On the movement of robot arms in 2-dimensional bounded regions

The classical mover´s problem is the following: can a rigid object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that a more general version of this problem involving objects with movable joints is PSPACE-complete, even for a simple tree-like structure. In this paper, we investigate a 2-dimensional mover´s problem in which the object being moved is a robot arm with an arbitrary number of joints. We reduce the mover´s problem for arms constrained to move within bounded regions whose boundaries are made up of straight lines to the mover´s problem for a more complex linkage that is not constrained. We prove that the latter problem is PSPACE-hard even in 2-dimensional space and then turn to special cases of the mover´s problem for arms. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle.