School of Mechanical Engineering and Automation Harbin Institute of Technology Shenzhen Graduate School Shenzhen, Guangdong Province, China

Transcription

1 2016 International Conference on Mechanics Design, Manufacturing and Automation (MDM 2016) ISBN: A New Decoupling Measurement Method of Five-axis Machine ools Geometric Errors Based on Cross Grid Encoder and DBB Zong-Peng WANG 1,a, Jian-Gang LI 2,b,* 1 School of Mechanical Engineering and Automation Harbin Institute of echnolog Shenzhen Graduate School Shenzhen, Guangdong Province, China 2 School of Mechanical Engineering and Automation, Harbin Institute of echnolog, Shenzhen Graduate School, Shenzhen KeLab for AMC&MAE a wang201438@126.com, b jiangang_lee@163.com *Corresponding author Kewords: Cross grid encoder, Double ball bar, Homogeneous coordinate transformation, Position-independent errors, Position-dependent errors. Abstract. In order to obtain all the geometric errors of five-axis CNC machine tool, a measuring strateg with cross grid encoder and double ball-bar (DBB) is proposed. According to the theor of multi-bod sstems, b using the homogeneous coordinate transformation method the model of geometric errors is established. he cross grid encoder is used to detect the translation geometric errors and the DBB is used to measure the rotation ones. Based on the kinematic chain and the properties of geometric errors, the translation geometric errors are measured before the rotation ones. For each axis, both PIGEs (position-independent errors) and PDGEs (position-dependent errors) are under consideration. hen a complete compensated model can be established. Introduction Except for the thermal error, the geometric errors are the most important factors affecting the machining accurac. About the measurement of geometric errors, man researches have been carried out. Most modeling methods are based on multi-bod sstem [1], different kinds of geometric errors can be modeled as matrices and are eas to analsis. In 1992, Soons et al using the modeling method based on rigid bod hpothesis established the comprehensive model including the rotation axes [2]. Kiridena et al established the quasi static errors model of the multi-axis machine tool based on the homogeneous coordinate transformation matrices [3]. Jianguo Yang et al applied the homogeneous coordinate transformation theor to the turning center to analsis the geometric and thermal errors [4]. sutsumi et al designed an deviations identifing methods with three kinds of simultaneous three-axis control motions for each rotar axis [5]. In 2007, Shengi Li summarized the modeling and measurement schemes of translation axes, but the method he used has some unfavorable factors that affecting the error identification [6]. Zargarbashi proposed a method to assess the axis motion errors of a trunnion-tpe A-axis using the magnetic double ball bar (DBB) as the measuring instrument. he method consisted of five DBB tests with a single setup for all of the tests and the exclusive motion of the trunnion axis during data acquisition [7]. Lee designed a series of measurement paths to increase the sensitivit of the geometric errors, and make a clear expiation that the position-dependent and position-independent errors could be modeled as nth-order polnomials and constants [8]. sutsumi et al analzed the measurement using a clindrical coordinate sstem and Cartesian coordinate sstem respectivel, and pointed out the advantages and disadvantages of the two methods [9]. In 2013, Lee et al studied the

2 position-independent geometric errors including the offset errors and squareness errors of rotar axes of a five axis machine tool with double ball bar [10]. And in the same ear, Lee proposed a robust and simple method using double ball-bar to measure position-independent geometric errors of a rotar axis involving single axis control during the measurement [11]. Above all, decoupling is main problem during the identification of the geometric errors. In this stud, a complete measuring strateg is proposed. For each axis, the PIGEs are alwas measured before the PDGEs. According to the kinematic chain, the translation geometric errors are detected and then the rotar ones. he Mathematical Model of Geometric Errors he research object is a five-axis machine tool with RR structural configuration as shown in Fig.1. Figure 1. Five-axis machine tool with a tilting table with two controlled axes. he machine tool can be divided into the cutter chain and the workpiece chain according the kinematic structure. Onl the Z-axis is in the cutter chain, the Y-axis, X-axis, B-axis and C-axis are in the workpiece chain. O X Z Y Z O W X C Z Y O C Y D B X Z O R X Y Figure 2. he coordinate frames of machine tool. Initiall, the coordinate sstem established as shown in Fig.2 below. he reference coordinate frame titled with R, the Y-axis coordinate frame Y, the X-axis one X, the Z-axis one Z and the B-axis one B, and the origins are the position coincides with the ideal intersection of B-axis and C-axis. he origins of C-axis coordinate located on the intersection of C-axis and the turntable, and the distance between the origins of reference coordinate frame is D. he x, and z direction of all coordinate sstems are the ideal direction of X-axis, Y-axis and Y-axis.

