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Chapter 6 Feasibility Study on Magnetically Levitated Planar Actuator This chapter proposes a conceptua玉design fbr a planar actuator having the same con丘guration for the孤agnetic drcuits as for the planar motion control so that the mover can be magneticaHy suspended. In addition, it presents a feasibility verification of motion・control characteristics by nUmeriCal analySiS. 171 6.Fe asibility Study on Magnetically Levitated Planar Actuator This chapter pre8ents a feasibMty veri丘cation as to whether a planar actuator can magnetica皿y suspend a mover, capable of 3・DOF motions on a pla ne, so as to further improve the(hive performance of a planar actuatOr. First, the planar actuator is redesigned so it can both suspend the mover and contro1 the planar motions. Then, the planar motion and magnetic suspension characteristics of the planar actuator are verified by nuエnerical analysis. 6.1.Conceptual De8ign of Magneticaly LeVitated Planar Actuator This section presents a compatibility veri丘cation of plallar motion and magnetic suspension, and then introduces a conceptual design fbr a planar actuator with a magnetically suspended mover. 6.1.1.De8ign Considerations The proposed planar actuator has spatially super皿posed magnetic circUits for the x−,アーandα一clirections, which are its most importa nt feature and enable the mover to travel over a wide movable area on a plane by exciting only two polyphase armature conductors. The magnetica皿y levitated planar actuator is also designed so that aU the magnetic ch℃Uits a re mutually superimposed, as in the fb皿owing methodology: (D Compatibility veri丘cation of both 3・DOF planar motion and magnetic− suspension controls of the planar actuator designed in Chapter 3. (ll) Redesign the planar actuator, without increasing the number of the armature conductors, so that planar motion a nd magnetic suspension are compatible if they are found not to be in(i). In order to design the planar actuator, a numerical analysis of 6−DOF driving fbrces for 6・DOF mover positions is performed. 172 6.1.2.6・DOF Force Ana lysi8 This section present8 an analytical model of面ving fbrces with 6 DOE and then presents the results of the analysis. (i) Analytical model fbr 6・DOF driving forces: The driving fbrces, inclu(ling the suspension forces, greatly depend on the siZe of the gap l〕etween the mover and armature condu伽rs, and therefbre this gap needs to be precisely contro且ed. Genera皿y, reducing this gap increases the driVing fbrces. If the mover is located below the stator, attraction forces to the stator are req血ed to suspend the mover.