Shaft Diameter Measurement Using Structured Light Vision

Transcription

1 Sensors 05, 5, ; do:0.3390/s Artcle OPEN ACCESS sensors ISSN Shaft Dameter Measurement Usng Structured Lght Vson Syuan Lu, Qngchang Tan * and Yachao Zhang College of Mechancal Scence and Engneerng, Jln Unversty, Changchun 300, Chna; E-Mals: syuan@mals.jlu.edu.cn (S.L.); zhangyc4@mals.jlu.edu.cn (Y.Z.) * Author to whom correspondence should be addressed; E-Mal: tanqc@jlu.edu.cn; Tel.: ; Fax: Academc Edtor: Vttoro M. N. Passaro Receved: 9 Aprl 05 / Accepted: 6 August 05 / Publshed: August 05 Abstract: A method for measurng shaft dameters s presented usng structured lght vson measurement. After calbratng a model of the structured lght measurement, a vrtual plane s establshed perpendcular to the measured shaft axs and the mage of the lght strpe on the shaft s projected to the vrtual plane. On the vrtual plane, the center of the measured shaft s determned by fttng the projected mage under the geometrcal constrants of the lght strpe, and the shaft dameter s measured by the determned center and the projected mage. Experments evaluated the measurng accuracy of the method and the effects of some factors on the measurement are analyzed. Keywords: mage measurement; shaft dameter; structured lght vson. Introducton Shafts are one of most mportant machne elements and the accuracy of machned shafts has a sgnfcant effect on the propertes of any mechancal devce that employs one. The accuracy of a machned shaft can be ensured usng a Computer Numercal Control lathe, but mechancal wear n a CNC lathe system, system deformaton, and the radal wear of the machnng tools wll reduce the accuracy of the machned shaft. If the varaton of the shaft dameter could be montored durng ts machnng, the dmensonal accuracy of the product could be ensured by the automatc compensaton system of the CNC lathe. Therefore, on-lne measurement of shaft sze s very mportant n machnng. A varety of noncontact machne vson methods for measurng the product dameters have been proposed [ 5]. These methods can be classfed as actve methods and passve methods. Passve

2 Sensors 05, measurement methods use a camera or two cameras to obtan shaft dameters [6,7]. In 008, Song et al. [8] proposed a method where both edges of a shaft were maged onto two cameras through two parallel lght paths as shown n Fgure [8]. Usng ths approach, the shaft dameter was obtaned by measurng the length of the photosenstve unts on the shadow feld of the two CCDs (Charge-Coupled Devce) and the dstance between the two parallel lght paths. Ths method can amplfy the measurement range whle ensurng the measurng accuracy; however, t s very dffcult to calbrate the poston of the two cameras. Fgure. The optcal prncple of the parallel lght projecton method wth double lght paths (wth permsson from [8]). In 03, Sun [9] used a camera to obtan shaft dameters, employng a theory of plane geometry n the cross-secton, whch s perpendcular to the axs of the shaft, as shown n Fgure [9]. Ths method can acheve a hgh measurng accuracy, but t requres the calbraton of the plane of the center lne of the shaft before the measurement. In practce, the poston of the center lne changes each tme the shaft s clamped by the jaw chucks of the lathe. Ths change of poston wll decrease the accuracy of measurement. Fgure. The orentatons of the shaft and the measurement plane n 3D space (wth permsson from [9]).

