Ji-Hong Li, Bong-Huan Jun, Pan-Mook Lee and Yong-Kon Lim

Maritime & Ocean Engineering Research Institute, KORDI,

Republic of Korea

In the past few decades, autonomous underwater vehicles (AUVs) have been playing one of most important roles in the applications ranging from scientific research, survey to industry and military operations. Today, there is an apparent trend that more and more underwater tasks are carrying out by cooperative operations of multiple AUVs instead of traditional method of using single AUV (Soura & Pereira, 2002; Edwards et al., 2004; Guo et al., 2004; Watanabe & Nakamura, 2005; Fiorelli et al., 2006). Multiple AUVs have cost-effective potential. However, a number of research efforts are still remained to be done before this advanced technology can be fully applied in the practice. And one of the efforts is about the efficient schooling scheme for these multiple underwater vehicles.

The history of the formation or cooperative control of multiple agent systems can be traced back to the 1980's. Reynolds (1987) introduced a distributed behavioural model for flocks of birds, herds of land animals, and schools of fishes. This model can be summarized as three heuristic rules: flock centring, collision avoidance and velocity matching. In the formation algorithm (Reynolds, 1987), each dynamic agent was modelled as certain particle system - a simple double-integrator system. This kind of agent model has been inherited in most of the following research works (Leonard & Fiorelli, 2001; Olfati-Saber & Murray, 2002, 2003; Fiorelli et al., 2006; Olfati-Saber, 2006; Do, 2007). Besides these works, another type of linear model was used in Smith et al. (2001), and certain nonlinear model was applied for underwater vehicles (Dunbar & Murray, 2002) and for wheel robots with terminal constraints (Fax & Murray, 2004). In both of Dunbar & Murray (2002) and Fax & Murray (2004), the nonlinear dynamics were all fully actuated.

In this chapter, we consider the schooling problem for multiple underactuated AUVs, where only three control inputs - surge force, stern plane and rudder are available for each vehicle's six degrees of freedom (DOF) motion. For these torpedo-type underwater flying vehicles, since there are non-integrable constraints in the acceleration dynamics, the vehicles do not satisfy Brockett's necessary condition (Brockett et al., 1983), and therefore, could not be asymptotically stabilizable to an equilibrium point using conventional time-invariant continuous feedback laws (Reyhanoglu, 1997; Bacciotti & Rosier, 2005). Moreover, these vehicles' models are not transformable into a drift-less chained form (Murray & Sastry, 1993), so the tracking method proposed in Jiang & Nijmeiner (1999) cannot be directly applicable to these vehicles. Recently, quite a number of research works have been carried out on the tracking of underactuated surface ships (Jiang, 2002; Do et al., 2002a, 2002b, 2004, 2005; Pettersen & Nijmeijer, 2001; Fredriksen & Pettersen, 2006). However, the presented tracking methods were all case-by-case that strongly depended on the ship's specifically simplified dynamics. Therefore, these tracking methods also cannot be directly applicable to the case of underwater vehicles. Since the sway and heave forces are unavailable, the most challenge in the tracking control is how to properly handle the vehicles' sway and heave dynamics in the position tracking. To deal with this problem, in this chapter, we introduce a certain polar coordinates transformation for the vehicle's velocities in the body-fixed frame. Through this coordinates transformation, each vehicle's dynamics can be transformed to a certain two inputs nonlinear strict-feedback form, according to which the proposed schooling scheme is derived.

For the torpedo-type underwater flying vehicles considered in this chapter, the pitch and yaw moments are proportional to the square of the vehicle's forward speed. From this point of view, the pitch and yaw moments are not exactly independent with the surge force. If the vehicle's forward speed is taken small value, then the pitch and yaw moments are also have to take small values, therefore, in this case we cannot fully excite the vehicle's pitch and yaw dynamics. To appropriately taking these three only available control inputs - surge force, pitch and yaw moments as independent ones, the vehicle's forward speed has to be guaranteed to take considerable magnitude. For this reason, in this chapter, firstly we assume that the vehicle's forward speed satisfies the above consideration. And the proposed schooling scheme, which is derived under this assumption, reversely can guarantee the assumption always to be fulfilled under certain initial conditions.

The common method of formation among the schemes presented so far is to apply certain potential function to conduct the agents' group behaviour. The potential function initially used in the robotics for mobile robot's motion planning (Latombe, 1991; Rimon & Koditschek, 1992), and recently widely applied in the formation of multiple agents systems (Leonard & Fiorelli, 2001; Olfati-Saber, 2006; Do, 2007). Aforementioned Reynolds's three heuristic rules of flock centring, collision avoidance, and velocity matching, which are also known as cohesion, separation, and alignment, are usually embodied by suitably selected potential functions. In Leonard & Fiorelli (2001), only 1 time differentiable function was used as potential for group formation, while p(p > 2) times differentiable one was applied in Do (2007) and a specific smooth potential was used in Olfati-Saber (2006). In this chapter, general form of smooth potential function is introduced and similar to Olfati-Saber (2006), the potential consists of three different components: one is for the interactions between vehicles, another is for group navigation, and the third is for obstacle avoidance. Unlike aforementioned previous works, in this chapter, the vehicle's orientation is also considered. Therefore, we have to discuss the vehicles orientation matching as well as their velocity matching. Proposed schooling scheme guarantees local minimum of the vehicles formation, and the group's velocity and orientation matching in terms of polar coordinates, while keeping obstacle avoidance.

The remainder of this chapter is organized as follows. In Section 2, the vehicles' kinematics and dynamics in the Cartesian frame are presented. Through certain polar coordinates transformation, the vehicle's model can be transformed to certain two inputs strict-feedback form. Proposed vehicles schooling rules are discussed in Section 3, and corresponding formation control laws are derived in Section 4. To illustrate the effectiveness of proposed schooling scheme, some numerical simulations are carried out and analyzed in Section 5. Finally, a brief summary and some of future works are discussed in Section 6.

Fig. 1. General framework for an AUV in the horizontal plane.

Consider a group of torpedo-type AUVs, where only surge force and yaw moment are available for each vehicle's three DOF horizontal motion1, see Fig. 1. To date over 400 true AUVs have been built (Westwood et al., 2007) and most of them such as REMUS AUVs (Prestero, 2001) and HUGIN AUVs (Marthiniussen et al., 2004) have this type of mechanical structure. For this kind of underactuated underwater vehicles, their horizontal kinematics and dynamics can be expressed as following (Fossen, 2002; Li & Lee, 2008)

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