且owever, the attraction fbrces are increased by reducing the gap, which makes the vertical motions of the mover unstable. Conversely, if the mover is located above the stator, repUlsion forces from the stator are req血ed to suspend the mover. The repulsion」e()rces are increased by reducing the gap, and so the vertical motions are stable. Therefbre, in this study, the mover of the magnetica皿y leVitated planar actuatOr is positioned on the stator. Figure 6.12・1 shows the analytica 1 mode1 for the driving forces. In this丘gure, the mover and p olyphase armature conductors for the x−or y−directions only are shown. A motmg 2・D且albach permanent・magnet array has the same structure as shown in Fig. 3.2.1・1,and fbur・pole・and・seven・segment magnetization with p ole・pitch length TpM=3 mm along the xi−andγ∫−clirections. Its dimensions are l l mm×11 mm×2mm, which are ahnost two−fifths the size of the magnet−array dimension shown in Fig.3.2.1−1. The ulthnate nmiaturization of the permanent・magnet mover enables higher accelerations to be generated using the same armature currents and且ux density as given in Subsection 3.3.1. Figure 6.1.2・2 shows an analyticaUy obtained且ux・density disUibution on the plane O.5 mm below the mover bottom for the x市and y.−directions. Figure 6.1.2−2 indicates that the permanent・magnet mover a lso generates a quasi・sinusoida1且ux density with a pitch length ofτ=2.1 mm in the xm− and ym−directions・On the other hand, pitch lengths of the meander・shaped armature conductors are equa1 to the pitch r(=2.1 mm). In the mover motions, there are 3−DOF rotations.且owever, this analysis deals with the rotations around only one axis(xm, y励, or zm). The rotational angles around the x.一, .Ym−, and zm−axes are referred to as roU an91e 7, pitch angle B, and yaw angleα, respectively」 The driVi ng forces acting on the mover can be calcUlated from the Lorentz force law with the same equations as Eqs.(3.3.1・1)一(3.3.1・8). 173 .十、 「一「 「.一 「 1 一 Armature conductors 1.・ forx−direCtional drive 1 川 ㌦ E 1 1 ∋鈎 i◇ 〉,s・〈う i 1 1 lli 1 a II:V 『 x ’” 2τ γ 3 2τ 丁 1 1 i }2τ 1丁 1 1 τ.T 1 1 遅 i 1 1 L=1 (a)Supplyingもhree・phase currents f()r the x−directional drive. Armature conductors β fory−direCtiona|driVe @ ス 1_.r O 〉・ ら.ら “. α fi コ ÷』 菰 ら・ 、σ .÷ L⊥____τ ら 今一輪「 γ=:□ ヴら ら. 匠三圭≡⇒…÷ 1 一一2r }T, 匡三≡丁巳≡、 L 3 ’i la ⊃ (b)SupPlying three・phase currents for the y一dlirectional drive. Fig. 6.1.2・1; Analytical model for 6・DOF driving forces。 174 500 400 600 ( 8 ) 喝” .≧ 8 8 300 400 200 200. 0 i 1100 D” ’ .一’ T. 0 一200 ,i一 一IOO き 口 一400, 一200 一600、 傷・陥 .‘ S6 一300 一400 ..・「 つ .....・r・.... ち〃−4−6 Dつ〇一 一罐ぽ Qヅ 一500 ツ Fig.6.1.2・2: Flux・density distribution on the plane O.5 mm below the 皿over bottom. (D) Analysis results fbr 6・DOF driving fbrces: Figure 6.12・3 shows the anaユysis results of the driVing f()rces&, E., Tr, T,., T. f()r the yaw angleαwhen the 4−and g−aXis currents for the x−directiona! drive are supplied(J、tr =1A, orし=1A), the air gap between the mover bottom and ai’mature conductors is O,5 mm, and the pitch and roll positions are not displaced(β=y=Odeg). Figure 6.1.2・3 indicates that the d−axis current generates the translational f()rces F. and torques T., and the q−axis current generates the translational forces Ev and torques Tr, T,.. The translational forces Er, F, and torques ny・ are a1皿ost constant, and the torques Tl. and T. are proportional to the yaw angleαwhen the yaw angleαny O deg. Because of the symmetric magnetization of the mover, the same driving forces can be generated every 180deg. In