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[资源] Nonlinear Control of Vehicles and Robots

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Notions, Background . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Short History . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Control Systems for Vehicles and Robots, Research Motivation . . 5
1.4 Outline of the Following Chapters . . . . . . . . . . . . . . . . . . 7
2 Basic Nonlinear Control Methods . . . . . . . . . . . . . . . . . . . . 11
2.1 Nonlinear System Classes . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 State Equation of Nonlinear Systems . . . . . . . . . . . . 12
2.1.2 Holonomic and Nonholonomic Systems . . . . . . . . . . 15
2.1.3 DifferentiallyFlatSystems . . . . . . . . . . . . . . . . . 24
2.2 Dynamic Model of Simple Systems . . . . . . . . . . . . . . . . . 30
2.2.1 Dynamic Model of Inverted Pendulum . . . . . . . . . . . 30
2.2.2 Car Active Suspension Model . . . . . . . . . . . . . . . . 33
2.2.3 The Model of the 2 DOF Robot Arm . . . . . . . . . . . . 35
2.3 Stability of Nonlinear Systems . . . . . . . . . . . . . . . . . . . 38
2.3.1 Stability Definitions . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Lyapunov Stability Theorems . . . . . . . . . . . . . . . . 40
2.3.3 BarbalatLemmas . . . . . . . . . . . . . . . . . . . . . . 47
2.3.4 Stability of Interconnected Passive Systems . . . . . . . . . 49
2.4 Input–Output Linearization . . . . . . . . . . . . . . . . . . . . . 54
2.5 FlatnessControl . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.7 SlidingControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.7.1 Sliding Control of Second Order Systems . . . . . . . . . . 65
2.7.2 ControlChattering . . . . . . . . . . . . . . . . . . . . . . 67
2.7.3 Sliding Control of Robot . . . . . . . . . . . . . . . . . . 70
2.8 Receding Horizon Control . . . . . . . . . . . . . . . . . . . . . . 71
2.8.1 Nonlinear Receding Horizon Control . . . . . . . . . . . . 72
2.8.2 Nonlinear RHC Control of 2D Crane . . . . . . . . . . . . 74
2.8.3 RHC Based on Linearization at Each Horizon . . . . . . . 76
2.9 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
xvii
xviii Contents
3 Dynamic Models of Ground, Aerial and Marine Robots . . . . . . . 81
3.1 Dynamic Model of Rigid Body . . . . . . . . . . . . . . . . . . . 81
3.1.1 Dynamic Model Based on Newton–Euler Equations . . . . 82
3.1.2 Kinematic Model Using Euler (RPY) Angles . . . . . . . . 84
3.1.3 Kinematic Model Using Quaternion . . . . . . . . . . . . . 85
3.2 Dynamic Model of Industrial Robot . . . . . . . . . . . . . . . . . 86
3.2.1 Recursive Computation of the Kinematic Quantities . . . . 87
3.2.2 Robot Dynamic Model Based on Appell’s Equation . . . . 89
3.2.3 Robot Dynamic Model Based on Lagrange’s Equation . . . 92
3.2.4 Dynamic Model of SCARA Robot . . . . . . . . . . . . . 94
3.3 Dynamic Model of Car . . . . . . . . . . . . . . . . . . . . . . . 98
3.3.1 Nonlinear Model of Car . . . . . . . . . . . . . . . . . . . 99
3.3.2 Input Affine Approximation of the Dynamic Model . . . . 102
3.3.3 Linearized Model for Constant Velocity . . . . . . . . . . . 103
3.4 Dynamic Model of Airplane . . . . . . . . . . . . . . . . . . . . . 104
3.4.1 CoordinateSystems forNavigation . . . . . . . . . . . . . 104
3.4.2 AirplaneKinematics . . . . . . . . . . . . . . . . . . . . . 108
3.4.3 Airplane Dynamics . . . . . . . . . . . . . . . . . . . . . 109
3.4.4 Wind-AxesCoordinateSystem . . . . . . . . . . . . . . . 111
3.4.5 GravityEffect . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.6 Aerodynamic Forces and Torques . . . . . . . . . . . . . . 113
3.4.7 Gyroscopic Effect of Rotary Engine . . . . . . . . . . . . . 116
3.4.8 StateEquationsofAirplane . . . . . . . . . . . . . . . . . 116
3.4.9 Linearization of the Nonlinear Airplane Model . . . . . . . 118
3.4.10 Parametrization of Aerodynamic and Trust Forces
andMoments . . . . . . . . . . . . . . . . . . . . . . . . 119
