# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 1 Introduction
Frequently, the contouring errors due to axis misalignment (also a relevant topic, investigated in our study) are involved in a bad P&A. The measurement of these errors and their compensation in the computer numerical control system of the printer is a major challenge in improving the additive fabrication performances. Thus, in [22] is proposed a simple compensation model for kinematic errors based on measurement results done on the printer using a Renishaw QC10 ball bar device (from Renishaw, UK). For the 3D printers based on parallel robotic systems, in [23; 24] some theoretical kinematic error models useful in automatic compensation are proposed. A volumetric experimental compensation technique for kinematic errors is exposed in [25]. In order to minimize the tracking errors of desired trajectories, a feed forward control procedure is proposed in [26]. This brief study of the literature reveals that few reports are focused on non-contact measurement methods of the errors produced by the kinematics of a 3D printer during complex motions of the extruder related to the print bed (especially 2D closed trajectories). Our approach on this paper was based on this obvious remark: the open-loop computer-aided control of each of three motions of a printer (usually 3D printers don't use feedback control related to real position) is vulnerable to some uncontrollable (constant or variable) phenomena generated by mechanical parts (elastic deformations in toothed belts, mechanical backlashes, hysteresis behaviour, errors in lead screw threads, variation of friction forces in axes carriages, mechanical wear, structural vibrations induced by stepper motors, etc.). Because the 3D printing is achieved by deposition of material layer by layer (in a horizontal plane), an important characterization of P&A for any printer should be done by the P&A displacement, position and, especially, of 2D complex closed trajectories with the printer running in absence of the printing process. On this line of thinking we consider that a circular trajectory (of the extruder system related to the print bed) is one of the best theoretical approaches mainly because two simultaneous, strictly correlated linear motions are involved (on the \(x\) and \(y\)-axis). With a constant speed on the circular trajectory, these motions should be described by two almost identical harmonic evolutions (except the \(\pi/2\) phase shift between), with periodic changes of the position, direction, speed and acceleration. An important argument for our approach based on circular trajectories is this one: the 2D circular trajectories are systematically involved in ISO standards methods for P&A evaluation of CNC manufacturing systems (ISO 230-4, [27]). The benefits of this approach with 2D circular trajectories in P&A evaluation is confirmed by the work from [22]. In addition, from an experimental point of view, it is appropriate to work with circular trajectories generated by two simultaneous, simple harmonic linear motions (cosine motion of the extruder system on the \(x\)-axis, sine motion of the print bed on the \(y\)-axis) measured with a computer-assisted experimental setup based on two optical (non-contact) position sensors. We consider that this is a better approach, in contrast with the measurement system proposed in [22], which uses a ball bar measurement device placed between the extruder and the print bed. Because of this device, the relative motion of the extruder related to the print bed is not totally free. The Section 2 of this paper presents the computer-aided experimental measurement setup, the Section 3 presents the theoretical and experimental considerations and comments related with the results (signal processing and P&A estimation, based mainly on real trajectory circular fitting), and the Section 4 is dedicated to conclusions and future work.