56844726.pdf - Precision Navigation for Aerospace Applications by Andrew K Stimac B.S Mechanical Engineering Massachusetts Institute of Technology 1999

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Unformatted text preview: Precision Navigation for Aerospace Applications by Andrew K. Stimac B.S. Mechanical Engineering Massachusetts Institute of Technology, 1999 SUBMITTED TO THE DEPARMENT OF MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2004 ©2004 Andrew K. Stimac. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. This work sponsored by the Department of the Air Force under Air Force contract #F19628-00-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government. Signature of Author: ____________________________________________________ Department of Mechanical Engineering May 7, 2004 Certified by: ___________________________________________________________ David L. Trumper Associate Professor of Mechanical Engineering Thesis Supervisor Accepted by: __________________________________________________________ Ain A. Sonin Chairman, Department Committee on Graduate Students Precision Navigation for Aerospace Applications by Andrew K. Stimac Submitted to the Department of Mechanical Engineering on May 7, 2004 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering Abstract Navigation is important in a variety of aerospace applications, and commonly uses a blend of GPS and inertial sensors. In this thesis, a navigation system is designed, developed, and tested. Several alternatives are discussed, but the ultimate design is a loosely-coupled Extended Kalman Filter using rigid body dynamics as the process with a small angle linearization of quaternions. Simulations are run using real flight data. A bench top hardware prototype is tested. Results show good performance and give a variety of insights into the design of navigation systems. Special attention is given to convergence and the validity of linearization. Thesis Supervisor: David L. Trumper Title: Associate Professor of Mechanical Engineering 2 Biographical Note Andrew Stimac is from Seattle, Washington and holds a Bachelor’s Degree in Mechanical Engineering from MIT in 1999. This thesis was conducted in conjunction with MIT Lincoln Laboratory, and is in partial fulfillment for a Master’s Degree in Mechanical Engineering at MIT in 2004. Andrew’s research interest is the intelligent control of mechanical systems. This includes modern control theory with particular attention to nonlinear systems and adaptive techniques. Andrew’s recreational activities include sand volleyball, alpine skiing, and music. He now lives in Acton, Massachusetts with his wife Amy. Acknowledgements I give special recognition to those who have inspired, educated, and assisted me in this project. Above all, I would like to thank my advisors Prof. David Trumper and Anthony Hotz for their guidance. I am also deeply indebted to Deborah Blanchard and Leonas Bernotas at MIT Lincoln Laboratory for the excellent engineering that made this project possible. Finally, I am grateful to my MIT instructors Prof. Harry Asada and Prof. Jean-Jacques Slotine for their enlightening illustration of the subject matter. 3 Table of Contents 1. Introduction............................................................................................................................. 9 1.1 Summary of Procedure ................................................................................................. 10 1.2 Summary of Key Results .............................................................................................. 11 2. Theoretical Derivation .......................................................................................................... 13 2.1 Navigation..................................................................................................................... 13 2.1.1 Coordinate Systems .............................................................................................. 13 2.1.2 Attitude Representation ........................................................................................ 17 2.1.3 Gravity Model....................................................................................................... 25 2.1.4 Rigid Body Dynamics........................................................................................... 25 2.1.5 Measurements ....................................................................................................... 26 2.1.6 Initial Alignment................................................................................................... 27 2.2 The Kalman Filter ......................................................................................................... 31 2.2.1 Continuous-Discrete Formulation......................................................................... 33 2.2.1.1 Problem Statement and Assumptions ............................................................... 33 2.2.1.2 Minimum Conditional Error Variance.............................................................. 34 2.2.1.3 Minimum Unconditional Error Variance.......................................................... 36 2.2.2 Nonlinear Extension.............................................................................................. 42 2.2.2.1 Linearization ..................................................................................................... 43 2.2.2.2 Measurement Pre-Compensation ...................................................................... 45 2.2.2.3 Offloading the Estimation Error ....................................................................... 47 2.2.3 Data Forgetting ..................................................................................................... 47 2.2.4 Convergence ......................................................................................................... 50 2.2.4.1 Observability..................................................................................................... 50 2.2.4.2 Covariance Analysis ......................................................................................... 57 2.2.4.3 The Effect of Modeling Errors.......................................................................... 58 2.2.4.4 Nonlinear Convergence .................................................................................... 60 2.2.4.5 Convergence with Data Forgetting ................................................................... 60 2.2.5 Implementation Techniques.................................................................................. 