where (xt, yt), i = 1,---,n denotes the position of the ¿th vehicle and y is yaw angle, all in the earth-fixed frame; ut, vt and rt denote the velocities each in the surge, sway and yaw
1 For the convenienceof discussion, in this chapter we only consider the vehicle's three DOF motion in the horizontal plane instead of its full six DOF motion.
directions in the vehicle's body-fixed frame; fu.(•), f(•), fr.(•) e C1 are the vehicle's nonlinear dynamics including hydrodynamic damping, inertia (including added mass terms) and gravitational terms in the surge, sway and yaw directions; surge force Tui and yaw moment Tri are two available control inputs with nonzero constant gains bui and bri . For this torpedo-type underwater flying vehicles, their yaw moments zri is proportional to the square of the forward speed, in other word, zri x u2Sri where Sri is the vehicle's rudder angle (Fossen, 2002), see Fig. 1. From this point of view, yaw moment Tri is not exactly independent of surge force Tui. Moreover, if the vehicle's forward speed takes value too small, then the yaw moment is also forced to take small value. Therefore, in this case, we could not fully excite the vehicle's yaw dynamics. In order to appropriately taking these two control inputs zui and Tri as independent ones, in the remainder of this chapter, we make the following assumption on each vehicle's dynamics.
Assumption 1. For each vehicle in the considering group, its dynamics satisfies the following conditions.
C1. ut > umin > 0 , where umin is a design parameter.
C2. For bounded u, and r, v, is also bounded and have | v, |< vmdx with vmdx a known
Remark 1. At first glance, in the above assumption, the condition that vmdx is known seems too restrictive in terms of control engineering. However, in the case of underwater vehicles, it becomes very reasonable. For underwater flying vehicles, because of the effect of hydrodynamic damping terms, which is usually proportional to the square of the vehicle's corresponding speed (Fossen, 2002; Newman, 1977), they only have to take limited magnitude of velocities under the limited thrust force. Therefore, in practice, given a torpedo-type flying vehicle, it's maximum forward and sway speeds and yaw angular velocity are all easy to bring out through certain simple experiments such as basin test. From this point of view, it is reasonable for us to design the parameter umin such that umin > pvmdx . As aforementioned, since the sway force is unavailable, the most difficulty for the control of (1) is how to properly handling the vehicle's sway dynamics. To deal with this problem, firstly we introduce a polar coordinates transformation which is defined in the vehicle's body-fixed frame as following (see Fig. 1)
where y/ai = drctdn(vi / ut) is a polar angle and also called as the sideslip angle (Fossen, 2002). Since ut > 0 , it is easy to verify that y/ai is defined and smooth in the domain (-0.5n, 0.5n) . Differentiating the first equation of (2) and further according to the relationships of ui = uii cos yai and vi = un sin yai, we can get uli = ui C0SWai + Vi SinWai. (3)
Using polar coordinates uu and y/u , vehicle's kinematics can be rewritten as xi = un cos ij/n, yi = uh sin ^. (4)
For the convenience of discussion, in the remainder of this chapter, we will call uu and j as the ith vehicle's velocity and heading.
Combining with (3) and (4), the vehicle's model (1) can be rewritten as following form
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