Error analysis of a flexure hinge mechanism induced by machining imperfection

ELSEVIER

Error analysis of a flexure hinge mechanism induced by machining imperfection Jae W. Ryu and Dae-Gab Gweon Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Taejon, South Korea Modeling of manufacturing tolerances for m a c h i n i n g of a monolithic flexure hinge m e c h a n i s m is presented. This m o d e l i n g uses a computer-based m e t h o d to generate the equations of m o t i o n for the mechanism and to predict the i n d u c e d m o t i o n errors for various types of machining imperfections. This paper describes the m o d e l i n g and the quantitative analysis of the m o t i o n errors using a well-known simple c o m p o u n d linear spring as an example. Based on the simulation results of the example problem, effects of the m a c h i n i n g imperfection types on m o t i o n errors are generalized. The simulation demonstrates that the imperfection of the center position and the size of a m a c h i n e d hole provide an in-plane m o t i o n error X, Y, Oz. In addition, the m a c h i n i n g error in the perpendicularity of the hole with respect to the plate also provides an out-of-plane parasitic error Z, 0x, 0v. © 1997 Elsevier Science Inc.

Keywords: error analysis, flexure hinge mechanism, high precision mechanism, m a c h i n i n g imperfection

Introduction Translational stages a n d / o r m o t i o n guides with piezoelectric actuation and monolithic flexure hinge mechanisms have been popular for such precision mechanical scanning systems as scanning tunneling microscopes (STM). 1-3 Such mic r o m o t i o n stages utilizing the flexure hinge mechanism can have many advantages: negligible backlash and stick-slip friction; s m o o t h and continuous displacement; adequate for magnifying the o u t p u t displacement of piezo actuator; and inherently infinite resolution. In ideal cases, the monolithic flexure-hinged translational stages a n d / o r the m o t i o n guides can be designed to provide pure straight line motions. In actual cases, however, the motions include parasitic errors caused by m a c h i n i n g imperfections. It is possible that these parasitic errors are signifiAddress reprint requests to Dr. D-G. Gweon, KAIST, Department of Mechanical Engineering, 373-1, Kusong-Dong, Yusong-Gu, Taejon 305-701, South Korea. E-mail: [email protected] Precision Engineering 21:83-89, 1997 © 1997 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010

cantly influenced by the degree of machining tolerances. For example, Jones 4 provided the simple formula of the Z-axis angular error 0 z caused by the positioning errors of the flexure hinge in the X- (motion axis) and Y-axes. He considered a simple linear spring (parallel four-bar mechanism) as the example problem. Smith et al. ~ described the design methodology for the monolithic flexure hinge mechanism with manufacturing tolerances. However, to d e t e r m i n e adequate machining tolerances, the design still requires a more generalized m e t h o d and also information about the effects of the different types of manufacturing tolerances on the m o t i o n of the hinge mechanism. The objective of this study is to predict the parasitic errors quantitatively for various manufacturing tolerances and to provide a better design for the mechanism. Therefore, in this study, the modeling procedures for the manufacturing imperfections arising from geometric errors during the machining for the hinge m e c h a n i s m are described. The modeling uses a computer-based m e t h o d that automatically generates the equa0141-6359/97/$17.00 PII S0141-6359(97)00059-7

Ryu a n d Gweon: Error analysis o f flexure hinge m e c h a n i s m Sl

~)

2 x~

4: secondary platform

z

x

y'

.

~)

3®(

[_J 1 : primary platform

Figure 1 Simple compound linear spring (moving bodies: 1, 2 . . . . . 6, flexure hinges: •, ® . . . . . ®) tions of m o t i o n for a flexure hinge m e c h a n i s m 5 a n d estimates the m o t i o n errors quantitatively for various m a n u f a c t u r i n g tolerances. T h e m o d e l i n g p r o c e d u r e s are d e m o n s t r a t e d using a well-known simple c o m p o u n d linear spring as the example. Finally, the relationships between the manufacturing tolerances and the m o t i o n errors o f the mechanism are discussed.

