Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar

Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar

Accepted Manuscript Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball ...

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Accepted Manuscript Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar Hong-jian Xia, Wei-chao Peng, Xiang-bo Ouyang, Xin-du Chen, Su-juan Wang, Xin Chen PII:

S0890-6955(17)30113-X

DOI:

10.1016/j.ijmachtools.2017.07.006

Reference:

MTM 3276

To appear in:

International Journal of Machine Tools and Manufacture

Received Date: 14 May 2017 Revised Date:

16 July 2017

Accepted Date: 21 July 2017

Please cite this article as: H.-j. Xia, W.-c. Peng, X.-b. Ouyang, X.-d. Chen, S.-j. Wang, X. Chen, Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar, International Journal of Machine Tools and Manufacture (2017), doi: 10.1016/j.ijmachtools.2017.07.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar Hong-jian Xia*, Wei-chao Peng, Xiang-bo Ouyang, Xin-du Chen, Su-juan Wang, Xin Chen Guangdong Provincial Key Laboratory for Micro/Nano Manufacturing Technology and Equipment, School of Electromechanical Engineering, Guangdong University of Technology, Guangzhou, China

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Correspondence to:School of Electromechanical Engineering, Guangdong University of Technology, No. 100 Waihuan Xi Road, Guangzhou Higher Education Mega Center, Panyu District, Guangzhou, P.R China. Post Code: 510006 E-mail address: [email protected] (H.-j. Xia)

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Running Title: Identification of geometric errors of rotary axis via kinematic analysis

ACCEPTED MANUSCRIPT Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar Abstract

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Accuracy identification of geometric errors of rotary axis is important for error compensation of the multi-axis machine tool. However, it is not easy because of the influence of setup error of measurement instrument, and there exists angular errors and displacement errors need to be identified simultaneously. In this paper, a decoupled method based on double ball bar is proposed to identify the geometric errors of rotary axis including both position independent geometric error (PIGE) and position dependent geometric error (PDGE). A formulation for ball bar measurement is derived from kinematic analysis to reveal the relationship between sensitivity direction and setup position of ball bar. The angular errors and displacement errors can be identified separately when choosing the appropriate setup position and direction of ball bar. It can effectively reduce the interaction influence between them and improve the accuracy. To identify the 4 PIGEs, a method by averaging the measured results of ball bar to compute the eccentricity and slant of rotary axis is proposed. And, the PDGEs are leaved to mainly describe the oscillation of geometric errors of rotary axis. It is useful to correct the deviation error resulting from setup error of ball bar. For the identification of 6 PDGEs, by means of adjusting setup position and direction of ball bar, they can be identified one by one along the sensitivity direction of ball bar. Furthermore, in order to reduce the influence of setup error of ball bar, the sensitivity analysis is performed to obtain the sensitivity of measured results of ball bar with respect to setup error. According to the sensitivity characteristics of setup error, a method is presented to correct the PDGEs. Finally, several numerical simulations and experiments are conducted to verify the theoretical model and the proposed identification method. The results from the simulations and experiments demonstrate that the method can identify the geometric errors of rotary axis effectively and accurately.

Keywords: Geometric errors; Rotary axis; Multi-axis machine tool; Kinematic analysis;

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Sensitivity analysis; Double ball bar

1 Introduction

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In the field of aerospace and optical engineering, due to the requirement for free-form surface and strict accuracy of machined part, the multi-axis precision machine tool with rotary axis providing the ability to machine the part with geometric complexity and high accuracy is increasingly popular [1]. For the machine tools, the machining accuracy is affected by many errors such as geometric error, thermal error, force-induced error, and so on. Among them, the geometric error of machine tool is the primary source [2]. Fortunately, the geometric error resulting from imperfect geometry and dimensions of machine components as well as their configuration in the machine's structural loop is systematic or repeatable [3]. Accuracy measurement of geometric error of machine axis is prerequisite to effectively compensate geometric errors and improve the accuracy of machine tool. For the translational axis, the laser interferometer is able to measure geometric errors directly. On many commercial three translational axis machine tool, the numerical compensation for translational positioning error is supplied [4]. But for the multi-axis machine tool, the presence of rotary axis make it difficult to measure the geometric error directly. According to ISO 230-7 [5],

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the geometric errors of rotary axis can be categorized into position independent geometric error (PIGE) and position dependent geometric error (PDGE) composed of ten error components. Recently, several indirectly methods are proposed with a ball bar [6,7], R-test [8,9], tough trigger probe [10,11] and tracking interferometer to identify the geometric errors of rotary axis. Because the ball bar can be set up on machine tool directly and auxiliary parts is not necessary when measuring, it is suitable to measure the geometric errors of rotary axis on multi-axis machine tool with simultaneous control of translational axes. In the past years, some studies on measurement and identification for the geometric errors of rotary axis using double ball bar have been conducted. Lee [12] proposed a method to measure and compensate for PIGEs and PDGEs of two rotary axes on the five-axis machine tool with a titling rotary table using double ball-bar measurement. It avoided the effect of translational axis by remaining stationary when identifying geometric errors of the titling rotary table and defined the PDGEs as zero at the initial conditions to reduce the effect of positioning errors of ball bar. Xiang [13] used a double ball bar by five patterns to measure PDGEs of rotary axes on five-axis machine tools. The ball bar was regarded as a high-precision displacement sensor along the sensitive direction. Chen [14,15] proposed a comprehensive geometric error measurement and identification method for rotary axis of a tilt table using the double ball bar by two steps. The condition number of identification matrix was analyzed to optimize three measurement points of ball bar to improve the identification accuracy. Furthermore, an adjustment method based on electrode clamp was applied to control the setup error of ball bar. Fu [16] presented a six-circle method of ball bar based on differential motion matrix to identify simultaneously all ten error components including the angular errors and displacement errors of each rotary axis. It also analyzed the effect of measurement results caused by the setup error of double ball bar. Jiang [17] investigated the use of double ball bar to identify the PIGEs in rotary axes of a five-axis machine tool. The test was performed with only one axis moving with and without an extension bar to simplify the error analysis. The PIGEs was identified by a least squares fitting method. Xiang [18] proposed a method based on screw theory to measure, model and compensate both PIGEs and PDGEs of five-axis machine tool. The geometric errors of two rotary axes were decoupled into 12 PDGEs and 8 PIGEs to be identified with ball bar measurements. Lee [19] also proposed a robust method using double ball bar to measure PIGEs of rotary axis by least squares method only involving single axis control. The measurement error and setup error of double ball bar were also considered during the measurement. The analysis of standard uncertainty was conducted to quantify the confidence interval of the measurement result. Ding and Huang [20] compared two different geometric error models used to identify and compensate geometric errors of rotary axis and analyze their relationship. According to the previous studies on the identification of geometric errors of rotary axis, it is important to reduce or control the influence of setup error of ball bar. However, the setup error is inevitable and is troublesome to be adjusted during measurement, and they are usually larger than the geometric errors of rotary axis. If the setup error is not restricted or corrected, it is easy to conceal the identified geometric errors, which will result in failure of identification. On the other hand, the geometric errors of rotary axis include angular errors and displacement errors. If the angular and displacement errors are identified simultaneously, the measurement errors resulting from the setup error of ball bar will be amplified because the elements of identification matrix are not dimensionally homogeneous which condition number is large. It will reduce the identification

