# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 2 Marine Vehicle Model  
A model of the considered planar vehicle in the North-East-Down (NED) frame is shown in Figure 1.  
In this frame, the position and the orientation of the vehicle are represented by the vector \(\eta=[x,y,\psi]^{T}\). The velocities in the body frame are \(v=[u,v,r]^{T}\) (the surge velocity, the sway velocity, and the yaw rate, respectively). Moreover, the vector \(V=[V_{x},V_{y},0]^{T}\) describes the ocean current in the NED frame. In the body frame, the ocean current vector  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & \multicolumn{3}{c}{**Asymmetry**} & \multicolumn{2}{c}{**Disturbances**} \\ \hline  & **No** & **y Axis** & **x y Axis** & \\ \hline [PERSON] et al. [10] & + & \(-\) & \(-\) & none \\ [PERSON] et al. [13] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [22] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [23] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [25] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [27] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [29] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [31] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [32] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [33] & + & \(-\) & \(-\) & variable \\ [PERSON] et al. [34] & + & + & \(-\) & variable \\ [PERSON] et al. [38] & + & + & \(-\) & constant \\ [PERSON] et al. [39] & + & + & \(-\) & variable \\ [PERSON] et al. [41] & + & + & \(-\) & variable \\ [PERSON] [43] & + & + & \(-\) & none \\ [PERSON] et al. [44] & + & + & \(-\) & variable \\ [PERSON] et al. [46] & + & + & \(-\) & variable \\ [PERSON] and [PERSON] [47] & + & + & \(-\) & variable \\ [PERSON] [48] & + & + & + & variable \\ \hline \hline \end{tabular}
\end{table}
Table 1: Examples of tracking controls solutions.  
Figure 1: Model of underwater vehicle in horizontal motion.  
is expressed as \(\
u_{c}=R^{T}(\psi)V\). The model of motion of an AUV moving horizontally is given as [1; 2]:  
\[\dot{\eta}=R(\psi)\
u_{r}+V, \tag{1}\] \[M\dot{\
u}_{r}+C(\
u_{r})\
u_{r}+D\
u_{r}=\tau, \tag{2}\]  
where \(\
u_{r}=[u_{r},v_{r},r]^{T}=\
u-\
u_{c}\) is the vector of relative velocities in the body frame, \(D\) means the linear damping coefficients matrix, and \(\tau\) is the control vector containing the applied force along the surge motion and the applied torque around the \(z\) axis. The matrices in the above equations are:  
\[R(\psi)=\left[\begin{array}{ccc}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{array}\right],\ \ \ \ M=\left[\begin{array}{ccc}m_{11}&0&m_{13}\\ 0&m_{22}&m_{23}\\ m_{13}&m_{23}&m_{33}\end{array}\right]\] \[C(\
u_{r})=\left[\begin{array}{ccc}0&0&c_{13}\\ 0&0&c_{23}\\ -c_{13}&-c_{23}&0\end{array}\right],\ \ \ \ D=\left[\begin{array}{ccc}d_{11}&0&d_{13}\\ 0&d_{22}&d_{23}\\ d_{31}&d_{32}&d_{33}\end{array}\right],\ \ \ \ \tau=\left[\begin{array}{c}\tau_{u}\\ 0\\ \tau_{r}\end{array}\right]. \tag{3}\]  
The elements of the matrices are as follows: \(m_{11}=m-X_{\dot{u}},\ m_{22}=m-Y_{\dot{\
u}},\ m_{13}=m_{31}=-my_{g}-X_{t},\ m_{23}=m_{32}=mx_{g}-Y_{t},\ m_{33}=j_{z}-N_{t},\ c_{13}=-m_{22}v_{r}-m_{23}r,\ c_{23}=m_{11}u_{r}-m_{13}r,\) and \(d_{11}\), \(d_{22}\), \(d_{23}\), \(d_{32}\), and \(d_{33}\) are constant hydrodynamic damping coefficients. The equation replacing (1) can be written in the following form:  
\[\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\psi}\end{array}\right]=\left[\begin{array}{c}u_{r}\cos\psi-v_{r}\sin \psi+V_{x}\\ u_{r}\sin\psi+v_{r}\cos\psi+V_{y}\\ r\end{array}\right]. \tag{4}\]