where q, = (x,, yi) is the ith vehicle's coordinate and qvk = (xVk, yvk) is the kth virtual vehicle's coordinate with m the number of virtual vehicles needed to conduct the group geometry, aa, ba, ap, bp> 0 are design parameters with aa < ba and ap< bp, and || • || denotes the vector Euclidean norm.

The potential Va in (9) is presenting the interactions between vehicles. From (9), it is easy to see that the desired geometry of the vehicles group is a certain net of regular triangles with vehicles located at the vertices and all side lengths are equal to aa . If any two vehicles are apart from each other more than ba, then there is not any cohesion between them. For (9) to be applicable, for each vehicle, all other vehicles' information including position and velocity information should be available. However, in some practical cases, this may not be available. Instead, only the information of the vehicles in its neighbour is available. In this case, the potential function can be chosen as

where N (qi) denotes the neighbour of the ith vehicle and is defined as

N(qt) = q :|| qj -qt ||< ba, j * i, j = 1,-,n}. (12)

In order to make Va be smooth, in (11), it should be chosen that aa = ba . In (10), the potential function Vp is for the interactions between vehicles and virtual vehicles. Here the virtual vehicles are introduced to construct the geometry of the vehicles schooling. Moreover, these virtual vehicles are used to guide the group navigation, which will be discussed in details in the next subsection. In other word, these virtual ones lead the group to follow a given desired motion. From this point of view, these virtual vehicles are also called as virtual leaders. Different arrangements of these virtual leaders can lead to different geometry of the schooling, see Fig. 3.

In the literature, group navigation is usually led by certain leaders, which can be some specific actual agents (Guo et al., 2004; Edwards et al., 2004) or some virtual ones (Leonard & Fiorelli, 2001; Olfati-Saber, 2006). The group movement can be guided through properly designing the reference paths for these virtual leaders. As aforementioned, in this chapter we apply the virtual leader concept to guide the group navigation.

Assumption 2. All virtual leaders move with the same velocity ulv and same heading yh . Moreover, we have uh .

In fact, these virtual leaders can take any complicated motions, which in turn can lead to various geometry of the schooling. However, the focus of this chapter is taken on that to propose a stable schooling for a group of underactuated underwater vehicles. Therefore, for the convenience of discussion, in this chapter we only consider a simple case where all virtual leaders take the same velocity and heading. Corresponding potential function is chosen as following

From (13), it is easy to see that the purpose of potential Vgn is to force the group to keep velocity and heading matching in terms of uu and j .

All obstacles considered in this chapter are position fixed. Modelling of these obstacles is as Fig. 4. Inside the circle centred at the ith vehicle qt with radius by, each obstacle block is modelled as the point from which to qt is the shortest. In Fig. 4, B1 and B2 which are two parts of the same obstacle B are considered to be independent and modelled as two different points qt 2 and qt 3. Also, it is notable that the same obstacle such as B in Fig. 4 can be modelled as different points according to different vehicles.

Fig. 4. Obstacle modelling.

Vehicle's obstacle avoidance is also guided by the following potential functions

where Q (qi) is a subgroup of obstacle points defined as following

From obstacle avoidance point of view, there is not any need to have cohesion between vehicles and obstacles. For this reason, in (14), we design the parameters such that ay= by . As aforementioned, in this chapter the group navigation is guided by the virtual leaders. Therefore, the vehicles' obstacle avoidances also have to be strongly related to the motion of virtual leaders. For this reason, we make the following assumption on the motion of the virtual leaders.

Assumption 3. All virtual leaders are designed to satisfying the following conditions.

C6. For any given obstacle, after a certain period of time, all virtual leaders always move away from this obstacle.

C7. After a period of time such that min II qvk - qt II> bp + by , if one or more vehicles are still trapped by obstacles, in other word, Vk , II qv k - q, II> bp with q, the trapped vehicle, then all virtual leaders (smoothly) stop movement so that qv k = 0 . Otherwise, C6 still satisfies.

Remark 4. From C7 in Assumption 3, we can see that the obstacle avoidance scenario introduced in this chapter cannot guarantee all vehicles to flee from any given obstacles. Instead, we only try to guarantee the vehicles to collision-free with obstacles. For example, as in C7, if one or more vehicles are trapped by obstacles, then we force these vehicle to stop movement so as to avoid collision with obstacles. In fact, obstacle avoidance is still being a complicated and open issue in the practical robotics. How to guide the group to move through the obstacles is out of the scope of this chapter.

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