# Analysis of Dynamic Behavior of ParReEx Robot Used in Upper Limb Rehabilitation
## 3 Dynamic Analysis in MSC.ADAMS of ParReEx-Wrist Robot
The material properties of level gears are shown in Table 2. Further, on the level gear, the contact parameters between the tooth flanks are set, as shown in Table 3, namely: contact stiffness, contact exponent and damping coefficient. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Gear 1** & **Gear 2** \\ \hline No. of teeth & 39 & 39 \\ Orientation of rotation axis & 0, 90, 0 & 90, 90, 0 \\ Face width (mm) & 25.4 & 25.4 \\ Back cone distance (mm) & 10 & 15 \\ Pitch angle (\({}^{\circ}\)) & 19.747 & 70.253 \\ Pitch apex & 0 & 0 \\ Bore radius (mm) & 2 & 2 \\ Mean spiral angle (\({}^{\circ}\)) and direction & 35, LH & 35, RH \\ Outer trans. module (mm) & 4.535 & 4.535 \\ Tooth depth and width & \multicolumn{2}{c}{Data Type I-ISO23509} \\ Profile shift coefficient & 0.505 & \(-0.505\) \\ Dedendum factor & 1.25 & 1.25 \\ Thickness mod. coef. & \(3.7\times 10^{-2}\) & \(-5.5\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bevel gear geometric parameters. Figure 4: Kinematic joints defined in the ADAMS dynamic model. The law of MOTION 1, which corresponds to the flexion-extension movement, has a variation of \(\pm 50^{\circ}\), with one step of the flexion movement performed in 5.7 s.  
The law of MOTION 2 corresponds to ulnar-radial motion; it is specified on the same 5.7 s step, but the amplitude is 30 degrees. MOTION 1: STEP (time, 0.0, 0.0d, 5.7, \(-50.0\)d) + STEP (time, 5.7, 0d, 11.4, 50.0d) + STEP (time, 11.4, 0.0d, 17.1, \(-50.0\)d) + STEP (time, 17.1, 0d, 22.8, 50.0d). MOTION 2: STEP (time, 0.0, 0.0d, 5.7, 30.0d) + STEP (time, 5.7, 0d, 11.4, \(-30.0\)d) + STEP (time, 11.4, 0.0d, 17.1, 30.0d) + STEP (time, 17.1, 0d, 22.8, \(-30.0\)d). The mannequin-rehabilitation robot assembly is dynamically analyzed in MSC.ADAMS with the two modules of the ParReEx-wrist robot (flexion/extension and radial-ulnar motion) active simultaneously for a period of 22.8 s. Figure 6 presents the two extreme operating positions of the ParReEx robot corresponding to the combined flexion-extension and ulnar-radial movements. In the first phase, a dynamic simulation with the specified laws of motion and without considering the friction in the kinematic joints of the robot is performed. The results obtained for this simulation--meaning the variations of the motor torque calculated by numerical simulation in ADAMS when simultaneously performing both types of movement: flexion-extension and radial-ulnar motion of the wrist--are presented in Figure 7.  
Figure 5: Law of motion imposed in ADAMS on ParReEx-wrist robot motor Joint E. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Stiffness (N/mm\({}^{2}\)) & \(5\times 10^{5}\) \\ Contact exponent & \(1.1\) \\ Damping coefficient & \(5000\) \\ Penetration (mm) & \(0.01\) \\ Static friction coefficient & \(0.11\) \\ Static friction transition velocity (mm/s) & \(0.2\) \\ Dynamic friction coefficient & \(0.10\) \\ Dynamic friction transition velocity (mm/s) & \(0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bevel gear contact parameters. Analyzing the graph in Figure 7a computed for the flexion-extension motion, it is found that the maximum value of the actuation moment of this flexion/extension module is 0.7436 Nm, while the minimum value is near 0 Nm. The motor torque obtained by numerical simulation for the radial-ulnar motion (Figure 7b) has a maximum value around 3.4748 Nm and a minimum of 2.47 Nm. In conclusion, the radial-ulnar movement requires an actuation moment greater than about four times the moment required for the flexion-extension movement, so the flexion-extension module will be the most structurally loaded. For the two modules, the graphs of the connecting forces that develop in Joints A, B, R1, R2 and C are presented. Figure 8 shows the orientation of the local reference axis systems attached to the kinematic torques of the ParReEx robot for clear identification of the components of the connecting forces.