Modeling and Dynamics Control for Distributed Drive Electric Vehicles 9783658322120, 3658322128

Table of contents :
Abstract
Contents
List of Figures
List of Tables
1 Introduction
1.1 Background
1.1.1 Problems of Conventional Vehicles
1.1.2 Electric Vehicles Development
1.2 Distributed Drive Electric Vehicles
1.3 Cut-in Points in this Research
1.3.1 Stability Based Research
1.3.2 Energy-Saving Based Research
1.4 Research Scope and Objectives
1.5 General Outline of the Book
2 Literature Review
2.1 Vehicle State and Tire-Road Friction Coefficient Estimation
2.1.1 Vehicle State Estimation
2.1.2 Tire-Road Friction Coefficient Estimation
2.2 Direct Yaw Moment Control
2.2.1 Fuzzy Logic Based Yaw Moment Control
2.2.2 PID-Based Direct Yaw Moment Control
2.2.3 LQR-Based Direct Yaw Moment Control
2.2.4 Sliding Mode Based Direct Yaw Moment Control
2.3 Torque Allocation
2.3.1 Rule-Based Control Allocation
2.3.2 Objective Function-based Control Allocation
3 Distributed Drive Electric Vehicle Model
3.1 Vehicle Modeling
3.1.1 Reference Coordinate Systems
3.1.2 Vehicle Equivalent Mechanical Model
3.1.3 Model Assumptions
3.1.4 Vehicle Motion Equations
3.2 Subsystem Modeling
3.2.1 Steering System Model
3.2.2 Wheel Motion Model
3.2.3 Tire Model
3.2.4 Electric Motor
3.2.5 Friction Brake Model
3.3 Driver Model
3.3.1 Target Trajectory and Velocity
3.3.2 Preview-Point Searching Algorithm and Lateral Motion Control
3.3.3 Longitudinal Motion Control
3.4 Vehicle and Driver Model Parameters
3.5 Chapter Summary
4 Vehicle State and Tire-Road Friction Coefficient Estimation
4.1 Vehicle Modeling for State Estimation
4.1.1 Measurement, Control Input, and State Vectors
4.1.2 Planar Vehicle Model
4.1.3 Tire Force Calculation
4.2 Hierarchical Estimation Algorithm Design
4.3 Vehicle State Estimation Based on UKF
4.4 Hybrid Estimator Design for Tire-Road Friction Coefficient
4.4.1 GRNN-Based Estimator Design
4.4.2 Bayesian Theorem-based Estimator Design
4.5 Chapter Summary
5 Direct Yaw Moment Controller Design
5.1 Control Variables
5.2 Desired Value of Control Variables
5.3 Feedforward Control
5.3.1 2-DOF Model Based on Actual Cornering Stiffness
5.3.2 Linear Cornering Stiffness Estimation
5.4 Feedback Control
5.4.1 Conventional Sliding Model Controller
5.4.2 Adaptive Sliding Model Controller
5.5 Chapter Summary
6 Stability-Based Control Allocation Using KKT Global Optimization Algorithm
6.1 Objective Function Description for Stability-Based Torque Allocation
6.2 KKT-Based Global Optimization Algorithm
6.3 Active-Set Allocation Algorithm
6.4 Chapter Summary
7 Energy-Efficient Toque Allocation for Traction and Regenerative Braking
7.1 Introduction
7.2 Energy-Efficient Traction Allocation
7.3 Energy-Efficient Braking Torque Allocation
7.3.1 Braking Stability
7.3.2 Optimization Boundary for Regenerative Braking Force Distribution
7.3.3 Optimal Front-Rear Braking Force Distribution Coefficient β
7.4 Conventional Strategy for Traction and Regenerative Braking Allocation
7.5 Chapter Summary
8 Simulation and Verification on the Proposed Model and Control Strategy
8.1 Simulation and Verification of Vehicle Simulation System
8.1.1 Simulation and Verification on 9-DOF Vehicle Dynamics Model
8.1.2 Simulation and Verification on Driver Model
8.2 Simulation and Analysis of the State and TRFC Estimation Algorithm
8.2.1 Acceleration and Brake Maneuver
8.2.2 Double Lane Change Maneuver
8.3 Simulation and Analysis of the Direct Yaw Moment Controller
8.3.1 Simulative Experiment on a Low-g Maneuver
8.3.2 Simulative Experiment on a High-g Maneuver
8.4 Simulation and Analysis of the Stability-based Torque Allocation
8.4.1 Initial Settings of the Simulative Experiments
8.4.2 Simulation Results and Analysis
8.5 Simulation and Analysis of the Energy-efficient Toque Allocation
8.5.1 Energy-Saving Assessment Criteria
8.5.2 Simulation Results and Analysis
8.6 Chapter Summary
9 Conclusions and Future Work
9.1 Summary and Conclusions
9.2 Future Works
References

Citation preview

Xudong Zhang

Modeling and Dynamics Control for Distributed Drive Electric Vehicles

Modeling and Dynamics Control for Distributed Drive Electric Vehicles

Xudong Zhang

Modeling and Dynamics Control for Distributed Drive Electric Vehicles

Xudong Zhang Beijing, China Dissertation Technische Universitaet Berlin, 2017

ISBN 978-3-658-32212-0 ISBN 978-3-658-32213-7 (eBook) https://doi.org/10.1007/978-3-658-32213-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Carina Reibold This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Abstract

In the recent years, due to the improvements on electric motors and motor control technology, alternative vehicle power system layouts have been considered. One of the latest is known as distributed drive electric vehicles (DDEVs), which consist of four motors that integrated into each drive and can be independently controllable. Such an innovative design provides a packaging advantages, including simple and compact chassis layout, short transmission chain, fast and accurate torque response, and so on. Based on these advantages and features, this book takes stability and energy-saving as cut-in points, and conducts investigations in the following aspects. Detailed vehicle dynamics model is typically of primary foundation for the vehicle stability and energy-saving control research. A closed-loop “humanvehicle-road” system is developed to reflect the human driver’s behavior and vehicle dynamics characteristics. Next, we propose a vehicle state and tire-road friction coefficient estimation algorithm. A hierarchical structure is adopted. An upper estimator is developed based on unscented Kalman filter to estimate vehicle state information, while a hybrid estimation method is applied as the lower estimator to identify the tire-road friction coefficient using general regression neural network (GRNN) and Bayes’ theorem. Subsequently, a direct yaw moment control algorithm is developed addressing the stability issue. The algorithm consists of a feedforward plus feedback component to calculate the desired external yaw moment to achieve the desired vehicle motion. The feedforward control aims at compensating the effect caused by the variation of tire linear cornering stiffness during the tire’s life cycle. Adaptive sliding mode control (ASMC) is used as the feedback component to make the controller robust against systematic uncertainties.

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Abstract

In order to systematically allocate the desired traction and direct yaw moment among four motors, a global optimization algorithm based on a Karush-KuhnTuckert (KKT) conditions is presented. An optimization term is introduced from the perspective of the tire grip margin to construct a stability objective function. According to KKT conditions, the nonlinear objective function is transformed into the eigenvalue problem, thereby making the solving process independent of the initial guess of the optimal solutions. Afterwards, to guarantee the global minima acquisition, two phases of optimization are designed. DDEVs provide great possibilities not only for the improvement of the vehicle dynamics, handling, safety but also energy-saving. An energy-efficient torque allocation (EETA) scheme is proposed for the improvement of traction efficiency and braking energy recovery. In traction conditions, the traction distribution is developed using an objective function of minimizing power loss of four electric motors. In braking conditions, aiming at guaranteeing the braking stability and recapturing the braking energy as much as possible, the changeable distribution of braking torque is obtained based on the ideal front-rear braking force distribution curve, while complying with braking regulations of Economic Commission for Europe (ECE). The proposed allocation scheme does not rely on the complex online computation. It is obtained via an offline optimization procedure and utilized for online allocation by simple interpolation. At last, to verify the effectiveness of the aforementioned vehicle modeling, estimation algorithm, and control systems, simulative experiments based on different maneuvers are carried out. The results demonstrate that the proposed stability and energy-saving control strategy can enhance the vehicle stability and energy efficiency effectively.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Problems of Conventional Vehicles . . . . . . . . . . . . . . . . . . . 1.1.2 Electric Vehicles Development . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.1 Key Advantages of Electric Vehicles . . . . . . . . . . 1.1.2.2 Policy Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.3 Current Status of Electric Vehicles . . . . . . . . . . . . 1.2 Distributed Drive Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Cut-in Points in this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Stability Based Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Energy-Saving Based Research . . . . . . . . . . . . . . . . . . . . . . . 1.4 Research Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 General Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 2 4 5 6 11 11 14 15 16

2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Vehicle State and Tire-Road Friction Coefficient Estimation . . . . 2.1.1 Vehicle State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.1 Non-Model-Based and Kinematic-Model-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Dynamic-Model-Based Approaches . . . . . . . . . . . 2.1.2 Tire-Road Friction Coefficient Estimation . . . . . . . . . . . . . . 2.1.2.1 Cause-based Estimation Method . . . . . . . . . . . . . . 2.1.2.2 Effect-based Estimation Method . . . . . . . . . . . . . . 2.2 Direct Yaw Moment Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Fuzzy Logic Based Yaw Moment Control . . . . . . . . . . . . .

19 19 19

20 20 22 22 23 24 25

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2.2.2 PID-Based Direct Yaw Moment Control . . . . . . . . . . . . . . . 2.2.3 LQR-Based Direct Yaw Moment Control . . . . . . . . . . . . . . 2.2.4 Sliding Mode Based Direct Yaw Moment Control . . . . . . 2.3 Torque Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Rule-Based Control Allocation . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Objective Function-based Control Allocation . . . . . . . . . . . 2.3.2.1 Objective Function Based on Tire Workload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.2 Objective Function Based on Motor Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.3 Other Torque Allocation Methods Based on Objective Function . . . . . . . . . . . . . . . . . . . . . . .

27 28 30 33 33 34 35 37 38

3 Distributed Drive Electric Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Vehicle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Reference Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Vehicle Equivalent Mechanical Model . . . . . . . . . . . . . . . . . 3.1.3 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Vehicle Motion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subsystem Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Steering System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wheel Motion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.1 “Magic Formula” Tire Model . . . . . . . . . . . . . . . . 3.2.3.2 Dugoff Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Electric Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Friction Brake Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Driver Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Target Trajectory and Velocity . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Preview-Point Searching Algorithm and Lateral Motion Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Longitudinal Motion Control . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Vehicle and Driver Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41 42 44 44 45 45 45 46 48 48 50 51 52 53 53

4 Vehicle State and Tire-Road Friction Coefficient Estimation . . . . . . . 4.1 Vehicle Modeling for State Estimation . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Measurement, Control Input, and State Vectors . . . . . . . . . 4.1.2 Planar Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Tire Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 65 67 68 69

54 58 60 61

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4.2 Hierarchical Estimation Algorithm Design . . . . . . . . . . . . . . . . . . . 4.3 Vehicle State Estimation Based on UKF . . . . . . . . . . . . . . . . . . . . . 4.4 Hybrid Estimator Design for Tire-Road Friction Coefficient . . . . 4.4.1 GRNN-Based Estimator Design . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Bayesian Theorem-based Estimator Design . . . . . . . . . . . . 4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 70 76 76 79 82

5 Direct Yaw Moment Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Control Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Desired Value of Control Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 2-DOF Model Based on Actual Cornering Stiffness . . . . . 5.3.2 Linear Cornering Stiffness Estimation . . . . . . . . . . . . . . . . . 5.4 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Conventional Sliding Model Controller . . . . . . . . . . . . . . . . 5.4.2 Adaptive Sliding Model Controller . . . . . . . . . . . . . . . . . . . . 5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 86 87 95 96 97 100 102 103 105

6 Stability-Based Control Allocation Using KKT Global Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Objective Function Description for Stability-Based Torque Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 KKT-Based Global Optimization Algorithm . . . . . . . . . . . . . . . . . . 6.3 Active-Set Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Energy-Efficient Toque Allocation for Traction and Regenerative Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Energy-Efficient Traction Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Energy-Efficient Braking Torque Allocation . . . . . . . . . . . . . . . . . . 7.3.1 Braking Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Optimization Boundary for Regenerative Braking Force Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Optimal Front-Rear Braking Force Distribution Coefficient β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conventional Strategy for Traction and Regenerative Braking Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 110 113 117 119 121 123 125 129 130 133 134 138 139

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8 Simulation and Verification on the Proposed Model and Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Simulation and Verification of Vehicle Simulation System . . . . . . 8.1.1 Simulation and Verification on 9-DOF Vehicle Dynamics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1.1 Quasi-linear Maneuver . . . . . . . . . . . . . . . . . . . . . . 8.1.1.2 Nonlinear Maneuver . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Simulation and Verification on Driver Model . . . . . . . . . . . 8.1.2.1 Road With Five Consecutive Curves . . . . . . . . . . 8.1.2.2 Double Lane Change (DLC) Maneuver . . . . . . . . 8.2 Simulation and Analysis of the State and TRFC Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Acceleration and Brake Maneuver . . . . . . . . . . . . . . . . . . . . 8.2.2 Double Lane Change Maneuver . . . . . . . . . . . . . . . . . . . . . . 8.3 Simulation and Analysis of the Direct Yaw Moment Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Simulative Experiment on a Low-g Maneuver . . . . . . . . . . 8.3.2 Simulative Experiment on a High-g Maneuver . . . . . . . . . 8.4 Simulation and Analysis of the Stability-based Torque Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Initial Settings of the Simulative Experiments . . . . . . . . . . 8.4.2 Simulation Results and Analysis . . . . . . . . . . . . . . . . . . . . . . 8.5 Simulation and Analysis of the Energy-efficient Toque Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Energy-Saving Assessment Criteria . . . . . . . . . . . . . . . . . . . 8.5.2 Simulation Results and Analysis . . . . . . . . . . . . . . . . . . . . . . 8.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 142 143 145 150 150 151 153 153 155 162 163 164 173 174 175 179 180 181 186

9 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 189

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

List of Figures

Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5

Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure

1.6 1.7 1.8 1.9 1.10 1.11 2.1 2.2 2.3 2.4 2.5

Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4

Air pollution in Los Angles (left) and Beijing (right) [8], [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smart-Grid schematic diagram [13] . . . . . . . . . . . . . . . . . . . . Structure comparison between IDVs and DDEVs . . . . . . . . SIM-WIL, Professor Hiroshi Shimizu’s research team at Keio University, 2012 [34] . . . . . . . . . . . . . . . . . . . . . . . . . Three generations PIVO comparison: Upper left: PIVO I (2005); upper right: PIVO II (2007); bottom: PIVO III (2011) [35, p. 3] . . . . . . . . . . . . . . . . . . . . . . . . . . . Michelin Active Wheel [36] . . . . . . . . . . . . . . . . . . . . . . . . . . Structure and components of ESP system [44] . . . . . . . . . . . Classification of direct yaw moment control system . . . . . . Overtaking through oncoming lane . . . . . . . . . . . . . . . . . . . . Motor efficiency map [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . Control system diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the fuzzy logic control system [120] . . . . Structure of fuzzy PI controller [124] . . . . . . . . . . . . . . . . . . Diagram of LQR control system [128] . . . . . . . . . . . . . . . . . Optimal preview acceleration driver model [131] . . . . . . . . Vehicle response to single sine wave steering input (a) on a dry surface; (b) on a slippery surface [147], [148] . . . Schematic diagram of 9-DOF vehicle model . . . . . . . . . . . . Relative position of the three coordinate systems . . . . . . . . Vehicle equivalent structure model and vehicle coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wheel force condition schematic . . . . . . . . . . . . . . . . . . . . . .

