this paper, which presents randomized, algorithmic techniques for path planning that are particular suited for problems that involve dierential constraints.

shortest path, kinodynamics, polyhedral obstacles Abstract: We consider the following problem: given a robot system, nd a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety-margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in R 3). This algorithm di ers from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error term �(2) to polynomially bound the computational complexity of our algorithm � and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error term, the algorithm returns a solution that is-close to optimal and requires only a polynomial (in ( 1)) amount of time. We extend the results of [1] in two ways. First, we modifyittohalve the exponent inthe polynomial bounds from 6d to 3d, so that that the new algorithm is O c d N 1 3d, where N is the geometric complexity of the obstacles and c is a robot-dependent constant. Second, the new algorithm nds a trajectory that matches the optimal in time with an factor sacri ced in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments. The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.

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Abstract — We introduce the notion of kinematic controllability for second-order underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collision-free trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by timeoptimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collision-free trajectory planning for a class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.

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Since differential constraints which restrict admissible velocities and accelerations of robotic systems are ignored in path planning, solutions for kinodynamic and non-holonomic planning problems from classical methods could be either inexecutable or inefficient. Motion planning with differential constraints (MPD), which directly considers differential constraints, provides a promising direction to calculate reliable and efficient solutions. A large amount of recent efforts have been devoted to various sampling-based MPD algorithms, which iteratively build search graphs using sam-pled states and controls. This thesis addresses several issues in analysis and design of these algorithms. Firstly, resolution completeness of path planning is extended to MPD and the first quantitative conditions are provided. The analysis is based on the relationship between the reachability graph, which is an intrinsic graph representation of a given problem, and the search graph, which is built by the algorithm. Because of sampling and other complications, there exist mismatches between these two graphs. If a solution exists in the reachability graph, resolution complete algorithms must con-struct a solution path encoding the solution or its approximation in the search graph

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xvii 1 Introduction 2 1.1 Motivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Previous Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2.1 Obstacles Representation : : : : : : : : : : : : : : : : : : : : : 4 1.2.2 Configuration Space : : : : : : : : : : : : : : : : : : : : : : : 5 1.2.3 Dynamic Planning : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2.4 Optimal Planning : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.2.5 Missile Guidance : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.2.6 Differential Game Theory : : : : : : : : : : : : : : : : : : : : 9 1.2.7 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3 Objective : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.4 Outline of the Approach : : : : : : : : : : : : : : : : : : : : : : : : : 12 1.5 Organization : : : : : : : : : : : : : : : : : ...

Complexity theory has used a game theoretic notion, namely alternation, to great advantage in modeling parallelism and in obtaining lower bounds. The usual definition of alternation requires that transitions be made in discrete steps. The study of differential games is a classic area of optimal control, where there is continuous interaction and alternation between the players. Differential games capture many aspects of control theory and optimal control over continuous domains. In this paper, we define a generalization of the notion of alternation which applies to differential games, and which we call "continuous alternation". This approach allows us to obtain the first known complexity theoretic results for open problems in differential games and optimal control. We concentrate our investigation on an important class of differential games, which we call polyhedral pursuit games. Pursuit games have application to many fundamental problems in autonomous robot control in the presence of ...

Abstract — Vehicles that cross lanes of traffic encounter the problem of navigating around dynamic obstacles under actuation constraints. This paper presents an optimal, exact, polynomial-time planner for optimal bounded-acceleration trajectories along a fixed, given path with dynamic obstacles. The planner constructs reachable sets in the path-velocity-time (PVT) space by propagating reachable velocity sets between obstacle tangent points in the path-time (PT) space. The terminal velocities attainable by endpoint-constrained trajectories in the same homotopy class are proven to span a convex interval, so the planner merges contributions from individual homotopy classes to find the exact range of reachable velocities and times at the goal. A reachability analysis proves that running time is polynomial given reasonable assumptions, and empirical tests demonstrate that it scales well in practice and can handle hundreds of dynamic obstacles in a fraction of a second on a standard PC. I.

) Pankaj K. Agarwal Therese Biedl y Sylvain Lazard z Steve Robbins x Subhash Suri -- Sue Whitesides k Abstract Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let P be a convex polygon with n vertices. We study the collision-free, optimal path-planning problem for B moving between two configurations inside P (a configuration specifies both a location and a direction of travel). We present an O(n 2 log n) time algorithm for determining whether a collision-free path exists for B between two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles. Center for Geometric Computing, Computer Science Depar...