Lecture 08: Animation

1. Lecture 08:Animation COMP 175: Computer Graphics March 4, 2014

2. Topics in Animation • Physics (dynamics, simulation, mechanics) • Particles • Rigid bodies (collisions, contact, etc.) • Articulated bodies (constraints, robotics) • Deformable bodies (fracture, cloth, elastic materials) • Natural phenomenon (fluid, water, fire, etc.) • Character dynamics (motion, skin, muscle, hair, etc.) • Character Animation • Motion capture • Motion synthesis (locomotion, IK, procedural motion, retargeting) • Motion playback • Artificial Intelligence • Behavioral animation • Simulation of crowds (flocks, herds, crowds) • Video game agents • Camera control

3. Key Framing • Traditionally used in cel animation • Animator specifies the positions and orientations at various key points • In between frames are then interpolated. • This used to be done by hand • Now this is mostly done by computers • John Lasseter, SIGGRAPH 87, “Principles of Traditional Animation Applied to 3D Computer Graphics”

4. Interpolating Key Frames • Splines are typically used to interpolate the positions of objects

5. Splines • The easiest spline to use is the cubic spline. • As the name suggests, cubic means an equation with a power of 3, which also means that we need 4 numbers to define the equation • E.g., for a linear curve, y = ax + b, there are two unknowns (a and b), so we need two sets of equations. For quadratic function, y = ax^2 + bx + c, we will need three sets of equations, etc. • With key framing, how do we get a “smooth” curve between any two points?

6. Cubic Splines • General form of a cubic spline: • q(t) = a + bt * ct^2 + dt^3 • q’(t) = b + 2ct + 3dt^2 • Given: • q(0) = S (starting point) • q’(0) = Vs (velocity at point S) • q(1) = G (goal point) • q’(1) = Vg (velocity at end point G)

7. Cubic Spline • Plug in the numbers: • q(0) = a + b(0) + c(0) + d(0) = S • => a = S • q’(0) = b + 2c(0) + 3d(0) = Vs • => b = Vs • Two remaining variables (c, d), two equations. Solve for c and d: • q(1) = a + b(1) + c(1) + d(1) = G • q’(1) = b + 2c + 3d = Vg

8. Cubic Spline • Plug in a = S, and b = Vs • q(1) = S + Vs + c + d = G ---- (eq1) • q’(1) = Vs + 2c + 3d = Vg ---- (eq2) • Solve for c first, (eq1)*3 – (eq2) • => c = -3S + 3G – 2Vs – 2Vg • Solve for d, (eq2) – ((eq1) * 2) • => d = 2S – 2G + Vs + Vg

9. Cubic Spline • Putting it all together: • q(t) = ( 1S + 0G + 0Vs + 0Vg) + ( 0S + 0G + 1Vs + 0Vg) * t + (-3S + 3G - 2Vs - 1Vg) * t * t + ( 2S - 2G + 1Vs + 1Vg) * t * t * t • Test this by plugging in t = 0 and t = 1

10. Splines in 3D • Note that the above equation solves for the cubic spline in one dimension, but you can generalize it to 3D. • This is because that since x, y, and z or orthogonal, we can compute the forces to x, y, and z independently.

11. Generalized Motion • Animators in the past didn’t have access to computers, so they had to “simulate” what our eyes see when perceiving motion. • These are known as stretch and squash

12. Cartoon Physics

13. How to Obtain the Key Frames? • Generating key frames continues to be an important question for animator because different effects require different techniques

14. Key Framing in Character Animation • Skeletons and Skinning • Skeletons: a collection of bones (rigid bodies) and joints • Skinning: associating each point of the mesh to an affine combination of bone/joint locations • Real-time deformation by applying skinning weights to deformed skeleton

15. Skeletal Animation • Popular in movies / games and supported by a wide range of software such as Maya, 3D Max.

16. Kinematics • Forward Kinematics: • “Given a set of joint angles, what is the x,y,z position of my hand?” • Inverse Kinematics: • “Given the position of my hand, what is the pose (joint angles) of my character?” • Note that the solution might not be unique

17. Simple IK • In a simple case: that is, • 2D, • two links, • 1 fixed joint (can rotate by can’t move), and • 1 regular joint • We can solve for theta_knee using cos-1 because we know the length of link1 and link2. • Given theta_knee, we can find beta

18. Complex IK • In more complex cases that involves multiple (hierarchical) joints, the problem is much more difficult. • Again, this is partially because the solution space could be empty, or it could be large (i.e., solutions are not unique) • The common solution is to use an iterative approximation algorithm (such as gradient descent, simulated annealing, etc.)

