Volumetric error modeling and accuracy improvement by

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.15, No.6, 2021

© 2021 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2021jamdsm0065]Paper No.21-00187

Volumetric error modeling and accuracy improvement by

parameter identification of a compound machine tool

1. Introduction

Large-scale products such as aviation, automobile and mold have the structure features of complex curved surface,

and the demand for five-axis machining centers of high-precision is increasing. Most of these products are thin-walled

structures with large dimensional changes and require high precision for docking and assembly. The modeling and

compensation of geometric errors are important ways of improving the accuracy of the machining center.

The geometric error modeling of CNC machine tools is mostly based on rigid body kinematics combined with the

homogeneous transformation matrix method, and the single geometric errors of the coordinate axis is introduced to

establish the geometric error model of the machine tool. Ordinary three-axis machine tools must test 21 geometric errors

(Okafor and Ertekin, 2000). The number of test error parameters of a typical five-axis machine tool with dual rotary axes

is 52 (Schwenke et al., 2008). Fan et al. (2014) used the orthogonal polynomial regression method to obtain the spatial

geometric error distribution and kinematic model of a five-axis machine tool. Chebyshev polynomials were also applied

to fit position-dependent geometric errors, it contributed to the establishment of the integrated model and parameter error

identification in three-axis machine tools (Li et al., 2015; Aguado et al., 2012).

Recently, related research used screw theory to model and compensate for the geometric errors of machine tools.

Xiang et al. (2016) proposed the forward and reverse kinematic model of a five-axis machine tool based on screw theory

Yingchun WU*,** and Jianxin SHEN* *Nanjing University of Aeronautics and Astronautics

29 Yudao Street, Qinhuai District, Nanjing 210016, China E-mail: [email protected]

**Wuxi Vocational Institute of Arts and Technology 99 Jingyinan Road, Yixing 214206, China

Received: 17 May 2021; Revised: 28 July 2021; Accepted: 20 August 2021

Abstract This paper presents a systematic method for kinematic modeling and improving the positioning accuracy of a compound machine tool. With the configuration of model frames and the adjustment of the link offset parameters, the position and orientation of the tool center point (TCP) are measured conveniently by a laser tracker, and the forward kinematic solution of the machine model is provided. Through the Levenberg-Marquardt (L-M) algorithm combined with chi-square fitting, the maximum likelihood estimators of the model parameters are obtained. The identified parameters indicate a certain squareness error between the linear and adjacent rotary axes, and each coordinate axis has a certain angular error. The link length parameter also has a slight error. The calibration result shows that the average position and orientation error of the machine tool are 0.03999 mm and 6.571 × 10-4 rad respectively, which are 79.6% and 44.9% lower than the initial error, indicating that the volumetric accuracy of the machine tool has been greatly improved through parameter identification. Compared with rigid body kinematics or screw theory modeling, the Denavit-Hartenberg (D-H) combined with Hayati-Mirmirani (H-M) modeling method used by the compound machine tool has a clear geometric meaning of the model parameters, and there is no need to measure the single geometric error of the coordinate axis. It has the advantages of fewer modeling steps and short test time. The volumetric error modeling, accuracy measurement, and parameter identification proposed in this paper are beneficial in improving the volumetric accuracy of machine tools with special structure.

Keywords : Compound machine tool, Modeling, Volumetric error, Parameter identification, Laser tracker

1

2© 2021 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2021jamdsm0065]

Wu and Shen, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.15, No.6 (2021)

and expressed the geometric errors of machine tools through error motion twists. Zhong et al. (2019) used screw theory

to establish the kinematic model of a three-axis machine tool, and identified the squareness error to improve the

volumetric accuracy of the machine tool. Volumetric error compensation based on the zero reference model is also a

method of improving the positioning accuracy of a machine tool. The volumetric positioning error of the tool center point

(TCP) is measured by a laser tracker. Parameter identification and compensation table were adopted in five-axis machine

tools (Freeman, 2006; Creamer et al., 2017), and the method has a good error compensation effect.

The geometric error compensation methods for machine tools are divided into two categories: active and pre-

calibration error compensation methods (Ramesh et al., 2000). Active error compensation is the direct compensation of

monitoring errors during processing. The parameter identification of the error model and compensation table belongs to

pre-calibration error compensation. At present, the volumetric error compensation software of mainstream computer

numerical control (CNC) systems is mainly used for the geometric error compensation of three-axis machine tools, and

there is no compensation software suitable for five-axis machine tools that include rotary axes.

The D-H method is also a commonly used modeling method for robots and machine tools. Parameter identification

of kinematic model was generally adopted to improve the positioning accuracy of robots. The Hayati-Mirmirani (H-M)

method is used to deal with the singularity problem when adjacent joints are parallel. Commonly used kinematics

calibration methods include least square estimation, Levenberg-Marquardt (L-M), and Kalman filtering. Given that multi-

axis machine tools and robots have similar series open chain structure and the D-H method has few parameters, the

method is also suitable for the kinematic modeling of machine tools. Kiridena et al. (1993) drew the volumetric

positioning error distribution map of a five-axis machine tool using the D-H method. Mahbubur et al. (1996) studied the

factors affecting the positioning accuracy and error compensation of the D-H model of a five-axis machine tool. Tsai et

al. (2009) used a modified D-H notation to establish a mathematical model of a multi-axis serial machine tool. Zhu et al.

