Position-independent Geometric Errors Research Articles

A novel identification method for all geometric errors of rotary axis based on boundaryless constraint optimization method

For the redundancy in definition of position independent geometric errors(PIGEs) and position dependent geometric errors(PDGEs), it is difficult to identify all geometric errors of rotational axis simultaneously. A novel identification method using boundaryless constraint optimization method(BCOM) is established to improve PIGEs and PDGEs identification accuracy of rotary axis. For PIGEs, the two different dimension errors are identified separately based on least square method through appropriate measurement direction by double ball bar(DBB), and thus decrease condition number of identification matrix. For PDGEs, the B-spline curve is applied to fit the geometric errors and the objective function has been established. Several measurement paths are conducted and geometric errors are figured out by iterations of objective function. Furthermore, setup errors of DBB are investigated and discussed based on the sensitivity analysis. Finally, Simulations and experiments are conducted and their results demonstrate the effectiveness of the proposed method.

A new machining test to identify position-independent geometric errors of rotary axes for five-axis machine tools

A new machining test to identify position-independent geometric errors of rotary axes for five-axis machine tools

Geometric error compensation method using the Laser R-test

Traditional methods for measuring geometric errors in machine tools, including interferometry and Double Ball Bar (DBB), are known to be expensive and time-intensive. Consequently, a non-contact calibration system called the “Laser R-test” has been developed. This innovative system is designed to measure both position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) efficiently. Since its development in 2000, this tool has been instrumental in analyzing eccentricity errors, angular position errors, and simultaneous trajectory errors. Through extensive research, it has been determined that the total error in a five-axis machine tool can be controlled to below 40 µm after compensating for eccentricity parameters and angular position errors. However, reducing this error to below ± 10 µm is challenging, primarily due to wobble errors in the orientation of the rotary axis without compensating. In this study, a new methodology based on Laser R-test and Rodrigues’ rotation formula has been developed to establish a PIGE model of rotary axis. Based on the methodology, the 8 PIGEs can be analyzed by measuring 5 coordinate positions. The compensation of 8 PIGEs in the rotary axis is completed within 30 min using the inspection path. Compatibility with ISO-10791–6 standards for BK1, BK2, and BK4 path tests is confirmed, validating the compensation effects. A precision of below ± 10 µm is achieved, with inspection time reduced by over 50%. This system can complete multiple errors by simply using the different paths. This greatly reduces the setup time for future users, enhancing its commercial applicability.

Adaptive Identification of the Position-independent Geometric Errors for the Rotary Axis of Five-axis Machine Tools to Directly Improve Workpiece Geometric Errors

Adaptive Identification of the Position-independent Geometric Errors for the Rotary Axis of Five-axis Machine Tools to Directly Improve Workpiece Geometric Errors

An efficient method for measuring and identifying geometric and dynamic errors in dual five-axis machine tools

The dual five-axis machining machine is state-of-the-art equipment designed for producing thin-walled aerospace parts like skins and panels. It comprises two five-axis machine tools with mirrored synchronous motions: one bi-rotary machine for milling and the other machine for follow-up support. The high absolute accuracy of the five-axis machine tools themselves and the high synchronous motion accuracy between the two machines are both crucial to ensure machining accuracy. This paper proposes an efficient method to measure and identify geometric and dynamic errors of dual five-axis machine tools. Using the R-test combined with three measurement trajectories to measure the absolute motion error of two five-axis machine tools and the synchronous motion error between the two machines. The designed trajectories include the main rotary axis linkage conditions and can fully reflect the primary motion forms of the dual five-axis machine tool. The main position-independent geometric errors and rotational axis linkage dynamic errors are identified by converting quasi-static and dynamic measurement data from three different trajectories into a unified reference coordinate system. The entire process requires only one instrument installation, effectively avoiding time-consuming repetitive installations and calibration processes while reducing the impact of installation errors on results. The effectiveness of the proposed method for measuring and identifying the geometric and dynamic errors is analyzed combined with Monte Carlo simulation. Experiments were carried out on a dual five-axis mirror milling system (MMS) with a 5000 mm X-stroke. Both simulations and experiments confirmed that the proposed method is efficient in quickly identifying key error sources of the MMS.

