# Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots
## 2 Materials and Methods
### Cosserat Rods Deformation Modelling  
Before solving the position problem of a mechanism formed by slender flexible elements, it is necessary to have a kinematic model of a single rod, which determines the deformation that it takes when it is subjected to a set of external loads. The Cosserat model is a classical approach to this problem, whose mathematical expression may be different depending on whether the rod is discretized in finite elements [39] or whether the continuous expression of the rod is considered [11], the latter being more accurate. In this work, we consider the simplification for the quasi-static case. The shape of a deformed rod is defined by a parametric Cartesian curve in space \(\mathbf{p}(s)\in\mathbb{R}^{3}\) that depicts the centroid of each cross-section, and the rotation matrix \(\mathbf{R}(s)\in\mathrm{SO}(3)\) expressing the orientation of a local frame attached to the section. The \(x\)- and \(y\)-axes are placed on the principal cross-section axes, while the \(z\)-axis is on the tangent to the deformed rod, see Figure 1. Both position and orientation are functions of a scalar reference arc-length parameter \(s\) over the finite interval, \(s\in[0,L]\), where \(L\) is the length of the rod in the reference configuration. It is assumed that cross-sections remain planar. By differentiating both position and orientation functions with respect to the arc length \(s\), the variables that define the longitudinal deformation \(\mathbf{v}\) and the bending deformation \(\mathbf{u}\) between cross sections of the bar are obtained:  
\[\mathbf{p}^{\prime} =\mathbf{R}\mathbf{v} \tag{1}\] \[\mathbf{R}^{\prime} =\mathbf{R}\widehat{\mathbf{u}} \tag{2}\]  
where \(\widehat{\mathbf{u}}=\mathbf{U}(s)\) is a skew-symmetric matrix of the form  
\[\mathbf{U}(s)= \begin{bmatrix}0&-u_{z}(s)&u_{y}(s)\\ u_{z}(s)&0&-u_{x}(s)\\ -u_{y}(s)&u_{x}(s)&0\end{bmatrix}  \tag{3}\]  
The constitutive relationship between strain and stress can be defined depending on the mechanical behaviour of the material. The classical Cosserat rod model assumes that  
Figure 1: Deformed shape of Cosserat rod. there is no cross-section deformation, thus the constitutive relationship can be established between the internal forces \(\mathbf{n}\) and moments \(\mathbf{m}\) and the magnitudes \(\Delta\mathbf{v}\) and \(\Delta\mathbf{u}\):  
\[\mathbf{n} =\mathbf{R}\mathbf{K}_{SE}\Delta\mathbf{v}=\mathbf{R}\mathbf{K}_{ SE}(\mathbf{v}-\mathbf{v}_{\circ}) \tag{4}\] \[\mathbf{m} =\mathbf{R}\mathbf{K}_{BT}\Delta\mathbf{u}=\mathbf{R}\mathbf{K}_ {BT}(\mathbf{u}-\mathbf{u}_{\circ}) \tag{5}\]  
where \(\mathbf{v}_{\circ}\) and \(\mathbf{u}_{\circ}\) are the values at free stress reference condition, and \(\mathbf{K}_{SE}\) and \(\mathbf{K}_{BT}\) are stiffness matrices for shear-extension and bending-torsion, respectively. To establish the equations of equilibrium of forces and moments for every infinitesimal piece of the rod \(ds\), internal forces and moments, and applied distributed forces \(\mathbf{f}(s)\) and moments \(\mathbf{l}(s)\), are considered.