2 3 7 7

8 9 12 12 13 15 18 26 28 30 31 36 41 42 43 47

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List of Figures

Figure 3.5 Figure 3.6 Figure 3.7 Figure Figure Figure Figure

3.8 4.1 4.2 4.3

Figure Figure Figure Figure Figure Figure Figure

4.4 4.5 4.6 4.7 4.8 4.9 5.1

Figure 5.2 Figure 5.3

Figure 5.4

Figure 5.5 Figure 5.6

Figure 5.7

Figure 5.8 Figure 6.1

Motor torque external characteristics and efficiency map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coordinate points distribution of the target trajectory . . . . . Diagram of preview point searching and lateral motion control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Driver model framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar vehicle model and coordinate systems . . . . . . . . . . . . Block diagram of the proposed hierarchical estimator . . . . . Example for mean and covariance propagation: (a) actual; (b) first-order linearization (EKF); (c) UT . . . . . . . . Block diagram of UKF estimation . . . . . . . . . . . . . . . . . . . . . Structure of the hybrid estimator for TRFC . . . . . . . . . . . . . GRNN architecture used for TRFC estimation . . . . . . . . . . . Process of GRNN establishment . . . . . . . . . . . . . . . . . . . . . . Selection of the smoothing factor . . . . . . . . . . . . . . . . . . . . . Bayes-based estimation process . . . . . . . . . . . . . . . . . . . . . . . Sideslip angle diagram on a high friction road. (a) lateral force; (b) yaw moment . . . . . . . . . . . . . . . . . . . . . . . . Sideslip angle diagram on a low friction road. (a) lateral force; (b) yaw moment . . . . . . . . . . . . . . . . . . . . . . . . The relationship between the desired yaw rate response and vehicle velocity (solid lines denote front wheel rotation curves and dotted lines designate road adhesion coefficient limit curves) . . . . . . . . . . . . . . . . . . . . . . Comparison of the response of unmodified and modified 2-DOF vehicle models during the J-turn maneuver. (a) steering angle input; (b) lateral acceleration; and (c) yaw rate response . . . . . . . . . . . . . . . . . Structure of the proposed integrated model matching control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nominal yaw velocity from the full four-wheel model with nonlinear new and worn summer and winter tires (steering wheel angel 60°) [206] . . . . . . . . . . . . . . . . . . . . . . Nominal yaw velocity from the full four-wheel model with nonlinear summer and winter tires, with worn tires on the front axle and new tires on rear axle (steering wheel angel 60°) [206] . . . . . . . . . . . . . . . . . . . . . . Weight gain coefficient p . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control architecture of the proposed allocation . . . . . . . . . .

52 54 56 59 66 71 72 75 76 78 79 79 81 88 89

92

93 94

95

96 98 109

List of Figures

Figure 6.2 Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4 Figure 7.5 Figure Figure Figure Figure Figure

7.6 7.7 7.8 7.9 7.10

Figure 7.11 Figure 7.12 Figure 7.13

Figure 7.14 Figure 7.15 Figure 8.1 Figure 8.2

xiii

Block diagram of the proposed KKT-based global optimization algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vehicle geometry and architecture . . . . . . . . . . . . . . . . . . . . . Motor torque external characteristics and efficiency map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity and acceleration profile of NEDC . . . . . . . . . . . . . . Traction energy consumption per km on different driving cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traction energy consumption per km of different speed regions for NEDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal traction distribution coefficient . . . . . . . . . . . . . . . . Schematic of the presented traction allocation . . . . . . . . . . . Forces acting on the vehicle during braking . . . . . . . . . . . . . I-curve and ECE curve for braking force distribution . . . . . Optimization boundary for regenerative braking force distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal front-rear braking force distribution coefficient calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal front–rear braking force distribution coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braking energy consumption and percentage of different speed regions (blue bar and red square denote braking energy consumption and percentage, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the proposed regenerative braking allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of the conventional regenerative braking allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulative experimental platform . . . . . . . . . . . . . . . . . . . . . Simulation results of the vehicle responses to the quasi-linear maneuver. a Simulation results of hand wheel angle input. b Simulation results of vehicle longitudinal velocity. c Simulation results of lateral acceleration. d Simulation results of yaw rate. e Simulation results of vehicle body sideslip angle. f Simulation results of vehicle trajectory . . . . . . . . . . . . . . .

116 124 125 126 127 127 129 130 131 133 134 135 136

137 138 139 142

144

xiv

List of Figures

Figure 8.3

Figure 8.4

Figure 8.5 Figure 8.6

Figure 8.7 Figure 8.8 Figure Figure Figure Figure Figure Figure

8.9 8.10 8.11 8.12 8.13 8.14

Figure 8.15 Figure Figure Figure Figure Figure Figure

8.16 8.17 8.18 8.19 8.20 8.21

Simulation results of the vehicle responses to the nonlinear maneuver. a Simulation results of hand wheel angle input. b Simulation results of vehicle longitudinal velocity. c Simulation results of lateral acceleration. d Simulation results of yaw rate. e Simulation results of vehicle body sideslip angle. f Simulation results of vehicle trajectory . . . . . . . . . . . . . . . Tracking performance of the exemplary maneuver: (a) simulation results of the pre-defined trajectory tracking; (b) simulation results of the pre-defined velocity tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lateral acceleration of the vehicle . . . . . . . . . . . . . . . . . . . . . Tracking performance of the DLC maneuver: (a) simulation results of the pre-defined trajectory tracking; (b) simulation results of the pre-defined velocity tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Target velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input signals of the acceleration and brake maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured signals with Gaussian noises . . . . . . . . . . . . . . . . . Estimated results and errors of the longitudinal speed . . . . . Estimated results and errors of the lateral speed . . . . . . . . . Estimated results and errors of the yaw rate . . . . . . . . . . . . . Estimated results and errors of tire lateral forces . . . . . . . . . Tire-road friction coefficient estimation (The red dotted line is estimated value; continuous black line is reference value.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input signals of the double lane change maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured signals with Gaussian noises . . . . . . . . . . . . . . . . . Estimated results and errors of the longitudinal speed . . . . . Estimated results and errors of the lateral speed . . . . . . . . . Estimated results and errors of the yaw rate . . . . . . . . . . . . . Estimated results and errors of tire lateral forces . . . . . . . . . Tire-road friction coefficient estimation (The red dotted line is the estimated value; continuous blue line is the reference value.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

150 151

152 154 155 155 156 156 157 157

158 159 159 160 160 161 161

162

List of Figures

Figure 8.22

Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure

8.23 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 8.32 8.33 8.34 8.35 8.36

Figure 8.37 Figure 8.38 Figure 8.39 Figure 8.40 Figure 8.41 Figure 8.42 Figure 8.43 Figure 8.44 Figure 8.45 Figure 8.46 Figure 8.47 Figure 8.48

xv

Relationship between slip ratio and normalized longitudinal tire force under different friction coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hand wheel steering angle input for a low-g maneuver . . . Yaw rate for a low-g maneuver . . . . . . . . . . . . . . . . . . . . . . . Yaw rate error compared with desired value . . . . . . . . . . . . . External yaw moment for a low-g maneuver . . . . . . . . . . . . Estimation results of the tire linear cornering stiffness . . . . Driving torque allocation on each wheel . . . . . . . . . . . . . . . . The tracking performance of the target trajectory . . . . . . . . Hand wheel steering input for a high-g maneuver . . . . . . . . Vehicle lateral acceleration comparison . . . . . . . . . . . . . . . . . β-γ phase plane comparison . . . . . . . . . . . . . . . . . . . . . . . . . . External yaw moment input for a high-g maneuver . . . . . . . Driving torque allocation on each wheel using IMMC . . . . Driving torque allocation on each wheel using CSMC; . . . . Target generalized force, including the driver desired traction and direct yaw moment . . . . . . . . . . . . . . . . . . . . . . . Distributed torque on each wheel based on KKT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed torque on each wheel based on AS method . . . Tracking performances of KKT-based and AS allocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tire grip margin of KKT-based and AS methods and the improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the generalized force tracking performance of KKT-based and AS methods . . . . . . . . . . . . Time consumption comparison between KKT-based and AS methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torque output of front motors under EETA and CTA . . . . . Torque output of rear motors under EETA and CTA . . . . . . Comparison of traction and regeneration power under EETA and CTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power improvement in traction and regenerative braking conditions by EETA . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of traction energy consumption and the decreasing rate of different speed regions . . . . . . . . . . . . . . . Comparison of regenerated energy and the improving rate of different speed regions . . . . . . . . . . . . . . . . . . . . . . . .

163 165 165 166 166 167 168 169 170 170 171 171 172 172 174 175 176 176 178 179 180 182 183 183 184 184 185

List of Tables

Table 1.1 Table 2.1

Table 3.1 Table Table Table Table

3.2 4.1 7.1 7.2

Table 7.3 Table Table Table Table Table Table Table

8.1 8.2 8.3 8.4 8.5 8.6 8.7

Table 8.8 Table 8.9 Table 8.10

Comparison of distributed electric drive vehicles and integrated drive vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean motor power losses for tested maneuvers under the even allocation scheme and the efficient wheel torque allocation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of the vehicle target trajectory and velocity information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters used for vehicle and driver modeling . . . . . . . . . . Ranges of input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . NEDC characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of traction and braking energy consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braking energy consumption in the speed range less than 10 km/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Road surface input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Road surface input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vehicle parameters for a low-g maneuver . . . . . . . . . . . . . . . . RMS error of cornering stiffness estimation . . . . . . . . . . . . . . Vehicle parameters for a high-g maneuver . . . . . . . . . . . . . . . Working performance comparison . . . . . . . . . . . . . . . . . . . . . . Tire grip margin comparison between IMMC and CSMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative comparison of the implementation performance for the target generalized force . . . . . . . . . . . . . Comparison of simulation results . . . . . . . . . . . . . . . . . . . . . . Comparison of simulation results . . . . . . . . . . . . . . . . . . . . . .

11

37 53 60 77 126 130 137 154 158 164 167 168 169 173 180 185 185

xvii

1

Introduction

1.1

Background

1.1.1

Problems of Conventional Vehicles

Since Ford launched the car assembly line in 1913, the cost of automobile production has greatly reduced and cars have gradually become a popular commodity. With the increase in car ownership, the conventional internal combustion engine vehicles (ICVs) were confronted with two serious problems: the energy crisis and air pollution. The gasoline and diesel consumed by ICVs are refined from petroleum, which is also known as “black gold.” Petroleum is a precious and non-renewable resource. According to the data published by the U.S Energy Information Administration, in 2014 the proven crude oil reserve is 1,655.561 billion barrels1 . In 2013, the petroleum consumption per day was as high as 91,253 thousand barrels2 . Therefore, the proven oil reserves of the earth will be exhausted in 50 years. In the 1950s, the United States formally proposed automobile exhaust pollution. After encountering strong ultraviolet radiation in the atmosphere, hydrocarbons and nitrogen oxides in car exhaust generate a new type of pollutant, namely photochemical smog, which can induce severe damage to health and lead to huge economic losses [3], [4]. In order to control the environmental pollution caused by car exhaust, governments worldwide have continuously developed more stringent emission standard [5], [6] and automobile manufacturers are also performing research into new forms of energy technology. Nevertheless, vehicle exhaust remains 1 U.S. 2 U.S.

Energy Information Administration (2015) [1]. Energy Information Administration (2015) [2].

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 X. Zhang, Modeling and Dynamics Control for Distributed Drive Electric Vehicles, https://doi.org/10.1007/978-3-658-32213-7_1

1

2

1

Introduction

a main source of air pollution, and in some regions is the primary source [7]. The severity of automobile exhaust pollution in Los Angeles and Beijing is shown as Figure 1.1.

Figure 1.1 Air pollution in Los Angles (left) and Beijing (right) [8], [9]

1.1.2

Electric Vehicles Development

1.1.2.1 Key Advantages of Electric Vehicles Forced by various problems arising from conventional ICVs, governments, automobile manufacturers and research institutions worldwide are making great efforts to push the EV3 forward in order to address the current plight of resources and the environment. Compared to the ICV, the EV has many advantages. It is easier for EV to be equipped with various clean-energy supply systems, for example the fuel cell or super capacitor, which may significantly reduce the fossil oil consumption and environmental pollution. Especially in the city, conventional ICVs may produce a great deal of exhaust when idling at traffic lights or during traffic jams. However, EVs do not have emissions, which effectively reduce the inhalable particle matter, the harmful gas, and the urban photochemical smog. Although the vehicle manufacture and power generation will also cause

3 There

are two types definition of the electric vehicle in the world: the narrow sense and the broad sense. “The narrow sense electric vehicle” means the plug-in, pure-battery drive and fuel-cell electric vehicle. “The broad sense electric vehicle” includes the hybrid vehicle (petroleum-electric hybrid and gas-electric hybrid). The electric vehicle in this article means “the narrow sense electric vehicle”.

