Toggle PRM: Simultaneous Mapping of C-free and C-obstacle - A Study in 2D - Jory Denny and Nancy M. Amato Abstract— Motion planning is known to be difficult. Prob- abilistic planners have made great advances, but still have difficulty for problems that require planning in narrow passages or on surfaces in Cspace. This work proposes Toggle PRM, a new methodology for PRMs that simultaneously maps both free and obstacle space. In this paper, we focus on 2 DOF problems and show that mapping both spaces leads to increased sampling density in narrow passages and to improved overall efficiency as compared to previous sampling based approaches. I. I NTRODUCTION In robotics, one challenging problem is planning a valid (e.g., collision-free) path for a movable object (robot) in an environment. This has been extensively studied and is commonly called Motion Planning [20]. Motion planning is not only used in robotic applications but even such domains as gaming/virtual reality [13] and bioinformatics [22]. Sampling-based planners [10] were a major breakthrough in motion planning. These algorithms were able to solve many previously unsolved problems, especially for high- dimensional configuration space (C space ). While these meth- ods have been shown to be probabilistically complete, narrow passages, or small volume regions of C free , remain difficult for them to map. In particular, it has been shown that the volume of such passages impacts the efficiency of a sampling-based planner [8]. The intuition is that the smaller the relative volume of the corridor, the more difficult it is to generate samples in it. As discussed in Section II, there have been many variants proposed which aim to address this weakness of sampling-based planners [1], [3], [7], [23]. This paper introduces a new approach to sampling-based planning. In the new method, called Toggle PRM, both free space, C free , and collision space, C obst , are mapped in an integrated, coordinated fashion. The intuition behind this strategy is that each connection attempt (local planner result), whether successful or not, provides important information about the connectivity of both spaces, and moreover, wit- nesses to unsuccessful connections between configurations in one space provide useful configurations in the other space. For example, a failed connection between two collision configurations on either side of a narrow passage would lead to the discovery of a configuration in the narrow This research supported in part by NSF Grants EIA-0103742, ACR- 0081510, ACR-0113971, CCR-0113974, ACI-0326350, CRI-0551685, CCF-0833199, CCF-0830753, by the DOE, Chevron, IBM, Intel, HP, and by King Abdullah University of Science and Technology (KAUST) Award KUS-C1-016-04. J. Denny and N. M. Amato are with the Parasol Lab., Dept. of Computer Science and Engineering, Texas A&M Univ., College Station, Texas, 77843- 3112, USA {jdenny,amato}@cse.tamu.edu passage. Moreover, the probability of finding such a narrow passage configuration would not depend on the volume of the passage, but only on the fact that a connection was attempted between configurations in the bounding obstacles. In this paper, we focus on 2 DOF problems and show that this paradigm shift from only mapping the free space to the coordinated mapping of both free and collision spaces results in significant improvements to the effectiveness and efficiency of sampling-based planners. Particular contribu- tions of this work include: • Introducing a method, Toggle PRM, that maps C free and C obst . • Sketching theoretical arguments that explain increased sampling of configurations inside narrow passages for arbitrary two-dimensional problems. • Providing experimental results that confirm that Toggle PRM sampling in narrow passages is less dependent on the volume of the narrow passage and of the volume of the obstacles surrounding the narrow passage. The above establishes that Toggle PRM addresses one of the major challenges of previous sampling-based methods for 2 DOF problems. We also provide experimental results that indicate Toggle PRM has similar benefits for higher DOF problems. II. PRELIMINARIES AND RELATED WORK In this section, preliminaries of motion planning and related work are discussed. After providing preliminary definitions, an overview of the various known PRM variants is given. Secondly, work that utilizes collision information to guide sampling is surveyed. These are usually more complex algorithms utilizing machine learning methods. Lastly, meth- ods which model C space to predict collisions are described. A. Preliminaries A robot is a movable object whose position and orientation can be described by n parameters, or degrees of freedom (DOFs), each corresponding to an object component (e.g., object positions, object orientations, link angles, link dis- placements). Hence, a robot’s placement, or configuration, can be uniquely described by a point (x 1 ,x 2 , ..., x n ) in an n dimensional space (x i being the ith DOF). This space, consisting of all possible robot configurations (feasible or not) is called configuration space (C space ) [15]. The subset of all feasible configurations is the free C space (C free ), while the union of the unfeasible configurations is the blocked or obstacle C space (C obst ). Thus, the motion planning problem becomes that of finding a continuous trajectory in C free