Uncertainty assessment of machine tool squareness error identification using on-machine measurement

Identification and compensation of geometric errors in the machine tool are widely performed to increase machining accuracy. Periodic verification of the geometric errors, as introduced in ISO 230-2, helps monitor the machine tool accuracy during continued manufacturing operation and detect accuracy degradation early on. While interferometry-based error identification techniques are commonly applied, on-machine measurement (OMM), with its increasing availability to machine tools, can be used to identify geometric errors as an alternative to traditional interferometry-based techniques. As geometric errors also contribute to OMM error, assessment of the uncertainty of error identification is essential to ensure the reliability of the identification. This work presents a method to evaluate the uncertainty in the OMM-based geometric error identification process by Monte Carlo simulation. The error ellipse model, which represents OMM errors with better accuracy, is utilized to improve the identification uncertainty. The squareness errors, as position-independent geometric errors that contribute to machining inaccuracy, are taken as an example to demonstrate the presented method. The influence of the artifact setup errors on the error identification is also investigated. A series of experiments are conducted to evaluate the uncertainty of the OMM-based identification method. The geometric error obtained with the presented method is found to deviate by less than 10% from that obtained with an interferometry-based commercial instrument. The uncertainty obtained from the proposed Monte Carlo simulation method matches well with the uncertainty results obtained by the Guide to the Expression of Uncertainty in Measurement method and repeated measurements.

R-test-based identification method for geometric errors of rotary axes in a five-axis machine tool with a rotary table and tilting head

This paper uses an R-test method to identify the 20 geometric errors of the two rotary axes in a five-axis machine tool with a tilting head and a rotary table. The Position-Independent Geometric Errors (PIGEs) result in deviations of the actual axis from the ideal axis, leading to the overall deviation of the set points on the circular trajectory. Therefore, the actual circular trajectory can be fitted by measuring the actual coordinates of the set points, and the actual axis can be further solved to establish the solution equation of PIGEs. Further, the volumetric error model for the set points is constructed, and the unknown motion matrices associated with the linear axes in the volumetric error model are adjusted, allowing for the determination of Position-Dependent Geometric Errors (PDGEs). This method can identify geometric errors of the two rotary axes by measuring six sets of circular trajectories under three setups.

Calibration of link errors in rotary axes of five-axis machine tools using a six-dimensional motion measuring device

In this study, the position-independent geometric errors (PIGEs) of five-axis machine tools are identified and calibrated using a Stewart–Gough platform (SGP) based Motion Measuring Device (MMD). As the MMD of micrometer-level accuracy is cable of six-dimensional pose measurements, a closed kinematic chain is established by installing the upper and lower plate of the MMD to the spindle and rotary worktable of the machine tool, respectively. Particularly, the sensor-mounting misalignments, which are inevitable but mostly ignored in the previous PIGE identification methods, are formulated and identified together with the PIGEs. Thus, the sensor-mounting misalignments will not affect the accuracy of PIGE calibration. The proposed method is verified by simulation and experimental results. The synchronous motion test (SMT) demonstrates that the five-axis machine tool owns a high accuracy of 5 micrometers in the tested motion range after PIGE calibration.

Position-independent geometric error compensation of a non-orthogonal five-axis machine tool using a simplified algebraic algorithm

Geometric errors are the key error sources that affect the machining quality of a five-axis machine tool. In the case of a non-orthogonal five-axis machine tool (NOFAMT), the modelling and compensation of geometric errors can be quite complex. This paper proposes a simplified modelling algorithm to derive the analytical expressions for the position-independent geometric error (PIGE) compensation of a NOFAMT without generating higher-order error terms. First, the geometric error modelling was conducted based on the homogeneous transformation matrix (HTM). Then, algebraic expressions for PIGE compensation were obtained according to the actual inverse kinematics and a simplified algorithm. By using the simplified algorithm, the efficiency of deriving algebraic expressions for geometric error compensation can be greatly enhanced. The PIGE compensation’s numerical control (NC) codes can be directly computed using these algebraic expressions. Finally, the effectiveness and efficiency of the new algorithm were verified by numerical analysis and virtual machining. The detailed results were compared with those obtained using the differential method, and both sets of results showed the same trend.

