## 5 Experiment and Results
### Results and Discussion  
Robot end tool trajectories and interaction forces of the four groups are shown in Figure 6. CAC-\(i\) corresponds to the training set Train-\(i\), which represents the results of the human-robot interaction experiment under constant admittance, and TPVAC-\(i\) is experiment results with predictor-\(i\). CAC-1, CAC-2, TPVAC-1 and TPVAC-2 were completed by Tester 1, and the rest of the experiments were completed by Tester 2. From the qualitative analysis of the data in Figure 6, the following conclusions can be drawn: (a) The direction of the interactive force is somewhat advanced relative to the direction of  
Figure 5: Experimental setup with designated task path. the robot end tool, especially in right-angle. (b) Compared with the constant admittance control (CAC), the amplitude of the human-robot interaction force under TPVAC is smaller, but the direction of the interaction force changes more frequently, indicating that the tester can perceive the change of the parameters and adjust it in time. (c) The habits of the two testers are slightly different. Tester 1 has a larger interaction force relative to Tester 2 and the steering is smoother. Figure 6: Robot end tool trajectories (black lines) and interaction forces (red arrows) of the four groups of comparative experiments. It is proved that our proposed variable admittance control method can effectively help the operator to make the robot end tool move along the task path. In this paper, the error between the actual trajectory and the task path is used to demonstrate the effectiveness of the method. Robot end tool trajectories versus task path of the four groups of comparative experiments are shown in Figure 6(a),b. The trajectory errors are shown in Figure 6(c). Compared with CAC, the deviation between the actual trajectory and the task path is far below with TPVAC, indicating that the TPVAC method can improve the robot's operational performance. After using TPVAC, the trajectory errors decreased by 51-67%, as shown in Table 2. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Indictors**} & **CAC-** & **TPVAC-** & **Reduction-** & **CAC-2** & **TPVAC-** & **Reduction-** & **CAC-3** & **TPVAC-** & **Reduction-** & **CAC-4** & **TPVAC-** & **Reduction-** \\  & **1** & **1** & **1** & **2** & **2** & **2** & **CAC-3** & **3** & **3** & **CAC-4** & **4** & **4** \\ \hline Trajectory & 6.1 & 2.0 & 67\% & 4.6 & 1.6 & 65\% & 4.5 & 2.2 & 51\% & 5.3 & 2.2 & 58\% \\ error (nm) & 0.90 & 0.71 & 21\% & 1.11 & 0.71 & 36\% & 0.62 & 0.43 & 31\% & 0.68 & 0.28 & 59\% \\ MA-force & 5.6 & 4.3 & 23\% & 6.7 & 3.6 & 46\% & 3.1 & 2.1 & 32\% & 4.7 & 2.6 & 45\% \\ (N) & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The average values and reduction rates of each group of curves. Figure 7: Robot end tool trajectory versus task path and trajectory error. Two indicators were adopted to quantitatively evaluate the compliance improvement of the TPVAC method. One is the moving average of the square of the jerk (MAS-jerk), which is to evaluate the compliance of operation. Another is the moving average of the interaction force (MA-force), which is to evaluate the magnitude of interaction force. The calculation method of the two indicators is shown in Equations (8) and (9). The reason for choosing the moving average of the parameters is that the trajectory within the task path is a N-shaped line, and the tester has constantly changing accelerations and decelerations. \[\text{MAS}-\text{jerk}(t)=\frac{1}{T}\int_{t-T}^{t}\Big{(}\lvert\text{jerk} \rvert_{x}^{2}+\lvert\text{jerk}\rvert_{x}^{2}\Big{)}\text{d}t \tag{8}\]  
\[\text{MA}-\text{force}(t)=\frac{1}{T}\int_{t-T}^{t}\sqrt{\text{force}_{x}^{2}+ \text{force}_{y}^{2}\text{d}t} \tag{9}\]  
The MAS-jerk and MA-force curves are shown in Figure 8. For each group of comparative experiments, the MAS-jerk and MA-force under TPVAC are less than those under CAC.