Consider a group of n underwater vehicles, all of which have the same model as (1). Similar to the Reynolds's (1987) three heuristic rules that are flock centring, collision avoidance and velocity matching, the schooling rules proposed in this chapter can be summarized as: geometry of schooling, group navigation, and obstacle avoidance. As aforementioned, the group behaviour is conducted by suitably selected potential functions.

Unlike the previous works (Leonard & Fiorelli, 2001; Olfati-Saber, 2006; Do, 2007), in this chapter we consider the following general form of smooth potential function. Definition 1 (Smooth potential function). A scalar function fp (Z, a, b), where Z e[0, +<») and a and b are constants with 0 < a < b , is called a smooth potential function, if it satisfies the following conditions.

C3. fp (Z,a,b) is smooth respect to Ze [0, + <x>) with monotonically decreasing at Z e [0, a) and monotonically increasing at Z e [a, + <x>).

C4. If a < b , then fp (Z,a,b) has a global minimum of zero at Z = a with df / dZ = 0 for

C5. If a = b , then fp (Z,a,b) = 0 for VZ > a . Remark 2. If b < +<» , then fp (Z,a,b) is said to have a finite cut-off (Olfati-Saber, 2006). This kind of feature plays an important role in the group formation (Leonard & Fiorelli, 2001; Olfati-Saber, 2006; Do, 2007). On the other hand, in Olfati-Saber (2006), the potential function took a finite value when Z ^ 0+ , while it took +» in Leonard & Fiorelli (2001) and Do (2007). From a collision-free point of view, the latter one seems to have its own benefit. However, the finite value case as in Olfati-Saber (2006) may be more natural in practical flocking or schooling.

Remark 3. There are many functions satisfying C3~C5. For example, fp (Z, a, b) = cfa(z-a)p(T/b)dT, (6)

where c > 0 is a constant and p(-) is a smooth bump function taken as following p(S) =

otherwise where h e (0,1) is a design parameter. It is easy to verify that when Z ^ 0+ , fp ^ ca2/2 which is a finite value in (6) while fp in (7). Moreover, in (6), if b ^ , then the potential function becomes a general quadratic form as c(Z - a)2 /2 . For another example of smooth potential and bump functions, refer to Olfati-Saber (2006). Functions (6) and (7) are depicted in Fig. 2.

3.1 Geometry of schooling

The schooling geometry is constructed according to the following two kinds of potential functions n n

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