Experimental Assessment of MEMS INS Stochastic Error Model

Transcription

1 Experimental Assessment of MEMS INS Stochastic Error Model Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Peng-Yu Chen, B.S Graduate Program in Civil Engineering The Ohio State University 2015 Master s Examination Committee: Dr. Dorota Grejner-Brzezinska, Advisor Dr. Alper Yilmaz Dr. Charles Toth

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3 Abstract In recent years, the Global Positioning System (GPS) and the Inertial Navigation System (INS) integrated systems or, Inertial Measurement Unit (IMU), have become a core positioning technique due to their complementary characteristics. In the past, GPS was considered the primary sensor that contributed to the GPS/INS integrated system s solution, while INS only played an aiding role in bridging GPS outages. The evolution of the navigation philosophy, due to a broad availability of inexpensive inertial sensors, has made the IMU a primary navigation sensor. While GPS is still the key to generate a reliable integrated solution, its role is to focus on proper estimation of the IMU errors. Though GPS has been outperforming itself in recent years, the system is highly dependent on environment, and infrastructure, which is outside the users control. Hence, in order to enable the GPS/INS integrated system to operate in GPS challenged environments, it is crucial for INS to perform properly when GPS is not available. Since the drifts and biases of the IMU sensors caused by the random noise are the major components that affect the performance of the inertial navigation solution, this thesis is aimed at assessing the effect of customized stochastic error model on free inertial navigation solutions. In essence, if the stochastic processes assumed by the customized model can represent the behavior of the IMU sensor error better than a generic model formulated by the parameters obtained from the manufacturer s ii

4 specification sheet, there is an opportunity to improve the performances of the free inertial navigation solution. In this thesis, the Allan variance is applied to analyze the noise in the IMU measurements. Two methods: (1) slope reading and (2) model matching methods are used to estimate the coefficients of the customized stochastic error model. The results show that in most of the tested trajectories, a customized model improves the accuracy of the free inertial navigation between 40 % and 80% after a 30-second GPS gap. Despite the improvements with respect to the case of using the manufacturer s specifications, the accuracy of the solution after 30-second GPS gap obtained using customized model is insufficient for many land vehicle navigation applications. Considering 5 m as the loosely defined maximum position error for some land vehicle navigation applications, using the low-cost sensors tested here (micro-electro-mechanical, MEMS IMU). under such restriction, the solution estimated from the customized model was able to generate usable solutions for 5-11 seconds of the GPS outage, 2-3 seconds longer than using the default stochastic error model. As far as the attitude angles go, they were not improved in any of the tested trajectories even in situations when the position performance was improved. The model matching method performs slightly better than the slope reading approach; however, in terms of the overall magnitude of the drifts in MEMS IMUs, this small difference is insignificant. iii

5 Dedicated to my parents iv

6 Acknowledgments First of all, I would like to express my sincere gratitude to my advisor, Dr. Dorota Grejner- Brzezinska, for her continuous support, encouragement, advice, and patience through-out my study at The Ohio State University. I appreciate the knowledge Ive gained from working on this study and in her classes. I would like to thank Dr. Charles Toth for reviewing my thesis and his help in providing the data. I would like to thank Dr. Alper Yilmaz for being on my committee and reviewing the thesis. I would like to thank Dr. Allison Kealy from the University of Melbourne for her generosity in sharing the data, and her graduate student, Azmir Hasnur, for helping me to navigate this set of data. I would like to thank my fellow students in The Ohio State University for providing helpful and fruitful discussion with me. Especially, Jianzhu Huai, who spent his time to discuss with me the difficulties I encountered; and Greg Jozkow, who helped me out in collecting the stationary data and provided valuable technical suggestions. Last but not the least, I appreciate the unconditional support from my parents and family. v

7 Vita December 8, Born - Taipei, Taiwan June B.S. Geomatics Engineering, National Cheng Kung University August 2011 to May Research Assistant, Monitoring and Analysis of Crustal Deformation Lab, National Cheng Kung University January M.S Civil Engineering, The Ohio State University Fields of Study Major Field: Civil Engineering vi

8 Table of Contents Page Abstract Dedication Acknowledgments Vita List of Tables ii iv v vi x List of Figures xii List of Abbreviations xix 1. Introduction Background and Motivations Research Objectives Methodology Outline GPS/INS Integrated System Coordinate Frames and Coordinate Transformations Coordinate Frames Coordinate Transformations Introduction of INS INS Principle Navigation Equation in Arbitrary Frame Navigation Equation in N-frame IMU Error Analysis and Modeling vii

9 2.3.1 Methodology of Modeling Perturbation of the Navigation Equation Sensor Error Model GPS/INS Integration Kalman Filter Discrete Kalman Filter Extended Kalman Filtering Augmentation Stochastic Error Characterization and Modeling Stochastic Processes Review of Stochastic Processes Stochastic Error Models Literature Review of IMU Stochastic Error Modeling Methods Allan Variance Allan Variance Methodology Noise Identification Error Model Coefficient Estimation and Stochastic Error Modeling Error Model Coefficient Estimation MEMS IMU Stochastic Model Alternative Method for Error Model Coefficient Estimation Experiments and Results Experiment Set-Up Experiment with Stationary Data Microstrain 3DM GX Epson M-G Experiment with Kinematic Data Microstrain 3DM GX Epson M-G Conclusions and Recommendations Conclusions Recommendations Appendices 107 A. AVAR of Simulated IMU Measurements viii

10 B. Navigation Solution Error Plots C. Position RMS Plot Before GPS Outage References ix

11 List of Tables Table Page 2.1 Summary of the characteristics of GPS and INS Coefficient estimation of stochastic error contribution of MEMS IMU according to Allan Variance slope reading analysis. σ 2 (T ) is the Allan Variance; N is the A/VRW coefficient; B is the flicker noise coefficient; T c is the correlation time of first-order GM; q c is the noise amplitude of first-order GM; T peak is the cluster time corresponding to the peak point; σ(t peak ) is the Alla deviation corresponding to the peak point; K is the RRW coefficient; R is the rate ramp coefficient Duration of stationary data Kinematic data collection date, duration, length, and speed Manufacturer s specification of tested sensors (sf = scale factor and wn = white noise) Microstrain coefficient estimation from the slope reading method Microstrain coefficient estimation from the model matching method Epson coefficient estimation from the slope reading method Epson coefficient estimation from the model matching method Information of Trajectory 1 about the speed, the beginning epoch, the GPS solution quality 1 minute before GPS outage, and the resolve of state vector x

12 4.9 Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage A with respect to the reference GPS/INS trajectory based on 5-second 10-second, and 30-second GPS outage Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage B with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage C with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage Information of Trajectory 2 about the speed, the beginning epoch, the GPS solution quality 1 minute before GPS outage, and the resolve of state vector Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage A with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage B with respect to the reference GPS/INS trajectory based on 5 seconds, 10 seconds, and 30 seconds GPS outage Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage B without calibrating the flicker noise and RRW with respect to the reference GPS/INS trajectory based on 5-second, 10-second and 30-second GPS outages A.1 Microstrain coefficient estimation from the slope reading method A.2 Microstrain coefficient estimation from the model matching method. 109 A.3 Epson coefficient estimation from the slope reading method A.4 Epson coefficient estimation from the model matching method xi

13 List of Figures Figure Page 2.1 Illustration of i-frame, e-frame and n-frame (Chu et al., 2012). The i, e, n in the superscripts of each axis represent the i-frame, e-frame and n-frame, respectively; P is a point on the Earth with longitude and latitude being represented by λ and φ; O is the Earth center of mass; R is the vector pointing from O to P; ωe t is the angle that rotates the i-frame into the e-frame, where ω e is the Earth rotation rate and t is the time Stabilized platform process flow. a n is the accelerations in the n-frame measured by the accelerometers; ωib n is the gyro measurement, that is, angular rates of the b-frame with respect to the i-frame coordinated in the n-frame; g is the gravity calculated from the gravity model, given the coordinates of the moving objects; ẋ n is the velocity in n frame; φ, λ and h are longitude, latitude and height, respectively; η, χ and α indicate roll, pitch and yaw, respectively; denotes the integration Stabilized platform (King, 1998) Strapdown platform process flow. ωib b denotes the orientation rate of the moving body with respect to the i-frame with coordinates in the b-frame; Cb n is the DCM of transformation from b-frame to n-frame. The rest of the notation is the same as in Figure System model (IEEE-Std , 1997) Relation between c-frame and true frame (Benson, 1975). The subscript t denotes the true frame, c denotes the computer frame and p denotes the platform frame which is considered as the same to the b- frame in the thesis. φ is the phi-angle, ψ is the psi-angle and δθ is an angle between true frame and c-frame. The angles in the figure demonstrate the relationship only, not to scale xii

14 2.7 Illustration of error model used in this thesis. Physical signal is the true value applied to the sensor. The physical signal is first multiplied with the scale factor, and subsequently added with bias and stochastic errors through ; f is the output acceleration of an accelerometer; ω is the output angular rate of a gyro Kalman Filter loop (Jekeli, 2001, p. 215). Notation is consistent with equation (2.52) (2.57) a) Auto-correlation of white noise. Since white noise is only related to itself, the auto-correlation of white noise manifests as a peak at τ = 0. b) PSD of white noise without introducing Dirac delta function shows no power, which explains why Dirac delta is needed Time sequence of a random walk process repeated 1000 times Auto-correlation plot of a first-order Gauss Markov process Sample Allan deviation plot (IEEE-Std , 1997) Sample plot for angle/ velocity random walk (IEEE-Std , 1997). σ(t ) is the Allan deviation; N is the A/VRW coefficient; T is the cluster time Sample plot for flicker noise (IEEE-Std , 1997). σ(t ) is the Allan deviation; B is flicker noise coefficient; T is the cluster time Sample plot for correlated noise (IEEE-Std , 1997). σ(t ) is the Allan deviation; T c is the correlation time for the correlated noise; q c is the amplitude of correlated process noise; and T is the cluster time Sample plot for rate random walk (IEEE-Std , 1997). σ(t ) is the Allan deviation; K is the rate random walk coefficient; T is the cluster time Sample plot for rate ramp. σ(t ) is the Allan deviation; R is the rate ramp coefficient; T is the cluster time xiii

15 4.1 Trajectory 1 is collected by Microstrain 3DM GX45 IMU; 3 outages: A, B and C, marked with red bold lines, are introduced intentionally. The start and end of the trajectory are shown in the enlarged figure on the top right of Figure 4.1; the square denotes the starting position and the triangle denotes the ending position Trajectory 2 is collected by Epson M-G362 IMU; 2 outages: A and B, marked with red bold lines, are introduced intentionally. The start of the entire trajectory is shown in the enlarged figure on the top left of Figure 4.2; the square denotes the starting position, the triangle denotes the ending position, and the two cross marks denote the real GPS gaps Microstrain accelerometers Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for each axis of accelerometer Microstrain gyros Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for x and y accelerometer Epson accelerometers Allan deviation plot. The intersection of the - 1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for x and z axis accelerometer Epson gyros Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for each axis of accelerometer xiv

16 4.7 Trajectory 1, outage A plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively Trajectory 1, outage B plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively Trajectory 1, outage C plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively Trajectory 2, outage A plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively xv

17 4.11 Trajectory 2, outage B plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively Trajectory 2, outage B plot without flicker noise and RRW calibration during GPS outage. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by the model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds, and 30 seconds of GPS outage, respectively A.1 Microstrain 3DM GX45 accelerometers ((a), (c), (e)) and gyros ((b), (d), (f))allan variance plot of real data versus simulated data A.2 Epson M-G362 Allan variance plot of real data versus simulated data. ((a), (c), (e)) accelerometer x, y and z. ((b), (d), (f)) gyro x, y and z. 112 B.1 a) Trajectory 1, outage A horizontal error plot. b)trajectory 1, outage A orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method xvi

18 B.2 a) Trajectory 1, outage B horizontal error plot. b)trajectory 1, outage B orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method B.3 a) Trajectory 1, outage C horizontal error plot. b)trajectory 1, outage C orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method B.4 a) Trajectory 2, outage A horizontal error plot. b)trajectory 2, outage A orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method B.5 a) Trajectory 2, outage B horizontal error plot. b)trajectory 2, outage B orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method B.6 a) Trajectory 2, outage B horizontal error plot. b)trajectory 2, outage B orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method xvii

19 C.1 Trajectory 1, position RMS plot 20 seconds prior outage A. The vertical dashed line denotes the beginning of the GPS gap C.2 Trajectory 1, position RMS plot 20 seconds prior outage B. The vertical dashed line denotes the beginning of the GPS gap C.3 Trajectory 1, position RMS plot 20 seconds prior outage C. The vertical dashed line denotes the beginning of the GPS gap C.4 Trajectory 2, position RMS plot 20 seconds prior outage A. The vertical dashed line denotes the beginning of the GPS gap C.5 Trajectory 2, position RMS plot 20 seconds prior outage B. The vertical dashed line denotes the beginning of the GPS gap xviii

20 List of Abbreviations AIMS AIMS PRO AR ARMA A/VRW AVAR b-frame c-frame DCM DP e-frame EKF GM GPS i-frame IFOG IMU INS KF MEMS t-frame UAV PN PSD RF RLG RMS RRW w-frame Airborne Integrated Mapping System Airborne Integrated Mapping System Professional Auto-regressive Auto-regressive-Moving Average Angle/ Velocity Random Walk Allan Variance Body frame Computer frame Direct Cosine Matrix Direct Predictor ECEF (Earth Center Earth Fixed) frame Extended Kalman Filter Gauss-Markov Global Positioning System Inertial frame Interferometric Fiber Optic Gyro Inertial Measurement Unit Inertial Navigation System Kalman Filter Microelectromechanical systems True frame Unnamed Aerial Vehicle Personal Navigator Power Spectral Density Radio Frequency Ring Laser Gyro Root Mean Square Rate Random Walk Wander azimuth frame xix

21 Chapter 1: Introduction 1.1 Background and Motivations Inertial Navigation System (INS) (used interchangeably with IMU - Inertial Measurement Unit - for the rest of this thesis) was originally designed for rocket guidance (King, 1998). Its geodetic application, until recently, was mainly in determining coordinates of points that did not require high accuracy, that is at the level of tens of meters or less. Nevertheless, in recent years, methods based on INS usage that provide higher accuracy for geodetic purpose have been established. INS has started to play a more significant role in geodetic applications since the emergence of the Global Positioning System (GPS). Since 1993, when GPS reached its full operational capability, INS has been playing an important role in bridging GPS outages due to the complementary characteristics of the two systems (Bossler, 1992; Schwarz et al., 1993; El-Sheimy, 1996; Grejner-Brzezinska, 1997). Example complementary characteristics are (1) the operation principle; (2) the rate of data collection; (3) the error characteristics; and (4) the system degrees of freedom. The details will be provided in chapter 2. 1

22 INS had been playing a supportive role to GPS in the integrated system throughout past two decades, while recently, the importance of INS has increased and gradually become equally important to GPS in the GPS/INS integrated systems due to the quality improvements in the low-cost IMU and the decreasing cost of the sensors. Sensors with better quality have smaller errors in the measurements, which allows INS to generate usable free (unaided) navigation solutions for a longer period of time; the decrease in cost have broaden the application of INS. Both advances have made inexpensive INS a suitable device in many applications when collaborating with GPS. For example, a extensions of the Airborne Integrated Mapping System (AIMS) (Grejner-Brzezinska, 1998; Grejner-Brzezinska & Toth, 1998): personal navigator (PN) (Grejner-Brzezinska, Toth, & Moafipoor, 2007; Moafipoor, Grejner-Brzezinska, & Toth, 2008) and Unnamed Aerial Vehicle (UAV), are supported by the GPS/INS navigation system. The UAV can be used in emergency response, precision farming, hydrological mapping, etc; while PN can provide navigation solution to firefighters and rescuers. The UAVs have weight limitation; therefore, instead of using the heavy high-end IMUs, the vehicles are equipped with light weighted IMUs, such as MEMS IMUs. As for the application of the personal navigator, the PN also requires compact devices; hence, light weighted IMUs are the suitable sensors to apply to any portable platform. Due to the advances in IMU sensors, over the past decade, the philosophy of INS and GPS integration has reversed; GPS is no longer considered as a primary sensor in the integrated system, but becomes a support device for INS by providing high accuracy position and velocity to correct IMU errors. So, the focus is on correcting IMU errors; a shift from an earlier focus to simply bridge GPS gaps. Nevertheless, 2