Manufacture of a flexure hinge mechanism Figure 1 shows a simple c o m p o u n d linear spring with monolithic flexure hinges. T h e m e c h a n i s m provides a linear m o t i o n along the X-direction. It can be n o t e d that the simple linear spring (parallel four-bar m e c h a n i s m ) has i n h e r e n t Y-directional parasitic e r r o r for a given X-directional motion. However, the simple c o m p o u n d linear spring, shown in Figure 1, can eliminate the Ydirectional parasitic e r r o r by c o m p e n s a t i n g for the errors by a d d i n g one m o r e parallel four-bar m e c h anism to it. Consequently, the primary platform can have an X-directional motion. To describe the simple c o m p o u n d linear spring in the figure the moving bodies are d e n o t e d by n u m b e r s 1 - 6 (not circled). Note that the flexure hinges are also d e n o t e d by n u m b e r s 1-8 (circled), a n d each flexure hinge, as can be seen f r o m Figure 2, has 3 constructional p a r a m e t e r s (i.e., m i n i m u m hinge thickness t, hinge radius R, a n d hinge width b) affected by the manufacturing. In general, manufacturing o f a monolithic flexure hinge mechanism has several different m a c h i n i n g processes. T h e y include drilling a n d r e a m i n g processes for m a c h i n i n g holes. In addition, wire electrical discharge m a c h i n i n g (WEDM) is also used to cut out the blank. 84

Figure 2 system

Single-axis flexure hinge and coordinate

It is well known that the spring rates of a notch-type flexure hinge are the function of b, R, a n d t. T h e y can be calculated by Paros and Weisb o r d equations for a single-axis flexure hinge. 6 T h e r e f o r e , dimensional errors of the parameters o c c u r r i n g d u r i n g the m a c h i n i n g processes can result in m o t i o n errors. For example, the drilling process can provide dimensional errors to the hole diameters, the hole locations, and the perpendicularity o f the holes with respect to the plate, etc. To the contrary, the WEDM process does not m a k e a serious impact on the m o t i o n errors, because the m a c h i n i n g errors cause c h a n g e in the body shapes only.

Static analysis of a flexure hinge mechanism Let us assume that the bodies and the hinges in a flexure hinge m e c h a n i s m are rigid bodies and translational/rotational springs, respectively. T h e n , the flexure hinge m e c h a n i s m can be ass u m e d to be a spring-mass system. Disregarding the dynamics o f the system, the m o t i o n of the flexure hinge m e c h a n i s m can be given by the following.

Kx = F

(1) i" qN, f ] where, x = [ql'~ . . . . q . . . . , T is the system displacement vector, o i is the displacement vector . . . . . . r -r o f 0", i o r q i = rt x t,. y ,.tz ,.uz ~. ,'~z , % ,, ~u j ,~z~ 1 , r~ = rtte l . . . . . oz t , , f N , ] T.lS the system force vector, f i is the resolved force vector o f body i, and K is the system stiffness matrix whose c o m p o n e n t s can be calculated f r o m SEPTEMBER/DECEMBER 1997 VOL 21 NO 2/3

Ryu a n d Gweon: Error analysis of tlexure h i n g e m e c h a n i s m i

{ ~ Ti'bt¢Ti ..K,i= k:l -1,,~ p, -i'/%kTj

i=j i ~ .j

for for

(2)

(Bodies i a n d j are c o n n e c t e d by flexure h i n g e k) In E q u a t i o n (2), Nh is total n u m b e r of m o v i n g bodies, N', is the n u m b e r of flexure hinges conn e c t e d to body i, T~, a n d T~ are geometric constant matrices, where the connectivity points of the h i n g e k on bodies i a n d j are d e n o t e d by p' a n d ~, respectively, a n d kh is the stiffness matrix of the flexure h i n g e k, which is r e p r e s e n t e d with respect to the global c o o r d i n a t e system. 5 Generally, the h i n g e c o o r d i n a t e system shown in Figure 2 is n o t aligned with the global c o o r d i n a t e system. A hom o g e n e o u s transformation matrix R k f r o m the h i n g e c o o r d i n a t e system to the global coordinate system is o b t a i n e d by

il[cos 0 sin0l

[ cos
0 "

×

1

-sin +; 0

0

1

0

0

cos +;

-siOn ~b;]

0

sin q5'x

4,; J

= R<,:&4,R<,,

•A x y~

*i,/

0

0

0

0

0

Ay

0

0

0

Ay M.

