# An Analysis Methodology for Stochastic Characteristic of Volumetric Error in Multiaxis CNC Machine Tool

Authors: Qiang Cheng; Can Wu; Peihua Gu; Wenfen Chang; Dongsheng Xuan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Traditional approaches about error modeling and analysis of machine tool few consider the probability characteristics of the geometric error and volumetric error systematically. However, the individual geometric error measured at different points is variational and stochastic, and therefore the resultant volumetric error is aslo stochastic and uncertain. In order to address the stochastic characteristic of the volumetric error for multiaxis machine tool, a new probability analysis mathematical model of volumetric error is proposed in this paper. According to multibody system theory, a mean value analysis model for volumetric error is established with consideration of geometric errors. The probability characteristics of geometric errors are obtained by statistical analysis to the measured sample data. Based on probability statistics and stochastic process theory, the variance analysis model of volumetric error is established in matrix, which can avoid the complex mathematics operations during the direct differential. A four-axis horizontal machining center is selected as an illustration example. The analysis results can reveal the stochastic characteristic of volumetric error and are also helpful to make full use of the best workspace to reduce the random uncertainty of the volumetric error and improve the machining accuracy.


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## Body

## 1. Introduction

Along with rapid progress and development of science, technology and social economy, the machining accuracy of CNC machine tools is increasingly demanding. How to improve the accuracy of CNC machine tools has been gotten great attention [1]. To enhance the machining accuracy of CNC machine tools, error modeling is crucial to maximize the performance of machine tools [2], and robust and accurate volumetric error modeling is also the first step to correct and compensate these errors [3–5]. A volumetric error model, which is the relative error between the cutting tool and the work piece, is a system analysis implement, used when accuracy is an important measure of performance to predict and control the total error of a system or to achieve compensation.However, machining accuracy of the multiaxis synchronized machine is mainly affected by the geometric errors of the guide system, structure stiffness, thermal behavior and the dynamic response, and so forth. The geometric errors are those errors that exist in a machine on account of its basic design and those resulting from the inaccuracies built in during assembly and from the components used in the machine [6]. Permanent and systematic geometric errors are the most common type found in CNC machine tools. In precision machining, geometric errors of machine tools have considerable effect on geometrical and dimensional accuracies of machined features [7] and make up the major part of the inaccuracy of a machine tool [5, 6, 8]. As for an important component of volumetric error, geometric error characterization and mapping is one of the most important steps to find a universal kinematic model [9].Geometric error of the machine tool primarily comes from manufacturing or assembly defects misalignment of the machine’s axis and the position and straightness error of each axis. Because the errors of a drive or axis or the outcome of an assembly process are random at some level [10]; thus, the geometric errors vary at different locations instead of being constants and can be taken as a function of displacement [11]. The individual geometric errors can be interpreted as a deterministic value and a probabilistic distribution of random variation [12]. And if the stochastic variation of the error is within the desired tolerance volume, the deterministic error can be eliminated if the error can be predicted and compensated [2]. However, if the random deviation of errors is too large, then only a certain number of the parts produced on this machine will meet the specified tolerance requirement and the remaining parts will exceed the tolerance. Therefore, how to express and analyze the random characteristic of machine tools is of paramount importance to improve the machining accuracy.


### 1.1. Volumetric Error Modeling Methods Considering Geometric Error

Error modeling technique provides a systematic and suitable way to establish the error model. In recent years, many research work has been done on modeling of multiaxis machine tools to find out the resultant error of individual components in relation to tool and work piece point deviation. And the modeling methods of the geometric errors from different perspectives have experienced several developing phases including geometric modeling methods, error matrix methods (EMMs), quadratic model methods, mechanism modeling methods, rigid body kinematics methods [13, 14], and model methods based on the multibody system kinematics theory [15].Love and Scarr [16] analyzed the volumetric errors by determining the combined effects through trigonometric technique. In 1977, Schultschik [17] introduced the close vector chain technique. In the same year, Hocken et al. [18] developed a matrix translation method and presented a calibration technique. Ferreira, et al. [19, 20] proposed an analytical quadratic model for the prediction of geometric errors. Unlike the previously proposed models, this method allowed for the variation of errors along the machine’s joints, and the model is not true in most cases due to its limitations. In 1991, K. Kim and M. K. Kim [21] established volumetric error prediction model for 3-axis CNC machine tool with rigid kinematic model. Chen et al. [22] addressed the non-rigid body effects associated with the volumetric accuracy of a horizontal spindle machine tool. Kiridena and Ferreira [23] developed a kinematic model to compensate for both the position and orientation errors of a 5-axis machining center using the same convention.In 2003, Lin and Shen [24] presented the matrix summation approach for modeling the geometric errors of 5-axis machine tools. This approach breaks down the kinematic equation into six components including the ideal tool tip position under nominal axis motions and the contribution of the error motions of each axis. A generalized geometric error model associated with 5-axis machine was developed by Jha and Kumar [25] with its experimental verification. They presented a scheme to compensate the geometric errors and analyzed the effect on the accuracy of a cam profile. In 2007, Bohez et al. [26] researched into 5-axis milling machine and presented a new method to identify and compensate the systematic errors in a multiaxis machine tool. The mathematical model is based on a first-order rigid body model of the machine tool.In recent years, the multibody system (MBS) as for the movement with error of complicated machinery system was presented by Houston in the late 1970s [27], and the error modeling using the unique lower array to describe complex systems is more convenient and very suitable for computer to model machine tool [28], which describes the topological structure of multibody system simply and conveniently. Because it is suitable for computer description to model volumetric errors, many investigators have carried through error modeling research of multiaxis machine tool with MBS [29]. Fan et al. [30] proposed the kinematics of MBS by adding movement error and positioning error items, and a universal way of how to make a kinematic model of numerical control (NC) machine tools was presented. To date, it is the best method for geometric error modeling of machine tool [5].


### 1.2. Stochastic Nature Analysis Methods of Volumetric Error

The above research work mostly addressed the relationship between the geometric error and volumetric error without considering the stochastic characteristic of geometric error. However, it is not practical to try to manufacture a part to an exact dimension in a production environment because of the inaccuracies associated with machine tools and the apparent randomness of most manufacturing processes, originating from variation in material properties, dimensions, friction, and so forth [2]. During operation, all machine tools generate a certain amount of errors due to imperfect mechanical structures, errors in control systems, and environmental disturbances. For many production machines, the stochastic portion of the error is defined as repeatability that accounts for a significant portion of the total error [31], and two or three standard deviations of the repeat measurement error have been used as the random error [12].Up to the present, some researchers have become interested in the uncertainty analysis of errors [32–36]. Shen and Duffie [33] presented the uncertainty analysis method for coordinate referencing and used the uncertainty interval concept to describe the essential characteristics of uncertainty sources in coordinate referencing and coordinate transformation relationships. Qian and Kazerounian [34] presented a new perspective on the calibration of industrial robotics systems by describing robot position errors as time varying variation that can be separated into random system variation and assignable variation.Yau [35] proposed a new rotation vector that provides a more general mathematical basis for representing vectorial tolerances. Shin and Wei [32] placed great importance on the random error of machine tools and tried to express the random error of machine tools. Unfortunately, their results were limited because they only examined the boundary of the random error in two-dimensional space and could not extrapolate the random error into three-dimensional space. Practically no effort has been made to incorporate stochastic terms into the three-dimensional workspace of machine tools. Ahn and Cho [2] addressed the stochastic nature of the volumetric error of a machine tool and the importance of modeling stochastic errors and proposed a model helping the prediction of the tolerances of products caused by uncertainties of a machine tool with beta distribution. Andolfatto et al. [36] presented a method to evaluate contributions to the uncertainties for identified link errors of a five-axis machine tool in order to improve the link errors identification method by decreasing or removing its impact. Nevertheless, there is short of a universal method addressing the stochastic characteristic of volumetric error to multiaxis machine tool with comprehensive consideration of the geometric errors’ probabilistic characteristics. Because of complicated mathematical operations caused by many geometric errors, it is also essential to present an analysis method suitable for computer modeling and calculation.The rest parts of the paper are organized as follows. In the next section, the modeling of volumetric error for machine tool is given based on MBS theory. The third section presents the stochastic characteristic analysis modeling process of volumetric errors. Section four demonstrates the proposed method with a 4-axis machine tool. The final section contains the conclusions.


