Implicit Crowds: Optimization Integrator for Robust Crowd ... · IMPLICIT POWER-LAW CROWDS Problem:...

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Ioannis Karamouzas 1 , Nick Sohre 2 , Rahul Narain 2 , Stephen J. Guy 2 1 Clemson University 2 University of Minnesota IMPLICIT CROWDS: OPTIMIZATION INTEGRATOR FOR ROBUST CROWD SIMULATION

Transcript of Implicit Crowds: Optimization Integrator for Robust Crowd ... · IMPLICIT POWER-LAW CROWDS Problem:...

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Ioannis Karamouzas1, Nick Sohre2, Rahul Narain2, Stephen J. Guy2

1Clemson University

2University of Minnesota

IMPLICIT CROWDS:

OPTIMIZATION INTEGRATOR FOR ROBUST CROWD SIMULATION

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COLLISION AVOIDANCE IN CROWDS

Given desired velocities, how should agents navigate around each other?

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COLLISION AVOIDANCE IN CROWDS

Given desired velocities, how should agents navigate around each other?

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LOCAL COLLISION AVOIDANCE

Adding Realism

• Vision-based approaches [Ondřej et al. 2010,

Kapadia et al. 2012, Hughes et al. 2015, Dutra et al. 2017]

• Probabilistic approaches [Wolinski et al. 2016]

• …..

4[Wolinski et al. 2016] [Kapadia et al. 2012]

[Yu et al. 2012]

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LOCAL COLLISION AVOIDANCE

Force-based methods [Reynolds 1987,

1999; Helbing et al. 2000; Pelechano et al.

2007, …]

• Require very small time steps for

stability

Velocity-based methods [van den Berg

et al. 2008, 2011; Pettré et al 2011, …]

• Overly conservative behavior

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[8agent/ttc_0.25] ORCA

Δt = 0.1s

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DESIDERATA

We seek a generic technique for multi-agent navigation that

• guarantees collision-free motion

• is robust to variations in scenario, density, time step

• exhibits high-fidelity behavior

• can update at footstep rates (0.3-0.5 s)

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OUR CONTRIBUTIONS

1. General form of collision avoidance behaviorsd𝐯

d𝑡= −

𝜕𝑅 𝐱, 𝐯

𝜕𝐯supporting optimization-based implicit integration

2. Application to state-of-the-art power law model

𝑅 𝐱, 𝐯 ∝ 𝜏(𝐱, 𝐯)−𝑝

for practical crowd simulations

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I. OPTIMIZATION INTEGRATOR FOR CROWDS

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IMPLICIT INTEGRATION

𝐱𝑛+1 − 𝐱𝑛 = 𝐯𝑛+1∆𝑡,𝐌 𝐯𝑛+1 − 𝐯𝑛 = 𝐟𝑛+1∆𝑡

• Unconditionally stable, but

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•𝑑

𝑑𝑡

𝐱𝐯

=𝐯

𝐌−1𝐟(𝐱, 𝐯) ⟺

[Baraff and Witkin, 1998]

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IMPLICIT INTEGRATION

𝐱𝑛+1−𝐱𝑛 = 𝐯𝑛+1∆𝑡,𝐌 𝐯𝑛+1 − 𝐯𝑛 = 𝐟𝑛+1∆𝑡

• Unconditionally stable, but

• Need to solve a non linear system

• Slow (but we can use large time steps)

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•𝑑

𝑑𝑡

𝐱𝐯

=𝐯

𝐌−1𝐟(𝐱, 𝐯) ⟺

[Kaufman et al. 2014]

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OPTIMIZATION INTEGRATORS

As long as forces are conservative:

𝐟 𝐱 = −d𝑈 𝐱

d𝐱,

we can express backward Euler in optimization

form [Martin et al. 2011; Gast et al. 2015]:

𝐱𝑛+1 = arg min𝐱

1

2∆𝑡2𝐱 − 𝐱 𝐌

2 + 𝑈(𝐱)

Interpretation: tradeoff between conserving

momentum and reducing potential energy

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[Martin et al. 2011]

[Bouaziz et al. 2014]

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OPTIMIZATION INTEGRATORS

As long as forces are conservative:

𝐟 𝐱 = −d𝑈 𝐱

d𝐱,

we can express backward Euler in optimization

form [Martin et al. 2011; Gast et al. 2015]:

𝐱𝑛+1 = arg min𝐱

1

2∆𝑡2𝐱 − 𝐱 𝐌

2 + 𝑈(𝐱)

