1. CSCE 552 Spring 2010
Animation
By Jijun Tang

2. 2D Example of Transformingθ

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

4. What is Physics Simulation?
The Cycle of Motion:
Force, F(t), causes acceleration
Acceleration, a(t), causes a change in velocity
Velocity, V(t) causes a change in position
Physics Simulation: Solving variations of the above equations over time to emulate the cycle of motion

5. Concrete Example: Target Practice
Projectile Launch
Position, pinit
Under constant gravitional acceleration, projectile path is a parabola
Target

6. Errors
Exact
Euler

7. Overlap Testing Results
Useful results of detected collision
Pairs of objects will have collision
Time of collision to take place
Collision normal vector
Collision time calculated by moving object back in time
until right before collision
Bisection is an effective technique

8. Bisect Testing: Iteration V
Time right before the collision

9. Overlap Testing: Limitations
Fails with objects that move too fast
Thin glass vs. bulltes
Unlikely to catch time slice during overlap

10. Intersection Testing: Swept Geometry
Extrude geometry in direction of movement
Swept sphere turns into a “capsule” shape

11. Simplified Geometry
Approximate complex objects with simpler geometry, like this ellipsoid or bounding boxes

12. Minkowski Sum

13. Using Minkowski Sum

14. Reduce Number of Detections
O(n) Time Complexity can be achieved.
One solution is to partition space

15. Animation Overview Fundamental Concepts Animation Storage
Playing Animations
Blending Animations
Motion Extraction
Mesh Deformation
Inverse Kinematics
Attachments & Collision Detection
Conclusions

16. Animation Roles
Programmer – loads information created by the animator and translates it into on screen action.
Animator – Sets up the artwork. Provides motion information to the artwork.

17. Different types of animation
Particle effects (fire, smoke, etc)
Procedural / Physics
“Hard” object animation (door, robot)
“Soft” object animation (tree swaying in the wind, flag flapping the wind)
Character animation

18. 2D Versus 3D Animation Borrow from traditional 2D animation
Understand the limitations of what can be done for real-time games
Designing 3D motions to be viewed from more than one camera angle
Pace motion to match game genre

19. 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.

20. Fundamental Problems Volume of data, processor limitations
Mathematical complexity, especially for rotations.
Translation of motion

21. Fundamental Concepts Skeletal Hierarchy The Transform Euler Angles
The 3x3 Matrix
Quaternions
Animation vs Deformation
Models and Instances
Animation Controls

22. 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”

23. Example

24. 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

25. Examples

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

27. 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

28. 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

29. 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

30. Problems Euler Angles Are Evil Use matrix rotation
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

31. Rotation Matrix

32. 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

33. 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

34. 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

35. Illustration

36. 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!

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

38. 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

39. 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

40. Quaternions as Rotations

41. Quaternion to Matrix
To convert a quaternion to a rotation matrix:

42. 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

43. 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

44. Pose

45. 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

46. 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

47. 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

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

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

50. 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

51. 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

52. Key Frame Example
Recall we have done something in Flash

53. Linear Interpolation

54. 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

55. 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

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

57. 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

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

59. Bézier Variants Store 2F2-T2 instead of T2
Equals next segment T1 for smooth curves
Store F1-T1 and T2-F2 vectors instead
Same trick as above – reduces data stored
Called a “Hermite” curve
Catmull-Rom curve
Passes through all control points

60. Catmull-Rom Curve
Defined by 4 points. Curve passes through middle 2 points.
P = C3t3 + C2t2 + C1t + C0
C3 = -0.5 * P * P * P * P3 C2 = P * P * P * P3 C1 = -0.5 * P * P2 C0 = P1

61. 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

62. 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

63. 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

64. 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

65. 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.

66. 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

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

68. 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.

69. Quaternion Blending Normalizing lerp (nlerp) Many others:
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)

70. 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

71. Multi-way Blending Can use nested lerps Weighted sum associates nicely
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

72. 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

73. 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

74. 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