## 3 Design Procedure for a Control Strategy
### Design Non-Singular Fast-Terminal Sliding Variable  
Based on the TSMC design approach, a state variable termed as the NFTSM variable was previously designed, where the novel NFTSM variables are proposed from the tracking positional error as  
\[s_{i}=\dot{c}_{i}+h_{1i}sign[\zeta_{i}]+h_{2i}\zeta_{i}^{[\alpha_{i}]}, \tag{7}\]  
where \(h_{1i},\ h_{2i}\) are positive values, \(\alpha_{i}>1\), and the variable \(\zeta_{i}\) is selected as  
\[\zeta_{i}=e_{i}+\int\limits_{0}^{t}\Big{(}\Gamma_{1i}e_{i}^{[2-\theta_{1}]}+ \Gamma_{2i}e_{i}+\Gamma_{3i}e_{i}^{[\theta_{i}]}\Big{)}d\sigma, \tag{8}\]  
where \(e_{i}=q_{i}-q_{ir}\ (i=1,\ 2,\ \dots,\ n)\) is the tracking positional error, \(q_{ir}\) is described as the desired path value, \(\zeta_{i}\) is the sliding surface variable, \(\Gamma_{1i},\ \Gamma_{2i},\ \Gamma_{3i}\) are positive coefficients satisfying the relation \(4\Gamma_{1i}\Gamma_{3i}>\Gamma_{2i}^{2}\), \(0<\theta_{i}<1\ (i=1,\ 2,\ \dots,\ n)\) and \(e_{i}^{[\theta_{i}]}\) is as described in [47]  
\[e_{i}^{[\theta_{i}]}=|e_{i}|^{\theta_{i}}sign[e_{i}]. \tag{9}\]  
**Remark  
According to the SMC manner, once the state variable proceeds in sliding mode, the following constraints are imposed (refer to [16; 17; 18; 19; 20]):  
\[s_{i}=0\text{ and }\dot{\zeta}_{i}=0, \tag{10}\]  
\[\varsigma_{i}=0\text{ and }\dot{\zeta}_{i}=0. \tag{11}\]  
Combining Equation (10) constraints with Equation (7) yields  
\[\dot{\varsigma}_{i}=-h_{1i}sign[\varsigma_{i}]-h_{2i}\varsigma_{i}^{[\alpha_{ i}]}, \tag{12}\]  
and combining Equation (11) constraints with Equation (8) gives  
\[\dot{e}_{i}=-\Gamma_{1i}e_{i}^{[2-\theta_{i}]}-\Gamma_{2i}e_{i}-\Gamma_{3i}e_ {i}^{[\theta_{i}]}. \tag{13}\]  
It must be proved that once the second-order sliding motion takes place, i.e., \(s_{i}=0\), the first-order sliding motion takes place in finite-time, i.e., \(\varsigma_{i}=0\), and the state variable system of Equation (13) reaches zero in finite-time. The following theorems have been established for this proof. **Theorem 1**. : _Consider the dynamic system shown in Equation (12). The original point \(\varsigma_{i}=0\) is globally balanced in finite-time and the state variable of the system (10) converges to zero in finite-time \(T_{s_{i}}\leq\varsigma_{i}^{2}(0)/\sqrt{2}h_{1i}\)._  
**Proof. ** The positive-definite Lyapunov functional is investigated as  
\[V_{1}=\frac{\varsigma_{i}^{2}}{2}. \tag{14}\]  
With Equation (12), the time derivative of Equation (14) is computed as  
\[ \begin{array}{ll}\dot{V}_{1}&=\varsigma_{i}\Big{(}-h_{1i}sign[\varsigma_{i}] -h_{2i}\varsigma_{i}^{[\alpha_{i}]}\Big{)}\\ &=-h_{1i}|s_{i}|-h_{2i}\dot{s}_{i}^{[\alpha_{i}+1]}\\ &\leq-h_{1i}|s_{i}|\\ &=-\sqrt{2}h_{1i}V_{1}^{1/2}\end{array} . \tag{15}\]  
It can be seen that (15) has the form \(\dot{V}_{1}+\sqrt{2}h_{1i}V_{1}^{1/2}\leq 0\). Therefore, the defined finite-time is given by [48]:  
\[T_{s_{i}}\leq\frac{\varsigma_{i}^{2}(0)}{\sqrt{2}h_{1i}}. \tag{16}\]  
This completes the proof. \(\square\)  
**Theorem 2**. : _Consider the dynamic system (13). The original point \(e_{i}=0\) consists of globally balanced points in finite-time and the state variable of the system (13) as it converges to zero in finite-time \(T_{e_{i}}\leq T_{e_{i}}^{f}\). \(T_{e_{i}}^{f}\) is defined as_  
\[T_{e_{i}}^{f}=\frac{2}{(1-\theta_{i})}\left(\frac{\pi}{2}-\tan^{-1}\frac{ \Gamma_{2i}}{\sqrt{4\Gamma_{1i}\Gamma_{3i}-\Gamma_{2i}^{2}}}\right)\frac{1}{ \sqrt{4\Gamma_{1i}\Gamma_{3i}-\Gamma_{2i}^{2}}}. \tag{17}\]  
**Proof. ** The Lyapunov function candidate is investigated as  
\[V_{2}=e_{i}^{\ 2}. \tag{18}\]With Equation (13), the time derivative of Equation (18) is calculated as  
\[ \begin{array}{ll}\dot{V}_{2}&=2e_{i}\dot{e}_{i}\\ &=2e_{i}\Bigl{(}-\Gamma_{1i}e_{i}[^{2-\phi_{i}}]-\Gamma_{2i}e_{i}-\Gamma_{3i}e_{ i}[^{\phi_{i}}]\Bigr{)}\\ &=2\Bigl{(}-\Gamma_{1i}e_{i}[^{3-\phi_{i}}]-\Gamma_{2i}e_{i}^{2}-\Gamma_{3i}e_{ i}[^{1+\phi_{i}}]\Bigr{)}\\ &=2\Bigl{(}-\Gamma_{1i}V_{2}^{(3-\phi_{i})/2}-\Gamma_{2i}V_{2}-\Gamma_{3i}V_{2}^{( \phi_{i}+1)/2}\Bigr{)}\end{array} .