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accuracy. Furthermore, the identification methods above mentioned are usually applied on the five-axis machine tool with a titling rotary table, the ball bar can be set up on the origin of coordinate system of rotary axis to simplify the identification procedure. But for the other configuration type of multi-axis machine tool, the research work is relative little. In this study, a novel method is presented to identify the PIGEs and PDGEs of rotary axis using the double ball bar. A formulation is derived from the kinematic analysis to reveal the relationship between the sensitivity direction and setup position of ball bar. And then, the angular errors and displacement errors are identified separately by choosing the appropriate setup position and direction of ball bar. Such a method can eliminate the interaction influence between angular and displacement errors and improve the accuracy of identification. For the PIGEs of rotary axis, it is identified by averaging the measured results of ball bar to compute the eccentricity of rotary axis. The PDGEs are leaved to express the oscillation of geometric errors. It is useful to correct the deviation error resulting from setup error of ball bar. Furthermore, in order to reduce the influence of setup error of ball bar, a sensitivity analysis is performed to compute the sensitivity of measured results of ball bar with respect to setup error. Based on the characteristic of sensitivity, a correction method is applied for the identified PDGEs to improve the accuracy. The remainder of this paper is organized as follows. The geometric error model of rotary axis and structure of the multi-axis machine tool in this paper is defined in Section 2. In Section 3, kinematic model of rotary axis is given based on the multi-body theory, and measurement theory of ball bar is also analyzed. And then, the method for identifying the PIGEs and PDGEs of rotary axis is proposed in Section 4. In Section 5, in order to analyze the influence of setup error of ball bar, the sensitivity analysis is conduct. A method to correct the geometric errors of rotary axis is presented. In Section 6, to verify the identification method and sensitivity analysis, several simulation and experiments are performed. Finally, the conclusion of the proposed method are given in Section 7.

2 Geometric errors of rotary axis

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According to the ISO230-7, the geometric errors of rotary axis are categorized into 4 PIGEs and 6 PDGEs, altogether ten error components. The 6 PDGEs are the geometric errors resulting from the rotation of rotary axis. It includes 3 displacement errors and 3 angular errors. When the rotary axis rotates, the value of PDGEs will change. In this paper, the symbol ε * (⋅) and ξ * (⋅)

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represent displacement error and angular error of PDGEs respectively. The subscript represents the direction of error or the rotary axis of error, the character in brackets is the name of the motion axis. The 4 PIGEs are the location and posture errors of rotary axis average line which are caused by the inaccuracy of assembling and manufacturing. The symbol δ * * and S * * depict displacement error and angular error of PIGEs. The first character of subscript is the name of the axis corresponding to the direction of error, the second character of subscript is the name of the motion axis. If the rotary axis is B axis, the PIGEs and PDGEs can be represented as Fig.1.

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Fig.1. Geometric errors of rotary axis. (a) PIGEs. (b) PDGEs

The error components of PIGEs and PDGEs of B rotary axis can be written as Table 1. Table 1 The error components of geometric errors of rotary axis B

Displacement errors X Y Z

Angular errors Y

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Axis B

X

ε y (β )

ε z (β )

ξ x (β )

PIGEs

δ xB

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δzB

SxB

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PDGEs

ε x (β )

ξ y (β )

NA

Z ξ z (β ) S zB

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In this study, a five-axis machine tool configuration with a swivel head and a rotary table is considered, as shown Fig 2. On the workbench side, a linear Z-Axis is fixed on the bed of the machine system, and a rotary B-Axis is equipped on the Z-Axis. Table of workpiece is on the B-Axis. On the other side, a linear X-Axis is fixed on the bed of the machine tool, and a linear Y-Axis is on the X-Axis. A rotary C-Axis is assembled on the Y-Axis, the main spindle can be equipped on the C-Axis or B-Axis. This structural configuration is easy to be transformed into the Milling machine or Lathe machine. Actually, the method proposed in this paper can also be used for other multi-axis machine tool with different configuration to identify the geometric errors of rotary axis.

Fig.2. Structural configuration of five-axis machine tool

3 Kinematic analysis of rotary axis 3.1 kinematic model In the machine tool, the rotary axis is a typical kinematic pair of mechanism. The kinematic relationship of bodies connected by rotary axis is able to be obtained according to the kinematic theory [21]. For example, a pair of bodied connected by the rotary kinematic pair can be represented as Fig 3. Let body B i be inboard of body Bi +1 . The reference frame of body B i is

ACCEPTED MANUSCRIPT represented by (Oi X iYi Z i ) , it is located by position vector ri and is oriented by transformation matrix Ai relative to the reference frame (O0 X 0Y0 Z 0 ) of machine bed. (Oib X ibYi b Z ib ) is fixed on the body B i to define an ideal rotary axis, which is located by a constant vector

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s i ( i +1) and is oriented by a constant transformation matrix Ci′(i +1) . To describe the PIGEs, ( O i′ X i′Y i ′Z i′ ) is defined which is located by a constant vector ηi ( i +1) and is oriented by a

transformation matrix C i′′( i + 1 ) . To represent the motion of rotary axis, ( O i′′X i′′Y i ′Z′ i′′) is defined

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which is oriented by a transformation matrix Ai′(i +1) . For the PDGEs, the position errors is defined

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by the vector εi (i +1) , and the transformation matrix of orientation errors is defined by the Ai′′(i +1) between ( O i′′X i′Y ′ i ′Z′ i′′) and (Oi +1 X i +1Yi +1 Z i +1 ) . Therefore, if the position of inboard body B i is

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given, the origin of body B i +1 connected with B i by a rotary axis can be obtained as

Fig.3. Kinematic notations of rotary axis

ri +1 = ri + si ( i +1) + ηi ( i +1) + ε i ( i +1)

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1

The orientation transformation matrix of B i + 1 can be obtained as A i + 1 = A i C i′( i + 1 ) C i′′( i + 1 ) A i′( i + 1 ) A i′′( i + 1 )

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The origin of body B i + 1 can also be written as follows for computation.

ri +1 = ri + Ai ( s i′( i +1) + C i′( i +1) ( ηi′( i +1) + C i′′( i +1) Ai′( i +1) ε i′( i +1) ))