23 the low cost MEMS IMUs nowadays still suffer from various sensor noise, which is likely to contaminate the accuracy of the navigation solution, and as a result, cause a decrease in its positioning accuracy. Thus, if the stochastic errors of MEMS can be modeled properly or even be customized for a specific sensor, there is the possibility to improve the accuracy of the navigation solution and improve the performance of the system in environments that suffer from temporary GPS outages. As a result, increase the feasibility of MEMS IMUs in civilian applications, like UAV and PN. 1.2 Research Objectives Section 1.1 lays out a promising path for low-cost INS applications, and points out the accumulative character of INS errors can cause a decay of the solution with time. INS errors are classified into two categories: deterministic and stochastic. Deterministic errors are normally calibrated in the factory and are mostly removed from the raw measurement by the internal algorithm of the sensor implemented by the manufacturers. However, calibration is never perfect, and the remaining uncompensated deterministic errors are considered as some random processes with probability characteristic given in the specification sheet and augmented to the state vector for further estimation through the filtering techniques. Stochastic errors, referred to as the internal noise of the sensors, on the other hand, can not be compensated or removed, and will always impact the measurements randomly. Stochastic errors are often characterized as white noise with the magnitude specified by the specification sheets distributed by the manufacturers. However, the noise in low-cost IMUs may not simply be a white noise; moreover, amplitude of the 3

24 noise of every sensor of the same type may not be exactly the same, and they may vary with the environment as compared to the test environment in the laboratory. The remaining uncompensated deterministic errors and stochastic errors will both be modeled when solving the navigation solution in this thesis, but only stochastic errors are of interest here. The deterministic errors will be handled as explained in the end of the first paragraph in this section. The stochastic errors will first be characterized and eventually, they will be augmented to the error model with the coefficients determined from the characterization. The objective of this thesis is to assess the type and stability of the stochastic error models for selected MEMS IMUs and to investigate the level of improvements in terms of the solution accuracy when the proposed customized stochastic error model is applied in the navigation solutions. 1.3 Methodology To achieve the objectives of this thesis, the stochastic model of the sensors must be formulated and applied in the free inertial navigation solution of a moving vehicle, that is, during the periods of no GPS updates. To formulate the stochastic model, one must collect IMU stationary data for several hours; then, Allan Variance or an alternative method must be applied to the stationary data for stochastic error identification. Finally, identified errors will be modeled as some stochastic processes with the coefficients determined from Allan Variance or an alternative method. GPS/INS integrated system, in general, generates a navigation solution: 3 positions, 3 velocities and 3 orientations; however, several modifications of the system exist in both hardware and software to generate solutions at different levels of accuracy. In terms of hardware, two configuration exist: (1) stabilized system and (2) 4

25 strapdown system; details can be found in section 2.2. Software on the other hand, has four categories: (1) uncoupled integration which simply resets the solution of integrated system to the coordinates solved by GPS whenever GPS solution is available (2) loosely coupled integration which fuses GPS solutions (position, velocity) with INS error states by a filtering technique (3) tightly coupled integration, which integrates GPS raw measurements with INS error states through a filtering technique (4) ultra-tightly-coupled integration, which not only fuses GPS and INS similar to tightly coupled, but also provide a feedback navigation solutions to predict GPS code and carrier-phase to aid the carrier tracking loops of a GPS receiver. The navigation data collected in this thesis are from strapdown sensors Epson M- G362, a part of the field testing of the OSU AIMS system (Grejner-Brzezinska, 1998; Grejner-Brzezinska & Toth, 1998), and Microstrain 3DM GX45 from the Intelligent Positioning Group of University of Melbourne. The reference trajectory for the purpose of performance assessment is provided by GPS/INS loosely coupled Extended Kalman Filter (EKF); the stochastic error characterization is processed through Allan Variance; the determination of stochastic model coefficients is tested with two different methods: (1) reading from the slope of AV plot and (2) model matching method proposed by (Berman, 2012). 1.4 Outline The rest of this thesis is organized as follows: Chapter 2 explains principles of the GPS/INS integrated system. This chapter will first introduce the principle of INS and the dynamic equations in a designated reference frame, followed by the error analysis of IMU sensors in the n-frame. Subsequently, the filtering technique to fuse INS 5

26 and GPS is explained. Chapter 3 presents the stochastic modeling of MEMS IMU. The basic characteristic of the stochastic process is briefly reviewed and a literature review of the existing stochastic modeling method is presented. The Allan Variance method of stochastic identification and coefficient estimation will be discussed in detail. An alternative method for estimating coefficient is presented in the end of chapter 3. Chapter 4 shows the experiments and results for the experimental error model for the selected MEMS IMU sensors. Chapter 5 presents the conclusions and recommendations. 6

27 Chapter 2: GPS/INS Integrated System 2.1 Coordinate Frames and Coordinate Transformations Coordinate Frames Inertial navigation is based on Newton s second law of motion in the inertial frame (i-frame). However, the geodetic applications and civilian applications require coordinate representations referred to the earth surface; thus, the Earth Center Earth Fixed frame, navigation frame, and wander azimuth frame introduced in the following section allow transformations from the inertial frame to other frames that are more intuitive for humans to interpret. The Body frame, computer frame, and true frame are introduced for the purpose of mathematic derivation in section Inertial Frame Broadly speaking, the inertial frame can be defined wherever Newton s law of motion holds. In inertial navigation, the i-frame is defined to be a non-rotating frame with origin attached to the Earth center. The i-frame is a right handed system with the z axis parallel to the spin axis of the Earth and x axis pointing to the mean vernal equinox as illustrated in Figure 2.1. Earth Center Earth Fixed Frame 7

28 The Earth center Earth fixed frame (e-frame) is a right-handed Cartesian system that is fixed to the Earth with its origin coinciding with the center of mass. The z axis of the e-frame is parallel to the mean polar axis of the Earth and the x axis is pointing to the intersection of the Greenwich meridian and the equator, see Figure 2.1. Figure 2.1: Illustration of i-frame, e-frame and n-frame (Chu et al., 2012). The i, e, n in the superscripts of each axis represent the i-frame, e-frame and n-frame, respectively; P is a point on the Earth with longitude and latitude being represented by λ and φ; O is the Earth center of mass; R is the vector pointing from O to P; ω e t is the angle that rotates the i-frame into the e-frame, where ω e is the Earth rotation rate and t is the time. Navigation Frame The navigation frame (n-frame) is a local coordinate frame used to describe the local directions of the vehicle. The n-frame has its origin at the center of the IMU 8

29 sensor, and as a result, the n-frame moves with the vehicle. N-frame is a right-handed system with x-axis pointing toward geodetic north and z axis pointing downwards, parallel to the gravity direction. Wander Azimuth Frame When in high latitude region, where north direction is difficult to define, a wander azimuth frame (w-frame) serves as the alternative to the n-frame. The only difference between the n-frame and the w-frame is that the x axis of the w-frame is rotated α (yaw) counterclockwise around z axis from the x axis of the n-frame. Body Frame The body frame (b-frame) refers to the frame of the navigating vehicle. It is a local frame with the first axis defined by the forward direction of the vehicle; the second axis defined by the right direction; and the third axis defined by the downward direction. In this thesis, the b-frame will be considered the same as the sensor frame and platform frame (p-frame). Computer Frame The computer frame (c-frame) is a local level frame where the computed trajectory is in; once the trajectory is determined, this frame is known. True Frame The true frame (t-frame) is a local level frame where the true trajectory is in; since the true trajectory can never be known, this frame is an unknown frame. Figure 2.6 depicts the relation between c-frame, t-frame, and p-frame. 9

30 2.1.2 Coordinate Transformations Transformation between frames can be accomplished by either direct transformation matrix (DCM) or quaternion. DCM is represented by C t o, which indicates transformation from o-frame to t-frame. DCM is a 3 by 3 matrix calculated by equation (2.4), given a set of Euler angles [θ 1, θ 2, θ 3 ] that represent the degrees of rotations one would need to rotate about the first, second and third axis of the original coordinate frame. The rotation about each axis is given in equations (2.1) (2.3). Note that the DCM depends on the order of applied rotations; different sequences will yield different DCMs R 1 (θ) = 0 cosθ sinθ (2.1) 0 sinθ cosθ R 2 (θ) = cosθ 0 sinθ sinθ 0 cosθ (2.2) R 3 (θ) = cosθ sinθ 0 sinθ cosθ (2.3) C t o = R 3 (θ 3 )R 2 (θ 2 )R 1 (θ 1 ) (2.4) where R 1, R 2, and R 3 represent rotation around first, second, and third axis, respectively; θ denotes the rotation angle; Co t is a DCM as explained in the previous paragraph. Quaternion, on the other hand defines a transformation with a 4 by 1 vector, shown on the left hand side of equation (2.6). The four elements of the quaternion are the coefficients from equation (2.5) and are real numbers that can be calculated 10

31 from DCM by equation (2.6). Instead of rotating three times about the axes of the original frame, transformation performed by quaternion rotates only one time about the vector indicated by the last three elements of a quaternion, b ξ, c ξ and d ξ. The amount of rotation is given by the first elements of quaternion, a ξ. q = cos ξ 2 + cos ξ 2 (ib + jc + kd) = a ξ + ib ξ + jc ξ + k d ξ (2.5) 1 a ξ C1,1 + C 2,2 + C 3,3 b ξ 1 c ξ = 4a ξ (C 2,3 C 3,2 ) 1 4a ξ (C 3,1 C 1,3 ) d 1 ξ 4a ξ (C 1,2 C 2,1 ) where: ξ is the rotation angle. q is a quaternion. i, j, and k are imaginary units. b, c, d are real numbers. a ξ, b ξ, c ξ, d ξ are the four elements of a quaternion. (2.6) C is the matrix representation of DCM equivalent to the quaternion, and the numbers in the subscripts indicate the element of this DCM matrix. The DCMs given below are some common coordinate transformation matrices used in this thesis. ω e ie = [0, 0, ω e ] T Ci e = R 3 (ω e t) = cos ω e t sin ω e t 0 sin ω e t cos ω e t Cn e = R 3 ( λ)r 2 ( π 2 + φ) = (2.7) (2.8) sin φ cos λ sinλ cos φ cos λ sin φ sin λ cosλ cos φ sin λ cosφ 0 sinφ 11, C n e = C e nt (2.9)

32 C n b = R 3 ( α)r 2 ( χ)r 1 ( η) (2.10) where: ω e ie denotes the rotation rate between e-frame and i-frame coordinated in the e-frame. ω e is the earth rotation rate. C e i is the transformation matrix of i-frame to e-frame. t is the time interval. Cn e is the transformation matrix of n-frame to e-frame. Ce n is the transformation matrix of e-frame to n-frame; orthogonal to Cn. e λ is the longitude; φ is the latitude. C n b is the transformation matrix of b-frame to n-frame. η (roll), χ (pitch), and α (yaw) are the rotation angles of the first, second, and third axis of the body frame with respect to the respective n-frame axes. 2.2 Introduction of INS Inertial Navigation System is a system that consists of a set of IMU sensors including accelerometers, which measure accelerations, and gyroscopes, that sense rotation rates; the platform where IMU sensors are mounted; and the computer that carries out the algorithms that are implemented to solve for positions and velocities from accelerations of the platform coordinatized in a frame, whose orientation is controlled by gyroscopes (Jekeli, 2001, p. 101). Due to different mechanization approaches of the platform, INS can be classified into two categories: stabilized platform system and strapdown platform system. They both yield 3 dimensional coordinates: longitude, latitude, height; and 3 dimensional 12

33 attitudes: roll, pitch, yaw. In stabilized platform system, the system calculates the velocities of the object internally from accelerometer measurements and a chosen gravity model, that includes the gravitational acceleration and centrifugal acceleration of the Earth (see equation (2.29)). The velocities are integrated to output the 3 positions. As illustrated by the flowchart in Figure 2.2, while the platform rotates, the gyros of each axis mounted on the stabilized platform will output an angle calculated from the velocity of the respective axes internally in the system; this set of output angles [η, χ, α] are then tuned into a current that eventually triggers the roll, pitch and azimuth servo motor on the platform gimbals respectively to null the gimbal angle of the gyro. Figure 2.2: Stabilized platform process flow. a n is the accelerations in the n-frame measured by the accelerometers; ωib n is the gyro measurement, that is, angular rates of the b-frame with respect to the i-frame coordinated in the n-frame; g is the gravity calculated from the gravity model, given the coordinates of the moving objects; ẋ n is the velocity in n frame; φ, λ and h are longitude, latitude and height, respectively; η, χ and α indicate roll, pitch and yaw, respectively; denotes the integration. Figure 2.3 depicts the configuration of the gimbals and the servo motors. This mechanism allows the platform to remain parallel to the i-frame or local horizon; 13

34 hence the angular motion of the platform is isolated from the vehicle motion. Spacestabilized platforms are fixed in orientation with respect to the i-frame and local-level stabilized platforms are kept tangent to the local horizon. Figure 2.3: Stabilized platform (King, 1998). In the strapdown platform system, IMU sensors are contained in one box that is mounted on the moving platform. The box contains only the six sensors to measure the dynamic of the object, there is no gimbal inside to perform feedback mechanism in the sensor to slew the platform. Instead, the platform is securely tied to the vehicle, since there is no gimbal to keep the platform stabilized, strapdown systems are more computationally intensive and complex. As illustrated by the flowchart in Figure 2.4, each accelerometer outputs the acceleration of the platform caused by specific force in the b-frame (a b ) and each gyro outputs the angular rate of the platform with respect to the i-frame with coordinates in the b-frame (ωib b ). The velocities, positions and orientations of the moving object are calculated through computer separated from the IMU platform. 14

35 Figure 2.4: Strapdown platform process flow. ωib b denotes the orientation rate of the moving body with respect to the i-frame with coordinates in the b-frame; Cb n is the DCM of transformation from b-frame to n-frame. The rest of the notation is the same as in Figure 2.2 Since the strapdown system, as compared to the stabilized system is more costeffective, smaller in size, less in weight and power consumption, it is widely adopted nowadays for commercial applications. The data collected in this thesis are from strapdown INS mechanization INS Principle Newton s second law of motion is the foundation of inertial navigation. It is expressed in equation (2.11), which states that the derivative of linear momentum (P ) of a particle equals to the total applied force (F ) applied on it. However, being in the gravitational field, there exist kinematic (non-specific) force that causes particles to accelerate; thus, equation (2.11) is modified to take gravitational force (m g g) into consideration, as shown in equation (2.12). Under the assumptions that the mass m i remains constant throughout the motion, and m i = m g, equation (2.13) is obtained, which states that ẍ i, the acceleration of a particle in the i-frame, is the sum of a i, 15

36 the acceleration caused by the specific force, and g(x i ), the gravitational acceleration being applied to the particle. Equation (2.13) is integrated once to yield the velocity of the particle in the i-frame, and integrated twice to generate the position of the particle in the i-frame. Equation (2.13) is not confined to particles, it can be applied to any moving object in the i-frame; the acceleration of the moving object caused by specific force can be sensed by the IMU and the gravitational acceleration can be calculated based on the position of the object. It is important to note that, unlike the output accelerations of a space stabilized system, which are transformed into i-frame mechanically, specific forces sensed by three accelerometers in a strapdown system are in the b-frame and should be transformed into the i-frame computationally by a 3 by 3 transformation matrix Cb i, as shown in equation (2.14). C i b at a specific epoch is obtained by integrating equation (2.15). To integrate equation (2.15), an initial C i b and a skew symmetric matrix of the rotation rate between b-frame and i-frame (Ω b ib ) defined in equation (2.16) is needed. Initial transformation matrix C i b is determined by the angles of each axis between i-frame and b-frame, which can be calculated by integrating the angular rates (ωib b ) sensed by gyros; Ω b ib is formed from the angular rates (ωb ib ) sensed by gyros, by following equation (2.16). The differential equation of C i b listed in equation (2.15) can be expressed equivalently in the quaternions form, as listed in equation (2.17), and is often adopted when implementing the transformation to avoid singularity problem of the rotation vectors and also to avoid the complexity in integrating trigonometric functions, which occurs 16

37 when integrating equation (2.15). F = d dt (P ) = d dt (m iẋ i ) (2.11) m i ẍ i = F + m g g (2.12) ẍ i = a i + g(x i ) (2.13) a i = C i ba b (2.14) Ċ i b = C i bω b ib (2.15) Ω b ib = [ω b ib ] (2.16) q = 1 2 Aq (2.17) A = skew 4 (ω b ib) (2.18) q = [ a ξ b ξ c ξ d ξ ] T (2.19) 1 a ξ C1,1 + C 2,2 + C 3,3 b ξ 1 c ξ = 4a ξ (C 2,3 C 3,2 ) 1 4a ξ (C 3,1 C 1,3 ) d 1 ξ 4a ξ (C 1,2 C 2,1 ) (2.20) [k ] = skew 3 (k) = 0 k 3 k 2 k 3 0 k 1 k 2 k 1 0 (2.21) 17