0

0

Az F

0

Az N/~

0

0

0

0

0

0

0

0

0

aB .....

0

0

0

0

kt,( b, R, t) =

Aa M

±~

0

Ay

F_

Mr

1

ky 0

(5)

In Equation (5), each e l e m e n t except for A a / M x is calculated using Paros a n d Weisbord equations 6 for a single-axis flexure h i n g e (Figure 2) a n d A a / M x can be o b t a i n e d f r o m the torsional e q u a t i o n of a rectangular bar. 7 It can be n o t e d that the stiffness matrix of the flexure h i n g e can be r e p r e s e n t e d as the f u n c t i o n of the h i n g e h e i g h t b, h i n g e radius R, a n d h i n g e thickness t, a n d each h i n g e of the simple c o m p o u n d linear spring shown in Figure I has ~bx, ~b~,,a n d ~b~ of 0, 0, a n d rr/2, respectively.

(3) M o d e l i n g of machining errors

where ~b!~,6,"., a n d ~bk are the Eulerian angles f r o m h-th h i n g e c o o r d i n a t e system to the global coordinate system. T h e r e f o r e , the stiffness matrix h* is f o u n d as follows. K' =

[ 1 [.' 0] Rk

0

0

Rk kh 0

In this section, various types of m a c h i n i n g errors that can cause m o t i o n errors are generalized, a n d the equations of m o t i o n considering the machining errors are presented. E r r o r s in p l a t e t h i c k n e s s

I,

Rk

(4)

where k~ is the stiffness matrix of the flexure h i n g e spring k with respect to the h i n g e coordinate syst e m a n d can be r e p r e s e n t e d as follows.

For a given d i m e n s i o n a l error Ab in the plate thickness, the actual flexure h i n g e width b* after m a c h i n i n g is b - Ab~
(6)

It is clear that the spring rates of the m a c h i n e d h i n g e can be c h a n g e d because of the new value of h i n g e width b*. It also arrives because the d i m e n -

~

R+dR

l

t*3R=t-2dR PRECISION ENGINEERING

:R-dR

dR= t

t ,jR=t+2z/R

Figure 3 Change in the hinge thickness due to the hole radius variation 85

Ryu a n d Gweon: Error analysis o f flexure hinge m e c h a n i s m zJc

~c

+÷ i

~/C=0

~ +÷

zJc

zJc

+

[

++

++

/

t ~c=t-2zlc

zc= t

a¢=t+2Ac

(a) change in the hinge thickness

t

i

/..-"~

c=0

~

=-,de

i--Ay,#~=- Ac

Ay,~c=0

Z/yz, =

Ac

(b) shift of the hinge rotational axis z/0~= -2Ac/(2R*+t*)

AOz=O

dOz = 2 ~ c/(2R*+ t*)

x

2R*+ t*

2R*+t*

Figure 4 Various types of dimensional change due to the hole center variation

(c) rotation of the hinge about z-axis

sional error Ab causes a new system stiffness matrix Kab a n d resultant parasitic errors.

t + 2AR/> t*./> t - 2AR

(8)

Errors in the location of hole centers Errors in hole radius For a given dimensional error AR in the hole radius, the actual hole radius R* after drilling is R - A R < ~ R * ~ < R + AR

(7)

Note that the difference in the hole radius changes the value o f the hinge thickness from the n o m i n a l value (Figure 3). Therefore, for a given dimensional error AR in the hole radius, the actual hinge thickness ~ARafter m a c h i n i n g is 86

It is clear from Figure4a that for a given positional error Acin the location o f a hole center, the actual flexure hinge thickness ~c after machining is t - 2Ac ~< t*,.~< t + 2Ac

(9)

Figure 4b also

shows that the a m o u n t of the positional shifts o f the rotational axis can be given by - A c ~< Ax~c ~< Ac

