CSCE 552 Fall 2012

1. CSCE 552 Fall 2012 Animations By Jijun Tang

2. Animation terms • frame – an image that is displayed on the screen, usually as part of a sequence. • pose – an orientation of an objects or a hierarchy of objects that defines extreme or important motion. • keyframe – a special frame that contains a pose. • tween – the process of going “between” keyframes. • secondary motion – an object motion that is the result of its connection or relationship with another object.

3. Skeletal Hierarchy • The Skeleton is a tree of bones • Modelling characters • Often flattened to an array in practice • Each bone has a transform, stored relative to its parent’s transform • Top bone in tree is the “root bone” • Normally the hip • May have multiple trees, so multiple roots • Transforms are animated over time • Tree structure is often called a “rig”

4. Example

5. The Transform • “Transform” is the term for combined: • Translation • Rotation • Scale • Shear • Can be represented as 4x3 or 4x4 matrix • But usually store as components • Non-identity scale and shear are rare

6. Examples

7. Homogeneous coordinates • Four-dimensional space • Combines 3  3 matrix and translation into one 4  4 matrix

8. Translation • Translation matrix • Translates the origin by the vector T

9. Scale • Scale matrix • Scales coordinate axes by a, b, and c • If a = b = c, the scale is uniform

10. Rotation (Z) • Rotation matrix • Rotates points about the z-axis through the angle q

11. Rotations (X, Y) • Similar matrices for rotations about x, y

12. Three rotations about three axes Intuitive meaning of values Euler Angles

13. Euler Angles • This means that we can represent an orientation with 3 numbers • A sequence of rotations around principle axes is called an Euler Angle Sequence • Assuming we limit ourselves to 3 rotations without successive rotations about the same axis, we could use any of the following 12 sequences: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ

14. Using Euler Angles • To use Euler angles, one must choose which of the 12 representations they want • There may be some practical differences between them and the best sequence may depend on what exactly you are trying to accomplish

15. Interpolating Euler Angles • One can simply interpolate between the three values independently • This will result in the interpolation following a different path depending on which of the 12 schemes you choose • This may or may not be a problem, depending on your situation • Note: when interpolating angles, remember to check for crossing the +180/-180 degree boundaries

16. Problems • Euler Angles Are Evil • No standard choice or order of axes • Singularity “poles” with infinite number of representations • Interpolation of two rotations is hard • Slow to turn into matrices • Use matrix rotation

17. Rotation Matrix

18. 3x3 Matrix Rotation • Easy to use • Moderately intuitive • Large memory size - 9 values • Interpolation is hard • Introduces scales and shears • Need to re-orthonormalize matrices after

19. Quaternions • Quaternions are an interesting mathematical concept with a deep relationship with the foundations of algebra and number theory • Invented by W.R.Hamilton in 1843 • In practice, they are most useful to use as a means of representing orientations • A quaternion has 4 components

20. Quaternions on Rotation • Represents a rotation around an axis • Four values <x,y,z,w> • <x,y,z> is axis vector times sin(θ /2) • w is cos(θ/2) • Interpolation is fast

21. Illustration

22. Quaternions (Imaginary Space) • Quaternions are actually an extension to complex numbers • Of the 4 components, one is a ‘real’ scalar number, and the other 3 form a vector in imaginary ijk space!

23. Quaternions (Scalar/Vector) • Sometimes, they are written as the combination of a scalar value s and a vector value v where

24. Unit Quaternions • For convenience, we will use only unit length quaternions, as they will be sufficient for our purposes and make things a little easier • These correspond to the set of vectors that form the ‘surface’ of a 4D hypersphere of radius 1 • The ‘surface’ is actually a 3D volume in 4D space, but it can sometimes be visualized as an extension to the concept of a 2D surface on a 3D sphere

25. Quaternions as Rotations • A quaternion can represent a rotation by an angle θ around a unit axis a: • If a is unit length, then q will be also

26. Quaternions as Rotations

27. Quaternion to Matrix • To convert a quaternion to a rotation matrix:

28. Matrix to Quaternion • Matrix to quaternion is doable • It involves a few ‘if’ statements, a square root, three divisions, and some other stuff • Search online if interested

29. Animation vs. Deformation • Skeleton + bone transforms = “pose” • Animation changes pose over time • Knows nothing about vertices and meshes • Done by “animation” system on CPU • Deformation takes a pose, distorts the mesh for rendering • Knows nothing about change over time • Done by “rendering” system, often on GPU

