A uniform expression model for volumetric errors of machine tools

International Journal of Machine Tools & Manufacture 100 (2016) 93–104

Contents lists available at ScienceDirect

International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

A uniform expression model for volumetric errors of machine tools Zhenya He a,n, Jianzhong Fu b, Xianmin Zhang a, Hongyao Shen b a Guangdong Provincial Key Laboratory of Precision Equipment and Manufacturing Technology, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China b The State Key Lab of Fluid Power Transmission and Control, School of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China

art ic l e i nf o

a b s t r a c t

Article history: Received 13 May 2015 Received in revised form 16 October 2015 Accepted 21 October 2015 Available online 23 October 2015

This paper proposes a new method, a Self-adaptive Mathematical Expression Model (SMEM), to describe volumetric errors of machine tools based on Non-Uniform Rational B-Spline (NURBS). The NURBS parameters of expression model were optimized by an improved Genetic Algorithm (GA). Simulation method was adopted to verify the effectiveness of the parameter optimization method of the SMEM, and the measurement experiment and machining experiment with error compensation based on the SMEM were conducted on a five-axis machining center with a titling rotary table. It was found that the SMEM can be used to uniformly express the position-dependant error parameters. Compared with the traditional methods, which adopt largely discrete database tables or polynomial method, the presented method is more concise, accurate and robust. In addition, volumetric errors of any position among the workspace of the machine tools can be quickly obtained by searching the SMEM of error parameters. And volumetric error of tool paths and the actual surface of the machining parts also can be expressed by the SMEM. The accuracy of the linear measured paths can have a great improvement of 70.63% with error compensation based on the SMEM. The accuracy of the part's predicted machining precision using the SMEM was 76.34%, and the surface profile error of the part can be improved significantly, up to 61.29%, when the SMEM was used for error compensation. Therefore, the expression model established in this study is feasible and robust, and could be used to express error parameters and volumetric errors. Moreover, it could be used to predict machining precision of part before machining and provide the basis for error compensation. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Machine tools Volumetric errors Mathematical expression model NURBS Improved GA

1. Introduction With the rapid development of modern manufacturing, more and more scholars and researchers pay close attention to the volumetric errors of machine tools. For each moving axis, there are 6 degrees of error parameters. For a 3-axis machine tool, there are 21 volumetric error parameters, including 18 position-dependent error parameters and 3 position-independent error parameters (squareness errors) [1]. For a 5-axis machine tool, there are 30 position-dependent error parameters [2]. The position-dependent error parameters for an RRTTT-type 5-axis machine tool are shown in Tables 1 and 2. Currently, volumetric error compensation has become one of the most cost-effective ways to improve the accuracy and performance of machine tools. Vibration caused surface location error n Correspondence to: School of Mechanical and Automotive Engineering, South China University of Technology, Room 311, Building No.10, Wushan Campus, Guangzhou 510640, China. E-mail address: [email protected] (Z. He).

http://dx.doi.org/10.1016/j.ijmachtools.2015.10.007 0890-6955/& 2015 Elsevier Ltd. All rights reserved.

(SLE) and self-excited vibrations also. These two can also limit machining performance, since vibration free cutting is the first step to apply volumetric error compensation [3,4]. Error modeling and error measurement play a fundamental role in volumetric error compensation. Therefore, extensive research has been performed to establish efficient error modeling and measurement methods [5,6]. Among the existing modeling techniques, a widely adopted method, and likely the most appropriate method, is to assume that a machine tool can be regarded as a multi-body system (MBS) composed of a few rigid bodies. And a homogenous transfer matrix (HTM) is used to express the relationship between moving transmissions [2,7,8]. The most common measurement tools are the laser interferometer, rotary table with motor, and ball bar. The measurement methods for translation axis errors have been made available recently, as specified in ISO230. There are 9-line method, 12-line method, the laser diagonal method, and some others [9]. For rotation axis measurement, Lei and Hsu [10] developed a 3D probe-ball device that measures the overall positioning errors of a five-axis CNC machine. Tsutsumi and Saito [11] measured and identified the eight deviations relating to two rotary

94

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

Table 1 Position-dependant error parameters caused by the 3 translation axes of a five-axis machine tool. Translation axis

X

Linear displacement errors Straightness errors Roll, pitch and yaw angular errors

Y

Z

δ xx

δ yy

δzz

δ yx δzx εxx εyx εzx

δzy δ xy

δ xz δ yz

εyy εzy εxy

εzz εxz εyz

Table 2 Position-dependant error parameters caused by the 2 rotation axes of a five-axis machine tool. Rotation axis

A

C

Displacement errors

δ xa δ ya δza

δ xc δ yc δzc

Angular errors

εαa εβa εγa

εαc εβc εγc

axes in a 5-axis machining center based on ball bar. Zargarbashi and Mayer [12] developed a 3D Capball sensor to measure eight link errors on a fly measurement. Ibaraki et al. [13] presented a machining test to identify the kinematic errors on 5-axis machine tools. He et al. [2] presented a dual optical path measurement method to identify error parameters of the rotational axis without the error modeling of machine tools. It is necessary to mention that the several dozen error parameters of a machine tool, obtained by measurement and identification techniques, are the discrete values related to the position, distributed among the machine tool workspace in a machine coordinate system (MCS). Moreover, the error parameters have two main characteristics: nonlinearity and uncertainty. It causes traditional fitted methods, such as the polynomial method, least square method, or exponential method, not feasible enough to describe them uniformly and meet the precision requirement. During error compensation of machine tools, the relationship between the error modeling, error measurement and error compensation is shown in Fig. 1. To date, the error measurement and identification for multi-axis machine tools have not yet been sufficiently investigated [2]. Many of the methods available identified error parameters of rotation axis by using the error modeling of machine tools, which involved a complex calculation process. The data transmission of error parameters between the error modeling and error compensation is significant importance for implementing error compensation. Therefore, this paper focuses on the primary connective link – the expressions of the volumetric errors, in order to quickly calculate and effectively compensate the errors. Non-Uniform Rational B-Spline (NURBS) is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces [14]. It is a highly flexible and precise method for handling both analytic and modeled shapes. ISO 10303 (known as Standard for the Exchange of Product model

Volumetric error modeling

Volumetric error measurement and parameter identification

Connecting link ?

