Relaxation of Roadmap Methods of Robot Motion Planning MILOŠ ŠEDA Institute of Automation and Computer Science Brno University of Technology Technická 2, 616 69 Brno CZECH REPUBLIC Abstract: - In robot motion planning a robot should pass around the obstacles, from a given starting position to a given target position, touching none of them, i.e. the goal is to find a collision-free path from the starting to the target position. This task has many specific formulations depending on the shape of obstacles, allowable directions of movements, knowledge of the scene, etc. Research on path planning has yielded many fundamentally different approaches to its solution that can be classified as roadmap methods (visibility graph method, Voronoi diagram) and methods based on cell decomposition. In this work, an approach based on Voronoi diagrams is proposed, considering point, straight line and polygonal obstacles in a completely known scene. Approximating polygonal obstacles by sets of points makes it possible to apply roadmap methods to more complex problems. Key-Words: - motion planning, cell decomposition, roadmap method, visibility graph, Voronoi diagram 1 Introduction The task of planning trajectories of a mobile robot in a scene with obstacles, has received considerable attention in the research literature [2,6,11,13]. A robot is usually represented by a single point or a circle. There are three basic types of robot motion planning algorithms [9]. The first type is the potential field method. The goal has an attractive potential and the obstacles have a repulsive potential. The robot moves in the direction of the gradient of a potential field produced by the goal configuration and the obstacles. Unfortunately, this algorithm often converges to a local minimum in the potential field and therefore we will not deal with it. The second type is the cell decomposition method. Here, the scene is decomposed into cells and the outcome of the search is a sequence of adjacent cells between start and target from which a continuous path can be computed. The square cell decomposition can be used for 8-directional (horizontal, vertical and diagonal) robot motion in the plane with static rectangular obstacles. Unfortunately, this approach has many drawbacks such as combinatorial explosion, limited granularity and generating infeasible solutions as we briefly show in the next paragraph. This approach can be slightly improved using a case-based reasoning procedure [5]. The third type of motion planning algorithm is referred to as a roadmap method. The roadmap is built by a set of paths where each path consists of collision free area connections. There are several different methods for developing the roadmap, e.g. visibility graphs and Voronoi diagrams [7]. As these methods do not have the drawbacks of the previous ones, we will study them in more detail and try to combine them. 2 Roadmap Methods The visibility graph is a graph whose vertices include the start, target and vertices of polygonal obstacles [2,7]. Its edges are given by the edges of the obstacles and edges joining all pairs of vertices that can see each other. A visibility graph is shown in Fig. 1 and the shortest path between the starting position and the target in this graph is shown in its right-hand part. Here, the full line represents the shortest path for the point robot and the dashed line is its modification for a robot of nonzero size. A Voronoi diagram of a set of sites in the plane is a collection of regions that divide up the plane. Each region corresponds to one of the sites and all the points in one region are closer to the site representing the region than to any other site [1,3,9]. More formally, we can define Voronoi diagrams in mathematical terms. The distance d(p i , p j ) Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp642-647)