Force Control involves moving the end-effector of a robot manipulator on the surface of an object while ensuring that no other part of the manipulator collides with the object. Suppose C is a given contour to be followed. If the end-effector can move between two points a and b on C while meeting the collision avoidance requirement, we can say that a path exists between a and b. We begin by considering a planar manipulator and a circular contour and derive the necessary and sufficient conditions for a path to exist between a pair of points. By extending these ideas, sufficient conditions are derived for a noncircular contour. The advantages of a (kinematically redundant) 3-link planar manipulator over a 2-link manipulator are pointed out. Finally, we consider spatial manipulators and derive the necessary and sufficient conditions for the case where the contour lies on the surface of a sphere.

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