## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
: _There exist constants \(\alpha_{j},\beta_{j}\geq 1\) for \(j=1,\mathrm{r}\), such that the following inequalities hold:_  
\[\left(\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta-T_{j}(\theta))\| ^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}\leq\alpha_{j}\left(\int\limits_{t _{0}}^{t}\|\mathbf{e}_{j}(\theta)\|^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}, \tag{47}\] \[\left(\int\limits_{t_{0}}^{t}\|\mathbf{\dot{e}}_{j}(\theta-T_{j}( \theta))\|^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}\leq\beta_{j}\left(\int \limits_{t_{0}}^{t}\|\mathbf{\dot{e}}_{j}(\theta)\|^{2}\,\mathrm{d}\theta\right)^ {\frac{1}{2}}, \tag{48}\]  
_for all \(t\geq t_{0}\)._  
**Remark 4**. : _Conditions (47) and (48) are relatively uncommon, warranting a thorough analysis of the scenarios in which they are satisfied. First, assume that \(\mathbf{x}(t)\equiv\mathbf{0}\) for all \(t<t_{0}\) (see (50)) and consider the state definition in (44)._  
_Under these assumptions, it follows that \(\mathbf{e}_{\rm t}(t)=\mathbf{e}_{\rm r}(t)=\mathbf{0}\) for \(t<t_{0}\). Additionally, since \(\mathbf{s}_{\rm l}(t)=\mathbf{s}_{\rm r}(t)=\mathbf{0}\) for \(t<t_{0}\), (29) implies that \(\mathbf{\dot{e}}_{\rm l}(t)=\mathbf{\dot{e}}_{\rm r}(t)=\mathbf{0}\) for \(t<t_{0}\) as well._  
_Given this, consider rewriting (47) in the equivalent form:_  
\[\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta-T_{j}(\theta))\|^{2}\,\mathrm{d} \theta\leq\alpha_{j}^{2}\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta)\|^{2}\, \mathrm{d}\theta, \tag{49}\]  
_where \(\alpha_{j}\geq 1\).__If \(T_{j}(t)\equiv T_{j}\) is constant, it is evident that (49) holds with \(\alpha_{j}=1\), and thus, (47) is satisfied as well. Intuitively, a delay of the form \(T_{j}(t)=T_{j}+f_{\mathrm{d}j}(t)\), where \(f_{\mathrm{d}j}(t)\) is a high-frequency signal of small magnitude, should also satisfy (47) with a similar value of \(\alpha_{j}\). Indeed, simulation results consistently indicate that (47) and (48) hold for finite values of \(\alpha_{j}\) and \(\beta_{j}\), often without requiring large values._  
_Therefore, we assert that (47) and (48) are not ad hoc conditions but instead are satisfied by a wide range of commonly encountered time delay scenarios. However, assuming that \(\alpha_{j}\) and \(\beta_{j}\) are known a priori is less realistic. Developing a methodology to calculate these parameters remains an open area for future research. In the experimental results presented, the values \(\alpha_{j}=\beta_{j}=1\) were simply assumed for \(j=1,\mathrm{r}\)._  
The central contribution of this study is formalized in the following theorem. Theorem 1: _Let us consider the PID control law defined by (6)-(8), combined with the assumed human dynamics given in (9)-(11). Assume, without loss of generality, that \(\mathbf{A}_{\mathrm{r}}=\mathbf{O}\) and \(\mathbf{K}_{\mathrm{ar}}=\mathbf{O}\), where \(\mathbf{O}\) denotes the null matrix of appropriate dimensions, and that the parameter relationships specified in (34)-(36) are satisfied._  
_Under these conditions, the variables \(\boldsymbol{\hat{e}}_{i}\), \(\boldsymbol{\hat{q}}_{i}\), \(\boldsymbol{e}_{i}\), and \(\boldsymbol{\xi}_{i}\) defined in (24)-(25) remain bounded for all \(t\geq t_{0}\), and the following convergence properties hold:_  
\[\boldsymbol{\hat{e}}_{i},\boldsymbol{\hat{q}}_{i},\boldsymbol{e}_ {i} \rightarrow\mathbf{0}, \tag{50}\] \[\boldsymbol{\xi}_{i} \rightarrow-\mathbf{K}_{\mathrm{ii}}^{-1}g_{i}(\boldsymbol{q}_{ \mathrm{d}}), \tag{51}\]  
_as \(t\rightarrow\infty\), provided that:_  
1. _Assumptions_ 2 _and_ 4 _are satisfied._
2.