## 2 Dynamic Model of a Teleoperator System  
Consider a bilateral robotic system, consisting of a local (l) and a remote (r) manipulator, each with \(n\) degrees of freedom (DOF), assuming for simplicity that all joints are revolute. The dynamics of the bilateral system are described by the following equation [30]:  
\[\mathbf{H}_{i}(\mathbf{q}_{i})\ddot{\mathbf{q}}_{i}+\mathbf{C}_{i}(\mathbf{q}_{i},\dot{\mathbf{q}}_{i} )\dot{\mathbf{q}}_{i}+\mathbf{g}_{i}(\mathbf{q}_{i})=\mathbf{u}_{i}+\mathbf{\tau}_{hi}=\mathbf{u}_{\tau i}, \tag{1}\]  
where, for \(i=\mathrm{l},\mathrm{r}\), \(\mathbf{q}_{i}\in\mathbb{R}^{n}\) represents the vector of generalized joint coordinates, \(\mathbf{H}_{i}(\mathbf{q}_{i})\in\mathbb{R}^{n\times n}\) is the symmetric, positive-definite inertia matrix, \(\mathbf{C}_{i}(\mathbf{q}_{i},\dot{\mathbf{q}}_{i})\dot{\mathbf{q}}_{i}\in\mathbb{R}^{n}\) represents the vector of Coriolis and centrifugal torques, and \(\mathbf{g}_{i}(\mathbf{q}_{i})\in\mathbb{R}^{n}\) denotes the gravitational torque vector. The control torques applied to the joints are given by \(\mathbf{u}_{i}\in\mathbb{R}^{n}\), while \(\mathbf{\tau}_{hi}\in\mathbb{R}^{n}\) represents the torque applied by the human operator. Importantly, if \(\mathbf{\tau}_{\mathrm{hl}}\
eq\mathbf{0}\), then \(\mathbf{\tau}_{\mathrm{hr}}\equiv\mathbf{0}\), and vice versa, ensuring that the human operator interacts with only one manipulator at a time. Finally, \(\mathbf{u}_{\tau i}\) denotes the total torque applied to the system. The largest and smallest eigenvalues of a matrix are denoted by \(\lambda_{\text{max}}(\cdot)\) and \(\lambda_{\text{min}}(\cdot)\), respectively. By considering that only revolute joints are involved, the following properties can be established [30]:  
**Property 1**. : _It holds \(\lambda_{\mathrm{hi}}\|\mathbf{x}\|^{2}\ \leq\ \mathbf{x}^{\mathsf{T}}\mathbf{H}_{i}(\mathbf{q}_{i})\mathbf{x}\ \leq\ \lambda_{\mathrm{H}i}\|\mathbf{x}\|^{2}\ \forall\ \mathbf{q}_{i}\in\mathbb{R}^{n}\), \(\mathbf{x}\in\mathbb{R}^{n}\), and \(0<\lambda_{\mathrm{H}i}\leq\lambda_{\mathrm{H}i}<\infty\), with \(\lambda_{\mathrm{H}i}\triangleq\min\limits_{\mathbf{q}_{i}\in\mathbb{R}^{n}}\lambda_ {\min}\Big{(}\mathbf{H}_{i}(\mathbf{q}_{i})\Big{)}\) and \(\lambda_{\mathrm{H}i}\triangleq\max\limits_{\mathbf{q}_{i}\in\mathbb{R}^{n}} \lambda_{\max}\Big{(}\mathbf{H}_{i}(\mathbf{q}_{i})\Big{)}\). \(\lambda_{\min}\) and \(\lambda_{\max}\) denote the minimum and maximum eigenvalues of a matrix._  
**Property 2**. : _By using the Christoffel symbols of the first kind to compute \(\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i})\), the matrix \(\mathbf{H}_{i}(\mathbf{q}_{i})-2\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{\dot{q}}_{i})\) is skew-symmetric._  
**Property 3**. : _There exist positive constants \(0<k_{ci}<\infty\), such that \(\|\mathbf{C}_{i}(\mathbf{q}_{i},\mathbf{x})\|\leq k_{ci}\|\mathbf{x}\|\) holds \(\forall\ \mathbf{x},\mathbf{q}_{i}\in\mathbb{R}^{n}\)._  
**Property 4**. : _The vector of the generalized gravitational torques \(\mathbf{g}_{i}(\mathbf{q}_{i})\) satisfies_  
\[\|\mathbf{g}_{i}(\mathbf{q}_{i1})-\mathbf{g}_{i}(\mathbf{q}_{i2})\|\leq k_{gi}\|\mathbf{q}_{i1}- \mathbf{q}_{i2}\|\quad\forall\mathbf{q}_{i1},\mathbf{q}_{i2}\in\mathbb{R}^{n}, \tag{2}\]  
_for some positive constant \(k_{gi}\)._  
The time-varying delay from the local to the remote robot is denoted by \(T_{\mathrm{I}}(t)\), whereas the delay in the opposite direction is represented by \(T_{\mathrm{r}}(t)\). The following assumptions are considered:  
**Assumption 1**. : _The initial states of \(T_{\mathrm{I}}(t)\) and \(T_{\mathrm{r}}(t)\) are assumed to be zero or small constant values, i.e., \(T_{\mathrm{I}}(t_{0})=T_{\mathrm{r}}(t_{0})\approx 0\). This assumption ensures continuity in the delay dynamics, allowing for a smooth initialization of the system._  
**Assumption 2**.