Skeletal Motion, Inverse Kinematics

Ladislav Kavan

kavanl@cs.tcd.iehttp://isg.cs.tcd.ie/kavanl/IET

Ladislav Kavan 2

Animating SkeletonCharacter animation decomposed to skinning (skin deformation)

previous lesson animation of the skeleton

this lesson

We consider only rotational joints n ... the number of joints joint rotations described by matrices T(0), ..., T(n)

The task: Acquire T(0), ..., T(n) for each frame of the animation

Ladislav Kavan 3

Summary of Animation Methods direct control of rotations (forward kinematics)

tedious work - only key postures designed others computed automatically

indirect control: inverse kinematics specify a goal, rotations computed automatically more intuitive to use

motion capture records a real motion difficult to control (but possible!)

forward dynamics physics engines

Ladislav Kavan 4

Keyframe Animationvery old concept (Disney's studios)Idea: specify only important (key) frames others given by interpolation of key frames

Anything can be controlled via keyframes (not only rotation of joints)

variety of interpolation curves piecewise constant, linear, polynomial (spline)

For rotation: good interpolation of rotations essential! SLERP (recall quaternions) SQUAD (Spherical QUADratic interpolation)

Ladislav Kavan 5

Forward Kinematics (FK) used for keyframe design

originated in robotics

kinematic chain: a chain (path) of joints base: the fixed part of a kinematic chain end-effector: the moving part of a kinematic chain

Assume (wlog): each joint rotates only about one fixed axis (... only 1-DOF)

Replace the 3-DOF joints by three 1-DOF joints (with the same origin)

Ladislav Kavan 6

Forward Kinematics (FK)Input: state vector = (1, ..., k)T - joint rotations k is the number of DOFs of the kinematic chain

Task: compute position (and orientation) of the end-effector: x = f()

only position: 3-DOF end-effector, xR3

position and orientation: 6-DOF, xR6

first three coordinates: translation second three coordinates: rotation in scaled axis

representation (unit axis multiplied by angle of rotation)

Solution: f is concatenation of transformations computed in the part about skinning: matrix F

Ladislav Kavan 7

Inverse Kinematics (IK)Input: the end-effector (goal) x is given

typically 3 to 6-DOFs

Task: compute state vector = (1, ..., k)T such that f() = x, i.e. = f-1(x)

Problems: sometimes no solution (example?) solution may not be defined uniquely (example?)

infinite number of solutions (example?) f is non-linear

because of the sin and cos in rotations

Ladislav Kavan 8

IK: Analytical SolutionIdea: solve the system of equations f() = x for very difficult for longer kinematic chains used only for 6/7-DOF manipulators usually the fastest method (if possible)

Example: Human Arm Like (HAL) chain 7 DOFs: 3 shoulder, 1 elbow, 3 wrist fast analytic solution for given position and

orientation - IKAN library 1 DOF remains: parametric solution

the parameter is a "swivel angle"

Ladislav Kavan 9

IK: Non-linear OptimizationIdea: minimize function E() = f() - x solved by non-linear optimization (programming)

Advantages easy to incorporate joint limits

add conditions li i hi, where li and hi are the limits

allows more general goals (planar, half-space, ...)

Disadvantages non-linear optimization can give a local minimum

(instead of the global one) slow for real-time applications

Ladislav Kavan 10

IK: Numerical SolutionsSolve the problem for velocities (small displacements) local approximation of f by linear function - called

Jacobian

k

d

2

d

1

d

k

2

2

2

1

2

k

1

2

1

1

1

k1

fff

fff

fff

JJf

J

k ... number of joints (DOFs)d ... dimension of goal (end-effector, usually 3-6)

Ladislav Kavan 11

IK: Jacobian MethodsTypically: d=6 and f = (Px, Py, Pz, Ox, Oy, Oz)

(Px, Py, Pz) ... position

(Ox, Oy, Oz) ... scaled rotation axis

Then Ji, the i-th column of Jacobian is computed asTT

T

T

i

z

i

y

i

x

i

z

i

y

i

xi

)(O,

O,

O,

P,

P,

PJ

i

ii

ω

rω

where i is the axis or rotation of the i-th joint

ri is the end-effector vector w.r.t. the i-th joint

Ladislav Kavan 12

IK: Jacobian InversionOriginal equation: x = f()Using Jacobian: x = J()

