*2.3. Dynamics Modeling*

According to the dynamics Equation (13) of the single tugboat, the dynamics model of the STS can be presented by Reference [24]:

$$\begin{cases}(m\_0 + M\_{x0})\dot{\upsilon}\_0 = -\frac{1}{2}\rho C\_f S \upsilon\_0^2 + X\_p + X\_r - T\cos(\theta - \Psi)\_r\\(I\_{z0} + I\_{z0})\dot{\omega}\_0 = -\frac{1}{2}\rho L^2 dv\_0 \omega\_0 (0.45\lambda - \lambda^2)(1 + 0.3\tau) + N\_p + N\_r\\ \qquad - \frac{1}{2}TL\sin(\theta - \Psi).\end{cases} \tag{14}$$

$$\begin{cases} (m\_1 + M\_{x1})\psi\_1 = -\frac{1}{2}\rho \mathbb{C}\_f S v\_1^2 + T \cos \Psi\_\prime\\ (I\_{z1} + I\_{z1})\dot{\omega}\_1 = -\frac{1}{2}\rho L^2 dv\_1 \omega\_1 (0.45\lambda - \lambda^2)(1 + 0.3\tau) + \frac{1}{2}TL\sin \Psi\_\prime \end{cases} \tag{15}$$

where *T* is the towline tension. From (14), one has

$$\begin{cases} \dot{\upsilon}\_{0} = \frac{-\frac{1}{2}\rho C\_{f}S\upsilon\_{0}^{2} + X\_{p} + X\_{r} - T\cos(\theta - \Psi)}{(m\_{0} + M\_{x0})},\\ \dot{\omega}\_{0} = \frac{-\frac{1}{2}\rho L^{2}dv\upsilon\omega\_{0}(0.45\lambda - \lambda^{2})(1 + 0.3\tau) + N\_{P} + N\_{r} - \frac{1}{2}TL\sin(\theta - \Psi)}{I\_{x0} + I\_{z0}}. \end{cases} \tag{16}$$

According to (15), the towline tension *T* is expressed as

$$\begin{split} T^2 &= \left[ (m\_1 + M\_{x1})\dot{\upsilon}\_1 + \frac{1}{2}\rho C\_f S \upsilon\_1^2 \right]^2 + \left[ \frac{2}{L}(I\_{z1} + I\_{z1})\dot{\omega}\_1 \\ &+ \rho L d \upsilon\_1 \omega\_1 (0.45\lambda - \lambda^2)(1 + 0.3r) \right]^2, \end{split} \tag{17}$$

where *<sup>v</sup>*1, *<sup>ϕ</sup>*<sup>1</sup> can be obtained according to (8), (9), and *<sup>ω</sup>*<sup>1</sup> <sup>=</sup> *<sup>ϕ</sup>*˙ <sup>1</sup> <sup>=</sup> *<sup>ϕ</sup>*˙ <sup>0</sup> <sup>−</sup> ˙ *θ*.

It follows from (16) and (17) that the dynamics equation of the STS is ultimately formulated as

$$\begin{cases} \dot{v}\_0 = \frac{-\Delta\_2 v\_0^2 + u\_1}{\Delta\_1} \\ \dot{\omega}\_0 = \frac{-\Delta\_4 v\_0 \omega\_0 + u\_2}{\Delta\_3} \end{cases} \tag{18}$$

where Δ<sup>1</sup> = *m*<sup>0</sup> + *Mx*0, Δ<sup>2</sup> = <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>Cf <sup>S</sup>*, <sup>Δ</sup><sup>3</sup> <sup>=</sup> *Iz*<sup>0</sup> <sup>+</sup> *Jz*0, <sup>Δ</sup><sup>4</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>L*2*d*(0.45*<sup>λ</sup>* <sup>−</sup> *<sup>λ</sup>*2)(<sup>1</sup> <sup>+</sup> 0.3*τ*), *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *Xp* <sup>+</sup> *Xr* <sup>−</sup> *<sup>T</sup>*cos(*<sup>θ</sup>* <sup>−</sup> *<sup>Ψ</sup>*), and *<sup>u</sup>*<sup>2</sup> <sup>=</sup> *Np* <sup>+</sup> *Nr* <sup>−</sup> <sup>1</sup> <sup>2</sup>*TL*sin(*θ* − *Ψ*).

#### **3. Trajectory Tracking Control of the Ship Towing System**

In order to make the tugboat track a given target trajectory curve accurately, the target trajectory curve should be firstly converted into a speed target form so as to match the dynamics equation. As such, the original motion task is converted into a general trajectory tracking control problem of the tugboat. Then, two torque controllers can be designed from the forward and yaw speed subsystems, to achieve the given trajectory tracking task.
