Chapter 3 : Computer Animation

Transcription

1 Chapter 3 : Computer Animation Histor First animation films (Disne) 30 drawings / second animator in chief : ke frames others : secondar drawings Use the computer to interpolate? positions orientations shapes Or give the trajector eplicitl «Descriptive animation» The animator full controls motion 1. Descriptive animation Based on interpolation methods! To interpolate positions: splines curves Interpolation: Hermite curves or Cardinal splines Local control Made of polnomial curve segments degree 3, class C 1 Control point Spline curve Eercise Goal: animate a bouncing ball Describe a method for computing the trajector from the control points. How would ou animate the changes of speed? What is missing in this kinematic animation in terms of realism? Direct kinematics

2 Direct kinematics Direct kinematics Interpolating ke positions Interpolation curves Enable inflection points! (where C 0 onl) Speed control: Reparameterie the trajector «velocit curve» dist time Interpolation of orientations Choose the right representation! Rotation matri? Euler angle? Quaternion? Rotation matri Euler Angles Representation : orthogonal matri each orientation = 9 coefficients Interpolation : Interpolate coefficients one b one Re-orthogonalie and re-normalie Costl and inappropriate : M = k M 1 + (1-k) M 2 can be degenerate Impossible to approimate it b an orthogonal matri in this case Eemple: Ais, angle cos sin 0 Eercise: 0 sin cos M 1 = Id M 2 = rotation ais, = M for k=0.5? Representation : Three angles (,, ) Intuitive : R(V) = R, (R, (R, (V))) «Roll, pitch, aw» in flight simulators

3 Interpolating Euler Angles + more efficient : 3 values for 3 Degrees of Freedom (DoF) non-invariant b rotation, and un-natural result Problem with Euler Angles: gimbal lock Two or more aes aligned = loss of rotation DoF and 3 do the same! 1 Quaternions Quaternions Representation : q = (cos(/2), sin(/2)n) S 4 B analog: 1, 2, 3-DoF rotations as points on 2D, 3D, 4D spheres N Algebra of quaternions Generated b (1,i,j,k) where the neutral elt is = (1,0,0,0) and i 2 = j 2 = k 2 = ijk = = 1 ij = -ji = k jk = -kj = i ki = -ik = j Notation q = (q r, q p ) where q p R 3 p. q = (p r q r p p q p, p r q p + q r p p + p p q p ) q -1 = (q r, -q p ) / (q r 2 + q p.q p )

4 Quaternions Quaternions Used to represent rotations Rotation (, N): q = (cos(/2), sin(/2) N) N Interpolate quaternions? : splines on S 4 Interpolation method? q 5 q 3 q 2 S 4 Unit quaternion S 4 Appl a rotation R(V) = q. (0,V). q -1 Compose two rotations : p. q q 5 q 3 q 4 q 2 S 4 q 1 ω Linear + project non-uniform speed! Use spherical! q 4 q 1 The are essential for animation! Ees move with head Hands move with arms Feet move with legs Hierarchical structures 1 V 1 Hierarchical structures Generalied coordinates Vector of degrees of freedom (DoF) at each joint Eample Frame hierarch Root epressed in the world frame (translation + rotation) Relative rotation with respect to the parent

5 Forward kinematics Forward kinematics Method: Interpolate ke rotations Eercise : Controlling a ccling motion Define ke-rotations over time What is the main difficult? What would be the etra problem for a walking motion? Conclusion: Difficult to control etremities! (eample : foot position while ccling) In practice: Top-down set-up method Tr to compensate un-desired motions! Advanced method: Inverse kinematics Descriptive Models Animate Deformations Control of the end of a chain Automaticall compute the other orientations? 1 = f (q) 2 = f (????) Method from robotics Local inversions of a non-linear sstem = J q, with J ij i q j Jacobian matri Under-constrained sstem, pseudo-inverse : J + = J t (J J t ) -1 Eo: Show that (q = J + ) and (q = J + + (I-J + J) ) are solutions. What can (called secondar task ) be used for? J q generalied coordinates 2 1 Interpolate «ke shapes» Eample : «Disne effects» Change scaling, color k (u) = (u 3 u 2 u 1) M spline [k i-1 k i k i+1 k i+2 ] t [Lasseter 1987]

6 Descriptive Models Animate Deformations Animate a geometric model = animate its parameters Eo: Propose methods to animate this bee with Disne effects including: squash & stretch anticipation