Procedia CIRP 56 ( 2016 ) 293 – 298
2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the scientific committee of the 5th CIRP Global Web Conference Research and Innovation for Future Production doi: 10.1016/j.procir.2016.10.086
ScienceDirect
9th International Conference on Digital Enterprise Technology- DET2016 – “Intelligent Manufacturing in
the Knowledge Economy Era
A new algorithm for the identification of CNC geometric errors
Min
Wan
a,*, Yang Liu
a, Weihong Zhang
aa
School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China * Corresponding author. Tel./fax:+86-29-88495774.E-mail address: [email protected](M.Wan);[email protected](W.H.Zhang).
Abstract
This paper proposes a geometric error model for the three-axis machine tool using the homogeneous transformation matrix. An error-prediction model for the three-axis CNC machine tools is presented. This model takes the geometric errors into considera-tion, using the homogeneous coordinate transformation matrix(HTM) to analyze the geometric error model, which can change the tool’s complex position and orientation changes in space into matrix calculations to make the model simpler and more intui-tive. A simple method using the cubic polynomial functions to fit the geometric error components is proposed. Using this method, the high-order, non-linear spatial geometric problem can be transformed into algebraic equation. To large extent, the difficulty of calculation is reduced. The QC20 ball-bar is used to measure, analyze and separate all the geometric error components. A new kind of particle swarm algorithm based on the genetic algorithm is developed to solve the algebra equation.
© 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of the “9th International Conference on Digital Enterprise Technology - DET 2016.
Keywords: HTM; geometric error; QC20; particle swarm algorithm
1. Introduction
The machining accuracy of CNC machine tools has great influence on the product quality. Geometric errors and thermal errors hold a dominant position in the various factors affecting the machining precision of machine tools. Since among all the influencing factors, the geometric errors are relatively stable, insensitive to the environment and prone to error compensa-tion, they become the main research subject of the machine errors and adopting effective methods to reduce and compen-sate them, the machining precision of CNC machine tools can be improved effectively. It can be seen that it is of great sig-nificance to measure and analyze the geometric errors pre-cisely and effectively.
Up to now, there are mainly three kinds of kinematic mod-ule for machine tools, namely the homogeneous coordinate transformation matrix(HTM)[1], Denavit-Hartenberg nota-tion(D-H)[2] and the screw theory[3].
The accuracy of machine tools is affected by many error sources. These error sources include kinematic errors, thermo-mechanical errors, loads, dynamic forces and motion control
and control software caused errors. Among all the error sources, the geometric error is made up to about 30-40% of the total error, much attention has been drawn to the identifi-cation and compensation of geometric errors.
Hsu and Wang[4] summarized the compensation technique into three steps: (1)developing a kind of error model for ma-chine tools, (2)measuring errors using specific measuring de-vice and (3)conducting error compensation based on the estab-lished error model. Peng establishes the machine tool’s kine-matic model uses the HTM, proposes a universal post process-ing algorithm for multi-axis machine tool considerprocess-ing geomet-ric errors and analyzes the sensitivity of geometgeomet-ric errors[5]. Zargarbashi et Mayer propose an assemblage method to estab-lish the kinematic model[6], they estabestab-lish the coordinate sys-tem by the order of the machine component being mounted. Yang builds a general kinematics module for all possible con-figurations of five-axis machine tool based on screw theory[3] and using the module to identify the position independent geometric errors[7].
Zhu proposes an effective identification method which can identify the geometric error parameters at a number of discrete © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
measuring positions, and compensate them. Experiment show the machining accuracy of the machine tool can be improved by about 40-50% using the method proposed.
The measuring methods can be classified into two kinds roughly, ‘direct’ and ‘indirect’ method. Here ‘direct’ method means the analysis of single errors, on the contrary, ‘indirect’ method focus on the synthetic error and uses the specific mathematical model to separate the component errors. Gener-ally, indirect measurements require multi-axis motion of the machine under test. The laser interferometer is a measuring instrument on behalf on the ‘direct’ method, using the inter-ference principle to measure the displacement errors. With a variety of specific combinations of the measuring mirrors, the laser interferometer can identify the positioning errors, angular errors, straightness errors and perpendicular errors[8]. But the measuring process costs a lot of time and the calibration of the light is kind of difficulty. Ballbar can identify the synthetic error of the machine tool from the deviation between the ac-tual length and the standard length of the bar[9]. It is the typi-cal instrument of the ‘indirect’ method.
