A new 6-degree-of-freedom measurement method of X-Y stages based on additional information

Precision Engineering 37 (2013) 606–620

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A new 6-degree-of-freedom measurement method of X-Y stages based on additional information Zhenyu Gao a,b,∗ , Jinchun Hu a,b , Yu Zhu a,b , Guanghong Duan a,b a b

Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China The State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 16 January 2012 Received in revised form 19 December 2012 Accepted 22 January 2013 Available online 29 January 2013 Keywords: X-Y stages 6-Degree-of-freedom Additional measurement

a b s t r a c t In some occasions, high accuracy and real-time 6-degree-of-freedom (6-DOF) displacements should be measured so that the control system can regulate the X-Y stages’ attitude in real time. The accuracy and real-time property of measurement’s result depend on not only the sensors itself, but also the accuracy and effort of the computational algorithm using the sensors’ measured data. As the relation between sensors’ measured data and displacements is immensely complex and usually described as strong nonlinear coupling equation when 6-DOF displacements are all considered, computational accuracy and effort are difficult to be ensured simultaneously. This paper designs a 6-DOF displacements’ measurement setup for X-Y stages based on nine interferometers’ additional information, and derives the corresponding computational algorithm. For rotation, its range is usually very small and high accuracy computational results can be obtained using two interferometers’ differential computation; for translation, the closed form solutions without rotational displacement’s computational error’s transmission are derived by making full use of all additional interferometers’ information, so that the computational accuracy can be ensured. In addition, the algorithm has simple form and doesn’t involve iteration and transcendental function’s computation, so that it helps real-time computation. This algorithm can acquire the computational accuracy of 10−15 rad and 10−13 mm for rotational and translational displacement separately, which can be seen in the simulation result for a lithography’s wafer stage. © 2013 Elsevier Inc. All rights reserved.

1. Introduction X-Y stages are extensively used in lithography, Computer numerical control (CNC) machine, bio-chip technology, surface features measurement and other areas, providing a high accuracy and fast-moving loading platform for them. A rigid body moving in a specified plane (e.g., X-Y plane) will inherently have translational and rotational errors in 6 DOFs, because of mechanical guide’s error, environmental vibration and motion actuator’s error. As shown in Fig. 1, the X-Y stage is expected to only have displacements x and y along X- and Ydirection respectively. However, In addition to the error in x and y, the displacements z,  x ,  y , and  z in the remaining 4 DOFs also exist because of the factors listed above. In some demanding applications, the control systems need to regulate stages’ 6-DOF attitude in real time in order to ensure stages’ position and pose accurate. Therefore, it is a critical task to compute stages’ 6-DOF displacements with high accuracy and real-time property based on the sensors’ measured data. At present, the common computational algorithms of stages’ displacements can be divided into two categories. (i) One is deriving simple expressions between displacements and sensors’ readings through appropriate simplification based on sensors’ proper arrangement. Fan et al. [1] designed a 5-DOF measurement system for linear motor using three interferometers and two quadrant detectors. On this basis, the 6-DOF measurement system was built by adding an interferometer at the other direction [2]. Gao et al. [3] set up two 6-DOF measurement systems for precision linear air-bearing stage, which consisted of interferometers, autocollimators and capacitance probes. Liu et al. [4] proposed a 6-DOF measurement system for a linear motor using one interferometer and three quadrant detectors. In the computation of translational displacements, the above papers ignored the influence of stages’ rotations to sensors’ measurement result; for rotational displacements, first-order differential method was used. A more intuitive explanation can be seen in Fig. 2. Only taking the

∗ Corresponding author at: Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China. Tel.: +86 010 62797803; fax: +86 010 62797543. E-mail address: [email protected] (Z. Gao). 0141-6359/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.precisioneng.2013.01.006

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Fig. 1. The schematic diagram of 6-DOF displacements of X-Y stage.

Fig. 2. The schematic diagram of measurement and computation method for translational and rotational displacements when only motion along X-direction is considered.

motion along X-direction as example, sensor A is used to measure translational displacement along X-direction, and its reading is considered the translational displacement when incremental sensor like interferometer is used. Sensor B and C are used to measure the rotational displacement around X axis, and the ratio of their readings’ difference to their distance d is considered as the measured displacement  x . This algorithm has simple form and is appropriate for real time computation. However, its accuracy is low and can only be used in less demanding situations. (ii) The other is establishing the relations between sensors’ readings and the displacements firstly, and then using numerical iterative algorithm to compute 6-DOF displacements because the relations are always nonlinear equations. Trethewey et al. [5,6] proposed a 6-DOF measurement system consisting of laser head, tetrahedral target with three mirrored facets and Position Sensitive Detector (PSD). Park et al. [7] improved their setup and corresponding computation algorithm on this basis. The tetrahedral target is fixed on the stage, and a laser beam is emitted from the laser head located at the upright position and vertically incident on the top of the target. The three mirrored facets of the target split the laser beam into three sub-beams and each sub-beam is reflected and intercepted by a corresponding PSD, which can be seen in Fig. 3. The sub-beams are located at the centers of the PSDs after adjustment at the beginning, and they will deviate from the centers when stage’s displacements occurred. When computing displacements, the relations between PSDs’ readings and displacements are established firstly and then numerical iterative’s algorithm like Newton’s method is used. This method has more accurate sensors’ measurement model, but the convergence problem can be solved mainly by choosing a very small time step which requires fast computation and is not suitable for real-time applications. For the X-Y stages which require high accuracy and fast real-time computation for feedback, the above two methods are difficult to apply. For assurance of real-time calculation, the computational algorithm with simple form should be used. This often requires us to simplify the measurement model which will decrease computational accuracy. High accuracy and real-time property seem to be a contradiction, and one way to solve the contradiction is additional measurement [8]. This paper proposes a new 6-DOF measurement setup of X-Y stages based on nine interferometers using the idea of additional measurement, and derives the 6-DOF closed form solutions to achieve high accuracy and real-time computation simultaneously. Section 2 is problem description. The interferometer’s measurement model is established when stages are moving in 6 DOFs. Taking the lithography’s wafer stage as example, then typical setups’ computational accuracy and some problems are analyzed. Section 3 is the description and analysis of new measurement setup, which consists of new setup’s presentation, computational algorithm’s derivation and its accuracy’s analysis. The last section is conclusion.

Fig. 3. The schematic diagram of translational and rotational displacements’ measurement when laser head, tetrahedral target with three mirrored facets and PSD are used.

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Fig. 4. Schematic diagram of interferometers in every direction.

