## 2 Model
### Cosserat Beam  
As implied in the introduction, the Cosserat beam is an established modeling approach in soft robotics. It extends the Euler-Bernoulli beam to include large deformations, axial stretch, twist and shear, which is necessary in order to cover the full range of deformations of pneumatic soft robots. This brief introduction to the Cosserat beam theory is based on [PERSON] [13] and [PERSON] et al. [14]. Referring to Figure 1, the beam is represented by its centerline  
\[\mathbf{q}(s)=x(s)\mathbf{e}_{1}+y(s)\mathbf{e}_{2}+(s+z(s))\mathbf{e}_{3}, \tag{1}\]  
where \(s\) is the arc-length parameter and \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) is a global coordinate system. The global coordinate system is supplemented by a local coordinate system \(\{\mathbf{d}_{1},\mathbf{d}_{2},\mathbf{d}_{3}\}\), so-called directors, where \(\mathbf{d}_{1}\) and \(\mathbf{d}_{2}\) are the principal axes of inertia of the beam and  
\[\mathbf{d}_{3}(s)=\mathbf{d}_{1}(s)\times\mathbf{d}_{2}(s). \tag{2}\]  
An arbitrary vector can be transformed between the two coordinate systems by three successive rotations. The static deformation of the beam results from the equilibrium of forces and moments  
\[\mathbf{n}^{\prime}(s)+\mathbf{f}(s)=\mathbf{0}, \tag{3}\] \[\mathbf{m}^{\prime}(s)+\mathbf{q}^{\prime}(s)\times\mathbf{n}(s)+ \mathbf{l}(s)=\mathbf{0}, \tag{4}\]  
where \(\mathbf{f}\) (force) and \(\mathbf{l}\) (moment) are external line loads, \(\mathbf{n}\) is the contact force (shear \(n_{1/2}\), stretch \(n_{3}\)), and \(\mathbf{m}\) is the contact moment (bending \(m_{1/2}\), torsion \(m_{3}\)). The superscript \((\cdot)^{\prime}\) denotes the derivation with respect to the arc-length parameter \(s\). Figure 1: Coordinate systems of the Cosserat beam (based on [PERSON] et al. [14]). The magnitude of deformation is measured by \(\mathbf{v}\) (shear \(v_{1/2}\), axial stretch \(v_{3}\)) and \(\mathbf{u}\) (curvature \(u_{1/2}\), torsion \(u_{3}\)), where  
\[\mathbf{v}(s)=\mathbf{q}(s)^{\prime}, \tag{5}\] \[\mathbf{d}(s)^{\prime}=\mathbf{u}(s)\times\mathbf{d}(s). \tag{6}\]  
The contact forces and moments result from  
\[\mathbf{n}=\mathbf{K}(\mathbf{v}-\mathbf{v}_{0}), \tag{7}\] \[\mathbf{m}=\mathbf{J}(\mathbf{u}-\mathbf{u}_{0}). \tag{8}\]  
The subscript \((. )_{0}\) denotes the initial position of the beam, with \(\mathbf{v}_{0}=(0,0,1)\) and \(\mathbf{u}_{0}=\mathbf{0}\) representing a straight beam in the \(\mathbf{e}_{3}\)-direction. Assuming linear-elastic material behavior,  
\[\mathbf{K}=K_{ij}(\mathbf{d}_{i}\otimes\mathbf{d}_{j}) \tag{9}\] \[K_{ij}= \begin{bmatrix}S&0&0\\ 0&S&0\\ 0&0&E\end{bmatrix} ,\]  
where \(S\) is the shear stiffness and \(E\) is the extensional stiffness. Analogously,  
\[\mathbf{J}=J_{ij}(\mathbf{d}_{i}\otimes\mathbf{d}_{j}) \tag{10}\] \[J_{ij}= \begin{bmatrix}B&0&0\\ 0&B&0\\ 0&0&T\end{bmatrix} ,\]  
where \(B\) is the bending stiffness and \(T\) is the torsional stiffness. For ease of reading, the stiffnesses \(S\), \(E\), \(B\) and \(T\) are combined quantities which comprise material stiffness as well as geometrical aspects, e.g., the unit of the extensional stiffness \(E\) is \(\,\mathbf{N}\) and the unit of the bending stiffness \(B\) is \(\,\mathbf{N}\mathbf{m}^{2}\). Similar to others [3; 8; 9; 10; 11], in this study we restrict ourselves to extensional stiffness \(E\) and bending stiffness \(B\) when determining beam parameters and performing simulations. The shear stiffness \(S\) and the torsional stiffness \(T\) are considered infinite. An important aspect being related to pneumatic soft robots is the consideration of pressurization in the beam model. One opportunity is to assume an initial deformation induced by pressure [8; 10].