Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

This paper shows how to compute the nonholonomic distance between a point-wise car-like robot and polygonal obstacles. Geometric constructions to compute the shortest paths from a configuration (given orientation and position in the plane of the robot) to a position (i.e., a configuration with unspecified final orientation) are first presented. The geometric structure of the reachable set (set of points in the plane reachable by paths of given length #) is then used to compute the shortest paths to straight-line segments. Obstacle distance is defined as the length of such shortest paths. The algorithms are developed for robots that can move both forward and backward (Reeds&Shepp's car) or only forward (Dubins' car). They are based on the convexity analysis of the reachable set. Keywords--- Car-like robots, shortest paths, nonholonomic distance. I. Introduction Distance computation plays a crucial role in robot motion planning. Numerous motion planning algorithms rely on obstacle dis...

The shortest paths for a mobile robot are a fundamental property of the mechanism, and may also be used as a family of primitives for motion planning in the presence of obstacles. This paper characterizes shortest paths for differential-drive mobile robots, with the goal of classifying solutions in the spirit of Dubins curves and Reeds-Shepp curves for car-like robots. To obtain a well-defined notion of shortest, the total amount of wheel rotation is optimized. Using the Pontryagin Maximum Principle and other tools, we derive the set of optimal paths, and we give a representation of the extremals in the form of finite automata. It turns out that minimum time for the Reeds-Shepp car is equal to minimum wheel-rotation for the differential drive, and minimum time curves for the convexified Reeds-Shepp car are exactly the same as minimum wheel-rotation paths for the differential-drive. It is currently unknown whether there is a simpler proof for this fact. An earlier version of this work appeared in [8, 7]. 1

This paper shows how to compute the nonholonomic distance between a car-like robot of polygonal shape and polygonal obstacles. Adopting an optimal control point of view, we use transversality conditions to get information about the structure of paths that are admissible solutions. With this information, the problem of minimizing the length of a path that is, in general, function of three parameters, is reduced to that of minimizing a function of one variable, namely, the robot final orientation. To solve the problem, we decompose it into three subproblems and find su#cient families of shortest paths solving each of the subproblems. 1 Introduction Distance computation plays a crucial role in robot motion planning. Numerous motion planning algorithms rely on obstacle distance computation, e.g., skeletonization and potential fields methods [11]. The distance from a robot configuration to an obstacle is the length of the shortest feasible path bringing one point on the robot boundary in ...

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This paper presents the minimum-time sequences of rotations and translations that connect two configurations of a rigid body in the plane. The configuration of the body is its position and orientation, given by(x,y,θ) coordinates, and the rotations and translations are velocities (˙x, ˙y, ˙ θ) that are constant in the frame of the robot. There are no obstacles in the plane. We completely describes the structure of the fastest trajectories, and present a polynomialtime algorithm that, given a set of rotation and translation controls, enumerates a finite set of structures of optimal trajectories. These trajectories are a generalization of the well-known Dubins and Reeds-Shepp curves, which describe the shortest paths for steered cars in the plane. 1

The optimal trajectories are known analytically for only a few ground vehicles: steered cars (Dubins 1957; Reeds & Shepp 1990), and wheel-chair-like differential-drive vehicles (Balkcom & Mason 2002). This paper presents the analytical