(2014) applied the D-H and modified H-M methods in an aircraft assembly machine of the special structure, and

implemented kinematic calibration to satisfy the precision requirement of drilled holes.

the laser interferometer and the geometric error of the rotary axis is identified by the ball bar (Dassanayake et al., 2008),

the measurement steps are complicated and time-consuming. If the laser tracker is used to detect the volumetric error of

the machine tool and identify the model error parameters, the test takes a short time and the steps are simple. In addition,

the above-mentioned literatures mainly improve the position accuracy of the machine tool through modeling and

parameter identification, and involve little measurement and improvement of orientation accuracy.

Aircrafts have a complex structure and consists of millions of parts, which require many rivets to connect these parts.

The assembly quality of aircrafts is greatly affected by the accuracy of the rivet holes. Automatic drilling and riveting

equipment are the research and development hotspot of the current aviation manufacturing enterprises. Robot automatic

drilling and riveting and five-axis flex track drilling systems have been applied in the machining of Boeing 787 aircrafts,

which are developed by the Electroimpact Company of the United States (Devlieg, 2009; Malcomb, 2013). The

positioning accuracy of the automatic drilling and riveting equipment is the main factor that determines the drilling

accuracy of aircraft components.

In this paper, a kinematic modeling and calibration method for a compound machine tool is presented. The method

was validated by the measurement of a laser tracker and can effectively improve the position and orientation accuracy of

the machine tool. Section 2 introduces the structure and the application of the compound machine tool, explains the

coordinate frame definition and parameters of the machine model using the D-H and H-M methods, and provides the

forward solution of the kinematic model. Section 3 presents the steps of using the L-M algorithm and the model

parameters that must be identified. Section 4 discusses the establishment of machine coordinate frame using a laser

tracker and the distribution of measurement points. Section 5 presents identified kinematic model parameters, as well as

the volumetric positioning accuracy and influencing factors of kinematic model. Section 6 summarizes the contribution

of the paper.

2. Structure and kinematic model of the compound machine tool

2.1 Structure of the machine tool

The compound machine tool in this article is a newly developed five-axis machine tool, which is mainly used for the

processing and assembly of aircraft fuselages. Compared with the robot drilling system, the linear axis of the machine

2

In the above geometric error modeling method, when the single geometric error of the Linear axis is measured by



tool has better rigidity than the rotary axis of the robot, and the processing accuracy of the machine is less affected by

gravity and load. Figure 1 shows that the machine tool is equipped with a high-precision rotary table and a multi-function

end effector. The Y-axis realizes the vertical movement of the beam on the column. The X-axis performs the left and right

movement of the slideway on the beam. The Z-axis realizes the back and forth movement of the ram on the slideway.

The A-axis realizes the end effector swinging up and down through the linear motion of the linkage mechanism, and the

B-axis realizes the end effector rotating left and right through the linear motion of another linkage mechanism, as shown

in Fig. 1(b). After the product parts within the stroke of the rotary axis are processed, the high-precision rotary table can

be rotated to realize the conversion of the parts.

The pressure foot of the end effector can be driven by the cylinder to be extended and positioned on the fuselage.

The spindle located in the center of the pressure foot is driven by a servo motor to expand to realize processing tasks

such as hole drilling, countersinking, tool change, nail feeding and riveting. The end effector has an on-line detection

function that enables positional adjustment to achieve high normal accuracy between the tool axis and the fuselage

contour.

(a) Overall structure of the machine tool (b) Rotary axes and end effector

Fig. 1 Overall structure of compound machine tool

The machine has five axes with 5 degrees of freedom. The translation/rotation of the five axes [ , , , , ]X Y Z A B are

called the joint axes of the machine. The TCP vector in the machine base coordinate frame is recorded as [ , , , , , ]x y z i j k .

The TCP vector has six variables among which the tool direction is a unit vector. If any two of i, j, and k are known, other

one can be derived. The TCP vector has 5 degrees of freedom like the machine tool.

The homogeneous transformation matrix between the machine base frame and the TCP frame is set as

0 0 0 1

=

x x x x

y x y yBase

TCP

z x z z

n s a d

n s a d

n s a dT (1)

Though the tool axis coincides with the z axis of the TCP frame, and the TCP vector is

[ , , , , , ] [ , , , , , ]= x y z x y zx y z i j k d d d a a a (2)

When the specific value of [ , , , , ]X Y Z A B is arbitrarily provided, the position and orientation of the TCP frame

relative to the machine base frame can be obtained according to homogeneous transformation matrix Base

TCPT , which is the

forward solution of the machine kinematic model.

2.2 Kinematic model of the machine tool

Column

Product

Beam

Ram

Slideway

End effector

ZY

X

Rotary table

B

A

O

Pressure foot

A axis

B axis

ZY

X

B motion

A motion

O

3



When five-axis machine tools are modeled by rigid body kinematics and screw theory, the model has many error

parameters, and the error parameters of the linear and the rotary axes must be measured separately. For example, the error

parameters of the linear axis include positioning accuracy, straightness and squareness. The relevant measurement method

is described in ISO230-1 (2012). The measurement steps of the above-mentioned modeling method are complicated and

time-consuming, and geometric error compensation is difficult to apply on multi-axis machine tools with special

structures.