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If the rotary axis is B axis, the vectors and transformation matrixes can be written as

ηi′(i +1)

0   1 − SzB  cos( β ) δ xB      1 − SxB  , Ai′( i +1) =  0 =  0  , Ci′′(i +1) = SzB   0 δ zB   − sin( β ) SxB 1 

0 1 0

sin( β )  0  , cos( β ) 

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 1  =  ξz ( β )  - ξ y ( β )

- ξz ( β ) 1 ξx ( β )

ξy ( β )   - ξ x ( β ) 1 

εi′(i +1)

εx (β ) = ε y ( β )   ε z ( β ) 

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3.2 Double ball-bar measurement

(5)

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Φ dist = ( rw − rt ) T ( rw − rt ) − R 2

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The double ball bar is a precise instrument for measuring or identifying errors of mechanism system by sensing the change of distance between two points. When it is used to measure the errors of machine tool, one bar ball is usually clamped on the spindle by tool holder, the other is set on the work piece table on rotary axis, shown as Fig 4. As a result of the geometric errors of machine tool, the ball cup on the rotary table should deviate from the ideal position. There also exist errors on the bar ball fixed on the spindle because of the inaccuracy of the linear axes on machine tool. However, the errors of linear axes are easy to be measured and compensated directly using laser interferometers. The error of spindle ball can be controlled and neglected. Therefore, the double ball bar can be used to identify only the rotary axis errors by measuring the displacement deviation of two center points. Based on the kinematic method, the measurement of ball bar can be considered as distance function written as

Fig.4. Measurement of geometric errors of rotary axis using ball bar

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Where rw is the center position of ball cup on the work piece rotary table, rt is the center position of the ball cup fixed on the spindle, R is the nominal length of ball bar, shown as Fig 4. According to the above kinematic model, r w can also be expanded as ′ + A b C bw ′′ A bw ′ ( ε bw ′ + A bw ′′ r p′ ) rw = rb + A b η bw

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 Px  where rp′ =  Py  is the vector from the origin of rotary axis frame to the center position of    Pz  ball cup which is defined on the coordinate system of rotary axis. On the other side, if the errors of linear axes are compensated before measuring the geometric errors of rotary axis, they can be controlled and neglected. The center position of ball fixed on

ACCEPTED MANUSCRIPT spindle side can be gotten from the kinematic chain of work piece, assume that the rotary axis is an ideal one without the errors. It can be written with respect to the frame of machine bed as follows ′ ( r p′ + s ′R ) rt = rb + A b A bw

(7)

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 Rx    where s′R = Ry is the direction vector of the ball bar defined on the coordinate system of    Rz  ideal rotary axis frame without consideration of the geometric errors of rotary axis.

Substituting the Eq.(4) Eq.(6) and Eq.(7) into the Eq.(5) and neglecting the quadratic terms of geometric errors. The distance function can be written as T

ε x ( β )   Rx  ε ( β )  − 2  R   y   y ε z ( β )   R z 

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δ xB   Rx   Rx  0       T − 2 0  Abw (β ) Ry  − 2 Ry  Abw (β ) SzB δ zB   Rz   Rz   0 T

ξ y ( β )   Px   - ξ x ( β )   Py 

0 ξ x (β )

− SzB 0

0

  Pz  

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- ξ z (β )

 0   ξ z (β ) - ξ y ( β ) 

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Φ

dist

 Rx  = − 2  R y   R z 

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0   Px   − S xB  Abw ( β ) Py   Pz  0 

(8)

From the view of measurement of double ball bar, neglecting the quadratic terms of measurement result because it is relatively small, the distance function can also be represented as (9) Φ dist = ( R + ∆ R ) 2 − R 2 = 2 R ∆ R + ∆ R 2 = 2 R ∆ R

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where ∆R is the deviated displacement of ball bar resulting from geometric errors of rotary axis which can be read from the double ball bar. Simplifying the Eq.(8) and Eq.(9), the relationship between measurement data of ball bar and geometric errors of rotary axis can be obtained as

 Rx  δ xB   0    T R∆R ( β ) +  R y  ( Abw ( β )  0  -  Pz  Rz  δ zB  - Py T

 0   Pz - Py 

- Pz 0 Px

Py  ξ x ( β )   R x   - Px  ξ y ( β )  −  R y  0  ξ z ( β )   R z 

- Pz

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 Rx  R   y  R z 

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T

0 Px

T

Py   S xB   T - Px  Abw ( β )  0  ) =  SzB  0 

ε x ( β )  ε ( β )   y  ε z ( β ) 

(10)

On the right of Eq.(10), there are terms only about the PDGEs of rotary axis. They can be defined as follows.  Rx  PDGEs( β ) =  Ry   Rz 

T

 0   Pz - Py 

- Pz 0 Px

Py  ξ x ( β )  Rx   - Px  ξ y ( β ) −  R y  0  ξz ( β )   Rz 

T

ε x ( β )  ε ( β )  y  ε z ( β ) 

(11)

4 Identification of geometric errors of rotary axis For the geometric errors of rotary axis, there are 4 PIGEs and 6 PDGEs according to the ISO230-7. However, it is well known that when an object moves in 3D space, it has six degrees of

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freedom [21]. Accordingly, six errors are enough for describing the position and orientation of rotary table. From the view of kinematic theory, the 4 PIGEs are redundant. But for measurement and identification of geometric errors, the PIGEs are very necessary. Therefore, the geometric errors of rotary axis have multiple combination of PIGEs and PDGEs. For example, the geometric errors of single linear axis is shown as Fig 5. The geometric errors can be only considered as PDGEs, and PIGEs are set zeros. Or the PIGEs are used to define the deviation of average line of geometric errors, and the rest of geometric errors is expressed by PDGEs.

Fig.5. Two kinds of combination method of PIGEs and PDGEs of linear axis

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In this paper, the PIGEs are used to define the eccentric position and slant orientation of rotary axis by averaging the measured results of ball bar. The PDGEs are used to mainly express the oscillation of geometric errors. It is helpful to correct the deviated errors resulting from setup error of ball bar.

4.1 Identification of PIGEs of rotary axis

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There are 2 angular errors and 2 displacement errors need to be identified for the PIGEs of rotary axis. If they are identified simultaneously, the measurement errors resulting from the setup error of ball bar will amplified because the elements of identification matrix are not dimensionally homogeneous and its condition number is large. Actually, the double ball bar can be regarded as a high-precision displacement sensor along the axis direction of bar. If the errors are along the sensitive direction, the accuracy of measurement is high. On contrary, if the errors are perpendicular to the direction, it cannot be measured. Fig.6 shows sensitive and insensitive direction of double ball bar. Such a characteristics of double ball bar is helpful to decouple the measurement of geometric errors of rotary axis.