Nomenclature
CL cutter location
D-H Denavit-Hartenberg notation IKM ideal kinematic model RKM real kinematic model
HTM homogeneous transformation matrix
GA-PSO genetic algorithm based partical swarm optimiza-tion algorithm
2. Sources of geometric errors
( ) xx H direction of motion ( ) zx G ( ) yx G ( ) xx G ( ) y x H Y X ( ) zx H Z
Fig. 1. Component errors of X-axis according to ISO 230-1[10]
According to whether the geometric error is dependent to the position, it is divided into two categories as position de-pendent geometric errors and position indede-pendent geometric errors. This paper pays attention to the former. The machine tool’s errors include the position errors, the straightness errors,
the pitch errors, the yaw errors, the roll errors and the perpen-dicular errors between every motion axis according to ISO 230-1[10], as illustrated in Fig. 1(taking the X-slide as an ex-ample) and the meaning of the symbol is listed in Tab. 1. The component errors of the Y-slide and the Z-slide can be de-fined using the same method.
Table 1. Component errors of horizontal X-axis
Component errors of X-axis Notation
Position error Ɂx(x)
Horizontal straightness error motion of Y Ɂy(x)
Vertical straightness error motion of Z Ɂz(x)
Roll error motion around X ߝx(x)
Pitch error motion around Y ߝy(x)
Yaw error motion around Z ɂz(x)
Perpendicular error between X and Y axis ߛx,y
3. Kinematic module
As the base of the kinematic module, the reference and the motion coordinate systems should be carefully established in the beginning of building the kinematic module of the ma-chine tool. The coordinate systems of a mama-chine should not be defined by machine components, but by the movements of the axes[11]. A rational way of defining the coordinate system of the serial kinematic machine is as follows: [12]the reference coordinate system is fixed on the machine base, motion coor-dinate systems are established on every motion parts, i.e. the X-slide, the Y-slide and the Z-slide with respect to the refer-ence coordinate system. In this way, the whole machine tool’s kinematic loop can be divided into two parts by the cutting tool tip, which is the tool tip chain and the workpiece chain, as show in Fig. 2. X - axis Y - axis Z - axis tool tR P wR P R YT Y XT X wT Z tT R ZT w tE workpiece
tool tip chain
workpiece chain
reference coordinate system
Take the machine as a rigid body without any deformation in the process of machining. Based on the assumption of the rigid body, machine tool can be seen as a set of links con-nected in a chain by joints(revolute joints and prismatic joints), and the kinematic relation between different machine compo-nents can be represented by a 4ൈ4 HTM[1]. The form of HTM is the same as Eq. (1).
3 3 3 1 1 3 1 1 rotation translation perspective scaling u u u u
ª
º ª
º
«
» «
¬
»
¼
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¼
R D T F S (1)The whole kinematic module of the machine tool can be expressed as the multiplication of the HTM between different components. In this way, the coordinates of points in different local coordinate systems can be converted into the reference coordinate system.
According to whether the geometric errors are considered, two different kinematic models are built. The kinematic mod-el ignoring the geometric errors is taken as IKM ,the modmod-el considering geometric errors is called RKM. Take the X-axis as an example in the following discussion.
The HTM of the X-axis in IKM can be written as
1 0 0 0 1 0 0 (x) 0 0 1 0 0 0 0 1 m x
ª
º
«
»
«
»
«
»
«
»
¬
¼
Trans (2)The HTM considering the geometric errors of the X-axis can be written as(ignoring the high-order items).
1 ( ) ( ) ( ) ( ) 1 ( ) ( ) (x) ( ) ( ) 1 ( ) 0 0 0 1 z y x z x y y x z x x x x x x x x x H H G H H G H H G
ª
º
«
»
«
»
«
»
«
»
¬
¼
E (3)Using the assemblage method[6], take the Y-axis as the in-stallation reference, then there exists a perpendicular error between the X-axis and the Y-axis:
, , , cos( ) 0 sin( ) m m x y m x y m x y x x x y x x J J J ' u | ' u | (4)
Under this circumstance, the error transfer matrix can be updated as: , 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) (x) ( ) ( ) 1 ( ) 0 0 0 1 z y x z x y m x y y x z x x x x x x x x x x H H G H H G J H H G
ª
º
«
»
«
»
«
»
«
»
¬
¼
E (5)Finally, the HTM of the X-axis in RKM can be written as:
, (x) (x) (x) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) 0 0 0 1 z y m x z x y m x y y x z x x x x x x x x x x x H H G H H G J H H G u
ª
º
«
»
«
»
«
»
«
»
¬
¼
Y XT T Trans E (6)Using the same method, we can establish the RKM of the
Y-axis R
YT, Z-axis
R
ZTand the HTM between the X-axis and
the workpiece, the tool and the Z-axis respectively, they are used to establish machine tool’s whole kinematic model.
