# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 4 Conclusions and Future Work  
Some theoretical and experimental approaches related to the precision and accuracy (P&A) of a 3D printer, particularly for 2D circular trajectories, were achieved in this paper. The choosing of 2D circular trajectories was inspired from ISO standards methods for P&A evaluation of CNC manufacturing systems (ISO 230-4 [27]) due to some similarities in terms of motion control. The evolution of the simultaneous displacement on two theoretically orthogonal axes (\(x\) and \(y\)) during a repetitive 2D circular trajectory of the extruder system related to the print bed was simultaneously and continuously measured using a computer  
Figure 21: A zoomed in detail of Figure 20 (in B area). assisted setup based with two contactless optical sensors and a numerical oscilloscope (for sampling and data acquisition). The signals of description for \(x\) and \(y\)-motions were numerically processed in order to find out some motions characteristics involved in the evaluation of P&A for circular trajectories. First, a non-perpendicularity of \(x\) and \(y\)-axes (or an axis misalignment) were experimentally detected (with 0.893 degrees error) by means of the curve fitting (using a pure sine model) of the dominant harmonic components of each signal (\(x\) and \(y\)-motions signal). Because of this error, a circular programmed trajectory is executed as an elliptical one (typically for a non-orthogonal \(x0y\) coordinate system), a topic also confirmed by some supplementary signal processing results. Second, it was established that the \(x\) and \(y\)-motions are not simple pure harmonic motions. The residuals of previous curve fitting on each axis movement describe the deviation from pure harmonic shape. A procedure for finding a repetitive periodical pattern in the evolution of these residuals was established and applied with good results. The model of each repetitive pattern is useful in the amelioration of P&A by correction and compensation. The analytical description of the \(x\)-motion residual pattern was already established (by curve fitting with a sum of sine model); a future approach intends to do the same for \(y\)-motion residual. Third, a procedure of the description for an average 2D trajectory (an average of several successive theoretically identical circular trajectories) was established. A computer-aided procedure of fitting for closed curves (particularly a circular trajectory) was developed. The circular fitting of the 2D average trajectory was done related to the least square circle. Two conventional graphical descriptions of the circularity errors of the average trajectory were proposed (as trajectory fitting residuals TFR\({}_{1}\) and TFR\({}_{2}\)): first description being related to the least square circle, second related to the minimum circumscribed circle. The shape of these trajectories fitting residuals and the size of the surface delimited by them are useful in P&A evaluation of circular trajectories in order to verify that the 3D printer works properly and, especially, for systematic errors compensation purposes. For example, the non-perpendicularity of \(x\) and \(y\)-axes previously detected is mirrored in the shape of the average trajectory and, finally, in the shape of these two trajectory fitting residuals (with four similar lobes on TFR\({}_{1}\) and two similar lobes on TFR\({}_{2}\)). The deviation from the harmonic shape for \(x\) and \(y\)-motions is described on these trajectories. A future approach will be focused on finding a complete procedure of experimental research of P&A using high range/resolution non-contact displacement sensors placed on each of three axes. Some complex 3D curves will be used as test trajectories. A numerical interface between the experimental setup and the 3D printer will be developed in order to perform automated testing and errors compensation. These signal processing procedures are available to verify the P&A of 2 Dcircular trajectories on any other similar equipment (e.g., a 3D CNC manufacturing system). Conceptualization, M.H.