Fig.6. Sensitive and insensitive direction of double ball bar

Simplifying the Eq.(10), it can be written as

 Rx  δ xB   Rx    T R∆R(β ) + Ry  A ( β) 0  - Ry   Rz  δ zB   Rz  T

T

 0 - Pz Py  Sxb    T   0 - Px  A ( β) 0  = PDGEs( β)  Pz - Py Px Szb  0  

(12)

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- Pz 0 Px

 Px   Px  Py   is the skew matrix of vector  P  . If the setup vector  P  - Px   y  y  0   Pz   Pz 

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Rx    is parallel with the direction vector  R y  of ball bar, the slant direction resulting from the  R z  squareness errors of rotary axis is perpendicular to the sensitive direction of ball bar. The angular errors in PIGEs should be shielded which cannot be measured by the double ball bar. It is shown as Fig.6. Accordingly, the Eq.(12) is simplified as follows  Rx  δ xB    T R ∆ R ( β ) +  R y  A ( β )  0  = PDGEs ( β )  R z  δ zB  T

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(13)

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Substituting Eq.(4) into Eq.(13), the Eq.(13) can be written as

 Rx  cos(β ) − sin(β ) δ xB  R      = - R∆R( β ) + PDGEs( β )  z   sin(β ) cos(β )  δ zB  T

(14)

During the measurement, the bar ball on rotary table is rotated with the rotary axis, and the other ball is driven by the linear axes along another circular path of radius. The distance errors of both balls are read from the ball bar shown as Fig.7. Summing the measurement results of ball bar around β ∈ [0 π ] and substituting them into Eq.(14). The equation can be written as N

− ∑ (sin( β i )

π 

) π N  δ xB  N = ( PDGEs ( β i ) - R ∆ R ( β i )) π  δ zB  ∑ N i =1 (cos( β i ) )  ∑ N  i =1

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N π (cos( β i ) )  R x  ∑ N i =1 R   N π  z   (sin( β ) ) i ∑ N  i =1 T

i =1 N

(15)

Based on the integral theory of trigonometric functions, simplifying the Eq.(15) yields  Rx  0 − 2 δ xB  N π R    δ  = ∑ ( PDGEs( β i ) - R∆R( β i )) N  z  2 0   zB  i =1

(16)

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T

Fig.7. Measurement scheme for PIGEs identification

Therefore, summing the measurement results of ball bar around β ∈ [π

2 π ] , the equation

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 0 2 δ xB  2 N π   δ  = ∑ ( PDGEs( β i ) - R∆R(β i )) 2 0 N    zB  i = N +1

T

(17)

N

π

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Subtracting Eq.(16) and collecting the similar terms, the identification function of displacement errors of PIGEs can be represented as N 2N δ [R z - R x ] xB  = -∑ ( R∆ R ( β i ) π ) + ∑ ( R ∆R ( β i ) π ) + 4N 4N i =1 i = N +1 δ zB 

π

2N

∑ ( PDGEs (β i ) 4 N ) − ∑ ( PDGEs (β i ) 4 N ) i =1

(18)

i = N +1

If summing the measured results of ball bar from β ∈ [π / 2 3π / 2 ] to β ∈ [- π / 2 π / 2] ,

M

π

π

2M

j = M +1

(19)

TE D

j =1

M AN U

∑ ( PDGEs ( β i ) 4 M ) − ∑ ( PDGEs ( β i ) 4 M )

SC

the identification function of displacement errors of PIGEs can be also written as M 2M δ [Rx Rz ] xB  = -∑ ( R∆ R ( β j ) π ) + ∑ ( R∆ R ( β j ) π ) + 4M 4M j =1 j = M +1 δ zB 

EP

Fig.8. Distribution of PIGEs and PDGEs for identification

AC C

In order to simplify the identification procedure, in this paper, the PIGES are defined as the eccentric and slant orientation errors of rotary axis, and PDGEs are used to describe the oscillation of geometric errors which is shown as Fig.8. Assume that the average values of PDGEs are approximately equal to zero, the PIGEs can be obtained by averaging the measured results of ball bar. The displacement errors of PIGEs is written as follows Rz R  x

2N  N π π  - ∑ ( R∆ R ( β i ) ) + ∑ ( R∆R ( β i ) )  - R x  δ xB  4 N 4 N  i =1 i = N +1  = 2M π π  R z  δ zB   M β ( R ∆ R ( ) ) + ( R∆R ( β i ) ) ∑ ∑ i  4M 4 M  i = M +1  i =1

(20)

This hypothesis does not influence the comprehensive accuracy of the identified errors, because the PIGEs are redundant for representing the deviation of rotary axis.  Px    On the other hand, if the setup vector  Py  and the direction vector  Pz 

Rx  R   y  selected are not  R z 

ACCEPTED MANUSCRIPT parallel, as the second measurement pattern shown in Table 2, the measured errors of ball bar have  RPx   R x  the contribution of angular errors of rotary axis PIGEs. Define  RP y  =  R y       RPz   R z 

T

 0   Pz  - Py 

- Pz 0 Px

Py   - Px  , 0 

The Eq.(12) can be simplified as  RPx   S xb   Rx  δ xB        T T R ∆ R ( β ) -  RPy  A ( β )  0  = PDGEs ( β ) −  R y  A ( β )  0   RPz   S zb   R z  δ zB  T

(21)

RI PT

T

According to the above procedure, the identification function of angular errors of PIGEs can be written as

SC

(22)

M AN U

 RPz  RP  x

2N  N δ   π π ) − ∑ ( R∆R( β i ) ) + [Rz - Rx ] xB    ∑ ( R∆R ( β i ) - RPx   S xB   i =1 4 N i = N +1 4N δ zB   = M    2M  δ   RPz   S zB  π π ) − ∑ ( R∆R( β i ) ) + [Rx Rz ] xB   ∑ ( R∆R( β j ) 4M 4M j = M +1 δ zB    j =1

On the basis of the characteristics of sensitive direction of ball bar, by choosing the appropriate setup position and direction of ball bar, the angular and displacement errors in PIGEs are identified separately to eliminate the interactive influence between them.

4.2 Identification of PDGEs of rotary axis

EP

TE D

For the PDGEs of rotary axis, there are also 6 geometric errors including 3 angular errors and 3 displacement errors. Some researchers [2,15,16,19] applied the least squares method to identify the 6 PDGEs simultaneously. However, the condition number of identification matrix is usually large because the elements of matrix are not dimensionally homogeneous. The identified accuracy is sensitive to the measurement error of ball bar. In order to decouple the identification of displacement errors and angular errors, appropriate setup position and direction of ball bar are selected. A constructive method is proposed to identify the 6 PDGEs one by one. The measurement patterns and paths are given in Table.2.