2. Problem description Interferometer has the advantages like high resolution, high accuracy and large signal to noise ratio(SNR), and it is widely used in precision X-Y stages’ 6-DOF measurement. 2.1. 6-DOF measurement based on interferometer X-Y stages have relatively large movement’s ranges in X- and Y-directions and very small movement’s ranges in the other 4 DOFs. For x and y, long plane mirrors are usually mounted on the stage and laser beams are introduced into them vertically. For z, Loopstra et al. [9,10] proposed a setup that reflected the laser beam to Z-direction by mounting a plane mirror in the stage at an angle of 45◦ with XY plane. These setups can be seen in Fig. 4. In order to design and analyze the measurement setup, the relations between the interferometer’s reading and the stage’s displacements, which are called the interferometric measurement model should be established first. Two coordinate systems are set, O X Y Z is affixed to the X-Y stage and its origin is the center of top surface, and OXYZ is affixed to the frame. As shown in Fig. 5, these two coordinates coincide with each other at initial time 0 (shown in the left part), and the relationship between them when the stage moves at a certain moment t is shown in the right part. Stage’s displacements can be denoted by three translational parameters [x, y, z] and three rotational parameters [ x ,  y ,  z ], which are coordinates of O X Y Z ’s origin in OXYZ and rotations of the stage around OX, OY and OZ. For any rotation’s sequence, the coordinate transform matrix Q can be written as

⎡

A1

A2

A3

x

⎤

⎢ ⎥ ⎢ B1 B2 B3 y ⎥ ⎥ ⎥ ⎣ C1 C2 C3 z ⎦

Q =⎢ ⎢

0

0

0

(1)

1

where A1∼3 ,B1∼3 ,C1∼3 are direction cosines in transform matrix. At any moment, if the coordinate of one point in OXYZ is (x0 ,y0 ,z0 ) and it’s coordinate in O X Y Z is (x1 ,y1 ,z1 ), then they have the following relation: (x0 , y0 , z0 , 1)T = Q (x1 , y1 , z1 , 1)T

(2)

Using the above equation, the mirror plane’s equation in OXYZ at every pose can be derived. Combining the incident laser’s and mirror plane’s equations, their intersection point’s coordinate and reflected laser’s equation can be obtained. Then we can establish the relations between stage’s displacements and interferometer’s reading. The range of stages’ rotations is usually very small and its influences on laser beam’s direction can be neglected in some less demanding applications. But in some applications with high measurement accuracy, these influences cannot be neglected. Concretely speaking, for the interferometer whose beam strikes X- or Y- direction’s plane mirror, its measurement model’s derivation can be seen in Fig. 6.

Fig. 5. The relationship between the coordinate system OXYZ affixed to frame and the coordinate system O X Y Z affixed to stage.

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laser wafer stage

l1 interferometer l3

laser wafer stage

l2 interferometer

Fig. 6. The schematic diagram of measurement model’s derivation for the interferometer whose beam strikes X- or Y-direction’s plane mirror.

At initial time (shown in the upper part) the measurement path length is MPL1 = l1 , it becomes MPL2 = (l2 + l3 )/2 at a certain moment t after stage’s pose changed (shown in the lower part). Then interferometer’s reading is X = MPL2 − MPL1 =

l2 + l3 − l1 . 2

(3)

For the interferometer whose beam strikes Z direction’s plane mirror, its measurement model’s derivation can be seen in Fig. 7. At initial time (shown in the upper part) the measurement path length is MPL1 = l1 + l2 , it becomes MPL2 = (l3 + l4 + l5 + l6 )/2 at a certain moment t after stage’s pose changed (shown in the lower part). Then interferometer’s reading is Z = MPL2 − MPL1 =

l3 + l4 + l5 + l6 − (l1 + l2 ) 2

(4)

As shown in Fig. 6, the derivation process of the X-direction interferometer’s measurement model is listed in Appendix A. The remaining two directions’ are similar with it. The results are listed here: (1) If the intersection point of the interferometer X-direction’s incident laser and mirror’s plane at initial time is A(m/2,−v1 ,−w1 ), its measurement model is X=−

A21 − 1 2A21

−1

p−



A1 2A21

−1

B1 (v1 ) + C1 (w1 ) + xA1 + yB1 + zC1 + (1 − A1 )

m 2



(5)

(2) If the intersection point of interferometer Y-direction’s incident laser and mirror’s plane at initial time is B(u2 ,−n/2,−w2 ), its measurement model is Y=

1 − B22 2B22 − 1

q+

B2



2B22 − 1

−A2 (u2 ) + C2 (w2 ) + xA2 + yB2 + zC2 + (B2 − 1)

n 2



(6)

(3) If the intersection point of interferometer Z-direction’s incident laser and the beveled plane’s mirror at initial time is C(−u3 ,−n/2,−w3 ), its measurement model is n q Z = + K − h + W3 − − H 4 2 Y=

1 − B22 2B22

−1

q+

B2 2B22

−1



−A2 (u2 ) + C2 (w2 ) + xA2 + yB2 + zC2 + (B2 − 1)

n 2



l2 laser l1

wafer stage

interferometer

l4

l5

laser l6

interferometer

l3

wafer stage

Fig. 7. The schematic diagram of measurement model’s derivation for the interferometer whose beam strikes Z direction’s plane mirror.

(7)

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Fig. 8. A measurement setup to measure two translational displacements and three rotational displacements using five interferometers.

2.2. Typical measurement setup’s and computational error’s analysis In some applications with high measurement accuracy and real-time property like lithography’s wafer stage and mask stage, it has high requirement in displacements’ computational accuracy. Taking lithography with 80 nm overlay as example, it requires wafer stage’s tracking errors in X- and Y-directions to reach 2.5 nm [11]. Moreover, lithography’s production efficiency keeps improving, and wafer stage’s speed and acceleration continue to increase so that the requirement of displacement’s computation speed is continuously rising. Now wafer stages widely use interferometer for measurement systems. Brink et al. [12,13] proposed a wafer stage’s measurement setup to measure two translational displacements and three rotational displacements using five interferometers, and the positional relationship at initial time is shown in Fig. 8. The origin of the coordinate system O X Y Z affixed to wafer stage is the center of wafer’s top surface, and the z-axis of the coordinate system OXYZ affixed to lithography’s lens (frame) is its central axis. After alignment at initial time, these two coordinate systems coincide with each other. In structural layout, X-direction’s interferometer’s central axis coincide with x-axis and Y-direction interferometer’s center axis coincide with y-axis in coordinate system OXYZ. Wafer stage’s displacements are coordinates of wafer top surface’s center in OXYZ (expressed as [x, y, z]) and stage’s rotations around OX, OY and OZ (expressed as [ x ,  y ,  z ]) respectively. For the above measurement setup, the corresponding computational algorithm is [12,13]

