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This paper proposes a synchronous measurement system of four position-independent geometric errors (PIGEs) and two position-dependent geometric errors (PDGEs) for the rotary axis in five-axis machine tools. The measuring instruments used in this system include a touch-trigger probe and two standard calibration spheres. The main contribution of this paper simultaneously and directly measures four PIGEs and two PDGEs (axial and angular positioning errors) through only one measuring process. It is expected to improve the deficiencies of previous studies measuring PIGEs and PDGEs separately. The geometric errors of the C -axis of a mill-turn machine tool MT-540 were analyzed. The calculation process includes the establishment of the mathematical model of the machine and geometric error equations. The least squares method was used to solve the linear overdetermined system and calculate the values of the geometric errors. Before solving the geometric errors, the accuracy of the calculation process was verified by using the simulated method. The simulation results also confirm the feasibility of this measurement system. Then the data obtained from the experiments were used to calculate the geometric errors of the machine tool. During the experiments, the calibration procedure of the touch-trigger probe was calibrated. After the calibration procedure was completed, the mechanical coordinate values of the two standard calibration balls were measured. The geometric error equations were programmed on a computer, and the data of calibration spheres obtained from the experimental measurements were substituted into the program to calculate the four PIGEs and two PDGEs.

This paper presents a synchronous measurement methodology aimed at identifying four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) of the rotary axis in five-axis machine tools. Previous studies and literature have emphasized the challenge of simultaneously measuring and identifying PIGEs and PDGEs of the rotary axis in five-axis machine tools. Therefore, the primary objective of this paper is to propose a measurement methodology that can identify these errors simultaneously through a single measuring process. Compared to commercially available measuring instruments, this measurement system offers several advantages: it is easy to install, cost-effective, and can be applied to various types of five-axis machine tools. These benefits enable the establishment of a fast on-machine error measurement. The initial phase of the research involves establishing a mathematical model and computing the geometric error equations based on the specific type of machine tools in use. Subsequently, the difference between the ideal and actual center positions of the calibration sphere is determined by utilizing a touch-trigger probe while positioning the machine's rotary table at various angles. Finally, the experimental data is inputted into the mathematical algorithm to obtain the PIGEs and PDGEs of the rotary table. Post-experimentation, the PIGEs and PDGEs obtained through the proposed measurement method are incorporated into the controller as compensations. The feasibility of this approach is evaluated by measuring the volumetric errors of the machine tools both with and without compensation. The results demonstrate a significant reduction in the deviation of the volumetric errors, decreasing from 11.97 to 2.31 µm after compensation. This outcome underscores the potential of the proposed method for simultaneous measurement of geometric errors in the rotary axis of machine tools across various types and scenarios. Highlights This paper proposes a novelly synchronous measurement methodology for identifying the four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) of the rotary axis in five-axis machine tools. The measurement principle, geometric errors calculation algorithm as well as simulation and experiment verification of the proposed synchronous measurement methodology are presented. Based on experimental results demonstrating a reduction in volumetric error of up to 80.7%, it is evident that the proposed measurement method is both efficient and precise.

The geometric errors (GEs) of the spiral bevel gear milling machine will seriously affect the machining accuracy of the tooth surface and need to be compensated. In this paper, an innovative method for compensating the geometric error of CNC gear milling machine is proposed. This method describes the explicit relationship between the motion axis of the machine tool and the geometric errors, realizes the dynamic compensation by tooth, and improves the machining accuracy and machining efficiency of the tooth surface. Firstly, the actual forward kinematics model (FKM) is constructed based on the geometric errors module, and the error tooth surface with GEs is established. Secondly, the corresponding relationship between the machine setting parameters of the spiral bevel gear universal generation machine (UGM) and the CNC machine tool motion axis is given, and then the functional expression between the motion axis with geometric errors and the machine setting parameters is established, which is the inverse kinematics model (IKM). Then, the tooth surface error correction model is established according to the relationship between the machine setting parameters and the tooth surface errors. The compensation amount of the machine setting parameters obtained by the model is introduced into the IKM to obtain the GEs compensation model. Finally, the effectiveness of the geometric error compensation technique is verified by numerical analysis and experiments. The results show that the tooth surface errors, contact stress, and loaded transmission error after geometric errors compensation are significantly reduced, and the contact pattern meets the design requirements, which verifies the feasibility and effectiveness of the geometric errors compensation technology.

The accuracy of material extrusion-based 3D printers is greatly affected by the high temperature in the chamber due to the thermal-induced deformation of components. However, most existing measurement equipment cannot be applied to high-temperature environments, which hinders the corresponding error measurement. To address this issue, a geometric error detection system and identification algorithm based on binocular vision are proposed. Firstly, a corner detection algorithm and a ray-intersection binocular model are used to identify the three-dimensional displacement of the target. Secondly, an error separation and identification algorithm is proposed to identify 21 position-dependent geometric errors. Error measurement experiments are conducted on a 3D printer at room temperature and high temperature, respectively. The experimental results at room temperature are verified using a double-ball bar. Finally, an error compensation experiment is conducted to verify the effectiveness of error identification, which also shows the contribution of error motions of linear axes on the printing accuracy.

