across the coils and the change in the bias. This is a straight-forward relationship to include as part of a simulation model, but it does require that the temperature gradient across the coils be known for this effect to be included in the gyro simulation. Since the modeling of physical thermodynamic aspects of the sensor falls outside the scope of this study, the Shupe effect was excluded from the simulation model.

84 Chapter 3 Unified FOG Sensor Model Bias as function of temperature gradient deg/h dt/dt: Temperature change over time (Degrees Celcius/min) Figure 3.8: Theoretical contribution of the Shupe effect to changes in the FOG bias. Drift Terms Equations (3.4) and (3.5) define the drift error in the system. As the basic sensor output with no input excitation contains both deterministic and stochastic errors, a set of such measurements needs to be analyzed to determine the various drift error components. Bias A bias D F is always present in the sensor measurements. The bias for a particular sensor can be determined by taking the average of the sensor output with zero input rotation applied over an extended period of time. A set of raw output data for the gyro under discussion is presented in figure 3.9. This data-set was collected over a period of 6 hours following a period of settlement for warm-up effects. It can be seen that the mean value for this sensor is just below -15 deg/hour (-.42 deg/s), thereby indicating this to be the bias. The standard deviation (sigma in the plot) is about 1 deg/hour (.28 deg/s).

85 Chapter 3 Unified FOG Sensor Model Gyro output rate Gyro output Mean Sigma Gyro Rate (deg/h) Time (hours) Figure 3.9: Raw gyro output for the computation of the bias and the standard deviation of the noise. Stochastic Errors The random drift rate defined in equation (3.5) presents the stochastic error components. Gebre-Egziabher [36] indicates that, for the FOG used as part of the presented research, the only stochastic component that was present was the wideband noise, which is called the angle random walk component D RN. This component can be modeled as Gaussian white noise. The name of this error component is slightly misleading as it refers to the fact that, when this particular error component is integrated as part of the INS mechanization, the resulting angular value will exhibit a random walk behaviour. Gebre-Egziabher based his statement on an Allan variance analysis ([36],[54],[51]) of the gyro data to determine the error components. One would therefore expect it to be satisfactory to only model the wide-band noise component as representative of the stochastic noise present in the system. To verify this fact and to determine whether Gebre-Egziabher s observations are applicable to the sensor under development, an Allan Variance (AV) plot was generated for

86 Chapter 3 Unified FOG Sensor Model 7 the FOG 4. The Allan variance plot is presented in figure 3.1. As indicated in the IEEE Allan varriance 1 1 Allan Variance (deg/h) τ (hours) Averaging time Figure 3.1: Allan variance plot of raw gyro output data. FOG Standard [54], the presence of wide-band noise is usually indicated by a 1 2 slope on the AV plot for every decade increase along the horizontal axis, the plot will drop by half a decade along the vertical axis. Quantization noise D Q is characterized by a slope of -1 on the AV plot and rate random walk is defined by a slope of + 1. Using this 2 knowledge, it can be seen that the AV plot rather indicates the presence of quantization noise instead of the expected wide-band noise for a measurement period up to 1 hour, which far exceeds the duration of the expected continuous measurement time during the calibration. For averaging times longer than 1 hour, the AV plot is not that accurate, but there is an indication of rate random walk. As indicated in the IEEE FOG Standard [54], the normal Allan variance that was used to obtain the plot in figure 3.1 cannot be used to distinguish between quantization noise 4 A modified version of the process presented in appendix C of Gebre-Egziabher [36] was used to compute the Allan variance.

87 Chapter 3 Unified FOG Sensor Model 71 and wide-band noise. The result is that the Allan variance graph presented in figure 3.1 could therefore indicate the presence of quantization noise as warranted by the -1 slope, or it could point towards the presence of wide-band noise. One could use the power spectral density (PSD) or the Modified Allan variance to distinguish between the two types of noise. For the current analysis this step was considered to be unnecessary as both types of noise appear as normal Gaussian noise as can be confirmed from the histogram compiled in figure 3.11 from the data in figure 3.9. Using the properties obtained from figure 3.9, the noise in the system will be modeled as normal Gaussian noise with the standard deviations indicated in figure 3.9. The use of this approximation of the noise was therefore deemed to be sufficient for the current analysis Bias measurement noise histogram 2 Number of Measurements Distribution Figure 3.11: Histogram of raw gyro output for bias computation. Although it plays a smaller role in the gyro under test, a common stochastic error component in low-cost gyros is rate random-walk. This error term is modeled as a first order Gauss-Markov process that appears in the rate measurement produced by the gyro. 5 The validity of this approach to the noise modeling was confirmed with colleagues in the navigation industry.

88 Chapter 3 Unified FOG Sensor Model 72 The impact of such an error component on the calibration accuracy that can be obtained with the new method will be evaluated to determine the applicability of the new method to a wider class of sensors than the specific sensor under investigation. The random walk noise component that will be implemented as part of the simulation is based on the discrete simulation model for a first order Gauss-Markov model presented by Flenniken[31, 32] and Wall and Bevly[117]. In this model the future value of some random input ḃ is presented as a scaled portion of the previous value of this random input b through the expression where ḃ = b τ + ω gyro (3.52) τ ω gyro is the correlation time of the random walk component and is the wide band sensor noise of the gyro. The ω gyro term is driven by zero mean Gaussian noise with covariance σ 2 gyro (referred to as the random walk noise variance in the gyro spec sheets) and is defined by the expression ω gyro = 2σ 2 gyro T s τ (3.53) where T s is the system sample rate. The differential equation for the first order Gauss- Markov process presented in equation (3.52) can be solved through integration as b = ḃdt ( ) b = τ + ω gyro dt (3.54) Using Euler integration to obtain a discrete equivalent, it can be expressed as ( ) 1 b k+1 = b k + T s τ b k + ω k ( = 1 T ) s b k + T s ω k (3.55) τ where the ω k term refers to the previous value of the system driving noise. The noise can either be generated with Matlab s built-in randn function for Gaussian noise or if a larger

89 Chapter 3 Unified FOG Sensor Model 73 level of independence is required, the truerand 6 function can be used which is based on atmospheric noise. Combined Simulation Model Throughout this study on the modeling of the FOG, it was indicated that there are various error components that influence the measurement accuracy apart from the noise components. The primary measurement parameter is obviously the angular rate of rotation that is experienced by the sensor. It was shown that the relationship between the applied rotation and the measurement error can be modeled with relative ease and that an empirical model for this relationship could be established. In addition to the rotation, it was also seen that the environmental temperature influences the measurement accuracy of the sensor and that this relationship could be empirically described. The other parameters that were identified as having an impact on the system, but for which measurements were not available to model these relationships were the impact of temperature gradient, time dependent temperature gradient, vibration and magnetism on the sensor accuracy. It was therefore decided to base the empirical simulation model for the gyro on the available data and to develop the new calibration strategy on this model. The model can easily be extended should a more extensive set of measurements be available. The current model will consist of two inputs and one output, which makes it easy to visualize, but if more input parameters are added to the system, the relationship will not be as easy to visualize any more. The empirical simulation model used in the development of the new calibration strategy therefore consists of a combination of the functional descriptions of the various error sources presented in this chapter. Using equation (3.48) and substituting the functional error descriptions into the model, an error surface for the gyro (not including noise) is obtained as defined in figure This error surface will be regarded as the absolutely correct functional description of the system errors for the rest of this study. Error data will be generated using this model and it will be used as the reference to determine the accuracy of the new calibration strategy. Where required the noise components or the sensor error will be included as part of the error model. 6 Available from

90 Chapter 3 Unified FOG Sensor Model 74 Figure 3.12: Error surface used for the empirical error model. 3.4 Misalignment Computations The work presented up to this point in this chapter was focussed on the development of a gyro measurement and error model for a single gyroscope. Although the conversion from the sensor measurement axis to the IMU body axis was briefly mentioned in equation (3.31), the misalignment components of the gyro model were restricted to the transformation from the sensor input axis to the measurement axis within a single gyro. In a practical system the gyro is used as part of a set of three orthogonal sensors that are used to measure the rotational motion of the host vehicle. Knowledge of the orientation of the individual sensors within the IMU is of critical importance to resolve the sensor measurements into a unified set of inertial measurements. The focus of this section will therefore be on the description of the axes transformations (generally called the misalignment matrices) from the measurement and input axes of a single sensor to the combined set of measurements of an orthogonal set of sensors. 7 7 It is important to note that, for the systems considered here, the typical misalignment errors are on the order of 1 milliradian on both the input axis and the body axis. Within this context, we have that

91 Chapter 3 Unified FOG Sensor Model 75 It should be noted that an IMU consists of a cluster of gyros and a cluster of accelerometers, but since the focus of this study is only on gyros, the misalignment computation of accelerometers will not be discussed here. The method presented in this section could be considered to be somewhat artificial since misalignment calibration usually consists of the determination of the various error terms discussed here as a single parameter, but the objective of the derivation presented here is to systematically define how the misalignment angles influence the overall measurement accuracy of a set of gyros Axes Systems One could define four axes systems (reference frames) to describe the various angular displacements present in the system. The axes systems are the Input Axes (IA); the Reference Axes (RA); the Cluster Axes (CA); and the Body Axes (B). These reference frames will be discussed below. Gyro Reference Frames The IA for the gyros follows the definition in section As defined in the IEEE Std [55], this axis is defined as the axis(es) about which a rotation of the case causes a maximum output. Due to the completeness of the discussion in section 3.2.4, further discussion on the IA definition will not be pursued in this section. Similar to the IA, the RA (sometimes called the Input Reference Axes) for the gyros is also defined in section The RA is defined as the direction of an axis (nominally parallel to an input axis) as defined by the case mounting surfaces, or external case markings, or both. This axes system is therefore the mounting axis of the gyro. Due to the completeness of the discussion in appendix 3.2.4, further discussion on the RA definition will also not be pursued in this section. sin(θ) = θ =.1 and cos(θ) = These simplifications will be used during the derivations in this part of the document.

92 Chapter 3 Unified FOG Sensor Model 76 Cluster Axes The CA is an intermediate reference frame that is used in the definition of the misalignment matrix. It defines the combination of three single axis gyros into an orthogonal gyro cluster. Although this reference frame could be argued to be somewhat hypothetical, it will be defined for the purpose of this derivation as an orthogonal set of axes centred on the Centre of Gravity (CG) of the IMU. Body Axes The body axes reference frame is the orthogonal set of axes centred on the IMU CG and fixed to the IMU assembly. All measurements from the individual sensors must be resolved into this reference frame as it is used to define the complete set of measurements obtained from the IMU. When the IMU is mounted within the host platform (aircraft, missile, vehicle, etc.), the body axes will be used to determine the orientation of the IMU within the host platform. Figure 3.13 shows the body axes as an overlay on the complete IMU assembly for the system that was used for this project. This definition is specific to the system that was used and will be dependent on the IMU design General Misalignment Components An IMU consists of sensors (gyros and accelerometers) that are mounted together as a cluster within the frame of the IMU. Figure 3.13 shows a complete IMU assembly that shows all the sensors as mounted on the IMU frame. As it is desired to resolve all the individual sensor measurements into a complete set of measurements in the IMU reference frame or IMU body axes, it is necessary to develop a strategy whereby the respective measurements are unified into the reference frame that is fixed to the IMU assembly. Figure 3.14 contains a schematic presentation of the steps that need to be taken to determine the complete misalignment matrix. This strategy is general enough to be applicable to both the gyros and the accelerometers. These steps can be described as follows: Step 1: Transform the individual sensor measurements from the IA to the RA. This transformation describes the misalignment between the sensing axes and the casing axes of the sensor. The transformation matrix will be described by the notation T RA IA. Step 2: Transform the individual sensor measurements from the RA to the CA. This

93 Chapter 3 Unified FOG Sensor Model 77 Y B X B Z B Figure 3.13: IMU assembly diagram with proposed body axes overlay. transformation describes the orientation of the sensor within the CA. It combines the individual sensors into a set of measurements that is obtained from the cluster. The transformation matrix will be described by the notation T CA RA. Step 3: Transform the cluster measurements from the CA to the body axes. This transformation describes the insertion misalignment of the cluster within the IMU assembly relative to the IMU body reference frame. The transformation matrix will be described by the notation T B CA. By following these steps, the misalignment matrix can be easily recomputed whenever one of the components change due to the redesign of the IMU assembly or due to a general configuration change. From these steps and from the diagram in figure 3.14, we can also see that the complete transformation matrix from the sensor IA to the body axes can be defined by the following expression: T B IA = T B CAT CA RAT RA IA (3.56)

94 Chapter 3 Unified FOG Sensor Model 78 T B IA Misalignment Orientation Insertion Input axes (Sensor) T RA IA Reference axes (Casing) T CA RA Cluster axes (Triad) T B CA Body axes (IMU) Figure 3.14: Overview of the components of the misalignment matrices. The rest of this section will consist of the definition of these transformation matrices for the gyro Gyro Misalignment Matrix Derivation The transformation from the IA to the RA for a single FOG is presented in section Defining the misalignment terms between the IA and RA for the x-gyro to be φ x and φ y, for the y-gyro to be θ x and θ y and for the z-gyro to be ψ x and ψ y, we can define the IA to RA transformation matrices for the respective gyros to be T RA gia x = T RA gia y = T RA gia z = 1 φ y 1 φ x φ y φ x 1 1 θ y 1 θ x θ y θ x 1 1 ψ y 1 ψ x ψ y ψ x 1 (3.57) (3.58) (3.59) where the misalignment angles are defined according to figure 3.2. The lower case g was added to the transformation matrix descriptions to indicate the they describe the gyro misalignment terms. All transformation matrices applicable to the gyros will be defined in this way. The definition of these expressions defines step 1 in the computation of the gyro misalignment matrix.

95 Chapter 3 Unified FOG Sensor Model 79 Step 2 in the derivation of the gyro misalignment matrix consists of the definition of the orientation of the individual gyros within the cluster. The orientation of the respective sensing coils is shown in table 3.1. As was indicated in figure 3.2, the gyro sensing axis is the z-axis of the IA. The orientations shown in table 3.1 are defined to ensure that each axis of the CA is aligned with the sensing axis of a gyro. Table 3.1: Reference axes (RA) to cluster axes (CA) orientations for the gyros. Sense Axis Orientation Transformation Matrix X-gyro RA z RA y CA z RA x CA y x CA y CA z CA T CA gra x = x RA y RA x RA y RA z RA = CA x z RA Y-gyro RA y CA z RA x RA z CA y x CA y CA z CA T CA gra y = x RA y RA x RA y RA z RA = CA x z RA Z-gyro RA x RA z CA z RA y CA y x CA y CA z CA T CA gra z = x RA y RA x RA y RA z RA = CA x z RA The third step that must be taken in the derivation of the gyro misalignment matrix, is to define the insertion misalignment matrix. This matrix defines the misalignment

96 Chapter 3 Unified FOG Sensor Model 8 of each one of the gyros with respect to the IMU assembly (and therefore with respect to the body axes) when the gyros are inserted within the IMU assembly. As indicated in section 3.2.4, the gyro measurements are not affected by rotations about the z-axis of the IA. For this reason the exact same format could be followed for the definition of the insertion misalignment matrix as was followed in the formulation of the IA to RA transformation matrices defined in equations (3.57) to (3.59). Defining the misalignment terms for the x-gyro to be µ x and µ y, for the y-gyro to be η x and η y and for the z-gyro to be ξ x and ξ y (refer to figure 3.15), we can define the misalignment matrices for the respective gyros to be T B gca x = T B gca y = T B gca z = 1 µ y 1 µ x µ y µ x 1 1 η y 1 η x η y η x 1 1 ξ y 1 ξ x ξ y ξ x 1 (3.6) (3.61) (3.62) Using the complete set of transformation matrix formulations defined in steps 1 through 3, we can now compute the transformation matrix from sensor IA to IMU body axes (T B gia) as a single matrix. Such a formulation is useful as it will eliminate a significant amount of computations from having to be performed on the embedded processor. The strategy that will be followed in the development of the single matrix expression, is to convert all the sensor measurements to the body axes where these measurements can be added to provide a single set of measurements resolved in the IMU body axes. If we define the x-axis, y-axis and z-axis gyro measurements as resolved in the body axes to be Ω B x, Ω B y and Ω B z, respectively, and the sum of these measurements to be Ω B T ot, we can express the summed set of measurements as where Ω B T ot = Ω B x + Ω B y + Ω B z (3.63)

97 Chapter 3 Unified FOG Sensor Model 81 CA z µ x B z CA x η x B x µ x µ y CA y η x η y CA z B x b CA x µ y B y B y b CA y η y B z (a) x-gyro (b) y-gyro CA y ξ x B y ξ x ξ y CA x B z b CA z ξ y B x (c) z-gyro Figure 3.15: Misalignment angle definitions between the CA and the body-axis. Ω B x =, Ω B y =, Ω B z = (3.64) Ω x Ω y Ω z If we make use of equation (3.56), we can rewrite equation (3.63) in terms of the individual sensor measurements in the IA as follows Ω B T ot = T B gia x Ω IA x + T B gia y Ω IA y + T B gia z Ω IA z (3.65) or expanding the transformation matrices further into the individual terms, we can write it as

98 Chapter 3 Unified FOG Sensor Model 82 Ω B T ot = T B gca x T CA gra x T RA gia x Ω IA x + T B gca y T CA gra y T RA gia y Ω IA y + T B gca z T CA gra z T RA gia z Ω IA z (3.66) Using the definitions for the various transformation matrices that have been defined in this chapter, we can formulate the transformation matrices for the various axes as follows: T B gia x = T B gca x T CA gra x T RA gia x 1 µ y 1 1 φ y = 1 µ x 1 1 φ x µ y µ x 1 1 φ y φ x 1 φ y µ y + φ x 1 µ y φ x = 1 µ x φ y + µ x φ x µ x + µ y φ y 1 µ y φ x µ x φ y φ x µ y φ y µ y + φ x 1 1 µ x φ y (3.67) µ x 1 φ x µ y T B IA y = T B gca y T CA gra y T RA gia y 1 η y 1 = 1 η x 1 η y η x η y θ y η y θ x = θ y θ x η x η x θ x θ y 1 θ x θ y θ x 1 η y η x θ y η x θ x + 1 η x θ x + η y θ y 1 η y θ y θ y θ x η x 1 (3.68) η y 1 η x θ x

99 Chapter 3 Unified FOG Sensor Model 83 T B IA z = T B gca z T CA gia z 1 ξ y 1 1 ψ y = 1 ξ x 1 1 ψ x ξ y ξ x 1 1 ψ y ψ x 1 1 ψ y ξ y ψ y ξ x ψ y + ξ y = ψ x ξ y 1 ψ x ξ x ψ x ξ x gra z T RA ψ y ξ y ψ x + ξ x 1 ψ y ξ y ψ x ξ x 1 ψ y + ξ y 1 ψ x ξ x ψ y ξ y ψ x + ξ x 1 (3.69) The final expression in equations (3.67), (3.68) and (3.69) was obtained by defining the products of the incremental error angles to be negligible. If we substitute equations (3.67), (3.68) and (3.69) as well as equations (3.64) into equation (3.65), we obtain the following expression for the combined angular rate measurement. Ω B T ot = Ω x Ω y θ y + Ω z ( ψy + ξ y ) Ω x φ y + Ω y Ω z (ψ x + ξ x ) Ω x ( µy + φ x ) + Ωy (η x θ x ) + Ω z This expression can be broken up into the following equation 1 θ y ψ y + ξ y Ω B T ot = φ y 1 (ψ x + ξ x ) (µ y + φ x ) η x θ x 1 from which we can finally define the gyro misalignment matrix as T B gia = 1 θ y ψ y + ξ y φ y 1 (ψ x + ξ x ) (µ y + φ x ) η x θ x 1 (3.7) Ω x Ω y Ω z (3.71) (3.72) Equation (3.72) will be used to define the gyro misalignment matrix when the sensor compensation is computed as part of the general set of IMU computations. It is equivalent to the F Algn term that was used in the gyro description of Savage in equation (3.31). The derivation presented in this section was first performed through hand-computation and then entered into Matlab using the Symbolic Math Toolbox to verify the result. The

100 Chapter 3 Unified FOG Sensor Model 84 two approaches produced the same result. The value of the Symbolic Toolbox implementation of the matrix calculations should become evident whenever configuration changes to the IMU are performed that influence the misalignment computations or if some of the assumptions that were made as part of this derivation appear to be inaccurate. One area of possible change is the orientations of the gyro coils within the CA. The assumed orientations are presented in table 3.1. If a different set of orientations for the RA relative to the IA emerge as part of the system development process, the orientation matrices that define these orientations could simply be redefined in the Symbolic Math code. Matlab will generate the new gyro misalignment matrix that can be used as part of the sensor compensation process Practical Insight Misalignment Computations To end off the derivation of the gyro misalignment matrices, it is perhaps worth to consider the total process that was described in the preceding section and consider what this process is really saying. Consider figures 3.16 and 3.17 where the gyro casing and assembly are presented with all the axes overlaid. The first part of the misalignment computation consisted of the rotation of the measurements of the respective gyro input axes for the x, y and z gyros into the reference axes as shown in figure The reference axes described the axes system fixed to the casing of the gyros and the input axes define the actual coil axes around which the true rotations are measured. Through this transformation, the raw sensor measurement is resolved into a useful gyro measurement. The second rotation consisted on the rotation of the respective gyro measurements to align these with the cluster axes. The cluster axes is roughly aligned with the IMU assembly axes (also known as the body axes) and is used as an intermediate axes system to define the rotation. This set of rotations is used to define the orientation of the respective gyros (the casings depicted in figure 3.16) within the assembly. As can be seen in figure 3.17, the x gyro is mounted onto the IMU assembly so that the z axis (the axis about which a rotation is measured) of the x gyro is aligned to the x axis of the IMU assembly. In a similar way, the y gyro is mounted onto the IMU assembly so that the z axis (the axis about which a rotation is measured) of the y gyro is aligned to the y axis of the IMU assembly and the z gyro is mounted onto the IMU assembly so that the z axis (the axis about which a rotation is measured) of the z gyro is aligned to the

101 Chapter 3 Unified FOG Sensor Model 85 θ x IRA IA IA θ y IAX θ y θ x IRY IAY IRX Figure 3.16: Gyro casing with the axes for the misalignment computation between the input and the reference axes. z axis of the IMU assembly. The third rotation defines the misalignment between the cluster axes and the IMU assembly (the body axes). This set of misalignments define the accuracy with which the individual gyros can be repeatedly re-inserted into the IMU assembly after these sensors were removed for single-axis calibration. Equation (3.72) defines the sequence on rotations that was defined here. It should be noted that products of misalignment angles was considered to be negligible and therefore excluded from this expression. This fact therefore explains the absence of the µ x and η y terms from equation (3.72). Misalignment Calibration The above discussion should clarify the steps that need to be taken when the misalignment angles of the gyros or IMU need to be determined as part of the calibration process. If the gyros are sold as individual sensors, then it is only the IA to RA misalignment angles that need to be determined. The IEEE Std [54] presents a detailed description for the computation of these angles.

