# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 4 Workspace Analysis
### Obtaining the Useful WS
#### 4.2.2 DKP Singularities (\(\text{WS}_{\text{DKP}}\))  
In contrast to IKP singularities, DKP singularities take place inside the WS. At these singularities, the mechanism becomes uncontrollable, and the end-effector can move in an infinitesimal way without changing the value of the inputs. Thus, DKP singularities are critical and must be identified to avoid trajectories of the end-effector that reach or cross them. From a mathematical point of view, DKP singularities appear when the determinant of the DKP Jacobian matrix, \(|\mathbf{J}_{x}|\), is null. Hence, for a WS free of DKP singularities (\(\text{WS}_{\text{DKP}}\)), the sign of the \(|\mathbf{J}_{x}|\) in all discrete poses that belong to it must be the same. Mapping the sign of \(|\mathbf{J}_{x}|\) inside the WS is a common representation that provides a DKP singularity map. The boundaries between regions with opposite signs of \(|\mathbf{J}_{x}|\) coincide with the DKP singularity positions, as for a change in sign, a position with a zero value \(|\mathbf{J}_{x}|\) must be crossed. Figure 8 shows examples of this kind of representation, where the regions with the same color represent DKP singularity-free regions. Figure 6: Shape of the RRU-3 RSS translational WS for different \(\varphi\) and \(\theta\) values. Figure 7: Shape of the RRU-3 RSS rotational WS for different \(x\) and \(z\) values. Unless the regions with the same sign of \(|J_{x}|\) are connected in 4D space, the manipulator will only be able to work in a controlled manner inside one of them, which is given by the determinant of the Jacobian matrix at the initial position of the manipulator (\(|J_{x}|_{0}\)) (i.e., initial assembly mode). To check the connectivity between same-sign regions, a tridimensional representation can be used, wherein the regions with the same sign of \(|J_{x}|\) are depicted for different values of a third output variable. In Figure 9, an example of this kind of representation for the \((+-+-)\) working mode is shown. Despite its general use to trace DKP singularity locus, mapping the sign of \(|J_{x}|\) does not identify those singularities that may take place inside a region of the WS with the same sign (i.e., when a DKP singularity can be crossed without changing the sign of \(|J_{x}|\)). For the RRU-3 RSS PM, this happens in the translational WS corresponding to 8 of the 16 working modes (Group B in Table 2) when the mobile platform is parallel to the fixed platform (\(\varphi=\theta=0^{\circ}\)). In these cases, the DKP singularity positions can be identified directly by computing and mapping the value of \(|J_{x}|\) inside the WS, as can be observed in Figure 10, wherein the yellow bands (close to null value of \(|J_{x}|\)) enclose the DKP singularity locus, represented by red curves. Among the other 8 working modes that do not present this particularity, 6 of them (Group C in Table 2) present DKP singularities inside the whole translational WS when \(\varphi=\theta=0^{\circ}\). This issue rules out the use of these working modes in applications that require a null rotation of the end-effector. We found no analogous situations for the rotational WS; that is, a specific position of the mobile platform that generates a DKP singularity for all the possible orientation combinations. However, for those positions with a null horizontal translation (\(x=0\) m), the working modes can be grouped in pairs as they reach DKP singularities for symmetrical values of \(\varphi\) (see Figure 11). Figure 8: Translational and rotational WS\({}_{\text{DKP}}\) for two working modes of the RRU-3 RSS PM. Light-green-colored regions correspond to \(|J_{x}|>0\), whereas dark-green-colored regions correspond to \(|J_{x}|<0\). Figure 10: \(|J_{x}|\) values inside the translational WS when the mobile platform is parallel to the fixed one (\(\varphi=\theta=0^{\circ}\)) for the three groups of working modes (Group A, B and C).