Skeletons

1. Skeletons CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2004

2. Linear Algebra Review

3. Coordinate Systems • Right handed coordinate system

4. Vector Arithmetic

5. Vector Magnitude • The magnitude (length) of a vector is: • Unit vector (magnitude=1.0)

6. Dot Product

7. Example: Angle Between Vectors • How do you find the angle θ between vectors A and B? b θ=? a

8. Example: Angle Between Vectors b θ a

9. Dot Products with Unit Vectors 0 <a·b < 1 a·b = 0 a·b = 1 b θ a -1 < a·b < 0 a·b = -1

10. Dot Products with Non-Unit Vectors • If a and b are arbitrary (non-unit) vectors, then the following are still true: • If θ < 90º then a·b > 0 • If θ = 90º then a·b = 0 • If θ > 90º then a·b < 0

11. Dot Products with One Unit Vector • If |u|=1.0 then a·u is the length of the projection of a onto u a u a·u

12. *Example: Distance to Plane

13. Cross Product

14. Properties of the Cross Product area of parallelogram ab is perpendicular to both a and b, in the direction defined by the right hand rule

15. Example: Area of a Triangle • Find the area of the triangle defined by 3D points a, b, and c c b a

16. Example: Area of a Triangle c c-a b a b-a

17. Example: Alignment to Target • An object is at position p with a unit length heading of h. We want to rotate it so that the heading is facing some target t. Find a unit axis a and an angle θ to rotate around. t • • p h

18. Example: Alignment to Target a t t-p • θ • p h

19. Matrices • Computer graphics apps commonly use 4x4 homogeneous matrices • A rigid 4x4 matrix transformation looks like this: • Where a, b, & c are orthogonal unit length vectors representing orientation, and d is a vector representing position

20. Matrices • The right hand column can cause a projection, which we won’t use in character animation, so we leave it as 0,0,0,1 • Some books store their matrices in a transposed form. This is fine as long as you remember that: A·B = BT·AT

21. Transformations • To transform a vector v by matrix M: v’=v·M • If we want to apply several transformations, we can just multiply by several matrices: v’=(((v·M1)·M2)·M3)·M4… • Or we can concatenate the transformations into a single matrix: Mtotal=M1·M2·M3·M4… v’=v·Mtotal

22. Trigonometry cos2θ+ sin2θ= 1 1.0 sin θ θ cos θ

23. Laws of Sines and Cosines • Law of Sines: • Law of Cosines: b α γ c a β

24. Skeletons

25. Kinematics • Kinematics: The analysis of motion independent of physical forces. Kinematics deals with position, velocity, acceleration, and their rotational counterparts, orientation, angular velocity, and angular acceleration. • Forward Kinematics: The process of computing world space geometric data from DOFs • Inverse Kinematics: The process of computing a set of DOFs that causes some world space goal to be met (I.e., place the hand on the door knob…) • Note: Kinematics is an entire branch of mathematics and there are several other aspects of kinematics that don’t fall into the ‘forward’ or ‘inverse’ description

26. Skeletons • Skeleton: A pose-able framework of joints arranged in a tree structure. The skeleton is used as an invisible armature to manipulate the skin and other geometric data of the character • Joint: A joint allows relative movement within the skeleton. Joints are essentially 4x4 matrix transformations. Joints can be rotational, translational, or some non-realistic types as well • Bone: Bone is really just a synonym for joint for the most part. For example, one might refer to the shoulder joint or upper arm bone (humerus) and mean the same thing

27. DOFs • Degree of Freedom (DOF): A variable φ describing a particular axis or dimension of movement within a joint • Joints typically have around 1-6 DOFs (φ1…φN) • Changing the DOF values over time results in the animation of the skeleton • In later weeks, we will extend the concept of a DOF to be any animatable parameter within the character rig • Note: in a mathematical sense, a free rigid body has 6 DOFs: 3 for position and 3 for rotation