## 1.1. Volumetric Error Modeling Methods Considering Geometric Error

Error modeling technique provides a systematic and suitable way to establish the error model. In recent years, many research work has been done on modeling of multiaxis machine tools to find out the resultant error of individual components in relation to tool and work piece point deviation. And the modeling methods of the geometric errors from different perspectives have experienced several developing phases including geometric modeling methods, error matrix methods (EMMs), quadratic model methods, mechanism modeling methods, rigid body kinematics methods [13, 14], and model methods based on the multibody system kinematics theory [15].Love and Scarr [16] analyzed the volumetric errors by determining the combined effects through trigonometric technique. In 1977, Schultschik [17] introduced the close vector chain technique. In the same year, Hocken et al. [18] developed a matrix translation method and presented a calibration technique. Ferreira, et al. [19, 20] proposed an analytical quadratic model for the prediction of geometric errors. Unlike the previously proposed models, this method allowed for the variation of errors along the machine’s joints, and the model is not true in most cases due to its limitations. In 1991, K. Kim and M. K. Kim [21] established volumetric error prediction model for 3-axis CNC machine tool with rigid kinematic model. Chen et al. [22] addressed the non-rigid body effects associated with the volumetric accuracy of a horizontal spindle machine tool. Kiridena and Ferreira [23] developed a kinematic model to compensate for both the position and orientation errors of a 5-axis machining center using the same convention.In 2003, Lin and Shen [24] presented the matrix summation approach for modeling the geometric errors of 5-axis machine tools. This approach breaks down the kinematic equation into six components including the ideal tool tip position under nominal axis motions and the contribution of the error motions of each axis. A generalized geometric error model associated with 5-axis machine was developed by Jha and Kumar [25] with its experimental verification. They presented a scheme to compensate the geometric errors and analyzed the effect on the accuracy of a cam profile. In 2007, Bohez et al. [26] researched into 5-axis milling machine and presented a new method to identify and compensate the systematic errors in a multiaxis machine tool. The mathematical model is based on a first-order rigid body model of the machine tool.In recent years, the multibody system (MBS) as for the movement with error of complicated machinery system was presented by Houston in the late 1970s [27], and the error modeling using the unique lower array to describe complex systems is more convenient and very suitable for computer to model machine tool [28], which describes the topological structure of multibody system simply and conveniently. Because it is suitable for computer description to model volumetric errors, many investigators have carried through error modeling research of multiaxis machine tool with MBS [29]. Fan et al. [30] proposed the kinematics of MBS by adding movement error and positioning error items, and a universal way of how to make a kinematic model of numerical control (NC) machine tools was presented. To date, it is the best method for geometric error modeling of machine tool [5].


## 1.2. Stochastic Nature Analysis Methods of Volumetric Error

The above research work mostly addressed the relationship between the geometric error and volumetric error without considering the stochastic characteristic of geometric error. However, it is not practical to try to manufacture a part to an exact dimension in a production environment because of the inaccuracies associated with machine tools and the apparent randomness of most manufacturing processes, originating from variation in material properties, dimensions, friction, and so forth [2]. During operation, all machine tools generate a certain amount of errors due to imperfect mechanical structures, errors in control systems, and environmental disturbances. For many production machines, the stochastic portion of the error is defined as repeatability that accounts for a significant portion of the total error [31], and two or three standard deviations of the repeat measurement error have been used as the random error [12].Up to the present, some researchers have become interested in the uncertainty analysis of errors [32–36]. Shen and Duffie [33] presented the uncertainty analysis method for coordinate referencing and used the uncertainty interval concept to describe the essential characteristics of uncertainty sources in coordinate referencing and coordinate transformation relationships. Qian and Kazerounian [34] presented a new perspective on the calibration of industrial robotics systems by describing robot position errors as time varying variation that can be separated into random system variation and assignable variation.Yau [35] proposed a new rotation vector that provides a more general mathematical basis for representing vectorial tolerances. Shin and Wei [32] placed great importance on the random error of machine tools and tried to express the random error of machine tools. Unfortunately, their results were limited because they only examined the boundary of the random error in two-dimensional space and could not extrapolate the random error into three-dimensional space. Practically no effort has been made to incorporate stochastic terms into the three-dimensional workspace of machine tools. Ahn and Cho [2] addressed the stochastic nature of the volumetric error of a machine tool and the importance of modeling stochastic errors and proposed a model helping the prediction of the tolerances of products caused by uncertainties of a machine tool with beta distribution. Andolfatto et al. [36] presented a method to evaluate contributions to the uncertainties for identified link errors of a five-axis machine tool in order to improve the link errors identification method by decreasing or removing its impact. Nevertheless, there is short of a universal method addressing the stochastic characteristic of volumetric error to multiaxis machine tool with comprehensive consideration of the geometric errors’ probabilistic characteristics. Because of complicated mathematical operations caused by many geometric errors, it is also essential to present an analysis method suitable for computer modeling and calculation.The rest parts of the paper are organized as follows. In the next section, the modeling of volumetric error for machine tool is given based on MBS theory. The third section presents the stochastic characteristic analysis modeling process of volumetric errors. Section four demonstrates the proposed method with a 4-axis machine tool. The final section contains the conclusions.


## 2. Volumetric Error Modeling with Consideration of Geometric Error

### 2.1. Error Parameter Definition

It is well known that when an rigid object moves in 3D space, it has six degrees of freedom (DOF); accordingly, its position description has six errors [5]. Similarly, in a machine tool, when a component moves along an axis, there are six position dependent errors [37]. Taking Z-slideway as an example, the six geometric error parameters are shown in Figure 1, including positioning error, horizontal and vertical straightness errors, pitch error, yaw error, and roll error.Figure 1
Six parametric errors associated with a prismatic joint.In addition, as a multiaxis machine tool, a rotating shaft also has six geometric errors. As Figure2 for an example, there are six geometric errors of B rotating shaft, respectively, for three linear errors and three angle errors. The linear errors are two radial errors and an axial motion error, while three angle errors are an angular position error and the two tilt error of the angle errors. These errors are function of the nominal movement only and do not depend on the location of the other joints [9].Figure 2
Six parametric errors associated with a rotary joint.


