# Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots
## 3 Results
### Workspace Analysis of Parallel Continuum Delta Robots  
We will focus on the obtaining and analysis of the workspace of the parallel continuum Delta-type robots. In particular, the workspace of the rotational Delta and the Keops-Delta will be determined. Bear in mind that the vertical or horizontal Delta robots are quite similar to the Keops-Delta, differing only in the configuration of the linear actuators. The strategy followed to obtain the workspace of the flexible Delta manipulators is based on the discretization of this space and on a \"wavefront\" analysis in such a way that starting from a specific point, adjacent points are analyzed consecutively. To do this, initially, the workspace is divided into a series of \(k\) planes associated with a constant \(Z_{k}\) value, this coordinate representing the height of the centroid of the end-effector or moving platform; subsequently, for each of these planes, and starting from the central position \(x=y=0\) of the moving platform, the IK problem of each \"neighbouring\" point is solved, advancing on a wavefront towards increasing \(x,y\) values. When the advancement algorithm detects a singularity [41; 42], it will stop at that point and will try on the adjacent ones, so that the boundary of the workspace is delimited. In a last phase, the 3D space is formed with the set of the analyzed \(Z_{k}\) planes. Since the singularities delimit the advance of the wavefront, the spaces we obtain correspond to the so-called _aspects_ of the total workspace, i.e., workspaces that are completely free of any type of singularities. It is worth mentioning that for each point that is solved, not only the value of the associated inputs is obtained (remember that we are solving the IK) but that, since all the kinematic parameters associated with that pose are known, the direct and inverse kinematic Jacobians are also obtained numerically, as well as other relevant parameters such as the value of the parasitic angle, the latter being the angle of inclination of the moving platform about the Euler pole obtained from the \(\tilde{\mathbf{e}}\) quaternion. It is of special interest to assess this parasitic angle, since one of the objectives is precisely to be able to extract the zones of the workspace in which this angle is the smallest possible, thus ensuring a quasi-translational motion in this part of the workspace. Figure 6 shows the workspace corresponding to the Delta rotational flexible manipulator. On the one hand, in Figure 6a, we can see the workspace resulting from imposing a limitation on the actuation system, given in this case by the rotational motors and the range of motion allowed by the articulated parallelogram connecting the motors to the clamping element of the proximal points \(A_{i}\). The colour scale is based on the value of the parasitic angle of the moving platform, the zones in red being the ones associated with the lowest values. On the other hand, by restricting the maximum parasitic angle to \(10^{\circ}\), the practical operational workspace of the flexible rotational delta is obtained in Figure 6b. It can be seen how it constitutes a fairly cylindrical workspace, whose central zone has very small values of the parasite angle (even less than \(5^{\circ}\) if we keep within a radius of action of 0.2 m), while the peripheral zones have a greater angle of inclination, reaching \(10^{\circ}\) in the blue extremes. Regarding the flexible Keops-Delta robot, the corresponding workspace is shown in Figure 7. As before, the resultant workspace, constrained by the limitations of the actuators, is depicted in Figure 7a. In this case, as it will be explained in the next section, the limitation of the actuators is given by the maximum stroke allowed by the linear guides implemented in the prototype, which is 0.6 m. On the other hand, Figure 7b shows the cylindrical-shaped workspace that is obtained when the parasitic angle is limited to \(10^{\circ}\). Similar to the rotational Delta, a large central zone ensures parasite angle values of less than \(5^{\circ}\).