Physically Based Motion Transformation

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Physically BasedMotion Transformation

• Zoran Popovic
• Andrew Witkin
• SIGGRAPH 99

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Objective

• Transform previously generated motion while
  preserving its physical properties
• User-controlled editing process
• Good for re-use of highly detailed motions
• Map motion between characters with different DOFs
  (control complex systems with simpler ones)

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Related Work

• Physical (forward) dynamics
• Spacetime constraints
• Robot controller design
• Motion capture (and editing)
• Biomechanics

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Forward Dynamics

• Highly realistic (physically accurate)
• Determining muscle forces is difficult
• Changing one frame drastically affects all others

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Spacetime Constraints

• Helps with realism and controllability
• Specify pose constraints that must be met
• Specify objective function or metric of
  performance or style
• Minimize objective function while satisfying
  constraints
• Does not scale up to complex characters (poor
  time complexity)
• Sensitive to starting position of optimization
  (may not converge)

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Robot Controller Design

• Drive actuator forces based on environment
• A set of reflexes that control muscles which
  produce motion
• Controllers adjust to changing environment
• Determining controllers that produce realistic
  motion is difficult

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Motion Capture

• Get motion data from the real world
• Highly realistic
• Unstructured, uncorrelated motion
• Editing typically has no notion of dynamics
• Large motion deformations give unwanted artifacts

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Biomechanics

• Similarity in multi-legged locomotion of
  kinematically different animals
• Studies optimality of natural motion (reaffirms
  spacetime optimization)

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Algorithm Outline
Complex Model
Original Motion
Final Motion
Simplification
Reconstruction
Simplified Model
Motion Fitting
?
Spacetime Motion Model
Transformed Spacetime Motion
Spacetime Edit
Change motion parameters Introduce new pose
constraints Change character kinematics or
objective function
Remap the change in motion of the simplified
model onto the original complex model
Create abstract character model with minimal
DOFs Map input motion onto simplified model
Find spacetime optimization problem with solution
most closely matching simplified character motion
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Character Simplification

• Why?
• Improves performance convergence
• Captures fundamental movement properties

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Three Simplification Principles

• DOF removal
• e.g. Remove DOFs in linkages
• Node subtree removal
• Replace hierarchy with a single object
• Exploit symmetric movement
• e.g. Two legs with identical jump kinematics

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Mapping motion onto simplified model

• Overdetermined problem
• simplified character has much fewer DOFs
• Use Handles
• correlate properties between complex and
  simplified motion sequences

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Handles

• Functions that can be evaluated on both complex
  and simplified models.
• Measurements of body properties
• eg. positions, directions, distances

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Handles

• complex motion handles h0(q0(t))
• simplified motion handles hS(qS(t))
• Find motion of simplified character
• Ed h0(q0(ti)) - hS(qS(ti))2
• minimize Ed over qS(ti) for each frame ti
• At least one handle per DOF

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Spacetime Motion Fitting

• Must make the simplified motion dynamically
  correct (and realistic)
• Find the spacetime optimization problem most
  closely matching simplified motion

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Spacetime Motion Fitting

• Character has two kinds of DOFs - q(t)
• kinematic qk(t)) and muscle qm(t))
• Character is constrained by
• pose constraints Cp
• mechanical constraints Cm
• dynamics constraints Cd
• Optimization problem
• minimize objective function, E(q(t),t), over all
  DOFs, q(t), subject to
• Cp(q(t),t) 0 Cm(q(t),t) 0 Cd(q(t),t)
  0

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Muscles

• Biomechanically accurate models are too complex
• Use generalized muscle forces, Q
• apply accelerations directly onto DOFs
• minimum set of muscles for full range of motion
• unstable spacetime optimization with poor
  convergence
• Use damped generalized muscle force
• Velocity-dependent damping encourages smoothness

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Constraints

• Most constraints are determined by input motion
• Avoid non-essential constraints
• Simplification may introduce constraints
• Motion editing may introduce constraints

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Objective Function

• Motion in nature assumed to be optimal
• Original motion close to optimum - YAY!
• Two components
• deviation from original motion Ed
• muscle Smoothness
• Gradually decrease wd to zero

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Spacetime Edit

• Modify dynamic properties of the simplified model
  of the animation

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Spacetime Edit

• Change constraints
• positions, timings
• Add new constraints
• Change the character model
• Add new components to the objective function
• After editing, re-solve spacetime optimization
  problem
• Already close to solution. Converges quickly.

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Motion Reconstruction

• Construct final motion from original complex
  motion and simplified spacetime motions

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Motion Reconstruction

• Now have three sets of handles
• original motion handles h0(q0)
• spactime fit handles hs(qs)
• transformed spacetime handles ht(qt)
• Combine hf(qf) h0(q0) (ht(qt) - hs(qs))
• Solve for qf?
• Number of handles much smaller than DOFs
• problem is underdetermined

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Motion Reconstruction

• Formulate sequence of per-frame subproblems
• minimize Edm(q0,qf) over qf subject to
• C(q) 0
• hf(qf) h0(q0) (ht(qt) - hs(qs))
• Follow transformed handles and satisfy
  constraints while trying to remain close to
  original motion.
• Objective function measures deviation from
  original motion

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Motion Reconstruction

• Objective function for deviation
• Edd (qf q0)2
• produces undesirable results
• each DOF must be scaled carefully
• Use a different objective function
• Edm - Measures relative mass displacement between
  two poses
• Done per-frame, so resulting motion may appear
  non-smooth
• define smoothing intervals and use the smoothness
  objective function

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Limitations

• Best for high-energy, dynamic movement
• but non-realism not as big an issue in lethargic,
  kinematic movement
• The motion-fitting step is mostly manual
• Intuitive and amortized over large numbers of
  transformations
• Motion fitting affects what kinds of
  transformations can be done
• Final motion not absolutely physically correct,
  but preserves essential properties

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Results

• Fitting 15-20 minutes
• Transformation optimization 2 minutes