102 Chapter 3 Unified FOG Sensor Model 86 Z RAx x-gyro Y B y-gyro Z RAx X B z-gyro Z B Z RAz Figure 3.17: IMU assembly with the axes for the misalignment computation. For the case where the gyros are used as part of an IMU, there are two possibilities for the computation of the comprehensive misalignment angles. The option that is usually taken requires the use of a 3-axes rotation table (such as the one presented in figure 1.3) to compute a single misalignment description for the IMU that bundles all the components discussed here into one rotation matrix. If a high-accuracy 3-axes table is available this method of calibration is very accurate. The other option that can conceptually be taken is to break the misalignment calibration process into different steps whereby the IA to RA misalignment calibration is performed using the steps defined in IEEE Std [54] and the RA to CA orientation is defined from knowledge of the system. This leaves the CA to body axes misalignment to be determined. This angle should be highly repeatable since the impact of temperature on the mechanical structure is not significant. It will therefore be possible to compute this angle from the manufacturing process or determine it using laser alignment equipment. Thereafter the overall misalignment matrix can be determined through the combination of the various components. The accuracy of the computed matrix can be verified using a slight modification of the method presented in IEEE Std [54]. This last method

103 Chapter 3 Unified FOG Sensor Model 87 where the calibration is divided into different components is of particular interest in situations where expensive 3-axes rotation table are not available, but the accuracy of this process is doubtful and it is not the method of preference if a 3-axes table if available. 3.5 Conclusion In this chapter the mathematical foundations for the description of the various errors present in a FOG were developed. It was shown that the different gyro measurement models found in various publications are all equivalent. The measurement models were extended to define a unified gyro model equation and this model equation was used as the basis for the development of an empirical description of the errors present in the FOG on which this study is based. Together with the noise components present in the sensor, the empirical error model was used to create an error surface that is based on two input parameters with the gyro measurement error. This error surface will be used in the rest of this document to create representative measurements of the gyro under investigation. The last section of the chapter was used to present the orientation angles that need to be computed when a gyro is calibrated for use in both a single-sensor and an orthogonal set configuration.

104 Chapter 4 A New Calibration Method 4.1 Introduction In chapter 2.5 the four different aspects of the systems identification based approach to calibration was discussed. Referring to the SID strategy presented in figure 2.3, this means that the experimental design; model structure; criterion of fit; and the identification strategy blocks need to be defined as part of a comprehensive calibration strategy that is based on the principles of SID. The focus of this chapter is to address these components of the SID strategy and thereby develop a new approach to gyro calibration. The identified shortcomings in the research on inertial sensor calibration that was presented in chapter 2.5 will be incorporated into this new strategy. Due to practical reasons the definition of the model structure will be presented before the experimental design since the modeling strategy will directly influence the experiment that will be used to determine the parameters of the chosen model. 4.2 Calibration Strategy Before the different components of the new calibration strategy are discussed in detail, it is important to first define some guidelines for this new strategy. 88

105 Chapter 4 A New Calibration Method 89 The breakdown of the gyro misalignment error parameters was presented in detail in chapter 3.4. At the end of the discussion it was mentioned that the widely accepted practice concerning the calibration of the misalignment components is to bundle together all the misalignment parameters in the gyro cluster into a single misalignment parameter that is determined during calibration. The parameters are characterized using a three-axes calibration table or using a high-accuracy orthogonal mount (generally called a cube ) that is used on a single-axis rotation table. The misalignment computations constitute a relatively small component of the overall calibration time. The process to determine these parameters is well-defined and it was seen during the literature study in chapter 2 that a number of researchers have investigated alternative approaches to specifically address the misalignment calibration. For this reason the computation of the misalignment errors will be excluded from this investigation into an alternative calibration strategy. The focus of this investigation will rather be on a single axis strategy that will be used to perform the calibration. As part of this strategy the error parameters that influence a single gyro will be identified and a new calibration strategy will be developed to focus on the parameters. Throughout the rest of this document this approach to calibration will be referred to as a single-axis calibration strategy. As a motivation for the single-axis approach, the sensor level misalignment between the input and reference axis that was presented in chapter can be used. Referring back to equation (3.13), it can be seen that the z-component of the angular rate as resolved in the reference axis is effectively the same as the value measured in the input axis of the gyro since, for small misalignment angles, cos θ 1. This means that no measurement deviation will be detectable from the reference axis measurements that will be used in the calibration and that there should be an insignificant impact on the accuracy of the calibration of the other system errors if the sensor level misalignment parameters are simply ignored. During the definition of the empirical calibration model in chapter it was highlighted that only a two-dimensional sensor model was developed. The new calibration strategy will be based on this model as it is considered to be a straight-forward extension of the current research to include additional input parameters to the system. The current calibration problem is therefore defined for a two input and one output system, which also makes it relatively easy to visualize the calibration results.

106 Chapter 4 A New Calibration Method Model Selection From the discussions presented in sections one can see that the dominant contributing factor to the time it takes to calibrate a FOG IMU is the requirement to calibrate the system over the full range of environmental conditions. As the traditional calibration approach of only calibrating for the static environmental temperature is already considered to be too time-consuming, the addition of a requirement to calibrate over the other environmental domains presented in section could be considered as unrealistic, unless an alternative approach is followed. The new calibration strategy should be usable to improve the calibration accuracy and shorten the calibration time in accordance with the requirements defined in chapter 1. One possibility is to follow a combination of classical and modern (possibly neural network based) methods where the primary calibration parameters (presented in section 1.2.2) are determined using the classical approach while the non-deterministic and non-linear errors are characterized using the modern techniques. Such an approach would make use of the traditional temperature and rotational profiles presented in table 2.1 and figure 2.2. This approach would however still require excessive time on the rotation table. Another possibility would be to use a modified rotational sequence and temperature variation profile to generate training data from which all the calibration parameters (the constants, the non-linearities and perhaps even the noise components) could be determined using only the neural network. Although this could be considered to be a drastic change from the classic method it is the method that was implemented for this study. The approach that is suggested for this new calibration scheme is to use the multidimensional, non-linear function approximation properties of neural networks to generate a set of calibration parameters that can adequately address the full calibration domain. Such an approach to calibration was suggested by Guo et al. [42] where they simultaneously used temperature and measurement voltage information to estimate the temperature dependent bias and scale factor dependent errors. They used a statistical software package to estimate a multi-factor linear regression model that would fit the data and relatively good results were obtained from their calibration strategy. For the work presented in this thesis the calibration of the primary calibration parameters as well as the non-linear effects will be attempted using this multi-dimensional approach with a neural network as the function approximation tool instead of the two-factor multiple-order polynomial regression that was used by Guo. As identified in chapter 2.5, there have been a number of people who have investigated

107 Chapter 4 A New Calibration Method 91 the use of neural networks in the modeling or characterization of inertial sensors. However, almost all of these references presented the neural networks as a strategy to the modeling of the sensors without an in-depth investigation into the ability of the sensors to model the data and in some cases the errors that were estimated were contradictory to the fundamental properties of neural networks. It is therefore considered appropriate to use neural networks as the modeling strategy for the gyro calibration and to perform an indepth investigation into the ability of these structures to adequately model the errors in the sensors. Following the decision to make use of neural networks to perform the calibration of the gyros, the most important questions that need to be answered are what type of neural network will be used; and how will the network be trained. These are not simple questions to answer, but with the abundance of research that has been performed in the last 3 years, significant pointers exist to guide one towards the best possible solutions based on the available knowledge Network Architecture The term Network Architecture will be used in the current context to refer to the type of network that will be used to implement the calibration. The architectures that are usually used to perform regression tasks are Multi-layer Feedforward Networks (MFFN), Radial Basis Function Networks (RBFN) and Support Vector Machines (SVM). Each one of these architectures have particular advantages and disadvantages when it comes to the particular problem that is being addressed. MFFNs and SVMs are complex networks which usually require significant amounts of time for training (Bishop [9]). As discussed by Hong et al. [49], SVMs have been shown to always converge to a global optimum, but due to the fact that the training of SVMs addresses a global optimization problem, the training takes a long time and the resulting network is not as parsimonious as it could be. This fact was investigated in a study performed by Shen et al. [17, 19]. As indicated by Hunt et al. [52], RBFNs have the best function approximation ability and when used with suitable training strategies such as the one proposed by Chen et al. [16] sub-optimal, realistic networks are obtained that train much faster than other types of networks due to the limited degrees of freedom of this type of network. The approach that will therefore be followed in this paper is to follow a similar multi-dimensional calibration

108 Chapter 4 A New Calibration Method 92 strategy as that presented by Guo et al. [42], but to perform the function approximation using an RBFN. Broomhead and Lowe [12] were the first to use Radial Basis Functions (RBFs) as the nodes (neurons) inside a neural network. The architecture that they proposed became known as Radial Basis Function Neural Networks or simply Radial Basis Function Networks. RBFNs are neural networks having a relatively simple structure of three layers (an input, hidden and output layer) as presented in figure 4.1-(a). The input layer is connected to the hidden layer and the hidden layer consists of a number of radial basis function nodes (as presented in figure 4.1-(b) 1 ) with the outputs of this layer being summed by a layer of weights to form the output of the network. The radial basis functions are usually implemented as Gaussian functions that are defined as each having a different centre but the same variance (width). In some cases the variances of the RBF nodes are defined to be adjustable throughout the network, but this is not the norm. The training of the network therefore consists of the determination of just three sets of parameters, being the number of nodes or neurons, the location of these nodes and the output weights associated with the nodes. In the case where variable width nodes are used, the training process also needs to determine the width of each node. The training is performed in two stages with the number of nodes and center of the radial basis functions being determined during the first stage and the weights connecting the hidden layer to the output layer being determined during the second stage of training. The simple architecture and limited degrees of freedom result in very short training times for this type of network. The working of an RBFN for regression problems is quite intuitive. To illustrate this an example taken from the documentation of the Matlab Neural Network Toolbox will be used. Figure 4.2 presents a graph showing three nodes the one centred about (blue), one centred about 1.5 (red) and one centred about -2 (green). The magenta line in figure 4.2 was obtained from a weighted combination of the three identical nodes. The weight of node 1 was 1, node 2 was also 1 and node 3 was weighted by.5. It should be obvious that one should be able to approximate a more complex curve through the addition of more nodes and the selection of the correct weights for each node. The example presented here represents an intuitive description of the working of an RBFN, but it is important to take note of the fact that the network is not constrained to the approximation of just single-dimensional functions. A formal definition of an RBFN that utilizes Gaussian basis functions is provided by Orr [77] and by Orr et al. [79]. Using 1 Figure 4.1-(b) is taken from the Matlab Neural Network Toolbox User s Guide[43].

109 Chapter 4 A New Calibration Method 93 N 1 Input 1 W 1 W 2 Output N 2 Input n W k N k (a) Radial basis function neural network architecture. (b) Radial basis function. Figure 4.1: Radial basis function neural networks Weighted Sum of Radial Basis Transfer Functions Node 1 Node 2 Node 3 Weighted sum 1 Output a Input p Figure 4.2: Demonstration of the modeling ability of an RBFN for a single dimensional problem.

110 Chapter 4 A New Calibration Method 94 figure 4.1(a) as the starting point, it can be seen that RBFNs are defined as linear in the parameter networks, thereby indicating that the output of the network is a linear combination of the outputs of the hidden units. The network creates a model f(x) of the function y(x) and thereby maps the input vectors x R n to y R using a single hidden layer in the network. This mapping can formally be defined as f(x) = m w j h j (x) (4.1) j=1 where m is the number of hidden units in the network, w j is the weighting factor from the j-th node to the output and h j (x) is the transfer function of the hidden node j. This transfer function can be defined as h j (x) = φ (z j (x)) (4.2) where φ is the radial function that will be used as the basis for the network (in this case it is a Gaussian function e x2 ) and z j (x) is a function that defines the scaled distance from some point to the centre of the j-th node. We can define z j (x) as z j (x) = n (x k c jk ) 2 r 2 k=1 jk (4.3) where n is the number of input parameters, c jk is the centre of the j-th node from the k-th input and r jk is the scaling parameter for the node which defines the width or spread of the node. The formulation of c jk and r jk will become clear if equation (4.3) is expanded as follows and it is recognized that z j (x) = (x 1 c j1 ) 2 r 2 j1 + (x 2 c j2 ) 2 r 2 j2 h j (x) = e z j(x) (x n c jn ) 2 r 2 jn (4.4) (4.5) From equations (4.4) and (4.5) one can see that the addition of a second input will result in the Gaussian transfer function that describes the nodes becoming a bivariate Gaussian. Additional inputs will change this to a multi-variate Gaussian function. The Gaussian function associated with a particular node will have a separate centre and scaling ratio for each input into the network. For a single input to single output type of network, the regression will effectively be as defined in figure 4.2 where the network is approximating a curve that describes the relationship between the input and output parameters. For a two-input, one output system this relationship will become a surface which

111 Chapter 4 A New Calibration Method 95 can be visualized as the input parameters defining the x and y axes and the output being represented by the z-axis. For more than two inputs the functional relationship becomes a hypersurface that is difficult to visualize Network Training Strategy Following the decision on the type of network that will be employed, the next step is to consider the training strategy. For the application under consideration for this project, the network training can be done in one of two ways, being batch (off-line) training or on-line (sequential) training. For the purpose of the current research it was considered to be adequate to only consider the off-line training and rather look into the quality of the resulting network and its ability to be used in an alternative calibration strategy. On-line implementation of the network training will be a natural extension of the current research with a strong focus on the engineering value on the new calibration strategy. Regularization In the case where too many training parameters are used within the training structure of an RBFN the network will start to match the noise in the signal. This behaviour is known as overfitting and results in the network having a poor generalization ability to new input data. This situation was discussed by Broomhead and Lowe [12] and the extreme case was mentioned where a node was placed on every one of the training points, thereby producing a network that fits the training data perfectly, but that is of little practical use due to it lacking the ability to generalize to previously untrained inputs. For this reason it is desirable to produce a network that will not overfit the training data. Regularization (Orr [76] and Orr [78]) is a technique that is used to reach this goal. Consider the expression for the functional model of the network as f(x) = m w j h j (x) (4.6) j=1 where m is the number of hidden units in the network, w j is the weighting factor from the j-th node to the output and h j (x) is the transfer function of the hidden node j. Linear regression on a set of training data (x i, y i ) p, which consists of p sets of input-output data pairs, can be performed by solving the set of equations defined by y = Hw + e (4.7)

112 Chapter 4 A New Calibration Method 96 where y = [y 1... y p ] T is the p-dimensional set of sample outputs in the training data, H is the system design matrix, w is the vector of output weights of the network that needs to be determined and e is the vector of network estimation errors. The network estimation error is usually defined as the sum squared error or E = p (y i f(x i )) 2 = e T e (4.8) i=1 with the weights being determined from the least squares formula w = (H T H) 1 H T y (4.9) When the process of regularization is used to regulate the complexity of the model, a penalty term is added to the sum of squares error as p p E = (y i f(x i ) 2 + λ wj 2 (4.1) i=1 i=1 = e T e + λw T w (4.11) When this error is optimized, large components in the weight vector w are avoided. The parameter λ is known as the regularization parameter and the effect of this last expression is that the training accuracy of the network is optimized, but the tendency of the network to overfit is inhibited by penalizing large output weights. A number of different methods exist for the selection of the optimal value of λ and are included in the Orr-package [8] that is used to model the networks in this study. A detailed discussion of these optimization methods is beyond the scope of this study. Orr discusses these options in great detail in references [76] and [78]. Batch Training Batch training of RBFNs refer to the adjustment of the various network parameters under conditions where all the input and output measurements needed for the training to the network has already been sampled and the network is trained as a post-processing operation. Probably the most significant researcher in the area of batch training algorithms for RBFNs is Chen 2 and his colleagues from the School of Electronics and Computer Sciences at the University of Southampton in the UK. They developed one of the earliest RBFN 2

113 Chapter 4 A New Calibration Method 97 training strategies as the Orthogonal Least Squares (OLS) method (Chen et al. [15] and [16]). The 1991 article ([16]) presents a detailed description of the OLS method. Other training strategies for RBFNs include clustering strategies and Expectation Maximization (EM) based algorithms, both of which are extensively discussed by Nabney [69]. As Sundararajan et al. [19] indicates, the clustering methods (such as K-means clustering) are better suited to pattern recognition problems and do not perform well for function approximation problems. The OLS algorithm is the most widely adopted algorithm for the training of RBFNs due to its efficiency during training and its ability to train a compact network. A number of derivatives of OLS were developed through the years to optimize the training time and the compactness of the resulting network. Sundararajan et al. [19] presents a comprehensive study on the various RBFN training strategies developed before In a recent set of publications ([49, 17, 19]) Chen and his associates present an overview of the various RBFN training strategies and discuss which strategies are the most promising solutions at present. From the options they present, the Locally Regularized Orthogonal Least Squares using the Leave-One-Out criteria (LROLS-LOO) (Chen et al. [18],[17]) and the Orthogonal Forward Selection using the Leave-One-Out criteria (OFS-LOO) (Chen et al. [19]), appears to be the best solutions to the batch training problem in RBFNs. The LROLS-LOO algorithm is a very fast training strategy resulting in a very compact network, but the resulting network is not as parsimonious as those resulting from the OFS-LOO training. The OFS-LOO training algorithm takes slightly longer to train every node of the network when compared to the LROLS-LOO algorithm, but since it results in less nodes than the LROLS-LOO algorithm, the overall training time is equivalent. From the results observed from the referenced publications, the OFS-LOO algorithm is the best candidate to perform the network training and meet the objectives of a parsimonious network, but it was decided to rather implement a simpler algorithm with reduced complexity compared to the OFS-LOO algorithm. The objective of this thesis is to prove the hypothesis stated in chapter 1 and for this reason the less optimal, but easier to implement OLS algorithm was considered to be good enough to prove the concept. Since the newer training strategies are only optimizations of OLS, the resulting network can easily be optimized using these strategies once the ability of calibration using neural networks has been determined.

