As mentioned in the introduction I worked with others to produce a system which runs faster than ClearPath with very similar behavior (van den Berg et al., 2011). The resulting system, known as ORCA, approximates the ClearPath FVO with a linear constraint in velocity-space (see Figure 2.12). By using only linear constraints, the convex optimization described in Section 2.3.3 can be replaced by simple linear programing. The resulting algorithm has a 2-5X speed up over ClearPath depending on the simulation.
ORCA is well suited for large scale simulations such as the Tawaf simulation pictured below (Figure 2.13). Despite involving 25,000 agents ORCA is still able to run the simulation in real-time.
Subsequent results in this dissertation will build of the faster ORCA formulation rather than ClearPath.
Figure 2.12: Constructing the set of ORCA allowed velocities. vopt is the agent’s current velocity. ORCA forces agents to choose new velocities which avoid at least half the collision
u. In RCAP, only agents whoseTsightB|A < T TobsgenerateORCAconstraints.
Figure 2.13: Still from a real-time, ORCA simulation of Tawaf ritual involving 25,000 agents.
CHAPTER 3
PLEdestrians: A Caloric-Minimization Approach to Crowd Simulation
3.1 Introduction
The main role of collision avoidance techniques such as ClearPath or ORCA is to tell agents where they cannot go. An interesting question still is where to go. Specifically, ORCA and ClearPath will generally leave a large range of allowed velocities but only one will be taken at any timestep. Which velocity is chosen can sometimes be largely a matter of personal preference, but often there are simple, general guidelines which can naturally describe typical motion. This chapter explores one such guideline known as the Principle of Least Effort (PLE) for it’s analogous role to the principle of least action from the field of physics.
As a guiding principle for understanding human motion PLE dates back to at least the 1940s with Zipf’s classic book on human behavior (Zipf, 1965). The principle suggests that people will naturally achieve their goals in a manner which uses the least amount of effort possible. The least-effort principle has influenced the design of recent crowd modeling systems which tend to find paths which minimize an agent’s travel time, travel distance, congestion, acceleration or other factor which can be indirectly related to the concept of effort.
In this chapter, I propose to directly measure an agent’s effort as it’s expected caloric expenditure. Minimizing caloric expenditure precedent from the field of biomechanics, an area which extensively studies human motion. For example, (Inman et al., 1981) uses the minimum calorie principle to successfully explain human locomotion patterns. Likewise, minimum calorie analysis can correctly predict the approximate speed which individuals
generally travel (Whittle, 2002) and is valid across a wide range of body types (Browning and Kram, 2005).
3.1.1 Main Result
This chapter describes a unified system for simulating human trajectories based on a biomechancially inspired implementation of the principle of least effort. The heuristic of minimizing the total amount of calories consumed, combined with ORCA for collision avoidance, can completely specify a crowd simulation. I will show how to use PLE to inform both low-level collision avoidance, and high level path selection. The resulting system is shown to generate more efficient trajectories than other recent crowd simulation work. 3.1.2 Organization
The rest of this chapter is organized as follows. Section 3.2 gives an overview of some related work. Section 3.3 describes the mathematical foundation of the PLE model. Section 3.4 details how to implement PLE as part of a full crowd simulation. Sections 3.5 and 3.6 analyze the resulting simulations in terms of fulfilling the principle of least effort and qualitative behaviors. Section 3.7 proves a summary of the work and outlines some future directions.
Chapter 4 continues the analysis of the PLE approach to crowd simulation and focuses on measuring its accuracy at capturing actual human motion.
3.2 Previous Work
There is extensive literature on simulating crowd dynamics and distributed motion planning, the surveys of (Pelechano et al., 2008b) and (LaValle, 2006) cover these areas respectively. Recent work has included attempts to generate group behaviors (Bayazit et al., 2002; Kamphuis and Overmars, 2004) and and support real-time navigation of large numbers of agents (Gayle et al., 2009).
3.2.1 Least Effort in Crowd Simulation
Crowd simulation has often been formulated as a problem of minimizing some metric a for group of independent agents. Techniques such as distance based roadmaps, help agents find minimum distance paths to reach a goal. Recent methods have focused on minimizing various effort functions directly inspired by PLE by using cellular automata (Still, 2000; Sarmady et al., 2010) or phenomenological forces (Kagarlis, 2002). These methods aim to capture the large scale patterns of movement, and are not well suited for animations, which require high quality, smooth, collision-free motion.
Several techniques have been proposed specifically for animating large crowds. There is extensive work in this area and, at a broad level, many of them can be classified into five main categories:potential-basedwhich focus on modeling agents as particles with potentials and forces(Helbing and Molnar, 1995; Karamouzas et al., 2009),boid-likeapproaches based on the seminal work of Reynolds which create simple rules for velocities (Reynolds, 1987, 1999),geometricwhich compute collision-free paths using sampling (van den Berg et al., 2008a) or optimization (Guy et al., 2009), andfield basedwhich either compute fields for agents to follow (Yersin et al., 2005; Pettr´e et al., 2009; Chenney, 2004; Jin et al., 2008), or by generate fields based on continuum theories of flows (Treuille et al., 2006) or fluid models (Narain et al., 2009).
Additionally, other approaches for crowd simulation are based on cognitive modeling and behavioral (Shao and Terzopoulos, 2005; Yu and Terzopoulos, 2007) or sociological or psychological factors (Pelechano et al., 2007).
3.2.2 Motion Synthesis
Related to the Principle of Least Effort, the Principle of Minimum Energy governs the behaviors of many dynamic systems. In fact, energy minimization techniques have been extensively used for character animation and to synthesize motions like walking, running
etc. (Kang et al., 1998; Juang, 1999). Gait generation algorithms have also been proposed to minimize energy consumption (Channon et al., 1992; Roussel et al., 1998).