38 skew 4 (k) = where: 0 k 1 k 2 k 3 k 1 0 k 3 k 2 k 2 k 3 0 k 1 k 3 k 2 k 1 0 (2.22) The i and b in the subscripts and superscripts denote the i-frame and b-frame, respectively. The single number in subscripts represent the axes; a pair of numbers in the subscript of C represents the row and the column of the transformation matrix element. Variables with one dot on top represent their first-order time derivatives and two dots on top represent their second-order time derivatives. F, P, m i, m g C i b, and Ωb ib are the same as explained in the context of section x, a and g are the position, acceleration and gravitational acceleration vectors. ω b ib is the vector of angular rates sensed by the three gyroscopes which is the angular rates of the b-frame relative to the i-frame, coordinated in the b-frame. A is a 4 by 4 skew symmetric matrix of ω ib. q is the quaternion vector that serves as another way to represent transformation between frames. The relationship between DCM (C) and q is given in equation (2.6). The sign ( ) denotes the operation of a 3-dimensional skew matrix skew 3 (). k = [k 1 k 2 k 3 ] T is a vector used here to demonstrate the skew matrices. Notation representation is consistent throughout the thesis. To summarize, the dynamics of the i-frame strapdown system are represented by equation (2.13) and (2.17), repeated below for convenience. Once the initial position and orientation of the platform are given from external sources, INS is able to provide the velocities by integrating equation (2.13) once and provide the positions by 18

39 integrating equation (2.13) twice. The orientations can be determined by integrating equation (2.17) once Navigation Equation in Arbitrary Frame Unfortunately, the i-frame is not the most commonly used frame for objects moving on Earth where coordinates are often represented as ECEF Cartesian coordinates or longitude, latitude, and height. Similar differential equations can be formulated in other reference frames, such as e-frame, n-frame or w-frame, depending on where the users wish to determine the solution of the object for the application. Thus, differential equations for an arbitrary frame, a-frame, are derived in this section, so that it can be applied to different coordinate frames. Suppose this arbitrary frame is concentric with the i-frame, a set of differential equation similar to (2.13) can be derived by differentiating x i = Cax i a twice to yield equation (2.23). Equation (2.23) can be written into two first order differential equation as shown in (2.24) and (2.25), also known as the navigation equations, by introducing the velocity vector ẋ a (Jekeli, 2001, p. 125). ẍ a = 2Ω a iaẋ a ( Ω a ia + Ω a iaω a ia)x a + a a + g a (2.23) d dtẋa = 2Ω a iaẋ a ( Ω a ia + Ω a iaω a ia)x a + a a + g a (2.24) d dt xa = ẋ a (2.25) where: The superscripts indicate the frame, in which the variables is defined. One dot and two dots above a symbol represent the first order and second order 19

40 derivative of the variable, respectively. The same notation rules apply for the rest of this thesis. x a, a a and g a are the 3 by 1 position, acceleration and gravitational force vectors in the arbitrary frame. Ω a ia is the skew symmetric matrix of the angular rate ωia, a that represents the orientation rate of a-frame relative to the i-frame coordinated in the a-frame. Determination of ωia a depends on where the a frame is defined. See equation (2.7), if a = e, and refer to equation (2.33), if a = n Navigation Equation in N-frame The integrated GPS/INS system, AIMS (Grejner-Brzezinska, 1998; Grejner-Brzezinska & Toth, 1998) developed at OSU SPIN lab, and used in this thesis, forms the navigation equation in n-frame. However, the navigation equation cannot be generated by simply substituting a for n in equation (2.24). This is due to the fact that unlike the i-frame or the e-frame, which are fixed coordinate frames, the n-frame is a dynamic frame that is designated to define local directions for velocity vector (Jekeli, 2001, p. 127), as shown in equation (2.26). Therefore, when using the unified approach to form navigation equations in n-frame, after replacing the subscript a with n, the ẋ n in the equation should be replaced with equation (2.27), to subsequently arrive at equation (2.28), which is the navigation equation in n-frame. v n = C n e ẋ e (2.26) ẋ n = v n Ω n enx n (2.27) 20

41 dtẋn d = (2Ω n ie + Ω n en)v n + a n Ce n Ω n ieω n iex e + g n (2.28) where: v n is the velocity in n-frame. C n e is the transformation matrix of e-frame to n-frame. ẋ e is the velocity vector in the e-frame. ẋ n is the derivative of the position vector x n. Ω n en is the skew symmetric matrix of ω n en, which represents angular rate of n-frame relative to e-frame coordinated in n-frame. Ω n ie is the skew symmetric matrix of ω n ie, which represents angular rate of e-frame relative to the i-frame coordinated in n-frame. a n and g n are the acceleration and gravitational acceleration in n-frame, respectively. Knowing that C n e Ω n ieω n iex e equals to the centrifugal acceleration caused by the rotation of the Earth (Jekeli, 2001, p. 127), the sum of the last two terms C n e Ω n ieω n iex e + g n in equation (2.28) equals to the gravity vector g, which by definition is the sum of the gravitational acceleration vector and the centrifugal acceleration vector. With the fact that (2Ω n ie + Ω n en) = (Ω n in + Ω n ie) (Jekeli, 2001, p. 128), equation (2.28) can be simplified to equation (2.29); together with (2.27) and an attitude dynamic equation stated in equation (2.31), which defines the transformation between n-frame and b- frame, form the complete set of navigation equations that represent the dynamic of platforms in the n-frame: d dtẋn = (Ω n in + Ω n ie)v n + a n + g n (2.29) d dt xn = v n Ω n enx n (2.30) 21

42 q = 1 2 Aq (2.31) Ω n in = [ω n in ] (2.32) ω n in = ω n ie + ω n en (2.33) Ω n ie = [ω n ie ] (2.34) ω n ie = C n e ω e ie = [ cos φω e 0 sin φω e ] T (2.35) Ω n en = [ω n en ] (2.36) ω n en = [ (v n ) 2 (R M +h) cosφ (v n ) 1 (R N +h) (v n ) 2 (R M +h) sinφ ] T (2.37) v n = [ φ cos φ φ λ sin φ ] T (2.38) where: x n, ẋ n, v n, a n, and g n are the same as in equations (2.26) (2.28). q is the quaternion that defines the transformation from b-frame to n-frame, it can be calculated by integrating equation 2.31 as demonstrated in equation A is the 4 by 4 skew symmetric matrix of ωbn n. Ω n in is the skew symmetric matrix of ωin, n that is determined by equation (2.33). Ω n ie is the skew symmetric matrix of ωie, n that is determined by equation (2.35). Where C n e is defined in equation (2.9), φ and λ denotes the latitude and longitude, respectively, and ω e is the Earth rotation rate. Ω n en is the skew symmetric matrix of ω n en, that is determined by equation (2.37). 22

43 Where R M and R N are the radii of curvature of the ellipsoid in the Meridian plane and the prime vertical plane, respectively; h is the geodetic ellipsoid height of the position. The subscript of the vector v n denotes the axis. Once the dynamic of the platform is formulated, the navigation solutions can be generated by integrating the navigation equations. Integration can be performed in different orders, higher order algorithms lead to less integration errors. The approach shown below is the simplest one. The integration of acceleration is shown in equation (2.39); the integration of velocity requires extra effort since n-frame moves along with the platform; conceptually the position of the moving objects in n-frame is always [0, 0, 0]. It would be meaningless to calculate the position by integrating velocity in n-frame. In AIMS PRO, the velocity is transformed into e-frame and is integrated in e-frame as shown in equation (2.40). Equation (2.41) shows the integration of the quaternion. v n k = v n k 1 + ( (Ω n in + Ω n ie)v n + a n + g n ) t (2.39) x e k = x e k 1 + (C e n(k)v n k ) t (2.40) q k = exp( 1 2 tk t k 1 Adt)q k 1 (2.41) where k and k 1 denotes the epoch of a vector or a matrix; t is the data sampling interval between two measurements of the IMU sensor; Cn e is the n-frame to e-frame transformation matrix. The rest of the symbols are the same as in equations (2.29) (2.37). 23

44 2.3 IMU Error Analysis and Modeling Methodology of Modeling According to (IEEE-Std , 1997), modeling of any unit (the term unit can be referred to a sensor, a system or other devices), as illustrated in Figure 2.5, should contain the following four components: (1) a mathematical statement that describes the physical operation of the unit; (2) an error model consisting of a perturbation model and environmental sensitivities; (3) a stochastic model that describes the unit s internal random drift behavior; and (4) a measurement model constructed by a linear combination of the external measurement output and its additive measurement noise. Figure 2.5: System model (IEEE-Std , 1997). INS being a complicated system constructed by up to six sensors is no exception. As of the mathematical statement, differential equations (2.29) (2.31), which describe the dynamics of the system in n-frame, serve as an example. The perturbation 24

45 error model can be described by equation (2.42) or (2.43) and will be explained in the next section. The stochastic model for controlling internal sensor noise is discussed in chapter 3. The measurement model is shown in equation (2.50) for the GPS/INS loosely coupled system Perturbation of the Navigation Equation To fully understand how sensor errors propagate through the system into the navigation solution, error dynamic equations are used (Jekeli, 2001, p. 139). Error dynamic equations are derived by perturbing the navigation equations. In this case, perturbation means that the variables in the navigation equations are differentiated and the differentials represent the small differences between the true value of a variable and the approximate value integrated from the measurements. These small differences are known as the errors of the variables. Since we assume the errors are small, it is reasonable to neglect terms higher than the first order in the perturbed equations. By doing so, the error dynamic equations become linear; thus, they can be implemented using Extended Kalman Filter (EKF). Since this thesis generates solutions in n-frame, the perturbation model and the stochastic model of INS developed and discussed subsequently will be formed in the n-frame. Two approaches of perturbation have been presented in the literature; they are the psi-angle error model (Bar-Itzhak & Berman, 1998) and the phi-angle error model (Britting, 1971). To obtain the phi-angle error model, navigation equations (2.29) 25

46 (2.31) are perturbed with respect to the true frame to arrive at: δẋ n = δv n ω n en δx n δ v n = C n b δf b + C n b f b Φ n δ(ω n in + Ω n ie)v n (Ω n in + Ω n ie)δv n + δg n (2.42) Φ n = ωin n Φ n Cb n δωib b + δωin n where: The symbol δ denotes the perturbation of the variables; hence, variables associated with such sign represent the error terms. δẋ n, δ v n and Φ n are 3 by 1 vectors that denotes the velocity, acceleration and orientation rate errors, respectively. δx n, δv n and Φ n are the position, velocity and orientation error vectors. C n b is the transformation matrix of b-frame to n-frame. δf b and δω b ib refer to the accelerometer and gyro sensor errors. δg n is the gravity error in n-frame. The rest of the notation is same as in equations (2.29) (2.37). The psi-angle error model, unlike phi-angle error model, which perturbs the navigation with respect to the unknown true frame, perturbs the navigation with respect to the computer frame that is a known frame where trajectory is in. Figure 2.6 shows the relationship between the c-frame, true frame and platform frame. The psi-angle error model is derived by perturbing the navigation equations (2.29) (2.31) with respect to the c-frame. To generate the psi-angle error model, replace the superscript n in equation (2.42) with c (Benson, 1975) and neglect the error terms ωic c and ωie, c since by definition, c-frame is a known frame determined by the calculated trajectory, once the trajectory is known, ω c ic and ω c ie can be exactly determined. Therefore, ω c ic and ω c ie are errorless, δω c ic and δω c ie will not contribute to the velocity error in the 26

47 Figure 2.6: Relation between c-frame and true frame (Benson, 1975). The subscript t denotes the true frame, c denotes the computer frame and p denotes the platform frame which is considered as the same to the b-frame in the thesis. φ is the phi-angle, ψ is the psi-angle and δθ is an angle between true frame and c-frame. The angles in the figure demonstrate the relationship only, not to scale. psi-angle error model. The details of psi-angle error model are shown in equation (2.43). δẋ c = δv c ω c ec δx c δ v c = C c bδf b + C c bf b Ψ c (Ω c ic + Ω c ie)δv c + δg c (2.43) Ψ c = ω c ic Ψ c C c bδω b ib Equation (2.43) can be organized and expressed in matrix form : δẋ c ω c δ v c ec I 0 δx c δω = 0 (Ω c Ψ c ic + Ω c ie) Cb cf b δv c + 0 C c ib b 0 0 ωic c Ψ c b I δf b (2.44) Cb c 0 0 δg c where: Ω c ic is the 3 by 3 skew symmetric matrix of ω c ic, which can be obtained using the velocity vector (v c ) and the latitude (φ) of current epoch following equation (2.33). Ω c ie is the 3 by 3 skew symmetric matrix of ω c ie, which can be obtained using the Earth rotation rate (ω e ) and the latitude (φ) of current epoch following equation (2.35). C c b can be determined using the attitude of current epoch following equation (2.10). 27

48 The rest of the notation is the same as in equation (2.42); but, instead of the true frame, equations (2.43) and (2.44) are in the c-frame Sensor Error Model As can be seen from the integration equations (2.39) (2.41), if the initial conditions v n 0, x e 0 and q 0 input to the system are errorless, the errors that occur in the navigation solution will mainly be the accumulations of accelerometer errors, δa b, and gyro errors, δωib b, as the system proceed with time. Of course, the initial information is not likely to be errorless most of the time, and there will be gravity model errors, numerical computation errors, and model errors referring to the terms neglected during linearization. Nevertheless, they are relatively minor, as compared to the sensor errors. Thus, modeling sensor errors correctly is the priority concern here. In this section, models for accelerometers and gyros will be discussed. Since this thesis focus on the low-cost MEMS IMU, sensor error model introduced accordingly are specifically for MEMS IMUs, which mainly focus on the constant bias offset and the scale factor error. The errors in IMU measurements can be classified into two groups, they are (1) the deterministic errors and (2) the stochastic errors. The deterministic errors, also called systematic errors, are predictable. The input-output relationships are described by functions (e.g (2.45) and (2.46) ); therefore, it can be determined how each deterministic error contributes to the sensor output. Deterministic errors in the IMU sensors include (1) scale factor errors (2) misalignments (3) biases (4) temperature related variations (5) non-linearity errors (6) G-sensitivity effects (only for gyros) and (7) environmentally induced errors (Yi, 2007). Normally, deterministic errors are already 28

49 calibrated from the output of the sensors at the hardware level using calibrated coefficients determined by the manufacturer; however, inaccuracies in the deterministic errors often exist. Thus, in this thesis, the inaccuracies of the deterministic errors are considered to be some random processes that are estimated using EKF through augmenting the error terms into the state vector when estimating the navigation solution. The stochastic errors, also known as random errors, include (1) long-time correlated random bias, also called turn-on bias, (2) random drifts, for instance, the rate white noise, the flicker noise, the rate random walk, the rate ramp, the Markov noise and the sinusoidal noise, and (3) the quantization noise. Detailed information about their models is provided in chapter 4. In this thesis, the author takes two main deterministic error sources into consideration; they are scale factor errors and biases. Figure 2.7 illustrates how the two deterministic errors considered in this thesis and stochastic errors contribute to the sensor measurements. The physical analog signal, acceleration or orientation, is first digitalized; thus, the scale factor error is introduced at this point. Subsequently, an offset and internal stochastic errors of the IMU enters the measurements before the scaled signal is output. Scale Factor Scale factors are ratios that are used to convert a physical quantity input acting along the sensitive axis to a digital output. Due to the manufacturing imperfections and change of environment, scale factors may contain errors. Scale factor errors can be divided into three parts, a static part, a random drift part and a temperature varying part. The static part is considered deterministic; therefore, can be addressed 29

50 Figure 2.7: Illustration of error model used in this thesis. Physical signal is the true value applied to the sensor. The physical signal is first multiplied with the scale factor, and subsequently added with bias and stochastic errors through ; f is the output acceleration of an accelerometer; ω is the output angular rate of a gyro. by calibration. On the other hand, the random drift part is not deterministic and cannot be compensated. This remaining uncompensated error vary from run to run and is augmented into the state vector as a random constant for better estimation when solving for navigation solution. Bias Bias is the offset between the physical quantity input and the digital output. Unlike scale factor, bias has no relation with the input quantity. In other words, the sensors will provide a non-zero output even when there is no specific force acting on the sensors. Like scale factor errors, bias can be decomposed into three parts: a static part called bias offset, a random part called bias drift and a temperature varying part. The bias offset is the deterministic part and the bias drift refer to the stochastic part. Sensor Error Model Equation (2.45) and (2.46) describe the error model of a single accelerometer and a single gyro used in this thesis, respectively (Yi, 2007). δf b = δs f + b f + e f (2.45) δω b ib = δs ω + b ω + e ω (2.46) 30