According to the structural characteristics and work requirements of the machine tool, a systematic method

combining the D-H and H-M methods is used for modeling. Unlike the ordinary rotary axis, the A and B rotary axes are

realized by planar linkage mechanisms, and the two rotary axes do not intersect and there is a common perpendicular.

The error parameters of the rotary axis of the machine tool are difficult to define and measure. The D-H method is used

to model the kinematics of the compound machine tool with few error parameters, and the modeling of the parallel axes

can also be combined with the H-M method.

In the D-H kinematic model of the compound machine tool, the machine body can be regarded as a series of

connecting links, and the spatial transformation between adjacent links is described by the relative position between the

joint frames fixed on the connecting link. A simplified machine model with zero position is shown in Fig. 2. The machine

has three linear axes (i.e., X, Y, and Z) and two rotary axes (i.e., A and B). The moving directions of the X-, Y- and Z- axes

are perpendicular to each other. A- and B- axis are parallel to the corresponding linear axes.

Fig. 2 Kinematic model of the compound machine tool

To more easily illustrate the kinematic model of the machine tool, the machine link parameters L1–L4 are defined

through some auxiliary points:

(1) L1: This is the distance from point PA to point PB; point PA is one end point of the common perpendicular between

the rotary A- and B-axes and locates in the A-axis; point PB is one end point of the common perpendicular between the

A- and B-axes and locates in the B-axis.

(2) L2: This is the distance from point PB to point PC in the Y direction; point PC is one end point of the common

perpendicular between the B-axis and tool axis, and locates in the tool axis.

(3) L3: This is the distance from point PB to point PC in the X direction and the error caused by the low precision of

parts manufacturing and assembly.

(4) L4: This is the distance from point PC to point PD, and point PD is the tool tip point, which is the TCP.

To make the measurement data convenient for model parameter identification, the established machine model has

the coordinate axis directions of the base and the tool coordinate frames consistent. Through the link parameter data

processing of the conversion matrix, the origin of the base frame is coincident with the measurement reference point.

According to the machine structure, the following eight coordinate frames are established: machine tool base frame

0, X-axis frame 1, Y-axis frame 2, Z-axis frame 3, A-axis frame 4, B-axis frame 5, and TCP frame 7. To facilitate kinematic

modeling, an auxiliary frame 6 is inserted between the B-axis and TCP frames. From the machine base mark to the tool

coordinate system, the topological sequence of the machine tool is 0→1→2→3→4→5→6→7.

4



Each axis direction of the coordinate frame is shown in Fig.2, and the coordinate frames are described as follows:

(1) Machine base frame 0: When the machine axes are at zero position, the machine base frame coincides with the

TCP frame. Through parameter processing, the origin of the machine base frame can be moved and fixed at the TCP.

(2) Frame 1: This frame is fixed to the Y-axis.

(3) Frame 2: This frame is fixed to the X-axis.

(4) Frame 3: This frame is fixed to the Z-axis.

(5) Frame 4: This frame is fixed to the A-axis, the origin coincides with PA, the y4 axis coincides with line A BP P ,

and the z4 axis coincides with the A-axis.

(6) Frame 5: This frame is fixed to the B-axis, the origin coincides with point PB, the x5 axis coincides with line

A BP P , and the z5 axis coincides with the B-axis.

(7) Auxiliary frame 6: This frame is fixed to the B-axis, the origin coincides with point PC, and the z6 axis coincides

with line C DP P .

(8) TCP frame 7: The origin coincides with the spindle tool tip, and the coordinate axis direction is consistent with

the machine base frame.

Thus, the transformation between the adjacent links of the compound machine tool can be written, as shown in Table

1. According to the D-H method, link offset d is the distance from the origin o0 to the intersection of the axes x1 and z0.

The distance is measured along the z0 axis (Spong et al., 2005). When the direction is opposite to the z0 axis, the negative

value is taken, and then d5 = -L2 and d6 = -L4 in Table 1.

When [ , , , , ]=[0,0,0,0,0]X Y Z A B , Fig. 2 shows that origin coordinates of the TCP frame in the machine base frame

are x = L3, y = -L2, z = L1 - L4. When joint variable of the machine tool is at the zero position, the origin of the machine

base frame is moved to the TCP of this position. At this time, when the machine coordinate axis is moving, the TCP

relative to machine tool frame x0’o0’y0’ after the movement has no change in orientation compared with machine base

frame x0o0y0, and the position must be subtracted the fixed constant from the link offset of serial number (1)–(3), that is,

the origin coordinate of the TCP frame, which are L3, -L2, L1 - L4, respectively.

Taking the moved machine base frame x0'o0'y0' as the measurement standard, when the machine tool joint variables

change, the Cartesian position and orientation errors of the TCP can be conveniently measured and directly used to

identify the relevant parameters of the machine tool kinematic model.

