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

Prediction and identiﬁcation of rotary axes error of non-orthogonal ﬁve-axis machine tool Dongju Chen n, Lihua Dong, Yanhua Bian, Jinwei Fan College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, Ping Leyuan 100#, Chaoyang district, Beijing 100124, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 4 December 2014 Received in revised form 30 March 2015 Accepted 31 March 2015 Available online 3 April 2015

This paper proposes an efﬁcient and automated scheme to predict and identify the position and motion errors of rotary axes on a non-orthogonal ﬁve-axis machining centre using the double ball bar (DBB) system. Based on the Denavit-Hartenberg theory, a motion deviations model for the tilting rotary axis B and rotary C of a non-orthogonal ﬁve-axis NC machine tool is established, which considers tilting rotary axis B and rotary C static deviations and dynamic deviations that total 24. After analysing the mathematical expression of the motion deviations model, the QC20 double ball bar (DBB) from the Renishaw Company is used to measure and identify the motion errors of rotary axes B and C, and a measurement scheme is designed. With the measured results, the 24 geometric deviations of rotary axes B and C can be identiﬁed intuitively and efﬁciently. This method provides a reference for the error identiﬁcation of the non-orthogonal ﬁve-axis NC machine tool. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Non-orthogonal ﬁve-axis NC machine tool Tilting rotary axis Denavit-Hartenberg motion deviations Identiﬁcation algorithm

1. Introduction Five-axis machining centres with three orthogonal linear axes and two rotary axes are suitable for a machining workpiece with complicated shapes. With a rapid development of precision machining for the complexity component, recently, ﬁve-axis machining centres with advantages that include higher material removal rate, improved machining surface accuracy, and shorter effective machining time are extensively used in various manufacturing [1]. However, it is well-known that kinematic errors due to the increased number of synchronous motion axes, that the overall motion accuracy of a ﬁve–axis machine tool is often inaccurate, and that it can cause signiﬁcant errors on the tool position and orientation with respect to the workpiece. Hence, the machined accuracy of the surface is often signiﬁcantly worse. To improve its motion accuracy, it is crucial to develop accurate and efﬁcient methodologies to calibrate its motion errors. However, the machine tool error (geometric errors, heat deformation errors and force deformation errors) increased inevitably as the processing shaft quantity increased [2]. Among them, the geometric error inﬂuences the positioning precision of the whole machine running time. According to the conventional classiﬁcation, geometric error is divided into position-dependent geometric errors and position-independent n

Corresponding author. Fax: þ86 10 67391617. E-mail address: [email protected] (D. Chen).

http://dx.doi.org/10.1016/j.ijmachtools.2015.03.010 0890-6955/& 2015 Elsevier Ltd. All rights reserved.

geometric errors [3]. For actual working conditions, geometric error that exists in every axis movement will directly lead to the positioning error in the machining process and then to machining error. Therefore, full and accurate identiﬁcation of the rotation axis error for the machine tool is a necessary condition. For ﬁve-axis machine tools, the measurement method for the linear axes has been more mature and perfect [4]. At present, the research for the identiﬁcation of the rotation axes error is mainly aimed at the TTTRR and RRTTT type machine tools, and most of them are identiﬁed for the rotation axes error with orthogonal series machine tool [5, 6, 7]. However, there is little research for the identiﬁcation of the geometric error from the tilting axis B of for the ﬁve-axis machine tool with a non-orthogonal type. Unlike the conventional ﬁve-axis machining centres, it is difﬁcult to intuitively understand the effect of the machine’s motion errors on the machining accuracy for ﬁve-axis machine tools with tilting rotary table types. Hence, the geometric errors for tilt axis B must be modelled and identiﬁed. Additionally, their inﬂuence on machining accuracy is predicted to improve the geometric accuracy of the machined workpiece. Although some of the latest papers [8–12] discuss the effect of kinematic errors on the machining error of ﬁve-axis machining centres with kinematic errors, they only consider a machine that has three orthogonal translational axes and two orthogonal rotary axes, and thus, they do not consider that the machine has a table or a spindle that was tilted by 45° for the non-

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

orthogonal type. There has also been numerous research works reported in the literature on the model of kinematic errors based on the DenavitHartenberg representation [13–14]. For the ﬁve-axis machine tools, this complex structure of the machining device that uses the Denavit-Hartenberg method to model the geometric errors is very intuitive and convenient. However, they [13] only calibrated the errors of linear axes X, Y and Z, and have not studied the geometric error of the rotary axis. The researched ﬁve-axis machine tools object is a machining cam, and tilting rotary axis B does not exist. The effect of kinematic errors on motion accuracy for various types of ﬁve-axis machines has been investigated by constructing their kinematic model [15–18] because it only describes the motion of the machine tool in an ideal case. A critical issue with this kinematic model is, however, that the inﬂuence of the machine’s error sources on the geometric accuracy of the machined workpiece is not fully understood by machine tool builders. The identiﬁcation of kinematic errors based on the measurement of the machine’s motion error and the telescoping double ball bar (DBB) measuring device have been applied to identify kinematic errors on ﬁve-axis machines [19–24]. The most common model of the ﬁve-axis machines is a tilting rotary table type [9, 23, 25], and some used DBB just to measure the position-independent geometric errors for rotary axes A and C. Rotary tilting axis B does not exist, and the position-independent geometric errors cannot describe the radial error motion or the run-out error of the rotary axes. Lei and Hsu [26] applied a 3D probe ball device to identify some of the kinematic errors. Bringmann and Knapp [27] showed the ‘R-test’ measurement device, where the three-dimensional displacement of the tool centre is measured by using a sphere attached to the spindle tip and four displacement sensors pointed to the centre of the sphere. However, during the measurements, the positioning variability caused by the simultaneous control of two linear axes mainly affects the results of this method. Finding the error sources is inconvenient, and the signal processing from the three sensors is cumbersome. In this article, a model of geometric deviations generated by using tilting rotary axis B and rotary C of the non-orthogonal ﬁveaxis NC machine tool with the Denavit-Hartenberg theory is presented. The objective of this paper is to propose the application of the ball bar to the identiﬁcation of motion errors associated with tilting rotary axes B and C intuitively. From the measured results in three patterns, the detailed motion errors of rotary C is obtained in Fig. 14. This is different from the published research, and it gives the real-time errors with the rotary angular position. This study will focus on the error identiﬁcation of rotary axes for the nonorthogonal ﬁve-axis machine tool using a DBB system. It will be

75

experimentally demonstrated on a commercial ﬁve-axis machining centre (Fig. 1).

2. Errors of DBB system description The absolute accuracy of a circular radius for the DBB path depends on the uncertainty of the bar length and the accuracy of the measurement sensor. The accuracy of the displacement sensor is calibrated during manufacturing according to relevant standards and with a scale factor to compensate for each data reading. However, the total length of the bar (including extensions) varies with the installation layout of the ball bar and the wear caused by use. Therefore, the absolute length of the bar should be collected, and the calibrated instrument must be used to calibrate the initial length of the bar (as well as the extensions). In addition, there is the positioning accuracy problem of the ball base when installing the ball bar, and the feature of an eccentric elliptical trajectory is formed in the measurement with error values from 20 to 50 μm. In addition, the measured results are interfered, and the error becomes the major measurement error and reduces the measuring accuracy of the ball bar. Our own analysis software Renishaw ballbar5 of the ball bar revises the eccentricity value in a certain manner, and the analysis in this paper is based on the raw data. It is necessary to prepare the program to separate the install error from the collected raw data, and the eccentric error caused by the inaccurate position of ball base install is eliminated from the raw measurement data. We assume that the distance of the ball bar that deviates from the ideal position is Le, and the angle that deviates from the ideal point is rl. The relationship between the displayed bar length Lm and the actual measured length L of the ball bar is:

Lm =

L2 + Le2 − 2LeL cos(αl − βl )

(1)

The transformed relationship between the actual position and the ideal position of the ball base can be represented by:

⎡cos( − γ ) l ⎢ ⎢sin( − γl) I AT = ⎢ ⎢0 ⎢⎣0

− sin( − γl) 0 xl ⎤ ⎥ cos( − γl) 0 yl ⎥ ⎥ 0 1 0⎥ 0 0 0 ⎥⎦

(2)

At a certain point, the coordinate system of the precision ball of the ball bar is established as follows:

x = L cos αl

(3)

y = L sin αl

(4)

In the I coordinate system, the following relationship is satisﬁed,

(x − xl )2 + (y − yl )2 = L m

(5)

Eq. (3) and (4) are substituted into Eq. (5), and Eq. (6) is obtained.

