# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 6 Experimental Validation
### Experimental Results  
Three different tests were performed with varying motion time: the first one was carried with motion time \(t_{f}=0.55\) s, the second one with \(t_{f}=0.6\) s, and the last one with \(t_{f}=0.7\) s. These experimental tests aimed to calculate the error between the simulated and experimental trajectories of the robot and the swept areas. Figures 14-16 show the trajectories of the end-effector (solid trajectories), of the back-end (counterbalancing mass) of the second link (dashed trajectories), and of the swept areas, and make comparisons between simulated and experimental results. There is a good agreement between the numerical and experimental trajectories in every test. The largest differences between calculated and measured values take place in the zone of the protuberance, especially when the motion time is short.  
Figure 13: Experimental setup: (**a**) the Portescap 35 NT2R 82 426 SP motor equipped with an incremental optical encoder LIKA Electronic I58-L-5000 ZCU16R; (**b**) the 2-DOF robot.  
Figure 14: Numerical and experimental trajectories with \(t_{f}=0.55\) s. Blue and orange lines show the numerical and experimental trajectories, respectively. Solid and dashed lines show the tip and the back-end of the end-effector, respectively.  
Figure 15: Numerical and experimental trajectories with \(t_{f}=0.6\) s. Blue and orange lines show the numerical and experimental trajectories, respectively. Solid and dashed lines show the tip and the back-end of the end-effector, respectively.  
The swept areas calculated from numerical simulations and estimated from the experimental data are compared in Table 5. There is always a good agreement between the numerical and the experimental values, with a maximum error of 1.28%.  
In addition, a comparison between the area swept by a 1-DOF robot (with link length \(a_{2}^{\prime}=a_{2}+a_{3}\)) and the one swept by the prototype 2-DOF underactuated robot is made. The results are summarized in Table 6. The area swept by the underactuated robot is smaller than the one swept by the 1-DOF robot to perform the same test. The reduction increases when the motion time decreases. This is an advantage in many robotic applications, particularly in environments where space is limited or where the robot must interact with other objects or people.  
All the experimental results presented in this section were retrieved with the torsional spring working in the linear range, as assumed in the mathematical model. For very short motion times, the torques applied by the actuators may result in very large rotations of the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Motion Time (s)** & **1-DOF Area (m2)** & **2-DOF Area (m2)** & **Reduction (\%)** \\ \hline
0.55 & 0.0693 & 0.0544 & 21.50 \\
0.6 & 0.0693 & 0.0558 & 19.48 \\
0.7 & 0.0693 & 0.0611 & 11.83 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Area reduction using the 2-DOF underactuated robot.  
Figure 16: Numerical and experimental trajectories with \(t_{f}=0.7\,\mathrm{s}\). Blue and orange lines show the numerical and experimental trajectories, respectively. Solid and dashed lines show the tip and the back-end of the end-effector, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Motion Time (s)** & **Numerical Area (m2)** & **Experimental Area (m2)** & **Error (\%)** \\ \hline
0.55 & 0.0537 & 0.0544 & 1.28 \\
0.6 & 0.0564 & 0.0558 & 1.07 \\
0.7 & 0.0605 & 0.0611 & 0.95 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of area estimation.  
nonactuated joint, which, in turn, may lead to the nonlinear behavior of the torsional spring. Therefore, the choice of the torsional spring has to take into account the compatibility of the range of torsion with linear elastic behavior.