Cascade Normal Forms for Underactuated Mechanical Systems

In this paper, we introduce cascade normal forms for underactuated mechanical systems that are convenient for control design. These normal forms are partially linear which results from a well-known fact that underactuated systems can be partially linearized using a change of control [12]. The difficulty arises when the new control appears both in the linear and nonlinear subsystems. We introduce a method for decoupling these two subsystems by applying a change of coordinates that transforms the dynamics of the system into a cascade normal form with the property that control of the overall system reduces to control of its nonlinear subsystem. Under a symmetry condition on the inertia matrix of the system, this transformation can be obtained explicitly from the Lagrangian. This eventually leads to classification of underactuated systems. We provide several applications and two detailed examples of complex underactuated systems, namely, the Acrobot and the Rotating Pendulum.