51 where δs f and b f are the scale factor error and the bias offset of a accelerometer, respectively; e f represents the accelerometer noise, which is the sum of all stochastic errors in acceleration measurement; likewise, δs ω and b ω are the scale factor error and the bias offset of a gyro, respectively; e f represents the gyro noise, which is the sum of all stochastic errors in the angular rate. 2.4 GPS/INS Integration As discussed in section 2.2, the IMU sensor errors introduce errors into the navigation solution; a perturbation error model helps analyze the effect of the sensor errors but is unable to estimate the errors and make corrections to the sensor output. As such, a independent measurements of the designated trajectory is required to aid the IMU in terms of reducing the effect of sensor errors on the navigation solution. GPS, as mentioned briefly in chapter 1, is a navigation technique complementary to INS. They have different operation principles, varying error characteristic and different data resolutions, which results in many distinct yet complementary characteristics between the two (see Table2.1). In terms of operation principle, GPS is a RF-based navigation and time transfer system. It relies on space segment and control segment to obtain broadcast ephemeris and measurements to the satellites to perform lateration to calculate user s coordinates. Thus, line of sight to satellites in space is required. Since GPS signals may be blocked or interfered with when traveling to Earth, the availability of GPS is highly dependent on the environment in which GPS receiver operates. INS, on the other hand, operates autonomously; therefore, the availability of the solution does 31

52 not depend on the environment. In terms of error characteristic, errors in GPS measurements do not drift with time. The errors in GPS measurements may be similar between consecutive epoch, but the errors are not passed on from one epoch to the next (with the exception of incorrect ambiguity integers). While INS drifts with time, the measurement from one epoch is affected by the previous measurements. In terms of the solution performance, the errors in GPS solution do not accumulate with time, since the solution is provided on the epoch by epoch basis with the observations to multiple satellites; but, the accuracy of GPS solution depends on the baseline between the user and the reference station. The errors in INS solution on the other hand do accumulate, since INS solution is obtained by integration. Table 2.1 summarizes the advantages and disadvantages of both system. Table 2.1: Summary of the characteristics of GPS and INS. System Advantages Disadvantages GPS INS Measurements do not drift. High long term accuracy. No initial position needed. Errors in solution do not accumulate. Operates autonomously, Does not require infrastructure. Less dependent on environments. 6 degrees of freedom. High data rate. Require infrastructure. (space and control segment). Highly dependent on environments. 3 degrees of freedom. Low data rate. Measurements drift easily. Low accuracy. Requires accurate initial position. Errors in solution accumulate with time. Due to these complementary characteristics, INS and GPS are integrated to serve as a navigation technique. Integration of the two systems are performed at different levels to reach the accuracy and robustness one would need for a specific application. Fusion of the two system can be accomplished by four approaches: (1) uncoupled 32

53 integration, (2) loosely-coupled integration, (3) tightly coupled integration, and (4) ultra-tightly coupled integration. The loosely-coupled integration is most often applied due to its simple architecture. 2.5 Kalman Filter Filtering is used to perform integration of IMU measurements with external observations such as GPS, camera or any other unconventional sensors to estimate the navigation solution and to better quantify the INS accumulative system errors to limit the navigation error growth. As a result, regardless of the integration scheme, a filtering technique is required to augment the IMU sensor error terms into the state vector and use external observations (e.g GPS) for the optimal estimation of not only the navigation solution but also the system s errors. Kalman Filter, based on the paper Kalman published in 1960 (Kalman, 1960), is most widely used to integrate external observations with INS due to its on-line characteristic and its ability to remove the noisy measurements while retaining the useful measurements with the weighting matrix in the algorithm. Kalman filter inherits the concept of Bayesian estimation, which is estimating the probability density of the states from one epoch to the next by using the probability density from the previous epoch and a given probability density of the external measurement at current epoch (Jekeli, 2001, p. 199). It is an algorithm that operates recursively on input data to provide minimum mean square error estimates of the system states of a linear dynamic system (Yi, 2007). See (Jazwinski, 1970) for a review of KF. 33

54 2.5.1 Discrete Kalman Filter To demonstrate the application of KF in inertial navigation, consider a continuous linear dynamic system like equation (2.47), where x(t) is the state vector of random variables depending on time t; w(t) represents a continuous zero-mean Gaussian white noise; F and G are matrices that govern the system dynamic. ẋ(t) = F (t)x(t) + G(t)w(t) (2.47) Since the INS dynamics based on Newton s second law of motion represented by a set of first order differential equations are continuous in nature, it is reasonable to apply equation (2.47) to represent dynamic equations (2.43). To estimate the state vector x(t), one must relate the external measurements (e.g GPS positions) to the system states through forming a linear measurement model. However, in reality, the output of INS free navigation solutions and external measurements from GPS are both in discrete form, hence, one must discretize equation (2.47) and generate a discrete linear model as shown in equation (2.48). Equation (2.48) together with the linearized measurement model shown in equation (2.50) and an initial condition (2.51) complete the set up for KF. The process of KF can be divided into three stages: (1) prediction (2) computing Kalman gain and (3) update. Equations (2.52) (2.57) demonstrate the optimal unbiased linear estimate of the unknown state vector, given an initial state vector (x 0 ) and its initial covariance matrix (P 0 ). Figure 2.8 illustrates an algorithmic flowcharts of the Kalman filter loop process. 34

55 Linearized system model: x k = Φ(t k, t k 1 )x k 1 + u k, (2.48) ε(u k u T k ) = Θ = tk t k 1 Φ(t, t )G(t )ε(w(t )w(t ))G(t )Φ(t, t )dt dt G k Q tg T k (2.49) Linearized measurement model: y k = H k x k 1 + v k, v k N(0, R k ) (2.50) Initial conditions: ˆx 0, ˆx 0 N(0, P 0 ) (2.51) Prediction: ˆx k = Φ(t k, t k 1 )ˆx k 1 (2.52) ŷ k = H ˆx k 1 (2.53) ˆP k = Φ(t k, t k 1 )P k 1 Φ(t k, t k 1 ) T + Θ (2.54) Compute Kalmain gain: K k = P k HT k (H k P k HT k + R k ) 1 (2.55) Update: ˆx k = ˆx k + K k(y k ŷ k ) (2.56) P k = (I K k H k )P k 1 (I K kh k ) T + K k R k K T k (2.57) 35

56 where: x is the state vector of the system; the subscripts k, k 1 and 0 denote the epochs. Φ(t k, t k 1 ), approximated by Φ(t k, t k 1 ) = I + F t, is the state-transition matrix used to predict state vector from epoch k 1 to k and is assumed to be constant during interval t between two epoch. u k is a vector that contains Gaussian, zero-mean, discrete white noise. ε is the expectation operator. w is continuous white noise. Θ represents the covariance matrix of the vector u k. G(t) and G k are well defined continuous and discrete matrices that describe the contribution of the white noise from epoch to epoch in the system. Q is the a matrix contain elements that describes the amplitude of the white noise. y is the observation vector formed by external measurement (e.g the position difference between GPS solutions and free navigation solutions). H is the measurement matrix. v is the measurement noise vector with covariance matrix R. P is the covariance matrix of its corresponding state vector. The negative sign in the superscripts represent the predicted state of the vector or matrix. The ˆ above variables indicates the estimated value of vectors and matrices from MMSE. Refer to (Gelb, 1974) for more details about optimal estimation. K is the Kalman gain matrix. I is the identity matrix. 36

57 Figure 2.8: Kalman Filter loop (Jekeli, 2001, p. 215). Notation is consistent with equation (2.52) (2.57) Extended Kalman Filtering The Kalman Filter plays a role in estimating the state vector, which refers to the navigation solution and errors in INS sensors. As long as GPS updates are available, the trajectory will be controlled by GPS updates. The solutions will drift between GPS updates or during GPS gaps. Recall from the previous section, when linearizing a non-linear system, higher order terms are neglected based on the assumption that they are small. However, if the indicated solution is drifting away from the trajectory obtained from GPS, this assumption will not hold. As a result, it will not be suitable to apply KF to the system. The solution to fix this is to apply Extended Kalman Filter, also known as closed loop estimation. The idea of EKF is that the navigation solution calculated by integration is adjusted with the error estimates, so that after 37

58 the correction, the solution indicates the true trajectory. As a result, ˆx k, the error state, is always kept at zero, equations (2.52) and (2.56) become: ˆx k = Φ(t k, t k 1 )ˆx k 1 = 0 (2.58) ˆx k = ˆx k + K k(y k H ˆx k 1 ) = K ky k (2.59) The navigation solution at epoch k is corrected with ˆx k recursively. Notation is the same as in section Augmentation One can easily discover the similarity between equations (2.44) and (2.47). Though being similar, equation (2.44) here cannot yet be considered as the linear dynamic model in equation (2.47.), since w(t) in equation (2.47) is specified to be a zero mean, Gaussian white noise, but, the accelerometer error and gyro error terms, as discussed in section 2.3.3, are not. It is important to model the system correctly, so that the variables can be optimally estimated by the filtering technique. Hence, in order to form (2.44) to match with the linear dynamic model, the sensor deterministic errors listed in equation (2.45) and (2.46) are augmented into the state vector, so that the error dynamic equations can be modified to be comparable with the linear dynamic model expressed in equation (2.47). In AIMS, the state vector includes: 9 navigation solution errors (3 for position, 3 for velocity and 3 for attitude), 6 scale factor errors (3 for accelerometers and 3 for gyros), 6 biases (3 for accelerometers and 3 for gyros) (Yi, 2007). Equations (2.60) (2.68) provide algorithmic details. ẋ xvψ F 11 F 12 F 13 x xvψ [ ] [ ] ẋ f = G wf + 0 w ẋ ω 6 1 ω F F 33 x f x ω 21 1 (2.60) 38

59 F 11 = F xx F xv 0 F vx F vv F vψ 0 0 F ψψ 9 9 (2.61) F xx = C n e 0 (ωen) n 3 (ωen) n 2 (ωen) n 3 0 (ωen) n 1 (ωen) n 2 (ωen) n 1 0 (C n e ) T, F xv = C n e I 3 (2.62) F vx = g+(f n ) 3 R N g+(f n ) 3 R N g+(f n ) 3 R N (2.63) F vv = 0 (ωie n + ωin) n 3 (ωie n + ωin) n 2 (ωie n + ωin) n 3 0 (ωie n + ωin) n 1 (ωie n + ωin) n 2 (ωie n + ωin) n 1 0 (2.64) F vψ = 0 (f n ) 3 (f n ) 2 (f n ) 3 0 (f n ) 1 (f n ) 2 (f n ) 1 0 (2.65) F ψψ = 0 (ωin) n 3 (ωin) n 2 (ωin) n 3 0 (ωin) n 1 (ωin) n 2 (ωin) n 1 0 (2.66) F 12 = 0 0 F vbf F vsf 0 0, F vbf = Cb n, F vsf = Cb n 9 6 (f b ib ) (f b ib ) (f b ib ) 3 (2.67) F 13 = F ψbω F ψsω, F ψbω = Cb n, F ψsω = Cb n 9 6 (ω b ib ) (ω b ib ) (ω b ib ) 3 (2.68) F 22 = 0 6x6, F 33 = 0 6x6 (2.69) where x xvψ, x f and x ω are the state sub-vectors of the system that represent navigation 39

60 solution errors, accelerometer sensor errors (3 biases and 3 scale factors) and gyro sensor errors (3 biases and 3 scale factors), respectively. F 11, F 12, F 13, F 22 and F 33 are the dynamic matrices that govern how error terms transit from epoch to epoch. G is the matrix that governs how stochastic errors contribute to the state vector. F xx, F xv, F vx, F vv, F vψ and F ψψ are detailed sub-matrix components of F 11. F vbf and F vsf are detailed sub-matrix components of F 12. F ψbω and F ψsω are detailed sub-matrix components of F 13. The symbols in the subscripts of F show the two states involved in the dynamics. (e.g F xv represents how current position errors (x) are affected by the velocity errors (v) from the previous epoch). F 12, F 13, F 22 and F 33 depend on the coordinate frame of the system and how the users wish to characterize the stochastic errors. If the sensor biases and scale-factor errors are modeled as random constant processes, the detailed corresponding sub-matrices are shown in equation(2.67) (2.69). fib b and ωb ib are the accelerometer and gyro measurements, respectively. The subscripts outside the bracket represent the measurements of a specific axis. g is the gravity. f n is the accelerometer measurements in n-frame. R N is the radius of curvature of the the ellipsoid in the prime vertical. 40

61 Chapter 3: Stochastic Error Characterization and Modeling The stochastic errors in MEMS IMUs are normally modeled as a stochastic process or a combination of several stochastic processes. For instance, the Gausss Markov (GM) process, white noise process, random constant or random walk are some commonly used processes for IMU stochastic modeling (Nassar, 2003). In order to describe stochastic processes mathematically, coefficients that indicate the amplitude of the noise process and the correlation between current epoch and previous epochs are needed. One way of determining these coefficients is through the Autoregressive (AR) process and Autoregressive-moving average (ARMA) (Nassar, 2003), which determine the parameters of the model by using estimation techniques. Another way is to collect long duration stationary data, so that noise that may impact the measurements can be analyzed based on different time domain by techniques, such as Allan Variance (AVAR) (Hou, 2004; Yi, 2007) and power spectral density (PSD) (IEEE-Std , 1997; Yi, 2007). The other way is simply obtain the coefficients of the stochastic error model from manufacturer specification data sheet; however, in terms of low cost MEMS IMU, most of the time the manufacturer specification sheet provides only the coefficients to model white noise but not flicker noise, rate random walk or rate ramp. Once the coefficients are determined, the models formed by the coefficients are then augmented with other error states to create an expanded 41

62 dynamic system, so that error states can be estimated using EKF. In summary, the essential idea of all stochastic error modeling method is to estimate the variance and covariance of the noise so that the noise are well characterized in the system. The stochastic modeling approach adopted in this study is Allan Variance (Allan, 1966). To determine the model using this approach, first, one needs to identify the types of errors through the AVAR plot; then, one needs to model the errors with the combination of white noise, random walk and 1st order GM process using the parameters obtained from the AVAR plot. In this chapter, the three stochastic processes used in this thesis to model IMU sensors are first introduced, followed by a brief literature review of several stochastic modeling techniques. Then, the Allan Variance methodology and its analysis of MEMS inertial sensors is presented. The augmentation of EKF dynamic equations and state vector using parameters obtained from AVAR is explained at the end of this chapter. 3.1 Stochastic Processes Review of Stochastic Processes Random process (stochastic process) is a collection of random variables representing the random value of a system in terms of a deterministic parameter, such as time or space (Jekeli, 2001, p. 172). Contrary to deterministic variables, random variables refer to those that cannot be described by functions; thus, generally speaking being indeterminable. The value of a random variable can only be predicted based on the probability this variable is subject to. Though being predictable, the actual value is governed by chance and depends only on the outcome of an experiment which may vary from time to time.the random process that is of interest in this thesis is the 42

63 stochastic errors that appear in the output of the MEMES IMUs. This section is aimed to explain the statistical terms that will be used to establish statistical model in describing the randomness of the noise. Note that, due to the complexity of the sensors, the models are only approximations which can never represent the true random error processes of the system. However, they seem to serve as a reasonable approach of characterizing sensor stochastic errors. Auto-covariance Since random variables cannot be exactly determined, they are often described by the first and second order moments, which refer to the mean and the variance of a single variable, respectively. The auto-covariance is a function that describes the covariance of the process at pairs of time arguments, see equation (3.1). The autocorrelation represents similar information to auto-covariance function, but the value is not centered with respect to their respective mean values, see equation (3.2). C x (t, t + τ) = ε((x(t) µ xt )(x(t + τ) µ xt+τ )) (3.1) = ((x(t) µx t )(x(t + τ) µx t+τ )f(x(t), x(t + τ))dx(t)dx(t + τ) = R x (t, t + τ) µx t µx t+τ where: C x (t, t + τ) is the auto-covariance that represents the covariance between process x at time t and t + τ. ε() is the expectation operator. τ is the time interval between two time instances. x is a time dependent random process. µ xt and µ xt+τ are the expected values of the process at t and t + τ, respectively. 43

