# 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
Define \(\Delta\xi_{3}^{*}=\xi_{3}-\xi_{3_{d}}\), \(\Delta\xi_{1}^{*}=\xi_{1}-\xi_{1_{d}}\), \(\Delta\xi_{1_{1}}=\xi_{1_{1}}-\xi_{1_{d}}\), \(\Delta\xi_{4}=\xi_{4}-\xi_{4_{d}}\), \(\Delta\xi_{2}=\xi_{2}-\xi_{2_{d}}\), \(\Delta\xi_{2_{l}}=\xi_{2_{l}}-\xi_{2_{dl}}\), where \(\xi_{i_{l}}=\int_{0}^{t}\xi_{i}(\sigma)d\sigma\), where \(i\in\{1,2,1_{d},2_{d}\}\) are the integrals of signals \(\xi_{i_{l}}\). Then, in the controller (34), there are two inputs \(\mu_{1}\) and \(\mu_{2}^{*}\), namely:  
\[\mu_{1} =-k_{v_{x}}\Delta\xi_{3}-k_{p_{x}}\Delta\xi_{1}-k_{x_{L}}\Delta \xi_{1_{l}}+\dot{\xi}_{3_{d}}, \tag{35}\] \[\mu_{2}^{*} =-k_{v_{y}}(l+\text{Y}_{23})\Delta\xi_{4}-k_{p_{y}}\Delta\xi_{2}- k_{y}\Delta\xi_{2_{l}}+\dot{\xi}_{4_{d}}, \tag{36}\]  
where \(k_{v_{x}}\), \(k_{v_{y}}\), \(k_{p_{x}}\), \(k_{p_{y}}\), \(k_{l_{x}}\), and \(k_{l_{y}}\), which are positive real constant gains, are used. The first signal (35) is the same as in [38], but the second is different, which is key to completing the control task. This algorithm guarantees the achievement of the control objectives (33). However, from the decomposition method used, it follows that \(\tau_{u}^{*}=(N_{1}^{-1}+N_{3}^{-1}\text{Y}_{13}^{2})\tau_{u}+N_{3}^{-1}\text{Y} _{13}\tau_{r}\) and \(\tau_{r}^{*}=N_{3}^{-1}\text{Y}_{13}\tau_{u}+N_{3}^{-1}\tau_{r}\). This relationship can be used for the input signals normalization and is given in the following form:  
\[\left[ \begin{array}{c}\tau_{u}\\ \tau_{r}\end{array} \right]=s_{f}\left[ \begin{array}{cc}N_{1}&-N_{1}\text{Y}_{1 3}\\ -N_{1}\text{Y}_{13}&N_{3}+N_{1}\text{Y}_{13}^{2}\end{array} \right]\left[ \begin{array} []{c}\mu_{1}\\ \mu_{2}^{*}\end{array} \right], \tag{37}\]where \(s_{f}\) means a scaling factor (a constant value). In addition, the following relationships arise from \(\tau_{u}=s_{f}N_{1}\left(\mu_{1}-Y_{13}\mu_{2}^{*}\right)\) and \(\tau_{r}=s_{f}\left(N_{3}\mu_{2}^{*}-N_{1}Y_{13}(\mu_{1}-Y_{13}\mu_{2}^{*})\right)\). If the values \(N_{1}\) and \(N_{3}\) are too large or too small, the control signal becomes ineffective. The benefit of such normalization is that the action of the control algorithm will be directly related to the dynamics of the vehicle. Note that, in [38], the control gains are selected without reference to the vehicle dynamics. _Comment_. Theorem 1 of [38] is only partially satisfied for the control algorithm in terms of QV due to the different equations of motion. To relate this to the results of the original work, a revised theorem was proposed based on a vehicle model analogy. In the reference, the desired total relative velocity \(U_{d}\) is defined. This quantity and its time derivative are limited, that is, \(U_{d}\in\left\langle U_{d}^{min},U_{d}^{max}\right\rangle\), and \(\dot{U}_{d}\in\left\langle U_{d}^{min},\dot{U}_{d}^{max}\right\rangle\). **Modified theorem**. An underactuated marine vehicle is described by the model (A4)-(A6) and (A12)-(A17) (cf. Appendix A). The point of position of the hand is \(h=[\xi_{1},\xi_{2}]^{T}=[x+l\cos\psi,y+l\sin\psi]^{T}\) (where \([x,y]^{T}\) is the position of the pivot point of the vehicle, \(l\) is a positive constant, and \(\psi\) is the yaw angle of the vehicle). Then, the velocity \(U_{d}=((\xi_{r3\lambda_{d}}^{2}+\xi_{r4\lambda_{d}}^{2})^{2})^{1/2}>0\) and \(\phi_{1}=\arctan(\xi_{r4\lambda_{d}}^{2}/\xi_{r3\lambda_{d}}^{2})\), the crab angle (the angle between the vehicle track and the longitudinal axis of the vehicle), are defined. Provided that assumptions 1-8 (including their modifications) are met, and if:  
\[0<U_{d}^{max}<Y_{2}/Y_{1}, \tag{38}\] \[k_{v_{2}},k_{v_{3}},k_{p_{2}},k_{p_{3}},k_{1_{2}},k_{l_{3}}>0,\] (39) \[l>\max\{m_{22}/m_{23},-X_{2}/Y_{2}\},\] (40) \[\dot{U}_{d}^{max}\leq\frac{2\min\{\dot{a}(\dot{d}-\xi),\dot{b}\} }{\frac{Y_{1}U_{d}^{max}}{l}+2\big{(}Y_{1}-\frac{X_{1}-1}{l}\big{)}},\quad b= Y_{2}+\frac{X_{2}}{l}, \tag{41}\]  
then, the controller described by Equation (34), where \(Y_{23}=0\), and with new input signals (35) and (36), is guaranteed to achieve the control objectives (33).