## 3 Investigation of the Workspace Volume
### Analytic Determination of the PKM Workspace Volume  
Considering the kinematic structure of the six DOF PKM, the three PR(RPR)RS share predefined support planes. The actuated (RPR) kinematic chain from the PR(RPR)RS kinematic chain imposes a fixed radius between the last joint of the chain, the spherical joint S, and the first revolute joint R of the chain. Depending on the total stroke of the actuated prismatic joint P from the (RPR) chain, a minimal and maximal radius are obtained, meaning that the PR(RPR)RS chain has a circular workspace, as shown in Figure 3, inside the support plane, centered in the revolute center of the first revolute joint R [20]. Given the fact that each of the three PR(RPR)RS kinematic chains defines an individual support plane that is actuated by the first prismatic joints of the previously mentioned kinematic chains, the workspace of the PKM structure is generated by translating the resulting section, as shown in Figure 4, among the prismatic axes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\varphi\)1 & 60\({}^{\circ}\) \\ \(\varphi\)2 & 60\({}^{\circ}\) \\ \(\varphi\)3 & 60\({}^{\circ}\) \\ d & 90 mm \\ link1 & 684 mm \\ link2 & 492.75 mm \\ r1 & 397.5 mm \\ r2 & 1172.5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PKM structure configuration parameters. Figure 3: Representation of the workspace of each PR(RPR)RS upper view. The resulting section of the workspace is constructed by the intersection of four curves, as shown in Figure 5, which resulted from the projection on the normal plane to the prismatic axis of the workspaces of the three PR(RPR)RS kinematic chains. The resulting shape is constructed on the intersection of only four ellipses by considering the symmetry of the structure that has the same tilt angle for the support plane of the first and last PR(RPR)RS kinematic chains. The four ellipses that determine the section are as follows:  
\[e_{1}:\ \frac{y^{2}}{R_{1}^{2}\cdot\sin(\varphi_{1})^{2}}+\frac{z^{2}}{R_{1}^{2}}=1 \tag{1}\]  
\[e_{2}:\ \frac{y^{2}}{R_{2}^{2}\cdot\sin(\varphi_{1})^{2}}+\frac{z^{2}}{R_{2}^{2}}=1 \tag{2}\]  
Figure 4: Representation of the resulting section of the workspace. Figure 5: Representation of the four ellipses used to construct the section of the workspace (\(e_{1}\)—red, \(e_{2}\)—orange, \(e_{3}\)—black, \(e_{4}\)—blue). \[e_{3}:\ \frac{y^{2}}{R_{1}^{2}}+\frac{z^{2}}{R_{1}^{2}\cdot\sin(\varphi_{2})^{2}}=1 \tag{3}\]  
\[e_{4}:\ \frac{y^{2}}{R_{2}^{2}}+\frac{z^{2}}{R_{2}^{2}\cdot\sin(\varphi_{2})^{2}}=1 \tag{4}\]  
where  
* \(R_{1}\) and \(R_{2}\) are the minimal and maximal radiuses imposed by the (RPR) chains for the PR(RPR)RS chains;
* \(\varphi_{1}\) is the tilt angle of the first and last support plane;
* \(\varphi_{2}\) is the tilt angle of the second support plane. For the determination of the workspace volume, it is required to determine the area of the cross section of the workspace. Considering that the section of the workspace is constructed by the intersection of four ellipses centered in the same point, the resulting geometrical shape for the section of the workspace has two-fold symmetry among the major axis of the coordinate system (OY and OZ). In the case in which the values for the tilt angles \(\varphi_{1}\), \(\varphi_{2}\) and \(\varphi_{3}\) are equal, the cross section gains symmetry, as shown in Figure 6, among the four major axes (OY, OZ, axis Y = Z and axis Y = -Z). Considering the symmetries of the cross section, the determination of the area of the cross section for the reachable workspace, as shown in Figure 7, was implemented by using the formula for the area of an ellipse sector.