Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators

We consider the following problem: given a robot system, find a minimal-time trajectory from a start state to a goal state, while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [CDRX] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds. This algorithm differed from previous work in three ways: it is possible (1) to bound the goodness of the approximation by an error term ε (2) to polynomially bound the running time (complexity) of our algorithm; and (3) to express the complexity as a polynomial function of the error term. We extend these results to d-link, revolute-joint 3D robots will full rigid body dynamics. Specifically, we first prove a generalized trajectory-tracking lemma for robots with coupled dynamics bounds. Using this result we describe polynomial-time approximation algorithms for Cartesian robots obeying L2 dynamics bounds and open kinematic chain manipulators with revolute and prismatic joints; the latter class includes most industrial manipulators. We obtain a general &Ogr;(n2 (log n)(1/ε6d-1) algorithm, where n is the geometric complexity. The algorithm is simple, but the new game-theoretic proof techniques we introduce are subtle, and employ tools from disparate parts of computational geometry, robotics, and dynamical systems.