Sliding mode control of underactuated multibody systems and its application to shape change control

In this article, we introduce an approach based on sliding mode control to design full state feedback controllers for stabilisation of underactuated non-linear multibody systems. We define first order sliding surfaces as a linear combination of actuated and unactuated coordinate tracking errors. Lyapunov stability analysis guarantees that all system trajectories reach and remain on the sliding surfaces. However, stability of the sliding surfaces depends on the equilibrium manifold. If the system has isolated equilibrium points, it is linearly controllable and asymptotic stability can be guaranteed under certain conditions. Otherwise, the control system fails Brockett's necessary condition for existence of a smooth stabilising feedback. In the latter case, if the total momentum is conserved, the closed-loop control system will be marginally stable. Consequently, a procedure is proposed to achieve an asymptotically stable discontinuous control law through sliding surface redefinition and shape changes. It is proposed that repetitive application of shape changes will lead to asymptotic convergence of the system to the desired configuration. Simulation results are presented for an inverted pendulum as an example of a system with isolated equilibrium points and an existing communication satellite as an example of shape change control. In both cases, the control is shown to be effective and robust with respect to uncertainties and disturbances.