Integrated Vehicle Dynamics and Control
1. Edition May 2016
376 Pages, Hardcover
Practical Approach Book
ISBN:
978-1-118-37999-8
John Wiley & Sons
A comprehensive overview of integrated vehicle system dynamics exploring the fundamentals and new and emerging developments
This book provides a comprehensive coverage of vehicle system dynamics and control, particularly in the area of integrated vehicle dynamics control. The book consists of two parts, (1) development of individual vehicle system dynamic model and control methodology; and (2) development of integrated vehicle dynamic model and control methodology. The first part focuses on investigating vehicle system dynamics and control according to the three directions of vehicle motions, including longitudinal, vertical, and lateral. Corresponding individual control systems, e.g. Anti-lock Brake System (ABS), Active Suspension, Electric Power Steering System (EPS), are introduced and developed respectively.
Particular attention is paid in the second part of the book to develop integrated vehicle dynamic control system. Integrated vehicle dynamics control system is an advanced system that coordinates all the chassis control systems and components to improve the overall vehicle performance including safety, comfort, and economy. Integrated vehicle dynamics control has been an important research topic in the area of vehicle dynamics and control over the past two decades. The research topic on integrated vehicle dynamics control is investigated comprehensively and intensively in the book through both theoretical analysis and experimental study. In this part, two types of control architectures, i.e. centralized and multi-layer, have been developed and compared to demonstrate their advantages and disadvantages.
* Integrated vehicle dynamics control is a hot topic in automotive research; this is one of the few books to address both theory and practice of integrated systems
* Comprehensively explores the research area of integrated vehicle dynamics and control through both theoretical analysis and experimental study
* Addresses a full range of vehicle system topics including tyre dynamics, chassis systems, control architecture, 4 wheel steering system and design of control systems using Linear Matrix Inequality (LMI) Method
Preface xi
1 Basic Knowledge of Vehicle System Dynamics 1
1.1 Traditional Methods of Formulating Vehicle Dynamics Equations 1
1.1.1 Newtonian Mechanics 2
1.1.2 Analytical Mechanics 3
1.2 Dynamics of Rigid Multibody Systems 3
1.2.1 Birth and Development 3
1.2.2 Theories and Methods of Multi-Rigid Body System Dynamics 5
1.2.3 An Example of the Application of Multi-Rigid Body Dynamics Method in Vehicle System Modeling 8
1.3 Flexible Multibody Dynamics 12
References 13
2 Tyre Dynamics 15
2.1 Tyre Models 15
2.1.1 Terminology and Concepts 15
2.1.2 Tyre Model 17
2.2 Tyre Longitudinal Mechanical Properties 19
2.2.1 Tyre Rolling Resistance 20
2.2.2 Road Resistance 21
2.2.3 Tyre Slip Resistance 23
2.2.4 Overall Rolling Resistance of the Tyres 23
2.2.5 Rolling Resistance Coefficient 24
2.3 Vertical Mechanical Properties of Tyres 26
2.4 Lateral Mechanical Properties of Tyres 29
2.5 Mechanical Properties of Tyres in Combined Conditions 30
References 32
3 Longitudinal Vehicle Dynamics and Control 33
3.1 Longitudinal Vehicle Dynamics Equations 33
3.1.1 Longitudinal Force Analysis 33
3.1.2 Longitudinal Vehicle Dynamics Equation 34
3.2 Driving Resistance 35
3.2.1 Aerodynamic Drag 36
3.2.2 Ramp Resistance 36
3.2.3 Inertial Resistance 37
3.3 Anti-lock Braking System 38
3.3.1 Introduction 38
3.3.2 Basic Structure and Working Principle 38
3.3.3 Design of an Anti-lock Braking System 40
3.4 Traction Control System 48
3.4.1 Introduction 48
