## 2 Materials and Methods
### The Design of the Forward-Rotation Subsystem  
For convenience, the system has been reorganized into the following form (24). \[\left\{ \begin{array}{l}\ddot{\zeta}_{1}=\frac{1}{\Omega}\Big{(}m_{13}c_{32} \dot{\zeta}_{2}-m_{33}c_{13}\dot{\zeta}_{3}-m_{13}g_{3}\Big{)}+\frac{m_{33}}{ \Omega}\tau_{1}+D_{1}\\ \ddot{\zeta}_{2}=\frac{1}{m_{22}}\Big{(}-c_{22}\dot{\zeta}_{2}-c_{23}\dot{\zeta }_{3}\Big{)}+\frac{1}{m_{22}}\tau_{2}+D_{2}\end{array} \right., \tag{24}\]  
The state variables in the system described by Equation (24) are highly coupled. To address this issue and to expand the system's asymptotic stability domain, a hierarchical sliding mode controller was designed. The controller's primary objective is to utilize an input control law that can simultaneously control both system variables \(\zeta_{1}\) and \(\zeta_{2}\), thereby, mitigating the problem of system coupling [38]. Having obtained the expected forward velocity (\(v_{d}\)) and angular velocity (\(w_{d}\)) from Equation (11), the error between the actual and expected values can be defined as follows:  
\[\left\{ \begin{array}{l}e_{xv}=\zeta_{1d}-\zeta_{1}=x_{r}\cos\theta_{r}+y_{r} \sin\theta_{r}-x_{v}\\ e_{\eta}=\dot{\zeta}_{1d}-\dot{\zeta}_{1}=v_{d}-v\\ e_{\theta}=\dot{\zeta}_{2d}-\dot{\zeta}_{2}=\theta_{r}-\theta\\ e_{w}=\dot{\zeta}_{2d}-\dot{\zeta}_{2}=w_{d}-w\end{array} \right., \tag{25}\]  
To design the sliding mode control error tracking scheme for the \(v\)-\(w\) subsystem, two mutually independent first-layer sliding mode surfaces were initially constructed. The equations used to create these slide surfaces are as follows:  
\[s_{1}=\gamma_{1}e_{xv}+e_{v},\ \gamma_{1}>0, \tag{26}\]  
\[s_{2}=\gamma_{2}e_{\theta}+e_{w},\ \gamma_{2}>0, \tag{27}\]  
The results of deriving Equations (26) and (27) are presented below. \[\dot{s}_{1}=\gamma_{1}\dot{e}_{xv}+\dot{e}_{v}=\gamma_{1}\dot{e}_{xv}+\ddot{ \zeta}_{1d}-\ddot{\zeta}_{1}=0\dot{s}_{2}=\gamma_{2}\dot{e}_{\theta}+\dot{ \epsilon}_{w}=\gamma_{2}\dot{e}_{\theta}+\ddot{\zeta}_{2d}-\ddot{\zeta}_{2}=0, \tag{28}\]  
According to [PERSON]'s equivalent control theory, the equivalent control laws for \(\zeta_{1}\) and \(\zeta_{2}\) are as follows:  
\[\left\{ \begin{array}{l}\tau_{eq1}=-\frac{\Delta}{m_{33}}\Big{(}\gamma_{1} \dot{e}_{xv}+\ddot{\zeta}_{1d}+\hat{D}_{1}\Big{)}+\frac{1}{m_{33}}\Big{(}m_{13} c_{32}\dot{\zeta}_{2}-m_{33}c_{13}\dot{\zeta}_{3}-m_{13}g_{3}\Big{)}\\ \tau_{eq2}=m_{22}\Big{(}\gamma_{2}\dot{e}_{\theta}+\ddot{\zeta}_{2d}\Big{)}+c _{22}\dot{\zeta}_{2}+c_{23}\dot{\zeta}_{3}-\hat{D}_{2}\end{array} \right. \tag{29}\]  
The second sliding surface can be expressed as a linear combination of the first sliding surface. \[s_{3}=\gamma_{3}s_{1}+\gamma_{4}s_{2},\gamma_{3},\gamma_{4}>0, \tag{30}\]  
To control \(\zeta_{1}\) and \(\zeta_{2}\), the equivalent control law must be included at the same time to control and enter their designed sliding surface, respectively. Therefore, the total control law is shown in the following equation. \[\tau_{u}=\tau_{eq1}+\tau_{eq2}+\tau_{sw}, \tag{31}\]\(\tau_{sw}\) is the switching law of the converging slide surface phase, and the expressions are as follows. \[\tau_{sw}=\frac{-\Big{(}\gamma_{3}\frac{m_{22}}{\Omega}\tau_{eq2}+\gamma_{4} \frac{1}{m_{22}}\tau_{eq1}+\eta_{1}o(s_{3})\text{sgn}(s_{3})+\mu_{1}s_{3}\Big{)} }{\gamma_{3}\frac{m_{22}}{\Omega}+\gamma_{4}\frac{1}{m_{22}}}, \tag{32}\]  
To mitigate the jitter phenomenon of the system, the isokinetic and exponential terms of the sliding mode control are improved, where \(\eta_{1}\) and \(\mu_{1}\) are the isokinetic and exponential terms of the previous design convergence law, and both are positive constants; \(o(s_{3})\) and \(\text{sgn}(s_{3})\) are the adaptive and symbolic functions designed in the previous paper. To prove that the designed controller is stable, the Lyapunov function is chosen as follows. \[V_{3}=\frac{1}{2}s_{3}^{2}>0 \tag{33}\]  
The derivative of \(V_{3}\) for time is given by the following expression.