a framework for planning multiple paths in free space - CiteSeerX

A FRAMEWORK FOR PLANNING MULTIPLE PATHS IN FREE SPACE

Bonny Banerjee and B. Chandrasekaran Laboratory for AI Research, Dept. of Computer Science & Engineering The Ohio State University, Columbus, OH 43210, USA {Banerjee | Chandra}@cse.ohio-state.edu

ABSTRACT Path planning, a topic of much interest in military planning, is largely treated as the task of finding the best path with respect to some criterion such as length, travel time, etc, for which efficient algorithms are already available. Military planning requires understanding enemy intentions and devising unexpected plans to fox the enemy which calls for not a best path, but a number of alternative paths. We study the problem of computing multiple paths with different properties, such as all paths with at most L loops, in free space among polygonal obstacles using a framework of Voronoi diagram. The complexity of the algorithms have been analyzed. We show that the Voronoi diagram, though widely used, is inadequate to represent certain important properties of representative paths in free space. Further, we show how this framework might be applied in three different military problems – entity reidentification, ambush analysis, and rapid re-routing in urban operations. 1. INTRODUCTION Path planning is a topic of much interest in military decision making. Much of the literature on path planning treats the task as one of finding the best path, where best is with respect to some criterion such as length, travel time, fuel costs, etc. However, quite often the real need is not for a best path, but for a number of alternative paths, in some cases not even optimal in any of the senses above. For example, in planning courses of action (COAs), the commander might be interested in paths that would be unexpected by the enemy, such as the famous ”Hail Mary” operation during the 1991 Gulf War, in which the coalition forces chose an especially long path to get behind Iraqi lines. Similarly, in understanding enemy intentions based on observations, the question is often not only, ”What is the best path for the enemy?” but ”What are the routes the enemy might have followed that can explain certain observations?” Path planning can be done in two different substrates – free space and graphs. A graph has the notion of nodes and links while free space has the concept of

homotopy classes. Two paths are considered to belong to the same homotopy class if they share the same starting and ending points, and one can be continuously deformed into the other without crossing any obstacle. It is often useful to interchange between the two substrates, especially from free space to graphs, when solving path planning problems. While this has been done a number of times in the past, the approaches have been ad hoc and a formal characterization is still missing. In this paper, we study a framework of Voronoi diagram for systematically transforming from free space to graphs and back. We also identify the corresponding properties of the paths in the two substrates, and prove that a path from any homotopy class can be generated from the Voronoi diagram of polygonal obstacles. We introduce the qualitative notion of representative paths which, as we will see, lets us talk intuitively about modifying a path to avoid small obstacles, such as potholes as opposed to mountains, without changing the nature of the path drastically. We also present algorithms for generating all paths in undirected graphs that satisfy different criteria, such as all simple paths or all paths with at most L loops, analyze the complexities of the algorithms, and discuss whether all the paths obtained from graphs necessarily retain their properties when transformed to free space. We illustrate the applications of the framework by solving problems involving path planning in the military domain. The rest of the paper is organized as follows. In the next section, we describe the framework of Voronoi diagram, including the algorithms for generating multiple paths between two points avoiding polygonal regions. Section 3 illustrates how the framework is used to solve three different military problems – entity re-identification, ambush analysis, and rapid re-routing in urban operations. Finally, we end with related work and conclusion. 2. THE FRAMEWORK 2.1 Voronoi diagram of regions One of the well known free space path planning methodologies is the roadmap-based method where the planning agent has complete knowledge of the obstacles,

starting and ending points. In such cases, the Voronoi diagram is an attractive way to obtain the roadmap. A Voronoi diagram of a set of polygonal regions S = {R1 , R2 , ...Rn } in finite 2D space is a partition of the space into polygonal Voronoi cells {V1 , V2 , ...Vn } such that the distance between any point P in the cell Vi and region Ri ∈ S is less than the distance between P and region Rj ∈ S, j 6= i. The Voronoi edges consist of locus of points that are equidistant from neighboring regions. The points where more than two Voronoi edges meet are called Voronoi vertices; they are equidistant from all the neighboring regions. Computation of a Voronoi diagram of n polygonal regions requires O(N log2 N + nN ) time where N is the number of points on the periphery of the regions (Srinivasan and Nackman, 1987; Meshkat and Sakkas, 1987). It can be shown using Euler’s formula that the size of a Voronoi diagram is linear in the number of regions. More precisely, if |v| and |e| are the number of Voronoi vertices and edges respectively, then for n ≥ 3, |v| ≤ 2n − 5 and |e| ≤ 3n − 6. Unless otherwise stated, we will refer to Voronoi diagram of regions by the term ”Voronoi diagram” throughout the rest of the paper. A Voronoi diagram, such as the one shown in Fig. 1(b) for the data set in Fig. 1(a), has both topological and geometric properties. On one hand, it is a graph structure with the Voronoi vertices being the nodes and Voronoi edges being the links. On the other hand, the Voronoi vertices are located at specific positions and the Voronoi edges retain the property that any point lying on it is equidistant from neighboring regions. A simplified graph structure can be obtained from a Voronoi diagram by stripping the Voronoi diagram off its geometrical properties, i.e. replacing the Voronoi edges by links, as shown in Fig. 1(c). An algorithm for obtaining the simplified graph from a Voronoi diagram has to visit each Voronoi vertex, thereby incurring a computational complexity of O(n). The problem of generating multiple paths in free space can be transformed into the problem of generating multiple paths in graphs provided there exists an algorithm for obtaining a suitable graph that retains the free space properties and an algorithm for obtaining the free space paths from the paths in graphs. Intuitively, a free space path refers to a representative path unless otherwise stated (such as shortest path). A path is considered a good representative path of a homotopy class if it is devoid of unnecessary turns. A homotopy class has a number of good representative paths but it has a unique shortest path. Representative paths maintaining maximum distance from obstacles at all points have been used extensively for path planning in free space, especially in robotics (Thrun et al., 1998) and military applications (Forbus et al., 2002; Banerjee and Chandrasekaran, 2004).

(a) Data set

(b) Voronoi diagram

(c) Simplified graph

(d) Shortest path

(e) Second shortest path

(f) Third shortest path

(g) Fourth shortest path

(h) 1000th shortest path

Fig. 1.

Illustrating the steps of our algorithm.

2.2 A free space path from any homotopy class can be derived from the Voronoi diagram We observe that the wide usage of Voronoi diagram for path planning in free space for the last few decades is due to the fact that it retains some of the useful free space properties. In particular, we claim that a representative path from any homotopy class can be derived from the Voronoi diagram. A path π can be derived from the Voronoi diagram M if for each point on π, there exists a corresponding but not necessarily unique point on M and M can be traced along the sequence of points thus

obtained to form a continuous path homotopic to π. In the following we will present a proof of this claim. Proof: There exists a unique shortest path in each homotopy class. Given the unique shortest path πH,shortest of a homotopy class H, a transform ζ produces a path πM i.e. ζ(πH,shortest ) = πM , if the normal at any point Pi on πH,shortest intersects M at a point PMi without previously intersecting any region Rk ∈ S or M . Since any path in H can be transformed to πH,shortest by shortening, any path in H can be derived from M if πH,shortest can be derived from M . The proof will be complete if we can prove the following: 1. Given the unique shortest path πH,shortest of a homotopy class H, ζ will produce at least one point PMi on M for each point Pi on πH,shortest . 2. Pi