Measurement of Three-Dimensional Rotational and Translational. Displacements using a Multi-Facet Mirror

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1 Mesurement of Three-Dimensionl Rottionl nd Trnsltionl Displements using Multi-Fet Mirror Won Shik Prk nd Hungsuk Cho Dept. of Mehnil Engineering, Kore Advned Institute of Siene nd Tehnolog 373-1, Kusong-dong, Yusong-gu, Tejeon , Kore ABSTRACT A mesurement sstem tht n mesure the si-degree-of-freedom motions of ritrr ojets is proposed. The mesurement sstem utilizes speil mirror looking like tringulr prmid nd hving three refletive lterl surfes, hih is to e mounted on the ojets of interest. One lser em illumintes the top of the mirror, the mirror reflets nd splits the em into three ems. These refleted ems re deteted three position-sensitive detetors (PSD), respetivel. Bsed on the signl outputs of the PSDs, the three-dimensionl position nd orienttion of the mirror n e omputed, hih, in turn, determines the three dimensionl position nd orienttion of the ojet. This pper proposes to options in ppling the mesurement priniple, one of hih hs its lser em soure fied in spe, nd the other ontrols the trnsltionl motion of the lser em soure to trk the mirror. In this pper, the priniple of the mesurement is proved to e vlid through theoretil nlsis nd eperiments. And the dvntges nd disdvntges of oth options re disussed. Keords: 3-dimensionl ojet pose, 6-DOF motion, 3-fet mirror, position-sensitive detetor (PSD) 1 INTRODUCTION Optil instruments suh s lser interferometers [1,2], lser doppler virometers [3], et. for mesuring miro or nno sle motions re idel used in vriet of industril res. These instruments, hih re sill for 1-D mesurement, sho ver high preision, even the order of nnometers in trnsltion. For higher dimensionl mesurement suh s 2- or 3-D position plus rottion, it is neessr to integrte severl lser interferometers nd lser doppler virometers, hih is ver diffiult tsk sine it eomes signifintl more diffiult to mesure the motions of n ojet ith inreses of even one-dof. Though there hve een gret dels of demnds for the mesurement methods to otin multi-degree-of-freedom, onl fe methods hve een developed for some limited pplitions. Lee et l. [4] investigted the possiilit of their ne optil sstem onsisting of lser soure, four qud-detetors, Prof. Hungsuk Cho(orrespondene), E-mil: hsho@l.kist..kr, WWW: Tel: , F: Optomehtroni Sstems IV, George K. Knopf, Editor, Proeedings of SPIE Vol (2003) 2003 SPIE X/03/$15.00

2 four em splitters, et. to mesure 6-DOF simultneousl. Their method mounts four em splitters on the ojet of interest to form light pth emitted from lser soure to four qud-detetors. Bsed on the preision of their quddetetors, the estimted the trnsltionl nd rottionl resolutions to e 0.05 µm nd 0.25 µrd, respetivel. But the mounting of four em splitters leds to lrge dditive eight of the ojet, thus the dnmi hrteristis of the ojet m e lrgel ffeted. In this pper, ne mesurement method is proposed tht n simultneousl mesure the 3-D position nd orienttion of rigid odies ith the preision order of mirometers nd mirordins. The sensor sstem utilizes lser soure, speill frited mirror, nd three PSDs. The speill frited mirror, lled three-fet mirror in this pper, looks like tringulr prmid hving n equilterl ross-setion. We mount this mirror on the ojets of interest, illuminte the mirror ith He-Ne lser em, nd then detet the ems refleted from the mirror ith PSDs. From the output signls of the PSDs, e n lulte the 3-D position nd orienttion of the three-fet mirror, thus enling us to otin the 3-D position nd orienttion of the ojets. This method utilizes smll nd light mirror, thus the mirror ffets the dnmi hrteristis of the ojet muh less thn tht of Lee et l. s method [4]. z lser soure z L lser em z s φ l PSD C z O su-em s su-em z s v l Lser em su-em ojet s τ z z O PSD B Q ψ ζ o o S ( ψ, ζ ) R ( ψ, ζ ) l o P (, ) p l l s M M M l t s P O PSD A PSD A O Figure 1 Shemti dring of the sensing sstem Figure 2 3-fet mirror nd PSD A 2 PRINCIPLE OF SENSOR SYSTEM 2.1 Sstem onfigurtion The shemti onfigurtion of the proposed sensor sstem is shon in Figure 1. In the figure, the sensor sstem is omposed of mirror of prmidl shpe, lser soure, nd three position-sensitive detetors (PSDs). The mirror hs n equilterl tringulr ross-setion, hih is referred to s three-fet mirror sine it hs three refletive lterl surfes. Its refletive surfes re inlined 45 to its ottom surfe, respetivel, nd the three ottom edges mke Pro. of SPIE Vol