(10)

- A c ~< Ay~c<~ Ac

(11)

SEPTEMBER/DECEMBER 1997 VOL 21 NO 2/3

Ryu a n d Gweon: Error analysis of flexure hinge m e c h a n i s m

~z

z

y

x

y

(a) rotation about x axis

x

(b) rotation about y axis

In addition, n o t e that the hinge with the e r r o r Ac causes the rotation of the hinge about z-axis with the a m o u n t o f di0= (Figure 4c). T h e value o f 2i0= can be given by 2Ac 2Ac ~< 2X0=~< 2R* + t* 2R* + t*

(12)

In addition, Ac and A~b provide the rotations o f the hinge about the x-, y-, a n d z-axes of the hinge coordinate system A0~, AOy, a n d di0z. From Equation (4), the resultant stiffness matrix with respect to the global coordinate system caused by the rotation of the hinge is

w h e r e R* is f r o m Equation (7) and t* = tz~,e + t*Ac

Perpendicularity of hole

centers

W h e n the m a c h i n e d hole c e n t e r is n o t perpendicular to the plate (xy-plane of the hinge coordinate system), it can result in rotations of the hole with respect to the x-axis a n d / o r y-axis o f the hinge coordinate system. F r o m Figure 5, for a given angular e r r o r A~b f r o m the n o m i n a l position of a hole center, the resultant rotations of the hinge with respect to x-axis a n d / o r y-axis can be given by -Aq5 ~< A0~ ~< dish

(13)

-A4~ ~< A0,, ~< A4)

(14)

Figure 5 Rotations of the hinge induced by inclined hole center axis

k'k=

[,: o.] [,. ' k~ 0

~,, = k,,(b*, R*, t*) Table 1

(15)

(161

where a h o m o g e n e o u s rotational matrix R *k, [see Equation (3)], can be written as follows.

R *k= R,~ ao~.zR,~ Ao~,,R~, ,,o~

(17)

As m e n t i o n e d before, there can be positional shifts o f the origin of hinge axis for the given positional errors o f the hole Ac. It also shifts the position o f the connectivity points of the hinge. T h e r e f o r e , from Equations (2) and (16), the new system stiffness matrix K* after considering the m a c h i n i n g errors can be given by the following.

System stiffness matrix after machining From Equations (6) t h r o u g h (9), for the given m a c h i n i n g tolerances dib, AR, a n d Ac, the mac h i n e d hinge has b*, R*, t* ( = ~,e + ~ ) , a n d the following stiffness matrix, which is different f r o m the o n e evaluated using the n o m i n a l hinge parameters b, R, t.

n*']

for

i= j

for

i 4=j

(18)

Note that bodies i a n d j are c o n n e c t e d by the flexure hinge k of which the actual connectivity points on bodies i a n d j are d e n o t e d by p*~ and q.l, respectively, as the result of the resultant shifts o f the hinge origin.

Construction parameter

b

R

t

L

S,

S2

H

E

G

10.0 mm

1.5 mm

1.0 mm

20.0 mm

50.0 mm

30.0 mm

10.0 mm

72 GPa

27 GPa

PRECISION ENGINEERING

87

Ryu and Gweon: Error analysis of flexure hinge mechanism A¢ (mrad) 10~ 0

A¢ (mrad)

5

10

I~'~!'~L"L"!!'"!IL"~L"~"

o !!!!!!!!!!!!!!!::!!!!!!UA!!!!!!::!!! "~ 10

de (mrad)

1~-~0 5 10 u iiiii~i~iiiii~ii~iiiii~ii_:ii~i~iiii !!!!!::!!!!!!!::!!!!!::!!

1~-~0 5 U :~!~ii~iiii~i !!!!!!!f!!!!!!!!!!!!!!!!!!!

10 ! i ' m ' ' ' ' ; " =

10

-2

"~ 10-~

:.m,==-

...

10"3 " ~ 10*

/U

"---" 10-3 10-~g Z " ............ !.................... 0 0.05 0.1 Ab, AR, Ac (mm)

.... Ab ............

