# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 3 Controller in Terms of Quasi-Velocities  
### Differences Compared to the Original Trajectory-Tracking Algorithm  
The proposed algorithm is based on the control idea developed in [38]. However, since the QV controller does not meet some of the conditions of the original controller, it is necessary to refer to assumptions 1-8, i.e., (A1)-(A8) given in the reference work. Assumptions (A1), (A3), (A4), (A7), (A8) remain unchanged. (A1) says that only surge, sway, and yaw are taken into account. In (A3), only the hydrodynamic damping is assumed to be linear. According to (A4), the ocean current in the inertial frame \(V=[V_{x},V_{y}]^{T}\) is non-rotational, constant, and bounded (\(\exists V_{max}>0\) under condition \((V_{x}^{2}+V_{y}^{2})^{1/2}\leq V_{max}\)). The desired trajectory, defined as \(\Gamma(t)=\{(\xi_{1_{d}}(t),\xi_{2_{d}}(t),\xi_{3_{d}}(t),\xi_{4_{d}}(t))|t\in R ^{+}\}\) with \(\dot{\xi}_{1_{d}}=\xi_{3_{d}},\dot{\xi}_{2_{d}}=\xi_{4_{d}}\), is considered. (A7) says that there are constant values for signal limitation (m-minimum value, M-maximum value), such that \(\xi_{3_{d}}(t)\in\langle\xi_{3}^{m},\xi_{3}^{M}\rangle\), \(\xi_{4_{d}}(t)\in\langle\xi_{4}^{m},\xi_{4}^{M}\rangle\), \(\dot{\xi}_{3_{d}}(t)\in\left\langle\xi_{3}^{m},\xi_{3_{d}}^{M}\right\rangle\), \(\dot{\xi}_{4_{d}}(t)\in\left\langle\hat{\xi}_{4_{d}}^{m},\hat{\xi}_{4_{d}}^{M}\right\rangle\). According to (A8), the following total relative velocity is chosen \(U_{d}=(\xi_{3_{d}}^{2}+\xi_{4_{d}}^{2})^{1/2}>0\), where \(\xi_{3_{d}}=\xi_{3_{d}}-V_{x}\) and \(\xi_{\pi_{4_{d}}}=\xi_{4_{d}}-V_{y}\). In addition, to overcome disturbances from ocean currents, the vehicle's propulsion motors must provide sufficient power. Assumptions (A2), (A5), and (A6) have been modified. Assumption (A2) of symmetry on the port and starboard sides is extended because, in the proposed approach, the vehicle may be asymmetric about two axes. The relationship between the model considered in [38] and in this work was discussed in Section 2.3. (A5) The body-fixed coordinate frame b is at a point \((x_{p}^{*},0)\) (the distance \(x_{p}\) is defined from the center of gravity along the vehicle center line). The point \((x_{p}^{*},0)\) is understood as the pivot point, that is, such that \(M^{-1}Bf=[\tau_{u},0,\tau_{r}]^{T}\). However, Equation (2) is slightly different from that in [38] due to the different location of the center of mass. According to the modification (A6), conditions \(Y_{1}>0\) and \(Y_{2}>0\) are satisfied (these quantities are calculated in Appendix A), which applies to the situation of unstable sway dynamics. Formally, the control objective is as follows:  
\[\lim_{t\to\infty}(\xi_{1}-\xi_{1_{d}}(t))=0,\ \lim_{t\to\infty}(\xi_{2}-\xi_{2_{d}}(t))=0,\] \[\lim_{t\to\infty}(\xi_{3}-\xi_{r3_{d}}(t))=0,\ \lim_{t\to\infty}(\xi_{4}-\xi_{r4_{d}}(t))=0. \tag{33}\]  
**Proposed quasi-velocity controller (QVC)**. To maintain the same idea of changing variables as in [38], it was assumed that \(u_{r}=\xi_{3}\cos z_{1}+\xi_{4}\sin z_{1}\), \(v_{r}=-\xi_{3}\sin z_{1}+\xi_{4}\cos z_{1}-z_{2}l\), and \(r=z_{2}\). The change in input to linearize the external dynamics can be written as follows:  
\[\left[ \begin{array}{c}\tau_{u}^{*}\\ \tau_{r}^{*}\end{array} \right]=\left[ \begin{array}{cc}\cos\psi&-(l+\text{ Y}_{23})\sin\psi\\ \sin\psi&(l+\text{Y}_{23})\cos\psi\end{array} \right]^{-1}\left[ \begin{array}{ c}-F_{\xi_{3}}(z_{1},\xi_{3},\xi_{4})+\mu_{1}\\ -F_{\xi_{4}}(z_{1},\xi_{3},\xi_{4})+\mu_{2}^{*}\end{array} \right]. \tag{34}\]  
The control input vector \(\mu=[\mu_{1},\mu_{2}^{*}]^{T}\) solves the problem under consideration, that is, tracking the desired trajectory \(\Gamma(t)\).