3 Sensors 05, Actve measurement methods generally use one or more cameras and lasers to acheve accurate measurements of shaft dameters. In 00, Sun et al. [0] used a par of lne-structured lasers and cameras to measure shaft dameters, as shown n Fgure 3 [0]. The lghts was projected onto the ppe, then captured by two cameras, and the dameter of a steel ppe could be determned by fttng the ellpse wth the two developed arcs. It s dffcult to ensure that the two arcs are on the same plane, so the accuracy of the method s lmted. In 00, Lu et al. [] reported a method for measurng shaft dameters usng a lne-structured laser and a camera, as shown n Fgure 4 []. The coordnates of the lght strpe projected on the shaft were obtaned usng a novel gray scale barycenter extracton algorthm along the radal drecton. The shaft dameter was then obtaned by crcle fttng usng the generated coordnates. If the lne-structured lght s perpendcular to the measured shaft, the shaft dameter can be obtaned by fttng a crcle. Thus, ths method s lmted by the measurement envronment. Fgure 3. Sketch of the measurement system (wth permsson from [0]). Fgure 4. Mathematcal model of the system (wth permsson from []). In ths study, a method for measurng shaft dameter s proposed usng a lne-structured laser and a camera. Based on the drecton vector of the measured shaft axs, whch s calbrated before the measurement, a vrtual plane s establshed perpendcular to the axs and the mage of a lght strpe on the shaft s projected to the vrtual plane. The shaft dameter s then determned from the projected mage on the vrtual plane. To mprove the measurng precson, the center of the projected mage was determned by fttng the projected mage usng a set of geometrcal constrants. Ths report s organzed as follows: Secton calbrates the structured lght system. Secton 3 outlnes the model of the proposed method. Secton 4 reports the expermental results used to test the measurng accuracy and nfluences of several factors. Secton 5 provdes the study s conclusons.

4 Sensors 05, Calbraton of the Structure Lght Measurng System.. Calbraton of Camera Parameters The calbraton of the measurement system employed world coordnate systems (O WX WY WZ W), a camera coordnate system (OCXCYCZC), a mage coordnate system (oxy), and a pxel coordnate system (Ouv), whch were establshed as shown n Fgure 5. Fgure 5. Calbraton model of the lne structured lght. Based on pnhole projecton and lens dstorton models, the mappng from the world coordnates to the pxel coordnates can be expressed as reported by Zhang [] and shown n Table, where R s the rotaton matrx; T s translaton vector; k, k, p, and p are the coeffcents of radal and tangental α γ u0 dstortons; A = 0 v β 0 s the nteror camera parameters; ZW = 0 n the model. A nonlnear 0 0 functon can be establshed by mnmzng the dstance between the calculated pxel coordnates of the corner ponts n the calbraton board and the actual pxel coordnates. The functon was solved usng the Levenberg-Marquardt algorthm [3]. Table. Coordnate transformaton of camera model. Transformaton From world coordnates to camera coordnates From camera coordnates to mage coordnates Introduce the dstorton model, where, r =x d+y d, From mage coordnates to pxel coordnates Equatons XC XW Y C R Y = W + T Z C xu X c y = u Zc Yc d d + ( + d) ( + d) + d d x u 4 xd p xy p r x ( kr kr ) y = + + u y + d p r y px y u α γ u0 xd v 0 v 0 y = β d 0 0 () () (3) (4)

5 Sensors 05, Calbraton of Structured Lght Parameters The calbraton model of the structured lght s shown n Fgure 5. Here, AB s the ntersecton lne between the lght plane and the calbraton board, P s a pont on AB. The pxel coordnate of P can be extracted as descrbed by Steger [4], and camera coordnates of P are determned usng Equatons () (4). Dfferent ntersecton lnes can be obtaned by turnng the board, so P j s the jth pont on the ntersecton lne when the board s n the th poston. Settng the equaton of the lght plane under OCXCYCZC: bx C + by C + bz 3 C 000 = 0 (5) The parameters b, b, b3 can be determned by the objectve functon: n k bx Cj+ by Cj+ bz 3 Cj = (6) = j= mn where n s number of board turns, k s the number of sample ponts on the ntersecton lne when the board s n the th poston, and (X Cj,Y Cj, Z Cj) are camera coordnates of P j. Accordng to the prncple of least squares, the coeffcents of Equaton (5) can be solved as: n k n k n k n k X cj X cjy cj X cjz cj X cj = j= = j= = j= b = j= n k n k n k b n k = 000 X cjy cj Y cj Y cjz cj Y cj = j= = j= = j= b = j= 3 n k n k n k n k X cjz cj Y cjz cj Z cj Z cj = j= = j= = j= = j= (7) Because the camera parameters and coordnates of the sample ponts have errors, the coeffcents obtaned by Equaton (7) wll be used as ntal values for optmzng the lght plane. The objectve functon of the optmzed plane s establshed by the dstances between the ponts and the plane: n k bx Cj + by Cj + bz 3 Cj 000 mn (8) = j= b + b + b 3 The parameters of the plane can be obtaned usng the Levenberg-Marquardt algorthm. In the experment, when the constant term of Equaton (5) s, the parameters of the plane are small and the denomnator of Equaton (8) s around 0. To mprove the optmzaton results, the constant term of Equaton (5) s set at 000 n the paper. 3. The Prncple and Model of the Measurement The model for measurng the shaft dameters s shown n Fgure 6. A shaft s clamped at two centers, mages of the shaft are captured by a CCD camera, п s the lght plane, and MN s the axs of the shaft. The strpe ab s formed by projectng the lght plane п onto the measured shaft. The pxel coordnates P ( u, v ) = of ab strpe centers are extracted by Steger s algorthm. The vrtual plane п p p,,..., N perpendcular to the measured shaft s created and the normal vector of п s the drectonal vector of the

