Computer Animation Algorithms and Techniques Kinematic Linkages

Hierarchical Modeling Define motion of one object relative to another Constrained motion - reduce dimensionality, simplify specification Linked appendages with joints Solar system Motion on a surface Local coordinate frames for simpler motion specification Data structure to facilitate keeping relative transformations

Solar system http://www.earth-live.net/tag/sun-earth-moon-system/

Linked appendages Joint Link End Effector

Forward & Inverse Kinematics images courtesy of Domin Lee

Revolute & Prismatic Joints

Complex Joints

Hierarchical structure

Tree structure

Tree structure

Tree structure Static

Relative movement

Tree structure Changable

Bifurcation in Model

Bifurcating tree

Traverse Tree Top to bottom traversal Accumulate transformations during traversal Transformation at node only for data Save transformation before following link Restore when traversing sibling link

traverse(arcPtr,matrix) { ; get transformations of arc and concatenate to current matrix matrix = matrix*arcPtr->Lmatrix ; concatenate location matrix = matrix*arcPtr->Amatrix ; concatenate articulation ; process data at node nodePtr = arcPtr->nodePtr ; get the node of the arc push(matrix) ; save the matrix matrix = matrix * nodePtr->matrix ; ready for articulation articulatedData = transformData(matrix,dataPtr) ; articulate the data draw(articulatedData); ; and draw it matrix = pop() ; restore matrix for node’s children ; process children of node if (nodePtr->arcPtr!= NULL) { ; if not a terminal node nextArcPtr = nodePtr->arcPtr ; get first arc emanating from node while (nextArcPtr != NULL) { ; while there’s an arc to process push(matrix) ; save matrix at node traverse(nextArcPtr,matrix) ; traverse arc matrix = pop() ; restore matrix at node nextArcPtr = nextArcPtr->arcPtr ; set next child of node }

Inverse kinematics Position and orient end effect Automatically calculate interior joint angles Only simple configurations can be solved analytically Often, solution must be iterative

Inverse kinematics Analytic method Iterative: Jacobian Iterative solutions Pseudo-inverse adding more control alternative Jacobian using the transpose cyclic coordinate descent

Inverse Kinematics

Inverse Kinematics - Analytic

Inverse Kinematics - Analytic

Inverse Kinematics - Numeric Determine affect of joint i on end effector

Inverse Kinematics - Numeric desired direction of end effector to goal

Inverse Kinematics - Numeric joint-effect vectors: gi

Singularities

IK - Jacobain

IK - goal out of reach Solutions with and without damping

IK - with control term all joints biased to 0 top: joint 2 has higher gain bottom: joint 3 has higher gain

IK - Pull goal to end effector

IK - transpose of the Jacobian

IK - Cyclic Coordinate Descent