114 Chapter 4 A New Calibration Method Experimental Design The focus of this section is to address the Experimental Design block in the SID strategy presented in figure 2.3. The trajectory used to generate the calibration data is one of the key aspects of the alternative calibration approach. The data gathering blocks in the SID strategy presented in figure 2.3 is addressed by means of the trajectory selection. For some systems it is possible to randomly excite the system to generate a dataset from which the system can be characterized, but for the gyro calibration process this is not possible. It is not practical to make the test equipment (rate tables and environmental chambers) to randomly visit sample points. For this reason the data-points need to be collected along a trajectory traced out by the sequential adjustment of the environmental parameters. The focus of this section is to address the process of designing such a trajectory Training Trajectory Neural networks are known to be poor extrapolators. What this means is that the network cannot make good predictions in areas outside the domain in which it was trained. Furthermore, it should also be noted that, to be able to model a particular aspect of any error domain, it is important to provide data from that domain to the network during the training phase. When a system is described using normal mathematical expressions, the functional relationships used to describe the system usually define different features of varying complexity across the complete functional space. This is not true for a neural network. The general statement is that if it is not seen by the network, it is not modeled by the network. When designing the training trajectory from this perspective, the importance of presenting all the important features within the functional space of the gyro to the network during training is highlighted. A second component to be kept in mind during the development of a training trajectory is the cost of the trajectory. As part of the problem statement in chapter 1, it was stated that one of the objectives of the alternative calibration strategy should be to reduce the calibration time and therefore the financial cost associated with calibration. From this perspective one could define a cost function for the calibration with the two parameters to be balanced being accuracy against calibration time. One could gather huge amounts of data and perform an almost perfect calibration (high accuracy), but this would result in an excessive calibration cost as the time spent to gather the data would result in significant financial expenses. On the other hand, it is possible to gather a reduced amount of data

115 Chapter 4 A New Calibration Method 99 in a short time (low financial cost), but it could potentially result in reduced accuracy. Since the design of the experiment constitutes a key step in the SID process, one would expect the authoritative texts on Systems Identification to discuss this step in great length, but this is unfortunately not the case. Ljung [65] approaches the subjects from a frequency domain perspective instead of working in the time domain. Ljung simply prescribes that the experiment must be maximally informative and that it should have adequate dynamic richness, but these aspects are discussed in very theoretical terms. Jategaonkar [56] refers to the fact that the frequency domain criteria of Ljung is used in the field of aircraft characterization, but that a significant amount of the characterization work in the time domain is often based on experimental trajectories that attempt to optimize the observability of the parameters to be identified. In these cases heuristically defined flight trajectories are often used to improve the informational richness of the experiment. Jategaonkar emphasizes the fact that prior knowledge of the plant to be identified (the aircraft in his case) is of primary importance for the experimental designer to be effective in his task. This perspective is supported by Nelles [73] and it is Nelles who perhaps presents the most practical guidelines on the design of the characterization experiment. There are a couple of important aspects that he highlights: If the behaviour of the process is not discernable from the data (the particular dynamic component is not observable), then it is not possible to incorporate that aspect of the system into the derived model without the inclusion of prior knowledge directly into the model. In general this would mean that, if the particular behaviour is not discernable, it will not be represented in an automatically generated model of the system such as a neural network. Few tools exist and little research has been conducted to generate more general tools and practical guidelines on this aspect of the design of the systems identification experiment. The experimental design is highly dependent on the expertise of the engineer who will be designing the experiment. From these comments one can therefore conclude that the design of the excitation signal should be based on the following guidelines: It must cover the complete functional space of this system-under-test, meaning that it should cover the whole operational domain of input and output signals in terms of the frequency components, dynamic response of amplitudes of these signals.

116 Chapter 4 A New Calibration Method 1 Care must be taken to ensure that any specific system behaviour that should be incorporated into the final model is triggered by the excitation signal so that it can be incorporated into the model. The knowledge of a systems expert is essential to the design of the signal as the system requirements of the final model and the capabilities of the test equipment needs to be considered during the design of the experiment. To use an example from the area of flight control, the expert will know what dynamics should be present in the model to design an accurate flight controller, but the expert will also know what the mechanical limitations of the aircraft are so that an experiment will not be designed that could damage the aircraft. A number of automatic trajectory generation strategies have been investigated that could use these guidelines to automatically generate the best trajectory. Areas from the fields of Design of Experiments (DoE) (Atkinson [5] and Cohn [23]), active learning (Abdullah and Allwright [1] and Thrun [114]) and exploration such as that which is used as part of the robotic problem of Simultaneous Localization and Mapping (SLAM) ([15]) were investigated to be used in the generation of an optimal training trajectory. The option to define the training trajectory through a process of observability enhancement of the calibration parameters in a way similar to that discussed by Jategaonkar [56] was also considered. It was finally decided that, while these approaches could result in some very interesting results, it would distract the focus from the prime objective of this research, which is the investigation into the use of neural networks for the development of an alternative calibration strategy. It was decided to first prove the new calibration concept using a heuristically designed training trajectory and postpone the investigation into optimal trajectory generation strategies for follow-up work to this research project. The initial design of a heuristic training trajectory was focussed on the reduction of the time needed to gather the calibration data. The objective was to determine whether it is possible to define a minimum time trajectory and then evaluate this trajectory against the training accuracy of the network. If it proved that the accuracy of the resulting network was not sufficient, the complexity of the training trajectory could be increased (thereby increasing the calibration time) to reduce the residual calibration error. Classical Calibration Trajectory The trajectory for the gather of data under the classical calibration strategy is presented in table 2.1 and figure 2.2. The exact sequence could differ between different references,

117 Chapter 4 A New Calibration Method 11 but it essentially consists of the gyro or IMU being taken through a particular stepped temperature sequence. It usually means that an environmental chamber is mounted within the gimbals of a 3-axis rotation table. An example of such a setup is presented in figure 4.3. At each measurement step in the temperature sequence the system would be left for the ambient temperature and the internal temperature within the sensors to stabilize. Once a stable temperature has been reached, the sequence of rotations presented in table 2.1 would be executed with the additional option of these rotations being executed for different rates of rotation. Following the execution of these steps, the temperature would be adjusted to a new measurement point and upon stabilization of the ambient and internal temperatures, the sequence of rotations would be repeated. This is a very time consuming trajectory to execute, but it results in the calibration constants being determined very accurately. The temperature and angular rate dependence of the bias and scale factors would usually be determined using a polynomial description of the average variation of these parameters as a function of the environmental conditions over the complete measurement space. The time-cost associated with temperature commands is the dominant factor in the cost-assessment of the trajectory design. Typical environmental chambers can only achieve single-digit temperature adjustment rates with the rates being measured in units of degrees Celsius per minute ( C/min). A value of 5 C/min is usually considered to be quite a high temperature adjustment rate. Using this rate as the benchmark 3 it is possible to determine an approximate time-cost for a typical calibration exercise. The specified operational temperature range for a gyro is usually from -6 C to +8 C. During the calibration 2 C incremental steps are taken with 3 minutes being allowed for the ambient and internal temperatures to stabilize and every rotational sequence that is performed at a stable temperature point taking approximately 3 minutes to be completed. Using these figures, one can deduce that a typical calibration exercise can take 8 hours using this sequence! If multiple rates of rotation are used at every measurement point or smaller temperature incremental steps are taken the time needed to gather the calibration data can even double. 3 The facilities used to perform the calibration for this research had an environmental chamber with a 5 C/min capability.

118 Chapter 4 A New Calibration Method 12 Figure 4.3: 3-axis rotation table (motion simulator) with environmental chamber mounted on the inner axis. New Calibration Trajectory Taken that a single-axis calibration strategy (as presented in section 4.2) is being developed, one could use a single-axis rate-table to perform the calibration. This would mean that the single-axis table can be mounted within the environmental chamber, which will allow for the experiments needed for this study to be performed at a relative low cost. An

119 Chapter 4 A New Calibration Method 13 Figure 4.4: An example of a single-axis rate table and an indication of how it is mounted within an environmental chamber. example of a single-axis rate table 4 and a similar table mounted within an environmental chamber is presented in figure 4.4. The typical inertia of a single axis of a rate table is quite low which means that the rate of rotation of the table can be varied quite rapidly. The delay for the table to reach new rate commands will usually be a few seconds or even milliseconds in some cases. For the purpose of the calibration exercise one could therefore consider the rate change in the table to happen instantaneously. The cost associated with a rate change could thus be defined to be negligible. There are a couple of variations that can potentially be incorporated into a new calibration trajectory. These changes are primarily linked to the working of the neural network. Churchland [22] presents a very insightful explanation of the functioning of a neural network. He indicates that, when a system is described in terms of mathematical expressions, the various parameters of the functional description needs to be defined to complete the description. A neural network does not attempt to replicate the functionality of a system by learning the mathematical relationship that can be used to describe the system. A network rather performs a mapping from the input to the output domain without any knowledge of the mathematical relationship. This mapping is embedded into the structure of the network during the training process. As indicated by Churchland, 4 Images obtained from the website of Ideal Aerosmith (

120 Chapter 4 A New Calibration Method 14 the mapping is therefore an abstraction of the functional relationship present within the input/output data-pair presented as the training data with the functional relationship being extracted from the set of discrete data-points presented to the network. To obtain a comprehensive and accurate mapping of the underlying functional relationship in the system it is necessary to create a training dataset of points that are evenly distributed across the functional space. It may be that a uniform distribution of the datapoints across the complete functional space is not available. Since the network can only model functional features when presented with data describing it, it is therefore necessary to at least ensure that the important aspects that are needed as part of the functional mapping be included within the training data. From this description of the internal working of a network, it is highlighted that the characterization process for a neural mapping differs somewhat from that which is followed for a mathematical functional description. The classical calibration trajectory was specifically defined with the aim of the identification of the various parameters within the mathematical relationship that describes the errors in the system. In this trajectory a significant number of redundant measurements are included that are meant to eliminate noise through averaging. Since the neural network s internal structuring (its ability to capture the functional relationship in the data) is based on the examples of the input-output relationship in the system that is presented to the network during training, one can follow a completely different type of trajectory. The objective of any set of training data that is generated should be to cover the system s functional space as comprehensively as possible. Redundancy is not required and in some cases redundant measurements are actually detrimental to the training accuracy of the network as it may result in the network being over-trained. In the case of the neural network the presence of noise in the system is handled through the process of regularization which ensures that the network does not incorporate the noise into its internal structure. From this perspective one can see that the first change that could be incorporated into the design of a new calibration trajectory is to make use of a continually varying temperature profile that does not make use of steps of static temperature at which the rotational data is gathered. This would result in a sweep of the temperature domain rather than the stepped approach. Using the value of 5 C/min for a typical environmental chamber and an operational environment that ranges from -6 C to +8 C, this would result in 28 minutes that is needed to complete a single sweep of the temperature domain. For the classical approach a single traversal of the temperature domain was used, so the

121 Chapter 4 A New Calibration Method 15 starting point will be to attempt to only make use of a single sweep across the temperature range 5. Since the neural network creates a non-linear mapping from the input data, which can consist of more than one parameter, to the output data, a typical training set should consist of a set of all the input parameters with the associated output. This leads one to the second modification that can be made to the classical calibration trajectory, which is to adjust the angular rate of rotation and the environmental temperature simultaneously. For the classical trajectory this was not possible as it would have resulted in the influence of the one parameter not being discernable from the other. This is not a problem for the neural network as the simultaneous adjustment of parameters simply brings the training data to a new operating point from which the multi-dimensional input-output mapping can be performed. For the system under investigation temperature data was sampled in the range from -2 to 6 C and angular rate measurements were taken from -1 to 1 degrees per second. The empirical error model was based on these measurements and it was consequently decided to constrain the calibration trajectory to this domain as well. A practical calibration trajectory can easily be extended to the required calibration range. A final aspect which needs to be considered during the design of a calibration trajectory is the thermal lag associated with the gyro packaging. As discussed in chapter 3, the coils of the FOG are not usually left open to direct environmental exposure, but are usually packaged within some kind of casing for environmental protection. The protective casing could vary between a casing per gyro such as is seen in figure 3.16 to a single external casing for all the inertial sensors together. Depending on the specific application, additional thermal isolation can also be added to the packaging to further isolate the sensors from temperature fluctuations. The impact of such thermal isolation is that the temperature at the gyro coils will lag the environmental temperature. This lag could range from a couple of seconds to a couple of minutes or even longer in some cases where the sensors are thermally controlled. Since final production of the sensor-under-test has not commenced at the time of writing, final thermal isolation data was not available. From experience gained in the field of inertial navigation and by working with sensors in the class of the 5 It should be noted that this concept by itself is not new and is generally referred to as a thermal ramp test (Titterton and Weston [115] and Aggarwal et al. [4]). It is the combination of the thermal ramp with some of the other components that creates the novel calibration trajectory.

122 Chapter 4 A New Calibration Method 16 current sensor, it was found that a typical thermal delay should be around 5 minutes 6. This value will be used throughout the rest of this document as a representative figure of this parameter. More realistic values could be obtained at a later stage, but the impact of a thermal delay on the system should be determined from this illustrative value. Since the thermal isolation will cause the internal temperature of the sensor to lag the environmental temperature changes by 5 minutes, any new calibration trajectory will have to be designed by taking this delay into consideration. The impact on the calibration trajectory is not too dramatic since the internal temperature of the sensor will just take 5 minutes longer to reach the required temperature, but it will reach it. The most important impact should therefore be that, if a thermal ramp is used for an up and down profile, the temperature will have to be kept constant at the one endpoint of the profile before it is adjusted in the other direction. This will ensure that the gyro temperature reaches the reference temperature value. Having made all these comments on the thermal delay, one could also ask whether this property of the system really impacts the ability of the neural network to perform the required calibration action. It could be argued that the network is used to map the operational space to the sensor error and that no temporal aspects of this relationship are incorporated into the network. The network effectively generates the expected measurement error for the temperature and rotational rate experienced at the sensor at that moment. The actual environmental temperature is therefore of no real significance as long as there is a sensor mounted next to the gyro inside the casing. If the temperature sensing is performed outside the gyro casing it will mean that the complexity of the neural network will increase with the temperature measurement history of the external sensor also having to be provided as an input to the network. For the current research project the general assumption will be made that there is a temperature sensor mounted inside the gyro casing. This is indeed the case for the specific gyro on which this study is based. The result of this assumption is that the network can be trained without taking the thermal delay into consideration and still be representative of the actual gyro. For this reason the impact of the thermal delay will be excluded from the calibration strategy developed in this study. 6 These values were discussed and confirmed with colleagues at Denel Dynamics.

123 Chapter 4 A New Calibration Method 17 Single Temperature Scan with Rate Cycling Based on the former statement that the angular rate adjustment effectively carries no cost while the temperature adjustments are very costly, the first trajectory that was investigated used a single temperature scan from the upper temperature limit to the lower limit or from the lower limit to the upper limit. Multiple cycles of the angular rate (from the lower limit to the upper limit and back) was performed during this single temperature scan. The trajectory traces the functional space like a inkjet-style printer where there is rapid motion in the lateral direction with relatively slow, unidirectional motion along the longitudinal direction. A very low cost addition was made to the scanning trajectory through the inclusion of an additional set of rate-scans at constant temperature at the beginning and at the end of the temperature cycle. The impact of these additional scans was to ensure that the domain boundaries at the lowest and highest temperatures are properly covered by the training data. The cost associated with the minute or two that is needed to scan from the upper to the lower angular rate limit is so low compared to the accuracy increase, that this extension to the trajectory was considered to be justifiable. It was found that these boundary areas were regions of high estimation error when the additional scans were not included. The training data was created by taking the temperature and angular rate parameters from the trajectory and computing the angular error associated with these measurement points from the empirical error model developed in chapter 3. An example of a single-scan training trajectory is presented in figure 4.5. The red circles on the graph indicate the first 1 samples and the green circles the last 1 samples. Figure 4.6 gives perhaps a better indication of the trajectory used to create the training data. The end sections used to augment the training data is visible at the temperature boundaries. The training data as a function of the sampling points is presented in figure 4.7. The flat sections at both ends of the temperature curve represents the boundary extensions that was added to the trajectory. The particular trajectory presented here only consists of 4 angular rate cycles for the purpose of making it easier to visualize on a graph. Dual Temperature Scan with Rate Cycling From figure 4.6 one can see that there are areas within the functional space that are not covered by the single temperature scan with the angular rate cycles with the result that

124 Chapter 4 A New Calibration Method 18 Training data for a single temperature scan trajectory.1.5 ω err ω meas Temperature 4 6 Figure 4.5: Single temperature scan training trajectory. the network cannot create an accurate map of the functional space in these areas. The areas of concern are in the respective troughs and peaks of the angular rate scans. There are two approaches which can be used to solve the resulting holes in the data. The one approach is to reduce the scan rate of the temperature while the cycling rate of the angular rate of rotation is being kept constant. This will result in more angular rate cycles being performed during the single temperature scan with the respective troughs and peaks of the angular rate scans being much closer together. The slower temperature scan rate will necessarily increase the cost associated with the trajectory. The alternative solution to the data coverage problem is to implement multiple temperature scans and to ensure that the angular rate cycles of every successive temperature scan are out of phase with the last scan cycle to ensure that the uncharted areas are covered by the successive scans. With this approach the temperature scan rate is kept at its maximum while the time cost for every successive addition of a temperature scan is increased by 28 min. An example of an up-down or two-cycle temperature scan is presented in figure 4.8. The cost of this trajectory is effectively 56 minutes. The red circles once again indicate the first 1 samples in the trajectory and the green circles the last 1. The ability of

125 Chapter 4 A New Calibration Method 19 Figure 4.6: Sample locations for the single scan training trajectory. this type of trajectory to fill in the holes in the data can probably be better observed from figure 4.9. If this graph is compared to figure 4.6 it can be seen that the dual scan creates measurements in areas that the single scan missed out on. The training data as a function of the sampling points are presented in figure 4.1. The flat sections at both ends and in the centre of the temperature curve represent the boundary extensions that was added to the trajectory. For the purpose of comparison to the single-scan trajectory, the particular trajectory presented here once again consists of 4 angular rate cycles per temperature scan for the purpose of making it easier to visualize on a graph. Something which only became visible with this trajectory was the periodic nature of the sample locations along the temperature axis. This fact was highlighted purely by chance as a result of the specific parameter configuration that was used for the generation of the training trajectory. When 1 samples were taken for a 4, 8, 16 or 32 angular rate cycle trajectory, it resulted in the training data portraying a periodic pattern along the angular rate axis which appears as lines along the temperature axis as indicated in figure The somewhat awkward viewing angle on the plot was used to indicate the periodic nature of the sample locations along the temperature axis. The result of this periodic pattern was that training accuracy was reduced as there were regular lines within

126 Chapter 4 A New Calibration Method 11 1 Single scan temperature and angular rate training data Measurements ω Temp Samples Figure 4.7: Temperature and angular rate data for the single scan training trajectory. the functional space that were not covered by the training data. To solve this problem, the number of samples was simply increased to 1111 to ensure that such periodic patterns were eliminated from the training data. As will be discussed later on, the adjustment reduced the calibration error as the coverage of the functional space by sample points was improved. Figure 4.12 shows the impact on the sample location with the adjustment. An a-periodic nature of the sample locations along the temperature axis can be observed from the graph. Although this phenomenon could be of academic importance, it does highlight the importance of the sample location and the experimental designer should keep the relationship between the shape of the trajectory and the sampling rate in consideration Test Trajectory The traditional approach to the testing of a neural network is to have a set of training data and a separate set of data that is used to test the accuracy of the network that

127 Chapter 4 A New Calibration Method 111 Training data for a double temperature scan trajectory.1.5 ω err ω meas Temperature 4 6 Figure 4.8: Dual temperature scan training trajectory. was trained. If the system under investigation was a single input single output (SISO) network this effectively would have meant that a training curve would have been provided with the network estimating this curve. The way the current problem is structured with a two input one output set of data, the trained network is trying to estimate the error surface presented in section Although the training data is still a trajectory, it is a trajectory across this error surface and, using the training trajectory, the network is trying to approximate the error surface. From this perspective one can therefore take two approaches to testing the estimation accuracy of the network. The first is to follow the traditional approach where a testing trajectory is used. For this system this trajectory is very similar to the training trajectory since it also criss-crosses the system s error space as a function of the angular rate and temperature. The difference is that the testing trajectory needs to be independent of the training trajectory in that it should test the estimation accuracy of the network at locations on the error space that were not part of the training data. For a real-life calibration experiment this is probably the most practical option that is available to evaluate the accuracy of the network. For the current investigation there is another option for the evaluation of the network s

128 Chapter 4 A New Calibration Method 112 Figure 4.9: Sample locations for the dual scan training trajectory. accuracy. The empirical error model that was used to create the test data is available as the absolutely accurate reference model. This error model can be used to create an error surface and the error surface that is approximated by the neural network can be compared to this surface to determine the accuracy of the estimated error surface. This is a somewhat artificial approach, but with the objective of the study being the evaluation of neural networks for use in gyro calibration, one will be able to obtain an absolute idea of the ability of the network to estimate the error surface. The use of a reference error surface for network evaluation could prove to be practical when it comes to the testing of a batch of gyros. Since the accuracy of the calibrated gyros is the primary objective, it could be considered viable to perform a proper calibration exercise to develop a reference error surface such as the one that is used in this study to create the test data. The data used in the development of this reference model for the set of gyros could be collected using a comprehensive characterization exercise of a reference gyro in a way similar to the one performed during the development of the empirical simulation model, or one could generate a very finely grained sweeping trajectory across the complete functional domain of the gyro. From this set of data one could create a similar 2-D interpolation or even a reference RBFN that can be used to benchmark the

129 Chapter 4 A New Calibration Method Double scan temperature and angular rate training data Measurements ω Temp Samples Figure 4.1: Temperature and angular rate data for the dual scan training trajectory. subsequent calibration results. The creation of a reference surface does carry additional cost, but with the possible reduction one could expect from the calibration of a single gyro and the fact that, using the single-axis approach, one can calibrate more than one gyro at a time, this cost could be offset across the whole batch of sensors. For this study both the test trajectory approach and the use of a reference error surface will be used to evaluate the calibration network. It will become apparent that more knowledge can be gained from the use of the error surface than simply using the test trajectory since the surface will cover the complete functional space whereas the testing trajectory can only test the network at a number of discrete measurement points. A typical test trajectory that can provide good coverage of the functional space is presented in figures 4.13, 4.14 and This particular trajectory was defined to contain 4 angular rate cycles per temperature cycle and 4 complete temperature cycles as well. The cost of generating such a trajectory could therefore be about = 224 minutes. The