Volumetric error compensation Improvement of the machining precisions Fig. 1. The relationship of the error modeling, error measurement and error compensation of CNC machine tools.

data, STET) used NURBS as its foundational geometry representation for free-form curves and surfaces to exchange data between different CAD/CAM systems. Currently, NURBS curve interpolation technology has been systematically studied for NC controllers. Yau et al. [15] presented a real-time look-ahead NURBS interpolator implemented on a PC-based controller in order to solve the problems of traditional linear short NC segments. Lei et al. [16] used the basis functions of setting NURBS path to approximate the positioning deviation function, and inserted a new knot at the mean position of the knot interval to improve the accuracy of compensated NURBS path for PC-based CNC controller. Yeh and Su [17] presented a real-time NURBS curve interpolator for highspeed and high-accuracy curve machining. It considered the confined contour error, acceleration/deceleration planning and the machine dynamic response simultaneously. According to the high flexibility of NURBS, and the NURBS interpolator technology for NC controllers, which is the last step for machining or compensated machining, has been studied by previous researchers, this paper will use the NURBS to establish the expression model for error parameters and volumetric error of machine tools. Hence, this paper presents a new expression method based on NURBS to express errors parameters and volumetric errors of machine tools for error compensation. To this end, Section 2 introduces the theory of Non-Uniform Rational B-Spline (NURBS) curve and surface fitting. Section 3 presents a new expression method, a Self-adaptive Mathematical Expression Model (SMEM), to describe the position-dependant error parameters and volumetric errors of machine tools based on NURBS. An optimal method based on improved Genetic Algorithm (GA) is presented to determinate the knot vectors and weights of the SMEM. Section 4 introduces the applications of the SMEM in error compensation of machine tools. To verify the feasibility and effectiveness of the SMEM, two experiments of error compensation on a five-axis machine tool is conducted in Section 5, following with experiment results analysis.

2. NURBS curve and surface fitting 2.1. NURBS curve and surface A pth-degree NURBS curve is defined as n

C (u) =

∑i = 0 wi Pi Ni, p (u) n

∑i = 0 wi Ni, p (u)

, (1)

where {wi } are the weights, {Pi } are the control points, the number of the control points is (n + 1), and {N i, p (u)} are the normalized B-spline basis functions of p degree. The i th basis function N i, p (u) defined on a knot vector U = {u0 , ui , ⋯ , un + p + 1} based on de Boor algorithm is recursively defined by

⎧ ⎧ 1 IF ui ≤ u < ui + 1 ⎪ Ni,0 (u) = ⎨ ⎩ 0 OTHERWISE ⎪ ⎪ ⎨ Ni, k (u) = u − ui Ni, k − 1 (u) + ui + k + 1 − u Ni + 1, k − 1 (u) . ui + k − ui ui + k + 1 − ui + 1 ⎪ ⎪ 0 ⎪ S.T. =0 ⎩ 0

(2)

Namely N i, p (u) is a step function, equal to zero everywhere except on the haft-open interval u = [ui , ui + 1), the weighted control points determine the shape of the curve, the number of the knot vectors is (n + p + 2), (p + 1) is defined as the number of nearby control points that influence any given point on the curve. For an open curve, generally the prescribed number of multiple knots on the initial and final points is (p + 1), in order to make sure

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

the initial and final control points correspond, respectively, to the initial and final data points,

U = {  a, ... , a, up + 1, ... , un ,  b, ... , b } . p+1

(3)

p+1

In this paper, we assume that a = 0, b = 1. A p × qth-degree NURBS surface is defined as m

m

n

∑i = 0 ∑ j = 0 wi, j Ni, p (u) Nj, q (v)

3. SMEM based on NURBS

, (4)

where {wi, j } are the weights, {Pi, j } are the control points, {N i, p (u)} and {N j, q (v )}are the normalized B-spline basis functions defined on the non-periodic knot vectors U = {0, 0, ⋯ , up + 1, ⋯ , um , 1, ⋯ , 1} and V = {0, 0, ⋯ , uq + 1, ⋯ , un , 1, ⋯ , 1}. Here, p and q are the degree in the u- and v -direction. 2.2. NURBS fitting based on least square We assume that a set of discrete data to be fitted is {Q k }, (k = 1, 2, ⋯ , m), the parameter of the Q k on the pth NURBS curve is u¯ k , and the number of control points Pi is (n + 1). m

Q k (u) ≈ C (u¯ k ) =

∑ Pi Ri, p (u¯ k )i ,

(k = 0, 1, 2, ⋯, m).

(5)

i=0

The matrix form is expressed as follows:

⎡ R 0, p (u¯ 0 ) ⎢ ⎢ R 0, p (u¯1) ⎢ ⋮ ⎢ R ⎣ 0, p (u¯ m )

⋯ Rn, p (u¯ 0 ) ⎤ ⎡ P0 ⎤ ⎡ Q 0 ⎤ ⎥⎢ ⎥ ⎢ ⎥ ⋯ Rn, p (u¯1) ⎥ ⎢ P1⎥ ⎢ Q 1 ⎥ = ⎥ ⎢ ⋮ ⎥ ⎢ ⋮ ⎥, ⋱ ⋮ ⎥⎢ ⎥ ⎢ ⎥ … Rn, p (u¯ m )⎦ ⎣ Pn ⎦ ⎣ Q m ⎦

(6)

where the rational functions are defined as

Ri, p (u) =

wi Ni, p (u) n

∑i = 0 wi Ni, p (u)

. (7)

Eq. (6) can be expressed as RP = Q , when m > n, the number of vector equations is greater than the unknown number (n + 1). These are over-determined equations. Least Square Approximation is adopted to solve the equations. Its objective function can be expressed by

min

‖J‖p = ‖Q − RP‖p ,

(p ≥ 1).

