The multiple robot extension is unique from other planning problems in that typically the DoFs between different robots can be decoupled. This leads to two different classi- fications of planners for multiple robots: centralized and decoupled. There is extensive literature on path planning for multiple agents in robot motion planning and virtual environments [LaV06a].
Several works have realized the various benefits of each planning model, and com- bined them into hybrid approaches. For instance, [CRL03] combines a centralized plan- ner for intra-group coordination while using a decoupled planner for inter-group plan- ning. Both types of planners have been applied to a number of situations such as flocking and shepherding [BLA02d,LRMA05].
2.4.1
Centralized Planners
The centralized approach to planning for multiple robots considers the robots as a single system, rather than independent entities. In this case, much of the work previously mentioned can be directly applied [BLA02b, FTT99, KO04, LD04, PLT05, SKG05,
SAC+07].
However, direct application of these ideas may also lead to complications. If robots are in close proximity to each other, this may result in narrow passages in the C-space. For instance, it may be difficult to generate a random sample such that a robot in the center of a cluster moves in unison with the surrounding robots. Li et al. identify and accommodate for groups in planning is one method aid in this problem and improve performance [LC03b]. Aronov et al gave centralized solutions for specific cases, such as for pairs or triples of roots in a low density workspace [AdBvds+99]. Several other algorithms have been proposed to extend the roadmap-based methods to dynamic envi- ronments and multiple agents [GO07,LG07, PLT05, ZKB07]. Again, in many of these cases, they have only been applied to relatively simple environments composed of a few robots and restricted obstacles. These approaches may not scale well to environments with a large number of independent agents.
The primary advantage of centralized methods is in their theoretical completeness. Additional tasks such as coordination are not necessary since it is automatically solved by the planner. However, again this comes at the price of complexity.
2.4.2
Decoupled Planners
In the decoupled approach, motion planning for multiple robots usually takes two stages. In the first stage, a path is found for each robot without consideration of the paths of the other robots. In the next step, paths are adjusted to try to allow for robot-robot interactions, usually through some sort of coordination, such as velocity tuning. This is often referred to as fixed-path coordination and is essentially adjusting velocities along the paths such that no collisions occur. It is often done by exploring acoordination space which represents the parameters along each robot path. In this approach, completeness is lost since it is possible that no solution can be found in coordination space of the predetermined paths even though a path may exist in C-space.
Prioritized planning for multiple robots is a variant of the decoupled approach. The idea is that each robot is assigned a priority ahead of time. In the order determined by priority, each robot plans a path while treating the robots whose paths have already been determined as dynamic obstacles [ELP86]. Choice of the priorities can have a large impact on the performance of the algorithm [vdBO05]. Some planners also search through a space of prioritizations [BBT02].
Alternatively, other decoupled schemes such as coordination graphs [LGP05], in- cremental planning [SI06], integration with replanning [BTK07], and velocity obstacles [vdBLM08] can help to ensure that no collisions occur along the paths even in cases with complex dynamics.
Another class of decouple planners allows agents to move independently and in the reactive style of potential fields. [Kha85, War90, GC02, BBGN07]. They can handle large dynamic environments, but suffer from ‘local-minima’ problems and may not be able to find a collision-free path when one exists [LaV06a]. Often these methods do not give any kind of guarantees on their behavior. Other route planning algorithms are based on path or roadmap modification, which allow a specified path for an agent to move or deform based upon obstacle motion. These methods include Elastic Bands [QK93] and Elastic Roadmaps [YB06]. Social potential fields alter the navigation function such that the resulting motion reflects social behaviors [RW99]. Our work is most closely related to this approach. Our approach bears some close resemblance to these techniques, but RDR is lazily updated to deal with dynamic obstacles and we present many new techniques to plan collision-free paths for multiple agents simultaneously using RDR.
2.4.3
Crowd Dynamics and Human Agents Simulation
Many different approaches have been proposed for modeling movement and simulation of multiple human agents or crowds or individual pedestrians. [ALA+01, SS01, ST05,
decomposition (discrete vs continuous), stochastic vs deterministic, etc.
Discrete methods
Discrete methods rely on a sampling of the environment or of the agents. Some common approaches include:
Agent-based methods: These are based on seminal work of Reynolds [Rey87] and can generate fast, simple local rules that can create visually plausible flocking behavior. Numerous extensions have been proposed to account for social forces [CBS+05], psycho- logical models [POSB05], directional preferences [SGC04], sociological factors [MT97], tactical behaviors [Ree03], etc. Velocity obstacles have been utilized for improving local avoidance in dense crowds [vdBPS+08].
Different techniques for collision avoidance have been developed based on grid-based rules [LMM03] and behavior models [TT94]. Most agent-based techniques use local collision avoidance techniques and cannot give any guarantees on the correctness of global behaviors. Although, in many situations these resulting motion matches how people navigate (e.g. people packing dense in a narrow corridor).
Cellular Automata methods: These methods model the motion of multiple agents by solving a cellular automaton. The evolution of the cellular automata at next time step is governed by static and dynamic fields [HLB+00]. While these algorithms can capture emergent phenomena, they are not physically based.
Particle Dynamics: Computing physical forces on each agent is similar to N-body particle system [SS01,HBW03]. Sugiyama et al. [SNH01] presented a 2D optimal veloc- ity (OV) model that generalizes the 1D OV model used for traffic flow. Our formulation is built on some of these ideas and we elaborate them in Section 6.5.2.
Continuous Methods
The flow of crowds or multiple agents can be formulated as fluid flows. At low densities crowd flow is like gases, at moderate densities it resembles fluid flow, and at high densities crowd has been compared to granular flow [HBJW05]. Most recently, a novel approach for crowd simulation based on continuum dynamics has been proposed by Treuille et al. [TCP06]. We compare our approach with these methods in Section 7.