Table 1 Kinematic parameters and initial values of the machine tool model

Link [number]

Link length

ia [mm]

Link twist

i or i [deg]

Link offset

id [mm]

Joint angle

i [deg] Identify

Sign Initial

values Sign

Initial

values Sign

Initial

values Sign

Initial

values

0→1 [0] 0a 0 0 -pi/2 0d 0 0 0 No

1→2 [1] 1a 0 1 pi/2 1d Y+L2 1 pi/2 Yes

2→3 [2] 2a 0 2 pi/2 2d X－L3 2 pi/2 Yes

3→4 [3] 3a 0 3 pi/2 3d Z－L1+L4 3 0 Yes

4→5 [4] 4a L1 4 pi/2 4d 0 4 pi/2+A Yes

5→6 [5] 5a L3 5 pi/2 5d -L2 5 pi/2+B Yes

6→7 [6] 6 0

6d -L4 Yes 6 0

After defining the coordinate frame of each axis in the machine tool, the transformation matrix between adjacent

coordinate frames can be achieved according to the link parameters and the translation/rotation of each axis, as shown in

Eq. (3)－(6). In addition, the z6 axis of the auxiliary frame 6 is collinear with the z7 axis of the TCP frame 7. To avoid the

singular problem of the kinematic model caused by the parallel adjacent joints, the H-M method is used here, and the

conversion for the nearly parallel prismatic joints is shown in Eq. (4), where is the rotary angle around the y7 axis of

the TCP frame (Hayati and Mirmirani, 1985).

5



0

1 0 0 0 0( , ) ( , ) ( , ) ( , )

1 0 0 0

0 0 1 0=

0 1 0 0

0 0 0 1

=

−

Rot z Trans z d Trans x a Rot xT

， 1

2

2

0 0 1 0

1 0 0 0

0 1 0

0 0 0 1

= +

Y LT (3)

2

3

3

0 0 1 0

1 0 0 0

0 1 0

0 0 0 1

− = − −

X LT ， 3

4

1 4

1 0 0 0

0 0 1 0

0 1 0

0 0 0 1

− = − +

Z L LT ，

1

14

5

sin 0 cos sin

cos 0 sin cos

0 1 0 0

0 0 0 1

− − =

A A L A

A A L AT ，

3

35

6

2

sin 0 cos sin

cos 0 sin cos

0 1 0

0 0 0 1

− − = −

B B L B

B B L B

LT ，

6

7 6 6 6

4

( , ) ( , ) ( , )

1 0 0 0

0 1 0 0

0 0 1

0 0 0 1

=

= −

Trans z d Rot x Rot y

L

T

(4)

The transformation from the machine base frame to the TCP frame is

0 0 1 2 3 4 5 6

7 1 2 3 4 5 6 7=T T T T T T T T (5)

Among them,

0

7 13 sin= =i BT ， 0

7 23 sin cos= = −j A BT ， 0

7 33 cos cos= =k A BT (6)

0

7 14 3 3 4cos sin= = − + −x X L L B L BT

0

7 24 1 2 2 3 4sin cos sin sin sin cos= = − + − + +y Y L A L L A L A B L A BT

0

7 34 1 1 2 3 4 4cos sin cos sin cos cos= = − + − − + −z Z L L A L A L A B L L A BT

According to Eq. (6), the TCP vector is the forward kinematic solution of the machine tool, and the inverse solution

of the machine model can be conveniently obtained after transformation. When the joint variables are

[ , , , , ] [0,0,0,0,0]=X Y Z A B , the TCP vector is [ , , , , , ] [0,0,0,0,0,1]=x y z i j k , and the TCP frame coincides exactly with

the machine base frame.

3. Parameter identification of the machine tool kinematic model

3.1 Actual identified parameters of the kinematic model

The joint angles and link parameters of serial numbers 1 to 6 in Table 1 must be calibrated theoretically, wherein X,

Y, Z, A, and B are the coordinate values. Then, it is considered that the coordinate value read by the encoder has no error.

rotary axis must also be calibrated. The initial parameters values of the rotary axis can be measured by the laser tracker:

L1= 205.51 mm, L2= 397.01 mm, L3= 1.81 mm, L4= 368.28 mm.

6

When the mechanical parts have manufacturing and assembly errors, the fixed structure parameters L1 L4 of the –



Figure 2 indicates that the axes zi-1 and zi between the frames 1, 2, 3, and 4 intersect. According to the D-H method,

the axis xi is perpendicular to the plane formed by the axes zi-1 and zi, and the positive direction can be chosen at will. In

this case, the parameter ai is zero. At this time, the link length parameters (i.e., a1 = 0, a2 = 0, and a3 = 0) in Table 1 do

not require parameter identification. Link twist parameters 6 and 6 only affect the orientation of the TCP of the

machine tool and not the position of the TCP. When the TCP measurement data include the position and the orientation,

link twist parameters 6 and 6 can be identified.

When the position and orientation of the end effector is measured (Schroer et al., 1997), the maximum number

(Everett et al., 1988) of independent kinematic model parameters of generic serial robots is expressed as Eq. (7):

N = 4R + 2P + 6 (7)

In Eq. (7), N is the maximum number of independent model parameters, R is the revolute joints number, and P is the

prismatic joints number. For the compound machine tool, R = 3, P = 2, then N = 20. The number of model parameters in

Table 1 is consistent with Eq. (7). In the numerical simulation of parameter identification of the machine tool kinematic

model, the four parameters (i.e., a1, a2, a3, and d4) analyzed above cannot be accurately identified. Therefore, the actual

calibrated independent link parameter of the machine tool model is 16, the model parameter errors can be fully identified

and the accuracy is high at this time. The kinematic model of the compound machine tool proposed in this research could

satisfy the requirements of continuity, integrity, and minimum parameters.

