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International Journal of Machine Tools & Manufacture 48 (2008) 338–349 www.elsevier.com/locate/ijmactool

Modeling and measurement of active parameters and workpiece home position of a multi-axis machine tool Psang Dain Lin, Chian Sheng Tzeng Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan Received 12 December 2006; received in revised form 3 August 2007; accepted 7 October 2007 Available online 12 October 2007

Abstract The complex structures of a multi-axis machine tool may produce inaccuracies at the tool tip caused by dimensional errors in the machine’s link parameters. This paper addresses two important issues for precision machining: (1) which link parameters (denoted as active parameters) of a machine tool can affect the machining accuracy of a workpiece and (2) how to measure the active parameters by using a grinding wheel as a measuring probe. To achieve this, a modiﬁed Denavit–Hartenberg (D–H) notation is introduced to model a multi-axis machine tool. The NC data equations are then derived in terms of the machine’s link parameters. It is found that the link parameters of a machine tool can be divided into two types: active and nonactive parameters. The prerequisite for obtaining an accurately machined workpiece is to have correct values of the active parameters and the workpiece home position. Based on the developed NC data equations of a multi-axis machine tool, this paper also addresses the technique of using a grinding wheel as a measuring probe to determine the active parameters and the workpiece home position. Experimental results are also given with illustrative examples. r 2007 Elsevier Ltd. All rights reserved. keywords: Multi-axis machine tool; Measurement; Home position

1. Introduction In earlier efforts, a large quantity of work was devoted to the study of the various problems in the ﬁeld of machine tools; these problems include post-processing, error analysis, on-line measurement, and virtual machine tool hardware and software realization. The commonly employed mathematical tools are vector notation [1], screw theory [2], and homogeneous coordinate notation [3]. Three decades ago, most papers were focused on error analysis of threeaxis machine tools by using vector notation. The most famous example may be the work of volumetric accuracy evaluation presented by Schultschik [1]. Recently, there has been a growing trend toward the study of multi-axis machine tools due to the deﬁnite advantages they offer in comparison to the three-axis machining process. Multi-axis machine tools are often preferred for their increased productivity, accuracy, and ﬂexibility in contrast to their Tel.: +886 6 275 7575; fax: +886 6 235 2973.

E-mail address: [email protected] 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.10.004

three-axis counterparts. To study a multi-axis machine tool, both the position and orientation of a link must be modeled. As a result, homogeneous coordinate notation is the most widely employed mathematical tool for design and analysis purposes. In 1989, Lin [4] modiﬁed the Denavit–Hartenberg (D–H) notation to perform error analysis on multi-axis machines. In 1993, Kiridena and Ferreira [5] used the D–H notation [6] to develop kinematic models for TTTRR, RTTTR, and RRTTT ﬁve-axis machines. Lin and Chu [7] applied the D–H notation to generate NC data equations in cam production. Later, Mahbubur et al. [8] also used D–H notation to study the positioning accuracy of ﬁve-axis milling machines. Recently, some work has been done in regard to the ﬁeld of error analysis and compensation for multi-axis machine tools. Wang and Ehmann [9,10] developed a method used to measure the machines’ position errors. Wang et al. [11] presented an efﬁcient error compensation method to increase a CNC machine’s accuracy. The complex structures of a multi-axis machine tool may lead to inaccuracies at the tool tip caused by the

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Nomenclature (xyz)w the coordinate frame built in a workpiece (xyz)t the coordinate frame imbedded in a grindingwheel j Ai the pose matrix of (xyz)i with respect to (xyz)j rworkpiece the radius of a workpiece twheel the thickness of a grinding-wheel rwheel the radius of a grinding-wheel yi, bi, ai, hi, ai the link parameters of link i from the modiﬁed D–H notation

dimensional errors in machine’s link parameters. In the past, various error models have been reported in the ﬁelds of robotics and machine tools. The error models in robotics usually include the following error components: (1) the sixdegree-of-freedom error motions (i.e., three small linear error motions and three small angular error motions) between joint elements and (2) the errors in link parameters. In contrast, the error model of a three-axis machine tool only includes 21 error components [1]: three linear displacement errors, three vertical straightness errors, three horizontal straightness errors, three roll angular errors, three pitch angular errors, three yaw angular errors, and three squareness errors. Unlike the work presented by Schultschik [1], this paper addresses the effects of link parameters on the coordinates of the machined surface from the viewpoint of NC data equations. It is known that different modeling techniques may include different parameters. Thus, the question of ‘‘which link parameters of a machine tool will affect the machining accuracy of a workpiece’’ is an important issue. To resolve this problem, this paper introduces a new, modiﬁed D–H notation in Section 2 to model a multi-axis machine tool to generate NC data equations. This modiﬁed D–H notation is different from the previous one. Compared with the conventional D–H notation [6], the proposed modiﬁed D–H notation can characterize both the position and orientation of a link with respect to its neighboring links. Inverse kinematics [3] is then used to generate the NC data equations. The inverse kinematic equations provide the machine’s link variables in terms of the listed link parameters. However, only some of the kinematic parameters appear in the NC data equations. Therefore, in Section 3, we divide a machine’s kinematic parameters into two types: active and nonactive parameters. The active parameters are those appearing in the expressions of the NC data equations and have to be measured accurately when a workpiece is clamped on to the machine. To investigate the effects of individual parameters on the problem of machining accuracy, the coordinates of the machined surface are also expressed in terms of the active parameters and the workpiece home position in Section 3. It is found that the prerequisite to assure the quality of the

