Skeletal Motion, Inverse Kinematics

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Skeletal Motion, Inverse Kinematics

• Ladislav Kavan
• kavanl_at_cs.tcd.ie
• http//isg.cs.tcd.ie/kavanl/IET

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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

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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

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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)

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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)

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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

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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

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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"

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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

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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)
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IK Jacobian Methods

• Typically d6 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

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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(?)

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IK Jacobian Inversion

• Problems with inversion of J(?)
• J(?) rectangular (square only if kd)
• 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

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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

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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)

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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)

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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

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IK Cy
IK Cyclic Coordinate Descent (CCD)

• In the i-th step ?1, ..., ?i-1, ?i1..., ?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)

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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

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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

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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