1. Character Animation Forward and Inverse Kinematics
CSE 3541
Matt Boggus

2. Overview Concepts Defining forward and inverse kinematics
Example calculations for forward kinematics
Inverse kinematics algorithms
Other topics
Ragdoll physics
Dynamics

3. Simplifies motion specification
Hierarchical Modeling
Parent-child relationship
Simplifies motion specification
Relative motion
Constrains motion
Reduces dimensionality

4. Modeling & animating hierarchies
3 aspects
Linkages & Joints – the relationships
Data structure – how to represent such a hierarchy
Converting local coordinate frames into global space

5. Terms Joint – allowed relative motion & parameters
Joint Limits – limit on valid joint angle values
Link – object involved in relative motion
Linkage – entire joint-link hierarchy
End effector – most distant link in linkage
Pose – configuration of linkage using given set of joint angles

6. Forward vs. Inverse Kinematics
Forward Kinematics
Input: joint angles
Output: link positions and orientations
end effector position
Inverse Kinematics
Input: end effector
Output: joint angles

7. Joints – relative movement
Every joint allowing motion in one dimension is said to have one degree of freedom (DOF)
Example: flying (Six DOF)
x, y, and z positions (prismatic or translation)
roll, pitch, and yaw (revolute or rotation)

8. Hierarchical structure

9. Forward Kinematics: A Simple Example
Forward kinematics map as a coordinate transformation
The body local coordinate system of the end-effector was initially coincide with the global coordinate system
Forward kinematics map transforms the position and orientation of the end-effector according to joint angles
Example from Jehee Lee

10. A Chain of Transformations

11. Thinking of Transformations
In a view of global coordinate system

12. Thinking of Transformations
In a view of global coordinate system

13. Thinking of Transformations
In a view of global coordinate system

14. Thinking of Transformations
In a view of global coordinate system

15. Thinking of Transformations
In a view of global coordinate system

16. Thinking of Transformations
In a view of body-attached coordinate system

17. Thinking of Transformations
In a view of body-attached coordinate system

18. Thinking of Transformations
In a view of body-attached coordinate system

19. Thinking of Transformations
In a view of body-attached coordinate system

20. Thinking of Transformations
In a view of body-attached coordinate system

21. Real-time forward kinematics
QWOP

22. Inverse kinematics Given goal position for end effector
Compute internal joint angles
Simple linkages  analytic solution
Otherwise  numeric iterative solution

23. IK solution uniqueness
Zero One
Two Many
Images from

24. Inverse kinematics – spaces
Configuration space
Reachable workspace

25. Analytic Inverse Kinematics
Given target position (X,Y)
Compute angle with respect to origin of linkage
Solve for angle of rotation for each joint
See reference text for solution and derivation

26. IK - numeric
Solve iteratively – numerically solve for step toward goal
Remember: a pose is the position and orientation of all links for given a set of joint angles
- Compute the desired change from this pose
Vector to the goal, or
Minimal distance between end effector and goal
- Compute set of changes to joint angles to make that change

27. Inverse Kinematics - Numeric
Given
Current configuration
Goal position
Determine
Goal vector
Positions & local coordinate systems of interior joints (in global coordinates)
Solve for change in joint angles & take small step, optionally clamp acceleration or clamp velocity
Repeat until:
Within epsilon of goal
Stuck in some configuration
Taking too long

28. Given the values for each xi, we can compute each yi
IK math notation
Given the values for each xi, we can compute each yi

29. IK – chain rule
The change in output variables (y) relative to the change in input variables (x)

30. Inverse Kinematics - Jacobian
Desired motion of end effector
Unknown change in articulation variables
The Jacobian is the matrix relating the two: describing how each coordinate changes with respect to each joint angle in our system

31. Inverse Kinematics - Jacobian
Change in position
Change in orientation
Change in articulation variables
Jacobian

32. If J is not square, use of pseudo-inverse of Jacobian
Solving
If J square, compute inverse, J-1
If J is not square, use of pseudo-inverse of Jacobian

33. Another example linkage

34. IK – compute positional change vectors induced by changes in joint angles
Instantaneous positional change vectors
Desired change vector
One approach to IK computes linear combination of change vectors that equal desired vector

35. Jacobian psuedocode
while ( Distance(endEffector,goal) > threshold && time < timeLimit) { vel = goal – endEffector; j = ComputeJacobian(linkage); pj = ComputePseudoinverse(j); deltaTheta = pj * vel; rotateJoints(deltaTheta,linkage); }
In addition to the reference text, should you want to implement this I also recommend

36. IK - singularity
Some singular configurations are not so easily recognizable

37. IK – cyclic coordinate descent
Heuristic solution
Consider one joint at a time, from outside in
At each joint, choose update that best gets end effector to goal position
In 2D – pretty simple
Goal
axisi Ji
EndEffector

38. Cyclic-Coordinate Descent
- Starting with the root of our effector, R, to our current endpoint, E.
- Next, we draw a vector from R to our desired endpoint, D
- The inverse cosine of the dot product gives us the angle between the vectors: cos(a) = RD ● RE

39. Cyclic-Coordinate Descent
Rotate our link so that RE falls on RD

40. Cyclic-Coordinate Descent
Move one link up the chain, and repeat the process

41. Cyclic-Coordinate Descent
The process is basically repeated until the root joint is reached. Then the process begins all over again starting with the end effector, and will continue until we are close enough to D for an acceptable solution.

42. Cyclic-Coordinate Descent

43. Cyclic-Coordinate Descent
We’ve reached the root. Repeat the process

44. Cyclic-Coordinate Descent

45. Cyclic-Coordinate Descent

46. Cyclic-Coordinate Descent

47. Cyclic-Coordinate Descent

48. Cyclic-Coordinate Descent

49. Cyclic-Coordinate Descent
We’ve reached the root again. Repeat the process until solution reached.

50. IK – cyclic coordinate descent
Other orderings of processing joints are possible
Because of its procedural nature
Lends itself to enforcing joint limits
Easy to clamp angular velocity

51. Dynamics – Featherstone equations

52. Dynamics – Featherstone equations
Forces + linkage constraints
For the details of the equations, see

53. Approximating dynamics
Archived reference of 2003 Gamasutra article:
Archived IO interactive slides:
Human particle/stick model
Video example
Not shown (points can be matched/rigged to 3D character model)

54. Stick position constraint solving
Figure and code from reference in previous slide

55. Stick angle constraint solving
Figure and code from reference in previous slide
(x2 – x1) ∙ (x2 – x1) > d
-OR-
(x2 – x0) ∙ (x1 – x0) < a
Where a and d are constants, representing limits

56. Additional slides

57. Derivation for analytical IK example

58. Complex Joints

59. IK – 3D cyclic coordinate descent
EndEffector
Goal Ji
axisi
Projected goal
First – goal has to be projected onto plane defined by axis (normal to plane) and EF
Second– determine angle at joint