## 4 Designing Sensors for Soft Robots
### Testing Procedures  
During testing, we mechanically forced the flexible module to assume the bending angles of \(\theta\) = \(\pm 30^{\circ}\), \(\pm 60^{\circ}\) and \(\pm 90^{\circ}\), with direction angle \(\beta\) = \(0^{\circ}\). Data from the light intensity modulated sensors were recorded, with channel lengths deduced from changes in reflected light intensity. As a benchmark for validation, the rigid section of the overall system was fixed to a flat tabletop, so that its main axis was parallel to that of the table. This was to keep any bend solely to the \(x\)-\(z\) plane. A ruler, attached to the top of the flexible module, enabled measurement of the bending angle (see Figure 11), testing angles from \(-90^{\circ}\) to \(+90^{\circ}\) at \(30^{\circ}\) intervals. Multiple tests (five) for each angle were conducted and the computed averages were used for further elaboration. At each angle, the centres of the bending circle were experimentally determined by finding the intersection between the ruler's projection line and the \(x\)-axis, which was marked out on the tabletop. In order to set each angle, the flexible module was manually manipulated, using the attached ruler [38]. In order to assess how well the proposed model can predict bend in the flexible module, the changes in wire lengths, \(l_{1}\), \(l_{2}\) and \(l_{3}\) for each tested bending angle are obtained (see Table 1), and then used in conjunction with Equations (2) and (7), to calculate the normal vector \(n_{2}\). Equations (3)-(6) were then employed to determine the corresponding bending circle centre, \(c\), as well as angles \(\theta\) and \(\beta\). This approach allowed the comparison between the sensor data and the tabletop ground truth measurements. A comparison of the experimental and sensor predicted measurements is shown in Table 2. What is most apparent is that at a constant curvature of \(\theta_{m}\) = (\(\pm 30^{\circ}\), \(\pm 60^{\circ}\)), there is close agreement between results. At \(\theta_{m}\) = (\(\pm 90^{\circ}\)) there is, however, increased deviation. We also find that the estimated bending radius \(R\) and angle \(\theta_{s}\) have good accuracy for \(\theta_{m}\) = (\(\pm 30^{\circ}\), \(\pm 60^{\circ}\)) with averaged errors of (4.1 mm, 3.9 mm) for \(R\) and (1.50, 4.50) for \(\theta_{s}\). This increases to 11 mm for \(R\) and 8.30 for \(\theta_{s}\) at testing angles of (\(\pm 90^{\circ}\)). Additionally noteworthy is the good estimation of angle \(\beta\), which corresponds to the orientation of the flexible section with respect to the \(x\)-axis (see Table 2 and Figure 11). Analysis of the discrepancies certainly suggests that this sensor system does show promise as a tool to estimate three-dimensional bending curvature. There are potential extraneous reasons  
Figure 11: Test rig for the evaluation of the bending sensor performance. The ruler attached at the top of the flexible module allowed measuring the bending angle. During calibration, the angular ruler positions were related to the fibre-optic measurements. After calibration the fibre-optic measurements were a good predictor of bending angles [38]. for the errors in \(R\) and \(\theta_{s}\) estimation, either because of the way in which the tests were conducted (partly manually) or because the theoretical assumption relating to constant curvature may be less applicable when working with higher bending angles--certainly a point that needs further investigation. The relatively wide channel diameter of 3.5 mm, around a wire that is just 0.25 mm in diameter, could also add to the inaccuracy of the results. In this instance, the reason would be that while minimal amount of bend in the channel would not immediately lead to a corresponding length change in the wire, as there is too much 'wiggle-room' within the chamber. Future designs would need to look at reducing channel diameter and checking that the material used in the channel lining is sufficiently flexible.