### 2.2. Volumetric Error Modeling Based on MBS

#### 2.2.1. Structure Diagram and Topological Graph

In this research, a 4-axis precision horizontal machining center, whose 3-dimension digital structure model is shown in Figure3, is selected as an example to describe the proposed model. It is composed of a bed, x-, y-, z- three axis motion components, a spindle, and a workbench because it has three prismatic joints and one rotary joint, so there 24 position dependent systematic geometric errors. And among the four joints, there are 6 position independent systematic geometric errors, including three perpendicularity errors, two parallelism errors and one offset error. Table 1 shows all the geometric errors.Table 1
Geometric errors of precision horizontal machining center.
Component
Error term
Symbol
Component
Error term
Symbol
X-prismatic  joint
Positioning error
Δ
x
x
B-rotary joint
X direction run-out error
Δ
x
B
Y direction of straightness error
Δ
y
x
Z direction run-out error
Δ
z
B
Z direction of straightness error
Δ
z
x
Axial error
Δ
y
B
Rolling error
Δ
α
x
Around theX axis turning error
Δ
α
B
Britain swing error
Δ
β
x
Positioning error
Δ
β
B
Yaw error
Δ
γ
x
Around theZ axis turning error
Δ
γ
B
Y-prismatic joint
X direction of straightness error
Δ
x
y
Orientation error between joints
X, Y-axis perpendicularity error
Δ
γ
x
y
Positioning error
Δ
y
y
X, Z-axis perpendicularity error
Δ
β
x
z
Z direction of straightness error
Δ
z
y
Y, Z-axis perpendicularity error
Δ
α
y
z
Rolling error
Δ
α
y
B, Y-axis parallelism error in XY plane
Δ
γ
x
B
Britain swing error
Δ
β
y
B, Y-axis parallelism error in ZY plane
Δ
α
z
B
Yaw error
Δ
γ
y
B, Y-axis offset error in Z axis direction
Δ
z
B
y
Z-prismatic joint
X direction of straightness error
Δ
x
z
Z-prismatic joint
Rolling error
Δ
α
z
Y direction of straightness error
Δ
y
z
Britain swing error
Δ
β
z
Positioning error
Δ
z
z
Yaw error
Δ
γ
zFigure 3
Structure diagram of precision horizontal machining center. 0 bed; 1X-prismatic joint; 2 Y-prismatic joint; 3 spindle; 4 Z-prismatic joint; 5 B-rotary joint.


#### 2.2.2. Topological Structure and Lower Body Array

On the basis of the theory of MBS, a multiaxis machine tool can be abstracted into a multibody system. As shown in Figure4, the machine base 0 is expressed into B0, 1 into B1 body, along the direction away from the body B1. According to the natural growth sequence, the bodies are sequentially numbered from one branch to another branch. Figure 4 gives the machine topology diagram. Table 2 is the lower body array of the selected precision horizontal machining center, and Table 3 depicts the degrees of freedom between each unit to the constraints, wherein “0” means no freedom of movement and “1” means one freedom of movement.Table 2
Lower body array of precision horizontal machining center.
Topic bodyj
1
2
3
4
5
6
L
0
(
j
)
1
2
3
4
5
6
L
1
(
j
)
0
1
2
0
4
5
L
2
(
j
)
0
0
1
0
0
4
L
3
(
j
)
0
0
0
0
0
0Table 3
Adjacent body freedom of the precision horizontal machining center.
Adjacent body
X
Y
Z
α
β
γ
0-1
1
0
0
0
0
0
1-2
0
1
0
0
0
0
2-3
0
0
0
0
0
1
0-4
0
0
1
0
0
0
4-5
0
0
0
0
1
0
5-6
0
0
0
0
0
0Figure 4
Topological graph of precision horizontal machining center.


#### 2.2.3. Generalized Coordinates Setting and Characteristic Matrix

In order to facilitate the precision of machine tool modeling, coordinate system is required to be setted specially. Consider the following settings.(1) In MBS theory, right-handed Cartesian coordinate systems are built on the inertial components and all moving parts. These coordinates are called generalized coordinates. In the generalized coordinates, coordinate system on the inertial body is called the reference coordinate system, and coordinate system on other moving body is called moving coordinate system. Three orthogonal matrices of each coordinate system are named x-, y-, z-axis, respectively, according to the right-hand rule. (2) Each x-, y-, and z-axis of the different coordinate systems parallel correspondingly. (3) The tool coordinate system is at the end of the spindle center, and Table 4 lists the characteristic matrices of the precision horizontal machining center.Table 4
The characteristic matrices of precision horizontal machining center.
The adjacent body
Body ideal static, motion characteristic matrix
Body static, kinematic error  characteristics matrix
0-1X axis
T
01
p
=
I
4
×
4
Δ
T
01
p
=
I
4
×
4
T
01
s
=
[
1
0
0
x
0
1
0
0
0
0
1
0
0
0
0
1
]
Δ
T
01
s
=
[
1
-
Δ
γ
x
Δ
β
x
Δ
x
x
Δ
γ
x
1
-
Δ
α
x
Δ
y
x
-
Δ
β
x
Δ
α
x
1
Δ
z
x
0
0
0
1
]
1-2Y axis
T
12
p
=
I
4
×
4
Δ
T
12
p
=
[
1
-
Δ
γ
x
y
0
0
Δ
γ
x
y
1
0
0
0
0
1
0
0
0
0
1
]
T
12
s
=
[
1
0
0
0
0
1
0
y
0
0
1
0
0
0
0
1
]
Δ
T
12
s
=
[
1
-
Δ
γ
y
Δ
β
y
Δ
x
y
Δ
γ
y
1
-
Δ
α
y
Δ
y
y
-
Δ
β
y
Δ
α
y
1
Δ
z
y
0
0
0
1
]
2-3 tool
T
23
p
=
[
1
0
0
x
t
d
0
1
0
y
t
d
0
0
1
z
t
d
0
0
0
1
]
Δ
T
23
p
=
I
4
×
4
T
23
s
=
I
4
×
4
Δ
T
23
s
=
I
4
×
4
0-4Z axis
T
04
s
=
I
4
×
4
Δ
T
04
p
=
[
1
0
Δ
β
x
z
0
0
1
-
Δ
α
y
z
0
-
Δ
β
x
z
Δ
α
y
z
1
0
0
0
0
1
]
T
04
s
=
[
1
0
0
0
0
1
0
0
0
0
1
z
0
0
0
1
]
Δ
T
04
s
=
[
1
-
Δ
γ
z
Δ
β
z
Δ
x
z
Δ
γ
z
1
-
Δ
α
z
Δ
y
z
-
Δ
β
z
Δ
α
z
1
Δ
z
z
0
0
0
1
]
4-5B axis
T
45
p
=
I
4
×
4
Δ
T
45
p
=
[
1
-
Δ
γ
z
B
0
0
Δ
γ
z
B
1
-
Δ
α
x
B
0
0
Δ
α
x
B
1
Δ
z
B
y
0
0
0
1
]
T
45
s
=
[
cos
B
-
sin
B
0
0
sin
B
cos
B
0
0
0
0
1
0
0
0
0
1
]
Δ
T
45
s
=
[
1
-
Δ
γ
B
Δ
β
B
Δ
x
B
Δ
γ
B
1
-
Δ
α
B
Δ
y
B
-
Δ
β
B
Δ
α
B
1
Δ
z
B
0
0
0
1
]Supposing that tool forming point in the tool coordinate system coordinate is(1)Pt=[PtxPtyPtz1]T.
And the work-piece forming point in the work-piece coordinate system coordinate is
(2)Pw=[PwxPwyPwz1]T.
When the machine tool moves in ideal form, that means that the machine tool is without error, the tool forming point and work-piece forming point will overlap together, and we can get
(3)[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]Pt=[∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s]Pw.
And the ideal forming function of tool forming point in work-piece coordinate system is
(4)Pt=[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]-1×∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)sPw.
Machining accuracy is finally determined by relative displacement error between the tool forming point of machine and work piece. During the actual machining process, the actual position of cutting tool forming point will inevitably deviate from the ideal location, which thereby results in volumetric error. The comprehensive volumetric error caused by the gap between actual forming point and ideal forming point can be written as
(5)E=(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w))Pw-(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t))×(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t)pTLk(t)Lk-1(t)s)-1×(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s)Pw.
With the characteristic matrices in Table 4, the comprehensive volumetric error model of a precision horizontal machining center can be gotten with (5). E reflects the deviation of the tool forming point between ideal location and actual location in work-piece coordinate system. If each geometric error value in the model is known, the volumetric error of machine tool can be obtained. As for other multiaxis machine tool, the general volumetric error model can be established in the similar way.