1.1 Background

3

air pollution [10], the British Department of Transport stated that the total emission produced during the electric car’s power generation process is at least 40% less than that produced by ICVs [11]. Furthermore, new energy can be utilized by electric vehicles. Currently, an increasing number of multiple accesses to electrical energy exist. Apart from the traditional coal-fired generated power, developments have occurred in a series of novel environmentally-friendly power sources, such as wind power, solar power, hydroelectric power, tidal power and nuclear power. Thus, EVs are able to easily adapt to new energy sources and therefore contribute to environmental protection. In addition, EVs can be easily integrated with smart grid. The smart grid is a fully automatic power transmission network which can observe and control each user’s and grid’s node. The characteristics of the smart grid guarantee that the information and energy that flow from the power plant to the end users is two-way between every node during the entire transmission [12]. It is therefore possible to charge the car during the night when energy consumption is at its lowest, and regulate the energy system comprehensively in the entire region with the power stored in cars during the day if power supply in some locations is inadequate. In this way, electricity can be fully utilized at all times, so that power is efficiently saved and comprehensively deployed. Figure 1.2 shows the comprehensive deployment of the smart grid.

Figure 1.2 Smart-Grid schematic diagram [13]

4

1

Introduction

1.1.2.2 Policy Support As the electric car has many advantages and forms a solution to the problems caused by the ICVs, many governments have introduced policies and regulations to encourage the development of EVs and the related industries [14]–[17]. The American government has introduced numerous policies concerning new EVs and automotive fuel standards; they have simultaneously purchased energysaving cars for daily use. The government has also given financial support to develop the infrastructure required for new sources of energy, which have further led the development of the car to become “smaller in size”, “produce lower amounts of emissions” and “have a lower consumption”. In 2007, the United States (US) Congress passed the “Energy Independence and Security Act”; this Act proposed that by 2025 the scale of investigation into clean energy and energy efficiency technologies will reach US$190 billion, US$20 billion of which will be devoted to EVs and other advanced- technology vehicle fields. The US Executive Office of the President, National Economic Council and the Office of Science and Technology Policy jointly issued “Strategy for American Innovation: Driving towards Sustainable Growth and Quality Jobs.” This publication stated that the government would provide a total of US$750 billion in tax credits in order to stimulate customer purchase of EVs. A further US$2 billion was allocated to the development of battery technologies and to the accessories industry, in order for the lightest, cheapest and maximum-efficiency batteries could be immediately produced. Countries in the European Union have also introduced relevant policies to encourage the growth of EVs [18], [19]. The German government commenced the “National Electric Vehicles Development Program” in 2009; this program focused on giving impetus to the application of the driving motor, control unit, energy transformation unit and energy storage device, as well as provided e500 million to support a battery research center and eight electric vehicle pilot city projects by the end of 2011. Recently, the German ministries of economics, transport and environment reportedly agreed on a new EV support program, the total funding of which is expected to be 1.3 billion euros. Starting in July 2016, e 5,000 incentive will apply to EVs for personal use, while commercial EVs will receive a e 3,000 incentive. Under this program, the government also plans to finance 15,000 new charging stations across the country and to subsidize battery research [20]. In December 2015, the British Department for Transport announced that the plugin car grant was extended until the end of March 2018 to encourage more than 100,000 UK motorists to purchase new energy automobiles. The total funding is expected to be £400 [21]. Norway has made the largest efforts, introducing the

1.1 Background

5

most preferential policies to popularize EVs in Europe. The Norwegian government has released a number of encouraging policies to spread the use of EVs, including relief of registration tax, parking fees, license fees, tolls, VAT and other taxes. Furthermore, the Norwegian Congress has recommended that these policies be extended until 2017 or until registered EVs reach 50,000 [22]. “Made in China 2025,” published by China’s State Council, has clearly pointed out that developing EVs is a very important strategic move which will be able to ease the environmental and energy pressure and push the technological innovation, transformation and upgrading of the automobile industry forward. This document details that the Chinese government will be increasingly supportive of new energy cars and that the policy is sustainable [23]. On August 19th of the same year the Ministry of Finance, State Administration of Taxation and the Ministry of Industry jointly issued the “Catalogue of the Models of Energy-saving and New-energy Vehicles Enjoying Vehicle and Vessel Tax Reduction and Exemption.” This notice contains 1,734 new energy car models that can enjoy a tax reduction. Meanwhile, there are also varying degrees of convenience policies for new energy cars’ license plates and traffic restriction controls [24]. Furthermore, “Guiding Opinions on Accelerating the Promotion and Application of New Energy Vehicles, ” produced by the office of the State Council, proposed a target requiring that the charging infrastructure system, which ought to be intelligent, highly efficient and to allow every car to be accompanied by a charging pile, will be basically constructed by 2020, and that the system can satisfy the charging demand of over five million cars [25].

1.1.2.3 Current Status of Electric Vehicles The technology behind EVs is not new. Early in 1834, Thomas Davenport invented the EV, the battery of which could not be recharged. Therefore, due to the limitation of the battery as well as the motor control technology, the development of EV lagged far behind that of the conventional ICVs. However, the problems that conventional cars bring are increasingly serious, and the significance of EVs is becoming more prominent in pace with these problems. Therefore, research and development in the technology of EVs is being paid an increasing amount of attention [26], [27]. The American Tesla Company, founded in 2013, has enjoyed a fast growth rate. Its MODELS 85D reaches a driving range of 528 kilometers; the MODELS P85D’s fastest 100 kilometers per hour acceleration time is 3.0 seconds. On January 22th 2014, Nissan declared on its official website that the cumulative sales volume of the pure EV “Leaf” had achieved 100,000 in the global market. Many outstanding electric cars have emerged in Germany, for example the 2014 BMW

6

1

Introduction

i3. A pure EV, it innovatively used “Life Drive” carbon fiber passenger cell structure and provides many pure electric car supporting services before, during and after the driving process [28]. With the progress of EV technology, the electric car market has also seen some development [29]. In 2013, the global electric car production exceeded 241,000, a year-on-year growth of 44%; sales exceeded 186,000, a year-on-year rise of 55%. Electric car ownership is also fast increasing, and at more than 350,000 has shown a year-on-year increase of 51%. The US, Japan, France, China, the Netherlands, Germany, Britain, Norway, Canada and Sweden are the top 10 countries in the EV field; these 10 countries have sold a total of 145,000 EVs, resulting in 78% of the global sales of EVs [30].

1.2

Distributed Drive Electric Vehicles

Traditional ICVs and the majority of EVs in the market are integrated drive vehicles (IDVs), which means that the driving power is generated by one engine or motor and transferred to each wheel through the transmission system. However, with the improvements of the electric motor and motor controller technology, many possible changes to the power train configuration have been proposed. One of the latest configurations is known as the distributed drive electric vehicle (DDEV), which employs four motors that are attached to each wheel and independently controlled. This configuration has many advantages, such as quick and accurate torque response, easier torque and RPM acquisition, or independent control for one single motor, and these advantages provide a broad prospect for vehicle dynamic and energy efficiency improvement. Figure 1.3 shows the structure comparison between IDVs and DDEVs. Professor Hiroshi Shimizu’s research team at Keio University, Japan, released the in-wheel motor [31] driving electric passenger car “KAZ” in 2001, which is lithium-battery powered, and also the in-wheel motor driving EV “Eliica” in 2009. In 2011, his team studied the performance and power of the in-wheel motor EV. Its first experimental car, “SIM-LEI,” has a range of 333 kilometers when fully charged [32]. On March 28th 2012, the team introduced the second car, “SIM-WIL” (Figure 1.4), the driving range of which is up to 351 kilometers [33]. An increasing number of vehicle companies then began performing research into the distributed drive electric car, for example the Volvo Recharge and Ford F150, which are equipped with motors provided by the Protean-Electric Company. The Volvo Recharge debuted at the 2007 Frankfurt auto show, and the Ford

1.2 Distributed Drive Electric Vehicles

7

Figure 1.3 Structure comparison between IDVs and DDEVs

pick-up F150 was first exhibited at the 2008 Special Equipment Market Association. Furthermore, the BYD Company has also declared they are developing a distributed drive electric SUV, the S9, stating “Challenge the integrated drive car with the distributed driving way.”

Figure 1.4 SIM-WIL, Professor Hiroshi Shimizu’s research team at Keio University, 2012 [34]

8

1

Introduction

As the populations in cities grow, so does the number of cars, meaning that the city space becomes increasingly limited. Therefore, the compact design of cars has become a trend. As the DDEV’s wheels are driven directly by the attached motors, the DDEV is able to avoid using mechanical transmission components. In this way, the interior space is apparently increased and the car size is effectively reduced, such as that seen in the PIVO produced by Nissan. This car can hold two or three passengers, its wheels are separately driven, and the angles of the cabin and wheels are more flexible, which allows the driver to face forward at all times, making parallel parking and narrow space parking possible. To date, there have been three generations of PIVO. A comparison of these three generations is shown in Figure 1.5.

Figure 1.5 Three generations PIVO comparison: Upper left: PIVO I (2005); upper right: PIVO II (2007); bottom: PIVO III (2011) [35, p. 3]

In addition, many component suppliers have also commenced related research, such as the wheel assembly “Michelin Active Wheel” for DDEVs and the “Proteandrive” solution developed by the Protean Electric Company. Figure 1.6 shows the “Michelin Active Wheel” assembly. Compared to the conventional ICV and IDEV, the key advantages of the DDEV are summarized into following three points.

1.2 Distributed Drive Electric Vehicles

9

Figure 1.6 Michelin Active Wheel [36]

1. Achieve optimal torque distribution The driving force distribution of the traditional ICV is in accordance with a fixed ratio between the front and rear axles, and approximately equal between the two wheels on the same axle due to the limitation of the differential in general cases. In recent years, BMW and Honda have developed their own four-wheel drive control system, the driving force in which can be distributed in different ratios between the two rear-axle wheels; however, the system requires very complicated electronically controlled clutches and is expensive [37]. The DDEV has the potential to adjust torque distribution within the rated range. Each wheel is equipped with a small but powerful motor, which can separately generate the driving or braking torque. Therefore, the optimal driving pattern can be adapted according to the current driving condition and road situation, and the control system gives different commands to each wheel to fulfill the most suitable torque distribution. When the road condition is good and the driving process is steady, the system will choose the economic mode to meet the energy-saving requirements. However, when facing complex conditions such as wet and slippery road, the safety-based distribution pattern can be applied to guarantee vehicle stability and handling under these critical circumstances. 2. Motor torque response is quick, accurate and easily measurable The driving and braking force control system of conventional ICVs consists of an internal combustion engine, gear box, differential and braking system.

10

1

Introduction

The system’s response speed is slow and affected by the delay of the actuators, hydraulic system and electromagnetic valves. For the Vehicle Dynamics Control (VDC) system and similar systems, the real delay time could reach 50–100 ms [38]; the actual control effect will obviously be impacted by the delay. In contrast, for the DDEV, the torque response of the motor attached to the wheel is quick and accurate. The response time of the motor torque is only several millisecond, which is only 1/10–1/100 of the engine and hydraulic braking system response time. Moreover, the uncertainty regarding the torque generated by the motors is much smaller than that of the conventional internal combustion engine and hydraulic braking system. Furthermore, a motor can precisely generate both driving and braking torques, enabling the integration of antilock brake system (ABS) and the traction control system just by software. In addition, relatively accurate motor output can be measured from the motor current. Therefore, a simple “driving force monitor” can be designed which can be used for the real-time observation of the driving force between the tire and road. This advantage will contribute greatly to application of new technologies based on road condition estimation. For example, when road surface conditions become poor, the control system is able to react before the driver. 3. A motor can be integrated to each wheel The traditional ICV chassis arrangement is frequently limited by the size and structure of the engine and transmission system, also limiting the interior space. In contrast, in the DDEVs, each wheel is driven by a small but powerful motor, and high-voltage cables connect the battery and motor which makes possible a more flexible design of the DDEV’s mass distribution and more reasonable axle load allocation. Additionally, this can enlarge the vehicle inside space because mechanical transmission components are not involved4 . Furthermore, it evidently improves the transmission efficiency due to a short transmission chain. The efficiency of the traditional gasoline car is approximately 15% and the diesel car approximately 25%, whereas efficiency of the normal EV reaches at least 80% [26]. The distributed drive power is directly delivered from each motor to the attached wheel and the energy transmission distance obviously becomes shorter; therefore, the efficiency of the DDEV efficiency is the highest at up to 90% [39]. The comparison between the distributed electric vehicle and the integrated drive vehicle is exhibited in Table 1.1 [40]. 4 Feiqiang

et. al. (2009).

1.3 Cut-in Points in this Research

11

Table 1.1 Comparison of distributed electric drive vehicles and integrated drive vehicles ICV

IDEV √

DDEV √

Low

Intermediate

High

Interior space

Fair

Fair

Good

Handling

Fair

Fair

Good

Transmission system

Transmission chain

Transmission chain

Short or without transmission chain

Unsprung mass

Low

Low

High

Cost

Low

Intermediate

High

Energy regeneration

×

Transmission efficiency

1.3

Cut-in Points in this Research

The optimal driving force allocation as well as precise and quick motor torque response can be easily achieved for DDEVs. Based on these advantages and features, we can start the research from the following two points.

1.3.1

Stability Based Research

Vehicle stability is a very important precondition of assuring safety during the driving process. Currently, the VDC system has been widely applied. This is sometimes called the Electronic Stability Control [41]–[43] or Electronic Stability Program (ESP). The VDC system provides each wheel with independent braking torque in order to correct the yaw moment and prevent vehicles from spinning and drifting out in critical situations. This is able to significantly reduce the number of car accidents. As shown in Figure 1.7, the VDC system is usually equipped with steering sensors, wheel sensors and lateral acceleration sensors, among others (Figure 1.8). The VDC system is realized through Direct Yaw Moment Control (DYC) [45], [46], which is an active stability technology. DYC generates a vehicle proper yaw moment by controlling the torque output of the inner and outer wheels, thereby improving the vehicle’s stability. The torque output can be divided into a braking torque and driving torque; therefore, there are three forms of DYC: the brakebased DYC, drive-based DYC and integrated DYC.

