# A Review of Autonomous Tugboat Operations for Efficient and Safe Ship Berthing
## 4 Autonomous Tugboat Element Technologies
### Modeling Technology
Although this study did not use a cluster of multiple tugboats to perform berthing operations, it is significant in that it is the first study to model ship characteristics, the dynamics of the towline, and the dynamics of the tugboat and the ship being towed while considering the towline. [PERSON] [28] proposed a work in 2020 which uses a simplified dynamic equation to model the system, assuming point contact between the ship and each tugboat, with independent variables expressing the speed constraints at the contact point. This was the first time that multibody dynamics, involving the dynamics of both a ship and a tugboat, were considered. The cooperative operation and control of multiple tugboats were demonstrated by  
Figure 15: Schematic description of the system composed of tugboats and mothership proposed by [PERSON] [22]. The tugboats were modeled as thrusters. modeling the mothership and several tugboats as one system and controlling independent variables using a simplified dynamic equation, excluding the contact force term. In 2021, [PERSON] proposed a hybrid system concept, as shown in Figure 17, where continuous and noncontinuous dynamic models interacted in a combination of a discrete-time dynamic model that considered the position movement of the tugboat and a continuous-time dynamic model that represented the tugboat control. In Figure 17, the mode of the tugboat is switched by the value of \(q\), and \(q_{1}\) and \(q_{2}\) mean the towing mode and the berthing mode, respectively. When the mode is switched, the position of the tugboat is changed, and the control input \(\tau\) is set to 0 during the changing time T. Moreover, \(x(t)\) and \(u(t)\) mean state variable and control input variable, respectively. In addition, the modeling of environmental disturbances in the paper was dominated by the influence of wind near the port. A control technique using the PPO (Proximal Policy Optimization), a reinforcement learning-based algorithm, was applied. (i) the mothership being towed by the tug, (ii) independent motion of the tug and the mothership, and (iii) the tug pushing the mothership. For each situation, a modular approach was adopted to construct the respective models by rearranging coefficient and component vectors in block matrices. Specifically, in the context of the pushing situation, a mathematical model was formulated to represent the multibody dynamics of the tug and mothership (Figure 18; Equations (1)-(8)). \[ \begin{bmatrix}A_{0}&\cdots&A_{pn0}\\ \vdots&\ddots&\vdots\\ A_{pn0}&\cdots&A_{pn}\end{bmatrix}\begin{bmatrix}B_{0}\\ \vdots\\ B_{pn}\end{bmatrix} = \begin{bmatrix}C_{0}\\ \vdots\\ C_{pn}\end{bmatrix}  \tag{1}\]  
Equation (1) is an equation organized as a matrix equation by considering the equation for the ship and the geometrical positional relationship for n tugboats as a linear system of equations. \[B_{pi}=[\dot{u}_{0}\quad\dot{v}_{0}\quad\dot{r}_{0}\quad\dot{u}_{eo}\quad\dot {v}_{oo}]^{T} \tag{2}\]  
\[C_{0}= \begin{bmatrix}(m_{i}+m_{yi})v_{i}r_{i}+m_{i}x_{Gi}r_{i}^{2}+X_{Hi}+(1-t _{i})T_{i}cos(\delta_{i})\\ -(m_{i}+m_{xi})u_{i}r_{i}+Y_{Hi}-T_{i}sin(\delta_{i})\\ -m_{i}x_{Gi}u_{i}r_{i}+N_{Hi}-x_{Ti}T_{i}sin(\delta_{i})\\ u_{i}r_{i}sin(\psi_{i})+v_{i}r_{i}cos(\psi_{i})\\ -u_{i}r_{i}cos(\psi_{i})+v_{i}r_{i}sin(\psi_{i})\\ r_{0}^{2}l_{pi}cos(\psi_{0}+\psi_{pi})-t_{i}^{2}a_{pi}cos(\psi_{i})\\ r_{0}^{2}l_{pi}sin(\psi_{0}+\psi_{pi})-t_{i}^{2}a_{pi}sin(\psi_{i})\end{bmatrix}  \tag{4}\]  
Equation (2) to (4) consists of the component vector \(B_{0}\) of the vessel, each containing five variables related to the motion of the vessel, and the component vector \(B_{pi}\) of each tugboat, each containing seven variables related to the motion and pushing force of the tugboat; when n tugboats push the ship, a matrix equation of size \(5+7\)n is solved. The constant vector consists of the component vector \(C_{0}\) of the ship and the component vector \(C_{pi}\) of each tugboat.