# Dynamic Modeling and Response Analysis of an Optical Machining Robot with Prismatic Joint Clearance
## 4 Dynamics Simulation
\begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value** \\ \hline Mass of UP branched-chain and moving platform \(m_{m}/\)kg & 138.239 \\ Inertia matrix of UP branched-chain and moving platform \(J_{m}/\)kg\(\cdot\)m\({}^{2}\) & diag[35.509,35.478,1.452] \\ Circumcircle radius of universal joints \(U_{i}\,r_{s}/\)m & 0.4 \\ Circumcircle radius of universal joints \(U_{k}\,r_{m}/\)m & 0.15 \\ Length of cylinder block components \(L_{a,i}/\)m & 0.6 \\ Mass of cylinder block components \(m_{a,i}/\)kg & 70.8 \\ Inertia matrix of cylinder block component \(J_{a,i}/\)kg\(\cdot\)m\({}^{2}\) & diag[14.264,14.598,0.203] \\ Length of telescopic rod components \(L_{b,i}/\)m & 1.2 \\ Mass of telescopic rod components \(m_{b,i}/\)kg & 29.85 \\ Inertia matrix of telescopic rod component \(J_{c,i}/\)kg\(\cdot\)m\({}^{2}\) & diag[5.94,5.94,0.02] \\ Length of rotating rod components \(L_{c,i}/\)m & 0.09 \\ Mass of rotating rod components \(m_{c,i}/\)kg & 1.3 \\ Inertia matrix of rotating rod component \(J_{c,i}/\)kg\(\cdot\)m\({}^{2}\) & diag[0.004,0.004,0.004] \\ External load/kg & 118.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Structural parameters of the 3 UPRU+UP parallel manipulator. \begin{table}
\begin{tabular}{l l l l} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Young’s modulus & 200 Gpa & static friction coefficient \(\mu_{s}\) & 0.80 \\ Restitution coefficient \(c_{e}\) & 0.9 & sliding friction coefficient \(\mu_{d}\) & 0.16 \\ Baumgarte \(\alpha_{b}\) & 5 & MATLAB integration step & 0.0001s \\ Baumgarte \(\beta_{b}\) & 1 & MATLAB integral function & ode45 \\ \hline \end{tabular}
\end{table} 
Table 3: Dynamic simulation parameters. with the growth in the clearance size, the fluctuations in the velocity curve become more conspicuous. Z-direction vibration amplitude is relatively large, 0.15, 0.19, and 0.26 m/s, respectively. As shown in Figure 9g-i, the clearance has a large effect on the acceleration of the moving platform. The larger the clearance is, the larger the acceleration peak is, and the amplitude of the sudden change in acceleration in the Z direction is 0.21, 0.32, and 0.61 m/s\({}^{2}\), respectively. This results in greater vibration and instability during the operation of the mechanism. The reason behind this is that the enlarged gap size causes frequent alterations in the contact state between the slider and the guideway. Moreover, the fluctuations and amplitudes of the clearance collision force increase, thus influencing the stability of the mechanism's movement. Figure 9: Dynamic responses of the moving platform with different clearance sizes: (**a–c**) displacement, (**d–f**) velocity, (**g–i**) acceleration. Collision and sliding at the gap inevitably generate friction. Figure 10 shows the dynamic response of the parallel manipulator with friction coefficients of 0.02, 0.05, and 0.08, and a clearance of 0.1 mm is simulated and solved. As shown in Figure 10a-c, when the clearance size gets larger, the displacement error of the moving platform becomes more significant, and the displacement of the moving platform in X and Y directions is very small, but the displacement deviation in Z direction becomes larger, and, with the increase in clearance, the displacement deviation in Z direction is 2.2 mm, 3.3 mm, and 5.4 mm, respectively. As shown in Figure 10d-f, it can be seen that, with the increase in the friction factor, the vibration amplitude changes slightly, and the Z-direction vibration amplitude is relatively large, the Z-direction amplitude being 0.08, 0.13, and 0.20 m/s, respectively. From Figure 10g-i, it can be observed that the friction factor has a large influence on the acceleration of the moving platform, the fluctuation amplitude of the acceleration curve changes with the change in the friction factor, and the fluctuation amplitude of the acceleration curve in the Z direction is 0.19, 0.32, and 0.42 m/s\({}^{2}\), respectively. Contact force in prismatic joints increases with increasing coefficient of friction.