## Tracking and Regulation Control of an Underactuated Surface Vessel with Nonintegrable Dynamics## IntroductionOver the past decade, many researchers have studied the control problem for underactuated systems with nonintegrable constraints. The majority of this research has targeted nonholonomic systems (i.e., systems with nonintegrable velocity constraints), such as wheeled mobile robots and the general chained-form system (for a survey of research that has targeted tracking and regulation control of nonholonomic systems see [2], [5], [4], [9], [10], [11], [19], [20], and the references within). However, motivated by the challenging theoretical aspects and numerous practical applications, researchers have also attacked underactuated systems with nonintegrable dynamics (e.g., surface vessels, twin rotor helicopters, underwater vehicles, V/STOL aircrafts, etc.). For example, in [17], Reyhanoglu et al. provides a detailed discussion on the controllability and the stabilizability of underactuated mechanical systems with nonintegrable dynamics. The conclusion from this discussion is a result similar to Brockett's condition [1] for nonholonomic systems. That is, Reyhanoglu et al. illustrated that underactuated systems with nonintegrable dynamics cannot be asymptotically stabilized by a continuous, time-invariant feedback law. In [12], Pettersen et al. showed that underactuated surface vessels cannot be asymptotically stabilized by either continuous or discontinuous time-invariant feedback laws. In addition, Pettersen et al. [12] proposed a time-varying feedback controller for an underactuated surface vessel that contained explicit time-periodic sinusoidal terms (similar in structure to [19]) to obtain local exponential regulation. In [13], Pettersen et al. modified the continuous time-varying feedback law of [12] to design a controller that also locally exponentially regulates the position and orientation of an underactuated surface vessel.In addition to the regulation problem, several controllers have also been proposed for the tracking control problem. Specifically, in [8], Godhavn utilized a continuous time-invariant state feedback controller to achieve global exponential position tracking provided the desired surge velocity is always positive; however, due to the control structure, the orientation of the surface vessel is not controlled. In [14], Pettersen et al. proposed a tracking controller that achieved global exponential practical stability (i.e., global exponential stability of an arbitrarily small neighborhood of the desired trajectory) of an underactuated surface vessel. In [15], Pettersen et al. proposed a continuous time-invariant control law that obtained semi-global exponential position and orientation tracking, provided the desired angular trajectory remains positive. That is, Pettersen et al. proved semi-global exponential position and orientation tracking for a class of desired trajectories (i.e., a straight line or a sinusoidal trajectory cannot be tracked). In this paper, we design a continuous time-varying tracking controller that yields global uniformly ultimately bounded (GUUB) position/orientation tracking. Specifically, we first manipulate a reference model generator and the dynamic model of an underactuated surface vessel into a form that allows a Lyapunov-based control structure to be developed. That is, motivated by the dynamic oscillator designed in [4] by Dixon et al. for wheeled mobile robots, a time-varying dynamic oscillator is constructed that globally exponentially forces the position/orientation tracking error to a neighborhood about zero that can be made arbitrarily small. The new result is facilitated by fusing a filtered tracking error transformation with the dynamic oscillator design. In addition, since the only restriction we place on the desired trajectory is that the reference generator remain bounded, it is straightforward to illustrate that the proposed controller also yields a GUUB result for the regulation problem. The paper is organized as follows. In Section 2, we present the kinematic and dynamic model for an underactuated surface vessel and then transform the open-loop tracking dynamics into a more convenient form for the subsequent controller development and the stability analysis. In Section 3, we present the proposed GUUB tracking control design. The corresponding closed-loop error system is given in Section 4 while the stability analysis is given in Section 5. An extension that illustrates that the proposed tracking controller also solves the regulation problem is given in Section 6. In Section 7, we illustrate that the proposed tracking/regulating controller for underactuated surface vessels can also be applied to other underactuated systems with nonintegrable dynamics such as twin rotor helicopters. Simulation results are provided in Section 8 in order to illustrate the performance of the proposed tracking controller. Concluding remarks are presented in Section 9. ## Simulation Results:Select and click to view some of the simulation plots: ## Publication:For more information concerning this research, please refer to the following publication: A. Behal, W. E. Dixon, D. M. Dawson, and Y. Fang, "Tracking and
Regulation Control of an Underactuated Surface Vessel with
Nonintegrable Dynamics'', |