Interpretation: tradeoff between maintaining

velocity and reducing potential energy

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[Bouaziz et al. 2014]

[Martin et al. 2011]

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OPTIMIZATION INTEGRATORS

𝐱𝑛+1 = arg min𝐱

1

2∆𝑡2𝐱 − 𝐱 𝐌

2 + 𝑈(𝐱)

Why this is good:

• Simple and fast algorithms (e.g. gradient descent, Gauss-

Seidel) can be given guarantees

• Highly nonlinear forces can be used without linearization

• Lots of recent advances in optimization for data mining,

machine learning, image processing, …

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[Fratarcangeli et al. 2016]

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NON-CONSERVATIVE FORCES

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Conservative potentials 𝑈(𝐱) can only model position-dependent forces

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NON-CONSERVATIVE FORCES

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Crowd forces depend both on positions and velocities!

Conservative potentials 𝑈(𝐱) can only model position-dependent forces

• Humans anticipate

• People anticipate future trajectories of others [Cutting et al. 2005, Olivier et al. 2012; Karamouzas

et al. 2014]

• Brains have special neurons for estimating collisions [Gabbiani 2002]

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ANTICIPATORY FORCES

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Hypothesis: Multi-agent interactions can be expressed as

𝐟 𝐱, 𝐯 = −𝜕𝑅 𝐱, 𝐯

𝜕𝐯where 𝑅 is an anticipatory potential that drives agents away from high-cost

velocities

(This is analogous to dissipation potentials [Goldstein 1980] in classical mechanics)

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OPTIMIZATION INTEGRATOR FOR

NON-CONSERVATIVE FORCES

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Anticipatory forces

𝐯𝑛+1 = arg min𝐯

1

2𝐯 − 𝐯 𝐌

2 + 𝑅(𝐱 + 𝐯∆𝑡, 𝐯)∆𝑡

• First-order accurate, equivalent to previous formula if 𝑅 = 0

• 𝑈(∙) + 𝑅(∙)∆𝑡 is analogous to the “effective interaction potential” in symplectic

integrators [Kane et al. 2000; Kharevych et al. 2006]

• Simple interpretation: tradeoff between conserving momentum, reducing 𝑈,

and reducing 𝑅

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OPTIMIZATION INTEGRATOR FOR

NON-CONSERVATIVE FORCES

Anticipatory + conservative forces

𝐯𝑛+1 = arg min𝐯

1

2𝐯 − 𝐯 𝐌

2 + 𝑈(𝐱 + 𝐯∆𝑡) + 𝑅(𝐱 + 𝐯∆𝑡, 𝐯)∆𝑡

• First-order accurate

• 𝑈(∙) + 𝑅(∙)∆𝑡 is analogous to the “effective interaction potential” in symplectic

integrators [Kane et al. 2000; Kharevych et al. 2006]

• Simple interpretation: tradeoff between maintaining velocity, reducing 𝑈, and

reducing 𝑅

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]:

𝐟𝑖𝑗 = −𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗 ⟺ 𝑅𝑖𝑗 = 𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗2

Velocity obstacles [van den Berg et al 2011]:

𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗 ⟺ 𝑅𝑖𝑗 = ቊ∞ if 𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗

0 otherwise

Power law model [Karamouzas et al 2014]:

𝑅𝑖𝑗 ∝ 𝜏(𝐱𝑖𝑗 , 𝐯𝑖𝑗)−𝑝

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]:

𝐟𝑖𝑗 = −𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗 ⟺ 𝑅𝑖𝑗 = 𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗2

Velocity obstacles [Fiorini and Shiller 1998]:

𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗 ⟺ 𝑅𝑖𝑗 = ቊ∞ if 𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗

0 otherwise

Power law model [Karamouzas et al 2014]:

𝑅𝑖𝑗 ∝ 𝜏(𝐱𝑖𝑗 , 𝐯𝑖𝑗)−𝑝

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𝐱𝑖

𝐱𝑗

𝑣𝑥

𝑣 𝑦

VO

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]:

𝐟𝑖𝑗 = −𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗 ⟺ 𝑅𝑖𝑗 = 𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗2

Velocity obstacles [Fiorini and Shiller 1998]:

𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗 ⟺ 𝑅𝑖𝑗 = ቊ∞ if 𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗

0 otherwise

What is 𝑅𝑖𝑗 for humans?

