Control of manipulators with free-joints via the averaging method

A manipulator with free joints is a class of under-actuated mechanisms. Control of such systems is one of major topics in robotics and control engineering. In this paper, we apply the averaging method to manipulators connected by free joints and describe their behaviors in response to periodic inputs. We analyze 2R and 3R free-joint manipulators and show that each system has invariant manifolds defined by an energy-like conserved quantity. We also develop a control method via feedback modulation of the input amplitude to reach a desired invariant manifold. The effectiveness of the method is verified by computer simulations.