3 he ranslation Parts he PIGEs of translations axes are the squareness errors of the three translation axes. And the PDGEs of each axis are the deviations of six degrees of freedom when the axis is moving. he PIGEs are listed in Fig.3, and the X-axis PDGEs are shown as an example in the same figure. z x x d e dz x x ez x ex x d x x x _real _ideal z_ideal z_ideal x z_real xz z_real z x x Figure 3. he PIGEs and PDGEs of translation axes. In Fig.3(1) the dx(x) is the position error, the d(x) and dz(x) are the radial errors, and the ex(x), e(x) and ez(x) are angular errors. he are all depending on the moving distance of X-axis. In Fig.3(2), the three angles express the squareness error of each two translation axes. For the X-axis, Y-axis and Z-axis, the ideal homogeneous transformation matrices are written as Eq.(1). he x, and z means ideal movement distance of X-axis, Y-axis and Z-axis respectivel. he matrix ab_ideal describes the coordinates based on the a-coordinate frame transform to the b-coordinate frame, the subscript x,, z and r corresponding to each coordinate sstem r _ ideal = x = , x _ ideal = z , zr _ ideal (1) As for the PIGEs of translation axes, the transformation matrices are written in Eq.(2). 1 - x 0 0 x r _ PIGE = , 1 0 z xz 0 zr _ PIGE = z xz (2) he matrices express the PDGEs of translation axes are written in Eq.(3). Where, the ex(x) means the angular error around the X-axis when X-axis moving and the dx(x) means the position error in the X-axis direction. According the above principle, the meaning of the other terms can be understood easil.

4 1 -e z() e () d x() e () 1 e () d () z x r _ PDGE =, e () e x() 1 d z() e z(z) e (z) d x(z) e(z) z 1 e(z) z d(z) zr _ PDGE = e (z) e x(z) 1 d z(z) e z(x) e (x) d x(x) e z(x) 1 e z(x) d (x) x _ PDGE = e (x) e x(x) 1 d z(x) (3) he Rotation Parts he PIGEs of one rotar axis consist of two offset errors and two parallelism errors. he PDGEs of rotar axes just like the translation ones. aking C-axis as an example to show the PDGEs in Fig.4(1) and the PIGEs in Fig.4(2). Figure 4. PDGEs and PIGEs of C-axis. he PIGEs can be described in Eq.(4) and the PDGEs in Eq.(5). 1 b 0 xb 1 0 b b bx _ PIGE =, 0 b 1 z b c xc 0 1 c c cb _ PIGE = c c (4) 1 -e z(b) e (b) d x(b) 1 -e z(c) e (c) d x(c) e e z(c) 1 e z(c) d (c) z(b) 1 e z(b) d (b) cb _ PDGE = bx _ PDGE =, e e (c) e x(c) 1 d z(c) (b) e x(b) 1 d z(b) (5) And the ideal motion of B-axis and C-axis can be written as Eq.(6).

5 cos b 0 sin b bx _ ideal = sin b 0 cos b cos c sin c 0 0 sin c cos c 0 0 = , cb _ ideal (6) he Complete Model In the cutter chain of machine tool, the coordinate of the center of cutter in cutter coordinate sstem can be transformed to reference coordinate sstem with Eq.(7). tr zr _ PDGE zr _ PIGE zr _ ideal tz In the workpiece chain of machine tool, the coordinate in the workpiece coordinate sstem can be transformed to reference coordinate sstem with Eq.(8). wr r _ PDGE r _ PIGE r _ ideal x _ PDGE x _ idealbx _ PDGE bx _ PIGE bx _ idealcb _PDGEcb_PIGEcb_idealwc (7) hen, the relationship of the two chains is established under the consideration of all geometric errors. he Measuring Strateg For each axis, PIGEs are alwas having more remarkable influence than PDGEs, so the PIGEs are measured before the PDGEs. Measurement of Geometric Errors of ranslation Axes he cross grid encoder is used during the measurement of translation axes errors. According to the nature of the error and the feature of the instrument, the PIGEs can be measured with the method in Fig.5. he figure describes the squareness error between X-axis and Y-axis, in which the x_ideal and _ideal represent the ideal direction of X-axis and Y-axis, the x_real and the _real represent the real directions. he direction of the real axis is the directions of measuring data since the influence of PDGEs. (8) Figure 5. Measurement of PIGEs in x-plane.