12

1

Introduction

Figure 1.7 Structure and components of ESP system [44]

Brake-based DYC

Direct Yaw Moment Control

Drive-based DYC

Integrated DYC

ABS-based Integrated Drive Vehicle

(1) Integrated Drive Vehicles based on Active Differential (2) Distributed Drive Electric Vehicles

Figure 1.8 Classification of direct yaw moment control system

The brake-based DYC is the most common control method for the traditional integrated drive vehicle. The ABS system separates brake cylinders, which makes it easy to control each wheel through independent braking actuators [47]. Consequently, various brake-based DYC systems have been developed and commercialized, which aims to make the vehicle follow the driver’s command and keep the vehicle stable under emergency situations. However, the limitations of brake-based DYC are also apparent, e.g. these systems mainly operate when the handling limit is approaching, but on normal driving conditions, they are unable to generate corrective yaw moment continuously [48]–[50]. Besides, drivers always have the feeling of intrusiveness due to the sudden declarations and loss of vehicle speed led by the activation of these control systems. For example, when drivers want to accelerate, the compulsory deceleration has the possibility of resulting in car accidents due to the inappropriate longitudinal response which doesn’t meet the driver’s intention. As shown in Figure 1.9, the car enters the oncoming lane

1.3 Cut-in Points in this Research

13

when there is an obstacle or a slow-moving vehicle in front, and needs to accelerate to avoid the oncoming car. However, if the auto is controlled by brake-based DYC, it will be forced to slow down, which may cause a serious crash.

Figure 1.9 Overtaking through oncoming lane

For the integrated drive vehicle, drive-based or integrated DYC systems can solve the above problem. These systems make use of active differential technology, through which the driving torque can be separately allocated to the left or right driving axle. In this way, the yaw moment is generated to enhance the handling. Compared to the brake-based DYC, the active differential is able to control the driving torque continuously under different driving circumstances instead of functioning only in critical situations. The active differential device usually consists of a traditional differential and two electronically controlled clutches. Two clutches distribute the driving torque, which is generated by the engine and transmitted via the input shaft to the differential and then to the left and right driving axles. However, it also has the following shortcomings. The vehicle mass and production costs greatly increase due to two electronic clutches and related sensors, and the differential structure becomes very complicated. Furthermore, the clutch is controlled by an electro-hydraulic or electromagnetic system, meaning a relatively lower response speed compared to the pure electric signal control system. The energy loss caused by the clutch friction is also not negligible. Finally, when the speed difference between the left and right axles is too great, it is not possible to transfer engine torque in order to make the torque distribution reasonable. According to the DDEV’s advantages mentioned in Section 1.2, it’s much easier for the DDEV to utilize the driving or integrated DYC control mode in comparison with conventional vehicles. The DYC control system of the DDEV can continuously provide corrective yaw moment to enhance the stability and handling during the entire driving process, without influencing the driver’s speed

14

1

Introduction

sense. At the same time, it is a complete active safety system, which can not only solve the vehicle stability problem but also can observe important vehicle conditions. Neither is the system’s performance affected by the speed difference between left and right wheels, nor is there energy loss caused by the clutch friction. The motor can generate both forward and backward torque, and the backward torque can assist the traditional braking system and aid in power regeneration [51]. A further advantage that should be noted is that the distributed drive can be easily realized through programs alone instead of the need for extra hardware for the DDEV. In other words, although even for the lowest cost basic DDEV, it can be equipped with a high performance VDC system [52].

1.3.2

Energy-Saving Based Research

Apart from the requirement for stability, it is also very important for the DDEV to focus on the energy-saving control. The battery energy density is lower than combustion engines, and therefore the key problem is how to utilize the energy efficiently. The main energy transfer of the DDEV is the motor driving system. If the efficiency of this part can be improved, this will in turn improve the entire energy-saving characteristics of the vehicle. Otherwise, the motor driving system will cause additional heat, which not only damages motor performance but also leads to the extra heat load reducing the motor’s use life. Thus, it is necessary to systematically improve DDEVs efficiency for the perspective of vehicle dynamics. Figure 1.10 is the motor efficiency map. This is an equal-efficiency curve which reflects the continuous mechanical torque/speed characteristic. Based on this figure, the conclusion can be drawn that the efficiency characteristic is quite different in different working conditions and we can qualitatively analyze how the distribution method between the wheels influences the energy savings. 1. Under the condition that low torque and high speed are required, if the driving torque is equally distributed to the four wheels, each wheel only has a low torque output. This may result in poor efficiency in driving power utilization. However, if the allocation mode is changed into to front-wheel drive, each driving wheel’s torque will be doubled and the vehicle efficiency will be improved as well [54]. 2. When there is a high torque requirement in low speed situations, such as in the start-up process, the torque can be distributed to four wheels. According to Figure 1.10, the efficiency point changes from the low efficiency zone of

1.4 Research Scope and Objectives

15

Figure 1.10 Motor efficiency map [53]

low speed and high torque to the high efficiency zone of low speed and lower torque, which is able to promote the system efficiency. Consequently, during the entire driving process, the real-time traction distribution among the four wheels is necessary for the improvement of the vehicle energysaving performance.

1.4

Research Scope and Objectives

Now most of the vehicle control systems are developed based on the ICV dynamic characteristics and, thus, cannot be directly applied on the DDEV. Considering the advantages of DDEVs, the main research objective is to develop a robust and comprehensive vehicle control system for DDEVs with a wide using range. As we know, in passenger vehicle applications, we do not have the luxury to choose the driving condition. It is dependent on driver behavior, road, traffic, even weather conditions. Thus we cannot expect that one single control approach is able to achieve satisfactory performance for all situations. For example, a commuting driving may involve low speed and frequently stop-go driving conditions. In such situations, an energy-saving control would be a preferred choice. On the contrary, at a relatively high speed or due to the poor road or weather conditions, stabilitybased control has to be applied preferentially.

16

1

Introduction

Furthermore, for both energy-saving and stability control, each effective control decision depends on an accurate knowledge of the current vehicle states, such as velocity, acceleration, yaw rate, as well as road-tire friction coefficient. Therefore, vehicle state estimation is very necessary and must be established before the control system design. In addition to the above mentioned, we also need to build the full vehicle model of DDEVs as a simulation platform in order to study the vehicle dynamic and kinematic characteristics. Finally, due to the redundant power train configuration, torque allocation is required to realize the control target. We will conduct our research into how to manipulate multi-actuators synthetically, including selecting appropriate optimization objectives and the corresponding solutions.

1.5

General Outline of the Book

This book consists of 9 chapters. Chapter 1 introduces the research background, the research scope and approaches. Then in Chapter 2, a comprehensive literature research on vehicle state estimation, DYC algorithm and torque allocation is presented. In Chapter 3, a full DDEV model including a 9 DOF vehicle motion model, wheel motion model, friction brake model, motor model, and “MF” and Dugoff tire models is established as the platform to investigate control strategies. Besides, a driver model based on “preview-follower” theory is built to form a closed-loop “human-vehicle-road” system with the full vehicle model. Finally, the validity and accuracy of the proposed model is verified in MATLAB/Simulink environment. In Chapter 4, we will present a vehicle state and tire-road friction coefficient (TRFC) estimation method based on a combination of an unscented Kalman filter (UKF) estimator, general regression neural network (GRNN) and Bayes theorem, in which UKF estimator severs for vehicle dynamic states estimation; GRNN and Bayes theorem are applied for TRFC estimation under small and large excitation as a hybrid estimator, respectively. Finally, the vehicle state estimator and TRFC estimator are able to simultaneously communicate and correct each other throughout the whole estimation process. Chapter 5 takes both yaw rate and side-slip angle as the control variables for vehicle stability. A modified reference model based on the linear 2-DOF bicycle model and tire/road conditions was presented to generate the desired yaw rate. Then, in order to adequately track the desired responses, a novel integrated model matching controller (IMMC) is designed using a feedforward-feedback

1.5 General Outline of the Book

17

compensator. This controller can well cope with the parametric and systematic uncertainties. In Chapter 6, a global optimization algorithm for stability torque allocation is proposed based on Karush-Kuhn-Tucker (KKT). In the objective function of the stability torque allocation, an optimization term is introduced from the perspective of the tire grip margin. Using KKT conditions, the nonlinear programming problem is transformed into a common eigenvalue problem. By this way, we can make the solving process independent of the initial guess of the objective vector. The optimization function is solved leaving out the physical constraint to finish the preliminary optimization. Subsequently, the secondary optimization is conducted under the realistic boundary. Through comparing and excluding, the global optimal solutions can be ultimately obtained. In Chapter 7, an energy-efficient torque allocation (EETA) scheme is developed for the improvement of traction efficiency and braking energy recovery. The proposed allocation scheme does not rely on the complex online computation. An offline calculation based on the motor efficiency map is applied, which ensures the real time performance and makes it easy to implement the algorithm on real vehicles. In Chapter 8, this book verifies the accuracy of the proposed DDEV dynamics model using the high-fidelity CarSim embedded model. On the basis of the DDEV model, simulation research and analysis on the aforementioned vehicle state and TRFC estimation algorithm, integrated model-matching yaw moment control, KKT-based torque allocation, and energy-efficient torque allocation, are performed, successively, to validate their effectiveness and robustness. Chapter 9 concludes the entire study and presents the potential directions for future work. Finally, in the appendix detailed formulas, and complementary tables and figures are included. The proposed control system diagram is shown in Figure 1.11, and it will be presented in detail from Chapter 3 to Chapter 7.

18

Figure 1.11 Control system diagram

1

Introduction

2

Literature Review

In order to achieve the improvement of vehicle stability and economy, DDEV needs to rationally distribute the traction and corrective yaw moment on four independent drive motors on the basis of the accurate estimation of vehicle state and tire-road friction coefficient. Therefore, in this chapter, literature review is carried out from the following three aspects: vehicle state and tire-road friction coefficient estimation, direct yaw moment control techniques, and torque allocation.

2.1

Vehicle State and Tire-Road Friction Coefficient Estimation

2.1.1

Vehicle State Estimation

Safety and energy conservation are two eternal themes for automotive design. Over the past few decades, active safety control has become one of the most effective accident-avoidance technologies [55]. Anti-lock braking system (ABS) [56], traction control system (TCS) [57] and electronic stability program (ESP) etc., all these key components of the active safety control system [58] heavily relies on the accurate knowledge of vehicle states, such as vehicle velocity, yaw rate, side-slip angle, tire forces and so on. Furthermore, not only for safety, vehicle state information is also crucial for the energy management system of electric vehicles [59]–[61]. However, for technical or cost reasons, most of the vehicle dynamics states cannot be measured directly. Notable attempts have been made to work out different estimation approaches. Currently, existing state estimation algorithms are classified into three categories: “non-model-based”, “kinematic-model-based” (KMB) and “dynamicmodel-based” (DMB) approaches. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 X. Zhang, Modeling and Dynamics Control for Distributed Drive Electric Vehicles, https://doi.org/10.1007/978-3-658-32213-7_2

19

20

2

Literature Review

2.1.1.1 Non-Model-Based and Kinematic-Model-Based Approaches “Non-model-based” approaches mainly include fuzzy logic [62], support vector machine [63] and artificial neural network etc. [64], where a vehicle is regarded as a black box and the nonlinear relationship between inputs and outputs is mapped based on a mass of training data [65], [66]. The accuracy of these methods strongly depends on the quality and quantity of the training data [67]. Meanwhile, because of without vehicle models, it is difficult to give out a convincing mathematical explanation of the mapping relationship [68]. Besides, for the KMB approach, it doesn’t contain physical parameters of the vehicle, and therefore this kinematic approach is not affected by parametric uncertainties. A. Hac and M. D. Simpson [69] developed a preliminary estimation of the yaw rate using kinematic relationships, and subsequently fed this initial estimate into a nonlinear observer to generate the final estimate of yaw rate. J. Farrelly and P. Wellstead [70] proposed a vehicle lateral velocity estimator using both physical and kinematic modeling. As for the kinematic part, the estimator has been shown to provide satisfactory performance. However, the drawbacks of the KMB approaches are also apparent. This method only works when the yaw rate is non-zero, because the kinematic model is unobservable when the yaw rate is zero. Additionally, the estimates produced by the kinematic approach highly depend on the sensor accuracy and are relatively noisy [70], the latter of which may affect the use of the estimated states.

2.1.1.2 Dynamic-Model-Based Approaches Unlike the aforementioned methods, DMB approaches include specific and detailed vehicle mathematical model as well as physical parameters, such as mass, yaw moment inertia, CoG (center of gravity) position and so on. The DMB estimation design can be carried out using sliding mode observer [71], Luenberger observer [72], Kalman filter (KF) and its extensions [73], [74], and moving horizon estimation (MHE) [75], etc. We will provide a more detailed review of the KF approaches, since the study of this book is in this area. Kalman filter and its extensions The Kalman filter is widely applied for linear system estimation under the assumption of Gaussian-distributed state and measurement noise. In the nonlinear case of automobile field, the extended Kalman filter (EKF) [76], [77] and unscented Kalman filter (UKF) take up a large percentage. M. Pengov [78] compared EKF and a higher gain observer, the results of which demonstrated the EKF has more robustness and accuracy. T. A. Wenzel et al. [79] proposed the dual EKF, which

2.1 Vehicle State and Tire-Road Friction Coefficient Estimation

21

makes two Kalman filter running in parallel, to estimate vehicle states and parameters simultaneously. However, the main drawback of the EKF is Jacobian matrices calculation, which requires costly computation. Besides, EKF only employs the first order Taylor expansion on nonlinear system, which may lead to great error or even divergence of the filter if the model is seriously nonlinear. Addressing these issues, the UKF utilizes a deterministic sampling technique known as the unscented transform (UT) to pick a minimal set of sample points (called sigma points) around the mean, which is a derivative-free alternative to EKF and meanwhile avoids the expensive update of the Jacobian matrix on each iteration. Meanwhile, UKF is able to achieve higher-order Taylor series expansion accuracy [80], [81]. Thus, UKF should be more suitable for vehicle state estimation application in consideration of the high nonlinearity of vehicle dynamics, particularly during the critical conditions. H. Hamann and J. K. Hedrich [82] developed a robust method to estimate the tire forces using UKF. Simulation results demonstrated a high convergence rate and good stability properties of the UKF-estimator. Based on the piecewise linear tire model, H. Ren et al. [83] achieved vehicle state estimation with UKF. S. Antonov et al. [84] constructed an UKF observer for the vehicle state using an advanced vertical tire load calculation method. Moving horizon estimation Apart from the Kalman filter methods, MHE is another powerful method which is able to provide good solution to nonlinear estimation systems. At each sampling time, MHE estimates the states or parameters by minimizing a cost function over the previous finite time horizon. In addition, constraints can be added to this optimization problem in a very natural way, which makes it possible for MHE to handle constrained processes very well [85]. M. Zanon [86] estimated the friction coefficient in autonomous driving using MHE. T. Kraus [87] proposed an MHEnonlinear model predictive control (NMPC) framework to control field vehicles using an adaptive nonlinear kinematic model. Observer-based approaches Luenberger observer is a full-state observer based on modern control theory. N. Ding et al. [88] designed an extended Luenberger observer to estimate vehicle sideslip angle based on local linearization and the Routh-Hurwitz criterion. Simulation results and analysis demonstrated that the designed observer was with good precision in the estimation of vehicle sideslip angle and good robustness against parameter variations. The sliding-mode observer is a nonlinear observer based on the variable structure control theory [89], [90]. Xu Li et al. [91] presented a virtual sensor based on the nonlinear sliding-mode observer, which could provide vehicle

22

2

Literature Review

state measurement information for the fusion system. Simulation results shows that the proposed nonlinear sliding-mode observer had a better the effectiveness and robustness compared with EKF estimation.

