A method for identifying and predicting the geometric errors of a rotating axis

To improve the machine tool accuracy, an integrated geometric error identification and prediction method is proposed to eliminate the positioning inaccuracy of tool ball for a double ball bar (DBB) caused by the rotary axis’ geometric errors in a multi-axis machine tool. In traditional geometric errors identification model based on homogenous transformation matrices (HTM), the elements of position-dependent geometric errors(PDGEs) are defifined in the local frames attached to the axial displacement, which is inconvenient to do redundance analysis. Thus, this paper proposed an integrated geometric error identification and prediction method to solve the uncertainty problem of the PDGEs of rotary axis. First, based on homogeneous transform matrix (HTM) and multi-body system (MBS) theory, The transfer matrix only considering the rotary axes is derived to determine the tool point position error model. Then a geometric errors identification of rotary axis is introduced by measuring the error increment in three directions. Meanwhile the geometric errors of C-axis are described as position-dependent truncated Fourier polynomials caused by fitting discrete values. Thus, The geometric error identification is converted to the function coefficient. Finally, the proposed new prediction and identification model of PDGEs in the global frame are verified through simulation and experiments with double ball-bar tests.


Introduction
Five-axis CNC machine tools provide greater productivity, better flexibility, and less fixture time than three axis machining centers, because the cutting tool can approach the workpiece from any direction. However, the two rotary axes would bring in additional geometric errors subsequently, such as squareness between a rotary axis and a translational axis [1].
Therefore, an accurate and efficient measurement method of geometric errors is a key prerequisite for the improvement of five-axis machine tool accuracy [2]. If there is an effective method to improve or compensate for those deviations, the machining performance of multi-axis machines will be improved drastically.
In the past decades, some researches have been taken to measure and identify the errors that are inherent to a rotary axis of a multi-axis machine tool. Tsutsumi and Saito proposed a calibration method using the simultaneous four-axis control technique for five-axis machining centers with a tilting rotary table. The eight deviations were estimated by the observation equations from the ball bar measured results [3]. Zargarbashi identified eight link errors of a five-axis machine through measuring the tool center position deviations at a number of five-axis poses by using a ball-bar sensor [4]. Zargarbashi and Mayer presented a method consisting of five double ball bar tests to evaluate five trunnion axis motion errors including axial, radial and tilt errors.
Some new instruments have also been designed and adopted for high efficiency indirect measurement of geometric errors, such as the R-test, Capball, 3D probe-ball or a touch trigger probe [5][6][7][8][9]. In recent years, geometric erorrs have been identified more conveniently and accurately based on new measuring instuments. However, It must be remembered that the measuring instruments themselves do not have the ability to identify geometric error elements, and their direct function is limited to measuring tool center position errors related to the workpiece. Therefore, the key factor to identify the geometric error is to correctly establish the mathematical model between the cutting tool processing point and the worktable processing point. At present, the commonly used error modeling methods are based on homogenous transformation matrix (HTM) [10][11][12] or screw theory [13][14]. A global description of rigid body motion is allowed with the screw theory, this is the advantage and difference from HTM method. However, the geometric errors defined in the Cartesian space must be converted to twists in the three-dimensional space via screw theory, which makes the calculation of screw theory relatively intricate compared to HTM method [15]. In this paper, A geometric error identification and compensation for the rotating axis of a five-axis machine tool is proposed based on HTM method.
Due to the complexity of the machine tool structure, geometric errors are commonly classified as position dependent geometric errors (PDGEs) and position independent geometric errors (PIGEs). PDGEs are mainly caused by imperfections of components, such as the straightness errors of the guide ways [16], while PIGEs are mainly caused by the imperfect assembly of parts, such as joint misalignments, angular offset and rotary axes separation errors [17]. Over the past few decades, many researches have reported the geometric errors identification methods for translational axes, such as 9-line method, the 15-line method and 12-line method by using the laser interferometer. Sepcially, the 9-line method has been recognized as a common method to detect translational errors according to ISO 230-1 (2012) [18]. While the error identification of the rotar axis yhas not been unified. Zhang et al. proposed a novel DBB measuring method, in which only the C-axis rotated. Unfortunately, it could only evaluate five PDGEs of the C-axis [19]. Similarly, Lei et al. presented a particular circular test path, which caused the two rotary axes only to move simultaneously and kept the other three linear axes stationary [20] However, some others may have such a capacity, require complex mathematical formulations and espacially involve a complicated measurement process to identify the error parameters at present. Therefore, in this paper, a new method for error identification of rotary axis is proposed, which is effective and uncomplicated to improve the accuracy of error compensation. Firstly, the geometric error truncated Fourier  which are usually produced during the assembly process [21]. For the fiveaxis CNC machine tools, whose configurations are depicted in Fig. 1, there are three translation axis(X,Y,Z) and two rotary axis (B,C) which is TTTRR Machine tool structure. Related parameters of rotary axis geometric errors are described in Table1.In these error parameters, PDGEs. For example, σ ! (C) represent the displacement error of the C axis in the X direction and xc represent the square errors between X aixis and C axis shown in Fig. 1 and Fig. 2 respectively.
According to Ref [13] and Ref [15], the position-independent geometric errors are caused by the deviation between the actual machine tool installation and the ideal installation. Thus, in this paper, these positionindependent geometric errors are regarded as a fixed size impact to related Accuracy of shaft movement. However, the PDGs are discribed as errors to the movement of the axis. Based on previous studies, these positionindependent geometric errors can be described as a function, which is more convenient to study geometric errors and more accurate for error Where f(θ) and θ denote geometric errors function and angles of rotation respectively. A i represent the coefficient.
is represented by the position error matrix and motion error matrix between adjacent bodies.
Therefore, the position error can be expressed as 6

