Computer Animation. Rick Parent

Transcription

1 Algorithms and Techniques Kinematic Linkages

2 Hierarchical Modeling Relative motion Parent-child relationship Simplifies motion specification Constrains motion Reduces dimensionality

3 Modeling & animating hierarchies 3 aspects 1. Linkages & Joints the relationships 2. Data structure how to represent such a hierarchy 3. Converting local coordinate frames into global space

4 Some terms Joint allowed relative motion & parameters Joint Limits limit on valid joint angle values Link object involved in relative motion Linkage entire joint-link hierarchy Armature same as linkage End effector most distant t link kin linkage Articulation variable parameter of motion associated with joint Pose configuration of linkage using given set of joint angles Pose vector complete set of joint angles for linkage Arc of a tree data structure corresponds to a joint Node of a tree data structure corresponds to a link

5 Use of hierarchies in animation Forward Kinematics (FK) animator specifies values of articulation variables global transform for each linkage is computed Inverse Kinematics (IK) animator specifies final desired global transform for end effector (and possibly other linkages) Values of articulation variables are computed

6 Forward & Inverse Kinematics

7 Joints relative movement

8 Comple Joints

9 Hierarchical structure

10 Tree structure

11 Tree structure

12 Tree structure

13 Relative movement

14 Relative movement

15 Tree structure

16 Tree structure

17 Implementation note Nodes & arcs NODE Pointer to data Data transformation Pointer to arcs ARC Transform of one net node relative to parent node Articulation transform Pointer to node

18 Implementation note Representing arbitrary number of children with fied-length data structure Use array of pointers to children In node arcptr[] Node points to first child Each child points to sibling Last sibling points to NULL In node: arcptr for 1 st child In arc: arcptr for sibling

19 Tree traversal L A dm traverse (arcptrmatri) { // concatenate arc matrices matri = matri*arcptr->lmatri matri = matri*arcptr->amatri; // get node and transform data nodeptr=acrptr->nodeptr push (matri) matri = matri * nodeptr->matri adata = transformdata(matridataptr) draw(adata) matri = pop(); } // process children If (nodeptr->arcptr!= NULL) { netarcptr = nodeptr-> arcptr while (netarcptr!= NULL) { push(matri) traverse(netarcptrmatri) matri = pop() netarcptr = netarcptr->arcptr } }

20 OpenGL Single linkage glpushmatri(); For (i=0; i<numdofs; i++) { } glpopmatri(); glrotatef(a[i]ais[i][0] ais[i][1] ais[i][2]); if (linklen[i]!= 0.0) { draw_linkage(linklen[i]); gltranslatef(0.0linklen[i]0.0); } OpenGL concatenates matrices A[i] joint angle Ais[i] joint ais linklen[i] length of link

21 Inverse kinematics Given goal position (and orientation) for end effector Compute internal joint angles If simple enough => analytic solution Else => numeric iterative solution

22 Inverse kinematics - spaces Configuration i space Reachable workspace Detrous workspace

23 Analytic inverse kinematics 2 2 L1 + L2 ( X + Y θ2 = a cos( 2L L ) )

24 IK - numeric If linkage is too comple to solve analytically E.g. human arm is typically modeled as or linkage Solve iteratively numerically solve for step toward goal Desired change from this specific pose Compute set of changes to the pose to effect that change

25 IK math notation IK math notation ) ( f y = ) ( ) ( ) ( f y f y f y = = ) ( ) ( f y f y = = ) ( ) ( f y f y = = ( 6) y 6 f 6 ( ) X Y = F

26 IK chain rule dy fi d fi 2 fi 3 fi 4 fi 5 fi = dt 1 dt 2 dt 3 dt 4 dt 5 dt 6 i 1 6 dt Y & F = X X&

27 Inverse Kinematics - Jacobian Y& = F X X& V = J ( θ )θ & Desired motion of end effector Unknown change in articulation variables The Jacobian is the matri relating the two: it s a function of current variable ibl values

28 Inverse Kinematics - Jacobian V = J ( θ )θ & V = [ v v v ω ω ω ] & θ = [& θ ] 1 & θ2 & θ3 & θ4 & θ5 & θ6 y z y z Change in position Change in orientation J = p p p K θ 1 θ2 θ6 p y K θ 1 K α z α z θ 1 θ 6 Change in articulation variables Jacobian

29 IK computing the Jacobian an (need to convert to global coordinates) Change in orientation Change in position Only valid instantaneously

30 IK - configuration

31 IK compute positional change vectors induced by changes in joint angles Instantaneous positional change vectors Desired change vector One approach to IK computes linear combination of change vectors that equal desired vector

32 IK compute position and ais of joints Set identity matri for (i=0; i<numdofs; i++) { record_transformed_joint(i) glrotate(angle[i]ais[i][0]ais[i][1]ais[i][2]); append_rotation(angle[i]ais[i][0]ais[i][1]ais[i][2]); if (linklen[i]!= 0) { draw_linkage(linklen[i]); gltranslatef(0.0linklen[i]0.0); append _ translation(0linklen[i]0); } } record_endeffector();

33 IK append rotation If joint ais is: one of major aes: 3 cases ofsimple rotation Arbitrary ais angle-ais ais to matri conversion IK append translation Form translation matri Matri Transformed coordinate system Position Transforms ais of rotation

34 IK record joint information Joint position last column of matri Joint coordinate system upper left 33 submatri Joint ais transform local joint ais vector by matri

35 IK - singularity Some singular configurations are not so easily recognizable Near singular configurations are also problematic why?

