## 4 FLOS Algorithm and ASIIBC
### Floss with Feso  
Considering the term containing the uncertain side-slip angle \(\beta\) in Formula (6) as an uncertain state, we define new state \(g\) as \(g=u\cos(\psi-\phi)\tan\beta\). Therefore, side-slip angle \(\beta=\arctan(g/u\cos(\psi-\phi))\). To observe the side-slip angle \(\beta\) in finite time, a finite-time converge observer is designed as  
\[\left\{ \begin{array}{l}\hat{y}_{\epsilon}=\hat{g}+u\sin(\psi-\phi_{p})-\dot {\phi}_{p}x_{\epsilon}-l_{1}f_{1}(\widetilde{y}_{\epsilon})\\ \dot{\hat{g}}=-l_{2}f_{2}(\widetilde{y}_{\epsilon})\end{array} \right. \tag{13}\]  
where \(\hat{y}_{\epsilon}\) is the observation of \(y_{\epsilon}\), \(\widetilde{y}_{\epsilon}=\hat{y}_{\epsilon}-y_{\epsilon}\). \(\hat{g}\) is the observation of \(g\). \(l_{1}\), \(l_{2}\) are positive adjustable parameters, \(f_{1}(\widetilde{y}_{\epsilon})\), \(f_{2}(\widetilde{y}_{\epsilon})\) can be expressed as  
\[\left\{ \begin{array}{l}f_{1}(\widetilde{y}_{\epsilon})=k_{1}|\widetilde{y}_ {\epsilon}|^{\frac{1}{2}}sign(\widetilde{y}_{\epsilon})+\widetilde{y}_{ \epsilon}\\ f_{2}(\widetilde{y}_{\epsilon})=\frac{3k_{1}}{2}|\widetilde{y}_{\epsilon}|^{ \frac{1}{2}}sign(\widetilde{y}_{\epsilon})+\frac{k_{2}^{2}}{2}sign( \widetilde{y}_{\epsilon})+\widetilde{y}_{\epsilon}\end{array} \right. \tag{14}\]  
Take \(\widetilde{y}_{\epsilon}\) derivation for time \(t\) and substitute it into the second line of Formulas (6) and (13), which yields  
\[\left\{ \begin{array}{l}\dot{\widetilde{y}}_{\epsilon}=\hat{g}+u\sin(\psi- \phi)-\dot{\phi}x_{\epsilon}-l_{1}f_{1}(\widetilde{y}_{\epsilon})-u\sin(\psi- \phi)-\dot{\phi}x_{\epsilon}-g\\ =\widetilde{g}-l_{1}f_{1}(\widetilde{y}_{\epsilon})\\ \dot{\widetilde{g}}=-l_{2}f_{2}(\widetilde{y}_{\epsilon})-\dot{g}\\ \dot{f}_{1}(\widetilde{y}_{\epsilon})=\left(\frac{1}{2}\widetilde{y}_{\epsilon }^{-1/2}sign(\widetilde{y}_{\epsilon})+1\right)(\widetilde{g}-l_{1}f_{1}( \widetilde{y}_{\epsilon}))\end{array} \right. \tag{15}\]  
**Remark 2**. : \(\beta\) _is slowly time-varying, so \(\hat{\beta}\approx 0\), \(\dot{g}\approx u\sin(\psi-\phi)(\dot{\psi}-\dot{\phi})\tan\beta\). \(u\), \(\dot{\psi}\), \(\dot{\phi}\) are bounded, so \(\left|\dot{g}\right|\leq h_{\max}\). \(h_{\max}\) is a positive constant._  
**Theorem 1**. : _The observer satisfies Formula (13), the observation error can converge in finite time T, where \(T\leq V(\widetilde{y}_{\epsilon}(0))^{1/2}/\left[\frac{1}{2}(\frac{k_{1}}{2_{ \min}}-\sigma_{1})\right]\), to the neighborhood of the equilibrium point \((0,0)\), where \(\sigma_{1}\) is a constant and \(\sigma_{1}\in\left(0,\frac{k_{1}}{2_{\min}}\right)\)._  
Figure 5: Control structure of the CDR on the ground and water surface. FLOS is used to plan the yaw angle and linear velocity of the robot. Speed controller and yaw angle controller are used to track the desired yaw angle and linear velocity. The proof of Theorem 1 can be found in Appendix A. In contrast to the idealized assumptions for the uncertainty term \(g\) in [38], we adopt a simpler and clearer approach to prove the convergence of the observer. The FLOS is designed as:  
\[\psi_{d}=\phi_{p}-\arctan\left(\tan\hat{\beta}+\frac{y_{e}+k_{y}|y_{e}|^{P}sign(y _{e})}{\Delta}\right) \tag{16}\]  
\(\Delta\) is the forward-looking distance. where \(0<p<1\), \(k_{y}\) is the positive adjustable parameter. The speed \(u_{p}\) is designed to make the convergence of the position error \(x_{e}\), the virtual variable \(u_{p}\) is  
\[u_{p}=k_{1}x_{e}+k_{2}|x_{e}|^{p}sign(x_{e})+u\cos(\psi-\phi_{p})+u\sin(\psi- \phi_{p})\tan\hat{\beta} \tag{17}\]  
where \(0<p<1\), yaw angle satisfies \(\psi\approx\psi_{d}\) by control, so \(\psi-\phi_{p}=\psi_{d}-\phi_{p}\).