6 Sensors 05, axs MN. Although the poston of the axs changes slghtly every tme a shaft s clamped, the drecton of the axs does not change. Thus, the equaton of the vrtual plane п can be obtaned n OCXCYCZC (see Appendx A for the detals): bx C + by C + bz 3 C 000 = 0 (9) The arc cd s formed by the vrtual plane п ntersectng the measured shaft. (a) (b) Fgure 6. The measurement model of the shaft. (a) global vew, (b) local vew. Snce the lght plane п s not perpendcular to the shaft, the cross-secton s an ellpse and the shaft dameter can be obtaned by fttng the ellpse wth strpe centers P. The ellptc equaton s: x Axy By Cx Dy E = 0 (0) The parameters of Equaton (0) can be obtaned by the least squares method, and the length of the mnor axs d s the measured shaft dameter: d = ( ACD BC D + 4 BE A E) ( A 4 B)( B+ A + ( B) + ) () Snce the vrtual plane п s perpendcular to the shaft, the cross-secton s a crcle and the dameter can be obtaned by fttng the crcle wth ponts Q, whch are P projectng on the vrtual plane п. The crcle equaton s x y ax by c = 0 () The parameters of Equaton () can be obtaned by the least squares method, and the crcle dameter d s the measured shaft dameter: d a + b 4 = c (3) Snce the pxel coordnates of the extracted strpe centers have errors, from the numercal analyss, the measurng accuracy of the shaft dameter by crcle fttng s better than by ellpse fttng (see Appendx B for the detal).

7 Sensors 05, The mage coordnates P u = ( x, ) T u y u =,,3... N of P can be acheved usng Equatons (3) and (4), so the equatons of OCP are: X = Y y Z x P u ZP P = u P The camera coordnates P = ( X,, P YP Z P) =,,3... N can be obtaned usng Equatons (5) and (4): (4) X P = 000 xu / ( b xu + b yu + b3) YP = 000 yu / ( b xu + b yu + b3) ZP = 000 / ( b xu + b yu + b3) (5) A set of lnes P Q are made by passng through the extracted centers P and parallel to the measured shaft axs, as shown n Fgure 6b. Snce the axs MN s perpendcular to the plane п, the drecton vectors of the lnes are the normal vector of п, ( b, b, b ). Thus, the equatons of lnes P Q are: 3 x X p y Yp z Zp = = = t b b b3 x X p y Yp z Zp = = = t b b b3 N N N x X p y Yp z Zp = = = t b b b3 N (6) Each lne P Q has an ntersecton pont Q on the plane п. The camera coordnates Q ( x, y, z ) = of the ntersecton ponts are obtaned usng Equatons (9) and (6), respectvely.,,3..., N Fgure 7. Oe XeYeZe and OC XCYCZC. To facltate the crcle fttng, a coordnate system (Oe XeYeZe) s establshed on the plane п, as shown n Fgure 7. In Oe XeYeZe, XeYe plane s the plane п, OeZe axs s perpendcular to the plane п, and the orgn of coordnate Oe s the ntersecton pont of the plane п wth OCZC axs. Addtonally, camera coordnates of the orgn of Oe (0,0,000 / b ) can be obtaned by Equaton (9). Thus, 3

