IMPLICIT CROWDS: OPTIMIZATION INTEGRATOR FOR ROBUST CROWD SIMULATION

Transcription

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

2 COLLISION AVOIDANCE IN CROWDS Given desired velocities, how should agents navigate around each other? 2

3 COLLISION AVOIDANCE IN CROWDS Given desired velocities, how should agents navigate around each other? 3

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

5 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 [8agent/ttc_0.25] Δt = 0.1s ORCA 5

6 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 ( s) 6

7 OUR CONTRIBUTIONS 1. General form of collision avoidance behaviors dv R x, v = dt v supporting 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 7

8 I. OPTIMIZATION INTEGRATOR FOR CROWDS 8

9 IMPLICIT INTEGRATION d dt x v = v M 1 f(x, v) x n+1 x n = v n+1 t, M v n+1 v n = f n+1 t Unconditionally stable, but [Baraff and Witkin, 1998] 9

10 IMPLICIT INTEGRATION d dt x v = v M 1 f(x, v) x n+1 x n = v n+1 t, M v n+1 v n = f n+1 t Unconditionally stable, but Need to solve a non linear system Slow (but we can use large time steps) [Kaufman et al. 2014] 10

11 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]: [Bouaziz et al. 2014] x n+1 = arg min x 1 2 t 2 x x M 2 + U(x) Interpretation: tradeoff between conserving momentum and reducing potential energy [Martin et al. 2011] 11

12 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]: [Bouaziz et al. 2014] x n+1 = arg min x 1 2 t 2 x x M 2 + U(x) Interpretation: tradeoff between maintaining velocity and reducing potential energy [Martin et al. 2011] 12

13 OPTIMIZATION INTEGRATORS Why this is good: x n+1 = arg min x 1 2 t 2 x x M 2 + U(x) 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, [Fratarcangeli et al. 2016] 13

14 NON-CONSERVATIVE FORCES Conservative potentials U(x) can only model position-dependent forces 14

15 NON-CONSERVATIVE FORCES 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] Crowd forces depend both on positions and velocities! 15

16 ANTICIPATORY FORCES Hypothesis: Multi-agent interactions can be expressed as R x, v f x, v = v where R is an anticipatory potential that drives agents away from high-cost velocities (This is analogous to dissipation potentials [Goldstein 1980] in classical mechanics) 16

17 OPTIMIZATION INTEGRATOR FOR NON-CONSERVATIVE FORCES Anticipatory forces v n+1 = arg min v 1 2 v 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 17

18 OPTIMIZATION INTEGRATOR FOR NON-CONSERVATIVE FORCES Anticipatory + conservative forces v n+1 = arg min v First-order accurate 1 2 v v M 2 + U(x + v t) + R(x + v t, v) t 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 18

19 SOME EXISTING MODELS Alignment behavior in boids [Reynolds 1987]: f ij = w x ij v ij R ij = w x ij v ij 2 Velocity obstacles [van den Berg et al 2011]: v ij VO x ij R ij = ቊ if v ij VO x ij 0 otherwise Power law model [Karamouzas et al 2014]: R ij τ(x ij, v ij ) p 19

20 SOME EXISTING MODELS Alignment behavior in boids [Reynolds 1987]: f ij = w x ij v ij R ij = w x ij v ij 2 x j Velocity obstacles [Fiorini and Shiller 1998]: v ij VO x ij R ij = ቊ if v ij VO x ij 0 otherwise v y VO Power law model [Karamouzas et al 2014]: R ij τ(x ij, v ij ) p v x x i 20

21 SOME EXISTING MODELS Alignment behavior in boids [Reynolds 1987]: f ij = w x ij v ij R ij = w x ij v ij 2 Velocity obstacles [Fiorini and Shiller 1998]: v ij VO x ij R ij = ቊ if v ij VO x ij 0 otherwise What is R ij for humans? Power law model [Karamouzas et al 2014]

