# Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots
## 2 Materials and Methods
### Analysis of Flexible Manipulators with Cosserat Rods  
Let us consider now a parallel mechanism of flexible rods (see Figure 2), where a rigid end-effector is connected to \(N\) flexible rods \(A_{i}B_{i}\), controlled by some actuators. By attaching a moving frame to the end-effector at reference point \(\mathcal{P}\), its orientation can be defined either using a rotation matrix \(\mathbf{R}_{EE}\) or a unit quaternion \(\mathbf{\check{e}}\). The position vector of the rods' attachment-points \(B_{i}\) expressed in the end-effector frame is \(\mathbf{r}_{i}\). The position vector of the rods' proximal-ends \(A_{i}\) expressed in the fixed frame is \(\mathbf{a}_{i}\). Bear in mind that to obtain a feasible pose of the flexible mechanism, there are two conditions that must be satisfied by the deformed shapes of rods: a geometrical assembly with the rigid end-effector, and the static equilibrium of the system. Regarding the first condition, we have to equate the coordinates of points \(B_{i}\) at the end-effector with those of the distal-ends of the rods. Thus,  
\[\mathbf{p}_{\mathcal{P}}+\mathbf{R}_{EE}\mathbf{r}_{i}=\mathbf{a}_{i}+\mathbf{ p}_{i}(L_{i})\ \ \ i=1\ldots N \tag{15}\]  
where \(\mathbf{p}_{i}(L_{i})\) is the location of the distal-end of the rod from the proximal-end obtained from the deformation model described in Section 2.1. Figure 2: Schematic representation of a general PCM. Also, geometrical conditions depending on how rods' distal ends are joined to the end-effector must be applied. The simplest connection, through spherical joints, makes a null moment appear at the distal-ends. Bearing in mind that \(\mathbf{u}=\mathbf{K}_{BT}^{-1}\mathbf{R}^{T}\mathbf{m}\), we can state the following:  
\[\mathbf{u}_{i}(L_{i})=\mathbf{0}\quad i=1\dots N \tag{16}\]  
Regarding static equilibrium conditions, these are stated for the whole device, which means that the end-effector and all the rods' values are highly coupled in the set of equations. The end-effector is subjected to external load \(\mathbf{F}_{ext}\) at \(\mathcal{P}\) and \(\mathbf{M}_{ext}\), and to reaction forces and moments at \(B_{i}\) attachments due to the rods' deformation, namely \(\mathbf{n}_{i}(L_{i})\) and \(\mathbf{m}_{i}(L_{i})\), respectively. And so, the following static equilibrium can be stated:  
\[ \begin{split}\sum_{i=1}^{N}\mathbf{n}_{i}(L_{i})-\mathbf{F}_{ext }=\mathbf{0}\\ \sum_{i=1}^{N}[(\mathbf{a}_{i}+\mathbf{p}_{i}(L_{i}))\times \mathbf{n}_{i}(L_{i})+\mathbf{m}_{i}(L_{i})]-\\ -\mathbf{p}_{\mathcal{P}}\times\mathbf{F}_{ext}-\mathbf{M}_{ext }=\mathbf{0}\end{split}  \tag{17}\]  
In summary, the position problem involves the following equations: the geometric conditions on the position of the distal-ends (15) at each rod; the geometric conditions on the attachment of the distal-ends to the end-effector (16); and the static equilibrium conditions on the end-effector (17), resulting in a total of \(6N+6\) conditions. If the orientation of the end-effector \(\mathbf{R}_{EE}\) was expressed with a unit quaternion \(\mathbf{\bar{e}}\), another condition has to be added, namely the unity condition of the quaternion:  
\[e_{0}^{2}+e_{1}^{2}+e_{2}^{2}+e_{3}^{2}=1 \tag{18}\]  
This results in a problem with \(6N+7\) conditions. In the solving process, which follows an iterative numeric approach, a vector of residuals \(\mathbf{g}_{res}\) is created with a chosen order to lower the computational cost, namely  
\[\mathbf{g}_{res}=\left\{ \begin{array}{c}\mathbf{p}_{i}(L_{i})-\mathbf{p}- \mathbf{R}_{EE}\mathbf{r}_{i}+\mathbf{a}_{i}\\ \mathbf{u}_{i}(L_{i})\\ \vdots\end{array} \right\} \tag{19}\]  
Due to the integration involved in each rod, the solution must be approached as a boundary value problem. For each rod, the vector of dependent variables at \(s=0\), \(\mathbf{y}_{i}(0)\), is introduced for integration along \(L_{i}\) with Runge-Kutta. Some of those variables may be known data from boundary conditions while some are guess values for unknowns.