3.5 Dynamic Model of Surface and Underwater Ships . . . . . . . . . 121
3.5.1 RigidBodyEquationofShip . . . . . . . . . . . . . . . . 121
3.5.2 Hydrodynamic Forces and Moments . . . . . . . . . . . . 123
3.5.3 RestoringForces andMoments . . . . . . . . . . . . . . . 124
3.5.4 BallastSystems . . . . . . . . . . . . . . . . . . . . . . . 126
3.5.5 Wind, Wave and Current Models . . . . . . . . . . . . . . 126
3.5.6 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . 130
3.5.7 Dynamic Model in Body Frame . . . . . . . . . . . . . . . 130
3.5.8 Dynamic Model in NED Frame . . . . . . . . . . . . . . . 131
3.6 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 Nonlinear Control of Industrial Robots . . . . . . . . . . . . . . . . . 135
4.1 Decentralized Three-Loop Cascade Control . . . . . . . . . . . . . 135
4.1.1 Dynamic Model of DC Motor . . . . . . . . . . . . . . . . 135
4.1.2 Design of Three-Loop Cascade Controller . . . . . . . . . 138
4.1.3 Approximation of Load Inertia and Disturbance Torque . . 143
4.2 Computed Torque Technique . . . . . . . . . . . . . . . . . . . . 144
4.3 Nonlinear Decoupling in Cartesian Space . . . . . . . . . . . . . . 145
4.3.1 Computation of Equivalent Forces and Torques . . . . . . . 146
Contents xix
4.3.2 Computation of Equivalent Joint Torques . . . . . . . . . . 147
4.3.3 Robot Dynamic Model in Cartesian Space . . . . . . . . . 147
4.3.4 Nonlinear Decoupling of the Free Motion . . . . . . . . . . 148
4.4 Hybrid Position and Force Control . . . . . . . . . . . . . . . . . 149
4.4.1 Generalized Task Specification Matrices . . . . . . . . . . 150
4.4.2 HybridPosition/ForceControlLaw . . . . . . . . . . . . . 151
4.5 Self-Tuning Adaptive Control . . . . . . . . . . . . . . . . . . . . 152
4.5.1 Independent Parameters of Robot Dynamic Model . . . . . 152
4.5.2 Control and Adaptation Laws . . . . . . . . . . . . . . . . 154
4.5.3 Simulation Results for 2-DOF Robot . . . . . . . . . . . . 156
4.5.4 IdentificationStrategy . . . . . . . . . . . . . . . . . . . . 156
4.6 Robust Backstepping Control in Case of Nonsmooth Path . . . . . 158
4.6.1 Gradient Update Laws for Speed Error . . . . . . . . . . . 159
4.6.2 Control of 2-DOF Robot Arm Along Rectangle Path . . . . 160
4.7 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5 Nonlinear Control of Cars . . . . . . . . . . . . . . . . . . . . . . . . 169
5.1 Control Concept of Collision Avoidance System (CAS) . . . . . . 169
5.2 PathDesignUsingElasticBand . . . . . . . . . . . . . . . . . . . 170
5.3 Reference Signal Design for Control . . . . . . . . . . . . . . . . 172
5.4 Nonlinear Dynamic Model . . . . . . . . . . . . . . . . . . . . . 174
5.5 DifferentialGeometryBasedControlAlgorithm . . . . . . . . . . 175
5.5.1 External State Feedback Design . . . . . . . . . . . . . . . 176
5.5.2 Stability Proof of Zero Dynamics . . . . . . . . . . . . . . 178
5.5.3 SimulationResultsUsingDGAMethod . . . . . . . . . . 181
5.6 Receding Horizon Control . . . . . . . . . . . . . . . . . . . . . . 182
5.6.1 NominalValues andPerturbations . . . . . . . . . . . . . . 184
5.6.2 RHCOptimizationUsingEndConstraint . . . . . . . . . . 186
5.7 StateEstimationUsingGPSandIMU . . . . . . . . . . . . . . . . 189
5.8 SimulationResultswithRHCControl andStateEstimation . . . . 192
5.9 Software Implementations . . . . . . . . . . . . . . . . . . . . . . 192
5.9.1 Standalone Programs . . . . . . . . . . . . . . . . . . . . 193
5.9.2 QuickPrototypeDesignforTargetProcessors . . . . . . . 195
5.10 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6 Nonlinear Control of Airplanes and Helicopters . . . . . . . . . . . . 199
6.1 Receding Horizon Control of the Longitudinal Motion
of anAirplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.1.1 Robust Internal Stabilization Using Disturbance Observer . 201
6.1.2 High Level Receding Horizon Control . . . . . . . . . . . 203
6.1.3 Simulation Results with External RHC and Internal
Disturbance Observer . . . . . . . . . . . . . . . . . . . . 208