(8)

The Gaussian orthogonal equation is expressed by T

R RP = RT Q.

(9)

Hence, the solution for Eq. (9) can be expressed as follows:

P = (RT R)−1RT Q.

(10)

The coefficient matrix can be calculated by Eq. (10), where namely the control points are obtained. For the NURBS surface fitting, let us assume that the discrete data to be fitted are (m + 1) × (n + 1) array points {Q k, l }, (k = 1, 2, ⋯ , m ; l = 1, 2, ⋯ , n), and the two vector directions are u and v , respectively. The data points can be expressed by m

Q k, l ≈ S (u¯ k , v¯l ) =

n

∑ ∑ Pi, j Ri, p (u¯ k ) Rj, q (v¯l )

n

⎡

m

⎤

∑ ⎢ ∑ Pi, j Ri, p (u¯ k ) ⎥ Rj, q (v¯l ) j=0

⎢⎣ i = 0

⎥⎦

n

=

∑ Cj (u¯ k ) Rj, q (v¯l ). j=0

To suitable for the characteristics of nonlinearity and uncertainty, a Self-adaptive Mathematical Expression Model (SMEM) based on NURBS is presented to describe the error parameters of machine tools. 3.1. Parameters of SMEM According to Section 2.2, it can be seen that the most important step of NURBS curve (surface) fitting is the reverse calculation of control points. This is necessary to determinate the parameters, including the parameterization method of the discrete data, the number of control points, knot vectors, weights and degree. These parameters directly affect the precision of the fitting curve or surface. The cubic NURBS curve is most widely used, and when the degree is greater than 3 the calculation is more complicated. Hence, throughout this paper, we assume that the degree p = 3. There are four kinds of the common parameterization methods [18].

⎧ u¯ 0 = 0, u¯ m = 1 ⎪ ⎪ Q i − Q i−1 k . ⎨¯ ⎪ ui = u¯ i − 1 + λ i⋅ m k ⎪ ∑ j=0 Q j − Q j−1 ⎩

(11)

(12)

(1) when λ i = 1, k = 0, it is a uniform method; (2) when λ i = 1, k = 1, it is a chord length method; (3) when λ i = 1, k = 1/2, it is a centripetal method; and (4) when λ i ≠ 1, k = 1, it is a correction centripetal method. Empirically, the centripetal parameterized method is the most typical parameterization method. Therefore, this paper selects the centripetal method as the parameterized method. The precision of the fitting curve or surface can be improved by increasing the number of the control points. However, with the increasing number of control points, the amount of calculations sharply increases. Hence, the number of the control points should be as small as possible, while the precision of curve or surface meets the requirements. Many researchers have been carried out to study the knot division method. One of the common methods is the uniform knots method (UKM), as follows:

⎧ U = {0, 0, ⋯, up + 1, ⋯, un , 1, ⋯, 1} ⎪ ⎪ ⎨ j ⎪ uj + p = , (j = 1, 2, ⋯, n − p) ⎪ n−p+1 ⎩

(13)

Piegl and Tiller [14] divided the parameter space (m + 1) into (n + 1) segments,

⎧ U = {0, 0, ⋯, up + 1, ⋯, un , 1, ⋯, 1} ⎪ ⎨ i = int (j⋅γratio ), αi = j⋅γratio − i , ⎪ ⎩ uj + p = (1 − αi ) u¯ i − 1 + αi u¯ i , (j = 1, 2, ⋯, n − p)

i=0 j=0

=

Similarly, the NURBS surface fitting process can be seen as NURBS curve fitting twice. First, by reversely calculating in the v -direction, the results are control points {Cj (u¯ k )}. Then Cj (u¯ k ) are considered the discrete points for the next reserve calculation in the u-direction. Finally, the control points {Pi, j } of the NURBS surface are obtained.

n

∑i = 0 ∑ j = 0 wi, j Pi, j Ni, p (u) Nj, q (v)

S (u, v) =

95

(14)

where, the scale factor γratio = (m + 1) /(n − p + 1), the method records as equal partition method (EPM). Yeh and Su [17] presented the divided method of the knot vector as

96

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

⎧ U = {0, 0, ⋯, up + 1, ⋯, un , 1, ⋯, 1} ⎪ ⎪ m− n+ p + j − 1 ⎨ . 1 ⎪ uj + p = ∑ u¯ i , (j = 1, ⋯, n − p) ⎪ m n p − + ⎩ i=j

Assume that the population size is N .

(15)

The method records as Local average method (LAM) in this paper. However, those above methods are hard to reasonably divide the knot vector when the data are nonlinear and uncertain. Hence, a new knot divided method will be discussed. The weight wi just affects the curve shape in the interval [ui , ui + p + 1] ⊂ [up , un + 1] when the other weights and the knot vectors are unchanged. Hence, adjusting the weights, in moderation, can be used to change the curve shape for improvement of fitting precision. Therefore, to improve the precision of the SMEM, an optimal method based on improved Genetic Algorithm (GA) is presented to determinate the knot vectors and weights of the NURBS. 3.2. Parameters optimization based on GA 3.2.1. Improved GA Genetic Algorithm (GA) was first presented by Prof. J.H. Holland [19]. It is a search heuristic that mimics the process of natural selection, placing it in the category of evolutionary algorithms [20]. The evolution process of GA is a robust and efficient process which solves for optimization and search problems. Therefore, GA was adopted to optimize the parameters of NURBS for the SMEM, including the knot vectors and weights. There are three steps in the optimization process based on GA, including selection, crossover and individual mutation. In order to avoid population degradation and ensure that each iteration result is evolutionarily populated, standard GA was improved to suit our process. The principle of the improved GA is that it saves all optimal solutions selected among individual parents and children after crossover (mutation) as the next generation, to ensure that each iteration result is evolutionary, rather than saving the individual children directly. 3.2.2. Knot vectors optimization The objective function f (U ) of NURBS curve fitting is expressed by

