Models and motion planning

We study the complexity of the motion planning problem for a bounded-reach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simple-cover complexity. We show that the maximum complexity of the free space of a robot with f degrees of freedom in the plane is Θ(n ^f/2 + n) for uncluttered environments as well as environments with small simple-cover complexity. The maximum complexity of the free space of a robot moving in a threedimensional uncluttered environment is Θ(n^2f/3 + n). All these bounds fit nicely between the Θ(n) bound for the maximum free-space complexity for low-density environments and the Θ(n^f) bound for unrestricted environments. Surprisingly-because contrary to the situation in the plane-the maximum free-space complexity is Θ(n^f) for a three-dimensional environment with small simple-cover complexity.