3 equl ngle of 60 to eh other. As shon in Figure 2, one lser em illumintes the top of the three-fet mirror, the refletive surfes M, M, nd M split the lser em into three ems, hih re lled suems in this pper. And the suems impinge the sensitive res of the PSDs, from hih the PSDs detet the positions of the spots of the suems s ell s the inident light poers. If the three-fet mirror moves in n, the positions of the em spots on PSDs nd their inident light poers vr in some orresponding. If there eists some proper reltionship eteen the displements of the mirror nd the PSDs outputs, e m utilize this optil priniple to estimte the three-dimensionl displements of the mirror from the PSDs outputs. In the folloing setions, e onstrut mthemtil model of the reltionship nd sed upon this, theoretill hek if the proposed priniple is vlid. Figure 3 shos the plnr vie of the sensor sstem ith its lser soure omitted for presenttion. Here, the lser em is kept vertil to the referene plne of the sensor sstem, hih is defined three origins of the PSDs. As shon in the figure, three PSDs re loted t the sme distne R PSD from the origin of the sensor sstem o, round hih the origins of the PSDs mke n equl ngle 120 ith eh other. In the previous stud [5], it hs een presented tht the ngles eteen the PSDs θ nd θ re not so importnt, ut the distne from the enter to the PSDs R PSD ffets the ngulr resolution of the sensor. In this pper, the issues out the sstem prmeters nd their effets re not presented, ut e fous on the priniple nd its vritions tht properl dpt to given prolems. 2.2 Sensor model The PSDs re illuminted the suems nd provide the output signls out the 2-D positions nd inident light poers of the em spots. Within the em spots, irrdine distriutions look like pie-piees s shon in Figure 2. If the three-fet mirror trnsltes or rottes, the irrdine distriutions on the PSDs vr in generl so tht the PSDs provide the hnges in the 2-D oordintes nd inident light poer of the em spots. In this setion, the mthemtil reltionship eteen the 3-D pose of three-fet mirror nd the PSD outputs is derived, hih is funtion tht lultes the PSD outputs from given 3-D position nd orienttion of three-fet mirror. The derivtion is sed on light pth tring from the soure to the destintion, in hih lser em is regrded s set of stright lines. For simpliit of the model, diffrtive effet is negleted nd on ount of spe onsidertion, the derivtion for one detetor, PSD A, is presented in this pper. The sstem model epressed s J f relting the three-dimensionl pose of three-fet mirror to the outputs of three PSDs n e ξ = J f ( τ ) (1) here ξ nd τ represent the output nd input of the sensor, respetivel, defined s follos. ξ = ψ ζ Φ ψ ζ Φ ψ ζ Φ [ ] T (2) 86 Pro. of SPIE Vol. 5264

