## 2 Preliminaries and Problem Formulation
### Path Following Error Dynamic  
In order to converge to the desired path, a general and effective guidance law for the path following control problem is a look-ahead LOS guidance law. Figure 1 shows the geometrical illustration of the LOS guidance law for the MSV. The desired path is defined as \(X_{p}(\vartheta)=\left[x_{p}(\vartheta),y_{p}(\vartheta)\right]^{T}\), where \(\vartheta\) represents the path variable. The path tangent angle is calculated by  
\[\psi_{p}(\vartheta)=\arctan 2(y^{\prime}_{p}(\vartheta),x^{\prime}_{p}(\vartheta)) \tag{7}\]  
where \(x^{\prime}_{p}=\frac{\partial x_{p}}{\partial\vartheta}\) and \(y^{\prime}_{p}=\frac{\partial y_{p}}{\partial\vartheta}\). The along-track error \(x_{e}\) and the cross-track error \(y_{e}\) can be written as follows:  
\[\left[ \begin{array}{c}x_{e}\\ y_{e}\end{array} \right]=\left[ \begin{array}{cc}\cos\psi_{p}&\sin\psi_{p}\\ -\sin\psi_{p}&\cos\psi_{p}\end{array} \right]\left[ \begin{array}{c}x-x_{p}\\ y-y_{p}\end{array} \right] \tag{8}\]  
The differentiation of \(x_{e}\) can be obtained:  
\[\dot{x}_{e}= \dot{x}\cos\psi_{p}+\dot{y}\sin\psi_{p}-\dot{x}_{p}\cos\psi_{p}- \dot{y}_{p}\sin\psi_{p} \tag{9}\] \[+\dot{\psi}_{p}\underbrace{\left(-(x-x_{p})\sin\psi_{p}+(y-y_{p}) \cos\psi_{p}\right)}_{\text{cross-track error $y_{e}$}}\]  
Then, substituting (4) into (9) yields  
\[\dot{x}_{e} =u\cos(\psi-\psi_{p})-v\sin(\psi-\psi_{p})+\dot{\psi}_{p}y_{e}- \dot{\vartheta}\sqrt{{x^{\prime}_{p}}^{2}+{y^{\prime}_{p}}^{2}} \tag{10}\] \[=U\cos(\psi-\psi_{p}+\beta)+\dot{\psi}_{p}y_{e}-\dot{\vartheta} \sqrt{{x^{\prime}_{p}}^{2}+{y^{\prime}_{p}}^{2}}\]  
where \(U=\sqrt{u^{2}+v^{2}}\) denotes the speed of the vessel and \(\beta=\arctan 2(v,u)\) is the vessel's sideslip angle. Similarly, the differentiation of \(y_{e}\) can be obtained:  
\[\dot{y}_{e}= -\dot{x}\sin\psi_{p}+\dot{y}\cos\psi_{p}+\dot{x}_{p}\sin\psi_{p}- \dot{y}_{p}\cos\psi_{p} \tag{11}\] \[-\dot{\psi}_{p}\underbrace{\left((x-x_{p})\cos\psi_{p}+(y-y_{p}) \sin\psi_{p}\right)}_{\text{along-track error $x_{e}$}}\]  
and substituting (4) into (11) results in,  
\[\dot{y}_{e} =u\sin(\psi-\psi_{p})+v\cos(\psi-\psi_{p})-\dot{\psi}_{p}x_{e} \tag{12}\] \[=U\sin(\psi-\psi_{p}+\beta)-\dot{\psi}_{p}x_{e}\]  
Figure 1: The geometrical illustration of LOS guidance for a MSV. Finally, similar to [27], the derivatives of errors become  
\[\left\{ \begin{array}{l}\dot{x}_{e}=U\cos(\psi-\psi_{p}+\beta)+\dot{\psi}_{p}y_{e }-\dot{\theta}\sqrt{{x^{\prime}_{p}}^{2}+{y^{\prime}_{p}}^{2}}\\ \dot{y}_{e}=U\sin(\psi-\psi_{p}+\beta)-\dot{\psi}_{p}x_{e}\end{array} \right. \tag{13}\]  
Considering the mathematical model of an underactuated MSV with the external disturbances, model parametric uncertainties, position constraint and input saturation, the control objectives of this paper are as follows: (1) Design the FT-LOS guidance law to ensure that the position error will not exceed the error constraint, and converge into a small neighborhood around zero in finite time. (2) The finite-time attitude constraint controller \(\tau_{r}\) and the finite-time velocity constraint controller \(\tau_{u}\) are designed to make the underactuated MSV track the desired path at the desired speed in finite time. In addition, it has faster convergence speed and better disturbance rejection performance. (3) All error signals of the whole closed-loop control system can converge into a small neighborhood around zero in finite time, and position errors meet the constraint requirements, that is to say, \(|x_{e}|<k_{x}(t)\), \(|y_{e}|<k_{y}(t)\), where \(\left\{k_{x}(t),k_{y}(t)\right\}\) are differentiable time-varying error constrains. To facilitate the controller design, we make the assumption as follows:  
**Assumption 3**. : _The initial position errors satisfy the prescribed constraints \(|x_{e}(t_{0})|<k_{x}(t_{0})\), \(|y_{e}(t_{0})|<k_{y}(t_{0})\)._  
**3. Finite-Time Path Following Control Algorithm Design**  
In this section, the finite-time path following control scheme is proposed to realize the control objectives.