AC C

Table 2 Measurement patterns for identifying geometric errors of rotary axis B

1st Measurement pattern

2nd Measurement pattern

3rd Measurement pattern

4th Measurement pattern



5th Measurement pattern

RI PT

ACCEPTED MANUSCRIPT

6th Measurement pattern

0 0     Choose s′R = R as the direction of ball bar and define rp′ = Py 3 as the setup position      0   0 

T

εx ( β ) ε ( β )  y   ε z ( β ) 

M AN U

0 PDGEs ( β ) = −  R   0 

SC

of ball center on rotary table which is shown as the 3rd measurement pattern in Table 2. Substitute them into Eq.(11), it can be simplified as

(23)

The direction of ball bar is coincided with the sensitive direction of displacement error ε y ( β ) , and the effect of other errors in PDGEs are shielded. The PDGEs(β ) can be obtained by

TE D

substituting Eq.(11) into Eq.(10). Therefore, the displacement error ε y ( β ) can be identified individually by the 3rd measurement pattern when the rotary axis rotates a circle. 3 ε y ( β ) = − PDGEs ( β )

(24)

R   Choose s′R = 0 as the direction of ball bar. It is the sensitive direction of displacement    0 

EP



R

AC C

error ε x ( β ) , angular errors ξ y (β ) and ξ z (β ) . Substituting it into Eq.(11), the equation is rearranged and simplified as

PDGEs( β) = R(Pyξz ( β) - Pzξy ( β) − εx ( β))

(25)

In order to identify those geometric errors, the 2nd, 4th and 5th measurement patterns are

 Px 2   Px 4   Px 5  P  P    conducted. Define  y 2  ,  y 4  and  Py 5  as three setup positions of ball center on rotary  Pz 2   Pz 4   Pz 5  table and set Pz 2 = Pz 4 , Py 2 = Py 5 and Py 2 ≠ Py 4 , Pz 2 ≠ Pz 5 . The geometric errors can be identified separately as follows

ACCEPTED MANUSCRIPT  ( PDGEs 4 ( β ) - PDGEs 2 ( β )) ξ z ( β ) = R ( Py 4 - Py2 )  5 2 ( PDGEs ( β ) - PDGEs ( β )) ξ y ( β ) = R ( Pz 2 - Pz5 )  2  ε x ( β ) = Py 2ξ z ( β ) − Pz2ξ y ( β ) - PDGEs ( β ) R 

 Rx    Choose s′R = Ry as the direction of ball bar and make it parallel with the setup position of    Rz 

RI PT



(26)

SC

 Px    ball center rP′ = Py . The 1st measurement patterns is selected. Substituting the direction of ball    Pz  bar into Eq.(11) yields

(27)

M AN U

PDGEs( β ) = − Rxε x ( β ) − Ryε y ( β ) − Rz ε z ( β )

Because the geometric errors ε x ( β ) and ε y ( β ) had been identified, the ε z ( β ) can be identified as follows

ε z (β ) = −

(28)

R z1

0   Choose s′p = 0 as the direction of ball bar. It is the sensitive direction of displacement    R 

TE D



( R x1ε x ( β ) + R y1ε y ( β ) + PDGEs 1 ( β ))

error ε x ( β ) and angular error ξ x ( β ) and ξ z ( β ) . Substituting the direction of ball bar into

EP

Eq.(11), the equation is simplified as

PDGEs( β ) = R( Pxξy ( β ) - Pyξ x ( β )) − Rε z ( β )

(29)

AC C

Because the angular error ξ z ( β ) and angular error ε z ( β ) have been identified. It is enough to identify the angular error ξ x ( β ) by the 6th measurement pattern as shown in Table 2.

Py 6    Define rp′ = Py 6 as the setup position of ball center on rotary table. The geometric error can be    Pz6  identified as follows ξx (β ) =

( RPx 6ξ y ( β ) − Rε z ( β ) − PDGEs 6 ( β )) ( RPy 6 )

(30)

Since the double ball bar can only measure the errors along the sensitivity direction, by means of choosing the appropriate setup position and direction of ball bar, the PDGEs of rotary axis are identified separately. It effectively reduces the interactive influence of angular error and

ACCEPTED MANUSCRIPT displacement error and improve the robustness for measurement error of double ball bar. For the rotary axis C, its geometric errors can also be identified as above.

5 Sensitivity analysis of setup error and correction

 1  ∂rw = A( β )(  ξ z - ξ y

M AN U

SC

RI PT

As stated early, in order to identify the geometric errors of rotary axis, it is necessary to setup ball cup on the rotary axis table and clamp another one on the spindle. However, the setup error of ball cup are inevitable during measurement, and they are usually larger than the geometric errors of rotary axis. If the setup error is not restricted or corrected, it is easy to conceal the geometric errors identified and result in failure of identification. For the setup error on the spindle, it is easy to be measured and compensated because they can be measured on coordinate system of machine bed [15]. Furthermore, the errors of linear axis are also easy to be measured and compensated directly using laser interferometers. Accordingly, the error of spindle ball can be controlled and restricted by adjusting linear axes. But for the setup error on rotary axis table, it needs to be measured on the coordinate system of rotary axis table. The coordinate system is an imaginary one which is positioned and oriented by PIGEs of rotary axis. Consequently, the setup error on rotary axis table is difficult to be measured and compensated directly. In order to reduce the influence of setup error of ball bar, a sensitivity analysis is performed to compute the sensitivity of measured results of ball bar with respect to setup error on the rotary table. Assuming the setup position on the rotary table as variables, applying the variation theory to Eq.(6) yields - ξz 1

ξ y   ∂Px   0  - ξ x  ) ∂Py  +  Szb

ξx

− Szb 0

1   ∂Pz   0

S xb

0   ∂Px   − S xb  A( β ) ∂Py   ∂Pz  0 

(31)

TE D

Also, the variation of distance constraint based kinematic analysis can be obtained

0  Px  ∂Px  δ xB  ε x   0 - ξz ξy   0 − Szb   T      dist T T ∂Φ = 2(rw − rt ) ∂rw = 2∂Py  ( A  0  + ε y  + ( ξz 0 - ξx  + A Szb 0 − Sxb  A)Py ) ∂Pz  δ zB  ε z  - ξy ξx  0 Sxb 0  0   Pz  T

0  ∂Px  1 0 0  0 - ε z ε y  ∂Px   Rx   0 − Szb       T   (0 1 0 +  ε z 0 - ε x )∂Py  - 2Ry  A Szb 0 − Sxb  A∂Py  0 0 1 - ε y ε x  0 0  ∂Pz   Rz  Sxb 0  ∂Pz  T

