## 4 Description of Subsystems
### The NFTSMC and TDE Subsystems
#### 4.2.2 The NFTSMC and TDE Systems Design  
First a nonsingular terminal sliding surface is defined as [20; 27]. \[\mathbf{s}=\mathbf{e}_{c}+\mathbf{K}_{1}\mathbf{e}_{c}^{p_{1}/q_{1}}+\mathbf{K }_{2}\dot{\mathbf{e}}_{c}^{p_{2}/q_{2}} \tag{15}\]  
where \(\mathbf{K}_{1}\) and \(\mathbf{K}_{2}\) are diagonal design matrices, \(1<p_{1}/q_{1}<2\), \(1<p_{2}/q_{2}<2\), and \(p_{1}\), \(p_{2}\), \(q_{1}\), and \(q_{2}\) are positive odd integers. As discussed above, the implementation of (8) requires the computation of \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\), which is in light of (7) highly computationally intensive. Thus, an estimate of \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\) is needed. We assume that \(L\) is the smallest obtainable time between which \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\) remains almost unchanged, such that an estimate of \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\), namely \(\mathbf{\dot{H}}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\) is equal to \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})_{l-L}\) which is \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\) evaluated at \((t-L)\). In other words, we obtain [22] the following:  
\[\mathbf{\dot{H}}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})=\mathbf{H}( \mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})_{l-L} \tag{16}\]  
where \(\ddot{\mathbf{q}}_{l-L}\) can be computed as follows. \[\ddot{\mathbf{q}}_{l-L}=\frac{\frac{\mathbf{q}_{l}-\mathbf{q}_{l-L}}{L}-\frac {\mathbf{q}_{l-L}-\mathbf{q}_{l-2l}}{L}}{l}=\frac{\left(\mathbf{q}_{l}-2 \mathbf{q}_{l-L}+\mathbf{q}_{l-2L}\right)}{l^{2}} \tag{17}\]From (6) and (16), we obtain the following. \[\hat{\mathbf{H}}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})=\mathbf{\tau}_{t-L}- \overline{\mathbf{M}}\ddot{\mathbf{q}}_{t-L} \tag{18}\]  
Consequently, (8) can be expressed as follows. \[\mathbf{\tau}=\hat{\mathbf{H}}(\mathbf{q},\dot{\mathbf{q}}\ddot{\mathbf{q}})+ \overline{\mathbf{M}}\mathbf{u} \tag{19}\]  
We proceed to develop a control \(\mathbf{u}\) as follows. \[\mathbf{u}=\ddot{\mathbf{q}}_{d}+\frac{q_{2}}{p_{2}}[\mathbf{K}_{2}(\mathbf{I} +\mathbf{\alpha}\mathbf{C})]^{-1}\Big{[}\big{|}\dot{\mathbf{e}}_{c}\big{|}^{p_{2}/q _{2}-1}\Big{]}^{-1}\bigg{(}1+\frac{p_{1}}{q_{1}}\mathbf{K}_{1}|\mathbf{e}_{c}| ^{p_{1}/q_{1}-1}\bigg{)}\dot{\mathbf{e}}_{c}+\mathbf{K}\mathbf{s}+\mathbf{K}_ {sw}\text{sign}(\mathbf{s}) \tag{20}\]  
where \(\mathbf{K}\) and \(\mathbf{K}_{sw}\) are diagonal design matrices, \(\text{sign}(\mathbf{s})=\left(\text{sign}(\mathbf{s}_{1}),\dots,\text{sign}( \mathbf{s}_{n})\right)^{T}\in\mathbb{R}^{n}\). Thus, replacing \(\mathbf{u}\) in (19) by (20) and using (18), we obtain the following. \[\mathbf{\tau}=\underbrace{\big{(}\mathbf{\tau}_{t-L}-\overline{\mathbf{M}}\ddot{ \mathbf{q}}_{t-L}\big{)}}_{\text{TDE}}+\underbrace{\overline{\mathbf{M}}\bigg{[} \ddot{\mathbf{q}}_{d}+\frac{q_{2}}{p_{2}}[\mathbf{K}_{2}(\mathbf{I}+\mathbf{ \alpha}\mathbf{C})]^{-1}\Big{[}\big{|}\dot{\mathbf{e}}_{c}\big{|}^{p_{2}/q_{2 }-1}\Big{]}^{-1}\bigg{(}1+\frac{p_{1}}{q_{1}}\mathbf{K}_{1}|\mathbf{e}_{c}|^{ p_{1}/q_{1}-1}\bigg{)}\dot{\mathbf{e}}_{c}+\mathbf{K}\mathbf{s}+\mathbf{K}_ {sw}\text{sign}(\mathbf{s})\bigg{]}}_{\text{NFTSM control}} \tag{21}\]  
As indicated above in Equation (21), CKCM input \(\mathbf{\tau}\) consists of two main components: a TDE-based input and an NFTSM input. The TDE-based component minimizes the impact of the unknown CKCM dynamics while the NFTSM component forces the cross-coupling errors \(\mathbf{e}_{c}\) and tracking error \(\mathbf{e}\) to converge to zero asymptotically.