## 2.1. Error Parameter Definition

It is well known that when an rigid object moves in 3D space, it has six degrees of freedom (DOF); accordingly, its position description has six errors [5]. Similarly, in a machine tool, when a component moves along an axis, there are six position dependent errors [37]. Taking Z-slideway as an example, the six geometric error parameters are shown in Figure 1, including positioning error, horizontal and vertical straightness errors, pitch error, yaw error, and roll error.Figure 1
Six parametric errors associated with a prismatic joint.In addition, as a multiaxis machine tool, a rotating shaft also has six geometric errors. As Figure2 for an example, there are six geometric errors of B rotating shaft, respectively, for three linear errors and three angle errors. The linear errors are two radial errors and an axial motion error, while three angle errors are an angular position error and the two tilt error of the angle errors. These errors are function of the nominal movement only and do not depend on the location of the other joints [9].Figure 2
Six parametric errors associated with a rotary joint.


## 2.2. Volumetric Error Modeling Based on MBS

### 2.2.1. Structure Diagram and Topological Graph

In this research, a 4-axis precision horizontal machining center, whose 3-dimension digital structure model is shown in Figure3, is selected as an example to describe the proposed model. It is composed of a bed, x-, y-, z- three axis motion components, a spindle, and a workbench because it has three prismatic joints and one rotary joint, so there 24 position dependent systematic geometric errors. And among the four joints, there are 6 position independent systematic geometric errors, including three perpendicularity errors, two parallelism errors and one offset error. Table 1 shows all the geometric errors.Table 1
Geometric errors of precision horizontal machining center.
Component
Error term
Symbol
Component
Error term
Symbol
X-prismatic  joint
Positioning error
Δ
x
x
B-rotary joint
X direction run-out error
Δ
x
B
Y direction of straightness error
Δ
y
x
Z direction run-out error
Δ
z
B
Z direction of straightness error
Δ
z
x
Axial error
Δ
y
B
Rolling error
Δ
α
x
Around theX axis turning error
Δ
α
B
Britain swing error
Δ
β
x
Positioning error
Δ
β
B
Yaw error
Δ
γ
x
Around theZ axis turning error
Δ
γ
B
Y-prismatic joint
X direction of straightness error
Δ
x
y
Orientation error between joints
X, Y-axis perpendicularity error
Δ
γ
x
y
Positioning error
Δ
y
y
X, Z-axis perpendicularity error
Δ
β
x
z
Z direction of straightness error
Δ
z
y
Y, Z-axis perpendicularity error
Δ
α
y
z
Rolling error
Δ
α
y
B, Y-axis parallelism error in XY plane
Δ
γ
x
B
Britain swing error
Δ
β
y
B, Y-axis parallelism error in ZY plane
Δ
α
z
B
Yaw error
Δ
γ
y
B, Y-axis offset error in Z axis direction
Δ
z
B
y
Z-prismatic joint
X direction of straightness error
Δ
x
z
Z-prismatic joint
Rolling error
Δ
α
z
Y direction of straightness error
Δ
y
z
Britain swing error
Δ
β
z
Positioning error
Δ
z
z
Yaw error
Δ
γ
zFigure 3
Structure diagram of precision horizontal machining center. 0 bed; 1X-prismatic joint; 2 Y-prismatic joint; 3 spindle; 4 Z-prismatic joint; 5 B-rotary joint.


### 2.2.2. Topological Structure and Lower Body Array

On the basis of the theory of MBS, a multiaxis machine tool can be abstracted into a multibody system. As shown in Figure4, the machine base 0 is expressed into B0, 1 into B1 body, along the direction away from the body B1. According to the natural growth sequence, the bodies are sequentially numbered from one branch to another branch. Figure 4 gives the machine topology diagram. Table 2 is the lower body array of the selected precision horizontal machining center, and Table 3 depicts the degrees of freedom between each unit to the constraints, wherein “0” means no freedom of movement and “1” means one freedom of movement.Table 2
Lower body array of precision horizontal machining center.
Topic bodyj
1
2
3
4
5
6
L
0
(
j
)
1
2
3
4
5
6
L
1
(
j
)
0
1
2
0
4
5
L
2
(
j
)
0
0
1
0
0
4
L
3
(
j
)
0
0
0
0
0
0Table 3
Adjacent body freedom of the precision horizontal machining center.
Adjacent body
X
Y
Z
α
β
γ
0-1
1
0
0
0
0
0
1-2
0
1
0
0
0
0
2-3
0
0
0
0
0
1
0-4
0
0
1
0
0
0
4-5
0
0
0
0
1
0
5-6
0
0
0
0
0
0Figure 4
Topological graph of precision horizontal machining center.


### 2.2.3. Generalized Coordinates Setting and Characteristic Matrix

In order to facilitate the precision of machine tool modeling, coordinate system is required to be setted specially. Consider the following settings.(1) In MBS theory, right-handed Cartesian coordinate systems are built on the inertial components and all moving parts. These coordinates are called generalized coordinates. In the generalized coordinates, coordinate system on the inertial body is called the reference coordinate system, and coordinate system on other moving body is called moving coordinate system. Three orthogonal matrices of each coordinate system are named x-, y-, z-axis, respectively, according to the right-hand rule. (2) Each x-, y-, and z-axis of the different coordinate systems parallel correspondingly. (3) The tool coordinate system is at the end of the spindle center, and Table 4 lists the characteristic matrices of the precision horizontal machining center.Table 4
The characteristic matrices of precision horizontal machining center.
The adjacent body
Body ideal static, motion characteristic matrix
Body static, kinematic error  characteristics matrix
0-1X axis
T
01
p
=
I
4
×
4
Δ
T
01
p
=
I
4
×
4
T
01
s
=
[
1
0
0
x
0
1
0
0
0
0
1
0
0
0
0
1
]
Δ
T
01
s
=
[
1
-
Δ
γ
x
Δ
β
x
Δ
x
x
Δ
γ
x
1
-
Δ
α
x
Δ
y
x
-
Δ
β
x
Δ
α
x
1
Δ
z
x
0
0
0
1
]
1-2Y axis
T
12
p
=
I
4
×
4
Δ
T
12
p
=
[
1
-
Δ
γ
x
y
0
0
Δ
γ
x
y
1
0
0
0
0
1
0
0
0
0
1
]
T
12
s
=
[
1
0
0
0
0
1
0
y
0
0
1
0
0
0
0
1
]
Δ
T
12
s
=
[
1
-
Δ
γ
y
Δ
β
y
Δ
x
y
Δ
γ
y
1
-
Δ
α
y
Δ
y
y
-
Δ
β
y
Δ
α
y
1
Δ
z
y
0
0
0
1
]
2-3 tool
T
23
p
=
[
1
0
0
x
t
d
0
1
0
y
t
d
0
0
1
z
t
d
0
0
0
1
]
Δ
T
23
p
=
I
4
×
4
T
23
s
=
I
4
×
4
Δ
T
23
s
=
I
4
×
4
0-4Z axis
T
04
s
=
I
4
×
4
Δ
T
04
p
=
[
1
0
Δ
β
x
z
0
0
1
-
Δ
α
y
z
0
-
Δ
β
x
z
Δ
α
y
z
1
0
0
0
0
1
]
T
04
s
=
[
1
0
0
0
0
1
0
0
0
0
1
z
0
0
0
1
]
Δ
T
04
s
=
[
1
-
Δ
γ
z
Δ
β
z
Δ
x
z
Δ
γ
z
1
-
Δ
α
z
Δ
y
z
-
Δ
β
z
Δ
α
z
1
Δ
z
z
0
0
0
1
]
4-5B axis
T
45
p
=
I
4
×
4
Δ
T
45
p
=
[
1
-
Δ
γ
z
B
0
0
Δ
γ
z
B
1
-
Δ
α
x
B
0
0
Δ
α
x
B
1
Δ
z
B
y
0
0
0
1
]
T
45
s
=
[
cos
B
-
sin
B
0
0
sin
B
cos
B
0
0
0
0
1
0
0
0
0
1
]
Δ
T
45
s
=
[
1
-
Δ
γ
B
Δ
β
B
Δ
x
B
Δ
γ
B
1
-
Δ
α
B
Δ
y
B
-
Δ
β
B
Δ
α
B
1
Δ
z
B
0
0
0
1
]Supposing that tool forming point in the tool coordinate system coordinate is(1)Pt=[PtxPtyPtz1]T.
And the work-piece forming point in the work-piece coordinate system coordinate is
(2)Pw=[PwxPwyPwz1]T.
When the machine tool moves in ideal form, that means that the machine tool is without error, the tool forming point and work-piece forming point will overlap together, and we can get
(3)[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]Pt=[∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s]Pw.
And the ideal forming function of tool forming point in work-piece coordinate system is
(4)Pt=[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]-1×∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)sPw.
Machining accuracy is finally determined by relative displacement error between the tool forming point of machine and work piece. During the actual machining process, the actual position of cutting tool forming point will inevitably deviate from the ideal location, which thereby results in volumetric error. The comprehensive volumetric error caused by the gap between actual forming point and ideal forming point can be written as
(5)E=(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w))Pw-(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t))×(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t)pTLk(t)Lk-1(t)s)-1×(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s)Pw.
With the characteristic matrices in Table 4, the comprehensive volumetric error model of a precision horizontal machining center can be gotten with (5). E reflects the deviation of the tool forming point between ideal location and actual location in work-piece coordinate system. If each geometric error value in the model is known, the volumetric error of machine tool can be obtained. As for other multiaxis machine tool, the general volumetric error model can be established in the similar way.


