Introduction

The research on underwater systems has gained an immense interest during the last decades with applications taken place in many fields such as exploration, investigation, repair, construction, etc. Hereby, control of underwater systems has emerged as a growing field of research. Underwater vehicles, in fact, accounted for 21% of the total number of service robots by the end of 2004, and are the most expensive class of service robots (UNECE/IFR, 2005). Typically, underwater vehicles can be divided into three underwater systems, namely, the manned submersibles, remotely operated vehicles (ROV) and autonomous underwater vehicles (AUV). ROVs and AUVs are mostly utilized in the oil and gas industries, and for scientific and military applications. AUVs, especially are of great importance due to their ability to navigate in abyssal zones without necessitating a tether that limits the range and maneuverability of the vehicle. However, their autonomy property directly affects the design of the control system. This requires advanced controllers and specific control schemes for given tasks.

Almost all AUVs are six degrees-of-freedom (DOF) systems, and various types of actuator configurations are available in the industry for the vehicles ranging from fully-actuated vehicles to underactuated ones. The vehicle of interest here falls into the class of underactuated AUVs. Any mechanical system having fewer actuators than its degrees of freedom is defined as an underactuated system. Some examples of underactuated systems include manipulators; (Arai et al., 1998), (Oriolo & Nakamura, 1991), (Yabuno et al., 2003), marine vehicles; (Reyhanoglu, 1997), (Pettersen & Egeland, 1996), space robots; (Tsiotras & Luo, 1997), and the examples given in (Fantoni & Lozano, 2002).

Controlling all of the DOF of underactuated mechanical systems is an arduous task compared to the fully actuated systems since the mathematical analysis of the system renders it difficult. Determining whether an underactuated system is controllable is one of these difficulties encountered. Control synthesis is also another challenge in this field and is still accepted as an open problem. The techniques used for fully actuated systems cannot be used directly for underactuated systems. However, there are some potential benefits over fully actuated systems depending on the efficiency of control and the task. In case of actuator failures, a fully actuated mechanical system falls into the class of underactuated systems and might still be controlled if a successful control scheme can be designed. Besides that, reduction of the weight and cost, and the increase of reliability can be considered as advantages of underactuated systems. On the other hand, underactuation may take place by design as in helicopters, ships, underwater vehicles, satellites, hovercrafts, etc. Control problem of underactuated systems has been generally studied as a control problem of a class of nonholonomic systems, although the relation between underactuated systems and nonholonomic systems has not been clear yet (Kolmanovsky & McClamroch, 1995). Nonholonomic systems are known as the mechanical systems of nonholonomic constraints which cannot be integrated to obtain the equations describing the position of the system. Control of nonholonomic systems poses a difficult problem requiring a special control approach depending on the nature of the mechanical system, and the modelling of nonholonomic systems as state equations is another difficulty (Sampei et al., 1999). In spite of having six effective DOF, the vehicle has one controllable DOF since it has just one actuator (the propeller). The vehicle does not have any other control element save for the thrust provided by the propeller. The propeller produces the main thrust. Consequently, the reaction of the body to the load torque of the propeller produces a moment with respect to its rotational axis. Thus, the vehicle is considered underactuated because it has fewer actuators than the degrees of freedom of the system. The vehicle is a nonlinear system: all equations of motion of the system include coupled terms. Some equations of the motion of the system appear as second-order nonholonomic constraints, and they cannot be integrated to obtain position. Therefore, such underwater vehicle is pertained as a nonholonomic system.

In control of underactuated autonomous underwater vehicles (UUVs), optimal control approach has not been widely applied. Jeon et al. (Jeon et al., 2003) proposed an optimal linear quadratic controller for a 6-DOF underwater vehicle with four thrusters to distribute the thrust optimally. Additionally, some motion planning approaches were discussed in (B0erhaug et al., 2006) and (Bullo & Lynch, 2001).

Fukushima (Fukushima, 2006) proposes a novel control method of solving optimal control problems including nonlinear systems. In this study, this control method is used. His method proposes a wide range of applicability and simplicity. It can be applied to both linear systems and nonlinear systems including the systems to be controlled in real-time. Although this method has similarities with the classical optimal control theory, it can be seen as a radical contribution in the control engineering field, rather than the extension of the existing optimal control theory.

Fukushima's method is fundamentally based on the employment of the energy generation, storage and dissipation of the controlled system. The total system energy stored in the system boundary is the sum of each energy. The criteria function consists of the control-performance, which is determined for a given task, the input energy, and the energy equation. First derivatives of the energy equation and the performance measures constitute a scalar function. The minimization of the scalar function yields the optimal control law. The necessary condition for the minimization is the Euler equation. The use of energy equation in the criteria function enables the optimal control law to have efficient dissipation characteristics. Obtaining the control law is a simple process and is not much mathematically involved. As one of the important properties of his control method, the control-performance can be of any form. There is no restriction in determining it, whereas the classical optimal control theory works well with performance measures of quadratic form.

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