## 4 Results
### Maximize the Sphere in the Useful Workspace of the 2 PRU-1 PRS  
We know that the geometry parameters that affect the workspace are the radius of the mobile platform (R), the radius of the fixed platform (H) and the length of the limbs (L), while the radius of the limbs and the thickness of the mobile platform do not affect the workspace. Accordingly, we obtain the optimum H, L and R combination to obtain the useful workspace with the biggest sphere in it. The first step is to define a set of GP combinations that will be studied, the _StudyVariables_. We define the ranges of the geometry parameters, shown in Table 1.  
We define the number of discretizations for each GP to be 10, so we obtain the 1000 GP possible combinations. Optimizing all of them would result in a very high computational cost, so we apply the restrictions that the geometry of the manipulator has to satisfy. As we observe in Figure 5, the 2 PRU-1 PRS manipulator can have two configurations depending on the relation between the value of the radius of the mobile platform and the radius of the fixed platform. Figure 5a shows the configuration with the radius of the mobile platform smaller than the radius of the fixed platform, while Figure 5b shows the configuration with the radius of the mobile platform bigger than the fixed one. In this work, we study the first configuration; thus, the _StudyVariables_ have to fulfill the condition H \(>\) R.  
Since H, L and R form a triangle, they must satisfy the triangle in equality. That is, the sum of the values of the radius of the mobile platform and the length of the limbs has to be greater than the radius of the fixed platform, H \(<\) L + R. Thus, there are two conditions that we have to apply when defining the suitable GP combinations:  
1. The radius of the mobile platform is smaller than the radius of the fixed platform, R \(<\) H;
2. The sum of the radius of the mobile platform and the length of the limbs is larger than the radius of the fixed platform R \(+\) L \(>\) H.  
By applying these restrictions, we obtain only 529 suitable GP combinations. The number of GP combinations to check has thus been reduced by 47.1% from the 1000 initial GP combinations to the 529 suitable GP combinations. Similarly, we obtain the \(StudyPoints\). The workspace represents the poses that the the end-effector of the manipulator can reach. In this case, since our outputs are two rotationsand one translation, the points in the workspace will be defined by the translation along Z-axis and the rotations about X and Y axes. Thus, _StudyPoints_ is the set of points which lie in the three-dimensional space bounded by the ranges given by Table 2. We discretize each axis into 60 points, obtaining a cubic grid of size 60 \(\times\) 60 \(\times\) 60. Therefore, the total number of _StudyPoints_ is 216,000. We also define the ranges of the linear guides (LG) and the spherical joint (SJ) as seen in Table 3.  
In order to have an idea of the regularity of the useful workspace and be able to determine which solution is the best one, we obtained the useful workspace and the biggest sphere in it for each suitable GP combination. The useful workspace denotes the region of the workspace free of singularities of the inverse and direct kinematic problems and where, additionally, the physical restrictions of the spherical joint and the linear guides are fulfilled. As already mentioned, we will also study the regularity of the useful workspace by obtaining the biggest sphere in it. Figure 6 shows the steps we will follow to obtain the useful workspace of the manipulator and the biggest sphere (S\({}_{max}\)) in it for each GP combination. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Description** & **Value** \\ \hline H (m) & Radius of the fixed platform & (0.3, 0.75) \\ \hline L (m) & Length of the limbs & (0.2, 0.65) \\ \hline R (m) & Radius of the mobile platform & (0.2, 0.65) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GP ranges for the optimization process.