4 τ = [ t t t γ β α ] T. (3) z Here the omponents of ξ re the output signls provided the PSDs. (, ) the em spot impinging on PSD A ith respet to the oordinte sstem the inident light poer on the sme detetor. Similrl, ( ψ, ζ ), ψ ζ represents the 2-D oordinte of o defined on PSD A nd Φ, ( ψ, ζ ), nd PSD B nd C ith respet to their orresponding oordinte sstems o nd o. τ omponents of three-fet mirror. t, Φ represents Φ represent the outputs of onsists of the 6-DOF t, nd t z re the oordintes of the top of the mirror ith respet to the referene frme o nd γ, β, nd α re the roll, pith, nd ngles, respetivel. The definition of these si omponents orresponds to the reltive liner nd ngulr displement eteen the ojet frme o s defined t the verte of the mirror nd the referene frme o. It is noted tht the sensor input τ inludes (, ), hih is the 2- D oordinte of the lser em enter defined s the intersetion eteen the lser em enter nd -plne. This is represented ith respet to the referene frme o. The reson h τ inludes (, ) is sed on theoretil nlsis to e eplined in the folloing setion nd to present tht J f provides one-to-one mpping eteen ξ nd τ. PSD B o θ = R PSD =250mm o o PSD A D(J f )/D(τ) θ = PSD C o ε l Figure 3 Plnr vie of the sensor sstem Figure 4 Joins of J f J f hs eight input vriles nd nine output vriles. For one-to-one mpping eteen input vriles nd output vriles, it is neessr tht there should e onstrint tht redues the numer of independent output vriles. It is ovious tht the sum of the inident light poers on the PSDs is onstnt or rther the sme s the poer output of the lser soure if the ttenution t the mirror surfe is negligile. Thus, hih the numer of independent output vriles redues to eight. Φ +Φ +Φ = P, (4) To derive the model tht predits the PSDs output, ξ, from the mirror pose plus the lser em position, τ, e Pro. of SPIE Vol

5 (7) should (1) identif the shpe of the em spot, (2) otin the irrdine distriution, nd (3) lulte the mss nd enter of mss of the irrdine distriution ithin the spot for eh PSD. These proedures re to e desried in the folloings. Shpes of em spots on the detetors Figure 2 shos sitution, in hih suem refleted from M impinges on PSD A to form spot resemling piee of pie, P. In the figure, the dotted ellipse L is the imginr ross-setion of the lser em s if the lser em ere not split the three-fet mirror ut normll refleted plnr mirror hose pose is the sme s tht of M. S is the enter of the dotted ellipse P, l nd l L nd lso the projeted imge of the enter of the lser em. To edges of, re the projeted imges of the edges of the 3-fet mirror, l nd l. L inluding the r of P is defined s the projeted imge of the oundr of the lser em defined em dimeter φ l. Q, the intersetion of l nd l, is the point projeted from the top of the mirror, s shon in the figure. If the enter of lser em oinides ith the mirror top, then Q nd S re loted t the sme position. R is the light entroid of P, hih is the reding of the 2-D position of the em spot on PSD A. More detils on the predition of the P inluding R, Q nd S re referred to the previous literture [5, 6]. Intensit distriutions on the detetors First, e model the originl lser em of Gussin intensit distriution from soure hose diretion vetor is vl nd the position of soure point is. As desried ove, the diretion of the lser em is kept vertil. Thus, n e epressed s v l v = [0 0 1] l T (5) The intensit distriution ithin the em is distriuted in Gussin funtionl form given 2 8P 8r Ir () = ep 2 2 πφl φl. (6) Here, P represents the poer of the lser em, φ l the dimeter of the lser em, nd r the rdil distne from the enter is of the lser em to n point of interest. As shon in Figure 2, the irulr ross-setionl profile of the Gussin em is someht hnged into n elliptil one, hih depends on the ngle of inidene of the suem. The irrdine distriution ithin the elliptil em profile of P n e epressed s follos: p I( r) = I( r) p here is the unit surfe norml vetor of PSD A nd p = [ p p p ] T z is the diretion vetor of the suem 88 Pro. of SPIE Vol. 5264