61 2.2.5.1 Multirate Measurements and Dropouts............................................................. 62 2.2.5.2 Modeling of Colored Noise .............................................................................. 62 2.2.5.3 Numerical Techniques ...................................................................................... 65 2.2.5.4 Filter Health Monitoring ................................................................................... 66 2.3 Application of the Kalman Filter to Navigation ........................................................... 68 2.3.1 Linearization ......................................................................................................... 74 2.3.1.1 Dynamics .......................................................................................................... 74 2.3.1.2 Measurements ................................................................................................... 77 2.3.1.3 Noise Modeling................................................................................................. 79 2.3.2 Summary of Equations.......................................................................................... 81 2.3.3 Navigation Observability ...................................................................................... 83 2.3.4 Additional Considerations .................................................................................... 88 3. Simulation ............................................................................................................................. 91 3.1 Overview....................................................................................................................... 91 3.2 Filter Implementation.................................................................................................... 94 3.3 Covariance Analysis ..................................................................................................... 96 4 3.3.1 Earth Stationary .................................................................................................... 97 3.3.2 Aircraft Flight Path ............................................................................................. 103 3.3.3 Booster Flight Path ............................................................................................. 105 3.4 Simulated Data............................................................................................................ 108 3.4.1 Nominal Operation with Initial Errors................................................................ 109 3.4.2 Unexpected Noise and Disturbance .................................................................... 113 3.4.3 Additional Trajectories ....................................................................................... 120 3.4.4 Omitting Bias States ........................................................................................... 123 3.4.5 Measurement Dropouts....................................................................................... 125 3.5 Flight Data Simulation................................................................................................ 129 3.5.1 Aircraft................................................................................................................ 129 3.5.2 Booster Rocket.................................................................................................... 133 3.6 Initialization ................................................................................................................ 135 4. Hardware............................................................................................................................. 137 4.1 Description of Hardware Components........................................................................ 137 4.2 Software Implementation............................................................................................ 139 4.3 Efficiency Techniques ................................................................................................ 140 4.3.1 Memory Map ...................................................................................................... 141 4.3.2 Custom Matrix Operation Routines .................................................................... 141 4.3.3 Processing Measurement One at a Time............................................................. 143 4.3.4 Reduction of Model Complexity ........................................................................ 144 4.4 Results......................................................................................................................... 144 5. Discussion ........................................................................................................................... 150 6. Recommendations............................................................................................................... 152 6.1 Immediate Capabilities ............................................................................................... 152 6.2 Direct Improvements .................................................................................................. 153 6.3 Future Objectives ........................................................................................................ 155 7. Conclusion .......................................................................................................................... 158 Appendix – IMU Process Model ................................................................................................ 159 References................................................................................................................................... 162 5 List of Figures Figure 1-1: Benchtop Testing Environment ................................................................................ 10 Figure 1-2: Hardware Components.............................................................................................. 10 Figure 2-1: Geodetic and Geocentric Coordinates [20]............................................................... 14 Figure 2-2: Euler Angle Representation ...................................................................................... 19 Figure 2-3: Model of Colored Noise............................................................................................ 62 Figure 2-4: Colored Noise with Varying Filter Rate ................................................................... 64 Figure 2-5: Colored Noise with Equal Variance.......................................................................... 