## 3 Methods
### Simulation  
Using the _bvp4c_ solver in _Matlab R2021a_, we implemented the static case of the Cosserat beam (Equations (11) and (12)) with the parameters in Table 1. Since the actuators consist of components with different weights, the beam is divided into six regions, as shown in Table 2. Gravitational forces corresponding to the weight of individual regions are considered as a line load in the \(z\)-direction. Some of the regions overlap at the real actuator, e.g., the chambers are slightly inserted to the end caps. The regions are chosen such that the distribution of weight of two overlapping components combined in one region is approximately balanced. According to Equation (29), the bending stiffness depends on the axial stretch ratio \(\frac{L_{c}}{L_{c}\cdot 0}\) of the actuator, which is represented by \(v_{3}\) in the Cosserat beam model. We assumed infinite extensional stiffness and bending stiffness for the stiff components of the actuator, and an infinite shear stiffness and torsional stiffness for all components.  
According to Equations (11) and (12), line loads along the chambers (regions 2 and 4 of the beam, Table 2) are applied due to pressurization. Additionally, counteracting jump conditions are applied at the ends of the chambers. For every position of the three-dimensional measurements, we know the pressurization, from which we calculate the force \(F_{k}\) induced at the individual chambers. The axial force \(F\) is equal to the sum of forces \(F_{k}\) of the individual chambers. The moments \(M_{1}\) and \(M_{2}\) correspond to the forces \(F_{k}\) of the individual chambers and their distance to the centerline in the \(x\)-direction and \(y\)-direction, respectively.  
The absolute error of a simulation is the Euclidean distance between the tip position in the experiment and in the simulation. The relative error is determined with two different reference lengths: the length of the actuator, as is common in the literature [3; 8; 10; 12; 23], and the deflection of the tip in the experiment [11], as a more refined reference method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(L_{c,0}\) in \(\mathbf{m}\) & \(L_{0}\) in \(\mathbf{m}\) & \(E=3\cdot E^{I}\) in \(\mathbf{N}\) & \(E^{II}\) in \(\mathbf{N}\) & \(E^{III}\) in \(\mathbf{N}\) & \(r^{I}\) in \(\mathbf{m}\) \\ \hline Actuator 1 & \(44.95\cdot 10^{-3}\) & \(105.9\cdot 10^{-3}\) & & & & \\ \hline Actuator 2 & \(44.8\cdot 10^{-3}\) & \(105.6\cdot 10^{-3}\) & 200 & 0 & 0 & \(\frac{20}{\sqrt{3}}\cdot 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the actuators used in the simulation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Region** & **Components** & **Weight in kg** & **Length in m** & **Annotation** \\ \hline
1 & End cap 1 & \(3.97\cdot 10^{-3}\) & \(3.5\cdot 10^{-3}\) & \\ \hline
2 & Chamber + rings (\(L_{c,0}\)) & \(25\cdot 10^{-3}\) & \(44.95\cdot 10^{-3}\) & Actuator 1 \\ \cline{2-5}  & & & \(44.8\cdot 10^{-3}\) & Actuator 2 \\ \hline
3 & Connector cap & \(8.26\cdot 10^{-3}\) & \(9\cdot 10^{-3}\) & \\ \hline
4 & Chamber + rings (\(L_{c,0}\)) & \(25\cdot 10^{-3}\) & \(44.95\cdot 10^{-3}\) & Actuator 1 \\ \cline{2-5}  & & & \(44.8\cdot 10^{-3}\) & Actuator 2 \\ \hline
5 &
\begin{tabular}{c} End cap 2 + mounting \\ plate marker \\ \end{tabular} & \(13.97\cdot 10^{-3}\) & \(7.5\cdot 10^{-3}\) & \\ \hline
6 & Marker & \(11.4\cdot 10^{-3}\) & \(17.2\cdot 10^{-3}\) & No load applied \\ \cline{2-5}  & Marker + load & \(113.8\cdot 10^{-3}\) & \(34.9\cdot 10^{-3}\) & Load applied \\ \hline \hline \end{tabular}
\end{table}
Table 2: Weight and dimensions of actuators used in the simulation, from fixed end to free end (each two alternatives for regions 2, 4 and 6).  
The error of the latter is higher since the deflection is typically lower than the length of the actuator.