Position-dependent Geometric Errors Research Articles

Synchronous Measurement and Verification of Position-Independent Geometric Errors and Position-Dependent Geometric Errors of Rotary Axes on Five-Axis Machine Tools

Synchronous Measurement and Verification of Position-Independent Geometric Errors and Position-Dependent Geometric Errors of Rotary Axes on Five-Axis Machine Tools

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.

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.

Modeling and compensation of enhanced volumetric error of machine tools containing crosstalk errors

The error compensation method is an economical and effective method to improve the machining accuracy of machine tools. Conventional volumetric error modeling of machine tools is based on the rigid body kinematics model using 21 terms of geometric errors and assumes the geometric error of one axis is independent of other axes. However, if the error motion does not satisfy this assumption in serial machine tools, it may significantly influence the overall machine tool accuracy. This paper reveals the existence of mutual crosstalk of angular errors in coupling axes by experiments with equipping the five-degree-of-freedom (5-DOF) embedded sensor module in each axis. A new on-machine measurement method is proposed for detecting crosstalk errors of the coupling axes covering the whole coupling area. It is confirmed that the position-dependent geometric errors (PDGEs) of each axis should be modified by two-dimensional dependence. Therefore, extended from the normal volumetric error model (NVEM) that is commonly used nowadays, a novel enhanced volumetric error model (EVEM) that contains crosstalk's PDGEs is proposed. The real-time error compensation controller (RECC) embedded in CNC system kernel has been developed in collaboration with the vender by writing in the application function. The effectiveness and feasibility of the EVEM and RECC are verified by non-cutting and cutting experiments. Through comparison tests with laser interferometer on body diagonals, the accuracy improving rate of EVEM is about 10 % higher than NVEM. The real-cutting tests on the ISO10791 sample are verified by a CMM and the results show the accuracy improvement of EVEM is better than NVEM by about 20 %.

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.

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.

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.

A vision-based error identification method for separating geometric errors of rotating axes in five-axis platforms

In this paper, a programmable identification method for visually measuring and identifying position-independent and position-dependent geometric errors (PIGEs and PDGEs) of rotary axes in compact five-axis platforms is presented. Firstly, the kinematic model of the five-axis platform is established via screw algebraic theory. Then, considering the measurement data diversity, the orientation errors identification mechanism for PIGEs is proposed. The mechanism optimally integrates the coordinate search and the least-squares method. Additionally, a least-squares sample regression model is developed to identify the positional deviation of the rotation axis. After identifying PIGEs, the PDGE sources are identified based on the right multiplication principle. Furthermore, the error estimation accuracy of the proposed method is investigated. Finally, a five-axis dispensing platform is employed to validate the identification accuracy and evaluate its feasibility.

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%.

Synchronous measurement and verification of position-independent geometric errors and position-dependent geometric errors in C-axis on mill-turn machine tools

Synchronous measurement and verification of position-independent geometric errors and position-dependent geometric errors in C-axis on mill-turn machine tools

Kinematic modeling and error parameter identification of automated fiber placement machine based on measured data

This paper presents a novel kinematic modeling and error parameter identification method for a six-axis gantry automated fiber placement (AFP) machine. Multi-body system theory is used to model the kinematics of the AFP machine, and then use the homogeneous transformation matrix to represent the transfer relationship of the coordinate system. Based on 54 geometric errors of the AFP machine, a volumetric error transfer model is established. To eliminate the training bias caused by sampling randomness, 10-fold cross-validation combined with the Levenberg–Marquardt method is used to train the kinematic model of the AFP machine. Based on the coordinates of random points in space measured by a laser tracker, all the position-dependent geometric errors and position-independent geometric errors of the linear and rotary axes can be identified simultaneously. After the error parameters are identified, the average error of the random points in the X-direction is reduced from 0.72 to 0.08 mm, a drop of 88.9%. Corresponding experiments are carried out on the body diagonals and the winglet placement path. The experimental results show that the volumetric error transfer model and error parameter identification method proposed in the article can accurately predict the volumetric errors of the AFP machine, and can also meet the requirements of the AFP machine placement accuracy.

