## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points  
Once the human operator generates the desired constant value \(\mathbf{q}_{\rm d}\in\mathbb{R}^{n}\) for both the remote and local manipulators, the system is expected to achieve the following equilibrium:  
\[\lim_{t\to\infty}\mathbf{q}_{\rm r},\mathbf{q}_{1}=\mathbf{q}_{\rm d}, \tag{13}\]  
despite the presence of time-varying delays in the communication channel and the absence of explicit gravity compensation. To understand how (13) can be achieved using a standard PID controller, let us assume that the manipulators exchange position data without any predetermined final value \(\mathbf{q}_{\rm d}\). This assumption implies that the human operator is not interacting with either end-effector, i.e., \(\tau_{\rm h}=\mathbf{0}\) for \(i=1,{\rm r}\). Substituting (6) into (1), we obtain the following closed-loop dynamics:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]  
Together with (7) and (8), this describes the closed-loop dynamics of the system, whose equilibrium is characterized by \(\mathbf{\dot{q}}_{i}=\mathbf{\dot{e}}_{i}=\mathbf{0}\), and the integral term satisfies:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]  
for some constant vector \(\mathbf{q}_{\rm ci}\). At equilibrium (\(\mathbf{\dot{q}}_{i}=\mathbf{0},\mathbf{\ddot{q}}_{i}=\mathbf{0},\mathbf{\dot{e}}_{i}=\mathbf{0}\)), the integral term of the PID controller precisely compensates for the gravitational torque \(\mathbf{g}_{i}(\mathbf{q}_{\rm ci})\). Here, \(\mathbf{q}_{\rm ci}\) denotes the equilibrium position of manipulator \(i\). To determine \(\mathbf{q}_{\rm ci}\), consider that at equilibrium, from (5), we have:  
\[\mathbf{\dot{e}}_{i}=\mathbf{0}\quad\Rightarrow\quad\mathbf{q}_{i}(t)=\mathbf{q}_{j}\Big{(}t -T_{j}(t)\Big{)}=\mathbf{q}_{\rm ci}. \tag{16}\]  
From this relationship, the following conclusions can be drawn:  
1. \(\mathbf{q}_{\rm c1}=\mathbf{q}_{\rm cr}\equiv\mathbf{q}_{\rm c}\), and
2. \(\mathbf{q}_{\rm c}\) is _not_ unique. As a direct consequence, an infinite number of equilibrium points exist. The main challenge is to determine whether (13) holds when the human operator moves an end-effector to achieve the desired position \(\mathbf{q}_{\rm d}\). Since \(\mathbf{q}_{\rm ci}=\mathbf{q}_{\rm c}\) is identical for both manipulators, the error definition \(\mathbf{e}_{i}\), as presented in (12), can be applied with the  
Figure 1: Teleoperation system with time-varying delays composed by two manipulators and a human operator interacting with the local robot. specific desired value \(\mathbf{q}_{\rm c}=\mathbf{q}_{\rm d}\). By substituting this relationship into (5), the expression is rewritten as:  
\[\dot{\mathbf{e}}_{i}(t) = \mathbf{q}_{i}(t)-\mathbf{q}_{\rm d}-\mathbf{q}_{j}\Big{(}t-T_{j}(t)\Big{)}+\bm {q}_{\rm d}\] \[= \mathbf{e}_{i}(t)-\mathbf{e}_{j}\Big{(}t-T_{j}(t)\Big{)}.\]  
This equation redefines \(\dot{\mathbf{e}}_{i}(t)\) in terms of the human-induced error, as defined in (12), and the delayed error, described in (5). This reformulation enables an analysis of the interaction between the errors of both manipulators while explicitly accounting for communication delays. From (17), it follows that if \(\mathbf{e}_{i}\to\mathbf{0}\) and \(\mathbf{e}_{j}\to\mathbf{0}\) for \(i,j=1,\mathrm{r}\), then \(\dot{\mathbf{e}}_{i}\to\mathbf{0}\) as well.