⎧ ¯i) − Q i ) ⎪ min ( max C (u 0≤ i ≤ m ⎨ , ⎪ ⎩ S.T. error ≤ ε

(2) Fitness function. Fitness function should satisfy the following conditions: normalization (single-valued, continuous, monotonic and nonnegative,) rationality and generality. The fitness function of the knot optimization is expressed as

Fit (f (U )) =

where error represents the fitting error, and ε represents the given allowance error. Assuming that the parameter domain of NURBS is equally divided into M segments, and (n − p) knots are chosen among the (M − 1) alternative knots, this becomes a combinational optimin−p zation problem CM − 1. The fitted precision is directly affected by the divided parameter M . When M → ∞, the solution tends toward global optimization. This requires greater computing time, however, to complete the search process. (1) Coding operator. Binary encoding is adopted to code the population, as shown in Fig. 2. We put knots on the points where a gene equals 1, and not on the points where a gene equals 0.

(17)

where c is the conservative estimates of the objective function, and λ is the penalty factor. The greater the value of fitness function is, the better the chromosome is. (3) Selection operator. A roulette wheel selection is used to select which object reproduces. The reproduced number of the i th population can be calculated by

Ni = N

Fit (fi ) N

∑i Fit (fi )

. (18)

(4) Crossover operator. Crossover operation is a primary method of generating new individuals, as it effectively determines the capability of the global search of GA. These operations include one-point crossover, double-point crossover, uniform crossover, arithmetic crossover, and others. The double-point crossover method is adopted to recombine the individuals in this paper, where the crossover probability Pc equals 0.7. (5) Mutation operator. For binary coding, the mutation is that the gene randomly selected changes from 1 to 0, or from 0 to 1. The mutation probability Pm is about 0.001–0.01. (6) Termination condition. This paper adopts two termination conditions, the first being that when the relative error of the fitness value, (relative to the average value of the N neighborhoods fitness values,) was not greater than η , the iteration would terminate. 1 N

Fit (fi ) − 1 N

i

i

∑i − N Fit (f j )

∑i − N Fit (f j )

× 100% ≤ η. (19)

The other condition is that when the number of iterations was greater than that of the given maximum iteration Ncount , the iteration would terminate.

count > Ncount . (16)

1 , 1 + c + λ⋅f (U )

(20)

To verify the effectiveness of the knot optimization method based on GA, a compared simulation of a set of data was carried out. Fig. 3 shows the cubic NURBS fitted curves based on the different knot division methods, including the UKM, EPM, LEM, and GA method. The maximum error and average error of the fitted curve based on GA division method is 1.733 and 0.727, respectively. Comparing the respective precision values of four fitted curves (Table 3), it can be seen that the precision of the fitted curve based on improved GA method is much more favorable than the others. The convergence of the optimal solution is shown in Fig. 4. When the number of control points is 10, the divided parameter M = 13, and the solution was converged after 9 iterations in a computing time of t = 4.745 s. The knot vectors of the optimal solution are

{0 0 0 0 0.3077 0.4615 0.5385 0.6154 0.6923 0.9231 1 1 1 1} .

Fig. 2. Coding strategy of the knot vectors.

Another comparison simulation was carried out between traditional combination global search and the improved GA search method. The results are shown in Table 4. The maximum errors of fitting curves by improved GA are consistent with those of the combination global search method, with a maximum difference of

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

0

-10

0

50 100 150 Axis position (mm)

discrete points fitted curve knots

Error ( m)

20 10

0

-10

0

50 100 150 Axis position (mm)

200

10

0

-10

200

0

50 100 150 Axis position (mm)

200

discrete points fitted curve knots

20

Error ( m)

Error ( m)

10

discrete points fitted curve knots

20

Error ( m)

discrete points fitted curve knots

20

97

10

0

-10

0

50 100 150 Axis position (mm)

200

Fig. 3. The cubic NURBS fitting curves based on different knot division methods (p = 3, n = 9) : (a) UKM, (b) EPM, (c) LEM, and (d) GA. Table 3 The precisions of the fitting curve based on the different knot division methods (p = 3, n = 9) . Division method

UKM

EPM

LPM

GA

Emax Eaverage

5.025 2.034

4.814 1.732

6.918 2.418

1.713 0.727

(1) Coding and decoding. When population coding, the realnumber encoding is adopted. It is different from the optimization of knot vectors using binary encoding. Assuming that the range of the weight is [T1, T2 ] and the length of the binary string is k , the resolution δ can be expressed by

0.4

0.35

Fittness

3.2.3. Weights optimization Similarly to the knot vectors optimization based on improved GA, the optimization process of NURBS weights includes coding, determination fitness function, selection, crossover and mutation. All the principles are similar to the knots optimization, with the exception of the coding process.

δ=

2k − 1

.

(21)

The corresponding relations between the binary strings and weights are shown as follows:

0.3

0000⋯000 = 0 0000⋯001 = 1 0000⋯010 = 2 ⋮ ⋮

maximum average

0.25

0

T2 − T1

20

40

60

80

100

→ T1 → T1 + δ → T1 + 2δ ⋮

1111⋯111 = 2k − 1

Generation Fig. 4. The convergence of the knot optimal solution (p = 3, n = 9) .