⎧ Y1 − Y2 ⎪ x = ⎪ ⎪ e ⎪ ⎪ ⎪ ⎪ X3 − (X1 + X2 )/2 ⎪ ⎪ y = ⎪ ⎪ c ⎪ ⎪ ⎨ X −X z =

2

1

a ⎪ ⎪  −(X + X + X ) b + c    ⎪ ⎪ 1 2 1 2 3 ⎪ ⎪ x + +  =  wh 1 − y ⎪ ⎪ 3 2 2 y ⎪ ⎪ ⎪      ⎪ ⎪ 1 ⎩ y = Y1 + Y2 − x d + e + wh 1 − 2 2

2

2

(8)

x

In the above formula, X1 ,X2 ,X3 ,Y1 ,Y2 are five interferometers’ readings, and the remaining geometric parameters’ meanings can be seen in Fig. 8. The positive or negative sign of tilt with reference to initial position is agreed by the “right-hand rule”. Taking 65 nm lithography with ˚300 mm wafer as example, its wafer stage’s displacements can be considered within these ranges x,y∈[−200 mm,200 mm], z∈[−0.5 mm,0.5 mm],  x ,  y ,  z ∈[−5 ␮rad,5 ␮rad] In addition, taking Agilent E1827A and E4399A as 2-axis and 3-axis interferometer [14], geometric dimensions of measurement system when dead path is neglected can be considered as follows: a = 26 mm, b = 9.5 mm, c = 33 mm, d = 9.5 mm, e = 26 mm; m = 500 mm, n = 500 mm, p = 200 mm, q = 200 mm, wh = 0.75 mm The above algorithm’s computational errors can be seen in Fig. 9. For rotation, when three translations are at their maximum ranges, the computational result may have their maximum error. The trends of rotational displacement  x ,  y , and  z ’s computational error over the other two rotational displacements display in (a)–(c). For translation, when three rotations are at their typical poses(1 ␮rad), the trends of translational displacement x and y’s computational error over x and y display in (d)–(e). From the above figures, we can see that rotational displacements’ computational errors are at 10−15 rad range. However, translational displacements’ computational errors are larger and can reach submicron range. So large error is mainly caused by stage’s rotation around Z-axis( z ), and the algorithm shown in Eq. (8) doesn’t eliminate it. When  z is zero, the rest of the error can still reach a few nanometers, which is still relatively large and cannot meet actual lithography’s demand. The reason of this situation can be summarized as two points: (1) This algorithm ignores the influence of stage’s rotations on laser path’s change; (2) When compensating Abbe error, a rough estimation using low-order Taylor expansion causes error.

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Fig. 9. The computational errors of Eq. (8)’s algorithm. (a) x = y = 200 mm, z = 0, the trend of  x ’s computational error over  y and  z ; (b) x = y = 200 mm, z = 0, the trend of  y ’s computational error over  x and  z ; (c) x = y = 200 mm, z = 0, the trend of  z ’s computational error over  x and  y ; (d)  x =  y =  z = 1 ␮rad, the trend of x’s computational error over x and y; (e)  x =  y =  z = 1 ␮rad, the trend of y’s computational error over x and y.

The above setup can only measure wafer stage’s 5-DOF displacements. For 6 DOFs, in theory six interferometers can achieve measurement needs. Combining 5-DOF measurement setup (shown in Fig. 8) and the setup for measuring z (shown in Fig. 7) as the 6-DOF measurement setup which can be seen in Fig. B1, the rotational displacements are also computed by the algorithm shown in Eq. (8), and translational displacements are computed by their closed form’s solutions in order to get high accuracy’s computational result. The specific process can be seen in Appendix B. There are still some problems in translational displacements’ computation: (1) Expressions contain a large number of nonlinear terms and have complex form, so that the computational effort is relatively large. (2) Rotational displacements’

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Fig. 10. Schematic diagram of additional measurement.

computational errors are transmitted in translational displacements’ computation and accumulated, and the maximum error reaches nanometer range which can be seen in simulation result. (3) A1–3 ,B1–3 ,C1–3 in expressions should be computed, which are different for different rotation’s sequence. Errors can be made when one rotation’s sequence is assumed. In the design of stage’s 6-DOF measurement system, computational algorithm should be linearized as much as possible based on the premise that computational accuracy is high enough, and computational error should not be mutually transmitted. Beyond this, in many cases the stage does not follow the Abbe principle for the metrology loop because of many other restrictions. The Abbe error is difficult to compensate because interferometer’s reading contains the coupled term of various errors, which can be seen from interferometer’s measurement model. Some simplified compensation method will cause computational error, which can be seen from the formula of solving x and y in Eq. (8). Therefore, we must study appropriate measurement setup and propose corresponding computational algorithm with high accuracy and simple form, which is the problem needed to solve in this paper. 3. A new measurement setup and displacements’ computation 3.1. New measurement setup The following laws can be discovered through studying interferometer’s measurement model: (1) Additional measurement can reduce the difficulty in computation. Through the analysis in Section2.1, if the intersection point of interferometer Y’s incident laser and Y direction’s mirror plane at initial time is B(u2 ,−n/2,−w2 ), its measurement model can be expressed by Eq. (6). As shown in Fig. 10, if arranging another interferometer Y’ in Y-direction which intersection point with mirror’s plane at initial time is B’(u3 ,−n/2,−w2 ), its measurement model is Y =

1 − B22 2B22 − 1

q+

B2 2B22 − 1



−A2 (u3 ) + C2 (w2 ) + xA2 + yB2 + zC2 + (B2 − 1)

n 2



(9)

Subtracting Eq. (6) from Eq. (9), the following result can be obtained: A2 B2 Y − Y = u3 − u2 2B22 − 1

(10)

So the nonlinear term A2 B2 /(2B2 2 −1) can be replaced by (Y−Y’)/(u3 −u2 ), and obviously the measurement model can be simplified greatly. (2) Positive and negative error’s compensation method can reduce the difficulty in computation. In some occasions, usually two measures are taken before and after sensor’s adjustment. One measurement error is positive and the other is negative, system error will be reduced to some extent if taking the mean of two readings as measurement result. Using this principle, as shown in Fig. 11, arranging another interferometer Y which is symmetric with Y about the Z axis of the fixed frame, its measurement model can be expressed as 

Y =

1 − B22 2B22 − 1

q−

B2 2B22 − 1



−A2 (u2 ) + C2 (wh + w2 ) + xA2 + yB2 + zC2 + (1 − B2 )

n 2

(11)

Adding up Eqs. (6) and (11), the following result can be obtained: 1 − B22 B2 (B2 − 1) n Y + Y  = q+ 2 2B22 − 1 2 2B22 − 1



 

(12)



So the nonlinear term 1 − B22 / 2B22 − 1 surement model can be simplified greatly.







q + B2 (B2 − 1)/ 2B22 − 1

(n/2) can be replaced by (Y + Y  )/2, and obviously the mea-

Fig. 11. Schematic diagram of positive and negative error compensation method.

