IMPLICIT CROWDS: OPTIMIZATION INTEGRATOR FOR ROBUST CROWD SIMULATION - - PowerPoint PPT Presentation

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

1Clemson University 2University of Minnesota

IMPLICIT CROWDS:

OPTIMIZATION INTEGRATOR FOR ROBUST CROWD SIMULATION

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]
• …..

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

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

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 behaviors

d𝐰 d𝑢 = − 𝜖𝑆 𝐲, 𝐰 𝜖𝐰 supporting optimization-based implicit integration

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

𝑆 𝐲, 𝐰 ∝ 𝜐(𝐲, 𝐰)−𝑞 for practical crowd simulations

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• I. OPTIMIZATION INTEGRATOR FOR CROWDS

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IMPLICIT INTEGRATION

𝐲𝑜+1 − 𝐲𝑜 = 𝐰𝑜+1∆𝑢, 𝐍 𝐰𝑜+1 − 𝐰𝑜 = 𝐠𝑜+1∆𝑢

• Unconditionally stable, but

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• 𝑒

𝑒𝑢

𝐲 𝐰 = 𝐰 𝐍−1𝐠(𝐲, 𝐰) ⟺

[Baraff and Witkin, 1998]

IMPLICIT INTEGRATION

𝐲𝑜+1−𝐲𝑜 = 𝐰𝑜+1∆𝑢, 𝐍 𝐰𝑜+1 − 𝐰𝑜 = 𝐠𝑜+1∆𝑢

• Unconditionally stable, but
• Need to solve a non linear system
• Slow (but we can use large time steps)

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• 𝑒

𝑒𝑢

𝐲 𝐰 = 𝐰 𝐍−1𝐠(𝐲, 𝐰) ⟺

[Kaufman et al. 2014]

OPTIMIZATION INTEGRATORS

As long as forces are conservative: 𝐠 𝐲 = −

d𝑉 𝐲 d𝐲 ,

we can express backward Euler in optimization form [Martin et al. 2011; Gast et al. 2015]: 𝐲𝑜+1 = arg min

𝐲

1 2∆𝑢2 𝐲 − ෤ 𝐲 𝐍

2 + 𝑉(𝐲)

Interpretation: tradeoff between conserving momentum and reducing potential energy

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[Martin et al. 2011] [Bouaziz et al. 2014]

OPTIMIZATION INTEGRATORS

As long as forces are conservative: 𝐠 𝐲 = −

d𝑉 𝐲 d𝐲 ,

we can express backward Euler in optimization form [Martin et al. 2011; Gast et al. 2015]: 𝐲𝑜+1 = arg min

𝐲

1 2∆𝑢2 𝐲 − ෤ 𝐲 𝐍

2 + 𝑉(𝐲)

Interpretation: tradeoff between maintaining velocity and reducing potential energy

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[Bouaziz et al. 2014] [Martin et al. 2011]

OPTIMIZATION INTEGRATORS

𝐲𝑜+1 = arg min

𝐲

1 2∆𝑢2 𝐲 − ෤ 𝐲 𝐍

2 + 𝑉(𝐲)

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]

NON-CONSERVATIVE FORCES

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Conservative potentials 𝑉(𝐲) can only model position-dependent forces

NON-CONSERVATIVE FORCES

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Crowd forces depend both on positions and velocities!

Conservative potentials 𝑉(𝐲) 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]

ANTICIPATORY FORCES

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Hypothesis: Multi-agent interactions can be expressed as 𝐠 𝐲, 𝐰 = − 𝜖𝑆 𝐲, 𝐰 𝜖𝐰 where 𝑆 is an anticipatory potential that drives agents away from high-cost velocities (This is analogous to dissipation potentials [Goldstein 1980] in classical mechanics)

OPTIMIZATION INTEGRATOR FOR NON-CONSERVATIVE FORCES

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Anticipatory forces 𝐰𝑜+1 = arg min

