## 5 Stability Analysis
\[\mathbf{\Omega}=\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q }}+\mathbf{G}(\mathbf{q})+\mathbf{F}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)} +\mathbf{\tau}_{d}-\big{(}\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)} \dot{\mathbf{q}}\big{)}_{t-L}-\big{(}\mathbf{G}(\mathbf{q})\big{)}_{t-L}- \big{(}\mathbf{F}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\big{)}_{t-L}-(\mathbf{ \tau}_{d})_{t-L} \tag{36}\]  
The friction term \(\mathbf{F}\) in (36) can be divided as \(\mathbf{F}=\mathbf{F}_{V}+\mathbf{F}_{C}\), where \(\mathbf{F}_{V}\) denotes that the viscous friction is continuous, and \(\mathbf{F}_{C}\) denotes that the Coulomb friction is bounded and discontinuous at velocity reversal [25]. Next, we divide \(\mathbf{\Omega}\) into continuous term and discontinuous term to obtain the following:  
\[\{\mathbf{K}_{\text{ss}}\}_{ii}>|\epsilon_{i}| \tag{28}\]0where the following is the case. \[ \begin{array}{l}\Omega_{\text{con}}\triangleq\mathbb{C}(\mathbf{q}\dot{\mathbf{a }})\dot{\mathbf{q}}+\mathbb{G}(\mathbf{q})+\mathbf{F}_{V}+\mathbf{\tau}_{d}-( \mathbb{C}(\mathbf{q}\dot{\mathbf{a}})\dot{\mathbf{a}})_{t-L}-(\mathbb{G}( \mathbf{q}))_{t-L}-(\mathbb{F}(\mathbf{q}\dot{\mathbf{a}}))_{t-L}-(\mathbb{ \tau}_{d})_{t-L}\\ \Omega_{\text{discon}}\triangleq\mathbb{F}_{\mathbb{C}}-(\mathbb{F}_{\mathbb{C }})_{t-L}\end{array} \right\} \tag{38}\]  
If \(\mathbb{C}\big{(}\mathbf{q}\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}+\mathbb{G}( \mathbf{q})+\mathbb{F}_{V}+\mathbf{\tau}_{d}\) is continuous and bounded, then \(\mathbf{\Omega}_{\text{con}}=\mathbf{O}\big{(}L^{2}\big{)}\), where \(\mathbf{O}\) is used to describe the error term in an approximation to a mathematical function [30]. In addition, the discontinuous term \(\mathbf{\Omega}_{\text{discon}}\) is described as follows. \[\mathbf{\Omega}_{\text{discon}}\leq\left\{ \begin{array}{l}\mathbf{b},\text{ at velocity reversal}\\ 0,\text{ otherwise.}\end{array} \right. \tag{39}\]  
Thus, \(\mathbf{\Omega}\) is bounded by the following:  
\[\mathbf{\Omega}\leq\mathbf{b}+\mathbf{O}\Big{(}L^{2}\Big{)} \tag{40}\]  
for a sufficient small \(L\), where \(\mathbf{b}\) is a constant vector. The approximation error can be made small by reducing sampling time \(L\). From (30), the delayed nonlinear term is given by the following. \[\tilde{\mathbf{q}}_{t-L}=\mathbf{u}_{t-L}-\varepsilon_{t-L} \tag{41}\]  
Substituting (41) to (35) yields the following  
\[ \begin{array}{rl}\mathbf{M}\varepsilon&=\big{[}(\mathbf{M}-\overline{ \mathbf{M}})\mathbf{u}-(\mathbf{M}-\overline{\mathbf{M}})\tilde{\mathbf{q}}_ {t-L}+(\mathbf{M}-\mathbf{M}_{t-L})\tilde{\mathbf{q}}_{t-L}+\mathbf{\Omega} \big{]}\\ &=\big{(}\mathbf{M}-\overline{\mathbf{M}}\big{)}\varepsilon_{t-L}+\big{[}( \mathbf{M}-\overline{\mathbf{M}})(\mathbf{u}-\mathbf{u}_{t-L})+(\mathbf{M}- \mathbf{M}_{t-L})\tilde{\mathbf{q}}_{t-L}+\mathbf{\Omega}\big{]}\end{array}  \tag{42}\]  
Therefore, from (42), \(\varepsilon\) can be determined as follows:  
\[\varepsilon=\mathbb{E}\varepsilon_{t-L}+\mathbb{E}\mathbf{\mu}_{1}+\mathbf{ \mu}_{2} \tag{43}\]  
where the following is the case. \[\mathbb{E}=\mathbf{I}-\mathbf{M}^{-1}\overline{\mathbf{M}},\text{ }\mathbf{\mu}_{1}=\mathbf{u}-\mathbf{u}_{t-L},\text{ }\mathbf{\mu}_{2}=\mathbf{M}^{-1}\big{[}(\mathbf{M}-\mathbf{M}_{t-L})\tilde{ \mathbf{q}}_{t-L}+\mathbf{\Omega}\big{]} \tag{44}\]  
For a sufficiently small-time delay \(L\), \(\mathbf{\mu}_{1}\) and \(\mathbf{\mu}_{2}\) are bounded. There is a conformal mapping on the complex plane from continuous-time to discrete-time [31]. In the discrete time domain, (43) is represented as follows. \[\varepsilon(k)=\mathbb{E}(k)\varepsilon(k-1)+\mathbb{E}(k)\mathbf{\mu}_{1}(k)+ \mathbf{\mu}_{2}(k) \tag{45}\]  
We assume \(\|\mathbb{E}\|<1\) by properly selecting \(\overline{\mathbf{M}}\)[31]. Thus, the eigenvalues of \(\mathbb{E}(k)\) reside inside a unit circle [32]. As a result, (45) is asymptotically bounded with bounded function \(\mathbf{\mu}_{1}\) and \(\mathbf{\mu}_{2}\). Therefore, \(\|\mathbb{E}\|<1\) implies the boundness of the \(\varepsilon\) in (43).