Effects of Viscous Damping on Differential Flatness-Based Control for a Class of Under-Actuated Planar Manipulators

For under-actuated manipulators, not all joint trajectories are dynamically feasible and it is difficult to characterize feasible joint trajectories analytically. Differential flatness, if applicable, provides a systematic approach to plan and control feasible trajectories for such systems. Previously, Agrawal and Sangwan (2008) have formulated a methodology to design under-actuated planar manipulators to be differentially flat. This methodology assumed that the joints in the manipulator have no dissipation. In real systems, however, these joints have dissipative effects (viscous damping, coulomb friction etc.). This letter studies effects of viscous damping at unactuated joints on differential flatness-based control for the class of underactuated manipulators presented by Agrawal and Sangwan in 2008. It is shown that in presence of viscous damping the original flat outputs lose relative degree and a stable internal dynamics is induced. Hence, the original outputs are no longer flat outputs. However, because of the stable internal dynamics induced they are still useful in applications where only output control is needed and full-state control is not desired. It is also shown that although the original set of outputs lose relative degree with viscous damping, the system remains differentially flat with another set of flat outputs. Hence, full-state control can still be performed with a new set of flat outputs in presence of viscous damping. Simulation results for controllers based on the two scenarios mentioned above are presented with a 3-DOF manipulator.