# 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  
This subchapter is dedicated to a virtual dynamic study in ADAMS of the actuation moments and the connection forces in the kinematic Joints A, B, C, R1 and R2 considering the friction in the kinematic joints--a case that simulates the real situation and is different from the previous simulation in which simplistic hypotheses were used. In principle, we are interested in simulating the dynamics of the robot in the context of its use by a patient. Therefore, in all dynamics studies, we consider this context. To define the parameters required for the study of dynamics considering friction, existing data in the literature were used [50]. The following values were adopted for this study: static friction coefficient of 0.2 and dynamic friction coefficient of 0.15 (Table 9). Figure 13: Connecting forces in rotational Joint C: (**a**) in the x-direction; (**b**) in the y-direction; (**c**) in the z-direction; and (**d**) the resultant. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter**Component** & **Joint C** **Fx** & **Joint C** **Fy** & **Joint C** **Fz** & **Joint C** **MAG** \\ \hline Min & \(-20.9865\) & \(-144.671\) & \(15.4783\) & \(30.957\) \\ Max & \(-0.0759\) & \(-26.8092\) & \(85.5258\) & \(168.3648\) \\ Avg & \(-8.4707\) & \(-67.4532\) & \(38.9441\) & \(78.4077\) \\ RMS & \(11.2895\) & \(78.3566\) & \(45.2392\) & \(91.18\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Joint C connection force components. The relevant results for the connecting force in Coupling A and its components are also given in Table 11.  
We compare the results of the second simulation in the Discussion. The results obtained for Joint B in Case 2 are presented in Table 12. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter**Component** & **Joint A** & **Joint A** & **Joint A** & **Joint A** & **Joint A** \\ \hline Min & \(-3.7855\) & 99.3604 & \(-32.0278\) & 99.3604 \\ Max & 32.514 & 210.0206 & 129.7281 & 210.0206 \\ Avg & 15.0504 & 150.641 & 38.4032 & 150.641 \\ RMS & 19.5913 & 153.1624 & 65.5017 & 153.1624 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Joint A connection force components. Figure 15: Connecting forces in rotation Joint A computed with friction consideration: (**a**) in the x-direction; (**b**) in the y-direction; (**c**) in the z-direction; and (**d**) the resultant. To model the contact between the radial bearings and the circular guideway, we defined in ADAMS superior rolling couplings of the outer ring of the bearing on the circular guideway. The ParReEx robot has four bearings that ensure contact between the circular guide and the conjugate element. Figure 17 shows the numerical results obtained by numerical simulation in ADAMS. View for the connection forces in the R1 coupling. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter**Component** & **Joint B Ex** & **Joint B Fy** & **Joint B Fz** & **Joint B MAG** \\ \hline Min & 10.4354 & 0.1475 & 11.5174 & 11.5174 \\ Max & 39.1768 & 0.3846 & 30.0259 & 30.0259 \\ Avg & 24.8299 & 0.2639 & 20.6054 & 20.6054 \\ RMS & 26.8299 & 0.2767 & 21.6025 & 21.6025 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Joint B connection force components. Figure 16: Connecting forces in Joint B calculated assuming friction: (**a**) in the x-direction; (**b**) in the y-direction; (**c**) in the z-direction; and (**d**) the resultant. Table 13 shows the values of the components of the connection forces as well as the resulting reaction in the kinematic joint R1 for the second case study. Figure 18 shows the numerical results obtained by numerical simulation in ADAMS. View for the connection forces in the R2 coupling.