6 A ith respet to o, hih n e otined ppling the refletion l [7] to the diretion vetor of the originl lser em. And it should e noted tht r is not the rdil distne s used in Eq. (6), ut someht different term, lled modified rdil distnes in this pper nd defined s follos: p r z 2 2 o 2 o 2 = ( ψ) + ( ) 2 2 ζ 2 2 p + p p + pz p (8) Mss nd enter of mss of the irrdine distriution As desried ove, the lotions ( ψ, ζ ), ( ψ, ζ ), nd ( ψ, ζ ) re the enters of mss of the irrdine distriution ithin P, P, nd P. For PSD A, ( ψ, ζ ) n e omputed s follos: I ( r ) d dz z I ( r ) d dz P P ψ =, ζ = I ( r) ddz I ( r) ddz P P At this stge, e hve ompleted modeling the reltionship eteen the 3-dimensionl pose, t, t, tz, γ, β, nd α, nd the PSD outputs, ( ψ, ζ, Φ ), ( ψ, ζ, Φ ), nd ( ψ, ζ, Φ ). The 2-dimensionl position of the lser em (, ) tkes prt ith in the input side of the model, hile it is not so importnt s others. When tul mesurement is performed ith the proposed sensor sstem, it is neessr to perform n inversion of the ove model, i.e., lultion of t, t, tz, γ, β, nd α ith given ( ψ, ζ, Φ ), ( ψ, ζ, Φ ), nd ( ψ, ζ, Φ ). The inversion is performed through numeril, the Neton s method [8], through hih e hve suessfull otined the solution, t, t, tz, γ, β, nd α, from given ( ψ, ζ, Φ ), ( ψ, ζ, Φ ), nd ( ψ, ζ, Φ ) in the eperiments to e eplined in setion 3. (9) 2.3 Eistene of the inverse funtion of the sensor model In this setion, the eistene nd uniqueness of the inverse funtion of J f is emined. For the emintion, inverse funtion theorem [9] is pplied. For the purpose, it should e shon tht there eists set of funtions { f1,, f8} tht is suset of the PSD outputs { ψ, ζ, Φ, ψ, ζ, Φ, ψ, ζ, Φ } stisfing ( f1,, f8) ( t,, ) τ f1 f1 ( τ) ( τ) t = 0 f1 f1 ( τ) ( τ) t (10) for ever τ in mesurement rnge. From (4), it is epeted tht one of Φ, Φ, nd Φ is redundnt for unique τ if the poer of lser em P is onstnt, hih is resonle ondition if e use some properl stilized lser soure. Here, Φ is omitted nd then to hek the ondition in Eq. (10) s follos: Pro. of SPIE Vol

7 ( ψ,, ζ) ( t,, ) τ = ψ ψ ( τ) ( τ) t ζ ζ ( τ) ( τ) t. (11) Figure 4 shos the determinnt vlues in Eq. (11) ith respet to the reltive distne eteen the lser em enter nd the verte of three-fet mirror ε l defined s ε = {( t ) + ( t ) }. (12) 2 2 1/2 l In the evlution, the sstem prmeters re given s in Figure 3 nd Tle 1 eept lser poer P nd em dimeter φ l. Here the lser poer P is ssumed to e 1mW nd the em dimeter φ l is to e 1mm. And τ, the 6-DOF pose of three-fet mirror nd 2-dimensionl position of the lser em, hs een hosen rndomlithin the mesurement rnge ondition. As shon in Figure 4, the join is negtive definite over the rnge, thus the sensor model J f is proved to hve unique inverse funtion ithin the mesurement rnge. The numer of points rndoml emined is over , hose neighorhoods re epeted to suffiientl over the input spe. Tle 1 Speifitions of the eperimentl sstem Components He-Ne Lser 3-fet mirror PSD Speifitions Poer 0.3 mw Bem dimeter 0.85 mm Bem orienttion t 3-fet mirror Vertil (onst.) Azimuth of surfe norml 45 Longitudinl distne eteen surfe normls 120 Surfe fltness λ/10 t nm Size of sensitive re 13 mm 13 mm Rdil distne from the sstem origin 250 mm Longitudinl distne eteen PSDs 120 Lout Orienttions See Figure 3 3 EXPERIMENTS In order to verif the priniple nd effetiveness of the proposed mesurement method, e hve implemented n eperimentl sstem nd performed series of eperiments. The onfigurtion of the eperimentl sstem is shon s in Figure 3 nd Tle 1. Figure 5 shos the tul pperne of the eperimentl sstem nd loser look of threefet mirror. As shon in setion 2.3, the proposed method n estimte the 6-DOF of three-fet mirror from the PSD outputs, regrdless of the 2-D position of the lser em, if the em illumintes the top of the mirror. In this setion, 90 Pro. of SPIE Vol. 5264

