## Simulation studies

In this section, we carry out some simulation studies to illustrate the effectiveness of proposed vehicles schooling scheme. In the simulation, the vehicles group consists of three vehicles, and each of them is modelled as the six DOF nonlinear dynamics of ISiMI AUV (Lee et al., 2003), which has the similar mechanical structure as REMUS AUV (Prestero, 2001). For this ISiMI AUV, we use the saturation conditions as I zui I < 50N and I Sri I < n/6rad in the simulation. And all virtual leaders are assumed move with the same velocity and heading as u lv = 1.54m / s, = 0rad .

### 5.1 Straight line schooling

In this case, we can choose the three virtual leaders to locate at the vertices of a certain isosceles triangle as in Fig. 3 (a) with the initial positions as qv1(0) = (6,10), qv2(0) = (6, 26) and qv3 (0) = (34,18). Three vehicles' initial conditions are as: qx(0) = (0, 24), q2(0) = (12,12), q3 (0) = (28,16), y/1 (0) = 0.1rad, y/2 (0) = 0.05rad, ^3(0) = 0.15rad, and u1 (0) = u2 (0) = u3(0) = 0.5 m /s with all other variables taking zero values. Other design parameters are taken as: kui = 8, kvn = 3, kri = 12,, = 1,---,3 and Ya = 0.3, Yp = 0.5, Yu = 100, Yw = 0.2 .

For potential functions Va and Vp, both of them take the form as (6) and corresponding parameters are chosen as: aa = 12, ba = 20, ap= 10, bp = 20, ca= 2, Cp= 3 and h = 0.9 . Simulation results are shown in Fig. 5 and 6. Fig. 5 shows the vehicles group schooling in its straight line movement. From Fig. 6, we can see that there is not any collision between vehicles in the schooling.

Fig. 5. Schooling of the vehicles in a straight-line movement.
Fig. 6. Schooling geometry for a straight-line movement.

5.2 Triangular schooling

For triangular schooling, we locate the three virtual leaders as in Fig. 3 (b) with the initial positions taken as qv1(0) = (0, 40), qv2(0) = (0, 8) and qv3(0) = (20, 24). Three vehicles' initial conditions are chosen as q1(0) = (9, 39), q2(0) = (3,10), q3(0) = (10, 22), ^1(0) = 0.1ra<i, ^2(0) =

0.05rad, w3(0) = 0T5rad, and u\(0) = u2(0) = u3(0) = 0.5 m/s with all other variables taking zero values. Other design parameters are taken as kui = \2, kWi = 8, kri = \2, i = and

Same as in the straight line case, both of Va and Vp take the form as (6) with the parameters chosen as aa = \2, ba= \5, ap= 8, bp = 20, ca = 2, cp= 3 and h = 0.9 .

Corresponding simulation results are presented in Fig. 7 and 8. The vehicles schooling in the triangular movement is shown in Fig. 7 with no collision between any of two vehicles (see Fig. 8).

Fig. 7. Schooling of the vehicles in a triangular movement.
Fig. 8. Schooling geometry for a triangular movement.

5.3 Equilateral triangular schooling with obstacle avoidance

In this case, we consider an equilateral triangular schooling of the vehicles with obstacle avoidance. The obstacle is modelled as a circle located at qp = (40, !0) with radius as 3m. For the schooling, two virtual leaders are chosen as in Fig. 3 (c) with the initial positions taken as qv1(0) = (2>/3,10) and qv2(0) = (10>/3,10). The vehicles' initial conditions are taekn as qj(0) = (0,1), q2 (0) = (2,15), q3(0) = (7^3, 11), \y1 (0) = 0.1rad, y/2(0) = 0.05rad, (0) = 0.15rad and u1 (0) = u2(0) = u3(0) = 0.5ra / 5 with all other variables taking zero values. Other design parameters are taken as ku¡ = 18, kw¡ = 10, kr¡ = 12, i = 1,---,3 and ya = 0.15, yp= 0.2, yy = 0.15, yu = 240, rw = 6.

Both Va and Vp are also taken as the form as (6) with the parameters as aa = 12, ba = 30, ap = 4V3, bp = 30, aY= by = 6, ca = 4, cp= 5, cY = 90 and h = 0.9 .

Simulation results are depicted in Fig. 9~12. Fig. 9 shows the vehicles schooling in the equilateral triangular movement with obstacle avoidance. From Fig. 10, we can see that there is not any collision between vehicles. Fig. 11 presents the vehicles' velocity and heading matching in the schooling, and Fig. 12 shows the histories of proposed formation control laws for z,¡: and Sri.

Fig. 9. Schooling of the vehicles in an equilateral triangular movement with obstacle avoidance.
Fig. 10. Schooling geometry for an equilateral triangular movement with obstacle avoidance.
Fig. 11. Group velocity and heading matching.
Fig. 12. Histories of proposed formation control laws.