64 f(x(t), x(t + τ)) is the joint probability density function. R x (t, t + τ) is the auto-correlation function defined in the equation below: R x (t, t + τ) = ε(x(t)x(t + τ)) (3.2) where notation is the same as in equation (3.1) If the probability does not change with time, such random process is called stationary process. To be more specific, the auto-covariance of a stationary process is independent of the time instances in the process and it is invariant under the translation of time (Jekeli, 2001, p. 173). In other words, the joint probability density function f(x(t), x(t + τ)) is identical for every pair of time arguments separated by the same τ; therefore auto-covariance function only depends on the time interval. Power Spectral Density Auto-covariance is an informative yet simple descriptor for random processes in time domain. The time interval of the sensor output is closely related to the frequency of the process. Hence, it is also useful to express the process in terms of frequency domain. To do that, one need to apply Fourier transform to the correlation function (3.2) to generate the power spectral density function (3.3) that characterizes the random process in frequency domain. In fact, the stochastic parameters provided by the specification sheet are often expressed by PSD. Φ x (f) = R x (τ)e i2πτ dτ (3.3) where Φ x (f) denotes the PSD of process x; R x (τ) is a atuo-correlation function of variable x; and e i2πτ denotes the basic waves in time domain. White noise 44

65 A random process can be called white noise process if it is a stationary process and if the random variables in this process are all uncorrelated regardless of any time interval, τ. That is, variable at a specific time is only related to itself, resulting in equation (3.4), and is illustrated by the auto-covariance plot shown in Figure 3.1 (a). This character of white noise makes it difficult to describe the power of it, since the Fourier transform of the white noise correlation function will result in zero for all frequency, as illustrated in Figure 3.1 (b). Thus, Dirac delta function, δ(t), defined in equation (3.5) (Bracewell, 1986, p. 74) is introduced to allow white noise processes to have different amplitudes. The idea is that for a function g(t), when integrated together with the delta function, the integral in equation 3.5 holds. Consider the auto-correlation of a white noise process, w, defined in equation (3.6); after applying the Fourier transform, the PSD of the white noise, Φ w (f), is shown in equation (3.7), which shows that the PSD is a constant q at all frequencies, indicating that the power from all frequencies contributes the same magnitude to the process. R x (t, t + τ) = 0, τ 0 (3.4) δ(t) = 0, t 0; and δ(t) =, t = 0 g(t )δ(t t )dt = g(t) (3.5) R w (t, t + τ) = ε(w(t)w(t + τ)) = qδ(τ) (3.6) Φ w (f) = qδ(τ)e i2πτ dτ = qe 0 = q, for all f (3.7) where δ(t) is the Dirac delta function; g(t) is a function of time; t denotes a time instance; w represents the white noise process; q is a constant; e is the exponential function; f is the frequency. The rest is the same as in equations (3.2) and (3.3). 45

66 (a) (b) Figure 3.1: a) Auto-correlation of white noise. Since white noise is only related to itself, the auto-correlation of white noise manifests as a peak at τ = 0. b) PSD of white noise without introducing Dirac delta function shows no power, which explains why Dirac delta is needed Stochastic Error Models Some stochastic processes represented by linear differential equations (also known as shaping filters in some literature) are proven to be applicable in modeling the stochastic nature of many physical processes, including the IMU stochastic errors discussed here. Three of such processes are: random constant, random walk and Gauss Markov model. Random Constant Random constant is a process in which variables remain constant values over the process as time goes on; however, this constant value itself is a random variable. When one repeats a random constant process, the value of the constant may differ from process to process. But, if one do experiments to collect this value repeatedly, one may know the mean and variance of the process. Continuous random constant is described by the differential equation (3.9) or equation (3.10) for a discrete process; initial 46

67 condition is listed in (3.8); the auto-correlation of the process is given in equation (3.11). x(t 0 ) = x 0 (3.8) ẋ(t) = 0 (3.9) x k = x k 1 (3.10) R X (x t, x t+τ ) = σ 2 x = const, for all τ (3.11) where x is a time dependent random process ; t 0 is the initial epoch; x 0 is the initial value of the random constant process; k and k 1 in the subscripts denote the epoch of the process; σx 2 is the variance of the random constant determined by repeated laboratory or field experiments; the rest of the symbols are the same as in equation (3.1). Random Walk Random walk is a process driven by white noise, w(t). It is described by the differential equation (3.13) or discrete form (3.14) with initial condition listed in (3.12); its auto-correlation is given in equation (3.15), calculated by the amplitude of PSD, q, and the time interval between current epoch and the initial epoch, (t t 0 ). Since the navigation solutions are calculated by integrating the outputs of IMU sensors, it is easily discovered that random walk process is useful in modeling the errors that are introduced to the solutions when integrating the additive white noise in the accelerations or angular rates. The white noise of the sensor data has a zero mean; however, as shown in Figure 3.2, the angles integrated have wider distribution as time progress. Random walk model provides a quantitative statement about the 47

68 distribution of the angles or velocities (Stockwell, 2003). For instance, if the angle random walk of a gyro is reported to be 1deg/ sec, the distribution of the angle will be ±1 deg after a second and ±10 deg after 100 seconds. x(t 0 ) = 0 (3.12) ẋ(t) = w(t) or x(t) = t t 0 w(t )dt (3.13) x k = x k 1 + w k (3.14) R x (x t, x t+τ ) = t t+τ t 0 t 0 ε(w(t )w((t + τ) ))dt d(t + τ) (3.15) = q(t t 0 ), τ > t 0 where t 0 is the initial epoch; w(t) represents the time dependent white noise process; k and k 1 in the subscripts denote the epoch of the process; q is a constant representing the amplitude of PSD; τ is the time interval between two time instances; the rest of the symbols are the same as in equation (3.1). First-order Gauss Markov Process First-order Gauss-Markov process refers to a stochastic process that satisfies the condition of both Gaussian process and Markov property (Lamon, 2008). That is, if the random variable in the random process has a normal distribution (Guassian process) and if the probability distribution of the current variable depends only on the most recent previous state next to it (Markovian property), this random process can be considered a first-order Gauss-Markov process. Higher order (n th order) GM processes refer to those that depend on the n most recent states. A first order continuous GM process can be expressed as a first order differential equation shown in 48

69 Figure 3.2: Time sequence of a random walk process repeated 1000 times. (3.16); a discrete form is presented in equation (3.17); correlation of the process is described in (3.18). ẋ(t) = βx(t) + w(t), ε(w(t)w(t )) = 2σ 2 βδ(t t ) (3.16) x k = e β t x k 1 + 2βσ 2 tw k (3.17) R x (t, t + τ) = σ 2 e β τ (3.18) where β = 1 T c is the correlation time coefficient, inverse of the correlation time T c, that describes the degree of correlation between time arguments for a first-order GM process; σ is the standard deviation of the process noise in first-order GM process; t is the sample interval when discretizing; the rest of the symbols are the same as in equations (3.13) (3.15). 49

70 Figure (3.3) displays a correlation function of the first-order Gauss-Markov process. As shown in the figure, the covariance decreases exponentially as time interval increases, which indicates that the correlation between samples decreases. The covariance reaches zero when τ arrives at infinity. Gauss-Markov is widely applied due to its ability to fit large number of physical process with reasonable accuracy and it simplicity in mathematical description (Brown & Hwang, 1996, p ). Figure 3.3: Auto-correlation plot of a first-order Gauss Markov process. 3.2 Literature Review of IMU Stochastic Error Modeling Methods IMU sensors, especially gyros, are subject to drifts and random noise. For highend IMUs, such as navigation grade or tactical grade IMUs, these terms can be handled through tuning the filter using sensor specification given by the manufacturer. Low-cost sensors, on the other hand, are subject to more error sources and noisier 50

71 output signals; thus, using the information from sensor specification sheet may not be sufficient to model the noise and drift in MEMS based IMUs. This leads engineers to develop various approaches that are able to estimate the stochastic model coefficients of MEMS IMUs. Autoregressive models (Nassar, 2003; Park, 2004), ARMA, power spectral density (IEEE-Std , 1997; Yi, 2007) and Allan Variance (Hou, 2004; Yi, 2007) are some common approaches. In the AR model, current stochastic error is described as a finite linear combination of its past values and the current additive white noise. In other words, each stochastic signal at any time is predicted from the previous values of itself (Nassar, 2003). Solving the coefficients α j and β 0 in equation (3.19) is accomplished by minimizing the difference between the process and the estimated signal represented by the AR model using optimal prediction methods such as t he Yule-Walker method, the covariance method and Burgs method (Nassar, 2003). x(t) = p α j x(t j) + β 0 w(t) (3.19) j=1 where x is a random process; α j and β 0 are the coefficients of the AR model; t represents the time; w(t) is the white noise. Though AR is relatively easy to implement in EKF and shows improvements in the solution (Nassar, 2003; Park, 2004); the linear prediction based approaches limit its ability to identify noises from different sources. That is, it models the entire effect of stochastic errors with one model; but, cannot model each stochastic errors caused by different sources. ARMA combines autoregressive method with the moving average, which appears to be a linear regression of the current value of the process against the current and the previous white noise. It had been reported in (Seong, Lee, & Park, 2000) that 51

72 different ARMA orders are equivalent to a different combination of white noise, rate random walk, first-order GM correlated noise and quantization noise (refer to the section for terminology explanation). But the fact that ARMA involves solving the non-linear equation makes it less attractive in actual implementation. PSD discussed in section is another way to characterize and model stochastic errors. Since each stochastic error shows different power characteristics, the stochastic error model coefficient of each identified stochastic error can be extracted from the PSD plot to form the augmented dynamic equation that eventually is fed to the EKF. It is reported in (Yi, 2007) that PSD method is unable to distinguish between some low frequency noise, such as rate random walk and rate ramp which will be explained in section The major drawback of PSD method is that it requires some calculations for the parameters needed in dynamic equations; hence, it is less straightforward for non-specialist. 3.3 Allan Variance Allan Variance is a time domain analysis approach initially developed to study the frequency stability of oscillators. It was found useful in determining the character of the underlying random process of the data noise (IEEE-Std , 1997). As such, Allan Variance is introduced to characterize the noise properties of ring laser gyro in the 1980s and later on being applied to different grades of gyros as well as accelerometers (Hou, 2004) Allan Variance Methodology When using Allan Variance method, the uncertainty in the data is assumed to be generated by noise sources of specific character. The idea of Allan Variance is 52

73 to represent root mean square (RMS) random drift error as a function of averaging time (IEEE-Std , 1997). Since different noise appear in different cluster time (T ) on the AVAR plot, various random processes that exist in the data can be easily identified. The coefficient of each noise source can be estimated by reading the curve from AVAR plot or fitting a curve to the AVAR plot, which is calculated from a set of stationary data collected by the IMU sensors. According to (IEEE-Std , 1997), Allan Variance is calculated by taking half of the mean value of a series of squared difference between time averages of two adjacent clusters. When the stationary IMU measurements are gyro rates or accelerations, the Allan Variance, σ 2 (T ), of N samples of measurements with sampling interval τ 0 can be calculated as shown in equation (3.21). σ 2 (T ) = 1 2 (Ω k+t Ω k ) 2 (3.20) = N 2m 1 (Ω 2T 2 k+t Ω k ) 2 (3.21) (N m) k=1 Ω k = 1 T tk +T t k Ω(t)dt (3.22) Ω k+t = 1 T tk +2T t k +T Ω(t)dt (3.23) where: T = mτ 0 (m < N/2) is the cluster time; τ 0 is the sampling interval of the stationary measurement. Ω(t) is the time dependent acceleration or angular rate measurement. Ω k represents the average measurement between time t k and t k + T as shown in equation (3.22). 53

74 Likewise, Ω k+t represents the average measurement between time t k + T and t k + 2T. represents the ensemble average. If the output of the sensor is N samples of angles or velocities which is defined as the integral of orientation rates or accelerations as shown in equation (3.24), AVAR can be calculated by equation (3.25), which substitutes Ω k and Ω k+t in equation (3.21) with equation (3.27) and (3.28), respectively. t θ(t) = Ω(t )dt (3.24) σ 2 (T ) = 1 2T (θ k+2t 2θ k+t + θ k ) 2 (3.25) N 2m 1 = θ 2T 2 k+2t 2θ k+t + θ k ) 2 (N m) (3.26) k=1 Ω k = 1 T (θ k+t θ k ) (3.27) Ω k+t = 1 T (θ k+2t θ k+t ) (3.28) where θ represents the angle or velocity outputs from the sensors; Ω(t ) denotes the acceleration or angular rate of the sensor; the rest of the symbols are the same as in equations (3.21) (3.23). If not considering the correlated noise and the sinusoidal noise, and assuming that the existing random processes are all statistically independent, then, it can be shown that AVAR of IFOG gyro at any given cluster time, T, is the sum of AVAR due to the individual random processes at the same T. See equation (3.29) and the approximated equation of Allan deviation (3.30). Typically, Allan deviation, which is the square root of Allan Variance is used for plotting a log-log plot and representing values. For example, the sample plot of IFOG 54

75 Figure 3.4: Sample Allan deviation plot (IEEE-Std , 1997). gyros shown in Figure 3.4. But, the term Allan Variance is used when discussing this method. This thesis also adopt this convention for the following discussion. σ 2 (T ) = N 2 T + 2B2 ln2 π + K2 T 3 + R2 T Q2 T 2 (3.29) σ(t ) NT B 2ln2 π + K T 1 R T + Q 3T 2 2 (3.30) 3 where σ 2 (T ) is the AVAR; σ(t ) is the Allan deviation; N, B, K, R, and Q are the coefficients of white noise, flicker noise, rate random walk, rate ramp, and quantization noise that may exist in the IFOG measurements, respectively. Their values are summarized in Table Noise Identification How Allan Variance curve is related to the instrument noise depends on the understanding of the physics of the instrument. The noise terms identified from Allan Variance curve laid out in (IEEE-Std , 1997) are specific for IFOG; therefore, 55

76 it is not entirely applicable to MEMS IMU since both sensors are designed based on different principles. Luckily, except for the quantization noise, the rest of the noise terms can be applied to MEMS IMU measurements. The random noise that may be identified from a MEMS Allan deviation curve include: (1) angle/velocity random walk (2) flicker noise (3) rate random walk (4) rate ramp (5) exponentially correlated noise (6) sinusoidal noise. There are possibilities that noise created by certain sources out-shadows other noise, or certain noise in the sensor being tested is insignificant to the extent that it cannot be identified; thus, it is not guaranteed that every error terms mentioned above will appear in the Allan Variance plot of any specific sensor. For example, the sinusoidal noise cannot always be discovered in the AVAR curve. The Allan Variance calculated from equation (3.21) or (3.25) is related to PSD of the noise term in the collected stationary data. The relationship is shown in equation (3.31). Therefore, once the mathematic representation of PSD of a specific noise term is known, its AVAR expression can be derived. In this thesis, AVAR expression of each noise will be given; detail derivation from the PSD of the noise, S Ω (f), to AVAR of the noise, σ 2 (T ), can be found in (Hou, 2004). σ 2 (T ) = 4 0 S Ω (f) sin4 (πft ) (πft ) 2 df (3.31) where σ 2 (T ) is the Allan Variance; T is the cluster time; S Ω (f) represents the PSD; f is the frequency; sin4 (x) (x) 2 is a transfer function. Angle/ Velocity Random Walk Angle/velocity random walk (A/VRW), also known as white rate noise, refers to the white noise caused by thermo-mechanical events that appears in the output of gyros/accelerometers (Leland, 2005). That is, when sensors are operating, there is always a random value added to the signal. For instance, when the gyro is stationary, 56

77 Figure 3.5: Sample plot for angle/ velocity random walk (IEEE-Std , 1997). σ(t ) is the Allan deviation; N is the A/VRW coefficient; T is the cluster time. the output of the sensor should be zero; nevertheless, the output for static gyros in reality is never zero. This type of noise is a high frequency noise that can be characterized in the short cluster part of the AVAR plot. It is named velocity/angle random walk because when rate signals are integrated over time to generate the navigation solution, the white noise in the rate signals will turn into random walk errors in angles or velocities. Despite the terminology used, this type of noise is simply a white noise. Thus, it is modeled as a white noise process in the system. In order to represent a variable as A/VRW in the EKF, the variance, σ 2, of this noise is needed. The variance of this additive white noise is determined through the A/VRW coefficient, N. As indicated by equation (3.32), the Allan deviation of A/VRW versus T in a log-log plot has a slope of 1 2 which serves as the indication of the existence of A/VRW. The T versus Allan deviation relation is depicted in Figure 57