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613

Fig. 12. Nine interferometers’ measurement setup.

Combining the setup for measuring z and using the principles like “additional measurement” and “positive and negative error’s compensation method”, five interferometers’ setup shown in Fig. 8 can be expanded into a nine interferometers’ setup to achieve 6-DOF displacements’ measurement. In the above figure, interferometers X1 ,X2 ,X3 ,Y3 are X1 ,X2 ,X3 ,Y1 in Fig. 8 respectively. Y2 in Fig. 8 is expanded into two interferometers Y1 and Y2 in Fig. 12. In Fig. 12, Z1 is used to measure z, and Y4 , Z2 are symmetric with Y3 , Z1 about the Z axis of the fixed frame respectively. 3.2. The computational algorithm of 6-DOF displacements For the measurement setup shown in Fig. 12, similar to Eqs. (5)–(7), nine interferometers’ measurement model can be obtained and listed below: X1 = −

X2 = −

X3 = −

Y1 =

Y2 =

Y3 =

2A21 − 1 A21 − 1 2A21

−1

A21 − 1 2A21

−1

1 − B22 2B22

−1

1 − B22 2B22 − 1 1 − B22 2B22 − 1

Y4 = −

Z1 =

A21 − 1

p−

p−

p−

q+

q+

q+

B22 − 1 2B22 − 1



A1 2A21 − 1



A1 2A21

−1



A1 2A21

−1

B2 2B22



−1

B2



2B22 − 1 B2



2B22 − 1

q−

B1

B2 2B22 − 1

a 2

+ C1 (wh + b) + xA1 + yB1 + zC1 + (1 − A1 )

 a

B1 −

2

+ C1 (wh + b) + xA1 + yB1 + zC1 + (1 − A1 )

C1 (wh + b + c) + xA1 + yB1 + zC1 + (1 − A1 )

 r

−A2 −

−A2

2

r 2

m 2

(13)

m 2

(14)

(15)

+ C2 (wh + d) + xA2 + yB2 + zC2 + (B2 − 1)

+ C2 (wh + d) + xA2 + yB2 + zC2 + (B2 − 1)

C2 (wh + d + e + f ) + xA2 + yB2 + zC2 + (B2 − 1)



m 2

n 2

C2 (wh + d + e + f ) + xA2 + yB2 + zC2 + (1 − B2 )

n 2

n 2

(16)

(17)

(18)

n 2

(19)

1n 1 + k − h + (d + e + f + g) − q − H 22 2

⎧ ⎫ n 2 ⎪ ⎪ + k − h − wh + x(A3 − A2 ) + y(B3 − B2 ) + z(C3 − C2 ) ⎨ (B3 − B2 )(C3 − C2 ) ⎬ 2 1 +   2(B3 − B2 )2 (C3 − C2 )2 − 1 ⎪ ⎩ + (B3 − B2 )(C3 − C2 )3  (wh + d + e + f + g) − [(B3 − B2 )(C3 − C2 )] H + 1 n + q ⎪ ⎭ 2

2

(20)

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1n 1 + k − h + (d + e + f + g) − q − H 22 2

Z2 =

⎫ ⎧  n 2 ⎪ ⎪ + k − h − wh + x(A3 + A2 ) + y(B3 + B2 ) + z(C3 + C2 ) ⎨ − (B3 + B2 )(C3 + C2 ) ⎬ 2 1 +   2 2 2(B3 + B2 ) (C3 + C2 ) − 1 ⎪ ⎩ − (B3 + B2 )(C3 + C2 )3  (wh + d + e + f + g) + [(B3 + B2 )(C3 + C2 )] H + 1 n + q ⎪ ⎭ 2

(21)

2

In above nine equations, a, b, c, d, e, f, g, h, H, k, r, m, n, p, q, wh are geometric parameters related with the measurement setup shown in Fig. 12. Rotational displacements are still computed by Eq. (8)’s algorithm. For translational displacements, their closed form’s solutions are derived by making full use of “additional information” and “positive and negative error’s information”. The process of derivation can be seen in Appendix C. Finally, the algorithm flow is listed below: A1. Storing the constants according to the geometric parameters T1 =

1n 1 + k − h + d + e + f + g − q − H; T2 = wh + d + e + f + g; 22 2

T3 =

n n 1 + k − h − wh; T4 = + q; T5 = wh + b + c 2 2 2

T6 = wh + d +

 1

3 3 e + f ; T7 = 4 4 2

p−

 m 2

; T8 =

(22)

 1 2

p+

 m 2

A2. Reading interferometers’ reading X1–3 ,Y1–4 and Z1–2 in each servo-controlled cycle, computing and storing the following values: S1 = S5 =

X − X  2 1 a



(X2 + X1 )/2 − X3

; S2 =

c

; S3 =



1 4S12 + 4S22 + 1

; S4 =

S3 + 1 ; 2

S1 S3 S2 S3 Y1 − Y2 Y3 − (Y2 + Y1 )/2 ; ; S6 = ; S7 = ; S8 = S4 S4 r e+f



S9 =





1



; S10 =

S7 S9 S8 S9 S9 + 1 ; S12 = ; ; S11 = 2 S10 S10

(23)

4S72 + 4S82 + 1 S13 = S6 S11 − S4 S12 ; S14 = S4 S10 − S5 S11 ; S15 = (S13 − S10 )(S14 − S12 ); S16 = (S13 + S10 )(S14 + S12 ); S17 =

1 2

1 2

(X1 + X2 ) + X3



+ T7 −

1 S4 m T8 + + S2 T5 S3 S3 2

A3. Computing 6-DOF displacements using the following formulas:

⎧ x = S8 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y = −S2 ⎪ ⎪ ⎪ ⎪ z = S1 ⎪ ⎪ ⎪  2  ⎪ ⎨ 2S15 − 1 z=

