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1 Announcements Assignment on the web page: est login procedure NOW! 0//0
2 Forward and Inverse Kinematics Parent: Chapter 4. Girard and Maciejewski 985 Zhao and Badler 994 COMPUER ANIMAION 5-497/5-86 0/6/0
3 Kinematics ast two classes: how to interpolate positions/translations and orientations But we also need to set the keframes make the foot come into contact with the ball, for eample
4 Hierarchical Models Want structure for bod to be maintained Pivot joint Prismatic joint Ball and socket joint
5 Human Models Articulated figures are a horrible approimation
6 Hierarchical Models in Simulation Wane Wooten, Gatech
7 Kinematics he stud of motion without regard to the forces that cause it. β α A Forward: A f ( α, β Backward: α, Draw graphics Specif fewer degrees of freedom More intuitive control of dof Maintain contact with the environment Calculate desired joint angles for control β f ( A
8 z cos sin Forward Kinematics cos( sin( [ trans ][ rot ][ trans ][ rot ]
9 Wh Inverse Kinematics? Pick up an object or place feet on the ground Hard to do with forward kinematics Allow animator to set fewer parameters or at least get a good first approimation
10 User Control via Inverse Kinematics oint space Position all joints fine level of control Cartesian space Specif environmental interactions easil Most degrees of freedom computed automaticall
11 What do we need? Natural looking path or just a goal position? (time coherenc ocal or global solution?
12 Wh Inverse Kinematics Control Balance---keep center of mass over support polgon Control---position vaulter s hands on line between shoulder and vault Control---compute knee angles that will give the runner the right leg length
13 What do we need? Skeleton with,,3 degree of freedom joints Solve from root position to end effector position Arbitrar position constraints Direction/orientation constraints oint limits echniques for handling unconstrained sstem Adding constraints Heuristics to push solution into right part of space Optimization based on some criterion
14 Inverse Kinematics Closed Form Solution 80 ( cos cosine rule ( cos(80 cos cosine rule cos( cos cos( a a a
15 Methods Closed form onl for fairl simple mechanisms Iterative solutions Solutions No solution (outside workspace, too few dof Multiple solutions (redundanc Single solution
16 (, What makes IK hard?--redundancies a subspace..., { } n defined b if f ( X Add constraints to reduce redundancies Choose solution that is closest to current configuration Move outermost links the most Energ minimization Minimum time Natural looking motion???
17 What makes IK hard?--singularities Ill-conditioned near singularities High state space velocities for low cartesian velocities Goal
18 Reachable Workspace Again, not so simple for more comple mechanisms
19 Iterative Method Used in Girard and Maciejewski 985 Described in Parent Uses inverse of acobian to iterativel step all the joint angles towards the goal
20 he acobian f ij ( f acobian is changes of ( i j the ( is of is of nm where the dimension n (generall 6 dimension m (# of matri relating differential to differential ij changes th element of dof is ( acobian maps velocities in state space to velocities in cartesian space. acobian is dependent on state (and must be recomputed frequentl.
21 he acobian using Parent s notation [ ] n z n ij n z z v v V v v v V ω ω ω ω Μ Ο Κ ,,,,,,
22 IK and the acobian f ( f ( is of is of dimension dimension n m (generall 6 (# of dof t k k linearize about k An iterative solution goal
23 Inverting the acobian ( ( ( (.... where compute pseudo - inverse not square in general is V V V V nm
24 Using the acobian for IK Singularities cause the rank of the acobian to change acobian is valid onl for the configuration for which it was computed Pseudo inverse minimizes joint angle rates (locall Could minimize other quantities
25 Non-inear Optimization Zhao and Badler, OG 994 (pointer from sllabus Non-linear programming numerical method for finding the (local minimum of a non-linear function Objective function Constraints Non-linear optimization routine
26 Objective Function Position of end effector Or position and orientation Or aiming at For eample, ou might want the hand to slide along the tabletop or to keep the glass upright. ( ( ( ( p P p P (, ( (, ( vectors orthonormal pair of as a the orientation goal is, where ( ( (, e g e e e g e e g g e g e g e e P P P e e
27 Formulation minimize G( joint limits are inequalities Solution subject to use a standard numerical a a i i e ( and G ( P ( e G technique i i < b b u i i l i i
28 Summar of Kinematics Forward is straightforward Inverse usuall requires a numerical solution Ma not alwas get the right answer Frame-to-frame coherence (fi long segments Natural looking motion (how defined? If ou don t like the solution add more constraints hardl an elegant solution he stud of motion without regard to the forces that cause it--- so, what about those forces? Simulation---echnique #3
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