Abstract
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 three-dimensional uncluttered environment is Θ( n 2 f/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.
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