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1.INTRODUCTIONGeometric error measurement and uncertainty analysis of rotary axis of five-axis machine is a difficult research point in recent years. The rotary axis geometric error directly affects the machining quality of the five-axis machine1. According to the causes and manifestations of the errors, the geometric errors of the rotary axis are mainly divided into: position-dependent geometric errors (PDGEs) and position-independent geometric errors (PIGEs)2. PDGEs can be understood as the non-ideal motion of the axis, which varies with rotation and is represented by a position-dependent function. There are many PDGEs’ measurement methods. He used a laser interferometer to measure by a dual optical path measurement method. The identification process only relies on simple algebraic relationships without volume error modeling, which is easier to understand3. Wang used laser tracker to identify rotation axis errors based on the idea of spatial sequence multiple points4. Deng considered rigid body motion constraints and used tracking interferometer measurements to effectively separate the rotational axis PDGE from the PIGE5. A ballbar is a commonly used instrument for measuring geometric errors, which can identify error terms from measurement data based on a kinematic model. However, since the measurement results are not necessarily reliable, uncertainty analysis is required6. Current uncertainty evaluation methods can be divided into GUM7 and Monte Carlo method (MCM)8. MCM allows the evaluation of uncertainty without calculating the partial derivatives of the measurement model9, which exactly matches the characteristics of ballbar’s multiple input and output, so this study uses MCM to analyze the measurement uncertainty of the ballbar. Scholars often ignore tool setting error and translation axis motion error, and these two cannot be eliminated by compensation10. Therefore, both types of errors need to be considered when analyzing the measurement uncertainty of a ballbar. In view of the above situation, this study established a geometric error model and carried out measurement uncertainty analysis based on MCM. The results of the uncertainty evaluation show the feasibility of this approach. 2.GEOMETRIC ERROR MEASUREMENT OF ROTARY AXIS2.1Kinematic model of five-axis machineThe structure of the five-axis machine used in this study is shown in Figure 1. The machine has a coordinate systems with five axes: X(XCS), Y(YCS), Z(ZCS), A(ACS) and C(CCS). In addition, there are bed (MCS), spindle (SCS), cutting tool (TCS) and workpiece coordinate systems (WCS). According to the machine tool structure, the tool chain is MCS → YCS → XCS → ZCS → SCS → TCS and the workpiece chain is MCS → ACS → CCS → WCS. The distance between measuring ball’s actual position Pactual and theoretical position Pideal needs to be calculated. The the ball’s position Pactual_A on the table side can be calculated by Equation (1). Where, EA represents PDGEs error matrix, TA-ideal represents the ideal motion matrix, and P0 represents the measurement ball’s initial position. And the following study only takes the A-axis as an example for analysis. The length change of the ballbar ∆L is based on the kinematic model, which is obtained by measuring the A-axis, as shown in Equation (2). Where, H represents the distance between the ball on the table side and the center line of the A-axis. 2.2Modeling and measurement of PDGEsThe error decomposition of the A-axis is performed in the measurement modes of three directions, and then the corresponding ballbar length variations ∆lradial, ∆ltan, and ∆laxial are obtained11. According to the error decomposition formula in Equation (2), the identification of A-axis PDGEs can be calculated by Equation (3). During the measurement of A-axis PDGEs, the A-axis motion range is set from 0° to -90°, and the A-axis angle an is recorded every 15°. As shown in Equation (4), three measurement modes are performed under three different installation parameters, and Equation (3) is simplified to obtain the identification equations of PDGEs under different installation parameters. Where, Lm and Hm are different installation parameters at m-th test (m=1, 2, 3), and the subscript n of PDGEs corresponds to the A-axis motion angle an. (∆lradial)m,n, (Δltan)m,n, and (∆laxial)m,n represent the variation of ballbar length corresponding to the A-axis motion angle an for radial, tangential, and axial measurements under m-th installation parameters, respectively. By decoupling Equation (4), all PDGEs of the A-axis can be obtained, as shown in Equation (5). It is noticed here that there are two solutions for εxa in Equation 5(a). Different sine and cosine in the denominator will have different results, and the appropriate formula should be selected according to the angle. 