Integral Manifolds of Singularly Perturbed Systems with Application to Rigid-link Flexible-joint Multibody Systems

In this paper, we first review results of integral manifolds of singularly perturbed nonlinear differential equations. We then outline the basic elements of the integral manifold method in the context of control system design, namely, the existence of an integral manifold, its attractivity, and stability of the equilibrium while the dynamics are restricted to the manifold. Toward this end, we use the composite Lyapunov method and propose a new exponential stability result which gives, as a by-product, an explicit range of the small parameter for which exponential stability is guaranteed. The results are applied to the control problem of multibody systems with rigid links and flexible joints in which the inverse of joint stiffness plays the role of the small parameter. The proposed controller is a composite control law that consists of a fast component, as well as a slow component that was designed based on the integral manifold approach. We show that the proposed composite controller has the following properties: (i) it enables the exact characterization and computation of an integral manifold (ii) it makes the manifold exponentially attractive, and (iii) it forces the dynamics of the reduced flexible system on the integral manifold to coincide with the dynamics of the corresponding rigid system (i.e. the one obtained by making stiffness very large) implying that any control law that stabilizes the rigid system would stabilize the dynamics of the flexible system on the manifold. We finally present a detailed stability analysis and give an explicit range of the joint stiffness, in terms of system parameters and controller gains, for which the established exponential stability is guaranteed.