T2

Assuming a binary string is bk bk − 1bk − 2 ⋯b2 b1, the decoding formula can be express as k

0.057. With the increase of the divided parameter M , the computing time of the traditional combination global increased sharply, while the computing time of the improved GA search method changed just slightly. Therefore, we conclude that the presented method using improved GA to optimize the knot vectors is reliable and more efficient.

→

wi = T1 + ( ∑ bi ⋅2i − 1)⋅ i=1

T2 − T1 2k − 1

. (22)

(2) Chromosome. Assuming that the number of the control points is (n + 1)and the weights are {w0, w1, ⋯ , wn }, the length of the chromosome L = (n + 1) × k . The binary string of a chromosome is shown in Fig. 5.

Table 4 The comparisons of the results between the combination global search and the modified GA search method (p = 3, n = 9) . Parameter M

13

Evaluation Combination search Improved GA search Difference

Emax 1.656 1.713 0.057

20 t/s 0.891 4.754 3.863

Emax 1.460 1.489 0.029

25 t/s 53.438 4.575 48.863

Emax 1.391 1.444 0.053

33 t/s 587.312 6.521 580.791

Emax 1.242 1.264 0.022

t/s 3495.367 4.971 3490.396

98

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

...

...

...

...

Fig. 5. Binary string of a chromosome.

0.45

Fittness

0.4 0.35 0.3 maximun average

0.25 0.2

0

20

40 60 Generation

80

100

Fig. 6. The convergence of the weight optimizing process.

To verify the effectiveness of the weights optimization method based on GA, a computer simulation of weight optimization of the SMEM for a set of data was carried out. When knot vector U = {0 0 0 0 0.20 0.35 0.40 0.65 0.70 0.80 1 1 1 1}, p = 3, n = 9, the convergence of the weight optimizing process is shown in Fig. 6. The fitted curves, based on weights of the 1st generation and the 100th generation, are shown in Fig. 7. Here, it is clear that the precision of the fitted curve based on the 100th generation is greatly improved. The optimizing process of weights based on the improved GA is quite effective. 3.3. Algorithm procedure of SMEM The algorithm procedure flow chart of the SMEM is shown in Fig. 8. First, the parameters and the initial values are input, including the discrete data of the machine tool error parameters (obtained by error measurement), precision requirement ε , degree, and the initial value of control points number. In order to reduce computational load, only the lowest amount of control points which satisfy the given precision are used – so as an initial value, we set for a small number of control points. And then the discrete error parameters are parameterized by the centripetal method, and knot vectors are optimized based on improved GA. The control points are reversed based on least square method and the fitted curve is obtained, the corresponding points of the discrete error parameters on the fitted curve are sought for evaluation of fitted

discrete points curve: gen=1 curve: gen=100

Error ( m)

20 10 0 -10 0

50 100 150 Axis position (mm)

Fig. 7. The fitted curves based on the different weights.

200

Fig. 8. The algorithm procedure flow chart of the SMEM for error parameters of machine tools.

precision. If the accuracy of the fitted curve meets the requirement, move to the next section – weights optimization based on improve GA, or go to the process of the knot vector optimization. Optimization and judge process of weights are similar to that of knot vectors. Finally, the fitted results are evaluated, if the results meet the precision requirement, output the fitted curve and the parameters of the SMEM, and terminate the program, or increase the control number and recalculate knot vectors and the weights. When executing a volumetric error compensation for machine tools, the volumetric errors of certain positions among the machine tool workspace need to be obtained. To satisfy this, we need to obtain the volumetric errors based on curve or surface expressions of error parameters. In order to improve seeking speed, Binary Search Algorithm (BSA) [21] is adopted to search the SMEM for obtaining the corresponding errors parameters. And then the volumetric errors of these certain volumetric positions are calculated based on kinematic error modeling of machine tools and corresponding error parameters. To explain this search method, let us take the X-axis displacement error as an example. Assuming that a point on the X-axis is x0 , it is necessary to obtain the corresponding linear displacement error δxx . The flow chart for searching the error parameters from the expression of SMEM based on BSA for two dimension curve is shown in Fig. 9. It should be noted that the search method is suitable for multi-dimensional curves and surfaces.

4. Applications of the SMEM The flow chart of error compensation for machine tools based on SMEM is shown in Fig. 10. The SMEM plays an important part in error compensation. It is good for data transmission and rapid calculation by computer, and probably achievement of intelligence. The volumetric errors for error compensation during machining are not only related to the position of the workpiece and the cutting tool, but also to the orientation of the cutting tool. Therefore, volumetric errors should be calculated according to the ideal points and paths in the workspace of machine tools, which is supplied by the numerical control system for machining. According to the theory of NURBS in Section 2, it should be noted that the number of control points must be greater than degree, and the number of the discrete data for establishing the SMEM must be greater than the number of control points. In order to meet the requirement and ensure certain fitting precision, the

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

99

lnear dispaclement error ( m)

0 discrete points SMEM polynomial method

-2 -4 -6 -8 -10 -12

0

50

100 X-axis position (mm)

150

200

Fig. 11. The expression of X-axis displacement error parameters based on SMEM.

4.2. Expressions for volumetric errors of machine tools Fig. 9. Flow chart for the seeking the error parameters from the expression of SMEM based on BSA.

Besides the application in expression of the error parameters, the SMEM can be used to express volumetric errors of machine tools, such as to express the plane errors, the error of the volumetric paths, and the actual paths with the volumetric errors. (1) Coupling error expression of the machine tools Let us take the expression of XY-plane errors as an example. First, the XY-plane of the machine tool is meshed, and according the displacement of the nodes, the corresponding error parameters of those nodes are searched from the SMEM of error parameters. And then the volumetric errors of the nodes are calculated based on kinematic error modeling of machine tools and corresponding error parameters. Finally, the SMEM is used to express the volumetric errors of nodes along the Z-direction, as shown in Fig. 12. (2) Error expression of the volumetric paths

Fig. 10. Flow chart of error compensation for machine tools based on SMEM.

minimum value of the discrete data was set at 8 in this study.