130 Chapter 4 A New Calibration Method 114 Figure 4.11: Sample location for the linear up and down trajectory with 1 sample points and a NN scale of number of angular rate cycles was limited to 4 per temperature cycle for display purposes and to indicate the improved coverage of the functional space that can be achieved with such a trajectory. A more realistic testing trajectory consisting of 16 angular rate cycles per temperature cycle per is portrayed in figure 4.16 which shows the comprehensive coverage of the functional space which can be achieved with the right choice of trajectory. One could use one of these testing trajectories to generate the reference surface discussed earlier. One could even use this trajectory as a training trajectory due to its improved coverage of the functional space. With a total training time of just under 4 hours the cost is much higher than the proposed training trajectory, but still significantly less than the

131 Chapter 4 A New Calibration Method 115 Figure 4.12: Sample location for the linear up and down trajectory with 1111 sample points and a NN scale of time taken to collect the data under the classical calibration strategy. 4.5 Criterion of Fit Definition Background The focus of this section is to address the Choose Criterion of Fit block in the SID strategy presented in figure 2.3. Within the overall SID philosophy the Criterion of Fit serves as a yardstick to determine whether the output of the identification step resulted

132 Chapter 4 A New Calibration Method ω err ω meas Temperature 4 6 Figure 4.13: 3D view of the test trajectory consisting of 4 temperature cycles with 4 rate cycles per temperature cycle. in a good system description. Since SID is usually a black box process, it is only natural to expect that neural network based system characterizations (which are effectively also SID processes) should also make use of a well-defined Criterion of Fit. As identified in chapter 2.5.3, none of the studied references paid much attention to the definition of a Criterion of Fit whereby the suitability of any particular model to the calibration problem can be determined. One could argue that some of the presented research did not directly follow the SID process and therefore did not have an explicit Criterion of Fit, but some way of verifying that the derived model is a good system description would be expected Argument for a New Criterion of Fit There are a number of ways in which the goodness of fit of a neural network can be tested. The network is usually tested during training where the cross-validation strategy is used to ensure that the network fits the training data. The second level of testing is where a set of test data is used to determine the ability of the network to estimate the errors for input values that it has not seen before. These steps are usually included in one way or another as part of any neural network implementation. However, for this particular project it is on

133 Chapter 4 A New Calibration Method Temperature and angular rate training data Measurements ω Temp Samples Figure 4.14: Test trajectory training data consisting of 4 temperature cycles with 4 rate cycles per temperature cycle. a much wider level of testing where the ability of the network to accurately calibrate the sensor is determined. On this level of testing one needs to look at the specific application and often use expert knowledge from this field to judge the quality of the derived model. The traditional method of determining whether a gyro has been properly calibrated is to calibrate the gyro/imu, compute the calibration coefficients, implement these within the gyro/imu software and then recalibrate the gyro/imu. During the recalibration the sensor should then provide near-perfect results apart from the measurement noise. For the method under development here this process effectively means that the network will be trained using a training set and tested using another set of the same operational domain, but no real knowledge is gained regarding the ability of the network to eliminate the system errors. This is especially true in the case where noise plays a role, as it inevitably does, since it could be expected that the network will include some aspect of the noise into its derived model. As discussed by Titterton and Weston [115] and by Bose et al. [11], one way of testing the calibration accuracy of inertial sensors is to expose it to environmental conditions

134 Chapter 4 A New Calibration Method 118 Coverage of the functional space 1 5 ω meas ω meas Temperature 4 6 Figure 4.15: Coverage of the functional space for the test trajectory consisting of 4 temperature cycles with 4 rate cycles per temperature cycle. representative of its operational domain. Such an operational profile can be generated by mounting the test unit on a programmable rate table (as presented in figure 1.3 also known as a motion simulator) or it can be generated mounting it along with a highaccuracy reference IMU mounted on an aircraft or vehicle that is following a characteristic motion profile an option which is excessively expensive. The sensor should therefore be exposed to the type of angular rates that it will experience during operation and by comparing the compensated sensor output to the known angular rates generated by the rotation table or by the aircraft, the accuracy of the compensation can be judged. The exposure of the sensor to operational conditions for the post-calibration testing is important to ensure that the correct calibration description has been computed for all areas of the operational domain. However, using this verification approach does not result in a single figure of merit that can be used to define the accuracy of the gyro. Criteria such as mean square error (MSE), the mean error or the variance of the error are sometimes used to describe the deviation of the corrected sensor output from the reference measurement, but these methods do not really describe the accumulated error components. What is really needed is a more comprehensive method of defining the

135 Chapter 4 A New Calibration Method ω err ω meas Temperature 4 6 Figure 4.16: 3D view of the test trajectory consisting of 4 temperature cycles with 16 rate cycles per temperature cycle. accuracy of the compensated gyro in a similar way to some of the information criteria that is used during model selection of neural network training. Orr [77] and Nelles [73] present a number of model selection criteria (information criteria) that are generally used to train a neural network. By using these criteria during training, a network is obtained that fits the data well and still generalizes well with the objective to minimize the specific information criterion used for the training. The criterion is computed across the complete data-set used to train the network. In a similar way it should be possible to defined an information criterion that will describe the information content in the output of the calibrated gyro. An information criterion that is specifically developed for measuring the compensation accuracy of inertial sensors along the lines of those used for neural network training could be of much value, but the results would perhaps not be that intuitive. In chapter 1 it was stated that one of the key requirements of a new calibration strategy is that the calibration accuracy must be interpretable by a system expert. If a specific information criterion is defined to characterize the calibration accuracy it will not necessarily be transparent to the system expert and would also not necessarily convey much knowledge about the effectiveness of the calibration to this evaluator, even if the criteria is very effective. For

136 Chapter 4 A New Calibration Method 12 this reason it is suggested that a much more intuitive criterion from the field of inertial navigation be used to judge the calibration effectiveness. As mentioned before, Titterton and Weston [115] and Bose et al. [11] discuss a number of methods for the testing of inertial sensors and systems. One of these methods is the static navigation or navigation-in-place (Pittman and Roberts [81]) approach where the IMU is left in a static position while the measurements from the sensors are processed through the complete set of INS algorithms (as presented by Titterton and Weston [115] amongst others). This method is used to determine the impact of the residual sensor errors on the growth in the navigation solution (the computed position, velocity and orientation of the platform). As discussed in chapter 1, the accuracy of the combined INS is defined in terms of the equivalent degrees of arc (represented in nautical miles) that the position will deviate from the true position after one hour of unaided operation. The impact of the errors on the system performance is therefore presented as a single parameter the growth rate in the positional error. Although this method gives much insight into the impact that the sensor errors have on the navigation accuracy, it only determines the compensation accuracy in a very small part of the operational domain. A more comprehensive test will be to still perform the navigation processing of the sensor measurements, but then stimulate the sensors with the types of operational profiles that were previously discussed. By using the measurements generated from these profiles as the input values to the INS computations, a more comprehensive idea of the impact of the sensor errors can be determined and the errors are effectively presented in a single parameter (figure of merit) as the positional drift. Using the positional drift as an indication of the system performance is equivalent to the various information criteria that attempt to integrate the errors across a complete trajectory or profile to define how good it fits the requirements. During the processing of the gyro measurements by the INS equations, the measurements are effectively integrated three times and any measurement errors in the gyros accumulate into the overall positional error. The figure of merit that is obtained in this way is well-known to any expert in the field of inertial navigation and is readily interpretable. The positional drift can therefore be considered as a single parameter Criterion of Fit that describes the accuracy of the calibration that was performed on the sensor. As mentioned before, this figure of merit, which is defined using traditional navigation terminology, is especially important for the neural network calibration strategy since it bridges the gap between the abstract neural model of the system and the interpretation thereof in terms of inertial navigation systems.

137 Chapter 4 A New Calibration Method Practical Aspects of the New Criterion of Fit Following the development of the argument for a new Criterion of Fit in the preceding paragraphs, the question arises as to how such a Criterion of Fit will be implemented in practice. As it was discussed before, the Criterion of Fit entails the accumulated measurement errors that are represented in terms of the positional drift during the time of observation. The key element which was not discussed before is the aspect of the observation time. As discussed in chapter 1, the positional drift of an INS is defined in terms of nautical miles per hour, which suggests that the error accumulation test through the INS must be performed for a complete hour. Although this would be preferred, it is not always practical, since the usage requirements of a gyro ranges from mere seconds of operation in the case of an air-to-air missile to multiple hours in the case of an Unmanned Aerial Vehicle (UAV). It is therefore suggested that the following approaches be followed for the determination of the post-calibration sensor accuracy: If a motion profile is available for a mission that exceeds an hour, the sensor can be stimulated with the motion inputs particular to this profile using the motion simulator. By propagating the sensor measurements through the INS simulation up to one hour duration, the positional drift after one hour can be determine as an indication of the calibration effectiveness. In this case the classical nautical mile per hour figure of merit is obtained from the simulation. If the sensor is being developed for an operational environment with missions shorter than an hour, the extension of the motion simulator input beyond the true mission duration does not make much sense, but it could be done. If it is decided to not extend the profile to an hour, the resulting figure of merit would still describe the positional drift, but only for the duration of the profile. In this case one would use another guideline such as the expected system performance, or the performance of a high-accuracy reference system to determine whether post-calibration accuracy is acceptable. Although not as well-defined as the nautical mile per hour figure of merit, the performance indication obtained in this way should still be interpretable by the domain expert as it still represents the accumulated systems errors as a single-figure positional drift at the end of the test profile. A good starting point in both scenarios presented here would be to perform a precalibration INS propagation to obtain an indication of the system performance before the calibration and the improvement that was achieved by the calibration of the sensor.

138 Chapter 4 A New Calibration Method 122 Since noise plays a significant role in the measurement accuracy of FOGs, it is suggested that the pre-calibration and post-calibration analysis be performed using a Monte-Carlo analysis (refer to Bar-Shalom et al. [7]) to eliminate the skewing of a single measurement run by the measurement noise. For the comprehensive INS algorithm (which preferably needs to be mechanized in terms of local geographic axes - North, East, Down) 3-axes inertial sensor inputs needs to be provided as input. This implies three accelerometer and three gyro inputs are required. Without reducing the efficacy of the experiment to be performed, it is possible to take the accelerometer inputs and simply zero the North and East input while noisefree gravitational acceleration is provided as the Down input. This will result in the accelerometer measurements effectively indicating a static system and that no positional drift in the INS implementation will result from the accelerometer inputs. The 3-axes gyro measurements can be constrained in a similar way to make sure that it is only the gyrounder-test that impacts the positional drift and that it is only the motion input from the motion simulator that is propagated through the INS implementation. During the excitation of the gyro with the operational profile, it is important that the measurement axis of the sensor should point to either absolute West or absolute East. This is important since a sensor that is pointing either East or West will not be measuring any component of the Earth s rate of rotation. If the Earth s rate of rotation was measured, it would corrupt the propagation of sensor measurements in the INS software. By then setting the other two gyro inputs as zero and setting the measured data as the East or West sensor, it is possible to purely determine the impact of the residual sensor errors on the navigation performance. During the execution of the testing strategy that is presented here, there will effectively be three levels on which the system is tested. These are the comparison of the reference angular rates with the compensated rates (and perhaps the uncompensated rates when pre-calibration data is analyzed); the comparison of the true orientation (angular position) of the motion simulator with the one computed from the integrated measurements; and the comparison of the true system position (effectively (,,)) to the one that will be computed from the triple-integrated gyro measurements; with the final steps then acting as the suggested point of evaluation of the calibration accuracy.

139 Chapter 4 A New Calibration Method 123 The Criterion of Fit can be simply used as an indication of the system performance as presented here, or it can be used in a more extensive way as a measure to optimize the calibration architecture. There are a number of parameters that can be adjusted during the calibration of an RBFN. Some of these parameters are the type of basis functions that are used in the network, although most references indicate that this does not impact the training accuracy; the width (spread) of the system nodes if Gaussian basis functions are used; the option to make use of a fixed node width or a variable node width training strategy; the value of the regularization parameter and the strategy of adaptation of this parameter; and the model selection criteria that is used to terminate the network training. Although it will not be performed as part of this study, the Criterion of Fit can be used as part of a sensitivity analysis to determine the impact of these parameters on the network performance and even determine the best network configuration for the current application. 4.6 The Identification Strategy The definition of an identification strategy within the SID process was presented in chapter 2.3. It was indicated that the computation of the best model comprises the identification step. Within the context of the neural network based calibration strategy, this would point towards the training of the network, but the actual identification step consists of much more than just the training of the network. This section will therefore present the identification strategy and define it in terms of the specific training architecture used for the calibration. The architectures discussed here are used along with the network training method and the criterion of fit to define the final calibration (identification) strategy. There are two possible training architectures that will be discussed in this section. The first method is closely linked to the INS/GPS sensor integration architectures presented by Sharaf et al. [97], Sharaf and Noureldin [96], Noureldin et al. [74] and Semeniuk and Noureldin [95] and follows the structure of the classical complementary filter as defined by Bar-Shalom [7]. In this structure, which is defined in figure 4.17, the measurement (true

140 Chapter 4 A New Calibration Method 124 rate perturbed by errors) is applied as input to the network with the measurement error (the deviation from the true applied rate) being provided as the output in the training data. The network is therefore trained to take the error perturbed gyro measurement as the input signal and estimate the measurement error. This error can be subtracted from the measurement to obtain the actual rate that was applied to the system. This is the architecture that will be implemented for this study due to its resemblance to the complementary filter commonly used in navigation applications and to the approach used by Noureldin and his colleagues. The alternative approach to the training of the system is to apply the measurement (true rate perturbed by errors) as the training input and the true rate as the training output. The architecture for this implementation is presented in figure The network is therefore trained to estimate the corrected gyro measurement rather than the measurement error. There is no fundamental difference between the two approaches presented here and the approach used will depend on the application that it is used for. The data input and training structure of the RBFN is presented in figure The RBFN architecture presented in figure 4.17 is based on the INS/GPS sensor integration architectures presented by Sharaf et al. [97], Sharaf and Noureldin [96], Noureldin et al. [74] and Semeniuk and Noureldin [95]. In the INS/GPS sensor integration architectures on which figure 4.17 is based, the researchers included time as an input parameter as they were using their system to estimate a dynamic system. Since the calibration process discussed here addresses a static system, time is not an input parameter that needs to be explicitly defined as an input signal to the system as part of the training data. The temperature rate is included as an input parameter to the system in figures 4.17 and 4.18 as an indication of additional input parameters that could be presented as input signals. Since empirical temperature rate data were not available to include in the simulation model, the temperature rate was not included as part of the training data for the network. Apart from the fact that only angular rate and temperature will be considered as network inputs for this study, the network architecture presented here is general enough to accommodate any additional environmental inputs such as temperature rate, vibration or magnetism which could influence the measurements performed by the network. If these additional parameters are included as inputs to the training of the network, the resulting network will effectively become a hyper-surface which will be difficult to visualize. The architecture of the RBFN will, however, automatically accommodate such additional input signals and it could even be possible to still visualize the resulting network if only two parameters at a time are

141 Chapter 4 A New Calibration Method 125 used as input to the network after it has been trained. Applied angular rate(reference) - Gyro errors + Gyro errors (Training data) Measured gyro output Temperature Temperature rate RBFNN Estimated gyro errors - Compensated gyro + Figure 4.17: RBFN training and input data architecture. The operational architecture of the RBFN is presented in figure 4.19 where the same input parameters are provided to the network, but where the estimated gyro measurement errors are presented as the output. For the current implementation the network was trained by post-processing of the simulation data instead of attempting real-time implementation as was presented by Semeniuk and Noureldin [95]. One of the key elements of the research performed in the mentioned works of Noureldin and El-Sheimy is the addition of a de-noising strategy to remove noise in the training data. These strategies vary from a neural network based de-noising model (El-Rabbany [27] and Zhu [127]) to grey theory (Fan et al. [29], wavelet-based methods (El-Sheimy [28] and Chiang [2]), Empirical Mode Decomposition strategies (Qian et al. [82]) and a Recursive Least Square Lattice (RLSL) filter (El-Gizawy et al. [26]). The wavelet-based technique has resulted in dramatically improved sensor performance for lower cost sensors where the high-frequency noise played a significant role and the RLSL approach proved to be effective and well-suited to real-time implementations. For the higher cost sensors (the scenario presented by Semeniuk and Noureldin [95]) the measurement are often clean

142 Chapter 4 A New Calibration Method 126 Applied angular rate(reference) Gyro errors True rate (Training data) Measured gyro output Temperature Temperature rate RBFNN Estimated input rate Figure 4.18: RBFN training and input data architecture for the case where the network does not estimate the error, but the corrected rate. enough to not require additional filtering 7. The essence of all these methods is to reduce or eliminate the high-frequency noise to present the data that contains the trends to be estimated. In a number of cases such as Noureldin et al. [74], the wavelet-based filtering is used to pre-process the data for use in the training of RBFNs. Although the calibration architecture presented in this study is based on the above-mentioned research on neural network-based INS/GPS sensor integration, techniques such as regularization and the proper selection of the scaling (width) of the RBFN nodes will be investigated first to obtain a more elegant solution for the handling of noise in the training data. If this does not produce satisfactory results, a more comprehensive de-noising strategy will have to be implemented. 7 Based on personal communication with Noureldin on 29/1/29.

143 Chapter 4 A New Calibration Method 127 Applied angular rate(reference) Gyro errors Measured gyro output Temperature Estimated gyro errors Compensated gyro Temperature rate RBFNN - + Figure 4.19: RBFN operational data architecture. 4.7 The New Calibration Methodology From the various components discussed in this chapter, a new calibration methodology can be defined as follows: 1. Create a high-accuracy dataset to characterize the complete functional domain and create a reference error surface, or create a reference trajectory that can be used to test the network created during the calibration. 2. Define a trajectory that can be used to collect the characterization data for each gyro in the batch that is to be characterized. This trajectory is different to the reference trajectory defined above in that it should limit the calibration cost through the balancing of the calibration accuracy versus the time that is needed to perform the calibration. The generation of such a trajectory will most likely be an iterative process. 3. Define a short duration post-calibration trajectory that can be used to confirm the performance of the sensor after the calibration corrections was implemented onboard the sensor. This trajectory should cover the measurement extremes and some areas containing rapidly-varying areas within the error surface. 4. Execute the training trajectory and collect the reference angular rates and temperatures along with the parameters measured by the gyro electronics. Apart from the angular rate the measurements could include the internal temperature within the gyro, the temperature rate, vibration levels and magnetic field strength.

144 Chapter 4 A New Calibration Method Decide on the required output for the calibration architecture. The options being either the measurement error or the corrected rate. 6. Decide on the input parameters that will be used as part of the system calibration. At least angular rate and temperature should be included, but the other environmental parameters could be included if dependable measurements of these parameters are available from the sensor electronics. 7. Train the network. This could involve a significant number of iterations for the first attempts to obtain the optimal balance between the scale (spread) of the network nodes and the calibration accuracy. The Criterion of Fit defined in section 4.5 should be used to judge the calibration accuracy. 8. Test the trained network using the reference dataset or reference error surface. Use the Criterion of Fit defined in section 4.5 to judge the calibration accuracy. If the network performance is not satisfactory, the network training parameters should be adjusted and the network retrained and re-evaluated. 9. Optimize the network for embedded implementation. 1. Implement within the sensor software. The focus of the new calibration strategy is not to limit the number of nodes in the network, but rather the trajectory that is used to generate the training data, the use of a neural network to perform the calibration and the development of a Criterion of Fit that can be used to define the performance of the calibration network. The network architecture can always be optimized for online implementation using one of more optimal training strategies, so no attention was given to the development of a compact and computationally efficient network architecture. In addition to the steps defined here for the proposed new calibration strategy that a significant amount of planning and heuristic knowledge will still be needed as part of the planning phase to define the calibration process. This is not any different from the classical method and this is not the most time-consuming aspect of calibration. The really costly stage is the gathering of measurements for every gyro in a batch and it is here where the new strategy proposes significant savings.

145 Chapter 4 A New Calibration Method Conclusion In this chapter the systems identification steps presented in chapter 2 were interpreted from the perspective of gyro calibration. A number of areas within the whole calibration process that have not really been developed from the perspective of SID were identified in chapter 2 and these deficient areas were addressed during this study. The specific components of the calibration process that were developed were: the development of a new error modeling strategy using a neural network; the development of a new calibration experiment; the development of an explicit Criterion of Fit that can be used with the neural network model to evaluate the calibration accuracy; and the development of a calibration strategy for the gathering of the data and the implementation of the network within the sensor measurement path. By approaching gyro calibration from the perspective of systems identification, a number of improvements to the process were identified and a new strategy for FOG calibration was developed.