Power law model [Karamouzas et al 2014]

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]:

𝐟𝑖𝑗 = −𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗 ⟺ 𝑅𝑖𝑗 = 𝑤 𝐱𝑖𝑗 𝐯𝑖𝑗2

Velocity obstacles [Fiorini and Shiller 1998]:

𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗 ⟺ 𝑅𝑖𝑗 = ቊ∞ if 𝐯𝑖𝑗 ∈ VO 𝐱𝑖𝑗

0 otherwise

What is 𝑅𝑖𝑗 for humans?

Power law model [Karamouzas et al 2014]

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II. IMPLICIT CROWDS USING

THE POWER-LAW MODEL

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POWER-LAW MODEL

For each pair of agents:

• Compute time to collision 𝜏 𝐱, 𝐯

• Compute potential 𝑅(𝐱, 𝐯) ∝ 𝜏 𝐱, 𝐯 −𝑝

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Collisions occurs when

𝐱𝑖𝑗 + 𝐯𝑖𝑗𝜏 = r𝑖 + r𝑗

𝐱𝑖 𝐱𝑗

𝐯𝑖 𝐯𝑗

[Karamouzas et al. 2014]

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POWER-LAW MODEL

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𝐱𝑖 𝐱𝑗

𝐯𝑖 𝐯𝑗

For each pair of agents:

• Compute time to collision 𝜏 𝐱, 𝐯

• Compute potential 𝑅(𝐱, 𝐯) ∝ 𝜏 𝐱, 𝐯 −𝑝

Collisions occurs when

𝐱𝑖𝑗 + 𝐯𝑖𝑗𝜏 = r𝑖 + r𝑗

[Karamouzas et al. 2014]

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POWER-LAW MODEL

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𝐱𝑖 𝐱𝑗

𝐯𝑖 𝐯𝑗

For each pair of agents:

• Compute time to collision 𝜏 𝐱, 𝐯

• Compute potential 𝑅(𝐱, 𝐯) ∝ 𝜏 𝐱, 𝐯 −𝑝

Collisions occurs when

𝐱𝑖𝑗 + 𝐯𝑖𝑗𝜏 = r𝑖 + r𝑗

[Karamouzas et al. 2014]

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POWER-LAW MODEL

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𝐱𝑖 𝐱𝑗

𝐯𝑖 𝐯𝑗

For each pair of agents:

• Compute time to collision 𝜏 𝐱, 𝐯

• Compute potential 𝑅(𝐱, 𝐯) ∝ 𝜏 𝐱, 𝐯 −𝑝

Collisions occurs when

𝐱𝑖𝑗 + 𝐯𝑖𝑗𝜏 = r𝑖 + r𝑗

[Karamouzas et al. 2014]

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IMPLICIT POWER-LAW CROWDS

Problem: Apply power law potential to optimization-based backward Euler

Easy? Not quite…

• 𝑅 is discontinuous at boundary of collision cone

• 𝑅 becomes infinitely steep as agents graze past

Both phenomena cause numerical solvers to “get stuck”

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R

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IMPLICIT POWER-LAW CROWDS

Problem: Apply power law potential to optimization-based backward Euler

Easy? Not quite…

• 𝑅 is discontinuous at boundary of collision cone

• 𝑅 becomes infinitely steep as agents graze past

Both phenomena cause numerical solvers to “get stuck”.

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𝐱𝑗

𝐱𝑖

A CONTINUOUS TTC POTENTIAL

Discontinuity due to time to collision (finite if collision predicted, infinite if not)

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𝐱𝑗

𝐱𝑖

A CONTINUOUS TTC POTENTIAL

Discontinuity due to time to collision (finite if collision predicted, infinite if not)

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𝑣 𝑝

𝑣𝑡

0

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𝐱𝑗

A CONTINUOUS TTC POTENTIAL

Discontinuity due to time to collision (finite if collision predicted, infinite if not)

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𝑣 𝑝

𝑣𝑡

R

0

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A CONTINUOUS TTC POTENTIAL

Solution:

Let’s work with 1

𝜏(or, “imminence” of collision)

• Replace it with a continuous approximation, e.g., by linear extrapolation

• 𝑅 becomes 𝐶𝑝−1-smooth

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1/𝜏

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A CONTINUOUS TTC POTENTIAL

Solution:

Let’s work with 1

𝜏(or, “imminence” of collision)

