## 2 Preliminaries and Problem Formulation
### Ft-Los Guidance Law Design  
The FT-LOS guidance law is proposed to obtain the update law and the desired heading angle.  
The time-varying BLF will be integrated into the FT-LOS guidance to ensure the position error constraints. Define the time-varying BLF as follows:  
\[\begin{split} V_{1}&=\frac{1}{2}\log\frac{k_{x}^{2 }(t)}{k_{x}^{2}(t)-x_{e}^{2}}+\frac{1}{2}\log\frac{k_{y}^{2}(t)}{k_{y}^{2}(t)- y_{e}^{2}}\\ &=\frac{1}{2}\log\frac{1}{1-\xi_{x}^{2}}+\frac{1}{2}\log\frac{1} {1-\xi_{y}^{2}}\end{split} \tag{14}\]  
Figure 2: The block diagram of path following control.  
where \(\xi_{x}=\frac{x_{e}}{k_{x}(t)}\) and \(\xi_{y}=\frac{y_{e}}{k_{y}(t)}\).  
Taking the derivative of \(V_{1}\), we obtain  
\[\dot{V}_{1}= \frac{\xi_{x}}{1-\xi_{x}^{2}}\left(\frac{\dot{x}_{e}}{k_{x}(t)}- \frac{x_{e}\dot{k}_{x}(t)}{k_{x}^{2}(t)}\right)+\frac{\xi_{y}}{1-\xi_{y}^{2}} \left(\frac{\dot{y}_{e}}{k_{y}(t)}-\frac{y_{e}\dot{k}_{y}(t)}{k_{y}^{2}(t)}\right) \tag{15}\]  
Substituting (13) into (15), we obtain  
\[\dot{V}_{1}= \frac{x_{e}}{k_{x}^{2}(t)-x_{e}^{2}}\left(U\cos(\psi-\psi_{p}+ \beta)-\dot{\vartheta}\sqrt{{x_{p}^{\prime}}^{2}+{y_{p}^{\prime}}^{2}}-\frac{x _{e}\dot{k}_{x}(t)}{k_{x}(t)}\right)\] \[-\frac{y_{e}^{2}}{k_{y}^{2}(t)-y_{e}^{2}}\frac{k_{y}(t)}{k_{y}(t) }+\frac{y_{e}}{k_{y}^{2}(t)-y_{e}^{2}}U\sin(\psi-\psi_{p}+\beta) \tag{16}\] \[+\frac{x_{e}}{k_{x}^{2}(t)-x_{e}^{2}}\psi_{p}y_{e}\left(1-\frac{ k_{x}^{2}(t)-x_{e}^{2}}{k_{y}^{2}(t)-y_{e}^{2}}\right)\]  
Taking the time derivative of \(\psi_{p}\) yields  
\[\dot{\psi}_{p}=\dot{\vartheta}\frac{{y_{p}^{\prime\prime}}{x_{p}^{\prime}}-{x _{p}^{\prime\prime}}{y_{p}^{\prime}}}{{x_{p}^{\prime}}^{2}+{y_{p}^{\prime}}^{ 2}} \tag{17}\]  
Invoking (17) into (16), we have  
\[\dot{V}_{1}= \frac{x_{e}}{k_{x}^{2}(t)-x_{e}^{2}}\left(U\cos(\psi-\psi_{p}+ \beta)-\dot{\vartheta}\sigma\right)\] \[-\frac{x_{e}^{2}}{k_{x}^{2}(t)-x_{e}^{2}}\frac{\dot{k}_{x}(t)}{k_ {x}(t)}-\frac{y_{e}^{  
Substituting (19)-(22) into (18) results in  
\[\dot{V}_{1}= -\frac{x_{e}}{k_{x}^{2}(t)-x_{e}^{2}}\delta_{x}-\frac{x_{e}^{2}}{k_{ x}^{2}(t)-x_{e}^{2}}\frac{\dot{k}_{x}(t)}{k_{x}(t)}-\frac{y_{e}^{2}}{k_{y}^{2}(t)-y_{e}^ {2}}\frac{\dot{k}_{y}(t)}{k_{y}(t)}-\frac{y_{e}U}{k_{y}^{2}(t)-y_{e}^{2}}\frac{ \delta_{y}}{\sqrt{\delta_{y}^{2}+\Delta^{2}}}+\frac{y_{e}U\phi_{1}\psi_{e}}{k_{ y}^{2}(t)-y_{e}^{2}} \tag{23}\] \[= -\frac{k_{1}}{k_{x}^{2}(t)-x_{e}^{2}}\log\frac{k_{x}^{2}(t)}{k_{x }^{2}(t)-x_{e}^{2}}-\frac{k_{3}U}{(k_{y}^{2}(t)-y_{e}^{2})\sqrt{\delta_{y}^{2}+ \Delta^{2}}}\log\frac{k_{y}^{2}(t)}{k_{y}^{2}(t)-y_{e}^{2}}\] \[-\frac{k_{2}}{k_{x}^{2}(t)-x_{e}^{2}}\log^{\frac{3}{4}}\left( \frac{k_{x}^{2}(t)}{k_{x}^{2}(t)-x_{e}^{2}}\right)\ -\frac{k_{4}U}{(k_{y}^{2}(t)-y_{e}^{2})\sqrt{\delta_{y}^{2}+ \Delta^{2}}}\log^{\frac{3}{4}}\left(\frac{k_{y}^{2}(t)}{k_{y}^{2}(t)-y_{e}^{2} }\right)\] \[+\frac{y_{e}U\phi_{1}\psi_{e}}{k_{y}^{2}(t)-y_{e}^{2}}\]  
where  
\[\phi_{1}=\frac{(1-\cos\psi_{e})}{\psi_{e}}\frac{\delta_{y}}{\sqrt{\delta_{y}^ {2}+\Delta^{2}}}+\frac{\sin\psi_{e}}{\psi_{e}}\frac{\Delta}{\sqrt{\delta_{y}^ {2}+\Delta^{2}}} \tag{24}\]  
and \(\psi_{e}=\psi-\psi_{d}\).  
**Remark 1**.: _In (24), we have \(\left|\frac{1-\cos\psi_{e}}{\psi_{e}}\right|<0.73\), \(\left|\frac{\sin\psi_{e}}{\psi_{e}}\right|\leq 1\), \(\left|\frac{\Delta}{\sqrt{\delta_{y}^{2}+\Delta^{2}}}\right|\leq 1\) and \(\left|\frac{\delta_{y}}{\sqrt{\delta_{y}^{2}+\Delta^{2}}}\right|\leq 1\). Therefore, \(\phi_{1}\leq 1.73\) is bounded._