Local Calibration of a Weak Stiffness On-Machine Measuring Device with a New 3-D Calibrator


 This paper presents a new local calibration method for a weak stiffness on-machine measuring (OMM) device. First, the original volumetric error models’ union is established, and an equivalent dimensionality reduction algorithm is proposed to reduce the model’s dimensionality, thus, the novel equivalent volumetric error models’ union (EVEMU) is derived, which is more suitable for limited calibration data. For better flexibility and efficiency of error measurements, a new 3-D ball array calibrator is proposed, and its local calibration procedures can be executed directly in the 3-D space. Then, the smoothly clipped absolute deviation (SCAD) regularization algorithm with oracle property is introduced to identify equivalent coefficients as well as select the optimal volumetric error sub-model of EVEMU, which can closely represent the real error state of the weak stiffness OMM device. Experiments demonstrated that methods proposed in this paper are effective, and provide new insights for the error compensation of similar devices.

especially with the increasing demands for lightweight structures, some new weak stiffness measuring devices appear, which brings new challenges to the improvement of their accuracy. In the past few decades, the software-based error compensation method has been proven to be a cost-effective method that can improve the geometric accuracy of measuring devices [2]. and it can be achieved by following three steps [3]: (1) develop an error model for the machine; (2)use a measuring instrument to measure the errors; and (3) conduct error compensation using the error model.
As for the error modeling, Many effective error modeling methods have been proposed such as the homogeneous transformation matrix (HTM) [4], multi-body system (MBS) [5], screw theory [6], differential motion matrix method (DMM) [7] and elasto-geometrical error modeling method [8], etc. For the measurement of the error, the approaches can be distinguished into "direct" and "indirect" methods [9]. The "direct" method measures directly one or several geometric error items for one measurement setup. The ISO 230-1 [10] describes the main direct measurement methods that require precise measurement instruments, such as a laser interferometer with different combinations of optics kit. Nevertheless, most direct measurement methods require skilled operators, careful setting, and long measurement times, while for volumetric error compensation, the efficiency of the direct measurement is an issue [11]. On the other hand, the indirect method has become one of the main research fields due to its efficiency advantages. In particular, indirect measurement focuses on the tooltip location as the superposition of single geometric errors [11] and relies on multiple measurements in the machine work volume. In indirect methods, reference artifacts and laser tracker [12] are mainly utilized. In the past few decades, several reference artifacts have been developed, which have been presented in ISO 10360-2:2009 [13]. They can be classified into three categories: one-dimensional (1-D) artifacts that include step gauges, ball bars, and 1-D ball arrays; two-dimensional (2-D) artifacts that include ball plates and hole plates; and three-dimensional (3-D) artifacts that include 3-D grid artifacts and 3-D ball plates. In particular, Zhang et al. [14] proposed a geometric calibration method for a 3 linear axis coordinate measuring machine (CMM) based on a 1-D ball array, and several measurements were required to identify 21 geometric errors. Pahk et al. [15] developed a geometric error identification method based on ball bar measurements. A volumetric error model was constructed based on the geometric errors which were modeled as polynomial, and the ordinary least square (OLS) method was utilized to identify the coefficients of the respective geometric errors. Kunzmann et al. [16] proposed a ball plate-based calibration method, where the ball plate was fixed at different measurement positions, and then, the identified errors were separated through geometric relations. Trapet et al. [17] presented a calibration procedure using a hole plate, where the hole plate was measured in 4 positions in the working space, and 21 errors could be calculated separately. Liebrich et al. [18] proposed a 3-D ball plate for detecting machine tool errors. The ball plate could detect deviations of the X, Y, and Z coordinates at several points, which were used to identify a single geometric error with equations, or