339

Xm, Ym, Zm, Am, Bm, Cm the absolute machine coordinates measured by the optical linear or angular encodes X0, Y0, Z0, A0, B0, C0 the obtained absolute machine coordinates when the machine tool is at its workpiece home position X ¼ XmX0, Y ¼ YmY0, Z ¼ ZmZ0, A ¼ AmA0, B ¼ BmB0, C ¼ CmC0 the desired NC values relative to the workpiece home position

machined workpiece is to have accurate values of the active parameters and workpiece home position. Therefore, in Section 4, we address the method of using a grinding wheel as a measuring probe to determine a machine’s active parameters and the workpiece home position. Experimental results are also provided to prove the manufacturing errors by using redundant measurements. A position vector Pxi+Pyj+Pzk is written as a column matrix j P ¼ ½ Px Py Pz 1 T . The pre-superscript ‘‘j’’ of the leading symbol j P indicated that this vector is with respect to the coordinate frame (xyz)j. Given a point j P, its transformation k P is represented by the matrix product k P ¼ k Aj j P, where k Aj is a 4 4 matrix deﬁning the pose of a frame (xyz)j with respect to another frame (xyz)k. These notation rules are also applicable to unit directional vectors j n ¼ ½ nx ny nz 0 T . 2. Modiﬁed D–H notation This paper investigate the generation of NC data equations ﬁrst and then performs the measurements of these parameters and the workpiece home position by using a grinding wheel as a probe. To achieve these targets, one ﬁrst needs an efﬁcient mathematical tool to establish the machine’s ability matrix. Compared with vector notation, homogeneous coordinate notation is more efﬁcient for modeling multi-axis machine tools. This is due to vector notation’s lack of orientation information. From a kinematics perspective, a multi-axis machine such as a machine tool (see Fig. 1) can be regarded as a system of n links. Each link i (see Fig. 2) is connected to its neighboring links, i.e., i1 and i+1, by two joint elements, namely, J þ i and J iþ1 , respectively. Note that the two joint þ þ elements J i (J i joint element lying on link i) and J i (J i joint element lying on link i1), which have identical working surfaces, constitute the joint i (see Fig. 2). Therefore, in modeling a multi-axis machine, it is ﬁrst necessary to develop a mathematical representation of the position and orientation (referred to hereafter as the pose) of the axis of joint element J iþ1 relative to that of joint element J þ . To fully specify the pose of the axis of joint i element J , three translation parameters (namely bi, ai, iþ1 and hi) are required to specify its position, and two angular

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Fig. 1. Schematic illustration of six-axis orthogonal machine tool.

oi, located at the center point of joint element J iþ1 . In this coordinate frame, the zi-axis is aligned with the axis of J iþ1 , the xi-axis is parallel (or anti-parallel) to the vector pointing from qi to ei, and the yi-axis is aligned such that (xyz)i forms a right-handed coordinate frame. Further, in a serial orthogonal machine tool, the axes of these two joint elements, i.e., J þ i and J iþ1 , must be either orthogonal or parallel to one another in order to prevent crossing effects between neighboring joints. The orthogonal feature can be accomplished by specifying the twist angle as either ai ¼ 901 or 2701. Furthermore, the two axes of the joint elements, J þ i and J iþ1 , of a link may be aligned along the same direction. This feature leads to ai ¼ 01 or 1801. In other words, the links in a serial orthogonal machine tool should be of the following types: (1) ai ¼ 01, (2) ai ¼ 901, (3) ai ¼ 1801, or (4) ai ¼ 2701. Therefore, to fully characterize the pose of J iþ1 with respect to the axis of Jþ i , it is convenient to model the individual links using a 3D cuboid with a minimum circumscribed area and having the line segment oi1 oi as its center diagonal (see Fig. 2). The length (bi), width (ai), and height (hi) of this cuboid represent the characteristic dimensions of the corresponding link since the three axes of frame (xyz)i are parallel to the three sides of the cuboid. As a result, the pose of frame (xyz)i with respect to frame (xyz)i1 (or equivalently, the pose of link i with respect to link i1) can be deﬁned in terms of the length, width, and height of the cuboid as i1

Fig. 2. Coordinate frame ðxyzÞi assigned by modiﬁed D–H notation.

parameters (namely yi and ai) are necessary to deﬁne its orientation. In previous studies [7,8], the kinematics of multi-axis machines was analyzed using the conventional D–H notation [6]. However, for reasons of simplicity, this notation comprises just four parameters, i.e., yi, di, ai, and ai. In other words, one position parameter is excluded. As a result, the user is unable to manually choose the origin of a link coordinate frame at the desired position. Thus, the original D–H notation is only of limited use when analyzing the error characteristics of machine tool systems. To resolve this problem, the current study develops a modiﬁed D–H notation scheme, as described in the following. To model a multi-axis machine tool such as that illustrated in Fig. 1, it is ﬁrst necessary to number its constituent links sequentially, starting from the clamping link, labeled as ‘‘0’’, and ending at the spindle link, marked as ‘‘n’’ (n ¼ 6 in the current case). As shown in Fig. 2, each link i contains a common normal line segment, starting at qi and ending at ei, of the axes of joint elements J þ i and J . In order to describe the relationships between the iþ1 various links in the system, a coordinate frame (xyz)i is assigned to each link i (i ¼ 0, 1, 2, y, n) with its origin,

Ai ¼ Rotðz; yi Þ Transð0; 0; bi Þ Transðai ; 0; 0Þ Transð0; hi ; 0Þ Rotðx; ai Þ.

