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Physically Based Motion Transformation. Zoran Popović Andrew Witkin SIGGRAPH ‘99. Objective. Transform previously generated motion while preserving its physical properties User-controlled editing process Good for re-use of highly detailed motions

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Physically Based Motion Transformation

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Physically BasedMotion Transformation

Zoran Popović

Andrew Witkin

SIGGRAPH ‘99


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)


Related Work

  • Physical (forward) dynamics

  • Spacetime constraints

  • Robot controller design

  • Motion capture (and editing)

  • Biomechanics


Forward Dynamics

  • Highly realistic (physically accurate)

  • Determining muscle forces is difficult

  • Changing one frame drastically affects all others


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)


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


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


Biomechanics

  • Similarity in multi-legged locomotion of kinematically different animals

  • Studies optimality of natural motion (reaffirms spacetime optimization)


Algorithm Outline

Complex

Model

Original Motion

Final Motion

Simplification

Reconstruction

Motion Fitting

Simplified

Model

Δ

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


Complex

Model

Original Motion

Final Motion

Reconstruction

Simplification

Simplified

Model

Δ

Motion Fitting

Spacetime

Motion Model

Transformed

Spacetime Motion

Spacetime Edit

Character Simplification

  • Why?

    • Improves performance & convergence

    • Captures fundamental movement properties


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


Mapping motion onto simplified model

  • Overdetermined problem

    • simplified character has much fewer DOFs

  • Use Handles

    • correlate properties between complex and simplified motion sequences


Handles

  • Functions that can be evaluated on both complex and simplified models.

  • Measurements of body properties

    • eg. positions, directions, distances


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


Complex

Model

Original Motion

Final Motion

Reconstruction

Simplification

Simplified

Model

Δ

Motion Fitting

Spacetime

Motion Model

Transformed

Spacetime Motion

Spacetime Edit

Spacetime Motion Fitting

  • Must make the simplified motion dynamically correct (and realistic)

  • Find the spacetime optimization problem most closely matching simplified motion


Spacetime Motion Fitting

  • Character has two kinds of DOFs - q(t)

    • kinematicqk(t)) and muscleqm(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


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


Constraints

  • Most constraints are determined by input motion

    • Avoid non-essential constraints

  • Simplification may introduce constraints

  • Motion editing may introduce constraints


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


Complex

Model

Original Motion

Final Motion

Reconstruction

Simplification

Simplified

Model

Δ

Motion Fitting

Spacetime

Motion Model

Transformed

Spacetime Motion

Spacetime Edit

Spacetime Edit

  • Modify dynamic properties of the simplified model of the animation


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.


Complex

Model

Original Motion

Final Motion

Reconstruction

Simplification

Simplified

Model

Δ

Motion Fitting

Spacetime

Motion Model

Transformed

Spacetime Motion

Spacetime Edit

Motion Reconstruction

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


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


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


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:


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


Results

  • Fitting: 15-20 minutes

  • Transformation optimization: ~2 minutes


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