were used as direct input of a spatial compensation grid [19]. Viprey et al. [20] designed a multifunctional bar and developed a program for identifying 21 geometric errors of the linear axes. In addition, several new methods have also been adopted for the indirect measurement of 5-axis geometric errors, such as ball bars [21], R-test [22] and others [23]. For large machines, multilateration methods with the laser tracker, such as the 12-line method [24], are commonly used. As for error compensation, several numerical compensation strategies have been proposed. The final result compensation is a feasible method for a measuring device [25], which compensates the measured values by adding the compensation vectors to the values.
The above literature review indicates that a great deal of work has been conducted for error modeling, measurement and compensation, and has made remarkable achievements. However, these methods are mainly for machine tools or CMMs with relatively high stiffness, and relatively few investigations have been conducted for the accuracy improvement of weak stiffness measuring devices. In terms of error modeling, when there are many axes, the error model may have redundancy, methods to deal with the redundancy still deserve further development. Concerning the error measurements, most of the direct and indirect measurement methods with laser trackers rely on expensive optical measurement instruments and time-consuming operations. As for artifact-based indirect measurement methods, the existing 1-and 2-D artifacts have limited measurable dimensions, most of which need to be parallel to some axis or plane in order to measure explicit single geometric errors, thus, many measurements are required for measuring all geometric errors of a machine, so, these methods are relatively time-consuming, too. Some 3-D artifacts allow the measurement of volumetric errors, which can be used to identify single geometric errors or as direct input of the spatial compensation grid [19], however, these artifacts often have a large volume and lack flexibility. Therefore, the calibrator with better flexibility and efficiency is worth studying.
The prominent feature of the weak stiffness measuring device is "deformations" and the resulting "interactions" in this paper, as shown in Fig. 1, it means that the device may have large deformations at limited positions, which will affect the absolute positioning accuracy of the device, and as a result, their geometric errors may not be independent and perform like interacting with each other. Therefore, measuring explicit geometric errors of every axis and directly taking them into the volumetric error model may not be effective enough, the indirect method is more appropriate. Thus, we are no longer dedicated to identifying the explicit geometric errors in this paper, but to seeking a further equivalent solution in the intermediate solution space. First, we establish the original volumetric models union based on MBS [26], DMM [7], and response surface method [27], then, a new equivalent dimensionality reduction method is proposed, to construct the equivalent volumetric models union (EVEMU), which is more suitable for small sample measuring data. To improve the measurement flexibility and efficiency of artifacts as much as possible, a new 3-D ball array calibrator is proposed, it has 3 measurable dimensions, a more simple structure with a small volume compared to conventional 2-D or 3-D calibrators. furthermore, the calibration procedures are presented, and the concept of "local calibration" is introduced, which aims at improving the accuracy of specific local volumes, instead of calibrating the entire machine working volume. To weaken "interactions", we transformed the physical "interactions" problem into the multicollinearity problem of linear algebra, and the smoothly clipped absolute deviation (SCAD) algorithm with oracle property is introduced to deal with the multicollinearity and to select the optimal volumetric error sub-model from EVEMU at the same time. Local calibration experiments on a ZYX OMM device with weak stiffness are performed. The correctness and feasibility of the methods proposed in this paper are verified experimentally.
The rest of this paper is organized as follows: Section 2 introduces the EVEMU method; Section 3 proposes the 3-D ball array calibrator and its local calibration procedures; Section 4 introduces the SCAD-based identification and volumetric error compensation; Section 5 verifies experimentally the methods proposed in this paper, and Section 6 are conclusions.

