Modeling and parameter estimation of an unmanned ground vehicle with a continuously variable transmission Auby, Per Magnus Master’s Thesis, Spring 2016 Modeling and parameter estimation of an unmanned ground vehicle with a continuously variable transmission Auby, Per Magnus 2016-05-23 ii Abstract This thesis is about modeling of the longitudinal dynamics of an unmanned ground vehicle. This includes development of a longitudinal dynamics model for the vehicle under consid- eration as well as estimation of unknown parameters in the model. The parameters are estimated from measurements provided by the vehicle’s instrumentation. The proposed use of the model is to improve the vehicle’s control system. The unmanned ground vehicle is based a Polaris Ranger utility vehicle, which has a con- tinuously variable transmission (CVT). The CVT consist of a rubber belt and two variable radius pulleys, which allows the speed ratio of the transmission to be continuously varied. A mechanical control system regulates the speed ratio of the transmission based on the en- gine speed and torque load. A good model of the CVT and its mechanical control system is assumed to be vital for an accurate model of the vehicle. A mathematical model describing the engine, CVT, drive line, tires, brakes, body, and terrain dynamics is proposed. The proposed CVT model is a new model, but its accuracy is not yet well proven. Simulations of the model show the expected behavior. The proposed parameter estimation is based on maximum likelihood estimation. A state estimator based on the unscented Kalman filter (UKF) is used to calculate the likelihood of the observed measurements. The model parameters are optimized using the likelihood calculated by the UKF as criterion. To decrease the computation time, the parameter estimation uses a two stage approach where parameters unrelated to the drive line dynamics are estimated first using a separate set of observations and a lower order filter model. Estimation of parameters from simulated measurements gave reasonable results, but with non-negligible bias for some of the parameters. Simulations of the model with estimated parameters did not give results good enough to be useful for control system development directly. The simulations followed the general shape of the response of the experimental ob- servations, but generally with a different magnitude. The model with estimated parameters is accurate enough for use as a filter model for state estimation. As the parameter estimation gave slightly biased results even for simulated data, it is hard to quantify how much of the inaccuracy is caused by the bias in the parameter estimation algorithm and how much is caused by errors in the model. It is nevertheless believed that at least some of the inaccuracy is caused by errors in the model. Suggestions for improvements to the both the model and the parameter estimation algorithm are given, but due to lack of time they could not be implemented. It is concluded that the proposed model and parameter estimation approach have good potential to give satisfactory results, but that it will require additional work. iii iv Preface This thesis is was written as the final part of my two year master’s degree in electronics and information technology with specialization in cybernetics at the University of Oslo1. It contains work done between 2016-01-18 and 2016-05-23, at the University Graduate Center Kjeller (UNIK). The thesis is solely written by the author, though parts of the thesis is based on previous research by other authors. References to the sources of previous research are given where applicable. All illustrations are the authors original work. The authors back- ground is a bachelors degree in electronics and information technology with specialization in automation from Oslo and Akershus University College of Applied Sciences2. I would like to thank my supervisors from the Norwegian Defence Research Establish- ment (FFI): researcher Kim Mathiassen, researcher Magnus Baksaas, and researcher Marius Thoresen, as well as the rest of FFIs unmanned ground vehicle (UGV) project, for providing me the with the subject of this thesis: modelling and parameter estimation of the longitudi- nal dynamics of FFIs experimental UGV, OLAV. I would also like to thank my supervisor at UNIK, professor Oddvar Hallingstad. Magnus Baksaas has been of special help in gathering measurements from OLAV’s instru- mentation. Due to the experimental nature of OLAV this is a task requiring two persons: one person to steer the vehicle using the steering wheel and one person to electronically control the throttle and brake actuators using a gamepad; which was an interesting experience. Finally a special thanks to my girlfriend and family for supporting me, in particular with warm dinner when coming home after a long day of working on my thesis. Kjeller, 2016-05-23 Auby, Per Magnus 1Norwegian: Master i elektronikk og datateknologi med studierettning kybernetikk ved Univeristet i Oslo 2Norwegian: Høgskolen i Oslo og Akershus (HiOA) v vi Contents 1 Introduction 1 1.1 Motivation . 