## 3 Teleoperation Control Design with Wave Variable and Virtual Force Feedback
### Stability and Transparency Analysis  
A sufficient condition to prove the stability of teleoperation systems with time delays is that the adopted wave variable-based communication architecture is passive, for the leader and follower with PID controllers are passive [30]. According to the scattering operator theory, the passivity of teleoperation system can be guaranteed if and only if its scattering operator \(S(s)\) satisfies:  
\[\left\|S(s)\right\|\leq 1 \tag{28}\]  
where the scattering operator \(S(s)\) is defined by:  
\[S(s){=}\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}\left[H(s)-I\right]\left[H(s)+I\right]^{-1} \tag{29}\]  
where the hybrid matrix of the proposed teleoperation scheme can be defined by Equation (29) and derived from Equations (20) and (21) as Equation (31).  
Figure 7: Designed teleoperation control scheme for CHCM.  
\[\begin{bmatrix}F_{m}(s)\\ -\dot{X}_{s}(s)\end{bmatrix}=H(s)\cdot\begin{bmatrix}\dot{X}_{m}(s)\\ F_{s}(s)\end{bmatrix} \tag{30}\]  
\[H(s)=\begin{bmatrix}K\cdot b&e^{-sT}\\ -e^{-sT}\cdot\frac{\lambda}{s+\lambda}&\frac{1}{b}-\frac{e^{-2 nT}}{b}\cdot \frac{\lambda}{s+\lambda}\end{bmatrix} \tag{31}\]  
As implied by Formulas (29) and (31), when \(s\) is defined as \(j\omega\), the function \(e^{-j\omega T}\) will have a periodic impact on the norm of \(S(s)\) with a constant time delay \(T\), which means \(T\) does not exert extra influence but on the angular frequency of \(S(s)\) among \(\omega\) axis. The parameters \(K\) and \(b\) to be designed affect the passivity of teleoperation system largely, of which the values are limited to \((0,1)\) and \([1,+\infty)\), respectively, considering the influences on the tracking performance and force feedback with regard to Equations (20) and (21). By the iteration of different parameters in MATLAB, it is illustrated that \(\|S(s)\|\) is always no more than one with \(K\in[0.5,1)\).  
The simulation result, as shown in Figure 8, preferably indicates that the scattering matrix norm of proposed teleoperation scheme will remain not greater than one with obvious periodical characteristic, and the one-order low-pass in the communication channel enhances the system passivity by reducing the norm of \(S(s)\) with increasing frequency, where \(K\)=0.6, \(b=10\), \(T=1\), \(\lambda=20\).  
With guaranteed passivity, the transparency of the proposed teleoperation system is also improved regarding Equations (20) and (21). Concretely, the virtual force \(F_{s}\) will present slight and small change according to Equations (23) and (24) when the tracking error \(X_{sd}-X_{s}\) is almost time invariant or smoothly varying; therefore, the transformed desired velocity \(\dot{X}_{sd}\) of follower accurately reaches or gets close to the desired value with the filter eliminating noise of wave variables. With move-and-wait operation strategy adopted [39], the operator will have an exact perception of virtual force via stopping the leader, since \(F_{m}(t)\) is exactly equal to the virtual force \(F_{s}(t-T)\) when \(\dot{X}_{m}(t)=0\). Furthermore, a larger value is chosen for \(b\) when \(F_{s}\) shows rapid changes to suppress the distortion of \(\dot{X}_{sd}\). However, the parameter \(b\) has a reverse effect on force feedback and increases the force feedback error according to Equation (21), to cope with which the secondary parameter \(K\) is introduced with a small value. Thus,  
Figure 8: The simulation result of the scattering matrix norm.  
a good system transparency including good position-tracking performance and force feedback can be obtained simultaneously in the proposed teleoperation scheme with appropriate values of \(b\) and \(K\).