EVEMU
To demonstrate the theory proposed in this paper, a 3-axis weak stiffness OMM measuring device is introduced (Fig.   1). The OMM device is designed to measure the internal and external freeform surfaces of thin-wall workpieces for several machine tools in production lines, so, it has a long cantilever structure and is movable that can move to different machine tools for on-machine measurement, however, the movement of the OMM device may lead to changes in accuracy, consequently, a relatively effectively instant calibration method is required in production scene. The OMM device is composed of a tactile probe and three orthogonal linear axes, i.e., Z-axis, X-axis, and Y-axis. In addition, there is an external rotary A-axis of another machine tool, and the workpiece is fixed on the A-axis. During the inspection, the tactile probe inspects along the intersection line of the workpiece over the X-Y plane (Fig. 1). Then, the A-axis rotates at a certain angle, the OMM device inspects again along the intersection line and repeats the above steps until the required measured data are obtained. The regions formed by these intersecting lines are always within the measuring volume shown in Fig. 1, which is a local volume within the entire working volume of the OMM device, and will be studied in this paper. Since the A-axis is calibrated beforehand, this paper only focuses on the local calibration of the three-axis OMM device. Due to the long cantilever structure and lightweight requirements, the OMM device shows relatively weak stiffness property especially at limited positions of the device, which causes the measured values to deviate from the theoretical values and is shown in Fig.8, and we define a weak stiffness structure if its volumetric error exceeds 1mm in every movement of 300mm. Therefore, the instantly online error calibration or accuracy maintenance of the OMM device are important for its industrial application, and this paper focuses on solving this problem. Figure 1. Schematic illustration of the ZXY-type OMM device, it is designed to measure the internal and external freeform surfaces of thin-wall workpieces for several machine tools in production lines, due to its long cantilever structure and lightweight requirements, the OMM device shows the relatively weak stiffness property especially at limit positions of the device, and cause the measured values to deviate from the theoretical values, we define a weak stiffness structure if its volumetric error exceeds 1mm in every movement of 300mm. Therefore, the instantly online error calibration or accuracy maintenance of the OMM device are important for its industrial application; and the detailed cutaway view shows the measuring volume of local calibration, and this paper focuses on solving this problem

Original volumetric error models' union
The ideal homogeneous coordinate transformation matrix from the tactile probe coordinate system to the machine's 6 base coordinate system wt_ideal ideal = ( , , ) f x y z T can be derived via MBS theory [26], but due to PDGEs and PIGEs, the actual transformation matrix wt_actual T is not equal to wt_ideal T , wt_actual actual = ( , , , , , , ) f x y z     T X Y Z S can be established via DMM [7]. Then, the volumetric error model of the OMM device can be derived by neglecting the higherorder error terms (above the 2 nd order) and errors of the tactile probe based on wt_actual x y z    E P P T T P Jξ (2) where   where G is the fitting variable matrix of K , x G , y G and z G are row matrices of each polynomial's variable , , the "np" is the highest order of each polynomial, and should be set in advance, etc. " ABC K " like column matrices are coefficient matrices of each polynomial related to ξ , and the " ABCi k " like expressions represent a polynomial coefficient, while the first letter of the subscript represents the error type, in particular, "d" represents a position error and "e" represents an angle error. The second letter of the subscript represents the direction of the affected axis, the third one represents the axis that moves, and the last one represents the corresponding variable's order. = E JGK (4) By merging JG to A, as shown in Eq. (5), and Eq. (6) is obtained  A JG (5) where A is a new Jacobian matrix about K, and E becomes a generalized linear model.  E AK (6) In Eq. (3) to Eq. (6), the selection of "np" is important for the bias-variance performance of a model, inappropriate choice of it may cause overfitting or underfitting problems. Aguado et al. [28] observed that the polynomial order is limited by the physical behavior of the geometric errors, and a third-order polynomial is a good approximation choice. Fu et al. [29] demonstrated that, sometimes, second-or first-order models may be preferable than third-order ones for PDGEs fitting.
Thus, orders not greater than 3 are considered as comprehensive choices, so, "np" is set as 3. It should be noted that "np" in this paper just represents the highest order of each polynomial and the optimal order will be automatically selected by the measured data and identification algorithm, so, each polynomial of PDGEs is actually a union of its sub-polynomials whose orders are not greater than "np". As a result, Eq. (6) is actually a union of its possible volumetric error sub-models, also called the original volumetric error models' union in this paper.