 (Z1 − T1 ) −



2 −1 2S16

(Z2 − T1 ) −

  S16 − S15 S3 S6 S17 − T4 − S14 T3 − 1 − S62 T2 + H. S4 4S15 S16

⎪ ⎪ ⎪ 1   S S ⎪ ⎪ S3 S10 S17 S5 S9 1 S6 S10 5 12 ⎪ ⎪ x=− − Y3 + (Y2 + Y1 ) − Y4 + z (T6 + z) − ⎪ ⎪ S S 2S S 2 2 S S14 14 14 4 10 14 ⎪ ⎪ ⎪ 1   ⎪ ⎪ S 1 ⎪ ⎩ y = 92 Y3 + (Y2 + Y1 ) − Y4 − 2S8 (T6 + z) − 2S7 x 2S15

2S10

2

2S16

(24)

2

3.3. The analysis of algorithm’s computational accuracy For rotations, the maximum computational error of the algorithm is only 10−15 rad, which has been analyzed in Section 2.2. For translations, the computational algorithm has simple form and does not contain iteration and transcendental function, so that the computational effort is much smaller than the algorithm in Appendix B. According to statistics, 53 addition operations, 60 multiplications and 24 shift operations are included in the algorithm, and the requirement of real-time measurement can be met easily in general. It is the closed form solution of three translational displacements, that is displacements can be computed from interferometers’ readings directly in which corresponding various errors such as Cosine and Abbe errors are compensated comprehensively. In addition, it doesn’t contain rotational displacements’ computed results, so that rotational displacements’ computational errors are not accumulated in translational displacements’ computation. Therefore, no computational errors appear in theory. However, the solutions of S3 , S4 , S9 and S10 in Eq. (23) involve in the computation of finding the square root, and they should be approximated through Taylor expansions in practical applications which will cause error. Similar to previous analysis, we can analyze the computational error of three translations when first-order Taylor expansion was used to compute S3 , S4 , S9 and S10 . Taking the wafer stage with ˚300 mm’s wafer in 65 nm lithography as example, the motion ranges of displacements are still same as the values shown in Section 2.1, and the geometric parameters are a = 26.22 mm, b = 9.5 mm, c = 33 mm, d = 9.5 mm, e = 26 mm, f = 10 mm, g = 19.05 mm, h = 70 mm, H = 80 mm, k = 20 mm, r = 26.22 mm, m = 500 mm, n = 500 mm, p = 200 mm, q = 200 mm, wh = 0.75 mm

Z. Gao et al. / Precision Engineering 37 (2013) 606–620

615

-13

3

x 10

error(x)/mm

2 1 0 -1 -2 -3 -4 0

2000

4000

6000

8000

10000

12000

14000

16000

12000

14000

16000

12000

14000

16000

pose number 4

-13

x 10

error(y)/mm

2 0 -2 -4 -6 0

2000

4000

6000

8000

10000

pose number 6

-13

x 10

error(z)/mm

4 2 0 -2 -4 0

2000

4000

6000

8000

10000

pose number Fig. 13. Computational error of new measurement setup’s computational algorithm at 15,625 poses which traverses whole motion spaces. ((a) x; (b) y; (c)z).

Each displacement traverses its motion range and takes five values with equal interval, then we can obtain 56 = 15,625 poses. The computational error of x, y, and z at these poses can be seen in Fig. 13. From Fig. 13 we can find that in whole motion spaces, the maximum computational error of three translations is 10−13 mm when firstorder Taylor expansion is used. Moreover, higher accuracy can be obtained when second-order Taylor expansion is used. In addition, there is one thing that needs to be explained. A1–3 , B1–3 , C1–3 are direction cosines in transform matrix and their specific meanings are different for different rotation’s sequence. An advantage of the algorithm is that we don’t use their specific meanings and only mutual relationship between them are used, e.g., A21 + A22 + A23 = 1 B12 + B22 + B32 = 1 C12 + C22 + C32 = 1 A1 B1 + A2 B2 + A3 B3 = 0.

(25)

A1 C1 + A2 C2 + A3 C3 = 0 B1 C1 + B2 C2 + B3 C3 = 0 ...... These properties are determined by the transform matrix itself, and they are all established for any rotation’s sequence. Therefore, for any rotation’s sequence, the derivation of computational algorithm is correct and the same result can be obtained. The reason why our measurement system can obtain so high computational accuracy is that the algorithm obtains direction cosines without an assumed stacked

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Z. Gao et al. / Precision Engineering 37 (2013) 606–620

slide model and computed translational displacements from interferometers’ readings directly without using inappropriate simplifications existed in the other two algorithms, which corresponding to compensate the errors such as Abbe and Cosine errors comprehensively. 4. Conclusions Strictly speaking, a rigid body moving in X-Y plane will produce small displacements at other degrees of freedom. In some occasions, 6-DOF motion error cannot be neglected and control system was asked to adjust in real time. Therefore, measurement system is asked to measure 6-DOF displacements with high accuracy and high real-time, in which accuracy and effort of the computational algorithm directly impact on the accuracy and real-time of displacements’ measurement. When 6-DOF displacements are considered, the relation between sensors’ reading and displacements is immensely complex and usually described as strong nonlinear coupling equation. When numerical iterative algorithm is used to compute, the real-time property will be difficult to guarantee. However, to get good real-time property, simplified computational model will also introduce relatively large error. This paper takes ultra-precision X-Y stage such as lithography’s wafer stage as research subject, and the relation of sensors’ reading and displacements was established first when 6-DOF displacements were considered. On this basis, the measurement setup and their computational algorithm of typical 5-axis and 6-axis are analyzed. Rotations are computed by two interferometers’ differential computation and high accuracy result is obtained because their ranges are usually very small. For translations, the 5-axis setup’s algorithm has simple form, however, it ignores the influence of rotations on laser path’s change and a rough estimation using low-order Taylor expansion is used, so that relatively large error is caused. The closed form solutions of 6-axis setup can be deduced, but it also contains a large number of nonlinear terms which results relatively large computational effort, and rotational displacements’ computational error is transmitted and accumulated. Through analyzing the characteristic of interferometer’s measurement model, we can find that if arranging some additional interferometers at the same directions or arranging two interferometers which are symmetric with the Z axis of the fixed frame, the simple operations of these interferometers’ readings can greatly simplify interferometers’ measurement expressions so that the difficulty in computation can be reduced. On this basis, this paper designs a 6-DOF measurement setup based on nine interferometers and derives the corresponding computational algorithm. For rotations, they are computed by two interferometers’ differential computation and high accuracy computational result is obtained because their ranges are usually very small. For translations, the closed form solutions without rotational displacements errors’ transmission are deduced by making full use of additional information, so that computational accuracy can be guaranteed. In addition, the algorithm has simple form and not contains iteration’s or transcendental function’s computation, so that computational effort is relatively small and the real-time property can be meted. A simulation result of lithography’s wafer stage demonstrates this algorithm can obtain the computational accuracy of 10−15 rad and 10−13 mm for rotational and translational displacement separately. In the derivation of the computational algorithm, this paper assumes the alignment errors of mirrors do not exist. There is no doubt that these errors exist in many cases though precision manufacturing and assembly technology are used, and they will cause error to final measurement results. In addition to this, some random errors also exist which include environmental error, nonlinear error and so on, and they can affect the final measurement accuracy. Some calibration and compensation technology should be used to reduce these errors. For example, stringent environmental safeguards are used to reduce environmental error; self-calibration method is used to calibrate systematic error and compensate it. This content is not within the scope of the present paper and will be discussed in the subsequent paper. Acknowledgements This work was supported by the National Basic Research Program of China [grant number 2009CB724205] and National Natural Science Foundation of China [grant number 51175296]. Appendix A. Appendix The geometric relationship between X-direction’s interferometer and stage at initial time can be seen in Fig. 4. X-direction’s interferometer’s measurement model will be derived and remaining two directions’ derivations are similar with it. As shown in Fig. 4, at initial time 0 the measurement path length is MPL1 = l1 = p.

