DYNAMIC MODELLING FOR THE PATH TRACKING CONTROL OF A FOUR-WHEEL INDEPENDENT-DRIVE, FOUR-WHEEL INDEPENDENT-STEER AUTONOMOUS GROUND VEHICLE A Thesis Submitted to the Graduate Faculty of the North Dakota State University of Agriculture and Applied Science By Karl Emerson Klindworth In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Major Department: Mechanical Engineering November 2017 Fargo, North Dakota North Dakota State University Graduate School Title DYNAMIC MODELLING FOR THE PATH TRACKING CONTROL OF A FOUR-WHEEL INDEPENDENT-DRIVE, FOUR-WHEEL INDEPENDENT-STEER AUTONOMOUS GROUND VEHICLE By Karl Emerson Klindworth The Supervisory Committee certifies that this disquisition complies with North Dakota State University’s regulations and meets the accepted standards for the degree of MASTER OF SCIENCE SUPERVISORY COMMITTEE: Dr. Majura Selekwa Chair Dr. Annie Tangpong Dr. Mariusz Ziejewski Dr. Sumathy Krishnan Dr. Jacob Glower Approved: 11/16/17 Dr. Alan Kallmeyer Date Department Chair ABSTRACT Robots that can be reconfigured to perform more than one task would be to consumers. The Four-Wheel Independent-Drive, Four-Wheel Independent-Steer (4WD4WS) robot is well suited for the role of reconfigurable robot due to its extremely high maneuverability and torque control. However, the nonlinear dynamics in conjunction with complex kinematic constraints make the 4WD4WS structure an extremely difficult control problem. As a result of this many who model the 4WD4WS structure make simplifications that aren’t realistic for a reconfigurable consumer robot. A 4WD4WS robot is kinematically and dynamically modeled using both the front and rear path angles and their respective coordinates. High fidelity equations of motion, for robots of arbitrary width, length, and mass, undergoing arbitrary accelerations at arbitrary steering angles have been created that have the potential to increase the path tracking ability of 4WD4WS systems. Simulations show the model behaves realistically, but needs a controller. iii ACKNOWLEDGEMENTS I would like to thank Dr. Selekwa for all the hard work he has put in helping me with this project. I have worked with him for two and a half years and in that time he has taught me numerous lessons that I know will help me over my career, but also in life. He has become a very good friend of mine. I am hopeful that we may get to work together again in the future. I would like to thank Dr. Kallmeyer for mentoring me over the years. I have gotten to know him very well through the four classes I have taken from him as well as various social gatherings. It is because of teachers like him that I was excited to come back for my Master’s Degree. I would like to thank my parents for family for always being there for me. Whenever school got difficult I knew a trip home would give me encouragement to get back at the grind. I would not be who I am today without your encouragement. I would like to thank Joe Cluett. He spent countless hours programming different subsystems (IMU, GPS, WIFI…) for me while also teaching me how to program in C. While we didn’t get the system going, we set the next group up for success. I would like to thank all the members of the team that helped me with building the experimental system, including: Tyler Lane, Brady Goenner, Willem Bohrer, Sam, Andrew Schlangen, Nate Peterson, and Andy Narvesen. iv TABLE OF CONTENTS ABSTRACT ................................................................................................................................... iii ACKNOWLEDGEMENTS ........................................................................................................... iv LIST OF TABLES ......................................................................................................................... ix LIST OF FIGURES ........................................................................................................................ x LIST OF ABBREVIATIONS ...................................................................................................... xiii LIST OF SYMBOLS ................................................................................................................... xiv 1. INTRODUCTION ...................................................................................................................... 1 1.1. Background and Motivation ................................................................................................. 1 1.2. Robot Steering Systems ....................................................................................................... 5 1.2.1. Differential Steering System ......................................................................................... 7 1.2.2. Skid Steering Systems ................................................................................................... 8 1.2.3. Ackerman Two Wheel Steering Systems .................................................................... 10 1.2.4. Ackerman Four Wheel Steering Systems .................................................................... 12 1.2.5. Articulating Steering Systems ..................................................................................... 13 1.2.6. Synchro-Drive Steering Systems ................................................................................. 15 1.2.7. Omnidirectional Steering Systems .............................................................................. 16 1.2.8. Four-Wheel Independent-Drive/Four-Wheel Independent-Steer ................................ 18 1.3. Research Objectives ........................................................................................................... 20 1.3.1. Dynamic Modelling of a 4WD4WS Robotic Vehicle ................................................. 22 1.3.2. Development of a Control Algorithm based on the Formulated Dynamic Model .......................................................................................................................... 23 1.3.3. Numerical Validation of the Developed Control Algorithm ....................................... 24 1.3.4. Experimental Validation of the Developed Control Algorithm .................................. 24 1.3.5. Thesis Write-up of the Results .................................................................................... 24 2. THE EXPERIMENTAL 4WD4WS PROTOTYPE ................................................................. 25 v 2.1. Evolution of the Experimental 4WD4WS Prototype ......................................................... 25 2.2. Subsystems of the 4WD4WS Prototype ............................................................................. 28 2.2.1. Main Chassis and Wheel Suspension .......................................................................... 28 2.2.2. Power and Actuation ................................................................................................... 29 2.2.3. Sensor Measurement ................................................................................................... 30 2.2.4. Control System ............................................................................................................ 33 2.3. Architecture of the Robot Control System ......................................................................... 34 3. KINEMATIC AND DYNAMIC MODELLING OF THE 4WD4WS ROBOTIC SYSTEM ....................................................................................................................................... 36 3.1. Definition of Bodies and Coordinates ................................................................................ 36 3.2. Path Tracking: The Relationship between the Path and Chassis Position ......................... 37 3.3. Relationship between the Path and the ICR ....................................................................... 42 3.4. Relationship between the ICR and the Steering Angles .................................................... 44 3.5. Wheel Velocities and Body Yaw Rates ............................................................................. 48 3.6. Formulation of Constraints ................................................................................................. 51 3.6.1. Position Constraints ..................................................................................................... 52 3.6.2. Velocity Constraints .................................................................................................... 57 3.7. Formulation of the Generalized Forces .............................................................................. 61 3.8. Formulation of the Lagrangian of Motion .......................................................................... 66 4. NUMERIC SIMULATION ...................................................................................................... 71 5. EXPERIMENTAL SIMULATION .......................................................................................... 74 6. CONCLUDING REMARKS .................................................................................................... 76 6.1. Future Work ....................................................................................................................... 77 7. REFERENCES ......................................................................................................................... 78 APPENDIX ................................................................................................................................... 87