## 4 Results
### Maximize the Sphere in the Useful Workspace of the 2 PRU-1 PRS
\begin{table}
\begin{tabular}{l c} \hline \hline  & **Description** & **Value** \\ \hline \(\psi\) (\({}^{\circ}\)) & Rotation about X-axis & (\(-90\), 90) \\ \hline \(\theta\) (\({}^{\circ}\)) & Rotation about Y-axis & (\(-90\), 90) \\ \hline z (m) & Translation along Z-axis & (0.3, 0.8) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Output limits. Figure 5: Possible configurations: (**a**) R \(<\) H; (**b**) R \(>\) H.  
For each GP combination, we first solve the inverse kinematic problem for all the candidate-poses for the workspace, and we obtain the value of the linear guides. If those solutions are imaginary, they are not possible solutions for our workspace. We define this workspace as \(WS_{0}\). We next check which points of \(WS_{0}\) are in the region free of singularities and fulfill the restrictions of the spherical joint and the linear guides. The calculation of the inverse kinematic problem and the singularities is presented in [PERSON] et al. [24]. When a manipulator is in a singularity of the inverse kinematic problem, the end-effector is stuck and it cannot move in the direction of the 3 DOF. Mathematically, singularities related to the inverse kinematic problem appear when the determinant of the Jacobian matrix \(\mathbf{J}_{q}\) is null: Equation (9), as explained by [PERSON] [3], [PERSON] et al. [26], [PERSON] et al. [27] and [PERSON] and [PERSON] [28]. In practice, these singularities appear on the boundary of the workspace. We obtain the singularities of the 2 PRU-1 PRS PM. \[|\mathbf{J}_{q}|=0\longrightarrow\text{Singularity of the inverse kinematic problem} \tag{9}\]  
When the manipulator reaches a singularity of the direct kinematics problem, it will be able to move in an infinitesimal way without changing the value of the inputs. In other words, some degrees of freedom become uncontrollable. Mathematically, this happens when the determinant of the Jacobian matrix \(\mathbf{J}_{x}\) is null: Equation (10) [3]. We define \(|\mathbf{J}_{x}|_{0}\) as the Jacobian matrix determinant of the direct kinematic problem in the initial mobile platform position. For a workspace free of singularities of the direct kinematic problem, \(|\mathbf{J}_{x}|\) of all the points of the workspace must have the same sign. Thus, the Jacobian matrix determinant must have the same sign as \(|\mathbf{J}_{x}|_{0}\)[26; 27; 28; 29]. The set of points that fulfills this condition defines the \(WS_{1}\) workspace. \[|\mathbf{J}_{x}|=0\longrightarrow\text{Singularity of the direct kinematic problem} \tag{10}\]  
Figure 6: Steps to obtain the useful workspace and the \(\text{S}_{max}\) in it. The range of rotation of spherical joints is very limited. We have defined a range of \([-25^{\circ},25^{\circ}]\) since it is one of the biggest ranges we can find in commercial spherical joints. \(WS_{2}\) denotes the set of points of the \(WS_{0}\) that fulfill the spherical joint restriction. By solving the inverse kinematic problem, we obtain the values of the linear guides for all the _StudyPoints_. We checked if those values fulfill the limits of the real linear guides. Since in our case the linear guides have a displacement (\(\rho_{i}\)) from 0 to 0.3 m, linear guides have to fulfill Equation (11):  
\[\rho_{i}\in[\rho_{min},\rho_{max}] \tag{11}\]  
where \(\rho_{min}=0\) and \(\rho_{max}=0.3\). Accordingly, we define the useful workspace as the set of points that fulfill all the previous conditions and obtain the biggest sphere that fits in each useful workspace. Since _StudyPoints_ are a uniform grid, we can define the \"diameter of a sphere\" in _StudyPoints_ space to be number of points along the diameter. We denote by S\({}_{max}\) the largest sphere from the obtained set. The best GP combination is the one that results in the useful workspace containing the S\({}_{max}\). Note that we can have multiple useful workspace that contain the S\({}_{max}\). In that case, the best GP combination is the one that results in the S\({}_{max}\) that can be placed in the most number of positions.