L2 − 2(xl cos αl + yl sin αl)L + (xl2 + yl2 − L m2) = 0

(6)

The actual length of the ball bar can be calculated by this equation, as shown in Eq. (7). Through the install eccentricity of the ball base and the reading bar length, the actual bar length is calculated.

L = (xl cos αl + yl sin αl) + Fig. 1. Schematic installation error of ball bar.

L m2 − (xl cos αl + yl sin αl)2 = 0

(7)

From the above analysis and from the program for processing data, the installation error of the ball base is ﬁltered; thus, the

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measured result of the ball bar can be corrected, and the measured accuracy is improved.

Table 1 12 errors of rotary axis B and rotary axis C. Position error

3. Measurement and identiﬁcation error of the tilting-rotary axis B

δ x(b) δ y(b) δ z(b)

3.1. Error parameters of rotary axes B The structure of the non-orthogonal ﬁve-axis machine tool is shown in Fig. 2. Number 1 expresses the base of the machine tool, number 2 is the mobile table in the Y direction, number 3 is rotary axis C, number 4 is the mobile table in the X direction, number 5 is the mobile table in the Z direction, and number 6 is rotary axis B. Axis B and axis C are the rotary axes of the non-orthogonal ﬁveaxis machine tool DMP60U. There is a 45° angle between rotary axis B of the tilting head and axis Z of the machine tool. For the convenience of this study, the base coordinate system, i.e., the body 1 system, is selected at the zero position of the mobile table in the Y direction. The ideal coordinate system of body systems 2, 3, 4, and 5 is coincident with the base coordinate system. The ideal origin of the coordinate system for body 6 is at the intersection point of axis B and for its junction surface, the direction is in line with the base coordinate system of body 6, which rotates around axis X at þ45°. The origin of the coordinate system for the tool is at the intersection point of the spindle and its end surface, and the direction is in line with the base coordinate system. According to the object’s spatial degrees of freedom for motion, the object in the space with six degrees of freedom, namely, the object has six motion modes, including three translation modes and three rotation modes. Each axis of rotation will produce six position errors and six motion errors; thus, tilting rotary axis B has 12 geometric errors, as shown in Table 1. b1(c1) to b6(c6) express the six position errors, and bb1(cc1) to bb6 (cc6) denote the corresponding motion error. In Table 1, parameter δ is the line displacement error, and ε is the angular displacement error. The subscript is the error direction. About the six position errors of tilting rotary axis B, δx(b), δy(b) and δz(b) are the translation errors along the X, Y and Z directions, εx(b), εy(b) and εz(b) are the angle errors around axes X, Y and Z. In the paper, tilting rotary axis B of the non-orthogonal DMU60P type ﬁve-axis machine tool is researched, and the DBB system is used to measure and identify the position and motion

Fig. 2. Structure of the Five axes CNC DMU60P machine tool.

εx(b) εy(b) εz(b) δ x(c) δ y(c) δ z(c) εx(c)

Motion error

b1 b2 b3 b4 b5 b6 c1 c2 c3 c4 c5 c6

dδ x(b) dδ y(b) dδ z(b) dεx(b) dεy(b) dεz(b) dδ x(c) dδ y(c) dδ z(c) dεx(c)

εy(c)

dεy(c)

εz(c)

dεz(c)

bb1 bb2 bb3 bb 4 bb5 bb6 cc1 cc2 cc 3 cc 4 cc5 cc6

errors of the tilting rotary axis. Four measurement patterns are designed for a total of 8 time measurements, and the six position errors and six motion errors of tilting rotary axis B are identiﬁed completely. 3.2. First Measurement pattern in X direction Fig. 3 shows the initial condition of the ﬁrst measurement mode in the X direction. One ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the cushion block and magnetic base, and L is the initial length of the ball bar. Axis B is started after the completion of the arrangement. Through the linkage function of the multi-axis machine, the real-time position of the ball at the end of the cutting tool remains unchanged in the coordinate system on the worktable. It may be based on the Denavit-Hartenberg convention. The ideal coordinate of the two ball for the DBB ﬁxed in the tool end after the axis B rotation angle b is

Fig. 3. First measurement mode in X direction.

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

⎡ XTX ⎤ ⎡1 ⎢ ⎥ ⎢ XTY ⎥ ⎢ b6 T=⎢ = ⎢ XTZ ⎥ ⎢ − b5 ⎢⎣1 ⎥⎦ ⎢⎣ 0 ⎡1 ⎢ ⎢ bb6 ⎢ − bb5 ⎢⎣ 0 ⎡1 ⎢ ⎢0 •⎢ 0 ⎢ ⎣0 ⎡ t1 ⎤ ⎢ ⎥ ⎢ t2 ⎥ ⎢ t3 ⎥ ⎢ ⎥ ⎣1 ⎦

− bb6 1 bb4 0

− b6 1 b4 0

bb5 − bb4 1 0

b5 − b4 1 0

0⎤ ⎡⎢ cos(b) ⎥ 0⎥•⎢ − sin(b) 0⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣ 0

sin(b) 0 0⎤ ⎥ cos(b) 0 0⎥• 0 1 0⎥ ⎥ 0 0 1⎦

modes (measurement pattern 1 and pattern 2 in the X direction work), Eq. (10) is obtained as follows,

0⎤ ⎥ 0⎥ 0⎥ 1 ⎥⎦

⎤ 0 − tx ⎥ sin(a 0 ) ( − ty1 • cos(a 0 ) − tz1 • sin(a 0 ) ⎥ ⎥ − sin(a 0 ) cos(a 0 ) (ty1 • sin(a 0 ) − tz1 • cos(a 0 ) ⎥ ⎦ 0 0 1 0 cos(a 0 )

⎧ ΔLXty2, b = 0 − ΔLXty1, b = 0 ⎪ b5 + b6 = − ⎪ (ty2 − ty1)• sin(a 0) ⎨ ΔLXtz2, b = 0 − ΔLXtz1, b = 0 ⎪ ⎪ b5 − b6 = (tz2 − tz1)• sin(a ) ⎩ 0

(11)

b1 = − ΔLXb = 0 − ((J1 + J 2)• b5 + (J1 − J 2)• b6)

(12)

Parameters b5 and b6 can be solved according to Eq. (11), and b1 is obtained with Eq. (12).