146 Chapter 5 Implementation and Analysis 5.1 Introduction The objective of this chapter is to determine whether the new calibration strategy (that was proposed in chapter 4) can be used to calibrate a real FOG. If the SID structure is rigidly followed, then the tests performed here need to evaluate the ability of the network to model the errors in the system and the model structure that is needed; the experiment that is necessary to calibrate and test the system; the calibration strategy and the architecture of the calibration; and the use of the Criterion of Fit to judge the accuracy of the above-mentioned steps. However, not all these components need to be investigated to the same depth. The design of the experiments consisted of the generation of a new set of trajectories that can be used to create training data for the neural network. The validity of these trajectories needs to be evaluated, but the experimental design was focussed on the development of an argument to support the new experiment. The only evaluation that is therefore required with regards to the experimental design is to determine whether the training trajectory in conjunction with the neural network architecture has the ability to accurately model the error surface. The calibration strategy and the calibration architecture consist of the structure used to calibrate and compensate the sensors, but since both of these are inherent to the whole calibration process, it does not really require additional testing. This leaves the modeling ability of the neural network and the Criterion of Fit to be explicitly evaluated. This chapter will therefore focus on the in-depth analysis of the neural network s modeling 13

147 Chapter 5 Implementation and Analysis 131 ability and the use of the Criterion of Fit to judge the quality of the calibration. The modeling ability of the network will be evaluated under noise-free measurement conditions and under conditions where various levels and types of measurement noise are present in the system measurements. 5.2 Aspects of Implementation Neural Network Implementation All tests and implementations for the alternative calibration strategy were implemented in Matlab. The neural networks were implemented using the RBFN Toolbox for Matlab written by Mark Orr [77, 78, 8]. A detailed discussion on the Orr-package is available on Mark Orr s website ([8]) which contains comprehensive documentation on the software package from both a theoretical and practical perspective. There are a couple of aspects regarding the implementation that need to be highlighted here. Only Gaussian radial basis function nodes were used in the implementation of the network as it is the most commonly used option for RBFNs. The package contains two parameters namely the radius (conf.rad) and the scale (conf.scales) to define the width of the RBF nodes. The radius parameter was fixed to a value of 1 and the scale parameter was adjusted to experiment with its impact on the network estimation accuracy. Unless stated otherwise, the model selection criteria (conf.msc) was defined to be the Bayesian Information Criterion. Only the OLS training strategy was used and some experimentation with regularization performed as part of the network training Experimental Data-sets Training Data The neural networks used for this study was trained using the data obtained from the empirical gyro measurement model developed in chapter 3 and not using actual measurement data obtained from a motion simulator that was executing the calibration trajectories presented in chapter 4. This means that a number of training trajectories could

148 Chapter 5 Implementation and Analysis 132 be used during the training to determine the impact thereof on the training accuracy without incurring any expenses from time spent on the motion simulator. The training trajectories used include those presented in chapter 4 where changes in the number of data points and variations in the number of temperature and angular rate cycles were incorporated into the various training trajectories. Test and Validation Data During the definition of the Criterion of Fit in chapter 4, it was indicated that a motion simulator is required to generate the test trajectories that can be used to evaluate the calibration accuracy. Since the neural network obtained from the calibration was not implemented on an embedded computer as part of this study, it was not possible to obtain post-calibration data from the compensated gyro mounted on a motion simulator. For this reason three test trajectories were used to perform the calibration validation. These trajectories are presented in appendix C and consist of a short one (1 seconds duration) that is characteristic of an air-to-air missile, a medium length trajectory (5 seconds duration) that is typical of a medium range missile and a long trajectory (2 hours duration) that can be considered to represent a UAV. The presented trajectories consist of the three-axes angular rate, acceleration, orientation, velocity and position measurements for the specific vehicle. The data were generated using high-accuracy six degrees of freedom (6-DOF) flight simulators. The original set of training trajectories as overlaid onto the functional space covered by the calibration process discussed in this study is presented in figure 5.1. From these figures it is observed that the angular measurements for the short and medium trajectories exceed the angular rate limits that were used during the training of the calibration neural network. The angular rates experienced by the long trajectory fall within these bounds. The objective of using the test trajectories was to test the calibration ability of the network over a comprehensive part of the functional domain. Due to the time and cost involved, it was not an option to gather a new set of measurements over a larger operational domain to define a new error function. The option was therefore to either scale or clip the angular rate measurements of the test trajectories to the range from 1 /s to +1 /s. It was decided to rather clip the test trajectory than to scale it as it results in a good coverage of the functional space by the test trajectory. In the case of scaling the test trajectory only covered a very small part of the overall functional space. The functional space coverage of modified angular rates for the various test trajectories are presented in figure 5.2. The long trajectory presented here is the unchanged original trajectory, but is

149 Chapter 5 Implementation and Analysis 133 Temperature rate vs Angular Temperature rate vs Angular 3 ω x 3 ω x ω y ω y 2 ω z 2 ω z ω (deg/s) ω (deg/s) Temperature ( C) (a) Short trajectory Temperature ( C) (b) Short trajectory: zoomed. 2 Temperature rate vs Angular 2 Temperature rate vs Angular ω x ω x 15 ω y 15 ω y ω z 1 5 ω z ω (deg/s) ω (deg/s) Temperature ( C) (c) Medium trajectory Temperature ( C) (d) Medium trajectory: zoomed. 5 4 Temperature rate vs Angular.6 ω x ω y 5 4 ω x ω y Temperature rate vs Angular 3 ω z 3 ω z 2 2 ω (deg/s) 1 1 ω (deg/s) Temperature ( C) (e) Long trajectory Temperature ( C) (f) Long trajectory: zoomed. Figure 5.1: Functional space coverage by the various validation trajectories.

150 Chapter 5 Implementation and Analysis 134 included in this set of figures for the sake of completeness. Following the modification to the angular rates, a new set of reference trajectories was generated based on the acceleration vector of the original reference trajectory and the modified angular rate vector. The clipping of the angular rates results in the Euler angles, the velocity and the position of the new trajectory being different from the original trajectory. This is not of much concern as the new short and medium length reference trajectories are simply generated using the original acceleration in combination with the modified angular rates. 5.3 Calibration Accuracy Interpretation Criterion of Fit Implementation As mentioned in the previous section, the testing of the calibration strategy was performed using a simulation and not on a real gyro platform. Since the Criterion of Fit was defined in chapter 4.5 for the case where a motion simulator was used to calibrate the sensor and where the calibration network was implemented on the embedded hardware, some modifications to the Criterion of Fit implementation are required. The implementation presented in chapter 4.5 required that the motion simulator should be used to apply rotational motion to the sensor without it experiencing any lateral motion. With the proposed test trajectories being generated with 6-DOF flight simulators, the static condition is not valid and the implementation of the evaluation strategy needs to be adapted to fit the simulated test environment. The first option would be to zero some of the accelerometer and gyro inputs of the reference trajectories, but since the data sets were generated with a 6-DOF simulation, the motion of the vehicle is ingrained into the test data and simply zeroing some of the inputs would result in incoherent results when processing the modified data set through the INS simulation. The alternative and more realistic option is to only perturb a single axis at a time and leave all of the other axes absolutely correct. This is an option, but as shown in the results presented in table 5.1 1, the single axis perturbation effectively sums up to the 3-axes sum (although the addition is not exact in all cases). The potential problem with the single perturbation-axis approach is that the wrong selection of the axis to perturb and observe 1 For reasons of formatting the abbreviation traj was used as a replacement for trajectory.

151 Chapter 5 Implementation and Analysis Temperature rate vs Angular 15 Temperature rate vs Angular ω x ω x ω y ω z 1 5 ω y ω z ω (deg/s) ω (deg/s) Temperature ( C) (a) Short trajectory Temperature ( C) (b) Short trajectory: zoomed. 15 Temperature rate vs Angular 15 Temperature rate vs Angular ω x ω x ω.13 y ω z ω y ω z ω (deg/s) ω (deg/s) Temperature ( C) (c) Medium trajectory Temperature ( C) (d) Medium trajectory: zoomed. 5 4 Temperature rate vs Angular.6 ω x ω y 5 4 ω x ω y Temperature rate vs Angular 3 ω z 3 ω z 2 2 ω (deg/s) 1 1 ω (deg/s) Temperature ( C) (e) Long trajectory Temperature ( C) (f) Long trajectory: zoomed. Figure 5.2: Functional space coverage by the various validation trajectories after the angular rates were clipped to fit the calibration space.

152 Chapter 5 Implementation and Analysis 136 Table 5.1: Analysis of the method to be used for the Criterion of Fit implementation. Short Traj Medium Traj Long Traj Classic (m) NN (m) Classic (m) NN (m) Classic (m) NN (m) 3 axes axis: x axis: y axis: z Reference trajectory INS Add errors Traditional compensation INS Comparison Neural network compensation INS Figure 5.3: INS performance test strategy. could lead to the masking of some systematic error. It was therefore decided to use the three axes perturbation approach to perform the Criterion of Fit implementation. The method used was to perturb all three gyro measurements and propagate these through the INS simulation. This results in a somewhat exaggerated indication of the error, but it was considered to be the most representative method to use. To eliminate any systematic errors that could exist between the INS implementation and the 6-DOF simulation that was used to generate the reference trajectory, the angular rates and accelerations of the reference trajectory were processed by the INS simulation to create a baseline against which the performance of the system can be evaluated. The postcompensation angular rates were then combined with the accelerations of the reference trajectory and the combination was processed by the INS simulation to determine the impact of the residual compensation errors on the navigation performance of the system. The test strategy is presented in figure 5.3. A comprehensive discussion of the results presented in table 5.1 will be provided in the following sections. The reason for the presentation of these results in a somewhat pre-

153 Chapter 5 Implementation and Analysis 137 mature way was simply to support the argument for the Criterion of Fit implementation Compensation Accuracy Analysis After compensation of the raw gyro measurements the data will still contain a small component of residual bias. This bias will be present due to imperfect calibration and compensation techniques, but also due to thermal effects within the gyro s various components and subsystems. Every time that the sensor is switched on, the internal components will have a different thermal state, resulting in different operating points which will affect the bias. This bias variation between different periods of operation is called the turn-on to turn-on bias variation and is used to define the final sensor performance accuracy. Component-level thermal effects can only be addressed through component redesign or through the addition of a thermal control unit as discussed by Titterton and Weston [115]. The component of the residual bias due to these effects therefore cannot really be modeled and corrected under the current calibration/compensation strategy, but the impact of the calibration strategy on the rest of the residual bias can be determined. The test trajectories (defined in appendix C) will therefore be used to determine the impact of the proposed calibration strategy on the resulting system accuracy. The assumption will be made that there are no component-level thermal effect residual bias components present in the system and that the residual bias is entirely attributable to the accuracy of the compensation. The reference trajectories will be used to produce the reference angular rate and altitude values that will be perturbed with the gyro errors as taken from the empirical error model. Using this set of perturbed measurements, the impact of two types of compensation strategies will be used. The first strategy to correct for the gyro errors will be to use a classical polynomial type of correction to the gyro measurements. Under this approach a single polynomial is generated to compensate for the temperature variations and another polynomial is used to compensate for the angular rate variations. This is an approach that is generally used in industry. The second strategy will be to use the neural calibration model to compensate for the measurement errors. Traditionally the compensation of a FOG consists of the subtraction of the bias, the temperature dependent bias and the scale factor related measurement error as a function of temperature and angular rate from the raw measurements. The temperature dependent

154 Chapter 5 Implementation and Analysis 138 bias is usually defined as a polynomial expression. For the scale factor related measurement error as a function of temperature and angular rate two separate polynomials are defined. Using figure 3.5 as reference, the angular rate measurements over all temperatures will be averaged to compute the polynomial expression that will define the scale factor error s dependence on the angular rate. In a similar way the temperature measurements over all angular rates will be averaged to define the polynomial expression for the scale factor error s dependence on the temperature. This is an approach that is generally used in industry. Figure 5.4 presents the reference error surface, the polynomial-based expression and the error between the two surfaces in units of degrees per second and degrees per hour. From these graphs it appears that the classical polynomial approach will not have satisfactory accuracy to perform the compensation on the required level. 5.4 Noise-free Modeling Evaluation Before going into the complexity of trying to characterize the system with all error sources and the noise included, the noise will be excluded for the first round of tests. The intention of this investigation is to first conceptually verify whether the network can be used to characterize the sensor to the required accuracy. Once this has been verified, the more complex task of trying to characterize the deterministic sensor errors in the presence of noise will be considered. There is a second reason for investigating the calibration accuracy in the absence of noise. The gyro that was primarily used for this study has a particular noise component, but it is possible to reduce the stochastic errors that is present in the gyro output through hardware and firmware modifications. The noise in the measurements can also be reduced by using the noise reduction strategies discussed in chapter 4.6. By performing calibration without noise, the ability of the network to approximate the underlying system model can be evaluated more effectively Accuracy Analysis A preliminary exploration was performed into the ability of the neural network to perform the calibration of the gyro. The first task was to determine whether the network can in fact estimate the error surface. The network was trained using the training dataset defined in section with 1 sample points and a spread of It consisted of an updown temperature cycle with 32 angular rate cycles being performed during the complete

155 Chapter 5 Implementation and Analysis 139 (a) Reference error surface (b) Polynomial-based error surface (c) Comparison between polynomial and reference error surface in units of degrees per second (d) Comparison in units of degrees per hour. Figure 5.4: Reference error surface and polynomial-based error surface comparisons.

156 Chapter 5 Implementation and Analysis 14 temperature cycle. The number of sampling points was arbitrary with the spread of the nodes having been experimentally determined to give good results. The accuracy of the trained network relative to the training data was considered to be adequate as is presented in figure 5.5. A Gaussian error distribution with a number of outliers is therefore observed. It was decided to also test the accuracy of the network using the reference error surface method. The reference error surface is generated by evaluating the empirical error model defined in chapter 3 at every point on a grid which is indexed in terms of the operational temperature and angular rate range of the sensor. With temperature along the x-axis and angular rate along the y-axis, the z-axis will represent the reference measurement error from the gyro. Since the network can only process a vectorized stream of data for each input, the xy locations on the reference test surface was unpacked from a matrix to a vector format and processed by the network to determine the error estimate at each reference point. The output from the network was reformatted into the same matrix format for a point-by-point comparison with the reference error surface. The error surface generated by the network is presented in figure 5.6(a) with the reference error surface presented in figure 5.6(b). From a purely visual inspection it appears that the network can approximate the error surface Training error distribution: nodes = 269 Mean = e 6 σ = Measurements Error (deg/s) x 1 4 Figure 5.5: Training error histogram for the linear up and down trajectory with 1 sample points and a NN scale of 17.5.

157 Chapter 5 Implementation and Analysis 141 (a) Error surface created by the network. (b) Reference error surface. Figure 5.6: Comparison between the estimated and reference error surfaces. As an indication of the internal structure of the network, the location of the network nodes as a function of the input parameters was also investigated. The node locations overlaid onto a contour plot of the estimated network is presented in figure 5.7. Although it can be seen that the nodes are fairly evenly spaced throughout the operational space, no obvious additional information can be derived from this plot. From the two error surfaces presented in figures 5.6(a) and 5.6(b) it seems that that the network has the ability to estimate the error surface. It also appears that the estimation is very good, but to confirm the accuracy of the estimation, the reference surface was subtracted from the estimated surface on a point-by-point basis. The result is presented in figure 5.8. Figure 5.8 therefore indicates the difference between figures 5.6(a) and 5.6(b) in the same way in which figure 5.4(c) defined the difference between figures 5.4(a) and 5.4(b). When considered along with the plot of the estimation error surface as viewed long the temperature axis (figure 5.9), it can be seen that the worst estimation errors appear along the ridges on the error surface plot of figure 5.6(a). The distribution of the estimation errors with respect to the reference error surface is presented in figure 5.1. It can once again be seen that the distribution is predominantly Gaussian with a number of outliers. From the results of the preliminary exploration into the ability of the neural network to perform the calibration presented above it seems that the task can be performed by the network, but the accuracy of the estimation has not been determined yet. As part of the preliminary exploration a number of additional tests were executed. The results from these tests are presented in table 5.2. As previously discussed, it was found that

158 Chapter 5 Implementation and Analysis Node placement on the contour map: nodes = ω meas (deg/s) Temperature (deg C) Figure 5.7: Network node placement for the linear up and down trajectory with 1 sample points and a NN scale of a periodic distribution of the sample space can result from a particular selection of the parameters used to generate the calibration trajectory. The trajectory consisting of 1 sample points portrayed this periodic character. To eliminate this trend, the samples were increased to 1111 and the training was repeated. A significant reduction in the mean error was obtained. This can probably be ascribed to the improved coverage of the operational domain by the training data set. Through a trial and error process it was determined that the best estimation accuracy for the trajectory consisting of 1 sample points was obtained when the spread of the Gaussian node was selected to be Following the adjustment in the number of samples used to perform the training of the network to 1111, a quick exploration was made into the impact of a smaller spread parameter and more sample points on the network accuracy. From the results presented in table 5.2 it can be seen that both the adjustment in the spread of the nodes and the increase in the number of data points in the training data set have a significant impact on the estimation accuracy of the network. The output of the network presented here in fact has an accuracy that is such that the maximum errors are still almost an order of magnitude less than the standard deviation of the noise present in the real system. Although a real system will have noise present in

159 Chapter 5 Implementation and Analysis 143 Figure 5.8: 3D error surface for the linear up and down trajectory with 1 sample points and a NN scale of the measurements, the figures presented here act as an illustration of the ability of the network to accurately estimate the underlying error model with the proper selection of the number of samples, the type of trajectory and the scaling of the RBF nodes. Figure 5.11 presents the graphical representation of the first four tests (whose results were summarized in table 5.2) through a side view along the temperature axis for each of the estimation error surfaces generated during the tests. It is interesting to note that all of these plots visually appear very similar while their statistical properties vary significantly. As an additional investigation into the potential accuracy that can be obtained from noise-free training, the proposed test trajectory defined in figure 4.16 was used to train

160 Chapter 5 Implementation and Analysis 144 Figure 5.9: Side view of the 3D error surface for the linear up and down trajectory with 1 sample points and a NN scale of the network. One would expect the improved coverage of the operational domain to result in improved accuracy of estimation in the calibration network. The distribution of the network nodes across the operational domain, which is dependent on the locations of the training data points, should be more precise, resulting in an improved training accuracy. It would also be expected that the increased density of the training data could result in smaller spread parameters being used to reduce the accuracy of the training. Table 5.3 indicates the results for the sensitivity analysis that was performed on the network trained with this high-density set of training data. Figure 5.12 presents the view of the estimation error surface along the temperature axis for this set of tests. The important parameter to consider in table 5.3 is the standard deviation of the estimation error. It can be seen that the mean value of the 7.5 scale could be larger, but the standard deviation is much less than the case of the 12.5 scale. The estimated data therefore falls in a much smaller band around the mean for this scale value and it should result in improved sensor performance. When the results presented in table 5.3 are compared to those presented in table 5.2,

161 Chapter 5 Implementation and Analysis Surface comparison error distribution: nodes = 269 Mean = σ = Measurements Error (deg/s) x 1 4 Figure 5.1: Histogram of the error distribution onto the 3D error surface for the linear up and down trajectory with 1 sample points and a NN scale of it is interesting to notice that the significantly increased number of data points and the improved coverage of the operational domain in the high-density trajectory seems to not really make any difference to the estimation accuracy of the trained network. Training a network with this trajectory took much longer and the time needed to generate the test data will be four times longer than for the trajectory used to generate table 5.3, but the accuracy increase is marginal. It therefore appears that the simpler trajectory will be quite adequate to provide the training data for the problem at hand. The validity of this statement will only be determined after the evaluation of the neural network calibration by means of the Criterion of Fit metric. One aspect of the investigation into the modeling ability of the network that stands out are the ridges along the temperature axis that appear on the estimation error surface presented in figure 5.8. At first glance these features appear odd, but with some inspection the origin of the features can be identified. To assist in the inspection, a much finer grained representation of the measurement error surface (figure 5.13(a)), the estimation error surface (figure 5.13(b)) and the combination of the two (figure 5.14) is used. The estimation error surface from figure 5.13(a) was scaled to combine with the estimation error onto the single plot presented in figure From figure 5.8 it can be seen that the

162 Chapter 5 Implementation and Analysis 146 Table 5.2: Basic network estimation accuracy analysis. Test Mean ( /h) σ ( /h) Max ( /h) Min ( /h) Nodes 1 samples; spread: samples; spread: samples; spread: samples; spread: samples; spread: (a) 1 samples, scale: (b) 1111 samples, scale: (c) 1111 samples, scale: (d) 2222 samples, scale: Figure 5.11: Impact of parameter adjustments on the estimation accuracy.

163 Chapter 5 Implementation and Analysis 147 (a) scale: 25 (b) scale: 2 (c) scale: (d) scale: (e) scale: 7.5. Figure 5.12: Impact of parameter adjustments on the estimation accuracy for the network generated with the high-density training set.

164 Chapter 5 Implementation and Analysis 148 Table 5.3: Node width sensitivity analysis for high density training trajectory. Test Mean ( /h) σ ( /h) Max ( /h) Min ( /h) Nodes Spread: Spread: E Spread: Spread: Spread: whole of the estimation space exhibit the egg box type of error landscape that one would associate with a surface represented in terms of Gaussian radial basis functions. This is even present on the ridges. There appears to be a periodic component within the spacing of the various nodes on this estimation error surface. Taking this into consideration, the cause of the ridges can be determined from the investigation of figure On this plot it can be seen that the location of the ridges within the functional space correspond to the areas where the gyro error surface has the steepest gradient. The largest ridge is observed at the point of the steepest gradient. This argument is supported by the graphs presented in figures 5.11 and 5.12 where it can be seen that the ratio of the error surface peaks in the centre of the functional space and those along the ridges stay a constant ratio of about 3 irrespective of the width of the network nodes. From this it can be deducted that the gradient of the error surface will always cause the observed ridges, but that the magnitude thereof can be controlled by reducing the width of the nodes in the network Criterion of Fit Evaluation of Noise-free Network Following the various investigations into and analysis of the ability of the neural network to estimate the gyro measurement error surface, the Criterion of Fit needs to be used to judge the true calibration accuracy. Being well aware that real sensors exhibit significant levels of measurement noise, it is of value to determine the quality of the noise-free calibration accuracy that was achieved since it will determine whether it is possible to calibrate high quality, low noise gyros with the neural networks. It will also give an indication of the achievable inertial performance that can be obtained from the sensors if efficient noise reduction measures are put in place. As indicated earlier in this chapter, the number of training samples, the symmetrical distribution of the training samples, the specific training trajectory used and the width

165 Chapter 5 Implementation and Analysis 149 (a) Error surface (b) Estimation error surface Figure 5.13: High accuracy error surfaces used for the interpretation of the ridge-estimation anomaly.