19. A Quick Reference to Gradient Descend • The general idea behind gradient descend method is based on the assumption that: • Given a function f, you can quickly compute f(x) in g • However, it is not easy to directly compute g(x) • In addition, there’s an “error function” that tells you how “wrong” you are (and sometimes in what directionality the error is – that is, the gradient) • So, gradient descent becomes iterative in that: • The algorithm takes a random guess on x • Given the error feedback, compute a delta_x • Then compute f(x + delta_x) • And so on

20. Inverse Kinematics • Given an articulated figure, there are often constraints to the joints (e.g., how far they can bend) • The iterative approach can integrate these constraints when looking for the solution x

21. Motion Capture • Different technologies, but roughly categorized into two groups: • Passive: passive systems are usually camera based. This means that an actor would wear reflective markers on a body suit, and cameras would capture and compute depth information. The benefits are that it’s easy to set up; can scale up to multiple actors; can be used on anything (such as faces or non-human actors). Downsides are: noise, occlusion, cost (in hardware and software), fixed environment (in the lab) • Active: active systems do direct measurements of the joint angles. These include skeletal suits or flexors to measure forces.The benefits are almost exactly the opposite of the passive systems

22. Passive Systems

23. Active Systems

24. Hybrid • “Practical Motion Capture in Everyday Surroundings” by Vlasic et al., SIGGRAPH 07

25. Motion Graphs • Motion capture labs are still largely very expensive to set up (depth cameras could cost hundreds of thousands) • One approach to cut down the cost is to capture “snippets” of motions. For example, for a walk cycle, a gesture, etc. • Given a sufficiently large motion library, one can generate a plausible motion graph by stitching these motion snippets together • The “easy” challenge is in blending the motions to hide the seams • The “hard” challenge is to model intent

26. Motion Graph • How would you blend motions?

27. Motion Retargeting • The idea of motion map sounds great, but there are still some big issues: • Motion captured data only works on a 3D model of the exact same (real) person. • For example, think about how an infant (toddler, and troll) walks differently because of differences in body part proportions (http://research.cs.wisc.edu/graphics/Gallery/Retarget/final_render.htm) • And can we apply walking motion to, say, the Pixar Lamp? (Semantic Deformation Transfer -- http://www.mit.edu/~ibaran/sdt/)

28. Motion Synthesis • In addition to the retargeting issue, there is also the problem of “coverage”. Namely, what happens if there is a missing chunk of animation that you need but it isn’t in the library/database? • http://www.cs.cmu.edu/~aw/pdf/spacetime.pdf • http://www.cs.cmu.edu/~aw/mpg/luxo.mpg • http://graphics.cs.cmu.edu/nsp/papers/sig97.pdf • http://graphics.cs.cmu.edu/nsp/projects/spacetime/spacetime.html

29. Perception of Motion • “Plausible Motion Simulation for Computer Graphics Animation”, Barzel et al., SIGGRAPH 96 • http://www.ronenbarzel.org/papers/plausible/ • “Perceptual Metrics for Character Animation”, Reitsma et all, SIGGRAPH 02 • http://graphics.cs.cmu.edu/nsp/projects/perception/perception.html

30. Generalized Motion • The motion of any dynamic system can be described with the following equation: • = generalized forces • = generalized positions • = generalized velocities • = generalized accelerations • = mass or inertia coefficients • = gravity and velocity contributions to Q (e.g., centrifugal and coriolis forces)

31. Generalized Motion • This is kind of a fancy way of saying f = ma, but still has the problem of needing to work with the linear forces and the rotational forces (torque) separately. • Thankfully, we can put the two together in one package in matrix form.

32. Generalized Forces • Where: • = force, a vector in • = mass of the body • = mass matrix (a 3x3 matrix), scalar m times the 3x3 identity matrix • = inertia tensor of a body (a 3x3 matrix) • = gravity, a vector in • = linear acceleration of the body, a vector in • = angular velocity of the body, a vector in • = angular acceleration of the body, a vector in

33. Going back to the Generalized Form • Using the original (generalized form), this means that:

34. Example • In this example, • , and • But the other terms, and are much more complicated. • See additional handout for more detail

35. Physically Based Simulation • Rigid Bodies • Fast collision detection • Transference of forces • Fast computation for large number of objects • Deformable Bodies • Natural phenomenon • http://physbam.stanford.edu/~fedkiw/ • http://www.youtube.com/watch?v=nJWz0PaMlkI

36. Last Word • Remco’s Master’s thesis…