3.2 Parameter identification algorithm

The position and orientation of TCP in end effector of the machine tool are incorrect due to certain errors in the

various link and angle parameters of the machine tool model. According to the kinematic model of the compound machine

tool, the non-linear transformation between the volumetric positioning error of the machine tool and the parameter error

of each link can be obtained. Conversely, the parameter errors of the links could be solved iteratively according to the

positioning errors of the machine, and the parameters of the kinematic model are calibrated.

When drilling with the compound machine tool, the position accuracy of the tool points must be ensured, and certain

requirements must be satisfied for the normal accuracy between the tool axis and the fuselage contour. Here, the position

and orientation error of the TCP frame is used to identify the model parameters. The relationship between the position

and orientation P of the TCP and the kinematic parameters of the machine model is shown in Eq. (8).

= ( , , , , ) P F a d (8)

The volumetric error of the TCP caused by the model parameter error is recorded as P .

1 1 6 1 4

1 1 6 1 4

+

= + + +

= + + + + + +

F F F F FP a d

a d

F F F F FL L

L L

(9)

In Eq. (9), a refers to 4a and 5a in the machine model parameters, that is, the structural parameters ( 1L

and 2L ) of the rotary axis to be calibrated. In addition, d , , , and have the same agreement.

If the coordinates of n points in the machine tool workspace are measured, the transformation form of Eq. (9) is

= P J q (10)

7



1 1 1 1

1 6 1 4

1 1 1 1

1 6 1 4

1 1 1 1

1 6 1 4

1 1 1 1

1 6 1 4

1 1 1 1

1 6 1 4

1 1 1 1

1 6 1 4

1

=

x x x x

y y y y

z z z z

i i i i

j j j j

k k k k

nk nk

F F F F

L L

F F F F

L L

F F F F

L L

F F F F

L LJ

F F F F

L L

F F F F

L L

F F

6 1 4

nk nkF F

L L

， (11)

1 1 6 1 4=( ）Tq L L ， (12)

1 1 1 1 1 1=( ）T

x y z i j k nkP P P P P P P P (13)

Among them, q is a column vector composed of parameter errors of the machine model; ixP , iyP , izP ,

iiP , ijP , and ikP refer to the position and orientation errors of the Pi point in the X, Y, and Z directions; and ixF ,

iyF , izF , iiF , ijF , and ikF respectively refer to the parameter expressions of the position and orientation of the Pi

point in the X, Y, and Z directions, where i = 1–n.

In this way, the kinematic parameter error calibration of the compound machine tool is transformed into a fitting

problem of non-linear least squares, which can be solved iteratively using the L-M algorithm combined with chi-square

fitting. The identification parameters of the machine tool model are statistically maximum likelihood estimators. The chi-

square error criterion represents the goodness of fit, and the measurement error is introduced through the weight matrix,

and the standard deviation and correlation coefficient of the parameters can also be obtained, making the identified model

parameters more accurate and intuitive.

The flow chart of machine model parameter calibration is shown in Fig. 3, and the steps are explained as follows:

(1) According to the kinematic forward solution of the position and orientation measurement of the TCP and the

initial value of the structure parameter, the initial position and orientation error of the machine tool are obtained.

(2) Chi-squared error criterion 2 ( )q indicates the goodness of fit between the measured value and the fitted value

(Gavin, 2017), and weight matrix W is the inverse matrix of the measurement error covariance, and each link parameter

error q of the machine tool is calculated, where is the algorithmic parameter, and the initial value of the parameter

here is 0.01.

(3) Metric i is calculated and compared with specified threshold 4 . Different updating methods are used for the

damping coefficient, and the model parameters are updated when the conditions are met.

(4) When the iteratively calculated positioning error of the machine tool and parameter accuracy meet one of the

convergence conditions, the identified link parameter (+q ) is obtained, and the standard deviation ( q ) of the identified

parameter and the parameter correlation matrix qC are obtained.

In Fig. 3, 2

is the reduced chi-square error criterion.

2 2= / ( 1) − +n m (14)

In Eq. (14), m is the kinematic parameters number of the machine tool, and n is the number of volumetric

measurement points.

8



Fig. 3 Algorithm flow chart of kinematic parameter calibration of the machine tool

4. Volumetric accuracy measurement and verification

4.1 Establishment of machine tool base frame

To obtain the volumetric positioning accuracy of the compound machine tool, a machine base coordinate frame must

be established. In this study, a Leica AT901 laser tracker is used to measure the volumetric accuracy. The absolute

measurement accuracy of the tracker is 0.02 mm, which meets the experimental requirements. The data is measured using

the Spatial Analyzer (SA) software associated with the laser tracker. The SA software has many functions such as

establishing a coordinate frame and fitting the plane. The measurement site is shown in Fig. 4.