28. Example Joint Hierarchy

29. Joints • Core Joint Data • DOFs (N floats) • Local matrix: L • World matrix: W • Additional Data • Joint offset vector: r • DOF limits (min & max value per DOF) • Type-specific data (rotation/translation axes, constants…) • Tree data (pointers to children, siblings, parent…)

30. Skeleton Posing Process • Specify all DOF values for the skeleton (done by higher level animation system) • Recursively traverse through the hierarchy starting at the root and use forward kinematics to compute the world matrices (done by skeleton system) • Use world matrices to deform skin & render (done by skin system) Note: the matrices can also be used for other things such as collision detection, FX, etc.

31. Forward Kinematics • In the recursive tree traversal, each joint first computes its local matrix L based on the values of its DOFs and some formula representative of the joint type: Local matrix L = Ljoint(φ1,φ2,…,φN) • Then, world matrix W is computed by concatenating L with the world matrix of the parent joint World matrix W = L · Wparent

32. Joint Offsets • It is convenient to have a 3D offset vector r for every joint which represents its pivot point relative to its parent’s matrix

33. DOF Limits • It is nice to be able to limit a DOF to some range (for example, the elbow could be limited from 0º to 150º) • Usually, in a realistic character, all DOFs will be limited except the ones controlling the root

34. Skeleton Rigging • Setting up the skeleton is an important and early part of the rigging process • Sometimes, character skeletons are built before the skin, while other times, it is the opposite • To set up a skeleton, an artist uses an interactive tool to: • Construct the tree • Place joint offsets • Configure joint types • Specify joint limits • Possibly more…

35. Poses • Once the skeleton is set up, one can then adjust each of the DOFs to specify the pose of the skeleton • We can define a pose Φ more formally as a vector of N numbers that maps to a set of DOFs in the skeleton Φ = [φ1 φ2 … φN] • A pose is a convenient unit that can be manipulated by a higher level animation system and then handed down to the skeleton • Usually, each joint will have around 1-6 DOFs, but an entire character might have 100+ DOFs in the skeleton • Keep in mind that DOFs can be also used for things other than joints, as we will learn later…

36. Joint Types

37. Rotational Hinge: 1-DOF Universal: 2-DOF Ball & Socket: 3-DOF Euler Angles Quaternions Translational Prismatic: 1-DOF Translational: 3-DOF (or any number) Compound Free Screw Constraint Etc. Non-Rigid Scale Shear Etc. Design your own... Joint Types

38. Hinge Joints (1-DOF Rotational) • Rotation around the x-axis:

39. Hinge Joints (1-DOF Rotational) • Rotation around the y-axis:

40. Hinge Joints (1-DOF Rotational) • Rotation around the z-axis:

41. Hinge Joints (1-DOF Rotational) • Rotation around an arbitrary axis a:

42. Universal Joints (2-DOF) • For a 2-DOF joint that first rotates around x and then around y: • Different matrices can be formed for different axis combinations

43. Ball & Socket (3-DOF) • For a 3-DOF joint that first rotates around x, y, then z: • Different matrices can be formed for different axis combinations

44. Quaternions

45. Prismatic Joints (1-DOF Translation) • 1-DOF translation along an arbitrary axis a:

46. Translational Joints (3-DOF) • For a more general 3-DOF translation:

47. Other Joints • Compound • Free • Screw • Constraint • Etc. • Non-Rigid • Scale (1 axis, 3 axis, volume preserving…) • Shear • Etc.

48. Programming Project #1: Skeleton

49. Software Architecture • Object oriented • Make objects for things that should be objects • Avoid global data & functions • Encapsulate information • Provide useful interfaces • Put different objects in different files

50. Sample Code • Some sample code is provided on the course web page listed as ‘project0’ • It is an object oriented demo of a spinning cube • Classes: • Vector3 • Matrix34 • Tokenizer • Camera • SpinningCube • Tester