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

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

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

We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.

A sweepline algorithm for Voronoi diagrams

Given n points in the plane, we present algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points. Our algorithms work in linear space and time $O(kn\lg n + n\lg ^2 n)$. For the special case $k = 3$ we give $O(n\lg n)$ algorithms for these problems. Several related issues are discussed.

Finding extremal polygons

Algorithms for simulation and verification of Numerically Controlled (NC) machining programs are presented. Compared to NC simulation based on conventional solid modeling systems, these models are designed to give approximate results, but with a substantial decrease in computer time. The surfaces of the part are discretized into a Surface Point Set (SPS) with a point spacing dependent on cutting tool size and shape local surface curvature and the desired accuracy of the approximate simulation. The surface-surface intersection calculations of the solid modeling approach are replaced by the intersection of the surface of the tool movement envelope with straight lines emanating from the surface points. The methods are applicable to both 3 and 5 axis machining. Samples test cases are presented, and implementation and efficiency issues are discussed.

Approximate methods for simulation and verification of numerically controlled machining programs

We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result[13] takes O(N2) worst-case and O(Nlog2N) expected time. This paper presents an O(N log3N) time, O(N log N) space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Strong and Bentley[1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.

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Computing the Largest Empty Rectangle

We describe a method for simulating and verifying the correctness of Numerical Control (NC) programs. NC programs contain the sequence of cutting tool movements which machine raw stock into a finished object. Our method is based on a discrete approximation of the object by a set of points. A vector is passed through each of the points and machining is simulated by finding the intersections of tool movements with these vectors. We present a point-selection method and an analysis that shows that the error introduced by the approximation can be made as small as desired. The run time is inversely proportional to the allowable error and the size of the cutting tool, and directly proportional to the distance that the cutting tool moves.

Robert L. Scot Drysdale

Papers

Finding extremal polygons

Approximate methods for simulation and verification of numerically controlled machining programs

Computing the Largest Empty Rectangle

Discrete simulation of NC machining

Farthest line segment Voronoi diagrams