(8)

where a0 ¼45°,(tx, ty1, tz1)T indicate the axis B position of the ideal coordinate system in the base coordinate system 0 and (t1, t 2, t 3)T indicates the position of the ball bar stick in the cutting tool in the base coordinate system. For the convenience of this study, the research coordinate expression and the relative position changes the balls on both ends of the ball bar in the ideal coordinate system of axis B. For ﬁrst measurement mode in the X direction, (t1, t 2, t 3)T = (0, 0, H)T ,tx = 0. The position of two balls shows the state in the ideal coordinate system after the axis B rotating angle b. After simpliﬁcation of Eq. (8), the position array of the balls that is installed on the rotary table and the cutting tool in the ideal coordinate system of the tilting rotary axis B is J1 = (ty1 − t 2)• sin(a0)

3.3. Second measurement pattern in X direction Fig. 4 shows the second measurement pattern in the X direction based on the ﬁrst measurement pattern in the X direction. Tilting rotary axis B needs to rotate the angle 180° to realize measurement pattern 2 in the X direction. One ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the ball socket and the pad, and L is the initial length of the ball bar. In the same way, based on the Denavit-Hartenberg convention, the ideal coordinate of the two balls for the ball bar ﬁxed in the tool end after the axis B rotation angle b is

J 2 = (t 3 − tz1)• sin(a0) {XTX = b1 + (J1 + J 2)• b5 + ((J1 − J 2)(b6 + bb6) + (J1 + J 2)• bb5 + bb1)• cos(b) + (( − J1 − J 2)• bb4 + bb2)• sin(b) + (J 2 − J1)• sin(b)XTY = b2 + ( − J 2 − J1)• b4 + (bb2 + ( − J 2 − J1)• bb4)• cos(b) + (−bb1 + (−J1 + J 2)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b) + ( − J1 + J 2)• cos(b)XTZ = b3 + bb3 + ( − J1 + J 2)• bb4 + ((J1−J 2)• b5)• sin(b) + ((J 2−J1)• b4)• cos(b) + J 2 + J1

⎧ XWX = − sin(b)•(J1 − J 2) − L ⎪ ⎨ XWY = − cos(b)•(J1 − J 2) where ⎪ ⎩ XWZ = J1 + J 2 a = pi/4, t1 = 0, t2 = 0, t3 = H , tx = 0.; (w1, w2, w3)T = (0, L, H)T is the ideal coordinate system of the ball that is arranged on the worktable, i.e., the coordinate values in O-XYZ; (t1, t2, t3)T = (0, 0, H)T is the initial position vector of the ball on the cutting tool in the machine coordinate system, and (tx, ty1, tz1)T , ideal coordinate of the axis B in the worktable coordinate system , i.e. the coordinate value in O-XYZ. According to the distance between both ends of the ball bar,

ΔL =

77

2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L T

(9)

T

(YTX , YTY , YTZ) , (WX , WY , WZ) are used for Eq. (9), respectively. Then, the following equation is obtained by the simpliﬁcation of the above equation.

ΔLXty1, tz1 = − (b1 + (J1 + J 2)• b5 + ((J1 − J 2)•(bb6 + b6) +(J1 + J 2)• bb5 + bb1)• cos(b) + (( − J1 − J 2)• bb4 + bb2)• sin(b))

(10)

In the initial position, that is, when b ¼0, it can be observed that all of the dynamic error values are zero. This shows that when measured in the ty and tz modes, respectively, i.e., in the J1 and J2

Fig. 4. Second measurement mode in X direction.

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D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

cushion block in Fig. 4), i.e., M has two values M1 and M2, the measurement is under two modes, and the other experimental condition is unchanged. The measured data are then ﬁtted by the best circle, and the actual eccentricity amount of the ball trajectory ex1, ex2, ey1, ey2 is solved.

⎡ XTX ⎤ ⎢ ⎥ XTY ⎥ T=⎢ ⎢ XTZ ⎥ ⎢⎣1 ⎥⎦ 0⎤ ⎡⎢ cos(b) ⎥ 0⎥•⎢ − sin(b) 0⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣ 0 ⎡1 − bb6 bb5 0⎤ ⎢ ⎥ − bb4 0⎥• •⎢ bb6 1 ⎢ − bb5 bb4 1 0⎥ ⎢⎣ 0 0 0 1 ⎥⎦

⎡1 ⎢ = ⎢ b6 ⎢ − b5 ⎢⎣ 0

⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎣0

− b6 1 b4 0

b5 − b4 1 0

sin(b) 0 0⎤ ⎥ cos(b) 0 0⎥ 0 1 0⎥ ⎥ 0 0 1⎦

⎤ ⎡t ⎤ 0 − tx ⎥ 1 sin(a 0 ) ( − ty2 • cos(a 0 ) − tz2 • sin(a 0 ) ⎥ ⎢⎢ t2 ⎥⎥ • − sin(a 0 ) cos(a 0 ) (ty2 • sin(a 0 ) − tz2 • cos(a 0 ) ⎥ ⎢ t3 ⎥ ⎥ ⎢ ⎥ ⎦ ⎣1 ⎦ 0 0 1

⎡1 ⎢ ⎢0 ⎢1 ⎢ ⎢⎣0

0 cos(a 0 )

(13)

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L

2 , tz 2

0 0 1 −M2

(19)

(

b1 = ex1 − M1 • b5 =

)

ex2 − ex1 M2 − M1

b2 = ey1 + M1 • b4 =

ex2 − ex1 M2 − M1

ey2 − ey1 M2 − M1

ey2 − ey1 M2 − M1

3.4. Measurement pattern in Y direction

2

(14)

(YTX , YTY , YTZ)T ,(WX , WY , WZ)T are used for eq. (14), respectively. Then, the following equation is obtained by the simpliﬁcation of the above equation. ΔLXty

1 −M1

M1 ⎤ ⎡ b1 ⎤ ⎡ex1 ⎤ ⎥ ⎢ ⎥ 0 ⎥ ⎢ b2 ⎥ ⎢ey1 ⎥ ⎢ ⎥= • M2 ⎥⎥ ⎢ b4 ⎥ ⎢ex2 ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎥⎦ ⎣ b5 ⎦ ⎣ey2 ⎦

there

where a = pi/4, t1 = 0, t2 = 0, t3 = H , tx = 0.; (w1, w2, w3)T = (0, L, H)T is the ideal coordinate system of the ball that is arranged on the worktable, i.e., the coordinate values in O-XYZ; (t1, t2, t3)T = (0, 0, H)T is the initial position vector of the ball on the cutting tool in the machine coordinate system, and (tx, ty1, tz1)T is the ideal coordinate of axis B in the worktable coordinate system, i.e., the coordinate value in O-XYZ. According to the distance between both ends of the ball bar,

ΔL =

0 0

= − (b1 + (J1 + J 2)• b5 + ((J1 − J 2)•(bb6 + b6) + (J1 + J 2)• bb5 + bb1)• cos(b) + (( − J1 − J 2)• bb 4 + bb2)

(15)

• sin(b))

Then, the following equation is obtained by the simpliﬁcation of Eqs. (13) and (14): 2

(Tx − a)2 + (Ty − b) = 4J2

(16)

Fig. 5 shows the initial condition of the measurement model in the Y direction. One ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the cushion block and magnetic base, and L is the initial length of the ball bar. Through the linkage function of the multi-axis machine, the real-time position of the ball at the end of the cutting tool remains unchanged in the coordinate system on the worktable, and the length of the ball bar also remains unchanged. In the same way, based on the Denavit-Hartenberg convention, the ideal coordinate of the two balls for the ball bar ﬁxed in the tool end after the axis B rotation angle b is ⎡ YTX ⎤ ⎢ ⎥ YTY ⎥ T=⎢ ⎢ YTZ ⎥ ⎢⎣1 ⎥⎦

Wx = sin(b)• cos(θ)• L + sin(b)• J Wy = cos(b)• cos(θ)• L + cos(b)• J

0⎤ ⎡⎢ cos(b) ⎥ 0⎥•⎢ − sin(b) 0⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣ 0 ⎡1 − bb6 bb5 0⎤ ⎢ ⎥ − bb4 0⎥• •⎢ bb6 1 ⎢ − bb5 bb4 1 0⎥ ⎢⎣ 0 0 0 1 ⎥⎦

⎡1 ⎢ = ⎢ b6 ⎢ − b5 ⎢⎣ 0

where

J = ty + t3 • sin(θ)

(17)

a = b1 + b5 • M b = b2 − b4 • M

(18)

In the ideal case, the two balls of the DBB system in the XY coordinate plane work in a concentric movement around the origin. The radius of the ball is J, the length of the two balls is L • cos(θ), and M is the initial coordinate system of the ball, which is installed on the worktable in the Z direction. We obtain a conclusion that when one or more of w3, b1, b2 , b4 , b5 exist, the actual trajectory of the ball on the worktable with respect to the ideal trajectory appears eccentric, but the radius is unchanged. When b6 , bb1, bb2 , bb4 , bb5 is kept constant, only the radius of the trajectory circle of the motion ball is changed, and there is no eccentric effect. According to the measured values ΔL , motion trajectories of the ball on the worktable can be ﬁtted by using the best circle. The actual eccentricity e x,ey of the ball in the B axis coordinate system in the X、Y plane is solved as: ex ¼b1 þb3.M ey ¼b2-b4.M With two group values of H (with cushion block and without

⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎣0

− b6 1 b4 0

b5 − b4 1 0

sin(b) 0 0⎤ ⎥ cos(b) 0 0⎥ 0 1 0⎥ ⎥ 0 0 1⎦

⎤ ⎡ t1 ⎤ 0 − tx ⎥ ⎢ ⎥ sin(a 0 ) ( − ty • cos(a 0 ) − tz • sin(a 0 ) ⎥ ⎢ t2 ⎥ • − sin(a 0 ) cos(a 0 ) (ty • sin(a 0 ) − tz • cos(a 0 ) ⎥ ⎢ t3 ⎥ ⎥ ⎢ ⎥ ⎦ ⎣1 ⎦ 0 0 1 0 cos(a 0 )

(20)

where a = pi/4, t1 = 0, t2 = 0, t3 = H , tx = 0; (w1, w2, w3)T = (0, − L, H)T is the ideal coordinate system of the ball that is arranged on the worktable, i.e., the coordinate values in O-XYZ; (t1, t2, t3)T = (0, 0, H)T is the initial position vector of the ball on the cutting tool in the machine coordinate system, and (tx, ty, tz)T is the ideal coordinate of the axis B in the worktable coordinate system, i.e., the coordinate value in O-XYZ. With the simpliﬁed Eq. (20), the position array of the balls installed on the rotary table and the cutting tool in the ideal coordinate system of axis B are,

⎧ J1 = (ty − t )• sin(a ) 2 0 ⎨ ⎩ J 2 = (t3 − tz)• cos(a 0) ⎪

⎪

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

79

⎡ XTX ⎤ ⎢ ⎥ XTY ⎥ T=⎢ ⎢ XTZ ⎥ ⎢⎣1 ⎥⎦ 0⎤ ⎡⎢ cos(b) ⎥ 0⎥•⎢ − sin(b) 0⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣ 0 ⎡1 − bb6 bb5 0⎤ ⎢ ⎥ − bb4 0⎥• •⎢ bb6 1 ⎢ − bb5 bb4 1 0⎥ ⎢⎣ 0 0 0 1 ⎥⎦

⎡1 ⎢ = ⎢ b6 ⎢ − b5 ⎢⎣ 0

⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎣0

− b6 1 b4 0

b5 − b4 1 0

sin(b) 0 0⎤ ⎥ cos(b) 0 0⎥ 0 1 0⎥ ⎥ 0 0 1⎦

⎤⎡ t ⎤ 0 − tx ⎥ 1 sin(a 0 ) ( − ty1 • cos(a 0 ) − tz1 • sin(a 0 ) ⎥⎢⎢ t2 ⎥⎥ − sin(a 0 ) cos(a 0 ) (ty1 • sin(a 0 ) − tz1 • cos(a 0 ) ⎥⎢ t3 ⎥ ⎥⎢ ⎥ ⎦⎣1 ⎦ 0 0 1 0 cos(a 0 )

(23)

where a = pi/4, t1 = 0, t2 = 0, t3 = H , tx = 0.; (w1, w2, w3)T = (0, 0, H)T is the ideal coordinate system of the ball that is arranged on the worktable, i.e. the coordinate values in O-XYZ (t1, t2, t3)T = (0, 0, H + L)T is the initial position vector of the ball on the cutting tool in the machine coordinate system; and (tx, ty, tz)T is the ideal coordinate of the axis B in the worktable coordinate system, i.e., the coordinate value in O-XYZ. With the simpliﬁed Eq. (23), the position array of the balls installed on the rotary table and the cutting tool in the ideal coordinate system of axis B are, Fig. 5. Measurement mode in Y direction.

⎧YTX = b1 + (J1 + J 2)• b5 + ((J1 − J 2)(b6 + bb6) + (J1 + J 2)• bb5 + bb1)• cos(b) ⎪ +(( − J1 − J 2)• bb4 + bb2)• sin(b) + (J 2 − J1)• sin(b) ⎪ ⎪ ⎪YTY = b2 + ( − J 2 − J1)• b4 + (bb2 + ( − J 2 − J1)• bb4)• cos(b) ⎪ +(−bb1 + (−J1 + J 2)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b) + ( − J1 + J 2) ⎨ ⎪ • cos(b) ⎪ ⎪ ⎪ YTZ = b3 + bb3 + ( − J1 + J 2)• bb4 + ((J1−J 2)• b5)• sin(b) + ((J 2−J1)• b4)• ⎪ cos(b) + J 2 + J1 ⎩

⎧YWX = − sin(b)•(J1 − J 2) ⎪ ⎨YWY = − cos(b)•(J1 − J 2) + L • cos(a 0) ⎪ ⎩YWZ = J1 + J 2 − L • sin(a 0)

⎧ ZTX = b1 + (J1 + J 2)• b5 + ((J1 − J 2)(b6 + bb6) + (J1 + J 2)• bb5 + bb1)• cos(b) ⎪ +(( − J1 − J 2)• bb4 + bb2)• sin(b) + (J 2 − J1)• sin(b) ⎪ ⎪ ⎪ ZTY = b2 + ( − J 2 − J1)• b4 + (bb2 + ( − J 2 − J1)• bb4)• cos(b) + (−bb1 ⎨ +(−J1 + J 2)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b) + ( − J1 + J 2)• cos(b) ⎪ ⎪ ⎪ ZTZ = b3 + bb3 + ( − J1 + J 2)• bb4 + ((J1−J 2)• b5)• sin(b) ⎪ +((J 2−J1)• b4)• cos(b) + J 2 + J1 ⎩

⎧ ZWX = − sin(b)•(J1 − J 2) ⎪ ⎨ ZWY = − cos(b)•(J1 − J 2) − L • sin(a 0) ⎪ ⎩ ZWZ = J1 + J 2 − L • cos(a 0) According to the distance between both ends of the ball bar,

According to the distance between both ends of the ball bar,

ΔL = ΔL =

2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L

(24)

(21)

(YTX , YTY , YTZ)T 、(WX , WY , WZ)T are used for Eq. (17), respectively. Then, the following equation is obtained by the simpliﬁcation of the above equation.

(ZTX , ZTY , ZTZ)T ,(WX , WY , WZ)T are used for eq. (24), respectively. Then, the following equation is obtained by the simpliﬁcation of the above equation.

⎧ J1 = (ty − t )• sin(a ) 2 0 ⎨ ⎩ J 2 = (t3 − tz)• cos(a 0)

ΔLY = − (b2 + ( − J1 − J 2)• b4 + (bb2 + ( − J1 − J 2)• bb4)• cos(b)

⎪

+ ( − bb1 + (J 2 − J1)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b))

⎪

+ (b3 + bb3 + (J 2 − J1)• bb4 + ((J1 − J 2)• b5)• sin(b) + ((J 2 − J1)• b4)• cos(b))

2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L

(22)

ΔLZ = (b2 + ( − J1 − J 2)• b4 + (bb2 + ( − J1 − J 2)• bb4)• cos(b) +( − bb1 + (J 2 − J1)•(bb6 + b6)

3.5. Measurement pattern in Z direction

+ ( − J1 − J 2)• bb5)• sin(b)) + (b3 + bb3 +(J 2 − J1)• bb4 + ((J1 − J 2)• b5)• sin(b)

Fig. 6 shows the initial condition of the measurement pattern in the Z direction. One ball of the ball bar is placed on the rotary table, and another ball bar is set on the cutting tool. H is the total height of the cushion and magnetic, and L is the initial length of the ball bar. In the pattern (t1, t 2, t 3)T = (0, 0, H + L)T , tx = 0. In the same way, based on the Denavit-Hartenberg convention, the ideal coordinate of the two balls for the ball bar ﬁxed in the tool end after the axis B rotation angle b is

+ ((J 2 − J1)• b4)• cos(b))

(25)

So far, the expressions of the corresponding change for the ball bar length in four measurement patterns are obtained, and the processing parameters are ΔLY, ΔLX andΔLZ.