8 Sensors 05, translatons of the camera coordnate system are (0,0, 000 / b3 ) relatve to Oe XeYeZe and the camera coordnate Q can be changed nto Oe XeYeZe by Equaton (7): x e 0 0 cosψ 0 snψx 0 y = e 0 cos sn 0 0 y 0 θ θ + z e 0 sn cos sn 0 cos z θ θ ψ ψ 000 / b 3 where, ψ s the rotaton angle round OCYC, θ s the rotaton angle round OCXC, and Q e = ( x,, e ye ze) are the coordnates of Q n Oe XeYeZe. Normal vector n = ( b, b, b ) T of the plane п s converted by the coordnate rotaton nto 3 3 (0,0, b a ) T n Oe-XeYeZe, where, converson are substtuted nto Equaton (8): 3 (7) a = ( b) + ( b ) + ( b ). Thus, n values before and after the cosψ 0 snψb 0 0 cos sn 0 0 = θ θ b b3 / a 0 snθ cosθ snψ 0 cosψ b3 θ and ψ are obtaned by Equaton (8). In Oe-XeYeZe, because ponts Q are on п, the coordnates Q are Q (, e = xe ye,0). Snce arc cd s at a sde of the shaft and the coordnates of ponts Q on cd have errors, the accuracy of the shaft dameter whch s drectly measured by fttng ponts on cd s poor, as shown n Fgure 8. In Fgure 8, the red crcle s a cross-secton formed by the vrtual plane п ntersectng wth the shaft and Ow s the center of the red crcle. The crcle obtaned by fttng the black ponts on cd s blue wth ts center at O. To mprove the measurng accuracy, the center s determned by fttng black ponts under the geometrcal constrants of arc cd, and the geometrcal constrants for a crcle center are shown n Fgure 9. In Fgure 9a, A, B, C, D are the fttng ponts n the crcle and O s the center. Pont E s the ntersecton pont of AC and BD. From the geometrcal feature of a crcle, the followng relaton holds: AE* CE BE* DE r OE (8) = = (9) where r s radus of the shaft. Settng the center coordnate as (x0,y0), the coordnates of the ponts A, B, C, D are (xa,ya), (xb,yb), (xc,yc), (xd,yd). The coordnate (xe,ye) of pont E can be obtaned from ponts A, B, C, D. Thus, the frst objectve functon can be presented as: mn n = D + DD (0) D = ( x x ) + ( y y ) ( x x ) + ( y y ) r + ( x x ) + ( y y ) a e a e c e c e 0 e 0 e DD = ( x x ) + ( y y ) ( x x ) + ( y y ) r + ( x x ) + ( y y ) b e b e d e d e 0 e 0 e where the value of r s estmated when optmzng Equaton (0); n s number of the ponts.

9 Sensors 05, Fgure 8. Fttng crcle by lmted data. (a) (b) Fgure 9. Geometrcal features of a crcle. (a) constrant condton; (b) constrant condton. As shown n Fgure 9b, F s the mdpont of the chord L. Thus, lne OF s perpendcular to the chord L and can be presented as: n mn d + dd () = xa + xc ya + yc d = ( x0 )( xa xc) + ( y0 )( ya yc) xb + xd yb + yd dd = ( x0 )( xb xd) + ( y0 )( yb yd) Therefore, the center coordnate can be determned by Equatons (0) and (): mn n D + DD + d + dd () = Mnmzng Equaton () s a nonlnear mnmzaton problem, whch can be solved by the Levenberg-Marquardt algorthm. The ntal value of the center s obtaned by fttng the crcle wth the ponts n cd. Fnally, the shaft dameter s obtaned by the dstance between the ponts n cd and the optmzed center:

10 Sensors 05, d n ( x X0) + ( y Y0) = (3) = where d s the dameter of shaft, (x,y) are coordnates of the ponts n cd, (X0,Y0) s the coordnate of the optmzed center, and n s the number of fttng ponts. 4. Experments and Analyss Experments were conducted to assess the utlty of the proposed method. The expermental equpment employed s shown n Fgure 0 and the man parameters of the equpment are shown n Table. The nteror parameters of the camera were calbrated by the method presented n Secton. Two shafts were used n the experments; shaft was a four-segment shaft and the shaft surface s reflecton was reduced usng a specal process, as shown n Fgure 0a. Shaft was a seven-segment shaft, as shown n Fgure 0b. The shafts dameters were measured usng a mcrometer wth a resoluton of μm. n (a) (b) Fgure 0. Expermental equpment for measurng shafts: (a) shaft ; (b) shaft. Table. Expermental equpment parameters. Equpment Mold NO. Man Parameters CCD camera JAI CCD camera Resoluton: Lens M084-MP Focal length: 5 mm Lne projector LH Wavelength: 650 nm Mold plane CBC75mm-.0 Precson of the grd: μm Frst, the mages of the shafts were captured usng the camera, as shown n Fgure. The dameters of the shafts were measured usng the method descrbed n Secton 3. The measurement data are lsted n Tables 3 and 4, and the measurement error was less than 5 μm.

11 Sensors 05, (a) (b) Fgure. Images of strpes by camera: (a) shaft ; (b) shaft. Table 3. Measurement results for shaft (mm). NO. 3 4 Measurement results Known values Errors Table 4. Measurement results for shaft (mm). NO Measurement results Known values Errors To compare the proposed method wth the other methods, the dameters of the shafts were obtaned by drectly fttng the ellpse and crcle. The data for these methods are shown n Tables 5 and 6. The measurng accuracy of the present method was found to be better than that of the other two methods. Table 5. Comparson of the measured dameters n shaft (mm). NO. D Fttng Ellpse Fttng Crcle Present Method D Errors D Errors D3 Errors Mean error To test the nfluence of the angle between the lght plane and cross-secton of the shaft on the proposed method, the shaft dameters were measured by lght planes wth three dfferent lght angles, as shown n Fgure. Measurement results are lsted n Tables 7 and 8.

12 Sensors 05, Table 6. Comparson of the measured dameters n shaft (mm). NO. D Fttng Ellpse Fttng Crcle Present Method D Errors D Errors D3 Errors Mean error (a) (b) (c) Fgure. Images of strpes at three dfferent angles. From Tables 7 and 8, the lght angle appears to have very lttle effect on the measurng accuracy of the proposed method. Table 7. Errors for dfferent angles n shaft (mm). NO. 3 4 Mean Error a/errors b/errors c/errors Table 8. Errors for dfferent angles n shaft (mm). NO Mean Error a/errors b/errors c/errors In order to test the nfluence of nose on the proposed method, nose wth a varance of 0.0 and a mean of zero was added to the strpe mages of shaft and shaft, as shown n Fgures 3 and 4. The center coordnates of the strpe mages wth the added nose were extracted by Steger s algorthm [4]. The shaft dameters can be determned respectvely by the ellpse, crcle, and the proposed method fttng the center coordnates. The results are lsted n Tables 9 and 0.

13 Sensors 05, (a) (b) Fgure 3. Images of strpe on shaft. (a) normal; (b) added nose. (a) (b) Fgure 4. Images of strpe on shaft. (a) normal; (b) added nose. Table 9. Measurement results for shaft (mm). Fttng Ellpse Fttng Crcle Present Method D D Errors D Errors D3 Errors Normal Added nose Δ D Table 0. Measurement results for shaft (mm). Fttng Ellpse Fttng Crcle Present Method D D Errors D Errors D3 Errors Normal Added nose Δ D Here, the shaft dameters D were obtaned by a mcrometer, and Δ D s change of the shaft dameter measurements before and after the added nose. As seen n Tables 9 and 0, the nose has lttle effect on the measurng accuracy of the proposed method.