78 3.5 σ 2 (T ) = N 2 T σ(t ) = N T (3.32) where σ 2 (T ) is the Allan Variance; σ(t ) is the Allan deviation; N is the angle/velocity random walk coefficient (see equation (3.29) and 3.30); T is the cluster time. Angle/velocity random walk is modeled as a white noise in this thesis. White noise, as explained in equation (3.7), requires a constant value to represent the power of it. N 2 is the constant that serve this purpose. Equation (3.33) shows how N is applied to describe the correlation between white noise process of two epochs. R w (t, t + τ) = N 2 δ(t) (3.33) where R w (t, t + τ) is the auto-correlation of the white noise; N is the angle/velocity random walk coefficient (see equation (3.29) and 3.30); δ(t) is the Dirac delta function. Flicker Noise Flicker noise, also known as bias instability in the INS community, is caused by the random flickering of the electronics or other components. It is a noise component with power spectra proportional to the inverse of spectral frequency, 1/f; thus, also referred to as 1/f noise. Flicker noise shows up as a low frequency noise, since the additive white noise dominates the higher frequency, other noise is overshadowed by additive white noise at high frequency spectrum. (3.34). The AVAR of the flicker noise derived from the rate noise PSD is shown in equation σ 2 (T ) = 2B2 π [ln 2 sin3 x 2x 2 (sin x + 4x cos x) + C i(2x) C i (4x)] (3.34) C i (x) = x cos t dt (3.35) t 58

79 σ 2 (T ) 0 for T 1 f 0 (3.36) σ 2 (T ) 0.664B for T 1 f 0 (3.37) where σ 2 (T ) is the Allan Variance; B is the bias instability coefficient (see equation (3.29) and 3.30); f 0 is the cutoff frequency; x equals πf 0 τ; C i is the cosine-integral function shown in equation (3.35). It is seen from equation (3.36) that when T is much smaller than the inverse of cutoff frequency, Allan deviation of flicker noise will approach 0; while (3.37) shows that when cluster time, T, is much larger than cutoff frequency, it will be approximately 0.664B. As depicted in Figure 3.6, Allan deviation of equation (3.34) starts with a slope of +1 for f 0 1 T and reach a segment that has slope of 0 with constant value at 0.664B for f 0 1. Therefore, flicker noise is T identified by the flat segment in the log-log plot. Figure 3.6: Sample plot for flicker noise (IEEE-Std , 1997). σ(t ) is the Allan deviation; B is flicker noise coefficient; T is the cluster time. 59

80 Flicker noise is modeled by the first-order Gauss-Markov process in this thesis. In order to do that, one needs to know the correlation time in addition to the magnitude of flicker noise which is represented by the coefficient B. Thus, the correlated (Markov) noise in AVAR is introduced here to help determining the correlation time (Yuksel & Kaygisiz, 2011). According to (IEEE-Std , 1997), the AVAR of correlated noise derived from rate PSD is shown in equation (3.38), and is illustrated in Figure 3.7. σ 2 (T ) = (q ct c ) 2 T [1 T c 2T (3 4e T Tc + e 2T Tc )] (3.38) where σ 2 (T ) is the Allan Variance; q c is the noise amplitude ; T c is the correlation time; T is the cluster time of an AVAR. As shown in Figure 3.7, there exists a peak in the plot of the correlated noise. Here, denote the peak point as (T peak, σ(t peak )). Through the relation between σ(t peak ), q c and T c listed in equation (3.40), and the relation between T peak and T c shown in equation (3.39); the correlation time and noise amplitude of such noise can be estimated. T peak = 1.89 T c = T peak T c 1.89 (3.39) σ(t peak ) = 0.437q c Tc q c = σ(t peak) T c (3.40) where T peak is the cluster time corresponding to the peak point; T c is the correlation time; σ(t peak ) is the Allan deviation corresponding to the peak point; q c is the noise amplitude. According to (Brown & Hwang, 1996), the PSD of a first-order GM process is: Φ x (f) = 2σ2 β 4π 2 f 2 + β (3.41) 60

81 Figure 3.7: Sample plot for correlated noise (IEEE-Std , 1997). σ(t ) is the Allan deviation; T c is the correlation time for the correlated noise; q c is the amplitude of correlated process noise; and T is the cluster time. where Φ x (f) is the PSD of the first-order GM process; σ is the standard deviation of the first-order GM process noise and β is the inverse of correlation time; f refers to the frequency. According to (IEEE-Std , 1997), the PSD of correlated Markov process is: S Ω (f) = (q c T c ) (2πfT c ) 2 (3.42) where S Ω (f) is the PSD of the Markov process; f is the frequency; q c is the noise amplitude ; T c is the correlation time. By dividing equation (3.42) by T 2 c, it can be verified that: q c = 2σ 2 β (3.43) 61

82 Equation (3.16), which describes the first-order GM process: ẋ(t) = βx(t) + w(t), ε(w(t)w(t )) = 2σ 2 βδ(t t ) In order to model the flicker noise with a first-order GM process by coefficient estimation T c and q c, replace β with the inverse of correlation time, T c, and replace 2σ 2 β with the q 2 c, arriving at equation (3.44). ẋ(t) = 1 T c x(t) + w(t), ε(w(t)w(t )) = q 2 c δ(t t ) (3.44) where x is the time dependent random process; w is the white noise process; T c is the correlation time for the correlated noise; q c is the amplitude of correlated process noise; δ(t t ) is the Dirac delta function. Rate Random Walk Rate random walk (RRW) error is a random process of uncertain origin (IEEE- Std , 1997). Due to the terminology used, it is easy to confuse RRW error with A/VRW error. Note that the A/VRW does not accumulates in the output acceleration or angular rate measurements; the white noise that enters the output at each epoch is integrated into navigation solution but does not sum up from one measurement to the next, while RRW error in the output does. Therefore, the rate random walk error in the output angular velocity or acceleration is likely to grow larger in the long run. Being a gradually growing error in the sensor output, it is characterized as a low frequency noise. Thus, RRW manifests itself on the right hand side of the AVAR plot, which represents highly correlated noise and on the left side of the PSD plot, which represents low frequency noise. As indicated by equation (3.45), the Allan deviation (σ(t )) of RRW errors versus cluster time (T ) in a log-log plot shown in Figure 3.8 has a slope of 1, which serves 2 62

83 Figure 3.8: Sample plot for rate random walk (IEEE-Std , 1997). σ(t ) is the Allan deviation; K is the rate random walk coefficient; T is the cluster time. as the indication of the existence of RRW in the measurements. σ 2 (T ) = K2 T 3 σ(t ) = K T 3 (3.45) Where σ 2 (T ) is the Allan Variance; σ(t ) is the Allan deviation; K is the rate random walk coefficient (see equation (3.29) and 3.30); T is the cluster time of AVAR. Rate random walk is modeled as a random walk process, which is described in equation (3.13) and (3.15), repeated here for convenience. ẋ(t) = w(t), differential equation for RRW R x (x t, x t+τ ) = t t+τ t 0 t 0 = q(t t 0 ), τ > t 0 ε(w(t )w((t + τ) ))dt d(t + τ), auto-correlation function for RRW The rate random walk coefficient, K (see equation (3.29) and 3.30), determined from the AVAR curve is the coefficient that quantify the power of the white noise in rate 63

84 random walk. Thus, the error model of RRW based on AVAR will be: ẋ(t) = w(t), R x (x t, x t+τ ) = (K t) 2 δ(t) (3.46) where x is a random process; w is the white noise process; R x (x t, x t+τ ) is the autocorrelation function of the process; t is the measurement sample interval; δ(t) is the Dirac delta function. Rate Ramp For finite length of data, rate ramp is reported to act as a deterministic error rather than random noise due to the fact that ramp noise present a slow changes over a long period of time (IEEE-Std , 1997). Equation shows the AVAR derived from PSD and Figure 3.9 shows the sample plot of rate ramp. σ 2 (T ) = R2 T 2 2 (3.47) Where σ 2 (T ) is the Allan Variance; R is the rate ramp coefficient (see equation (3.29) and 3.30) and T is the cluster time of AVAR. Quantization Noise The quantization noise mentioned in (IEEE-Std , 1997) refers to the angle quantization noise that appears in the output of rate integrating IMUs when such IMUs calculate angles or velocities increments from angular rates or accelerations. It does not refer to the quantization errors that occur when converting continuous analog signals to discrete digital signals in MEMS IMU. Therefore, the quantization noise coefficient, Q, defined in (IEEE-Std , 1997) is not taken into consideration when modeling MEMS IMU. 64

85 Figure 3.9: Sample plot for rate ramp. σ(t ) is the Allan deviation; R is the rate ramp coefficient; T is the cluster time. 3.4 Error Model Coefficient Estimation and Stochastic Error Modeling Error Model Coefficient Estimation Noise identification of the sensor is followed by the estimation of the coefficient N, T c, q c, K, R (see equation (3.29) and 3.30) that are used in the stochastic modeling. Typically, (1) slope matching and (2) regression technique are the two approaches used for error model coefficient estimation. In this section, slope matching is introduced. As listed in Table 3.1, coefficient of A/VRW, flicker noise, RRW and rate ramp can simply be estimated by reading the value of each of their fitted slopes at a specific cluster time, T. 65

86 Table 3.1: Coefficient estimation of stochastic error contribution of MEMS IMU according to Allan Variance slope reading analysis. σ 2 (T ) is the Allan Variance; N is the A/VRW coefficient; B is the flicker noise coefficient; T c is the correlation time of first-order GM; q c is the noise amplitude of first-order GM; T peak is the cluster time corresponding to the peak point; σ(t peak ) is the Alla deviation corresponding to the peak point; K is the RRW coefficient; R is the rate ramp coefficient. Noise term Allan Variance Slope Coefficients White noise σ 2 (T ) = N 2 T 1 2 N = σ 2 (1) Flicker noise 0 B = σ 0.664, T c = T peak 1.89, q c = σ(t peak) T c Rate random walk σ 2 (T ) = K2 T K = σ 2 (3) Rate ramp σ 2 (T ) = K2 T R = σ 2 ( 2) MEMS IMU Stochastic Model To create the customized stochastic model for a IMU sensor, the coefficients estimated from the previous section should be input to the corresponding Q and Φ matrix (see equation (2.48 and 2.49)) elements according to the stochastic model each noise is subject to. In addition, state transition matrix, Φ, needs to be expanded accordingly. Recall the augmented model presented in section equation (2.60), where the 21 by 1 state vector is: x 1 = [ δx n δv n δφ n b f δs f b ω δs ω ] T where δx n, δv n and δφ n are the position, velocity and attitude error vectors, respectively; b f and b ω are the zero offset bias vectors of accelerometers and gyros, respectively ; δs f and δs ω are the scale factor error vectors of accelerometers and 66

87 gyros, respectively. Each one is a 3 by 1 vector, corresponding to the x, y and z axes of IMU. The next step is to, augment the state vector in equation (2.60) with the flicker noise, d f,rb andd ω,rb ; rate random walk, d f,rk andd ω,rk ; and rate ramp and d f,rr, d ω,rr terms for all three axes of accelerometers and gyros: x 2 = [ d f,rb, d ω,rb, d f,rk, d ω,rk, d f,rr, d ω,rr ] T (3.48) combining the 21 by 1 vector x 1 and the 18 by 1 vector x 2, the augmented state of the customized model will become a 39 by 1 vector: x = [ x1 x 2 ] (3.49) The error dynamic differential equation augmented with stochastic noise is given in equation (3.50): d dt [ x1 x 2 ] [ ] [ ] [ ] [ ] F1 F = 2 x1 G1 G + 2 w F 4 x G 4 w 2 (3.50) F 2 = Cb n Cb n Cb n C n b C n b C n b (3.51) diag( 1 T c ) acc diag( 1 T c ) gyro F 4 = (3.52) G 2 = Cb n Cb n Cb n Cb n Cb n Cb n (3.53)

88 G 4 = I (3.54) where: F 1 is the 21 by 21 error dynamic matrix in equation (2.60). F 2, given in equation (3.51), is formed by zeros and the b-frame to n-frame transformation matrix Cb n. F 4, given in equation (3.52), is formed by zeros and the inverse of correlation time, T c, of the flicker noise. G 1 is the 21 by 6 noise matrix listed in equation (2.60); G 2 and G 4 is given in equation (3.55) and (3.54) The 24 by 24 auto- and cross-power spectral density matrix, Q, that is applied in equation (2.49) is: Q = diag(n 2 acc, N 2 gyro, q 2 c,acc, q 2 c,gyro, (K acc t) 2, (K gyro t) 2, (R acc t 3 2 ) 2, (R gyro t 3 2 ) 2 ) (3.55) where N is the A/VRW coefficient; q c is the amplitude of the first-order GM process noise; K is the rate random walk coefficient; and R is the rate ramp coefficient; t is the sample interval. Equation (3.50) is applied when all four types of noise are identified. For real IMU sensor stationary data, not all four coefficients may be identified. Thus, when the noise is not identified, one simply removes the noise from the state vector. 3.5 Alternative Method for Error Model Coefficient Estimation It is stated in (Berman, 2012) that slope reading approach for estimating IMU noise coefficients discussed in section 3.4 can be effective in characterizing the white 68

89 noise in the sensor, but may not be effective for the Markov process (flicker noise), especially for MEMS IMU and short stationary data due to the fact that flat region which identifies the flicker noise appears in longer cluster time. Indeed, as will be seen in section 4.2, the AVAR curve does not always shows a exact flat region at the bottom of the curve; thus, makes estimating the coefficients of flicker noise difficult and uncertain. In addition, since the flat region may appear in longer cluster time (hundreds thousands seconds); therefore, according to the equation (3.21), which calculates AVAR based on adjacent clusters, it is recommended that stationary data should be twice of a thousand of seconds to generate a reliable AVAR curve that is able to cover longer cluster time domain and identified noise besides a/vrw. However, it is not very efficient to collect such long period of stationary data; hence, (Berman, 2012) proposed a model matching approach to replace the slope reading approach in estimating the coefficients of white noise and flicker noise. It is stated in (Berman, 2012) that instead of using the local slopes of Allan Variance, the method is proposed to search for a entire model for the sensor. The idea of this model matching approach is to solve for the white noise and Markov noise coefficients by minimizing the residual of direct predictor (DP) calculated between real data and the DP from analytic data. The concept of direct predictor is based on interpreting Allan Variance from a different angle. Recall Allan Variance from equation (3.21): σ 2 (T ) = 1 2 (Ω k+t Ω k ) 2 = N 2m 1 (Ω 2T 2 k+t Ω k ) 2 (N m) k=1 The term Ω k+t Ω k was explained as the difference between measurement averages of two adjacent clusters k k +T and k +T +1 k +2T. Since the data collected for 69

90 AVAR analysis is stationary, another interpretation is to consider Ω k as the estimated mean value of measurement, that is, the drift in measurements during time k k+t ; then (Ω k+t Ω k ) rewritten in (3.56) can be considered as the prediction error of the drift between time interval k + T + 1 k + 2T ; therefore, σ 2 (T ) is called direct predictor. Ω k+t Ω k = 1 n = 1 n k+2n i=k+n+1 k+2n i=k+n+1 Ω(i) 1 n k+n i=k+1 (Ω(i) Ω k ), Ω(i) (3.56) n = T t where Ω(i) is the sensor measurement at epoch i; n is the number of data points in cluster time T ; t is the sample interval of the measurement. Up to this point, the interpretation is discussed based on real sensor data. Now, assume the stationary IMU data consist of a white noise process, x wn, and a first-order GM process, x GM, which can be simulated by equation (3.57) and (3.58), respectively. Simulate a set of data with these two processes for infinite data length to model the real sensor data. x wn (k) = x wn (k 1) + v(k) (3.57) x GM (k) = 1 T c x GM (k 1) + w(k 1) (3.58) x = x wn (k) + x GM (k) (3.59) where x wn is the white noise process; x GM is the first-order GM process; x is the combination of white noise and first-order GM process; v is the additive white noise of the white noise process; w is the process noise of first-order GM process; k and k 1 refers to the epoch of the process. 70

91 The DP of the simulated white noise process is listed in equation (3.60) and the DP of simulated first-order GM process is given in equation (3.61). Since the real data are modeled with the sum of the two simulated process, as shown in equation (3.59), the DP for this infinite length data is calculated with equation (3.62). Detail derivation can be found in (Berman, 2012). σx 2 wn (T ) = N 2 2n (n + 2 S(µ2, m)) (3.60) σx 2 GM (T ) = σ2 2n [(S(α, 2 n)αm 1 µ m 1 S( α T c µ, m))2 (3.61) m 1 + (1 α 2 ) [S(α, n)α m i 1 µ m 1 i S( α T c µ, m 2 i)]2 i=1 m+n 1 + (1 α 2 ) S(α, m + n i) 2 ] i=m σ x = σ 2 x GM (T ) + σ 2 x wn (T ) (3.62) where σ 2 x wn (T ) and σ 2 x GM (T ) are the DP of the white noise process and the first-order GM process, respectively; N is the amplitude of the additive white noise; σ is the standard deviation of the process noise in first-order GM; T c is the correlation time of the first-order GM process. n is the number of data points in cluster time T, m = n, µ = 1, β = 1, α = e t Tc, and S(q, n) is a sum of geometric series with ratio q: S(q, n) = 1 + q q n = 1 qn 1 q (3.63) Given the DP value calculated from finite real sensor data, the goal is to search for parameters σ (rad/sec), N, (rad/ sec) and T c (sec) that will bound this DP calculated from finite real data. To achieve this goal, the idea is to minimize the DP 71