3.4.2 Control Techniques of TCS 49
3.4.3 TCS Control Strategy 51
3.4.4 Traction Control System Modeling and Simulation 53
3.5 Vehicle Stability Control 54
3.5.1 Basic Principle of VSC 55
3.5.2 Structure of a VSC System 56
3.5.3 Control Methods to Improve Vehicle Stability 59
3.5.4 Selection of the Control Variables 60
3.5.5 Control System Structure 64
3.5.6 The Dynamics Models 64
3.5.7 Setting of the Target Values for the Control Variables 67
3.5.8 Calculation of the Nominal Yaw Moment and Control 68
Appendix 75
References 75
4 Vertical Vehicle Dynamics and Control 77
4.1 Vertical Dynamics Models 77
4.1.1 Introduction 77
4.1.2 Half-vehicle model 78
4.2 Input Models of the Road's Surface 81
4.2.1 Frequency-domain Models 81
4.2.2 Time Domain Models 83
4.3 Design of a Semi-active Suspension System 84
4.3.1 Dynamic Model of a Semi-active Suspension System 85
4.3.2 Integrated Optimization Design of a Semi-active Suspension System 87
4.3.3 The Realization of the Integrated Optimization Method 88
4.3.4 Implementation of the Genetic Algorithm 90
4.3.5 LQG Controller Design 91
4.3.6 Simulations and Result Analysis 92
4.4 Time-lag Problem and its Control of a Semi-active Suspension 95
4.4.1 Causes and Impacts of Time-lag 96
4.4.2 Time-lag Variable Structure Control of an MR (Magneto-Rheological) Semi-active Suspension 98
4.4.3 Simulation Results and Analysis 103
4.4.4 Experiment Validation 108
4.5 Design of an Active Suspension System 110
4.5.1 The Dynamic Model of an Active Suspension System 111
4.5.2 Design of the Control Scheme 112
4.5.3 Multi-objective Mixed H2/H infinity Control 114
4.5.4 Simulation Study 116
4.6 Order-reduction Study of an Active Suspension Controller 119
4.6.1 Full Vehicle Model with 7 Degrees of Freedom 122
4.6.2 Controller Design 124
4.6.3 Controller Order-reduction 125
4.6.4 Simulation Analysis 129
References 133
5 Lateral Vehicle Dynamics and Control 135
5.1 General Equations of Lateral Vehicle Dynamics 135
5.2 Handling and Stability Analysis 137
5.2.1 Steady State Response (Steady Steering) 137
5.2.2 Transient Response 140
5.2.3 The Frequency Response Characteristics of Yaw Rate 144
5.3 Handling Stability Evaluations 144
5.3.1 Subjective Evaluation Contents 144
5.3.2 Experimental Evaluation Contents 144
5.4 Four-wheel Steering System and Control 145
5.4.1 Control Objectives of the Four-wheel Steering Vehicle 146
5.4.2 Design of a Four-wheel Steering Control System 146
5.4.3 Multi-body Dynamics Modeling of a Four-wheel Steering Vehicle 150
5.4.4 Simulation Results and Analysis 152
5.5 Electric Power Steering System and Control Strategy 152
5.5.1 EPS Model 154
5.5.2 Steering Torque Model of the Steering Pinion 155
5.5.3 The Estimation Algorithm of the Road Adhesion Coefficient 159
5.5.4 Design of the Control Strategy 160
5.5.5 Simulation and Analysis 163
5.5.6 Experimental Study 165
5.6 Automatic Lane Keeping System 167
5.6.1 Control System Design 167
5.6.2 Desired Yaw Rate Generation 168
5.6.3 Desired Yaw Rate Tracking Control 171
5.6.4 Simulation and Analysis 173
5.6.5 Experimental Verification 175
References 180
6 System Coupling Mechanism and Vehicle Dynamic Model 183
6.1 Overview of Vehicle Dynamic Model 183
6.2 Analysis of the Chassis Coupling Mechanisms 184
6.2.1 Coupling of Tyre Forces 184
6.2.2 Coupling of the Dynamic Load Distribution 185
6.2.3 Coupling of Movement Relationship 185
6.2.4 Coupling of Structure Parameters and Control Parameters 186
6.3 Dynamic Model of the Nonlinear Coupling for the Integrated Controls of a Vehicle 186
6.4 Simulation Analysis 191
6.4.1 Simulation 191
6.4.2 Results Analysis 192
References 199
7 Integrated Vehicle Dynamics Control: Centralized Control Architecture 201
7.1 Principles of Integrated Vehicle Dynamics Control 201