New machining test for identifying geometric and thermal errors of rotary axes for five-axis machine tools

This paper presents a new machining test for identifying the geometric and thermal errors of rotary axes for a five-axis machine tool. A disc-shaped test piece containing 12 rectangular grooves is designed and processed. A coordinate-measuring machine is used to measure the rectangular grooves to identify four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) of the rotary axis. On-machine measurements are performed under heating and cooling conditions to identify the six thermal errors. The identification principle and analytical solution of each geometric and thermal error are provided, and an uncertainty analysis is performed. Compared with two classical machining tests, the results of the proposed method are in good agreement; the agreement degrees of the PIGE, PDGE, and thermal error are 90.9%, 91.3%, and 85.1%, respectively. The identification principle and analytical solution of each error are simple and not affected by linear axes, thereby improving accuracy.

Sensitivity analysis of inverse multilateration based on tracking interferometer measurement for a bi-rotary head five-axis machine tool

This paper proposes a novel inverse multilateration that improves the accuracy by sensitivity analysis and calibration of rotary axes’ position-independent geometric errors. In recent years, the tracking interferometer measurement with inverse multilateration has been widely adopted in the assessment and compensation of the five-axis machine tool. It uses the bi-rotary head as the steering mechanism to reduce cost, which also introduces errors from rotary axes. In this paper, a sensitivity analysis for position-independent geometric errors of rotary axes is proposed to reveal the error propagation mechanism and identify the main error sources. An optimization for the number and distribution of measured points is investigated based on the result of sensitivity analysis. Monte Carlo simulations and experiments are carried out on a bi-rotary head five-axis machine tool to verify the proposed method. The results show that the accuracy and efficiency of inverse multilateration measurement have been significantly improved.

Defocus compensation scheme for femtosecond laser direct writing of large-area non-periodic micro-structures

The optical path error, the position-independent geometric error, and the form and position tolerance of the substrate will lead to position deviations of the laser focus resulting in the inconsistency of the large area micro-structure and limiting the processing range of the femtosecond laser fabrication platform. To realize continuous defocus compensation along the processing trajectory, an autofocus scheme consisting of an optimized depth from focus (DFF) method based on grid division and the bilinear interpolation algorithm is proposed in this paper. The optimized DFF method includes the Laplace-DWT evaluation function combining the advantages of spatial domain and frequency domain sharpness evaluation function, an adaptive focusing window selection method, and an adaptive step length three-step hill-climbing search method. The optimized DFF method was used to obtain the best focus position of each grid point. Compared with the traditional DFF method, the optimized algorithm has higher focusing precision and avoids the disadvantage of time-consuming focusing. Then, to compensate for interpolating points of the processing trajectory, the bilinear interpolation algorithm was introduced to achieve online focusing during two-photon lithography processing. Finally, the feasibility and effectiveness of the auto-focusing scheme were validated by experiments. Through defocus compensation, the sharpness evaluation value of the focus image on the processing trajectory improved by 90%, which indicated the successful focus of the laser beam on the workpiece surface. Moreover, a high-precision and large-area encoded micro-polarizer array with consistent structure were fabricated successfully. This paper provides a simple way to identify and compensate the defocus errors online with high accuracy and efficiency. The proposed method can improve the machining accuracy and quality of the femtosecond laser direct writing.