2.1.2

Tire-Road Friction Coefficient Estimation

The existing estimation methods can be roughly divided into two categories: cause-based estimation method and effect-based estimation method. Cause-based methods try to measure factors that lead to changes in friction and then attempt to predict friction coefficient based on past experience or friction models. Effectbased methods, on the other hand, measure the effects that road leads to the vehicle or tires during driving. They then attempt to extrapolate what the limit friction will be based on the vehicle and tires responses [92].

2.1.2.1 Cause-based Estimation Method The cause-based estimation method focuses on the physical characteristics of the tire-road interface. It establishes in advance the estimation model of relative factors on tire-road friction coefficient, directly measures relative factors, and then estimates the coefficients. The factors affecting adhesion coefficients can be divided into three categories: vehicle parameters such as speed, camber angle, and wheel load; tire parameters such as tire material and type, tire pressure and tread depth; road parameters such as slippery conditions (dry, wet, or icy road and others) and road type (asphalt, cement, and gravel and so on) [93]. Many of these parameters are easily obtained, for instance, speed, wheel load, tire type and camber angle. However, the measurement on the road parameters will require special sensors. Y. Sasada et al. [94] developed a road surface sensing system, combining infrastructure sensors installed at roadside with sensors mounted on road patrol vehicle, as well as data from the sensors on the user’s vehicle. F. Holzmann et al. [95] introduced a new methodology for the road friction coefficient estimation by using a camera and a microphone. The algorithm extracts the patterns corresponding of the different road surface friction coefficients depending on the general luminance. These patterns will be matched on the current specimens to deduce the friction coefficient along the road ahead and a confidence value. Yamada [96] conducted a study of the road surface condition detection technique based on the image taken by TV camera attached to the rearview mirror of a vehicle. Their method measured the reflection of the wet road using the optical sensors to estimate the water thickness of the road. M. Riehm

2.1 Vehicle State and Tire-Road Friction Coefficient Estimation

23

[97] presented a method for detection of road surface ice formation based on remote temperature measurements with infrared thermometers. Freezing events were detected based on the temperature dynamics that result from the exothermic reaction as water freezes. Experimental measurements in a climate chamber and in field conditions showed that ice formation often causes a distinct temperature pattern, which is easily identified and distinguished from other temperature fluctuations. However, it is noted that all of these methods need additional expensive sensors, which is the main disadvantages of cause-based estimation approaches.

2.1.2.2 Effect-based Estimation Method Effect-based estimation method focuses on measuring and monitoring vehicle responses, such as tire-tread deformation, tire and vehicle dynamics effects. U. Eichhorn et al. [98]–[100] investigated a method for monitoring and predicting the tire-road friction coefficient using a tire-tread deformation sensor. Experiments have shown that the proposed method can provide accurate estimated friction coefficient. Even so, it still has some disadvantages that it requires a sophisticated instrumented tire with a self-powered, wireless data link to the vehicle. Therefore, it is difficult to be applied on production vehicles for some time in the future. From the perspective of cost and estimation effect, the method based on vehicle dynamic response has received extensive attention in recent years. The method measures the tire and entire vehicle dynamic response and recognizes the corresponding tire-road friction coefficient. K. Li et al. [101] presented an on-board road condition monitoring system, in which the road condition is classified into four grades, normal (μmax ≥ 0.5), slippery (0.3 ≤ μmax < 0.5), very slippery (μmax < 0.3), and rough surface (gravel). A non-linear curve fitting technique is adopted to estimate the maximum tire–road friction coefficient using the “magic formula” tire model. Experimental results demonstrated the feasibility of the road condition monitoring system for detecting slippery and rough road surfaces in close to real-time. L. Ray et al. [102]–[104] applied extended Kalman filtering (EKBF) and Bayesian hypothesis selection to estimate the road friction coefficient of friction on smooth road surfaces. Force estimates are compared statistically with those that result from a nominal tire model to select the most likely friction coefficient from hypothesized values. Both simulation results and results of applying estimation methods to field test data show the good convergence and accuracy of friction coefficient estimates.

24

2

Literature Review

T. Shim et al. [105] proposed model-based road friction estimation approach, which used the reference vehicle model and estimated the road friction value based error signals between the responses of an actual car and a vehicle model. The tire model requires friction information to produce the tire force, the response of the proposed vehicle model can be directly compared to the actual vehicle response to determine the road friction. The verification using actual measurements for different road and driving conditions demonstrated that the proposed approach can well indicate the road surface changes. Bin Huang et al. [106] proposed a limited-memory adaptive extended Kalman Filter (LM-AEKF) to estimate tire-road friction coefficient. The longitudinal and lateral accelerated speed and yaw rate were taken as measurement values of system with Gaussian white noise; corner of steering wheel and normalized force of each wheel were taken as the control input to system. Then, the current tire-road friction coefficient was estimated in real time. Alexandre M. Ribeiro et al. [107] deals with the tire—road friction coefficient estimation problem through the knowledge of lateral tire force. A time delay neural network (TDNN) is adopted for the proposed estimation design. The TDNN aims at detecting road friction coefficient under lateral force excitations avoiding the use of standard mathematical tire models, which may provide a more efficient method with robust results. In addition to the friction estimation work above, which works based force and slip ratio information. Yasui [108] pointed that in some conditions, the selfalignment torque is even more sensitive to the road condition compared with the tire force. Y. Hsu [109] introduced a new algorithm to estimate tire cornering stiffness and coefficient of friction based on the model of aligning moment. P. Luque et al. [108] developed a real-time algorithm to estimate forces and aligning torque in the tires using an extended Kalman filter and neural networks techniques.

2.2

Direct Yaw Moment Control

Direct yaw moment control (DYC), with a function of measuring real-time motion parameters, works to ensure a yaw moment that is determined based on the errors between the measured and the expected parameters, and is then realized by actuators. This allows the realization of the purpose of guaranteeing a stable motion state of the vehicle. For conventional vehicles, this purpose can be realized only through collaboration between the engine and the braking system, which, however, would result in both slow response time and imprecise control

2.2 Direct Yaw Moment Control

25

result. The driving torque at each wheel of the DDEV is independently controlled and can synchronously follow the expected torque. This greatly improves the vehicle stability and safety. A number of DYC methods have been proposed in recent years. Among these, the widely applied methods are based on fuzzy logic controller (FLC) [110], proportional integral derivative (PID) controller [111], [112], linear quadratic regulator (LQR) [113]–[115] and sliding mode controller (SMC) [116], [117]. Detailed information of the above control methods will be presented in the following sections.

2.2.1

Fuzzy Logic Based Yaw Moment Control

Fuzzy logic control (FLC) has become the most prevalent control technique in this field of vehicle control, due to its robustness against disturbances and uncertainties [118]. The vehicle system has strong uncertainties and nonlinearities, the stability of which could be frequently influenced by diverse factors, such as structural parameters, speed, road adhesion coefficient, crosswinds and operations of the drivers. The FLC is described in vague linguistic terms that reflect the subjective assessment of vehicle stability and handling. Therefore, it is a suitable option for vehicle control system design. A yaw moment controller based on FLC was proposed by B.L. Boada et al. [119]. The advantages of fuzzy methods are their simplicity and their good performance in controlling nonlinear systems. Both yaw rate and sideslip angle were set as the control objectives. The actual yaw rate was directly measured by a gyroscope and the sideslip angle was obtained from an estimator based on the 2-DOF linear vehicle model. The developed controller generated the extra yaw moment that was achieved through the difference in the brake forces between the front wheels, so that the target values of yaw rate and sideslip angle were followed by the vehicle. The simulation results show the effectiveness of the proposed control method under different cornering steering maneuvers. However, it should be noted that the effect of the FLC on snow is reduced in compare with dry surfaces. In addition, F. Tahami et al. proposed a fuzzy logic driver-assist stability system for all-wheel drive EVs [120]. The structure of the control system is shown in Figure 2.1. Unlike in other studies, in this book the yaw reference is generated by a well-trained neural network instead of a simplified vehicle model. The neural network as a feedforward component is trained by sine-wave steering with varying speed. Neural reinforcement learning algorithms have been proved to be very useful in situations in which it is not possible to give the system explicit

26

2

Literature Review

instructions regarding how exactly it should go about improving its performance. Furthermore, the presented FLC is used to maintain the yaw rate at a desired value, which has the input signals of the loop error e = r − rd and the error rate of change, the control output of which is the deviation of the applied torque to the motors. The investigation with the computer simulation proved the effects of the control system on improving vehicle performance under severe maneuvering conditions and preventing the vehicle from falling into a catastrophic motion.

Figure 2.1 Block diagram of the fuzzy logic control system [120]

The FLC has a multitude of applications related to DYC. Its structure is relatively static and can lead to barely satisfactory results. FLC does not need a precise mathematical model of the control objective. In addition, it is of great robustness regarding changes to road conditions as well as to the nonlinearity of the contact between tire and road. However, rich experience is always required when designing both the fuzzy rules and the membership function, and the controller also needs to be repeatedly debugged. Furthermore, fuzzy processing with simple control rules will result in a low control precision and poor dynamic quality. In other words, although a high precision requires increasing the magnitude, it will enlarge the search range of rules, lower the speed of decision making and even invalidate the real-time control [121].

2.2 Direct Yaw Moment Control

2.2.2

27

PID-Based Direct Yaw Moment Control

The PID controller is commonly used in industrial control systems [122]. A PID controller continuously calculates an error value as the difference between a desired set point and a measured process variable. The controller attempts to minimize the error over time through adjustment of a control variable. Since the PID controller is a feedback control system with constant parameters, and has no direct knowledge of the process, the overall performance is thus reactive and a compromise; optimal control cannot in general be provided [123]. Furthermore, when PID controllers are used alone, they can give a poor performance when PID loop gains must be reduced. They also have difficulties in the presence of nonlinearities such as vehicle systems, may tradeoff regulation versus response time, do not react to changing process behavior, and lag in response to large disturbances. The alternative improvement method for the control effectiveness is the combination of the PID and an additional controller. Y. Dei Li et al. proposed a fuzzy PI control method, taking into account the complicated vehicle system that was susceptible to outside disturbance [124]. Compared with conventional PI control, the fuzzy PI control can regulate the proportional and integral parameters and improve the response of the system. Moreover, it possesses virtues such as a simple structure, better adaptability and faster response times. The structure of the fuzzy PI controller is shown in Figure 2.2. The fuzzy PI controller consists of a fuzzy controller and PI controller [125]. The input of the PI controller is the yaw rate error e and its rate of change e, while the output is the corrective yaw moment Mz . The sideslip angle error E is the input of the fuzzy controller, and the output is the new proportionality coefficient and integral coefficient, which are applied to the regulation of the PI controller. The entire control unit works cooperatively, and the yaw moment generated by the longitudinal force of each tire impels the yaw rate to reach its desired value. The vehicle handing and stability on low friction road is significantly improved. H. Zhang and J. Wang proposed a generalized PI control method when examining EVs as the study objective [126]. Since it is difficult to obtain an analytic solution for the PI gains, an augmented system was developed to tune the modified PI gains. The PI control is then converted into a state-feedback control for the augmented system. By analyzing the stability and the energy-to-peak performance of the augmented system, the generalized PI control law was defined. Simulation results show the control system’s robustness varies with the longitudinal velocity. The application of PID control is quite flexible. Normally, a good control effect could be obtained under certain working conditions. Nevertheless, it is hard for

28

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Literature Review

Figure 2.2 Structure of fuzzy PI controller [124]

a conventional PID control, which possesses fixed parameters, to adapt to the variable road surface conditions and working conditions. With the combination of the conventional PID controller and other controllers, a better control effect may be enabled.

2.2.3

LQR-Based Direct Yaw Moment Control

The theory of the LQR concerns the operating of a dynamic system at minimum cost. In this case, the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the linear quadratic (LQ) problem. M. Mirzaei presented a new strategy for the minimum usage of external yaw moment based on optimal LQ method to maintain the vehicle actual motions, yaw rate and sideslip angle, close to their desired responses with a minimum external yaw moment [127]. Firstly, a newly desired model for vehicle handling based on the linear 2-DOF and tire/road conditions is presented to be tracked by the DYC system. Since the responses of the linear (or limited linear) and actual nonlinear models of the vehicle at the beginning of the maneuvers are close, the tracking errors and consequently the required external yaw moment will be the lowest when the maneuvers start. Then, in order to adequately track the proposed desired responses using a minimum external yaw moment, the optimal LQ method will be applied to design the yaw moment controller. The LQ optimal problem is

2.2 Direct Yaw Moment Control

29

formulated as follows equation (2.1): J=

1 2

 0

tf

[wb (β − βd )2 + wr (r − rd )2 + wu M2z ] dt

(2.1)

This equation is very easy to solve and implement. The optimality of this control law provides the possibility of reducing the external yaw moment to as low a level as possible, at the cost of some admissible tracking errors. Furthermore, the different DYC control types can be easily derived by designating one of the state weight factors, wb or wr , to be zero. However, the vehicle itself is a strongly nonlinear system and parametric uncertainties are also unavoidable. If the vehicle stability control system is only based on a simplified 2-DOF linear model with fixed parameters, the robustness of the proposed controller is unable to be guaranteed. Z. Yu et al. presented that, if the vehicle stability control systems are only based on the simplified linear vehicle model that ignores the nonlinear characteristics of the tire, the robustness against parameter uncertainties cannot be guaranteed [128]. Therefore, considering the strong nonlinearity of the tires under critical driving conditions, a novel control system adopting gain-scheduling control (GSC) and the LQR theory is proposed to regulate the yaw motion of the vehicles. The basic idea of the algorithm is the key parameter for vehicle stability control, i.e., the cornering stiffness of the front and rear tires, is obtained via online estimation. The vehicle model is then linearized at the equilibrium point using the estimated parameters. Based on the linearized vehicle model, a linear optimal controller is designed (in Yu et al., the LQR algorithm based on model following control (MFC) is adopted). The parameters of the controller are instantaneously updated by the estimated cornering stiffness. In this way, a so-called GSC, which is adaptive to environmental disturbances as well as to the parameter uncertainties of the model, was developed. As shown in Figure 2.3, the total yaw moment is the sum of yaw moment feedforward output M f f and optimal state feedback output M f b . MzT = M f f + M f b

(2.2)

Furthermore, the cornering stiffness of the front and rear tires, which is a major parameter of the controller, is estimated online.

