Implicit Crowds: Optimization Integrator for Robust Crowd Simulation IOANNIS KARAMOUZAS, Clemson University NICK SOHRE, University of Minnesota RAHUL NARAIN, University of Minnesota STEPHEN J. GUY, University of Minnesota (a) (b) Le inset (c) Right Inset Fig. 1. (a) A challenging crowd simulation scenario with multiple flows interacting simultaneously. By reformulating implicit integration to allow velocity-based energy functions, we can enable numerically stable crowd simulations even in these dense multi-directional interactions. Our method leads to smooth, collision-free motion for a wide range of time steps sizes. (b-c) We perform implicit integration by minimizing a global energy function, which is visualized here by taking slices with respect to the velocities of the two highlighted agents. White crosses indicate optimal new velocities for the highlighted agents. Large multi-agent systems such as crowds involve inter-agent interactions Additional Key Words and Phrases: Crowd simulation, implicit integration, that are typically anticipatory in nature, depending strongly on both the physics-based animation positions and the velocities of agents. We show how the nonlinear, anticipa- ACM Reference format: tory forces seen in multi-agent systems can be made compatible with recent Ioannis Karamouzas, Nick Sohre, Rahul Narain, and Stephen J. Guy. 2017. work on energy-based formulations in physics-based animation, and propose Implicit Crowds: Optimization Integrator for Robust Crowd Simulation. a simple and eective optimization-based integration scheme for implicit ACM Trans. Graph. 36, 4, Article 136 (July 2017), 13 pages. integration of such systems. We apply this approach to crowd simulation by DOI: hp://dx.doi.org/10.1145/3072959.3073705 using a state-of-the-art model derived from a recent analysis of human crowd data, and adapting it to our framework. Our approach provides, for the rst time, guaranteed collision-free motion while simultaneously maintaining 1 INTRODUCTION high-quality collective behavior in a way that is insensitive to simulation Simulating the motion of multiple intelligent agents interacting with parameters such as time step size and crowd density. ese benets are each other, such as in crowds, ocks, or trac, is an important task demonstrated through simulation results on various challenging scenarios and validation against real-world crowd data. in computer animation. In many dierent elds which study hu- man motion, early methods for simulation used force-based models CCS Concepts: •Computing methodologies ! Physical simulation; Con- [Helbing and Molnar´ 1995; Reynolds 1987] inspired by physics, and tinuous simulation; could take advantage of improvements in physics-based animation. Over the past years, both multi-agent modeling and physics-based is work has been supported in part by the National Science Foundation, under grants animation have seen many powerful advances, leading to highly CHS-1526693 and CNS-1544887. sophisticated techniques in both elds. However, as the two elds Rahul Narain and Stephen J. Guy are joint last authors. have largely developed independently, they have diverged enough Author’s addresses: I. Karamouzas, School of Computing, Clemson University, Clemson, SC; email: [email protected]; N. Sohre, R. Narain, and S.J. Guy, Department of that recent numerical techniques from physics-based animation Computer Science and Engineering, University of Minnesota, Twin Cities, MN; email: cannot directly be applied to modern multi-agent models. In this fsohre007,narain,[email protected]. work, we seek to remove this barrier by building a new connection Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed between the two elds, enabling robust and ecient simulation of for prot or commercial advantage and that copies bear this notice and the full citation intelligent, anticipatory agent behavior. on the rst page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permied. To copy otherwise, or A key advancement in physics-based animation has been the de- republish, to post on servers or to redistribute to lists, requires prior specic permission velopment of numerically robust simulation techniques, which give and/or a fee. Request permissions from [email protected]. consistent and stable results across a variety of scenarios, simulation © 2017 Copyright held by the owner/author(s). Publication rights licensed to ACM. 0730-0301/2017/7-ART136 $15.00 conditions, and even time step sizes. In particular, implicit integra- DOI: hp://dx.doi.org/10.1145/3072959.3073705 tion schemes such as backward Euler are exceptionally stable even ACM Transactions on Graphics, Vol. 36, No. 4, Article 136. Publication date: July 2017. 