92 error between real finite data and simulated infinite data M D (σ, N, T c ) as shown in equation (3.64). Using the nonlinear least square method, the unknown σ, N, T c can be estimated. K τ min { (σ 2 (T i, M D (σ, N, T c )) DP (T i )) 2 } (3.64) σ,n,t c i=1 where notation is the same as in equations (3.60) and (3.61). There are total of four DP types introduced in (Berman, 2012), they are DP type 0 that is explained here; DP type 1 that based on predictor calculated over a fixed cluster time T ; DP type 2 that based on predictor calculated from the KF estimation of nominal model; DP type 3 that based on DP calculated by optimal estimator and optimal predictor of a nominal model, see (Berman, 2012) for details. The parameters m, n, and µ input to equation (3.61) and (3.62) will depend on the chosen DP type. After testing all DP types, no differences in the estimated results were found; hence, the simplest DP type 0 is applied in this thesis. 72

93 Chapter 4: Experiments and Results 4.1 Experiment Set-Up The experiment is separated into two stages for each tested IMU sensor. In the first stage of the experiment, the long duration stationary data are collected to estimate the coefficients for the customized stochastic error model with the methods explained in Chapter 3. In the second stage of the experiment, IMU and GPS kinematic data are collected by vehicles equipped with IMU sensors and GPS antenna/receiver. GPS positioning solution is calculated using RTKLIB ( and the integrated solution is solved by AIMS PRO (recently updated version of AIMS software) loosely coupled integrated system. Two kinematic trajectories were acquired. Trajectory 1 originally has no GPS gaps. Trajectory 2, though has a few gaps, the real gaps were not included in the two sections being tested. In order to evaluate the performance of the proposed customized stochastic error model, a few GPS outages are introduced intentionally to the GPS kinematic data. During the introduced outages, no external GPS information can contribute to the estimation of the state error vectors; hence, the system uses the state updated by the most recent GPS updates and the performance of the solution depends entirely on the performance of the 73

94 INS error model. Since the deterministic error model for zero offset biases and scale factor errors are consistent in customized error model and default error model (the error model based on the manufacturers specification sheets, which only models rate white noise), the performance of the customized error model and default error model can be compared. The navigation solutions of the kinematic data provided by the GPS/INS loosely coupled integrated system serve as the true reference trajectory. Two MEMS IMUs are involved in the experiment; these are: Microstrain 3DM GX45 and Epson M-G362. Each sensor collects one long duration stationary data and one kinematic data set. GPS kinematic data are collected along with each IMU kinematic data to provide the integrated solutions. The data from Epson M-G362 were collected by the AIMS PRO system from The Ohio State University; the data from Microstratin 3DM GX45 were collected by the Intelligent Positioning Group (IPOS) from the University of Melbourne. Figure 4.1 shows the trajectory of the kinematic data collected by IPOS lab from University of Melbourne and Figure 4.2 shows the the trajectory collected by the AIMS PRO system at The Ohio State University. Table 4.1 shows the duration of the static data and Table 4.2 summarizes the information of kinematic data. The manufacturer s specification about the errors are listed in Table

95 Figure 4.1: Trajectory 1 is collected by Microstrain 3DM GX45 IMU; 3 outages: A, B and C, marked with red bold lines, are introduced intentionally. The start and end of the trajectory are shown in the enlarged figure on the top right of Figure 4.1; the square denotes the starting position and the triangle denotes the ending position. Figure 4.2: Trajectory 2 is collected by Epson M-G362 IMU; 2 outages: A and B, marked with red bold lines, are introduced intentionally. The start of the entire trajectory is shown in the enlarged figure on the top left of Figure 4.2; the square denotes the starting position, the triangle denotes the ending position, and the two cross marks denote the real GPS gaps. 75

96 Table 4.1: Duration of stationary data. sensor duration (hour) Microstrain 3DM GX45 7 Epson M-G362 8 Table 4.2: Kinematic data collection date, duration, length, and speed. sensor trajectory collection date duration length of trajectory (second) (km) Microstrain 3DM GX Epson M-G Table 4.3: Manufacturer s specification of tested sensors (sf = scale factor and wn = white noise). sensor name Microstrain 3DM GX45 Epson M-G362 Accelerometer Gyroscope Accelerometer Gyroscope deterministic characteristic sf = 0.05% bias = 0.04 mg sf = 0.05% bias = 18 deg/hr sf = ±0.5% bias <0.1 mg sf = 0.005± 0.5% bias = 3 deg/hr stochastic characteristic wn = 80 µg/ hz wn = 0.03 deg/s/ hz wn = 0.1 mg/ hz wn = deg/s/ hz 4.2 Experiment with Stationary Data This section shows the results of the first stage of the experiment where Allan Variance is applied to the stationary data to identify the noise and the two methods: (1) slope reading and (2) model matching are used to estimate the coefficients the the identified noise. In order to verify that the coefficients are indeed realistic and to compare two different coefficient estimation methods, two sets of stationary data are 76

97 simulated, respectively, with the same duration as the real data using the coefficients estimated from the two methods. Allan Variance calculation is applied to the simulated data, so that they can be compared with the Allan Variance curve from the real data. Note that the model matching method is only capable of estimating the coefficient of additive white noise and flicker noise; therefore the rate random walk coefficients estimated from reading the slope will be applied to the simulating stationary data and solving kinematic solutions in the second stage, when using coefficients estimated from the model matching method. Simulation is conducted by applying coefficients to the mathematic description of the random processes that represent the noise based on a given data rate and data length. For example, the additive white noise is simulated by applying the angle/velocity random walk coefficient, N (see equation (3.29) and 3.30), to equation (4.1) with the random terms, w(k), generated through Matlab function, randn; the flicker noise is simulated using the first order Gauss-Markov by applying correlation time and standard deviation of noise to equation (4.2). Rate random walk is simulated according to equation (4.3). x wn (k) = N t w(k) (4.1) x GM (k) = e t Tc x(k 1) + B 2 (1 e 2 t Tc )w(k) (4.2) x rw (k) = x(k 1) + K tw(k) (4.3) X(k) = x wn (k) + x GM (k) + x rw (k) (4.4) where x GM (k), x GM (k), and x rw (k) are the white noise processes, first-order GM process, and random walk process, respectively; X(k) is the sum of the three processes. 77

98 The rest of the notation is the same as in chapter 3. The process can be summarized as follows: the three noise components are summed up as shown in equation (4.4), to generate simulations for long duration static data. To lessen the effect of finite data length, repeat the simulation 10 times for coefficients of each axis; calculate the AVAR for the ten repeated simulations and eventually average the 10 AVAR to arrive at an Allan Variance curve for simulated data Microstrain 3DM GX45 The Allan Variance curves of the Microstrain accelerometer measurements are shown in Figure 4.3. The rate white noise shown by the -0.5 slope line dominates the short cluster time. For y and z axis accelerometers, it dominates the cluster time up to 10 seconds; x axis shows slightly shorter dominance than the other two axes; it is 1 second. The flicker noise follow the additive white noise, which appears in the section where the slopes are around zero. Y and z axis show roughly the same magnitude at the bottom of the curve; while x axis has larger value and shows little but identifiable flat region. All axes show the rate random walk by slopes approximately +0.5 at the right hand side of the curves; however, each axis enters the rate random walk region at a different cluster time, and each of them shows a distinct level of rate random walk. The Allan Variance of three accelerometers end with the slope approximately +0.5, thus, showing no rate ramp. Unfortunately, except for the z axis accelerometer, the model matching method cannot estimate the white noise and the first-order GM coefficients of the accelerometer correctly, see Appendix A and Figures A.1a, A.1c, A.1e for details. It can possibly be explained that during the cluster time where first-order GM is expected to appear, 78

99 Figure 4.3: Microstrain accelerometers Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for each axis of accelerometer. Microstrain x and y axis accelerometers are not behaving close to a first-order GM process; thus, the AVAR curves do not present exactly the way first-order GM process does, which makes it difficult for the algorithm to fit the assumed model to the AVAR curve calculated through real stationary data. Therefore, the model matching method cannot generate an estimation close to the noise of the real measurements. As a result, the coefficients for the Microstrain accelerometers estimated from the slope reading method, listed in Table 4.4, will be used as a substitute for model matching method throughout the experiment. For the Microstrain gyro measurements, shown in Figure 4.4, the Allan Variance curves from three axes are consistent. The curves with slopes around -0.5 between 79

100 0.01 second and 10 seconds cluster time show that rate white noise exists in every axes of the Microstrain gyro measurements. They are followed by the flicker noise, which can be discovered in the longer cluster time by the flat region around hundreds of seconds. The rate random walk appears at the end of the x and z axis curve with slopes around +0.5; however, the y axis curve appears to be a curvature, it fits neither slope +0.5 nor +1; thus, one can conclude that there is no rate random walk and no rate ramp in the y axis gyro of the tested Microstrain IMU. Rate ramp did not appear for the gyros either; nevertheless, it is unsure whether it is due to the instability of the long stationary data or the fact that such noise does not exist in the gyro measurements. Figure 4.4: Microstrain gyros Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for x and y accelerometer. The coefficients of the three noise components: angle random walk, flicker noise and rate random walk are estimated and listed in Tables (4.4) and (4.5). The model 80

101 matching method works, as expected, on the Microstrain gyro measurements, the simulation data of both methods are shown in figure A.1b, A.1d, A.1f in Appendix A. Table 4.4: Microstrain coefficient estimation from the slope reading method. Sensor: Microstrain 3DM 45 VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) m/s 2 / s X axis acc Y axis acc Z axis acc ARW (N) deg/s s flicker noise (B) deg/s flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) deg/s/ s X axis gyro Y axis gyro Z axis gyro Table 4.5: Microstrain coefficient estimation from the model matching method. Sensor: Microstrain 3DM 45 VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise correlation time sec X axis acc Y axis acc Z axis acc ARW (N) flicker noise (B) flicker noise correlation time sec deg/s s deg/s X axis acc Y axis acc Z axis acc

102 4.2.2 Epson M-G362 The Allan Variance curves of the Epson accelerometer measurements are plotted in Figure 4.5. Three AVAR curves of the accelerometers show the rate white noise with the -0.5 slopes, the z axis is slightly smaller in magnitude. Flicker noise can be identified by the flat region from 1 second 100 seconds for all the axes; note that y axis presents a larger value. At the end of the AVAR curve the x and z accelerometers show slopes approximately 0.5, but the AVAR curve of y accelerometer drops; therefore, rate random walk is modeled in the x and z accelerometer but not in the y accelerometer. Since the AVAR curve of all axes do not have slope around 1, rate ramp of the Epson accelerometers are not considered in the model. Similarly as shown in Figures A.2a, A.2c and A.2e, when estimating coefficients Figure 4.5: Epson accelerometers Allan deviation plot. The intersection of the - 1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for x and z axis accelerometer. 82

103 of the Epson accelerometers using the model matching method, the method fails to generate coefficients for white noise and flicker noise that are comparable to slope reading method in x and z axis. Therefore, only the coefficients from the slope reading method listed in Table 4.6 will be used for the accelerometers. The Allan Variance curves of the Epson gyro measurements are given in Figure 4.6. The curves of the Figure 4.6: Epson gyros Allan deviation plot. The intersection of the -1/2 slope with the dash line that denotes 1 second cluster time is the value of the coefficients of velocity random walk; the intersection of the 1/2 slope with the dash line that denotes 3 seconds cluster time is the value of the coefficients of RRW for each axis of accelerometer. three axes are consistent. Thus, the rate white noise identified by the -0.5 slope and flicker noise identified by the flat region appear at the bottom of the curve can be considered to be the same value for all three axes. The end of the AVAR curve shows a positive slope. The slopes of x and y axes of gyro AVAR are larger than for the z 83

104 Sensor: Epson Table 4.6: Epson coefficient estimation from the slope reading method. VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) m/s 2 / s X axis acc Y axis acc Z axis acc ARW (N) deg/s s flicker noise (B) deg/s flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) deg/s/ s X axis gyro Y axis gyro Z axis gyro Table 4.7: Epson coefficient estimation from the model matching method. Sensor: Epson VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise correlation time sec X axis acc Y axis acc Z axis acc ARW (N) flicker noise (B) flicker noise correlation time sec deg/s s deg/s X axis gyro Y axis gyro Z axis gyro

105 axis, but all slopes are not large enough to be considered as the identification of rate ramp; thus, they are considered the rate random walk. Figures A.2b, A.2d and A.2f display the model matching method for the Epson gyros. Refer to Tables 4.6 and 4.7 for the coefficients estimated with the slope reading method and the model matching method. 4.3 Experiment with Kinematic Data This section describes the second stage of the experiment, where the coefficients estimated from stage one are input to their corresponding elements in the Q matrix of the stochastic errors to form a customized stochastic error model for each sensor, and the navigation solution is estimated under no GPS updates conditions. The effects of the customized error model on the navigation solution are discussed by comparing the solution from the customized error model with the solution where the manufacturer specification sheet model are used; meanwhile, the performance of the two noise coefficient estimation methods are compared by evaluating the level of improvements in the navigation solution. Since this thesis aim to discuss the applications in land vehicle, the comparison of the solution will consider only the horizontal positions (E,N) and the 3 orientations Microstrain 3DM GX45 Trajectory 1 was collected by the Microstrain 3DM GX45 IMU bound on land vehicle on August, 20, 2014 with an average speed of 36 km/hr. According to Figure 4.1, three 30-second outages, A, B and C, are introduced. The intention of choosing this three sections is to test the stochastic model under different types of trajectory; such as a 90-degree turn during outage A, a straight trajectory during outage B, and a 85

106 ramp trajectory in outage C. Table 4.8 summarizes the trajectory details and indicates whether state vector has been properly resolved before the outage begins. Refer to Appendix B for the position RMS plot before GPS gap begins. As of the result, a trajectory plot and a table that shows the horizontal errors and orientation errors are presented to summarize how the Microstrain 3DM GX45 standalone performs under 5-second, 10-second, and 30-second of GPS outage, based on the customized stochastic model and the default error model. Plots showing the detail of horizontal errors and orientation errors can be found in Appendix B. Table 4.8: Information of Trajectory 1 about the speed, the beginning epoch, the GPS solution quality 1 minute before GPS outage, and the resolve of state vector. Outage A Outage B Outage C Average speed (km/hr) Beginning epoch of the outage (sec) The % of fix/float GPS solutions 1 minute before the outage begins 100/0 90/10 100/0 State vector resolved before GPS outage begins yes yes yes According to the results from the first stage of the experiment, the customized stochastic error model of the Microstrain takes A/VRW (6 states), flicker noise (6 states) and RRW (6 states) into consideration. Augmenting the three noise components of each axis into the original 21 by 1 state vector of error dynamic model, forms a 39 by 1 state vector for customized Microstrain IMU dynamic system. Note that according to the analysis of the Microstrain stationary data in 4.2.1, no RRW coefficients are estimated for y axis gyros; therefore, zero is input to its corresponding element in the Q matrix of the stochastic errors. 86

107 Outage A Figure 4.7 plots the solutions generated for outage A in trajectory 1. Since according to Table 4.8, the state vector is properly resolved, there is no problem to apply the values of the state vector estimated by the most recent GPS update to calibrate the measurements during outage A. As can be observed from Table 4.9, the Figure 4.7: Trajectory 1, outage A plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively. roll error of the solutions estimated using the three error models is small, no bigger than 1 degree. The pitch error of the solutions estimated using the three error models is at the same level for the first 10 seconds of outage A, but the error of the solution 87

108 estimated from the default error model doubled after 30 seconds gap. In terms of the heading, the error of the solution obtained from customized error model stay smaller than 2.5 degrees while the performance of the default error model is 50 % worse. Regarding the positioning performance, after 30 seconds GPS gaps, the maximum horizontal position error of the solution estimated by the default error model is 44 m; while it is 14 m for the customized error model. Though the customized error model improves the horizontal solution by 70 %; still, the level of accuracy of the solution after the 30second outage is not acceptable for navigation application. Now, if we consider a horizontal error within 5 m as an acceptable quality for some navigation applications, the error of the solution obtained from the default error model for outage A exceeds 5 m after 9 seconds; while using the customized error model allows the system to achieve the same performance with 11-second gaps. 88

