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? a b θ=?

8. Example: Angle Between Vectors a b θ

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

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

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

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

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. p h t

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

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 = B T ·A T

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·M 1 )·M 2 )·M 3 )·M 4 … Or we can concatenate the transformations into a single matrix: M total =M 1 ·M 2 ·M 3 ·M 4 … v’=v·M total

22. Trigonometry 1.0 cos θ sin θ θ cos 2 θ + sin 2 θ = 1

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

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 (φ 1 …φ N ) 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 1. Specify all DOF values for the skeleton (done by higher level animation system) 2. Recursively traverse through the hierarchy starting at the root and use forward kinematics to compute the world matrices (done by skeleton system) 3. 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 = L joint (φ 1,φ 2,…,φ N ) Then, world matrix W is computed by concatenating L with the world matrix of the parent joint World matrix W = L · W parent

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 = [φ 1 φ 2 … φ N ] 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...

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

51. Project 1 Assignment Load a skeleton from a ‘.skel’ file and display it in 3D All joints in the skeleton should be 3-DOF rotational joints Compute world matrices yourself and use glLoadMatrixf() Due: Monday, Jan 19, 11:59pm

52. Sample Skel File balljoint root { [data for root] balljoint head { [data for head] [children of head] } balljoint leg_l { [data for leg] [children of leg] } [more children of root] }

53. Skel File Data Tokens offsetx y z(joint offset vector) boxminx y z(min corner of box to draw) boxmaxx y z(max corner of box to draw) rotxlimitmin max(x rotation DOF limits) rotylimitmin max(y rotation DOF limits) rotzlimitmin max(z rotation DOF limits) posex y z(values to pose DOFs) balljoint name { }(child joint)

54. Possible Object Breakdown One should consider making objects (classes) for the following: DOF Joint Skeleton

55. Common Routines Many classes will need functions for some or all of the following: Constructor / destructor Initialize Load Update (move things, pose, animate…) Draw Reset

56. What is a DOF? Data Value Min, max Functions SetValue() (can clamp value at the time of setting) GetValue() SetMinMax()…

57. What is a Joint? Data Local & World matrices Array of DOFs Tree data (child/sibling/parent pointers, etc.) Functions Update() (recursively generate local matrix & concatenate) Load() AddChild() Draw() Note: One could also make a Joint base class and derive various specific joint types. In this case, it would be a good idea to make a virtual function for MakeLocalMatrix() that the base traversal routine calls

58. What is a Skeleton? Data Joint tree (might only need a pointer to the root joint) Functions Load Update (traverses tree & computes joint matrices) Draw

59. Tree Data Structures The skeleton requires only the most basic N-tree data structure The main things the tree needs: An easy way to add a new child to a node A way for each node to iterate through its children There are several options: Use STL Implement a tree data structure Store sibling & first child pointers in the Joint itself

60. *Tree Data Structure

61. *Sibling/Child Tree

62. Sibling & Child Pointers class Joint { Joint *Sibling,*Child; … // other data public: bool Load(Tokenizer &t); bool AddChild(Joint &c); void Update(Matrix &parent); void Draw(); … // other functions };

63. AddChild() bool Joint::AddChild(Joint &c) { c.Sibling=Child; Child=&c; return true; } Ideally, it should do some error checking as well (such as verifying that c isn’t already a child of another node) Note that Child & Sibling must be set to 0 initially in the constructor! Also, notice that this routine adds c as the first child on the list (essentially adding them in reverse order)

64. Update & Draw void Joint::Update(Matrix &parent) { … // Compute LocalMatrix … // Compute WorldMatrix if(Sibling) Sibling->Update(parent); if(Child) Child->Update(WorldMatrix); } void Joint::Draw() {.. // Do some OpenGL if(Sibling) Sibling->Draw(); if(Child) Child->Draw(); }

65. Load bool Skeleton::Load(const char *file) { Tokenizer token; token.Open(file,"skel")); token.FindToken("balljoint")); // Parse tree Root=new Joint; Root->Load(token); // Finish token.Close(); return true; }

66. bool Joint::Load(Tokenizer &t) { token.FindToken("{")); while(1) { char temp[256]; token.GetToken(temp); if(strcmp(temp,"offset")==0) { Offset.x=token.GetFloat(); Offset.y=token.GetFloat(); Offset.z=token.GetFloat(); } else // Check for other tokens else if(strcmp(temp,"balljoint")==0) { Joint *jnt=new Joint; jnt->Load(token); AddChild(*jnt); } else if(strcmp(temp,"}")==0) return true; else token.SkipLine();// Unrecognized token }

67. Notes About Sibling/Child Trees Hard to implement a tree with less code A bit on the ‘quick and dirty’ side though Not abstracting the tree structure away means that code will be duplicated if other classes need to be in their own trees The double recursion in the traversal routines can tend to lead to unnecessarily deep runtime stacks (stack overflows)