Keyframing Keyframing Motion of objects is described as a function - PDF document

04-05-2017 Keyframing Keyframing Motion of objects is described as a function of time from a set of key object positions ( keyframes ). s t ( ) • Keyframes are drawn by skilled animator. • Computer generates in-betweens using interpolation . Lecture 10 Slide 2 6.837 Fall 2003 1

04-05-2017 Keyframing Complex movements may require multiple keys. Interpolating Positions ( x , y , t ), i  0, , n Given positions: i i i x t ( ) x     C ( ) t   i C ( ) t   find curve such that   i y t ( ) y     i ( x , y , t ) 2 2 2 u ( x , y , t ) C ( ) t 0 0 0 0 ( x , y , t ) 1 1 1 Slide 4 2

04-05-2017 Linear Interpolation ( x , y , t ) 2 2 2 ( x , y , t ) 0 0 0 ( x , y , t ) 1 1 1   x t ( )  x 1  t  x t t =0 and =1 t Simple problem: 0 1 0 1 Any t i :  t  t x t  t   x , t   t , t 1 0   0 1 0 1 t  t t  t  1 0 1 0 x t ( )     t t t t  x  x , t   t , t 2 1     1 2 1 2 t  t t  t  2 1 2 1 Lecture 10 Slide 5 6.837 Fall 2003 Polynomial Interpolation ( x , y , t ) 2 2 2 ( x , y , t ) 0 0 0 parabola ( x , y , t ) 1 1 1 An n-degree polynomial can interpolate any n+1 points. The n+1 coefficients of an n-degree polynomial can be obtained from the n+1 points. Lecture 10 Slide 6 6.837 Fall 2003 3

04-05-2017 Spline Interpolation For large number of points (N), many polynomials of small degree can be used. Polynomials of small degree are faster and easier to control. t t t   x t 8-degree spline spline vs. polynomial polynomial Lecture 10 Slide 7 Spline Interpolation A cubic polynomial between each pair of points: x t ( )  c  c t  c t 2  c t 3 0 1 2 3 4 parameters ( degrees of freedom ) for each spline segment. Input parameters: 4 points 2 points + 2 directions Lecture 10 4

04-05-2017 Interpolating Key Frames • Interpolation is not fool proof. • The splines may undershoot and cause interpenetration. • The animator must solve these types of side-effects. Lecture 10 Slide 12 6.837 Fall 2003 Keyframing Keyframe animation can be applied to different parameters Parameter to interpolate: position, orientation, deformation, lights, camera, opacity? 5

04-05-2017 Physics Simulation 6

04-05-2017 Physics Simulation Particles Rigid bodies Deformable bodies Fluid dynamics Vehicle dynamics Characters Definitions Kinematics : The study of motion without consideration of the underlying forces Dynamics : Study of physical motion (or more abstractly, the study of change in physical systems) Forward Dynamics : Computing motion resulting from applied forces Inverse Dynamics : Computing forces required to generate desired motion Mechanics, Statics, Kinetics 7

04-05-2017 Particles Particula  p p , p , p x y z p ( t ) t ) p ( t + 8

04-05-2017 Kinematics of Particles Position : x Velocity: v = dx/dt a = dv/dt = d 2 x/dt 2 Acceleration:   x v dt Aproximações númericas    Exato se v 0 não varia em [0, t] x ( t ) x v t 0 0       x ( t ) x ( t t ) v ( t ) t Exato se v(t) não varia em  t 9

04-05-2017 Aceleração Acceleration a=a 0     v a dt a t v Velocity 0 0 1      2 x v dt a t v t x Position 0 0 0 2 Aproximações númericas 1      Exato se a não varia em [0, t] 2 x ( t ) x v t a t o 0 0 2 1          2 x ( t ) x ( t t ) v ( t ) t a ( t ) t 2 Exato se a(t) não varia em  t 10

04-05-2017 Forces Multiple forces can add up to a single total force: Forces cause change in momentum (accelerations)   f f total i  a f / M total Gravity Gravity near Earth’s surface is constant: f=m.g (g = -9.8 m/s 2 ) Gravity for distant objects: (G=6.673 × 10 -11 m 3 /kg·s 2 ) f=Gm 1 m 2 /r 2 11

04-05-2017 Spring-Damper Spring: f = -kx k=spring constant x=distance from rest state Damper: f=-cv c=damping factor v=velocity along spring axis Spring-damper: f=-kx-cv Aerodynamic Drag Drag force: f=(1/2) ρa c c d v 2 ρ=fluid density a c =cross sectional area c d =coefficient of drag (geometric constant based on shape of object, usually between 0 and 1, but can be higher) v=velocity of the object relative to velocity of the fluid Note: for simple cases, (1/2)ρa c c d is constant 12