136:2 • Ioannis Karamouzas, Nick Sohre, Rahul Narain, and Stephen J. Guy for numerically challenging problems, while variational integrators (2) We combine this new integration approach with a potential oer impressive long-term conservation of energy and momenta. function derived from a recent analysis of anticipation in Recent advances in physics-based animation have made these time human crowds, which we treat in the same way as a dissi- integration schemes faster and more robust by formulating them in pation function. e resulting implicit crowd simulation terms of optimization, leading to performance speedups for implicit technique allows robust simulation of anticipatory crowd variational integrators [Kharevych et al. 2006] and highly robust dynamics. and parallelizable solvers for backward Euler integration [Bouaziz Our implicit method yields crowd simulations with several valu- et al. 2014; Gast et al. 2015]. ese practical benets are made possi- able properties: stability across a very large range of time steps, ble by working directly with the underlying energy function that scenarios, and densities; guaranteed collision-free motion; smooth, determines the dynamics of the system. ecient trajectories for all agents; and anticipatory behavior be- In a similar fashion, many modern techniques for procedural tween agents leading to the emergence of collective phenomena. animation of intelligent systems model agents as individual entities each trying to minimize an energy or cost function, such as the distance to the goal or a cost based on proximity to other agents. 2 RELATED WORK However, in practice these techniques tend to use simple time inte- 2.1 Crowd Simulation gration schemes such as forward Euler, which require small time Many approaches exist to simulate crowds, including techniques steps to maintain stability [Karamouzas et al. 2014; Pelechano et al. based on continuum dynamics [Hughes 2002; Treuille et al. 2006], 2007; Reynolds 1999]. e necessity of small time steps brings with data-driven approaches [Charalambous and Chrysanthou 2014; Ju it several issues that widely aect elds such as crowd simulation. et al. 2010; Lerner et al. 2007], as well as methods for interactive For example, it is oen necessary to carefully tune simulation pa- crowd authoring [Kim et al. 2014; Normoyle et al. 2014]. Below, rameters such as the size of the time step for each new scenario. If we highlight some prior work on local navigation that is highly the time step is too small, the simulation becomes computationally relevant to our paper. We also refer the reader to Sections 6 and 7 for expensive; if it is too large, collisions, jiering, and discontinuous comparisons of our approach with existing local collision avoidance motion can be observed. e magnitude of inter-agent interaction schemes. forces can be adjusted to smooth out motion to an extent, but dense In general, local navigation methods can be classied into force- scenarios oen still require impractically small time steps to be based approaches and velocity-based models. Force-based methods able to produce smooth motion. Even when these simulations are represent humans as particles and model their interactions using computationally feasible, very small time steps directly give rise to physical forces. Two of the most popular force-based methods are frequent changes in velocity, which can make it dicult to apply Reynolds’s boids model [1987], which captures ocking behavior character animations to simulated trajectories. Consequently, many using separation, alignment, and cohesion forces, and the social existing crowd simulations are fragile and require deep familiarity force model of Helbing et al. [2000] which uses a mixture of soci- with the underlying methods to adapt to new scenarios. ological and physical forces to model pedestrian interactions. In In this paper, our goal is to allow the application of optimization- both approaches, the forces depend only on the separation of the based integration schemes to enable robust simulation of large sys- agents and, hence, can lead to simulation artifacts such as oscil- tems of intelligent agents, such as crowd simulation. A key chal- lations and backward movement. Since these two seminal works, lenge in applying these physics-based animation techniques to the many approaches have been proposed to address such issues, in- simulation of intelligent systems is the non-physical nature of the cluding techniques to control individual agents in dense crowds dynamics. e entities in these systems do not follow force laws [Pelechano et al. 2007], and more recently anticipatory models that based on conservative