Jacobian maps small changes to small x changes

Assume the Jacobian can be inverted: = J()-1xIK algorithm

Input: current posture , goal xg, current end-effector position xc

Repeat until xc close to xg:

1. x = k(xg - xc) // k ... small constant

2. compute J()-1

3. = + J()-1 x4. xc = f()

Ladislav Kavan 13

IK: Jacobian InversionProblems with inversion of J():1. J() rectangular (square only if k=d)2. J() singular (rank-defficient)

Ad 1) (Moore-Penrose) pseudo-inversion compute from Singular Value Decomposition (SVD)

Ad 2) serious problem: configuration singular Example: fully outstretched kinematic chain near singularity: large velocities & oscilation SVD detects singularity

Ladislav Kavan 14

Singular Value Decomposition (SVD)Theorem: Any real mn matrix M can be written out

as M = UDVT, where U ... mn with orthonormal columns D ... nn diagonal (singular values) V ... mm orthonormal

Properties: M regular iff D contains no zeroes if Di is D with non-zero elements inverted, then

UDiVT is the pseudo-inverse matrix efficient and robust algorithms for SVD exist

implemented for example in LAPACK

Ladislav Kavan 15

IK: Jacobian TransposeIdea: force acts on the end-effector and pushes it to the

goal configuration

Principle of virtual work: work = force distance, work = torque angle

FTx =T (work = work)x = J() (Jacobian approximation)

FT J() =T (putting together) FT J()=T

= J()TF (transposing)

Ladislav Kavan 16

IK: Jacobian TransposeInstead of = J()-1xuse simply = J()Tx

interpret as , and F as x

The algorithm: the same as Jacobian Inverse, just use transpose instead of inversion

Comparison with the Jacobian Inverse: (pseudo-)inversion more accurate - less iterations transposition faster to compute transposition: physically based

intuitive control (rubber band)

Ladislav Kavan 17

IK: Cyclic Coordinate Descent (CCD)Idea: change only one joint angle i per step others constant: analytical solution possibleInput: Pc - current end-effector position

Pg - goal end-effector position

(u1c, u2c, u3c) - orthonormal matrix: curr. orientation

(u1g, u2g, u3g) - orthonormal matrix: goal orientation

Objective: minimize

3

1j

2

g

2

g 1,)(E jcjc uuPP

Ladislav Kavan 18

IK: CyIn the i-th step: 1, ..., i-1, i+1..., k constant

Minimize E() = E(1, ..., k) only w.r.t. i

Pic - current end-effector position w.r.t. joint i

Pig - goal end-effector position w.r.t. joint i

Pic'(i) - Pic rotated about joint i's axis with angle i

Minimization of w.r.t. i equivalent to

Maximization of g1(i) =

IK: Cyclic Coordinate Descent (CCD)

2

g cPP

)(', iicig PP

Ladislav Kavan 19

(u1c'(i), u2c'(i), u3c'(i))

orthonormal matrix of the current orientation rotated by angle i about the i-th joint's axis

Minimization of w.r.t. i equiv. to

Maximization of

Position and rotation together: g(i) = wpg1(i) + wog2(i)

wp resp. wo is weight of position resp. orientation goal

IK: Cyclic Coordinate Descent (CCD)

23

1jg 1,

jcj uu

3

1jigi2 )(',)(g jcj uu

Ladislav Kavan 20

The function g(i) can be maximized analytically

(complicated formulas, but fast to evaluate)

CCD Algorithm:i = 1;Repeat until E() close to zero:1. compute i giving maximal g(i)

2. i = (i % k) + 1

CCD has no problems with singularities converges well in practice

IK: Cyclic Coordinate Descent (CCD)

Ladislav Kavan 21

IK: Modern TrendsCharacter animation: joint limits importantFirst idea: impose limits on individual DOFs (angles) too simple to express real joint range

Advanced joint limits quaternion field boundaries:

human motion recorded using motion-capture rotations of joint (shoulder) converted to quaternions boundaries expressed on unit quaternion sphere