# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 4 Fractional Order Hierarchical Sliding Mode Controller
### Stability Analysis of Each Surface  
**Theorem 1**. : _For the dynamic system of a spherical robot which satisfies the constraint of Equation (14), and the control parameters \(\alpha\), \(\beta\), and \(\chi\) satisfy \(\alpha>1\), \(\beta>0\), \(\chi>1\)/2, the first-level sliding surface described in Equations (20) and (21) are asymptotically stable, and the second sliding surface Equation (27) is uniformly bounded by the control laws defined in Equations (35) and (36)._  
Proof. : Choose the Lyapunov function as  
\[V_{1}=\frac{1}{2}S^{2}+\frac{1}{2}\hat{S}^{2} \tag{37}\]The derivative of Equation (37) as  
\[ \begin{split}&\dot{V}_{1}=S\dot{S}+\hat{S}\dot{\hat{S}}\\ &=S[\eta(k_{1}D^{\lambda}e_{2}+k_{2}D^{\mu+1}e_{2}+f_{1}+b_{1}(\tau +\Delta\tau)+g_{1}\tau_{f}-\dot{\hat{\tau}}_{2d})+\\ &\xi(k_{3}D^{\lambda}e_{4}+f_{2}+b_{2}(\tau+\Delta\tau)+g_{2} \tau_{f}-\dot{x}_{4d})]+\hat{S}\dot{\hat{S}}\\ &=S[-\eta b_{1}\tau_{1}-\xi b_{2}\tau_{2}+(\eta b_{1}+\xi b_{2}) \tau+(\eta b_{1}+\xi b_{2})\Delta\tau]+\hat{S}\dot{\hat{S}}\end{split}  \tag{38}\]  
Equation (38) needs to take the following two cases into account. **Case 1. **_If \(\left|\hat{S}\right|\geq\rho\), by putting Equation (36) into Equation (38), one can obtain_  
\[ \begin{split}&\dot{V}_{1}=S\dot{\hat{S}}+\hat{S}\dot{\hat{S}}\\ &=S[\eta(k_{1}D^{\lambda}e_{2}+k_{2}D^{\mu+1}e_{2}+f_{1}+b_{1}( \hat{\tau}+\Delta\tau)+g_{1}\tau_{f}-\dot{\hat{\tau}}_{2d})+\\ &\xi(k_{3}D^{\lambda}e_{4}+f_{2}+b_{2}(\hat{\tau}+\Delta\tau)+g_ {2}\tau_{f}-\dot{x}_{4d})]+\hat{S}\dot{\hat{S}}\\ &=S[-\eta b_{1}\hat{\tau}_{1}-\xi b_{2}\hat{\tau}_{2}+(\eta b_{1} +\xi b_{2})\hat{\tau}+(\eta b_{1}+\xi b_{2})\Delta\tau]-\\ & S(\eta b_{1}+\xi b_{2})\Delta\tau-\frac{1}{2}\Delta\tau^{2}-k_{3 }\hat{S}^{2}+\hat{S}\Delta\tau\\ &\leq-\alpha S^{2}-\beta|S|-\frac{1}{2}\Delta\tau^{2}-\chi\frac{ 1}{2}\hat{S}^{2}+\frac{1}{2}\hat{S}^{2}+\frac{1}{2}\Delta\tau^{2}\\ &\leq-\alpha S^{2}-(\chi-\frac{1}{2})\hat{S}^{2}-\beta|S|\\ &\leq-u_{1}V_{1}\end{split}  \tag{39}\]  
_where \(u_{1}=\min\{\alpha,\chi-\frac{1}{2}\}\)._  
**Case 2. **_If \(\left|\hat{S}\right|<\rho\), one can obtain_  
\[ \begin{split}&\dot{V}_{1}=S\dot{\hat{S}}+\hat{S}\dot{\hat{S}}\\ &=S[-\eta b_{1}\hat{\tau}_{1}-\xi b_{2}\hat{\tau}_{2}+(\eta b_{1} +\xi b_{2})\hat{\tau}+(\eta b_{1}+\xi b_{2})\Delta\tau+\hat{S}]\\ &=-\alpha S^{2}-\beta|S|+S(\eta b_{1}+\xi b_{2})\Delta\tau+S\dot{ \hat{S}}\\ &\leq-\alpha S^{2}-\beta|S|+\frac{1}{2}S^{2}+\frac{1}{2}[(\eta b_ {1}+\xi b_{2})\Delta\tau]^{2}+\frac{1}{2}S^{2}+\frac{1}{2}\hat{S}^{2}\\ &\leq-(\alpha-1)S^{2}-\frac{1}{2}\hat{S}^{2}+\frac{1}{2}[(\eta b _{1}+\xi b_{2})\Delta\tau]^{2}\\ &\leq-u_{2}V_{1}+\frac{1}{2}[(\eta b_{1}+\xi b_{2})\Delta\tau]^{ 2}\end{split}  \tag{40}\]  
_where \(u_{2}=\min\{\alpha-1,\frac{1}{2}\}\)._  
Thus, one can conclude that the second sliding surface \(S\) is uniformly ultimately bounded. When \(\eta\), \(\xi\) satisfy the condition \(\eta\), \(\xi>0\), the convergence of the function is independent of the specific value of \(\eta\), \(\xi\). Considering the following two sliding surfaces:  
\[S_{1}=\eta_{1}s_{1}+\xi s_{2} \tag{41}\]  
\[S_{2}=\eta_{2}s_{1}+\xi s_{2} \tag{42}\]  
where \(\eta_{1}\), \(\eta_{2}\), \(\xi\) are positive constant and \(\eta_{1}\
eq\eta_{2}\). Here, one supposes that \(|S_{1}|\geq|S_{2}|\)  
\[ \begin{split}\int_{0}^{t}(S_{1}^{2}-S_{2}^{2})d\sigma& =\int_{0}^{t}[\eta_{1}^{2}s_{1}^{2}-\eta_{2}^{2}s_{1}^{2}+2\xi s_{1 }s_{2}(\eta_{1}-\eta_{2})]d\sigma\\ &=\int_{0}^{t}[-(\eta_{1}-\eta_{2})^{2}s_{1}^{2}+2(\eta_{1}-\eta_ {2})s_{1}S_{1}]d\sigma>0\end{split}  \tag{43}\]  
From Equation (43)  
\[\int_{0}^{t}(\eta_{1}-\eta_{2})^{2}s_{1}^{2}d<\int_{0}^{t}2(\eta_{1}-\eta_{2}) s_{1}S_{1}d\sigma \tag{44}\]Then  
\[\int_{0}^{t}s_{1}^{2}d\sigma<\int_{0}^{t}\frac{2s_{1}S_{1}}{(\eta_{1}-\eta_{2})}d \sigma<\int_{0}^{t}\frac{2||s_{1}||_{\infty}||S_{1}||_{\infty}}{(\eta_{1}-\eta_{ 2})}d\sigma \tag{45}\]  
The actuator directly drives the inner suspension, so one can see that \(x_{3}\) and \(\dot{x}_{4}\) are bounded.