14 Sensors 05, Conclusons A method for the measurement of shaft dameters s proposed whch s based on a structured lght vson measurement. A vrtual plane s establshed perpendcular to the measured shaft axs and an equaton for the vrtual plane s determned usng a calbrated model of structured lght measurement. To mprove the measurement accuracy, the center of the measured shaft can be obtaned by fttng the projected lght strpe mage on the vrtual plane under the geometrcal constrants. Usng specfc expermental condtons, measurement errors of the method were less than 5 μm and the angle between the structured lght plane and the measured shaft barely affected the measurement accuracy. Acknowledgments The work descrbed n ths paper s partally supported by the Foundaton of Jln Provncal Educaton Department under Grant 059. Author Contrbutons Syuan Lu proposed the algorthm, desgned and performed experments, and wrote the ntal manuscrpt. Qngchang Tan conducted the analyss of the algorthm. Yachao Zhang performed experments. Conflcts of Interest The authors declare no conflcts of nterest. Appendx A Snce the vrtual plane п s perpendcular to the measured shaft, the drecton vector of the axs s the normal vector of п. As shown n Fgure A, the calbraton board s fxed by fxtures, and they are clamped by the two centers and the connectng lne of the two centers s the axs of the shaft MN. Turnng the calbraton board, the mages of the board can be captured by a camera and MN s the ntersectng lne of the calbraton board planes. (a) (b) Fgure A. The model of the calbratng axs. (a) sketch map; (b) mage of equpment.

15 Sensors 05, Based on Equaton (), Equaton (A) can be obtaned: j j XC t X W j j Rj YC t = YW j j ZC t 3 Z W (A) Set to d d d R D d d d j j j j 3 j j j = j = 3 j j j d3 d3 d 33, where, Rj and tj are the rotaton matrx and translaton vector of the calbraton target at the jth poston and can be obtaned by the method provded n Secton, j =,. Snce world coordnate systems O j WX j WY j WZ j W are bult on the calbraton target plane, Z j w = 0. The equaton of the calbraton target plane at the jth poston s: d X + d Y + d Z = d t + d t + d t (A) j j j j j j j j j 3 C 3 C 33 C The normal vector of the plane s Sj ( d j 3, d j 3, d j 33) the drecton vector of MN can be represented as: =. MN s the ntersectng lne of the two planes, so (,, ) ( d , , ) SMN = dx dy dz = S S = d d d d d d d d d d d (A3) As shown n Fgure 6, Or s a pont n п and set Or s the ntersecton pont of MN and п, and the camera coordnate of Or s Or = (Xr,Yr,Zr). The equaton of п n OCXCYCZC s ( ) ( ) d ( X X ) + d Y Y + d Z Z = 0 (A4) x C r y C r z C r Thus, equaton parameters of the vrtual plane can be obtaned usng Equatons (9) and (A4): b = 000 dx ( dx Xr + dy Yr + dz Zr) b = 000 dy ( dx Xr + dy Yr + dz Zr) b3 = 000 dz ( dx Xr + dy Yr + dz Zr) (A5) Appendx B Q e are the ponts for crcle fttng, and the coordnates Q e = ( x, e y e) =,,3..., N can be obtaned by Equaton (7). The coordnates Q e are substtuted nto Equaton (). Then Equaton () s handled by the prncple of least squares as: A x = b (B) N N N N 3 ( xe) xy e e xe ( xe) + xe( ye) a = = = = N N N N where,, 3, x A = xy e e ( ye) ye b = ( xe) y e + ( y ) = b e. = = = = c N N N N xe ye ( xe) + ( ye) = = = = Usng x obtaned by Equaton (B), the measured shaft dameter can be obtaned by Equaton (3).