Fig. 4 Volumetric accuracy measurement by laser tracker

1[ ]T T

iq J WJ I J W P − = +

2 ( )= Tq P W P

1 10*i i + =

1i iq q q+ = +

1iq q+

+=

4i

2 2( )- ( )=

( )i T T

i

q q q

q q J W P

+

+ 1 /10i i + =

1max TJ W P

1[ ] / ( * )T T

q p PC J WJ −=

1([ ] )T

q diag J WJ −=

NO

YES

NO

YES

= ( )P P F q −

2max /q q orConvergence

criteria

Parameters update

Identified parameters

Updatecriteria

Initial errorStep

calculation

Dampingcoefficient

Improvement metric

or 2

3

Laser tracker

SMR

End effector

P P’

Olx

ly

ZX

Y

9



The machine base frame could not be read directly from the laser tracker. Here, the construction method is used to

establish the machine base frame. The steps are explained as follows:

(1) The machine coordinate axes are returned to the zero position, the spindle feed motor is controlled so that the

tool tip is at the drilling positioning point, the spherically mounted retro reflector (SMR) of the laser tracker is installed

at the end of the spindle, the coordinates of the center of the SMR at this time are measured, and this point is recorded as

O.

(2) The X-axis is moved, and the laser tracker measures the motion trajectory of the SMR and can fit a straight line,

Meanwhile, the laser tracker measures the SMR trajectory, and the straight line fitting the trajectory is ly.

(3) Point O is set to the origin of the machine base frame. First, the Z-axis direction is defined as the cross product

(lx, ly) with lx and ly. The line parallel with lx and passing through point O is taken as the X-axis. To ensure that the three

base vectors of the machine base frame are perpendicular to each other, the right-hand rule can be used to define the Y-

axis.

The measurement coordinate frame is transferred to the machine base coordinate frame by the SA software. In this

way, the measured value of the TCP is the Cartesian coordinate value in the machine base frame.

(4) To measure the volumetric accuracy of the TCP, the orientation of the TCP is measured by changing the extension

position of the spindle. After measuring the position of the tool tip P of the drilling position, the spindle of the machine

tool is moved a certain distance inward, and the position of the tool tip P’ at this position is measured according to the

same steps. In the SA software, the corresponding measurement points before and after are connected to form a straight

line PP’. This line represents the axis of the spindle, and the angle between the three coordinate axes of the machine base

frame is the orientation of the TCP.

4.2 Measurement points distribution and accuracy verification

The accuracy of the kinematic machine model is verified by the cross-validation method, which is measured the

distance between the measurement points of the laser tracker and the model prediction points. To avoid the light blocking

of laser tracker SMR due to the wide orientation change of the measurement points, the point distribution uses half the

stroke of the coordinate axis, and the stroke of each axis of the machine tool is shown in Table 2. The machine tool in

this study is a functional verification machine, and the stroke is a fraction of the actual industrial application machine.

The measurement points are divided into two parts. A total of 100 randomly distributed points of model identification

are generated in each axis coordinate stroke of the machine tool, and 20 randomly distributed points are used to verify

After repeated measurements of the measuring point, the standard deviations of the position and orientation

measurement of the TCP in the three coordinate axis directions can be obtained, namely, 0.03 =x mm , 0.07 =y mm ,

0.05 =z mm , 0.3 =i rad , 0.5 =j rad , and 0.1 =k rad , which are used for weight matrix W calculation and

machine model parameter identification.

Table 2 Each axis range of the compound machine tool

Coordinate axis Type Range [mm or deg]

X Prismatic -560－0

Y Prismatic -1280－0

Z Prismatic -480－0

A Revolute -14－14

B Revolute -14－14

According to the volumetric error of the identification points, the machine model parameters are identified by the L-

M algorithm. By comparing the difference between the theoretical position and orientation of the verification points of

correction machine model and the actual measured position and orientation, the improvement of the volumetric accuracy

10

which is recorded as lx. The axes of the machine tool are returned return to the origin. Only the Y-axis is moved.

the model accuracy. The spatial position and orientation distribution of the measuring points are shown in Fig. 5. The

starting point of the vector line in Fig. 5 represents the position of the measuring point, and the direction arrow is the

orientation of the measuring point.



of the machine model is verified after parameter identification.

Fig. 5 Position and orientation of the identification and verification points in the working space

5. Identified parameters and analysis

5.1 Calibration results

Parameter identification obtains an estimated value of model parameters. Then, the difference between the theoretical

pose of the sampling point calculated by the recognized kinematic model and the actual measured pose is minimized.

According to the position and orientation error of the identification points, the correct model parameters of the compound

machine tool can be calculated by using the L-M algorithm. The results are shown in Table 3.

Table 3 shows that the actual machine model has a certain degree of deviation in the joint angles and the link

parameters, which reflects the geometric error of the machine tool structure. The standard deviation of the machine model

parameter error represents the influence of the changes in the volumetric error measurement on the model parameters,

and its value represents the accuracy of parameter identification, which is the measurement uncertainty of model

parameters. Among them, the identification of link twist 6 , 6 , and joint angle 3 is close to 0, which respectively

indicate the angular error of the TCP around the X axis, Y axis and Z axis. Due to the influence of the orientation

measurement method of the TCP, the ratio of the standard deviation of the above three parameters to the identified value

is relatively large, resulting in low identification accuracy of the three parameters.

