*4.3. Robustness of the Proposed Controller*

Since the controllers are designed by considering the sliding mode control and inverse dynamics adaptive control methods simultaneously, it has highly robust. Moreover, by using the dynamical tracking target, even though the forward speed and yaw rotation speed error subsystems are unstable due to uncertain factors, the towed ship is also able to achieve satisfactory tracking performance.

Assume that the forward speed error subsystem is subject to an uncertain factor, which is given by

$$\dot{\mathbf{Y}} = \mathbf{A}\_1 \mathbf{Y} + \mathbf{B}\_1 u\_1(t) + \eta(t) + d(t),$$

where *d*(*t*)=(10.2*y*1, 10.2*y*2)<sup>T</sup> is an uncertain factor.

As shown in Figures 7 and 8, although forward speed and yaw rotation speed errors are large and even divergent, the actual motion trajectory of the tugboat almost coincides with the target curve. The main reason is that the relative curvature error which is obtained by dividing the actual yaw rotation speed by the actual forward speed is small via the dynamical tracking target. Therefore, as long as the relative curvature error is

small enough, the accurate tracking of the target trajectory curve can still be guaranteed. Moreover, the towed ship can also obtain satisfactory tracking performance by means of an appropriate steering coefficient, such as *μ* = −20. Otherwise, there will be a large deviation from the target trajectory curve, such as *μ* = −16, as depicted in Figure 8.

(**b**) Yaw rotation speed error of the tugboat

**Figure 7.** Actual motion speed error of the tugboat under the uncertain factor *d*(*t*).

**Figure 8.** Actual motion trajectory curve under the uncertain factor *d*(*t*).