36 Inverse Kinematics - Numeric Given Current configuration Goal position/orientation Determine Goal vector Positions & local coordinate systems of interior joints (in global coordinates) Jacobian V = J ( θ )θ & Is in same form as more recognizable : A = b Solve & take small step or clamp acceleration or clamp velocity Repeat until: Within epsilon of goal Stuck in some configuration Taking too long

37 Solving If J square compute inverse J -1 If J not square usually under-constrained: more DoFs than constraints Requires use of pseudo-inverse of fjacobian V = J & θ J T V = J T J & θ ( T = J J ) 1 T J V ( T 1 T J J ) J J & θ J + V = θ &

38 Solving Avoid direct computation of inverse by substitution solving A=B form then substituting back V J V J T T = J & θ β = & θ T = JJ β β = & θ

39 IK Jacobian solution

40 IK Jacobian solution - problem When goal is out of reach Bizarre undulations can occur As armature tries to reach the unreachable Add a damping factor

41 IK Jacobian w/ damped least squares J T 1 θ & ( ) = J J T Undamped form: V Damped form with user parameter: θ & = J ( JJ T + λ 2 I ) T 1 V

42 IK Jacobian w/ control term Physical systems (i.e. robotics) and synthetic character simulation (e.g. human figure) have limits on joint values IK allows joint angle to have any value Difficult (computationally epensive) to incorporate hard constraints on joint values Take advantage of redundant manipulators - Allow user to set parameter that urges DOF to a certain value Does not enforce joint limit constraints but can be used to keep jit joint angles at mid-range values

43 IK Jacobian w/ control term IK Jacobian w/ control term 1 ) ( z I J J V J θ + = + + & 2 ) ( ) ( ci i i z z I J J V J θ θ α θ = + ) ( = = + z I J J J V J V θ & 0 ) ( ) ( = = + z V z J J JJ V 0 0 = = V z V Change to the pose parameter in the form of the Change to the pose parameter in the form of the control term adds nothing to the velocity

44 IK Jacobian w/ control term All bias to 0 Top gains = { } Bottomgains={01 { } 05} & θ = J z + V + ( J = α ( θ θ i ( i ci + ) J 2 I ) 1 z

45 IK alternate Jacobian Jacobian formulated ltdto pull llthe goal toward the end effector Use same method to form Jacobian but use goal coordinates instead of end-effector coordinates

46 IK Transpose of the Jacobian Compute how much the change vector contributes to the desired change vector: Project tjoint change vector onto desired dchange vector Dot product of joint change vector and desired change vector => >Transpose of fthe Jacobian

47 IK Transpose of the Jacobian p = θ & J T V v α z p p y z y v v V θ θ θ z y T p p J K K = y V ω ω = 2 α θ z T p J K z ω 6 θ 6 θ z p

48 IK cyclic coordinate descent Heuristic solution Consider one joint at a time from outside in At each joint choose update that best gets end effector to goal position In 2D pretty simple ais i J i Goal EndEffector

49 IK cyclic coordinate descent In 3D a bit more computation is needed

50 IK 3D cyclic coordinate descent Goal ais i J i Projected goal EndEffector First goal has to be projected onto plane defined by ais (normal to plane) and EF Second determine angle at joint

51 IK cyclic coordinate descent 3D Other orderings of processing joints are possible Because of its procedural nature Lends itself to enforcing joint limits Easy to clamp angular velocity

52 Inverse kinematics - review Analytic method Forming the Jacobian Numeric solutions Pseudo-inverse of the Jacobian J + with damping J + with control term Alternative Jacobian Transpose of the Jacobian Cyclic Coordinate Descent (CCD)

53 Inverse kinematics - orientation Change in orientation at end-effector is same as change at joint ais i J i EF

54 Inverse kinematics - orientation How to represent orientation (at goal at end-effector)? How to compute difference between orientations? How to represent desired change in orientation in V vector? How to incorporate into IK solution? Matri representation: M g M ef Difference M d = M ef -1 M g Use scaled ais of rotation: θ(a a y a z ): Etract quaternion from M d Etract (scaled) ais from quaternion E.g. use Jacobian Transpose method: Use projection of scaled djoint ais onto etracted t ais