INTRODUCTION TO ROBOTICS MECHANICS, PLANNING, AND CONTROL F. C. Park and K. M. Lynch Contents 1Preview 1 2 Configuration Space 9 2.1DegreesofFreedomofaRigidBody................ 10 2.2DegreesofFreedomofaRobot................... 13 2.2.1 RobotJoints......................... 13 2.2.2 Gr¨ubler’sFormula...................... 14 2.3 Configuration Space: Topology and Representation ........ 19 2.3.1 ConfigurationSpaceTopology................ 19 2.3.2 Configuration Space Representation ............ 21 2.4ConfigurationandVelocityConstraints............... 25 2.5TaskSpaceandWorkspace..................... 28 2.6Summary............................... 31 2.7Exercises............................... 33 3 Rigid-Body Motions 51 3.1Rigid-BodyMotionsinthePlane.................. 53 3.2RotationsandAngularVelocities.................. 59 3.2.1 RotationMatrices...................... 59 3.2.2 AngularVelocity....................... 65 3.2.3 Exponential Coordinate Representation of Rotation . 68 3.3Rigid-BodyMotionsandTwists................... 77 3.3.1 HomogeneousTransformationMatrices.......... 77 3.3.2 Twists............................. 83 3.3.3 Exponential Coordinate Representation of Rigid-Body Mo- tions.............................. 90 3.4Wrenches............................... 93 3.5Summary............................... 95 3.6Software................................ 97 3.7NotesandReferences......................... 98 3.8Exercises............................... 99 i CONTENTS 4 Forward Kinematics 117 4.1ProductofExponentialsFormula..................120 4.1.1 First Formulation: Screw Axes Expressed in Base Frame . 120 4.1.2 Examples...........................122 4.1.3 Second Formulation: Screw Axes Expressed in End-Effector Frame.............................127 4.2 The Universal Robot Description Format .............130 4.3Summary...............................135 4.4Software................................137 4.5Exercises...............................138 5 Velocity Kinematics and Statics 149 5.1ManipulatorJacobian........................155 5.1.1 SpaceJacobian........................155 5.1.2 BodyJacobian........................159 5.1.3 Relationship between the Space and Body Jacobian . 161 5.1.4 AlternativeNotionsoftheJacobian............162 5.1.5 InverseVelocityKinematics.................163 5.2StaticsofOpenChains........................163 5.3SingularityAnalysis.........................164 5.4 Manipulability ............................169 5.5Summary...............................172 5.6Software................................173 5.7NotesandReferences.........................174 5.8Exercises...............................175 6 Inverse Kinematics 191 6.1AnalyticInverseKinematics.....................193 6.1.1 6RPUMA-TypeArm....................193 6.1.2 Stanford-TypeArms.....................196 6.2NumericalInverseKinematics....................198 6.2.1 Newton-RaphsonMethod..................198 6.2.2 OptimizationBasics.....................199 6.2.3 NumericalInverseKinematicsAlgorithm.........201 6.3InverseVelocityKinematics.....................205 6.4ANoteonClosedLoops.......................207 6.5Summary...............................207 6.6Software................................208 6.7NotesandReferences.........................208 6.8Exercises...............................209 7 Kinematics of Closed Chains 217 7.1InverseandForwardKinematics..................218 7.1.1 3×RPRPlanarParallelMechanism.............219 7.1.2 Stewart-GoughPlatform...................221 7.1.3 GeneralParallelMechanisms................222 Park and Lynch ii 16:36 September 20, 2016 CONTENTS 7.2DifferentialKinematics........................223 7.2.1 Stewart-GoughPlatform...................224 7.2.2 GeneralParallelMechanisms................225 7.3Singularities..............................227 7.4Summary...............................231 7.5NotesandReferences.........................232 7.6Exercises...............................234 8 Dynamics of Open Chains 241 8.1 Lagrangian Formulation .......................242 8.1.1 BasicConceptsandMotivatingExamples.........242 8.1.2 GeneralFormulation.....................246 8.1.3 UnderstandingtheMassMatrix..............248 8.1.4 Lagrangian Dynamics vs. Newton-Euler Dynamics ....250 8.2DynamicsofaSingleRigidBody..................251 8.2.1 ClassicalFormulation....................251 8.2.2 Twist-WrenchFormulation.................255 8.3Newton-EulerInverseDynamics...................257 8.3.1 Derivation...........................257 8.3.2 Newton-EulerInverseDynamicsAlgorithm........260 8.4DynamicEquationsinClosedForm.................260 8.5ForwardDynamicsofOpenChains.................263 8.6DynamicsinTaskSpaceCoordinates................264 8.7ConstrainedDynamics........................266 8.8RobotDynamicsintheURDF...................267 