(A.1)

Assuming at one moment t, the 6-DOF displacements of the stage are [x, y, z,  x ,  y ,  z ]. The rotation matrix is shown in Eq. (1) and its inverse matrix is

⎡

Q

−1

A1

B1

C1

−A1 x − B1 y − C1 z

⎤

⎢ ⎥ ⎢ A2 B2 C2 −A2 x − B2 y − C2 z ⎥ ⎢ ⎥ =⎢ ⎥ ⎣ A3 B3 C3 −A3 x − B3 y − C3 z ⎦ 0

0

0

(A.2)

1

This moment, if the coordinate of one point in OXYZ is (x0 ,y0 ,z0 ) and it’s coordinate in O’X’Y’Z’ is (x1 ,y1 ,z1 ), then these two coordinates have the following relation: (x1 , y1 , z1 , 1)T = Q −1 (x0 , y0 , z0 , 1)T .

(A.3)

The equation of X-direction’s plane mirror in O’X’Y’Z’ is x1 =

m 2

(A.4)

Z. Gao et al. / Precision Engineering 37 (2013) 606–620

617

Substituting Eq. (A.4) into Eq. (A.3), the equation of X-direction’s plane mirror in OXYZ can be obtained and listed here: m = A1 x0 + B1 y0 + C1 z0 − A1 x − B1 y − C1 z 2

(A.5)

The equation of X-direction’s interferometer’s incident laser in OXYZ is



y0 = −v1

(A.6)

z0 = −w1

Substituting Eq. (A.6) into Eq. (A.5), the intersection point of X-direction interferometer’s incident laser and mirror’s plane at time t can be obtained. Its X-direction’s coordinate is x0 =

m/2 + B1 v1 + C1 w1 + A1 x + B1 y + C1 z A1

(A.7)

As shown in Fig. 6, the length of l2 is l2 =

m m/2 + B1 v1 + C1 w1 + A1 x + B1 y + C1 z +p− 2 A1

(A.8)

From Eq. (A.5), we can find that the unit vector of the plane normal is (A1 ,B1 ,C1 ). Therefore, the cosine value of the angle  between it and the incident laser is cos  = A1

(A.9)

According to the reflection theorem, the angle between l2 and l3 is 2. Therefore, the length of l3 is l3 =

l2 cos 2

(A.10)

The measurement path length at time t is MPLt =

1 + cos 2 l2 + l3 = l2 2 2 cos 2

(A.11)

According to interferometer’s measurement principle, the X-direction’s interferometer’s reading is X = MPLt − MPL1 = −

A21 − 1 2A21

−1

p−



A1 2A21

−1

B1 (v1 ) + C1 (w1 ) + xA1 + yB1 + zC1 + (1 − A1 )

m 2

(A.12)

Appendix B. Appendix Similar to the results shown in Eqs. (5)–(7), the measurement models of the six interferometers shown in Fig. B1 can be obtained. From X1 , X2 and X3 ’s measurement models, we can get the following expression: xA1 + yB1 + zC1 =



2A1 −

1 A1

1 1 2

2

(X1 + X2 ) + X3

 

− A1 −

1 A1





p − C1 wh + b +

From Y2 , and Y3 ’s measurement models, we can get the following expression: xA2 + yB2 + zC2 =



2B2

1 B2

1 2

1

(Y2 + Y3 ) −

B2





− B2 q − C2 wh + d +

1 c 2



− (1 − A1 )

m . 2

(B.1)



1 n e − (B2 − 1) . 2 2

Fig. B1. The 6-DOF measurement setup which combines 5-DOF measurement setup (shown in Fig. 8) and the setup of measuring z (shown in Fig. 7).

(B.2)

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Z. Gao et al. / Precision Engineering 37 (2013) 606–620

From Y1 , Y2 and Y3 ’s measurement models, we can get the following expression: A3 x + B3 y + C3 z =



2B2 −

 2 (B3 − B2 ) −

−

 +

1 B2

1



2



1 (−B2 + B3 ) (−C2 + C3 )

2





1

1 B3 − − 2 B 2 (B3 − B2 ) (C3 − C2 )

+1

q+



1 C2 e + 2

(Y2 + Y3 ) − C2 d −

(B3 − B2 ) (C3 − C2 )



(k − h) −



1

2 (B3 − B2 ) −

1

2 (B3 − B2 ) −

(−B2 + B3 ) (−C2 + C3 )

1 2 (B3 − B2 ) − + 2 C − C2 3 (B3 − B2 ) (C3 − C2 ) 1

2

2

Y1 − B3

n 2



+ (C3 − C2 )

(d + e + f )

(B.3)



H − (C3 − 1) wh

We can get translational displacements’ closed form’s solutions which list below from the simultaneous equations based on the above three equations: xˆ =



2A21

1 1

−1

2

+ (A1 B1 + A2 ) + (A3 − 2A2 )

 +

2

(X1 + X2 ) + X3





+



 1

2(B3 − B2 )2 (C3 − C2 )2 − 1 (B3 − B2 ) (C3 − C2 )



  1 1

B1 2A21 − 1 A1

2



2

2

(X1 + X2 ) + X3

− [(B3 − B2 ) C2 + 2B2 C2 ] d +

+







  1

2



  1 1

C1 2A21 − 1 A1

2

+ (C3 − 1) wh +

2

e +

(X1 + X2 ) + X3





B2



C3 (B3 − B2 )2 (C3 − C2 )2 − 1 (B3 − B2 ) (C3 − C2 )



+



(B3 + B2 ) 2B22 − 1 B2



2

2 (B3 − B2 ) −





−





p − C12 b +

A1

q (B.5)