16 Sensors 05, The coordnates of ponts for ellpse fttng are P (, t = xt y t) =,,3..., N. To facltate the ellpse fttng, a new coordnate system s establshed on the structured lght plane п, and the camera coordnates P can be converted nto (, Pt = xt y t) =,,3..., N n the coordnate system. The coordnates P t are substtuted nto Equaton (0). Then Equaton (0) s handled by the prncple of least squares as: where, A x = b (B) A = x y x y x x y x N N N N N 3 ( xt) ( yt) xt( yt) ( xt) yt xt( yt) xtyt = = = = = N N N N N xt( yt) ( yt) xt( yt) ( yt) ( yt) = = = = = N N N N N ( t) t t( t) ( t) t t t = = = = = N N N N N 3 xt( yt) ( yt) xtyt ( yt) yt = = = = = N N N N N xy t t ( yt) xt yt = = = = =, N 3 ( xt) yt = N ( xt) ( yt) = N 3 = ( xt ) = N ( xt) yt = N ( xt ) = b, x A B = C. D E Usng x obtaned by Equaton (B), the measured shaft dameter can be obtaned by Equaton (). Snce the matrxes (A, A) and the vectors (b,b) n Equatons (B) and (B) consst of the coordnates of the lght strpe centers, the matrxes and vectors are nfluenced by errors of the lght strpe centers coordnates, whch show that x and x are naccurate. From Numercal Analyss (Seventh Edton) by Burden and Fares [5], the nfluence s analyzed as shown below. Consderng the errors of the lght strpe centers coordnates, matrx A and vector b n Equaton (B) are expressed as: A = A+δ A ; b = b +δ b where δ A and δ b are caused by errors of the lght strpe centers coordnates. Thus, x n Equaton (B) becomes x = x +δ x, and Equaton (B) can be expressed as: ( A +δ A)( x +δ x) = ( b +δ b) (B3) ( A +δ A ) x + ( A +δa ) δ x = b +δ b ( A +δa ) δ x =δb δa x (B4) Set ( A +δ A) s an nvertble matrx, and from Equaton (B4) we get: δ x = ( A+δA) ( δb δa x) (B5) In order to analyze the error of the fttng crcle equaton s parameters, the norm of Equaton (B5) s: δx ( A+δA) δb δa x (B6) where δ x s error of the crcle equaton s parameters x obtaned by the crcle fttng. In the same way, the error of the equaton s parameters x for the ellpse fttng can be obtaned by Equaton (B) δx ( A +δa) δb δa x (B7)

17 Sensors 05, From Equatons (B) and (B), ( A+δ A) and ( A+δ A) can be calculated by the lght strpe centers coordnates. When the measured shaft dameter s mm, ( A +δ A) and ( A +δ A ) can be calculated by the coordnates of the lght strpe centers. ( A+δ A) = 7.04 ( A +δ A) = So δx 7.04 δb δa x (B8) δx δb δa x (B9) From Equatons (B8) and (B9), δb δa x must be very small f δx δ x. From Equaton (B4), δb δa x cannot be a zero vector. So, a very small δb δa x requres that the absolute values of elements n the matrces δ A and the vector δ b must be very small. Ths shows that the errors of the lght strpe centers coordnates for the ellpse fttng must be smaller than the ones for the crcle fttng. On the other hand, f the coordnate errors of the lght strpe centers for the crcle fttng are the same as those for the ellpse fttng, the error of the crcle equaton parameters obtaned by the crcle fttng s smaller than the one of the ellpse equaton parameters obtaned by the ellpse fttng. From Equatons () and (3), the measurng accuracy of the shaft dameter by crcle fttng s better than by ellpse fttng. When the measured shaft dameter s mm, ( A +δ A ) = 9.03 and ( A A) δ =, the same results can be obtaned. References. Takesa, K.; Sato, H.; Tan, Y.; Sata, T. Measurement of dameter usng charge coupled devce (CCD). CIRP Ann. Manuf. Technol. 984, 33, Butler, D.J.; Forbes, G.W. Fber-dameter measurement by occluson of a Gaussan beam. Appl. Opt. 998, 37, Lemeshko, Y.A.; Chugu, Y.V.; Yarovaya, A.K. Precson dmensonal nspecton of dameters of crcular reflectng cylnders. Optoelectron. Instrum. Data Process 007, 43, Xu, Y.; Sasak, O.; Suzuk, T. Double-gratng nterferometer for measurement of cylnder dameters. Appl. Opt. 004, 43, Song, Q.; Zhu, S.; Yan, H.; Wu, W. Sgnal processng method of the dameter measurement system based on CCD parallel lght projecton method. Proc. SPIE 007, 689, do:0.7/ We, G.; Tan, Q. Measurement of shaft dameters by machne vson. Appl. Opt. 0, 50, Sun, Q.; Hou, Y.; Tan, Q. A Planar-Dmensons Machne Vson Measurement Method Based on Lens Dstorton Correcton. Sc. World J. Opt. 03, 03, 6.

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