166 Chapter 5 Implementation and Analysis 15 Figure 5.14: Combined high accuracy error surfaces used for the interpretation of the ridge-estimation anomaly.

167 Chapter 5 Implementation and Analysis 151 Table 5.4: Criterion of Fit interpretation of noise-free training results without any thermal delay included. Trajectory Classical Neural network 1 seconds m.8 m 5 seconds m.19 m 1 hour m (5.12 ) 226 m (.2 ) of the RBF nodes all play a role in the ability of the generated network to estimate the gyro errors. In this section the Criterion of Fit will be used as a metric to determine the true estimation accuracy of the network as a function of all of these training parameters. This sensitivity analysis will be performed for the case where there is no delay in the environmental temperature measurements experienced by the sensor. Once the impact of the training parameters on the calibration network has been determined, the impact of temperature delays will be evaluated. The complete sensitivity analysis will be repeated for the temperature delay investigation. No Temperature Delay Initial Investigation The first Criterion of Fit evaluation of the noise-free compensation was performed with no temperature delay included. The means that it is assumed that the gyro experiences all environmental temperature changes instantaneously. These results are presented in table 5.4. A network consisting of 1111 samples with a scale of 17.5 was used to perform the compensation for the these trajectories. The neural network was trained using the lower density calibration trajectory as defined in table 5.2. The starting temperature for each trajectory was 1 C which is equivalent to an altitude of approximately 75 m above sea level. The results presented in table 5.4 indicate that the compensation accuracy of the neural network approach is at least two orders of magnitude better than the classical approach of angular rate and temperature dependant calibration for all three trajectories considered. The short trajectory contains the highest dynamics of the available test trajectories and could typically represent the flight of a short-range missile. The 1.24 m spatial error for the classical approach is usually considered to be acceptable for a short range missile as the missile will probably hit the target or get close enough to the target to ensure a lethal impact. The spatial error of.8 m that is achieved by the neural network based strategy

168 Chapter 5 Implementation and Analysis 152 can therefore also be considered to be acceptable for such a trajectory as it should result in the missile hitting the target in the absence of other errors. The medium length trajectory is representative of a medium range missile that is exposed to lower angular rates than the short-range missile s trajectory. The combined spatial error of 1.47 m for the classical approach could be considered to be acceptable for such a trajectory as it indicates that the missile can hit its target after 5 seconds of flight without additional sensors having to be used to compensate for the drift in the sensor measurements. The combined spatial error of.19 m that is achieved by the neural network based strategy can therefore also be considered to be acceptable for such a trajectory. The long trajectory has a duration of 125 minutes with quite a complex flight path that is typical of a UAV on a reconnaissance mission. The results for the Criterion of Fit evaluation of the classical compensation and the neural network based compensation presented in table 5.4 seem to indicate ridiculously large errors, but some closer inspection does provide more insight. The residual gyro drift is defined in units of degrees (of circumference of the Earth) per hour as defined in chapter 1. Roughly speaking one degree of circumference is equal to 111 km or 6 nautical miles. Although this is a single trajectory and the shape of the trajectory could shadow some systematic errors, one can get an indication of the residual positional drift in the sensor from a particular compensation technique by looking at the positional drift after 1 hour. The spatial positional error from the classical technique is m after 1 hour, which is roughly equivalent to a drift of 5 degrees in that hour. The neural network based compensation caused a spatial drift of 226 m after 1 hour which indicates a system performance.2 degrees per hour. The.2 degrees per hour residual error is characteristic of a gyro that will be used for a UAV mission and points towards a system that can depend purely on the INS for navigation for periods up to an hour. The 5 degrees per hour error resulting from the classical calibration method means that the UAV that is using this sensor will have to use a hybrid GPS/INS navigation system to ensure accurate navigation. Training Parameter Sensitivity Analysis Following the initial investigation into the network performance and the discovery that the classical compensation technique produces significantly worse Criterion of Fit results, a more in-depth investigation into the performance of the neural network based calibration is required. The results presented in tables 5.2 and 5.3 indicated that the number of samples (training points), the spread (width) of the RBF nodes and the specific training trajectory all impact the training

169 Chapter 5 Implementation and Analysis 153 error. This section will therefore present the results from an investigation into the impact of these parameters on the calibration performance of the system by using the Criterion of Fit metric. The starting temperature for all the trajectories in all the simulations was 5 C. It was also indicated that the location of the training data points over the trajectory (a symmetrical or more random distribution) plays a role, but this aspect will not be investigated explicitly here. All training data sets will consist of an arbitrary number of samples to eliminate such symmetry concerns. Tables 5.5, 5.6 and 5.7 contain the results from the sensitivity analysis simulations that were performed. The column headings in these tables are abbreviated for reasons of compactness and used to indicate the width of the basis function nodes (Node width), the results for the standard (low density) training trajectory with 1111 training data points (Std-1111 ), the standard training trajectory with 2222 data points (Std-2222 ) and the high density training trajectory with 1111 training data points (Dense-1111 ). The content from these tables are presented in a graphical form in figures 5.15, 5.16 and These simulations were performed using Matlab and for higher sample numbers, smaller node widths and the dense training trajectory the memory requirements become excessive and caused Matlab to stop functioning. This is the reason for the x at the bottom right of tables 5.6 and 5.7 since a valid result could not be generated for these configurations. A more comprehensive set of results could therefore not be obtained, but the results presented here are still considered to be informative enough to derive the design criteria for a practical neural calibration strategy. Table 5.5: Evaluation of training parameter impact on the Criterion of Fit metric for the short trajectory. Node width Std-1111 (m) Std-2222 (m) Dense-1111 (m)

170 Chapter 5 Implementation and Analysis 154 Table 5.6: Evaluation of training parameter impact on the Criterion of Fit metric for the medium trajectory. Node width Std-1111 (m) Std-2222 (m) Dense-1111 (m) x Table 5.7: Evaluation of training parameter impact on the Criterion of Fit metric for the long trajectory. Node width Std-1111 (m) Std-2222 (m) Dense-1111 (m) x

171 Chapter 5 Implementation and Analysis Short trajectory 1111 std 2222 std 1111 dense.1 CoF Node width Figure 5.15: Criterion of Fit evaluation as a function of the network node width and the training trajectory for the short evaluation trajectory Medium trajectory 1111 std 2222 std 1111 dense.3 CoF Node width Figure 5.16: Criterion of Fit evaluation as a function of the network node width and the training trajectory for the medium evaluation trajectory.

172 Chapter 5 Implementation and Analysis 156 x Long trajectory 1111 std 2222 std 1111 dense 4 CoF Node width Figure 5.17: Criterion of Fit evaluation as a function of the network node width and the training trajectory for the long evaluation trajectory. The increase in the number of samples or the use of the denser trajectory resulted in a significant increase in the computational cost associated with the generation of the training network, but it was clear that more training data points and data points that constituted a better coverage of the functional space resulted in improved network performance. Looking at the graphs, one tends to expect that more data points and a more thorough coverage of the functional space by the training trajectory will provide even lower Criterion of Fit figures, but looking at the minimum Criterion of Fit values for each training set used on any specific test trajectory presented in the tables discussed here, one needs to ask the question whether the first network did not actually constitute a structure that that was good enough to ensure the required performance. Although the higher complexity training sets results in smaller Criterion of Fit values, the overall performance gained is questionable. There may be a small percentage increase that was gained by using these training sets, but the numerical increase does not seem significant. Also, given that there will inevitably be some component of noise present in the system, the differences will most likely be obscured by the noise. The bottom-line in all of this is that the simpler training trajectory appears to be adequate for the task at hand. During the initial tests performed in section it appeared like the high density

173 Chapter 5 Implementation and Analysis 157 training did not really make a significant impact on the performance of the trained network. The results presented in tables 5.2 and 5.3 presented a pure analysis of the mean and standard deviation of the training error when the trained network was evaluated using a test data set. Comparing the content of those tables with the results presented in this section clearly highlights the inadequacy of using just the mean and standard deviation of the training error as a metric to judge the system performance. It was mentioned before that just training a neural network on gyro measurements does not constitute a structure that can be used to perform inertial sensor compensation. The use of an applicable Criterion of Fit now clearly becomes an indispensable part of any successful neural calibration strategy as underlying functional and systematic characteristics that are not visible during the network training come to the fore during the evaluation of the calibration network by the Criterion of Fit metric. Temperature Delay The thermal delay was excluded during training of the network since the network does not learn any temporal temperature component, but a 5 minutes delay was included for the purpose of analysis. With the temperature at the gyro coil being delayed by 5 minutes relative to the outside temperature, the impact of this delay is that the temperature is maintained at the initial value for the first 5 minutes of the flight before it starts to follow the 5 minutes delayed height profile of the vehicle. One would perhaps have thought that the constant temperature experienced by the gyro would result in the same or even improved results as compared to the case where no delay was present in the system. This appears not to be the case. It was clear that it does make a difference since the temperature is held constant for the whole duration of the flight in the case of the short and medium trajectories and for a very small part of the long trajectory. To visualize the impact of the starting point, one needs to look at the coverage of the functional space by the three trajectories as presented in figure 5.1. The left hand column of graphs is a pure overlay of the trajectory onto the contour plot of the error surface with the right hand column presenting a zoomed version of the original graph to indicate the magnitude of the angular rate and temperature variations. It is clear that the short trajectory has the most severe angular rate deviations, but the smallest variation in temperature while the long trajectory has the largest temperature variation with the smallest angular rate deviation. The medium trajectory represents a balance between the two since it presents good angular rate coverage at the same time as which it is traversing along the temperature axis. If the 5 minutes delay is added to the

174 Chapter 5 Implementation and Analysis 158 temperature experienced by the FOG, the short and medium trajectories will effectively stay at a single temperature value with the angular rate still varying. As mentioned before, the temperature for the long trajectory will be varying throughout the flight. Since the first simulations of thermal delay indicated unexpected results, the impact of the starting temperature on the Criterion of Fit simulations was investigated for both the cases where the temperature delay is included and where it is excluded. The simulation results using the Criterion of Fit strategy for all three trajectories at various starting temperatures are presented in table 5.8. The x that is inserted in some places in these columns in an indication that no results were obtained for that particular simulation run. The reason for this is that the temperature exceeded the calibrated range of 2 C to 6 C which resulted in invalid calibration corrections for both the classical and the neural methods and therefore in invalid simulation results. Looking at the results presented here, it can be seen that the tests reported on in table 5.4 was performed at a starting temperature of 1 C or 75 m above sea level. Both the neural and the classical calibration methods perform consistently for all testing trajectories for different starting points along the temperature axis. The results for the Criterion of Fit simulation that contains the 5 minute delay are presented in table 5.9. It can be seen that there is a small difference between the results presented in these two tables. To gain some insight into the impact that the thermal delay has on the calibration accuracy, the results presented in tables 5.8 and 5.9 was compared with table 5.1 containing the direct difference between tables 5.8 and 5.9 and table 5.11 presents the difference as a percentage change. Even though the percentage change for the neural compensation looks quite significant for different conditions, closer investigation indicates that the numeric variation for these cases are mostly insignificant. Looking at tables 5.8 and 5.9, it can be seen that there is no clear pattern in the measurement errors as a function of different thermal starting points. The reason for this is probably due to the nature of the calibration error surface presented in figure 5.8. In this figure it can be seen that the calibration error is represented by a 3-D egg box type of surface that contains various peaks and troughs. The thermal delay causes the short and medium length trajectory to fall onto different points of this surface which causes the accumulated error across the complete trajectory to differ for different starting points. Since the long trajectory far exceeds the thermal delay in duration, it is not limited to a single point on the temperature axis and it still traverses the total error surface as it would without the thermal delay. The variations due to the inclusion of the thermal delay recorded here also seem to represent a significant percentage variation, but is not really numerically

175 Chapter 5 Implementation and Analysis 159 Table 5.8: Noise-free Criterion of Fit evaluation of trajectories at different starting temperatures without any thermal delay. Short Traj (m) Medium Traj (m) Long Traj (m) Temp Classic NN Classic NN Classic NN 2 C x x x x x x 1 C x x x x C x x x x 5 C x x C C Table 5.9: Noise-free Criterion of Fit evaluation of trajectories at different starting temperatures with a 5 minute thermal delay. Short Traj (m) Medium Traj (m) Long Traj (m) Temp Classic NN Classic NN Classic NN 2 C x x 1 C x x C x x 5 C C C significant. For the classical calibration it was important to hold the temperature constant since this allowed for an accurate thermal polynomial to be applied to the measurements. On high accuracy gyros the sensor are often packaged within a temperature controlled housing to ensure that the operational temperature of the coils do not deviated from the calibrated working point. With the neural network, the accurate 2D model means that the constant temperature is not a requirement and that it can even be a problem since it can result in reduced performance due to the local minimum problem. For the current investigation it is important to include the thermal delay as it is a reflection of the true environmental conditions experienced by the sensor, but the non-smooth character of the estimation

176 Chapter 5 Implementation and Analysis 16 Table 5.1: Changes in the noise-free Criterion of Fit evaluation of the trajectories due to the addition of a 5 minute thermal delay. Short Traj (m) Medium Traj (m) Long Traj (m) Temp Classic NN Classic NN Classic NN 2 C x x x x x x 1 C x x x x C x x x x 5 C.3 x x C C Table 5.11: Percentage change in the noise-free Criterion of Fit evaluation of the trajectories due to the addition of a 5 minute thermal delay. Short Traj (%) Medium Traj (%) Long Traj (%) Temp Classic NN Classic NN Classic NN 2 C x x x x x x 1 C x x x x C 1-19 x x x x 5 C x x C C

177 Chapter 5 Implementation and Analysis 161 error surface rather requires that the exact temperature of the sensor be used to compute the compensation. This said, it should also be noted that the deviation due to the thermal delay is numerically (not as a percentage) insignificant for all of the trajectories and the impact can easily be reduced through the adjustment of some of the training parameters to reduce the network estimation error Conclusion and Discussion: Noise-free Training In this section it was determined that a neural network can be used to approximate the gyro measurement error surface. It was also determined that the bivariate Gaussian nodes are well-suited to the approximation of the three-dimensional error surface used to describe the sensor errors. It was shown that there are a number of variables such as the node width, the type of training trajectory and the number of training samples that have a significant impact on the training accuracy. Probably the most important aspect of the investigation into noise-free training is that it acts as an indication that the neural network can estimate the underlying error surface in conditions where the measurement noise can either be eliminated through design, or where it can be filtered out from the measurements. This aspect acts as an indication that the network can be used to calibrate high-quality, low-noise sensors. It was found that the visual inspection of the error surface and the considerations of different mean and standard deviation values for the estimation error surface do not represent solid criteria to judge the quality of the calibration network. The network performance needs to be judged using the Criterion of Fit metric since there are a significant number of parameters that influence the system performance and makes these simply tools of evaluation ineffective. To quantify the estimation accuracy of the network during the calibration exercise, the neural calibrated network output was processed using the Criterion of Fit technique previously defined. It was observed that the calibration strategy can generate results that are in line with the expectations of the calibration accuracy for high accuracy inertial sensors. 5.5 Modeling Evaluation in the Presence of Noise This section will consist of an investigation into the ability of the network to still approximate the error surface in the presence of various degrees of stochastic error components

178 Chapter 5 Implementation and Analysis 162 within the measurements. It can also be seen as an investigation into the ability of the network to calibrate lower quality sensors Basic Investigation into Noise-handling Ability One of the key challenges in the training of neural networks is to ensure that the network will learn the characteristics of the underlying process without fitting the noise during the training. This ability is called generalization and it ensures that the network can accurately estimate the output for previously unseen data in the presence of system noise. The deterministic nature of the various trajectories used to map the error surface is such that, given the assumption that the training trajectory sufficiently covers the error surface, the testing trajectory should not contain any unmodeled aspects of the system. The inclusion of the noise adds a component of uncertainty to the data which varies between different trajectories. With the addition of noise to the training data one effectively determines whether the training strategy has the ability to truly estimate the underlying properties of the error surface through the noise Monte Carlo Analysis Monte Carlo Analysis Overview Monte Carlo (MC) simulations are used to evaluate the performance of an algorithm through simulation when the performance cannot be determined through analytical techniques, or when noise is present in the system and the impact of the noise on the output cannot be determined through analytical means (Bar-Shalom et al. [7], Robert and Casella [85]). It consists of the execution of a number of successive simulations (runs) of the same system, each with different initial conditions. The performance of the system is determined by evaluating the statistical properties of the output (the mean and standard deviation) over the whole set of simulations. Based on the Central Limit Theorem (Bar-Shalom et al. [7]) the results will gradually converge to a Gaussian distribution. This discussion immediately brings to mind the question about the number of measurements that are a required to validate the Monte Carlo process. The fundamental question to ask is how may simulations need to be performed to ensure that the measurements will have a Gaussian distribution? Since only uniform Gaussian noise is added to the training and the testing data and the whole calibration system does not contain any additional source of noise, it is not expected that the output will portray a different type of noise distribution, but it is important to still take the formal steps to validate

179 Chapter 5 Implementation and Analysis 163 Reference trajectory INS Comparison Add errors Add noise In every run Neural network compensation INS Close when n runs completed New simulation run Figure 5.18: Description of the Monte Carlo simulation strategy used. this assumption since the non-linear neural network can potentially modify the statistical properties of the data. The procedure that is usually taken to determine the number of MC runs needed is to use a Goodness-of-fit test such as the Anderson-Darling test for normality discussed in great detail in Stephens and D Agostino [17]. This test is valid for more than 2 runs and provides a level of confidence (usually a 95% confidence level) that the dataset belongs to a normal distribution. The way that this test will be used is to perform a number of MC runs (at least 25) and then evaluate the Anderson-Darling test. If the test indicates that the dataset is normal, then the particular experiment (consisting of a set of MC runs) can be terminated and the mean and standard deviation of the output dataset can considered to be descriptive of the output of the MC experiment. If the Anderson-Darling test does not indicate normality after the completion of the initial number of MC runs, the experiment will be continued (more runs will be performed) until the test indicates that the output dataset has a normal distribution. For the system under investigation the measurement noise will be zero mean Gaussian white noise. The implementation of the Monte Carlo strategy for this study is presented in figure The reference trajectory was propagated through the INS simulation and the errors was added to the reference trajectory in the same way as for the noise-free case. The difference is that the measurement noise was added to the already perturbed measurements after the additions of the system errors. Since two successive noise measurements are not the same, two successive Criterion of Fit evaluations of the gyro measurements will also not be the same. The objective of the Monte Carlo simulations is therefore to determine the performance of the neural network compensation in the presence of noise by looking at the mean and standard deviation of the Criterion of Fit output with respect to the reference trajectory.