## 2.2.1. Structure Diagram and Topological Graph

In this research, a 4-axis precision horizontal machining center, whose 3-dimension digital structure model is shown in Figure3, is selected as an example to describe the proposed model. It is composed of a bed, x-, y-, z- three axis motion components, a spindle, and a workbench because it has three prismatic joints and one rotary joint, so there 24 position dependent systematic geometric errors. And among the four joints, there are 6 position independent systematic geometric errors, including three perpendicularity errors, two parallelism errors and one offset error. Table 1 shows all the geometric errors.Table 1
Geometric errors of precision horizontal machining center.
Component
Error term
Symbol
Component
Error term
Symbol
X-prismatic  joint
Positioning error
Δ
x
x
B-rotary joint
X direction run-out error
Δ
x
B
Y direction of straightness error
Δ
y
x
Z direction run-out error
Δ
z
B
Z direction of straightness error
Δ
z
x
Axial error
Δ
y
B
Rolling error
Δ
α
x
Around theX axis turning error
Δ
α
B
Britain swing error
Δ
β
x
Positioning error
Δ
β
B
Yaw error
Δ
γ
x
Around theZ axis turning error
Δ
γ
B
Y-prismatic joint
X direction of straightness error
Δ
x
y
Orientation error between joints
X, Y-axis perpendicularity error
Δ
γ
x
y
Positioning error
Δ
y
y
X, Z-axis perpendicularity error
Δ
β
x
z
Z direction of straightness error
Δ
z
y
Y, Z-axis perpendicularity error
Δ
α
y
z
Rolling error
Δ
α
y
B, Y-axis parallelism error in XY plane
Δ
γ
x
B
Britain swing error
Δ
β
y
B, Y-axis parallelism error in ZY plane
Δ
α
z
B
Yaw error
Δ
γ
y
B, Y-axis offset error in Z axis direction
Δ
z
B
y
Z-prismatic joint
X direction of straightness error
Δ
x
z
Z-prismatic joint
Rolling error
Δ
α
z
Y direction of straightness error
Δ
y
z
Britain swing error
Δ
β
z
Positioning error
Δ
z
z
Yaw error
Δ
γ
zFigure 3
Structure diagram of precision horizontal machining center. 0 bed; 1X-prismatic joint; 2 Y-prismatic joint; 3 spindle; 4 Z-prismatic joint; 5 B-rotary joint.


## 2.2.2. Topological Structure and Lower Body Array

On the basis of the theory of MBS, a multiaxis machine tool can be abstracted into a multibody system. As shown in Figure4, the machine base 0 is expressed into B0, 1 into B1 body, along the direction away from the body B1. According to the natural growth sequence, the bodies are sequentially numbered from one branch to another branch. Figure 4 gives the machine topology diagram. Table 2 is the lower body array of the selected precision horizontal machining center, and Table 3 depicts the degrees of freedom between each unit to the constraints, wherein “0” means no freedom of movement and “1” means one freedom of movement.Table 2
Lower body array of precision horizontal machining center.
Topic bodyj
1
2
3
4
5
6
L
0
(
j
)
1
2
3
4
5
6
L
1
(
j
)
0
1
2
0
4
5
L
2
(
j
)
0
0
1
0
0
4
L
3
(
j
)
0
0
0
0
0
0Table 3
Adjacent body freedom of the precision horizontal machining center.
Adjacent body
X
Y
Z
α
β
γ
0-1
1
0
0
0
0
0
1-2
0
1
0
0
0
0
2-3
0
0
0
0
0
1
0-4
0
0
1
0
0
0
4-5
0
0
0
0
1
0
5-6
0
0
0
0
0
0Figure 4
Topological graph of precision horizontal machining center.