A Theoretical and Experimental Identification with Featured Structures for Crucial Position-Independent Geometric Errors in Ultra-Precision Machining

In ultra-precision machining (UPM), position-independent geometric errors (PIGEs), i.e., squareness errors, have a crucial impact upon the form accuracy of a machined surface. Accordingly, more research work has been conducted in PIGE identification, to improve the form accuracy. However, the general identification methods were developed without consideration of the specific squareness errors for crucial PIGEs under the form errors of the machining process. Therefore, a new method with featured structures was proposed, to identify crucial PIGEs in UPM. Firstly, a volumetric error model was developed for PIGEs, to discuss the relationship between squareness errors and their resulting machining form errors. Secondly, following the developed model, some featured structures have been proposed with their machining form errors, to significantly indicate crucial PIGEs. Finally, a series of UPM and measuring experiments were conducted for the featured structures, and then their machining form errors were measured and extracted with specific squareness errors for the identification of crucial PIGEs. The theoretical and experimental results revealed that the proposed method is simple and efficient with the featured structures to accurately identify crucial PIGEs in UPM. Significantly, the study offers a deep insight into high-quality fabrication in UPM.

Geometric error calibration and analysis of rotary axes on a five-axis dispensing machine

Geometric rotary axis errors are one of the main factors affecting the dispensing positioning accuracy of five-axis vision dispensing machines. Hence, the accurate detection and compensation of these error terms is an important and challenging problem. In this study, a decoupling identification method for both position-dependent geometric errors and position-independent geometric errors of a rotary dispensing table is proposed based on vision measurement technology. According to this strategy, the screw theory and exponential product formula are used to establish a kinematic model of the end-effector, followed by the definition of the mapping relationship between position-independent geometric errors and axis posture. Then, to separate the coupling error terms of the rotating axes, a step-by-step identification strategy is proposed based on the least squares method. Furthermore, a coordinate search-based error compensation algorithm is developed, which avoids the inverse singular value problem and is insensitive to the error model. Finally, the effectiveness of the proposed identification algorithm and compensation algorithm is experimentally validated.

Simultaneous identification for geometric error of dual rotary axes in five-axis machine tools

For the rotary axes, the position-dependent geometric errors (PDGEs) and position-independent geometric errors (PIGEs) affect their rotational accuracy concurrently. Therefore, aiming at analyzing this coupling effect of geometric errors, a novel method is presented to identify the PDGEs and PIGEs of dual rotary axes simultaneously. The identification model is established based on the geometric error model and actual distances between the two retroreflectors, by which all geometric errors of two rotary axes can be obtained simultaneously. The identification results are validated by predicting volumetric errors, in which the deviations between the predicted and measured VEs are within 21.7 μm in three directions. Moreover, the root mean square error between the predicted and measured volumetric errors is 6.8 μm, which indicates that the proposed method can effectively accomplish the simultaneous identification of geometric errors. This identification method facilitates the further study of the coupling effect of PDGEs and PIGEs.

Non-redundant identification of geometric errors in multilateration of rotary axis

The multilateration measurement acquires sufficient data for the identification of the position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) of the rotary axis. However, because of redundancy in the used intermediate parameters, existing identification methods suffer from a loss of accuracy and precision caused by the relatively lower stability to the influences of random errors. Meanwhile, this may result in a more complicated process if additional constraints are added to reinforce stability. This paper proposes a non-redundant method for identifying both the PIGEs and PDGEs of the rotary axis. Firstly, the identification model, in which the rotating component pose errors are fully and non-redundantly described by the twists, is established to identify the pose errors caused by geometric errors. Then, by deriving the relationship between the pose error twists and the PIGE & PDGE parameters, the sequential decoupling algorithm applying the least squares condition, together with identification models, for both the PIGEs and PDGEs is developed. Comparative simulations and experiments are carried out. With no redundancy, the proposed method exhibits smaller discrepancies between the identified and preset test lengths in the cone path test simulation, yielding an average improvement of 15.73% and 56.97% in two identification modes, respectively. Furthermore, the method demonstrates smaller discrepancies between the predicted and measured test data in the cross-validation experiment with an average improvement of 80.21% and 86.87% in the two modes, respectively. These results jointly verify the improvement in accuracy for identifying the PIGE and PDGE parameters. In addition, an overall reduction of the PIGE and PDGE uncertainties is observed in the Monte-Carlo simulations, which verifies the precision improvement.