This paper presents a double ballbar method with an extension fixture to identify the position-independent geometric errors of three-axis machine tools with respect to their entire workspace by conducting face- and body-diagonal length tests. To extend the length of the double ballbar to the required nominal length in the face- and body-diagonal directions, an extension fixture is designed and manufactured using a fused-deposition-modeling 3D printer to ensure that it is lightweight. Inherent position-independent geometric errors can be calculated from measured lengths by using homogeneous transformation matrices, which are multiplied under the assumption of small values, to determine the volumetric error produced by a machine tool. The relationships between the position-independent geometric errors, roll-pitch-yaw errors, and measured lengths can be derived according to the definition of the straightness errors, based on the end-point of the reference straight line. Finally, the position-independent geometric errors, with analysis of measurement uncertainties, can be identified by substituting the values of the roll-pitch-yaw errors, measured by a multi-axis calibrator, into the derived relations. The results can then be validated by re-measuring the face- and body-diagonal lengths with error compensation. The main advantage of the proposed approach is that it can be used to identify the inherent position-independent geometric errors of machine tool workspaces. This could contribute to reducing volumetric errors in machine-tool workspaces by allowing compensation for measurement errors, thus making machine tools more effective.

For achieving high precision and effectiveness in five-axis computer numerical control (CNC) machine tools, the geometrical accuracy of the rotary axes is a crucial performance criterion. Furthermore, recent advancements in commercial CNCs for machine tools have enabled the numerical compensation for all parameters of geometric errors within rotary axes. As a result, this paper initially delves into the evolution of ISO standards concerning the accuracy testing and error definition in machine tools. Subsequently, the classifications of the rotary axis’s geometric errors in five-axis machine tools are described in this paper. Moreover, this paper comprehensively reviews various measurement schemes aimed at identifying the geometric errors of rotary axes. These measurement schemes are categorized based on the measurement instruments or technologies employed. Finally, it is essential to emphasize that this paper offers an overview of diverse measurement theories and technologies pertaining to geometric errors in rotary axes. The primary aim is to contribute to the progression of geometric error measurement and compensation in five-axis machine tools.

A novel elasto-geometric model is introduced that simultaneously estimates joint compliances and geometric error parameters by employing the loaded double ball bar apparatus. The model parameters are estimated from tests at different force levels by distinguishing between errors that change with the applied force (compliance effect) from those that do not (geometric effects). At lower forces, the geometric errors are dominant while at higher forces compliance errors dominate. Using all data to build a single global geometry and compliance set of parameters (global constant compliance model), the radial volumetric variations due to geometric errors and compliance are estimated at 0.019 mm and 0.046 mm, respectively, making compliance dominant by more than three times. The impact of dominant and non-dominant equivalent global compliance C XXX , C YYY , C XYX , C CXY , C CYY , and C CCY on the loaded circular test readings at the highest force level of 742 N are predicted to be around 0.045, 0.034, 0.00058, 0.0022, 0.0014, and 0.0045 mm peak-to-peak, respectively. The impact of loaded geometric parameters E XX1 , E YY1 , E YX2 , E XY2 , E C(0Y)X , E Xt0 , and E Yt0 on the loaded circular test readings is predicted to be around 0.019, 0.014, 0.0074, 0.012, 0.00017, 0.0076, and 0.0012 mm peak-to-peak, respectively. The dominant global compliances are C XXX and C YYY at 0.0619 and 0.0461 μ m / N , respectively.

Reconfigurable machine tool (RMT) is designed for mass customization machining and the structure is reconfigured unpredictably due to the wide range of requirements. After reconstruction, accuracy of the new machine tool must be guaranteed firstly, which puts forward higher requirements of flexibility for error definition and modeling. Therefore, a digital and structure-adaptive geometric error definition and modeling method is presented to quickly respond to the structure changes of RMT. The highlight of this method is that the definition and modeling of geometric errors can be realized automatically. Firstly, a coding method is proposed to express the machine tool component with structure and motion information so that the geometric error definition and modeling can be computerized. Then, common expression of the geometric errors considered kinematic and structural attributes is presented. The identification coefficient matrix of geometric error is defined and calculated by using an assignment algorithm according to the configuration tree. At last, geometric error modeling modules are defined and sequential multiplication calculation is presented to establish the geometric error model automatically. Three typical examples are illustrated to verify the correction of digital method. It is the basis of intelligent error compensation of RMT under the market environment of changing demands.

Identification of, and compensation for, geometric errors is a cost-effective way to reduce the volumetric errors of five-axis machine tools and thus reduce workpiece geometric errors. An adaptive identification method is introduced to directly reduce workpiece geometric errors. We determined the relation between the root-sum-square values of geometric error sensitivity coefficients and workpiece geometric errors. Then, an optimal measurement path minimizing those values was adaptively determined to identify position-independent geometric errors of the rotary axis. We applied our method to improve the radial deviation of the cone-shaped ISO 10791-7 testpiece, as an example. The radial deviations were 22.6 and 27.6 μm in the counterclockwise (CCW) and clockwise (CW) directions, respectively, after compensating for the position-independent geometric errors identified using a common measurement path. These values improved by 27% and 17% to 16.4 and 22.9 μm in the CCW and CW directions, respectively, after compensating for the position-independent geometric errors identified using the optimal measurement path, thus confirming the validity of our approach.

Robotic positioning accuracy is critically affected by both geometric and non-geometric errors. To address this dual error issue comprehensively, this paper proposes a novel two-stage compensation framework. First, a Memory based red billed blue magpie optimizer (MRBMO) is employed to identify and compensate for geometric errors by optimizing the geometric parameters based on end-effector observations. This memory-guided evolutionary mechanism effectively enhances the convergence accuracy and stability of the geometric calibration process. Second, a tuned graph neural network (AMPSO-GNN) is developed to model and compensate for non-geometric errors, such as cable deformation, thermal drift, and control imperfections. The GNN architecture captures the topological structure of the robotic system, while the adaptive momentum PSO dynamically optimizes the network’s hyperparameters for improved generalization. Experimental results on a six-axis industrial robot demonstrate that the proposed method significantly reduces residual positioning errors, achieving higher accuracy compared to conventional calibration and compensation strategies. This dual-compensation approach offers a scalable and robust solution for precision-critical robotic applications.