Time Optimal Spacecraft Attitude Control with Considerations of Failure in AOCS & Payload Components

Transcription

1 Student Research Paper Conference Vol-2, No-58, July 2015 Time Optimal Spacecraft Attitude Control with Considerations of Failure in AOCS & Payload Components 1 M.Ibbtisam Asim, 2 Hassaan Bin Jalil Dept. of Aeronautics & Astronautics Institute of Space Technology Islamabad, Pakistan 1 ibbtisam@gmail.com 2 Hassaan431@hotmail.com Abstract A time optimal and fault tolerant attitude control scheme of a satellite is developed with the aid of one degree of freedom spacecraft simulator called TableSat. TableSat emulates the dynamics, sensing and actuation capabilities required for spacecraft attitude control. Sensing capabilities include two gyros; one for attitude control and second to aid the first in fault tolerant control scheme. Two computer fans provide actuation capabilities to TableSat in clockwise and anticlockwise direction. A step by step approach is adopted in control scheme by starting from basic PDcontroller to Luenberger observer, leading to time optimal control i.e. LQR (Linear Quadratic Regulator). Attitude control system of TableSat includes angular rate control and heading control or yaw control. The former is achieved by Luenberger observer and LQR and later is done by PD controller. Different cases are included in fault tolerant scheme, to decrease the vulnerability of system to faults. Real time results of these controllers are obtained on the pc in the form of graphs using Bluetooth as a communication system between TableSat s flight processor and the pc. actuators. The goal was to accomplish fault-tolerant and time optimal control of TableSat. TableSat has the ability to switch between rate control and heading or yaw control or reference tracking. TableSat s software makes it fault tolerant which guarantees stability during faulty situations. The fault tolerant scheme consists of two parts, i.e. fault detection and control reconfiguration. Fault detection requires information about faults in Attitude and Orbit Control Systems (AOCS), processed measurements, diagnostic actions, and ongoing mission-related actions since these actions can affect the probabilities of failures [2]. In the case of TableSat, AOCS includes sensors (gyros) and actuators (fans). Whereas control reconfiguration decisions require error or fault conditions, information is required about current logic (switch) and control configuration, current faults, ongoing mission-related actions, and mission status [2]. I. INTRODUCTION TableSat platform is a single degree-of-freedom tabletop satellite, actuated by two thrusters (computer fans) imposing clockwise and anti-clockwise torques respectively [1] (illustrated in figure A-1). The purpose of TableSat is to serve as a spacecraft model for actual applications of controls system theory and as a research facility. It helps students and scientists in gaining practical understanding of tangible sensors and actuation systems similar to those present in actual Aerospace systems. Though TableSat is restricted to one degree-offreedom studies, but numerous guidance, controls and navigation methodologies established in it, offer experiences extending from rate control and (attitude) trajectory tracking to nonlinear sensor calibration and spacecraft dynamics compensation. The entrenched software of TableSat provides solid real-time multi-threaded implementation, aiding convincing if limited-scale simulating of software empowered control system of spacecraft. The hardware and software systems of TableSat are described in this paper, concentrating on the development of control algorithms, fault tolerance and sensor calibration of TableSat. Sensor calibration and development of control laws was made challenging by substantial drifts in TableSat s gyros, as well as dynamically changing friction properties of Figure I-1: TableSat Underneath, the researchers have initially introduced the design of TableSat system, from electronics parts to mechanical system. Then they have briefly explained the baseline technique of sensor calibration and control law development. TableSat offers numerous challenges for Aerospace students with dynamics, controls and navigation interests. Sensors and control scheme software of TableSat 284