ð1Þ

Fig. 1 illustrates the assigned coordinate frames (xyz)i (i ¼ 0–6) for the six-axis machine tool considered for illustration purposes in the current study. Fig. 3(a)–(h) illustrates the cuboids of each of the major links. The corresponding kinematics parameters of the machine tool are summarized in Table 1. Eq. (1) deﬁnes the pose of link i with respect to link i1 since frame (xyz)i (i ¼ 0, 1, 2, y, n) is embedded in link i. Trans and Rot are translation and rotation operators, respectively, and are deﬁned in Appendix. bi and yi are the link variables of prismatic and revolute joints, respectively. 3. NC data equations To investigate the generation of NC data equations, a speciﬁc machine and grinding wheel should be studied since it depends strongly on the type of machine tool and grinding wheel. The parametrical expression of the grinding wheel, t P ¼ xðhÞCv xðhÞSv zðhÞ 1 T , (2) in this study is given in Eq. (A1) of Ref. [12]. h and v are the length and angle parameters that deﬁne the shape of the grinding wheel having (xyz)t as its coordinate frame. The machine used is an Ewag CNC six-axis machine tool (see Fig. 1). This machine consists of three linear motions

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341

Fig. 3. Links of machine tool and their cuboids. Table 1 Link parameters of the modiﬁed D–H notation i1

0

Ai

A1 A2 2 A3 3 A4 4 A5 5 A6 1

yi

bi

ai

hi

ai (1)

Range of link variable

y1 y2 0 901 0 y6

b1 b2 b3 b4 b2 b6

a1 a2 a3 a4 a5 a6

h1 h2 h3 h4 h5 h6

90 180 90 90 0 90

Npy1pN 1201py2p1201 0pb3p240 mm 0pb4p360 mm 0pb5p240 mm 751py6p1131

and three angular motions in three mutually perpendicular directions along the machine coordinate frame (XYZ). These motions (referred to as absolute machine coordinates Xm, Ym, Zm, Am, Bm, Cm, according to ISO standards) are measured by the optical linear or angular encodes embedded in the machine axes and can be read out from its control panel screen. The ability matrix of this machine tool can then be obtained by using w At ¼ w A0 0

A1 1 A2 2 A3 3 A4 4 A5 5 A6 6 At , where i1 Ai ði ¼ 1; 2; . . . ; nÞ is given by Eq. (1).w A0 is the pose matrix of the clamping link frame (xyz)0 with respect to the workpiece frame (xyz)w (see Fig. 3(a)). It can be w A0 ¼ Transð0; 0; b0 Þ, if the workpiece possesses a large enough tolerance to allow it to be clamped arbitrarily in the clamping link. However, if the workpiece is moved off the machine tool and reclamped for re-machining, accurate measurements have to be performed in order to obtain a correct w A0 . The pose matrix of the grinding wheel frame (xyz)t with respect to the spindle link frame (xyz)6 is given by 6 At ¼ Rot ðz; 180 Þ Rotðy; 90 Þ Transð0; 0; d t Þ (see Fig. 3(h)). Now, the ability matrix of the machine is determined by 2

Ix 6 Iy 6 w At ¼ w A0 0 A1 1 A2 2 A3 3 A4 4 A5 5 A6 6 At ¼ 6 4 Iz 0

Jx Jy

Kx Ky

Jz 0

Kz 0

3 Px Py 7 7 7 Pz 5 1 (3)

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where

Eq. (4) states that the grinding wheel rotates along the axis of the workpiece with an angular velocity, O, to generate the ﬂutes. The parameters of Eqs. (4)–(6) are obtained from numerical simulations using CAD/CAM software. Eqs. (4)–(6) can be written in the following homogeneous matrix form if Trans and Rot of Appendix are used to perform the matrix product: 2 3 I x J x K x Px 6 I y J y K y Py 7 6 7 w At ¼ 6 (7) 7 4 I z J z K z Pz 5

I x ¼ Sy1 Sy6 þ Cy1 Sy2 Cy6 I y ¼ Cy1 Sy6 þ Sy1 Sy2 Cy6 I z ¼ Cy2 Cy6 J x ¼ Cy1 Cy2 J y ¼ Sy1 Cy2 J z ¼ Sy2 K x ¼ Sy1 Cy6 Cy1 Sy2 Sy6

0

K y ¼ Cy1 Cy6 Sy1 Sy2 Sy6 K z ¼ Cy2 Sy6 Px ¼

ðb3 h1 b2 a4 a5 ÞSy1 ðh2 h3 þ b4 h5 Þ Cy1 Sy2 þ ða2 þ a3 þ h4 þ b5 þ b6 ÞCy1 Cy2 ða6 d t ÞSy1 Cy6 þ ða6 d t ÞCy1 Sy2 Sy6 þh6 Sy1 Sy6 þ h6 Cy1 Sy2 Cy6 þ a1 Cy1

Py ¼

ðb3 h1 b2 a4 a5 ÞCy1 ðh2 h3 þ b4 h5 Þ Sy1 Sy2 þ ða2 þ a3 þ h4 þ b5 þ b6 ÞSy1 Cy2 þða6 d t ÞCy1 Cy6 þ ða6 d t ÞSy1 Sy2 Sy6 h6 Cy1 Sy6 þ h6 Sy1 Sy2 Cy6 þ a1 Sy1

Pz ¼

ðh2 h3 þ b4 h5 ÞCy2 ða2 þ a3 þ h4 þ b5 þb6 ÞSy2 þ ða6 d t ÞCy2 Sy6 þ h6 Cy2 Cy6 b0 þ b1

y1, y2, b3, b4, b5, and y6 are the link variables of this machine. In a machining process, the pose matrix, w At , of the grinding wheel frame (xyz)t with respect to the workpiece frame (xyz)w should be given by either CAD-based off-line planning or manual design while also taking into consideration collision avoidance, volume removal rate, process visibility, etc. In this paper, a designed two-ﬂuted, thinned drill [12] is used as an illustrative example. Its main geometry includes two ﬂutes and two ﬂanks with two notches at the drill tip to shorten its chisel edges. The required pose matrices, w At , to manufacture the ith ﬂute, the ith ﬂank, and the ith notch (i ¼ 1, 2) are given, respectively, by w