109 Table 4.9: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage A with respect to the reference GPS/INS trajectory based on 5-second 10-second, and 30-second GPS outage. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

110 Outage B The outage B of the trajectory 1 solved for with different stochastic error models is depicted in Figure 4.8b. As shown in Table 4.8, the state is properly resolved before outage B begins; therefore, the system uses the values estimated by the most recent GPS updates to calibrate the measurements during outage B. Figure 4.8: Trajectory 1, outage B plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively. According to Table 4.10, the roll and pitch errors, calculated by differencing the solution with the reference trajectory, did not show significant difference between default error model and customized error model. The heading errors of all three 90

111 tested models are not large, maximum 0.5 degree ; the heading errors solved using both customized stochastic error models are especially small, it stays below 0.3 degree for the entire 30 seconds outage. As a result, the direction of three trajectories drift a little, as plotted in Figure 4.8. Table 4.10: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage B with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

112 The horizontal position errors on the other hand grow as outage interval increases, as indicated in Table After 30 seconds of GPS gap, the maximum horizontal position error of the solution estimated by the default error model is 188 m; while it is 98 m for the customized error model. The customized error model reduces the error in horizontal position by 50 % after 30 seconds of GPS outage. The horizontal position error stays below 5 m for 3 seconds for the solution obtained by the default error model, and 4.3 seconds for the solution obtained by the customized error model. Outage C Figure 4.9: Trajectory 1, outage C plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using customized error model with coefficients determined by slop reading method; and represents the solution obtained using customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively. 92

113 The solutions obtained for outage C in trajectory 1 are shown in Figure 4.9, and the information about outage C is summarized in Table 4.8. As can be observed on Table 4.11, the orientation error of each axis performs differently. In average, the roll solution estimated by the customized error model have errors smaller than 2 degrees after 30 seconds of GPS gap; it is 50 % better than the solution estimated by the default error model. While the pitch solution estimated from the customized error model have errors approximately 5.5 degrees, but in this direction the default error model is by 25 % better. The trend of the heading error is similar to the pitch error. In terms of horizontal position error, the maximum error is 125 m and 265 m, respectively, for the solution obtained by the customized error model and the default error model after 30 seconds of GPS outage. Again, since meter level of errors are not applicable for navigation purpose, considering a hypothetical 5 m horizontal position error limit (which is sufficient for some applications), the solution obtained by the default error model is able to stay under 5 m horizontal error for 3 seconds of GPS outage; while the system can provide the same performance under 5.5 seconds of outage when using the customized error model. 93

114 Table 4.11: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 1 outage C with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

115 4.3.2 Epson M-G362 Trajectory 2 was collected using the Epson M-G362 IMU on August, 20, 2014 with an average speed around 30 km/hr. As shown in Figure 4.2, two 30-second outages, A and B, are introduced. Outage A involves one 90-degrees turn and outage B is a straight trajectory. Table 4.12 provides the basic summary characteristics of the outages. See Appendix B for the position RMS plot before the GPS gap begins. In terms of the results, one trajectory plot (Figure ) and a table (Table ) are presented for each outage to summarize the performance of the Epson M-G362 customized stochastic error model. Table 4.12: Information of Trajectory 2 about the speed, the beginning epoch, the GPS solution quality 1 minute before GPS outage, and the resolve of state vector. Outage A Outage B Average speed (km/hr) Beginning epoch of the outage (sec) The % of fix/float GPS solutions 1 minute before the outage begins 0/100 0/100 State vector resolved before GPS outage begins yes yes According to the results from the first stage of the experiment, the customized stochastic model of Epson includes A/VRW (6 states), flicker noise (6 states) and rate random walk (6 states). Augmenting the identified noise to the original 21 by 1 state vector will generate a 39 by 1 state vector for the customized Epson IMU dynamic system. Note that no RRW coefficients are estimated for accelerometer y as given in Table 4.6; therefore, zero is input to the corresponding element in the Q matrix. 95

116 Outage A The outage A of trajectory 2 obtained using different stochastic error models is depicted in Figure The information about outage A is listed in Table It can be seen in Figure 4.10 that the trajectory estimated by the default error model starts to drift away from reference trajectory immediately when GPS signal becomes unavailable, but not the case for the solution estimated by the customized error model. Figure 4.10: Trajectory 2, outage A plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively. According to Table 4.13, the magnitude of roll and pitch error are at the same level for the solutions solved by different error models after 30 seconds of GPS outages; while the heading error of the solution obtained from the default error model is 6 degree larger than the solution from the customized error model. As listed in Table 96

117 4.13, the error of the horizontal position obtained from the customized error mode is 38 m after 30 seconds GPS gap, and the error is doubled when using the default error model. The horizontal position error stays below 5 m for 2.6 seconds GPS outage when using the default error model, and 6 seconds when using the customized error model. Table 4.13: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage A with respect to the reference GPS/INS trajectory based on 5-second, 10-second, and 30-second GPS outage. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

118 Outage B The outage B of trajectory 2, generated using different stochastic error models, is plotted in Figure The basic information about this outage is given in Table According to Table 4.14, the customized error model did not improve the accuracy of the orientation solutions. On the contrary, the orientation solutions obtained from the customized error model have larger error than the orientation solutions obtained from the default error model. Figure 4.11: Trajectory 2, outage B plot. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds and 30 seconds of GPS outages, respectively. 98

119 Regarding the errors in the horizontal position, the customized error model is affective in improving the accuracy of the horizontal position for the first 10 seconds; nevertheless, after 10 seconds GPS gap, the error starts to increase to the extent that it surpass the errors of the default error model solution. After 30 seconds of GPS outage, the horizontal position error of the solution obtained from the customized error model is 240 m; while it is 72 m for solution obtained from default error model. It is unclear why the errors in horizontal positions obtained from the customized error model starts to grow larger than the errors in the solutions obtained from the default error model. However, when the flicker noise and RRW states solved by the most recent GPS updates are not used in the IMU measurement calibration during the GPS outage, the accuracy of the orientation solution and the horizontal position both improve significantly, as can be seen in Figure As shown in Table 4.15, the maximum horizontal position errors decrease from 240 m to 80 m during 30 seconds of GPS gap when not using the flicker noise and RRW values to calibrate the IMU measurement. The average error in the heading after 30 seconds of GPS gaps decreases from 5 degrees to 1.5 degrees. 99

120 Figure 4.12: Trajectory 2, outage B plot without flicker noise and RRW calibration during GPS outage. The black trajectory is the reference solution provided by loosely coupled GPS/INS system; + denotes the solution obtained using default stochastic error model; is the solution obtained using the customized error model with coefficients determined by slop reading method; and represents the solution obtained using the customized error model with coefficients determined by the model matching method. The square marks the starting point of GPS outage, the triangles mark the positions of the true trajectory after 5 seconds, 10 seconds, and 30 seconds of GPS outage, respectively. 100

121 Table 4.14: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage B with respect to the reference GPS/INS trajectory based on 5 seconds, 10 seconds, and 30 seconds GPS outage. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

122 Table 4.15: Free inertial navigation solution (horizontal position and orientations) error of Trajectory 2 outage B without calibrating the flicker noise and RRW with respect to the reference GPS/INS trajectory based on 5-second, 10-second and 30- second GPS outages. Error model Default Customized (slope reading) Customized (model matching) 5 seconds 10 seconds 30 seconds mean max std mean max std mean max std Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree) Horizontal position (meter) Roll (degree) Pitch (degree) Yaw (degree)

123 Chapter 5: Conclusions and Recommendations 5.1 Conclusions As indicated in Chapter 1, the objective of this thesis is to examine the availability of the customized stochastic error model of MEMS IMU and to investigate the amount of impact of the customized stochastic error model on the navigation solution under GPS denied conditions. According to the experiments and the results presented in section 4.3, it is shown that in 4 out of 5 outages, the horizontal position solutions obtained using the customized error model present better performance than the solutions obtained using a default error model. The improvements in horizontal position after 30 seconds GPS gap is between 40% and 70 % when using customized stochastic model. Considering the applicability to a land vehicle navigation application, applying the default error model allows the system to maintain horizontal position errors smaller than 5 m for 2 to 9 seconds, depending on the trajectory; while using the customized error model allows the same horizontal position performance for 5 seconds to 11 seconds. The customized error model provides the system a possibility to navigate more reliably for a 2 to 3 seconds longer GPS gap. The outage B from trajectory 2 is an exception, the customized error model did not improve the solutions in this outage like it did in other GPS outages; however, 103

124 when the flicker noise and RRW are not used in calibrating the IMU measurements, the customized error model improves the solution. Since as indicated in Table 4.8, the state vector was properly estimated before the outage begins, the possibility that the system uses the incorrect calibration values can be eliminated. Hence, the other possibility may be the unstablility of the flicker noise and rate random walk in Epson IMU measurement that causes this effect. In other word, when the flicker noise and rate random walk are unstable, even if these states are properly resolved by the most recent GPS update, the estimated value may have significant difference with the correct value in the measurements. In this case, if the estimated flicker noise and RRW are used, it might mis-calibrate the measurements. In terms of the orientation solution, the impact of the customized stochastic error model is not consistent between each tested outage. In most cases, the solutions estimated from the customized error model have smaller errors; however, the pitch error of outage B and outage C from trajectory 1 and the heading error of outage C in trajectory 1 perform otherwise. But since the magnitude of the roll and pitch errors are within 2 to 3 degrees, the differences in roll and pitch errors are insignificant for land vehicle applications. Regarding the two stochastic error model coefficient estimation approaches: slope reading and model matching, it is concluded that the model matching method is not applicable to the accelerometers used in this thesis, but works well with the gyros. The data length required to estimate the coefficients is less demanding than using the slope reading method. In terms of the solution performance, sub-meter to meter difference occur between two methods; but, considering the level of drift in the tested 104

125 MEMS IMUs (about 10 m after 10 seconds, and over 50 m after 30 seconds), the differences can be ignored. 5.2 Recommendations 1. Based on the experience gained by analyzing the outage B in trajectory 2 it can be stated that the customized model improves the solution only when the flicker noise and RRW are not used to calibrate the measurements during a GPS gap. Thus, a further study of the relationship between the stability of these two noise types and their calibration performances when GPS gap occurs is recommended. 2. As the MEMS IMU market expands, with the particular emphasis on the close-totactical grade MEMS IMU that began entering the market, it is recommended to test more MEMS sensors under the conditions of temporary GPS gaps using customized stochastic error model in order to assess their applicability to various applications. 3. According to the experiment with kinematic data, the accuracy performance of different trajectory dynamics using identical IMUs are not at the same level. Hence, it is recommended to test the customized model under varying trajectory dynamics, various lengths of GPS gaps, and various lengths of the calibration period prior to the gap. 4. In this thesis, the application of GPS/INS integration using carrier phase data is tested. More tests may be needed to see if the pseudorange-based simple GPS receivers can benefit from the integration with inexpensive MEMS IMUs, as pseudorangebased GPS receivers are more cost effective for many civilian applications. 105

126 5. The data used for stochastic error assessment were static; additional tests may be carried out to verify if error models can be derived on-the-fly, using dynamic data, to reflect the impact of the sensor dynamic on the stochastic error model itself. 106

127 Appendix A: AVAR of Simulated IMU Measurements In this appendix, the AVAR of the simulated static IMU measurements generated from the error coefficients estimated by the slope reading method and the model matching method is presented. Table A.1 and A.2 list the error coefficients of the Microstrain 3DM GX45 determined by the two methods. Figures A.1a A.1f plot the AVAR for the 6 hours of simulated IMU measurements, and the AVAR for the 6 hours real IMU measurements for each axis. As shown in the figure, the 3 accelerometer and 3 gyro AVAR curve of the simulated measurements from the slope reading method has the same pattern as the AVAR curve of the real measurements; thus, this indicates that the error coefficients model the noise properly and can be input to the Q matrix. For the model matching method, 3 gyro and the z accelerometer AVAR curve of such simulated IMU measurements show the same pattern as real data; however, it is not the case for x and y accelerometers. Therefore, this indicates that the x and y accelerometer error coefficients estimated from the model matching method is not applicable to the Q matrix. Table A.3and TableA.4 lists the error coefficients of Epson M-G362 determined by the two methods. Figure A.2a Figure A.2f plot the AVAR of 6 hours simulated IMU measurements and the AVAR of 6 hours real IMU measurements for each axis. As can be observed from the figure, the 3 gyro AVAR curves of the simulated measurement 107

128 based on the slope reading method follow the same pattern as the AVAR curve of the real measurements; therefore the error coefficients of 3 gyros estimated using slope reading method can be used. While the 3 accelerometer AVAR curve of the simulated measurement based on slope reading method has the same pattern as the AVAR curves of the real accelerometer measurements in the short cluster time section, but is different in the region between the -1/2 slope and the flat region due to the fact that the flat region of the accelerometer AVAR of real measurements is either too wide, like the AVAR of y and z accelerometer measurements, or the flat region cannot be easily identified, like the x accelerometer AVAR curve. In terms of the AVAR of the simulated measurements generated from error coefficients estimated by model matching method, the curve of the 3 gyros are consistent with each of their AVAR of the real IMU measurements, which indicates the gyro error coefficients obtained using model matching method can be used for all 3 axes. However, the accelerometer error coefficients estimated using the model matching method cannot be used since the AVAR curve of y and z accelerometer is different from the AVAR curve of the real measurements. The x axis though matches the AVAR curve of the real measurements, but the first-order GM noise is not fitting the flat region at the bottom section of AVAR curve. Thus, it is not consider as estimating the coefficients correctly. 108

129 Table A.1: Microstrain coefficient estimation from the slope reading method. Sensor: Microstrain 3DM 45 VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) m/s 2 / s X axis acc Y axis acc Z axis acc ARW (N) deg/s s flicker noise (B) deg/s flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) deg/s/ s X axis gyro Y axis gyro Z axis gyro Table A.2: Microstrain coefficient estimation from the model matching method. Sensor: Microstrain 3DM 45 VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise correlation time sec X axis acc Y axis acc Z axis acc ARW (N) flicker noise (B) flicker noise correlation time sec deg/s s deg/s X axis acc Y axis acc Z axis acc

130 Sensor: Epson Table A.3: Epson coefficient estimation from the slope reading method. VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) m/s 2 / s X axis acc Y axis acc Z axis acc ARW (N) deg/s s flicker noise (B) deg/s flicker noise amplitude (q c ) flicker noise correlation time sec RRW (K) deg/s/ s X axis gyro Y axis gyro Z axis gyro Table A.4: Epson coefficient estimation from the model matching method. Sensor: Epson VRW (N) m/s 2 s flicker noise (B) m/s 2 flicker noise correlation time sec X axis acc Y axis acc Z axis acc ARW (N) flicker noise (B) flicker noise correlation time sec deg/s s deg/s X axis gyro Y axis gyro Z axis gyro

131 (a) (b) (c) (d) (e) (f) Figure A.1: Microstrain 3DM GX45 accelerometers ((a), (c), (e)) and gyros ((b), (d), (f))allan variance plot of real data versus simulated data. 111

132 (a) (b) (c) (d) (e) (f) Figure A.2: Epson M-G362 Allan variance plot of real data versus simulated data. ((a), (c), (e)) accelerometer x, y and z. ((b), (d), (f)) gyro x, y and z. 112

133 Appendix B: Navigation Solution Error Plots In section 4.3, the tables compare the free inertial navigation solution error of each trajectory with respect to the reference GPS/INS trajectory based on 5 seconds, 10 seconds, and 30 seconds of GPS outages using different stochastic error models. Appendix B shows the horizontal error plot and the orientation error plot of the entire 30 seconds GPS outage when estimating solution with three different stochastic error models: the default stochastic error model, the customized stochastic error model with coefficients determined by the slope reading method, and the customized stochastic error model with coefficients determined by the model matching method. 113

134 (a) (b) Figure B.1: a) Trajectory 1, outage A horizontal error plot. b)trajectory 1, outage A orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method. 114

135 (a) (b) Figure B.2: a) Trajectory 1, outage B horizontal error plot. b)trajectory 1, outage B orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method. 115

136 (a) (b) Figure B.3: a) Trajectory 1, outage C horizontal error plot. b)trajectory 1, outage C orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method. 116

137 (a) (b) Figure B.4: a) Trajectory 2, outage A horizontal error plot. b)trajectory 2, outage A orientation error plot. + denotes the error of the solution obtained using default stochastic error model; is the error of the solution obtained using customized error model with coefficients determined by slop reading method; and represents the error of the solution obtained using customized error model with coefficients determined by model matching method. 117