Robot motion planning and the single cell problem in arrangements

Dan Halperin
1994 Journal of Intelligent and Robotic Systems  
Robot motion planning has become a central topic in robotics and has been studied using a v ariety of techniques. One approach, followed mainly in computational geometry, aims to develop combinatorial, non-heuristic solutions to motion-planning problems. This direction is strongly related to the study of arrangements of algebraic curves and surfaces in low-dimensional Euclidean space. More speci cally, the motion-planning problem can be reduced to the problem of e ciently constructing a single
more » ... ell in an arrangement of curves or surfaces. We p r e s e n t t h e basic terminology and the underlying ideas of the approach. We describe past work and then survey a series of recent results in exact motion planning with three degrees of freedom and the related issues of the complexity and construction of a single cell in certain arrangements of surface patches in three-dimensional space. In this pure formulation of the problem, we are only interested in the geometric aspects of the motion. We ignore many issues, such as acceleration, speed, uncertainty or incompleteness in the geometric data, control strategies for executing the motion, etc. A comprehensive o verview of problems and techniques in robot motion planning can be found in a recent book by Latombe La]. Several surve y s o n t h e t o p i c h a ve also been published, e.g., Sh89], Ya]. A special approach to solving motion-planning problems, called exact motion planning, is followed by computer scientists, mainly in the eld of computational geometry. The approach is exact (non-heuristic) in the sense that it aims to nd a solution whenever one exists and otherwise report that no solution exists. In this framework, the quality of an algorithm that solves a motion-planning problem is measured by its asymptotic running time and storage requirements, which in turn are measured as functions of the combinatorial complexity of the problem input, e.g., numb e r o f o bstacle features and robot features, the cost of the (algebraic) representation of each feature, etc. There are other criteria to evaluate the performance of motion-planning algorithms, for example, the clearance between the path produced for the robot motion and the obstacles (larger clearance meaning safer motion in the presence of uncertainty), or the length or other cost of the generated path. However, in this survey we will con ne ourselves to the most prevailing criteria in theoretical computer science, namely the time and space requirements of the algorithm. Much of the study of exact motion planning is carried out in the con guration space of the problem. The con guration space of a motion-planning problem with k degrees of freedom is kdimensional and every point in it represents a possible placement of the robot in the physical space. We distinguish three types of points in the con guration space of a speci c problem according to the placements of the robot that they represent: free placements, where the robot does not intersect any obstacle, forbidden placements, where the robot intersects the interior of an obstacle, and semi-free placements, where the robot is in contact with the boundary of an obstacle, but does not intersect the interior of any obstacle. The collection of all the points in the con guration space that represent semi-free robot placements partitions the con guration space into free r egions and forbidden regions. In a motionplanning problem with two degrees of freedom, for instance, the points representing semi-free placements of the robot lie on several curves in a two-dimensional con guration space. We will call these curves constraint curves. Note that each such curve is induced by the contact of a robot feature and an obstacle feature. We are therefore interested in studying the partitioning of 2D space by a collection of constraint curves (for motion-planning with two degrees of freedom), and similarly the partitioning of a three-dimensional con guration space by constraint surfaces (for motion-planning problems with three degrees of freedom). (The entire discussion applies to problems with more degrees of freedom as well, but this survey only deals with problems with two o r three degrees of freedom.) This is the point where robot motion planning overlaps a basic study in computational geometry, namely, the study of the combinatorial structure of arrangements of algebraic curves or surfaces in low-dimensional Euclidean spaces. The connection between robot motion planning and the study of arrangements has been noted
doi:10.1007/bf01258293 fatcat:frsnyobyznh75aqirjjhy5p3te