# Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots
## 3 Results
### Experimental Validation of the Keops-Delta Prototype  
Figure 9 shows the experimental prototype of the parallel continuum Keops-Delta that has been built to perform experimental tests. The flexible rods that make up each of the parallelograms are Nitinol elastic rods of length 0.7 m, with the proximal ends vertically clamped to the linear tables (see detail in Figure 10a), and the distal-ends connected to  
the spherical joints located on the mobile platform by means of some 3D printed auxiliary pieces (see Figure 10b). Figure 10: (**a**) Flexible rod clamped to the linear guide; (**b**) Connection to the moving platform through spherical joints. Figure 9: Experimental prototype of parallel continuum Keops-Delta robot. The prototype combines commercial elements and parts of our own design, which have been 3D printed in our laboratory. The main commercial elements are the aforementioned Nitinol rods of 0.7 m and the following elements from Igus: drylin ZLW linear guides of 0.6 m stroke, stepper motors NEMA23 XL and GFSM-AG spherical joints. For the experimental testing, we have built a graphical user interface (GUI) using App Designer in MATLAB R2023b, as shown in Figure 11. Among all the tasks that have been programmed, the main ones are as follows: the user can move the robot either by giving the values of the linear guides (that is, FK solving), or by establishing the coordinates of the moving platform's centroid (IK solving); also, any predefined trajectory can be directly uploaded to the GUI, and performed; the evolution of main parameters along a certain specified motion can be visualized, such as graphics showing each \(\rho_{i}\), trajectory of the moving platform, quaternions' values, parasitic orientation Euler angle, and so on. To compare the theoretical results from the MATLAB code with the experimentally obtained results, an MTi-680 sensor from Xsens/Movella (Henderson, NV, USA), has been installed at the center of the moving platform to measure inclination, as shown in Figure 10b. Subsequently, the experimental tests that have been carried out will be explained. The motors we used in these tests limit the velocity to 1.17 m/s with a torque of 0.5 Nm, and 0.12 m/s with the maximum torque of 3.1 Nm. Our maximum speed in the tests was 0.1 m/s, but its effect is under study and the velocity can be modified. In Figures 12-14, the theoretical results are plotted in red and the experimental sensed ones are depicted in blue. The first test consists in performing a vertical motion along the \(z\)-axis maintaining null the \(x\) and \(y\) coordinates of the centroid of the platform. The theoretical and experimental results associated with the quaternions and the corresponding Euler angle, which establishes the inclination of the moving platform with respect to the horizontal plane, can be seen in Figure 12.  
Figure 12a shows the comparison among the theoretical values of the quaternions (plotted in red) and the real sensed ones (depicted in blue). The three \(e_{i}\) quaternion components, with \(i=1,2,3\), represent the rotation around X, Y and Z axes, respectively. As a vertical motion along the \(Z\)-axis produces a pure translational motion of the moving platform, in the kinematic model the theoretical values are 0 for the three \(e_{i}\) quaternions, along a value \(e_{0}=1\). In the experimental test, the real sensed quaternion components are  
Figure 11: Graphical user interface in App Designer. not exactly zero, the blue curves showing a certain small variation. However, take note that the order of magnitude in the results is \(10^{-3}\). Figure 12b represents the Euler angle corresponding to those quaternions. As expected, the real sensed value (in blue) is not exactly zero, but the resultant error varies in the approximate range \((0.5^{\circ},0.75^{\circ})\).