6 Appling the same method in z-plane and the zx-plane the other two PDGE can be obtained. For the PIGEs of each translation axis, the methods are similar. So onl the measurement of X-axis will be described in detail. he PDGEs measurement strateg of X-axis is shown in Fig.6. Choosing three lines in the working space, onl moving the X-axis and measuring the deviation in the x, and z direction through different installation positon. Figure 6. Measurement of the PDGEs of X-axis. he Eq.(9) can be deduced. he deviation in each direction not onl has relationship with PDGEs but also the PIGEs. Now the PIGEs have been detected, so the can be considered to be constants. x d (x) e (x) e (x) z 1 x z 1 1 d (x) e (x) z e (x) x x 1 x 1 z 1 x 1 z d (x) e (x) x e (x) x 1 z 1 x 1 xz 1 x d (x) e (x) e (x) z 2 x z 2 2 d (x) e (x) z e (x) x x 2 x 2 z 2 x 2 z d (x) e (x) x e (x) x 2 z 2 x 2 xz 2 x d (x) e (x) e (x) z 3 x z 3 3 d (x) e (x) z e (x) x x 3 x 3 z 3 x 3 z d (x) e (x) x e (x) x 3 z 3 x 3 xz 3 Since there are onl six PDGEs in each axis, six equations are enough to solve the answer. After integration the Eq.(10) can be obtained. hen in ever position, the PDGEs can be calculated b using Eq.(10). (9) x z1 1d x (x) z1 0 x 1 d(x) x x 1 z x 0 d(x) x z xz 1 x e x(x) x 2 e(x) xx 2 x e(x) z 0 (10)

7 For the translation axes, the decoupling onl reflects in that the PIGEs should be measured before the PDGEs. hen all the geometric errors can be obtained successfull. Measurement of Geometric Errors of Rotar Axes During the measurement of rotar axes, all geometric errors of translation axes have been obtained. So in the equations of rotar axes, the translation errors have been compensated before and been neglected for simplifing. According the order of kinematic chain, the PIGEs of B-axis are measured in the first place. And before the presentation of measurement strateg, some constants should be introduced. First one Dz is the distance between the center of workpiece-ball and the intersection of B-axis with C-axis in the z-direction. he second one D1 is the distance between the center of workpiece-ball and the intersection of C-axis with worktable in the z-direction too. And the last one D=Dz-D1. In this paper, the measurement of B-axis PIGEs designed as shown in Fig.7. For each mode in the measurement strateg, the motion of B-axis should be associated with two translation axes because the structure of the rotar table. he mode (1), (2) and (3) are axial, tangential and radial direction measurement respectivel. (1) (2) (3) Figure 7. Measurement of B-axis PIGEs. B using the kinematic relation between the cutter ball and the workpiece ball, the Eq.(11) can be deduced which establish the relation between the variation of the DBB s length and the PIGEs, in the equation the PDGEs of B-axis are not taken into account but the PIGEs of C-axis are under consideration. he two and above error terms have been neglected in the equations. R (D D ) (D D ) cosb (D D ) sin b 1 c z c z b z b hen for other two modes, the Eq.(12) and Eq.(13) can be derived. R x (D D x ) cosbdsinb 2 b z 1 c c (11) (12) R z ( x D ) sinbdcosb 3 b c 1 c B observing these three equations, the PIGEs of B-axis and C-axis are coupled and reflect in the DBB s length. But b using the least square method, the PIGEs of C-axis do not impact (13)