6.2 Backstepping Control of an Indoor Quadrotor Helicopter . . . . . 213
6.2.1 Dynamic Model of the Quadrotor Helicopter . . . . . . . . 215
6.2.2 Sensor System of the Helicopter . . . . . . . . . . . . . . . 217
xx Contents
6.2.3 State Estimation Using Vision and Inertial Measurements . 226
6.2.4 Backstepping Control Algorithm . . . . . . . . . . . . . . 230
6.2.5 Embedded Control Realization . . . . . . . . . . . . . . . 236
6.3 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7 Nonlinear Control of Surface Ships . . . . . . . . . . . . . . . . . . . 245
7.1 ControlSystemStructure . . . . . . . . . . . . . . . . . . . . . . 245
7.1.1 Reference Path Design . . . . . . . . . . . . . . . . . . . . 247
7.1.2 Line-of-Sight Guidance . . . . . . . . . . . . . . . . . . . 247
7.1.3 Filtering Wave Disturbances . . . . . . . . . . . . . . . . . 248
7.1.4 StateEstimationUsingIMUandGPS. . . . . . . . . . . . 249
7.2 Acceleration Feedback and Nonlinear PD . . . . . . . . . . . . . . 254
7.3 Nonlinear Decoupling . . . . . . . . . . . . . . . . . . . . . . . . 255
7.3.1 Nonlinear Decoupling in Body Frame . . . . . . . . . . . . 255
7.3.2 Nonlinear Decoupling in NED Frame . . . . . . . . . . . . 256
7.4 Adaptive Feedback Linearization . . . . . . . . . . . . . . . . . . 257
7.5 MIMO Backstepping in 6 DOF . . . . . . . . . . . . . . . . . . . 259
7.6 ConstrainedControlAllocation . . . . . . . . . . . . . . . . . . . 262
7.7 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.8 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8 Formation Control of Vehicles . . . . . . . . . . . . . . . . . . . . . . 269
8.1 Selected Approaches in Formation Control of Vehicles . . . . . . . 269
8.2 Stabilization of Ground Vehicles Using Potential Field Method . . 270
8.2.1 Low Level Linearizing Controller . . . . . . . . . . . . . . 270
8.2.2 HighLevelFormationController . . . . . . . . . . . . . . 272
8.2.3 PassivityBasedFormationStabilization . . . . . . . . . . 275
8.3 Simulation Results for UGVs . . . . . . . . . . . . . . . . . . . . 276
8.4 Stabilization of Marine Vehicles Using Passivity Theory . . . . . . 277
8.4.1 Problem Formulation for Synchronized Path Following . . 278
8.4.2 ControlStructure . . . . . . . . . . . . . . . . . . . . . . 279
8.4.3 Stability Proof Based on Passivity Theory . . . . . . . . . 280
8.5 SimulationResults forUMVs . . . . . . . . . . . . . . . . . . . . 283
8.6 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
9 Modeling Nonsmooth Nonlinearities in Mechanical Systems . . . . . 291
9.1 Modeling and Stability of Nonsmooth Systems . . . . . . . . . . . 291
9.1.1 Modeling and Stability of Switched Systems . . . . . . . . 292
9.1.2 Modeling, Solution and Stability of Differential Inclusions . 295
9.2 Static Friction Models . . . . . . . . . . . . . . . . . . . . . . . . 298
9.2.1 Stick–Slip Motion . . . . . . . . . . . . . . . . . . . . . . 301
9.2.2 Friction-Induced Dead Zone . . . . . . . . . . . . . . . . . 303
9.3 Dynamic Friction Models . . . . . . . . . . . . . . . . . . . . . . 304
9.3.1 Classic Dynamic Friction Models . . . . . . . . . . . . . . 304
9.3.2 Modified and Advanced Dynamic Friction Models . . . . . 308
9.4 Piecewise Linearly Parameterized Friction Model . . . . . . . . . 310
Contents xxi
9.4.1 Parameter Equivalence with the Tustin Model . . . . . . . 312
9.4.2 ModelingErrors . . . . . . . . . . . . . . . . . . . . . . . 313
9.4.3 Incorporating the Dynamic Effects . . . . . . . . . . . . . 313
9.5 Backlash in Mechanical Systems . . . . . . . . . . . . . . . . . . 314
9.6 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
10 Mechanical Control Systems with Nonsmooth Nonlinearities . . . . . 319
10.1 Switched System Model of Mechanical Systems with Stribeck
Friction and Backlash . . . . . . . . . . . . . . . . . . . . . . . . 319
10.2 MotionControl . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
10.2.1 Stabilizing Control . . . . . . . . . . . . . . . . . . . . . . 322
