# Suitability of UR5 Robot for Robotic 3D Printing
## 4 Methodology and Conditions for Measuring the UR5 Collaborative Robot
### Assessing the Drift Test of Unidirectional Pose Repeatability (dRP)  
In the present test (dRP), a total of 49 measurements were performed at measurement point P1, with the maximum load of 100% and the maximum robot motion speed of 100%, under unchanged conditions as in the previous measurement, in order to determine the magnitudes of the changes in the unidirectional repeatability of the wrist end-member position over a specified time. The summary data (dRP) of the results from the calculated values are presented in Table 2.  
When comparing the drift of the unidirectional pose repeatability, we used the value specified by the manufacturer, namely, the value of the unidirectional pose repeatability of \(\pm\)0.1 mm. Since none of the measured values exceeded the prescribed value, it can be concluded that the robot \"passed\" the test.  
The drift of the unidirectional repeatability (dRP) is derived from the following equation  
\[\text{dRP}=\left|\text{RP}_{t=0}-\text{RP}_{t=T}\right| \tag{8}\]  
Figure 6: Drift of the unidirectional pose accuracy (dAP) of the UR5 robot.  
Figure 7: Drift of the unidirectional pose accuracy (dAP\({}_{a}\), dAP\({}_{b}\), dAP\({}_{c}\)) of the UR5 robot.  
whereas the value of RP as the radius of the sphere whose center is the barycenter is based on relations (9), (10), and (11).  
\[RP=\overline{l}+3S_{l};\overline{l}=\frac{1}{n}\sum\limits_{j=1}^{n}l_{j};\ l_{j}=\sqrt{\left(x_{j}- \overline{x}\right)^{2}+\left(y_{j}-\overline{y}\right)^{2}+\left(z_{j}- \overline{z}\right)^{2}} \tag{9}\]  
\[S_{l}=\sqrt{\frac{\sum\limits_{j=1}^{n}\left(l_{j}-\overline{l}\right)^{2}}{n- 1}} \tag{10}\]  
where (\(x_{j}\), \(y_{j}\), \(z_{j}\)) are the actual (measured) values. We calculated the barycenter coordinates of the points reached in n repetitions of the same position using the following equations:  
\[\overline{x}=-\frac{1}{n}\sum\limits_{j=1}^{n}x_{j}\quad\overline{y}=\frac{1} {n}\sum\limits_{j=1}^{n}y_{j}\quad\overline{z}=\frac{1}{n}\sum\limits_{j=1}^{ n}z_{j} \tag{11}\]  
The resulting drift of unidirectional orientation repeatability (\(\mathrm{dRP_{a}}\), \(\mathrm{dRP_{b}}\), \(\mathrm{dRP_{c}}\)) is derived from the following equations:  
\[\mathrm{dRP_{a}}=\left|\mathrm{RP_{a_{t}=0}-RP_{a_{t}=T}}\right|;\ \mathrm{dRP_{b}}=\left| \mathrm{RP_{b_{t}=0}-RP_{b_{t}=T}}\right|;\ \mathrm{dRP_{c}}=\left|\mathrm{ RP_{c_{t}=0}-RP_{c_{t}=T}}\right| \tag{12}\]  
where (\(\mathrm{RP_{a}}\), \(\mathrm{RP_{b}}\), and \(\mathrm{RP_{c}}\)) are derived from Equation (13) of the individual variances in the angles (\(3S_{a}\), \(3S_{b}\), \(3S_{c}\)) around the means (\(\overline{a}\), \(\overline{b}\), \(\overline{c}\)), with (\(S_{a}\), \(S_{b}\), \(S_{c}\)) being the standard deviations derived from  
\[\mathrm{RP_{a}}=\pm 3S_{a}=\pm 3\sqrt{\frac{\sum\limits_{j=1}^{n}\left(a_{j}- \overline{a}\right)^{2}}{n-1};\ \mathrm{RP_{b}}=\pm 3S_{b}=\pm 3\sqrt{\frac{\sum \limits_{j=1}^{n}\left(b_{j}-\overline{b}\right)^{2}}{n-1}};\ \mathrm{RP_{c}}=\pm 3S_{c}=\pm 3 \sqrt{\frac{\sum\limits_{j=1}^{n}\left(c_{j}-\overline{c}\right)^{2}}{n-1}}} \tag{13}\]  
where (\(a_{c}\), \(b_{c}\), \(c_{c}\)) are the angles of the programmed position; (\(a_{j}\), \(b_{j}\), \(c_{j}\)) are the angles of the \(j\)-th achieved position.  
Figure 8 presents a plot of the calculated drift values of the unidirectional pose repeatability (dRP) in three mutually perpendicular directions during a measurement lasting 8 h. The calculated average value is \(\mathrm{dRP}=0.030831588<0.1\) mm.  
Figure 9 shows a plot of the calculated drift values of the unidirectional pose repeatability (\(\mathrm{dRP_{a}}\), \(\mathrm{dRP_{b}}\), and \(\mathrm{dRP_{c}}\)) in two mutually perpendicular directions. The graphical representation shows that the average drift values of unidirectional repeatability are \(\mathrm{dRP_{a}}=0.036974234<0.1\) mm, \(\mathrm{dRP_{b}}=0.030677601<0.1\) mm and \(\mathrm{dRP_{c}}=0.038800076<0.1\) mm.  
Figure 8: Drift of the unidirectional pose repeatability (\(\mathrm{dRP}\)) of the UR5 robot.