8 the identification of PIGEs of B-axis. So the four PIGE of B-axis can be identified b the data of DBB. Secondl, the PIGEs of C-axis should be measured. he measurement of C-axis PIGEs is different from the B-axis because onl C-axis motion can be achieved. As shown in Fig.8, during the measurement of C-axis PIGEs the C-axis is controlled to do a circular motion in the different plane with difference in the z-direction. (1) (2) Figure 8. Measurement of C-axis PIGEs. hrough the kinematic relation between the cutter ball and the workpiece ball, the Eq.(14) and Eq.(15) can be derived. In this case, the PIGEs of B-axis influence the identification of C-axis. So b the designed order, the PIGEs of B-axis are detected before and can be regard as constants. R ( x x ) coscsinc 1 c b c R ( x x L ) cosc [ L ( )] sinc 2 c b c c b c he Eq.(14) can identif the two offset errors and the Eq.(15) can obtain the angular error b the least square method. Lastl, the PDGEs should be measured. he measurement of B-axis and C-axis are ver similar, so onl the measurement of C-axis PDGEs will be illustrated. he measurement strateg are designed as shown in Fig.9 (14) (15) (1) (2) (3) Figure 9. Measurement of C-axis PDGEs. According to the kinematic relationship between the two balls, the Eq.(16), (17) and (18) can be deduced, the equations express the relationship between DBB s length and geometric

9 errors in x, and z direction. In the measurement of PDGEs, the PIGEs should be taken into account as the equations shown. R ( x x D ) dcxd ec x b c z c z R [D ( ) ] dc D ecx z b c c z R z dcz z b here are six PDGEs of C-axis, so six equations are needed. hough observation, the Unique coefficient in the equations is just about the fixed position of balls in the z-direction. So just like the measurement strateg in Fig.8, the balls should be raised in the z-direction. As for the limitation of the method, the angle positioning error cannot be obtained. he other five PDGEs can be calculated b the Eq.(19). (16) (17) (18) R x D d(c) z1 x C11 R D d(c) z1 0 C 21 R d(c) z z C 31 R D e(c) C x2 z2 x 12 R D e(c) 2 z2 0 C 22 (19) Where, C 11 ( xb xc D z1c ) C 21 [D z1( b c) c] C31 zb C 12 ( xb xc D z2c ) C 22 [D z2( b c) c]. he last column vector in the Eq. (19) is consists of PIGEs of B-axis and C-axis, the PDGEs and PIGEs are coupled. Since the PIGEs of two rotar axes are measured before, the vector can be regard as constant vector and the PDGEs of C-axis in different angle of rotation can be calculated. hen all the geometric errors of machine tool have been measured in the strateg before except for the angle position error of rotar axis. As the rotar table adapts the absolute encoder, the neglected error in fact also can be ignored. he Simulation and Experiment In this section, the simulation is mainl carried out in the identification of B-axis PIGEs and C-axis PIGEs. he other errors calculated b equations are not simulated. he ab.1 lists some simulation parameters.

10 able 1. he simulation parameters of PIGEs. PIGEs Given value PIGEs Given value b rad x b mm b rad zb mm c rad x c mm c rad c mm In the simulation, the noise of measurement is under consideration. he noise is presumed among ±0.002mm. hrough the simulation of B-axis axial mode b Matlab as shown in Fig.10, the b can be identified as and the b can be identified as he deviation ratios are 6% and 1%. he magnified part can reveal the deviation between ideal path and simulation path clearl. Figure 10. he simulation of axial mode of B-axis PIGEs. Similarl, the measurement of C-axis PIGEs are simulated as shown if Fig.11. Figure 11. he measurement simulation of C-axis PIGEs. he identified value of and are and he deviation are small enough to show that the identification c methods c are effective both B-axis and C-axis. he next part is some of the experimental results. he Fig.12 and Fig.13 show the measured data of translation axes. Figure12. he measurement of squareness between X-axis and Z-axis.

11 In the Fig.12, the (1) shows the measurement of z-direction and the (2) shows the measurement of x-direction. he Fig.13 shows the three lines measurement of X-axis PDGEs. he data sampling interval is chosen 5mm because the screw pitch is 5mm. he PDGEs in ever 5mm can be calculated b using the measuring data and Eq.10. (1) Line1 measurement in x and direction (2) Line1 measurement in z direction (3) Line2 measurement in x and direction (4) Line3 measurement in x direction Figure 13. Measurement of X-axis PDGEs. Conclusion B using the measuring strateg proposed in the paper, not onl the PIGEs but also the PDGEs of ever axis can be measured in the specified measurement order. And most of all, the coupling among the geometric errors of machine tool can be measured in the separated methods. his strateg can be applied to the CNC machine tool in an structure onl making a corresponding change in the kinematic chain. From the simulation as well as the experimental results, the validit of the proposed method was clarified. Acknowledgements his work is financial supported b Shenzhen science and technolog plan projects, Shenzhen Basic research plan (No. JCYJ ).

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