# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 4 Numerical Simulation
### Performance Evaluation for Percussive Riveting  
For the application scenario as illustrated in Figure 1, the robot can be subject to forces and torques in different directions and due to different processes within its operating workspace. As an example, we demonstrate the evaluation of vibration characteristics for robotic percussive riveting. This operation is completed by the robot with a percussive rivet gun and the principle of percussive riveting is illustrated in Figure 8. A rivet is first inserted into pre-drilled mating holes on the panels. Then the simulated robot drives a percussive rivet gun and another supporting robot (see Figure 8) drives a bucking bar to the riveting spot. Once the rivet gun and the bucking bar are aligned to the normal direction of the panels, the robot stops and activates the gun, and then the hammer of the gun repeatedly hits the rivet head. The impact forces cause the rivet to deform. When the diameter of the rivet becomes equal to the diameter of the pre-drilled holes, the panels are joined. For motion planning of the robotic riveting operations, operational space trajectory planning is used to avoid interference during the motion. Note that since the robot is stationary when its end-effector is subject to dynamic loading during its riveting operations, the trajectory between every two riveting spots does not affect the vibration characteristics if the forced vibration can be damped out during the motion from one spot to the next. During percussive riveting, the end-effector is subject to repetitive impact forces normal to the panel. If the work frame is established on the panel at the riveting spot and with the \(z^{\prime}\) axis normal to the panel, then the process force and torque can be written in the work frame as below:  
\[\mathbf{F}^{{}^{\prime}}=\left[0,0,F(t)\right]^{T}\quad\mathbf{\tau}^{{}^{ \prime}}=0 \tag{25}\]  
The percussive impact force \(F(t)\) is plotted in Figure 9a, where the amplitude \(F_{m}\) is 550 N, the force frequency \(\omega_{v}\) is about 62.8 rad/s (10 Hz), and \(T_{d}\)/\(T\) is 0.08. The force \(F(t)\) is approximated by Fourier expansion using (22), with the Fourier coefficients drawn in Figure 9b for \(k=0,\ \pm 1,\ \pm 2,\ \dots,\ \pm 20\), i.e.,  
\[F(t)\approx\sum_{k=-20}^{20}F_{k}e^{jk\omega_{v}t} \tag{26}\]  
Figure 8: Principle of percussive riveting. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Indices** & **\(\mathbf{\omega}\) = 20 Hz** & **\(\mathbf{\omega}\) = 30 Hz** & **\(\mathbf{\omega}\) = 40 Hz** & **\(\mathbf{\omega}\) = 50 Hz** & **\(\mathbf{\omega}\) = 60 Hz** & **\(\mathbf{\omega}\) = 70 Hz** & **\(\mathbf{\omega}\) = 80 Hz** \\ \hline Mean value of \(\eta_{\text{max}}\) (\(10^{-5}\) m/N) & 2.2786 & 4.3875 & 6.3616 & 3.2873 & 2.2236 & 1.4321 & 1.2691 \\ Maximum value of \(\eta_{\text{max}}\) (\(10^{-5}\) m/N) & 4.5850 & 13.100 & 15.770 & 8.0613 & 7.2806 & 5.8410 & 4.7531 \\ Mean value of \(\eta_{\text{max}}\) (\(10^{-5}\)N) & 2.2145 & 3.9172 & 5.7468 & 3.3286 & 2.2313 & 1.5149 & 1.4133 \\ Maximum value of \(\eta_{\text{max}}\) (\(10^{-5}\)/N) & 3.4936 & 9.7238 & 13.869 & 9.0598 & 5.0308 & 4.4893 & 4.0502 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean and maximum values of both indices over workspace with different excitation frequencies. Note that the force in (26) can be in different directions relative to the inertial frame for different parts with transformation (21). The robot is used for the assembly in a cuboid-shaped operation workspace as drawn in Figure 10. The operation workspace can be determined by Points \(P_{k}\) with \(k\) from 1 to 8. The coordinates of \(P_{1}\) is [1, 0.5, 1]\({}^{T}\) (in m) and the length, height, and width of the operation workspace are \(P_{1}\)\(P_{2}\) = 1 m, \(P_{1}\)\(P_{4}\) = 0.8 m and \(P_{1}\)\(P_{5}\) = 0.5 m, respectively.