The Pennsylvania State University The Graduate School College of Engineering MODELING AND SIMULATION OF A SIX WHEEL DIFFERENTIAL TORQUE STEER VEHICLE A Thesis in Mechanical Engineering by Madhu Soodhanan Govindarajan Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2010 The thesis of Madhu Soodhanan Govindarajan was reviewed and approved by the following: H. J. Sommer III Professor of Mechanical Engineering Thesis Advisor E.R. Marsh Professor of Mechanical Engineering K.A.Thole Professor and Head of Mechanical and Nuclear Engineering ii ABSTRACT Autonomous Ground vehicles (AGV) are often required to automatically follow a pre-defined trajectory. They are used extensively for various tasks in unstructured environments because of their ability to navigate and perform well in such conditions. AGV‟s can be modeled and various simulations can be performed on these models to obtain helpful data. This will help us understand their behavior better under adverse conditions. Modeling of a six wheeled differential torque system is presented in this thesis. Differential torque steer vehicles, in order to follow a curved path must skid and have lateral tire motion which makes control at a kinematic level insufficient. Thus a dynamic model was developed in this work. This thesis is aimed at developing a mathematical model in MATLAB that will provide useful results about the performance of the vehicle and also validate the simulations done using Adams. iii Table of Contents LIST OF TABLES ..................................................................................................................... vi LIST OF FIGURES ................................................................................................................... vii NOTATIONS .............................................................................................................................. x CHAPTER 1 ................................................................................................................................ 1 INTRODUCTION ....................................................................................................................... 1 1.1 Modeling and Simulation ................................................................................................. 1 CHAPTER 2 ................................................................................................................................ 4 LITERATURE REVIEW ............................................................................................................ 4 2.1 Fundamentals of Steering ................................................................................................. 4 2.2 Fundamentals of Skid Steering ........................................................................................ 6 2.3 Previous Skid Steer Modeling .......................................................................................... 7 CHAPTER 3 ................................................................................................................................ 9 PHYSICAL PARAMETER DETERMINATION ...................................................................... 9 3.1 Measuring Mass Moment of Inertia with Torsional Pendulum ....................................... 9 3.2 Mass moment of inertia calculation of axle shaft .......................................................... 10 3.3 Mass moment of inertia calculation of wheel: ............................................................... 11 3.4 Mass moment of inertia calculation of sprockets: .......................................................... 12 3.5 Mass moment of inertia calculation of chain 1: ............................................................. 14 3.6 Mass moment of inertia calculation of chain 2: ............................................................. 14 CHAPTER 4 .............................................................................................................................. 15 ADAMS SIMULATIONS AND RESULTS ............................................................................ 15 4.1 Simulations ..................................................................................................................... 17 4.1.1 Straight line maneuvers .......................................................................................... 17 4.1.1.1 On flat road..................................................................................................... 17 4.1.1.2 Accelerating on flat road ................................................................................ 18 4.1.1.3 Accelerating on a ramp .................................................................................. 19 4.1.1.4 Running over an oblique bump ...................................................................... 21 4.1.1.5 Running over a straight bump ........................................................................ 22 4.1.2 Circular motion ........................................................................................................... 24 iv 4.1.2.1 On flat road......................................................................................................... 24 4.1.2.2 On a ramp ........................................................................................................... 27 4.1.3 Lane change maneuver ................................................................................................ 29 4.1.4 J- Turn maneuver ........................................................................................................ 31 CHAPTER 5 .............................................................................................................................. 34 MATLAB MODELING AND SIMULATION ........................................................................ 34 5.1 Tire Modeling ................................................................................................................. 36 5.2 Friction Consideration .................................................................................................... 37 5.2.1 Mechanics of Force Generation ............................................................................ 37 5.3 Dynamic Equations ........................................................................................................ 39 5.4 Simulations ..................................................................................................................... 42 5.4.1 Straight line maneuvers ......................................................................................... 42 5.4.1.1 On flat road .................................................................................................. 42 5.4.1.2 Accelerating on flat road .............................................................................. 42 5.4.1.3 Accelerating on a ramp ................................................................................. 43 5.4.2 Lane change maneuver ........................................................................................... 44 5.4.3 J- turn maneuver ..................................................................................................... 47 CHAPTER 6 .............................................................................................................................. 50 CONCLUSION AND FUTURE WORK .................................................................................. 50 6.1 Scope for Future Research ............................................................................................ 51 REFERENCES .......................................................................................................................... 52 APPENDIX A Model parameters ............................................................................................. 54 APPENDIX B MATLAB code ................................................................................................. 57 v LIST OF TABLES Table 3.1 Time taken for 20 oscillations of the wooden platform for the wheel………………..11 Table 3.2 Time taken for 20 oscillations of the metal platform for the sprockets..……………..13 vi LIST OF FIGURES Fig 1.1 Organization of thesis……………………………………………………………………..3 Fig 2.1 Types of steering………………………………………………………………………….5 Fig 2.2 Clutch/Brake steering system……………………………..………………………………6 Fig 2.3 Differential steering system………………………………..……………………………...7 Fig 2.4 Planetary Gear steering system……………………………….…………………………..7 Fig 3.1 Torsional pendulum……………………………………….………...…………………..10 Fig 4.1 Tractive effort vs longitudinal slip………………………………………………………17 Fig 4.2 Cornering force vs slip angle………………………………......………………………..17 Fig 4.3 Input torque applied to the wheels (Straight line on a flat road)…….………………….18 Fig 4.4 Distance travelled by the vehicle (Straight line on a flat road)…………………………19 Fig 4.5 Input torque applied to the wheels (Accelerating on flat road)……….………………...19 Fig 4.6 Distance travelled by the vehicle (Accelerating on flat road)………...………………...20 Fig 4.7 Input torque applied to the wheels (Accelerating on a ramp)………….………………..21 Fig 4.8 Distance travelled by the vehicle (Accelerating on a ramp)…………...………………..21 Fig 4.9 Input torque applied to the wheels (Running over an oblique bump)…………………..22 Fig 4.10 Distance travelled by the vehicle (Running over an oblique bump)……...………..….22 Fig 4.11 Path traced by the vehicle (Running over an oblique bump)……………...….……….23 vii Fig 4.12 Input torque applied to the wheels (Running