In the IKM, the CL is supposed to coincide with the de-signed point in the workpiece, but as for the existence of the geometric errors, they won’t coincide in most instances. This kind of situation affects the machining precision and the sur-face quality.
The coordinate of the designed point in the reference
coor-dinate system PwRand the coordinate of CL point in the
refer-ence coordinate system PtRcan be expressed using the product
of the HTM as follows: wR w tR t P V P V u u u u u R Y X Y X w R Z Z t T T T T T (7)
The deviation between the two points is:
w t t w E V V u u u u u tR wR R Z R Y X Z t Y X w P P T T T T T
JJG JJG
(8) Where w ( , , , 0)tE ' ' 'x y z is the deviation between the two
points in the reference coordinate system. 4. Ballbar’s principle of measurement
The Ballbar measuring system is made up of an extension bar, a mounting base and a measuring ball bar. The function of the three parts is: The extension bar is designed to have a three point supported magnetic ball socket, on which the Ballbar’s first precision ball can be mounted. The base is to fix on the worktable with the magnet at its bottom. Inside the base, there is a central bar with a three point supported mag-netic ball socket, on which the Ballbar’s second precision ball can be mounted. Inside the ball bar, there is a grating ruler to detect the displacement of the length change of the bar. The measuring signal is sent to computer via bluetooth.
In the process of the Ballbar measurement, the two preci-sion ball of Ballbar should be attached to the extenpreci-sion bar and the central bar of the magnetic base respectively, and then the kinematic chain of the three-axis CNC machine tool is closed at the center point of the second precision ball.
Due to the exist of the geometric errors, the actual length of the Ballbar can be expressed as
2 2 2 2
(r 'r) (x 'x) (y 'y) (z 'z) (9)
Rearranging Eq. (9), we can obtain the measuring principle of the Ballbar as:
1
( )
r x x y y z z
r
' ' u ' u ' u (10)
Where οr is the deviation of the Ballbar’s length, οx, οy, οz is the error component in the X, Y, Z direction.
Experiments show that the geometric error components are position-dependent, which means the value of the geometric errors in the working volume is different from place to place instead of a constant value. We need to take this phenomenon into consideration. A simple method is using the cubic poly-nomial function to fit the geometric error components.
According to[12], the straightness error is caused by the angular error around the axis perpendicular to the moving axis. They can be seen as the integration of the roll error in the cor-responding direction. Taking the X-axis as an example, the positioning error and the straightness error along the Y-axis can be expressed as˖
2 3 1 2 2 3 3 ( ) x x x x x dxx dxx dxx r r r G (11) 2 3 4 1 2 2 3 3 0 ( ) ( ) 2 3 4 x y z x x x
x x dx ezx ezx ezx
r r r
G
³
H (12)Substitute the related equation into the Eq. 10, we can ob-tain an equation with 36 coefficients related to the deviation of the Ballbar’s length οr.
The Renishaw QC20 Ballbar is used to identify the synthe-sized error οr of the three axis vertical CNC milling machine GX 710PLUS. In the measurement of the machine tool’s error, the backlash error should be firstly compensated and then use the Ballbar to measure the synthetic error of the machine tool because the backlash error will influence the result of the measurement. From the measurement, we can obtain an equa-tion with 21 component errors in total but with thousands group of data. This forms a statically indeterminate equation with no precise solution. Due to this situation, the genetic algorithm based particle swarm algorithm(GA-PSO) is devel-oped to get the optical solution.
5.The GA-PSO algorithm
The GA-PSO algorithm is a kind of particle swarm algo-rithm based on the genetic algoalgo-rithm. The mean idea of the algorithm is integrating the crossover operation of genetic algorithm into the particle swarm algorithm’s position up-dating procedure to get a better global search ability. The ex-periment data measured by Ballbar is solved by this algorithm. The flow chart of the GA-PSO algorithm is shown as Fig. 3.
The crossover operator follows the following rule:
1 1 (1 ) ( ) (1 ) , 0 1 ( ) i j i t t t j j i j t t t t i t x x x f x x x x f x D D D D D (13)
The inertia weight of the algorithm is changed from 0.5 to 0.9 dynamically so as to enhance the global search ability in the beginning of the algorithm and the local search ability in the later period of the algorithm.