8.9Actuation,Gearing,andFriction..................268 8.9.1 DCMotorsandGearing...................270 8.9.2 ApparentInertia.......................273 8.9.3 Friction............................276 8.9.4 Joint and Link Flexibility ..................277 8.10Summary...............................277 8.11Software................................280 8.12NotesandReferences.........................282 8.13Exercises...............................283 9 Trajectory Generation 289 9.1Definitions...............................289 9.2Point-to-PointTrajectories.....................290 9.2.1 Straight-LinePaths.....................290 9.2.2 TimeScalingaStraight-LinePath.............292 9.3PolynomialViaPointTrajectories.................297 9.4Time-OptimalTimeScaling.....................299 9.4.1 The (s, s˙)PhasePlane....................301 9.4.2 TheTime-ScalingAlgorithm................304 9.4.3 AssumptionsandCaveats..................306 9.5Summary...............................307 Park and Lynch iii 16:36 September 20, 2016 CONTENTS 9.6Software................................308 9.7NotesandReferences.........................308 9.8Exercises...............................310 10 Motion Planning 315 10.1OverviewofMotionPlanning....................315 10.1.1TypesofMotionPlanningProblems............316 10.1.2PropertiesofMotionPlanners...............317 10.1.3MotionPlanningMethods..................318 10.2 Foundations ..............................319 10.2.1ConfigurationSpaceObstacles...............319 10.2.2 Distance to Obstacles and Collision Detection .......323 10.2.3GraphsandTrees.......................324 10.2.4GraphSearch.........................325 10.3CompletePathPlanners.......................328 10.4GridMethods.............................329 10.4.1 Multi-Resolution Grid Representation ...........331 10.4.2GridMethodswithMotionConstraints..........332 10.5SamplingMethods..........................337 10.5.1TheRRT...........................338 10.5.2ThePRM...........................342 10.6VirtualPotentialFields.......................344 10.6.1APointinC-space......................344 10.6.2NavigationFunctions.....................347 10.6.3WorkspacePotential.....................348 10.6.4WheeledMobileRobots...................349 10.6.5UseofPotentialFieldsinPlanners.............349 10.7NonlinearOptimization.......................349 10.8Smoothing...............................351 10.9Summary...............................352 10.10NotesandReferences.........................354 10.11Exercises...............................356 11 Robot Control 359 11.1ControlSystemOverview......................360 11.2MotionControl............................360 11.2.1 Motion Control of a Multi-Joint Robot with Velocity Input361 11.2.2 Motion Control of a Single Joint with Torque or Force Input363 11.2.3 Motion Control of a Multi-joint Robot with Torque or Force Input ..........................374 11.2.4TaskSpaceMotionControl.................376 11.3ForceControl.............................378 11.4HybridMotion-ForceControl....................380 11.4.1NaturalandArtificialConstraints.............380 11.4.2AHybridController.....................382 11.5ImpedanceControl..........................384 Park and Lynch iv 16:36 September 20, 2016 CONTENTS 11.5.1ImpedanceControlAlgorithm................386 11.5.2AdmittanceControlAlgorithm...............387 11.6OtherTopics.............................387 11.7Summary...............................388 11.8Software................................390 11.9NotesandReferences.........................391 11.10Exercises...............................392 12 Grasping and Manipulation 399 12.1ContactKinematics.........................400 12.1.1First-OrderAnalysisofaSingleContact..........401 12.1.2ContactTypes:Rolling,Sliding,andBreakingFree....402 12.1.3MultipleContacts......................405 12.1.4CollectionsofParts.....................406 12.1.5OtherTypesofContacts...................407 12.1.6PlanarGraphicalMethods..................409 12.1.7FormClosure.........................413 12.2ContactForcesandFriction.....................418 12.2.1Friction............................418 12.2.2PlanarGraphicalMethods..................420 12.2.3ForceClosure.........................423 12.2.4DualityofForceandMotionFreedoms...........428 12.3Manipulation.............................428 12.4Summary...............................434 12.5NotesandReferences.........................435 12.6Exercises...............................437 13 Wheeled Mobile Robots 455 13.1TypesofWheeledMobileRobots..................455 13.2OmnidirectionalWheeledMobileRobots..............457 13.2.1Modeling...........................457 13.2.2MotionPlanning.......................461 13.2.3FeedbackControl.......................462 13.3NonholonomicWheeledMobileRobots...............462