2 provide challenging real-world experiences in understanding and acclimatizing mathematical models to challenging sensor and actuator behaviors complicated to completely characterize with discrete state models. II. TABLESAT ELECTROMECHANICAL SYSTEMS Figure A-1 displays a picture of TableSat with germane components labeled. TableSat spins in the horizontal plane around a pole with low but nontrivial friction, minimized by using deep groove bearing. The hardware is mounted on a 30- cm diameter disc, which is balanced on a rod/pipe with the help of a bearing that acts as TableSat s rotation point. It is pictured in Figure B-1 and B-2. Figure II-1: TableSat Mounting Assembly (Downside View) real systems, is subject to nonlinearities such as friction and actuator saturation. Therefore, part of the model development includes finding ways to linearize the system by compensating for and reducing the effects of those nonlinearities in the system response. To a first approximation, TableSat is assumed to have the following equations of motion [3]: Where is the TableSat moment of inertia, is the TableSat angular velocity, is the speed of the positive fan, is the speed of the negative fan, is moment arm, is the TableSat friction and could be a function of, is the fan speed to force constant, is the voltage applied to the positive fan, is the voltage applied to the negative fan, α is the fan time constant, fan voltage to change in speed constant, and, are the frictions of fans, could be functions of respectively. TableSat Moment of Inertia is calculated using Creo 2.0. All the parts of the TableSat are separately modeled in Creo Parametric with their respective dimensions and densities, then these parts are assembled in Creo Assembly to calculate the total moment of inertia of the TableSat. Creo model is shown in figures below. Inertia was found out to be; Figure II-2: TableSat Mounting Assembly A pair of unidirectional computer fans delivers clockwise and anticlockwise actuation capabilities. Two digital MEMEs technology rate gyros measures angular velocity. An onboard flight processor, Arduino UNO based on the ATmega328 microcontroller featuring 16MHz clock speed, communicates to a ground station, laptop, via a Bluetooth interface, HC-06. TableSat exhibits nonlinear dynamics due to friction effects, including off-axis wobble at certain actuation magnitudes [1]. TableSat helps in understanding rate control and pointing or heading control with single or multiple sensor measurements. A. TableSat Modelling One of the core objectives of the TableSat research project is to get a model of the TableSat system which can be used to design and test controllers and estimators. Ideally, the model would be linear so that linear controllers can be designed and will behave as predicted by theory. However, TableSat, like all Figure C-1: TableSat Creo Model (Upside View) 285

3 The parameters involved in above transfer function can be calculated as; Figure C-2: TableSat Creo Model (Downside View) The linear model of TableSat is created so that it can be used to design controllers and state estimators for the system. Of course, a linear model is only valid if the system itself can be considered linear. To compensate for the friction characteristics of TableSat, static and dynamic frictions of both fans and TableSat s hardware are calculated and introduced as dead bands in the onboard software of TableSat, so nonlinearities in the system can be virtually eliminated, which implies that;, and can be neglected in the TableSat equations of motions. In addition, because the static fan friction has been virtually eliminated, the two separate fan equations can be reduced to one equation, representing a single bi-directional fan. The simplified equations of motion are then: B. Fan Time Constant The fan time constant, α is the time it takes for the fans to reach 63% of their steady state output force for a given commanded voltage [4]. In other words, it is a measure of how quickly the fans can respond to a change in input. Fan time constant was determined experimentally with the help of a stop watch. Both the team members did this experiment at the same time to remove human errors. A total of 30 readings were taken, and average was calculated, so it comes out to be; C. Voltage to Change in Fan Speed Constant To determine a nominal estimate of, some data from fan s datasheet is taken. For a constant applied voltage, the change in fan speed will be equal to zero when the fans reach a steady state speed. Consider the fan equation; For constant voltage it will become; From fan s datasheet; Which can be written in state space form as: So after calculations it becomes; The transfer function is calculated; D. Fan Speed to Fan Force Constant From fluid dynamics, it is known that the thrust force of a fluid exiting a propulsion unit can be calculated as follows [5]: Where are output and input mass flow rates of the fluid respectively. are output and input 286

4 velocities of the fluid, respectively. For TableSat, the fluid being moved is air, and it is assumed assume that the input air velocity is zero, which implies that the air coming into the fan is stationary. Strictly speaking, if TableSat has a non-zero angular velocity, this assumption would be false. However, since the input air velocity would be much less than the output air velocity, the assumption is still reasonable. Based on the above assumption, the second term in the above equation can be neglected. The output mass flow rate and fluid velocities can be calculated using the following equation [6]: Its root-locus, bode plot and response to step command are shown below; Where is defined above, ρ is the density of the fluid being moved by the propulsive unit, is the volumetric flow rate of the propulsion unit, and A is the area through which the fluid is being moved. So; (From fan datasheet) So it becomes; (From fan datasheet) Figure II-3: Rootlocus and Bode plot of TableSat is calculated experimentally when the TableSat fans are spinning at their maximum speed, it comes out to be; So using these values fan speed to fan force constant comes out to be; Where is the moment arm i.e. distance between fan center point to the TableSat pivot point. So putting these values in the above transfer function; Figure II-4: Respond of TableSat to step command 287