End
Initialize the particles' position and speed
Search the local and global extreme value among all the particles Calculate the particles' fitness
Do the crossover operation according to the crossover rate
Update the position and speed
Recalculate the fitness and update the local and global extreme
Fit the terminal condition?
Yes
No
Fig. 3. The flow chart of the GA-PSO algorithm
min max max min min
,
w 0.5, 0.9w w w w w m (14)Using the GA-PSO algorithm, the coefficients of the poly-nomial are listed in the table(taking the X-axis as an example) and the corresponding picture shows the algorithm get con-verged in the fourth generation.
6. Results
Using the Ballbar to measure the synthetic geometric error of the machine tool in the XY-plane, YZ-plane, ZX-plane respectively. The length of the bar is taken as 100mm and the backlash error is compensated before the Ballbar measure-ment. The motion in the Z direction can be taken as zero when do the experiments in the XY-plane, the motions in the X and Y direction can be taken as zero related to the experiment in the YZ-plane and ZX-plane. The average value of three times experiments is taken as the final value of the Ballbar’s devia-tion in order to reduce the random error. The cubic polynomi-al coefficients is solved using the proposed polynomi-algorithm.
Fig. 5 .The changes of fitness function value with iterations Tab 2. The coefficients calculated using the GA-PSO algorithm Coefficient Value Coefficient Value
ezx1 0.4949 exz1 0.8136 ezx2 1.0679 exz2 0.4686 ezx3 -0.0102 exz3 0.2476 ezy1 0.1839 exy1 0.7847 ezy2 0.3242 exy2 0.1702 ezy3 0.2848 exy3 0.4163 dxx1 0.0721 dzz1 -0.0058 dxx2 -0.0714 dzz2 0.1251 dxx3 -0.0810 dzz3 1.0341 ɀx,y -0.0205 ɀy,z -0.2095 eyz1 -0.6543 eyx1 1.6785 eyz2 0.1914 eyx2 0.5844 eyz3 -0.3241 eyx3 0.9171 ɀx,z 0.0926
Fig.5 shows the changes of fitness function value with inte-ractions. It can be seen that the fitness function value arrives the minimum point in the seventh generation as to the mea-surement data of XY-plane and the minimum value almost remains the same no matter how many times we carry on the calculation. So the proposed algorithm has fast convergence
rate and high location accuracy. The data of the YZ-plane and the ZX-plane is solved with the same method. Tab 2 lists the coefficients of the cubic polynomial functions. The coeffi-cients make the prediction and compensation of the geometric errors possible.
Fig. 6 .The geometric errors of X axis with the change of position
Fig. 7 .The geometric errors of Y axis with the change of position
Fig. 8 .The geometric errors of Z axis with the change of position
0 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
the changes of fitness function value with iterations
iterations fi tness f unc ti on v a lu e -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -1.5 -1 -0.5 0 0.5 1 1.5 2
The position of X axis(unit m)
˄˅ T he ge om et ri c errors of X ax is uni t m \um Position error
Straightness error in Y direction Straightness error in Z direction Pitch error Roll error -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -1.5 -1 -0.5 0 0.5 1 1.5
The position of Y axis(unit m)
˄˅ T he geo m e tr ic err o rs of Y ax is u n it m \u m Position error
Straightness error in X direction Straightness error in Z direction Pitch error Roll error -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -1 -0.5 0 0.5 1 1.5 2
The position of Z axis(unit m)
˄˅ T he geom et ri c errors of Z ax is u n it m \u m Position error
Straightness error in X direction Straightness error in Y direction Pitch error
Based on the measuring results, the machine tool’s geome-tric errors are predicted using the coefficients shown in Tab 2. as Figs. [6-8]. The results in Figs [6-8] indicate that among all the CNC machine tool’s geometric errors, the angular devia-tions take a large account of the overall geometric error, far more than the translation deviations. Because the accumula-tion of the errors, each geometric error component reaches the maximum value in the extreme limit of motion. The position error of Z axis can’t be ignored since it is almost equal to the angular deviation.
7. Conclusion
In this paper, a new algorithm for the identification of geometric errors is proposed. The algorithm is a kind of par-ticle swarm algorithm integrating the crossover operation of the genetic algorithm into the particle swarm algorithm’s po-sition updating procedure to get a better global search ability. The synthetic error measured by the Ballbar is separated and calculated using the algorithm proposed. Based on this, the geometric errors in the machine tool’s working volume is predicted.This work will be a reference for the compensation of geometric errors.
8. Acknowledgements
This research has been supported by the National Natural Science Foundation of China under Grant No. 11272261 and the Program for New Century Excellent Talents in University under Grant No. NCET-12-0467.
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