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 T(nf/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 three-dimensional uncluttered environment is T(n2f/3+n). All these bounds fit nicely between the T(n) bound for the maximum free-space complexity for low-density environments and the T(nf) bound for unrestricted environments. Surprisingly—because contrary to the situation in the plane—the maximum free-space complexity is T(nf) for a three-dimensional environment with small simple-cover complexity.