# Open Loop Position Control of Soft Hydraulic Actuators for Minimally Invasive Surgery
## 3 Materials and Methods  
With reference to Figure 1, the parameters used to manufacture the actuators are shown in Table 1. The value of \(L_{0}\) was calculated before the operating range was defined, and an ideal actuator of this length would have been capable of 25 mm contraction. A total of 2 mm had also been added to the length to compensate for the width of the laser welded seams.  
An important step towards achieving predictable contraction was to adjust for the difference between the theoretical volume and the real-world volume in the manufactured actuators. Note that this calibration step is distinct from the calibration step completed at the start of every test in order to zero the volume. The calibration to zero the volume is described with the pump hardware in Section 3.2.  
The actuator was filled at a constant volume flow rate up to a gauge pressure of 75 kPa, while the contraction was monitored using an optical tracking system (Optitrack). The contraction was then plotted against the volume inside the actuator as shown in Figure 2.  
The theoretical curve plotted in Figure 2 represents the ideal volume/length curve for an actuator with the geometry described in Table 1. The calibrated curve is defined by multiplying the theoretical volume by a factor, \(k_{v}\), which in this case was equal to 0.815. The calibrated expression for volume of the actuator as a function of membrane curvature is therefore  
\[V_{c}(\theta)=k_{v}\frac{L_{0}^{2}}{n_{L}}\bigg{(}\frac{d_{c}}{3}+\frac{D_{s}} {2}\bigg{)}\bigg{(}\frac{\theta-\cos\theta\sin\theta}{\theta^{2}}\bigg{)} \tag{3}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter Name** & **Value** \\ \hline \(n_{L}\) & 3 \\ \(L_{0}\) & 70.80 mm \\ \(D_{s}\) & 20 mm \\ \(d_{c}\) & 15 mm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Actuator design parameters.  
Note also that the actuator stopped contracting after it had been filled to a certain level. The contraction at which this happens for this specific actuator is approximately 18 mm, and this would inform the operating range of the actuators.  
Two different considerations affected the minimum and maximum contractions and meant that constraints were placed on each in the control system. Firstly, it was observed that the maximum contraction, \(L_{Cmax}\), could not be reached, and this may be because the membrane material had non-zero bending stiffness, non-zero thickness, and finite Young's modulus. Secondly, when the actuator is contracted by only a small amount, the volume flow rate has to be high to achieve even low contraction speeds. Therefore, the minimum contraction was not set to zero to allow for reasonable actuation speeds when the actuators were used in antagonistic or parallel arrangements and the speeds must'match'.  
In antagonistic and parallel arrangements, it may be the case that one actuator is contracted a little and can only move slowly, while another is significantly contracted and is able to move quickly due to the high volume change to length change sensitivity in this state. The actuation speed is therefore limited by the slower actuator as they must move together.  
Consider that when a theoretical actuator is half full it will have contracted by only approximately 10% of its full potential stroke; conversely, when a theoretical actuator has contracted by half of its stroke, it will have to be filled to approximately 90% of its maximum volume.