Robustness of a flatness based controller against parametric uncertainties for a class of under-actuated planar manipulators

Abstract

Differential flatness provides a systematic approach to plan and control feasible trajectories for underactuated nonlinear systems. Previously [1], the authors have developed a methodology to design a class of under-actuated planar manipulators to be differentially flat. This paper investigates robustness of this flatness based control with respect to parametric uncertainties. Existing literature for robustness in this regard addresses fully actuated systems and underactuated systems where the flat output state does not depend on system parameters. In both these scenarios, the flat output state (with respect to which the controller is designed) is directly available from sensor measurements. However, in general for an under-actuated differentially flat system, the flat output state depends on the measurable original system state through an invertible transformation containing system parameters. In such a scenario with parametric uncertainty, the flat output states can not be measured exactly. This paper addresses robustness against parametric uncertainties in this scenario for a specific class of planar under-actuated manipulators. It is shown that if the transformation is known accurately and only the dynamic model is uncertain a robust control strategy can guarantee exponential stability. It is also shown that in the presence of uncertainties in the transformation, the tracking error can be bounded from above. The effectiveness of this robust control strategy is demonstrated through simulations with a 3-DOF robot.