QUATERNION BASED ATTITUDE ESTIMATION TECHNIQUE INVOLVING THE EXTENDED KALMAN FILTER A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Ishwor Gautam May 2019 QUATERNION BASED ATTITUDE ESTIMATION TECHNIQUE INVOLVING THE EXTENDED KALMAN FILTER Ishwor Gautam Thesis Approved: Accepted: ______________________ ___________________ Advisor Department Chair Dr. Celal Batur Dr. Sergio Felicelli ______________________ ____________________ Committee Member Dean of the College Dr. Ajay Mahajan Dr. Craig Menzemer _______________________ ____________________ Committee Member Dean of the Graduate School Dr. Siamak Farhad Dr. Chand Midha ___________________ Date ii ABSTRACT This thesis illustrates the application of the Extended Kalman filter for online estimation of attitude of a body. The accuracy of controlled attitude largely depends on the performance of the estimation algorithm. In this thesis, the extended Kalman filter (EKF) algorithm consisting of quaternion based state representation is used. The EKF algorithm utilizes gyroscope reading for priori estimation and measurements reading from the accelerometer and the magnetometer to correct the states. In simple terms, the extended Filter is used as the estimation tool by fusing the data from the gyroscope, the accelerometer and the magnetometer. A device that combines the gyroscope, accelerometer and magnetometer is called inertial measurement unit (IMU). The non- accurate scaling, sensor misalignment and non-zero biases of IMU devices are eliminated by proper calibration. The sensors utilized in the estimation have noise and biases which results in propagation of error in time. The noise and biases should be eliminated to get the accurate estimates. In this work, the EKF algorithm with some modification in state equation and in the Kalman filter gain is implemented for both the steady state and the body acceleration conditions. The estimation of the modified EKF is compared with the estimation technique used by VECTORNAV, a well-known commercial IMU. The modified EKF performed well compared to VECTORNAV in steady state condition. However, under body acceleration the modified EKF did not perform as well as what VECTORNAV did. The attitude estimation technique discussed in this thesis is less expensive and easy compared to those used in missile and aircraft guidance. The algorithm discussed in this thesis can be well implemented in navigation of robots and drones for home applications. iii ACKNOWLEDGEMENT I would like to express my sincere gratitude and appreciation to my advisor, Dr. Celal Batur, for all his valuable suggestions, motivation and support that he has provided during my time as a Master’s student at the department of Mechanical Engineering, The University of Akron. Whenever I was in a trouble or had a question related to my research work, it was easy for me to stop by his office. Dr. Batur is truly a mentor and it was a great pleasure working with him. I appreciate the contribution of Parker Hannifin Corporation for providing the Torsional Plant needed for project implementation. I wish to extend my sincere thanks to the members of my thesis defense committee, Dr. Ajay Mahajan and Dr. Siamak Farhad, for their time and valuable advice on this study. I would like to acknowledge Mr. Clifford G Bailey, for his technical support for solving hardware and software related problems. Sincere appreciation is also given to Mrs. Stacy Meier and Ms. Shannon Skelton for their assistant to my graduate study. I would also like to acknowledge a number of friends at the University of Akron, Alexander Sorin, Hari Poudyal and Sudeep Adhikari for their support in my research work. I also want to express my appreciation to my girlfriend for her love, support and always cheering me up. Last but not the least, I would like to give special thanks to my parents and sisters for their unconditional love, support and motivation for my inspiration. iv TABLE OF CONTENTS Page LIST OF FIGURES……………………………………………………………… viii LIST OF TABLES……..……………………………………………………… xii CHAPTER I. INTRODUCTION…………………………………………………………... 1 1.1 Introduction of Inertial Navigation Systems……………………………. 1 1.2. Literature Review……………………………………………………… 11 1.3. Thesis Outline…………………………………………………………. 15 II. EULER ANGLES, ROTATION MATRIX AND QUATERNION……………16 2.1 Introduction……………………………………………………………. 16 2.2 Euler Angles…………………………………………………………… 17 2.3. Rotation Matrix………………………………………………………... 21 2.4. Gimbal Lock…………………………………………………………… 27 2.5. Quaternions……………………………………………………………. 29 2.5.1. Quaternion Algebra…………………………………………………. 29 2.5.2. Quaternion Properties………………………………………………. 30 2.5.3. Rotation Matrix and Quaternion conversion……………………….. 30 2.5.4. Euler Angles and Quaternion conversion………………………….. 