𝐰 1 2 𝐰 − ෤

𝐰 𝐍

2 + 𝑆(𝐲 + 𝐰∆𝑢, 𝐰)∆𝑢

• First-order accurate, equivalent to previous formula if 𝑆 = 0
• 𝑉(∙) + 𝑆(∙)∆𝑢 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 𝑉,

and reducing 𝑆

OPTIMIZATION INTEGRATOR FOR NON-CONSERVATIVE FORCES

Anticipatory + conservative forces 𝐰𝑜+1 = arg min

𝐰 1 2 𝐰 − ෤

𝐰 𝐍

2 + 𝑉(𝐲 + 𝐰∆𝑢) + 𝑆(𝐲 + 𝐰∆𝑢, 𝐰)∆𝑢

• First-order accurate
• 𝑉(∙) + 𝑆(∙)∆𝑢 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 𝑉, and

reducing 𝑆

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]: 𝐠𝑗𝑘 = −𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘 ⟺ 𝑆𝑗𝑘 = 𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘

2

Velocity obstacles [van den Berg et al 2011]: 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 ⟺ 𝑆𝑗𝑘 = ቊ∞ if 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 0 otherwise Power law model [Karamouzas et al 2014]: 𝑆𝑗𝑘 ∝ 𝜐(𝐲𝑗𝑘, 𝐰𝑗𝑘)−𝑞

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SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]: 𝐠𝑗𝑘 = −𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘 ⟺ 𝑆𝑗𝑘 = 𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘

2

Velocity obstacles [Fiorini and Shiller 1998]: 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 ⟺ 𝑆𝑗𝑘 = ቊ∞ if 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 0 otherwise Power law model [Karamouzas et al 2014]: 𝑆𝑗𝑘 ∝ 𝜐(𝐲𝑗𝑘, 𝐰𝑗𝑘)−𝑞

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𝐲𝑗 𝐲𝑘 𝑤𝑦 𝑤𝑧

VO

SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]: 𝐠𝑗𝑘 = −𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘 ⟺ 𝑆𝑗𝑘 = 𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘

2

Velocity obstacles [Fiorini and Shiller 1998]: 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 ⟺ 𝑆𝑗𝑘 = ቊ∞ if 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 0 otherwise What is 𝑆𝑗𝑘 for humans? Power law model [Karamouzas et al 2014]

SOME EXISTING MODELS

Alignment behavior in boids [Reynolds 1987]: 𝐠𝑗𝑘 = −𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘 ⟺ 𝑆𝑗𝑘 = 𝑥 𝐲𝑗𝑘 𝐰𝑗𝑘

2

Velocity obstacles [Fiorini and Shiller 1998]: 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 ⟺ 𝑆𝑗𝑘 = ቊ∞ if 𝐰𝑗𝑘 ∈ VO 𝐲𝑗𝑘 0 otherwise What is 𝑆𝑗𝑘 for humans? Power law model [Karamouzas et al 2014]

• II. IMPLICIT CROWDS USING

THE POWER-LAW MODEL

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POWER-LAW MODEL

For each pair of agents:

• Compute time to collision 𝜐 𝐲, 𝐰
• Compute potential 𝑆(𝐲, 𝐰) ∝ 𝜐 𝐲, 𝐰 −𝑞

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Collisions occurs when 𝐲𝑗𝑘 + 𝐰𝑗𝑘𝜐 = r𝑗 + r𝑘 𝐲𝑗 𝐲𝑘 𝐰𝑗 𝐰

𝑘

[Karamouzas et al. 2014]

POWER-LAW MODEL

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𝐲𝑗 𝐲𝑘 𝐰𝑗 𝐰

𝑘

For each pair of agents:

• Compute time to collision 𝜐 𝐲, 𝐰
• Compute potential 𝑆(𝐲, 𝐰) ∝ 𝜐 𝐲, 𝐰 −𝑞

Collisions occurs when 𝐲𝑗𝑘 + 𝐰𝑗𝑘𝜐 = r𝑗 + r𝑘 [Karamouzas et al. 2014]