When the volumetric paths of machine tools are known, the volumetric errors of the paths can be obtained by searching the error parameters expressions and calculating based on kinematic error modeling, and the actual path can be predicted by adding the volumetric error vector. For example, the volumetric errors on the XZ-plane diagonal path along the Y-direction were calculated and expressed by the SMEM, as shown in Fig. 13(a). Similarly, the actual paths of the volumetric paths also can be obtained by adding the volumetric error vectors. The actual path of a spiral path, considering its volumetric errors, is shown in Fig. 13(b). -3

4.1. Expressions for error parameters of machine tools

x 10

0

5

-2 0

error (mm)

For a 3-axis machine tool, there are 18 error parameters related to its positions. The SMEM can be used to express the relationship between error parameters and positions, which are obtained by measuring and identifying technology. For this study, the error parameters of machine tools comes form Ref. [2]. The expression of X-axis displacement error parameters based on SMEM is shown in Fig. 11. The expression adopts 12 control points to fit the 40 discrete points. Comparing with large database table and polynomial method, the SMEM is more concise, generally favorable and accurate to record error parameters. For a 5-axis machine tool, its position-dependant error parameters can also be expressed by the SMEM, respectively.

x 10 2

-3

-4 -6

-5

-8

-10

-10 0

-15 0

50

100

100 150

Y-axis position(mm)

200 X-axis position (mm)

Fig. 12. The expression of XY-plane errors based on SMEM (z = 0) .

-12

100

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

Fig. 13. The error expressions of the volumetric paths based on SMEM: (a) XZ-plane diagonal path, (b) spiral path.

8

40 20

6

0 4

-20 -50

50

50

2

X-axis position (mm)

x 10

0

0

0

Z-axis position (mm)

40 20 0 -20 -50

50

-50

Y-axis position(mm)

-3

0

X-axis position (mm)

50

-50

Y-axis position(mm)

Error ( m) -1

5

-2 0

-3 -4

-5

-5 -6

-50

50

0

-7

0 -50 50 X-axis position (mm) Y-axis position(mm)

Z-axis position (mm)

Z-axis position (mm)

10

Z-axis position (mm)

Z-axis (mm)

ideal surface actual surface

30 20 10 0 -10 -50

50 0

0 50

X-axis position (mm)

-50

Y-axis position(mm)

Fig. 14. Application of the SMEM in prediction of machining precision and error compensation for a saddle-style bicycle seat: (a) the surface of the saddle model, (b) the volumetric error vectors of the saddle surface, (c) the expression of volumetric error along Z-axis direction based on the SMEM, and (d) the expressions of the actual surface of the saddle based on the SMEM.

NNP

PNP

PPP Z

Laser beam

NPP

Y

X Fig. 15. Laser measurement experiment: (a) Laser measurement paths, (b) the measurement along diagonal line of the machine tool workspace.

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

4

x 10

101

-3

0.02

2

0.01

Errors (mm)

Errors (mm)

0 -2 -4 xx

-6

0

PPP

-0.01

PPP comp. NPP

-0.02

xx comp.

NPP comp.

yy

-8

NNP

yy comp. -10

NNP comp.

-0.03

zz

PNP

zz comp. -12

0

PNP comp.

50

100

150

200

Position (mm)

-0.04

0

50

100

150

200

250

300

350

Position (mm)

Fig. 16. The results of measurement experiments with and without error compensation: (a) lines alone each translational axis, (b) body diagonal lines.

Table 5 Comparison of the measured results of laser measurement experiment (μm). Maximum error Without comp. With comp. Comp. rate (%)

δ xx 10.3 1.7 83.50

δ yy 8.0 2.2 72.50

δzz 10.7 1.9 82.24

PPP 12.8 6.2 51.56

NPP 25.3 5.7 77.47

NNP

PNP

19.7 5.5 72.08

6.9 3.1 55.07

which are amplified by a factor of 2000. The volumetric errors expression of the saddle based on the SMEM is shown in Fig. 14(c). By adding the volumetric error to ideal tool paths, the actual machining surface can be rebuilt based on the SMEM, as shown in Fig. 14(d). By comparing the ideal and actual surfaces, the surface profile error of the part can be predicted. This demonstrates the method's ability to predict the necessary precisions of machining parts. In addition, it can be used to create error compensation by reversing the error vectors as the compensation values, which will be introduced in the next section. Therefore, the SMEM could be used to express the positiondependent error parameters. It is good for searching error parameters so that the volumetric errors of any position among the machine tool workspace can be quickly calculated. Moreover, the SMEM could be used to express the volumetric errors of machine tools and the actual paths or surfaces considered the volumetric errors, which could be used to compensate volumetric errors and predicted the machining precision of parts before machining.

5. Experiments and results

Fig. 17. The saddle part machining on a five-axis machining center.

4.3. Prediction of machining precision and error compensation During machining, when the blank is fixed on the machine table and the workpiece coordinates are set, the volumetric errors of the parts can be calculated by searching the expressions of error parameters based on the designed tools' paths, and then actual machining surface can be rebuilt according to the SMEM. By comparing the ideal and actual surfaces, the surface profile error of the part can be predicted by comparing the ideal and actual surfaces before machining. Moreover, it can be used to provide the basis for volumetric error compensation. Let us take the machining of a saddle-style bicycle seat as an example (Fig. 14). The surface of the saddle-style bicycle seat is shown in Fig. 14(a). The volumetric errors, corresponding to discrete points of the tool paths, were calculated by searching the error parameter expressions (Fig. 14(b)). The blue lines with arrows represent the volumetric error directions and the scales,

