# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 2 Experimental Setup  
The experimental research was done on an Anet A8 3D printer [28], previously used in additive manufacturing for 230 operating hours. The 3D printer has the print bed movable on the \(y\)-axis, while the extruder system moves along the \(x\) and \(z\)-axes independently. On the \(x\)-axis, motion is numerically controlled (in open loop) via a stepper motor and a toothed belt (similarly on the \(y\)-axis); on the \(z\)-axis, the motion is controlled (in open loop) with two synchronized stepper motors with screw-nut systems. The absolute displacement measurement on the \(y\)-axis is done with a non-contact ILD 2000-20 [29] laser triangulation sensor (from MICRO-EPSILON MESSTECHNIK GmbH & Co, Ortenburg, Germany) with 20 mm measurement range (1 \(\upmu\)m resolution and 10,000 s\({}^{-1}\) sampling rate) with the laser beam placed perpendicularly on the target (the print bed) as Figure 1 indicates. Here the point of incidence is placed in the center of the red rectangle. An identical sensor is firmly placed on the print bed, with the laser beam placed perpendicularly to the extruder system (which moves as a sensor target in \(x\)-direction), as Figure 2 indicates. Figure 1: A partial view of the setup with optical sensor on \(y\)-axis. 1-optical sensor; 2-print bed; 3-the extruder system; 4-toothed belt for \(y\)-axis displacement; 5-first screw for \(z\)-axis displacement. The signals generated by these sensors are two voltages proportional with the displacements (with 1.975 mm/V as proportionality factor for the \(x\)-axis sensor and 2.0185 mm/V for the \(y\)-axis sensor) related to the middle of the measurement range. These signals are simultaneously numerically described (by sampling and data acquisition)with a PicoScope 4824 numerical oscilloscope (from PicoTechnology UK, 8 channels, 12 bits, 80 MS/s maximum sampling rate, 256 MS memory) and delivered in numerical format to a computer for processing and analysis. Figure 3 presents a scheme of the computer-assisted experimental setup with only the movable parts of the 3D printer involved in 2D circular trajectories (the extruder and the print bed), the optical sensors for \(x\) and \(y\)-motion non-contact measurement, the power supply for sensors, the numerical oscilloscope and the computer. Using an appropriate programming of the drive system written in G-code, the 3D printer was programmed to generate some identical repetitive circular trajectories of the extruder system related to the print bed (with 8.987 mm radius, each one covered in 8.987 s) for 50 s (for almost 5.5 complete trajectories). There are not special reasons to have this  
Figure 3: A scheme of the computer-assisted experimental setup. Figure 2: A partial view of the setup with optical sensor on \(x\)-axis; 6-optical sensor; 7-toothed belt for \(x\)-axis displacement; 8-s screw for \(z\)-axis displacement. coincidence of values for radius and time, except the fact that these values should be accurately found by curve fitting (with the sine model) of the motions involved in the achieving of circular trajectory. However, these values assure a relatively small speed on the 2D circular trajectory. These trajectories are generated using two theoretical pure harmonic motions (\(x\)-motion accomplished by the extruder system, \(y\)-motion accomplished by the print bed) on the \(x\) and \(y\)-axes (both having a period \(T\) of 8.987 s), experimentally revealed by the computer measurement setup, as a detail in Figure 4 indicates. Here the blue-colored curve describes the motion on the \(x\)-axis; the one colored in red describes the motion on \(y\)-axis, both with 20,000 s\({}^{-1}\) sampling rate. The choosing of this sampling rate is based on this argument: it should be at least two times bigger than the sampling rate of the sensors (10,000 s\({}^{-1}\)). The sampling rate of the sensors acts as the Nyquist frequency for the sampling rate of the oscilloscope. It is not difficult to remark the resources of these two simultaneous evolutions for P&A evaluation.