# A Review of Autonomous Tugboat Operations for Efficient and Safe Ship Berthing
## 4 Autonomous Tugboat Element Technologies
### Modeling Technology
Additionally, \(X_{H},Y_{H}\), and \(N_{H}\) inside the matrix are expressed to represent the hydrodynamic surge force, sway force, and yaw moment due to the maneuvering motions, respectively; \(m_{0}\) is the mass and \(m_{x0}\) and \(m_{y0}\) are the added masses in the surge and sway of the ship, respectively. \(I_{zGO}\) is the moment of inertia, and \(I_{zGO}\) is the added moment of  
Figure 18: Coordinate systems in the case of pushing assistance [30]. inertia. The \(x_{Ti}Tisin(\delta_{i})\) term gives an azimuth thruster located at \((x_{Ti},0)\) of \(m_{i}-x_{ij}y_{i}z_{i}\), representing the assumed thrust. Lastly, \(\dot{u},\dot{v},\dot{v}\), and \(\dot{r}\) are the surge, sway accelerations, and angular acceleration in the yaw direction, respectively, and \(F_{ai}\), \(F_{bi}\) means the force at the contact point. \[A_{0}= \begin{bmatrix}m_{0}+m_{x0}&0&0&0&0\\ 0&m_{0}+m_{y0}&m_{0}x_{G0}&0&0\\ 0&m_{0}x_{G0}&I_{zG0}+I_{zG0}+m_{0}x_{G0}^{2}&0&0\\ cos(\psi_{0})&-sin(\psi_{0})&0&-1&0\\ sin(\psi_{0})&cos(\psi_{0})&0&0&-1\end{bmatrix}  \tag{5}\]  
\[A_{pi}= \begin{bmatrix}m_{0}+m_{xi}&0&0&0&0&0\\ 0&m_{i}+m_{yi}&m_{i}x_{Gi}&0&0&0&0\\ 0&m_{i}x_{Gi}&I_{zGi}+I_{zi}+m_{i}x_{Gi}^{2}&0&0&0&0\\ cos(\psi_{i})&-sin(\psi_{i})&0&-1&0&0\\ sin(\psi_{i})&cos(\psi_{i})&0&0&-1&0&0\\ 0&0&a_{pi}sin(\psi_{i})&-1&0&0&0\\ 0&0&-a_{pi}cos(\psi_{i})&0&-1&0&0\end{bmatrix}  \tag{6}\]  
\[A_{pi}= \begin{bmatrix}0&0&0&0&cos(\psi_{i}-\psi_{0})&-sin(\psi_{i}-\psi_{0}) \\ 0&0&0&0&0&sin(\psi_{i}-\psi_{0})&cos(\psi_{i}-\psi_{0})\\ 0&0&0&0&l_{pi}sin(\psi_{i}-\psi_{0}-\psi_{pi})&l_{pi}cos(\psi_{i}-\psi_{0}- \psi_{pi})\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}  \tag{7}\]  
\[A_{pi}= \begin{bmatrix}0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&-l_{pi}sin(\psi_{0}+\psi_{pi})&1&0\\ 0&0&l_{pi}cos(\psi_{0}+\psi_{pi})&0&1\end{bmatrix}  \tag{8}\]  
Finally, Equations (5) to (8) represent a \(5\times 5\)\(A_{0}\) matrix composed of ship variables, a \(7\times 7\)\(A_{pi}\) matrix composed of variables of the ith tugboat, and the geometric dependence between the ship and the ith tugboat, meaning \(A_{pi}\)(\(5\times 7\)) and \(A_{pi}\)(\(7\times 5\)) columns. The \(A_{0}\), \(A_{pi}\) and \(A_{pi}\), \(A_{pi}\) matrices are arranged diagonally, respectively. Then, it is assumed that there is no dependency between the tugboats, and an \(A_{pi}\) matrix representing direct interference from the j-th tugboat to the i-th tugboat is set to a zero matrix. System modelings that include both tugboats and motherships were conducted initially in the form of thrusters and considered as a single system, which was different from the actual operation method of berthing, which includes pushing and pulling mechanisms of multiple tugboats. However, as research progressed, modeling techniques that incorporated environmental disturbances and the use of a lowline were developed. In the most recent study [29], different multibody dynamic modeling techniques were introduced for both the towing and berthing modes, and modeling techniques for autonomous tugboat swarm and motherships for automatic berthing were developed. Future studies should consider the dynamics of towlines, individual tugboats, and motherships.