w

w

ð4Þ

By equating the corresponding elements of the ability function matrix w At , Eq. (3), and the designed pose matrix w At of the grinding wheel, Eq. (7), one can solve for the link variables y1, y2, b3, b4, b5, and y6. In determining the angles, we always use the four-quadrant inverse tangent function tan 21, which is a two-argument function. On examining the entries (3, 1) and (3, 3) of Eq. (3), we have qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cy2 ¼ K z I 2z þ K 2z =jK z j, where the leading sign of the root is determined by the sign of Kz/|Kz|, when Kz6¼0. From the above expression for Cy2 and entry (3, 2) of Eq. (3), we can solve for y2 as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 0 K z I 2z þ K 2z A. y2 ¼ tan 21 @J z ; (8a) jK z j From entries (2, 2) and (1, 2) of Eq. (3), we obtain expressions for Cy1 and Sy1 to solve for y1: 1 J y J x y1 ¼ tan 2 ; . (8b) Cy2 Cy2 From Eq. (3), we can obtain Sy6 ¼ Sy1IxCy1Iy and Cy6 ¼ Sy1KxCy1Ky to solve for y6: y6 ¼ tan 21 ðSy1 I x Cy1 I y ; Sy1 K x Cy1 K y Þ.

(8c)

We can solve for the link variables b3, b4, and b5 by equating the 4th column vectors of Eqs. (3) and (7) as b3 ¼ Px Sy1 Py Cy1 h6 Sy6 þ ða6 d t ÞCy6 þ ðh1 þ b2 þ a4 þ a5 Þ,

ðh2 h3 h5 Þ,

ðb0 b1 ÞSy2 ða2 þ a3 þ h4 þ b6 Þ. ð5Þ

At ¼ Rotðz; 180 ði 1ÞÞ Transð38:554, 4:918; 29:01Þ Rotðz; 172:459 Þ Rotðy; 38:117 Þ Rotðx; 96:743 Þ.

1

ð8dÞ

ð8eÞ

b5 ¼ Px Cy1 Cy2 þ Py Sy1 Cy2 Pz Sy2 a1 Cy2

At ¼ Rotðz; 50 þ 180 ði 1ÞÞ Transð0; 5:613; 104:815Þ Rotðy; 24 Þ Transð0; 0; 108:607Þ,

0

b4 ¼ Px Cy1 Sy2 Py Sy1 Sy2 Pz Cy2 þ a1 Sy2 ðb0 b1 ÞCy2 þ ða6 d t ÞSy6 þ h6 Cy6

At ¼ Rotðx; 180 Þ Transð0; 0; 16:576OtÞ Rotðz; Ot þ 180 ði 1ÞÞTransð50:469 þ 0:1034Ot; 0; 0Þ Rotðx; 26:5 ÞRotðy; 12 Þ,

0

ð6Þ

ð8fÞ

Note that at y2=7901 (i.e., Kz=0), the link variables y1 and y6 become dependent. At this singular conﬁguration, the conditions for uniqueness and smoothness of solutions may be violated. Thus, the solutions of link variables have to be checked numerically in order to preserve the smoothness property.

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One has to notice that the link variable, bi or yi, of the modiﬁed D–H notation is measured along the zi1 axis of (xyz)i1 (i=1,2, y, 6), and not the axis of the machine frame (XYZ). Therefore, there may be a sign difference when the machine’s variables (i.e., y6, y2, y1, b5, b3, and b4) of the modiﬁed D–H notation are converted into the respective absolute machine coordinates (i.e., Am, Bm, Cm, Xm, Ym, Zm) of the machine frame (XYZ). Keeping this in mind and after carefully observing the frames (XYZ) and (xyz)i in Fig. 1, we ﬁnd that there is a sign difference when b3 and b4 are converted into Ym and Zm, respectively. Furthermore, it is very common that one physical quantity is measured using two scales from different datum points. One example is temperature, which is measured using both the Celsius scale and its Kelvin equivalent by K ¼ C1+2731. Similarly, there are two measurements, the link variable along zi1 (i ¼ 1, 2, y, 6) axis of the modiﬁed D–H notation and the absolute machine coordinates of the linear/rotary motion along the corresponding axis of frame (XYZ), in each axis in Fig. 1. Their conversions are given by the following equations: Am þ A0 ¼ Am þ 90 ¼ y6 0

(9a)

Bm þ B ¼ Bm þ 0 ¼ y2

(9b)

C m þ C 0 ¼ C m þ 270 ¼ y1

(9c)

X m þ X 0 ¼ b5

(9d)

Y m þ Y 0 ¼ b3

(9e)

0

Z m þ Z ¼ b4

343

Fig. 4. The virtual measurement of the workpiece home position.

respect to workpiece 2 1 0 0 6 0 1 0 6 w At ¼ 6 4 0 0 1 0

0

frame (xyz)w is given by 3 0 07 7 7 05

0

(10)

1

By substituting the corresponding entry of Eq. (10) into Eqs. (8a)–(8f), the values of the machine’s six link variables when the machine tool is at the workpiece home position are obtained as y10=2701, y20=01, b30=h6+(h1+b2+ a4+a5), b40=(b0–b1)+(a6–dt)+(h2+h3+h5), b50= a1(a2+a3+h4+b6), and y60=901. Consequently, from Eqs. (8a) to (8f), we have the following expressions of the NC values of the workpiece home position: A0 þ 90 ¼ y60 ¼ 90

(11a)

B0 ¼ y20 ¼ 0

(11b)

C 0 þ 270 ¼ y10 ¼ 270

(11c)

X 0 þ X 0 ¼ b50 ¼ a1 ða2 þ a3 þ h4 þ b6 Þ

(11d)

Y 0 þ Y 0 ¼ b30 ¼ h6 ðh1 þ b2 þ a4 þ a5 Þ

(11e)

