Efficient algorithms for planning purely translational collision-free motion in two and three dimensions

1. Introduction In this abstract we present a collection of results representing recent progress in the design and analysis of efficient algorithms for planning purely transla-tional collision-free motion of rigid objects moving in two-or three-dimensional space amidst a collection of obstacles whose geometry is known to the system. (The results reported here were obtained in collaboration with K. Motion planning problems of this kind are particularly favorable because they involve only two or three (translational) degrees of freedom, and because they do not have to consider rotational degrees of freedom, which tend to make the structure of the free configuration space of the moving system more complex to analyze. Besides obvious applications for mobile (or flying) autonomous systems , these problems are significant for two reasons: First they constitute the simplest of all motion planning problems, so careful analysis of their complexity is called for before moving to more difficult and complex problems. Second, these translational problems often arise as subtasks in motion planning algorithms for systems with additional rotational degrees of freedom (as in [LS], [KS]). In spite of their relative simplicity, precise analysis of the combinatorial structure of the configuration space of such a translating system leads to many deep and difficult geometric problems, whose efficient solutions require development and application of sophisticated tools from computational geometry. It is the purpose of this abstract to review these prob-l e m , describe the techniques used in their solution, present sharp upper bounds on the complexity of these solutions, and mention some open problems that arise in connection with certain variants and extensions of the translational motion planning problem. 2. Two-Dimensional Problems Let B be a rigid k-sided polygonal object free to translate in the plane amidst a collection of polygonal goal is to calculate the free configuration space FP of B , consisting of all free placements of B (i.e. placements in which B does not intersect any obstacle). Having calculated FF, we can then decompose it into its arcwise connected components, so that, given any pair of free placements Z1, 22 of B, we can determine whether they lie in the same connected component of FP, in which case collision-free translational motion of B from Z1 to Zz is possible. (We can use FP for other applications, e.g. preprocess FP so that we capa determine in logarithmic time whether a given "query" placement of B is free …

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