Adapt ive Tracking Control of Underactuated Surface Vessels I

In this paper, we design a continuous, t ime-varying tracking controller for an underactuated surface vessel in the presence of uncertainty associated with the hydrodynamic damping coefficients. A Lyapunov-based approach is used to ensure that the posi t ion/orientat ion tracking error is ultimately confined to a ball that can be made arbitrarily small. The result is achieved via the judicious design of a dynamic oscillator m conjuction with two state transformations. We also illustrate how the proposed tracking controller yields the same stability result for the stabilization problem. 1 I n t r o d u c t i o n Over the past few years, control of underac tua ted systems with nonintegrable dynamics has become an active topic of research. Typical examples of such problems include surface vessels, twin rotor helicopters, underwater vehicles, and Vert ical /Convent ional TakeOff and Landing aircraft. The active interest in this area of research stems from the fact tha t the systems under consideration do not satisfy Brocke t t ' s necessary condition [2] for s tabil izat ion to the origin via continuous t ime-invariant s ta te feedback. For a detai led discussion on the controllabil i ty and the stabi l izabil i ty of underac tua ted mechanical systems with nonintegrable dynamics, we refer the reader to [15]. Wi th regard to related work for underactuated systems, Morin and Samson [7] proposed a continuous t ime-varying feedback law tha t locally p-exponentially stabil ized an underac tua ted spacecraft. In [9], Pettersen et al. demons t ra ted tha t a large class of underactuated vehicles cannot be asymptot ical ly stabil ized by either continuous or discont inuous t ime-invariant feedback laws. In addit ion, Pet tersen et al. [9] proposed a continuous t ime-varying feedback controller for the surface vessel tha t locally exponential ly stabilized the averaged system to the origin. In addit ion to the s tabi l izat ion problem, several controllers have also been proposed for the tracking control problem. Specifically, in [4], Godhavn utilized a continuous t ime-invariant s ta te feedback controller to achieve global exponential posit ion tracking for a class of t rajectories provided the desired surge velocity is always positive; however, due to the control structure, the orientat ion of the surface vessel is not controlled. In [11], Pet tersen et al. proposed a kinematic t racking controller tha t achieved global exponential pract ical stabi l i ty 1This work is supported in part by the U.S. NSF Grants DMI9457967, DMI-9813213, EPS-9630167, ONR Grant N00014-99-10589, a DOC Grant, and an ARO Automative Center Grant. 2Corresponding author (i.e., global exponential s tabi l i ty of the averaged system to an arbi t rar i ly small neighborhood of the desired t ra jectory) of an underac tua ted surface, vessel. In [12], Pet tersen et al., proposed a continuous t ime-invariant control law tha t obtained semi-global exponential posit ion and orientation tracking, provided the desired angular t ra jec tory remains positive. Tha t is, Pet tersen et al. proved semi-global exponential posit ion and orientat ion tracking for a class of desired t ra jector ies (i. e., a straight line or a sinusoidal t rajectory cannot be tracked). In [1], Behal et al., proposed a continuous, t ime-varying tracking controller tha t globally exponential ly forces the posi t ion/or ienta t ion tracking error of an underac tua ted surface vessel to a neighborhood about zero tha t can be made arbi t rar i ly small (i.e., global uniformly u l t imate ly boundedness). Recently, a few researchers have examined robust and adaptive control s t rategies for the underac tua ted surface vessels. In [10], Pettersen et al. modified the continuous t ime-varying feedback law of [9] to design a controller tha t locally p-exponentially regulates the average posit ion and orientation of an underac tuated surface vessel. The control law contains explicit sinusoidal terms and is robust to variations in the hydrodynamic damping coefficients. A similar kind of s t ra tegy was employed by Pet tersen and Egeland in [13] to obtain a robust controller for the a t t i tude stabil ization of an Autonomous Underac tua ted Vehicle. Recently, Pettersen and Nijmeijer [14] presented an adapt ive controller for stabilizing an underac tua ted Surface Vessel in the presence of a disturbance; however, the unknown disturbance was assumed to be a constant. In this paper, we design a continuous t ime-varying tracking controller tha t forces the posi t ion/or ienta t ion tracking error to an arbi t rar i ly small neighborhood about zero in the presence of uncer ta inty in the hydrodynamic damping coefficients. Specifically, we first manipula te a reference model generator and the dynamic model of an underactuated surface vessel into a form tha t facilitates a Lyapunov-based adapt ive control structure. Motivated by the result in [1] and the uncertainty in the damping matr ix, we then design a desired dynamic oscillator tha t fosters asymptot ic tracking of a t ransformed system while simultaneously adapt ing for the unknown parameters . We then proceed to prove how the tracking error in the original system is ul t imately confined to a ball of arbi t rar i ly small radius. In addition, since the only restr ict ion we place on the desired t ra jec tory is tha t the reference generator remain bounded, it is s traightforward to i l lustrate tha t the proposed controller also yields a similar result for the regulat ion problem.