5 In order to use the above dynamics model of TableSat in microcontroller, the above derived continuous transfer function of TableSat is converted into discrete time transfer function [7] using a sample time of 0.1 seconds [8]. The discrete time transfer function of TableSat comes out to be; Based on the above linear dynamics model of TableSat controllers are designed in MATLAB and Simulink to find the values of gains required to control the TableSat hardware. At first PD controller is designed in Simulink [9], its results are displayed later on. Figure II-6: Luenberger Observer for TableSat Figure II-5: PD Controller for TableSat Then Luenberger observer [10] is designed in Matlab and Simulink as in Figure C-6 given next. This Simulink model is implemented in TableSat flight processor by the following equations [11]; III. TABLESAT ONBOARD SOFTWARE The controllers designed above are coded in C++ and then implemented on TableSat by uploading them on flight processor (Arduino Uno) using Arduino IDE (Integrated Development Environment), an open-source Arduino Software used to write code and upload it to the board [12]. The baseline software used for calibration provides the offsets of gyros which are employed in control law development that initializes the hardware, and executes a loop, nominally at 10 Hz. Every execution cycle reads both gyros data sequentially, applies calibration offsets to convert this data into meaningful results in radian/seconds, then calculates inputs for actuators based on a reference position fixed by the operator. 288

6 Calibration of sensors and development control laws were accomplished in two stages. First, a code is developed to calculate offsets of both gyros, then these offsets are used in every loop of control law to achieve steady rotation rates and yaw angles. The overall TableSat software is summarized in the below flowchart. fault free, so in this case, the simplest idea of fault tolerance is adapted. The fault tolerance part consists of 2 sub-parts i.e. fault detection and control reconfiguration. In fault detection part, TableSat s flight processor keeps on checking whether there is a fault in a sensor. If it finds any fault, the control reconfiguration mode takes over and shift the system to second gyro, while it keeps on checking if the faulty gyro is working correctly or not, as soon as it works correctly, the system goes back to fault detection mode where it takes the readings from both gyros and uses the average value. Three types of gyro errors are considered in fault detection routine; Controller implementation part is explained below, where from to 0.03 is the allowed dead band [13] of TableSat both in both heading control (in radians) and angular rate control (in radians/sec). a. If a gyro shows no rate even in the presence of a disturbance b. If a gyro shows random meaning less values c. If a gyro shows rates exactly opposite to the disturbance In all the above 3 cases primary gyros readings are taken as reference and secondary gyro is compared to the primary gyro, to locate faults in it. Other sensors can also be added for better fault detection like magnetometer for north tracking, but in every case some reference has to be fixed and considered as 289

7 Figure IV-2: LQR Controlled TableSat's Response to Step Command IV. TABLESAT RESULTS A. Matlab and Simulink Based Modelling Results The result of the PD controller designed above is shown below; B. Real Time TableSat Results Luenberger observer equations are implemented in the TableSat s microcontroller in angular rate control setup. To make the system time optimal, gain matrix K in observer equations is replaced by the optimal gains calculated from LQR in Matlab. After giving a disturbance to TableSat in rate control setup, real time data is obtained from Bluetooth to PC and graphs are plotted. Blue bars in the graph below represent angular rate of TableSat. Time is not shown in these graphs because it varies based on the magnitude of the applied disturbances. Figure IV-1: PD Controlled TableSat's Response to Initial Disturbance One of the main goals of the researchers is to make the TableSat a time optimal system, to accomplish this, Linear Quadratic Regulator (LQR) controller is used to find optimal gains [14] in Matlab by deploying the above dynamics model of TableSat. The resultant time optimal response of TableSat to step command is; Figure IV-3: LQR Controlled TableSat's Response to a Disturbance The control input for fans calculated by the combination of Luenberger observer and LQR controller as a result of the angular rate shown in above graph (due to an initial disturbance) is graphed below; 290