3.MEASUREMENT UNCERTAINTY ANALYSISThe influence of tool setting error, ballbar installation error and translational axis motion error on measurement uncertainty is considered. On this basis, an identification model Y=f (X1, X2, …, Xn) considering multiple error sources is established. Here the output Y is the PDGEs identification result and the input Xn is the source of uncertainty. For the A-axis, the measurement model of the PDGEs can be expressed by Equation (6) after substituting the uncertainties introduced by the tool setting error, ballbar installation error, and translational axis motion error into Equation (3). Where, (∆ρradial)m,n, (∆ρtan)m,n, (∆ρaxial)m,n, (∆xt_axis)m,n, (∆yt_axis)m,n and (∆zt_axis)m,n are calculated by Equation (7). ∆xt_axis, ∆yt_axis, and ∆zt_axis are the errors in three directions caused by the translational axes motion error, respectively; ∆h is the error caused by the tool setting error; ∆ρradial, ∆ρtan, and ∆ρaxial are the ballbar length variations in radial, tangential, and axial measurements caused by the installation error. 4.EXPERIMENTAL VERIFICATION4.1PDGEs identification and uncertainty evaluationThe PDGEs measurement experiments of the rotary axis are carried out on an AC rotary table five-axis machine. The measurement process is shown in Figure 2. In this study, the experiments of various geometric errors when an is 60° and the geometric errors εxa at different motion angles of the A-axis are carried out. The identification result is shown in Figure 3. Figure 3.Identification results with the same rotation angle and εxa with different rotation angles.  According to the uncertainty evaluation model of PDGEs, the tool setting error, ballbar installation error and translational axis motion error are sampled in the distribution interval with a sample size of M times in Monte Carlo test. For the inclusion probability p = 0.95, M is chosen to be 106 times in this study. Substituting the above uncertainty sources into the uncertainty evaluation model, the M times output results of six PDGEs with different motion angles are obtained, which constitute the distribution of PDGEs output. The PDGEs output is sorted by non-decreasing order to obtain the identification results of PDGEs of A-axis under the influence of uncertainty sources, as shown in Figure 4. The results show that the uncertainty of the certain errors is very large compared to the geometric error itself. Due to the uncertainties vary at different angles, the mean values of the uncertainties of each geometric error of A-axis at different angles are shown in Table 1. Figure 4.Identification results of A-axis PDGEs considering measurement uncertainty. Error bars represent the estimation uncertainty: (a) Position error δxa; (b) Position error δya; (c) Position error δza; (d) Angle error εxa; (e) Angle error εya; (f) Angle error εza.  Table 1.The mean value of the uncertainty evaluation for different angles of A-axis.
4.2Analysis of uncertainty evaluation resultsWhen the sine and cosine term in the denominator tend to zero, it leads to large measurement uncertainty and result to large deviation from the true value. Therefore, for the A-axis, when an is close to -90°, the equation containing the sine term is used to identify εxa, which can effectively reduce the measurement uncertainty of the angular positioning error. As shown in Figure 5, εxa of the A-axis under -60° and -75° is significantly reduced compared with that before optimization, and the uncertainty represented by the error bars tend to be stable under different angles. The standard uncertainty of εxa is reduced from 9.38″ to 6.43″ and the inclusion interval is narrowed from (ε - 18.40, ε + 18.40) to (ε – 12.61, ε + 12.61), showing a decrease of about 30%. 5.CONCLUSIONS(1) The rotary axis’s PDGEs error model considering tool setting error, translational axis error and installation error is established. (2) The translational axis motion error and tool setting error are the main uncertainty sources in the PDGEs measurement uncertainty. (3) The PDGEs standard uncertainty of AC axis is stable, which verifies the validity of this study and can reflect the quality of the measurement results. ACKNOWLEDGEMENTSThis research is sponsored by Key Research and Development Program of Ningbo (2021Z077). REFERENCESFu, G., Gong, H., Fu, J., Gao, H. and Deng, X.,
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