80

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

⎧ΔLY = − (b2 + ( − J1 − J 2)• b4 + (bb2 + ( − J1 − J 2)• bb4)• cos(b)+ ⎪ ⎪ ( − bb1 + (J 2 − J1)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b)) ⎪ ⎪ + (b3 + bb3 ⎪+ J − J • bb + J − J • b • 1) 4 (( 1 2) 5) sin(b) + ((J 2 − J1)• b4)• cos(b)) ⎪ (2 ⎪ΔLZ = (b2 + ( − J1 − J 2)• b4 + (bb2 + ( − J1 − J 2)• bb4)• cos(b) ⎪ ⎪+ ( − bb1 + (J 2 − J1)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b))+ ⎨ ⎪ (b3 + bb3 + (J 2 − J1)• bb4 + ((J1 − J 2)• b5)• sin(b) + ((J 2 − J1)• b4)• ⎪ ⎪ cos(b)) ⎪ΔLXtz = − (b1 + (J1 + J 2)• b5 + ((J1 − J 2)•(bb6 + b6) 1 ⎪ ⎪+ (J1 + J 2)• bb5 + bb1)• cos(b) + (( − J1 − J 2)• bb4 + bb2)• sin(b)) ⎪ ⎪ΔLXtz 2 = − (b1 + (J1 + J 2)• b5 + ((J1 − J 2)•(bb6 + b6) ⎪ ⎩+ (J1 + J 2)• bb5 + bb1)• cos(b) + (( − J1 − J 2)• bb4 + bb2)• sin(b))

(26)

In the initial position, that is, when b ¼0, it can be observed that all of the dynamic error values are zero, and Eq. (26) is given as follows,

⎧ ΔLZb = 0 − ΔLYb = 0 = (b2 + ( − J1 − J 2)• b4) ⎪ 2 cos(a 0) ⎪ ⎪ ΔLY b = 0 + ΔLZb = 0 ⎪ = (b3 + (J 2 − J1)• b4) ⎪ 2 cos(a 0) ⎨ ⎪ ΔLXty , tz , b = 0 = − (b1 + (J11 + J 21)• b5 + (J11 − J 21)• b6) 1 1 ⎪ ⎪ ΔLX ty2 , tz2, b = 0 ⎪ ⎪ ⎩ = − (b1 + (J12 + J 22 )• b5 + (J12 − J 22 )• b6)

(27)

For the ﬁrst expression in Eq. (27), when measured in a different ty pattern, parameters b4 and b2 can be calculated

⎧ ⎪ ⎪ b4 = ⎨ ⎪ b2 = ⎪ ⎩

ΔLZ 2b = 0 − ΔLY 2b = 0 2 cos(a 0)

−

ΔLZ1b = 0 − ΔLY1b = 0

Fig. 6. Measurement mode in Z direction.

2 cos(a 0)

(ty1 − ty2) • sin(a 0) ΔLZb = 0 − ΔLYb = 0 2 cos(a 0)

+ (J1 + J 2)• b4

Error parameter b3 can also be solved by using the above result in the second expression of equation (27).

b3 =

ΔLYb = 0 + ΔLZb = 0 − (J 2 − J1)• b4 2 cos(a 0)

For the third expression in Eq. (27), when measured in different ty and tz modes, Eq. (28) can be obtained, and parameters b5 and b6 can be calculated. Error parameter b1 can also be solved by using the above result in Eq. (22).

⎧ ΔLXty2, b = 0 − ΔLXty1, b = 0 ⎪ b5 + b6 = − ⎪ (ty2 − ty1)• sin(a 0) ⎨ ΔLXtz2, b = 0 − ΔLXtz1, b = 0 ⎪ ⎪ b5 − b6 = (tz2 − tz1)• sin(a ) ⎩ 0

(

ΔLZ − ΔLY 2 cos(a 0)

− (b2 + ( − J1 − J 2)• b4)

N = − ΔLX − b1 + (J1 + J 2)• b5 A = (bb2 + ( − J1 − J 2)• bb4)• cos(b)

(28)

)

(

(29)

If assuming

M=

⎧ ΔLXty2, b = 0 − ΔLXty1, b = 0 1 ΔLXtz2, b = 0 − ΔLXtz1, b = 0 − (ty2 − ty1) • sin(a ) ⎪ b5 = 2 (tz2 − tz1) • sin(a 0) 0 ⎪ ΔLXty2, b = 0 − ΔLXty1, b = 0 ΔLXtz 2, b = 0 − ΔLXtz1, b = 0 ⎪ 1 ⎪ b6 = 2 − (tz2 − tz1) • sin(a0) − (ty2 − ty1) • sin(a0) ⎪ ⎪ ⎪ b1 = − ΔLXty1, tz1, b = 0 − (ty1 − t 2 + t 3 − tz1) ⎨ ΔLXty2, b = 0 − ΔLXty1, b = 0 1 ΔLXtz 2, b = 0 − ΔLXtz1, b = 0 ⎪ •2 − (ty2 − ty1) • sin(a ) (tz2 − tz1) • sin(a 0) ⎪ 0 ⎪ ⎪ −(ty1 − t 2 − t 3 + tz1) ⎪ ΔLXty2, b = 0 − ΔLXty1, b = 0 ΔLX − ΔLX 1 ⎪ • 2 − (tztz22,−b =tz01) • sin(tza1, b)= 0 − (ty2 − ty1) • sin(a ) ⎪ 0 0 ⎩

( (

⎧ ΔLZ − ΔLY − (b2 + ( − J1 − J 2)• b 4) ⎪ ⎪ 2 cos(a 0) ⎪ = ((bb2 + ( − J1 − J 2)• bb 4)• cos(b) ⎪ ⎪ ⎨ +( − bb1 + (J 2 − J1)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b)) ⎪ ⎪ −ΔLX − b1 + (J1 + J 2)• b5 ⎪ ⎪ = (((J1 − J 2)•(bb6 + b6) + (J1 + J 2)• bb5 + bb1)• cos(b) ⎪ ⎩ +(( − J1 − J 2)• bb 4 + bb2)• sin(b))

B = ( − bb1 + (J 2 − J1)•(bb6 + b6) + ( − J1 − J 2)• bb5)• sin(b) There is

)

⎧ M = A • cos(b) + B • sin(b) ⎨ ⎩ N = A • sin(b) − B • cos(b)

)

⎧ A = M • cos(b) + N • sin(b) ⎨ ⎩ B = M • sin(b) − N • cos(b)

)

In the same way, according to the measured data in different ty and tz condition, motion error parameters bb4, bb2, bb5, bb6 and bb1 can be calculated.

By now, the parameters b1, b2, b3, b4, b5, and b6 have been solved by using them in Eq. (26) and by obtaining Eq. (29).

