CS-184: Computer GraphicsLecture #18: Forward and Inverse KinematicsProf. James O’BrienUniversity of California, BerkeleyV2008-S-18-1.012TodayForward kinematicsInverse kinematicsPin jointsBall jointsPrismatic joints23Forward KinematicsArticulated skeletonTopology (what’s connected to what)Geometric relations from jointsIndependent of display geometryTree structureLoop joints break “tree-ness”34Forward KinematicsRoot bodyPosition set by “global” transformationRoot jointPositionRotationOther bodies relative to rootInboard toward the rootOutboard away from root45Forward KinematicsA jointJoint’s inboard bodyJoint’s outboard body56Forward KinematicsA bodyBody’s inboard jointBody’s outboard jointMay have several outboard joints67Forward KinematicsA bodyBody’s inboard jointBody’s outboard jointMay have several outboard jointsBody’s parentBody’s childMay have several children78Forward KinematicsInterior jointsTypically not 6 DOF jointsPin - rotate about one axisBall - arbitrary rotationPrism - translation along one axis89Forward KinematicsPin JointsTranslate inboard joint to local originApply rotation about axisTranslate origin to location of joint on outboard body910Forward KinematicsBall Joints Translate inboard joint to local originApply rotation about arbitrary axisTranslate origin to location of joint on outboard body1011Forward KinematicsPrismatic Joints Translate inboard joint to local originTranslate along axisTranslate origin to location of joint on outboard body1112Forward KinematicsComposite transformations up the hierarchy1213Forward KinematicsComposite transformations up the hierarchy1314Forward KinematicsComposite transformations up the hierarchy1415Forward KinematicsComposite transformations up the hierarchy1516Forward KinematicsComposite transformations up the hierarchy1617Inverse KinematicsGivenRoot transformationInitial configurationDesired end point locationFindInterior parameter settings1718Inverse KinematicsEgon Pasztor1819Inverse KinematicsA simple two segment arm in 2DSimple System: A Two Segment ArmWarning: Z!up Coordinate System1920Inverse KinematicsDirect IK: solve for the parametersDirect IK: Solve for and 2021Inverse KinematicsWhy is the problem hard?Multiple solutions separated in configuration spaceWhy is this a hard problem?Multiple solutions separated in configuration space2122Inverse KinematicsWhy is the problem hard?Multiple solutions connected in configuration spaceWhy is this a hard problem?Multiple solutions connected in configuration space2223Inverse KinematicsWhy is the problem hard?Solutions may not always exist2324Inverse 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...2425Inverse KinematicsRecall simple two segment arm:Simple System: A Two Segment ArmWarning: Z!up Coordinate System2526Inverse KinematicsWe can write of the derivativesSimple System: A Two Segment Arm2627Inverse KinematicsSimple System: A Two Segment ArmDirection in Config. Space2728Inverse KinematicsThe Jacobian (of p w.r.t. !)Example for two segment arm2829Inverse KinematicsThe Jacobian (of p w.r.t. !) 2930Inverse KinematicsSolving for and 3031Inverse KinematicsSolving for and Is the Jacobian invertible?3132Inverse 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)3233Inverse KinematicsMore complex systemsMore complex joints (prism and ball)More linksOther criteria (COM or height)Hard constraints (joint limits)Multiple criteria and multiple chains3334Inverse KinematicsSome issuesHow to pick from multiple solutions?Robustness when no solutionsContradictory solutionsSmooth interpolation Interpolation aware of constraints3435Inverse KinematicsPrism Joints}}3536Inverse KinematicsBall Joints3637Inverse KinematicsBall Joints (moving axis)That is the Jacobian for this joint{3738Inverse KinematicsBall Joints (fixed axis)That is the Jacobian for this joint{3839Inverse KinematicsMany links / jointsNeed a generic method for building JacobianMany Links/JointsWe need a generic method of building Jacobian12a32b3940Inverse KinematicsCan’t just concatenate individual matricesMany Links/Joints12a32b4041Inverse KinematicsMany Links/JointsTransformation from body to worldRotation from body to world4142Inverse KinematicsMany Links/JointsNeed to transform Jacobians to commoncoordinate system (WORLD)2b312a2b34243Inverse KinematicsMany Links/JointsNote: Each row in the aboveshould be transposed....4344Suggested ReadingAdvanced Animation and Rendering Techniques by Watt and WattChapters 15 and
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