## 2 Materials and Methods
### Optimal Synthesis Problem and Its Formulation
\begin{array}{cc}\mathbf{n}_{i}&\mathbf{Z} \cdot(\mathbf{R}_{\Phi}\cdot\mathbf{B}_{1})\times\mathbf{n}_{i}\end{array}  \right]^{\text{T}}\text{, \ }i=1.4\text{,} \tag{6}\]  
\(\mathbf{B}_{1}\) is the vector containing the anchor points coordinates (\(\mathcal{B}_{i}\)) in the local frame. Figure 5: Geometric representation of the planar robot. Thus, the expression of the cable tensions vector \(\mathbf{T}\) can be deduced using Equation (5) as follows:  
\[\mathbf{T}=\mathbf{T_{p}}+\mathbf{T_{h}}=\left(\mathbf{J^{T}}\right)^{+}\left( \mathbf{M}\tilde{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}-\mathbf{f_{g}}-\mathbf{ F_{ext/O_{L}}}\right)+\lambda\ \text{Null}\left(\mathbf{J^{T}}\right) \tag{7}\]  
where \(\mathbf{T_{p}}\) and \(\mathbf{T_{h}}\) are the particular and the homogenous solutions, \(\left(\mathbf{J^{T}}\right)^{+}\) is the Moore-Penrose pseudoinverse and \(\lambda\) is an arbitrary scalar. The cable tensions must be bounded between a minimum positive and a maximum value in order to avoid slack and over-tensioned cables. This condition forms the first problem constraint, which is formulated as follows:  
\[0<T_{min}\leq T_{i}(j)\leq T_{max},\ \ i=1..4,\ \ j=1..n \tag{8}\]  
where \(n\) is the number of points composing each trajectory, \(T_{i}(j)\) is the tension of the \(i^{th}\) cable at the \(j^{th}\) position. The second constraint concerns the collisions between the cables and the end-effector. to prevent this issue, the angle \(\alpha_{i}\) between the mobile platform and the \(i^{th}\) cable is computed for each position of the prescribed trajectories. \(\alpha_{i}\) must remain higher than a limit angle \(\alpha_{lim}\). The formulation of the collision constraint is given as follows:  
\[\alpha_{i}=\cos^{-1}\left(\mathbf{n_{i}}\cdot\frac{\mathbf{m_{i}}}{\|\mathbf{ m_{i}}\|}\right)>\alpha_{lim},\ \ \ \ \ \ i=1..4, \tag{9}\]  
where \(\mathbf{m_{i}}=-\mathbf{m_{q}}=\boldsymbol{B_{1}B_{4}}\), \(\mathbf{m_{2}}=-\mathbf{m_{3}}=\boldsymbol{B_{2}B_{3}}\), and \(\mathbf{n_{i}}\) is the unit vector along the \(i^{th}\) cable as illustrated in Figure 5.  
The last constraint concerns the location of the points \(\mathcal{P}_{i}\). They must be located on the edges of the square forming the robot fixed frame. This constraint facilitates the robot reconfiguration when any other trajectory, included in the robot workspace, is selected. The \(4\times 3\) Jacobian matrix \(\mathbf{J}\) of the considered robot includes components of different physical units since the latter has mixed DOFs. To have a meaningful value of the condition number, the Jacobian matrix must be normalized allowing assessment of the closeness of a pose to a singularity. In order to settle the dimensional inhomogeneity, several methods have been suggested based on dividing the rotational elements of the Jacobian matrix \(\mathbf{J}\) by a conventional length \(L\). [PERSON] et al. defined \(L\) in [PERSON] et al. (2016) as a nominal length represented by the distance between the origins of the global and the local frames. [PERSON] introduced in [PERSON] et al. (2016) the notion of natural length, which is the value of \(L\) that minimizes the condition number. This length is approximated to the radius of the end-effector [PERSON] et al. (2016). The latter method was adopted in this paper. The homogenous Jacobian matrix, \(\mathbf{J_{h}}\), of size \(4\times 3\), is calculated as given by Equation (10), where \(\epsilon\) is the side length of the mobile platform. \[\mathbf{J_{h}}=\mathbf{J}\cdot diag\left(1,\ 1,\ \frac{2}{\epsilon\sqrt{2}} \right), \tag{10}\]  
Three criteria are considered for the optimization problem formulation, namely minimizing the tension in each cable and maximizing both the elastic stiffness and the dexterity of the end-effector.