DYNAMIC MODELING AND CONTROL OF SPACECRAFT ROBOTIC SYSTEMS USING DUAL QUATERNIONS A Dissertation Presented to The Academic Faculty By Alfredo Valverde Salazar In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Aerospace Georgia Institute of Technology May 2018 Copyright c Alfredo Valverde Salazar 2018 DYNAMIC MODELING AND CONTROL OF SPACECRAFT ROBOTIC SYSTEMS USING DUAL QUATERNIONS Approved by: Dr. Panagiotis Tsiotras, Advisor Dr. David S. Bayard School of Aerospace Engineering Guidance and Control Analysis Georgia Institute of Technology Group Jet Propulsion Laboratory Dr. Marcus J. Holzinger School of Aerospace Engineering Dr. Glenn Lightsey Georgia Institute of Technology School of Aerospace Engineering Georgia Institute of Technology Dr. Patricio A. Vela School of Electrical and Computer Date Approved: April 6, 2018 Engineering Georgia Institute of Technology Nothing in life is to be feared, it is only to be understood. Marie Curie To my mother, who taught me that perfection is not a myth, and to my wife, who stands by my side rain or shine. ACKNOWLEDGEMENTS First and foremost, I would like to thank my mother Rosario Salazar and my wife Bailey Valverde, whose unconditional love has no limits, and whose support has never wavered. Taty, Guille, Amis, Rosita, Mari, Laura, Francisco, Tim, Ms. Carol, you are the best family anyone could ever ask for, and each one of you has contributed greatly to me becoming a better person. Thank you for supporting my dreams and enabling me in every possible way. Professor Panagiotis Tsiotras thank you for being an admirable professional and an even better advisor. Your desire to pursue the highest levels of research has shaped me in ways I could have only wished for when I joined the DCSL. Thank you for providing me, and those who surround you, the opportunity to learn by your side. I would also like to thank the members of my defense committee for their valuable insight and remarkably keen eye: Dr. Marcus J. Holzinger, Dr. Patricio A. Vela, Dr. Glenn Lightsey and Dr. David S. Bayard. In one way or the other, you all contributed enormously to the work contained in this dissertation, and even that might still be an understatement. Nuno Filipe and Chiluwata Lungu, as my most impactful non-faculty mentors through- out my journey at Georgia Tech, I would like to thank you for setting the bar so high. Esteban Q., Eduardo B., María José O., Benjamin H., Oliver D., Alex S., Luke N.: thank you for unparalleled levels of friendship. Alex D., thank you for an amazing friendship and for your help with Clifford algebras, most of which I have incorporated in this disserta- tion. Mehregan D. and Matthew K-S.: thank you for your patience, your persistence, your determination, your hard work, and an uncountable amount of sleepless nights in the lab. As this space is finite, many people who have played an important role in my life have not been explicitly mentioned, including family, friends, teachers, coaches, mentors, and role models. To each and every person I have come across in my life, contributing to shaping me into the individual I am today: thank you! v TABLE OF CONTENTS Acknowledgments . v List of Tables . xi List of Figures . xiii Chapter 1: Introduction . 1 1.1 Motivation . 1 1.2 Literature Review . 5 1.2.1 Dual Quaternions in Robotics . 5 1.2.2 Dynamic Modeling and Control of Rigid Bodies in 6-DOF Motion . 8 1.2.3 Dynamic Modeling of Spacecraft-Mounted Robotic Manipulators . 12 1.3 Research Focus and Contributions . 23 Chapter 2: Dual Quaternions . 27 2.1 Clifford Algebras . 27 2.2 Quaternions . 29 2.3 Dual Quaternions . 33 2.3.1 Wrench Notation and Transformations Using Dual Quaternions . 40 2.3.2 Dual Inertia Matrix, Dual Momentum and 6-DOF Rigid Body Dy- namics . 43 vi Chapter 3: Dual Quaternions in Robotics . 50 3.1 Dual Quaternion Notation . 50 3.2 Product of Exponentials Formula in Dual-Quaternion Form . 51 3.3 Denavit-Hartenberg Parameters in Dual Quaternion Form . 54 3.4 Convex Constraints Using Dual Quaternions . 57 3.5 Example: Forward Kinematics with an Inertially Fixed Base . 61 3.6 Example: Forward Kinematics of a Floating Double Pendulum with End- Effector . 67 Chapter 4: Pose Tracking and Dual Inertia Estimation Using Concurrent Learn- ing with Dual Quaternions . 70 4.1 Concurrent Learning . 