## 4 Verification Experiment of Impact Force
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Theoretical Value/N** & 42.1 & 43.3 & 44.9 & 47.1 & 49.7 & 52.9 & 56.5 & 60.7 & 65.3 & 70.4 & 75.9 \\
**Experimental Value/N** & 39.8 & 41.3 & 42.1 & 44.8 & 48.2 & 50.3 & 53.5 & 57.9 & 62.6 & 66.2 & 70.8 \\
**Identification Error/\%** & 5.4 & 4.6 & 6.2 & 4.8 & 3.1 & 4.9 & 5.3 & 4.9 & 4.1 & 5.9 & 6.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative data of the normal impact force. Figure 9: Normal impact force comparison results. the proportional valve. The comparison between the experimental data and the theoretical data of the friction force when the internal pressure was 2.05 Mpa is shown in Figure 11. The actual friction force was greater because the vibration generated during the impact process and the accuracy error of the experimental device had a nonlinear effect on the friction force. The experimental results of sliding friction obtained during the experiment are shown in Figure 12. When the internal pressure was in the range of 1.5-1.9 MPa, the friction force was obviously nonlinear. There are several reasons for this, including because the rubber material is elastic material, generating different degrees of vibration during the experiment as the impact force changed. Stress and strain were not synchronized due to the viscoelasticity of the rubber material. With the change of the internal pressure of the impact end-effector, the contact area changed during the experiment. These reasons are difficult to describe accurately with current theoretical research, and friction is difficult to control. Therefore, this range of pressure when designing flexible end-effectors or performing mechanical impact active detumbling should be avoided. The research purpose of this paper was to build a more accurate mathematical model to describe the impact force, but through numerical calculations and experiments, accurate theoretical predictions and practically controllable pressure ranges were found, providing guidance for practical use in engineering. The experiments showed that the friction coefficient was not constant and the identification error of the friction force was 22.6%, which did not meet the requirements of accuracy. Figure 11: Comparison of experimental friction. Figure 10: Friction test. The experiments showed that the rubber ball slipped during the sliding process, causing its volume to deform, as shown in Figure 13. The red dashed line in Figure 13 represents the slipping process. In the sliding process, a frictional force \(f_{disp}\) was generated, which prevented slipping. The \(f_{disp}\) is as follows:  
\[f\text{disp}= (a+a\exp(bs)[c\sinh(ds)-e\cosh(ds)])F_{n}=\mu F_{n}, \tag{9}\]  
where \(s\) is the sliding displacement, \(F_{n}\) is the load, and \(\mu\) is the coefficient. Each coefficient was fitted using experimental data as  
\[\left\{ \begin{array}{l}a=0.74769-0.01684F_{n}\\ b=-0.13056-0.01728F_{n}+5.35714 exp^{-4}{F_{n}}^{2}\\ c=-1.04239+0.00583F_{n}-4.07238 exp^{-4}{F_{n}}^{2}\\ d=0.010619+0.01829F_{n}-7.75 exp^{-4}{F_{n}}^{2}\\ e=2.69364-0.05815F_{n}+0.00712 exp^{-4}{F_{n}}^{2}\end{array} \right.. \tag{10}\]  
In summary, the corrected friction is as follows:  
\[f=\frac{\pi}{4}K\sigma_{0}\frac{F_{n}}{p_{r}}\tan\delta+\frac{\pi F_{n}}{10} \bigg{(}\frac{3p_{\text{c}}}{E^{*}}\bigg{)}^{\frac{2}{3}}\tan\delta+\mu F_{n}. \tag{11}\]  
The comparison of the actual value, the theoretical value, and the corrected value of the friction force are shown in Figure 14. The comparative data are shown in Table 4. The identification error of the friction force was less than 6.9%. Figure 12: Friction force comparison results. Figure 13: Results of sliding deformation. The corrected friction force considered the large deformation of the impact ball structure during sliding, which resulted in a larger contact area and was also in line with the calculation method of the actual friction force.