*3.1. Dynamical Tracking Target*

In this subsection, we will solve the problem of converting the target curve to an appropriate speed tracking target so as to match the dynamics Equation (18). For a target trajectory curve **r**˜(*t*)=(*x*˜(*t*), *y*˜(*t*))T, the speed form of the target trajectory curve is expressed as [21]

$$\begin{cases} \overline{v}\_0 = \sqrt{\vec{x}^2 + \vec{y}^2} \\ \omega\_0 = \dot{\varphi}\_0 = \frac{\dot{x}\ddot{y} - \dot{y}\dddot{x}}{\dot{x}^2 + \dot{y}^2} = k(t)\overline{v}\_{0\prime} \end{cases} \tag{19}$$

where *<sup>k</sup>*(*t*) = ˙*x*˜ ¨ *<sup>y</sup>*˜<sup>−</sup> ˙*y*˜ ¨ *x*˜ ( ˙*x*˜2+ ˙*y*˜2) 3 2 is the relative curvature of the target trajectory curve. We note that the relative curvature is the key point of the target trajectory curve. If the relative curvature is tracked very well, the tugboat can follow the target trajectory curve precisely. On this basis, the target trajectory curve **r**˜(*t*) can be further improved into a dynamical tracking target form as

$$\begin{cases} \mathfrak{v}\_0 = \dot{\phi}(t),\\ \tilde{\omega}\_0 = \dot{\tilde{q}}\_0 = k(s(t))v\_{0,} \end{cases} \tag{20}$$

where *v*<sup>0</sup> stands for the actual forward speed of the tugboat, and *φ*˙(*t*) represents an appropriate forward speed target which is given by

$$
\dot{\phi}(t) = l\beta^2 t e^{-\beta t}.\tag{21}
$$

In (21), *l* is the length of the target curve, *β* is an appropriate parameter according to actual needs [21].

It can be seen from (20) that the target trajectory curve can be converted into a speed target form with the relative curvature. Combined with the dynamics model, two torque controllers can be designed to implement the tracking task of the target trajectory curve. In fact, there are two main advantages by using the dynamical tracking target. First, by choosing an appropriate forward speed target, the initial speed error is equal to zero, which can significantly reduce the position error caused by the accumulated speeds errors. Second, the yaw rotation speed target depends on the actual forward speed, which can be adjusted from moment to moment. Moreover, no matter how large the actual forward speed error is, as long as the curvature tracking error is small enough, satisfactory tracking

performance can still be achieved. As a consequence, the idea of dynamical tracking target can fundamentally solve the problem of accurate trajectory tracking.
