Provably Good Approximation Algorithms for Optimal Kinodynamic Planning: Robots with Decoupled Dynamics Bounds 1

We consider the following problem: given a robot system, nd a minimal-time tra-jectory 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 diiers 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 modify it to halve the exponent in the 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 sacriiced 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.