EVEMU based on equivalently dimensionality reduction
When calibration, A in Eq. (6) will be constructed from series of measured data, and K is the parameter vector to be identified. Before identifying K, it is necessary to check the redundancy of Eq. (6) [30], because if Jacobian matrix A is not column full rank, K has no unique solution. When using the iterative method to solve K, different initial values may produce different solutions. Although some analytical methods can solve its general solution [31], if there is a method which can equivalently reduce the dimensionality of Eq. (6), then, the equivalent A is column full rank, the corresponding equivalent K's solution will be unique, for the same amount of observed data, the dimensionality reduced equivalent model will relatively improve the data utilization and the confidence of the solution [32], especially when the dimensionality is significantly reduced with limited measured data. Assuming that there is a dimensionality reduction matrix rh 

W and its
Moore-Penrose inverse matrix † nr  W , and they satisfy the relationship shown in Eq. (7). Then, the dimensionality reduction can be realized.
Where mh  A and h1  K are same with those in Eq. (6), their subscripts represent the number of rows and columns, Based on Eq. (7) and Eq. (8), the E can also be represented by the new mr  X and r1  β in Eq. (9). Comparing Eq.
(9) with Eq. (7), the column dimensionality of mh  A and the row dimensionality h1  K is equivalently reduced from "h" to "r", and thus Eq. (9) is called the equivalent volumetric error models' union (EVEMU) in this paper. m r r 1   EXβ (9) To construct the appropriate    [31].
For the OMM device in this paper, based on Eq. (6) and "np" = 3, Compared with the original volumetric error models' union (

3-D ball array calibrator and its local calibration procedures
In Eq. (9), a series of   ,, x y z E E E data is required. Artifact-based calibrators are often used to acquire such observations in calibration. Among them, standard balls are commonly used since their centers can act as spatial references [14,35]. In this paper, to improve the local measuring efficiency and flexibility, a new 3-D ball array calibrator and its local calibration procedures are proposed (Fig. 2). the new calibrator is mainly inspired by the 1-D ball array [14,35] and ball bar [21], and its overall layout is similar to the ball bar and machine checking gauge system of Renishaw [36] to some extent.

3-D ball array calibrator
The 3-D ball array calibrator is composed of two 1-D standard ball arrays [14,35] with an L-type structure. According to data statistics, when measuring the volume, the bending shape of most intersecting lines can be generally approximated as the shape of the letter "L". Therefore, the L-type structure is closer to the workpiece surface, and does not increase the complexity of the calibrator's structure significantly. There are 9 standard balls (Ball1-Ball9;  (2). Subsequently, the same center coordinates of the standard balls and normal vectors of the standard blocks can be measured by the OMM device, and the measured values are treated as the actual values corresponding to actual P in Eq. (2).
Since the standard balls and blocks are fixed on the calibrator, the corresponding center coordinates and normal vectors are theoretically constant relative to the calibrator itself, irrespective of being measured at different positions or by different instruments, which is the theoretical basis of local calibration. However, due to various error sources, in practice, there will be deviations E   ,, x y z E E E between ideal and actual values. A method for establishing a uniform coordinate system (CS) is defined to compare the errors. The method is based on the following steps, which is also indicated in Fig. 3:

1) Position of Origin: Coincide with the center coordinate of Ball1
2) Orientation of Z-axis z V : same as the normal vector of Block1 As it can be seen in the above steps, the origin position refers to Ball1; thus, Ball1 is called the "position tracing ball", which establishes a 3-degree-of-freedom (DOF) position reference. The orientation of the uniform CS refers to Block1 and Block2; thus, the two blocks are called "orientation tracing blocks", which establish another 3-DOF orientation reference. In general, the uniform CS establishes a 6-DOF reference with regard to Ball1, Block1, and Block2. Consequently, the origin point error becomes zero relative to itself. For pre-calibration or local calibration, the OMM device and the pre-calibration instruments measure coordinate values of balls and blocks on the same calibrator in their own machine CS. After performing the above-mentioned procedure for establishing the same uniform CS, the coordinate values measured by different machines are represented in the same uniform CS. Then, the deviation between their values can be quantitatively compared. Consequently, the procedure for establishing a uniform CS is also a procedure that transforms the coordinate values from the measuring machine CS into the uniform CS. As it can be seen in Fig. 3 (11): (11) where 1_ are the 1 , , , have the same center coordinate of Ball1, there will be only one zero-error point in the measuring volume, which will produce a stable traceability chain to SI units. To this end, a center fixture was designed to keep the center coordinate of Ball1 the same when the right side of the calibrator moves at different positions, the overall layout is similar to the machine checking gauge system of Renishaw [36]. As it can be observed in Fig. 2, Ball1 is in contact with three circumferentiallyarranged and uniformly-distributed ruby balls in front of the fixture. One end of a high-strength cable is consolidated with Ball1, while the other end is tightened by an extension spring. The tension of the extension spring can be adjusted by the tension adjusting nut, to keep Ball1 in contact with the ruby balls all the time. When the right side of the calibrator is placed at a different position, the position of the left-side center fixture is fixed all the time, and the center coordinates of Ball1 can remain unchanged, since the center of Ball1 is restrained by the three ruby balls. The fixture limits the three translational DOFs of the calibrator, but provides a stable traceability chain. 2) Measuring order: It is set according to the distance between Ball1 and other blocks or balls from near to far; thus, the measuring order is Ball1, Block1, Block2, Ball2, to Ball9 in turn.

Local calibration procedures
3) Ball measurement: To measure multiple discrete points on a spherical surface, the points should cover as much of the sphere surface as possible. Then, the center coordinates of the balls can be fitted by the least square method according to ISO 10360-6:2001 [37].

4)
Block measurement: To measure multiple points scattered on the upper surface of the block, these points should cover as much area as possible. Then, the normal vectors of the blocks can be fitted by the least square method [37]. Compared to conventional calibrators, such as 1-D ball arrays [14], 2-D hole plates [17], or 3-D ball plates [18], the proposed calibrator has three measurable dimensions, a relatively simple structure, and a small volume at the same time.
There is no need to be fixed strictly along with some axes or planes by several times, the proposed calibration procedures are executed directly in the 3-D space, and the efficiency and accuracy can be flexibly controlled by the density of the calibration positions. It does not rely on expensive optical instruments such as laser tracker in use, most parts of the calibrator could be mature industrial products; therefore, its cost is relatively low and has the potential for large-scale application.