With reference to Fig. 2, the link twist identification of 1 indicates that the Y- and X-axes have a certain squareness

error in the XY plane, and joint angle identification 1 indicates that the Y-axis has a certain rotary angle error in the XZ

plane, that is, the roll angular error around the Y-axis. The link twist identification of 2 indicates a certain squareness

error in the XZ plane between the X- and Z-axis, and 2 indicates that the X-axis has a certain angular error in the YZ

plane. The identification of 3 reflects a certain squareness error in the XZ plane between the A rotary axis and the Z-

axis, and the angular error ( 3 ) of the Z-axis in the YZ plane is relatively large. 4 reflects a relatively large squareness

error between the B- and A-axes in the XY projection plane, and 4 reflects that the A-axis has a certain angular error in

the YZ plane. 5 indicates a certain squareness error between the spindle axis and the B-axis on the YZ projection plane,

and the B-axis also has a certain angular error ( 5 ) on the XZ plane. Link twists 6 and 6 represent the angular

errors of the TCP around the X- and Y-axes, respectively. Link parameters identification L1－L4 indicate the parts

manufacturing error and component assembly errors of the machine tool, and the L2 error is relatively large.

In addition, 2 and 4 indicate the rotations in the YZ plane, 3 and 5 indicate the rotations in the XZ plane,

and 3 and 4 are both rotations in the XY plane. The correlation coefficients of the two pairs of parameters derived

from parameter correlation matrix qC are high, both exceeding 0.9.

By comparing the position and orientation of the identification points calculated according to the identified model

11



parameters with the measured position and orientation, the residual at the identification points of the machine model can

be obtained, as shown in Fig. 6. Among the position error residuals, the X-axis residual is large, the Y-axis residual is

small, and the Z-axis residual is the smallest. For the orientation error residual, the residual of the Y-axis direction is

large, the residual of the X-axis direction is small, and the residual of the Z-axis direction is the smallest. The results are

basically consistent with the trend of the standard deviation of the TCP orientation measurement.

Table 3 Identified kinematic parameters of the machine model

Parameter name Initial value

[deg or mm]

Identified value

[deg or mm]

Standard deviation

[deg or mm]

Standard

deviation/Identified

value [%]

1 90 90.0140 0.001278 0.0014

1 90 90.0016 0.001278 0.0014

2 -90 -89.9936 0.001965 0.0022

2 90 90.0089 0.001507 0.0017

3 90 89.9880 0.02946 0.033

3 0 0.1318 0.06797 51.6

4 90 90.0249 0.06811 0.076

4 90 89.9937 0.001599 0.0018

5 90 89.9951 0.0006360 0.0007

5 90 90.1719 0.01399 0.016

6 0 0.063 0.1027 163

6 0 -0.1432 0.1041 72.7

1L

205.51 205.6519 0.04512 0.022

2L

397.01 395.1598 0.03560 0.009

3L

1.81 1.7646 0.1121 6.35

4L

376.68 376.8609 0.02833 0.0075

Fig. 6 Residual of identification points of the kinematic model

The position and orientation errors of the verification points before and after parameter identification of the machine

axes of the machine tool have been significantly reduced by 94.5%, 99.0% and 90.9% respectively. In addition, the range

of the Z-direction is also decreased significantly, followed by the range of the Y-direction, and that of the X-direction

12

model are shown in Fig. 7–8 and Table 6 respectively. After calibration, the average position error of the X-, Y- and Z-



decreased slightly. After calibration, the average orientation errors of the X and Y directions of the TCP of the machine

tool are greatly reduced by 80.6% and 93.0% respectively, and the average Z-direction errors are basically unchanged.

The range of the orientation error in the X and Y directions of the TCP are basically unchanged, and range of the

orientation error in the Z direction is reduced by 55.8%.

Among the main technical indicators of the compound machine tool, the positioning accuracy of the X-axis, Y-axis

and Z-axis are respectively less than 0.08 mm, 0.06 mm and 0.03 mm. It can be seen from Fig. 7 that after calibration,

the maximum X-axis positioning error is 0.07 mm, the Y-axis positioning error is 0.037 mm, and the Z-axis positioning

error is 0.021 mm, which meets the accuracy requirements of the machine tool.

The comparison of the integrated volumetric position error (2 2 2p x y z = + + ) in the three directions (X, Y,

and Z) is shown in Fig. 9 and Table 4. The above chart shows that the average of integrated position error is reduced from

0.1962 mm to 0.03999 mm, and the error is reduced by 79.6%. The standard deviation is reduced from 0.05673 mm

before calibration to 0.01669 mm after calibration, and the error is reduced by 70.6%. In addition, the average of the

integrated orientation error (2 2 2o i j k = + + ) is reduced from 1.193 × 10-3 rad to 6.571 × 10-4 rad, and the error

is reduced by 44.9%; the standard deviation is reduced from 6.482 × 10-4 rad before calibration to 3.296 × 10-4 rad after

calibration, and the error is reduced by 49.2%.