72 4.2 Adaptive Control with Continuous Concurrent Learning . 75 4.3 Adaptive Control with Discrete Concurrent Learning . 82 4.4 Concurrent Learning as a Contributing Factor to Meet Persistency of Exci- tation Conditions . 87 4.5 Numerical Results . 91 4.6 Conclusions . 97 Chapter 5: Dynamic Modeling of A Robotic Arm on a Satellite Using Dual Quaternions . 100 5.1 Dynamic System Modeling - Decoupled Formulation . 100 5.1.1 Frame Assignment to Each Variable . 103 5.1.2 Constraints . 104 5.1.3 Attitude Kinematics . 108 5.1.4 Position Kinematics . 109 vii 5.1.5 State Definition . 109 5.2 Dual Quaternion Formulation . 110 5.2.1 Wrenches . 110 5.2.2 Robot Geometry in Dual Quaternions . 111 5.2.3 Robot Dynamic Equations in Dual Quaternion Form . 114 5.2.4 Relative Kinematics and Joint Motion . 115 5.2.5 Derivation of Kinematic Constraints Using Dual Quaternions . 118 5.2.6 Dual Quaternion Kinematics . 120 5.2.7 Dual Quaternion Formulation Implementation . 122 5.3 Results . 129 Chapter 6: Dynamics Framework Generalization . 138 6.1 Variable Definition and Conventions . 138 6.2 Kinematics . 146 6.3 Dynamics . 147 6.4 Locking or Prescribing Joint Motion . 151 6.5 Framework Summary . 154 Chapter 7: Two-Manipulator Satellite . 156 7.1 System Architecture and Frame Definition for Satellite with Two Manipu- lators . 156 7.2 Auxiliary Matrices for Two Arm Architecture . 156 7.3 Variable Definition for Two Arm Architecture . 160 7.4 Kinematics for Two Arm Architecture . 162 viii 7.5 Dynamics for Two Arm Architecture . 165 7.5.1 Matrices S11 and S22 . 165 7.5.2 Matrix S12 ..............................165 7.5.3 Matrix S21 ..............................167 7.5.4 Vector B1 ...............................168 7.5.5 Vector B2 ...............................170 7.6 Numerical Results . 170 Chapter 8: Robotic End Effector Control . 176 8.1 Differential Dynamic Programming . 176 8.2 Derivation of Differential Dynamic Programming . 176 8.3 Control Update Step . 184 8.4 End-Effector Kinematics . 185 8.5 End-Effector Pose Stabilization Using DDP . 186 8.6 Objective Function Design for End-Effector Pose Tracking . 188 8.7 End-Effector Pose Tracking . 189 Chapter 9: Estimation of Mass Properties for Spacecraft-Mounted Manipulator 203 9.1 Incorporation into the Concurrent Learning Framework . 203 9.2 Adaptive Estimation of Dual Inertia . 206 9.3 Algorithm Description . 208 9.4 Aggressive Estimation . 210 9.5 Numerical Simulations . 212 ix Chapter 10:Conclusions . 220 10.1 Future Work . 223 Appendix A: Expansion of Translational Constraint Encoded in Dual Quater- nion for Joint 0 . 227 A.1 Derivation . 227 References . 243 Vita ........................................... 244 x LIST OF TABLES 2.1 Matching of Clifford algebra elements and quaternion algebra elements. 30 2.2 Quaternion Operations . 31 2.3 Matching of Clifford algebra elements and dual quaternion algebra elements. 35 2.4 Dual Quaternion Operations . 36 2.5 Unit Dual Quaternion Operations . 37 3.1 Screw (ξi) for revolute and prismatic joints. 66 5.1 Frame assignment for relevant variables. 103 5.2 Average and variance of run times for 100 simulations for two different dynamics formulations using MATLAB. 133 5.3 Average and variance of run times for 100 simulations for two different dynamics formulations using SIMULINK. 134 6.1 Generalized coordinates ΓJi for joint Ji depending on its type. 143 6.2 Form of reduced reaction wrenches for different joint types. 145 6.3 Form of actuation wrenches for different joint types. 146 6.4 Mapping matrix from angular velocity to generalized coordinates. 147 7.1 P (·) and N(·) functions applied to the columns of C for two-arm satellite topology. 158 7.2 N(·) function applied to the rows of C for two-arm satellite topology. 160 xi 7.3 Joint characterization for two-arm spacecraft. 161 8.1 Cost function parameters. 187 8.2 Parametrization parameters for iteration dependence. 192 xii LIST OF FIGURES 2.1 Wrench interpretation. 41 3.1 Screw motion parametrized by θ and s. ................... 51 3.2 Denavit-Hartenberg parameters. 54 3.3 Line-of-sight constraint during grappling. 62 3.4 Approach slope constraint. 63 3.5 Upper-and-lower bounds constraint. 64 3.6 General attitude constraint with respect to inertial directions. 64 3.7 Robot arm configuration. 65 3.8 Robot arm configuration. ..