7.2 Integrated Control of Vehicle Stability Control Systems (VSC) 204
7.2.1 Sideslip Angle Control 204
7.2.2 Estimation of the Road Adhesion Coefficient 218
7.3 Integrated Control of Active Suspension System (ASS) and Vehicle Stability Control System (VSC) using Decoupling Control Method 226
7.3.1 Vehicle Dynamic Model 227
7.3.2 2-DOF Reference Model 228
7.3.3 Lateral Force Model 229
7.3.4 Integrated System Control Model 229
7.3.5 Design of the Decoupling Control System 230
7.3.6 Calculation of the Relative Degree 230
7.3.7 Design of the Input/Output Decoupling Controller 232
7.3.8 Design of the Disturbance Decoupling Controller 233
7.3.9 Design of the Closed Loop Controller 233
7.3.10 Design of the ASS Controller 233
7.3.11 Design of the VSC Controller 234
7.3.12 Simulation Investigation 236
7.3.13 Experimental Study 240
7.4 Integrated Control of an Active Suspension System (ASS) and Electric Power Steering System (EPS) using H Control Method 240
7.4.1 Vehicle Dynamic Model 243
7.4.2 EPS Model 243
7.4.3 Design of Integrated Control System 245
7.4.4 Simulation Investigation 246
7.5 Integrated Control of Active Suspension System (ASS) and Electric Power Steering System (EPS) using the Predictive Control Method 249
7.5.1 Designing a Predictive Control System 249
7.5.2 Boundary Conditions 250
7.5.3 Simulation Investigation 251
7.6 Integrated Control of the Active Suspension System (ASS) and Electric Power Steering System (EPS) using a Self-adaptive Control Method 253
7.6.1 Parameter Estimation of a Multivariable System 253
7.6.2 Design of the Multivariable Generalized Least Square Controller 254
7.6.3 Design of the Multivariable Self-adaptive Integrated Controller 255
7.6.4 Simulation Investigation 255
7.7 Integrated Control of an Active Suspension System (ASS) and Electric Power Steering System (EPS) using a Centralized Control Method 256
7.7.1 Centralized Controller Design 256
7.7.2 Simulation Investigation 259
7.8 Integrated Control of the Electric Power Steering System (EPS) and Vehicle Stability Control (VSC) System 264
7.8.1 Interactions Between EPS and VSC 264
7.8.2 Control System Design 264
7.8.3 Dynamic Distribution of Tyre Forces 265
7.8.4 Design of a Self-aligning Torque Controller 267
7.8.5 Simulation Investigation 270
7.9 Centralized Control of Integrated Chassis Control Systems using the Artificial Neural Networks (ANN) Inverse System Method 271
7.9.1 Vehicle Dynamic Model 272
7.9.2 Design of the Centralized Control System 273
7.9.3 Simulation Investigation 278
References 281
8 Integrated Vehicle Dynamics Control: Multilayer Coordinating Control Architecture 283
8.1 Multilayer Coordinating Control of Active Suspension System (ASS) and Active Front Steering (AFS) 283
8.1.1 AFS Model 284
8.1.2 Controller Design 285
8.1.3 Simulation Investigation 289
8.2 Multilayer Coordinating Control of Active Suspension System (ASS) and Electric Power Steering System (EPS) 291
8.2.1 System Modeling 291
8.2.2 Controller Design 295
8.2.3 Simulation Investigation 298
8.3 Multilayer Coordinating Control of an Active Suspension System (ASS) and Anti-lock Brake System (ABS) 299
8.3.1 Coordinating Controller Design 300
8.3.2 Simulation Investigation 301
8.4 Multilayer Coordinating Control of the Electric Power Steering System (EPS) and Anti-lock Brake System (ABS) 303
8.4.1 Interactions between the EPS System and ABS 304
8.4.2 Coordinating Controller Design 305
8.4.3 Simulation Investigation 306
8.5 Multi-layer Coordinating Control of the Active Suspension System (ASS) and Vehicle Stability Control (VSC) System 308
8.5.1 System Model 308
8.5.2 Multilayer Coordinating Controller Design 308
8.5.3 Simulation Investigation 313
