# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 3 Experimental Results and Discussion
On this subject, Figure 10 presents a detail with \(x_{aAr}\) and \(x_{Ar}\) evolutions located in the area A of Figure 8 with \(n=105\) (105 components in sum of sine model from Equation (3)). The strong variation of displacement depicted here is likely the result of structural vibrations of the 3D printer induced by the stepper motor. The \(x_{Ar}\) and \(x_{aAr}\) evolutions are certain arguments that the experimental setup is able to describe the P&A of \(x\)-motion. Moreover, the analytical model from Equation (3) and Table 1 helps (at least in theoretical terms) to compensate for the errors of \(x\)-motion. Figure 11 presents the evolution of \(y_{Ar}\) with \(k=5\) (for five completely circular trajectories, each sample of \(y_{Ar}\) being an average of \(k\) correlated samples of \(y_{r}\)). As with \(x\)-motion, it is obvious that the \(y\)-motion also has a well-defined periodic recurrent pattern (the same period \(T\) as \(x_{Ar}\)), having systematic errors. In other words, \(y_{r}\) (and \(y\)-motion as well) is well correlated with itself. Unfortunately, it was not possible to find an acceptable analytical model \(y_{aAr}\) with harmonic components (similar to the \(x_{aAr}\) model for \(x_{Ar}\) based on Equation (3)). A very high number of harmonic components (\(n\)) is necessary in this sum of sine model. A future approach intends to identify a more appropriate analytical model. There are strong repetitive irregularities on \(y\)-motion (and \(y_{r}\) and \(y_{Ar}\) as well) revealed in A, B areas of Figures 8 and 11. Likely they are irregularities generated by a suddenly releasing of a mechanical stress inside the toothed belt used to produce the \(y\)-motion. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(j\) & \(a_{xj}\)[mm] & \(b_{xj}\)[rad/s] & \(c_{xj}\)[rad] & \(j\) & \(a_{xj}\)[mm] & \(b_{xj}\)[rad/s] & \(c_{xj}\)[rad] \\ \hline
**1** & 0.203 & 0.3379 & 1.969 & **9** & 0.001888 & 6.191 & \(-\)2.542 \\
**2** & 0.004801 & 4.885 & 0.3713 & **10** & 0.002353 & 119.3 & 1.698 \\
**3** & 0.004398 & 1.383 & 1.758 & **11** & 0.1188 & 121.8 & 2.263 \\
**4** & 0.003755 & 2.083 & 1.377 & **12** & 0.001696 & 44.14 & \(-\)3.301 \\
**5** & 0.004055 & 4.163 & \(-\)2.608 & **13** & 0.00148 & 120.4 & \(-\)0.3449 \\
**6** & 0.004279 & 3.47 & \(-\)2.578 & **14** & \(-\)0.1166 & 121.8 & 2.283 \\
**7** & 0.1981 & 0.3481 & \(-\)1.225 & **15** & 0.001224 & 123.3 & \(-\)2.064 \\
**8** & 0.001892 & 42.6 & \(-\)2.541 & **16** & 0.0004954 & 118.6 & \(-\)2.431 \\ \hline \end{tabular}
\end{table} 
Table 1: The values of \(a_{xj}\), \(b_{xj}\) and \(c_{xj}\) involved in \(x_{aAr}\) model from Equation (3) with \(n=16\). Figure 9: The evolution of an analytical model \(x_{aAr}\) (with \(n=16\)) of the average residual \(x_{Ar}\). It is interesting now to examine the average trajectory generated by the 3D printer with \(x_{a}(t)+x_{A\prime}(t)=x_{e}(t)\) as \(x\)-motion and \(y_{a}(t)+y_{A\prime}(t)=y_{e}(t)\) as \(y\)-motion. It is obvious that this trajectory is not a perfect circle (at least because the wrong phase shift between \(x_{a}\) and \(y_{a}\)). A computer program was developed in order to find out the description of the least square circle (the coordinates \(x_{c}\), \(y_{c}\) of center and the radius \(R_{c}\) as well) by circular fitting. This program is available for the fitting of any closed curve with known analytical description. The circular fitting supposes to find out the values \(x_{c}\), \(y_{c}\) and \(R_{c}\) for which a fitting criterion \(\varepsilon\) described in Equation (4) reaches a minimum value. \[\varepsilon=\sum_{i=1}^{N}\Bigl{[}(x_{ei}-x_{c})^{2}+(y_{ei}-y_{c})^{2}-R_{c} ^{2}\Bigr{]} \tag{4}\]  
In Equation (4), \(N\) is the number of samples \(x_{ei}\) or \(y_{ei}\) (\(N\) = 179,740) used for \(x_{e}\)-motion or \(y_{e}\)-motion description of the average trajectory. If the average trajectory is a perfect circle, then a perfect fitting produces a value \(\varepsilon=0\) for the fitting criterion.