## 2.2.3. Generalized Coordinates Setting and Characteristic Matrix

In order to facilitate the precision of machine tool modeling, coordinate system is required to be setted specially. Consider the following settings.(1) In MBS theory, right-handed Cartesian coordinate systems are built on the inertial components and all moving parts. These coordinates are called generalized coordinates. In the generalized coordinates, coordinate system on the inertial body is called the reference coordinate system, and coordinate system on other moving body is called moving coordinate system. Three orthogonal matrices of each coordinate system are named x-, y-, z-axis, respectively, according to the right-hand rule. (2) Each x-, y-, and z-axis of the different coordinate systems parallel correspondingly. (3) The tool coordinate system is at the end of the spindle center, and Table 4 lists the characteristic matrices of the precision horizontal machining center.Table 4
The characteristic matrices of precision horizontal machining center.
The adjacent body
Body ideal static, motion characteristic matrix
Body static, kinematic error  characteristics matrix
0-1X axis
T
01
p
=
I
4
×
4
Δ
T
01
p
=
I
4
×
4
T
01
s
=
[
1
0
0
x
0
1
0
0
0
0
1
0
0
0
0
1
]
Δ
T
01
s
=
[
1
-
Δ
γ
x
Δ
β
x
Δ
x
x
Δ
γ
x
1
-
Δ
α
x
Δ
y
x
-
Δ
β
x
Δ
α
x
1
Δ
z
x
0
0
0
1
]
1-2Y axis
T
12
p
=
I
4
×
4
Δ
T
12
p
=
[
1
-
Δ
γ
x
y
0
0
Δ
γ
x
y
1
0
0
0
0
1
0
0
0
0
1
]
T
12
s
=
[
1
0
0
0
0
1
0
y
0
0
1
0
0
0
0
1
]
Δ
T
12
s
=
[
1
-
Δ
γ
y
Δ
β
y
Δ
x
y
Δ
γ
y
1
-
Δ
α
y
Δ
y
y
-
Δ
β
y
Δ
α
y
1
Δ
z
y
0
0
0
1
]
2-3 tool
T
23
p
=
[
1
0
0
x
t
d
0
1
0
y
t
d
0
0
1
z
t
d
0
0
0
1
]
Δ
T
23
p
=
I
4
×
4
T
23
s
=
I
4
×
4
Δ
T
23
s
=
I
4
×
4
0-4Z axis
T
04
s
=
I
4
×
4
Δ
T
04
p
=
[
1
0
Δ
β
x
z
0
0
1
-
Δ
α
y
z
0
-
Δ
β
x
z
Δ
α
y
z
1
0
0
0
0
1
]
T
04
s
=
[
1
0
0
0
0
1
0
0
0
0
1
z
0
0
0
1
]
Δ
T
04
s
=
[
1
-
Δ
γ
z
Δ
β
z
Δ
x
z
Δ
γ
z
1
-
Δ
α
z
Δ
y
z
-
Δ
β
z
Δ
α
z
1
Δ
z
z
0
0
0
1
]
4-5B axis
T
45
p
=
I
4
×
4
Δ
T
45
p
=
[
1
-
Δ
γ
z
B
0
0
Δ
γ
z
B
1
-
Δ
α
x
B
0
0
Δ
α
x
B
1
Δ
z
B
y
0
0
0
1
]
T
45
s
=
[
cos
B
-
sin
B
0
0
sin
B
cos
B
0
0
0
0
1
0
0
0
0
1
]
Δ
T
45
s
=
[
1
-
Δ
γ
B
Δ
β
B
Δ
x
B
Δ
γ
B
1
-
Δ
α
B
Δ
y
B
-
Δ
β
B
Δ
α
B
1
Δ
z
B
0
0
0
1
]Supposing that tool forming point in the tool coordinate system coordinate is(1)Pt=[PtxPtyPtz1]T.
And the work-piece forming point in the work-piece coordinate system coordinate is
(2)Pw=[PwxPwyPwz1]T.
When the machine tool moves in ideal form, that means that the machine tool is without error, the tool forming point and work-piece forming point will overlap together, and we can get
(3)[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]Pt=[∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s]Pw.
And the ideal forming function of tool forming point in work-piece coordinate system is
(4)Pt=[∏j=n,Ln(t)=0j=1TLj(t)Lj-1(t)pTLj(t)Lj-1(t)s]-1×∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)sPw.
Machining accuracy is finally determined by relative displacement error between the tool forming point of machine and work piece. During the actual machining process, the actual position of cutting tool forming point will inevitably deviate from the ideal location, which thereby results in volumetric error. The comprehensive volumetric error caused by the gap between actual forming point and ideal forming point can be written as
(5)E=(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w))Pw-(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t))×(∏k=n,Ln(t)=0k=1TLk(t)Lk-1(t)pTLk(t)Lk-1(t)s)-1×(∏u=n,Ln(w)=0u=1TLu(w)Lu-1(w)pTLu(w)Lu-1(w)s)Pw.
With the characteristic matrices in Table 4, the comprehensive volumetric error model of a precision horizontal machining center can be gotten with (5). E reflects the deviation of the tool forming point between ideal location and actual location in work-piece coordinate system. If each geometric error value in the model is known, the volumetric error of machine tool can be obtained. As for other multiaxis machine tool, the general volumetric error model can be established in the similar way.


## 3. Stochastic Characteristic Modeling of Volumetric Error

### 3.1. Mean Value Modeling of Volumetric Error

Just as aforementioned, geometric errors are caused by the inaccuracies built in during original manufacturing, assembly, and in-service stage of component, and are stochastic variable with uncertainty characteristic [5]. Consequently, measured at different points, one individual geometric error may display different values, and each geometric error parameter can be expressed as a function of the position of the axis (x,y,z) [38]. Because the variation of geometric error is small compared with the error itself, it is convenient to divide the geometric error into a constant portion and a variable portion [2]. Supposing that g*is an arbitrary geometric error, it can be expressed as follows:
(6)g*=gc+gv,
in which, gc denotes the constant portion, and gv denotes the variable portion. By means of several measurements at corresponding points in the workspace and statistical analysis to the gotten sample data, the constant portion gc and the variable portion gv of geometric error g* can be described by mean and variance. Next, this section will discuss how to get the probability characteristic of total volumetric error with that of individual geometric error.According to (5), the explicit mean analysis model of volumetric error for precision horizontal machining center can be expressed simply as
(7)E=E(G,Pw,U,Uw,Ut),
where E=[Ex,Ey,Ez,1]T represents the volumetric error vector; G=[g1,g2,…,gn]T represents a vector consisting of n geometric errors; Pw=[pwx,pwy,pwz,1]T represents the work-piece forming point vector in the work-piece coordinate system; U=[x,y,z,B]T represents the machine tool motion axis position vector; Uw=[xw,yw,zw,1]T represents the vector of work-piece coordinate position; Ut=[xt,yt,zt,1]T represents the vector of tool coordinate position.With (7), when the random variable in matrix are in their mean values, we can calculate the mean value of volumetric error. Ex,Ey, and Ez are the volumetric error components in the x-, y-, and z-directions, respectively, and the resultant modulus of the volumetric error vector can be gained by the following equation [39]:
(8)Emod=Ex2+Ey2+Ez2.


### 3.2. Variance Modelling of Volumetric Error

When (7) is expanded in Taylor series about the nominal value of each random variable and the second- and higher-order terms are neglected, we can get
(9)E(G,Pw,U,Uw,Ut)=E(Gμ,Pwμ,Uμ,Uwμ,Utμ)+∂E∂Gμ(G-Gμ)+∂E∂Pwμ(Pw-Pwμ)+∂E∂Uμ(U-Uμ)+∂E∂Uwμ(Uw-Uwμ)+∂E∂Utμ(Ut-Utμ).
And (9) can be further written as
(10)ΔE=∂E∂GμΔG+∂E∂PwμΔPwμ+∂E∂UμΔU+∂E∂UwμΔUw+∂E∂UtμΔUt,
where ΔE=E(G,Pw,U,Uw,Ut)-E(Gμ,Pwμ,Uμ,Uwμ,Utμ), ΔG=G-Gμ, ΔPw=Pw-Pwμ, ΔU=U-Uμ, ΔUw=Uw-Uwμ, ΔUt=Ut-Utμ. ∂E/∂Gμ, ∂E/∂Pwμ, ∂E/∂Uμ, ∂E/∂Uwμ, ∂E/∂Utμ are Jacobian matrix and are all 4×n matrixes. Taking ∂E/∂Gμ as an example, it has the following formation:
(11)∂E∂Gμ=[∂Ex∂g1μ∂Ex∂g2μ⋯∂Ex∂gnμ∂Ey∂g1μ∂Ey∂g2μ⋯∂Ey∂gnμ∂Ez∂g1μ∂Ez∂g2μ⋮∂Ez∂gnμ00⋯0].
Other Jacobian matrixes also have homologous form, and the elements in matrixes are mean values of every random variable.Supposing thatA1=∂E/∂Gμ, A2=∂E/∂Pwμ, A3=∂E/∂Uμ, A4=∂E/∂Uwμ, A5=∂F/∂Utμ, (10) can be changed into
(12)ΔE=A1ΔG+A2ΔPw+A3ΔU+A4ΔUw+A5ΔUt.
Here, A1,A2,A3,A4,A5 are called sensitivity coefficient matrices and reflect the sensitivity of volumetric error to respectively. According to that, the stochastic process and the derivative process are irrelevant [40], and based on (12), volumetric error covariance matrix of the multiaxis machine tool are obtained
(13)VE=A1VGA1T+A2VPwA2T+A3VUA3T+A4VUwA4T+A5VUtA5T,
in which
(14)VE=[VexxVexyVexz0VeyxVeyyVeyz0VezxVezyVezz00000],Veij=Cov(ei,ej)=ρijσeiσej={ρij=1i=j,1≤i≤3,1≤j≤3-1<ρij<1i≠j,1≤i≤3,1≤j≤3.
Because the systematic geometric error components in G are independent [3, 41] actually, VG can be expressed as VG=diag{VG1,VG1,…,VGn} and reflects the probability characteristic of geometric errors. Likewise, VPw,VU,VUw,VUt are also described as follows: VPw=diag{VPwx,VPwy,VPwz,0}, VU=diag{VUx,VUy,VUz,0}, VUw=diag{VUwx,VUwy,VUwz,0}, VUt=diag{VUtx,VUty,VUtz,0}. Therefore, if the VG,VPw,VU,VUw,VUt are known, VE can be calculated with (13) and is a diagonal matrix. It can be seen that since the individual error is stochastic variable in (13), the volumetric errors also have to be stochastic, and their stochastic behaviour can be calculated based on those of the individual stochastic variable. Hereto, with (7) and (13), the mean value and variance value, which reflect the stochastic characteristic of the volumetric error of multiaxis machine tool, can be acquired.