B3 (h − k − d − e − f + H + Y1 )

2



1 c 2



+ (−C1 + A1 C1 )



1 1 (Y2 + Y3 ) − (C3 + C2 ) C2 d + e 2 2

1 H C3 − C2







(C3 + C2 ) B22 − 1

+ C3 h − k + (C3 − C2 ) (−d − e − f ) +



B2



1

C1 A21 − 1

B2



(B3 + B2 ) B22 − 1

+

2

(B3 − B2 ) (C3 − C2 )

1 H C3 − C2



B1 A2 − 1   1 1 1 p − B1 C1 b + c (Y2 + Y3 ) − A1 2 2

B3 (B3 − B2 )2 (C3 − C2 )2 − 1



+

2



q

B2

(B3 − B2 ) (C3 − C2 )



(C3 + C2 ) 2B22 − 1



m



(A3 − 3A2 ) B22 − 1

+ B3 h − k + wh + (C3 − C2 ) (−d − e − f ) +



− A1 − A21





2



+

 

(B.4)

n m  2 − (B1 − A1 B1 ) − A1 + C12 − B2 + 2 2





(h − k − d − e − f + H + Y1 )

2 (A3 − 2A2 ) (B3 − B2 )2 (C3 − C2 ) − 1

(B3 − B2 ) (C3 − C2 )

zˆ =

B2



  1 1 (Y2 + Y3 ) − A21 − 1 p − A1 C1 b + c 2 2

n 1 − [(A3 − A2 ) C2 + 2A2 C2 ] d + e + (A3 − 2A2 ) (h − k) + wh + (C3 − C2 ) (−d − e − f ) + H 2 2 C3 − C2



yˆ =



(A3 − 3A2 ) 2B22 − 1





q+

m n + (C2 − B1 C1 ) 2 2

(B.6)



C3 2(B3 − B2 )2 (C3 − C2 )2 − 1 (B3 − B2 ) (C3 − C2 )



2

(h − k − d − e − f + H + Y1 )

Appendix C. Appendix From the expressions of X1 , X2 and X3 , we can obtain the following result: 2A21 − 1 A1

1 2

1 2

(X1 + X2 ) + X3



A21 − 1

A1 (1 − A1 ) m A1 C1 + + p+ 2A21 − 1 2 2A21 − 1 2A21 − 1



1 wh + b + c 2

!

= −xA1 − yB1 − zC1

(C.1)

The following expression can be deduced from Z1 and Z2 :





2(B3 − B2 )2 (C3 − C2 )2 − 1 (B3 − B2 ) (C3 − C2 )

 − =

2

2



2(B3 + B2 ) (C3 + C2 ) − 1 (B3 + B2 ) (C3 + C2 )

Z1 −

1 n

Z2 −

22

+k−h+d+e+f +g−

1 n 22

+k−h+d+e+f +g−

 (B3 + B2 ) (C3 + C2 ) − (B3 − B2 ) (C3 − C2 ) n 2 (B3 − B2 ) (C3 − C2 ) (B3 + B2 ) (C3 + C2 )





1 q−H 2

2



+ q + 2C3

n



2



1 q−H 2

 

+ k − h − wh



+2 C32 + C22 (wh + d + e + f + g) − 2A1 C1 x − 2B1 C1 y + 2 C32 + C22 z − 2H

(C.2)

Z. Gao et al. / Precision Engineering 37 (2013) 606–620

Both sides of the equal sign in Eq. (C.1) multiply 2C1 and we can obtain 2C1

2A21 − 1

1

1 2

A1

(X1 + X2 ) + X3

2





A21 − 1

A1 (1 − A1 ) m A1 C1 + p+ + 2A21 − 1 2A21 − 1 2 2A21 − 1

619

1 wh + b + c 2

! (C.3)

= −2A1 C1 x − 2B1 C1 y − 2C12 z Subtracting Eq. (C.3) from Eq. (C.2), x and y can be eliminated and the following result can be obtained:





2(B3 − B2 )2 (C3 − C2 )2 − 1 (B3 − B2 ) (C3 − C2 )





2

2

2(B3 + B2 ) (C3 + C2 ) − 1

−

(B3 + B2 ) (C3 + C2 ) 2A21 − 1

−2C1

1 2

A1

1 2

1 n

Z1 −

+k−h+d+e+f +g−

22

1 n

Z2 −

(X1 + X2 ) + X3

+k−h+d+e+f +g−

22



A21 − 1

+

2A21

−1

 (B3 + B2 ) (C3 + C2 ) − (B3 − B2 ) (C3 − C2 ) n

=

2 (B3 − B2 ) (C3 − C2 ) (B3 + B2 ) (C3 + C2 )



1 q−H 2

2

 2

p+





1 q−H 2



A1 (1 − A1 ) m A1 C1 + 2A21 − 1 2 2A21 − 1

+ q + 2C3

n

2



wh + b +



1 c 2

!

(C.4)

+ k − h − wh

+2 C32 + C2 (wh + d + e + f + g) + 2z − 2H The expression for solving z can be obtained with further simplification:



z=



2(B3 − B2 )2 (C3 − C2 )2 − 1 2 (B3 − B2 ) (C3 − C2 )



2(B3 + B2 )2 (C3 + C2 )2 − 1

−

2 (B3 + B2 ) (C3 + C2 )

−C1 − −



2A21

 −1 1 1

A1

2

2

1 n

Z1 −

(X1 + X2 ) + X3

22

1 n

Z2 −

C32

+ C22



1 q−H 2

1 q−H 2



A1 (1 − A1 ) m A1 C1 + + p+ 2A21 − 1 2A21 − 1 2 2A21 − 1

n

2



+ q − C3

n

2





A21 − 1

(B3 + B2 ) (C3 + C2 ) − (B3 − B2 ) (C3 − C2 ) 4 (B3 − B2 ) (C3 − C2 ) (B3 + B2 ) (C3 + C2 )



+k−h+d+e+f +g−

22



+k−h+d+e+f +g−



1 wh + b + c 2

!

