# Preliminary Analysis of a Lightweight and Deployable Soft Robot for Space Applications
## 4 Results  
The experimental tests were carried out with constant inflation pressure value, considering pressures in the p = 10-60 kPa range. During the tests, the first wrinkle has occurred at the distance \(l_{1}=30\) mm from the fixed joint, so it can be considered in the link model as the length of the first body. Having set the distance \(h=500\) mm for each test, through trigonometric considerations, the angular deflection \(\theta\) and the torque \(\tau\) can be calculated, assuming small-angle approximation, from the recorded force \(F\) and the position set \(v_{UR}\). Results are plotted in Figure 9. According to the experimental data, the stiffness characteristic below the theoretical wrinkling moment value, given by (2), can be approximated as linear. The inflatable link linear model can be considered valid until \(\tau<M_{w}\). Consequently, for each test, a linear regression of the points below wrinkling conditions is performed to evaluate the stiffness \(k\). The average torque of the points over wrinkling  
Figure 8: Static test for stiffness estimation, experimental set-up. Figure 7: Cantilever beam, experimental set-up. conditions \(M_{max}\) allows to trace a characteristic for the entire range of the examined angles, assuming \(M_{max}\) as a saturation value. Results are summarized in Table 3, where the estimated stiffness \(k\) with its standard deviation \(\sigma_{k}\), the theoretical wrinkling moment \(M_{w}\) and the average maximum experimental torque \(M_{max}\) are reported. The experimental maximum moment \(M_{max}\) follows the trend of the theoretical wrinkling moment \(M_{w}\). The internal pressure \(p\) establishes the wrinkling moment, and so the maximum torque the link can support. On the other hand, the estimated \(k\) values do not show a clear correlation with the pressure, and their variations in the range 155.89-187.30 Nm/rad could be due to the quality of the support. This result is consistent with the prediction from Equation (1). Using the average of the stiffness values \(\overline{k}=172.71\) Nm/rad, a characteristic of the link is built, making it possible to estimate the unknown Young's modulus \(E\) of the PVC coated fabric, knowing the features of the link, and setting the pressure. The computed values of the Young's modulus result in the range E = 136-140 MPa for pressures from 10 kPa to 60 kPa, hence its dependence on pressure in this range can be considered negligible. In Figure 10 the configurations of the soft robot during a movement of the EE from a starting point \(\mathbf{x}_{start}\) to an end point \(\mathbf{x}_{d}\), following a planned trajectory, using the elastostatic inverse kinematics algorithm, are shown. The simulation utilizes as stiffness the average of experimental stiffness values \(\overline{k}\) and as external load a sample value \(\mathbf{f}_{e}=[0,\ 10,\ -15]\) N.  
Figure 9: Experimental data and linear model on varying pressures for the considered inflatable link. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{p}\):[**kPa**] & \(k\)**[Nm/rad]** & \(\mathbf{\sigma_{k}}\)**-[Nm/rad]** & \(\mathbf{M_{max}}\)**-[Nm]** & \(\mathbf{M_{w}}\)**-[Nm]** \\ \hline
10 & 182.56 & 3.37 & 4.99 & 4.11 \\
20 & 187.30 & 1.48 & 9.45 & 8.21 \\
30 & 155.89 & 2.66 & 14.99 & 12.32 \\
40 & 174.42 & 6.22 & 18.77 & 16.42 \\
50 & 169.44 & 5.51 & 21.76 & 20.53 \\
60 & 166.66 & 4.99 & N/A & 24.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results from the inflatable link characterization. The robot model effectively follows the planned trajectory, setting the rotations of the motor joints to compensate the deformations of the inflatable links. This aspect is underlined in Figure 11, where the values of motor joints angles computed by the elastostatic algorithm are compared with those defined by a rigid body reverse kinematic algorithm. During the simulation, the maximum moments acting on the links are identified and compared to the experimental data to find the minimum internal pressure required to ensure the validity of the model.