30

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Literature Review

Figure 2.3 Diagram of LQR control system [128]

2.2.4

Sliding Mode Based Direct Yaw Moment Control

SMC (sliding mode control) can achieve the rapid convergence of control variables by designing the sliding mode surface and arrival conditions, without involving a precise mathematical model of the control objective [129], [130]. SMC is also able to enhance the system robustness regarding parametric uncertainties and external disturbances, making it highly applicable to vehicle systems that have high nonlinearity and uncertain parameters. Y. Chen et al. developed a novel direct yaw moment controller based on a driver model using SMC [131]. This article claimed that in an emergency, due to panic, the driver may make an incorrect manoeuver. Therefore, in this book, the driver’s intention is not simply obtained by means of measuring the steering angle. Instead, as shown in Figure 2.4, an optimal preview acceleration driver model is applied to predict the driver’s behavior. This can, to a large extent, reflect the driver’s intention. For the SMC, the yaw rate alone was taken into consideration as the parameter to be limited. Thus, the sliding mode surface is defined as follows: s = ψ˙ − ψ˙ des

(2.3)

Where ψ˙ represents the actual yaw rate and ψ˙ des the desired yaw rate. The desired yaw moment can be obtained and written as follows:

2.2 Direct Yaw Moment Control

31

Figure 2.4 Optimal preview acceleration driver model [131]

M yaw_des = Mˆ yaw_des − ηs

(2.4)

Where M yaw_des indicates the desired yaw moment and η represents the SMC constant. The majority of the sliding mode direct yaw moment controllers possess fixed controller parameters [132]. In order to enhance the robustness against disturbances and uncertainties, C. Fu and M. Hu proposed a new SMC scheme, which contains the two varying controller gains kˆ1 and kˆ2 . The gains are updated according to equations (2.5) and (2.6). Their initial values are both set to zero, indicating that no prior knowledge is required.   k˙ˆ1 = −|s|a f l f

(2.5)

k˙ˆ2 = −|s||ar |lr

(2.6)

Where s represents the error between the actual and the desired yaw rates, and a f and ar stand for the front and rear tire slip angles, respectively. l f and lr denote the distances from the front and rear axles to the center of mass, respectively. Finally, the corrective yaw moment Mz is given as:  s    Mz = Cˆ f α f l f − Cˆ r αr lr + Iz r˙ ∗ + α f l f kˆ1 + |αr |lr kˆ2 sat ∅

(2.7)

The simulation results show that the vehicle equipped with the proposed DYC scheme closely tracks the ideal yaw rate and vehicle path, and considerably outperforms the vehicle with the conventional SMC. Taking the vehicle longitudinal dynamics into consideration, R. Tchamna and I. Youn proposed a new DYC controller based SMC [133]. Most conventional vehicle stability controllers operate on the basis of many simplifying assumptions,

32

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Literature Review

such as constant longitudinal velocity and a small sideslip angle. β≈

vy vx

(2.8)

β˙ ≈

v˙ y vx

(2.9)

This proposed approach achieved the vehicle stability without neglecting its longitudinal dynamics and without making simplifying assumptions about its motion. Let the vehicle sideslip angle be given by the following expression: β = ar ctan

vy vx

(2.10)

Where vx and v y denote the vehicle longitudinal and lateral velocities, respectively. Its derivative, assuming the variation of vx and v y , is given by the following expression:    v˙ x 2 −1 v˙ y ˙ β = 1 + tan β − tanβ vx vx

(2.11)

This expression is utilized in the SMC. The sliding mode surface for yaw rate and sideslip integration consists of the yaw rate error and the vehicle sideslip, assuming zero desired vehicle sideslip. This is given as the following expression: s = ωz − ωzd + ρβ

(2.12)

Both the conventional braking controller and the braking controller based on SMC were constructed by the authors, and their performances were compared in different manoeuvers. The simulation results of the two different controllers show that the proposed algorithm can maintain a lower sideslip angle than the conventional algorithm, proving that the proposed controller is more effective than conventional vehicle stability controllers.

2.3 Torque Allocation

2.3

33

Torque Allocation

DDEVs are equipped with multiple power units, with each one be controllable and of rapid and accurate response compared with internal combustion engines and hydraulic systems. Real-time driving force allocation should be realized based on the current conditions of vehicle and road surface, guaranteeing vehicle safety and dynamic property. Due to the redundant power train configuration, the longitudinal driving performance and the lateral stabilizing performance will be improved. Therefore, the study on driving force control allocation is of major significance to the comprehensive promotion of dynamic performances of DDEVs [134]. The control allocation methods can be classified into two categories: rulebased [135]–[137] and objective function-based approaches. We first review some results of rule-based control allocation research. Then we focus special attention on the category of objective function-based methods, since the proposed approach of this article is in this area.

2.3.1

Rule-Based Control Allocation

The emphasis of the approach is to create allocation rules, apply the rules and finally achieve control objectives. This control allocation is a method of good robust. Osborn [138] proposed two parallel PI controllers for torque distribution; in which, one PI controller determined the front and rear axle torque distribution coefficient based on the deviation of yaw rate, while the other PI controller determined the left and right wheels torque distribution coefficient according to the lateral acceleration deviation. For 6 × 6 in-wheel motor driven vehicles, Jackson and Andrew [139] put forward a fuzzy controller for torque distribution. For torque output of the motors at both the right and left sides of the vehicle they created 49 fuzzy rules according to the yaw rate deviation eωz and its change rate e˙ωz . However, regarding of the ipsilateral motors, the torque distribution is prorated simply. It can be seen that all the above applications have the common limitation. In the control methods they proposed proportional allocation is applied more or less, which reduces the computation effort and enhances the system robustness, however does not give full play to the independently controllable advantages of electric wheels. Y. Chen et al. [140] developed a differential longitudinal force distribution rule. According to the different contribution of different electric wheels to the yaw moment, the driving torque is firstly distributed to the wheel with the largest contribution. When this wheel reaches the tire adhesive limit, then the torque

34

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Literature Review

outputs of the other wheels are adjusted successively with a descending order according to the contribution of each electric wheel. Focusing on the drive capability shortage and instability induced by failure motor(s), a rule-based allocation strategy was proposed [141]–[143] based on DDEVs. Subject to the designed rule, once motor failure occurs, the failed and the contralateral motors are shut down simultaneously to guarantee the vehicle safety. However, this allocation approach is not on the basis of the current vehicle state and it lacks of the consideration of the response speed of the motor, weakens the longitudinal driving performance of the vehicle. W. Chu [144] presented a rule-based coordinated controller aiming at the motor failure issue of DDEVs. The designed controller could apply different torque allocation rules according to different motor failure situations and vehicle states. Trial results verify that, this allocation method can enhance the longitudinal driving capability of the vehicle in low velocity and small steering angle situations, and improve the lateral stability in high velocity or large steering angle situations. Koichi Kondo et al. [145] proposed a driving mode (FWD/4WD) shift rule for a hybrid electric vehicles. On different conditions, different driving modes (FWD/4WD) are adopted in order to improve the vehicle fuel economy; under the normal driving condition, FWD should be adopted, while 4WD is adopted at the starting out stage. Besides, on the conditions that the front wheel skid or the vehicle is making a small turning, the torque is distributed to front and rear wheels such that the vehicle stability is improved. During deceleration, in order to recover energy as much as possible, then four-wheel regeneration mode should be applied.

2.3.2

Objective Function-based Control Allocation

As for DDEVs, driving and braking forces of 4 wheels are both independently controllable [146]. If AFS/4WS or other active steering systems are equipped, then the system redundancy can be further increased. In other words, there are more actuation degrees of freedom then necessary to achieve the desired vehicle motions, which may lead to infinite sets of feasible solutions for the motor output. Therefore, the essence of the objective function-based allocation is to select the optimal solution from the infinite sets of feasible solutions to meet the certain expected performance indicators.

2.3 Torque Allocation

35

2.3.2.1 Objective Function Based on Tire Workload The tire workload is a reflection of the tire stability margin and widely selected as the optimizing index in vehicle stability control. It is defined as

ε=

2 + F2 Ft_i j s_i j

Fz_i j

(2.13)

Taking the tire-road friction coefficient in consideration, the tire workload can be accurately expressed as

ε=

2 + F2 Ft_i j s_i j

μ · Fz_i j

(2.14)

Mokhiamar and Abe [147], [148] proposed a vehicle dynamics integrated control algorithm, which minimizes the weighted sum square of the workload of four wheels. The objective function is given as follows: min J =



Ci j

2 + F2 Ft_i j s_i j 2 Fz_i j

(2.15)

where C ij represents the weighting coefficient of the tire ij. Abe selects three sets of weight allocated coefficients to conduct simulation experiments (Figure 2.5): Case1 Ci j = 1 Case2 C f l = C f r = 2; Crl = Crr = 1; Case3 C f l = C f r = 1; Crl = Crr = 2; As can be seen, vehicle with optimum control case (3) achieves higher stability limit compared to the other cases. This is because increasing the weighting coefficient of the rear tires means increasing its margin to produce shear force. Consequently, the stabilizing yaw moment produced by the rear tires increased results in the improvement of the vehicle stability. On the other hand, increasing the weighting coefficient of the front tires leads to the increase of the vehicle responsiveness, which sometimes causes unstable motion especially in severe driving situations, like case (2). In addition, to avoid the effect led by the lateral force on the optimization results simplifications were made on equation (2.16) in [149] as

36

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(a)

Literature Review

(b)

Figure 2.5 Vehicle response to single sine wave steering input (a) on a dry surface; (b) on a slippery surface [147], [148]

min J =



Ci j

2 Ft_i j 2 μ2 · Fz_i j

(2.16)

However, the above objective function based on the weighted sum square cannot guarantee that every tire is fully used. Sometimes, one tire is already nearing its limit while the workload of others is relatively low, which impedes the further improvement of the vehicle stability. Addressing this drawback, Hori and Nishihara [150] proposed a minimax control allocation that minimizes the utilization of the tire with maximum workload, the objective function of which is given as   2 2 min J = max ε 2f l , εrl , ε 2f r , εrr

(2.17)

Besides, the objective function as shown in the equation (2.18) was presented in [151]. This optimization formulation is aimed at achieving uniform utilization of tire forces, which is able to make full use of four tires and ensure all tires have the same wear condition. Nevertheless, totally six constraint equations need to be satisfied for this approach. Therefore, it only can be implemented on the DDEV with 4WS.

2.3 Torque Allocation

37

min J =

  2  εi2j − E εi2j

(2.18)

2.3.2.2 Objective Function Based on Motor Power Loss A. Pennycott et al. [152] introduced a torque allocation method, which can reduce the loss of motor power losses for a 4WD electric vehicle. In this study, the optimal wheel torque distribution for minimal power losses from electric motor drives is proposed in an offline optimization procedure and then approximated using a simple function for online control allocation. The optimization problem can now be expressed as   min J = Ploss γ , ωm , τwtot

(2.19)

where γ is the torque output ratio; ωm is the motor speed, and τwtot denotes the total wheel torque demand for one side of the vehicle (either left or right). Table 2.1 shows mean motor power losses for tested maneuvers under the even allocation scheme and the efficient wheel torque allocation scheme. Although the energy-efficient scheme has little impact during the straight-ahead maneuver at 50 km/h, the motor power losses at 100 km/h are 3.1% lower when the optimization-based scheme is applied. Furthermore, the motor power losses when the optimal allocation scheme is used are lower than those induced by the even distribution approach by 5.8% for the ramp steer maneuver and 4.5% for the sequence of step steer maneuvers respectively.

Table 2.1 Mean motor power losses for tested maneuvers under the even allocation scheme and the efficient wheel torque allocation scheme Maneuvers

Mean motor power loss (kW) Even allocation

Efficient allocation

Straight-ahead maneuver at 50 km/h

0.519

0.518

Straight-ahead maneuver at 100 km/h

1.28

1.24

Ramp maneuver

2.59

2.44

Sequence of step steer maneuvers

2.92

2.79

38

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Literature Review

2.3.2.3 Other Torque Allocation Methods Based on Objective Function J. Yamakawa et al. [153] proposed an optimal torque allocation algorithm using the changeable principle to minimize the dissipation energy produced by the tires on the contact with the ground. It can reduce the tire slippage and transmit motor torques to the ground efficiently. J. Tjønnas and A. Johansen [154], [155] proposed a dynamic control allocation algorithm that takes actuator constraints and uncertainty in the tire-road friction model into consideration. The objective function is divided into two parts, namely J(u) = J1(u) + J2(u), where the function J1(u) represents the actuator penalty and the function J2(u) is a barrier function of the actuator constraints. J1 (u) = u T u u

J2 (u) = −ωu

4  i=1

ln(λxi − λxmin )− = −ωu

(2.20) 4 

ln(λxi + λxmax )

(2.21)

i=1

where u stands for wheel slip in the longitudinal wheel direction. λxmax and λxmin are wheel-slip constraints, and u and wu are weighting parameters. The reduction of actuators workload is achieved through the objective function J1, and wheel slipping or locking is limited by the objective function J2. H Zhou et al. [156] proposed a hierarchical control system to meet the driving command of driver and keep the vehicle lateral stability. In the upper layer, a nonlinear model predictive control is implemented to solve the nonlinear multiinput multioutput, over-actuated problem. The controller is based on a nonlinear three degree-of-freedom model with nonlinear tire model, considering wheel slips as virtual control input. In the lower layer, the wheel slips are manipulated by a PID controller for generating driving and regenerative braking torques of the independent motors. This proposed controller is tested in a hardware-in-the-loop system with four typical maneuvers, which are constant velocity, accelerating, decelerating, and low road adhesion coefficient situations to show different driver command. The results show that the driver command of longitudinal and lateral motion control are both satisfied.

