—(uh cosvh - Xi) + -f^-(uh smvh -Vi) || qvki || || qvki ||
= Z [a xi (ulv cosVlv - uli cosVli) + A Vi (ulv sin Vlv - uli sin Vli ) + A O
+ Yuulvi (ulv - uii) + YwVlvi (Wh - Vli)] (Axi cosVli +A Vi sinVii )uhi + 2uh I A ^ cosVlv +Vli - Axi ivn^^V I sin^
+ A o + Yuu lvi(u lv - fui C0SVai - Li sVnVai - bui C0SVai Tui ) + YwVlvi (Vlv at -ari + eri )] (18)
where eri = ari - rt with ari a stabilizing function (Krstic et al., 1995) for virtual input r, Axi and A O are defined as following n x , m x ,
A * = 2 Ya Z dfp (Hqj, H,aa, ba)-j. + Yp Z dfp (Uqvki Up, bp j*i || qji || k=1 || qvki ||
ipi ipi uiv cosViv +7-uiv svnVi.
y and Ayi is defined to have the same form with Ax, with only x displaced by y .
According to (18), in this step we choose the control laws as following
Tui = Ki sec Wai [ulv - fui C0S Wai - fvi Sln Wai + Y~u (kuiulvi + AC0S Wli + Ayi Sln Wli )] (21)
uJ A ,cos Wiv + W -A ,sm Wiv + WA Sm(Wl"/2) W'riv' J y' 2 x' 2 ) Whi/2
where kui, kWi > 0 are design parameters.
Remark 5. According to (2), we can see that Wai contains the acceleration terms it and v . However, since fui (•), fvi (•), fri (•) e C2, it is not difficult to verify that the surge force control law (21) is differentiable, and this means that it and v are also differentiable. Consequently, we can conclude that Wai is also differentiable and so is ali . Substituting (21) and (22) into (18), get
Step 2. Rewrite the final equation of (5) as following eri =ari - fri - briTri . (24)
Now, consider the following Lyapunov function candidate l n 2
Differentiating (25) and substituting (23)~(24) into it, finally we can have
V2 =Z [- kuiuhi - kWiVhi + YWeriWlvi + eri (ari - fri - briTri ) + Ao ] . (26)
According to (26), the control law for Tri is chosen as
where kri > 0 is a design parameter. Substituting (27) into (26), finally we can have
According to Assumption 3, after a certain period of time such that min || qvk - qip ||> bp + by , if all vehicles are still following the virtual leaders (this means || qj ||< bp), then fp(|| qipi ||,ay,by) = 0 , Vi = —,n and Vp e Q(qi). Otherwise, we have ulv = 0 . In both cases, it is easy to verify that Ao = 0 . Consequently, after a certain period of time, we always have V2 < 0 , and V2 = 0 if and only if ulvi = Wivi = eri = 0 , Vi = !,•■■,n .
Theorem. Consider the schooling for multiple underactuated underwater vehicles whose kinematics and dynamics can be expressed as (1) with Assumption 1~3. If we choose the formation control laws as (21) and (27), then the schooling asymptotically converges to a
certain local minimum and all vehicles included in the group asymptotically move with the same velocity and heading while keeping obstacle avoidance.
Remark 6. Consider the case where one or more vehicles are trapped by obstacles. If we design Vp to take finite cut-off (bp < +< ), which is different from Olfati-Saber (2006) and Do (2007), then after II qvkj II>bp with q, trapped vehicles, we have f (\Iqvkj II,ap,bp) = 0 . This means that after a certain period of time, only two kinds of potentials Va and Vy are remained to restrict the behaviour of the trapped vehicles. Therefore, we can guarantee the trapped vehicles to collision-free with obstacles even without C7 in Assumption 3. Remark 7. So far, we derive the schooling algorithms under the condition ut > umin > pvmax as in Assumption 1. In fact, substituting (21) into (3), we have uhi = -yUulkuiuhi - Yu-1 (a xi cos ^ + A y, sin ) . (29)
On the other hand, if we take the smooth potential function as the form of (6), then it is not difficult to verify that dfp is bounded. Further according to (19), we can see that Axi and Ayi are also bounded. Therefore, through proper selection of weighting factors ya, Yp and yr, we can make Axi cos^ft +Ayi sin^ft arbitrarily small. Consequently, further through proper selection of ku, , it is easy to get
Therefore, if we design the virtual leaders' motions such that u lv > ^vj^ + uj^ + U with U > 0 a design parameter, then it is not difficult to verify that Vt > 0, have ui(t)>umin under the initial condition Iuhi (0) I < U . In other word, assumption of ui(t)>umin always can be guaranteed in practice under suitable selection of initial conditions. Meanwhile, this means that the proposed formation control method only guarantees local stability of the schooling. Here it is notable that Iuhi(0) I<U is a certain sufficient condition for u(t)>umin, not a necessary one.
Remark 8. Since polar coordinate transformation does not satisfy to be a diffeomorphism, usually uhi, y/lvi ^ 0 does not guarantee the same convergence properties of ut ^ uv, vt ^ vv and rt ^ rv, where uv, vv, rv are the virtual leader's velocity components. This may be one of disadvantages to applying polar coordinate transformation. However, in practical applications, it is much more difficult to design the motion of virtual leaders in the Cartesian frame. For example, for given uv and rv, because of high nonlinearity, it is difficult to directly calculate vv through vv = fv (uv, vv, rv). On the contrary, in the corresponding polar frame, we can easily bring out the reference paths for virtual leaders from given u h and .
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