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.

This paper focuses on the collision-free coordination of multiple robots with kinodynamic constraints along specified paths. We present an approach to generate continuous velocity profiles for multiple robots; these velocity profiles satisfy the dynamics constraints, avoid collisions, and minimize the completion time. The approach, which combines techniques from optimal control and mathematical programming, consists of identifying collision segments along each robot's path, and then optimizing the robots' velocities along the collision and collision-free segments. First, for each path segment for each robot, the minimum and maximum possible traversal times that satisfy the dynamics constraints are computed by solving the corresponding two-point boundary value problems. The collision avoidance constraints for pairs of robots can then be combined to formulate a mixed integer nonlinear programming (MINLP) problem. Since this nonconvex MINLP model is difficult to solve, we describe two related mixed integer linear programming (MILP) formulations, which provide schedules that give lower and upper bounds on the optimum; the upper bound schedule is designed to provide continuous velocity trajectories that are feasible. The approach is illustrated with coordination of multiple robots, modeled as double integrators subject to velocity and acceleration constraints. An application to coordination of nonholonomic car-like robots is described, along with implementation results for 12 robots.

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this paper is a novel non-uniform approximation method for the kinodynamic motion-planning problem. The kinodynamic motion-planning problem is to compute a collision-free, minimum-time trajectory for a robot whose accelerations and velocities are constrained. Previous approximation methods are all based on a uniform discretization in time space. On the contrary, our method employs a non-uniform discretization in configuration space (thus also a non-uniform one in time space). Compared to the previously best algorithm of Donald and Xavier, the running time of our algorithm reduces in terms of 1=", roughly from O((1=")

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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

. The first main result of this paper is a novel nonuniform discretization approximation method for the kinodynamic motion-planning problem. The kinodynamic motion-planning problem is to compute a collision-free, time-optimal trajectory for a robot whose accelerations and velocities are bounded. Previous approximation methods are all based on a uniform discretization in the time space. On the contrary, our method employs a nonuniform discretization in the configuration space (thus also a nonuniform one in the time space). Compared to the previously best algorithm of Donald and Xavier, the running time of our algorithm reduces in terms of 1/#, roughly from O((1/#) 6d-1 ) to O((1/#) 4d-2 ), in computing a trajectory in a d-dimensional configuration space, such that the time length of the trajectory is within a factor of (1 + #) of the optimal. More importantly, our algorithm is able to take advantage of the obstacle distribution and is expected to perform much better than the analyt...

This paper surveys some recent theoretical work in incorporating the dynamics of robot models in motion planning algorithms. Such work is often referred in the literature as kinodynamic motion planning. We first give an overview of the problem in general and how it differs from motion planning with pure kinematics. We then describe three model algorithms, each representing one of three broad categories of existing methods to attack problems involving kinodynamic motion planning. For each algorithm, we review the problem considered, assumptions made, the key ideas, experimental results, and possible extensions. Computational complexity, as well as other issues relevant to practical implementations are discussed, including areas of future research. 1 Introduction Motion planning algorithms attempt to find a sequence of actions to move a system from an initial to a goal configuration [7]. The seminal paper by Schwarth and Shahir on the generalized piano movers' problem sparked a great de...

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