04-05-2017 Friction Static friction: f ≤ f n μ s Dynamic friction: f = f n μ d f n =normal force μ s =coefficient of static friction μ d =coefficient of dynamic friction Force Fields Generic force fields can be created that use arbitrary rules to define a force at some location: f=f(x) Examples: vortex, attractors, turbulence, torus… 13

04-05-2017 Collisions: Impulse Momentum: p = m.v [kg.m.s -1 ] or [N.s] Impulse: J= Δp An impulse is a finite change in momentum Impulses are essentially large forces acting over a small time Modified momentum update:  p/m = p 0 /m+f Δt/m+J/m p=p 0 +f Δt+J  v = v 0 +a Δt+J/m Particle Simulation 1 Particle struct { Vector3d Velocity; Vector3d Position; float Mass; } UpdateParticle(float dtime) { Force = ComputeTotalForce(); Acel = Force/Mass; Velocity = Velocity + Acel * dtime; Position = Position + Velocity * dtime; } 14

04-05-2017 Particle Simulation 2 Particle struct { Vector3d Momentum; Vector3d Position; float Mass; } UpdateParticle(float dtime) { Impulse = ComputeTotalImpulse(dtime); Momentum = Momentum + Impulse; Velocity = Momentum / Mass; Position = Position + Velocity * dtime; } Integration Explicit Euler method: v=v 0 +a Δ t x=x 0 +v Δ t Optimize for: Other methods: • Stability Implicit Euler • Accuracy Runge-Kutta Crank-Nicholson • Convergence Multipoint • Performance Leapfrog DuFort-Frankel Adams, Adams-Moulton, Adams-Bashforth 15

04-05-2017 Rigid Bodies Angular Speed • The average angular speed,         ω avg , of a rotating rigid f i   avg t t t f i object is the ratio of the angular displacement to the time interval • The instantaneous angular    d    lim speed is defined as the t    0 t dt limit of the average speed as the time interval approaches zero • 16 • •

04-05-2017 Instantaneous Angular Acceleration • The instantaneous angular acceleration is defined as    d    lim t the limit of the average    0 t dt angular acceleration as the time goes to 0 • SI Units of angular acceleration: rad/s² Torque Torque, t , is tendency of a force to rotate object about some axis • F is the force t  Fd d is the lever arm (or moment arm) • Units are Newton m • Torque is vector: Direction determined by axis of twist t  Fd t  Fr sin  Φ is the angle between F and r 17

04-05-2017 Offset Forces Torque resulting from offset force: τ =r × f Total force:   f res f i Total torque:  t   ( r f ) res i i Rotational Inertia • Rotational inertia or moment of inertia I is the resistance of an object to changes in its rotational motion. • Rotational inertia I is the rotational equal of mass M . • Rotational inertia depends on: • Mass • Distribution of that mass around the axis of rotation. I = ⌠ ⌡ r 2 dm Units: kg·m 2 18

04-05-2017 Moments of Inertia Rotational Inertia For arbitrary axis:     I I I  xx xy xz     I  I I I  yx yy yz      I I I  zx zy zz   0  T I A I A    2 2 I ( ) y z dm xx   I 0 0   xx   I ( xy ) dm  xy I 0 0 I 0   yy     0 0 I zz A=3x3 orientation matrix 19

04-05-2017 Torque and Rotational Inertia • Torque T is the application of a force which changes the angular acceleration of the object. Newton’s Second Law for Rotation:  I   T Angular Momentum Rigid body L  I  L  mvr  m  r 2 Point particle Analogy between L and p Angular Momentum Linear momentum L = Iw p = mv t = dL/dt F = dp/dt t = I .  F = m.a Conserved if no net Conserved if no net outside outside torques forces 20

04-05-2017 Angular Momentum L = I  ω = A  I 0  A T  ω L =angular momentum I =rotational inertia ω=angular velocity A =3x3 orientation matrix Rigid Body Simulation RigidBody struct { Vector Position; Vector Velocity; Vector Orientation; Vector AngVelocity; float Mass; Matrix RotationInertia; } UpdateRigidBody(float dtime) { Force=ComputeTotalForce(); Torque=ComputeTotalTorque(); Aceleration = Force / Mass; Velocity = Velocity + Aceleration * dtime; Position = Position + Velocity * dtime; AngAceleration = Torque * Inverse(RotationalInertia); AngVelocity = AngVelocity + AngAceleration * dtime; Orientation = Orientation + AngVelocity * dtime; } 21