22 SOME EXISTING MODELS Alignment behavior in boids [Reynolds 1987]: f ij = w x ij v ij R ij = w x ij v ij 2 Velocity obstacles [Fiorini and Shiller 1998]: v ij VO x ij R ij = ቊ if v ij VO x ij 0 otherwise What is R ij for humans? Power law model [Karamouzas et al 2014]

23 II. IMPLICIT CROWDS USING THE POWER-LAW MODEL 23

24 POWER-LAW MODEL [Karamouzas et al. 2014] For each pair of agents: Compute time to collision τ x, v Compute potential R(x, v) τ x, v p Collisions occurs when x ij + v ij τ = r i + r j v i v j x i x j 24

25 POWER-LAW MODEL [Karamouzas et al. 2014] For each pair of agents: Compute time to collision τ x, v Compute potential R(x, v) τ x, v p Collisions occurs when x ij + v ij τ = r i + r j v i v j x i x j 25

26 POWER-LAW MODEL [Karamouzas et al. 2014] For each pair of agents: Compute time to collision τ x, v Compute potential R(x, v) τ x, v p Collisions occurs when x ij + v ij τ = r i + r j v i v j x i x j 26

27 POWER-LAW MODEL [Karamouzas et al. 2014] For each pair of agents: Compute time to collision τ x, v Compute potential R(x, v) τ x, v p Collisions occurs when x ij + v ij τ = r i + r j v i v j x i x j 27

28 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 R Both phenomena cause numerical solvers to get stuck 28

29 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. 29

30 A CONTINUOUS TTC POTENTIAL Discontinuity due to time to collision (finite if collision predicted, infinite if not) x j x i 30

31 A CONTINUOUS TTC POTENTIAL Discontinuity due to time to collision (finite if collision predicted, infinite if not) x j v p 0 v t x i 31

32 A CONTINUOUS TTC POTENTIAL Discontinuity due to time to collision (finite if collision predicted, infinite if not) x j R v p 0 v t 32

33 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 C p 1 -smooth 1/τ 33

34 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 C p 1 -smooth 1/τ 34

35 MAINTAINING SEPARATION Grazing trajectories make R badly behaved Add some distance-based repulsion U ij 1 x ij r ij Continuous collision detection: replace distance x ij with minimum distance over the time step Alternative approach to repulsion: add uncertainty to time to collision computation [Forootaninia 2017] 35

36 MAINTAINING SEPARATION Grazing trajectories make R badly behaved Add some distance-based repulsion U ij 1 x ij r ij Continuous collision detection: replace distance x ij with minimum distance over the time step Alternative approach to repulsion: add uncertainty to time-to-collision computation [Forootaninia et al. 2017] 36

37 III. ANALYSIS AND RESULTS 37

38 THEORETICAL ANALYSIS Implicit integration + continuous PowerLaw potential Guaranteed collision-free motion Smooth (C 2 -continuous) trajectories 38

39 THEORETICAL ANALYSIS Implicit integration + continuous PowerLaw potential Guaranteed collision-free motion Smooth (C 2 -continuous) trajectories Collision-free proof v n+1 minimizes 1 2 vn+1 v M 2 + U(x n+1 ) + R(x n+1, v n+1 ) t R is infinite for a colliding state. U is infinite for a tunneling step. So these cannot be minima. 39

40 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) 40

41 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) 41

42 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) 42

43 FIDELITY TO HUMAN DATA Comparison to human crowds [Charalambous et al. 2014] Implicit Δt=0.4s ORCA Δt=0.4s 43

44 TIME STEP STABILITY (1.5x playback speed) Implicit Δt=0.4s 44

45 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 45

46 PERFORMANCE [fig: performance graph] Δ Δ 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) Future work: Improve performance via local-global alternating minimization techniques [Narain et al. 2016] [Liu et al. 2017] 46

47 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? Δt=0.1s Δt=1s Δt=4s 47

48 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? Δt=0.1s Δt=1s Δt=4s 48

49 FUTURE WORK Applications to LOD systems and footstep-based animation engines Applications to nonlinear dissipation forces in physics-based animation [Xu and Barbic 2017] [Zhu et al. 2015] 49

50 THANK YOU 50