42 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 18, NO. 1, FEBRUARY 2002 Path Coordination for Multiple Mobile Robots: A Resolution-Complete Algorithm Thierry Siméon, Stéphane Leroy, and Jean-Paul Laumond, Member, IEEE Abstract—This paper 1 presents a geometry-based approach for multiple mobile robot motion coordination. The problem is to co- ordinate the motions of several robots moving along fixed inde- pendent paths to avoid mutual collisions. The proposed algorithm is based on a bounding box representation of the obstacles in the so-called coordination diagram. The algorithm is resolution-com- plete but it is shown to be complete for a large class of inputs. De- spite the exponential dependency of the coordination problem, the algorithm efficiently solves problems involving up to ten robots in worst-case situations and more than 100 robots in practical ones. Index Terms—Coordination diagram, mobile robots, multiple robots, path coordination. I. INTRODUCTION T HIS paper addresses the following problem: consider mobile robots sharing the same workspace and planning their paths independently; given such paths, we want to devise an algorithm deciding whether coordinated motions exist for the mobile robots along their own paths, so that each robot can reach its own goal without colliding with the other ones. The problem is known as the multiple robot path coordination problem [12]. A. Path Coordination Versus Path Planning Multiple robot path coordination and path planning are two related issues in robot motion planning. In multiple robot path planning, the robot paths are not computed a priori. A solution to the multiple robot path planning problem is a collision-free path in the Cartesian product of the configuration spaces of all the robots. A solution to the problem exists iff the start and goal configurations belong to the same connected component of the global collision-free configuration space. Searching such a space is a combinatorially difficult problem [8]. Complete and exact centralized algorithms are therefore limited to simple problem settings involving two or three simple robots (e.g., [21]). Potential field techniques (e.g., [22]) have been proposed for more complex problems. In [2], a randomized search is combined with potential fields to centrally plan the motions of several translating robots. To face this complexity, several authors have investigated de- coupled schemes. The decoupled approach was introduced in Manuscript received November 16, 2000; revised August 30, 2001. This paper was recommended for publication by Associate Editor A. Maciejewski and Editor S. Hutchinson upon evaluation of the reviewers’ comments. T. Siméon and J.-P. Laumond are with LAAS-CNRS, 31077 Toulouse, Cedex 4, France. (e-mail: nic@laas.fr; jpl@laas.fr). S. Leroy is with the Rational Software Corporation, BP 10-31312 Labége Cedex, France (e-mail: sleroy@rational.com.) Publisher Item Identifier S 1042-296X(02)01776-7. 1 This paper is built upon work published separately in [19] and [15]. [9] to solve problems with multiple moving objects: the method first plans a path among the stationary obstacles and then tunes the velocity along the path to avoid collisions with the moving obstacles. Such a decoupled approach has been further revis- ited: the prioritized planning scheme proposed in [7] assigns priorities to each robot and sequentially computes paths in a time-varying configuration space, given the paths computed for the higher priority robots. In [23], prioritization is combined with potential fields to resolve possible conflicts. Issues for se- lecting priorities are discussed in [4]. Some prioritized scheme is also used in the decentralized approach proposed in [1] for controlling the execution of a large fleet of autonomous mobile robots. The path coordination problem as such was addressed in [17] where the notion of coordination diagram was first introduced for two robots. This diagram represents placements along each robot path at which mutual collisions might occur. The coordi- nation space for two manipulators is analyzed in [3] and [6]. An algorithm based on dynamic programming was proposed more recently in [10] to find optimal strategies for three robots. This paper also introduced the idea of roadmap coordination that im- poses weaker constraints than path coordination on the robot motions. A probabilistically complete planner based onto this roadmap coordination scheme [20] was applied to problems in- volving up to five robots. From another point of view, cooperation-oriented approaches are based on local information (potential methods) (see, for in- stance, [18] and [5] for a recent overview). Path coordination is out of the scope of these methods. B. Objective, Approach, and Contribution Our objective is to solve practical problems involving a large fleet of mobile robots. Fig. 1 shows a coordination problem with 150 robots solved by the algorithms described in the paper. The proposed technique consists of searching an -dimensional co- ordination diagram. The main contribution is an efficient al- gorithm that we propose to explore the coordination diagram without computing the exact shape of the obstacles. With re- spect to the previous works above, we do not use any regular grid representation. We propose instead an implicit model of the diagram obstacles by an adequate approximation of two-dimen- sional (2-D) diagrams represented by a set of bounding boxes. The algorithm is resolution–complete but it is shown to be com- plete for a large class of inputs. Despite the exponential depen- dency of the coordination problem on the number of robots, the model we propose allows us to efficiently solve problems in- volving up to ten robots in complex situations where most of the robots interfere. Such efficacy combined with partitioning 1042–296X/02$17.00 © 2002 IEEE