1. Animation Following “Advanced Animation and Rendering Techniques” (chapter 15+16) By Agata Przybyszewska

2. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics/inverse kinematics

3. Keyframing in 2D Skilled animator draws frames Less skilled draws inbetweens

4. Keyframing in 3D Skilled animator draws important keyframes Computer generates in-betweens using interpolation Rigid body motion not enough…

5. Keyframing We can move points in 3D translation, rotation We can set keyframes (= parameter to interpolate): –position, –orientation, –deformation, –lights, –camera, –opacity?

6. Speed control We want a constant velocity interpolation from A to B Then we can control our own velocities Interpolation methods linear, cubic, polynomial

7. Keyframing? Arc length parametrization -> Speed curves -> control of movement Problem: uniform steps in parameter don’t correspond to uniform path distances Solution: parametrization by arclength

8. Overview Keyframing –Parametrizing by arclength –Rotations –Quaternion interpolation Kinematics/inverse kinematics

9. Transformations Every 3D transformation is a composition of –Rotation –Scaling –Translation We can compose transformations P – point of the model P’ – point after transformation

10. Homogenous coordinates Use homogenous coordinates in graphics Use 4 by 4 matrix to represent: scaling, translation, rotation

11. Gimbal lock Gimbal = device used for holding a gyroscope Illustrates problem with interpolating Euler angles Gimbal lock is a basic problem with representing 3D rotations using Euler angles

12. Axis Angle Eulers rotation theorem Any orientation can be represented by a 4-tuple angle, vector(x,y,z) where angle – amount to rotate vector – axis of rotation We can interpolate angle and axis seperately

14. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics/inverse kinematics

20. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics –Articulated figures –Forward kinematics –Inverted kinematics

21. Kinematics Study of movement without regards to the forces that cause it

22. Kinematics Inbetweening –Inverse kinematics or dynamics Articulated figure –Structure of rigid liks connected at joints Degrees of freedom (D0F) – number of independent position variables End effector –End of chain of links

23. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics –Articulated figures –Forward kinematics –Inverted kinematics

24. Articulated figures Suitable for humanoid figures Want the structure of the body to be maintained Horrible approximation

25. Articulated figures Inbetweening –Compute joint angles between computer frames

26. DH notation Represents state of articulated structure Attach coordinate frame to each link Transformation matrix between consecutive coordinate frames

27. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics –Articulated figures –Forward kinematics –Inverted kinematics

28. Forward kinematics Joint motion can be specified by spline curves

29. Forward kinematics

30. Example Walk cycle Hip joint orientation

31. Example Walk cycle Ankle joint orientation

32. Overview Keyframing –Parametrizing by arclength –General kinematics –Rotations –Quaternion interpolation Kinematics –Articulated figures –Forward kinematics –Inverse kinematics

33. Inverse kinematics What if animation knows position at the end? Pick up object from plate, put object in place

34. Inverse kinematics End effector positions can be specified by spline curves

35. Inverse kinematics Solution for more complex structures –Find best solution (ie. Minimize energy in motion) –Non linear optimization

36. What makes IK hard? Ill conditioned near singularities –High state velocities for low cartesian velocities Redundancies –Add constraints to reduce redundancies Find “closest” solution: –Minimize time –Minimize energy –Natural looking motion

37. IK and the Jacobian Use inverse Jacobian Jacobian maps velocities in state space to velocities in cartesian space Iteratively step all the joint angles toward the goal

38. Computing IK Computational problems –Singularities – change rank –Jacobian only valid for given configuration Non linear optimization –Nummerical programming: Method for finding (local) minimum of function

39. Summary of kinematics Forward kinematics –Specify conditions (joint angles) –Compute positions and end-effectors Reverse kinematics –“Goal directed” motion –Specify goal positions of end effectors –Compute conditions required to achieve goals For many tasks inverse kinematics provides easier specification, but is more computationally difficult