EP

T

AC C

 Rx  - 2Ry   Rz 

The variation of distance measurement by ball bar can be computed as ∂Φ dist = 2 R∂ ( ∆R ( β ))

(32)

(33)

Substituting Eq.(33) into Eq.(32), and neglecting the terms with geometric errors because they are relative small, the variation of measurement result of ball bar with respect to the variation of setup position on rotary table can be gotten as ∂ ( ∆ R ( β )) = -

Rx

R

× ∂ Px -

Ry R

× ∂ Py -

Rz

R

× ∂ Pz

(34)

RI PT

ACCEPTED MANUSCRIPT

Fig.9. Relationship between setup error and measurement error

EP

TE D

M AN U

SC

From the above equation, it is evident that the setup error is directly superimposed to the measurement result along the direction of ball bar. It is closely mixed with the measured result, so it is difficult to directly separate it from the geometric errors of rotary axis. However, it is found that the setup error is added along the radial direction and is independent from the rotary angle. Accordingly, it is also called as radial offset error shown as Fig.9. The characteristic is helpful to control and correct the geometric errors identified of rotary axis. For example, the data is measured along both opposite directions can effectively counteract the effect of setup error. For the identification of PIGEs of rotary axis in this paper, the errors identified are average values measured around a circle or on both symmetrical parts as Eq.(20) and Eq.(22). Therefore, the effect of setup error can be removed by neutralizing the measurement results on both symmetrical parts. But for the PDGEs, it is not appropriate, because the PDGEs are also counteract if they are measured on both opposite directions. Therefore, the setup errors of ball bar cannot be totally eliminated during identification.

AC C

Fig.10. The offset correction of PDGEs

Since the effect of setup error on rotary table for PDGEs identification is a constant offset error, it is always along the radial direction. Some researches [16,20] set the initial values of PDGEs to zeros to reduce the offset error. These methods can improve the identification accuracy, but they are sensitive to the identification method of PIGEs. In this paper, the PIGEs of rotary axis are identified by averaging the measured result of ball bar and the PDGEs are used to mainly depict the oscillation of geometric errors. Therefore, a correction method of PDGEs is presented based on the hypothesis that the average values of PDGEs are approximately equal to zero. Because the oscillation amplitude of geometric errors of rotary axis on multi-axis machine tool is smaller than PIGEs, and they usually distributed on the both side of ideal trajectory. Accordingly, this correction method is more stable. It can be considered as a constraint condition to effectively reduce the offset error resulting from the setup error of ball bar cup on rotary axis table. The corrected PDGEs can be written as

ACCEPTED MANUSCRIPT 2N  c ε * ( β ) = ε * ( β ) − ∑ ε * ( β i ) / 2 N (35) i =1  2N c ξ * ( β ) = ξ * ( β ) − ∑ ξ * ( β i ) / 2 N  i =1 Where ε * ( β ) and ξ * ( β ) represent the identified displacement error and angular error of B rotary axis respectively. ε *c ( β ) and ξ*c ( β) are the corrected PDGEs. The correction method can

be shown as Fig.9.

RI PT

6Simulation and experiment 6.1 Identification simulation

M AN U

SC

In order to validate the foregoing analysis and to test the accuracy of identification for geometric errors of rotary axis. A simulation example is conducted and demonstrates how the approach presented is implemented. The rotary table is shown as Fig.2 which is put on the five-axis machine tool including three linear axes and two rotary axes. The B rotary axis is selected to test. The displacement errors of PIGEs are given as 0.1mm and 0.2mm, and the angular errors of PIGEs are given as 3mrad and 4mrad. The setup positions and directions of ball bar of measurement patterns are given as Table 3 respectively. The PDGEs of rotary axis are given as Fig.11. To simulate the actual situation, some geometric errors come from the measurement results of several other instruments. Table 3 The setup position and direction of ball bar of measurement patterns (mm)

Patterns Position

1st

2nd

(60,50,50)

4th

Displacement Errors (um)

5th

(60,50,50)

(0,50,0)

(60,50,60)

(50,40,50)

(100,0,0)

(0,100,0)

(100,0,0)

(100,0,0)

TE D

Direction (64.7,53.9,53.9)

3rd

6th

(60,50,50)

7th (20,30,20)

(0,0,100) (-57.7,57.7,-57.7)

XTError YTError ZTError

20 10 0

-10 -20

-160

-120

-80

-40

0

40

80

120

160

XRError YRError ZRError

0.5

0.0

AC C

Angular Errors (mrad)

EP

Rotational Angle (deg)

-0.5 -160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Fig.11. The given displacement errors and angular errors of rotary axis

st

Firstly, the 1 and 2nd measurement patterns are applied to identify the PIGEs of rotary axis. The displacement errors of PIGEs are gotten as 0.0974mm and 0.2078mm, and the angular errors of PIGEs are identified as 3mrad and 3.7mrad. The identified PIGEs are slightly different from the given PIGEs because the PIGEs are identified based on the hypothesis that the PIGEs mainly describe the off-center and slant of rotary axis. Since the PIGEs is redundant, it does not influence the comprehensive accuracy. And then, the rest four measurement patterns are also applied to identify the PDGEs of rotary axis. The identification result of PDGEs is shown as Fig.12. It is found that the identified PDGEs are also different from the given PDGEs. The reason is that the

ACCEPTED MANUSCRIPT

Displacement Errors (um)

PDGEs are identified based on the above identified PIGEs. XTError YTError ZTError

15 10 5 0 -5 -10 -15 -20

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Angular Errors (mrad)

0.3 0.0 -0.3 -0.6 -160

-120

-80

-40

0

40

80

120

160

SC

Rotational Angle (deg)

RI PT

XRError YRError ZRError

0.6

Fig.12. The identified displacement errors and angular errors of rotary axis Real Distance Errors Predicted Distance Errors

200

0

-200

-400

-160

-120

M AN U

Distance Errors (um)

400

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Residual Errors

0.4

0.2

TE D

Residual Errors (um)

0.6

0.0

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Fig.13. Comparison between real distance errors and predicted ones and residual errors

AC C

EP

To verify the comprehensive accuracy of identification method for geometric errors of rotary axis, the 7th measurement pattern in Table 3 is selected to compare the measurement results using the given geometric errors and identified ones. The distance deviation between real point and predicted point is shown as Fig.13. The maximum error is 0.50um. It shows that the proposed method for identifying the PIGEs and PDGEs of rotary axis is effective and accurate. Furthermore, to validate the sensitivity analysis, the difference method is implemented to compute the sensitivity of measurement result with respect to the setup error of ball bar on rotary table. The perturbation value of setup error for difference computation is define as 0.1mm. The relative error between the difference method and analytical sensitivity method using infinite norm of matrix is 0.83%. The comparison result shows that the analytic sensitivity is effective. Finally, the identification of PIGEs and PDGEs with setup error is conducted to test the correction method. The setup error of ball cup on rotary table is set as (0.1mm, 0.2mm, 0.3mm). The identification results without correction are shown as Fig.14. It is found that there exists the offset errors of PDGEs resulting from the setup error. The distance deviation between real point and predicted point using the identified geometric error is given as Fig.15. The comprehensive error is large. The maximum value is 118.6um. The identified geometric errors almost are concealed by the setup error of ball bar.