8.6 Multilayer Coordinating Control of an Active Four-wheel Steering System (4WS) and Direct Yaw Moment Control System (DYC) 315
8.6.1 Introduction 315
8.6.2 Coordinating Control of DYC and 4WS 316
8.6.3 Simulation Investigation 320
8.7 Multilayer Coordinating Control of Integrated Chassis Control Systems 321
8.7.1 Introduction 321
8.7.2 Controller Design 322
8.7.3 Simulation and Experiment Investigations 327
8.8 Multilayer Coordinating Control of Integrated Chassis Control Systems using Game Theory and Function Distribution Methods 330
8.8.1 Structure of the Chassis Control System 331
8.8.2 Design of the Suspension Subsystem Controller 331
8.8.3 Design of the Steering Subsystem Controller 332
8.8.4 Design of the Braking Subsystem Controller 333
8.8.5 Design of the Upper Layer Controller 333
8.8.6 Simulation Investigation 335
References 337
9 Perspectives 339
9.1 Models of Full Vehicle Dynamics 339
9.2 Multi-sensor Information Fusion 340
9.3 Fault-tolerant Control 340
9.4 Active and Passive Safety Integrated Control Based on the Function Allocation Method 341
9.5 Design of System Integration for a Vehicle 344
9.6 Assumption about the Vehicle of the Future 345
References 346
Index 347
1 Basic Knowledge of Vehicle System Dynamics 1
1.1 Traditional Methods of Formulating Vehicle Dynamics Equations 1
1.1.1 Newtonian Mechanics 2
1.1.2 Analytical Mechanics 3
1.2 Dynamics of Rigid Multibody Systems 3
1.2.1 Birth and Development 3
1.2.2 Theories and Methods of Multi-Rigid Body System Dynamics 5
1.2.3 An Example of the Application of Multi-Rigid Body Dynamics Method in Vehicle System Modeling 8
1.3 Flexible Multibody Dynamics 12
References 13
2 Tyre Dynamics 15
2.1 Tyre Models 15
2.1.1 Terminology and Concepts 15
2.1.2 Tyre Model 17
2.2 Tyre Longitudinal Mechanical Properties 19
2.2.1 Tyre Rolling Resistance 20
2.2.2 Road Resistance 21
2.2.3 Tyre Slip Resistance 23
2.2.4 Overall Rolling Resistance of the Tyres 23
2.2.5 Rolling Resistance Coefficient 24
2.3 Vertical Mechanical Properties of Tyres 26
2.4 Lateral Mechanical Properties of Tyres 29
2.5 Mechanical Properties of Tyres in Combined Conditions 30
References 32
3 Longitudinal Vehicle Dynamics and Control 33
3.1 Longitudinal Vehicle Dynamics Equations 33
3.1.1 Longitudinal Force Analysis 33
3.1.2 Longitudinal Vehicle Dynamics Equation 34
3.2 Driving Resistance 35
3.2.1 Aerodynamic Drag 36
3.2.2 Ramp Resistance 36
3.2.3 Inertial Resistance 37
3.3 Anti-lock Braking System 38
3.3.1 Introduction 38
3.3.2 Basic Structure and Working Principle 38
3.3.3 Design of an Anti-lock Braking System 40
3.4 Traction Control System 48
3.4.1 Introduction 48
3.4.2 Control Techniques of TCS 49
3.4.3 TCS Control Strategy 51
3.4.4 Traction Control System Modeling and Simulation 53
3.5 Vehicle Stability Control 54
3.5.1 Basic Principle of VSC 55
3.5.2 Structure of a VSC System 56
3.5.3 Control Methods to Improve Vehicle Stability 59
3.5.4 Selection of the Control Variables 60
3.5.5 Control System Structure 64
3.5.6 The Dynamics Models 64
3.5.7 Setting of the Target Values for the Control Variables 67
3.5.8 Calculation of the Nominal Yaw Moment and Control 68
Appendix 75
References 75
4 Vertical Vehicle Dynamics and Control 77
4.1 Vertical Dynamics Models 77
4.1.1 Introduction 77
4.1.2 Half-vehicle model 78
4.2 Input Models of the Road's Surface 81
4.2.1 Frequency-domain Models 81
4.2.2 Time Domain Models 83
4.3 Design of a Semi-active Suspension System 84
4.3.1 Dynamic Model of a Semi-active Suspension System 85
4.3.2 Integrated Optimization Design of a Semi-active Suspension System 87
4.3.3 The Realization of the Integrated Optimization Method 88
4.3.4 Implementation of the Genetic Algorithm 90
4.3.5 LQG Controller Design 91
4.3.6 Simulations and Result Analysis 92