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

⎧ ΔLZty2 − ΔLYty2 ΔLZty1 − ΔLYty1 − + (ty2 − ty1) • b4 − ΔLXty2 + ΔLXty1 + (ty2 − ty1) • b5 ⎪ 2 cos(a0) 2 cos(a0) ⎪ bb4 = − 2 • (ty2 − ty1) • sin(a0)cos(b) ⎪ ⎛ ΔLZty1 − ΔLYty1 ⎞ ⎪ ⎜⎜ − (b2 + (−ty1 + t 2 − t 3 + tz1) • b4) − ΔLXty1 − b1 + (ty1 − t 2 + t 3 − tz1) • b5⎟⎟ ⎪ 2 cos(a0) ⎪ ⎝ ⎠ ⎨ bb2 = 2 • cos(b) ⎪ ⎪ − (ty1 − t 2 + t 3 − tz1) ⎪ ΔLZty2 − ΔLYty2 ΔLZty1 − ΔLYty1 ⎪ − + (ty2 − ty1) • b4 − ΔLXty2 + ΔLXty1 + (ty2 − ty1) • b5 2 cos(a0) 2 cos(a0) ⎪ • ⎪ 2 • (ty2 − ty1) • sin(a0)cos(b) ⎩

When b equates 90° and 270°, Then, based on Eq. (30), the motion error parameter bb3 can be solved

ΔLY + ΔLZ = 2 cos(a 0) (b3 + bb3 + (J 2 − J1)• bb4 + ((J1 − J 2)• b5)• sin(b) + ((J 2 − J1)• b4)• cos(b))

(30)

81

T

Where (Tx, Ty, Tz) is the position vector of the ball ﬁxed in the tool in which rotate angle c around axis C in the machine tool coordinate system, (w1, w2, w3)T = (w1, 0, H)T is the initial coordinate system of the ball in which arranged on the worktable, i.e., the coordinate values in O-XYZ; and (t1, t2, t3)T = ((w1 − L), 0, H)T is the initial position vector of the ball on the cutting tool in the machine coordinate system. ⎡1 ⎢ c6 W=⎢ ⎢ −c 5 ⎢⎣ 0

−c 6 1 c4 0

c5 −c 4 1 0

⎡1 ⎢ cc 6 •⎢ ⎢ −cc 5 ⎢⎣ 0

⎡ c1 ⎤ ⎢ cos(c) ⎥ c 2 ⎥ ⎢ sin(c) • c 3⎥ ⎢ 0 ⎢ ⎥ 1 ⎦ ⎣0 −cc 6 1 cc 4 0

− sin(c) 0 0 ⎤ ⎥ cos(c) 0 0 ⎥ ⎥ 0 1 0⎥ 0 0 1⎦ cc 5 −cc 4 1 0

cc1 cc 2 cc 3 1

⎤ ⎡ w1 ⎤ ⎥ ⎢w ⎥ ⎥•⎢ 2 ⎥ ⎥ ⎢ w3 ⎥ ⎥⎦ ⎢⎣1 ⎥⎦ (32)

T

ΔLY + ΔLZ bb3 = − (b3 + (J 2 − J1)• bb4 + ((J1 − J 2)• b5)• sin(b) 2 cos(a 0) + ((J 2 − J1)• b4)• cos(b)) According to the above measured result in the four different modes in the X, Y and Z direction (measurement mode in the X direction including ﬁrst and second kind), the position error parameters b1 to b6 and motion error parameters bb1 to bb6 can be solved totally.

4. The geometric error identiﬁcation of the axis of rotation axis C From the above introduction, there are a total of 12 geometric errors on the C axis, including six position errors and six movement errors, as shown in Table 1. c1 to c6 express the six position errors, and cc1 to cc6 denote the corresponding motion errors. For these 12 geometric error measurements with the machine tool RTCP function, three types of modes and a total of six time measurements from the DBB system are designed to collect data. Finally, the identiﬁcation of the geometric error is realized.

Where (Wx, Wy, Wz) is the position vector of the ball on the worktable in the machine tool coordinate system in which the rotate angle c is around axis C. ΔL is the measured length of the ball bar and shows the changes of the relative length between the two balls in the process of the movement. By processing eqs (31) and (32), the ideal position array of the two balls on the workpiece and the cutting tool in the coordinate system of axis C are obtained, respectively.

⎧ W = c1 + w 3 • c5 + (w1 + cc1 + w 3 • cc5)• cos(c) ⎪ x ⎪ − (cc 2 + w1 •(cc 6 + c 6) + w 3 • cc 4)• sin(c) ⎪ ⎨ Wy = c 2 − w 3 • c 4 + (w1 + cc1 + w 3 • cc5)• sin(c) ⎪ ⎪ + (cc 2 + w1 •(cc 6 + c 6) + w 3 • cc 4)• cos(c) ⎪ ⎩Wz = c 3 + cc 3 − w1 •(cc5 + c5 • cos(c) − c 4 • sin(c))

(33)

⎧ T = t1 • cos(c) ⎪ x ⎨ Ty = t1 • sin(c) ⎪ ⎩ Tz = t 3

(34)

The distance of both end balls of the ball bar is

ΔL =

4.1. The radial circle measurement pattern

2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L

(35)

We can obtain Fig. 7 shows the initial conditions of measurement, one ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the cushion block and magnetic base, and dr is the initial length of the ball bar. With the RTCP function of the machine tool, synchronous movement of the axes X, Y and C can be realized, i.e., the concentric movement for both ends of the ball bar around the rotation axis C is realized, and the length of the ball bar remains constant. However, in the process of actual measurement, due to the existence of the geometric error, both ends of the ball bar cannot achieve the expected movement. So, the relative distance is not a constant value. In this case, the length of the ball bar can be measured. According to the theory of the multi-body system, the ideal coordinate of the two ball for the ball bar ﬁxed in the worktable and tool end after the axis C rotation angle c is

⎡T ⎤ ⎡cos(c) ⎢ x⎥ ⎢ ⎢T ⎥ sin(c) T = ⎢ y⎥ = ⎢ ⎢0 T z ⎢ ⎥ ⎢ ⎢⎣1 ⎥⎦ ⎣0

− sin(c) 0 0⎤ ⎡t1⎤ ⎥ ⎢ ⎥ cos(c) 0 0⎥ ⎢t2 ⎥ • 0 1 0⎥ ⎢t3⎥ ⎥ ⎢ ⎥ 0 0 1 ⎦ ⎣1 ⎦

ΔL − a • cos(c) − b • sin(c) = cc1 + w 3 • cc5

(36)

For the initial condition c ¼ 0, the dynamic error can be observed as cc1 = 0, cc5 = 0, and with the above equation we obtain

ΔLc = 0 − (c1 + w 3·c5) = 0 Thus, for the measuring value ofΔL under the different heights, we can solve the value of the parameters c5 and c1 as shown in the following type

⎡1 w 31⎤⎡c1⎤ ⎡ΔL1c = 0 ⎤ ⎥ ⎢⎣ ⎥⎢ ⎥ = ⎢ 1 w 32⎦⎣c5⎦ ⎣ΔL2c = 0 ⎦ ⎧ ⎪ c1 = ΔL1c = 0 − w 31 • ⎨ ⎪ c5 = ΔL2c = 0 − ΔL1c = 0 ⎩ w32 − w31

ΔL2c = 0 − ΔL1c = 0 w32 − w31

With Eqs. (31), (32) and (35), the following formula can be obtained 2

(Wx − a)2 + (Wy − b) = w12

(31)

where

(37)

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D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

achieve the expected movement. So, the relative distance is not a constant value. In this case, the length of the ball bar can be measured. According to the theory of the multi-body system, the ideal coordinate of the two ball for the ball bar ﬁxed in the worktable and tool end after the axis C rotation angle c is

⎡T ⎤ ⎡cos(c) ⎢ x⎥ ⎢ ⎢T ⎥ sin(c) T = ⎢ y⎥ = ⎢ ⎢0 T z ⎢ ⎥ ⎢ ⎢⎣1 ⎥⎦ ⎣0

− sin(c) 0 0⎤ ⎡t1⎤ ⎥ ⎢ ⎥ cos(c) 0 0⎥ ⎢t2 ⎥ • 0 1 0⎥ ⎢t3⎥ ⎥ ⎢ ⎥ 0 0 1 ⎦ ⎣1 ⎦

(39)

T

where(Tx, Ty, Tz) is the position vector of the ball ﬁxed in the tool in which rotate angle c around axis C in the machine tool coordinate system, (w1, w2, w3)T = (w1, 0, H)T is the ideal coordinate system of the ball which arranged on the worktable, i.e., the coordinate values in O-XYZ; and (t1, t2, t3)T = (w1, 0, (H + ds))T is the initial position vector of the ball on the cutting tool in the machine coordinate system.

