COMP 175 COMPUTER GRAPHICS. Lecture 10: Animation. COMP 175: Computer Graphics March 12, Erik Anderson 08 Animation

Transcription

1 Lecture 10: Animation COMP 175: Computer Graphics March 12, /37

2 Recap on Camera and the GL Matrix Stack } Go over the GL Matrix Stack 2/37

3 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 3/37

4 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 4/37

5 Interpolating Key Frames } Splines are typically used to interpolate the positions of objects 5/37

6 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? 6/37

7 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) 7/37

8 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 8/37

9 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 9/37

10 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 10/37

11 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. 11/37

12 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 12/37

13 Cartoon Physics 13/37

14 How to Obtain the Key Frames? } Generating key frames continues to be an important question for animator because different effects require different techniques 14/37

15 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 15/37

16 Skeletal Animation } Popular in movies / games and supported by a wide range of software such as Maya, 3D Max. 16/37

17 Kinematics } Forward Kinematics: } Given a set of joint angles, what is the x,y,z position of my hand? } This is basically your assignment 3 (and lab 3) with SceneGraph } 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 17/37

18 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 18/37

19 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.) 19/37

20 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 20/37

21 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 21/37

22 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 nonhuman 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 22/37

23 Passive Systems 23/37

24 Active Systems 24/37

25 Hybrid } Practical Motion Capture in Everyday Surroundings by Vlasic et al., SIGGRAPH 07 25/37

26 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 26/37

27 Motion Graph } How would you blend motions? 27/37

28 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 ( r.htm) } } And can we apply walking motion to, say, the Pixar Lamp? (Semantic Deformation Transfer /37

29 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? } } } } me/spacetime.html 29/37

30 Perception of Motion } Plausible Motion Simulation for Computer Graphics Animation, Barzel et al., SIGGRAPH 96 } } Perceptual Metrics for Character Animation, Reitsma et all, SIGGRAPH 02 } rception.html 30/37

31 Rotations and Quaternions } Gimbal Lock problems with using Euler rotations } } (real world example, Apollo 13): } Solution in Graphics: Quaternions } Gnarly math, but the general idea is that the rotation happens in 4D (w, i, j, k) } W represents rotation and is a real number, i, j, k represents an axis of rotation and are imaginary such that: i " = j " = k " = ijk = 1 } on 31/37

32 Generalized Motion } The motion of any dynamic system can be described with the following equation: Q = H q q + C q, q } Q = generalized forces } q = generalized positions } q = generalized velocities } q = generalized accelerations } H q = mass or inertia coefficients } C q, q = gravity and velocity contributions to Q (e.g., centrifugal and coriolis forces) 32/37

33 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. 33/37

34 Generalized Forces f τ = M 0 0 I } Where: } f = force, a vector in R ; } τ = force, a vector in R ; 34/37 x ω + mg ω Iω } m = mass of the body } M = mass matrix (a 3x3 matrix), scalar m times the 3x3 identity matrix } I = inertia tensor of a body (a 3x3 matrix) } g = gravity, a vector in R ; } x = linear acceleration of the body, a vector in R ; } ω = angular velocity of the body, a vector in R ; } ω = angular acceleration of the body, a vector in R ;

35 Going back to the Generalized Form } Using the original (generalized form), this means that: Q = H q q + C q, q } Q = f τ } q = x ω } H q = M 0 0 I } C q, q = mg ω Iω 35/37

36 Example } In this example, } q = X GHIJ Y GHIJ θ GHIJ θ M θ ", and f GHIJO f GHIJP } Q = τ GHIJ τ M τ " } But the other terms, H q and C q, q are much more complicated. 36/37

37 Physically Based Simulation } Rigid Bodies } Fast collision detection } Transference of forces } Fast computation for large number of objects } Deformable Bodies } Natural phenomenon } ford.edu/~fedkiw/ } e.com/watch?v=njw z0pamlki 37/37