SUMMARYA design methodology is presented for tracking control of second-order chained form systems. The methodology separates the tracking-error dynamics, which are in cascade form, into two parts: a linear subsystem and a linear time-varying subsystem. The linear time-varying subsystem, after the first subsystem has converged, can be treated as a chain of integrators for the purposes of a backstepping controller. The two controllers are designed separately and the proof of stability is given by using a result for cascade systems. The method consists of three steps. In the first step we apply a stabilizing linear state feedback to the linear subsystem. In the second step the second subsystem is exponentially stabilized by applying a backstepping procedure. In the final step it is shown that the closed-loop tracking dynamics of the second-order chained form system are globally exponentially stable under a persistence of excitation condition on the reference trajectory. The control design methodology is illustrated by application to a second-order non-holonomic system. This planar manipulator with two translational and one rotational joint (PPR) is a special case of a second-order non-holonomic system. The simulation results show the effectiveness of our approach.

A controller is presented for point stabilization of the extended chained form system. The extended chained form system can not be stabilized by means of continuous pure-state feedback. Moreover, it can not be exponentially stabilized by smooth feedback, since the linearization around equilibrium points is uncontrollable. In this paper, a controller is given that ρ-exponentially stabilizes the system. The controller is derived by using a combined averaging technique for homogeneous systems and a backstepping approach. The controller is illustrated in a benchmark example by application to the V/STOL aircraft without gravity.

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Abstract. This paper addresses point stabilization for the extended chained form (ECF), a control system that may be used to model a number of mechanical underactuated systems. A control law is proposed, based on well-known hybrid open-loop/feedback techniques, which exponentially stabilizes the origin of a dynamic extension of the ECF and ensures a degree of robustness to additive disturbance terms that may represent, for instance, model uncertainties. Numerical simulations are included to illustrate the performance of the presented stabilizers.

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