computer animation algorithms and...

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Computer AnimationRick Parent

Computer AnimationAlgorithms and Techniques

Kinematic Linkages


Hierarchical ModelingRelative motion Parent-child relationship

Constrains motion Reduces dimensionality

Simplifies motion specification

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Modeling & animating hierarchies

3 aspects1. Linkages & Joints – the relationships2. Data structure – how to represent such a hierarchy3. Converting local coordinate frames into global space


Some termsJoint – allowed relative motion & parametersJoint Limits – limit on valid joint angle valuesLink – object involved in relative motionLinkage – entire joint-link hierarchyArmature – same as linkageEnd effector – most distant link in linkageArticulation variable – parameter of motion associated with jointPose – configuration of linkage using given set of joint anglesPose vector – complete set of joint angles for linkage

Arc – of a tree data structure – corresponds to a jointNode – of a tree data structure – corresponds to a link

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Use of hierarchies in animation

Forward Kinematics (FK)animator specifies values of articulation variables

Inverse Kinematics (IK)animator specifies final desired global transform for

end effector (and possibly other linkages)

global transform for each linkage is computed

Values of articulation variables are computed


Forward & Inverse Kinematics

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Joints – relative movement


Complex Joints

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Hierarchical structure


RootLeafParentChild

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Tree structure


Tree structure

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Tree structure


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Relative movement


Relative movement

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Tree structure


Treestructure

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Tree traversal

traverse (arcPtr,matrix){

// concatenate arc matricesmatrix = matrix*arcPtr->Lmatrixmatrix = matrix*arcPtr->Amatrix;

// get node and transform datanodePtr=acrPtr->nodePtrpush (matrix)matrix = matrix * nodePTr->matrixaData = transformData(matrix,dataPTr)draw(aData)matrix = pop();

// process childrenIf (nodePtr->arc != NULL) {

nextArcPtr = nodePTr->arcwhile (nextArcPtr != NULL) {

push(matrix)traverse(nextArcPtr,matrix)matrix = pop()nextArcPtr = nextArcPtr->arc

}}

}

L A

d,M

NOTE: Node points to first childEach child points to siblingLast sibling points to NULL


OpenGLSingle linkage

glPushMatrix();For (i=0; i<NUMDOFS; i++) {

glRotatef(a[i],axis[i][0], axis[i][1], axis[i][2]);if (linkLen[i] != 0.0) {

draw_linkage(linkLen[i]);glTranslatef(0.0,linkLen[i],0.0);

}}glPopMatrix();

A[i] – joint angleAxis[i] – joint axislinkLen[i] – length of link

OpenGL concatenates matrices

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Inverse kinematics

Given goal position (and orientation) for end effector

Compute internal joint angles

If simple enough => analytic solutionElse => numeric iterative solution


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Inverse kinematics - spaces

Configuration spaceReachable workspaceDextrous workspace


Analyticinverse kinematics

Note: typos in the book

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•XY

(x,y)L1L2

(0,0)

•

•


X=4, y=1

Sqrt(16+1) = sqrt(17) = 4.12, 13.8 = acos(4/4.12)

L1 = 3L2 = 2Theta1 = 9+16+1-4= 22/(6*4.12)=22/24.7, acos(.89) = 27.1+ 34.5 = 61.6

•

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L1 = 3L2 = 2

Theta2 = -(9+4-17)/12=4/12= 1/3, acos(1/3) = 70.5


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IK - numericIf linkage is too complex to solve analytically

Desired change from this specific poseCompute set of changes to the pose to effect that change

Solve iteratively – numerically solve for step toward goal

E.g., human arm is typically modeled as 3-1-3 or 3-2-2 linkage

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IK math notation


IK – math notation

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Inverse Kinematics - Jacobian

Desired motion of end effector

Unknown change in articulation variables

The Jacobian is the matrix relating the two: it’s a function of current avar values


Inverse Kinematics - Jacobian

Change in position

Change in orientation

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IK – computing the Jacobian

One only valid instantaneously – what does that mean?

Convert to global coordinates

Change in positionChange in orientation


IK - configuration

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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


Cross((0,0,1),E) = (E.y,E.x,0)

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IK - singularity

Some singular configurations are not so easily recognizableNear singular configurations are also problematic – why?


Inverse Kinematics - NumericGiven• Current configuration• Goal position/orientation

Determine• Goal vector• Positions & local coordinate systems of interior joints (in global coordinates)• Jacobian

Solve & take small step – or clamp acceleration or clamp velocity

Repeat until:• Within epsilon of goal• Stuck in some configuration• Taking too long

Is in same form as more recognizable :

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Solving

If J not square, usually under-constrained: more DoFs than constraints Requires use of pseudo-inverse of Jacobian

If J square, compute inverse, J-1

Avoid direct computation of inverse by solving Ax=B form


IK – Jacobian solution

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IK – Jacobian solution - problem

When goal is out of reachBizarre undulations can occurAs armature tries to reach the unreachable

Add a damping factor


IK – Jacobian w/ damped least squares

Undamped form:

Damped form with user parameter:

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IK – Jacobian w/ control term

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 joint angles at mid-range values

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 expensive) to incorporate hard constraints on joint values



Change to the pose parameter in the form of the control term adds nothing to the velocity

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All bias to 0Top gains = {0.1, 0.5, 0.1}Bottom gains = {0.1, 0.1, 0.5}


IK – alternate Jacobian

Jacobian formulated to pull the goal toward the end effector

Use same method to form Jacobian but use goal coordinates instead of end-effector coordinates

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IK – Transpose of the Jacobian

Compute how much the change vector contributes to the desired change vector:

Project joint change vector onto desired change vector

Dot product of joint change vector and desired change vector => Transpose of the Jacobian


IK – Transpose of the Jacobian

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IK – cyclic coordinate descent

Consider one joint at a time, from outside inAt each joint, choose update that best gets end effector to goal position

In 2D – pretty simple

EFGoal

Ji

axisi

Heuristic solution


IK – cyclic coordinate descent

In 3D, a bit more computation is needed

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IK – cyclic coordinate descent – 3D

EFGoal

Ji

axisi

First – goal has to be projected onto plane defined by axis and EF


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

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Inverse kinematics - review

Analytic methodForming the JacobianNumeric solutions

Pseudo-inverse of the JacobianJ+ with dampingJ+ with control termAlternative JacobianTranspose of the JacobianCyclic Coordinate Descent (CCD)


Inverse kinematics - orientation

Change in orientation at end-effector is same as change at joint

Ji

axisi

EF

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Inverse kinematics - orientationHow 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?

Matrix representation: Mg, Mef

Difference Md = Mef-1 Mg

Use scaled axis of rotation: B(ax ay az ):• Extract quaternion from Md• Extract (scaled) axis from quaternion

E.g., use Jacobian Transpose method: Use projection of scaled joint axis onto extracted axis

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