Reading. Topics in Articulated Animation. Character Representation. Animation. q i. t 1 t 2. Articulated models: Character Models are rich, complex

Transcription

1 Shoemake, Quaternions Tutorial Reading Topics in Articulated Animation 2 Articulated models: rigid parts connected by joints Animation They can be animated by specifying the joint angles (or other display parameters) as functions of time. Character Representation Character Models are rich, comple hair, clothes (particle systems) muscles, skin (FFD s etc.) q i q ( t ) i Focus is rigid-body Degrees of Freedom (DOFs) joint angles t 1 t 2 t 1 t 2 t 2 3 4

2 Simple Rigid Body Æ Skeleton Computing a Sensor Position vs. h,y h,z h,q h,f h,s h q t,f t,s t q c Forward kinematics uses vector-matri multiplication transformation matri is composition of all joint transforms between sensor/effector and root y Copyright Squaresoft 1999 q f,f f vw = v s z v s T(, y, z ) R( θ, φ, σ ) TR( θφσ,, )TR( θ ) TR ( θ, φ c ) h h h h h h t t t f f vs 5 6 Joints = Rotations To specify a pose, we specify the joint-angle rotations Each joint can have up to three rotational DOFs Euler angles An Euler angle is a rotation about a single Cartesian ais Create multi-dof rotations by concatenating Eulers Can get three DOF by concatenating: 1 DOF: knee 2 DOF: wrist 3 DOF: arm Euler-X Euler-Y Euler-Z 7 8

3 Singularities What is a singularity? continuous subspace of parameter space all of whose elements map to same rotation Why is this bad? induces gimbal lock - two or more aes align, results in loss of rotational DOFs (i.e. derivatives) Singularities in Action An object whose orientation is controlled by Euler rotation XYZ(q,f,s) ( θ φ σ) = ( θ) ( φ ) ( σ ) R,, R Ry Rz R(0,0,0) : Okay R(0, ± 90º, 0) : X and Z aes align 9 10 Eliminates a DOF Resulting Behavior In this configuration, changing q (X Euler angle) and s (Z Euler angle) produce the same result. No way to rotate around world X ais! y No applied force or other stimuli can induce rotation about world X-ais Ds (Z-rot) Df (Y-rot) Dq (X-rot) z The object locks up!! 11 12

4 Singularities in Euler Angles Cannot be avoided (occur at 0 or 90 ) Difficult to work around But, only affects three DOF rotations Other Properties of Euler Angles Several important tasks are easy: interactive specification (sliders, etc.) joint limits Euclidean interpolation (Hermites, Beziers, etc.) May be funky for tumbling bodies fine for most joints Quaternions But singularities are unacceptable for IK, optimization Traditional solution: Use unit quaternions to represent rotations S 3 has same topology as rotation space (a sphere), so no singularities History of Quaternions Invented by Sir William Rowan Hamilton in 1843 H = w+ i+ jy+ kz = = = = where i j k ijk 1 I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluions [the calculus] was for the close of the seventeenth. Hamilton [quaternions] although beautifully ingenious, have been an unmied evil to those who have touched them in any way. Thompson 15 16

5 Quaternion as a 4 vector Ais-angle rotation as a quaternion w w q = = y v z w w q = = y v z θ r cos( θ / 2) q = sin( θ / 2) r Unit Quaternions ( yz,, ) Quaternion Product w q = y z q = y + z + w = 1 w = + y + z ( ) q 1 w1 w2 ww 1 2 v1 v2 = v v wv + w v + v v w1 w2 w2 w1 v v v v q 3 q 2 q

6 Quaternion Conjugate Quaternion Inverse q * ( p ) * * ( pq) * w1 w1 = = v v 1 1 = p = q p * * * ( p+ q) = p + q * * * 1 q q= w w = / = / = /( w + ) v v q 1 q * q q 2 v v Quaternion Rotation Quaternion constraints Restricting the rotation cone θ p r 1 w 0 w qpq = v p v w p v = w v p p v wp v wp v = 0 = ww ( ) + ( ) + ( w ) p pv p v v v p p v What about a quaternion product qq 1 2? 23 r θ Restricting the rotation twist around an ais θ 1 cos( θ ) = q + 2 q tan( θ / 2) = q 2 2 y qz ais w 24

7 Matri Form Quaternions: What Works Simple formulae for converting to rotation matri w q = y z y 2z 2y + 2wz 2z 2wy 2 2 M = 2y 2wz 1 2 2z 2yz + 2w 2 2 2z + 2wy 2yz 2w 1 2 2y Continuous derivatives - no singularities Optimal interpolation - geodesics map to shortest paths in rotation space Nice calculus (corresponds to rotations) What Hierarchies Can and Can t Do Advantages: Reasonable control knobs Maintains structural constraints Disadvantages: Doesn t always give the right control knobs e.g. hand or foot position - re-rooting may help Can t do closed kinematic chains (keep hand on hip) Other constraints: do not walk through walls Procedural Animation Transformation parameters as functions of other variables Simple eample: a clock with second, minute and hour hands hands should rotate together epress all the motions in terms of a seconds variable whole clock is animated by varying the seconds parameter 27 28

8 Models as Code: draw-a-bug void draw_bug(walk_phase_angle, pos, ypos zpos){ pushmatri translate(pos,ypos,zpos) calculate all si sets of leg angles based on walk phase angle. draw bug body for each leg: pushmatri translate(leg pos relative to body) draw_bug_leg(theta1&theta2 for that leg) popmatri popmatri } Motion Editing Tools for transforming already eisting animations void draw_bug_leg(float theta1, float theta2){ glpushmatri(); glrotatef(theta1,0,0,1); draw_leg_segment(segment1_length) gltranslatef(segment1_length,0,0); glrotatef(theta2,0,0,1); draw_leg_segment(segment2_length) glpopmatri(); } Hard Eample In the figure below, what epression would you use to calculate the arm s rotation angle to keep the tip on the star-shaped wheel as the wheel rotates??? θ? 31