Fig. 7. Radial circle measurement model.

a = c1 + w3 • c5 b = c 2 − w3 • c 4 w3 is the initial coordinate system of the ball, which is arranged on the worktable in the Z direction. We know that the value of a and b are constant, when one or more the values from w3, c1, c2, c4, c5 exist. The motion trajectory of the ball in the worktable appears eccentric relative to the ideal trajectory, but the radius is unchanged. Therefore, according to the measured values ofΔL , the work of the station side of the ball and the motion trajectory of the ball on the worktable can be ﬁtted with the best circle. The actual eccentric of the motion trajectory of the ball in the axis C coordinate system in the XY planee x,ey can be solved, and the result is shown in Fig. 8. Take two different Hs, that is, a measurement under two modes. Other experimental conditions are kept constant. The measured data are then ﬁtted with the best circle, and the actual eccentricity of the two-ball trajectory ex1, ex2, ey1, ey2 can be solved

⎡1 ⎢ ⎢0 ⎢1 ⎢ ⎢⎣0

0 0 1 −w31 0 0 1 −w32

w31 ⎤ ⎡c1 ⎤ ⎡ex1 ⎤ ⎥ ⎢ ⎥ 0 ⎥ ⎢c 2 ⎥ ⎢ey1 ⎥ ⎥ ⎢ • = w32 ⎥⎥ ⎢c 4 ⎥ ⎢ex2 ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎦⎥ ⎣c5 ⎦ ⎣ey2 ⎦

(

ex2 − ex1 w32 − w31

−cc 6 1 cc 4 0

c5 −c 4 1 0

cc5 −cc 4 1 0

c1 ⎤ ⎡cos(c) ⎥ ⎢ c 2⎥ ⎢sin(c) • c 3⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣0

− sin(c) 0 0⎤ ⎥ cos(c) 0 0⎥ 0 1 0⎥ ⎥ 0 0 1⎦

cc1 ⎤ ⎡ w1 ⎤ ⎥ ⎢ ⎥ cc 2⎥ ⎢ w2 ⎥ • cc 3⎥ ⎢ w3 ⎥ 1 ⎥⎦ ⎢⎣1 ⎥⎦

)

ey2 − ey1 32 − w 31

ey2 − ey1

(40)

T

2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L T

(38)

ex2 − ex1 w32 − w31

c 2 = ey1 + w31 • w

−c 6 1 c4 0

Where (Wx, Wy, Wz) is the position vector of the ball on the worktable in the machine tool coordinate system in which the rotate angle c is around axis C ΔL is the measured length of the ball bar and, shows the changes of the relative length between the two balls in the process of movement. According to the distance of both end balls of the ball bar,

(41)

T

The result of (Tx, Ty, Tz) and (Wx, Wy, Wz) is used in the above equation. Then, ΔL =

c1 = ex1 − w31 •

c4 =

⎡1 ⎢ cc 6 •⎢ ⎢−cc5 ⎢⎣0

ΔL =

and

c5 =

⎡1 ⎢ c6 W=⎢ ⎢−c5 ⎢⎣0

L2 − 2 • L •(c 3 + cc 3) + 2 • L • w1 •(cc5 + c5 • cos(c) − c 4 • sin(c)) − L

w 1•(cc5 + c5 • cos(c) − c 4 • sin(c)) − ΔL = c 3 + cc 3

(42) (43)

For the initial condition c ¼ 0, the dynamic error can be observed as cc3 = 0, and with the above equation we obtain:

c 3 = w 1• c5 − ΔLc = 0

(44)

w31 − w32

4.3. Tangential circle measurement model 4.2. Axial circle measurement pattern Fig. 9 shows the initial condition of the measurement model in the axial direction, one ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the ball socket and the pad, and ds is the initial length of the ball bar. With the RTCP function of the machine tool, synchronous movement of the axes X, Y and C can be realized, i.e., the concentric movement for both ends of the ball bar around rotation axis C is realized, and the length of the ball bar remains constant. However, in the process of actual measurement, due to the existence of the geometric error, both ends of the ball bar cannot

Fig. 10 shows the initial conditions of measurement, one ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. H is the total height of the cushion block and magnetic base, and dt is the initial length of the ball bar. With the RTCP function of the machine tool, synchronous movement of the axes X, Y and C can be realized, i.e., the concentric movement for both ends of the ball bar around rotation axis C is realized, and the length of the ball bar remains constant. However, in the process of actual measurement, due to the existence of the geometric error, both ends of the ball bar cannot achieve the expected movement. So, the relative distance is not a

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

constant value. In this case, the length of the ball bar can be measured. According to the theory of the multi-body system, the ideal coordinate of the two balls for the ball bar ﬁxed in the worktable and tool end after the axis C rotation angle c is

⎡T ⎤ ⎡cos(c) ⎢ x⎥ ⎢ ⎢T ⎥ sin(c) T = ⎢ y⎥ = ⎢ ⎢0 T ⎢ z⎥ ⎢ ⎢⎣1 ⎥⎦ ⎣0

− sin(c) 0 0⎤ ⎡t1⎤ ⎥ ⎢ ⎥ cos(c) 0 0⎥ ⎢t2 ⎥ • 0 1 0⎥ ⎢t3⎥ ⎥ ⎢ ⎥ 0 0 1 ⎦ ⎣1 ⎦

(45)

T

where (Tx, Ty, Tz) is the position vector of the ball ﬁxed in the tool in which rotate angle c around axis C in the machine tool coordinate system, (w1, w2, w3)T = (w1, 0, H)T is the ideal coordinate system of the ball, which is arranged on the worktable, i.e., the coordinate values in O-XYZ; and (t1, t2, t3)T = (w1, − ds, H)T is the initial position vector of the ball on the cutting tool in the machine coordinate system.

⎡1 ⎢ c6 W=⎢ ⎢−c5 ⎢⎣0 ⎡1 ⎢ cc 6 •⎢ ⎢−cc5 ⎢⎣0

−c 6 1 c4 0

−cc 6 1 cc 4 0

c5 −c 4 1 0

cc5 −cc 4 1 0

c1 ⎤ ⎡cos(c) ⎥ ⎢ c 2⎥ ⎢sin(c) • c 3⎥ ⎢0 ⎢ 1 ⎥⎦ ⎣0 cc1 ⎤ ⎡ w1 ⎤ ⎥ ⎢ ⎥ cc 2⎥ ⎢ w2 ⎥ • cc 3⎥ ⎢ w3 ⎥ 1 ⎥⎦ ⎢⎣1 ⎥⎦

− sin(c) 0 0⎤ ⎥ cos(c) 0 0⎥ 0 1 0⎥ ⎥ 0 0 1⎦

(46)

Fig. 9. The measurement model in axial direction

Fig. 8. The trajectory of the radial circular inﬂuenced by the errors.

83

84

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87 T

Where (Wx, Wy, Wz) is the position vector of the ball on the worktable in the machine tool coordinate system in which the rotate angle c is around axis C. ΔL is the measured length of the ball bar and, shows the changes of the relative length between the two balls in the process of movement. According to the distance of the both ends balls of the ball bar 2

(Wx − Tx)2 + (Wy − Ty) + (Wz − Tz)2 − L

ΔL =

T

(47)

T

The result of (Tx, Ty, Tz) and (Wx, Wy, Wz) is used in the above equation, then ΔL =

(t2)2 − 2t2 •(cc 2 + w1 •(cc 6 + c 6) − c1 • sin(c) + c 2 • cos(c) − L

(48)

− w3 • c 5 • sin(c) − w3 • c 4 • cos(c) − w3 • cc 4)

cc 2 − w 3 • cc 4 = − ΔL + (c1 + w 3 • c5)• sin(c) − (c 2 − w 3 • c 4)• cos(c) − w1 •(cc 6 + c 6)

(49)

For the initial condition c ¼ 0, the dynamic error can be observed as cc 2 = 0, cc 4 = 0, cc 6 = 0, and with the above equation we obtain

ΔLc = 0 = w1 • c 6 + c 2 − w 3 • c 4

Fig. 10. Tangential circle measurement model.

