## 3 Finite-Time ANFTSMC Design
### Proof of Stability
\begin{array}{ll}\dot{V}_{A}&=s_{1}\dot{s}_{1}+s_{3}\dot{s}_{3}+s_{5}\dot{ s}_{5}-\sum\limits_{i=0}^{2}\frac{\gamma_{i1}\beta_{1}b_{2}}{\mu_{i1}}\tilde{a}_{i1} \dot{a}_{i1}-\sum\limits_{i=0}^{2}\frac{\gamma_{i2}\beta_{3}b_{4}}{\mu_{3}} \tilde{a}_{i3}\dot{a}_{i3}-\sum\limits_{i=0}^{2}\frac{\gamma_{i2}\beta_{5}b_{6 }}{\mu_{3}}\tilde{a}_{i5}\dot{a}_{i5}\\ &=s_{1}\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}\big{(}-c_{1}s_{1}-(\dot{a}_{01}+ \tilde{a}_{11}|e_{1}+\tilde{a}_{21}|e_{2}|+\delta_{1})sign(s_{1})+d_{\phi}\big{)} \\ &+s_{3}\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}(-c_{3}s_{3}-(\dot{a}_{03}+\tilde{ a}_{13}|e_{3}+\tilde{a}_{23}|e_{4}+\delta_{3})sign(s_{3})+d_{\theta})\\ &+s_{5}\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}(-c_{5}s_{5}-(\dot{a}_{05}+\tilde{ a}_{15}|e_{5}+\tilde{a}_{25}|e_{6}+\delta_{5})sign(s_{5})+d_{\psi}\big{)} \\ &-\sum\limits_{i=0}^{2}\frac{\gamma_{i1}\beta_{1}b_{2}}{\mu_{i1}}\tilde{a}_{i1 }-\sum\limits_{i=0}^{2}\frac{\gamma_{i2}\beta_{3}b_{4}}{\mu_{3}}\tilde{a}_{i3 }\dot{a}_{i3}-\sum\limits_{i=0}^{2}\frac{\gamma_{i2}\beta_{5}b_{6}}{\mu_{3}} \tilde{a}_{i5}\dot{a}_{i5}\\ &\leq\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}(-c_{1}s_{1}^{2}-\delta_{1}|s_{1}|)+ \beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}\big{(}-c_{3}s_{3}^{2}-\delta_{3}|s_{3}| \big{)}+\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}(-c_{5}s_{5}^{2}-\delta_{5}|s_{5}| )\\ &+\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}|s_{1}|(\tilde{a}_{01}+\tilde{a}_{11}|e_{1} +\tilde{a}_{21}|e_{2}|+\delta_{1})+\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}|s_{3}| (\tilde{a}_{03}+\tilde{a}_{13}|e_{3}|+\tilde{a}_{23}|e_{4}|+\delta_{3})\\ &+\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}|s_{5}|(a_{05}+\tilde{a}_{15}|e_{5}+ \tilde{a}_{25}|e_{6}+\delta_{5})-\sum\limits_{i=0}^{2}\frac{\gamma_{i1}\beta_{1 }b_{2}}{\mu_{i1}}\tilde{a}_{i1}\dot{a}_{i1}-\sum\limits_{i=0}^{2}\frac{\gamma_{ i2}\beta_{1}b_{4}}{\mu_{3}}\tilde{a}_{i3}\dot{a}_{i3}-\sum\limits_{i=0}^{2} \frac{\gamma_{i2}\beta_{5}b_{6}}{\mu_{3}}\tilde{a}_{i5}\dot{a}_{i5}\end{array}  \tag{36}\]  
Considering the adaptive law (25), it can be obtained as follows:  
\[\begin{array}{ll}\dot{V}_{A}&\leq\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}\big{(}-c_{ 1}s_{1}^{2}-\delta_{1}|s_{1}|\big{)}+\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}\big{(}-c_ {3}s_{2}^{2}-\delta_{3}|s_{3}|\big{)}+\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}\big{(}- c_{5}s_{5}^{2}-\delta_{5}|s_{5}|\big{)}\\ &+\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}|s_{1}|(\tilde{a}_{01}+\tilde{a}_{11}|e_{1} |+\tilde{a}_{21}|e_{2}|\big{)}+\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}|s_{3}|( \tilde{a}_{03}+\tilde{a}_{13}|e_{3}|+\tilde{a}_{23}|e_{4}|\big{)}\\ &+\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}|s_{5}|(\tilde{a}_{05}+\tilde{a}_{15}|e_{5} |+\tilde{a}_{25}|e_{6}|\big{)}-\gamma_{i1}\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}|s_{ 1}|(\tilde{a}_{01}+\tilde{a}_{11}|e_{1}|+\tilde{a}_{21}|e_{2}|)\\ &-\gamma_{i3}\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}|s_{3}|(\tilde{a}_{03}+\tilde{ a}_{13}|e_{3}|+\tilde{a}_{23}|e_{4}|)-\gamma_{5}\beta_{5}b_{6}|e_{6}|^{\beta_{5}-1}|s_{ 5}|(\tilde{a}_{05}+\tilde{a}_{13}|e_{5}|+\tilde{a}_{25}|e_{6}|)\\ &\leq-\delta_{1}\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}|s_{1}|-\delta_{3}\beta_{3}b_{4 }|e_{4}|^{\beta_{3}-1}|s_{3}|-\delta_{5}b_{5}b_{6}|e_{6}|^{\beta_{5}-1}|s_{5}| -(\gamma_{i1}-\beta_{1})\beta_{1}b_{2}|e_{2}|^{\beta_{1}-1}|s_{1}|(\tilde{a}_{0 1}+\tilde{a}_{11}|e_{1}|+\tilde{a}_{21}|e_{2}|)\\ &-(\gamma_{i3}-1)\beta_{3}b_{4}|e_{4}|^{\beta_{3}-1}|s_{3}|(\tilde{a}_{03}+ \tilde{a}_{13}|e_{3}+\tilde{a}_{23}|e_{4}|)-(\gamma_{5}-1)\beta_{5}b_{6}|e_{6}| ^{\beta_{5}-1}|s_{5}|(\tilde{a}_\[