30. Pose

31. Model • Describes a single type of object • Skeleton + rig • One per object type • Referenced by instances in a scene • Usually also includes rendering data • Mesh, textures, materials, etc • Physics collision hulls, gameplay data, etc

32. Instance • A single entity in the game world • References a model • Holds current states: • Position & orientation • Game play state – health, ammo, etc • Has animations playing on it • Stores a list of animation controls • Need to be interpolated

33. Animation Control • Links an animation and an instance • 1 control = 1 anim playing on 1 instance • Holds current data of animation • Current time • Speed • Weight • Masks • Looping state

34. Animation Storage • The Problem • Decomposition • Keyframes and Linear Interpolation • Higher-Order Interpolation • The Bezier Curve • Non-Uniform Curves • Looping

35. Storage – The Problem • 4x3 matrices, 60 per second is huge • 200 bone character = 0.5Mb/sec • Consoles have around 256-512Mb • Animation system gets maybe 25% • PC has more memory, but also higher quality requirements

36. Decomposition • Decompose 4x3 into components • Translation (3 values) • Rotation (4 values - quaternion) • Scale (3 values) • Skew (3 values) • Most bones never scale & shear • Many only have constant translation • But human characters may have higher requirement • Muscle move, smiling, etc. • Cloth under winds • Don’t store constant values every frame, use index instead

37. Keyframes • Motion is usually smooth • Only store every nth frame (key frames) • Interpolate between keyframes • Linear Interpolate • Inbetweening or “tweening” • Different anims require different rates • Sleeping = low, running = high • Choose rate carefully

38. Linear Interpolation

39. Higher-Order Interpolation • Tweening uses linear interpolation • Natural motions are not very linear • Need lots of segments to approximate well • So lots of keyframes • Use a smooth curve to approximate • Fewer segments for good approximation • Fewer control points • Bézier curve is very simple curve

40. Bézier Curves (2D & 3D) • Bézier curves can be thought of as a higher order extension of linear interpolation p1 p1 p2 p3 p1 p0 p0 p0 p2

41. The Bézier Curve • (1-t)3F1+3t(1-t)2T1+3t2(1-t)T2+t3F2 T2 t=1.0 T1 F2 t=0.25 F1 t=0.0

42. The Bézier Curve (2) • Quick to calculate • Precise control over end tangents • Smooth • C0 and C1 continuity are easy to achieve • C2 also possible, but not required here • Requires three control points per curve • (assume F2 is F1 of next segment) • Far fewer segments than linear

43. C0/C1/C2 The curves meet the"speed" is the same before and after the tangents are shared

44. Catmull-Rom Curve • Defined by 4 points. Curve passes through middle 2 points. • P = C3t3 + C2t2 + C1t + C0 • C3 = -0.5 * P0 + 1.5 * P1 - 1.5 * P2 + 0.5 * P3C2 = P0 - 2.5 * P1 + 2.0 * P2 - 0.5 * P3C1 = -0.5 * P0 + 0.5 * P2C0 = P1

45. Non-Uniform Curves • Each segment stores a start time as well • Time + control value(s) = “knot” • Segments can be different durations • Knots can be placed only where needed • Allows perfect discontinuities • Fewer knots in smooth parts of animation • Add knots to guarantee curve values: Transition points between animations

46. Looping and Continuity • Ensure C0 and C1 for smooth motion • At loop points • At transition points: walk cycle to run cycle • C1 requires both animations are playing at the same speed: reasonable requirement for anim system

47. Playing Animations • “Global time” is game-time • Animation is stored in “local time” • Animation starts at local time zero • Speed is the ratio between the two • Make sure animation system can change speed without changing current local time • Usually stored in seconds • Or can be in “frames” - 12, 24, 30, 60 per second

48. Scrubbing • Sample an animation at any local time • Important ability for games • Footstep planting • Motion prediction • AI action planning • Starting a synchronized animation • Walk to run transitions at any time • Avoid delta-compression storage methods • Very hard to scrub or play at variable speed

49. Delta Compression • Delta compression is a way of storing or transmitting data in the form of differences between sequential data rather than complete files. • The differences are recorded in discrete files called deltas or diffs. • Because changes are often small (only 2% total size on average), it can greatly reduce data redundancy. • Collections of unique deltas are substantially more space-efficient than their non-encoded equivalents.

50. Animation Blending • The animation blending system allows a model to play more than one animation sequence at a time, while seamlessly blending the sequences • Used to create sophisticated, life-like behavior • Walking and smiling • Running and shooting