Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

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Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Motion planning is one of the most important areas of robotics research. The complexity of the motion-planning problem has hindered the development of practical algorithms. This paper surveys the work on gross-motion planning, including motion planners for point robots, rigid robots, and manipulators in stationary, time-varying, constrained, and movable-object environments. The general issues in motion planning are explained. Recent approaches and their performances are briefly described, and possible future research directions are discussed.

Gross motion planning—a survey

A path-planning algorithm for the classical mover's problem in three dimensions using a potential field representation of obstacles is presented. A potential function similar to the electrostatic potential is assigned to each obstacle, and the topological structure of the free space is derived in the form of minimum potential valleys. Path planning is done at two levels. First, a global planner selects a robot's path from the minimum potential valleys and its orientations along the path that minimize a heuristic estimate of the path length and the chance of collision. Then, a local planner modifies the path and orientations to derive the final collision-free path and orientations. If the local planner fails, a new path and orientations are selected by the global planner and subsequently examined by the local planner. This process is continued until a solution is found or there are no paths left to be examined. The algorithm solves a much wider class of problems than other heuristic algorithms and at the same time runs much faster than exact algorithms (typically 5 to 30 min on a Sun 3/260). >

A potential field approach to path planning

This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R/sup 2//spl times/S/sup 1/ of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning. >

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A motion planner for nonholonomic mobile robots

A general and simple algorithm is presented which computes the set FP of all free configurations for a polygonal object I (with m edges) which is free to translate and/or to rotate but not to intersect another polygonal object E. The worst-case time complexity of the algorithm is O(m/sup 3/n/sup 3/ log mn), which is close to optimal. FP is a three-dimensional curved object which can be used to find free motions within the same time bounds. Two types of motion have been studied in some detail. Motion in contact, where I remains in contact with E, is performed by moving along the faces of the boundary of FP. By partitioning FP into prisms, it is possible to compute motions when I never makes contact with E. In this case, the theoretical complexity does not exceed O(m/sup 6/n/sup 6/ alpha (mn)) but it is expected to be much smaller in practice. In both cases, pseudo-optimal motions can be obtained with a complexity increased by a factor log mn. >

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A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles

A general and simple algorithm is presented which computes the set FP of all free configurations for a polygonal object I (with m edges) which is free to translate and/or to rotate but not to intersect another polygonal object E. The worst-case time complexity of the algorithm is O(m/sup 3/n/sup 3/ log mn), which is close to optimal. FP is a three-dimensional curved object which can be used to find free motions within the same time bounds. Two types of motion have been studied in some detail. Motion in contact, where I remains in contact with E, is performed by moving along the faces of the boundary of FP. By partitioning FP into prisms, it is possible to compute motions when I never makes contact with E. In this case, the theoretical complexity does not exceed O(m/sup 6/n/sup 6/ alpha (mn)) but it is expected to be much smaller in practice. In both cases, pseudo-optimal motions can be obtained with a complexity increased by a factor log mn.<<ETX>>

A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles extended abstact

We present a gênerai algorithm which computes an exact description of the set of ail placements for a polygon I (with m edges) which is free to translate and/or to rotate but not to intersect another polygon E (with n edges). The time complexity of our algorithm is O(mn log mn) which is close to optimal in the worst-case. Moreover, in some practical situations, the time complexity is only O (n log n). This algorithm is rather simple and has been implemented. It can be used as an efficient tool in several applications such as cutting stock, inspection and motion planning for a two dimensional robot admidst polygonal obstacles. Résumé. Cet article présente un algorithme général qui calcule une description analytique exacte de Vensemble des placements d'un polygone I (ayant m arêtes) libre de se déplacer en translation et rotation dans le plan sans intersecter un polygone E (ayant n arêtes). La complexité de l'algorithme est O (m n log mn) ce qui est proche de l'optimal dans le cas le pire. On montre de plus que la complexité réelle de Valgorithme, dans certaines situations pratiques est O (n log n). Valgorithme présenté est assez simple et a été implanté. Il recouvre un champ d'applications varié incluant les problèmes de découpe automatique, de conformité de pièces industrielles ou les problèmes de planifications de trajectoires pour un robot mobile plan évoluant au milieu d'obstacles polygonaux.

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Polygon Placement Under Translation and Rotation

We propose a method of evaluating signs of 2×2 and 3×3 determinants withb-bit integer entries using onlyb and (b+1)-bit arithmetic, respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and compared with other exact computation methods.

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Evaluating signs of determinants using single-precision arithmetic

IINTRODUCTION We investigate the problem of whether one or several Polygons I, . ..lk can be translated to fit inside another polygon E without overlapping each other. The containment problem for one polygon, say I, has been studied Previously by Chazelle (C), Baker, Fortune and Mahaney (BFM) and Fortune (F). Chazelle derived an algorithm that runs in time O(n+m) to solve lhis problem, in the case that both polygons E and I are convex. Here n is the number of edges of E and m the number of edges of I. The case that I is non-convex but E is convex, can be easily reduced to the previous one. The case that E is non-convex but I is convex is considered in @FM) and (F). The best algorithm is obtained by Fortune who presents an O(nm lognm) algorithm. This algorithm is surely close to optimal (in the worst-case sense) since the feasible region may have O(mn) vertices. This paper presents an algorithm that solve the general case where I and E may be non-convex and may have (non-convex) holes. The algorithm provides the set of all the possible placements. Its complexity is O(CiCeNlOgN) where N=cin+cem, Ci=l +number of concave vertices of I, c,=number of convex vertices of E which are not vertices of the convex hull of E plus number of edges of the convex hull of E which are not edges of E This algorithm is close to optimal. in the worst-case where c,=O(n) and ci=O(m), since the feasible region may have O(n2m2) vertices. In the case of rectilinear polygons the algorithm runs in O(n2m2) time and, thus, is optimal. The problem of coordinating the movements of several disks has been studied by Schwartz and Sharir (SS) and by Yap(Y). Yap has obtained an O(nk) algorithm for the coordination of k disks

Francis Avnaim

Papers

A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles

A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles extended abstact

Polygon Placement Under Translation and Rotation

Evaluating signs of determinants using single-precision arithmetic

Simultaneous containment of several polygons