(C.5)

+ k − h − wh

(wh + d + e + f + g) + H

From the expressions of Y1 , Y2 , Y3 and Y4 , we can obtain the following result: 2B22 − 1

1 

2B2

2



1 (Y2 + Y1 ) − Y4 2

Y3 +





− C2 wh + d +

3 3 e+ f 4 4



= xA2 + yB2 + zC2

Both sides of the equal sign in Eq. (C.1) multiply B2 and we can obtain B2

2A21 − 1

1 2

A1

1 2

(X1 + X2 ) + X3



+

A21 − 1 2A21 − 1

p+

A1 C1 A1 (1 − A1 ) m + 2A21 − 1 2 2A21 − 1



(C.6)

wh + b +

1 c 2

! (C.7)

= −xA1 B2 − yB1 B2 − zB2 C1 Both sides of the equal sign in Eq. (C.6) multiply B1 B1

2B22 − 1

1 

2B2

2

Y3 +



1 (Y2 + Y1 ) − Y4 2





− B1 C2 wh + d +

3 3 e+ f 4 4



= xA2 B1 + yB1 B2 + zB1 C2

The relations about x and z can be obtained after summing the above two equations: B2

2A21 − 1

+B1

1 2

A1 2B22 − 1

1 2

 A2 − 1 1

(X1 + X2 ) + X3 +

1 

2B2



2A21 − 1

1 Y3 + (Y2 + Y1 ) − Y4 2 2



p+

A1 C1 A1 (1 − A1 ) m + 2A21 − 1 2 2A21 − 1

− B1 C2



3 3 wh + d + e + f 4 4





wh + b +

1 c 2

! (C.9)

= −C3 x + A3 z

Thus, we can get the expression for solving x: x=−

B2 2A21 − 1 C3 A1

B1 2B22 − 1 − C3 2B2

1 

1 2

1 2

 A2 − 1 1

(X1 + X2 ) + X3 +



1 Y3 + (Y2 + Y1 ) − Y4 2 2

2A21 − 1

 B C  1 2 +

C3

p+

A1 (1 − A1 ) m A1 C1 + 2A21 − 1 2 2A21 − 1

3 3 wh + d + e + f 4 4

(C.8)

 A 3 +

C3



wh + b +

1 c 2

! (C.10)

z

Finally, the expression for solving y can be deduced from Eq. (C.6): y=

2B2 2 − 1 2B2 2

"

2B2 C2 1 [Y3 + (Y2 + Y1 )] − Y4 − 2 2B2 2 − 1



3 3 wh + d + e + f 4 4



−

2A2 B2 2B2 2 − 1

x−

2B2 C2 2B2 2 − 1

# z

.

(C.11)

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Z. Gao et al. / Precision Engineering 37 (2013) 606–620

Eqs. (C.5), (C.10) and (C.11) are formulas for solving three translations and they contain A1–3 ,B1–3 ,C1–3 which need to find out in advanced. A1–3 ,B1–3 ,C1–3 are elements of rotation matrix and they have the following properties:

⎧ 2 A1 + B12 + C12 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ A21 + A22 + A23 = 1 ⎪ ⎪ ⎪ ⎪ ⎨ A1 B1 + A2 B2 + A3 B3 = 0

(C.12)

⎪ A3 = B1 C2 − C1 B2 ⎪ ⎪ ⎪ ⎪ ⎪ B3 = C1 A2 − A1 C2 ⎪ ⎪ ⎪ ⎩ C3 = A1 B2 − B1 A2

Because of these properties, coupled with additional measurement information, A1 ,B1 ,C1 can be solved by the following formulas: S1 =

1 A1 B1 1 S2 = (X2 − X1 ) = a c 2A21 − 1

 S4 =

1 2



(X2 + X1 ) − X3 =

Y1 − Y2 1 A2 B2 S8 = = r e+f 2B22 − 1



S10 =

S3 =



1 4S12 + 4S22 + 1

= 2A21 − 1 (C.13)

S3 + 1 S1 S3 S2 S3 = B1 S6 = = C1 = A1 S5 = 2 S4 S4

A2 ,B2 ,C2 can be solved by the following formulas: S7 =

A1 C1 2A21 − 1



Y3 −

Y2 + Y1 2



=

B2 C2 2B22 − 1

S9 =



1 4S72 + 4S82 + 1

= 2B22 − 1 (C.14)

S9 + 1 S7 S9 S8 S9 = B2 S11 = = A2 S12 = = C2 2 S10 S10

Then A3 ,B3 ,C3 can be computed by the properties shown in Eq. (C.12). In summary, we can get the algorithm for solving 6-DOF displacements. Firstly, storing the constants according to geometric parameters, which can be seen in Eq. (22); Then, calculating and storing A1 ,B1 ,C1 and other items which associated with them, which is shown in Eq. (23); Finally, calculating 6-DOF displacements using the formulas shown in Eq. (24), which is obtained by substituting the above results into Eqs. (C.5), (C.10) and (C.11). References [1] Fan KC, Chen MJ, Huang WM. A six-degree-of-freedom measurement system for the motion accuracy of linear stages. International Journal of Machine Tools and Manufacture 1998;38(3):155-164. [2] Fan KC, Chen MJ. A 6-degree-of-freedom measurement system for the accuracy of X-Y stages. Precision Engineering 2000;24:15–23. [3] Gao W, Arai Y, Shibuya A, et al. Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage. Precision Engineering 2006;30:96–103. [4] Liu CH, Hsu WC, Hsu CC, et al. Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage. Review of Science Instruments 2005;76, 055110. [5] Trethewey MW, Sommer HJ, Cafeo JA. A dual beam laser vibrometer for measurement of dynamic structural rotations and displacements. Journal of Sound and Vibration 1993;164(1):67–84. [6] Bokelberg EH, Sommer HJ, Trethewey MW. A six-degree-of-freedom laser vibrometer, Part I: theoretical development. Journal of Sound and Vibration 1994;178(5):643–54. [7] Park WS, Cho HS. Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation. Optical Engineering 2002;41(April (4)). [8] Bonev IA, Ryu J. A new method for solving the direct kinematics of general 6-6 Stewart Platforms using three linear extra sensors. Mechanism and Machine Theory 2000;35:423–36. [9] Loopstra R, Heeze, Straaijer A. Interferometer system and lithograph apparatus including an interferometer system 2000. US006020964A. [10] Loopstra R. Method and apparatus for repetitively projecting a pask pattern on a substrate, using a time-saving height measurement 2001. US006208407B1. [11] Freriks HJM, Heemels WPMH, Muller GJ, et al. On the systematic use of budget-based design. In: Sixteenth annual international symposium of the international council on systems engineering (INCOSE), July. 2006. p. 8–14. [12] Brink MA, Stoeldraijer JMD, Linders HFD. Overlay and field by field leveling in wafer steppers using an advanced metrology system[C], SPIE. Integrated Circuit Metrology, Inspection, and Process Control VI 1992;1673. [13] Brink MA, Straaijer A. Lithographic apparatus for step-and-scan imaging of mask pattern with interferometer mirrors on the mask and wafer holders[P]. 2000;US006084673A. [14] Agilent laser and optics, user’s manual, vol. II; July 2007.