CS-184: Computer GraphicsLecture #17: Forward and Inverse KinematicsProf. James O’BrienUniversity of California, BerkeleyV2006-F-17-1.02TodayForward kinematicsInverse kinematicsPin jointsBall jointsPrismatic joints3Forward KinematicsArticulated skeletonTopology (what’s connected to what)Geometric relations from jointsIndependent of display geometryTree structureLoop joints break “tree-ness”4Forward KinematicsRoot bodyPosition set by “global” transformationRoot jointPositionRotationOther bodies relative to rootInboard toward the rootOutboard away from root5Forward KinematicsA jointJoint’s inboard bodyJoint’s outboard body6Forward KinematicsA bodyBody’s inboard jointBody’s outboard jointMay have several outboard joints7Forward KinematicsA bodyBody’s inboard jointBody’s outboard jointMay have several outboard jointsBody’s parentBody’s childMay have several children8Forward KinematicsInterior jointsTypically not 6 DOF jointsPin - rotate about one axisBall - arbitrary rotationPrism - translation along one axis9Forward KinematicsPin JointsTranslate inboard joint to local originApply rotation about axisTranslate origin to location of joint on outboard body10Forward KinematicsBall Joints Translate inboard joint to local originApply rotation about arbitrary axisTranslate origin to location of joint on outboard body11Forward KinematicsPrismatic Joints Translate inboard joint to local originTranslate along axisTranslate origin to location of joint on outboard body12Forward KinematicsComposite transformations up the hierarchy13Forward KinematicsComposite transformations up the hierarchy14Forward KinematicsComposite transformations up the hierarchy15Forward KinematicsComposite transformations up the hierarchy16Forward KinematicsComposite transformations up the hierarchy17Inverse KinematicsGivenRoot transformationInitial configurationDesired end point locationFindInterior parameter settings18Inverse KinematicsEgon Pasztor19Inverse KinematicsA simple two segment arm in 2DSimple System: A Two Segment ArmWarning: Z!up Coordinate System20Inverse KinematicsDirect IK: solve for the parametersDirect IK: Solve for and21Inverse KinematicsWhy is the problem hard?Multiple solutions separated in configuration spaceWhy is this a hard problem?Multiple solutions separated in configuration space22Inverse KinematicsWhy is the problem hard?Multiple solutions connected in configuration spaceWhy is this a hard problem?Multiple solutions connected in configuration space23Inverse KinematicsWhy is the problem hard?Solutions may not always exist24Inverse KinematicsNumerical SolutionStart in some initial configurationDefine an error metric (e.g. goal pos - current pos)Compute Jacobian of error w.r.t. inputsApply Newton’s method (or other procedure)Iterate...25Inverse KinematicsRecall simple two segment arm:Simple System: A Two Segment ArmWarning: Z!up Coordinate System26Inverse KinematicsWe can write of the derivativesSimple System: A Two Segment Arm27Inverse KinematicsSimple System: A Two Segment ArmDirection in Config. Space28Inverse KinematicsThe Jacobian (of p w.r.t. !)Example for two segment arm29Inverse KinematicsThe Jacobian (of p w.r.t. !) 30Inverse KinematicsSolving for and31Inverse KinematicsSolving for and Is the Jacobian invertible?32Inverse KinematicsProblemsJacobian may (will!) not always be invertibleUse pseudo inverse (SVD)Robust iterative methodJacobian is not constantNonlinear optimization, but problem is (mostly) well behavedProblems...Jacobian may (will) not be invertibleOption #1: Use pseudo inverse (SVD)Option #2: Use iterative methodJacobian is not constantNon!linear optimization... but problem is well behaved (mostly)33Inverse KinematicsMore complex systemsMore complex joints (prism and ball)More linksOther criteria (COM or height)Hard constraints (joint limits)Multiple criteria and multiple chains34Inverse KinematicsSome issuesHow to pick from multiple solutions?Robustness when no solutionsContradictory solutionsSmooth interpolation Interpolation aware of constraints35Inverse KinematicsPrism Joints}}36Inverse KinematicsBall Joints37Inverse KinematicsBall Joints (moving axis)That is the Jacobian for this joint{38Inverse KinematicsBall Joints (fixed axis)That is the Jacobian for this joint{39Inverse KinematicsMany links / jointsNeed a generic method for building JacobianMany Links/JointsWe need a generic method of building Jacobian12a32b40Inverse KinematicsCan’t just concatenate individual matricesMany Links/Joints12a32b41Inverse KinematicsMany Links/JointsTransformation from body to worldRotation from body to world42Inverse KinematicsMany Links/JointsNeed to transform Jacobians to commoncoordinate system (WORLD)2b312a2b343Inverse KinematicsMany Links/JointsNote: Each row in the aboveshould be transposed....44Suggested ReadingAdvanced Animation and Rendering Techniques by Watt and WattChapters 15 and
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