# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 4 Workspace Analysis
### Obtaining the Useful WS
#### 4.2.3. Collisions between KCs (WS\({}_{\text{KCs}}\))
\[\lambda =\big{(}\mathbf{B}_{j}-\mathbf{B}_{i}\big{)}\textsuperscript{{}^{T}}\cdot \frac{\mathbf{s}_{i}-n\cdot\mathbf{s}_{j}}{1-n^{2}} \tag{30}\] \[\gamma =\big{(}\mathbf{B}_{j}-\mathbf{B}_{i}\big{)}\textsuperscript{{}^{T}}\cdot \frac{\mathbf{s}_{j}-n\cdot\mathbf{s}_{i}}{1-n^{2}} \tag{31}\]  
Figure 12: Three parallelism cases to check the possibility of collision. Figure 11:0 Minimum distance (**a**) inside the limits of the links and (**b**) outside the limits. If the value of any of these two parameters is not within \([0,\ L_{i}]\) or \([0,\ L_{j}]\), the minimum distance must be recalculated, which will be denoted as \(d_{min}^{\ mod}\) and will coincide with the module of \(c^{mod}\)(see Figure 13b). To calculate \(d_{min}^{\ mod}\) we must know if the condition is not fulfilled due to surpassing the upper bound or not. For instance, if \(\lambda>L_{i}\), the point \(c_{i}\) is further from \(B_{i}\), than point \(A_{i}\). The new extreme point \(c_{i}\) will then coincide with \(A_{i}\), whereas if \(\lambda<0\), \(\mathbf{c}_{i}\) will coincide with \(B_{i}\). The same procedure applies for \(\gamma\) and \(\mathbf{c}_{j}\). This is because when \(d_{min}\) takes place outside of the link limits, the minimum distance closer to the value of \(d_{min}\) will take place in the extreme of the link, which is closer to \(\mathbf{c}_{i}\) or \(\mathbf{c}_{j}\). Finally, we must verify whether \(d_{min}^{\ mod}\) is bigger or smaller than the sum of the radii of the two links. If it is bigger, there is no collision, but if it is smaller, a collision takes place, and the studied discrete pose is not valid. This procedure is summarized in the flowchart shown in Figure 14, and it should be applied to check the collisions between the links of the four KCs of the proposed mechanism. Figure 11:1 Procedure to check the collisions between the links of different KCs. Three types of KC collisions must be checked: lower links collisions, upper links collisions and lower-upper link collisions. According to the KCs' arrangement in the RRU-3 RSS PM (see Figure 1), the lower links of the first and third KCs are the only ones that may collide as they are constrained to move inside the same \(XZ\) plane. In contrast, the upper links of all the KCs can collide with each other, as the upper links of the second and fourth KCs can move outside the \(XZ\) planes when a rotation \(\theta\) takes place. For the same reason, collisions between the lower and upper links of different KCs can occur. Collisions between links of the same KC will be considered in Section 4.2.5. In Figure 15, the WS\({}_{\text{KCs}}\) corresponding to two working modes are shown, wherein links with 0.01 m thickness were considered. Blue-colored regions represent the positions and orientations of the manipulator free of KCs collisions, whereas gray-colored regions illustrate the manipulator poses where at least one collision between KCs takes place. The initial position in which the mechanism is assembled determines which regions free of KCs collisions (WS\({}_{\text{KCs}}\)) are valid or not.