# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 4 Effect of Motion Time on Trajectories  
Motion time is a typical parameter of robot motion planning. To perform a parametric analysis, eight different motion times were considered, ranging from 0.45 s to 0.8 s, and considering stiffness and damping ratio values equal to \(k_{3}=2.6\cdot 10^{-3}\) Nm/rad and \(\zeta_{3}=0.03\), respectively. The corresponding trajectories of the end-effector are represented in Figure 8.  
Figure 8: End-effector trajectories calculated considering different motion times and the same stiffness value of last nonactuated joint (\(2.6\cdot 10^{-3}\) Nm/rad).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stiffness Value (Nm/rad)** & **2-DOF Area (m\({}^{2}\))** & **3-DOF Area (m\({}^{2}\))** & **Reduction (\%)** \\ \hline \(1.5\cdot 10^{-3}\) & 0.2065 & 0.1320 & 36.08 \\ \(2.5\cdot 10^{-3}\) & 0.2065 & 0.1468 & 28.91 \\ \(4.0\cdot 10^{-3}\) & 0.2065 & 0.1884 & 8.77 \\ \(5.0\cdot 10^{-3}\) & 0.2065 & 0.1989 & 3.68 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Area reduction using the 3-DOF underactuated robot considering the same motion time (\(t_{f}=0.6\) s).  
It is worth noting that the trajectories obtained with varying motion times are similar to the ones obtained in Section 3. This result suggests that the same end-effector trajectory can be obtained either by selecting the torsional stiffness of the nonactuated joint or by setting the motion time. This is an advantage, because it is possible to operate on two different parameters to achieve the same result.  
The joint variables of the robot are depicted in Figure 9. In this case, the joint variable \(q_{1}\) depends on motion time, like the other joint variables. It is interesting to notice that the maximum values of \(q_{3}\) obtained in these simulations are similar to the ones obtained considering different values of joint stiffness \(k_{3}\).  
Figure 10 depicts motor torques calculated considering different motion times and the same torsional stiffness of the nonactuated joint. In this case, both motor torques change their amplitude and shape when the motion time changes. Finally, Figure 11 shows the swept areas corresponding to four different motion times. The motion time has a scarce effect on the behavior of the system in the third and fourth quadrants, but the decrease in motion time causes a significant decrease in the swept areas in the first and second quadrants (in the central part of the motion).  
Figure 10: Motor torques calculated considering different motion times and the same stiffness value of the last nonactuated joint (\(2.6\cdot 10^{-3}\,\mathrm{Nm}/\mathrm{rad}\)).  
Figure 9: Joint variables of the robot calculated considering different motion times and the same stiffness value of last nonactuated joint (\(2.6\cdot 10^{-3}\,\mathrm{Nm}/\mathrm{rad}\)).  
Figure 11a,b depict a behavior similar to the one described in the case of stiffness variations. If motion time is short, large accelerations of the last link are required to meet the imposed end conditions; the result is a swept area in which the outermost point is not the tip of the end-effector, but the back-end of the third link (with the counterbalance). In Figure 11c,d the motion time is longer, and the motion of the third link does not cause the abovementioned phenomenon.  
Finally, the swept areas of the underactuated system are compared to those of a fully actuated system with two degrees of freedom (DOF), having \(a_{2}^{{}^{\prime}}=a_{2}+a_{3}\). The calculated results are reported in Table 4. When the underactuated robot is quickly moved, there is a reduction in swept area of about 35%.