3

Distributed Drive Electric Vehicle Model

List of Symbols Af

Windward area

Kr f

Front roll steer coefficient

a

Front semi wheelbase

K rr

Rear roll steer coefficient

ax

Vehicle longitudinal acceleration

K∅

Roll stiffness

ay

Vehicle lateral acceleration

l

Longitudinal wheelbase

B

Track width

m

Vehicle mass

b

Rear semi wheelbase

ms

Vehicle sprung mass

Cp

Drag coefficient

mu

Vehicle unsprung mass

C∅

Roll damping

p

Roll rate

c

Distance from sprung mass center S to vehicle coordinate origin(o) in x-direction

R

Tire rolling radius

d

Rolling resistance coefficient

r

Yaw rate

e

Distance from unsprung mass center Tm_i j U to vehicle coordinate origin (o)

Driving torque

Ft_i j

Tire longitudinal force

Tb_i j

Braking torque

Fs_i j

Tire side force

vd

Vehicle target velocity

Fx_i j

Tire force along x axis

vx

Longitudinal velocity

Fy_i j

Tire force along y axis

vy

Lateral velocity

Fz_i j

Tire vertical force

αi j

Tire slip angle

g

Acceleration due to gravity

β

Vehicle body side-slip angle

hg

Height of center of gravity above ground

∅

Roll angle

hs

Distance from c.g. of sprung mass to θ roll axis

Vehicle course angle

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 X. Zhang, Modeling and Dynamics Control for Distributed Drive Electric Vehicles, https://doi.org/10.1007/978-3-658-32213-7_3

39

40

3

Distributed Drive Electric Vehicle Model

Iw_i j

Tire equivalent inertia

ψ

Yaw angle

Ix

Resultant moment of inertia of sprung mass around x-axis

λi j

Tire slip ratio

Ix z

Products of inertia of the sprung mass with respect to xz-plane

δw

Hand wheel steering angle

Iz

Resultant yaw moment of inertia of the vehicle

δf

Front wheels steering angle input

i

Reduction ratio

δW f /r

Steering angle of front/rear wheels

i steer

Steering transmission ratio

ωi j

Tire angular velocity

K RS F

Ratios of front roll stiffness to the total stiffness

ρa

Air density

To validate the vehicle stability and energy-saving control system proposed in the subsequent chapters through comprehensive computer simulations, an accurate full vehicle model that closely reflects the vehicle dynamics in reality is required in order to establish the simulation model in the MATLAB/Simulink environment. The vehicle model should contain sub-models of vehicle body motion, wheel, tire, steering, motor and driver, which have following functions. 1. The model can fully describe vehicle dynamics and motion characteristics, including longitudinal, lateral, rolling and pitching motions and their relationships. 2. The driver model can be used to simulate actual driving operations, including velocity control and path tracking. 3. The driving system model can output the driving or braking torque according to the instructions from the driver or the vehicle control unit, and the electric wheels are independently controllable. In this chapter, a vehicle model that meets the requirements will be established as follows. Firstly, reference coordinate systems will be selected based on the objective of this study. Then, a vehicle equivalent mechanical model and some assumptions (see Section 3.1.3) will be applied to simplify the modeling process. Next, a nine degree of freedom (9-DOF) vehicle dynamic model will be established. This model will be integrated with a motor model, tire model as well as a driver model to allow for a closed-loop simulation of “human-vehicle-road.”

3.1 Vehicle Modeling

3.1

41

Vehicle Modeling

This study focuses not on DOF that are related to ride-comfort, such as pitching DOF and the DOF of the vertical movements of both sprung and unsprung mass, but rather on vehicle stability and energy-saving performance [157]. As indicated in earlier studies, a vehicle model that can fully reflect this stability must be composed of four DOF: longitudinal DOF, lateral DOF, yaw DOF and rolling DOF. It must also take into account factors such as tire nonlinearity and load transfer. Consequently, the vehicle nonlinear model is built, which includes 9-DOF, specifically 4-DOF in the vehicle body (longitudinal, lateral, yaw and rolling), 4-DOF in the wheels and 1-DOF in the steering angle of the front wheels (Figure 3.1).

l

a

b Ft_rl

y

Fs_rl ωrl

vy αrl

B ωrr

Ft_fl β z r

Fs_rr

Ft_rr

ωfl

Fs_fl

x

vx

Fs_fr

αrr

p ωfr

Ft_fr Figure 3.1 Schematic diagram of 9-DOF vehicle model

δ

αfl

αfr δ

42

3

3.1.1

Distributed Drive Electric Vehicle Model

Reference Coordinate Systems

Motions and forces about the vehicle must be described when modeling the vehicle. To increase the clarity of the description as well as simplify the modeling process, this book has introduced several coordinate systems: the global coordinate, vehicle coordinate and wheel coordinate systems. Figure 3.2 shows the relative positions of these coordinate systems.

v

y Y

y(t)

Ψ

β

x

o y yw

Ψ

xw v δ x

X O

x(t)

Global coordinate system Vehicle coordinate system Wheel coordinate system

Figure 3.2 Relative position of the three coordinate systems

1. Global coordinate system X OY Z The global coordinate system, also called the inertial coordinate system, is connected to the ground and is used to describe the absolute vehicle motion state, such as its position, velocity, acceleration and direction. The origin (O) of the coordinate is fixed to the ground. The positive direction of the X-axis is the direction of the initial motion of the vehicle and is always parallel to the ground. The positive direction of the Z-axis is vertical to the ground. The

3.1 Vehicle Modeling

43

positive direction of the Y-axis is determined by using the right-hand rule, as the global coordinate system is a right-handed system. 2. Vehicle coordinate systems The vehicle coordinate system is fixed to the vehicle, and is used to describe a vehicle that consists of sprung and unsprung mass. For the convenience of describing and deducing the motion equations, two vehicle coordinate systems (unsprung mass coordinate system xoyz and sprung mass coordinate system x  o y  z  ) are applied in this research. The motion of the unsprung mass xoyz is the same as that of the whole vehicle. It is worth noting that the origins o and o of the two coordinate systems are the same, and that the x -axis and the x-axis are same to the roll axis of the vehicle. Moreover, the origin (o) of the vehicle coordinate system is the projection point of the mass center of the vehicle onto the roll axis when the vehicle is static. The positive direction of the x-axis is the direction in which the vehicle is heading; the positive direction of the y-axis points to the left side of the driver; the positive direction of the z-axis is vertical to the ground. Figure 3.3 shows the relationships between the coordinate systems xoyz and x  o y  z  .

Figure 3.3 Vehicle equivalent structure model and vehicle coordinate systems

44

3

Distributed Drive Electric Vehicle Model

3. Wheel coordinate system xw ow yw z w The wheel coordinate system xw ow yw z w is mainly used to calculate tire forces. The origin (ow ) of this coordinate system is located at the center of the tire, and each wheel has its own coordinate system. The positive direction of the xw -axis is the direction in which the wheel is heading; the positive direction of the yw -axis is vertical to the rotation plane of the wheel and points to the left side of the wheel; and the positive direction of the z w -axis is vertical to the ground.

3.1.2

Vehicle Equivalent Mechanical Model

The vehicle equivalent mechanical model is shown in Figure 3.3. The rigid body in rolling motion, as shown in the upper part of Figure 3.3, represents the sprung mass of the vehicle, where S is the center of the sprung mass. When the vehicle is static, the vertical distance from point S to the x-axis is h s , and the distance from the origin o to the projection of point S onto the x-axis is c. The roll angle of the sprung mass is ϕ; the rolling rate is p; and the yaw rate is r. The lower part of Figure 3.3 shows the unsprung mass. U represents the center of the unsprung mass and locates on the negative x-axis, and the distance from U to the origin o is e.

3.1.3

Model Assumptions

This study places emphases on the vehicle stability and energy-saving performance with minor factors being ignored. For the sake of simplification, the following assumptions were made: 1. 2. 3. 4. 5. 6.

The The The The The The

steering angles of the two front wheels are the same; tires are always in contact with the ground; vehicle is running on a flat road; yaw rates of the sprung and unsprung masses are the same; roll-center heights of the front and rear axles are the same; sprung mass is symmetrical about plane x-z and y-z.

Based on the above assumptions, the dynamic vehicle model that has 4-DOF of the vehicle body, 4-DOF of the wheels and 1-DOF of the steering angle of the front wheels, was developed.

3.2 Subsystem Modeling

3.1.4

45

Vehicle Motion Equations

The equilibrium equation of the longitudinal forces acting on the entire vehicle is:    1 m v˙ x − v y r = Fx_i j − C p ρa A f vx2 2

(3.1)

The equilibrium equation of lateral forces acting on the entire vehicle is:    ˙ s= Fy_i j m v˙ y + vx r + m s ph

(3.2)

The moment equilibrium equation of the rolling motion of the entire vehicle is:   I x p˙ − I x z r˙ + m s h s v˙ y + vx r = m s gh s sin∅ − K ∅ ∅ − C∅ ∅˙

(3.3)

The moment equilibrium equation of the yawing motion of the entire vehicle is:     Iz r˙ − I x z p˙ =a Fy_ f l + Fy_ f r − b Fy_rl + Fy_rr  B  B Fx_ f l + Fx_rl − Fx_ f r + Fx_rr + 2 2

(3.4)

where Ft is the longitudinal force of the tire; Fs stands for the lateral force of the tire; K ∅ is the roll stiffness of the roll axis; C∅ denotes the roll damping of the roll axis; C p denotes the drag coefficient; ρa is the air density; and A f stands for the windward area of the vehicle; and j = fl, fr, rl and rr represent the front left wheel, front right wheel, rear left wheel and rear right wheel, respectively. The detailed equation derivation on the vehicle translational and rotational motion based on Newton’s laws of motion, D’Alembert’s principle and Euler’s laws of motion can be found in Appendix B.2.

3.2

Subsystem Modeling

3.2.1

Steering System Model

The steering angle of the front wheels can be calculated based on the signal of the steering wheel angle inputted by the driver, with consideration of the transfer characteristics of the steering system, as shown below:

46

3

δf =

Distributed Drive Electric Vehicle Model

δw i steer

(3.5)

Where δ f represents the input steering angle of the front left and right wheels; δw is the steering wheel angle; and i steer denotes the transmission ratio of the steering system. In consideration of the rolling motion of the vehicle, the total steering angle of both front and rear wheels can be expressed as: δW f = δ f + K r f · ∅

(3.6)

δW r = K rr · ∅

(3.7)

Where δW f and δW r are the steering angles of front and rear wheels, respectively; K r f and K rr stand for the front and rear steering coefficient; and ∅ represents the roll angle of the vehicle.

3.2.2

Wheel Motion Model

Differing from conventional IDVs, each wheel of the DDEV is independently controlled by a motor that receives varying signals from the control system. Thus, the rotational equation for each wheel is independent. That is to say, each wheel can have a different moment of inertia, driving/braking torque and angular velocity. Figure 3.4 shows the forces acting on the wheels. The rotation dynamics equation of the wheel can be expressed as: Iw_i j ·

dωi j = Tm_i j · i − Ri j · Ft_i j − d · Fz_i j − Tb_i j dt

(3.8)

Where Iw_i j is the moment of inertia of the wheel; ωi j represents the angular velocity of the wheel; dωi j /dt denotes the angular acceleration of wheel; Tm_i j is the driving torque and Tb_i j is the braking torque; i is the transmission ratio; Fz_i j stands for the normal force from the ground to the tire; d is the distance from the acting point of the normal force to the wheel axis; Ft_i j represents the longitudinal force acting on the tire; R is the actual rotation radius; F p stands for the reactive force from the driving shaft to the tire; and vw denotes central velocity along the headed direction.

3.2 Subsystem Modeling

47

Figure 3.4 Wheel force condition schematic

The longitudinal slip ratio λi j and slip angle αi j of each independent tire are also two important parameters of the tire motion model. There are several definitions of longitudinal slip. In this study, the Pacejka model [158] was employed, in which the longitudinal slip has a determined value in the interval (−∞,∞). When the rotational speed is higher than the longitudinal speed, the longitudinal slip ratio is positive, as calculated by: λi j = −

vw_i j − R · ωi j vw_i j

(3.9)

Generally speaking, the actual rolling radius R varies with rotational speed, load, road slope and curvature [159]. In this study, however, R is deemed a constant. The central velocity is calculated by [160]: 

vw_ f l vw_ f r

   1 = vx + Br cosδW f + v y + ar sinδW f 2     1 = vx − Br cosδW f + v y + ar sinδW f 2

(3.10)

(3.11)

48

3

Distributed Drive Electric Vehicle Model

   1 = vx + Br cosδW r − br − v y sinδW r 2     1 = vx − Br cosδW r − br − v y sinδW r 2 

vw_rl vw_rr

(3.12)

(3.13)

For each tire, there is a slip angle αi j , as shown in the following equations [160]:  α f l = δW f − arctan α f r = δW f − ar ctan αrl = δW r + arctan



vx + 21 Br   v y + ar

vx − 21 Br   br − v y 

αrr = δW r + arctan

3.2.3

ar + v y

vx + 21 Br br − v y vx − 21 Br

(3.14)

(3.15)

(3.16)

 (3.17)

Tire Model

The tire model plays an important role in vehicle dynamics modeling. In this study, the well-known “Magic Formula” tire model and Dugoff tire model, are employed [161], [162].

