## 3 Teleoperation Control Design with Wave Variable and Virtual Force Feedback
### Virtual Force Feedback Generation and Controller Design  
To realize the teleoperation control from leader to follower, namely the CHCM, considering the assumptions and irregular accessible range of the manipulator's motion, a 2-DOF Cartesian working space mapping method is proposed in this section. The motion of selected leader robot, Phantom Omni device, in \(XOY\) plane is transformed to desired position of follower in \(PTY\) plane, which can be written as:  
\[\begin{bmatrix}x_{sd}\\ y_{sd}\\ z_{sd}\end{bmatrix}=\begin{bmatrix}k_{x}&0&0\\ 0&0&0\\ 0&k_{y}&0\end{bmatrix}\begin{bmatrix}x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}+\begin{bmatrix}0\\ y_{0}\\ 0\end{bmatrix} \tag{22}\]  
where \(k_{x}\), \(k_{y}\) are scare factors and \(y_{0}\) is the \(y\) coordinate of \(PTY\) plane.  
To achieve the operator's perception of follower's motion state, a virtual force feedback generation algorithm is presented based on some presuppositions of working space and the position of leader and follower, which can be derived as:  
\[F_{s}=\left\{\begin{array}{ll}F_{e}&X_{s}\in X_{u}\\ F_{e}+F_{b}&others\end{array}\right. \tag{23}\]  
\[F_{e}=k_{f}\cdot(\frac{1}{1+e^{X_{sd}-X_{s}}}-0.5) \tag{24}\]  
\[F_{b}=k_{b}\cdot(X_{b}-X_{s})+c_{b}\cdot\vec{X}_{s}\cdot\vec{N} \tag{25}\]  
where \(k_{f}\) is the primary parameter to adjust the amplitude of virtual force during free motion; \(k_{b}\) and \(c_{b}\) are the spring modulus and damping coefficient of environment exceeding presupposed accessible unconstrained location \(X_{u}\), which is regarded as a spring-damper. \(X_{b}\) represents the boundary of \(X_{u}\) and \(\vec{N}\) denotes the unit normal vector of contact position of the follower and \(X_{b}\) pointing to \(X_{u}\). Remarkably, the parameters \(k_{b}\) and \(c_{b}\) are defined based on presupposed virtual boundary or known prior information about real environment and take values only when the position of follower \(X_{s}\) reaches or exceeds the setting boundary \(X_{b}\).  
As expressed in Equations (23)-(25), the generated virtual force \(F_{s}\) changes continuously and boundedly with tracking errors \(X_{sd}-X_{s}\) and presents more sensitivity with small tracking errors. Hence, the virtual force feedback can provide operators with the perception of tracking errors, and motion state of follower relative to the boundary \(X_{b}\).  
Based on the structures of CHCM, a teleoperation control scheme is proposed to achieve good tracking performance of follower, as shown in Figure 7. According to the inverse kinematics model of CHCM, the desired position \(X_{sd}\) is transformed to desired lengths \(L_{d}\) of driving cables of each joint module. By measuring the actual lengths of driving cables \(L_{s}\) with linear magnetic encoders, a closed-loop PID controller is designed as Equation (26) to realize accurate cable length control.  
\[u=K_{P}\cdot(L_{d}-L_{s})+K_{I}\cdot\int_{0}^{t}(L_{d}-L_{s})d\tau+K_{D}\cdot( \dot{L}_{d}-\dot{L}_{s}) \tag{26}\]  
where \(K_{P}\), \(K_{I}\), \(K_{D}\) are parameters of the PID controller, and the output of controller \(u\) determines the rotation speed of each actuator.  
Meanwhile, the virtual force \(F_{s}\) transmitted from the follower and the velocity of leader \(\dot{X}_{m}\) generate the feedback force \(F_{m}\) on the leader. Hence, the dynamics of the leader robot can be expressed as Equation (27), considering the leader as a particle with a mass of \(M_{m}\).  
\[F_{h}(t)-F_{m}(t)=M_{m}\cdot\ddot{X}_{m}(t),\dot{X}_{m}(t)=F_{h}/Z_{h} \tag{27}\]  
where \(F_{h}\) is the control force exerted by the operator and \(Z_{h}\) is the operator impedance coefficient.