0

0.05

Ab, ~R, Ac (mm)

(a)

"~ 10 !!!!!!!!!!!!!!!!!!!!i!!!!!!!i!!!fi!!!!!!!i ~::::::!!!!!!!!!!!!!!i!!!!!!!!!!!!!!!!!!!!!

1 0 .................... i .................... 0 0.05 0.1

(c)

A¢ (mrad)

i

10 ! ! ! ! ! ! ! l i l A C

101

! ................ "="... ". . . .

Ab, AR, Ac (mm)

A¢ (mrad)

~o 0 5 10 /U !!ii!!!!!!!!!!ii.:!!!!!!!!!!!!!!!!!!!!!!!!!

................ ?0

'~

A!!!!i!:::

!!~ :::!i:!!!!!!!! A¢!!!!! ~10_3 ............. :::::i::::::::: ...... ::::: ~-'~ ' I[~~:'!!!!!!!!!!!!!!!!!!!!!!!!i!!!!!!!!!!!!!!!!!!!!

(b)

A¢ (mrad)

loOO...................

0.1

.................... ~ i ~ ] ~ ! ~

10

.... iiiii~iii

10o 0

5

10

0.05

0.1

i:::::::::::::::::::::::::::::::: Z21111112111111111i AIIIIT ii .... ;i!:

"~ -2 ¢D 0

0.05

Ab, AR, Ac (mm)

0.1

10"30

0.05

Ab, AIR,Ac (mm)

(d) Figure

6

0.1

(e)

10"30

Ab, AR, Ac (mm) (f)

Simulation results of the motion errors for a given X directional movement of the primary

platform Simulation

results

Ideally the simple c o m p o u n d linear spring shown in Figure I is a plane motion mechanism and has only three degrees of freedom, X, Y, Oz. However, in this study, the model considers three additional degrees of freedom Z, Ox, and 0 v for the analysis on the parasitic motion errors. Note that the position of the bodies can be obtained by applying a static X-directional force to the primary platform of the simple c o m p o u n d linear spring. The size of the displacement vector x is (6 × 6) × 1, because there are six moving bodies in the system, and each body has six degrees of freedom. Similarly, the force vector F also has the same size (6 × 6) × 1, and the system stiffness matrix K has the size of (6 × 6) × (6 x 6). Table I shows the nominal value of the parameters and material properties applied to the simple c o m p o u n d linear spring example. Note that all the hinges have the same nominal values for b, R, and t. Figure 6 describes the calculated characteris-

88

tics behaviors of the motion errors for a given X-directional movement of the primary platform of the simple c o m p o u n d linear spring shown in Figure 1. In Figure 6, the six different directional motions have been considered. For example, consider Figure 6a. For the computation of the motion error for the given value of Ab, each hinge takes either b - Ab or b + Ab for the actual hinge width, and the X-directional motion error is computed by applying a X-directional static force to the p r i m a ~ platform. It can be noted that there can be 2 ' combinations for the computation of the maxim u m error. The maximum motion error is the largest value selected among the 2 N" n u m b e r of computed errors. The similar procedures have been applied for the computation of the maxim u m motion errors for the other types of the manufacturing tolerances. In addition, the maxim u m motion errors considered all the types of manufacturing tolerances simultaneously is also given by the graph with the letter A in the figure.

