An Asymptotically Stable Joint PD Controller for Robot Arms with Flexible Links Under Gravity

'The regulation problem for articulated mechanical arms is often solved by designing simple control laws which strongly exploit the physical properties of the system. It is well known that a rigid robot can be globally asymptotically stabilized around a given joint configuration via a P D controller on the joint errors, provided that gravity is exactly cancelled by feedback [l]. Under a mild condition on the proportional gain, this scheme can be simplified by performing only a constant gravity compensation at the desired configuration [2]. This result was extended in [3] t o the case of robots with elastic joints, under the further assumption that joint stiffness overcomes the gradient of the gravitational term. Asymptotic stability of a joint PD controller for robot arms with flexible links has been recently shown in the absence of gravity [4]. Inspired by the approach of [3], in this work we prove global asymptotic stability of a joznt PD controller, i.e. avoiding feedback from the elastic coordinates, with constant gravity compensation for the full nonlinear model of multilink flexible robots. A structural assumption about link elasticity is required and a mild condition on the proportional gain is derived. The proof goes through a classical Lyapunov argument.