1. Kinematics

2. ILE5030 Computer Animation and Special Effects2 Kinematics The branch of mechanics concerned with the motions of objects without regard to the forces that cause the motion Why kinematics? Hierarchical articulated model Posing a character

3. ILE5030 Computer Animation and Special Effects3 Degrees of Freedom (DOF) The minimum number of coordinates required to specify completely the motion of an object 6 DOF: x, y, z, raw, pitch, yaw pitch roll yaw

4. ILE5030 Computer Animation and Special Effects4 Degrees of Freedom in Human Model Root: 3 translational DOF + 3 rotational DOF Rotational joints are commonly used Each joint can have up to 3 DOF Shoulder: 3 DOF Wrist: 2 DOF Knee: 1 DOF 1 DOF 3 DOF

5. ILE5030 Computer Animation and Special Effects5 Revolute Joints 3 DOF joint gimbal ball and socket 2 DOF joint universal

6. ILE5030 Computer Animation and Special Effects6 Hierarchical Articulated Model Represent an articulated figure as a series of links connected by joints Enforce limb connectivity in a tree-like structure root

7. ILE5030 Computer Animation and Special Effects7 Joint Space vs. Cartesian Space Joint space space formed by joint angles position all joints—fine level control Cartesian space 3D space specify environment interactions

8. ILE5030 Computer Animation and Special Effects8 Forward and Inverse Kinematics Forward kinematics mapping from joint space to cartesian space Inverse kinematics mapping from cartesian space to joint space Forward KinematicsInverse Kinematics

9. ILE5030 Computer Animation and Special Effects9 Forward and Inverse Kinematics (cont.) Forward kinematics rendering Inverse kinematics good for specifying environment interaction good for controlling a character—fewer parameters

10. ILE5030 Computer Animation and Special Effects10 Forward Kinematics ? Base End Effector

11. ILE5030 Computer Animation and Special Effects11 Forward Kinematics by Composing Transformations End Effector Relative rotation from the local coordinate of parent link to the local coordinate of the child link

12. ILE5030 Computer Animation and Special Effects12 Tree Traversal T 1.1 T 1.2 T 2.1 T 2.2 T0T0 M = I M = T 0 M = T 0 *T 1.1 M = T 0 *T 1.1 *T 2.1 M = T 0 M = T 0 *T 1.2 M = T 0 *T 1.2 *T 2.2 M = T 0 *T 1.1

13. ILE5030 Computer Animation and Special Effects13 Inverse Kinematics Base End Effector=P

14. ILE5030 Computer Animation and Special Effects14 Redundancy in IK Our example 2 equations (constraints) 3 unknowns Multiple solutions exist! This is not uncommon! see how you can move your elbow while keeping your finger touching your nose

15. ILE5030 Computer Animation and Special Effects15 Other problems in IK Infinite solutions

16. ILE5030 Computer Animation and Special Effects16 Other problems in IK No solutions

17. ILE5030 Computer Animation and Special Effects17 Why is IK hard? Redundancy Natural motion joint limits minimum jerk style? Singularities ill-conditioned matrix shown later

18. ILE5030 Computer Animation and Special Effects18 Solving Inverse Kinematics analytic method inverse-Jacobian method optimization-based method example-based method

19. ILE5030 Computer Animation and Special Effects19 Analytic Method (X,Y) L1 L2 11 22 Goal

20. ILE5030 Computer Animation and Special Effects20 Analytic Method (X,Y) L1 L2 11 TT 180-  2 X Y

21. ILE5030 Computer Animation and Special Effects21 Cosine Law A B C 

22. ILE5030 Computer Animation and Special Effects22 Analytic Method (X,Y) L1 L2 11 TT 180-  2 X Y

23. ILE5030 Computer Animation and Special Effects23 Inverse-Jacobian method When linkage is complicated Iteratively change the joint angles to approach the goal position and orientation

24. ILE5030 Computer Animation and Special Effects24 Jacobian Jacobian is the n by m matrix relating differential changes of   to  differential changes of P (dP) Jacobian maps velocities in joint space to velocities in cartesian space

25. ILE5030 Computer Animation and Special Effects25 An Example of Jacobian Matrix

26. ILE5030 Computer Animation and Special Effects26 Linearize about  k  locally Small increments Iteratively Solving PP+dPP goal

27. ILE5030 Computer Animation and Special Effects27 Jacobian may not be invertible! Non-square matrix pseudo inverse Singularity causes infinite joint velocities occurs when any cannot achieve a given

28. ILE5030 Computer Animation and Special Effects28 Pseudo Inverse of the Jacobian

29. ILE5030 Computer Animation and Special Effects29 Null space The null space of J is the set of vectors which have no influence on the constraints The pseudoinverse provides an operator which projects any vector to the null space of J

30. ILE5030 Computer Animation and Special Effects30 A solution of this form …doesn’t affect the desired configuration But it can be used to bias The solution vector When put into this formula Like this After some manipulation, you can show that it…

31. ILE5030 Computer Animation and Special Effects31 Utility of Null Space The null space can be used to reach secondary goals Or to find natural pose / control stiffness of joints

32. ILE5030 Computer Animation and Special Effects32 Optimization-based Method Formulate IK as an nonlinear optimization problem Example Objective function Constraint Iterative algorithm Nonlinear programming method by Zhao & Badler, TOG 1994

33. ILE5030 Computer Animation and Special Effects33 Objective Function “distance” from the end effector to the goal position/orientation Function of joint angles : G(  ) Base End Effector Goal distance

34. ILE5030 Computer Animation and Special Effects34 Objective Function Position Goal Orientation Goal Position/Orientation Goal weighted sum

35. ILE5030 Computer Animation and Special Effects35 Nonlinear Optimization Constrained nonlinear optimization problem Solution standard numerical techniques MATLAB or other optimization packages usually a local minimum depends on initial condition joint limits limb coordination

36. ILE5030 Computer Animation and Special Effects36 Example-based Method Utilize motion database to assist IK solving IK using interpolation Rose et al., “Artist-directed IK using radial basis function interpolation,” Eurographics’01

37. ILE5030 Computer Animation and Special Effects37 Example-based Method (cont.) IK using constructed statistical model Grochow et al., “Style-based inverse kinematics,” SIGGRAPH’04 Provide the most likely pose based on given constraints

38. ILE5030 Computer Animation and Special Effects38 Videos “Style-based inverse kinematics”

39. ILE5030 Computer Animation and Special Effects39 Assignment 1 Keyframing using different interpolation approaches and orientation representations Due at 11:59 PM on Oct 15 (Sun) An ASF/AMC viewer is provided No need to be panic on FK implementation! See the course web for details Quick tutorial given by TA on Thursday

40. ILE5030 Computer Animation and Special Effects40 Notes on ASF/AMC format Developed by Acclaim ASF skeleton file defines tree hierarchy defines rotation axes of a joint represented in global coordinate defines bone length AMC motion file Euler angles

41. ILE5030 Computer Animation and Special Effects41 Example Results

42. ILE5030 Computer Animation and Special Effects42 Forward Kinematics A series of transformations on an object can be applied as a series of matrix multiplications : position in the global coordinate : position in the local coordinate

43. ILE5030 Computer Animation and Special Effects43 Hierarchical Articulated Model Represents a character ’ s skeleton as a series of links connected by joints connectivity constraints is enforced in a tree-like structure family of parent-child spatial relationships