the same way, the driving ferces resulting fi’om the d− and q−aXis currents for the )一directional drive I、tv., Ig.i. can be numerically analyzed, and are shown in Fig.6.1.2・4. From these results, the d−axis currents fbr the x−and〕・−directional drives i、lr and I,i、「 generate nearly equal translationa1丘)rces E. and torques r,, and therefbre cannot be uniquely determined f壬om the total translational forces F. and torques T,. In other 175 w・・d・,with・nly th・d−aXi・ ・urr・nt・伽th・x−andγ一由recti・na邉v・・ldu and 1、,, 2’DOF妙ing飴rces cann・t be c・nt・・U・d. Th・t・rqu…e・ulting丘・m th・g一舳 ・urr・nt・f・・th・x−and y−・lirecti・nal d亘ves伝and拓紺・・imila・ becau・e・f th。 symmetry of the actuator. When the yaw angleα=±24.7 deg and±45 deg,3・DOF translational fbrces ca皿ot b・g・n・・at・d rega・dless・f th・magnitudes・f・the・d−・and・9−aXt・・cur・en偽伝, lqx. Thi・i・ presumed to be caused by the magnetic field resUlting from magnet mover, which is t且ted an angle of 24.7 deg or 45 deg. The mover generates opposite magnetic poles every pitch lengthτin the .Mm−・lirecti・n・and・・th・magn・ti・p・le・at a p・・iti・n・and・5rdi・tant p・siti・n a1・ng th。 ルーdirection are mutuaUy opposite as shown in Fig.6.1.2・5. When tited byαb=23.6 deg (・1・・et・24・7・deg)・in・th・a−directi・n, th・m・v・・g・nerates・PP・・it・magneti, p。1。、 every 2τalong the x∫−direction as shown in Fig.6.1.2・5 because of geometry relation, as shown in the following equation: α・=・in−1 k2τ5τ〕=23・6deg・・…….…….……..………………...………………..……..……(6.・.2−・) Then, the same armature currents且ow every 2τ along the xs−direction. Therefbre, if the magnet mover generates a completely・sinusoidal magnetic丘eld distribution in the xm−and y.一一clirections, each phase current generates opposite translational fbrces every 2τ in the x.−direction du血g the yaw angleα=23.6 de g. Consequently, these opposite translational fbrces can be mutually offset. The error between the theoretically(23.6 deg)and analyt元cally(24.7 deg)obtained yaw angle is presumed to be caused by an incomplete sinusoidal magnetic field generated by the magnet mover. As mentioned in Subsection 3.2.1, the m血iaturized mover also generates a quasi・ sinusoida1且ux density in the xi−and yt−directions. When the mover is t江ted by 45 deg in theα一(lirection as shown in Fig.6.1.2・6, the且ux densities Bx,∂ソ, Bごbelow the mover are approxhnately expressed as fbUows: 輌・石)=−Bm(・・)・in〔。lti;−dx・〕…〔,: l………一…・・………一…・……・(6…2・2) ・y・ 輌・・』ω…〔π XlTPAd〕・i・〔 ,元ア1〕……………・……………・・…・………・・(6…2・3) 』・・Zs)=矯in〔π xlrPAイ〕・in〔。:, (6…2・4) So, armature currents flowmg through a lineみ(i=xor y, k=u, v, or w)in armature conductors, i」・k generate no translational fbrce l)ecause average of the且ux densities Bx,、Bv, B。 with respect to the必一direction is nearly equal to zero, that is, translational fbrce F, 176 shown in Eq.(3.3.1・3), is expressed as fbUows: /戸=・・/戸N・・S,,B・⑭.…………...………………....……….…(6.・2・5) F=− w怪・B)一…...………………..….……………..…………..…….……..(6.