Current Issues in Sampling-Based Motion Planning Stephen R. Lindemann Steven M. LaValle Dept. of Computer Science University of Illinois Urbana, IL 61801 USA {slindema, lavalle}@uiuc.edu Abstract In this paper, we discuss the field of sampling-based motion planning. In contrast to methods that con- struct boundary representations of configuration space obstacles, sampling-based methods use only information from a collision detector as they search the configuration space. The simplicity of this approach, along with in- creases in computation power and the development of ef- ficient collision detection algorithms, has resulted in the introduction of a number of powerful motion planning algorithms, capable of solving challenging problems with many degrees of freedom. First, we trace how sampling- based motion planning has developed. We then discuss a variety of important issues for sampling-based motion planning, including uniform and regular sampling, topo- logical issues, and search philosophies. Finally, we ad- dress important issues regarding the role of randomiza- tion in sampling-based motion planning. 1 Introduction In recent years, a number of motion planning algo- rithms have been introduced which have had remark- able success in solving challenging motion planning prob- lems, including ones with many degrees of freedom (DOFs). Examples include the Randomized Path Plan- ner (RPP), Probabilistic Roadmap planners (PRMs), Ariadne’s Clew, and Rapidly-exploring Random Trees (RRTs). Each of these methods, and many others, can be seen as belonging to a field we call sampling-based motion planning. We believe a fundamental distinction exists between sampling-based motion planners and ear- lier planners that built explicit representations of the ob- stacle boundary in the configuration space. While the problem of motion planning has been rec- ognized for many decades (e.g, Nilsson’s work in the late 1960s [56]), it can be argued that the epoch marking the beginning of modern motion planning was the in- troduction in 1979 of the idea of the configuration space by Lozano-P´ erez and Wesley [50]. In the configuration space, the robot is reduced to a point; hence, the motion planning problem becomes that of finding a path for a point from an initial point to a goal point in the con- figuration space. This contrasts with previous methods which planned directly in the workspace, using methods such as swept volumes to determine whether or not a path was feasible (i.e., did not collide with an obstacle). However, planning in C-space poses a problem: unlike the obstacles in the workspace, which are well-defined, how does one represent invalid configurations in C-space (C-space obstacles 1 )? Lozano-P´ erez and Wesley gave methods for constructing representations of C obs based on contact conditions between the robot and the obsta- cles, and their ideas formed the foundation for many mo- tion planning algorithms in the decade to follow. However, constructing explicit representations of C obs from the geometry of the problem has several disadvan- tages. The first complete, general motion planning algo- rithm, by Schwartz and Sharir, used cylindrical algebraic decomposition, whose running time was doubly exponen- tial in the degrees of freedom [62]. Canny introduced a roadmap algorithm which improved this to “only” singly exponential [17]. Both methods employ general-purpose techniques from computational real algebraic geometry [8], which are very difficult to implement correctly, es- pecially due to numerical considerations. Furthermore, the running times of these algorithms also grow quickly with the number of primitives in the obstacle and robot representations, as opposed to the true difficulty of a particular motion planning task. Most realistic motion planning models require thousands of primitives, which is well beyond what can be handled by the motion planning algorithms that work directly with algebraic constraints on the obstacle region. In some special cases, efficient combinatorial algorithms have been developed, but these usually apply to low-dimensional C-spaces and/or simpli- fied geometries. The PSPACE-hardness bound [61] pro- vides further discouragement from attempting to develop and use exact, combinatorial solutions, such as those in [17, 62], to solve motion planning problems that have many degrees of freedom and model primitives. In light of these difficulties, sampling-based motion planning has emerged over the past fifteen years as a way to avoid explicit constructions of C obs . By sampling, it is hoped that many practical, high-dimensional problems 1 The set of all invalid configurations is often denoted C obs and its complement, the set of all free configurations, as C free 1