Identification and compensation of position independent geometric errors of dual rotary axes for hybrid-type five-axis machine tool based on unit dual quaternions

This paper presents a novel synchronous motion trajectory based on the unit dual quaternion (UDQ) to identify the position-independent geometric errors (PIGEs) of the dual rotary axes of a hybrid five-axis machine tool using a Double ball-bar (DBB). A general irregular trajectory equipartition method is presented to deal with the asynchronous matching between DBB data sampling and trajectory points. The kinematic model is effectively simplified by redefining the PIGEs of linear/rotary axes. The installation errors of the DBB are obtained by circular/spherical fitting trajectories, which are brought into the error model for elimination. The simulation of the UDQ kinematic model combined with preset error value shows that the non-uniform variation of DBB motion will lead to the deviation of PIGEs decoupling results. A compensation strategy of major orientation errors based on the UDQ is proposed, which is verified by simulation and compensation experiments.

Dual quaternion-based kinematic modeling for decoupling identification of geometric errors of rotary axes in five-axis platforms

Identification of the geometric errors of rotary axes is important for deciding the machining accuracy of five-axis platforms. In this paper, a programmable identification method is proposed to decouple the position-dependent and position-independent geometric errors (PDGEs and PIGEs) of the rotation axis. First, the kinematics model of the five-axis platform is developed by dual quaternions theory, and then the physical significance of left multiplication and right multiplication of PDGEs error model is analyzed. The motion difference between the base coordinate system and its local coordinate system is compared by simulation analysis. The geometric relationship between rotation direction and motion elements is established based on the error model, which is used to design an optimization algorithm for identifying the orientation errors of PIGEs. Additionally, the effect of linear axis PIGEs is eliminated by equivalent replacement, and a sample regression model is developed to identify the offset errors of PIGEs. The PDGEs are separated considering the Motion First principle. Finally, the proposed model and identification method were verified on a five-axis dispensing platform.

A new method to identify the position-independent geometric errors in the rotary axes of five-axis machine tools

Position-independent geometric errors (PIGEs), caused by assembly imperfection, affect the machine tool's accuracy. Moreover, the rotary axes of five-axis machine tools introduce more PIGEs during parts machining, hence, the need to identify them to improve the accuracy. However, the identification of rotary axes centerlines and the use of extension bars, using double ball bar (DBB), introduce additional setup errors and uncertainties during error measurements and sometimes make the process tedious. This work proposes a new method to identify the eight (8) PIGEs in the rotary axes of a five-axis machine tool with a tilting table. The homogeneous transformation matrix (HTM) was used to construct the relative motion matrix to the established coordinate systems and then deduced the error model. Three different DBB installation setups were employed for the identification of the PIGEs for both rotary axes using the planar test, the recorded DBB length data was substituted into the model to obtain the PIGEs. The results show that the proposed PIGEs identification method is accurate and can improve the accuracy of the five-axis machine tool effectively, and this method has some advantages over some existing methods of identification in terms of simplicity which shows a good improvement in the manufacturing industry.

All position-independent and position-dependent geometric error measurement and identification of the precision of a horizontal boring machine tool

Geometric error is the main obstacle that affects the quasi-static accuracy of horizontal boring machine tools. To improve quasi-static performance, the error terms of all position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) must be measured and identified precisely. This study proposes a novel method for measuring and identifying the above two types of geometric errors based on double ballbar for the rotary axis. The error terms of the linear axis are identified from the measurement model based on the exponential product method and a preset measure trajectory, and the measure coordinate system is combined with the reference coordinate to lower the uncertainty in the identification accuracy. Then, the elastic net method and improved weight function IGG3-LTS method are established to identify the PIGEs and PDGEs of the rotary and linear axes, respectively. This method has the advantage that measurement with the built-in sensor method synchronously does not interfere with the running state of the machine tool, and neither the inaccuracy of reference-frame misalignment nor installation error of the measuring instrument affects the geometric error identification results of the linear axis. Moreover, all PIGEs and PDGEs, particular to the rotary axis of the horizontal boring machine tool, are determined accurately. The measurement and identification strategy is demonstrated by experimental comparison on a four-axis precision horizontal boring machine tool. The minimum compensation rate of each precision index of the test specimen is 40.0%, the average compensation rate is 52.7%, and the maximum compensation rate is 62.5%.