## 3 Main Result  
### Problem Statement  
The standard PID controller has been employed for decades in the control of robot manipulators due to its ability to compensate for gravitational effects without the need to explicitly compute the gravitational vector. Its key properties are as follows:  
(1) It can achieve position regulation, but does not ensure exact trajectory tracking. (2) It guarantees only local stability. For detailed proofs of these properties, refer to [28; 30; 39]. The aforementioned characteristics do not suggest the possibility of obtaining a stronger result for bilateral systems. Therefore, the primary objective of this work is to provide a stability proof demonstrating that these properties hold for bilateral systems with time-varying delays. The key distinction is that the desired (constant) position is specified by a human operator on either of the robots. Consider the following error definition:  
\[\mathbf{\hat{e}}_{i}=\mathbf{q}_{i}(t)-\mathbf{q}_{j}\Big{(}t-T_{j}(t)\Big{)}, \tag{5}\]  
where \(i=1\) implies that \(j=r\), and vice versa. In this expression, \(\mathbf{q}_{j}\Big{(}t-T_{j}(t)\Big{)}\) denotes the delayed position of manipulator \(j\), taking into account a communication delay \(T_{j}(t)\). The PID control laws are defined as follows:  
\[\mathbf{u}_{i} =-K_{\mathrm{pi}}\mathbf{\hat{e}}_{i}-K_{\mathrm{vi}}\mathbf{\hat{q}}_{i}- K_{\mathrm{ii}}\mathbf{\xi}_{\mathrm{ui}}, \tag{6}\] \[\mathbf{\hat{\xi}}_{\mathrm{ui}} =\mathbf{\hat{e}}_{i},\] (7) \[\mathbf{\xi}_{\mathrm{ui}}(t_{0}) =\mathbf{0}, \tag{8}\]  
where \(K_{\mathrm{pi}},K_{\mathrm{vi}},K_{\mathrm{ii}}\in\mathbb{R}^{n\times n}\) are positive-definite diagonal matrices. The integral term \(\mathbf{\xi}_{\mathrm{ui}}\), initialized at zero, accumulates based on the relative error \(\mathbf{\hat{e}}_{i}\). Note that a desired velocity of zero is assumed, simplifying the control law. However, the error definition (5) does not account for the human operator's intention to move the robots in a specific direction. Therefore, it is assumed that the torque applied by the human operator satisfies the following condition [16]. **Assumption 3**. : _The torque exerted by the human operator can be modeled as a proportional-integral (PI) controller in the position space, i.e.,_  
\[\mathbf{\tau}_{\mathrm{hi}} =-K_{\mathrm{ai}}\mathbf{e}_{i}-K_{\mathrm{ii}}\mathbf{\xi}_{\mathrm{hi}}, \tag{9}\] \[\mathbf{\hat{\xi}}_{\mathrm{hi}} =A_{i}\mathbf{e}_{i},\] (10) \[\mathbf{\xi}_{\mathrm{hi}}(t_{0}) =\mathbf{0}, \tag{11}\]  
_where_  
\[\mathbf{e}_{i}=\mathbf{q}_{i}-\mathbf{q}_{\mathrm{d}}, \tag{12}\]  
_with \(\mathbf{q}_{\mathrm{d}}\in\mathbb{R}^{n}\) denoting the desired constant position specified by the operator for the \(i\)-th end-effector. In this context, \(\mathbf{K}_{\mathrm{ai}},\mathbf{A}_{i}\in\mathbb{R}^{n\times n}\) are positive-definite diagonal matrices for \(i=1\), while \(\mathbf{K}_{\mathrm{ai}}=\mathbf{A}_{i}=\mathbf{O}\) for \(i=r\), or vice versa._  
**Remark 2**. : _Assumption 3 can be viewed as a specific instance of the widely accepted assumption of human passivity [5], given that a PI control law inherently represents a passive system. It is worth noting that this assumption is not novel, as it has also been employed in [16], for example. For a more comprehensive discussion, the interested reader is referred to [16]. Some authors consider human dynamics to be represented by a PID controller [40]; however, since the operator is never intended to control the local (or remote) end-effector independently but always in conjunction with the PID controller, providing a more elaborate description of their behavior becomes unnecessary. Finally, it is important to highlight that the same integral gain \(\mathbf{K}_{\mathrm{ii}}\) is assumed for both the human operator and the PID control law. Naturally, this is impractical, and to avoid an unrealistic condition, the gain \(\mathbf{A}_{i}\) has been introduced in (10) and (11), making the effective integral gain for the human equal to \(\mathbf{K}_{\mathrm{ii}}\mathbf{A}_{i}\).