*4.1. A Comparison between the Dynamical Target and Statical Target*

On the one hand, the statical tracking target (19) is expressed by

$$\begin{cases} \begin{aligned} \sigma\_0 &= \sqrt{\overline{\mathfrak{x}}^2 + \overline{\mathfrak{y}}^2} = 80 \cos(\frac{t-\pi}{4}),\\ \omega\_0 &= \frac{\mathfrak{x}\overline{\mathfrak{y}} - \overline{\mathfrak{x}}\overline{\mathfrak{y}}}{\overline{\mathfrak{x}}^2 + \overline{\mathfrak{y}}^2} = \frac{1}{4}. \end{aligned} \tag{42}$$

If we use the statical tracking target (42) to design controller *u*<sup>1</sup> and *u*2, the initial speed error is not zero, given by

$$\begin{cases} \upsilon\_0(0) - \overline{\upsilon}\_0(0) = -80 \cos(-\frac{\pi}{4}) = -40\sqrt{2}, \\ \omega\_0(0) - \overline{\omega}\_0(0) = -\frac{1}{4}. \end{cases}$$

On the other hand, the dynamical tracking target (20) is expressed by

$$\begin{cases} \overline{v}\_0 = 80 \sqrt{2} t e^{-t}, \\ \overline{\omega}\_0 = k\_0(\mathbf{s}) v\_0(t). \end{cases}$$

The relative parameters of the towing system are set as *λ* = 1.3, *L* = 2.6 m, *τ* = 0.25, *S* = 3.3 m2, *ρ* = 1000 kg/m3 , *d* = 0.1 m, *Cf* = 0.063, *m*<sup>0</sup> + *Mx*<sup>0</sup> = 103 kg, *m*<sup>1</sup> + *Mx*<sup>1</sup> = 103 kg, *Iz*<sup>0</sup> <sup>+</sup> *Jz*<sup>0</sup> <sup>=</sup> <sup>30</sup> kg · <sup>m</sup>2, *Iz*<sup>1</sup> <sup>+</sup> *Jz*<sup>1</sup> <sup>=</sup> 30 kg · m2, *<sup>Q</sup>*<sup>1</sup> <sup>=</sup> diag(10, 100), *<sup>R</sup>* <sup>=</sup> 1, *<sup>k</sup>*<sup>2</sup> <sup>=</sup> 204, *<sup>G</sup>* <sup>=</sup> 1,*ε* = 0.1, *γ* = 0.3. Therefore, all the required quantities in the trajectory tracking controllers *u*<sup>1</sup> and *u*<sup>2</sup> are available in hand. Accordingly, the time histories of all state variables can be simulated.

As can be seen in Figure 3, by using the dynamical tracking target, both the initial forward speed and yaw rotation speed errors are smaller than the one using statical tracking target. Comparing Figure 4a with Figure 4b, we see that the actual motion trajectory of the tugboat deviates largely from the target curve by using the statical tracking target, whereas the actual motion trajectory of the tugboat coincides well the target curve via the dynamical tracking target. In other words, by using the dynamical tracking target, accurate trajectory tracking can be achieved as long as the curvature tracking error is controllable. Even though forward speed and yaw rotation speed errors are large, accurate tracking can be also maintained as long as the relative curvature is well tracked.

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

**Figure 3.** Actual speed error of the tugboat.

(**a**) Actual motion trajectory of the tugboat with statical target

(**b**) Actual motion trajectory of the tugboat with dynamical target

**Figure 4.** Actual motion trajectory curve of the tugboat by using different speed targets.