3.2.3.1 “Magic Formula”Tire Model “Magic Formula” (MF) tire model is proposed by Professor H. B. Pacejka from Delft University of Technology. In general, a MF tire model describes the tire behavior for rather smooth roads (road obstacle wavelength is longer than the tire radius), up to frequency of 8 Hz [163]. The PAC2002, one of the latest “MF” tire models, provides more advanced tire-transient modeling using a contact mass at the contact point with the road. This results in a more realistic dynamic tire model response during large slip, low speed and standstill [164]. Therefore, this tire model can be applied for all generic vehicle handling and stability simulations, including:

3.2 Subsystem Modeling

49

• • • • • •

Steady-state cornering; Single- or double-lane change; Braking or power-off in a turn; Split-mu braking tests; J-turn or other turning maneuvers; ABS braking, when stopping distance is important (not for tuning ABS control strategies); and • Other common vehicle dynamics maneuvers on rather smooth roads (the wavelength of the road obstacles must be longer than the tire radius). When the Pacejka2002 tire model is used, the longitudinal and lateral forces of the tire are calculated based on the thought that a correction factor is involved to correct the tire forces under complex conditions, with consideration of the constraints of “adhesion ellipse” on the longitudinal and lateral forces. The basic idea of using the Pacejka2002 tire model to calculate the longitudinal and lateral forces is that, firstly, the equations for pure longitudinal (lateral) slip working conditions are used to calculate the longitudinal (lateral) force; and secondly, that a correction factor is used to correct the longitudinal (lateral) force under combined slip working conditions (i.e., the constraints of “adhesion ellipse” on the longitudinal (lateral) force are considered) [165]. The basic form of the MF tire model is: (Ft , Fs , M) = Pac2002(λ, α, μ, Fz ) y(x) = D · sin(C · ar ctan{B · x − E · (B · x − ar ctan(B · x))})

(3.18) (3.19)

Y (x) = y(x) + Sv

(3.20)

x = X + Sh

(3.21)

Where Y (x) stands for the longitudinal (lateral) force or aligning torque; X is the slip angle or longitudinal slip rate; B represents a stiffness factor; C is a shape factor; D denotes a peak value; E is a curvature factor; Sh stands for the horizontal shift; and Sv is the vertical shift. See the Appendix B.1 for detailed values.

50

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Distributed Drive Electric Vehicle Model

3.2.3.2 Dugoff Tire Model Although it can accurately describe the tire’s behavior, the modification of the “MF” tire model’s parameters are difficult. Sometimes, however, we need to use different tire parameters to verify our control algorithms, and hence a Dugoff tire model is also applied in this research. Dugoff et al. [161] developed this nonlinear tire model based on the friction ellipse idea in the following form, Cx λ Ft = 1−λ f (S) C y tan(α) Fs = 1−λ f (S)

f (S) =

S(2 − S), S ≤ 1 1, S > 1

(3.22)

where

 √ μFz 1 − εvx λ2 + tan2 α (1 − λ) S= 2 C x2 λ2 + Cα2 tan2 α

(3.23)

and μ is the nominal friction coefficient and ε is the velocity dependency factor. Besides, in Dugoff tire mode tire side force lag is another important aspect and is modeled as a first order time lag, sl F˙sl = Fs τ−F sl Csl R τsl = vx

(3.24)

where τsl and Csl denote side force lag time constant and side force lag constant, respectively. In the two tire models above, to calculate these parameters, the slip ratio λ, the tire slip angle α and the normal reactive force Fz are required, where the slip ratio λ and the tire slip angle α have been described in Section 3.3.2. As for Fz calculation, a quasi-static lateral load transfer due to both lateral acceleration and roll angle should be considered. The front roll stiffness ratio, K RS F , determines the front-rear distribution of the total lateral load transfer [160]. The normal force Fz_i j acting on each tire changes with the vehicle longitudinal or lateral load. These formulae are expressed as: Fz_ f l =

  h g ay mg b ax h g ms · e − + K RS F − · sin∅ 2 l g l B g m·B

(3.25)

3.2 Subsystem Modeling

mg 2

mg a = 2 l

mg a = 2 l

Fz_ f r = Fz_rl Fz_rr

3.2.4

51

 h g ay b ax h g − − K RS F − l g l B g  h g ay ax h g + + (1 − K RS F ) g l B g  h g ay ax h g + − (1 − K RS F ) g l B g

ms · e · sin∅ m·B

 (3.26) 

−

ms · e · sin∅ m·B

−

ms · e · sin∅ m·B

(3.27)  (3.28)

Electric Motor

For DDEVs, each wheel is individually driven by a PMSM (permanent-magnet synchronous motor) through a fixed reduction gear, which is one of the widely applied configurations. The electric motor model is described by E = Rm I + L m

dI + Eb dt

(3.29)

where E b , Rm and L m are the back EMF (Electromotive force), armature resistance and inductance, respectively. The other parameters can be further defined as E b = K v ωm

(3.30)

Tm = K T I

(3.31)

ωm = β R ω

(3.32)

Moreover, the motor current and the terminal voltage have to always subject to I ≤ Imax ,

E ≤ E max

In addition, the motor torque external characteristics and the contour efficiency map are shown in Figure 3.5, which is modified according to the test results from [166]. For calculation simplification, it is assumed that the motor efficiency of regenerative braking condition is identical with that of the driving condition.

52

3 100

Distributed Drive Electric Vehicle Model

5 0.8

0. 7

0.9

0. 6

5

0. 9

0. 75

0. 8

0.9 5

80

8 0.

0.95

0.85

0. 8

5

40

0.9

0.8

0.9 0.85 0.8

Torque(N*m)

20

0.75

0.85 0.8

0

0.

0.85

-20

75

0.6 5

0.8

0.8

0.6

5 0.6

0.7

0.75

0.85

0.7 5

0.9

0.

0. 9

0. 9 5

0. 9

5 0. 8 0. 8 75 0.

0. 7

-40

0. 7

0. 7 5

60

0. 5

5

0.7

8

0.85

-60 0.9 5

500

1000

1500

8

0.8 5

2000

0.9

2500

3000 3500 RPM(r/min)

5

0.

0. 7

0. 6

-100

0.7

-80

4000

4500

5000

5500

6000

Figure 3.5 Motor torque external characteristics and efficiency map

Motor power is the function of torque output, rotational speed, and motor efficiency, thus yielding Pm =

Tm ·n m η(Tm ,n m ) ,

driving condition Tm · n m · η(Tm , n m ), regenerative conditon

(3.33)

where Pm and n m denote the motor power and rotational speed, respectively. The motor efficiency η can be looked-up in Figure 3.5 according to the motor torque and motor rotational speed signals. And these two signals can be obtained from motor controllers.

3.2.5

Friction Brake Model

It should be noted that the regenerative braking torque provided by the motor is quick but limited. Therefore, it is necessary to complement the motor braking torque with friction brake system In this study, it is assumed that electro-mechanic

3.3 Driver Model

53

brakes are equipped to enable a continuous modulation of the braking torque [167]. Here the torque response is approximated as a first-order system [168]. Tf 1 e−δ f s (s) = T f∗ τfs + 1

(3.34)

where T f∗ is the reference friction brake output; T f is the actual output torque; τ f is the time constant; and δ f is the pure delay.

3.3

Driver Model

In this section, a driver model will be established as follows. At first, the target trajectory and velocity will be described using a discrete data table. Then, a preview-point searching algorithm and the vehicle directional controller are presented, respectively. At last, the vehicle speed is controlled based on sliding mode control.

3.3.1

Target Trajectory and Velocity

In this study, we take the target trajectory and velocity as the known information. The technology related trajectory planning is not our research focus. The target trajectory is described using a series of road coordinates in the global coordinate system. The vehicle velocity at the corresponding global coordinate is also the input parameter for the proposed driver model. Therefore, we utilize a table of discrete data set to represent the target trajectory and velocity information as shown in Table 3.1. Table 3.1 Table of the vehicle target trajectory and velocity information Sequence number

X-displacement

Y-displacement

Velocity

s1

X1

Y1

u1

s2

X2

Y2

u2

: :

: :

: :

: :

sn-1

X n-1

Y n-1

un-1

sn

Xn

Yn

un

54

3

Distributed Drive Electric Vehicle Model

Remark: As illustrated in Figure 3.6, the distribution of discrete points should change with the road curvature. For the small curvature part, the discrete coordinate points could be relatively sparse to reduce the computation effort. On the contrary, for the high curvature part, the coordinate points should be denser to increase the accuracy of the road description.

Figure 3.6 Coordinate points distribution of the target trajectory

3.3.2

Preview-Point Searching Algorithm and Lateral Motion Control

The lateral motion control algorithm of the proposed driver model is developed based on “preview-follower” theory [169] using the vehicle lateral acceleration as feedback variable. The driver model is designed, to a large extent, to reflect a human driver’s behavior on the directional control providing a hand wheel input that minimizes the error  f p between the target trajectory F and the vehicle position as shown in Figure 3.7. There are two coordinate systems in Figure 3.7, where (X,Y) and (x,y) represent the point in global coordinate system and vehicle coordinate system, respectively. These two coordinates can be transformed mutually by

3.3 Driver Model

55



 x(i) y(i)

 =

 cos() sin() sin() cos()

X (i) − X (0) Y (i) − Y (0)

 (3.35)

When we have the current vehicle position coordinate (X p0 ,Y p0 ),  f p can be calculated according to the proposed preview-point searching algorithm, which includes the following three steps. 1. Transform the current vehicle position and the data table of target trajectory (described in Section 3.3.1) from the global coordinate system to the vehicle coordinate system. 2. Find a point, which is the closet one behind the current vehicle position, in the data table; and take it as the starting point for the preview point searching. As demonstrated in Figure 3.7, we assume that s0 is the starting point for the last searching and the vehicle velocity is always positive. In this case, we need to search for a point starting from sn , which satisfies the following equation, x(sn ) · x(sn+1 ) ≤ 0

(3.36)

Then obviously, sn is the closet point behind the current vehicle position. It will be taken as the starting point for the preview point searching. 3. Find two adjacent points of the preview point in the transformed trajectory data table using equation (3.37), 

   x(sn+m ) − vx τ p · x(sn+m+1 ) − vx τ p ≤ 0

(3.37)

It is apparent that the preview point is located in the middle of the points sn+m and sn+m+1 . Therefore,  f p is calculated by linear interpolation as  f p = y(sn+m ) +

 y(sn+m+1 ) − y(sn+m )  vx τ p − x(sn+m ) x(sn+m+1 ) − x(sn+m )

(3.38)

Furthermore, the current vehicle lateral displacement as Y p0 in the global coordinate system; meanwhile the lateral displacement of the preview point is Y pm . Through the coordinate system transformation using equation (3.35), we obtain, 1 y pm = y p0 + v y τ p + a y τ p 2 2

(3.39)

56

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Distributed Drive Electric Vehicle Model

Y spm

y

Preview Point

sn+m+1

sn+m sp0 sn+1

sn

1 a yτ 2

F

Δfp 2 p

vyτ p

s0

Ψ

(Xp0,Yp0)

x

vx*τp X

O Figure 3.7 Diagram of preview point searching and lateral motion control

namely, 1  f p = vy τ p + ay τ p 2 2

(3.40)

In this work, the preview time τ p is assumed to be a short and known parameter. Thus, the ideal lateral acceleration can be given in the following form, ay ∗ =

 2   f p − vy τ p τp2

Besides, the lateral acceleration response gain is expressed as

(3.41)

3.3 Driver Model

57

G ay =

ay δw

 = s

li steer



vx 2  1 + K vx 2

where K denotes the understeer gradient. Then, we obtain the ideal hand wheel angle input, δw

∗

    2li steer 1 + K vx 2   f t + τ p − Y (t) − v y τ p = 2 2 vx τ p

(3.42)

However, it should be noted that in practice, the driver has his physiological limitations and the vehicle is a nonlinear system with parametric uncertainties. It is unrealistic to control the vehicle direction just using the calculated steering wheel angel δw ∗ from equation (3.42). The driver’s physiological limitations are mainly embodied in the response delay, including the neural response delay and action response delay. The former is usually expressed by a pure delay as the following transfer function ex p(−td s)

(3.43)

where td is the neural response time constant, usually between 0.2 s to 0.6 s. The action delay describes the driver’s operation process, which is written as a first-order inertia 1 1 + th s

(3.44)

where th is the action response time constant, usually between 0.05 s to 0.20 s. The ideal hand wheel input δw ∗ should be adjusted to δw ∗∗ via the physiological limitation equation (3.43) and (3.44). However, due to the parametric uncertainties and the nonlinearity of the vehicle dynamics, the actual vehicle lateral acceleration under δw ∗∗ is not equal to the ideal lateral acceleration a y ∗ , which will reduce the tracking accuracy to the target trajectory. In order to overcome this issue, in this study, the vehicle lateral acceleration is taken as the feedback variable to minimize the lateral tracking error. The correction to the hand wheel steering angle based lateral acceleration feedback is given as,   H · ay ∗ − ay δw = (1 + th s)

(3.45)

58

3

Distributed Drive Electric Vehicle Model

where H is the correction coefficient. The hand wheel steering angle input to the vehicle is finally calculated as, δw = δw + δw ∗∗

3.3.3

(3.46)

Longitudinal Motion Control

The longitudinal velocity vx can be controlled through the driver desired torque Td based on sliding mode control. Define a switching function s as follows s = vx − vd

(3.47)

Then, s˙ = υ˙ x − υ˙ d     = m1 Fti j cos δi j − Fsi j sin δi j − 21 C p ρa A f vx 2 − m s h s pr + v y r − υ˙ d (3.48) The exponential reaching law is selected and given as s˙ = −ks − εsgn(s)

(3.49)

where ε and k are strictly positive constants. By adjusting these two it can be achieved to guarantee the dynamic quality of the process of sliding mode reaching and weaken the chattering existed in the SMC method. Besides, based on equation (3.8), we have Iw ·

 dωi j dt

= Td · i − R ·



Ft_i j − d ·



Fd_i j

(3.50)

Substituting equation (3.50) into equation (3.49), yields 

Td =

1 2 2 C p ρa A f vx +m s h s pr

  d· Fdi j +Iw · + i

dωi j dt

  +m [υ˙ d −v y r −ks−εsgn(s)] ·R i

(3.51)

3.3 Driver Model

59

In addition, to reduce high-frequency chattering due to the generally nondeterministic switching control signal, a saturation boundary layer is applied. sat(s/φ) is used instead of sgn(s), where φ is a positive constant which determines the thickness of the boundary layer and sat denotes a saturation function s/φ, | |s| ≤ φ sat(s/φ) = sgn(s), | |s| > φ Finally, the driver model framework including the target trajectory and velocity loading, preview-point searching. vehicle directional and speed control, is shown in Figure 3.8.

Lateral Motion Control based on Lateral Acceleration Feedback Lateral Acceleration Feedback

-

ay*

+

1/Gay

2/τS²

-

ĂĂĂĂĂ 1+ths

Δδw

δw* exp(-tds) δw** ĂĂĂĂĂĂ + 1+ths

δw

τs Gay=f(Vx)

Preview-point Searching Algorithm

+

-

Target Trajectory and Velocity V V

; ;