(50)

Three time measurements are executed. For the condition with different height H and w1, the equation W3 ¼H is in the above equation, and we can see that

⎡ w1 1 −H ⎤⎡ ⎤ ⎡ΔL w11, H1, c = 0 ⎤ 1 c6 ⎥ ⎥⎢ ⎥ ⎢ ⎢ 1 ⎢ w12 1 −H1 ⎥⎢c 2 ⎥ = ⎢ΔL w12 , H1, c = 0 ⎥ ⎥ ⎢ w1 1 −H ⎥⎣c 4 ⎦ ⎢ ⎣ 1 ⎢⎣ΔL w11, H2, c = 0 ⎥⎦ 2⎦

(51)

⎧ Δ L w1 , H , c = 0 − Δ L w1 , H , c = 0 2 1 1 1 ⎪c6 = w12 − w11 ⎪ ⎨ c 2 = ΔL w11, H1, c = 0 + w11 • c 6 + H1 • c 4 ⎪ ⎪ c 4 = ΔL w11, H1, c = 0 − ΔL w11, H2, c = 0 H2 − H1 ⎩ cc 6 =

ΔL w12 , H1, c = 0 − ΔL w11, H1, c = 0 w12 − w11

− c6

cc4 can be solved with different height H.

cc 4 =

ΔL w11, H2, c = 0 − ΔL w11, H1, c = 0 H2 − H1

− c5 + c 4

Fig. 11. The measuring principle of double ball bar.

cc 2 = − ΔL w11, H1 + (c1 + H1 • c5)• sin(c) − (c 2 − H2 • c 4)• cos(c) − w11 •(cc 6 + c 6) + H1 • cc 4 ⎡1 w 31⎤ ⎡ cc1 ⎤ ⎢⎣ ⎥•⎢ ⎥ 1 w 32⎦ ⎣ cc5⎦ ⎡ ΔL1 − (c1 + w 3 • c5)• cos(c) − (c 2 − w 3 • c 4)• 1 1 ⎢ ⎢ sin(c) =⎢ ⎢ ΔL2 − (c1 + w 32 • c5)• cos(c) − (c 2 − w 32 • c 4)• ⎢ ⎢⎣ sin(c) cc5 =

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

ΔL w1 , H − ΔL w1 , H − (H2 − H1)(c5 • cos(c) − c 4 • sin(c)) 2 1 2 1 H2 − H1

cc1 = ΔL w11, H1 − (c1 + H1 • c5)• cos(c) − (c 2 − H1 • c 4)• sin(c) − H1 • cc5 cc 3 = w1 1•(cc5 + c5 • cos(c) − c 4 • sin(c)) − ΔL w11, H1 − c 3 Compared with the research from Ref. [25], the thinking steps

Fig. 12. The error measurement curve in radial measurement pattern.

D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

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is ﬁnished. 5.2. Measurement of the geometric error of axis C The proposed method is applied to a non-orthogonal ﬁve-axis machine tool (DMU60P, DECKEL MAHO, Germany) with a DBB system, the DBB type is QC20-w, Renishaw, United Kingdom. One ball of the ball bar is placed on the rotary table, and another ball of the ball bar is set on the cutting tool. The nominal range of rotary axis C is given as [0°, 360°], and the measurement result is acquired at each 5°.

Fig. 13. The error measurement curve in tangential measurement pattern.

(1) Radial error measurement pattern Fig. 12 shows the length variation of the bar with the angle, and the nominal length of the bar L is 100 mm. Height H1 is 60 mm, and H2 is 117.285 mm. (2) Tangential error measurement pattern Fig. 13 shows the length variation of the bar with the angle and the measurement parameters. First case: the nominal length of the bar L1 is 150 mm, and height H1 is 60 mm. Second case: the nominal length of the bar L2 is 100 mm, and height H1 is 60 mm. Third case: the nominal length of the bar L1 is 150 mm, and height H2 is 112.285 mm. (3) Axial error measurement pattern Fig. 14 shows the length variation of the bar with the angle. The nominal length of the bar L1 is 250 mm, and the height H is 60 mm. 5.3. Identiﬁcation results of rotation axis C errors

Fig. 14. The error measurement curve in axial measurement pattern.

Table 2 Error parameters and eccentricity of axis C. Error parameters c1 c2 c4 c5

Eccentricity 78.7 μm 7.83 μm 5.42′′ 13.75′′

ex1 81.8 μm ey1 6.6 μm ex2 85.6 μm ey2 5.1 μm

are consistent with this paper, the centre shifts are identiﬁed by Eqs. (22) through (29) in Ref. [25], and we calculated Eq. (19) for axis B and eq. (38) for axis C. For axis B, we use four patterns to measure about axis C, and we only use three patterns to measure, at the same time, the detailed intuitive ﬁgure as given in Fig. 8.

5. Experimental veriﬁcation and identiﬁcation

According to the above measured value and the identiﬁcation of the axis for rotation axis C in Section 4, the motion trajectory of the ball on the worktable is ﬁtted with the best circle. The position errors of c1, c2, c4 and c5 are solved with the above equations, and the results are shown in Table 2. In addition, the centre offset value is obtained by the double ball bar measurement, which is consistent with the eccentricity resulting from the best circle ﬁtting, and Table 2 gives the corresponding eccentricity value. At the same time, the six motion errors from cc1 to cc6 are obtained, as shown in Fig. 15. From this ﬁgure, we can see not only the count of the variation of each motion error but also the real-time measurements with angle changes. Then, with the same procedure, we can solve the actual eccentricity amount for the ball in axis B coordinate system in the X, Y plane, as shown in Table 3. In this paper, the position error of rotary axes B and C for the DMP60U CNC machine tool is measured and identiﬁed. We provide a fast and feasible measurement and an identiﬁcation scheme. In Fig. 8, the results show intuitive information for the geometric errors of axis B, and this is useful for error source identiﬁcation. From the measured results in Fig. 14, the detailed motion errors of rotary C are obtained in Fig. 15, which is different from the published research. It gives the real-time errors with the rotary angular position. Different error term impacts are dissimilar on the trajectory of forming. Only the location of the error term will affect the trajectory of the circle eccentricity.

5.1. Measurement principle of the ball bar Fig. 11 shows the principle of the measurement of the ball bar. The measurement system consists of a magnetic base mounted on the worktable and a movable bar, and the movable bar is a high precision displacement sensor. When the length of the bar changes, the displacement signal is transformed into an inductive signal by using an internal detecting circuit and is imported into a PC machine through an interface, thus the measurement of the error

6. Conclusion Based on the Denavit-Hartenberg convention, a general model for the position and motion errors of the tilting rotary axis B and rotary C of the non-orthogonal ﬁve-axis machine tool DMU60P is established. Measurement schemes for tilting rotary axis B and rotary C are designed by analysing and concluding the geometric

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D. Chen et al. / International Journal of Machine Tools & Manufacture 94 (2015) 74–87

Fig. 15. Motion errors of rotation axis C.

Acknowledgments

Table 3 Error parameters and eccentricity of axis B. Error parameters b1 b2 b4 b5

eccentricity 55.6 μm 5.6 μm 19.3′′ 47.4′′

ex1 57.8 μm ey1 4.7 μm ex2 60.5 μm ey2 3.6 μm

error expression. In the identiﬁcation process, the variation of the bar length is an important key, and it affects the result of the error parameters. The error parameters b1 (c1) to b6 (c6) and bb1 (cc1) to bb6(cc6) are all calculated with the measurement patterns, and the results show intuitive information on the geometric errors of tilting axis B. This is useful for error source identiﬁcation. Finally, this study is veriﬁed by a DBB system to identify the errors of rotary axes B and C for the non-orthogonal ﬁve-axis machine tool. The results give the real-time error variation of the rotary axes with angular change, and the detailed motion errors of rotary axes are obtained. This method is universal and provides a reference for the error identiﬁcation of the non-orthogonal ﬁve-axis NC machine tool. Conﬂicts of Interest The authors declare no conﬂict of interest.

This project is supported by National Natural Science Foundation of China (Grant No. 51105005 and 51475010), Beijing Natural Science Foundation(Grant No. 3142005).

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