Feedback control of a planar manipulator with an unactuated elastically mounted end effector

We study feedback control laws that can stabilize an equilibrium position of an underactuated planar PPR robot manipulator. The manipulator operates in a horizontal plane, and it consists of two actuated prismatic links, an actuated revolute link, and an unactuated end effector that is elastically mounted on the third link. The fact that the end effector is unactuated and is elastically mounted on the manipulator, and thus can be controlled only through coupling with the manipulator dynamics, provides a major obstacle to control design for this problem. The control objective is to design a feedback controller that can achieve arbitrary positioning and arbitrary attitude of the end effector while suppressing any motion of the end effector relative to the third link. A nonlinear control model is constructed for this underactuated manipulator and a time-invariant discontinuous feedback law is constructed that achieves the objectives with exponential convergence rates. The effectiveness of the proposed feedback law is illustrated through simulations