4.4 Time-lag Problem and its Control of a Semi-active Suspension 95
4.4.1 Causes and Impacts of Time-lag 96
4.4.2 Time-lag Variable Structure Control of an MR (Magneto-Rheological) Semi-active Suspension 98
4.4.3 Simulation Results and Analysis 103
4.4.4 Experiment Validation 108
4.5 Design of an Active Suspension System 110
4.5.1 The Dynamic Model of an Active Suspension System 111
4.5.2 Design of the Control Scheme 112
4.5.3 Multi-objective Mixed H2/H infinity Control 114
4.5.4 Simulation Study 116
4.6 Order-reduction Study of an Active Suspension Controller 119
4.6.1 Full Vehicle Model with 7 Degrees of Freedom 122
4.6.2 Controller Design 124
4.6.3 Controller Order-reduction 125
4.6.4 Simulation Analysis 129
References 133
5 Lateral Vehicle Dynamics and Control 135
5.1 General Equations of Lateral Vehicle Dynamics 135
5.2 Handling and Stability Analysis 137
5.2.1 Steady State Response (Steady Steering) 137
5.2.2 Transient Response 140
5.2.3 The Frequency Response Characteristics of Yaw Rate 144
5.3 Handling Stability Evaluations 144
5.3.1 Subjective Evaluation Contents 144
5.3.2 Experimental Evaluation Contents 144
5.4 Four-wheel Steering System and Control 145
5.4.1 Control Objectives of the Four-wheel Steering Vehicle 146
5.4.2 Design of a Four-wheel Steering Control System 146
5.4.3 Multi-body Dynamics Modeling of a Four-wheel Steering Vehicle 150
5.4.4 Simulation Results and Analysis 152
5.5 Electric Power Steering System and Control Strategy 152
5.5.1 EPS Model 154
5.5.2 Steering Torque Model of the Steering Pinion 155
5.5.3 The Estimation Algorithm of the Road Adhesion Coefficient 159
5.5.4 Design of the Control Strategy 160
5.5.5 Simulation and Analysis 163
5.5.6 Experimental Study 165
5.6 Automatic Lane Keeping System 167
5.6.1 Control System Design 167
5.6.2 Desired Yaw Rate Generation 168
5.6.3 Desired Yaw Rate Tracking Control 171
5.6.4 Simulation and Analysis 173
5.6.5 Experimental Verification 175
References 180
6 System Coupling Mechanism and Vehicle Dynamic Model 183
6.1 Overview of Vehicle Dynamic Model 183
6.2 Analysis of the Chassis Coupling Mechanisms 184
6.2.1 Coupling of Tyre Forces 184
6.2.2 Coupling of the Dynamic Load Distribution 185
6.2.3 Coupling of Movement Relationship 185
6.2.4 Coupling of Structure Parameters and Control Parameters 186
6.3 Dynamic Model of the Nonlinear Coupling for the Integrated Controls of a Vehicle 186
6.4 Simulation Analysis 191
6.4.1 Simulation 191
6.4.2 Results Analysis 192
References 199
7 Integrated Vehicle Dynamics Control: Centralized Control Architecture 201
7.1 Principles of Integrated Vehicle Dynamics Control 201
7.2 Integrated Control of Vehicle Stability Control Systems (VSC) 204
7.2.1 Sideslip Angle Control 204
7.2.2 Estimation of the Road Adhesion Coefficient 218
7.3 Integrated Control of Active Suspension System (ASS) and Vehicle Stability Control System (VSC) using Decoupling Control Method 226
7.3.1 Vehicle Dynamic Model 227
7.3.2 2-DOF Reference Model 228
7.3.3 Lateral Force Model 229
7.3.4 Integrated System Control Model 229
7.3.5 Design of the Decoupling Control System 230
7.3.6 Calculation of the Relative Degree 230
7.3.7 Design of the Input/Output Decoupling Controller 232
7.3.8 Design of the Disturbance Decoupling Controller 233
7.3.9 Design of the Closed Loop Controller 233
7.3.10 Design of the ASS Controller 233
7.3.11 Design of the VSC Controller 234
7.3.12 Simulation Investigation 236
7.3.13 Experimental Study 240
7.4 Integrated Control of an Active Suspension System (ASS) and Electric Power Steering System (EPS) using H Control Method 240
7.4.1 Vehicle Dynamic Model 243
7.4.2 EPS Model 243
7.4.3 Design of Integrated Control System 245
7.4.4 Simulation Investigation 246