In order to verify the feasibility and reliability of the SMEM, measurement and machining experiments with error compensation for a 5-axis machining center with a titling rotary table were conducted. The discrete error parameters of the 5-axis machining center, which were obtained by laser measurement (see Ref. [2]), were described by the SMEM. The X-axis displacement errors were expressed as shown in Fig. 11. 5.1. Measurement experiment The measurement experiment was conducted as shown in Fig. 15. There were 7 laser measurement paths, including 3 linear lines along each translational axis and 4 body diagonal lines, as shown in Fig. 15(a). The measurement without the error compensation was carried out firstly, and then the measurement with the error compensation based on the SMEM was conducted. The measurement results were shown in Fig. 16. The measured results are compared in Table 5. It can be seen that the accuracy of the measured paths with the error compensation based on the SMEM is much better that without compensation. The compensation rate on each path has exceeded 51.56%, the average compensation rate is more that 70.63%, and the maximum is 83.50%. These clearly show that our SMEM is effective

102

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

Fig. 18. Flow chart of error compensation experiment for saddle part machining.

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

103

Table 6 Measurement data of the parts (mm). N

Part 1 (without comp.)

Ideal

Actual (part1) Actual (part2) Error (part1)

Error (part2)

1

X Y Z I J K

41.1605 9.9504 12.5899 0.4019 0.4552 0.7945

41.1582 9.9559 12.5952 0.4019 0.4552 0.7945

41.1573 9.9471 12.5878 0.4019 0.4552 0.7945

0.0023 0.0055 0.0053 0.0000 0.0000 0.0000

0.0032 -0.0033 0.0021 0.0000 0.0000 0.0000

2

X Y Z I J K

41.1641 7.7431 13.6831 0.3698 0.3622 0.8556

41.1601 7.7435 13.6868 -0.3698 0.3622 0.8556

41.1610 7.7455 13.6845 0.3698 0.3622 0.8556

0.0040 0.0055 0.0036 0.0000 0.0000 0.0000

0.0031 0.0024 0.0014 0.0000 0.0000 0.0000

3

X Y Z I J K

41.1621 5.5297 14.4729 0.3410 0.2634 0.9024

41.1577 5.5387 14.4820 -0.3410 0.2634 0.9024

41.1614 5.5327 14.4752 0.3410 0.2634 0.9024

0.0045 0.0090 0.0091 0.0000 0.0000 0.0000

0.0007 0.0030 0.0023 0.0000 0.0000 0.0000

372 X Y Z I J K

41.1602 3.3210 12.9081 0.2628 0.5907 0.7629

41.1677 3.3302 12.9152 0.2628 0.5907 0.7629

41.1636 3.3233 12.9112 0.2628 0.5907 0.7629

0.0076 0.0092 0.0071 0.0000 0.0000 0.0000

0.0034 0.0023 0.0032 0.0000 0.0000 0.0000

Part 2 (with comp.)

Fig. 19. The parts machined without and with error compensation.

…….

Table 7 Comparisons of the surface profile errors. Items

Prediction

Without comp.

With comp.

Surface profile errors (μm) Proportion (%)

7.1 76.34

9.3 –

3.6 61.29

Fig. 20. Part testing using the CMM.

in expressing the error parameters, and thus can be readily used to improve machining precision. 5.2. Machining experiment The machining experiment consisted of five parts: error modeling, error measurement and parameter identification, error compensation, part machining and part measurement. Firstly, part 1 of saddle without error compensation was machined, as shown in Fig. 17 Secondly, part 2 with error compensation was machined. The flow chart is shown in Fig. 18, which includes 8 panels. (1) The SMEM was established based on NURBS by using above optimization method (in panel 4 of Fig. 18). And the discrete error parameters of the machining center, which were obtained by laser measurement and identification in panel 2, were described by the SMEM in panel 5. (2) Before machining Part 2, the blank ( 100 × 80 × 40 mm) was fixed on the machine table and the workpiece coordinates were set to the same position as Part 1, as shown in panel 1. (3) Error parameters corresponding to discrete points of tool paths were obtained by searching the SMEM of error parameters and then volumetric errors of the part were calculated based on kinematic error modeling of machine tool. The error modeling was shown in panel 3.

(4) Volumetric errors of the saddle were expressed based on the SMEM, as shown in panel 6. By adding the volumetric errors to ideal tool paths, the actual machining surface was rebuilt based on the SMEM before machining, as shown in panel 7. And the surface profile error of the part was predicted by comparing the ideal and actual surfaces. (5) New tool paths were rebuilt according to the superposition of the discrete points of ideal paths and the opposite numbers of the corresponding volumetric errors. Part 2 was machined based on the new tool paths which reflected the volumetric errors compensation, as shown in panel 8. The parts with and without error compensation are shown in Fig. 19. Finally, the parts were measured using a gantry coordinate measuring machine (CMM) with Renishaw PH20 probing systems, as shown in Fig. 20. The 372 sampling points on the saddle surface of the parts were selected and measured. During measurement, the normal vectors of the saddle surface were treated as the feed directions for probing. The partial measurement data are shown in Table 6. The surface profile error of the part was considered to be its evaluation criteria, which is the minimum distance between the ideal surface and enveloping surface of actual surface by offsetting the ideal surface. The comparisons between prediction result and measurement results are shown in Table 7. The machining precision of the part by prediction before machining was 7.1 μm, and the actual precision of the part without compensation was 9.3 μm.

104

Z. He et al. / International Journal of Machine Tools & Manufacture 100 (2016) 93–104

The prediction accuracy of the surface profile was as high as 76.34%. After compensation, the surface profile error of the parts was reduced from 9.3 μm to 3.6 μm, and the precision of the part was improved by 61.29%. The SMEM is sensible and feasibility. It can used to predict the precision of the parts before machining, and provide the basis for error compensation in order to improve the precision of machining.