(9f)

Note that since conventional machine tools such as that in Fig. 1 are serial orthogonal machine tools, the constants A0 ¼ 901, B0 ¼ 01, and C0 ¼ 2701 can be obtained from the property of orthogonality between two neighboring axes. Contrastingly, constants X0 , Y0 , and Z0 cannot be determined from this property. However, their values are not of interest in the modeling process reported in this paper. The NC data values of a multi-axis machine refer to the position and orientation, which the machine has been instructed to hold a grinding-wheel by G-code instructions. Offsets are used to deﬁne a workpiece home that is different from the absolute machine coordinates Am, Bm, Cm, Xm, Ym, and Zm. This allows the programmer to set up a workpiece home position for multiple parts. To use offsets, the NC values (denoted as A0, B0, C0, X0, Y0, and Z0), with respect to (XYZ), of the desired workpiece home position must be stored by G54 to G59 prior to running a program that uses them. Referring now to Fig. 4, there is shown the machine tool at the workpiece home position, where y1 ¼ 2701, y2 ¼ 01, y6 ¼ 901 and the origin of the grinding-wheel frame (xyz)t coincides with the origin of the workpiece frame (xyz)w. At this workpiece home position, the pose matrix of the grinding-wheel frame (xyz)t with

Z0 þ Z0 ¼ b40 ¼ ðb0 b1 Þ ða6 d t Þ ðh2 þ h3 þ h5 Þ (11f) Referring to Fig. 1, the desired NC data equations of the A, B, C, X, Y, and Z axes can now be obtained from the corresponding differences between Eqs. (9) and (11), respectively: A ¼ Am A0 ¼ tan 21 ðSy1 I x Cy1 I y ; Sy1 K x Cy1 K y Þ 90 0 B ¼ Bm B0 ¼ tan 21 @J z ;

C ¼ C m C 0 ¼ tan 2

1

K z

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 I 2z þ K 2z A jK z j

J y J x ; Cy2 Cy2

270

ð12aÞ

(12b)

(12c)

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X ¼ X m X 0 ¼ Px Cy1 Cy2 þ Py Sy1 Cy2 Pz Sy2 þ a1 ð1 Cy2 Þ þ ðb1 b0 ÞSy2 Y ¼ Y m Y 0 ¼ Px Sy1 þ Py Cy1 h6 ð1 Sy6 Þ ða6 d t ÞCy6

w

ð12dÞ

CC m CBm ½a6 d t zðhÞðCC m SBm CAm þ SC m SAm Þ þ ½h6 þ xðhÞCvðCC m SBm SAm ð12eÞ

Z ¼ Z m Z 0 ¼ Px Cy1 Sy2 þ Py Sy1 Sy2 þ Pz Cy2 a1 Sy2 þ ðb1 b0 Þð1 Cy2 Þ þ ða6 d t Þð1 Sy6 Þ h6 Cy6

ð12fÞ

It is shown in Eqs. (12a)–(12f) that these six NC values are functions of the active parameters a1, b1b0, h6, and (a6dt), but not functions of nonactive parameters b2, b6, a2, a3, a4, a5, h1, h2, h3, h4, and h5. We now investigate the effects of the active parameters and workpiece home position on the problems of machining accuracy. By subtracting Eqs. (11d)–(11f) from Eqs. (9d)–(9f), we have XmX0a1=a2+a3+h4+b5+b6, (YmY0h6)=b3h1b2a4a5, and ZmZ0+b0b1 a6+dt=(h2h3+b4h5), respectively. After substituting the above equations and Eqs. (9a)–(9c) into Eq. (3), we obtain the pose matrix of the grinding wheel frame (xyz)t with respect to the workpiece frame (xyz)w as 2

CC m CAm SC m SBm SAm 6 6 SC m CAm þ CC m SBm SAm 6 w At ¼ 6 6 CBm SAm 4

Py ¼ ðY m Y 0 þ h6 ÞSC m ðZ m Z 0 þ b0 b1 a6 þ d t ÞCC m SBm ½X m X 0 a1 xðhÞSv

SC m CAm Þ a1 CC m , w

Pz ¼ ðZ m Z0 þ b0 b1 a6 þ d t ÞCBm ½X m X 0 a1 xðhÞSvSBm þ ½a6 d t zðhÞCBm CAm ½h6 þ xðhÞCvCBm SAm b0 þ b1 .

The sensitivity of individual parameters to the coordinates of the machined surface can be obtained by directly differentiating Eq. (14). To assure the quality of the machined workpiece, in addition to the requirement for good resolutions of optical linear/angular motions, the following three processes are needed: (1) the critical dimensions, such as radii, thickness, and chamfers of the employed grinding wheel t P ¼ ½ xðhÞCv xðhÞSv zðhÞ 1T must be inspected; (2) the values of the active parameters must be measured accurately; (3) accurate NC values of X0, Y0, and Z0 must be provided. In the following, a method of simultaneously measuring the active parameters and X0, Y0, and Z0 is developed.

