## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
To facilitate the analysis, we define the delayed error as:  
\[\bar{\mathbf{e}}_{j}=\mathbf{e}_{j}\Big{(}t-T_{j}(t)\Big{)}, \tag{18}\]  
and note that the time derivative of \(\bar{\mathbf{e}}_{j}\) can be expressed as:  
\[\dot{\bar{\mathbf{e}}}_{j}=\Big{(}1-\hat{T}_{j}(t)\Big{)}\;\dot{\mathbf{e}}_{j}\Big{(}t -T_{j}(t)\Big{)}. \tag{19}\]  
This expression is obtained by applying the chain rule to the time-shifted error \(\bar{\mathbf{e}}_{j}=\mathbf{e}_{j}(t-T_{j}(t))\). Specifically, differentiating \(\bar{\mathbf{e}}_{j}\) with respect to time introduces the factor \(\Big{(}1-\hat{T}_{j}(t)\Big{)}\) due to the time-varying delay \(T_{j}(t)\). This term accounts for the effect of the delay's rate of change on the overall error dynamics. Since \(\mathbf{q}_{\rm d}\) is a constant vector, we can rewrite (6)-(8) as:  
\[\mathbf{u}_{i} =-\mathbf{K}_{\rm pi}\mathbf{e}_{i}-\mathbf{K}_{\rm vi}\dot{\mathbf{e}}_{i}-\mathbf{K} _{\rm ii}\bar{\mathbf{\xi}}_{\rm ui}+\mathbf{K}_{\rm pi}\bar{\mathbf{e}}_{j}, \tag{20}\] \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]0 (21) \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]1  
The term \(\dot{\mathbf{\xi}}_{\rm ui}\) accumulates the difference between the instantaneous error \(\mathbf{e}_{i}\) and the delayed error \(\bar{\mathbf{e}}_{j}\). Initially, \(\bar{\mathbf{\xi}}_{\rm ui}\) is set to zero at \(t_{0}\), ensuring that the integral term builds up over time based on the dynamics of the errors. We now include the assumed dynamics of the human operator given by (9)-(11), resulting in a combined control law:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]2 \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]3 (24) \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]4  
The total torque applied to the system, \(\mathbf{u}_{\tau i}\), encompasses the contributions from the PID controller, \(\mathbf{u}_{i}\), and the torque applied by the human operator, \(\tau_{\rm hi}\). The dynamics of \(\bar{\mathbf{\xi}}_{i}\) incorporate the integral terms associated with both the controller, \(\dot{\bar{\mathbf{\xi}}}_{\rm ui}\), and the human operator, \(\dot{\bar{\mathbf{\xi}}}_{\rm hi}\). This combined behavior captures the interaction between the system's internal feedback mechanisms and the external inputs provided by the operator.