180 Chapter 5 Implementation and Analysis 164 The output of the simulation was interpreted in terms of an along-track and two crosstrack components to simulate the impact on a target-board. The x and y components of the cross-track error component are perpendicular to the velocity vector at the end of the trajectory while the along-track error component is in the direction of the end of trajectory velocity component and can be considered as a z error component. This breakdown of the errors into vector components was important to ensure that a true mean with the correct standard deviation was obtained for the system. For the noise-free cases the Criterion of Fit error was simply the error vector from the reference end point to the predicted trajectory end point and this value was a fixed value that did not vary between simulation runs. For the noisy case the predicted end point varies from trajectory to trajectory and it can be anywhere in a sphere around the reference end point. It was therefore necessary to make provision for both positive and negative values along the various axes to ensure that the statistics of the impact point do not get warped. An example of the breakdown of the error vector into its components are presented in figure The use of the cross-track and along-track decomposition of the end-of-trajectory error is of particular importance for the proper use of the Anderson-Darling tests. If only a single valued modulus of the error data was used, the underlying properties (the actual mean and standard deviation) of the error would have been warped. The computation of the modulus before the computation of the error statistics would result in a biasing of the true mean due to the modulus computation removing the sign of the error. By working in vectors and decomposing the errors into the various vector components, the signs of the errors are maintained and the true mean is computed. When the statistics of the three axes are therefore combined, once the Anderson-Darling test for all three axes prove valid, the mean and standard deviation obtained reflects the combined vectorized components. The standard deviation of the estimation error is computed from the expression (Topping [116]) σ tot = σ 2 x + σ 2 y + σ 2 z (5.1) where σ tot is the total standard deviation and σ x,y,z is the standard deviation of the various estimation error components. The ability of the network to effectively compensate the sensor measurements for different types of trajectories was proven in the first part of this chapter. Since the focus in this section is to determine whether the neural network can be used to compensate the sensor errors in the presence of noise in the training data and in the test data, only one test trajectory will be used to evaluate the performance of the network. It was decided to use the short trajectory with a 1 seconds duration in an attempt to reduce the time

181 Chapter 5 Implementation and Analysis 165 Figure 5.19: Breakdown of the end-of-trajectory error into along-track and cross-track components. required to perform the large number of simulations required as part of the Monte Carlo process. Monte Carlo Simulation Output Noise-free Test Data The first set of MC tests was conducted to determine whether it makes any difference if the test data (the data that the network needs to remove the errors from) contains noise or not and also whether the network can adequately calibrate a gyro in the presence of noise. Figure 5.2 presents the comparison of the mean error for the case where there was no noise in the test data (the red curve) and where there

182 Chapter 5 Implementation and Analysis 166 was noise in the test data (the blue curve). Figure 5.21 presents the standard deviation of the two testing scenarios. From these results it is clear that there is no real difference between including noise and excluding noise from the test data and that the difference that there is points towards the inclusion of noise resulting in slightly better calibration performance. To get an indication of the estimation accuracy when compared to the noise-free neural calibration, the noiseless network scenario was also added to the diagram (green curve). The difference between the Monte Carlo results and the noiseless value (expressed as a percentage in figure 5.22) may seem large, but is clearly numerically insignificant. Two additional simulations were performed to put these results in context. The first was to add a bias equivalent to 1 /h (the standard deviation of the noise included during the testing) to the uncorrupted measurements of the reference trajectory before processing it through the INS simulation. The second test consisted of the addition of white Gaussian noise with a standard deviation of 1 /h to the angular rates of the reference trajectory before processing the resulting angular rates through the INS simulation. The first test mentioned here indicated a.15 m end-of-trajectory error and the second test indicated a.3 m mean error with a standard deviation of.14 m. One would therefore expect the Criterion of Fit errors to be much smaller than the extreme case of.15 m, which it is as observed from figure 5.2. If the calibration errors were perfectly removed by the neural network one would expect only noise to remain in the system and then the mean error should be around.3 m. This is clearly not the case with the mean error from the Monte Carlo output being at least an order of magnitude more than.3 m. This therefore clearly points towards the fact that the network does not perform an exact error removal during compensation and this is supported by the green curve in figure 5.2. Considering the standard deviation difference between the Monte Carlo results and the second test case mentioned here, the increase in the standard deviation of the estimation error is computed from the expression (Topping [116]) σ inc = σ 2 MC σ2 just noise (5.2) where σ inc is the standard deviation increase, σ MC is the standard deviation of the estimation error from the Monte Carlo output and σ just noise is the standard deviation of the estimation error when pure noise was injected into the system. One can therefore conclude that the network generalizes well in the presence of noise since the mean values of the Monte Carlo runs in both cases were close to the results obtained under noiseless calibration conditions. Even the standard deviation of the residual

183 Chapter 5 Implementation and Analysis 167 noise is within the practical usage bounds that is expected from an inertial sensor used on a short-range missile. Combining figures 5.2 and 5.21 clearly points towards the fact that the larger network nodes lead to better calibration performance than the small nodes under the noisy conditions Mean comparison Noise free test Noisy test Noiseless.2 CoF mean Node width Figure 5.2: Monte Carlo mean error analysis. Noise Amplitude The impact of the noise amplitude on the system performance was investigated as a method to gauge the level to which the network can handle noise and therefore to determine its generalization ability. The amplitude of the measurement noise present during training and during testing was varied from.1 to 1 /h and the Criterion of Fit performance was evaluated using the Monte Carlo analysis. The particular levels of noise were considered to be representative of the noise levels that can be encountered in this type of FOG. The impact of the noise magnitude on the standard deviation of the Criterion of Fit error is presented in figure From the results it is clear that there is an almost linear relationship between the input noise amplitude (standard deviation) and the standard deviation of the Criterion of Fit error, but that the network has the ability to reject

184 Chapter 5 Implementation and Analysis Standard deviation comparison No noise Noise.55.5 CoF σ Node width Figure 5.21: Monte Carlo error standard deviation analysis. about 5% of the noise present in the system. The mean error obtained from the neural compensation and the pure-noise Criterion of Fit evaluation is presented in figure There are a number of aspects that can be observed from this figure. It is clear that the network compensation accuracy is on the same order of magnitude as the mean of a pure noise Criterion of Fit evaluation and that the network has actually rejected a significant amount of the noise during the compensation. The Criterion of Fit mean from.1 to about 5 deg/h was averaging about.1 m. When comparing this to the noise-free Criterion of Fit for the short trajectory presented in table 5.4 which indicated an error of.8 m, it is clear that the network was generalizing very well in the presence of noise. The relationship between the Criterion of Fit mean and the noise amplitude is not linear. The standard deviation for these test is clearly linear as shown in figure Even the pure noise evaluation has a linear growth in magnitude if the average slope is considered.

185 Chapter 5 Implementation and Analysis Percentage CoF variation Noise free test Noisy test CoF % change Node width Figure 5.22: Monte Carlo mean error deviation expressed as a percentage deviation of the noiseless case. To evaluate the last point, the impact of the network node width on the Criterion of Fit was evaluated under conditions where the noise amplitude was adjusted as before. The results for the evaluation of the mean is presented in figure 5.25 and for the standard deviation in figure Figure 5.26 shows the expected linear relationship between the noise amplitude (standard deviation) and the Criterion of Fit error standard deviation and indicates that larger node widths result in a reduced Criterion of Fit error standard deviation, thereby implying better system performance. Figure 5.25 is not that obvious to interpret. Figure 5.24 was generated for a node width of 17.5 and the value presented in table 5.4 was also obtained with this value. This should explain the correspondence between the values presented in these tables and graphs. From the graphs for the different node widths it is clear that the node width has a dominant impact on the system accuracy, but that the deviation is still within the bounds of the standard deviation variations (see figure 5.27). The non-linear relationship that is observed between the mean error and the noise amplitude is a repeat of the node sensitive Criterion of Fit error that was observed in the noise-free network evaluation.

186 Chapter 5 Implementation and Analysis Neural Just noise.5 CoF sigma Noise amplitude (deg/h) Figure 5.23: Impact of the noise amplitude on the Criterion of Fit performance standard deviation. Monte Carlo Results Interpretation The objective of this section was to determine the impact that noise has on the calibration accuracy by making use of Monte Carlo simulations with the implicit use of the Criterion of Fit metric. It was observed that the mean calibration error in the presence of noise is usually very close to the calibration error in the noise-free calibration data. The standard deviation of the Criterion of Fit error pointed towards the network having a significant noise rejection ability which has a slight dependence on the width of the nodes used in the network Investigation into the Training with Other Types of Noise A stochastic error component that is commonly encountered in low-cost gyros is noise that portrays a random-walk (RW) type of behaviour. Although the Allan Variance analysis that was performed for the gyro-under-test did not indicate the presence of RW noise, it is important to determine whether the new calibration strategy can still estimate the

187 Chapter 5 Implementation and Analysis Neural Just noise.1 CoF mean Noise amplitude (deg/h) Figure 5.24: Impact of the noise amplitude on the Criterion of Fit performance mean. true error surface in the presence of this noise component. The important question to ask with the inclusion of RW noise is whether the network will overtrain and start to estimate the noise in the training data as part of the system. RW noise is descriptive of a low frequency type of signal and it is to be expected that the network will not be able to reject noise of this character from the training data. Based on the analysis that was performed by Gebre-Egziabher [36], it can be expected that the correlation time of the RW noise can vary between 1 seconds for lower quality to 1 seconds for higher quality sensors. For the first analysis quite a long correlation time of 1 seconds was used to generate the RW noise. Since a 1 second sampling period was used, this effectively meant that 1 samples were used to generate the RW signal. The test trajectory instead of the reference error surface was used to evaluate the ability of the network to perform the calibration in the presence of a RW signal. If the estimation error of the test trajectory portrays the RW trend, it means that the network has actually learned the RW signal as part of the system and did not distinguish between that and the true error surface. It can clearly be seen from the results presented in figure 5.28 that the network is actually including the RW signal as part of the system that it is estimating.

188 Chapter 5 Implementation and Analysis 172 CoF mean Node width Noise amplitude (deg/h) Figure 5.25: Impact of the noise amplitude on the Criterion of Fit performance mean as a function of the network node width. The periodic nature of the estimation error directly correlates with the training time of approximately 23 seconds, thereby indicating that the network considered the noise to be part of the system that was to be approximated. To determine whether the failure to reject the noise can be ascribed to the duration of the training data (23 seconds) not being long enough to accommodate the 1 seconds time constant of the RW signal, the correlation time was reduced to 1 seconds and the test was repeated. The results are presented in figure 5.29 and it should be clear that the network is still unable to reject the noise. Figure 5.3 presents the estimation accuracy for the cases where regularization was included and excluded as a function of different scale values. A correlation time of 1 seconds was used to generate the RW signal. It should be clear that the inclusion of regularization has an insignificant impact on the ability of the network to reject the RW signal as noise. Various initial values for the regularization parameter was used in an attempt to increase the system performance, but this did not improvement the ability of the network to reject the RW noise. It can therefore be concluded that the only option to handle this problem is to consider some alternative noise reduction steps such as those presented

189 Chapter 5 Implementation and Analysis CoF std Node width Noise amplitude (deg/h) Figure 5.26: Impact of the noise amplitude on the Criterion of Fit performance standard deviation as a function of the network node width. in section 4.6. It is therefore not surprising that the network does not have the ability to estimate the RW noise component. The RW noise component is essentially a lowfrequency signal and the network does not have the ability to distinguish between this component and the true sensor error model. Due to the obvious inability of the network the handle the RW noise, the processing of the generated network through the Criterion of Fit was not deemed necessary Conclusion and Discussion: Training with Noise Since stochastic noise components are an integral part of FOGs and other low-cost gyros, it was deemed essential to investigate the performance of the calibration algorithm under conditions where wide-band noise and random-walk error components are present in the training signal. A number of conclusions can be drawn from the results obtained on noisy signal training. The network generalizes well through the noise as it did manage to generate a good approximation of the underlying error surface in the presence of noise. The

190 Chapter 5 Implementation and Analysis Impact of noise amplitude on CoF mean Impact of noise amplitude on CoF std CoF mean CoF std Noise amplitude (deg/h) (a) Mean Noise amplitude (deg/h) (b) Standard deviation Figure 5.27: Impact of the noise amplitude on the Criterion of Fit performance - normal plots. 4 Noise 15 NN estimation error: nodes = 58 4 Noise 2.5 NN estimation error: nodes = Deg/s Deg/s Deg/s 1 Deg/s Time Time Time Time (a) Noise (b) scale: 5 (c) Noise (d) scale: 25 4 Noise 2.5 NN estimation error: nodes = Noise 3.5 NN estimation error: nodes = Deg/s 1 Deg/s Deg/s 1 Deg/s Time Time Time Time (e) Noise (f) scale: 17.5 (g) Noise (h) scale: 12.5 Figure 5.28: Estimation error as a function of different scaling parameters when a random-walk noise source with a correlation time of 1 seconds is included in the training data.

191 Chapter 5 Implementation and Analysis Noise 15 NN estimation error: nodes = 6 6 Noise 4 NN estimation error: nodes = Deg/s Deg/s Deg/s Deg/s Time Time Time Time (a) Noise (b) scale: 5 (c) Noise (d) scale: 25 6 Noise 4 NN estimation error: nodes = Noise 5 NN estimation error: nodes = Deg/s 2 Deg/s 1 Deg/s 2 Deg/s Time Time Time Time (e) Noise (f) scale: 17.5 (g) Noise (h) scale: 12.5 Figure 5.29: Estimation error as a function of different scaling parameters when a random-walk noise source with a correlation time of 1 seconds is included in the training data. Criterion of Fit evaluation indicated adequate results which could be of practical use for the case where noise was present in the measurements and when compared to the noise-free case, the results were acceptable. The network cannot completely reject the wide-band noise in the training signal and does include a component of this noise in the derived sensor model. However, given the ratio between the mean error and that of the classical error and the standard deviation of the error and the classical error, the residual error components could be considered to be negligible. The ability of the network to accurately obtain a sensor model in the presence of wide-band noise is very dependent on the width of the RBF nodes constituting the network. A training strategy that optimizes for the node width can therefore be considered a necessity for a realistic implementation. The OFS-LOO algorithm of Chen et al. [19] could be feasible in this regard. The calibration network does not have the ability to reject a random-walk component in the training signal. It is suggested that additional de-noising strategies be implemented to address situations where random-walk noise components are

192 Chapter 5 Implementation and Analysis Noise 15 NN estimation error: nodes = 6 6 Noise 15 NN estimation error: nodes = Deg/s Deg/s Deg/s Deg/s Time Time Time Time (a) Noise signal (b) No scale = 5 regularization; (c) Noise signal (d) With regularization; scale = 5 6 Noise 4 NN estimation error: nodes = Noise 4 NN estimation error: nodes = Deg/s Deg/s Deg/s Deg/s Time Time Time Time (e) Noise signal (f) No scale = 25 regularization; (g) Noise signal (h) With regularization; scale = 25 6 Noise 4 NN estimation error: nodes = Noise 4 NN estimation error: nodes = Deg/s Deg/s Deg/s Deg/s Time Time Time Time (i) Noise signal (j) No regularization; scale = 17.5 (k) Noise signal (l) With regularization; scale = Noise 5 NN estimation error: nodes = Noise 5 NN estimation error: nodes = Deg/s Deg/s 1 Deg/s Deg/s Time Time Time Time (m) Noise signal (n) No scale = 12.5 regularization; (o) Noise signal (p) With regularization; scale = 12.5 Figure 5.3: Estimation error as a function of the inclusion of regularization when a random-walk noise source with a correlation time of 1 seconds is included in the training data.

193 Chapter 5 Implementation and Analysis 177 present. The noise reduction strategy can actually be applied to all the cases where noise was considered to reduce the impact of the wide-band noise on the system as well. 5.6 Conclusion This chapter consisted of the testing and implementation of the new calibration strategy for FOGs. In light of the results presented, the new strategy can be considered to be an adequate if not superior replacement to the traditional method of calibration. The comparison that was made here focussed on accuracy rather than calibration time. The traditional method that was tested against consisted of polynomial averaging for constant temperature and constant angular rate respectively to determine the various polynomial expressions of the system errors. As presented in table 5.4 it can be seen that the classical method presents usable, if somewhat out-of-spec results (using the long trajectory as a yard-stick), but the new calibration strategy surpasses the classical method by approximately two orders of magnitude in the noise-free case. These results clearly promote the new method as a candidate to replace the classical method. A true validation of the neural network calibration could only be achieved with a proper analysis of this method under noisy conditions. Monte Carlo simulations were used as a technique to perform the noise analysis. The classical method was excluded from this part of the testing since the initial tests were considered to be adequate indications of the performance that could be achieved using this strategy. It was observed that the network can act as an effective calibration tool in the presence of wide-band noise since its resulting estimation error (evaluated with the Criterion of Fit criteria) has a mean resembling the noise-free estimation error and a standard deviation that is around 5% of the standard deviation of the measurement noise present in the system. Tests were conducted to determine whether the new calibration strategy will be able to remove low frequency random walk type noise from the system, but the results were not positive.

194 Chapter 6 Results and Discussions 6.1 Introduction This chapter will act as a summary of the results obtained throughout the thesis and present a more comprehensive discussion of some of the results. For the reader who studied the dissertation extensively the results will be familiar, but since this chapter is to act as a summary of the research performed, it is considered in order to repeat some of the previous results. 6.2 Specific Results Literature Study In the literature study performed in chapter 2 it was identified that no previous research has been performed where calibration was considered to be a systems identification problem. It was observed that the systems identification approach is directly applicable to the gyro calibration problem. The literature survey identified the deficiencies in the published research towards the systems identification approach to calibration. The three dominant areas for further research were identified to be: An extensive redesign of the calibration experiment to reduce the calibration time and therefore the calibration cost. A proper investigation into the use of neural networks to act as the calibration modeling structure considering fundamental neural network concepts and the ability of the network to function in the presence of noise. 178

195 Chapter 6 Results and Discussions 179 The definition of a proper criterion of fit to define the network performance within the area of application. The definition of such a criterion was deemed to be a major stumbling block in the practical evaluation of a neural calibration strategy within the field of inertial navigation as it would allow a systems expert to evaluate the calibration that was performed. The impact of the literature study was to create of roadmap for the rest of the research to be performed. The main contribution made in this chapter was the presentation of calibration as a systems identification problem FOG Model Chapter 3 focused on the development of a simulation model for fiber optic gyros. The first step was to combine the various gyro error models presented in the literature into a unified error and simulation model. The effect of such a unified model was to define the equivalence of the various models, but also to create a basis from which the gyro under investigation can be simulated. The simulation was developed using data measured on the real gyro and an empirical error model was developed that would later act as the basis for the development and testing of the new calibration strategy. As a final part to this chapter the impact of misalignment and axis transformation on the overall modeling of the gyro within a three-axis orthogonal configuration was defined. The main contribution from this chapter was the development of a unified FOG error model; and the generation of an empirical simulation model for the FOG under development A New Calibration Method In chapter 4 the findings from the literature study were used to develop a novel calibration strategy. The suitability of radial basis function neural networks as a suitable architecture to model the gyro error parameters was established. As a direct companion to the neural network modeling approach a Criterion of Fit was developed to evaluate the normally abstract neural network in engineering terms within the field of inertial navigation. This process has not been documented in literature and is considered to be one of the major contributions of this study.

196 Chapter 6 Results and Discussions 18 Further to the neural modeling and Criterion of Fit concepts new calibration trajectories were presented that focussed on the reduction of the overall calibration time. It was shown that the overall calibration time could be reduced from a number of hours to around one hour. The final part of this chapter presented the strategy for training of the network and for implementing the network in an online configuration. The main contribution from this chapter was the development of a Criterion of Fit metric that can be used to determine the calibration quality of the neural network Implementation and Analysis Chapter 5 contains the implementation of the new calibration strategy and the testing thereof on the previously developed empirical calibration model. It consists of two parts being the implementation and evaluation of the calibration strategy on noise-free systems and the evaluation of the calibration performance on measurements that contain noise. Noise-free Implementation The first part of the noise-free implementation consisted of an investigation into the network that is obtained when trained under calibration conditions. The objective was to determine whether the neural network can approximate the three dimensional error function defined in chapter 3. It was determined that the network can seemingly approximate the error surface, but that the quality of the approximation is highly dependent on the width of the nodes in the neural network, the number of datapoints used during the training and the coverage of the functional space represented within the training data. Whilst the fundamental analysis presented above highlighted some aspects of the quality of the network, no real judgement of the accuracy of the network could be determined in this way. What was needed to truly judge the performance of the new calibration strategy was the application of the Criterion of Fit metric to the calibration process. This metric consisted of the training of the network on the noise-free error surface followed by the network removing the errors from reference angular rate data (the compensation process). The compensated angular rates were then propagated through an INS simulation and the position output was compared to that of a reference simulation generated with the correct (uncorrupted) angular rates. Three different reference trajectories were used for the evaluation: one of 1 seconds, one of 5 seconds and one of two-hour duration.

197 Chapter 6 Results and Discussions 181 The results, presented in table 5.4 indicated that the new calibration strategy generally provides errors that are two orders or magnitude smaller than those obtained with the classical, polynomial based, calibration strategy. Further analysis confirmed the sensitivity of the system to the network node width, the number of training data points and the type of training trajectory and highlighted the ineffectiveness of using the initial visual inspection to judge the network performance. As a final investigation into noise-free calibration the impact of the realistic thermal delay on the calibration performance was studied. It was observed that the delay does not have a significant impact on the system performance and that the impact is insignificant within the overall system performance. The most important observations made in this section were that: The radial basis function neural network can be used to model the errors in the gyro. The neural calibration strategy can estimate the underlying error surface in conditions where the measurement noise can either be eliminated through design, or where it can be filtered out from the measurements. This aspect acts as an indication that the network can be used to calibrate high-quality, low-noise sensors. The use of the Criterion of Fit metric is a critical component in the true evaluation of the neural calibration accuracy. Implementation in Noisy Environments From the lessons learned in the noise-free case the noise-included evaluations focussed directly on the Criterion of Fit evaluations of the system performance. Since noisy measurements require the use of Monte Carlo simulations to effectively evaluate the system performance, these simulations were used from the start in combination with the Anderson-Darling test for normality. The essence of these investigations were that: The calibration strategy can effectively estimate the underlying error function since the mean of the estimation error is very close to the estimation error for the noisefree case. This error varied somewhat as a function of the network node width, but the variation was never more than a factor 5. The resulting network had the ability to reject a large component of the measurement noise with a general value of 5% noise rejection being observed.