、.2−6) Figures 6.1.2・3 and 6.1.2・4 a180 indicate that a magnitude of torque 7>resulting 丘om the mover tiled by 24.7 deg is larger than that by 45 deg. Magnitudes of torques]㌦ and Ty resulting from the mover tiled by 45 deg are equal because且ux density resulting from the magnet mover is symmetrically disUibuted in the xs−and距directions. On the other hand, magnitudes of torques Tx and Ty resUlting from the mover tiled by 24.7 de g are not equal because of asymmetric dis旋ibution of the flux density in the x、−and Ys−directions. Figures 6.12・7 and 6.1.2・8 show the analysis results of the torques Tx,7}, T.. for the pitch angleβwhen the o』and g−axis currents are supplied(伝=1A,」伝=1A,レ=1A, or」lay=1A), the yaw and ro皿positions are not displaced(α=γ=Odeg). From these results, it can be seen that the d・−axis currents generate the torques Ty proportional to th・pit・h angl・βand th・g−axis c㎜・ntS 9・n・・ates th・瓠m・・t・・n・tant t・rques T,. Figure 6.12・9 shows schematic views of the generation of the torques 7}. The g−axis current for the y−directional drive also generates the tx)rques T. proportiona 1 to the pitch angleμ Figures 6.1.2−10 and 6.1.2・11 show the analysis results of the torques Tx, Ty, T. fbr the roU angleγwhen the o』and g−aXis currenbS are supplied(伝=1A, Igx=1A, Idy=IA, or lay=1A), the yaw and pitch positions are not displaced(α=β=Odeg). From these results, it can be seen that the d・−axis currents generates the torques]㌦proportional to the roU angle% and the g−axis currents generates the almost constant torques Tr. Figure 6・1・2’12 shows schematic views of the generation of the torques Tx. The 4−axis current fbr the x−directional drive also generates the torques T. proportional to the roU angle r. 177 20 40 1(7’,),、・24.7[>1(γ二)rt..b−1 2 30 15 邑lo 207 憲5 ξ 2 口 0 lo 0 0 \」/. 亘 &i ε一5 −io8 碧 宮 o −20e 匡’iO −15 1A・・ 一30 一20 一40 一90 −60 −30 0 30 60 90 Yaw angleα(deg) (a)Driving fbrces加m the 4−aXis current for the x−dii’ectional di’ive ld,=1A. 20 40 15 2 30 ε10 害5 で 2 日 0 20目 10る }ひ ’7’ 0 斥’ 言 ご 一10㎏ 呂 一20ぎ .9 −5 烏 冨 =−10 E 、T.r −15 一30 1( T.r)、,・4sl lCT,・),、−4sl 一20 一40 一90 −60 −30 0 30 60 90 Yaw angleα (deg) (b)Driving forces from the q−axis current fbr the x−directional driveち、,=1A. Fig.6.L2・3: Driving forces fbr yaw ang]eαat pitch and roll anglesβ=∼’=Odeg when the armature currents fbr theエーdh℃ctional drive are supplied. 178 20 40 1(z1二)、,−24.・[>iσ),、..4il 2 15 30 207 邑lo 巨 §5 扁 0 lo 2 旦 /.’\ 0 已 ㌧∨ ぎ一5 一ユ障 亘 .20ξ ‖−lo 占 一15 一30 一20 −40 一90 −60 −30 0 30 60 90 Yaw angleα (deg) (a)D亘ving fbrces丘om the∂−axis current for the y−directional drive ld),=1A, 20 40 「(Tx)。・45|k、Tv),、.451 15 30 乞 ε 10 に’ 8 L 20巨 で loる 5 ? 0 o『 肩 o = .10肖 一5 ’二: 6 芝 〔−10 L α=24.7deg 一20き 9 一15 一30 一20 一40 一90 −60 −30 0 30 60 90 Yaw angleα(deg) (b)Driving fbrces from the g−axis current fbr the J一directional drive 1亘、.ニ1A. Fig.6、12・4: Driving fbrces fol’yaw angleαat pitch and roll angles fi =y=Odeg when the armature current・s fc)r the .i,−directional drive are supplied. 179 一 「 一 「一「 .・ D1し. ’n ソ、,,1 1 一1 「一 Mutual「y opPosite poles for the)C〃−direction Opρosite driving ft)rces are offset ll の SnS 1 1 i li 句 ↓≧ 1 σ、 の 1 L A. 1 α 1 1 の の 1 の }1 1 2τ1 α=23.6deg llll ,1 ,” 1 E O 口§am竺三「「99/S一 [ ..J 一 Fig.6.1.2・5: Relation between pitch lengths of the meander shape and magnetic pole when the yaw angleα=23.6 deg. 1 il 一 1 x’ )㌔ Z oo の Z の Z Z の Z の Z の Z Z の @33.7τ b1 1 乃 1.