• Replace it with a continuous approximation, e.g., by linear extrapolation

• 𝑅 becomes 𝐶𝑝−1-smooth

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1/𝜏

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MAINTAINING SEPARATION

Grazing trajectories make 𝑅 badly behaved

• Add some distance-based repulsion 𝑈𝑖𝑗 ∝1

𝐱𝑖𝑗 −𝑟𝑖𝑗

• Continuous collision detection: replace distance 𝐱𝑖𝑗 with minimum distance

over the time step

Alternative approach to repulsion: add uncertainty to time to collision

computation [Forootaninia 2017]

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MAINTAINING SEPARATION

Grazing trajectories make 𝑅 badly behaved

• Add some distance-based repulsion 𝑈𝑖𝑗 ∝1

𝐱𝑖𝑗 −𝑟𝑖𝑗

• Continuous collision detection: replace distance 𝐱𝑖𝑗 with minimum distance

over the time step

Alternative approach to repulsion: add uncertainty to time-to-collision

computation [Forootaninia et al. 2017]

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III. ANALYSIS AND RESULTS

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THEORETICAL ANALYSIS

Implicit integration + continuous PowerLaw potential

• Guaranteed collision-free motion

• Smooth (C2-continuous) trajectories

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THEORETICAL ANALYSIS

Implicit integration + continuous PowerLaw potential

• Guaranteed collision-free motion

• Smooth (C2-continuous) trajectories

Collision-free proof

• 𝐯𝑛+1 minimizes 1

2𝐯𝑛+1 − 𝐯

𝐌

2+ 𝑈(𝐱𝑛+1) + 𝑅(𝐱𝑛+1, 𝐯𝑛+1)∆𝑡

• 𝑅 is infinite for a colliding state. 𝑈 is infinite for a tunneling step. So these

cannot be minima.

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COLLISION-FREE MOTION

• Comparisons to

– ORCA [van den Berg et al. 2011] (representative velocity-based approach)

– PowerLaw [Karamouzas et al. 2014] (non-continuous TTC + forward Euler)

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COLLISION-FREE MOTION

• Comparisons to

– ORCA [van den Berg et al. 2011] (representative velocity-based approach)

– PowerLaw [Karamouzas et al. 2014] (non-continuous TTC + forward Euler)

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COLLISION-FREE MOTION

• Comparisons to

– ORCA [van den Berg et al. 2011] (representative velocity-based approach)

– PowerLaw [Karamouzas et al. 2014] (non-continuous TTC + forward Euler)

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FIDELITY TO HUMAN DATA

Comparison to human crowds [Charalambous et al. 2014]

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Implicit

Δt=0.4sORCA

Δt=0.4s

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TIME STEP STABILITY

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Implicit

Δt=0.4s

(1.5x playback speed)

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TIME STEP STABILITY

Motion doesn’t change significantly with time step

Potential applications:

• Collision avoidance synced with footstep-based character animation

• Crowd timestep level of detail without affecting behavior

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PERFORMANCE

[fig: performance graph]

Cost is linear in number of agents,

increases slowly with time step size

(Still, 2x-10x slower than ORCA or PowerLaw on

a 6-core Intel Xeon E5-1650)

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Δ

Future work: Improve performance via

local-global alternating minimization techniques

Δ

[Liu et al. 2017]

[Narain et al. 2016]

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LIMITATIONS AND FUTURE WORK

• Can other recent crowd models be formulated via interaction energies

[Wolinksi et al. 2016, Dutra et al. 2017; …]?

• Incorporating asymmetrical interactions, e.g., leader-following behavior

• What is the Δt threshold where quality is maintained?

47Δt=0.1s Δt=1s Δt=4s

Page 48: Implicit Crowds: Optimization Integrator for Robust Crowd ... · IMPLICIT POWER-LAW CROWDS Problem: Apply power law potential to optimization-based backward Euler Easy? Not quite…

LIMITATIONS AND FUTURE WORK

• Can other recent crowd models be formulated via interaction energies

[Wolinksi et al. 2016, Dutra et al. 2017; …]?

• Incorporating asymmetrical interactions, e.g., leader-following behavior

• What is the Δt threshold where quality is maintained?

48Δt=0.1s Δt=1s Δt=4s

Page 49: Implicit Crowds: Optimization Integrator for Robust Crowd ... · IMPLICIT POWER-LAW CROWDS Problem: Apply power law potential to optimization-based backward Euler Easy? Not quite…

FUTURE WORK

• Applications to LOD systems and footstep-based animation engines

• Applications to nonlinear dissipation forces in physics-based animation

49

[Xu and Barbic 2017]

[Zhu et al. 2015]

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THANK YOU

50

https://www.cs.clemson.edu/~ioannis/implicit-crowds/