## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
Furthermore, the presence of \(\mathbf{K}_{\omega i}\) in (31) highlights its role in shaping the system's response through the auxiliary dynamics \(\mathbf{\omega}_{i}\), which provides an additional layer of control to enhance stability and performance. **Remark 3**. : _Given any arbitrary set of gains \(\mathbf{K}_{\rm pi}\), \(\mathbf{K}_{\rm vi}\), and \(\mathbf{K}_{\rm ii}\), it is not always feasible to determine an equivalent set of gains \(\mathbf{\Lambda}_{i}\) and \(\mathbf{K}_{\omega i}\) that satisfy (34)-(36). Therefore, it is necessary to select \(\mathbf{\Lambda}_{i}\) and \(\mathbf{K}_{\omega i}\) first, ensuring that the corresponding values of \(\mathbf{K}_{\rm pi}\), \(\mathbf{K}_{\rm vi}\), and \(\mathbf{K}_{\rm ii}\) can be derived accordingly for implementation._  
The previous analysis shows that the system described by (1), when in closed loop with (31)-(33), achieves equilibrium at \(\mathbf{e}_{i}=\mathbf{s}_{i}=\mathbf{0}\) and \(\mathbf{\omega}_{i}=-\mathbf{K}_{\omega i}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm d})\). This is equivalent to \(\dot{\mathbf{q}}_{i}=\mathbf{\dot{e}}_{i}=\mathbf{0}\), meaning that the tracking error vanishes and the desired position \(\mathbf{q}_{\rm d}\) is maintained. Additionally, the integral term \(\mathbf{\omega}_{i}\) compensates precisely for the gravitational torque \(\mathbf{g}_{i}(\mathbf{q}_{\rm d})\), as indicated in (28). This equilibrium can be verified by substituting (31) into (1), which yields:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]6where:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]7  
The term \(\mathbf{\phi}_{ai}\) encapsulates additional dynamic effects arising from delayed communication and system interaction. Specifically, it accounts for the dynamic coupling between the manipulators, incorporating contributions from the inertia matrix \(\mathbf{H}_{i}(\mathbf{q}_{i})\), Coriolis and centrifugal forces \(\mathbf{C}_{i}(\mathbf{q}_{i},\hat{\mathbf{e}}_{i})\), and delayed error dynamics \(\bar{\mathbf{e}}_{j}\). The factors \(\mathbf{B}_{i}\) and \(\mathbf{\Lambda}_{i}\) introduce proportional and scaling adjustments, ensuring alignment with the controller design. Furthermore, the assumption \(\hat{\mathbf{q}}_{\rm d}=\bar{\mathbf{q}}_{\rm d}=\mathbf{0}\) simplifies the formulation by eliminating variations in the desired trajectory. Together with (32)-(33), these equations describe the closed-loop system dynamics. This formulation integrates the auxiliary dynamics of \(\mathbf{\omega}_{i}\), which track the evolution of \(\mathbf{s}_{i}\), ensuring the system achieves stability and tracks the desired trajectory under the effects of communication delays and gravitational torques. Define now the state vector as:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]8  
where \(\mathbf{\omega}_{1}\) and \(\mathbf{\omega}_{r}\) are auxiliary variables introduced to track the evolution of the composite error dynamics \(\mathbf{s}_{1}\) and \(\mathbf{s}_{r}\), respectively. The variables \(\mathbf{e}_{1}\) and \(\mathbf{e}_{r}\) represent the position tracking errors for the local and remote manipulators. With this definition, the closed-loop dynamics of the system can be expressed as:  
\[\hat{\mathbf{y}}=\left[ \begin{array}{c}\mathbf{s}_{1}\\ -\mathbf{\Lambda}_{1}\mathbf{B}_{1}\mathbf{e}_{1}+\mathbf{s}_{1}+\mathbf{\Lambda}_{1}\bar{\mathbf{e}}_ {r}\\ -\mathbf{H}_{1}^{-}\{\mathbf{C}_{1}\mathbf{s}_{1}+\mathbf{K}_{\omega i}\mathbf{s}_{1}-\mathbf{\phi}_{ \rm d}+\mathbf{K}_{\omega i}\mathbf{\omega}_{1}+\mathbf{g}_{1}\}\\ \mathbf{s}_{r}\\ -\mathbf{\Lambda}_{r}\mathbf{B}_{r}\mathbf{e}_{r}+\mathbf{s}_{r}+\mathbf{\Lambda}_{r}\bar{\mathbf{e}}_ {1}\\ -\mathbf{H}_{r}^{-1}\{\mathbf{C}_{r}\mathbf{s}_{r}+\mathbf{K}_{\omega r}\mathbf{s}_{r}-\mathbf{\phi}_{ \rm ar}+\mathbf{K}_{\omega r}\mathbf{\omega}_{r}+\mathbf{g}_{r}\}\end{array} \right].