Computer Animation Animation Methods Keyframing Interpolation Kinematics Inverse Kinematics Slides courtesy of Leonard McMillan and Jovan Popovic Lecture 13 6 837 Fall 2002 Administrative Office hours Durand Teller by appointment Ngan Thursday 4 7 in W20 575 Deadline for proposal Friday Nov 1 Meeting with faculty staff about proposal Next week Web page for appointment Lecture 14 Slide 2 6 837 Fall 2002 Animation 4 approaches to animation Pros Cons Lecture 14 Slide 3 6 837 Fall 2002 Computer Assisted Animation Keyframing automate the inbetweening good control less tedious creating a good animation still requires considerable skill and talent Procedural animation ACM 1987 Principles of traditional animation describes the motion algorithmically applied to 3D computer animation express animation as a function of small number of parameteres Example a clock with second minute and hour hands hands should rotate together express the clock motions in terms of a seconds variable the clock is animated by varying the seconds parameter Example 2 A bouncing ball Abs sin t 0 e kt Lecture 14 Slide 4 6 837 Fall 2002 Computer Assisted Animation Physically Based Animation Assign physical properties to objects masses forces inertial properties Simulate physics by solving equations Realistic but difficult to control ACM 1988 Spacetime Constraints Motion Capture Captures style subtle nuances and realism You must observe someone do something Lecture 14 Slide 5 6 837 Fall 2002 Overview Keyframing and interpolation Interpolation of rotations quaternions Kimematrics articulation Particles x t Rigid bodies v t Deformable objects clothes fluids Lecture 14 Slide 6 6 837 Fall 2002 Kinematics vs Dynamics Kinematics Describes the positions of the body parts as a function of the joint angles Dynamics Describes the positions of the body parts as a function of the applied forces Lecture 14 Slide 7 6 837 Fall 2002 Now Dynamics ACM 1988 Spacetime Constraints Lecture 14 Slide 8 6 837 Fall 2002 Particle A single particle in 2 D moving in a flow field x1 Position x x 2 v 1 dx Velocity v v dt v 2 x2 The flow field function dictates particle velocity v g x t x t g x t x1 Lecture 14 Slide 9 6 837 Fall 2002 Vector Field The flow field g x t is a vector field that defines a vector for any particle position x at any time t x2 g x t x1 How would a particle move in this vector field Lecture 14 Slide 10 6 837 Fall 2002 Differential Equations The equation v g x t is a first order differential equation dx g x t dt The position of the particle is computed by integrating the differential equation t x t x t 0 g x t dt t0 For most interesting cases this integral cannot be computed analytically Lecture 14 Slide 11 6 837 Fall 2002 Numeric Integration Instead we compute the particle s position by numeric integration starting at some initial point x t0 we step along the vector field to compute the position at each subsequent time instant This type of a problem is called an initial value problem x2 x t 1 x t 2 x t 0 x1 Lecture 14 Slide 12 6 837 Fall 2002 Euler s Method Euler s method is the simplest solution to an initial value problem Euler s method starts from the initial value and takes small time steps along the flow x t t x t t g x t Why does this work Let s look at a Taylor series expansion of function x t d x t 2 d 2 x x t t x t t L 2 dt 2 dt Disregarding higher order terms and replacing the first derivative with the flow field function yields the equation for the Euler s method Lecture 14 Slide 13 6 837 Fall 2002 Other Methods Euler s method is the simplest numerical method The error is proportional to t 2 For most cases the Euler s method is inaccurate and unstable requiring very small steps x2 Other methods Midpoint 2nd order Runge Kutta Higher order Runge Kutta 4th order 6th order Adams Adaptive Stepsize Lecture 14 Slide 14 x1 6 837 Fall 2002 Particle in a Force Field What is a motion of a particle in a force field The particle moves according to Newton s Law d 2x f f ma 2 dt m The mass m of a particle describes the particle s inertial properties heavier particles are easier to move than lighter particles In general the force field f x v t may depend on the time t and particle s position x and velocity v Lecture 14 Slide 15 6 837 Fall 2002 Second Order Differential Equations Newton s Law yields an ordinary differential equation of second order d 2 x t f x v t 2 dt m A clever trick allows us to reuse the same numeric differentiation solvers for first order differential equations If we define a new phase space vector y which consists of particle s position x and velocity v then we can construct a new first order differential equation whose solution will also solve the second order differential equation x y v Lecture 14 d y d x dt v dt d dt m v f Slide 16 6 837 Fall 2002 Particle Animation AnimateParticles n y0 t0 tf y y0 t t0 DrawParticles n y while t tf f ComputeForces y t dydt AssembleDerivative y f y t ODESolverStep 6n y dy dt DrawParticles n y Lecture 14 Slide 17 6 837 Fall 2002 Particle Animation Reeves et al 1983 Start Trek The Wrath of Kahn Star Trek The Wrath of Kahn Reeves et al 1983 Lecture 14 Slide 18 6 837 Fall 2002 Particle Modeling Lecture 14 Reeves et al 1983 Slide 19 6 837 Fall 2002 Overview Keyframing and interpolation Interpolation of rotations quaternions Kimematrics articulation Particles x t Rigid bodies v t Deformable objects clothes fluids Lecture 14 Slide 20 6 837 Fall 2002 Rigid Body Dynamics We could compute the motion of a rigid body by computing the motion of all constituent particles However a rigid body does not deform and position of few of its particles is sufficient to determine the state of the body in a phase space We ll start with a special particle located at the body s center of mass x t v t Lecture 14 x t y t v t Slide 21 6 837 Fall 2002 Net Force f1 t f2 t x t v t f t fi t i Lecture 14 f3 t Slide 22 6 837 Fall 2002 Net Torque f2 t f1 t p1b t x t pb2 t v t pb3 t t pbi x t fi t i f3 t Lecture 14 Slide 23 6 837 Fall 2002 Rigid Body Equation of Motion x t v t d d R t t R t y t f t dt dt M v t t I t t M v t linear momentum I t t angular momentum Lecture 14 Slide 24 6 837 Fall 2002 Simulations with Collisions Simulating motions with collisions requires that we detect …
View Full Document
Unlocking...