SEPTEMBER/DECEMBER 1997 VOL 21 NO 2/3

Ryu and Gweon: Error analysis of flexure hinge m e c h a n i s m It can Be n o t e d that the m o t i o n errors can be increased as the m a n u f a c t u r i n g tolerances are increased. For example, from Figures 6a a n d b, for 0.05 m m of hole location error (Ac) gives 33% o f X-directional m o t i o n error a n d 0.5% of Y-directional m o t i o n error, respectively. It is also clear from the figures that 0.1 m m o f Ac gives 80% of X-directional m o t i o n error a n d 1.5% of Y-directional m o t i o n error, respectively. It also gives 0.17 arcsec/Ixm of rotational errors a b o u t Z-axis (0z) tbr the given X-directional m o t i o n (Figure 6j). In o t h e r words, tbr 0.05 m m of Ac, 50 n m of Ydirectional and 0.8 arcsec of 0 z rotational parasitic errors can be i n t r o d u c e d for 10 /xm of X-directional m o v e m e n t of the primary platform. T h e error in the hole radius AR gives the same patterns as the case of Ac. F r o m Figures 6b a n d f, however, it can be n o t e d that Ac has m o r e sensitive effect on Y-directional and 0 z rotational parasitic errors. T h e errors in the hole radius a n d location have small effects on the out-of-plane parasitic errors Z, 0x, Or. O n the contrary, it is evident from Figures 6c, d, e, the perpendicularity o f the hole center with respect to the XY-plane provides a significant effect on the out-of-plane parasitic errors that include Z-directional m o t i o n error and the tilt motion errors 0 x, 0 v. For example, 5 m r a d of A& provides 0.7% of Z-directional parasitic errors (Figure 6c) a n d it also provides 0.10 arcsec//xm and 0.06 arcsec//xm of tilt errors a b o u t X- and Y-axes, respectively. T h e simulation results also indicate that even a perfectly m a n u f a c t u r e d simple c o m p o u n d linear spring can have the rotational errors a b o u t the Z-axis (Figure 6]). It may be that the hinges numb e r e d @, ®, ®, a n d ® take axial tensile stresses (along the x-axis of the hinge c o o r d i n a t e system), a n d the hinges n u m b e r e d ®, @, ®, a n d ® take compressive stresses, respectively, w h e n a X-directional force is applied to the primary platform. As a result, the primary platform can rotate with respect to Z-axis. H e n c e , it can be c o n c l u d e d that a d o u b l e c o m p o u n d rectilinear spring is m o r e appropriate tbr a m o r e accurate m o t i o n because it can c o m p e n s a t e for such rotational error by add-

PRECISION ENGINEERING

ing o n e additional simple c o m p o u n d linear spring to the original mechanism.

Conclusions Effects of the m a n u f a c t u r i n g tolerances on the m o t i o n of a monolithic flexure hinge m e c h a n i s m have b e e n presented. This p a p e r provides a computer-based m o d e l i n g p r o c e d u r e that generates the equations of m o t i o n for the flexure hinge mechanism. It can predict the a m o u n t of the motion errors quantitatively. A simple c o m p o u n d linear spring has b e e n used to d e m o n s t r a t e the modeling p r o c e d u r e . T h e simulation demonstrates that machining errors in the position and size of the holes can have serious influences on the inplane m o t i o n error X, Y, Oz. O n the contrary, the machining error in the perpendicularity o f holes with respect to the plate has a serious influence on the out-of-plane parasitic errors Z, 0 x, 0p The m e t h o d p r e s e n t e d in this p a p e r can be applied to o t h e r types of monolithic flexure hinge mechanisms.

Acknowledgment This work was f u n d e d by the Korea Science and Engineering F o u n d a t i o n (KOSEF). T h e authors thank KOSEF for their financial s u p p o r t (94-020009-3).

References 1 Furukawa, E. and Mizuno, M. "Piezo-driven translation mechanisms utilizing linkages," Int J Japan Soc Prec Eng, 1992, 26, 54-59 2 Furukawa, E., Mizuno, M. and Hojo, T. "A twin-type piezodriven translation mechanism," Int J Japan Soc Prec Eng, 1994, 28, 70-75 3 Smith, S. T., Chetwynd, D. G. and Bowen, D. K. "Design and assessment of monolithic high precision translation mechanisms," J Phys E: Sci Instrum, 1988, 20, 977-983 4 Jones, R. V. "Parallel and rectilinear spring movements," J Sci Instrum, 1951, 28, 38-41 5 Ryu, J. R. "Six-axis ultraprecision positioning mechanism design and positioning control," Ph.D. diss., KAIST, Taejon, South Korea, 1997 6 Paros, J. M. and Weisbord, L. "How to design flexure hinge," Machine Design, 1965, 37, 151-157 7 Timoshenko, S. and Goodier, J. N. Theory of Elasticity, 3rd ed. New York: McGraw-Hill, 1970, pp. 309-313

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