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. Example

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

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

6. Rotation Matrix

7. 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

8. 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

9. 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

10. Illustration

11. 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!

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

13. 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

14. 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

15. Quaternions as Rotations

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

17. Pose

18. Triangles • Fundamental primitive of pipelines • Everything else constructed from them • (except lines and point sprites) • Three points define a plane • Triangle plane is mapped with data • Textures • Colors • “Rasterized” to find pixels to draw

19. Mesh

20. Vertices • A vertex is a point in space • Plus other attribute data • Colors • Surface normal • Texture coordinates • Whatever data shader programs need • Triangles use three vertices • Vertices shared between adjacent triangles

21. 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

22. 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

23. 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

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

25. 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

26. 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

27. 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

28. Linear Interpolation

29. 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

30. 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

31. 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

32. 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

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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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.

40. 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

41. Blending Animations • The Lerp • Quaternion Blending Methods • Multi-way Blending • Bone Masks • The Masked Lerp • Hierarchical Blending

42. The Lerp • Foundation of all blending • “Lerp”=Linear interpolation • Blends A, B together by a scalar weight • lerp (A, B, i) = iA + (1-i)B • i is blend weight and usually goes from 0 to 1 • Translation, scale, shear lerp are obvious • Componentwise lerp • Rotations are trickier – normalized quaternions is usually the best method.

43. Quaternion Blending • Normalizing lerp (nlerp) • Lerp each component • Normalize (can often be approximated) • Follows shortest path • Not constant velocity • Multi-way-lerp is easy to do • Very simple and fast • Many others: • Spherical lerp (slerp) • Log-quaternion lerp (exp map)

44. Which is the Best • No perfect solution! • Each missing one of the features • All look identical for small interpolations • This is the 99% case • Blending very different animations looks bad whichever method you use • Multi-way lerping is important • So use cheapest - nlerp

45. Multi-way Blending • Can use nested lerps • lerp (lerp (A, B, i), C, j) • But n-1 weights - counterintuitive • Order-dependent • Weighted sum associates nicely • (iA + jB + kC + …) / (i + j + k + … ) • But no i value can result in 100% A • More complex methods • Less predictable and intuitive • Can be expensive

46. Bone Masks • Some animations only affect some bones • Wave animation only affects arm • Walk affects legs strongly, arms weakly • Arms swing unless waving or holding something • Bone mask stores weight for each bone • Multiplied by animation’s overall weight • Each bone has a different effective weight • Each bone must be blended separately • Bone weights are usually static • Overall weight changes as character changes animations

47. The Masked Lerp • Two-way lerp using weights from a mask • Each bone can be lerped differently • Mask value of 1 means bone is 100% A • Mask value of 0 means bone is 100% B • Solves weighted-sum problem • (no weight can give 100% A) • No simple multi-way equivalent • Just a single bone mask, but two animations

48. Hierarchical Blending • Combines all styles of blending • A tree or directed graph of nodes • Each leaf is an animation • Each node is a style of blend • Blends results of child nodes • Construct programmatically at load time • Evaluate with identical code each frame • Avoids object-specific blending code • Nodes with weights of zero not evaluated

49. Triangles • Fundamental primitive of pipelines • Everything else constructed from them • (except lines and point sprites) • Three points define a plane • Triangle plane is mapped with data • Textures • Colors • “Rasterized” to find pixels to draw

50. Mesh