Adaptive Output Feedback Control of a Spheroidal Underactuated Underwater Vehicle V. Stepanyan, N. Hovakimyan, and C. A. Woolsey Abstract— Adaptive output feedback is applied to an un- certain model of an underactuated underwater vehicle. The vehicle is modelled as a neutrally buoyant, prolate spheroid moving in the longitudinal plane. The two inputs are thrust along the symmetry axis and pitch moment. A linear reference model, which has the same vector relative degree of the system, is designed by linearizing the vehicle dynamics about a trim condition. A linear dynamic compensator is developed for the reference model, causing it to track a given bounded reference command. The input is augmented with an adaptive element and this adaptive control law is applied to the nonlinear system. The adaptive control law is realized using a linearly parameterized neural network which operates over a finite sample of the input-output history. I. I NTRODUCTION Underwater vehicles have long motivated robust and adaptive control methods [1]. Precise motion control is re- quired for important applications such as synthetic aperture imaging, measuring ocean turbulence, and docking. Model- based control is limited, however, by the fact that underwa- ter vehicle model parameters are difficult to measure and are subject to change. For these reasons, robust and adaptive control techniques for underwater vehicles have been widely studied. Early, and even recent papers on adaptive control of underwater vehicles have focused on fully actuated vehicles. For example, Majdalani and Mrad have applied the compos- ite adaptive control technique to maintain the desired posi- tion and attitude of a fully actuated underwater vehicle in an uncertain and changing environment [2]. While vehicles designed to operate at low speeds are often fully actuated, this is rarely true for streamlined vehicles designed to swim efficiently at moderate speeds. Building on the work in [3], Morel and Leonessa considered the adaptive tracking problem for the directional motion of an underactuated underwater vehicle [4]. They considered a vehicle with a fixed thruster and a rudder or, equivalently, a vectored thruster. More recently, Do and Pan treated the more general problem of a vehicle with six degrees of freedom. They considered a vehicle with a fixed thruster and three control moments [5]. While these papers do address the important issue of underactuation, the resulting controllers require state feedback rather than output feedback. State feedback is often impractical in underwater vehicle applications. Department of Aerospace and Ocean Engineer- ing, Virginia Tech, Blacksburg, VA 24061-0203, USA. {vahrams,nhovakim,cwoolsey}@vt.edu In this paper, we consider adaptive output feedback con- trol of the longitudinal motion of an underwater vehicle with thrust and pitch moment as inputs. The directional motion control problem in the papers cited above is analogous, except for a gravitational moment in the pitch equation. The adaptive controller is designed from the perspective of augmenting a fixed gain linear design that is assumed to satisfy performance requirements in the absence of modeling errors. The linear model is constructed by lin- earizing the nonlinear dynamics around a trim condition. A linear output feedback dynamic compensator is constructed for the reference model to track asymptotically the given reference command. A neural network (NN) reconstructs the unknown dynamics from a finite history of available measurements as described in [6]. The adaptive laws are written in terms of the output of a linear observer for the nominal system’s error dynamics, as in [7]. Ultimate boundedness of the error signals can be shown through Lyapunov’s direct method [8]. Section II presents the kinematic and dynamic equations. The linear reference model and the linear control law are defined in Section III and the adaptive output feedback problem is formulated. In Section IV, we design the adap- tive control law which uses a NN to approximate the unknown dynamics from a finite history of the inputs and measurements. In Section V, we present simulation results for a small AUV. Section VI states some concluding remarks and plans for extending this work. II. EQUATIONS OF MOTION Consider an autonomous underwater vehicle (AUV) mod- eled as a rigid, spheroidal hull, a shape which adequately represents many existing AUV’s. The spheroid principal axes are taken as the axes of a body-fixed coordinate frame. Thus, the center of buoyancy (CB) is located at the body frame origin. The underwater vehicle is equipped with a single external thruster, which is aligned with the axis of symmetry, and with some means (e.g., tail fins) of providing control torques about the three body axes. The vehicle is assumed to be neutrally buoyant, with its center of gravity (CG) located at the point (0, 0,b) in the body frame (directly below the CB). While these latter two assumptions are convenient, they could be easily relaxed in the presentation that follows. For now, we consider only the vehicle’s motion in the longitudinal plane. Let x denote horizontal position of the vehicle with respect to an inertial frame and let z