Animation

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 methodslinear, 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 modelP’ – 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