10.2.2 Extension of the Control Law for Tracking . . . . . . . . . 326
10.2.3 SimulationResults . . . . . . . . . . . . . . . . . . . . . . 327
10.3 Friction and Backlash Induced Limit Cycle Around Zero Velocity . 330
10.3.1 Chaotic Measures for Nonlinear Analysis . . . . . . . . . . 333
10.3.2 SimulationMeasurements . . . . . . . . . . . . . . . . . . 334
10.4 Friction Generated Limit Cycle Around Stribeck Velocities . . . . 336
10.4.1 SimulationResults . . . . . . . . . . . . . . . . . . . . . . 339
10.4.2 Experimental Measurements . . . . . . . . . . . . . . . . 339
10.5 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11 Model Based Identification and Adaptive Compensation
of Nonsmooth Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 343
11.1 Friction and Backlash Measurement and Identification in Robotic
Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
11.1.1 FrictionMeasurement andIdentification . . . . . . . . . . 345
11.1.2 Backlash Measurement . . . . . . . . . . . . . . . . . . . 346
11.1.3 VelocityControl forMeasurements . . . . . . . . . . . . . 347
11.1.4 Experimental Measurements . . . . . . . . . . . . . . . . 349
11.2 Friction Measurement and Identification in Hydraulic Actuators . . 355
11.2.1 Mathematical Model of Hydraulic Actuators . . . . . . . . 356
11.2.2 FrictionMeasurement andIdentification . . . . . . . . . . 358
11.2.3 Experimental Measurements . . . . . . . . . . . . . . . . 359
11.3 Nonlinear Control of a Ball and Beam System Using Coulomb
Friction Compensation . . . . . . . . . . . . . . . . . . . . . . . . 363
11.3.1 Adaptive Friction Identification . . . . . . . . . . . . . . . 366
11.3.2 Nonlinear Control Algorithm for the Ball and Beam System 367
11.3.3 Experimental Evaluations . . . . . . . . . . . . . . . . . . 368
11.4 Adaptive Payload and Friction Compensation in Robotic
Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
11.4.1 Simulation Results—Adaptive Friction Compensation
in the Presence of Backlash . . . . . . . . . . . . . . . . . 377
11.4.2 Experimental Measurements . . . . . . . . . . . . . . . . 379
11.5 ClosingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
xxii Contents
12 Conclusions and Future Research Directions . . . . . . . . . . . . . 385
12.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.2 FutureResearchDirections . . . . . . . . . . . . . . . . . . . . . 387
Appendix A Kinematic and Dynamic Foundations of Physical Systems . 389
A.1 OrientationDescriptionUsingRotations andQuaternion . . . . . . 389
A.1.1 Homogeneous Transformations . . . . . . . . . . . . . . . 389
A.1.2 OrientationDescriptionUsingRotations . . . . . . . . . . 391
A.1.3 OrientationDescriptionUsingQuaternion . . . . . . . . . 393
A.1.4 Solutionof the InverseOrientationProblem . . . . . . . . 394
A.2 DifferentiationRule inMovingCoordinateSystem . . . . . . . . . 396
A.3 InertiaParametersofRigidObjects . . . . . . . . . . . . . . . . . 398
A.4 Lagrange, Appell and Newton–Euler Equations . . . . . . . . . . . 400
A.4.1 Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . 402
A.4.2 Appell Equation . . . . . . . . . . . . . . . . . . . . . . . 403
A.4.3 Newton–Euler Equations . . . . . . . . . . . . . . . . . . 404
A.5 Robot Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 406
A.5.1 Denavit-Hartenberg Form . . . . . . . . . . . . . . . . . . 406
A.5.2 DirectKinematicProblem . . . . . . . . . . . . . . . . . . 408
A.5.3 InverseKinematicProblem . . . . . . . . . . . . . . . . . 410
A.5.4 Robot Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 411
Appendix B Basis of Differential Geometry for Control Problems . . . . 417
B.1 Lie Derivatives, Submanifold, Tangent Space . . . . . . . . . . . . 417
B.2 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 422
B.3 Local Reachability and Observability . . . . . . . . . . . . . . . . 428
B.4 Input/Output Linearization, Zero Dynamics . . . . . . . . . . . . . 439
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

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