Experiment and identify geometric errors of rotary axis
The identification and experiment of rotating axis geometric error are introduced in this paper. In previous studies, the identification of translational axis geometric error has been clearly specified, but the identification of rotating axis error has not been clearly specified. It can be seen from Table 1 multi-body theory and the Houston transformation matrix which contains six the linear errors and four angle errors. In order to express more briefly, the five PDGEs of C-axis are identified by using DBB, which is taken as an example of rotary axis.

Linear errors
Angle errors Table 1 The PDGs of rotary axis

Measurement
This article selects the double pendulum milling planer head for testing machine shown in Fig. 4, with translational axis geometric error compensated, using double cue instrument and machine tool axis of rotation movement relationship, success is a small local branch geometric error of each axis of rotation.
The detection of the C axis is divided into three steps: c-x, c-y and c-z.
The detection mode of the cue meter is detected in the X, Y and Z directions respectively. As shown in Fig   If Eq.(11) and Eq.(12) are put into Eq(10), the result will be described as 11 Here the equation has four unknowns and the rank of the coefficient matrix is 2. Hence, the error relationship needs to be confirmed in the c-y direction.
(2) C-Y measurement It will not be detailed here,since the detection method of c-y is the same as that of c-x. By combining Eq.10 and Eq.15, the error parameters of ( ) concerning the position of the four terms of the c-axis. C-Y can be inversely obtained.  (

3) C-Z measurement
In the c-z detection mode, the rod length transformation detected is the axial displacement of C-axis.So can be obtained as 18 Thus, Based on obtained constant, the five PDGEs of C axis will be indentified totally.

Results of calculation and simulation
In the third part, a method of geometric error identification of rotation axis is proposed，which only thinks about picking two points at random.  Taking C axis as an example, its five PDGEs are represented by the following, 20 [ ]   In order to verify effectivenss of the proposed method, a TTTRR type five-axis planer milling machine tool was utilized to measure the geometric errors of C-axis as shown in Fig.8. The workspace of the translational axes to be measured was x * y * z = 2500 mm * 2000 mm * 500 mm. The rotational ranges of the swivelling heads to be measured were C [0 °, 360 °] and B[0 °, 180°]. In order to more accurately measure the geometric error, the room should be about 20°C constant temperature.
In the simulations, the identification accuracy of different sets of measurement points was compared. The result of identified PDGEs were shown in Fig. 9, which are fitted by third orders truncated Fourier series.
From these, it can be seen that for both the predicted set and the measured sets of points, the motion positions of C-axis covered the entire measurement workspace and ensured a valid identification accuracy obtained from these sets of measurement points. It is obvious that the predicted values gets close to the measured values by comparing with the measured results. Among them, the maxmium values of residual errors are 3.4um, 2.8um, 1.2um, 0.48udeg and 0.52udeg, which can describe the deviation from predicted and measured values, respectively. In addition, the values of R-square were about fitting curve shown in Table.2 that the maxmium value of R-square gets to 0.976. As a matter of fact, the identification method used in this paper possesses less term numbers, smaller fitting residuals, and higher fitting accuracy.
In order to further prove that the proposed method is feasibility, a experiement is conducted which results are shown in Fig. 10    In addition to the above, the following areas need to be studied in the future.
(1) In order to further verify the advancement of this method, error identification tests using the fourth and more higher orders are conducted.
(2) In this paper, the position error modeling and identification method for geometric errors of C-axis have been studied without considering thermal error and the cutting force of the machine tool. Thus, more error source factors should be considered into error identification in the future.