## Adaptive Control for Guidance of Underwater Vehicles

Mario Alberto Jordán and Jorge Luis Bustamante

Argentinian Institute of Oceanography (IADO-CONICET),

National University of the South (UNS), Bahía Blanca Argentina

### 1. Introduction

Underwater vehicles are extensively employed in the offshore industry, subaquatic scientific investigations and rescue operations. They are sophisticated mechanisms with complex nonlinear dynamics and large lumped perturbations. They can remotely be operated or eventually autonomously navigate along specified scheduled trajectories with geometric and kinematic restrictions for obstacle avoidance or time-optimal operations (Fossen, 1994). In a wide spectrum of applications, underwater vehicles are generally described by nonlinear and time-varying dynamics. For instance, dynamics with variable inertia and buoyancy arriving from sampling missions or hydrodynamics related to large changes of operation velocity or current perturbations in which laminar-to/from-turbulent transitions are involved in the hydrodynamics.

Due to the inherent nonlinear equations of motions, perturbed environments and complex missions, subaquatic vehicles require the guidance by means of complex controllers that usually involve automatic speed controls, dynamic positioning and tracking, and autopilot systems for automatic steering of depth and altitude. It is experimentally corroborated that adaptive techniques may provide superior trajectory tracking performance compared with the fixed model-based controllers (Smallwood & Whitcomb, 2003; 2004).

Many different adaptive and robust adaptive approaches for underwater vehicles have been discussed in the literature in the past 15 years to handle uncertainties related to the dynamics, hydrodynamics and external disturbances, see for instance Fossen & Fjellstad, 1995; Hsu et al., 2000, Antonelli et al., 2004; Wang & Lee, 2003; Do et al., 2004. However, the employment of novel high-performance nonlinear control design methodologies like backstepping (Krstic et al., 1995), passivity-based approaches (Fradkov et al., 1999) or sliding modes (Hsu et al., 2000) do not appear in the literature except as incipient applications, see for instance, Do & Pan, 2003; Li et al., 2004; Jordán & Bustamante, 2006; 2007; Conte & Serrani, 1999.

From previous theoretic results and some experimental corroborations, it seems that novel adaptive techniques can give rise to an improvement of the global performance in path tracking, above all when more precise manoeuvrability with a high celerity in motion is necessary in a changeable and uncertain subaquatic environment.

In this chapter we present a general adaptive approach based on speed-gradient techniques that are modified for complex time-varying dynamics. Moreover, the dominant vehicle dynamics and hydrodynamics together with the often neglected thruster dynamics are considered in a complete time-varying and nonlinear model.

The chapter is organized as follows. First a description of the vehicle dynamics and hydrodynamics under environmental perturbations in 6 degrees of freedom is given. Additionally, thruster dynamics is described as embedded in the dominant dynamics like a fast dynamics (parasitics). Then, the tracking and regulation problems are introduced in a general form as minimization of a energy cost functional involving positioning and kinematic errors. Afterwards, a design of a fixed controller is presented. The same methodology is extended to the adaptive case. Then, an extension of the adaptive system structure by means of a state/disturbance observer is developed. Afterwards, global convergence of positioning and kinematic path errors is proved in form of theorem results. Finally, the analysis of a selected case study of navigation in a complex sampling mission illustrates the achievable high performance of the presented control approach.

### 2. Vehicle dynamics

We consider a Lagrangian approach for describing the vehicle dynamics in 6 degrees of freedom (Fossen, 1994). Moreover, we shall develop here an extension of the usual model for time-varying vehicle dynamics.

2.1 Time-varying dynamics

Consider the Fig. 1. and let us define the generalized position of the vehicle with respect to an earth-fixed frame. This is denoted by the vector r\ = [x,y, z,ip,d,tl)]T indicating translations in the x direction (surge), in the y direction (sway), in the z direction (heave), and the so-called Euler angles: (roll) about the x axis, d (pitch) about the y axis, and ^

(yaw) about the ; axis, respectively. Similarly, but referred to a body-fixed frame, the generalized velocity vector v = [u, v, w,p, q, r]Tindicates linear rates u, v, w, along the main vehicle axis X, y, Z, respectively, and angular rates p, q, r, about the axis X, y', Z, respectively.

The start point for dynamics description is the kinematic and potential energies of the vehicle, termed T and V, respectively. Consider besides the earth-fixed frame. So, the Lagragian is

After applying the Lagragian equation it is valid particularly

where Tn is the generalized force applied to the vehicle in some arbitrary point O' (not necessarily the mass center G, see Fig. 1) and Pd is the dissipated energy which is related to a drag force term through

with Dy being the generalized drag matrix with respect to the earth-fixed frame. Moreover, the Lagragian is given by

where Trb is the kinematic energy of the rigid body, Tf the kinematic energy of the fluid and V fulfills dV drj

with gn being the generalized buoyancy force. Additionally, the total kinematic energy in (4) can be expressed as

with My being the generalized inertia matrix with respect to the earth-fixed frame. Thus

Besides

Finally, (2) can be written over again as

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