SCAD-based identification and compensation
As mentioned in Section 2, there are 45 2 possible sub-models of the EVEMU, the sub-model with optimal statistical significance is the optimal estimation of the actual model, which, from the statistical perspective, is treated as the actual sub-model of the OMM device in this paper. thus, the identification is required to select the optimal sub-model. The where E is an N-dimension vector, which is a flattened sequence of ( , , ) xi yi zi E E E . Thus, N is equal to 3n, where n represents the total number of balls to be calibrated. If a local calibration procedure for eight positions is adopted, then, n=72 and N=216. β is a Q-dimension vector of ks, Q is 45, X is a NQ  matrix, whose terms are obtained with respect to β. In actual measurements, apart from systematic component errors, there is also noise. Eq. (13) can be further expressed as Eq. (14):  E X β (15) where β is the estimation of the coefficients and the systematic components of ks as well, the parameter ε is an Ndimension residual vector which represents bias between actual and estimated values, and represents the random deviation identified by algorithms. The systematic components of volumetric error E can be estimated by Ê , as defined by Eq. (15).
OLS regression is commonly used to identify the parameters β for linear equations [39]. OLS can obtain an unbiased estimation of β by minimizing the sum of squares of the difference between the observed and estimated values of the sample. Its objective function is given as Eq. (16): Nevertheless, OLS may overfit data when it handles a model with high dimension and limited X data, since it cannot perform well in multicollinearity problems occurring in X. Even after the dimension reduction performed in Section 2, the number of dimensions is as high as 45. The multicollinearity problems are caused by the interaction among geometric errors of structures with weak stiffness. Increasing the number of measurements can overcome multicollinearity to some extent; however, additional measurements require additional local calibration time.
In the past few decades, several methods have been proposed to solve multicollinearity problems. Among them, several regularization algorithms, also called penalized least-squares regression algorithms, have been well developed [40]. Such algorithms retain the items that are most relevant to the coefficients and reduce or even eliminate the effects of miscellaneous coefficients. Breiman [41] proposed Ridge regressions with L2 penalty; Tibshirani [42] proposed the least absolute shrinkage and selection operator (LASSO) with L1 penalty, and Fan [43] proposed the SCAD method. Among them, LASSO and SCAD belong to regularized sparse algorithms, which eliminate some redundant coefficients and keep the most relevant ones. Ridge and LASSO models are biased regression methods, while SCAD is approximately unbiased [43]. SCAD also has demonstrated the so-called oracle property [43], which means that it performs well in identifying parameters and selecting the sub-model as if it knows the actual sub-model ahead in an asymptotic sense [44]. Thus, SCAD is a theoretically ideal method for model identification of the weak stiffness OMM device. OLS is also an unbiased method and has no sparsity ability, therefore, its model structure is fixed. However, SCAD can select the sub-model among 45 2 candidates, and has more possibilities. The objective function of SCAD is given as Eq. (17) (17) where ,  are tuning parameters which affect the identification performance; , ( ) p    is the nonconvex penalty function, and p  is a coefficient of β . To find the optimal  and  , a hybrid approach combining BIC, CV, and convexity diagnostics are introduced [44], which is implemented as a free available R package called ncvreg [44]. When there is a certain value in  , the ncvreg solver selects the optimal value of  based on the smallest averaged crossvalidation error (CVE) from a sequence of  values, and fits the SCAD at the same time by a coordinate descent algorithm. In order to automatically select  , the 2-D GridSearchCV is applied [45], which combines a 2-D grid search and the ncvreg solver. The steps are as follows: (a) A 2-D candidate parameter grid is set in advance, which includes When searching the optimal i  and i  , the optimal β can obtained at the same time; thus, the optimal equivalent volumetric error sub-model is also acquired. It is worth noting that, since the penalty function , ( ) p    is non-convex, the optimal solution obtained following the above method may still be a local optimal solution rather than the global one.
Nevertheless, as long as the CVE is small enough to meet the industrial demands, the CVE performance is feasible for industrial applications.
' 'î i i  P P E (19) 5 Experimental case study

Experimental setup
In this paper, a custom-made OMM device is introduced, which has been built for medium accuracy (Figs. 1, 7, and 8).
Its main specifications are listed in Table 1 This demonstrated that the calibrator is relatively stable and meets the medium precision application requirement.
Subsequently, a local calibration procedure at 8 positions was performed on the OMM device using the 3-D ball array calibrator. The local calibration at Position3 is as shown in Fig. 6, while the entire procedure is actually the same as that shown in Figs. 4(c) and (d), where the actual values of the calibrator corresponding to Fig. 3 were acquired. In order to verify the local calibration performance (Fig. 7), a Leica AT901-LR laser tracker with a maximum observed deviation of 0.005 mm was utilized. A spherical target reflector was mounted at the end of the OMM device and close to the tactile probe to reflect the laser emitted by the laser tracker; thus, the spatial coordinates of the target reflector could be measured. In any of the above experiments, the environmental temperature was controlled at 20±3℃, and the machine state and other ambient conditions were also maintained as similar as possible.   0.6μm 2.5μm 10 L / 750 