## 3.1. Mean Value Modeling of Volumetric Error

Just as aforementioned, geometric errors are caused by the inaccuracies built in during original manufacturing, assembly, and in-service stage of component, and are stochastic variable with uncertainty characteristic [5]. Consequently, measured at different points, one individual geometric error may display different values, and each geometric error parameter can be expressed as a function of the position of the axis (x,y,z) [38]. Because the variation of geometric error is small compared with the error itself, it is convenient to divide the geometric error into a constant portion and a variable portion [2]. Supposing that g*is an arbitrary geometric error, it can be expressed as follows:
(6)g*=gc+gv,
in which, gc denotes the constant portion, and gv denotes the variable portion. By means of several measurements at corresponding points in the workspace and statistical analysis to the gotten sample data, the constant portion gc and the variable portion gv of geometric error g* can be described by mean and variance. Next, this section will discuss how to get the probability characteristic of total volumetric error with that of individual geometric error.According to (5), the explicit mean analysis model of volumetric error for precision horizontal machining center can be expressed simply as
(7)E=E(G,Pw,U,Uw,Ut),
where E=[Ex,Ey,Ez,1]T represents the volumetric error vector; G=[g1,g2,…,gn]T represents a vector consisting of n geometric errors; Pw=[pwx,pwy,pwz,1]T represents the work-piece forming point vector in the work-piece coordinate system; U=[x,y,z,B]T represents the machine tool motion axis position vector; Uw=[xw,yw,zw,1]T represents the vector of work-piece coordinate position; Ut=[xt,yt,zt,1]T represents the vector of tool coordinate position.With (7), when the random variable in matrix are in their mean values, we can calculate the mean value of volumetric error. Ex,Ey, and Ez are the volumetric error components in the x-, y-, and z-directions, respectively, and the resultant modulus of the volumetric error vector can be gained by the following equation [39]:
(8)Emod=Ex2+Ey2+Ez2.


## 3.2. Variance Modelling of Volumetric Error

When (7) is expanded in Taylor series about the nominal value of each random variable and the second- and higher-order terms are neglected, we can get
(9)E(G,Pw,U,Uw,Ut)=E(Gμ,Pwμ,Uμ,Uwμ,Utμ)+∂E∂Gμ(G-Gμ)+∂E∂Pwμ(Pw-Pwμ)+∂E∂Uμ(U-Uμ)+∂E∂Uwμ(Uw-Uwμ)+∂E∂Utμ(Ut-Utμ).
And (9) can be further written as
(10)ΔE=∂E∂GμΔG+∂E∂PwμΔPwμ+∂E∂UμΔU+∂E∂UwμΔUw+∂E∂UtμΔUt,
where ΔE=E(G,Pw,U,Uw,Ut)-E(Gμ,Pwμ,Uμ,Uwμ,Utμ), ΔG=G-Gμ, ΔPw=Pw-Pwμ, ΔU=U-Uμ, ΔUw=Uw-Uwμ, ΔUt=Ut-Utμ. ∂E/∂Gμ, ∂E/∂Pwμ, ∂E/∂Uμ, ∂E/∂Uwμ, ∂E/∂Utμ are Jacobian matrix and are all 4×n matrixes. Taking ∂E/∂Gμ as an example, it has the following formation:
(11)∂E∂Gμ=[∂Ex∂g1μ∂Ex∂g2μ⋯∂Ex∂gnμ∂Ey∂g1μ∂Ey∂g2μ⋯∂Ey∂gnμ∂Ez∂g1μ∂Ez∂g2μ⋮∂Ez∂gnμ00⋯0].
Other Jacobian matrixes also have homologous form, and the elements in matrixes are mean values of every random variable.Supposing thatA1=∂E/∂Gμ, A2=∂E/∂Pwμ, A3=∂E/∂Uμ, A4=∂E/∂Uwμ, A5=∂F/∂Utμ, (10) can be changed into
(12)ΔE=A1ΔG+A2ΔPw+A3ΔU+A4ΔUw+A5ΔUt.
Here, A1,A2,A3,A4,A5 are called sensitivity coefficient matrices and reflect the sensitivity of volumetric error to respectively. According to that, the stochastic process and the derivative process are irrelevant [40], and based on (12), volumetric error covariance matrix of the multiaxis machine tool are obtained
(13)VE=A1VGA1T+A2VPwA2T+A3VUA3T+A4VUwA4T+A5VUtA5T,
in which
(14)VE=[VexxVexyVexz0VeyxVeyyVeyz0VezxVezyVezz00000],Veij=Cov(ei,ej)=ρijσeiσej={ρij=1i=j,1≤i≤3,1≤j≤3-1<ρij<1i≠j,1≤i≤3,1≤j≤3.
Because the systematic geometric error components in G are independent [3, 41] actually, VG can be expressed as VG=diag{VG1,VG1,…,VGn} and reflects the probability characteristic of geometric errors. Likewise, VPw,VU,VUw,VUt are also described as follows: VPw=diag{VPwx,VPwy,VPwz,0}, VU=diag{VUx,VUy,VUz,0}, VUw=diag{VUwx,VUwy,VUwz,0}, VUt=diag{VUtx,VUty,VUtz,0}. Therefore, if the VG,VPw,VU,VUw,VUt are known, VE can be calculated with (13) and is a diagonal matrix. It can be seen that since the individual error is stochastic variable in (13), the volumetric errors also have to be stochastic, and their stochastic behaviour can be calculated based on those of the individual stochastic variable. Hereto, with (7) and (13), the mean value and variance value, which reflect the stochastic characteristic of the volumetric error of multiaxis machine tool, can be acquired.