Vision-based measurement for decoupling identification of geometric errors of rotating axes for five-axis platform

In this paper, a method for identifying and decoupling geometric errors of rotation axes using vision measurement is proposed. Based on screw theory and exponential product formula, identification equations of position-dependent geometric errors (PDGEs) and position-independent geometric errors (PIGEs) of the rotation axes are established. The mapping relationships between the error twist and geometric errors are established. The error model provides the coupling mechanism of PDGEs and PIGEs. Furthermore, a progressive decoupling method is proposed to separate PDGEs and PIGEs without additional assumptions. The pose parameters, required for solving the identification equations, are obtained by visual measurement. Then, the error terms of PIGEs and PDGEs are determined. Lastly, the error calibration of the rotation axes is investigated, thus providing an average rotary table orientation error reduction of 28.1% compared to the situation before calibration.

Separation and compensation of geometric errors of rotary axis in 5-axis ultra-precision machine tool by empirical mode decomposition method

In machining of a microstructure on a free form surface, a five-axis ultra-precision machine tool has a great advantage in its flexibility and efficiency compared to a conventional machine tool. However, rotary axes in the five-axis machine tools are sensitive to geometric errors due to their complicated structures. Therefore, the geometric errors of the rotary axes must be identified and compensated. Due to the nonlinear coupling and inevitable mutual influence between position independent geometric errors (PIGEs) and position dependent geometric errors (PDGEs), it is difficult to separate and compensate PIGEs and PDGEs accurately and effectively through conventional modeling and compensation methods, and there is no unified standard to judge whether the results are accurate. This paper first proposed an empirical mode detection (EMD) method to identify PIGEs and PDGEs of the rotary axis of a five-axis ultra-precision machine tool directly and simultaneously without any objective function, which solves the problem of mutual influence between mixed errors that cannot be avoided by conventional methods. The geometric errors are regarded as the mixed signals related with the rotation angle of the rotary axis, which are brought into the EMD procedure to calculate the error components. Cutting experiments were carried out on an Al6061 workpiece, and the results were analyzed to verify the proposed method.

A reconfigurable mechanism model for error identification in the double ball bar tests of machine tools

A unified approach is presented in this paper for error identification of machine tools by means of double ball bar (DBB) tests and a reconfigurable mechanism model. The mechanism model consists of a kinematic chain describing motion of the tested object and a spherical-prismatic-spherical (SPS) chain describing motion of the DBB. Depending on the types of motion of a machine tool measured, five different mechanisms, designated as PP-SPS, PPP-SPS, R–SPS, RR-SPS, and PPRRR-SPS, are employed to deal with (i) two linear axes, (ii) three linear axes, (iii) one rotary axis, (iv) two rotary axes, and (v) three linear and two rotary axes, respectively. Treating the whole DBB test loop as a mechanism allows for the measured data of the DBB, the structure parameters of the mechanism and the motion parameters to be coupled together by the displacement equations. The structure parameters are obtained using the measurements with the help of kinematic synthesis of mechanisms and saddle point programming. The position-independent geometric errors and the mounting position of the DBB, and the error motions caused by the position-dependent geometric errors of the machine tool are also discerned successfully with the proposed mechanism concept for any DBB mounting position and direction. The experimental results confirm that the proposed mechanism approach is valid for error identification of machine tools composed of the linear and rotary axes.