Local calibration results
After pre-calibration, local calibration at 8 positions, and data preprocessing using Eq. It can be observed that, the farther away from Ball1, the larger the magnitude of the volumetric error vectors, while the volumetric error at Ball1 is zero. Figs. 8 (b), (c), and (d) depict the volumetric error components in the X, Y, and Z directions, respectively. As it can be seen in Fig. 8(b), for a certain position, x E generally increases with increasing X, and Ball9 had the largest x E . As it can be seen in Fig. 8(c), y E can be divided into two groups: one group contains Position1-Position4 with values lower than zero; the other group contains Position5-Position8 with values are greater than zero; the two groups are divided by Ball1. In general, the absolute values of y E also increase with increasing distance from Ball1. As it can be observed in Fig. 8(d), z E can also be divided into two groups similar to y E ; these two groups are also divided by Ball1. The reason behind the existence of two groups is that, since their reference origin is the center of Ball1, the continuously changing volumetric error will produce positive and negative values relative to the middle zero point. In fact, even for an OMM device with medium or low accuracy, the volumetric error values are not small enough. If the OMM device was highly rigid, such high error would not occur theoretically. The reason is that the OMM device has a relatively weak stiffness, especially along the X and Z axes. On the other hand, repeated local calibration experiments revealed that the repeatability of the OMM device prototype is less than ±0.008 mm, which demonstrated that error compensation is possible.

SCAD-based identification and volumetric error compensation
As mentioned in Section 4, the 2-D GridSearchCV was employed to select the optimal values of  and  . A large-and a small-scale 2-D grid were investigated, as shown in Fig. 9. For the large-scale 2-D grid (Figs. 9(a) and (b)), the dimension γ was set as a sequence from 2.0001 to 100, which was equally spaced into 50 values. More specifically, the minimum value of  was set to 2.0001, since in SCAD,  should be greater than 2 [44]. λ was set as a sequence of values from 1E-10 to 1E-1 with a length of 100, equally spaced on the log scale according to the default sequence of the ncvreg solver. For the small-scale 2-D gird (Figs. 9(c) and (d)), the γ was set as a sequence of values from 2.0001 to 10, with 50 equally spaced values as well. The parameter λ was set as a sequence of values from 1E-10 to 1E-1 with a length of 100, equally spaced on the log scale. Under the large-scale 2-D grid ( Fig. 9(a)), the minimum CVE was equal to 0.005842233 and its corresponding selected i  and i  were 20.00081633 and 0.0000259, respectively. Under the smallscale 2-D grid (Fig. 9(c)), the minimum CVE was equal to 0.005865556 and its corresponding selected i  and i  were 5.592387755 and 0.000021, respectively. Among both the large-and small-scale 2-D grids, the lowest CVE value was 0.
005842233 in the large-scale 2-D grid; thus, the optimal tuning parameters were i  =20.00081633 and i  =0.0000259.  (Table 3). There were model can be obtained (Eq. (14)), which is a sparse sub-model of the EVEMU Eq. (9).

Verification
In order to verify the accuracy compensation performance, a Leica AT901 Absolute laser tracker was used (Fig. 7). In order to fully cover the same measuring volume with local calibration, a larger cubic volume was measured. Taking the measuring accuracy and efficiency into consideration, the volume was uniformly discretized into a series of points. The measuring scale and discrete step length were as follows: X: -100 mm -800 mm with a discrete step of 100 mm; Y: -100 -400 mm with a discrete step of 100 mm; Z: -40 mm -80 mm with a discrete step of 20 mm. A target reflector was fixed at the end of the OMM device as close to the tactile probe as possible, and the end of the OMM device carried the target reflector to locate the discretized points one by one. Every time the OMM device located a measuring point, it remained still for a few seconds to filter out the possible deviation induced by mechanical vibration, and then, the laser tracker measured the spatial coordinate values of the target reflector. At each measuring point, the OMM device measured a coordinate value, and the laser tracker measure a spatial coordinate value as well. Similar to local calibration, the former is treated as the actual value and the latter as the ideal value. All values are measured in their own coordinate systems. Through coordinate transformation and data post-processing, the data measured by the laser tracker can be converted into the OMM device CS, and then, the values measured by the OMM device are subtracted to obtain the volumetric error. The points and corresponding volumetric error are illustrated in Fig. 11(a). To facilitate the comparison with the 2924 compensation vectors estimated by local calibration (Fig. 10), the volumetric error values measured by the laser tracker were linearly interpolated to the same 2924 points. The error reference origin was relocated to a position coincident with the center of Ball1, as shown in Fig. 9(b). This way, the volumetric error estimated by local calibration can be qualitatively and quantitatively compared with that measured by the tracker. that, similar to the data measured by the laser tracker, the magnitude of compensation vector estimated by OLS and SCAD also increased with increasing X. However, in both cases, a slight upward trend towards the upper right corner of the plot was observed, while the data of the laser tracker did not have such an obvious trend. In addition, the vector magnitude estimated by OLS was quite different from that of the laser tracker, while that estimated by SCAD was more similar. By comparing Fig. 12       The SCAD-based compensation is more effective than OLS for several reasons. First, the OMM device's volumetric error changes regularly in the measuring volume, and the regularity is the premise of using small samples to effectively estimate the population. Second, the EVEMU method offers a massive amount of possible volumetric error sub-models of resmag structures with weak stiffness, the EVEMU is basically derived from the physical structure of the OMM device and combined with the polynomial fitted PDGEs, incorporating both physical meaning and data correlation. Then, the 3-D ball array calibrator and its local calibration procedures effectively expose the volumetric error. Finally, the SCAD performs well in selecting the optimal volumetric error sub-model due to its approximately unbiased and oracle properties. It is worth noting that the experiments in Section 5 were repeated several times, and each time, a similar accuracy improvement performance was obtained, which supports the theory proposed in this study.

