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 + 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 Θ(n+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 ) bound for unrestricted environments. Surprisingly—because contrary to the situation in the plane—the maximum free-space complexity is Θ(n ) for a three-dimensional environment with small simple-cover complexity.