A general identification method for position-dependent geometric errors of rotary axis with single-axis driven

Position-dependent geometric errors (PDGEs) of rotary axis affect the accuracy of the multi-axis machine tool. However, many PDGE identification methods were not general enough and not applicable when the structure of the machine tool was limited or changed. In this paper, a general PDGE identification method with single-axis driven was proposed. Firstly, the comprehensive length change model of the double-ball bar (DBB) was established. The simplification was conducted through the constraint condition to obtain the identification matrix. Then, the effect of the installation errors of the DBB was analyzed and eliminated. Simulations were carried out to validate the correctness of the proposed method when considering the installation errors. Next 10 measurement patterns were determined according to the structure limitation of the small-size 5-axis machine tool. Totally 364 combinations with full rank identification matrix are available. Finally, the prediction experiments and analysis for standard deviation were conducted to further validate the effectiveness of the proposed method. It turned out that the small condition number of the identification matrix can more likely achieve high accuracy. And an improved combination using 5 patterns was proposed with the same accuracy achieved compared to that using a total of 10 patterns.

Highly efficient and accurate calibration method for the position-dependent geometric errors of the rotary axes of a five-axis machine tool

This paper proposes a calibration method for continuous measurements with a double ball bar (DBB) used to identify the position-dependent geometric errors (PDGEs) of the rotary axes of five-axis machine tools. The different DBB installation modes for the rotary axes of the spindle and workbench are established, and the same initial DBB installation position is used for multiple tests. This approach minimizes the number of required DBB installations, which increases the measurement efficiency of the PDGEs of the rotary axes and reduces installation errors. PDGEs identification based on the adaptive least absolute shrinkage and selection operator (LASSO) method is proposed. By assigning coefficients to the PDGEs polynomial, the ill-conditioned problem of the identification process can be effectively avoided, thereby improving the identification accuracy. The measurement and identification methods proposed in this paper are verified by experiments on a five-axis machine tool. After compensation, the PDGEs are obviously reduced and the accuracy indexes of the circular trajectory tests performed under multiaxis synchronous control are obviously improved.

Geometric errors identification considering rigid-body motion constraint for rotary axis of multi-axis machine tool using a tracking interferometer

The measurement and identification of the four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) in the rotary axis are necessary to reduce their contributions to the overall machining errors of a multi-axis machine tool. In this paper, a new geometric error identification method using a tracking interferometer is presented by considering the rigid-body motion constraint in multilateration. The rigid-body motion constraint is introduced to establish a new coordinate calculation model for the measurement points, and an identification process is presented to separately identify the PIGEs and PDGEs by deriving identification models based on established geometric error models. The main novelty of the proposed method lies in the consideration of the rigid-body motion constraint, which makes the identification more robust against random factors. Monte Carlo simulations and verifying experiments were performed for validation. The results show that the PIGEs and PDGEs in the rotary axis can be successfully separated and identified by the new method. Improved identification accuracy compared with the traditional method is achieved. The maximum angular positioning error was reduced by 84% after compensation. A lower uncertainty in the identified errors compared with those obtained by the traditional method and instrument software is achieved. The proposed method is validated by comparing the identification results with those obtained by the instrument software and laser interferometers.

Geometric error identification and compensation for rotary worktable of gear profile grinding machines based on single-axis motion measurement and actual inverse kinematic model

Gear profile grinding accuracy is often restricted by geometric errors of the rotary worktable. This paper proposes a novel approach to improve the geometric accuracy of the worktable based on single-axis motion measurement and the actual inverse kinematic model (IKM). The volumetric error model of gear profile grinding machines is first established based on homogeneous transformation matrices (HTMs) and machine kinematic chains, and the actual IKM based compensation method is then proposed to offset the position-dependent geometric errors (PDGEs) of the worktable. It directly gives explicit analytical expressions of compensated motion commands for efficient compensation. Besides, a novel PDGEs identification approach based on single-axis motion measurement with a double ball bar (DBB) is presented for the worktable. This approach has improved identification accuracy and stronger robustness as the interference error of non-targeted axes is avoided. The influence of the installation errors of the workpiece ball on measurement results and the PDGEs on gear grinding errors were discussed. The experiments verified the effectiveness of the identification approach, and the geometric accuracy tests of the worktable validated the compensation method.