# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 4 Workspace Analysis
### Obtaining the Useful WS
#### 4.2.5 Kinematic Joint Motion Range (WS\({}_{\text{joints}}\))  
Finally, the end-effector must be able to reach the set of discrete poses that define the WS, fulfilling the physical restrictions of the kinematic joints; that is, without surpassing their range of motion. WS\({}_{\text{joints}}\) is defined as the set of positions of the WS wherein the motion ranges of all the kinematic joints are not exceeded.  
The three types of kinematic joints (R, U and S) of the proposed manipulator allow relative rotation movements between the elements that they connect. Hence, the restrictions considered for the calculation of the WS\({}_{\text{joints}}\) are angular restrictions. In Figure 19, the angles involved in this calculation are depicted, where  
* \(\delta_{\text{C}}\): represents the angle rotated by the active R joints, which coincides with \(\beta_{i}\) values;
* \(\delta_{BR}\): is the relative angle between links of the RRU KC, which limits the rotation of the intermediate R passive joint around Y-axis;
* \(\delta_{BS}\): is the relative angle between links of RRS KC inside the \(XZ\) plane;
* \(\gamma_{BS}\): is the angle rotated by the intermediate S joint outside a plane parallel to \(XZ\);
* \(\delta_{AS}\): is the angle rotated by the upper S joint inside a plane parallel to \(uw\);
* \(\gamma_{AS}\): is the angle rotated by the upper S joint outside a plane parallel to \(uw\);  
Figure 17: Schematic representation of (**a**) non-collision and (**b**,**c**) collision positions.  
Figure 18: Translational and rotational WS\({}_{\text{platform}}\) for an infinite plane mobile platform. Orange-colored regions are free of collisions with the platform.  
* \(\delta_{AU}\): is the angle rotated about the first axis of the U joint;
* \(\gamma_{AU}\): is the angle rotated about the second axis of the U joint. It coincides with \(\theta\) value.
* \(\mathbf{A}^{\prime}\) and \(\mathbf{B}^{\prime}\) auxiliary points were used to define some of the angles. These auxiliary points are the projections of \(\mathbf{A}\) and \(\mathbf{B}\), in the \(XZ\) and \(uw\) plane, respectively.  
The range of values that each angle can take is variable, and it can be adjusted based on the design of the kinematic joints and the rest of the elements of the mechanism. For instance, if commercial kinematic joints are used, \(\delta_{i}\) and \(\gamma_{i}\) values will be defined by the manufacturer. If own-designed joints are used, the angular values will be constrained by design restrictions, such as collisions between links of the same KC and interferences with the fixed frame.  
To illustrate some examples, we assumed that all KCs present the same admissible motion ranges as those gathered in Table 3. For these ranges, the use of own-designed R and U joints and commercially available S joints was considered.  
Figure 20 shows the WSjoints for different working modes, using the values in Table 3. Note that the axes limits in the rotational representations are different to those of previous  
Figure 19: Representation of the angles involved in the kinematic joint motion range restrictions.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**RRS KCs** & **RRU KC** \\ \hline \(\delta_{C}\in[-20^{\circ},200^{\circ}]\) & \(\delta_{C}\in[-20^{\circ},200^{\circ}]\) \\ \(\downarrow\delta_{BS}|>20^{\circ}\) & \(\downarrow\delta_{BR}|>20^{\circ}\) \\ \(\delta_{AS}\in[10^{\circ},170^{\circ}]\) & \(\delta_{AU}\in[10^{\circ},170^{\circ}]\) \\ \(\gamma_{BS},\gamma_{AS}\in[-15^{\circ},15^{\circ}]\) & \(\gamma_{AU}\in[-90^{\circ},90^{\circ}]\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Example of the motion range values for each kinematic joint.  
rotational WS subsets. This is due to the limited area of WS\({}_{\text{points}}\) (non-gray colored regions) in comparison to the complete available WS.