Fig. 7 Position error of verification points before and after calibration

Fig. 8 Orientation error of verification points before and after calibration

13



Fig. 9 Volumetric position and orientation error before and after calibration

Table 4 Volumetric error of the machine tool before and after calibration

Error [mm or rad] Before calibration After calibration Error reduction [%]

x Average 0.1192 -0.00651 94.5

Range 0.1612 0.119 26.2

y Average 0.01455 -0.00015 99.0

Range 0.228 0.081 64.5

z Average 0.02154 0.001965 90.9

Range 0.544 0.027 95.0

i Average -3.47 × 10-4 6.73 × 10-5 80.6

Range 8.32 × 10-4 9.10 × 10-4 -9.38

j Average 1.10 × 10-3 7.66 × 10-5 93.0

Range 2.338 × 10-3 2.444 × 10-3 -4.53

k Average 3.05 × 10-6 3.55 × 10-6 -16.4

Range 3.73 × 10-4 1.65 × 10-4 55.8

Position error Average 0.1962 0.03999 79.6

Standard deviation 0.05673 0.01669 70.6

Orientation error Average 1.193 × 10-3 6.571 × 10-4 44.9

Standard deviation 6.482 × 10-4 3.296 × 10-4 49.2

5.2 Error analysis

Through parameter identification, the volumetric positioning accuracy of the machine tool is improved to some

extent. Though the positioning repeatability and straightness of the machine linear axis is good, but each axis is not

strictly vertical. A certain squareness error exists, and this error can be considered constant regardless of the coordinate

axis position. Then, the volumetric positioning accuracy of the machine tool is improved by identifying the parameter

error of the kinematic model.

In the parameter calibration algorithm, the value of the machine coordinate axis is considered highly precise and

error-free. When the rotary axis has an angular positioning error, it will have a certain influence on the positioning

accuracy of the machine tool correction model. The positioning error of A- and B-axes is not linearly distributed because

14



of the planar linkage mechanism (Wu and Shen, 2020). Among them, the positioning accuracy of the A axis is high, while

the positioning accuracy of the B axis is slightly worse.

The position and orientation accuracy of the TCP in the X-axis direction is mainly affected by the positioning

accuracy of rotary axis B, and the stroke range of the axis is the sensitive direction of the X-axis. Therefore, the average

position error is reduced by 94.5% after calibration, which greatly improves the position accuracy in the X-axis direction,

but the range shows nearly no improvement. The volumetric accuracy of the TCP in the Y-axis direction is mainly affected

by the positioning accuracy of axis A and hardly affected by the accuracy of axis B. Therefore, the average position error

is reduced by 94.5% after calibration, which greatly improves the positioning accuracy in the Y-axis direction. The range

of position error dropped by 64.5%, and the range of orientation error is basically unchanged. The volumetric accuracy

of the TCP in the Z-axis direction is affected by the positioning accuracy of two rotary axes (A and B), but the stroke

range of the rotary axes is insensitive to the Z positioning accuracy. Thus, its positioning error has been greatly reduced

by 90.9% after calibration. The range of position and orientation error are also reduced by 80.6% and 55.8% respectively,

which greatly improved the positioning accuracy in the Z-axis direction.

In addition, the orientation measurement is also affected by the measurement method, mainly because the distance

between the two TCP points of the measurement spindle axis is relatively short, and the orientation accuracy of the

machine tool is slightly improved. The projection value of the TCP vector in the X and Y directions is small, and the

measurement accuracy is not very high. Thus, the range of the X- and Y-axis directions before and after calibration does

not change greatly. The projection value of the TCP vector in the Z direction is large, and the measurement accuracy is

relatively high. Thus, the range of the Z-axis before and after calibration is significantly reduced by 55.8%.

6. Summary and Conclusion

This paper presents a systematic method of kinematic modeling for a compound machine tool. The position and

orientation error of the workspace was measured by a laser tracker, and the machine model parameters were identified

by the L-M algorithm and chi-square fitting. The calibration results show that the volumetric positioning accuracy of the

compound machine tool can be effectively improved by the modeling and parameter identification method. The

calibration results show that the average positioning error of the machine tool is 0.03999 mm, and the average orientation

error is 6.571 × 10-4 rad, which are respectively reduced by 79.6% and 44.9% from the initial error, indicating that the

modeling and parameter identification method can effectively improve the volumetric positioning accuracy of the

compound machine tool.

The machine tool structure is different from an ordinary five-axis machine tool. Using D-H combined with H-M

method for machine tool kinematic modeling, compared with commonly used rigid body kinematics or screw theory

modeling, there is no need to measure the single geometric error of the coordinate axis, which simplifies the modeling

steps, and has fewer model parameters and clear geometric meanings of the parameters.

In the machine tool kinematic model, by setting the auxiliary frame, the direction of the machine base and TCP

frames are consistent when the joint variable is zero. Furthermore, through the link parameter data processing, the

machine base frame origin coincides with the measurement datum, so that the TCP position and orientation measured by

the laser tracker can be directly used for the parameter identification of the machine tool model.

The identification parameter of the machine tool model is the maximum likelihood estimators. The identification

errors of the machine tool model indicate that the link twist and joint angle errors imply a certain degree of squareness

and an angular error between the linear and adjacent rotary axes, and the link length parameter also has a slight error.

The positioning accuracy of the machine model after parameter identification indicates that the positioning accuracy of

the rotary axis has a significant influence on the positioning accuracy of certain coordinate directions in the workspace.

Acknowledgments

This research is supported by the Funding of Jiangsu Innovation Program for Graduate Education (No.

KYLX15_02982015), the National Science and Technology Major Project of China (No. 2014ZX04001071), and Science

and Technology Innovation Special Fund of Yixing City (No. 2020SF07).

15



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