# Open Loop Position Control of Soft Hydraulic Actuators for Minimally Invasive Surgery
## 2 Soft Hydraulic Actuators  
### Length Change  
The soft actuators are formed from heat-sealed membranes that consist of a single chamber. When this chamber is pressurised or filled via the inlet, the volume of the chamber changes and the membranes, or walls, become curved. As a result of the two membranes bending, the distance between long edges of the rectangle is reduced as the curvature of the membranes increases, up to a limit. The distance between the edges, or the curvature dependent length, \(L(\theta)\), of the soft actuator is highlighted in Figure 1A and described by the equation  
\[L(\theta)=L_{0}\frac{\sin\theta}{\theta} \tag{1}\]  
where \(L_{0}\) is the length of the empty actuator, and \(\theta\) is half the central angle of the segment made when the actuator is contracted by some amount. This is as described in [19; 20]. In our design, we set the number of sub-chambers, which we will call \(n_{L}\), as three. The expression in (1) does not change as there are effectively \(n_{L}\) actuators of length \(L_{0}/n_{L}\).  
In practice, the sheet material that the soft actuators were made from did not behave exactly as described by the theory, and for this reason the original rectangular shape was redesigned. The laser welding manufacturing method from [9] was used, which enables rapid and low-cost manufacture. To take advantage of the manufacturing method's flexibility, a parametric design approach is followed that allows rapid customisation of the actuators. Figure 1 shows key parameters of the rectangular actuator design and our new design.  
Figure 1: Key dimensions of a single actuator chamber. (**A**) Cross section of a partially contracted actuator. (**B**) Dimensions of a typical rectangular Pouch Motor (left) and our actuator design (right). (**C**) Images of actuator in a test setup.  
When filled, the ends of the rectangular actuator tended to have an irregular shape, crinkling and buckling the membrane in complex ways. The material at the ends was seen to bend perpendicular to the major axis of the cylindrical central segment. Bending in this direction prevents contraction of the actuator, so improving this was part of the motivation for a new design. Another consequence is that the theoretical expression for the volume would not describe the real volume of the actuator.  
As such, the rectangular ends of the actuators were replaced with a triangular pattern, so that the ends would approximate conics when filled with actuation fluid. The purpose of this was to make the curvature from the base to the tip of the conic ends constant and therefore constant along the actuator's entire width.  
The new theoretical volume is found by integrating the cross-sectional area along the width of the conic end. The resulting volume of an actuator made from three sub-chambers with triangular ends, each of width \(d_{c}\), and a centre section of width \(D_{s}\), is given in the following equation:  
\[V_{c}(\theta)=\frac{L_{0}^{2}}{n_{L}}\bigg{(}\frac{d_{c}}{3}+\frac{D_{s}}{2} \bigg{)}\bigg{(}\frac{\theta-\cos\theta\sin\theta}{\theta^{2}}\bigg{)} \tag{2}\]  
Figure 1 highlights the parameters of the actuator. Using this actuator design, the membrane became less irregular at the ends, and the ends had a smooth conic shape when filled. This was done with the aim of improving durability and repeatability, because unpredictable buckling was less common with the new design.