Displacement Errors (um)

ACCEPTED MANUSCRIPT XTError YTError ZTError

300 270 240 210 180 150 120 90 -160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Angular Errors (mrad)

0.3 0.0 -0.3 -0.6 -160

-120

-80

-40

0

40

80

120

Rotational Angle (deg)

RI PT

XRError YRError ZRError

0.6

160

SC

Fig.14. The identified displacement errors and angular errors with setup error Real Distance Errors Predicted Distance Errors

200 0

M AN U

Distance Errors (um)

400

-200 -400

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Residual Errors

Residual Errors (um)

-110 -112 -114 -116 -118

-160

-120

-80

-40

0

40

80

120

160

TE D

-120

Rotational Angle (deg)

Fig.15. Comparison between real distance errors and predicted ones and residual errors with setup error

XTError YTError ZTError

20

Displacement Errors (um)

AC C

EP

If the correction step is added into the PDGEs identification, the identification result is gotten as Fig.16. The offset part of PDGEs is evidently reduced and controlled. And then, the corrected PDGEs are used to compare the distance deviation between real point and predicted point. The result is shown as Fig.17. The maximum distance deviation is 2um. It is clear that the correction method can effectively improve the comprehensive accuracy.

10 0 -10 -20 -160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

XRError YRError ZRError

Angular Errors (mrad)

0.6 0.3 0.0 -0.3 -0.6 -160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Fig.16. The corrected displacement errors and angular error of rotary axis

ACCEPTED MANUSCRIPT Real Distance Errors Predicted Distance Errors

400

Distance Errors (um)

300 200 100 0 -100 -200 -300 -400

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Residual Errors

1 0 -1 -2 -160

-120

-80

-40

0

40

Rotational Angle (deg)

80

120

RI PT

Residual Errors (um)

2

160

SC

Fig.17. Comparison between real distance errors and predicted ones and residual errors with correction

6.2 Identification experiment

EP

TE D

M AN U

In order to further test the identification method, an experiment is also conducted on a Moore Nanotech 350FG multi-axis machine tool with a Renishaw QC10 type double ball bar. Before the identification of geometric errors of rotary axis, the software compensated method is implemented to eliminate and control the influence of linear axis. And, a Mahr inductive lever-type probe is used to adjust the center of bar ball as close to the axis line of rotary table as possible by rotating the rotary table to reduce the initial setup error of ball bar shown as Fig 18. Furthermore, a two-freedom movement platform with micrometer resolution is used to accurately adjust the position of bar ball on rotary table when identification.

Fig.18. position adjustment of bar ball by lever-type probe

AC C

During measurement, the bar ball on rotary table is rotated with the rotary axis, another ball is driven by the linear axes along another circular path of radius. Therefore, the distance between both balls of ball bar are ideally kept constant. But, in fact, the distance should be changed when measurement, because there exist the geometric errors of machine tool. Since the geometric errors of linear axes have been controlled or eliminated by compensation, the data read from double ball bar can be considered as the resultant of geometric errors of rotary axis. To identify the PIGEs and PDGEs of rotary axis, the six measurement patterns are applied. Because the movement range of machine tool is 350mm × 150mm × 300mm, the 100mm extension bar is adopted. The setup positions and directions of ball bar of measurement patterns are defined as Table 4 respectively. The measurement procedures for identifying the rotary axis B are shown as Fig. 19. During the measurement, the distance errors of both balls are read from the ball bar at every 10 degrees when the rotary axis is rotating. There are 36 measurement results gotten in every measurement pattern. The measurement results of the 4th and 6th patterns are

ACCEPTED MANUSCRIPT shown as Fig 20. Table 4 The setup position and direction of ball bar for experiment (mm)

Patterns

1st

2nd

Position

(5,80.12,0)

(5,80.12,0)

(5,89.66,0)

(5,80.12,5)

(0,80.12,0)

(0,80.12,5)

(5,89.66,5)

(6.23,99.8,0)

(100,0,0)

(0,100,0)

(100,0,0)

(100,0,0)

(0,0,100)

(0,0,100)

4th

5th

6th

7th

SC

RI PT

Direction

3rd

Fig.19. Measurement procedures of the 1st measurement pattern and 2nd measurement pattern Pattern 4 Pattern 6

0.02

0.00

-0.02

-0.04

-160

-120

M AN U

Distance errors (mm)

0.04

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Fig.20. Measurement results of the 4th and 6th measurement patterns XTError YTError ZTError

6

TE D

Displacement Error (um)

8

4 2 0

-2 -4

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

XRError YRError ZRError

0.5

0.0

AC C

Angular Errors (mrad)

EP

1.0

-0.5

-160

-120

-80

-40

0

40

80

120

160

Rotational Angle (deg)

Distance Errors (um)

Fig.21. The identified displacement errors and angular error of rotary axis PDGEs Measured Distance Errors Predicted Distance Errors

12 6 0 -6

-12 -18 -160

-120

-80

-40

0

40

Rotational Angle (deg)

80

120

160

ACCEPTED MANUSCRIPT Residual Errors (um)

2

Residual Errors

1 0 -1 -2 -160

-120

-80

-40

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Fig.22. Comparison between measured distance errors and predicted ones and residual errors

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The displacement errors of PIGEs are identified as 7.3 um and 11.8 um, and the angular errors of PIGEs are gotten as 2.38 mrad and 0.23 mrad. The PDGEs of rotary axis are identified with correction and shown as Fig.21. The 7th measurement pattern in Table 4 is selected to compare the measurement result and identified one. The distance errors of the 7th measurement pattern are given as Fig.22. The absolute value between the maximum and minimum distance errors is 26.4um. The predicted distance errors are also obtained based on the identified PIGEs and PDGEs. The maximum deviation error between measured distance and predicted one is 3.3 um shown as Fig.22. Therefore, the proposed method can effectively and accurately identify the geometric errors of rotary axis. It is possible for further improving the machining accuracy of multi-axis machine tool with rotary axis.

