## 4 Experiment
### Experiment Results  
The experiment results of free motion (Set1) are shown in Figures 10 and 11, where the stylus is operated repletely with different control designs C1 and C2. To show the results of the practical teleoperation control experiments with virtual force feedback, the position signals and force signals of leader and follower robots are displayed by curves. \(x_{m}\) denotes the position of the leader robot in \(X\)-axis, measured by the internal encoders of Touch device and \(z_{m}\) denotes that in the \(Z\)-axis. Since there are no sensors in the motion subsystem of CHCM, its end-effector potions \(x_{m}\) and \(z_{m}\) are derived from actual cable lengths via established forward kinematics model, which are measured by the linear magnetic encoders of the driving structure of cables. Through the comparison of these position signals, the trajectory tracking performance of proposed teleoperation control scheme can be shown. \(F_{s,x}\) represents the virtual force feedback in the \(X\)-axis produced by the virtual force feedback generation algorithm proposed in this paper, and that with subscript \(z\) denotes signals in the \(Z\)-axis. \(F_{m,x}\) represents the force feedback signals transmitted to the leader robot through the modified wave transform structure. Through the comparison of the above signals, the system transparency can be verified. It is shown that CHCM could track the desired motion trajectory and reach an ideal position when the force feedback is close to zero, which indicates the proposed method of forward and inverse kinematics solution is proper. Specifically, Figures 10a,b and 11a,b show the response of CHCM among the \(X\)-axis or \(Z\)-axis during the free motion via control design C1 and C2, and that the position of follower could track the change of desired trajectory with a small error. Because of the similar expression of \(\dot{X}_{sd}\) and same value of \(b\) in C1 and C2, the position-tracking performance of C2 is approaching that of C1. During free motion, the force feedback among the \(X\)-axis or \(Y\)-axis is generated only by the position-tracking error, which presents a state of flux shown in Figures 10,d and 11,d. However, due to the influence of the velocity of leader \(\dot{X}_{m}\) caused by the wave variable architecture with regard to Equation (20), the transmitted force feedback to the leader is deteriorated in the control scheme C1, which may mislead the operator by providing imprecise perception of tracking errors, while the results of C2 show obviously better force feedback performance than that of C1 via introduction of \(K\). Figure 10: The results of Sett1-C1. The experiment results of motion in free and constrained location (Set2) are shown in Figures 12 and 13, where the end-effector of CHCM reaches and stays in constrained location, which is defined by a virtual boundary and described by a spring-damper. The operator moves the stylus of leader robot toward an arbitrary direction in free space. It presents similar results to Set1 including position-tracking and force feedback performance, when the  
Figure 11: The results of Set1-C2. Figure 12: The results of Set2-C1. theoretically guaranteed stability and improved transparency via appropriate values of the impedance parameter \(b\) and the gain coefficient \(K\), and a virtual force feedback generation algorithm is developed to provide operators with the perception of tracking errors and collision between the follower and the setting boundary. The comparative experiments were carried out and the results show that the seamless combination of derived kinematics model of CHCM and teleoperation scheme with modified wave variable architecture achieves good trajectory tracking performance of CHCM and provides sensible and improved force feedback during free and constrained motion. This work provides a design of a remote control for a cable-driven manipulator and accurate perception of a follower's motion state for operators. Conceptualization, Z.C. and W.S. ; methodology, Z.C. and W.S. ; software, L.Q.