## 3 Experimental Procedure and Data Analysis
### Dynamic Stiffness Analysis  
Based on the abovementioned approaches, all data are processed to obtain the dynamic stiffness (Table 4). The distributions of the dynamic stiffness are illustrated in Figure 8. Because the values of the dynamic stiffness vary significantly, a logarithmic coordinate is used for the illustration, which shows that all values obtained by the direct approach are higher than the values of the indirect approach. The direct approach gives constant excitation on the industrial robot, and each robot joint is activated to vibrate. Compared with the indirect approach, the viscous damping of the robot joint has a greater impact on the attenuation of vibration. Therefore, the direct approach can obtain relatively higher dynamic stiffness. The distributions show a similar trend of change in each direction. However, the two distributions have their own change ratios. As shown in Figure 8c, \(K_{e}\) has an obvious decrease from 14 Hz to 25 Hz. However, \(K_{m}\) has a slight decrease. Both excitation frequencies are close to the eigenfrequency of the robot in the Z direction, which means that the dynamic stiffness is more sensitive to the forces and excitation frequency. As the distribution under different conditions is rather random, there is no regularity in the relationship between \(K_{m}\) and \(K_{e}\). The ratio of \(K_{e}\) to \(K_{m}\) ranges from 3.2 to 90.8. The large difference between the two types of dynamic stiffness is possible because the excitation energy implemented on the robot is totally different. In addition, there are excitation frequencies close to the eigenfrequency of the robot. The dynamic stiffness in this situation is prone to being influenced by the external excitation conditions, which contain the excitation force, robot pose, damping ratio and measurement uncertainty of the measuring equipment. Nevertheless, more experiments that have the same excitation force under different excitation frequencies should be planned to confirm whether this results in constant variance. Regarding the values of the dynamic stiffness, different directions have their own characteristics under different excitation frequencies. For practical robotic machining, the dynamic stiffness of the direct approach is more meaningful for machining quality control. This apparently shows that the robot has a very low dynamic stiffness under excitation frequencies of 14 Hz and 25 Hz, while these two frequencies are close to the eigenfrequencies of the robot. In the external range of the eigenfrequency, both 40 Hz and 75 Hz have higher dynamic stiffness. Figure 7: Vibration displacement in time and frequency domain under 14 Hz excitation frequency. The dynamic stiffness at 40 Hz has slight variance in three directions. Its X direction has the lowest value of \(1.678\times 10^{6}\) N/m, which is comparable to the static stiffness of industrial robots, whose values usually range between \(10^{5}\)-\(10^{6}\) N/m [28,29]. However, the values from the indirect approach show a large difference between the Y and \(Z\) directions (Table 5). This indicates that the direct approach and indirect approach have no coherence to reflect the distribution of dynamic stiffness in each direction under the same excitation frequency. The rotation of the pneumatic spindle simultaneously brings excitation forces in two directions. This coupling effects affect the dynamic stiffness value. Therefore, the indirect approach is closer to the reality for this variance analysis. Figure 8: Dynamic stiffness distributions under two calculation approaches. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Frequency (Hz)** & \(\mathbf{X}\_{Ke}\) & \(\mathbf{Y}\_{Ke}\) & \(\mathbf{Z}\_{Ke}\) & \(\mathbf{X}\_{Km}\) & \(\mathbf{Y}\_{Km}\) & \(\mathbf{Z}\_{Km}\) \\ \hline
14.0 & 0.018 & 0.170 & 0.454 & 0.002 & 0.006 & 0.005 \\
25.0 & 0.100 & 1.235 & 0.162 & 0.031 & 0.042 & 0.005 \\
40.0 & 1.678 & 2.042 & 1.953 & 0.123 & 0.137 & 0.046 \\
75.0 & 0.982 & 19.790 & 6.054 & 0.087 & 0.606 & 0.593 \\ \hline \hline \end{tabular}
\end{table}