Obstacle Distances and Visibility for Car-Like Robots Moving Forward

This paper deals with distance computation between a car-like robot moving only forward and polygonal obstacles. We propose eecient geometric algorithms to compute the shortest paths to obstacles. We then derive the \visibility" domain in the presence of obstacles , i.e. the set of positions reachable from a starting connguration, by a collision-free shortest path unaffected by the presence of the obstacles. 1 History A car-like robot is deened by the following control system in R 2 S 1 : 8 < : _ x = cos u 1 _ y = sin u 1 _ = u 2 (1) where ju 2 (t)j ju 1 (t)j 1 are, respectively, the linear and angular velocity of the car (the lower bound on the turning radius is supposed to be 1). The study of the shortest paths for a car-like robot has already an history. The pioneering result has been achieved by Dubins who characterized the shape of the shortest paths when u 1 1 6] 1. More recently, Reeds and Shepp have provided a suucient family of 48 shortest paths when ju 1 j 1 9]. In both cases, optimal paths are constituted by a sequences of at most ve straight line segments or arcs of a circle of radius 1. Then the problem has been revisited from a control theory point of view: Sussmann and Tang 12] and Boissonnat, Cerezo and Leblond 1] independently provided a new proof of Reeds and Shepp's result. In addition Sussmann and Tang reduced the suucient family to 46 canonical paths. At the same time, Sou eres and Laumond, using these results, computed a synthesis of the shortest 1 Starting from this result, Cockayne and Hall have computed the accessibility set for this model 5]. paths, i.e. a partition of the manifold R 2 S 1 into cells reachable by only one type of shortest path (among the 46 ones) 10]. They also computed the exact shape of the shortest path metric, i.e., the shape of the domain in R 2 S 1 reachable by paths with bounded length 7]. The projection of that domain on R 2 corresponds to the accessibility domain in the plane; it has been computed by Sou eres, Fourquet and Laumond 11] who then extended the result of Cockayne and Hall 5] to the car moving also backward; the authors provide also a synthesis of the shortest path from …