# Analysis of Dynamic Behavior of ParReEx Robot Used in Upper Limb Rehabilitation
## 3 Dynamic Analysis in MSC.ADAMS of ParReEx-Wrist Robot
### Dynamic Analysis in MSC.ADAMS of the ParReEx-Wrist Robot Considering Joint Friction
Since in operation this element has the highest elasticity, it is very important to study the dynamics of the robot considering this circular guide as a deformable solid. Figure 20 shows that Vibration Mode 8 is flexural and occurs at 1538 Hz. Vibration modes consider the external connections of the element and can be flexural, torsional or a combination of the two. Numerical simulation in ADAMS considering the friction in joints and the flexibility of the circular guide element resulted in laws of variation of the torque necessary to operate the flexion-extension module as well as the radial-ul \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode Number** & **Frequency (Hz)** & **Mode Number** & **Frequency (Hz)** \\ \hline
7 & 597.685 & 16 & 1.299 \(\times\) 10\({}^{4}\) \\
8 & 1538.863 & 17 & 1.546 \(\times\) 10\({}^{4}\) \\
9 & 1828.696 & 18 & 2.315 \(\times\) 10\({}^{4}\) \\
10 & 2804.835 & 19 & 2.827 \(\times\) 10\({}^{4}\) \\
11 & 3835.677 & 20 & 2.987 \(\times\) 10\({}^{4}\) \\
12 & 4032.088 & 21 & 6.923 \(\times\) 10\({}^{4}\) \\
13 & 5993.852 & 22 & 8.053 \(\times\) 10\({}^{4}\) \\
14 & 6743.001 & 23 & 1.343 \(\times\) 10\({}^{5}\) \\
15 & 9246.1609 & 24 & 1.808 \(\times\) 10\({}^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Circular guide modal parameters. Figure 20: The deformable body in ADAMS and view of body deformation map. There is an increase in the motor torque required to operate the modules in this case as well as a steeper variation, similar to values determined experimentally [18]. In Table 17, the characteristic values for the actuation moments of the two modules of the robot are presented for case 3 of the study, i.e., considering the flexibility of the elements and the friction in the couplings. Figure 22 shows the laws of variation for the components of the connection forces in coupling A as well as for the resultant reaction. These results are obtained in case 3 of the simulation, i.e., the one in which we considered simultaneously the friction in the kinematic couplings and the flexibility of the elements. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter**Component** & **Motion 1 Torque** & **Motion 2 Torque** \\ \hline Min & 0.0013 & 2.431 \\ Max & 1.8086 & 5.5339 \\ Avg & 0.8143 & 4.3712 \\ RMS & 0.9316 & 4.4062 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Robot motion parameters in Case 3. Figure 21: The laws of variation of computed torque for: (**a**) the flexion-extension module and (**b**) the radial–ulnar module. Figure 22: _Cont._  
Table 18 shows the numerical results obtained for the components of the forces and the resulting reaction in joint A for simulation case 3. Figure 23 shows the laws of variation for the components of the connection forces in coupling B as well as for the resultant reaction, for the third simulation case. Table 19 shows the numerical results obtained for the components of the forces and the resulting reaction in joint B for simulation case 3. Figure 24 shows the laws of variation for the components of the connection forces in coupling R1 as well as for the resultant reaction, for the third simulation case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ParameterComponent** & **Joint A Fx** & **Joint A Fy** & **Joint A Fz** & **Joint A MAG** \\ \hline Min & \(-2.2236\) & 99.6565 & \(-42.8897\) & 115.1857 \\ Max & 62.8521 & 210.1286 & 98.7351 & 231.5797 \\ Avg & 30.9324 & 150.7614 & 17.3479 & 163.231 \\ RMS & 38.4147 & 153.2828 & 48.9964 & 165.4441 \\ \hline \end{tabular}
\end{table} 
Table 18: Joint A connection force components. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ParameterComponent** & **Joint B Fx** & **Joint B Fy** & **Joint B Fz** & **Joint B MAG** \\ \hline Min & 0.4806 & \(-0.8625\) & \(-1.6387\) & 0.4815 \\ Max & 9.1875 & 0.4868 & 1.0018 & 9.2307 \\ Avg & 8.8505 & 0.4033 & \(-0.4287\) & 8.91 \\ RMS & 8.853 & 0.4095 & 0.9483 & 8.9131 \\ \hline \end{tabular}
\end{table} 
Table 19: Joint B connection force components.