## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
Note that the equilibrium point of (1) in closed loop with (23) and (24) is characterized by:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]5 \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]6 (27) \[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]7At equilibrium, the integral term \(\mathbf{\xi}_{i}\) compensates exactly for the gravitational torque \(\mathbf{g}_{i}(\mathbf{q}_{\rm d})\), ensuring that the system remains stationary at the desired position \(\mathbf{q}_{\rm d}\). For the stability analysis, a similar approach to that described in [39] can be employed, which requires defining:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]8  
where \(\mathbf{s}_{i}\) represents a composite variable combining the tracking error dynamics \(\dot{\mathbf{e}}_{i}\), the delayed error term \(\mathbf{\varepsilon}_{j}\), and the proportional error term \(\mathbf{e}_{i}\) scaled by \(\mathbf{B}_{i}\), with:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{\ddot{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i} )\mathbf{\dot{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=-\mathbf{K}_{\rm pi}\mathbf{\dot{e}}_{i}-\mathbf{ K}_{\rm vi}\mathbf{\dot{q}}_{i}-\mathbf{K}_{\rm i}\mathbf{\xi}_{\rm ui}. \tag{14}\]9  
This formulation consolates the key dynamic components into a single variable, simplifying the stability analysis. This selection renders (23) and (24) equivalent to the following system dynamics:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]0 \[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]1 (32) \[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]2  
where \(\mathbf{\omega}_{i}\) serves as an auxiliary variable introduced to track the evolution of \(\mathbf{s}_{i}\). The dynamics of \(\mathbf{\omega}_{i}\), as defined by (32) and (33), facilitate the design and analysis of the system's stability properties. Specifically, \(\dot{\mathbf{\omega}}_{i}=\mathbf{s}_{i}\) establishes a direct relationship between the auxiliary variable and the composite tracking error \(\mathbf{s}_{i}\), while \(\mathbf{\omega}_{i}(t_{0})=\mathbf{0}\) ensures a consistent initial condition for the analysis. The equivalence described above holds provided that the following parameter relationships are satisfied:  
\[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]3 \[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]4 (35) \[\mathbf{\xi}_{\rm ui}=-\mathbf{K}_{\rm ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm ci}), \tag{15}\]5  
These parameter relationships, defined in (34)-(36), ensure a consistent interaction between the controller gains and the system dynamics. By linking the integral gain \(\mathbf{K}_{\rm ii}\), the proportional gain \(\mathbf{K}_{\rm pi}\), and the dynamic contributions introduced by \(\mathbf{\Lambda}_{i}\) and \(\mathbf{A}_{i}\), these relationships guarantee that the closed-loop system achieves stability under the specified conditions.