1. Skeletal Motion, Inverse Kinematics
Ladislav Kavan

2. Animating Skeleton Character 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 Animation very 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, xR3
position and orientation: 6-DOF, xR6
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 Solution
Idea: 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 Optimization
Idea: 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 Solutions
Solve the problem for velocities (small displacements)
local approximation of f by linear function - called Jacobian
k ... number of joints (DOFs)
d ... dimension of goal (end-effector, usually 3-6)
Ladislav Kavan

11. IK: Jacobian Methods Typically: 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 as
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 Inversion
Original equation: x = f()
Using Jacobian:  x = J() 
Jacobian maps small  changes to small x changes
Assume the Jacobian can be inverted:  = J()-1x
IK algorithm
Input: current posture , goal xg, current end-effector position xc
Repeat until xc close to xg:
x = k(xg - xc) // k ... small constant
compute J()-1
 =  + J()-1 x
xc = f()
Ladislav Kavan

13. IK: Jacobian Inversion
Problems with inversion of J():
J() rectangular (square only if k=d)
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 mn matrix M can be written out as M = UDVT, where
U ... mn with orthonormal columns
D ... nn diagonal (singular values)
V ... mm 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 Transpose
Idea: force acts on the end-effector and pushes it to the goal configuration
Principle of virtual work:
work = force  distance, work = torque  angle
FTx =T (work = work)
x = J() (Jacobian approximation)
FT J() =T (putting together)
FT J()=T
 = J()TF (transposing)
Ladislav Kavan

16. IK: Jacobian Transpose
Instead of  = J()-1x
use simply  = J()Tx
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 possible
Input:
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
Ladislav Kavan

18. IK: Cyclic Coordinate Descent (CCD) IK: Cy
In 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) =
Ladislav Kavan

19. IK: Cyclic Coordinate Descent (CCD)
(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
Ladislav Kavan

20. IK: Cyclic Coordinate Descent (CCD)
The function g(i) can be maximized analytically
(complicated formulas, but fast to evaluate)
CCD Algorithm:
i = 1;
Repeat until E() close to zero:
compute i giving maximal g(i)
i = (i % k) + 1
CCD has no problems with singularities
converges well in practice
Ladislav Kavan

21. IK: Modern Trends Character animation: joint limits important
First 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
Ladislav Kavan