SC m CBm

CC m SAm SC m SBm CAm

CC m CBm

SC m SAm þ CC m SBm CAm

SBm

CBm CAm

0 0 0 3 ðY m Y 0 þ h6 ÞCC m þ ðZ m Z 0 þ b0 b1 a6 þ d t ÞSC m SBm þ ðX m X 0 a1 ÞSC m CBm 7 ða6 d t ÞCC m SAm þ ða6 d t ÞSC m SBm CAm h6 CC m CAm h6 SC m SBm SAm þ a1 SC m 7 7 7 ðY m Y 0 þ h6 ÞSC m ðZ m Z0 þ b0 b1 a6 þ d t ÞCC m SBm ðX m X 0 a1 ÞCC m CBm 7 7 7 ða6 d t ÞSC m SAm ða6 d t ÞCC m SBm CAm h6 SC m CAm þ h6 CC m SBm SAm a1 CC m 7 7 7 7 ðZm Z 0 þ b0 b1 a6 þ d t ÞCBm ðX m X 0 a1 ÞSBm þ ða6 d t ÞCBm CAm 7 7 7 h6 CBm SAm b0 þ b1 5

ð13Þ

1 The machined surface w P is determined by the matrix product of w At and the working surfaces, given by Eq. (2), of the grinding wheel. Therefore, the machined surface w P is given by w

P ¼ w At t P ¼

h

w

Px

w

Py

w

Pz

1

iT

,

where w

Px ¼ ðY m Y 0 þ h6 ÞCC m þ ðZm Z 0 þ b0 b1 a6 þ d t ÞSC m SBm þ ½X m X 0 a1 xðhÞSv SC m CBm þ ½a6 d t zðhÞðSC m SBm CAm CC m SAm Þ ½h6 þ xðhÞCvðSC m SBm SAm þ CC m CAm Þ þ a1 SC m ,

(14)

4. Measurement of active parameters and workpiece home position A machine tool possessing three translational and two rotational axes (their link variables being y1, y2, b3, b4, and b5) is sufﬁciently ﬂexible to move the grinding wheel to the desired position and orientation. Therefore, in this section, to simplify the machining setting, the sixth axis of this machine is locked at Am ¼ 01 (y6 ¼ 901). In this case, Eqs. (12a)–(12f) are reduced to A ¼ A m A 0 ¼ 0 ,

(15a) 0

B ¼ Bm B0 ¼ tan 21 @J z ;

K z

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 I 2z þ K 2z A, jK z j

(15b)

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C ¼ C m C 0 ¼ tan 2

1

J y J x ; Cy2 Cy2

270 ,

345

(15c)

X ¼ X m X 0 ¼ Px Cy1 Cy2 þ Py Sy1 Cy2 Pz Sy2 þ a1 ð1 Cy2 Þ þ ðb1 b0 ÞSy2 ,

ð15dÞ

Y ¼ Y m Y 0 ¼ Px Sy1 þ Py Cy1 ,

(15e)

Z ¼ Z m Z0 ¼ Px Cy1 Sy2 þ Py Sy1 Sy2 þ Pz Cy2 a1 Sy2 þ ðb1 b0 Þð1 Cy2 Þ.

ð15fÞ

Note that from Eqs. (15a)–(15f), it is clear that after the sixth axis is locked at y6=901, the active parameters are a1 and (b1b0), but not a6dt. Therefore, the following important conclusion can be drawn: some active parameters will be become nonactive parameters when the axes of the machine tool are locked. As mentioned in Section 3, the workpiece home position (i.e., A0, B0, C0, X0, Y0, and Z0) must be stored by G54 to G59 prior to running the NC codes to machine the workpiece. One can assume that A0 ¼ B0 ¼ C0 ¼ 01 if it is a blank (or a machined workpiece having large tolerance for re-machining). The following addresses to the method of simultaneously measuring: (1) the active parameters a1 and (b1b0) and (2) the workpiece home position in terms of the NC values (X0, Y0, and Z0) by using a grinding wheel as a probe after the blank is clamped. This is different from an on-line measurement system. Special care is required in this manual measurement because the NC controller cannot generate a skip function to stop the machine when the grinding wheel touches the blank. This measurement methodology depends on the conﬁguration of the machine tool. Five readings can be obtained from the ﬁve measurements shown in Figs. 5–9. The ﬁrst measurement is illustrated in Fig. 5. In this ﬁgure, the origin of the grinding wheel frame (xyz)t, with respect to the workpiece frame (xyz)w, is at Py ¼ (rworkpiece+twheel) and the machine tool is positioned at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 901 (y2 ¼ 901), Am ¼ 01 (y6 ¼ 901). rworkpiece and twheel are, respectively, the radius of the workpiece and the thickness of the grinding wheel. If the absolute machine coordinate read from the control panel is Z1, the following equation is obtained from Eq. (15f): Z 1 Z 0 ¼ rworkpiece þ twheel a1 þ ðb1 b0 Þ.

Fig. 5. The ﬁrst measurement to obtain Z1.

(16)

The second measurement is shown in Fig. 6. In this ﬁgure, the origin of the grinding wheel frame (xyz)t with respect to the workpiece frame (xyz)w is positioned at Pz ¼ rwheel and the machine tool is positioned at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 901 (y2 ¼ 901), Am ¼ 01 (y6 ¼ 901). rwheel is the radius of the grinding wheel. If the absolute machine coordinate read from control panel is X2, the following equation is obtained from Eq. (15d):

the workpiece frame (xyz)w, is placed at Py ¼ rworkpiece þ twheel and the machine tool is positioned at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 901 (y2 ¼ 901), Am ¼ 01 (y6 ¼ 901). If the absolute machine coordinate read from control panel is Z3, the following equation is obtained from Eq. (15f):

X 2 X 0 ¼ rwheel þ a1 þ ðb1 b0 Þ.

Z3 Z0 ¼ rworkpiece þ twheel þ a1 þ ðb1 b0 Þ.

(17)

Fig. 7 shows the third measurement. In this ﬁgure, the origin of the grinding wheel frame (xyz)t, with respect to

Fig. 6. The second measurement to obtain X2.

