# Analysis of Dynamic Behavior of ParReEx Robot Used in Upper Limb Rehabilitation
## 3 Dynamic Analysis in MSC.ADAMS of ParReEx-Wrist Robot
Figure 6: The two extreme positions during the operation of the virtual model of the ParReEx-wrist robot, corresponding to combined flexion–extension and radial–ulnar motion. Figure 7: ADAMS simulation computed torque for: (**a**) the flexion–extension module and (**b**) the radial–ulnar module. Graphs of time variation for the components of the connecting force in Joint A in relation to the local reference system attached to this joint are shown in Figure 9a-c. Analysis of these graphs shows that the maximum value recorded along the Y-axis of the local reference system in Joint A is 111.7138 N, and the minimum is 60.8693 N, with an RMS (round mean square) equal to 95.1698 N (Figure 9b). The component along the X-axis reaches an absolute maximum value of 20.8049 N, and the minimum value is equal to 0.6315 N, with an RMS of 13.9479 N (Figure 9a). The component along the Z-axis reaches 88.6393 N with an RMS of 49.6273 N (Figure 9c). The value of the resulting force in this joint has a maximum of 142.1042 N and a minimum of 64.3436 N according to Figure 9d. Figure 8: Positioning of the axes of the kinematic joints of the ParReEx robot. Figure 9: Connecting forces in rotation Joint A: (**a**) in the x-direction; (**b**) in the y-direction; (**c**) in the z-direction; and (**d**) the resultant. These results are also presented in Table 4.  
Graphs of time variation for the components of the connecting force in Joint B are shown in Figure 10a-c. The maximum value reported for the X-axis component is 25.8542 N, and the minimum is 11.002 N, with an RMS equal to 20.1699 N (Figure 10a). The Y-axis component has a maximum value equal to 0.3771 N, and the minimum value is equal to 0.2556 N, with an RMS of 0.3118 N (Figure 10b) The component along the Z-axis reaches 29.4384 N with an RMS of 24.3394 N (Figure 10c). The value of the resulting force in this joint reaches a maximum of about 32.6581 N, a minimum of 30.7333 N and an RMS equal to 31.6122 N, as we can see in Figure 10d. These results are also shown in Table 5.  
There is a higher load distribution on Coupling A, i.e., the RMS value is 3.48 times higher for the force resulting from Joint A. To model the contact between the radial bearings and the rolling path of the circular guide, we defined in ADAMS primitives such as the upper rolling torque between the outer ring of the bearing on the rolling path of the circular guide. The ParReEx robot has four bearings in the structure that make contact between the circular guide and the adjacent element. Similar graphs are obtained for Joints R1 and R2 (Figures 11 and 12). The values of these results are presented in tabular form in Tables 6 and 7. For the situation in which the ParReEx robot is used by a patient and the frictional forces in the kinematic torques are not taken into account, the following results are obtained for the connecting forces in the Joint C, as presented in Figure 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter**Component** & **Joint R1 Ex** & **Joint R1 Fy** & **Joint R1 Ez** & **Joint R1 MAG** \\ \hline Min & \(-0.1407\) & \(-51.1415\) & \(-39.9417\) & \(35.325\) \\ Max & \(13.994\) & \(-32.1514\) & \(4.533\) & \(63.5811\) \\ Avg & \(7.2739\) & \(-44.6805\) & \(-15.7091\) & \(50.263\) \\ RMS & \(8.9958\) & \(45.1932\) & \(22.3775\) & \(51.226\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Joint R1 connection force components. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter**Component** & **Joint R2 Ex** & **Joint R2 Fy** & **Joint R2 Ez** & **Joint R2 MAG** \\ \hline Min & \(-14.4231\) & \(28.4593\) & \(-17.9002\) & \(36.502\) \\ Max & \(6.4731\) & \(61.2824\) & \(-19.281\) & \(64.5064\) \\ Avg & \(-5.6765\) & \(48.3102\) & \(-2.6405\) & \(21.6015\) \\ RMS & \(9.5789\) & \(49.7893\) & \(13.7125\) & \(52.39\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Joint R2 connection force components.