198 Chapter 6 Results and Discussions 182 Low frequency random walk signals, when presented as alternative noise sources, were recognized to be part of the system and were not rejected by the neural network. 6.3 Results Summary A new methodology for the calibration of fiber optic gyroscopes was presented in this thesis. The suggested method differs from the classical approach in that the sensors are calibrated using dynamically varying environmental conditions instead of calibrating the sensor under static conditions of rotational rate and temperature. The training accuracy was proven to be sensitive to the selection of the spread parameter. A training algorithm is required that optimizes the radial basis function spread against the training error. Radial Basis Function Neural Networks were used to perform the sensor calibration. This approach was tested through simulation and the viability thereof was confirmed. It was shown that the neural network can be used to calibrate the sensor for a multidimensional set of environmental conditions by estimating the gyro measurement error caused by these external influences. The ability of the network to estimate the underlying error surface in the presence of wide band noise was confirmed, although it was shown that random walk noise in the measurements cannot be removed by the network. The new calibration strategy appears to be a particularly good solution for lower class FOGs where random-walk measurement noise does not play a dominant role. For higher quality sensors where the error surface is not as non-linear as the one presented in this study the classical polynomial approach to calibration could be acceptable, although the neural network based calibration allows for the compensation in domains of operation that is not usually covered by the polynomial expressions. From the results presented here it can therefore be stated that the new strategy is effective in fulfilling the requirements presented in chapter 1: It requires significantly less time to gather the calibration data. The resulting accuracy is far superior to the classical method of calibration. The calibration results can be interpreted by a inertial navigation domain expert. 6.4 Research Hypothesis The hypotheses that were to be tested in this thesis are that

199 Chapter 6 Results and Discussions 183 it is possible to derive an alternative gyro calibration strategy; while retaining the inherent accuracy related to present calibration techniques; this strategy will address time constraints in present calibration strategies; and the new calibration strategy will still result in a dataset that can be interpreted by a domain expert. At the conclusion of this thesis it can be stated that an alternative gyro calibration strategy was derived; where the inherent accuracy of the present calibration strategies was not only maintained, but surpassed; and that the proposed strategy addressed the time constraints of the present strategy by presenting alternative calibration profiles; and that the proposed calibration strategy resulted in a dataset that can be interpreted by a domain expert. 6.5 Publications Throughout this research project the focus was not on publications but rather on the development of a practical and novel calibration strategy, but two publications have already been accepted during the completion of the study. The first was presented at the AIAA Guidance Navigation and Control Conference in Chicago in 29 [25]. This paper presented the strategy and the key concepts on which the research is based. The other publication was presented at IEEE Africon in 211 in Zambia [24]. It presented the unified FOG error model.

200 Chapter 7 Contributions and Future Work 7.1 Contributions The following contributions were made as part of this study: 1. The development of a unified FOG error model and the development of an empirical FOG simulation model. 2. A review of the classical calibration approach through the casting of the calibration problem as an SID problem. 3. The use of computational intelligence techniques to perform the calibration of gyros which included the development of an explicit Criterion of Fit metric that is used to evaluate the abstract neural network within the problem domain. 4. The development of a novel non-linear gyro calibration strategy. 5. The calibration of the sensors under simultaneous dynamic excitation of the full range of multi-dimensional environmental conditions. 7.2 Future Research There are a number of areas into which the research presented here can be expanded. Some of these areas could result in novel research outputs whereas some of the others listed here are more focussed on the engineering contributions of the work. 184

201 Chapter 7 Contributions and Future Work Extensive Testing Although this study presents a novel calibration strategy for FOGs and investigates this strategy is great detail, all the work presented here was performed on a simulation of the real system. Further work should gather more measurement data from possibly a range of sensors and test the calibration performance against a high accuracy rotation table. Such a study should also investigate the application of the current calibration strategy to lower quality gyros and to accelerometers. One of the key aspects that prohibits the practical implementation of the neural network strategy as a calibration and compensation method to inertial sensors is the computational requirements associated with the neural network implementation on embedded hardware. The use of parallel computational platforms such as FPGAs and GPUs (Graphical Processor Units) could be investigated as a solution to this problem Recursive Real-time Calibration Apart from only using the suggested new calibration strategy to calibrate a single gyro at a time, it is also proposed that the technique could be used in an incremental or recursive way to reduce total system calibration time for a particular series of gyros. A flow diagram of the calibration process is presented in figure 7.1. With this approach it is suggested that the data for the first gyro in a new series of gyros be captured and presented to the network for training of the calibration network. When the network has been trained to a satisfactory accuracy, the resulting network (consisting of the interconnecting weights, node centres and node variances) is saved to be used for the online correction of the gyro measurement as presented in figure When a new gyro is presented for calibration, the network is initialized with the structure (consisting of the interconnecting weights, node centres and node variances) of the previous network that was trained and the network is trained to the required accuracy. When calibration (network training) is completed, the network is once again saved for calibration and for initialization of the calibration network of the next sensor. There are a couple of possible advantages of this recursive neural network based calibration strategy. 1. When a whole batch of sensors have been calibrated using this strategy, it is possible that any new sensors would only require a small incremental training sequence as the recursive calibration strategy has already generated a median set of calibration parameters for the sensor model under calibration. If it is observed that too much

202 Chapter 7 Contributions and Future Work 186 calibration effort is required for a particular sensor unit, that unit could be rejected or simply marked as a lower quality device. It should be apparent that this would require a real-time or online calibration strategy where the sensor is calibrated (network is trained) as the data is generated instead of having to generate a complete set of data and then performing the calibration through post-processing. 2. The traditional calibration strategy uses a set of data recorded on the 3-axis table to compute the calibration parameters and subsequently requires that another set of rotations are being performed with the sensor mounted on the table to verify the calibration accuracy. If the measured data is used to calibrated the network in the new strategy, the network is continually evaluated during training to determine whether the training threshold has been met. The result is that, if an extensive set of training data has been applied, it would not be necessary to perform a new set of rotations to validate the calibration, but that a simple subset of the original measurements be used to test the network. 3. If the network is trained in real-time while the rotations is being performed, the option exists to interrupt the whole calibration process once the network has reached its training threshold, thereby possibly resulting in significant reduction of the calibration cost Low-cost Inertial Sensors Some of the most important developments in the area of inertial navigation systems in recent years have been in the development of low-cost inertial sensors based on MEMS (micro electro-mechanical sensors) technology. The cost of these sensors is such that most consumer communication devices (smart-phones, tablet computers) and a significant component of recreational sports equipment contain these devices. The problem is that the methods that are used to manufacture these sensors (gyros, accelerometers and magnetometers) results in significant measurement errors present in these sensors. These errors are both stochastic and non-linear in nature and the presence of these errors effectively keeps the low-cost sensors from being used in serious applications such as robotics or aircraft applications. It is foreseen that the calibration strategy that was developed as part of this study be used to develop a low-cost calibration strategy for these sensors that will reduce the measurement errors and make the sensors more useful. Such a calibration strategy should make use of low-cost calibration equipment or should even not be making use of any

203 Chapter 7 Contributions and Future Work 187 specialized calibration equipment at all, but could perhaps use the available localization and mapping (SLAM) or camera-based algorithms to replace the need for the specialized equipment. Another requirement of such a calibration strategy will be that it should be able to be executed on a desktop computer and that the compensation algorithm should be able to be executed on mobile platforms such as smartphones or tablet computers.

204 Chapter 7 Contributions and Future Work 188 Start with first gyro Yes Is this first gyro in sequence? No Init RBFNN with ZERO weight set Init RBFNN with weight set of last calibrated gyro RBFNN input data Train RBFNN for new gyro RFBNN meet accuracy criteria? Yes No Continue training Stop training Store RBFNN calibration parameters STOP Figure 7.1: Proposed recursive calibration procedure.

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218 Appendix A Rogers Rotation Matrix Computation Method Rogers [87] presents a method for the computation of the rotation matrices between axes that is simple and accurate. The steps taken as part of this recipe are: 1. The axis of rotation has only zeros and ones, with the 1 located on the axis of rotation. 2. The cosine terms should be located on the diagonal. 3. The sine terms should be located on the cross-diagonal. 4. The minus sign should be assigned to the sine term that falls outside the original quadrant. This method is graphically depicted in figure A.1. 22

219 Appendix A Rogers Rotation Matrix Computation Method 23 y 2 This axis falls outside original quadrant θ x 2 Original quadrant θ y 1 x 1 (2) - Diagonal y 1 y 2 y 3 = (3) - Cross-diagonal cosθ sinθ sinθ cosθ 1 x 1 x 2 x 3 (1) - 1 s on axis of rotation, s off this axis (4) - - sign on y 2 -axis outside original quadrant x 3, y 3 Figure A.1: Rogers rotation matrix method.

220 Appendix B Definitions of Terminology Generalization A concept in neural networks whereby the trained network effectively learns to approximate the underlying functional relationship in the training data in such a way that it can accurately predict outputs for inputs that were not included in the original training set. Over-fitting A concept observed in the training of neural networks that stands in direct contrast to the concept of generalization. An over-fitted network is obtained when the network was trained to incorporate the noise present in the training data into the model of the system. The network therefore does not accurately represent the underlying process dynamics, as the correct system model is corrupted with the noise that was present in the training data. Parsimony The concept of parsimony is also known as Occam s Razor and states that the model that is used to describe a system should as simple and compact as it could possibly be while still being adequate to represent the system. A parsimonious model is sometimes also referred to as a sparse model. Regularization A technique used in the training of neural networks whereby nonsmooth networks are penalized through the addition of a generalization term in the cost function that is minimized during the training of the network. The effect is that a smooth network is obtained that does not attempt to learn the noise in the training data but only the underlying function. 24

221 Appendix C Test Trajectories The test trajectories that was used to evaluate the performance of the calibration strategies are presented in this appendix chapter. C.1 Temperature Model The trajectories were generated with high-precision 6-DOF flight generation software, but the original trajectories did not include a temperature model. A simple model was defined based on the US Standard Atmosphere 1962 (SA) as presented by Rauw [83]. The model is strictly only applicable up to altitudes of 11 meters above sea level, but for the purpose of the work done here this constraint was ignored for the cases where the altitude of the vehicle in the test trajectory were to exceed 11 meters. In the SA the air temperature in the troposphere (the atmospheric layer up to 11 meters above sea level) decreases linearly with altitude. The air temperature T in units of Kelvin is defined as T = T + λh (C.1) where T λ h is the air temperature at sea level ( K), is the temperature gradient in the troposphere (-.65 K/m) and is the altitude above sea level in units of meter, Since the empirical gyro error model was defined in terms of the temperature being measured in units of C, the results obtained from equation (C.1) can be converted from 25

222 Appendix C Test Trajectories 26 units of Kelvin to C by subtracting from the value computed. Since the temperature model is linear in the altitude, the temperature profile per trajectory will be a direct reflection of the altitude profile.

223 Appendix C Test Trajectories 27 C.2 aps TurnAround48g 1 Body axis Euler angle rates Roll rate [ /s] time [s] 1 Pitch rate [ /s] time [s] 1 Yaw rate [ /s] time [s] Figure C.1: Test trajectory aps TurnAround48g: angular rates.

224 Appendix C Test Trajectories 28 6 Body axis accellerations 4 X [m/s 2 ] time [s] Y [m/s 2 ] time [s] 1 Z [m/s 2 ] time [s] Figure C.2: Test trajectory aps TurnAround48g: acceleration.

225 Appendix C Test Trajectories 29 2 Euler angles 1 Roll [ ] time [s] Pitch [ ] time [s] 2 1 Yaw [ ] time [s] Figure C.3: Test trajectory aps TurnAround48g: Euler angles.

226 Appendix C Test Trajectories 21 6 NED axis velocity VN [m/s] time [s] 6 4 VE [m/s] time [s] 1 5 VD [m/s] time [s] Figure C.4: Test trajectory aps TurnAround48g: velocity.

227 Appendix C Test Trajectories NED axis position 5 N [m] time [s] E [m] time [s] D [m] time [s] Figure C.5: Test trajectory aps TurnAround48g: position.

228 Appendix C Test Trajectories 212 Horizontal position displacement 1 5 North position [m] East position [m] Figure C.6: Test trajectory aps TurnAround48g: horizontal displacement.

229 Appendix C Test Trajectories 213 C.3 aps vl668 1 Body axis Euler angle rates Roll rate [ /s] time [s] 1 Pitch rate [ /s] time [s] 1 Yaw rate [ /s] time [s] Figure C.7: Test trajectory aps vl668: angular rates.

230 Appendix C Test Trajectories Body axis accellerations 2 X [m/s 2 ] time [s] 8 6 Y [m/s 2 ] time [s] 8 6 Z [m/s 2 ] time [s] Figure C.8: Test trajectory aps vl668: acceleration.

231 Appendix C Test Trajectories Euler angles 1 Roll [ ] time [s] 1 8 Pitch [ ] time [s] 15 1 Yaw [ ] time [s] Figure C.9: Test trajectory aps vl668: Euler angles.

232 Appendix C Test Trajectories NED axis velocity 2 VN [m/s] time [s] 6 4 VE [m/s] time [s] VD [m/s] time [s] Figure C.1: Test trajectory aps vl668: velocity.

233 Appendix C Test Trajectories NED axis position 6 N [m] time [s] 15 1 E [m] time [s] D [m] time [s] Figure C.11: Test trajectory aps vl668: position.

234 Appendix C Test Trajectories 218 Horizontal position displacement North position [m] East position [m] Figure C.12: Test trajectory aps vl668: horizontal displacement.

235 Appendix C Test Trajectories 219 C.4 aps UAVref data 2 Body axis Euler angle rates Roll rate [ /s] time [s] 15 1 Pitch rate [ /s] time [s] 15 1 Yaw rate [ /s] time [s] Figure C.13: Test trajectory aps UAVref data: angular rates.

236 Appendix C Test Trajectories Body axis accellerations 2 X [m/s 2 ] time [s] 1.5 Y [m/s 2 ] time [s] 6 8 Z [m/s 2 ] time [s] Figure C.14: Test trajectory aps UAVref data: acceleration.

237 Appendix C Test Trajectories Euler angles 2 1 Roll [ ] time [s] 15 1 Pitch [ ] time [s] 2 1 Yaw [ ] time [s] Figure C.15: Test trajectory aps UAVref data: Euler angles.

238 Appendix C Test Trajectories NED axis velocity VN [m/s] time [s] 6 4 VE [m/s] time [s] VD [m/s] time [s] Figure C.16: Test trajectory aps UAVref data: velocity.

239 Appendix C Test Trajectories x 14 NED axis position 2 1 N [m] 1 E [m] time [s] x time [s] D [m] time [s] Figure C.17: Test trajectory aps UAVref data: position.

240 Appendix C Test Trajectories 224 x 1 4 Horizontal position displacement North position [m] East position [m] x 1 4 Figure C.18: Test trajectory aps UAVref data: horizontal displacement.

241 Appendix D INS Simulation D.1 INS Model Overview The full set of INS navigation equations as presented by Titterton and Weston [115] was implemented for the purpose of this investigation. The local geographic navigation frame mechanization of the strapdown INS equations was selected. The block diagram of the equations is presented in figure D.1. The various blocks presented in this figure was practically implemented as follows: For the blocks labelled Body mounted accelerometers and Body mounted gyroscopes the accelerations and angular rates form the reference trajectory was used, respectively. The block labelled Attitude computer contained the following computations: The gyro rates correction using the Earth rotation rate (ω n ie) and the transportation rate (ω n en) was performed. For both of these, the elliptical Earth assumption was followed. The attitude was presented as quaternions and the propagation of the attitude quaternion as a function of time was performed The block labelled Resolution of specific force measurements performed the rotation of the measured specific forces in the body frame (f b ) to the navigation frame (f n ) using quaternions. The Navigation computer block contained the double integrators for taking the acceleration measurements to velocity and positions. 225

242 Appendix D INS Simulation 226 Local gravity vector Gravity computer Position information IMU Coriolis correction Body mounted accelerometers f b Raw accelerometer compensation f b f n Resolution of specific force measurements Navigation computer Σ Σ x n x n x n Position and velocity estimates Body mounted gyroscopes ω b ib Raw gyro compensation ω b ib Cn b Attitude computer Initial position and velocity estimates Transport rate ω n ie + ω n en Initial attitude estimates Complete INS Figure D.1: Local geographic navigation frame mechanization of the strapdown INS equations as presented in Titterton and Weston [115].

243 Appendix D INS Simulation 227 The Coriolis correction block contains this correction based on the elliptical Earth model. The Gravity computer block computed the local gravity vector based on the altitude and latitude. A deliberate omission from the implemented INS model (as presented in figure D.1), is the altitude stabilization loop. This loop usually consists of an additional feedback loop similar to the gravity and Coriolis correction loops in figure D.1, that is used to address the altitude instability, but it requires the addition of altitude sensors. Since the pure inertial performance of the calibrated sensor was investigated, the addition of the altitude loop was not considered. D.2 INS Model Implementation The INS equations discussed in the previous section was implemented using the Simulink environment of Matlab. It was developed using discrete building blocks and not using the Aerospace Blockset. The simulation was based on the supplied reference trajectories with sampling rate set according to the specific trajectory and was triggered using the supplied trajectory data. The Simulink model of the implemented INS as it was used for the model validation is presented in figure D.2 with the RefINS block being defined as described in figure D.3. The Quaternion Simulink Library 1 Version 1.4 developed by Jay St.Pierre at the University of Colorado was used to perform the quaternion computations. 1

244 Appendix D INS Simulation 228 C Initial attitude 1 Import trajectory data from workspace Init att (Euler) From Gyro From Accel Accel error Gyro error Position Init pos Init att error (Euler) Velocity C RefINS Acceleration Initial position (lat,lon,alt) ( ) Angular rates Initial attitude error1 Attitude 12:34 SimTime Digital Clock To Workspace14 Figure D.2: Simulink implementation of the simulated INS. NED pos NED vel NED accel quat Lat,Lon,Alt posout To Workspace3 velout To Workspace4 attout To Workspace5

245 Appendix D INS Simulation 229 This implementation uses the full navigation equations to propagate the INS error states. It is based on chapter 3 in Titterton and Weston 24. Local gravity vector Gn g lat h 6 Init pos Gravity correction lat 4 Accel error out coriolis alt velvec Coriolis correction Init att error (Euler) 1 7 Init att (Euler) 3 From Accel att E2Q q att E2Q q 1 x o s.5 Gain q1 q2 f_b q1q2 Quaternion Multiply1 q1q2 Qb2l q1 q2 Quaternion Multiply V q qvq* Quaternion Vector Rotation2 q* Quaternion Conjugate1 f_n Ql2b qvq* V q Quaternion Vector Rotation1 accel w_e rates Earth rates q* 1 s lat alt velvec vel C 1 s pos q q2euler att initpos posvec Quaternion Conjugate2 q2euler Qb2l 5 Gyro error 2 From Gyro Figure D.3: Simulink implementation of the simulated INS. posconvert Lat,lon,alt Lat Lon Alt 1 NED pos 2 NED vel 3 NED accel 4 quat 5 Lat,Lon,Alt

246 Appendix E Test Definition E.1 Introduction The aim of this chapter is to present a description of the equipment used and processed followed in the gathering of the dataset that was used to develop the empirical sensor model. The methods presented here are extracted from the Incomar FOG Performance Evaluation Report [91]. Please contact Incomar Sensors 1 to obtain a copy of the complete report. E.2 Equipment Figure E.1 presents the equipment used and the connections between the different units of equipment for the calibration of the gyros in this dissertation. Table E.1 presents the requirement specification for the various units of equipment that was used during the calibration. The experimental setup consisted of a rate table controlled by the gyro-test software. The rotation rate of the table is measured with an optical encoder which is divided into 2, indexes for 36deg revolution. The rate table contributes considerably to the scale factor errors if not enough table revolutions are used per scale factor measurements. Multiple rotations reduce the noise contribution of the rate table. These measurements act as the reference against which the measurements from the gyro are compared. The measurements from the gyro is relayed via the sliprings of the rotation table to the data logging computer. Although the temperature of the environmental champer can be controlled rea

247 Appendix E Test Definition 231 Table E.1: Test equipment specifications. Equipment Single axis rotation table Environmental chamber Independent temperature sensor Single-axis test fixture Gyro data logging software Rotation table control and monitoring software Environmental chamber control and logging software Specification 15 /s input rates with axes misalignment of less that 1 milli-radian. -4 C to +85 C with a rate of change of temperature of 5 C /min and a control resolution of.1 C. Accurate to.1 C over the operational range of the environmental test. Orthogonal test cube for the simultaneous testing of multiple sensors along the same axis with a fixture orthogonality misalignment of less than 1 milliradian. Software capable of interfacing to the gyros and simultaneously logging inputs from all the sensors as well as the temperature sensors mounted on the gyros. Software capable of controlling the rotation table and logging the actual speed of rotation. Software capable of controlling the environmental chamber and logging the temperature and rate of change of the temperature.

248 Appendix E Test Definition 232 Environmental chamber Environmental chamber control interface Gyro Temperature probe Gyro slipring connection Data logging computer Rotation table Axial encoder Sliprings Rotation table assembly Figure E.1: Equipment setup for the gyro calibration. sonably well, an additional high accuracy temperature probe was added to the chamber to increase the accuracy of the calibration. E.3 Measurement Strategy E.3.1 Temperature-Angular Rate Relationship Measurements was performed according to a strategy where the temperature measurements were made at -2,, 2, 25, 4 and 6 C. Once the temperature inside the chamber stabilized at the required temperature, gyro measurement data were collected at the input angular rates of -1, -8, -6, -4, -2, -1, 1, 2, 4, 6, 8 and 1 degrees per second. Data were collected for 1 minute (which is an industry standard as define in reference [35]) at each of the angular rate measurement points. E.3.2 Allan Variance Computation For the computation of the Allan variance and the sensor bias, it is necessary to orient the gyro so that the measurement axis points either directly East or directly West. Under these conditions the rotational motion of the Earth will not be detected by the sensor during the data collection process and all measurements will be purely due to the bias and the various drift components that are part of the sensor. For the sensor tested in this