(18)

Fig. 8 schematically describes the fourth measurement. In Fig. 8, the origin of the grinding wheel (xyz)t, with

ARTICLE IN PRESS P.D. Lin / International Journal of Machine Tools & Manufacture 48 (2008) 338–349

346

Fig. 9. The ﬁfth measurement to obtain Y5.

b1 b0 ¼

X2 X4 þ rwheel . 2

(21)

X0, Y0, and Z0 can be obtained by letting the origins of the workpiece frame (xyz)w and grinding wheel (xyz)t coincide (i.e., Px ¼ Py ¼ Pz ¼ 0; see Fig. 4). It must be noted that it is impossible to actually perform such measurements as shown in Fig. 4 due to the workpiece penetrating the grinding wheel in this ﬁgure. The following measurements, shown in Figs. 9 and 10, can determine X0, Y0, and Z0:

Fig. 7. The third measurement to obtain Z3.

(1) Fig. 9 illustrates the ﬁfth measurement; here, the machine tool is at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 01 (y2 ¼ 01), Am ¼ 01 (y6 ¼ 901), and the origin of the grinding wheel (xyz)t is positioned at Px ¼ rworkpiece+ rwheel with respect to (xyz)w. From Eq. (15e), if the absolute machine coordinate read from control panel is Y5, then we obtain Y5Y0 ¼ rworkpiece+rwheel. Therefore, Y0 can be obtained as follows: Y 0 ¼ Y 5 rwheel rworkpiece .

Fig. 8. The fourth measurement to obtain X4.

respect to the workpiece frame (xyz)w, is at Pz ¼ rwheel and the machine tool is positioned at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 901 (y2 ¼ 901), Am ¼ 01 (y6 ¼ 901). If the absolute machine coordinate read from the control panel is X4, the following equation is obtained from Eq. (15d): X 4 X 0 ¼ rwheel þ a1 ðb1 b0 Þ.

(19)

The two active parameters, a1 and b1b0, can respectively be obtained by subtracting Eq. (16) from Eq. (18), and Eq. (19) from Eq. (17), as a1 ¼

Z3 Z1 , 2

(20)

(22)

(2) Determination of Z0: Drive the machine tool to Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 01 (y2 ¼ 01), Am ¼ 01 (y6 ¼ 901). Position the origin of the grinding wheel (xyz)t, with respect to the workpiece frame (xyz)w, at Pz ¼ twheel (Fig. 10a) and record the absolute machine coordinate Z 6 from the control panel. We have Z6Z0 ¼ twheel from Eq. (15f). Then, Z0 can be determined from the following equation: Z 0 ¼ Z6 twheel .

(23)

(3) Determination of X0: Keep the machine tool at Cm ¼ 01 (y1 ¼ 2701), Bm ¼ 01 (y2 ¼ 01), Am ¼ 01 (y6 ¼ 901). Position the workpiece at both the right (Py ¼ rwheelrworkpiece) and left (Py ¼ rwheel+rworkpiece) sides (Fig. 10b) of the grinding wheel. Read the absolute

ARTICLE IN PRESS P.D. Lin / International Journal of Machine Tools & Manufacture 48 (2008) 338–349

machine coordinates, X7 and X8, from the control panel. Then we have X7X0 ¼ rwheel+rworkpiece and X8X0 ¼ rwheelrworkpiece from Eq. (15d). X0 is thus determined from Eq. (24):

X0 ¼

X7 þ X8 . 2

(24)

We performed several experiments on the six-axis machine tool while using the above eight measurements

to produce a thinned drill. Table 2 lists one set of recorded absolute machine coordinates and the obtained values a1, b1, X0, Y0, and Z0. Fig. 11 is an example showing the experimental results in which the drills were not properly ground. There is always a defect—a small ﬂat surface—on the tips of the produced drills. After a thorough examination, we found that the last three measurements (Z6, X7, and X8) are redundant measurements. This is due to the ﬁve measurements in Figs. 5–9 being sufﬁcient to solve for the two active parameters and three NC values (X0, Y0, and Z0) of the workpiece home position. Fig. 4 illustrates the virtual measurement used to determine the NC values X0, Y0, and Z0. From this ﬁgure and Eqs. (16)–(19), the following equations are obtained: X0 ¼

Fig. 10. The three measurements to obtain Z6, X7, and X8.

X 2 þ X 4 Z1 Z3 þ , 2 2

(25)

Fig. 11. The undercutting at the drill tip due to redundant measurements.

Table 2 The measured active parameters and the workpiece home position Unit: mm A ¼ 01, B ¼ 901, C ¼ 01

Z1 ¼ 139.8974 X2 ¼ 94.7198

a1 ¼ (Z3Z1)/2 ¼ 0.1547

A ¼ 01, B ¼ 901, C ¼ 01

Z3 ¼ 140.2068 X4 ¼ 94.6905

b1 ¼ (X2X4)/2+rwheel ¼ 45.4162

A ¼ 01, B ¼ 01, C ¼ 01

Y5 ¼ 282.2034 Z6 ¼ 103.5882 X8 ¼ 58.2051 X7 ¼ 58.5091

Y0 ¼ Y5(rwheel+rworkpiece) ¼ 223.9144 Z0 ¼ Z6twheel ¼ 112.5882 X0 ¼ (X7+X8)/2 ¼ 0.152

Note: rwheel ¼ 49.289, twheel ¼ 9.000, and rworkpiece ¼ 9.000.

347

ARTICLE IN PRESS P.D. Lin / International Journal of Machine Tools & Manufacture 48 (2008) 338–349

348

Fig. 12. A successful ground drill.