8 e present to kinds of vrition of the proposed method, one of hih uses lser em of fied position nd the other ontrols the lser em to trk the top of the mirror. The former is dvntgeous for fst ojets nd the ltter is for lrger mesurement rnge. lser soure mirror lser em 3-fet mirror PSD B PSD C 3-fet mirror 6-is stge PSD A Figure 5 Photogrph of the eperimentl sstem DOF pose estimtion ith fied lser em If e need to mesure some motion of smll rnge tht n e overed the ross-setionl re of the lser em, the lser em ould e etter to e fied t position hile its orienttion is lso fied s vertil. It is quite nturl to lote the lser em t the enter of the sensor sstem so tht (, ) = (0, 0). Figure 6 shos some mesurement rnges of the sensor sstem hen its lser em is fied t the enter, in hih eh plot presents mesurement rnge of some 2-DOF motion hile the rest 4-DOF omponents re set to zero. In the plot shoing the rnge of plnr motion, e n see the mesurement rnge is muh smller thn the lser em profile. This is euse s the top of the mirror moves outrd the em, some of the three PSDs ill get so ek inident light tht its signl-to-noise rtio gets too lo. Thus, the mesurement rnge in -diretion is limited the S/N rtio of the PSDs. On the other hnd, the other motions suh s z -trnsltion nd rottions re limited the size of sensitive re of PSD mm mrd mrd mm β (mrd) mrd α (mrd) mrd γ (mrd) β (mrd) Figure 6 Mesurement rnge ith fied lser em Pro. of SPIE Vol

9 Figure 7 - Figure 9 present some of our eperimentl results for single is motions nd multi-is motions, hile the errors in 6-DOF re lso presented. From the error plots, e n see tht the proposed method suessfull otins the trnsltionl nd rottionl displements from the PSD signls ith limited ross-il errors ε 0.15 ε 0.12 ε z 0.09 ε γ ε β ε α Mesurement error Mesurement error (mrd) Figure 7 Trnsltion in -is ε ε γ 15 ε ε β ε z 0.09 ε α α (mrd) Mesurement error Mesurement error (mrd) α (mrd) Figure 8 Rottion round z -is α (mrd) 0.20 α (mrd) Figure 9 Multi-DOF motions 92 Pro. of SPIE Vol. 5264

10 3.2 6-DOF pose estimtion ith em position djustment The proposed method n e esil modified to hieve lrger mesurement rnge in -diretion. The lser em soure is mounted on to-is trnsltion stge so tht the em n trk the top of three-fet mirror. In this method, e dopt feedk loop to djust the 2-D position of the lser em to three-fet mirror. The error signls inditing the offset eteen the enter of the lser em nd the top of three-fet mirror re otined ompring the inident light poer shres mong the PSDs. This is sed on tht (, ) ( t, t ) if Φ =Φ =Φ nd if γ, β, α < ε. (13) For emple, if ll of roll, pith, nd ngles re smller thn 10mrd, hih is degree, then e hve the trking error, the offset eteen the lser em enter nd the top of the mirror, less thn 0.89µm. For n tul feedk signl, e define some indies s follos: χ Φ +Φ os120 = +Φ os120 o o Φ +Φ +Φ, χ Φ sin120 = Φ sin120 o o Φ +Φ +Φ. (14) If χ is positive, e hve to djust the lser em in negtive -diretion, or if χ is negtive, the em should go in positive -diretion. And for χ, similr of djustment is used. And the mgnitude of χ nd χ represent the distne eteen the lser em enter nd the top of the mirror. Thus it n e used s the proportionl gin for P-ontrol lgorithm. Figure 10 shos the mesurement rnge in -diretion hieved through the mirror trking, in hih the sstem prmeters re the sme s those of the method using fied lser em s shon in Figure 3 nd Tle 1. As shon in the figure, the mesurement rnge in -diretion hs een remrkl enlrged. In this method, the mesurement rnge is not defined the S/N rtio of the PSDs ut the size of sensitive re of the PSDs mm mm Figure 10 Mesurement rnge in -plne motion ith em position djustment Figure 11 shos the eperimentl results for nd -diretion. As shon in the figure, the proposed method ith mirror trking hs suessfull mesured the motions in lrger rnge thn tht ith fied lser em, hile the Pro. of SPIE Vol