Conclusions
This paper primarily proposed an integrated local calibration method including error measurement, modeling, identification, and compensation for a weak stiffness measuring device. The EVEMU method was proposed for error modeling. A new 3-D ball array calibrator and its local calibration procedure were proposed for better error measurement flexibility and efficiency purposes, SCAD was introduced to identify and compensate the volumetric error for weakening the multicollinearity problem as well as dealing with the "interaction" problem. The verification experiment revealed that the SCAD-based local calibration approach is more effective than OLS. The main conclusions are as follows: 1. An equivalent dimensionality reduction algorithm is proposed for the construction of EVEMU, which equivalently reduced the highest dimension of the volumetric error model from 75 to 45, and the amount of corresponding possible sub-models is reduced from about 75 2 to 45 2 . the dimensionality reduced EVEMU is more suitable for limited calibration data. And due to the equivalent method, the error measurements can be performed directly in the 3-D space, and are not limited by identifying explicit geometric errors, but to seek the equivalent solution in the intermediate solution space. which relatively improves the calibration efficiency to a certain extent. The proposed method provides new insight and a theoretical tool for error modeling of multi-linear axes structures.

2.
A new 3-D ball array calibrator and its local calibration procedure are proposed. Compared to conventional calibrators, the new calibrator has three measurable dimensions, a simple structure and small volume at the same time. All the local calibration procedures are executed directly in the 3-D space instead of combining many measurements strictly along with some axes or planes, and dose not rely on expensive optical instruments such as laser trackers in application. The efficiency and accuracy of calibration can be flexibly controlled by the density of the calibration positions.
3. the SCAD algorithm with approximately unbiased and oracle properties is introduced to simultaneously identify equivalent coefficients and select the optimal volumetric error sub-model for weak stiffness OMM device. Experiments demonstrated that SCAD-based identification and compensation can improve the accuracy of the measuring volume by about 3-5 times (within the 25th -75th percentile interval), and by 2-10 times (within the entire percentile interval).
Compared to OLS, SCAD performs more accurately, which can more closely represent the real error state of the machine. OLS has an inferior performance, which can even worsen the error compensation.
In this paper, a novel relatively concise and effective error compensation solution for a weak stiffness OMM device.
The 3-orthogonal linear axes structures, which are widely used in 3-or 5-axis machine tools and CMMs or OMMs, are almost the foundation of the worldwide industry. This paper provides new possible accuracy improvement support for the design and manufacture of such machines. On the other hand, the averaged accuracy improvement of the OMM device is limited (not more than one order of magnitude) in this paper, so, the accuracy and stability of the 3-D ball array calibrator are still worth further improvement in future studies.