7 Conclusion

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In this paper, an identification method for position independent geometric error (PIGE) and position dependent geometric error (PDGE) of rotary axis is presented using the double ball bar. Because the ball bar can be considered as a high-precision displacement sensor along the sensitive direction, a formulation is derived from the kinematic analysis to reveal the relationship between the sensitivity direction and setup position of ball bar. By choosing the appropriate setup position and direction of ball bar, the angular and displacement errors are able to be identified separately. The condition number of identification matrix is evidently smaller than the one of simultaneous identification method, because the elements of matrix in simultaneous identification method are not dimensionally homogenous. It is more robust for the measurement errors of double ball bar resulting from the instrument error, setup error and so on. To identify the PIGEs, a method by averaging the measured results of ball bar to compute the eccentricity and slant of rotary axis is proposed. A hypothesis that PDGEs mainly describe the oscillation of geometric errors and have no effect on the off-center and slant of rotary axis is conducted. Because PIGEs are redundant for representing the deviation of rotary axis, it does not influence the comprehensive accuracy of identified errors. But, it is very useful to correct the PDGEs as a constraint condition. For the identification of PDGEs, the appropriate setup position and direction of ball bar are selected, they are identified one by one. It effectively eliminates the interaction influence between angular and displacement errors and improve the accuracy of identification. Furthermore, in order to decrease the influence of setup error of ball bar on rotary table, the sensitivity analysis is performed to obtain the sensitivity of measured results of ball bar with respect to setup error. According to the radial offset characterization of setup error of rotary table ball, the PIGEs identification in the proposed method is insensitive for them, and the identified PDGEs are corrected by the constraint condition to reduce the offset errors. Finally, several numerical simulations and experiments are conducted to verify the theoretical model and the proposed identification method. The results from the simulations and experiments demonstrate that the method can effectively and accurately identify the geometric errors of rotary axis.

ACCEPTED MANUSCRIPT However, for the identification of the PIGEs and PDGEs of rotary axis, several measurement patterns are necessary and it is troublesome to accurately adjust the setup error of double ball bar for every patterns. In the future study, in order to simplify the identification procedure, efforts should be devoted to the study of a method that needs only one or two setup positions of double ball bar.

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Acknowledgement This project is supported by National Nature Science Foundation of China (NO. 51405177, 51375185), Guangdong Innovative Research Team Program (NO.201001G0104781202) and the Science & Technology Planning of Guangdong Province (NO. 2015B010101013)

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8 Reference [1] L.B. Kong, C.F. Cheung, S. To, A kinematic and experiment analysis of form error compensation in ultra-precision, Int. J. Mach. Tools Manuf. 48 (2008) 1408-1419 [2] H. Schwenke, W. Knapp, H. Haitjema, Geometric error measurement and compensation of machines -- An update, Ann. CIRP. 57 (2008) 660-675 [3] R. Ramesh, M.A. Mannan, A.N. Poo, Error compensation in machine tools – a review Part I: Geometric, cutting-force induced and fixture-dependent errors, Int. J. Mach. Tools Manuf. 40 (2000) 1235-1256 [4] S. Ibaraki, W.G. Knapp, Indirect measurement of volumetric accuracy for three-axis and five-axis machine tools: A Review, Int. J. Autom. Technol. 6(2) (2012) 110-123 [5] ISO 230-7-2006, Test code for Machine Tools-Part 7 : Geometric Accuracy of axes of rotation, ISO, 2006 [6] K. Lee, S-H. Yang, Robust measurement method and uncertainty analysis for position-independent geometric errors of a rotary axis using a double ball-bar, Int. J. Precis. Eng. Manuf. 14 (2) (2013) 231–239 [7] M. Tsutsumi, A. Saito, Identification and compensation of systematic deviation particular to 5-axis machining centers, Int. J. Mach. Tools Manuf. 43(2003) 771-780 [8] C. Hong, S. Ibaraki, Non-contact R-test with laser displacement sensors for error calibration of five-axis machine tools, Precis. Eng. 37 (2013) 159-171 [9] S. Ibaraki, C. Oyama, H. Otsubo, Construction of an error map of rotary axes on a five-axis machining center by static R-test, Int. J. Mach. Tools Manuf. 51(2011) 190-200 [10] S. Ibaraki, T. Iritani, T. Matsushita, Calibration of location errors of rotary axes on five-axis machine tools by on the machine measurement using a touch-trigger probe, Int. J. Mach. Tools Manuf. 58(2012) 44-53 [11] Q.Z. Bi, N.D. Huang, C. Sun, Y.H. Wang, Identification and compensation of geometric error of rotary axes on five-axis machine by on-machine measurement, nt. J. Mach. Tools Manuf. 91(2015) 109-114 [12] K. Lee, S.H. Yang, Compensation of position-independent and position-dependent geometric errors in the rotary axes of five-axis machine tools with a titling rotary table, Int. J. Adv. Manuf. Technol. 85(2016) 1677-1685 [13] S.T. Xiang, J.G. Yang, Using a double ball bar to measure 10 position-dependent geometric errors for rotary axes on five-axis machine tools, Int. J. Adv. Manuf. Technol. 75(2014)559-572 [14] J.X. Chen, S.W. Lin, B.W. He, Geometric error measurement and identification for rotary

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table of multi-axis machine tool using double ballbar, Int. J. Mach. Tools Manuf. 77(2014) 47-55 [15] J.X. Chen, S.W. Lin, A ballbar test for measurement and identification the comprehensive error of tilt table, Int. J. Mach. Tools Manuf. 103(2016) 1-12 [16] G.Q. Fu, J.Z. Fu, Y.T. Xu, Accuracy enhancement of five-axis machine tool based on differential motion matrix: Geometric error modeling, identification and compensation, Int. J. Mach. Tools Manuf. 89(2015) 170-181 [17] X.G. Jiang, J. Cripps, A method of testing position independent geometric errors in rotary axes of a five-axis machine tool using a double ball bar, Int. J. Mach. Tools Manuf. 89(2015) 151-158 [18] S.T. Xiang, Y. Altintas, Modeling and compensation of volumetric errors of five-axis machine tools, Int. J. Mach. Tools Manuf. 101(2016) 65-78 [19] K. Lee, S.H. Yang, Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball bar, Int. J. Mach. Tools Manuf. 70(2013) 45-52 [20] S. Ding, X.D. Huang, Identification of different geometric error models and definitions for the rotary axis of five-axis machine tools, Int. J. Mach. Tools Manuf. 100(2016) 1-6 [21] E.J. Haug, Computer aided kinematics and dynamics of mechanical systems: Basic method. Allyn and Bacon, Boston, 1989

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Highlights  The relationship between sensitivity direction and setup position of ball bar for identification is revealed by a formulation based on

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kinematic analysis;  A decoupled identification method for displacement and angular geometric errors of rotary axis is proposed;

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 A corrected method for identifying geometric errors of rotary axis is

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presented to reduce the influence of setup error of ball bar.