64 Figure 2-6: Filter Implementation of Rigid Body Dynamics....................................................... 69 Figure 2-7: Tightly-Coupled and Loosely-Coupled Architectures.............................................. 70 Figure 2-8: Small Angle Orientation Error.................................................................................. 74 Figure 2-9: Equivalence of a Bias and a Small Angle Error........................................................ 87 Figure 3-1: Simulation Block Diagram........................................................................................ 92 Figure 3-2: Top Level Filter Architecture ................................................................................... 94 Figure 3-3: Data flow in the Extended Kalman Filter ................................................................. 95 Figure 3-4: Time Plot of Nominal Attitude and Position Accuracy (principal standard deviations)............................................................................................................................. 98 Figure 3-5: Accuracy under Varying Noise in the Gyroscopes (ω), Accelerometers (a), GPS Position (p), and GPS Velocity (v) ....................................................................................... 99 Figure 3-6: Accuracy using Fixed-to-Earth Measurement ........................................................ 100 Figure 3-7: Accuracy with Biases using the Fixed-to-Earth Measurement............................... 101 Figure 3-8: Accuracy with Biases without using the Fixed-to-Earth Measurement.................. 101 Figure 3-9: Effect of Latitude on Attitude Accuracy................................................................. 102 Figure 3-10: Effect the GPS Rate .............................................................................................. 102 Figure 3-11: Effect of the Forgetting Rate................................................................................. 102 Figure 3-12: Time plot of Nominal Attitude and Position Accuracy ........................................ 103 Figure 3-13: Accuracy under Varying Noise in the Gyroscopes (ω), Accelerometers (a), GPS Position (p), and GPS Velocity (v) ..................................................................................... 104 Figure 3-14: Accuracy under Varying Gyroscope Bias (ω) and Accelerometer Bias (a) ......... 104 Figure 3-15: Effect of the GPS Rate .......................................................................................... 105 Figure 3-16: Effect of Data Forgetting ...................................................................................... 105 Figure 3-17: Time Plot of Nominal Attitude and Position Accuracy ........................................ 106 Figure 3-18: Accuracy under Varying Noise in the Gyroscopes (ω), Accelerometers (a), GPS Position (p), and GPS velocity (v) ...................................................................................... 106 Figure 3-19: Accuracy under Varying Gyroscope Bias (ω) and Accelerometer Bias (a) ......... 107 Figure 3-20: Effect of GPS Rate................................................................................................ 107 Figure 3-21: Effect of Data Forgetting ...................................................................................... 107 Figure 3-22: Trajectory for Normal Operation Simulations ...................................................... 109 Figure 3-23: Attitude Error Response........................................................................................ 110 Figure 3-24: Body Rate Response ............................................................................................. 110 Figure 3-25: Torque Disturbance Response .............................................................................. 111 Figure 3-26: Position Response ................................................................................................. 111 Figure 3-27: Velocity Response................................................................................................. 111 Figure 3-28: Force Disturbance Response................................................................................. 111 Figure 3-29: Gyroscope Bias Response..................................................................................... 112 6 Figure 3-30: Accelerometer Bias Response............................................................................... 112 Figure 3-31: Gravity Model Bias Response............................................................................... 112 Figure 3-32: Chi-Squared Statistic............................................................................................. 112 Figure 3-33: 10x Underestimated Attitude Error....................................................................... 114 Figure 3-34: Benefit of Forgetting............................................................................................. 114 Figure 3-35: Attitude Error caused by Underestimated Position Error ..................................... 114 Figure 3-36: 10x Underestimated Position Error....................................................................... 114 Figure 3-37: Chi-Squared Statistic for Underestimated Initial Position Error .......................... 115 Figure 3-38: Attitude Accuracy with Underestimated Measurement Noise.............................. 116 Figure 3-39: Chi-Squared Statistic for Underestimated Measurement Noise ........................... 116 Figure 3-40: Attitude Response with Underestimated Process Noise (100x)............................ 117 Figure 3-41: Chi-Squared Statistic with Underestimate Process Noise (100x)......................... 117 Figure 3-42: Position Estimate with GPS Position Glitch at t = 10s ......................................... 118 Figure 3-43: Attitude Estimate with GPS Position Glitch at t = 10s ......................................... 118 Figure 3-44: Chi-Squared Statistic with GPS Position Glitch at t = 10s ................................... 118 Figure 3-45: Body Rate Estimate during Gyroscope Glitch (100x) .......................................... 118 Figure 3-46: Attitude Error with GPS Position Step Glitch ...................................................... 119 Figure 3-47: Position Error with GPS Position Step Glitch ...................................................... 119 Figure 3-48: Chi-Squared Statistic with GPS Position Step Glitch...........................................
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