31 2.5.5. Advantages of quaternions…………………………………………. 32 III. CALIBRATION……………………………………………………………… 36 3.1. Introduction…………………………………………………………… 36 v 3.2. Calibration of tri-axis accelerometer………………………………….. 36 3.3. Calibration of tri-axis magnetometer………………………………….. 45 3.4. Calibration of tri-axis gyroscope………………………………………. 48 3.5. Conclusion……………………………………………………………... 54 IV. EXTENDED KALMAN FILTER (EKF)…………………………………….. 55 4.1. Introduction……………………………………………………………. 55 4.2. The State Equation (f(.))…………………………………………………… 57 4.3. Measurement Function (h(.))………………………………………….. 60 4.4. Linearization…………………………………………………………… 60 4.5. Extended Kalman Filter Algorithm…………………………………… 63 4.6. Simulated performance of the Extended Kalman filter under steady state condition…………………………………………………… 64 4.6.2. Process and Measurement noise covariance matrix…………… 64 4.6.3. Simulated Results………………………………………………. 68 4.6.4 Discussion of Result……………………………………………. 69 4.7. Conclusion…………………………………………………………………. 71 V. MODIFIED EXTENDED KALMAN FILTER……………………………….. 72 5.1 Introduction………………………………………………………………… 72 5.2. Modified Extended Kalman Filter Algorithm……………………………... 77 5.3 Implementation of the Algorithm for the Steady State Conditions………… 78 5.4. Simulated performance of the Modified Extended Kalman filter under body acceleration……………………………………………………81 5.4.1. Introduction………………………………………….. 81 5.4.2. Low pass filter………………………………………. 84 5.4.3. Assignment of measurement noise covariance (R) with respect vi to error………………………………………………………………. 86 5.4.4. Simulated Results……………………………………………………... 87 5.4.5. Discussion of simulated results…………………………………… 90 5.5. Conclusion……………………………………………………………. 94 VI. CONCLUSION AND RECOMMENDATIONS…………………………….. 95 6.1. Conclusion………………………………………………………………… 95 BIBLIOGRAPHY……………………………………………………………….. 98 APPENDICES…………………………………………………………………… 102 APPENDIX A: PROCEDURE TO GET STEP RESPONSE………………… 102 APPENDIX B: MATLAB CODE FOR IMPLIMENTING THE EKF……… 105 vii LIST OF FIGURES Figure Page 1.1 Conventional mechanical gyroscope mounted in an airplane (before roll) ........................................................................................ 2 1.2 Conventional mechanical gyroscope mounted in an airplane (after roll) .......................................................................................... 3 1.3 Working Principle of MEMS gyroscope. ........................................................ 5 1.4 The inclination, declination and earth’s magnetic intensity ............................ 8 1.5 The charge flow in a conductor carrying current. .......................................... 10 1.6 The charge flow in a conductor carrying current subjected to the magnetic field................................................................................................. 10 1.7 Fleming’s left hand Rule ................................................................................ 10 2.1 Yaw, pitch and roll in an airplane .................................................................. 17 2.2 Example of 3-2-1 Euler angles set angle is rotation about the y-axis and the roll angle is rotation about the x-axis. .............................................. 18 2.3.a Reference frame with point A in Y-axis ....................................................... 18 2.3.b Rotation about Z-axis by -45o ........................................................................ 18 2.3.c Rotation about Y-axis by +90o ...................................................................... 19 2.3.d Rotation about X-axis by +90o ...................................................................... 19 2.4.a Reference frame with point A in Y-axis ....................................................... 20 2.4.b Rotation about Z-axis by -90o ....................................................................... 20 2.4.c Rotation about Y-axis by +90o ...................................................................... 20 2.4.d Rotation about X-axis by +45o ...................................................................... 20 viii 2.5 Body co-ordinate located on an airplane and reference co-ordinate on the earth .................................................................................................... 22 2.6 Rotation of point A by an angle in Z-Y-X Euler angle set ......................... 23 2.7 Rotation of point A by an angle in Z-Y-X Euler angle............................24 2.8 Rotation of point A by an angle in Z-Y-X Euler angle set. .......................... 25 2.9 Rotation of an airplane plane in by pitch angle about Y-axis ...................... 28 3.1 Uncompensated accelerometer reading for position P1 and P2....................... 42 3.2 Uncompensated accelerometer