CS-184: Computer Graphics Lecture #19: Forward and Inverse - PDF document

CS-184: Computer Graphics Lecture #19: Forward and Inverse Kinematics ! ! ! Prof. James O’Brien ! University of California, Berkeley ! ! ! V2015-S-19-1.0 1 19-Kinematics.key - April 13, 2015 Today • Forward kinematics ! • Inverse kinematics ! • Pin joints ! • Ball joints ! • Prismatic joints 2 2 19-Kinematics.key - April 13, 2015

Forward Kinematics • Articulated skeleton ! • Topology (what’s connected to what) ! • Geometric relations from joints ! • Independent of display geometry ! • Tree structure ! • Loop joints break “tree-ness” 3 3 19-Kinematics.key - April 13, 2015 Forward Kinematics • Root body ! • Position set by “global” transformation ! • Root joint ! • Position ! • Rotation ! • Other bodies relative to root ! • Inboard toward the root ! • Outboard away from root 4 4 19-Kinematics.key - April 13, 2015

Forward Kinematics • A joint ! • Joint’s inboard body ! • Joint’s outboard body 5 5 19-Kinematics.key - April 13, 2015 Forward Kinematics • A body ! • Body’s inboard joint ! • Body’s outboard joint ! • May have several outboard joints 6 6 19-Kinematics.key - April 13, 2015

Forward Kinematics • A body ! • Body’s inboard joint ! • Body’s outboard joint ! • May have several outboard joints ! • Body’s parent ! • Body’s child ! • May have several children 7 7 19-Kinematics.key - April 13, 2015 Forward Kinematics • Interior joints ! • Typically not 6 DOF joints ! • Pin - rotate about one axis ! • Ball - arbitrary rotation ! • Prism - translation along one axis 8 8 19-Kinematics.key - April 13, 2015

Forward Kinematics • Pin Joints ! • Translate inboard joint to local origin ! • Apply rotation about axis ! • Translate origin to location of joint on outboard body 9 9 19-Kinematics.key - April 13, 2015 Forward Kinematics • Ball Joints ! • Translate inboard joint to local origin ! • Apply rotation about arbitrary axis ! • Translate origin to location of joint on outboard body 10 10 19-Kinematics.key - April 13, 2015

Forward Kinematics • Prismatic Joints ! • Translate inboard joint to local origin ! • Translate along axis ! • Translate origin to location of joint on outboard body 11 11 19-Kinematics.key - April 13, 2015 Forward Kinematics • Composite transformations up the hierarchy 12 12 19-Kinematics.key - April 13, 2015

Forward Kinematics • Composite transformations up the hierarchy 13 13 19-Kinematics.key - April 13, 2015 Forward Kinematics • Composite transformations up the hierarchy 14 14 19-Kinematics.key - April 13, 2015

Forward Kinematics • Composite transformations up the hierarchy 15 15 19-Kinematics.key - April 13, 2015 Forward Kinematics • Composite transformations up the hierarchy 16 16 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Given ! • Root transformation ! • Initial configuration ! • Desired end point location ! • Find ! • Interior parameter settings 17 17 19-Kinematics.key - April 13, 2015 Inverse Kinematics Egon Pasztor 18 18 19-Kinematics.key - April 13, 2015

Inverse Kinematics • A simple two segment arm in 2D 19 Warning: Z − up Coordinate System 19 19-Kinematics.key - April 13, 2015 Inverse Kinematics • Direct IK: solve for the parameters 20 20 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Why is the problem hard? ! • Multiple solutions separated in configuration space 21 21 19-Kinematics.key - April 13, 2015 Inverse Kinematics • Why is the problem hard? ! • Multiple solutions connected in configuration space 22 22 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Why is the problem hard? ! • Solutions may not always exist 23 23 19-Kinematics.key - April 13, 2015 Inverse Kinematics • Numerical Solution ! • Start in some initial configuration ! • Define an error metric ( e.g. goal pos - current pos) ! • Compute Jacobian of error w.r.t. inputs ! • Apply Newton’s method (or other procedure) ! • Iterate... 24 24 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Recall simple two segment arm: 25 Warning: Z − up Coordinate System 25 19-Kinematics.key - April 13, 2015 Inverse Kinematics • We can write of the derivatives 26 26 19-Kinematics.key - April 13, 2015

Inverse Kinematics Direction in Config. Space 27 27 19-Kinematics.key - April 13, 2015 Inverse Kinematics The Jacobian (of p w.r.t. θ ) Example for two segment arm 28 28 19-Kinematics.key - April 13, 2015

Inverse Kinematics The Jacobian (of p w.r.t. θ ) 29 29 19-Kinematics.key - April 13, 2015 Inverse Kinematics Solving for and 30 30 19-Kinematics.key - April 13, 2015

Inverse Kinematics Solving for and 31 Is the Jacobian invertible? 31 19-Kinematics.key - April 13, 2015 Inverse Kinematics • Problems ! • Jacobian may (will!) not always be invertible ! • Use pseudo inverse (SVD) ! • Robust iterative method ! • Jacobian is not constant ! ! • Nonlinear optimization, but problem is (mostly) well behaved 32 32 19-Kinematics.key - April 13, 2015

Inverse Kinematics Jacobian is not always invertible ! • Use pseudo inverse (SVD) ! ! Computing a linear approximation ! • End effector only locally moves linearly ! • So iterate (choosing proper step size) and update Jacobian after each step ! • Choosing step size requires line search at each step ! • Choose some step size (say 5 degrees) and compute how to update joint parameters ! • Calculate distance of end effector from goal ! • If distance decreased take step ! • Is distance did not decrease set parameters to be half the current change and try again ! ! 33 33 19-Kinematics.key - April 13, 2015 Inverse Kinematics • More complex systems ! • More complex joints (prism and ball) ! • More links ! • Other criteria (COM or height) ! • Hard constraints (joint limits) ! • Multiple criteria and multiple chains 34 34 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Some issues ! • How to pick from multiple solutions? ! • Robustness when no solutions ! • Contradictory solutions ! • Smooth interpolation ! • Interpolation aware of constraints ! ! • Numerical evaluation of Jacobian 35 35 19-Kinematics.key - April 13, 2015 Inverse Kinematics Prism Joints } } 36 36 19-Kinematics.key - April 13, 2015

Inverse Kinematics Ball Joints 37 37 19-Kinematics.key - April 13, 2015 Inverse Kinematics Ball Joints (moving axis) { That is the Jacobian for this joint 38 38 19-Kinematics.key - April 13, 2015

Inverse Kinematics Ball Joints (fixed axis) { That is the Jacobian for this joint 39 39 19-Kinematics.key - April 13, 2015 Inverse Kinematics • Many links / joints ! • Need a generic method for building Jacobian 2 b 2 a 3 1 40 40 19-Kinematics.key - April 13, 2015

Inverse Kinematics • Can’t just concatenate individual matrices 2 b 2 a 3 1 41 41 19-Kinematics.key - April 13, 2015 Inverse Kinematics Transformation from body to world Rotation from body to world 42 42 19-Kinematics.key - April 13, 2015

Inverse Kinematics Need to transform Jacobians to common coordinate system (WORLD) 2 b 2 a 3 3 1 2 b 43 43 19-Kinematics.key - April 13, 2015 Inverse Kinematics Note: Each row in the above should be transposed.... 44 44 19-Kinematics.key - April 13, 2015

Suggested Reading • Advanced Animation and Rendering Techniques by Watt and Watt ! • Chapters 15 and 16 45 45 19-Kinematics.key - April 13, 2015