*2.2. Dynamics Modeling of a Single Ship*

In order to establish the dynamics model of the STS, we should first clarify the dynamics equation of a single ship, taking the single tugboat for example. As shown in Figure 2, an earthbound coordinate frame *O* − *XYZ* is used to describe the motion of the single ship in the horizontal plane. The body-fixed coordinate frame *O*<sup>0</sup> − *XbYbZb* centered at midship point *O*<sup>0</sup> of the single ship is used for better force analysis.

On the one hand, according to the kinematics equation in Reference [22] and neglecting the drifting speed, the dynamics equation of the tugboat is given by

$$\begin{cases} \mathfrak{x}\_{\mathfrak{b}} = m\_0 \psi\_{0\mathfrak{v}} \\ z\_{\mathfrak{b}} = I\_{z0} \dot{\omega}\_{0\mathfrak{v}} \end{cases} \tag{11}$$

where *xb* represents the component of the external force in the *Xb* direction, and *zb* denotes the component of the external moment of inertia in the *Zb* direction.

On the other hand, according to force analysis of the hull [23], one has

$$\begin{cases} x\_b = -M\_{x0}\dot{\upsilon}\_0 - \frac{1}{2}\rho C\_f S v\_0^2 + X\_p + X\_{r\prime} \\ z\_b = -J\_{z0}\dot{\omega}\_0 - \frac{1}{2}\rho L^2 dv\_0 \omega\_0 (0.45\lambda - \lambda^2)(1 + 0.3\tau) + N\_p + N\_{r\prime} \end{cases} \tag{12}$$

where *Xp* denotes the component force acting on the propeller along the *Xb*-axis, and *Np* represents the corresponding component of the inertia moment along the *Zb*-axis. *Xr* represents the component force on the rudder along the *Xb*-axis and *Nr* is the corresponding component of the inertia moment along the *Zb*-axis. *ρ* is the water density. *d* is the full load draft height of the tugboat. *S* is the hull wet area of the tugboat. *λ* is the aspect ratio of the rudder of the tugboat. *τ* is the trim value of the tugboat. And *Cf* is the coefficient of frictional resistance.

It follows from (11) and (12) that the desired dynamics equation of the single tugboat is expressed by

$$\begin{cases} (m\_0 + M\_{x0})\dot{v}\_0 = -\frac{1}{2}\rho \mathbf{C}\_f S v\_0^2 + X\_p + X\_{r\prime} \\ (I\_{z0} + I\_{z0})\dot{\omega}\_0 = -\frac{1}{2}\rho L^2 dv\_0 \omega\_0 (0.45\lambda - \lambda^2)(1 + 0.3\tau) + N\_p + N\_r \end{cases} \tag{13}$$

**Figure 2.** Force analysis of a single ship.
