# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 4 Fractional Order Hierarchical Sliding Mode Controller  
### Design of the Fractional Hierarchical Sliding Mode Controller  
The hierarchical sliding mode control method is usually used to design the controller for underactuated systems. Underactuated systems consist of serval subsystems. A first layer sliding surface is given for each subsystem, and then a second layer sliding surface is proposed. From Equation (3), the whole system is divided into an inner suspension system and a spherical shell system. Without considering the input saturation, the desired state of the system can be assumed to be:  
\[x_{d}=[x_{id}\quad x_{2d}\quad x_{3d}\quad x_{4d}]^{T} \tag{16}\]  
Therefore, the tracking error variables are written as  
\[ \begin{array}{l}e_{1}=x_{1}-x_{1d},e_{2}=x_{2}-x_{2d}\\ e_{3}=x_{3}-x_{3d},e_{4}=x_{4}-x_{4d}\end{array}  \tag{17}\]  
Without considering input saturation, the shell and pendulum subsystem are rewritten in Equations (18) and (19) respectively. \[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f_{1}(x)+b_{1}(x)\tau+g_{1}(x)\tau_{f}\end{array} \right. \tag{18}\]  
\[\left\{ \begin{array}{l}\dot{x}_{3}=x_{4}\\ \dot{x}_{4}=f_{2}(x)+b_{2}(x)\tau+g_{2}(x)\tau_{f}\end{array} \right. \tag{19}\]  
Based on hierarchical sliding mode control method, the first-layer sliding surface for the shell subsystem is given as:  
\[s_{1}=k_{1}D^{\lambda-1}e_{2}+e_{2}+k_{2}D^{\mu}e_{2} \tag{20}\]  
The first-layer sliding surface for the inner suspension system is as follows:  
\[s_{2}=k_{3}D^{\lambda-1}e_{4}+e_{4} \tag{21}\]  
where \(\lambda,\mu\in(0,1)\), \(k_{1}\), \(k_{2}\), and \(k_{3}\) are positive constants. To accelerate the response velocity, a fractional order operator is used in the two first-level sliding surfaces. The first-layer sliding surface in Equation (20) has a fractional \(PI^{\lambda}D^{\mu}\) structure, while the structure of Equation (21) is \(PI^{\lambda}\). Accordingthe definition of fractional integral in Equation (9), the fractional integral operator has one more weight function compared to the integer order operator. The weight function has a larger value in the initial stage, but it decreases over time [45]. This means that fractional-order integrals have larger integration coefficients in the initial stage and smaller integration coefficients afterwards. If the fractional-order integral sliding mode surface is simply added to the integral-order integral sliding mode surface to form a fractional-order integral sliding mode surface, it will, on the one hand, have a faster response than the integer-order integral sliding mode surface in the initial stage, but on the other hand, may cause greater overshoot. For this reason, the first-layer sliding surface adds a fractional differential term to reduce the overshoot and further improve the response speed. The integer differential term is sensitive to errors, a feature which makes it possible to reduce the overshoot and accelerate the response speed. This also will reduce the robustness of the system. Following _Property 2_, the fractional differential term can be treated as a fractional integral of the integer order differential term. This means that the robustness of the control system can be improved by adjusting the value of fractional differential order \(\mu\).