7.5 Integrated Control of Active Suspension System (ASS) and Electric Power Steering System (EPS) using the Predictive Control Method 249
7.5.1 Designing a Predictive Control System 249
7.5.2 Boundary Conditions 250
7.5.3 Simulation Investigation 251
7.6 Integrated Control of the Active Suspension System (ASS) and Electric Power Steering System (EPS) using a Self-adaptive Control Method 253
7.6.1 Parameter Estimation of a Multivariable System 253
7.6.2 Design of the Multivariable Generalized Least Square Controller 254
7.6.3 Design of the Multivariable Self-adaptive Integrated Controller 255
7.6.4 Simulation Investigation 255
7.7 Integrated Control of an Active Suspension System (ASS) and Electric Power Steering System (EPS) using a Centralized Control Method 256
7.7.1 Centralized Controller Design 256
7.7.2 Simulation Investigation 259
7.8 Integrated Control of the Electric Power Steering System (EPS) and Vehicle Stability Control (VSC) System 264
7.8.1 Interactions Between EPS and VSC 264
7.8.2 Control System Design 264
7.8.3 Dynamic Distribution of Tyre Forces 265
7.8.4 Design of a Self-aligning Torque Controller 267
7.8.5 Simulation Investigation 270
7.9 Centralized Control of Integrated Chassis Control Systems using the Artificial Neural Networks (ANN) Inverse System Method 271
7.9.1 Vehicle Dynamic Model 272
7.9.2 Design of the Centralized Control System 273
7.9.3 Simulation Investigation 278
References 281
8 Integrated Vehicle Dynamics Control: Multilayer Coordinating Control Architecture 283
8.1 Multilayer Coordinating Control of Active Suspension System (ASS) and Active Front Steering (AFS) 283
8.1.1 AFS Model 284
8.1.2 Controller Design 285
8.1.3 Simulation Investigation 289
8.2 Multilayer Coordinating Control of Active Suspension System (ASS) and Electric Power Steering System (EPS) 291
8.2.1 System Modeling 291
8.2.2 Controller Design 295
8.2.3 Simulation Investigation 298
8.3 Multilayer Coordinating Control of an Active Suspension System (ASS) and Anti-lock Brake System (ABS) 299
8.3.1 Coordinating Controller Design 300
8.3.2 Simulation Investigation 301
8.4 Multilayer Coordinating Control of the Electric Power Steering System (EPS) and Anti-lock Brake System (ABS) 303
8.4.1 Interactions between the EPS System and ABS 304
8.4.2 Coordinating Controller Design 305
8.4.3 Simulation Investigation 306
8.5 Multi-layer Coordinating Control of the Active Suspension System (ASS) and Vehicle Stability Control (VSC) System 308
8.5.1 System Model 308
8.5.2 Multilayer Coordinating Controller Design 308
8.5.3 Simulation Investigation 313
8.6 Multilayer Coordinating Control of an Active Four-wheel Steering System (4WS) and Direct Yaw Moment Control System (DYC) 315
8.6.1 Introduction 315
8.6.2 Coordinating Control of DYC and 4WS 316
8.6.3 Simulation Investigation 320
8.7 Multilayer Coordinating Control of Integrated Chassis Control Systems 321
8.7.1 Introduction 321
8.7.2 Controller Design 322
8.7.3 Simulation and Experiment Investigations 327
8.8 Multilayer Coordinating Control of Integrated Chassis Control Systems using Game Theory and Function Distribution Methods 330
8.8.1 Structure of the Chassis Control System 331
8.8.2 Design of the Suspension Subsystem Controller 331
8.8.3 Design of the Steering Subsystem Controller 332
8.8.4 Design of the Braking Subsystem Controller 333
8.8.5 Design of the Upper Layer Controller 333
8.8.6 Simulation Investigation 335
References 337
9 Perspectives 339
9.1 Models of Full Vehicle Dynamics 339
9.2 Multi-sensor Information Fusion 340
9.3 Fault-tolerant Control 340
9.4 Active and Passive Safety Integrated Control Based on the Function Allocation Method 341
9.5 Design of System Integration for a Vehicle 344
9.6 Assumption about the Vehicle of the Future 345
References 346
Index 347
