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

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

Art Gallery and Illumination Problems

Two points in a polygon are called visible if the straight line segment between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point of P is visible from some point of G. This problem has been shown to be NP-hard by Lee and Lin [71]. However, Chvatal showed that the number of points of G will never exceed [n/3] for a simple polygon of n sides [21]. This latter result is referred to as the art gallery theorem

Recent Results in Art Galleries

We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line.
Our analysis is doubly founded on a novel geometric structure called angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D, ${\cal AH}(D)$, to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of ${\cal AH}(D)$ cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight.

The Polygon Exploration Problem

Given a simple polygon in the plane with two distinguished vertices, s and g, is it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O (n log n) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n log n+k), where k is the size of the output, and we prove that this is optimal.

The two guards problem

Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest-- by perimeter or area--axis-parallel rectangle and the narrowest strip enclosing at least one site of each color.

Smallest Color-Spanning Objects

Given a simple polygon in the plane with two distinguished vertices, s and g ,i s it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O(n logn) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n logn +k), where k is the size of the output, and we prove that this is optimal.

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Suppose there are k types of facilities, e. g. schools, post offices, supermarkets, modeled by n colored points in the plane, each type by its own color. One basic goal in choosing a residence location is in having at least one representative of each facility type in the neighborhood. In this paper we provide algorithms that may help to achieve this goal for various specifications of the term “neighborhood”. Several problems on multicolored point sets have been previously considered, such as the bichromatic closest pair, see e. g. Preparata and Shamos [14, Section 5.7], Agarwal et al. [1], and Graf and Hinrichs [8], the group Steiner tree, see Mitchell [11, Section 7.1], or the chromatic nearest neighbor search, see Mount et al. [12]. Let us call a set color-spanning if it contains at least one point of each color. A natural approach to the above location problem is to ask for the center of the smallest color-spanning circle. For k = n this amounts to finding the smallest circle enclosing n given points. This problem can be solved in time O(n log n) by means of the farthest site Voronoi diagram [3], in time O(n) using Megiddo’s linear programming method [10], or in randomized time O(n) by Welzl’s minidisk algorithm [16]. A special case is k = 2, then the solution is given by the bichromatic closest pair, see above. For 2 < k < n one can solve the problem as follows. Let us generalize Voronoi diagrams in the following way. If p denotes a site of color c, we put all points of the plane in the region of p for which c is the farthest color, and p the nearest c-colored site, i. e., z belongs to the

Christian Icking

Papers

The Polygon Exploration Problem

The two guards problem

Smallest Color-Spanning Objects

The two guards problem

The Farthest Color Voronoi Diagram and Related Problems