## 5 Stability Analysis  
This section presents the stability analysis of the control scheme using the Lyapunov Theorem. Substituting \(\mathbf{\tau}\) in (6) by (21), using (9) and (14) and solving for \(\ddot{\mathbf{e}}_{c}\), after rearranging some terms, we obtain the following:  
\[\ddot{\mathbf{e}}_{c}=-\bigg{\{}\frac{q_{2}}{p_{2}}[\mathbf{K}_{2}]^{-1}\Big{[} \big{|}\dot{\mathbf{e}}_{c}\big{|}^{p_{2}/q_{2}-1}\Big{]}^{-1}\bigg{(}1+\frac{ p_{1}}{q_{1}}\mathbf{K}_  
In other to achieve the asymptotic stability of \(\dot{\mathbf{s}}\) about the equilibrium point \(\mathbf{s}=0\), the following conditions must be satisfied [28]:  
\[ \begin{array}{ll}\text{(a)}&\dot{V}<0\text{ for }\mathbf{s}\
eq 0;\\ \text{(b)}&\lim\limits_{|\mathbf{s}|\rightarrow\infty}V=\infty.\end{array} \]  
Condition (b) is obviously satisfied by \(V\). In (26), since \(p_{2}\) and \(q_{2}\) are positive integers and \(1<p_{2}/q_{2}<2\), there is \(\dot{\mathbf{e}}_{\epsilon_{i}}^{\left|p_{2}/q_{2}-1\right.>0}\) for \(\dot{\mathbf{e}}_{\epsilon_{i}}\
eq 0\)[29]. Thus, (26) can be presented as follows. \[\dot{V}=\mathbf{s}^{\text{T}}\bigg{\{}\frac{p_{2}}{q_{2}}(\mathbf{I}+\mathbf{ \alpha}\mathbf{C})\mathbf{K}_{2}\big{|}\dot{\mathbf{e}}_{\epsilon}\big{|}^{p_ {1}/q_{1}-1}[-\mathbf{K}\mathbf{s}-\mathbf{K}_{\text{ss}}\text{sign}(\mathbf{ s})+\epsilon]\bigg{\}}\leq\mathbf{s}^{\text{T}}[-\mathbf{K}\mathbf{s}- \mathbf{K}_{\text{ss}}\text{sign}(\mathbf{s})+\epsilon] \tag{27}\]  
The derivative of the candidate Lyapunov function (27) is negative definite if the following is the case:  
\[\{\mathbf{K}_{\text{ss}}\}_{ii}>|\epsilon_{i}| \tag{28}\]  
where \(\bullet_{ii}\) denotes \(i\)th diagonal element of \(\bullet\). Thus, if \(\epsilon\) is bounded, the stability condition (28) ensures that the time derivative of the candidate Lyapunov function is negative and the cross-coupling error is bounded. Using (19), (6) becomes the following. \[\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})-\bar{\mathbf{H}}( \mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})=\overline{\mathbf{M}}\big{(} \mathbf{u}-\ddot{\mathbf{q}}\big{)} \tag{29}\]  
Applying (29), the TDE error in (23) is given as follows. \[\epsilon=\mathbf{u}-\ddot{\mathbf{q}} \tag{30}\]  
From (5), the acceleration \(\ddot{\mathbf{q}}\) can be determined as follows. \[\ddot{\mathbf{q}}=\mathbf{M}^{-1}(\mathbf{q})\big{[}\mathbf{\tau}-\mathbf{C} \big{(}\mathbf{q},\ddot{\mathbf{q}}\big{)}\dot{\mathbf{q}}-\mathbf{G}( \mathbf{q})-\mathbf{F}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}-\mathbf{\tau}_{d} \big{]} \tag{31}\]  
Substituting (31) into (30) yields the following. \[\mathbf{M}\epsilon=\mathbf{M}\mathbf{u}+\big{(}\mathbf{C}\big{(}\mathbf{q}, \dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})+\mathbf{F} \big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}+\mathbf{\tau}_{d}-\mathbf{\tau}\big{)} \tag{32}\]  
Then, using (16) and (19), (32) becomes the following. \[\mathbf{M}\epsilon=\mathbf{M}\mathbf{u}+\big{(}\mathbf{C}\big{(}\mathbf{q}, \dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})+\mathbf{\tau}_{d} \dot{\mathbf{q}}\big{)}+\mathbf{\tau}_{d}-\overline{\mathbf{M}}\mathbf{u}- \mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})_{t-L}\big{)} \tag{33}\]  
From (7), the delayed nonlinear term can be derived as follows. \[\mathbf{H}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})_{t-L}=\big{(} \mathbf{M}_{t-L}-\overline{\mathbf{M}}\big{)}\ddot{\mathbf{q}}_{t-L}+\big{(} \mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}\big{)}_{t -L}+\big{(}\mathbf{G}(\mathbf{q})\big{)}_{t-L}+\big{(}\mathbf{F}\big{(} \mathbf{q},\dot{\mathbf{q}}\big{)}\big{)}_{t-L}+(\mathbf{\tau}_{d})_{t-L} \tag{34}\]  
Substituting (34) into (33) provides the following:  
\[\mathbf{M}\epsilon=\big{(}\mathbf{M}-\overline{\mathbf{M}}\big{)}\mathbf{u}- \big{(}\mathbf{M}_{t-L}-\overline{\mathbf{M}}\big{)}\ddot{\mathbf{q}}_{t-L}+ \mathbf{\Omega} \tag{35}\]  
where the following is the case.