1 1.2 Purpose and goals . 2 1.3 Contribution . 2 1.4 Outline . 2 2 Background 5 2.1 System identification . 5 2.1.1 Deterministic and stochastic state space models . 5 2.1.2 Parameter estimation . 6 2.2 The Unscented Kalman filter . 7 2.2.1 The unscented transformation . 8 2.2.2 UKF algorithm . 8 2.2.3 Computing the log likelihood from UKF state estimate . 10 2.3 Numerical solutions to ordinary differential equations . 11 2.4 Optimization . 11 2.4.1 Particle swarm optimization . 11 2.5 Bond graph modeling . 12 2.5.1 Ports and bonds . 12 2.5.2 Standard elements . 12 2.5.3 Modulated elements . 15 2.5.4 Deriving the equations of motion from a bond graph model . 15 2.6 Ground vehicle modeling . 17 2.6.1 Tire modeling . 17 2.6.2 Engine modeling . 18 2.6.3 Rubber Belt Continuously Variable Transmission . 19 3 Unmanned vehicle platform OLAV 23 3.1 Polaris Ranger XP 900 EPS . 23 3.2 Modifications . 23 3.2.1 Actuators . 23 3.2.2 Instrumentation . 25 4 Vehicle model 27 4.1 The intended use of the model and limitations . 27 4.2 Engine model . 27 4.3 Engine, CVT, and transmission . 29 4.3.1 Engine bond graph model . 30 4.3.2 Transmission and differential bond graph model . 30 4.3.3 CVT bond graph model . 32 4.3.4 Complete drive line model . 32 4.4 Brake system . 35 4.5 Tire model . 36 4.6 Vehicle dynamic model . 38 4.7 Complete vehicle model . 41 4.8 Measurement model . 42 vii viii CONTENTS 4.9 Simulation of the model . 44 5 Parameter estimation algorithm 49 5.1 Coast down experiment filter model . 49 5.2 Full model filter model . 51 5.3 Negative log likelihood criterion . 52 5.4 Optimization strategy . 52 6 Parameter estimation 53 6.1 Choosing parameters to be estimated . 53 6.1.1 Known and approximated parameters . 53 6.1.2 Parameters to be estimated . 55 6.2 Testing the parameter estimation algorithm with simulated data . 57 6.2.1 Tuning the unscented Kalman filters . 57 6.2.2 Parameter sensitivity . 59 6.2.3 Parameters estimated from simulated data . 59 6.2.4 State estimation and simulation using estimated parameters . 61 6.3 Experimental data used for parameter estimation . 61 7 Estimation results and model performance 65 7.1 Estimated parameters . 65 7.2 Simulations using the estimated parameters . 66 7.2.1 Coasting down hill . 66 7.2.2 Driving . 66 7.3 State estimation . 66 8 Discussion 73 8.1 Vehicle dynamics model . 73 8.2 Parameter estimation algorithm and filter models . 74 8.3 Further work . 75 9 Conclusion 77 Appendices 79 A MATLAB code 81 A.1 MATLAB implementation of the filter and simulation models . 81 A.2 Batch processing UKF . 88 A.3 Script for testing the UKF and full vehicle filter model . 89 A.4 Matlab code for parameter estimation . 92 B Derivation of the drive line equations 99 Glossary 101 Bibliography 103 List of Figures 2.1 Bond graph symbol for a bond. 12 2.2 Bond graph symbols for sources of effort and sources of flow. 13 2.3 Bond graph symbols for 0-junctions and 1-junctions. 13 2.4 Bond graph symbol for a resistive element. 13 2.5 Bond graph symbol for a capacitive element. 14 2.6 Bond graph symbol for a inertial element. 14 2.7 Bond graph symbol foe a transformer element. 15 2.8 Bond graph symbol foe a gyrator element. 15 2.9 Causality strokes for sources acting on a general one port element P. 15 2.10 0-junctions and 1-junctions with valid causality configuration and C- and I- elements with. 16 2.11 Typical curve generated by the magic formula for dry asphalt. 17 2.12 Tire forces, torques and velocities. 18 2.13 Simplified diagram of a rubber belt CVT . 20 2.14 Simplified diagram of primary pulley speed sensing device. 20 2.15 Simplified diagram of secondary pulley torque sensing device . 21 3.1 Picture of a standard Polaris Ranger XP 900 EPS . 24 3.2 Picture of off-road light autonomous vehicle (OLAV). 25 4.1 Bond graph model of Polaris Ranger subsystems. 28 4.2 Reduced bond graph model of vehicle subsystem. 29 4.3 Plot of engine torque model . 30 4.4 Simplified diagram of drive line components. 31 4.5 Bond graph model of engine. 31 4.6 Bond graph model of transmission and rear axle. 31 4.7 Equivalent bond graph model of transmission and rear axle without derivative causality.