1. COMP 175 | COMPUTER GRAPHICS Remco Chang1/3608 – Animation Lecture 08: Animation COMP 175: Computer Graphics March 10, 2015

2. COMP 175 | COMPUTER GRAPHICS Remco Chang2/3608 – 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 Topics in Animation

3. COMP 175 | COMPUTER GRAPHICS Remco Chang3/3608 – Animation  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” Key Framing

4. COMP 175 | COMPUTER GRAPHICS Remco Chang4/3608 – Animation  Splines are typically used to interpolate the positions of objects Interpolating Key Frames

5. COMP 175 | COMPUTER GRAPHICS Remco Chang5/3608 – Animation  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? Splines

6. COMP 175 | COMPUTER GRAPHICS Remco Chang6/3608 – Animation 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. COMP 175 | COMPUTER GRAPHICS Remco Chang7/3608 – Animation 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. COMP 175 | COMPUTER GRAPHICS Remco Chang8/3608 – Animation 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. COMP 175 | COMPUTER GRAPHICS Remco Chang9/3608 – Animation 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. COMP 175 | COMPUTER GRAPHICS Remco Chang10/3608 – Animation  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. Splines in 3D

11. COMP 175 | COMPUTER GRAPHICS Remco Chang11/3608 – Animation  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 Generalized Motion

12. COMP 175 | COMPUTER GRAPHICS Remco Chang12/3608 – Animation Cartoon Physics

13. COMP 175 | COMPUTER GRAPHICS Remco Chang13/3608 – Animation  Generating key frames continues to be an important question for animator because different effects require different techniques How to Obtain the Key Frames?

14. COMP 175 | COMPUTER GRAPHICS Remco Chang14/3608 – 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 Key Framing in Character Animation

15. COMP 175 | COMPUTER GRAPHICS Remco Chang15/3608 – Animation  Popular in movies / games and supported by a wide range of software such as Maya, 3D Max. Skeletal Animation

16. COMP 175 | COMPUTER GRAPHICS Remco Chang16/3608 – Animation  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 Kinematics

17. COMP 175 | COMPUTER GRAPHICS Remco Chang17/3608 – Animation  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 Simple IK

18. COMP 175 | COMPUTER GRAPHICS Remco Chang18/3608 – Animation  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.) Complex IK

19. COMP 175 | COMPUTER GRAPHICS Remco Chang19/3608 – Animation  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 A Quick Reference to Gradient Descend

20. COMP 175 | COMPUTER GRAPHICS Remco Chang20/3608 – Animation  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 Inverse Kinematics

21. COMP 175 | COMPUTER GRAPHICS Remco Chang21/3608 – Animation  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 Motion Capture

22. COMP 175 | COMPUTER GRAPHICS Remco Chang22/3608 – Animation Passive Systems

23. COMP 175 | COMPUTER GRAPHICS Remco Chang23/3608 – Animation Active Systems

24. COMP 175 | COMPUTER GRAPHICS Remco Chang24/3608 – Animation  “Practical Motion Capture in Everyday Surroundings” by Vlasic et al., SIGGRAPH 07 Hybrid

25. COMP 175 | COMPUTER GRAPHICS Remco Chang25/3608 – Animation  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 Motion Graphs

26. COMP 175 | COMPUTER GRAPHICS Remco Chang26/3608 – Animation  How would you blend motions? Motion Graph

27. COMP 175 | COMPUTER GRAPHICS Remco Chang27/3608 – Animation  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/) Motion Retargeting

28. COMP 175 | COMPUTER GRAPHICS Remco Chang28/3608 – Animation  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/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/spaceti me/spacetime.html Motion Synthesis

29. COMP 175 | COMPUTER GRAPHICS Remco Chang29/3608 – Animation  “Plausible Motion Simulation for Computer Graphics Animation”, Barzel et al., SIGGRAPH 96  http://www.ronenbarzel.org/papers/plausible/ http://www.ronenbarzel.org/papers/plausible/  “Perceptual Metrics for Character Animation”, Reitsma et all, SIGGRAPH 02  http://graphics.cs.cmu.edu/nsp/projects/perception/pe rception.html Perception of Motion

30. COMP 175 | COMPUTER GRAPHICS Remco Chang30/3608 – Animation Generalized Motion

31. COMP 175 | COMPUTER GRAPHICS Remco Chang31/3608 – Animation  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. Generalized Motion

32. COMP 175 | COMPUTER GRAPHICS Remco Chang32/3608 – Animation Generalized Forces

33. COMP 175 | COMPUTER GRAPHICS Remco Chang33/3608 – Animation Going back to the Generalized Form

34. COMP 175 | COMPUTER GRAPHICS Remco Chang34/3608 – Animation Example

35. COMP 175 | COMPUTER GRAPHICS Remco Chang35/3608 – Animation  Rigid Bodies  Fast collision detection  Transference of forces  Fast computation for large number of objects  Deformable Bodies  Natural phenomenon  http://physbam.stan ford.edu/~fedkiw/  http://www.youtub e.com/watch?v=nJW z0PaMlkI Physically Based Simulation

36. COMP 175 | COMPUTER GRAPHICS Remco Chang36/3608 – Animation  Remco’s Master’s thesis… Last Word