## 4 FLOS Algorithm and ASIIBC
### ASMIBC for Yaw Angle and Linear Velocity of the CDR
\[u_{1}=\frac{1}{g(x)}\bigg{(}-k_{1}z_{2}-f(x)-\dot{\eta}(x)+2\dot{\alpha}_{1}- \frac{\overline{x}_{2}^{2}-x_{2}^{2}}{\overline{x}_{2}^{2}}z_{1}\bigg{)}+ \frac{\overline{x}_{2}^{2}-x_{2}^{2}}{\overline{x}_{2}^{2}}l_{1}\dot{\alpha}_{1},\ u_{2}=-\frac{1}{g(x)}k_{2}sign(z_{2})\]  
Define Lyapunov function  
\[V_{2}(z_{2},\alpha_{1},\widetilde{\eta})=\frac{1}{2}z_{1}^{2}+\int_{0}^{z_{2}} \frac{\sigma\overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1 }\right)^{2}}d\sigma+\frac{1}{2}\widetilde{\eta}^{2} \tag{29}\]  
\(V_{2}\) takes the derivative of time \(t\), substitutes \(u_{1}\), \(u_{2}\), \(\alpha_{1}\) into it, and yields  
\[\dot{V}_{2}=-k_{0}z_{1}^{2}-k_{1}z_{2}^{2}\frac{\overline{x}_{2}^{2}}{ \overline{x}_{2}^{2}-x_{2}^{2}}-k_{2}z_{2}sign(z_{2})\frac{\overline{x}_{2}^{2} }{\overline{x}_{2}^{2}-x_{2}^{2}}+\widetilde{\eta}\bigg{(}\dot{\dot{\eta}}- \frac{z_{2}\overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-x_{2}^{2}}\bigg{)} \tag{30}\]  
The adaptive law is designed as  
\[\dot{\dot{\eta}}=\frac{z_{2}\overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-x_{2}^{2}} \tag{31}\]because \(sign(z_{2})=\frac{z_{2}}{|z_{2}|}\geq\frac{z_{2}}{|\xi||k_{k}|}\), so \(-sign(z_{2})\leq-\frac{z_{2}}{|\xi||k_{k}|}\). Substitute Formula (31) in to Formula (30)  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]0  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]1  
Define \(k_{2}=\frac{k_{2}|z_{2}|}{1-|\xi|}\), \(k_{3}\) is a positive constant, rewrite Formula (32)  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]2  
According to Lemma 2, \(0\leq\frac{1}{2}x_{2}^{2}\leq V_{2}\), and \(\dot{V}_{2}<0\). Based on Lyapunov stability theorem, when \(t\rightarrow\infty\), \(V_{2}\to 0\). It should be noted that, when \(z_{2}\to k_{h}\), \(\xi\to 1\), thus \(k_{2}\rightarrow\infty\). The robustness of the system can be improved by adjusting the gain of sliding mode \(k_{2}\). According to the controller design method described above, the yaw angle controller is design as  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]3  
\(e_{\psi}=\psi-\psi_{d}\), \(\psi_{d}\) is the desired yaw angle. \(k_{\psi}\) is the positive constant. \(r_{d}\) is the desired yaw rate. The yaw angle velocity controller is designed as  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]4  
where \(k_{r1},k_{r2},k_{r3}\) are positive constants, \(e_{r}=r-r_{d}\). \(\overline{r}\) is the maximum value of yaw angular velocity. \(I\) is the moment of inertia of the robot. \(l_{r}=\frac{\overline{r}^{2}}{2e_{r}}\ln\frac{\left(\overline{r}^{2}+r^{2} \right)\left(\overline{r}^{2}-r_{d}^{2}\right)}{\left(\overline{r}^{2}-r^{2} \right)\left(\overline{r}^{2}+r_{d}^{2}\right)}\), \(k_{r2}=\frac{k_{2}|e_{r}|}{1-|\xi_{r}|}\), \(\hat{\eta}_{r}=\int\frac{e_{r}\overline{r}^{2}}{\overline{r}^{2}-r^{2}}dt\). In our proposed algorithm, the yaw angle and the linear velocity are designed by FLOS. The linear velocity is a constant, so \(\dot{u}_{d}=0\). In this situation, the traditional IBC loses the ability to constrain the linear speed, thus the velocity controller is designed as  
\[V_{1}\big{(}z_{2},\dot{\alpha}_{1}\big{)}=V_{0}+\int_{0}^{z_{2}}\frac{\sigma \overline{x}_{2}^{2}}{\overline{x}_{2}^{2}-\left(\sigma+\alpha_{1}\right)^{2 }}d\sigma \tag{26}\]5  
where \(k_{u1},k_{u2},k_{u3}\) are positive constant, \(e_{u}=u-u_{d}\). \(m\) is the mass of the robot.