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Nonlinear Control of Vehicles and Robots
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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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