8 Figure IV-4: LQR Controlled TableSat's Response to a Disturbance To display a clear picture, both the above graphs (E-3 and E-4) are combine below in Figure E-5. It clearly shows how control input is varying exactly with the variation of angular rate of TableSat. Figure IV-6: LQR Controlled TableSat's Response to a Disturbance The graph (E-7) shows the overall behavior of TableSat subjected to a disturbance in rate control setup by combining the above 4 graphs (E-3, E-4, E-5, E-6). The horizontal green grid lines are for right hand side s vertical axis lines. Figure IV-5: LQR Controlled TableSat's Response to a Disturbance By using different set of conditions as mentioned in the control implementation flowchart given above, TableSat is also capable of reference tracking in the angular rate control setup mode, with dead band of -0.1 to 0.1 radians. Figure IV-7: LQR Controlled TableSat's Response to a Disturbance In a similar manner, TableSat is given a single disturbance in heading/yaw control setup, real time data is obtained from Bluetooth to PC and graphs are plotted. In the graphs E-8 and E-9, blue bars show angle and green line is its respective control input for fans calculated by PD controller. The dead band in this case is -0.1 to 0.1 radians (extreme cases). 291

9 Figure IV-8: PD Controlled TableSat's Response to a Disturbance The graph below shows the overall behavior of TableSat subjected to a single disturbance in heading or yaw control setup. with the capabilities of fault tolerant control and time optimal angular rate and heading/yaw control. This paper describes the TableSat system hardware and software, detailing the evolution of software from baseline calibration and control modes. Sensors and control laws have been shown to function adequately during nominal operation, and fault tolerant scheme facilitates fault detection and control system reconfiguration in cases where gyros fail or malfunction. Though TableSat has become a capable and reliable tool for research and education, future extensions can appreciably enhance performance in multiple directions. First, the gyros used currently are very cheap, causing drifting issues which are difficult to handle, these gyros can be replaced by better IMUs with gyros and magnetometers for better heading control and for adding more functions in fault tolerant part. Secondly the Bluetooth system used for communication purposes between PC and TableSat isn t stable i.e. on high data rates it fails and if some other Bluetooth device from surroundings tries to connect with it, it again creates connection problems. So it can be replaced by some better serial communication device. The fault tolerance can be extended to tolerate faults in actuators. Moreover, TableSat is currently working only for regulatory control, it can be extended to servo control. In the hardware perspective, TableSat can be extended to 2D or even 3D system. TableSat will continue to provide an invaluable educational embedded programming experience for students ranging from college freshmen to graduate students. Tighter integration of control systems and software engineering is becoming increasingly critical throughout the engineering of today s complex Aerospace systems. Our collaborative efforts with TableSat represent a small step in this direction. VI. REFERENCES Figure IV-9: PD Controlled TableSat's Response to a Disturbance V. CONCLUSIONS AND FUTURE WORK This paper has presented the TableSat tabletop satellite, a one-degree of freedom satellite simulation platform enriched [1] Ella M. Atkins, Jianliang Yi, Honguk Woo, James Browne, Aloysius Mok. The TableSat Platform and its Verifiable Control Software, AIAA Infotech@Aerospace Conference, Seattle, Washington, 6-9 April 2009.J. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, 1892, pp [2] Ali Nasir. Comprehensive Fault Tolerance and Science- Optimal Attitude Planning for Spacecraft Applications, University of Michigan, 2012.K. Elissa, Title of paper if known, unpublished. [3] Melissa F. Vess. System Modeling and Controller Design for A Single Degree of Freedom Spacecraft Simulator, University of Maryland, 2005.Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, Electron spectroscopy studies on magneto-optical media and plastic substrate interface, IEEE Transl. J. Magn. Japan, vol. 2, pp , August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982]. [4] Bela G. Liptak, Instrument Engineers' Handbook: Process Control and Optimization, vol. II, 2006, pp [5] C. Kleinstreuer, Modern Fluid Dynamics: Basic Theory and Selected Applications in Macro- and Micro Fluidics, vol. I, 2009, pp

10 [6] P.M. Whelan, M.J. Hodgeson, Essential Principles of Physics, 2nd Edition, [7] Gregory P. Starr, Introduction to Applied Digital Control, 2nd Edition, 2006, pp [8] Martin H. Weik, Communications Standard Dictionary, 2nd Edition, [9] Karl J. Astrom, Tore Hagglund, PID Controllers: Theory Design and Tuning, 2nd Edition, 1995, pp [10] Verica Radisavljevic-Gajic, Linear Observers Design and Implementation, ASEE Zone 1, [11] Katsuhiko Ogata, Modern Control Engineering, 4th Edition, 2002, pp [12] Alan G. Smith, Introduction to Arduino, 1st Edition, 2011, pp [13] D. V. S. MURTY, Transducers and Instrumentation, 2nd Edition, 2008, pp