1. Computer Animation Algorithms and Techniques
Kinematic Linkages

2. 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

3. Solar system

4. Linked appendages
Joint
Link
End Effector

5. Forward & Inverse Kinematics
images courtesy of Domin Lee

6. Revolute & Prismatic Joints

7. Complex Joints

8. Hierarchical structure

9. Tree structure

10. Tree structure

11. Tree structure
Static

12. Relative movement

13. Tree
structure
Changable

14. Bifurcation in Model

15. Bifurcating tree

16. 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

17. 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 }

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

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

20. Inverse Kinematics

21. Inverse Kinematics - Analytic

22. Inverse Kinematics - Analytic

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

24. Inverse Kinematics - Numeric
desired direction of end effector to goal

25. Inverse Kinematics - Numeric
joint-effect vectors: gi

26. Singularities

27. IK - Jacobain

28. IK - goal out of reach
Solutions with and without damping

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

30. IK - Pull goal to end effector

31. IK - transpose of the Jacobian

32. IK - Cyclic Coordinate Descent