# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 5 Simulation Study  
This section presents the results of numerical simulations in the MATLAB software, which were performed to demonstrate the effectiveness of the proposed novel fractional PI\({}^{\lambda}\)D\({}^{\mu}\) sliding mode controller (NFO-PID SMC). Table 1 lists the main physical parameters of the spherical robot. To test the performance of the controller, the desired angular velocity of the spherical robot was set to \(x_{2d}=10\) rad/s. State \(x_{1}\) is the rolling angle of the spherical robot; thus, the desired state \(x_{1d}\) was set to \(x_{1d}=10t\) rad. The shell system was driven by the eccentric moment produced by the rotation between the shell and pendulum subsystems. When the shell keeps moving at a constant velocity, the swinging-up angular velocity of the inner suspension relative to the shell should be zero [20]. This way, the desired state vector of the control system of Equation (3) can be determined.  
\[x_{d}=[10t\quad\overline{x}_{2d}\quad 0\quad 0] \tag{58}\]  
[PERSON] et al. [19] proposed an integer-type order sliding mode surface for spherical robot speed control. For comparison, an integer-order sliding controller (IO-SMC) based on the method proposed in this paper was implemented for the spherical robot system. The integer-type order sliding mode surface equation can be written as follows:  
\[s_{1}=k_{1}e_{1}+e_{2} \tag{59}\]  
\[s_{2}=k_{3}e_{3}+e_{4} \tag{60}\]  
Table 2 summarizes the parameters of the fractional-order hierarchical sliding mode controller with the PI\({}^{\lambda}\)D\({}^{\mu}\) sliding surface. In order to highlight the differences between the controllers, some parameters were kept the same for both controllers. Furthermore, two additional controllers without the filter were used by setting \(\varepsilon=0\). To verify the robustness of the four controllers, suppose the friction force \(\tau_{f}=0\) at time \(t=0\) s, and then set the friction force \(\tau_{f}=1\) N\(\cdot\)m at \(t=15\) s.  
Figure 2 shows the curves of the velocity response of the spherical shell, while Figure 3 shows the curves of the errors. Figure 4 shows the control output. Figures 5-7 show the time evolutions of all sliding mode surfaces. Table 3 summarizes the performance of the two controllers in terms of overshoot and adjustment time.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **\(M\)/kg** & **\(m\)/kg** & **\(\tau\)/m** & **\(l\)/m** & \(\tau_{\textbf{max}}\)/N\(\cdot\)m** & \(\tau_{\textbf{min}}\)/N\(\cdot\)m \\ \hline
**Value** & 2.5 & 8 & 0.15 & 0.09 & 2.3 & –2.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main physical parameters of the spherical robot.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameter** & \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(\eta\) & \(\zeta\) & \(\alpha\) & \(\beta\) & \(\lambda\) & \(\mu\) & \(\varepsilon\) \\ \hline
**Value** & 1 & 0.3 & 12 & 2 & 2 & 5 & 5 & 0.1 & 0.1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the controllers.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Controller** & **Overshoot (\%)** & **Adjustment Time (s)** \\ \hline NFO-PID SMC & 0 & 3.4 \\ IO-PI SMC & 35 & 6.1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performances of the controllers.  
Figure 4: The output of two controllers.  
Figure 3: Curves of error \(e_{2}\).  
Figure 2: Comparison of the curves of the velocity of shell response among four controllers: **(a)**. Time interval 0–30 s; **(b)**. Time interval 0–7 s.