11 mesured rnge is restrited ithin smller rnge thn tht shon in Figure 10 due to some limittion of eperimentl hrdre. (mesurement, mm) 0.3 men -σ 0.2 +σ min. 0.1 m error men -σ +σ min. m (mesurement, mm) 0.3 men -σ 0.2 +σ min. 0.1 m error 0.03 men -σ +σ 0.02 min. m Figure 11 Eperimentl results for the motions in nd -diretion ith mirror trking 3.3 Sensor evlution In generl, sensors n e evluted their mesurement rnge nd resolution. The sensor presented in this pper hs not een optimized ut n eperimentl setup. Hoever, e n roughl understnd its possiilit nd ppliilit to n prolems tht e m hve to solve. Tle 2 shos the mesurement rnges nd resolutions of the eperimentl sstem onstruted in this pper. In the mesurement method proposed in this pper, mesurement rnge nd resolution in 6-DOF m not e simpl presented s shon in the tle. The numers listed in the tle n e regrded s n outline on its performne. The mesurement rnge in eh is hs een tken s the mimum distne in the diretion of interest, hile other es ere set to their origin. Thus the rnges m vrhen other es re elsehere. The resolutions in eh is lso m vrith respet to the 6-DOF pose of the mirror. Thus to otin the most representtive quntit out the resolutions for si es, e otin numer of repeted mesurement for eh point in 6-DOF spe, lulte the stndrd devitions for eh point, nd finll tke the mimum of the stndrd devitions for eh is. The resolutions hve een defined, in this pper, s the tie s the mimum stndrd devitions for eh is, hih is, in other ord, root-men-squred error. 94 Pro. of SPIE Vol. 5264

12 Tle 2 Summr of eperimentl results z γ β α Rnge (mm, mrd) 0.36 / / Resolution (µm, µrd) Mesurement rnge hieved mirror trking, otherise fied lser em 4 CONCLUSIONS The proposed priniple hs een proved to e vlid in oth theoretil nd eperimentl s. In theoretil nlsis, e onstruted sensor model nd emined it hether its Join mtri is nonsingulr. And in eperiments, to different pprohes, depending on hether the position of the lser em trks the mirror or not, hve een emined. The one hose lser em is not ontrolled hs so tin mesurement rnge for in-plne motions tht it n over t most 0.25mm 0.25mm squre re ut dvntgeous to fst ojets of round 3kHz in ndidth. And the other, of hih the lser em trks the mirror, hs muh lrger mesurement rnge overing over 8mm 8mm for motions in -diretion ut not suitle for fst ojets euse it needs mehnil servoing. The mesurement preisions hieved in our eperiments re t most 2.8µm for trnsltion nd 13.0µrd for rottion. REFERENCES 1 L. M. Smith nd C. C. Doson, Asolute displement mesurement using modultion of the spetrum of hite light in Mihelson interferometer, Appl. Opt., Vol. 28, pp , S. Hosoe, Lser interferometri sstem for displement mesurement ith high preision, Nnotehnolog, Vol. 2, pp , J. R. Bell nd S. J. Rotherg, Lser virometers nd ontting trnsduers, trget rottion nd si degree-offreedom virtion: ht do e rell mesure? J. Sound Vi., Vol. 237, No. 2, pp , N. Lee, Y. Ci, nd A. Jonej, High-resolution multidimensionl displement monitoring sstem, Opt. Eng., Vol. 36, No. 8, pp , W. S. Prk, Mesurement of 6-dof motions of rigid odies through splitting lser em, Ph.D. thesis, Kore Advned Institute of Siene nd Tehnolog, W. S. Prk nd H. S. Cho, Mesurement of fine 6-degrees-of-freedom displement of rigid odies through splitting lser em: eperimentl investigtion, Optil Engineering, Vol. 41, No.4, pp , April M. Bss, E. W. Strlnd, D. R. Willims, W. L. Wolfe, Hndook of Optis, 2nd Ed., vol. 1, MGr-Hill, W. H. Press, S. A. Teukolsk, W. T. Vetterling, nd B. P. Flnner, Numeril Reipes in C, 2nd Ed., Cmridge Universit Press, Jerrold E. Mrden nd Anthorn J. Trom, Vetor lulus, 2 nd Ed., Freemn, Pro. of SPIE Vol