## 4. A Case Study

Figure5 is the newly designed and manufactured 4-axis precision horizontal machining center whose 3-dimension numerical model is shown in Figure 1. In this research, in order to emphatically analyze the influence of geometric error to the volumetric errors Pw, U, Uw, and Ut are supposed to have no error, and only aforesaid 30 geometric errors are considered. The six position-dependent geometric errors of each prismatic joint were measured by dual-frequency laser interferometer [39] and electronic level directly, and it was a tedious and time-consuming task [42]. The five position-dependent geometric errors of B rotary joint, including ΔxB,ΔyB,ΔzB,ΔαB, and ΔβB, were measured by dual-frequency laser interferometer with its lens accessories and gotten with calculation of geometric relation. Because ΔxB,ΔyB,ΔzB,ΔαB, and ΔβB are all possible to arouse the error of ΔγB, the last error ΔγB can be identified by the identification method in Su [43]. The perpendicularity errors were measured with dial indicator and flat ruler. The parallelism errors and offset error were measured with standard mandrel and dial indicator. Some calibrated pictures are shown in Figure 6. As for each position dependent geometric error, it was measured 10 times at different points in the similar way, and Figure 7 lists the measured values of the 6 DOF geometric errors position dependent on x-prismatic joint.Figure 5
A 4-axis precision horizontal machining center.Tests of straightness error; (a)Δxy error; (b) Δzy error.
(a)
(b)Position dependent onx-prismatic joint; (a) linear errors; (b) angular errors.
(a)
(b)With the statistical analysis of the gotten sample data, the stochastic characteristic of geometric errors can be obtained. Taking the positioning errorΔxx at x = 200 mm as example, Figure 8 presents the distribution test result because the 10 discrete sample data are very close to the inclined line and it can be concluded that the Δxx geometric error approximately belongs to normal distribution. In the same way, the probability characteristics of other geometric errors, including mean value and variance value, can be gained. Table 5 gives the probabilistic characteristic of Δxx at partial test points, including the mean value (represented by M) and variance value (represented by V).Table 5
Probabilistic characteristic ofΔxx (μm).
Point1
Point2
Point3
Point4
Point5
Point6
Point7
Point8
M
V
M
V
M
V
M
V
M
V
M
V
M
V
M
V
1.2
0.0120
1.3
0.3282
1.2
0.0561
1.5
0.0930
2.1
0.0378
−0.5
0.0000
1.4
0.0112
2.5
0.0974
Point9
Point10
Point11
Point12
Point13
Point14
Point15
Point16
M
V
M
V
M
V
M
V
M
V
M
V
M
V
M
V
2.9
0.0502
3.8
0.2430
4.0
0.1068
4.7
0.0325
3.9
0.0707
5.9
0.0428
4.1
0.0679
5.4
0.1735
Point17
Point18
Point19
Point20
Point21
Point22
Point23
Point24
M
V
M
V
M
V
M
V
M
V
M
V
M
V
M
V
5.5
0.0967
6.5
0.0921
6.9
0.0165
7.3
0.0161
8.2
0.0020
8.8
0.0000
8.2
0.1104
7.4
0.0232
Point25
Point26
Point27
Point28
Point29
Point30
Point31
Point32
M
V
M
V
M
V
M
V
M
V
M
V
M
V
M
V
6.6
0.0342
6.1
0.0548
6.7
0.0087
7.8
0.1096
8.2
0.1833
8.5
0.0005
7.6
0.1540
8.7
0.0016
Point33
Point34
Point35
Point36
Point37
Point38
Point39
Point40
M
V
M
V
M
V
M
V
M
V
M
V
M
V
M
V
8.1
0.0075
8.3
0.0237
8.8
0.0046
9.3
0.0657
9.5
0.0000
9.8
0.0084
8.7
0.0116
8.6
0.0016Normal distribution test curve ofΔxx.
(a)
(b)With (7), (8), and Table 5, the mean value of volumetric error in whole workspace can be calculated. Just as shown in Figure 9, the workspace of the machine center is 1000 × 900 × 900 (mm), in which the modulus of volumetric error vector is expressed with corresponding color according to numerical magnitude. It can be seen that the mean value of volumetric error fluctuates in the whole workspace greatly and displays a symmetric distribution with y-z plane (x=0) as symmetry plane, which reflects the symmetric structure of the machine center. When the spindle and worktable are close up to each other, the volumetric error decreases gradually, and the central region is with relatively higher machining accuracy than the region around. When the spindle moves in the forward direction of z-axis, the volumetric error reaches the biggest value nearly and lies in the range of 0.02 mm to 0.025 mm. So, during the machining process, the forward direction of z-axis should be avoided as much as possible, and on the contrary, the central region can be made full use of.Figure 9
Mean value distribution of volumetric error.As for the 24 position-dependent geometric errors, their relation curves with correspondingx-, y-, and z-axes can be fitted according to the sample data. Figure 10 depicts the fitted polynomial curvs of Δxx, and with the fitted polynomial function, the partial derivative matrix used in (13) can be gained. With (13) and Table 5, the variance value distribution of volumetric error in whole workspace can be obtained. In the similar way, the modulus of variance vector is expressed with corresponding color in Figure 11. It can be drawn that the variance of volumetric error in whole workspace does not fluctuate obviously, which means that machining performance of the new manufactured machine center is relatively stable. But it is worthy to note that a little of higher variances exist within the machining endpoints in the backward or forward direction along x-, y-, and z-axes, especially in the forward direction at x=500 mm and the backward direction at x=-500 mm, and the maximum value reaches about 2.5×10-9 mm. So, according to the results, the end installation and pretightening of individual axis should cause enough recognition and attention.Fitted polynomial curve ofΔxx.
(a)
(b)Figure 11
Variance value distribution of volumetric error.Therefore, with above analysis results of the random uncertainty of volumetric error, on the one hand, we can make full use of the machining performance of machining center by keeping away from the workspace with lower machining accuracy or bigger accuracy fluctuation. For example, advisable adjustment during the clamping process of work piece may be carried out according to the error distribution in Figures9 and 11. In this way, the machining points can be fallen into the workspace with higher machining accuracy as much as possible, and consequently the qualified rate of the finished products can be improved effectively. On the other hand, because the random uncertainty of systematic geometric errors is a combination influence of guide ways misalignment, flatness errors, link length error, angular error, gradual machine wear errors, static deflection of machine components, and so forth, the analysis results can be of guiding significance to help make better of the guideway or ballscrew precision in x-, y-, and z-axes. For instance, avoiding eccentricity installation of ballscrew, eliminating incorrect prestretching of ballscrew, replacing worn ballscrew, adjustment and collimation local guideway installation, replacing worn guideway, and other maintenance measures can be put into practice according to the probability distribution of geometric errors.


## 5. Conclusions

Conventional approaches about error modeling and analysis of machine tool few consider the probability characteristics of the geometric error and volumetric error systematically. However, the individual geometric error is variational and stochastic measured at different points, and therefore the resultant volumetric errors are aslo stochastic and uncertain. In order to investigate the stochastic characteristics of geometric error and volumetric error, further to make rational use of machine tool and improve its performance at lower cost, an analysis method has been developed in this research. Based on MBS theory, an mean value analysis model of the volumetric error of multiaxis machine tool is established. With probability statistics and stochastic process theory, a variance analysis model of volumetric error is established, which can be used to analyze the stochastic characteristic of the volumetric error of the whole 3-dimensional manufacturing workspace. A case study example on four-axis machine tool has been conducted to demonstrate the effectiveness of this method.Characteristics of this method are summarized as follows.(1)
Compared with the traditional analysis methods, the proposed method deems the individual geometric error and volumetric error as uncertainty variables and deals with their random stochastic characteristics with mean and variance.(2)
In addition, the proposed probability analysis method is based on MBS theory and in matrix, which can avoid complex mathematics operations caused by direct differential, and is suitable for computer to model and calculation.Despite the progress, a number of issues need to be further addressed to perfect the currently developed method. Our future research will focus on the following two aspects:(1) development of an effective approach to obtain the area probability characteristic of thermal error, which is caused by the thermal deformation of machine tool components and takes a large proportion of volumetric error, (2) development of a method to attain the probability distribution of load error aroused by the joint interface deformation between machine tool components, which is influenced by cutting force, clamping force and the weight of work-piece and fixture.



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*Source: 863283-2013-10-21.xml*