6. Conclusions During error compensation of machine tools, the several dozen error parameters obtained by measurement are discrete values related to the position. However, traditional methods, such as the largely database table or polynomial method, are not concise and robust enough to express the measurement results. It is not good for calculating the volumetric errors and implementing error compensation. Therefore, this paper has presented a new expression method called SMEM based on NURBS to describe error parameters of machine tools. And the improved GA was adopted to optimize the knot vectors and weights of the SMEM. The verification experiments were conducted on a five-axis machining center. The experimental results bring about the following major conclusions: (1) Compared with the traditional methods, it is more concise, universal and precise to express the several dozen error parameters related to the position using the SMEM, although the error parameters are nonlinear and uncertain. (2) The volumetric errors of any position among the machine tool's workspace not only can be quickly obtained by searching the SMEM of error parameters, but also can be expressed by the SMEM. (3) Considered the volumetric errors, the machining paths and surfaces can be rebuilt by the SMEM, and the machining precision of the parts can be predicted before machining. (4) The results have shown that the accuracy of linear measured paths by laser interferometer with compensation based on the SMEM was improved by 70.63%. The prediction value of the part's surface profile was as high as 76.34%, and after compensation, the surface profile error of the part improved by 61.29%. It is therefore reasonable to conclude that SMEM is feasible, robust and efficient. It can be used to express the error parameters and volumetric errors of machine tools, and also can be used to predict the machining precisions of parts and provide the basis for error compensation to improve machining precision.

Acknowledgments This work was financially supported by the Fundamental Research Funds for the Central Universities (No. 2015ZM043), the

National Natural Science Foundation of China (Nos. 91223201 and 51175461) and the Natural Science Foundation of Guangdong Province (No. S2013030013355).

References [1] H. Shen, J. Fu, Y. He, X. Yao, On-line asynchronous compensation methods for static/quasi-static error implemented on CNC machine tools, Int. J. Mach. Tools Manuf. 60 (2012) 14–26. [2] Z. He, J. Fu, L. Zhang, X. Yao, A new error measurement method to identify all six error parameters of a rotational axis of a machine tool, Int. J. Mach. Tools Manuf. 88 (2015) 1–8. [3] B.P. Mann, K.A. Young, T.L. Schmitz, D.N. Dilley, Simultaneous stability and surface location error predictions in milling, J. Manuf. Sci. Eng. 127 (2005) 446–453. [4] G. Stepan, J. Munoa, T. Insperger, M. Surico, D. Bachrathy, Z. Dombovari, Cylindrical milling tools: comparative real case study for process stability, CIRP Ann. Manuf. Technol. 63 (2014) 385–388. [5] S. Mekid, T. Ogedengbe, A review of machine tool accuracy enhancement through error compensation in serial and parallel kinematic machines, Int. J. Precis. Technol. 1 (2010) 251–286. [6] H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, F. Delbressine, Geometric error measurement and compensation of machines – an update, CIRP Ann. Manuf. Technol. 57 (2008) 660–675. [7] S. Zhu, G. Ding, S. Qin, J. Lei, L. Zhuang, K. Yan, Integrated geometric error modeling, identification and compensation of CNC machine tools, Int. J. Mach. Tools Manuf. 52 (2012) 24–29. [8] M.S. Uddin, S. Ibaraki, A. Matsubara, T. Matsushita, Prediction and compensation of machining geometric errors of five-axis machining centers with kinematic errors, Precis. Eng. 33 (2009) 194–201. [9] C.B. Bui, J. Hwang, C.-H. Lee, C.-H. Park, Three-face step-diagonal measurement method for the estimation of volumetric positioning errors in a 3D workspace, Int. J. Mach. Tools Manuf. 60 (2012) 40–43. [10] W. Lei, Y. Hsu, Error measurement of five-axis CNC machines with 3D probe– ball, J. Mater. Process. Technol. 139 (2003) 127–133. [11] M. Tsutsumi, A. Saito, Identification of angular and positional deviations inherent to 5-axis machining centers with a tilting-rotary table by simultaneous four-axis control movements, Int. J. Mach. Tools Manuf. 44 (2004) 1333–1342. [12] S. Zargarbashi, J. Mayer, Single setup estimation of a five-axis machine tool eight link errors by programmed end point constraint and on the fly measurement with Capball sensor, Int. J. Mach. Tools Manuf. 49 (2009) 759–766. [13] S. Ibaraki, M. Sawada, A. Matsubara, T. Matsushita, Machining tests to identify kinematic errors on five-axis machine tools, Precis. Eng. 34 (2010) 387–398. [14] L. Piegl, W. Tiller, The NURBS Book, Springer, Berlin Heidelberg, 1997. [15] H.-T. Yau, J.-B. Wang, C.-Y. Hsu, C.-H. Yeh, PC-based controller with real-time look-ahead NURBS interpolator, Computer-Aided Des. Appl. 4 (2007) 331–340. [16] W. Lei, M. Sung, NURBS-based fast geometric error compensation for CNC machine tools, Int. J. Mach. Tools Manuf. 48 (2008) 307–319. [17] S.-S. Yeh, H.-C. Su, Implementation of online NURBS curve fitting process on CNC machines, Int. J. Adv. Manuf. Technol. 40 (2009) 531–540. [18] H. Jung, K. Kim, A new parameterisation method for NURBS surface interpolation, Int. J. Adv. Manuf. Technol. 16 (2000) 784–790. [19] J.H. Holland, Adaption in Natural and Artificial Systems, The University of Michigan Press, Ann Arbor, MI, 1975. [20] J. Andre, P. Siarry, T. Dognon, An improvement of the standard genetic algorithm fighting premature convergence in continuous optimization, Adv. Eng. Softw. 32 (2001) 49–60. [21] A. Hatamlou, In search of optimal centroids on data clustering using a binary search algorithm, Pattern Recognit. Lett. 33 (2012) 1756–1760.