Z0 ¼

Z3 þ Z1 X 2 X 4 twheel rworkpiece rwheel . 2 2 (26)

The workpiece home position deduced from Eqs. (25) and (26) are X0=0.14005 and Z0=112.6360. These values are different from those (X0=0.1520 and Z0= 112.5882) of Eqs. (23) and (24). The error in the Z0 value from Eq. (23) results in the defect shown in Fig. 11. Furthermore, the error in X0 from Eq. (24) results in the manufactured drill being unsymmetrical in geometry and may introduce a wobble motion while drilling. Fig. 12 illustrates a successfully manufactured drill using Eqs. (25) and (26) to determine X0 and Z0. 5. Conclusions Different modeling approaches such as vector notation, screw theory, and the modiﬁed D–H notation for kinematics modeling may induce different parameters. Unlike the work presented by Schultschik [1], this paper addressed the effects of link parameters on the coordinates of the machined surface from the viewpoint of NC data equations. The following important conclusions can be drawn: (1) In this paper, a modiﬁed D–H notation was introduced for modeling a multi-axis machine tool. Compared with conventional D–H notation [6], it can completely deﬁne the position and orientation of a link with respect to its neighboring links.

(2) Based on the NC data equations, in this paper, machine parameters are divided into active and nonactive parameters. Active parameters are parameters that appear in the NC data equations. These parameters include some inherent machine parameters (e.g., parameter a1) as well as some related to the workpiece dimensions (e.g., parameter b1) or the grinding wheel dimensions (e.g., workpiece home position). (3) The coordinates of the machined surface are expressed in terms of above parameters. It is found that the sensitivity of individual parameters to the coordinates of the machined surface is in the form of a trigonometric function. This analysis is particularly useful in determining a machine’s tolerances at its design stage and also to characterize the performance of an existing machine. (4) From Eq. (14), it is found that small dimensional errors in the active parameters (e.g., a1 ¼ 0.1547) can degrade the accuracy of a workpiece machined on a multi-axis machine tool. (5) It can be concluded from Eqs. (15d)–(15f) (by setting y2 ¼ 0) that conventional machine tools with three linear axes with/without one rotary axis do not have active parameters. Machine operators only need to provide the NC values (i.e., X0, Y0, and Z0) of the workpiece home position before running the NC codes on such machine tools. Therefore, comment (4) is not valid for such machine tools. (6) A method for simultaneously measuring the active parameters and workpiece home position is one of the important topics for precision machining processes. The number of active parameters and the workpiece home position should be consistent with the number of measurements. Redundant or unnecessary measurements will cause over-cutting or under-cutting. Thus, the proposed methodology in this paper is very important for manufacturing activities on multi-axis machine tools. Acknowledgments The authors are thankful to the National Science Council of Taiwan for supporting this research under grant NSC 95-2221-E-006-272. Sincere thanks to the president, S.C. Hwang, and general managers, Randy Kuo and Z.M. Hwang, of Di Ku Diamond Enterprises Co., Ltd., who provided the Ewag six-axis tool-grinding machine and their valuable suggestions. Appendix 2

Cc 6 Sc 6 Rotðz; cÞ ¼ 6 4 0 0

Sc Cc

0 0

0

1

3 0 07 7 7 05

0

0

1

(A.1)

ARTICLE IN PRESS P.D. Lin / International Journal of Machine Tools & Manufacture 48 (2008) 338–349

2

Cc

6 0 6 Rotðy; cÞ ¼ 6 4 Sc 0 2

1 60 6 Rotðx; cÞ ¼ 6 40 0

0

1 0

0 Cc

07 7 7 05

0

0

0 Cc

0 Sc

Sc 0

Cc 0

2

3

Sc

0

(A.2)

1 3 0 07 7 7 05

(A.3)

1 3

1

0

0

tx

60 6 Transðtx ; ty ; tz Þ ¼ 6 40

1 0

0 1

ty 7 7 7 tz 5

0

0

0

1

(A.4)

References [1] R. Schultschik, The components of the volumetric accuracy, Annals of the CIRP 25 (1) (1977) 223–228. [2] O.R. Tutunea-Fatan, H.Y. Feng, Conﬁguration analysis of ﬁve-axis machine tools using a generic kinematic model, International Journal of Machine Tools and Manufacture 44 (11) (2004) 1235–1243. [3] R.P. Paul, Robot Manipulators: Mathematics. Programming and Control, MIT press, Massachusetts, 1982.

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[4] P.D. Lin, Error analysis, measurement and compensation for multiaxis machines, Ph.D. Thesis, Mechanical Engineering Department, Northwestern University, 1989. [5] V.S.B. Kiridena, P.M. Ferreira, Mapping of the effects of positioning errors on the volumetric accuracy of ﬁve-axis CNC machine tools, International Journal of Machine Tools and Manufacture 33 (3) (1993) 417–437. [6] J. Denavit, R.S. Hartenberg, A kinematic notation for lower pair mechanisms based on matrices, ASME Journal of Applied Mechanics 22 (1955) 215–221. [7] P.D. Lin, M.B. Chu, Machine tool settings for manufacturing of cams with ﬂat-face followers, International Journal of Machine Tools and Manufacture 34 (8) (1994) 1119–1131. [8] R. Md. Mahbubur, J. Heikkala, K. Lappalainen, J.A. Karjalainen, Positioning accuracy improvement in ﬁve-axis milling by postprocessing, International Journal of Machine Tools and Manufacture 37 (2) (1997) 223–236. [9] S.M. Wang, K.F. Ehmann, Measurement method for position error of a multi-axis machine—Part I: Principle and sensitivity analysis, International Journal of Machine Tools and Manufacture 39 (1999) 951–964. [10] S.M. Wang, K.F. Ehmann, Measurement method for position error of a multi-axis machine—Part II: Applications and experimental results, International Journal of Machine Tools and Manufacture 39 (1999) 1485–1505. [11] S.M. Wang, Y.L. Liu, Y. Kang, An efﬁcient error compensation system for CNC multi-axis machines, International Journal of Machine Tools and Manufacture 42 (2002) 1235–1245. [12] P.D. Lin, C.S. Tzeng, New method for determination of the pose of the grinding wheel for thinning drill points, International Journal of Machine Tools and Manufacture 47 (15) (2007) 2218–2229.

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