# Fast Fractional-Order Terminal Sliding Mode Control for Seven-Axis Robot Manipulator
## Appendix B  
The derivation of (27) and (28) are shown in this section. Using (25) and (26), we obtain  
\[-\mathbf{k}_{1}\mathbf{s}-\mathbf{k}_{2}sig(\mathbf{s})^{\mathbf{r}_{2}}-D_{0,t}^{\beta+\alpha-2}(\mathbf{k}_{3}sgn(\mathbf{s})+\overline{\mathbf{\Psi}})= D_{0,t}^{\alpha+\beta-2}\ddot{\mathbf{e}}+{}_{c}D_{0,t}^{\beta}(\boldsymbol{ \lambda}_{1}sig(\mathbf{e})^{\mathbf{r}_{1}}) \tag{12}\]  
As \(\ddot{\mathbf{e}}=\ddot{\mathbf{x}}-\ddot{\mathbf{x}}_{d}=\eta\overline{ \mathbf{u}}+\mathbf{\Psi}-\ddot{\mathbf{x}}_{d}\), it yields  
\[-\mathbf{k}_{1}\mathbf{s}-\mathbf{k}_{2}sig(\mathbf{s})^{\mathbf{r}_{2}}-D_{ 0,t}^{\beta+\alpha-2}(\mathbf{k}_{3}sgn(\mathbf{s})+\overline{\mathbf{\Psi}}) =D_{0,t}^{\alpha+\beta-2}(\eta\overline{\mathbf{u}}+\mathbf{\Psi})+{}_{c}D_{0,t}^{\beta}(\boldsymbol{\lambda}_{1}sig(\mathbf{e})^{\mathbf{r}_{1}}) \tag{13}\]As the semigroup property of the Riemann-Liouville derivative is limited, such as \(D_{0,t}^{2-\alpha-\beta}D_{0,t}^{\alpha+\beta-2}x(t)=x(t)\
eq D_{0,t}^{\alpha+ \beta-2}D_{0,t}^{2-\alpha-\beta}x(t)\)[8], taking the fractional derivative \(D_{0,t}^{2-\alpha-\beta}\) on both sides of (A7) yields  
\[D_{0,t}^{2-\alpha-\beta}(-\mathbf{k_{1}s}-\mathbf{k_{2}sig(s)^{r_{2}}})- \mathbf{k_{3}}sgn(\mathbf{s})-\widetilde{\mathbf{\Psi}}=\eta\overline{\mathbf{ u}}+\overline{\mathbf{\Psi}}+D_{0,t}^{2-\alpha-\beta}{}_{c}D_{0,t}^{\beta}( \boldsymbol{\lambda}_{1}sig(\mathbf{e})^{r_{1}})\] (A8)  
As \({}_{c}D_{0,t}^{\beta}(\boldsymbol{\lambda}_{1}sig(\mathbf{e})^{r_{1}})=D_{0,t }^{-\beta}\frac{d}{dt}(\boldsymbol{\lambda}_{1}sig(\mathbf{e})^{r_{1}})\) and \(\widetilde{\mathbf{\Psi}}=\overset{\boldsymbol{\alpha}}{\mathbf{\Psi}}- \overline{\mathbf{\Psi}}\), by substituting them for (A8), (27) is derived.  
However, due to the limited semigroup property of the Riemann-Liouville derivative, using the equivalent control input (27) for the real dynamics of the sliding surface (25) cannot obtain the desired dynamics (26). This can be found after substituting (27) for (25) as  
\[{}_{c}D_{0,t}^{\beta}\mathbf{s}=D_{0,t}^{\alpha+\beta-2}[D_{0,t}^{2-\beta- \alpha}(-\mathbf{k_{1}s}-\mathbf{k_{2}sig(s)^{r_{2}}})-D_{0,t}^{1}D_{0,t}^{- \alpha}D_{0,t}^{1}(\boldsymbol{\lambda}_{1}sig(\mathbf{e})^{r_{1}})- \widetilde{\mathbf{\Psi}}-\mathbf{k_{3}}sgn(\mathbf{s})]+{}_{c}D_{0,t}^{\beta }(\boldsymbol{\lambda}_{1}sig(\mathbf{e})^{r_{1}})\] (A9)  
The term \(D_{0,t}^{\alpha+\beta-2}D_{0,t}^{2-\beta-\alpha}(-\mathbf{k_{1}s}-\mathbf{k_{ 2}sig(s)^{r_{2}}})\) can be rewritten as follows, using Property 1.  
\[D_{0,t}^{\alpha+\beta-2}D_{0,t}^{2-\beta-\alpha}(-\mathbf{k_{1}s}-\mathbf{k_{ 2}sig(s)^{r_{2}}})=-\mathbf{k_{1}s}-\mathbf{k_{2}sig(s)^{r_{2}}}+D_{0,t}^{1- \beta-\alpha}[\mathbf{k_{1}s}+\mathbf{k_{2}sig(s)^{r_{2}}}]_{t=0}\frac{t^{1- \beta-\alpha}}{\Gamma(2-\beta-\alpha)}\] (A10)  
With the assumption that the initial error of velocity and acceleration is zero, i.e., \(\tilde{\mathbf{e}}(t=0)=\dot{\mathbf{e}}(t=0)=0\), the term \(D_{0,t}^{\alpha+\beta-2}[D_{0,t}^{1}D_{0,t}^{-\alpha}D_{0,t}^{1}(\boldsymbol{ \lambda}_{1}sig(\mathbf{e})^{r_{1}})]\) in (A9) is simplified as  
\[D_{0,t}^{\alpha+\beta-2}D_{0,t}^{1}D_{0,t}^{-\alpha}D_{0,t}^{1}(\boldsymbol{ \lambda}_{1}sig(\mathbf{e