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Unformatted text preview: Computer Animation Algorithms and Techniques Kinematic Linkages Objectives • Implement hierarchical kinematics. • Compare and contrast forward and backward kinematics • Explain the role of the Jacobian in IK. • Implement numerical iterative IK using the transpose of the Jacobian. Hierarchical Modeling Relative motion Parentchild relationship Constrains motion Reduces dimensionality Simplifies motion specification Modeling & animating hierarchies 3 aspects 1. Linkages & Joints – the relationships 2. Data structure – how to represent such a hierarchy 3. Converting local coordinate frames into global space Some terms Joint – allowed relative motion & parameters Joint Limits – limit on valid joint angle values Link – object involved in relative motion Linkage – entire jointlink hierarchy Armature – same as linkage End effector – most distant link in linkage Articulation variable – parameter of motion associated with joint Pose – configuration of linkage using given set of joint angles Pose vector – complete set of joint angles for linkage Arc – of a tree data structure – corresponds to a joint Node – of a tree data structure – corresponds to a link Use of hierarchies in animation Forward Kinematics (FK) animator specifies values of articulation variables Inverse Kinematics (IK) animator specifies final desired global transform for end effector (and possibly other linkages) global transform for each linkage is computed Values of articulation variables are computed Forward & Inverse Kinematics Joints – relative movement Complex Joints Inverse kinematics Given goal position (and orientation) for end effector Compute internal joint angles If simple enough => analytic solution Else => numeric iterative solution Inverse kinematics  spaces Configuration space Reachable workspace Dextrous workspace Analytic inverse kinematics ) 2 ) ( cos( 2 1 2 2 2 2 2 1 2 L L Y X L L a + + = θ IK  numeric If linkage is too complex to solve analytically Desired change from this specific pose to reach some target. to reach some target....
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 Winter '08
 Jones,M
 Dot Product, Jacobian

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