# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 3 Experimental Results and Discussion
With the area formula of a triangle from [30] (based on the vertices coordinates), the total area delimited by TFR\({}_{1}\) generated by \(x_{e}\) and \(y_{e}\) (Figure 12 or Figure 14) is calculated as 8515.3 \(\upmu\)m\({}^{2}\), while the total area delimited by TFR\({}_{1}\) generated by \(x_{a}\) and \(y_{a}\) (Figure 14) is 7624 \(\upmu\)m\({}^{2}\). A second conventional graphical description of the circularity error of the average trajectory (as a second trajectory fitting residual, TFR\({}_{2}\)) is available in polar coordinates (\(d_{i2}\), \(\alpha_{i2}\)) related to the minimum circumscribed circle (having the same center as the center of the least square circle), with \(d_{i2}\), \(\alpha_{i2}\) defined as:  
Figure 9:1  
In the first Equation from (6), \(R_{cc}=\min\left(\sqrt{\left(x_{ei}-x_{c}\right)^{2}+\left(y_{ei}-y_{c}\right) ^{2}}\right)\) is the radius of the minimum circumscribed circle. Figure 16 presents the TFR\({}_{2}\) of the average trajectory generated by \(x_{e}\) and \(y_{e}\) (\(R_{cc}=8.8434\) mm) and TFR\({}_{2}\) generated by \(x_{a}\) and \(y_{a}\) (with green color, \(R_{cc}=8.9171\) mm), with circular grid (with a 50 \(\upmu\)m increment on radius). Here the maximum value of distance \(d_{i2}\) is 246 \(\upmu\)m.  
A TFR\({}_{2}\) for a perfect circular average trajectory is a point placed in the origin of the least square circle. The existence of these two lobes on TFR\({}_{2}\) in Figure 16 is explicable if, in addition to the comments and simulation done in Figure 13, we take into account that the minimum circumscribed circle touches the elliptical trajectory in two symmetrical points. The TFR\({}_{2}\) evolution of a pure elliptical trajectory related to the minimum circumscribed circle is depicted in Figure 17 (by simulation). Figure 9:2 The evolution of TFR\({}_{2}\) generated by \(x_{e}\) and \(y_{e}\) and TFR\({}_{2}\) generated only by \(x_{a}\) and \(y_{a}\) (with green color). Some supplementary resources on the P&A of circular trajectories are revealed by circular fitting of the 2D curve generated only by \(x_{Ar}\)-motion (already described in Figure 8) and \(y_{Ar}\)-motion (already described in Figure 11), as Figure 18 indicates. The least square circle (14.2 \(\upmu\)m radius, with center at \(x_{c}=-3.3\)\(\upmu\)m and \(y_{c}=-1.9\)\(\upmu\)m) should also be considered as an indicator for P&A of the average trajectory. We should mention that the effect of strong repetitive irregularities on \(y\)-motion (\(y_{r}\) and \(y_{Ar}\)) already revealed in A, B areas on Figures 8 and 11 are also well described in Figures 14-16 and Figure 9:3 Moreover, the mirroring of these events A, B in Figure 14 or Figure 15 confirms the previously formulated hypothesis (the comments in Figure 11) that they are related by a sudden release of a mechanical stress inside the toothed belt used for \(y\)-motion. In Figure 11, a maximum positive peak from A is immediately followed by a minimum negative peak from B. Therefore, in Figure 14, these two peaks A, B are described as a single peak because of modulus in definition of \(d_{i1}\) (Equation (5)). As it is clearly indicated in Figures 8 and 11, there are strong vibrations on both motions (on \(x\) and \(y\)), with a negative effect on the P&A of circular trajectories. There are two different strategies available to reduce or to eliminate these vibrations. Figure 9:4 The evolution of TFR\({}_{2}\) generated by circular fitting on an elliptical trajectory (simulation). Figure 9:5 The evolution of the average trajectory generated by \(x_{Ar}\) and \(y_{Ar}\). The first strategy (probably as the better approach) is to use each stepper motor also as an actuator inside an open-loop active vibration suppression system. The second strategy is to use passive dynamic vibration absorbers or tune mass dampers as well [31] placed on the print bed and on the extruder system.