## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
\tag{40}\]  
In (40), the terms \(\mathbf{\Lambda}_{i}\mathbf{B}_{i}\mathbf{e}_{i}\) and \(\mathbf{\Lambda}_{i}\bar{\mathbf{e}}_{j}\) capture the proportional and delayed error corrections, while \(\mathbf{\phi}_{ai}\) encapsulates dynamic effects such as inertia, Coriolis forces, and delayed errors. The arguments for \(\mathbf{H}_{i}\), \(\mathbf{C}_{i}\), and \(\mathbf{g}_{i}\) have been omitted for simplicity. The equilibrium point of (40) is clearly given by \(\mathbf{e}_{i}=\mathbf{s}_{i}=\mathbf{0}\) and \(\mathbf{\omega}_{i}=-\mathbf{K}_{\omega i}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\rm d})\), as previously determined. This equilibrium reflects the condition where tracking errors vanish, and the gravitational torques are fully compensated by the integral terms. Consider the following change of coordinates:  
\[\mathbf{x}_{1} =\mathbf{\omega}_{1}+\mathbf{K}_{\omega i}^{-1}\mathbf{g}_{1}(\mathbf{q}_{\rm d}). \tag{41}\] \[\mathbf{x}_{4} =\mathbf{\omega}_{r}+\mathbf{K}_{\omega r}^{-1}\mathbf{g}_{r}(\mathbf{q}_{\rm d}). \tag{42}\]  
These coordinate transformations center the auxiliary variables \(\mathbf{\omega}_{1}\) and \(\mathbf{\omega}_{r}\) around their equilibrium values, simplifying the analysis by ensuring the equilibrium corresponds to \(\mathbf{x}_{1}=\mathbf{x}_{4}=\mathbf{0}\). Under this change of coordinates, the system becomes:  
\[\hat{\mathbf{x}}=\left[\begin{array}{c}-\mathbf{\Lambda}_{1}\mathbf{g}_{1}\mathbf{e}_{1}+\bm {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{x}_{1}+\mathbf{g}_{1}(\mathbf{q}_{1})-\mathbf{g}_{1}(\mathbf{q}_{ \rm d})\}\\ \mathbf{s}_{r}\\ -\where  
\[\mathbf{x} = \big{[} \begin{array}{cccccc}\mathbf{x}_{1}^{\rm T}&\mathbf{x}_{2}^{\rm T}& \mathbf{x}_{3}^{\rm T}&\mathbf{x}_{4}^{\rm T}&\mathbf{x}_{5}^{\rm T}&\mathbf{x}_{6}^{\rm T}\end{array}  \big{]}^{\rm T} \tag{44}\] \[\equiv \big{[} \begin{array}{cccccc}\mathbf{x}_{1}^{\rm T}&\mathbf{e}_{1}^{\rm T }&\mathbf{s}_{1}^{\rm T}&\mathbf{x}_{4}^{\rm T}&\mathbf{e}_{\rm r}^{\rm T}&\mathbf{s}_{\rm r}^{ \rm T}\end{array} \big{]}^{\rm T}.\]  
In the equivalent system defined in (43), the equilibrium point simplifies to \(\mathbf{x}=\mathbf{0}\), confirming its uniqueness. This reduction facilitates the stability analysis by consolidating all dynamic terms into a unified representation. The region of interest for stability analysis can be defined as:  
\[\mathcal{D} = \Big{\{}\mathbf{x}\in\mathbb{R}^{6n}\backslash\|\mathbf{x}\|\leq\epsilon \Big{\}}, \tag{45}\]  
where \(\mathcal{D}\) represents a bounded region around the equilibrium within which stability can be studied. The parameter \(\epsilon>0\) defines the radius of this region, reflecting the inherent local asymptotic stability properties of PID control. To facilitate the stability analysis and account for the dynamic interactions within the closed-loop system, the matrix \(\mathbf{K}_{vi}\) is defined as the sum of two diagonal positive-definite matrices:  
\[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]  
where \(\mathbf{K}_{vi1},\mathbf{K}_{vi2}\in\mathbb{R}^{n\times n}\). This decomposition provides additional flexibility in the tuning of the control gains, allowing \(\mathbf{K}_{vi1}\) and \(\mathbf{K}_{vi2}\) to be adjusted independently to address distinct aspects of the system's behavior. For instance, \(\mathbf{K}_{vi1}\) may be selected to enhance damping characteristics, while \(\mathbf{K}_{vi2}\) can be tailored to counteract specific dynamic effects or improve robustness under varying conditions. Moreover, this formulation aligns with the composite structure of the system, as it allows for separate consideration of contributions from intrinsic manipulator dynamics and external influences such as time delays or human interaction. As a result, the gain matrix \(\mathbf{K}_{vi}\) can be systematically tuned to ensure stability and performance across a broad range of operating scenarios. **Assumption 4**.