## 3 Experimental Procedure and Data Analysis
### Dynamic Stiffness Calculation with Modal Test Data  
A frequency response function is defined as the ratio of output to input. Three alternative estimators for the frequency response function, H1, H2 and H3, are available using a dual-channel analyzer [25]. The selection of these three estimators depends on whether there is noise in the input or output. Because impact excitation is adopted in the modal test, the noise in the input signal is negligible. As a result, H1 is used for estimating the frequency response function [26]. The equation is described below.  
\[H1=\frac{G_{AB}(f)}{G_{AA}(f)} \tag{4}\]  
where \(G_{AA}(f)\) is the auto spectrum and \(G_{AB}(f)\) is the cross spectrum. Here, using PULSE software, H1 can be directly obtained. The dynamic stiffness \(K_{m}\) is calculated by dividing \(f^{2}\) by the magnitude of \(H1(f)\)[27]. Here, \(f\) is the frequency value.  
\[K_{m}\ =\ \frac{f^{2}}{\left|H1(f)\right|} \tag{5}\]  
Figure 4: Interface of PULSE software in the vibration measurement.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & \multicolumn{2}{c}{**Sensing Ranges**} & \multicolumn{2}{c}{**Measurement Uncertainty**} \\ \hline \multirow{2}{*}{ATI Delta IP 65} & Fx, Fy & Fz & Fx, Fy & Fz \\  & 660 N & 1980 N & \(\pm\)0.66 N & \(\pm\)1.98 N \\ \hline \hline \end{tabular}
\end{table}
Table 3: Force sensor parameters.  
An arbitrary robot position configuration in the workspace (Figure 2) is selected for this test. First, the experimental modal test is performed on the end effector of the robot to obtain the accelerometer FRF of the robot. The accelerometer is fixed on the plane of the end effector (Figure 1. The characteristics of the robot are different according to the three directions (Figure 3). Hence, the FRF of each direction is measured separately when the impact force is executed in the same direction. Five impact times are planned for each direction to get the average accelerometer FRF. To reduce the influence of the impact force and impact direction on the experimental results, the FRF is only recorded for analysis when the minimum coherence value is larger than 0.95. Tests with different parameters are made to ensure the appropriate setting of the modal analysis system. The details of the parameter settings are stated in Table 4. The average times in the table concern the times of impact on the end effector. All the measurements are made in a static state without activating the motor.  
The impact modal tests are carried out based on the settings above. Figure 5 illustrates the FRF H1 in three directions. This shows that the profile of H1 in each direction has its own characteristics. The peaks of the curves, which represent the eigenfrequencies of the robot in each direction, are located at different frequency values. To compare the dynamic stiffness under different frequencies, four frequency values are chosen from low to high. The frequency of f1 = 14 Hz is close to the eigenfrequency of the X and Y directions. The frequency of f2 = 25 Hz is close to the eigenfrequency of the Z direction. The other two frequencies of f3 = 40 Hz and f4 = 75 Hz are away from the eigenfrequency.