POWER-LAW MODEL

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𝐲𝑗 𝐲𝑘 𝐰𝑗 𝐰

𝑘

For each pair of agents:

• Compute time to collision 𝜐 𝐲, 𝐰
• Compute potential 𝑆(𝐲, 𝐰) ∝ 𝜐 𝐲, 𝐰 −𝑞

Collisions occurs when 𝐲𝑗𝑘 + 𝐰𝑗𝑘𝜐 = r𝑗 + r𝑘 [Karamouzas et al. 2014]

POWER-LAW MODEL

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𝐲𝑗 𝐲𝑘 𝐰𝑗 𝐰

𝑘

For each pair of agents:

• Compute time to collision 𝜐 𝐲, 𝐰
• Compute potential 𝑆(𝐲, 𝐰) ∝ 𝜐 𝐲, 𝐰 −𝑞

Collisions occurs when 𝐲𝑗𝑘 + 𝐰𝑗𝑘𝜐 = r𝑗 + r𝑘 [Karamouzas et al. 2014]

IMPLICIT POWER-LAW CROWDS

Problem: Apply power law potential to optimization-based backward Euler Easy? Not quite…

• 𝑆 is discontinuous at boundary of collision cone
• 𝑆 becomes infinitely steep as agents graze past

Both phenomena cause numerical solvers to “get stuck”

28 R

IMPLICIT POWER-LAW CROWDS

Problem: Apply power law potential to optimization-based backward Euler Easy? Not quite…

• 𝑆 is discontinuous at boundary of collision cone
• 𝑆 becomes infinitely steep as agents graze past

Both phenomena cause numerical solvers to “get stuck”.

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𝐲𝑘 𝐲𝑗

A CONTINUOUS TTC POTENTIAL

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

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𝐲𝑘 𝐲𝑗

A CONTINUOUS TTC POTENTIAL

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

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𝑤𝑞 𝑤𝑢

∞

𝐲𝑘

A CONTINUOUS TTC POTENTIAL

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

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𝑤𝑞 𝑤𝑢

R

∞

A CONTINUOUS TTC POTENTIAL

Solution: Let’s work with

1 𝜐 (or, “imminence” of collision)

• Replace it with a continuous approximation, e.g., by linear extrapolation
• 𝑆 becomes 𝐷𝑞−1-smooth

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1/𝜐

A CONTINUOUS TTC POTENTIAL

Solution: Let’s work with

1 𝜐 (or, “imminence” of collision)

• Replace it with a continuous approximation, e.g., by linear extrapolation
• 𝑆 becomes 𝐷𝑞−1-smooth

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1/𝜐

MAINTAINING SEPARATION

Grazing trajectories make 𝑆 badly behaved

• Add some distance-based repulsion 𝑉𝑗𝑘 ∝

1 𝐲𝑗𝑘 −𝑠𝑗𝑘

• Continuous collision detection: replace distance 𝐲𝑗𝑘

with minimum distance

• ver 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 𝑆 badly behaved

• Add some distance-based repulsion 𝑉𝑗𝑘 ∝

1 𝐲𝑗𝑘 −𝑠𝑗𝑘

• Continuous collision detection: replace distance 𝐲𝑗𝑘

with minimum distance

• ver 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

• 𝐰𝑜+1 minimizes

1 2 𝐰𝑜+1 − ෤

𝐰 𝐍

2 + 𝑉(𝐲𝑜+1) + 𝑆(𝐲𝑜+1, 𝐰𝑜+1)∆𝑢

• 𝑆 is infinite for a colliding state. 𝑉 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=0.4s ORCA Δt=0.4s

TIME STEP STABILITY

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Implicit Δt=0.4s

(1.5x playback speed)

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]

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?

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Δt=0.1s Δt=1s Δt=4s

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?

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Δt=0.1s Δt=1s Δt=4s

FUTURE WORK

• Applications to LOD systems and footstep-based animation engines
• Applications to nonlinear dissipation forces in physics-based animation

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[Xu and Barbic 2017] [Zhu et al. 2015]

THANK YOU

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