・・lt、・.・・1 F=一Σ∫,セ、・・8)・−ll・,≒o ノ.k ” i の ∫.旭・.…∫ぴ・t?・・ iNo translational if。・ceg・・e・ati・・ l i 1 ㌦ 0 {,、・ Fig.6.1.2−6: Integration of且ux density B二along a line when the yaw angleα=45 deg, 180 1)・kin armature conductors 15 10 言 日 5 z 邑 已 O F・;””−5 Id.v 日 一10 ・15 一2 一15 一1 一〇5 0 05 1 1.5 2 Pitch angleβ(deg) (a)Driving forces due to the d−aXis current for the x−directional drive塩,=1A. 15 10 官 ξ 5 E ) 0 z / ’− р氏f一合一一〔←・一.fi− ・一・s!一・−fF−_fi_. t 巨’ 一5 P 一10 一15 一2 一1.5 一1 一〇5 0 0.5 1 15 2 Pitch angleβ (deg) (b)Driving forces due to the q−axi. s current for the x−directional driveんv=1A. Fig.6.L2−7: Driving fbrces fbr pitch angleβat yaw and 1’oll angles cr=〆=Odeg when the armature currents fbr the x−directional drive are supphed. 181 15 10 官 ∈ z T.v T. ,− 5 t l t ≡ ) φ 0 タ ー in△一・−tS. . 一合一 ・ −de ≡ −a合一 一 一 ^・ 口1 一5 fd)・=IA Tv A 一10 一15 .2 一1.5 一1 一〇.5 0 1 0.5 15 2 Pitch angleβ(deg) (a)Driving forces due to the d−axis current for the )」−directional drive ld、,=1A. 15 7’ 10 ?D 官 巳 5 Z ..... @ ...・....... @ . ε 自 0 @ τv . .A &”’ D5 己 ノ、、、・=1A 一.. .」. @ 1’, 一 一10 @ ..・「「 .」 一15 @ ‘ ‘ . 一2 一15 一1 @ . 一〇.5 0 0.5 1 1.5 2 Pitch angleβ (deg) (b)Driving forces due to the q−axis cui’rent for the〕r−directional drive∫1,t,.=1A. Fig.6.1.2−8: Driving fbrces fbr pitch angleβat yaw and roll anglesα=γ=Odeg when the armature currents for the s−directional drive are supplied. 182 Mover 、1’=.;’ 0 Disp[aced inβ T,,>0 CE) AUraction force d−axis current for.Y−drive/、t,>0 ・Y、 N S S N Stator Positive lorqtle generation (a)Generated torques T,,丘om the dLaxis current for the x−directional drive. )・つ∫ M。verろ・<0 ・瓢:;∵1驚:顯 。) ↓,tC .、 S N S N Stator Negative torque gelleration (b)Generated torques Ti. f士om the q−axis current for the x−directional ditve. )’=.1・、 Mover 71・>0 「 S Stato「 Positive torqしle generation (c)Generated torques T,.f士om the d−axis cun’ent f{)r the .v−directional drive. Mover Zi・=O .∼.’=.T’、 Disp「aced inβ o〈/ : ‘/−axis current fo「.1一drive/...、>0 『・ 0(N◎Tnagnetization) Stator No Tol’qLlc genen’atioll (d)Generated torques T,、 fi’om the q−axis current for the .i’−directiona1 drive. Fig.6.L2・9: Schematic views of generation of torques T,.. 183 15 10 官 il 5 /. z 邑 已 0 ・1・.5 ∫d.y 一10 一15 一15 一2 一1 1 一〇5 0 0.5 2 1.5 Roll angle/(deg) (a)Driving forces due to the d−aXis cum’ent for the .r−directional drive塩v=1A. 15 10 ... . ...... .. 官 巨 5 Z .. D. . @ ・ .・ ・ 7∵ ε 已 O .・ . A t’“・ D5 ... ご ∫甲= ・ 1 ... τ二 一10 .・. D . 一15 .2 .1.5 一1 一〇.5 0 05 1 1.5 2 Roll angle 1(deg) (b)DrMng fbrces due to the q−aXis current for the .r−directional drive lci.r=1A. Fig.6.12・10: Driving fbrces fol’roll angle/at}・aw and pitch anglesα=β=Odeg when the armature currents fbr the x−directional di’ive are supplied. 184 15 10 Tし. 71. E / / z日 5 ε 已 O 7−° 一 秩E・一;tw.】、.t..tr’記一・〔・・一. .挙. ィ一一{←・.一・Pt・C∴・Z= ti−・ ∫dv=IA. Tx 一10 一15 1 一2 一15 一1 一〇.5 0 0.5 1 1.5 2 Roll angle/(deg) (a)Driving forces due to the d−aXis current for the y−directional drive J,1、,=1A. 15 10 官 ξ 5 z ε 自 O \ 『→ …臼… ↑ r に」5 ・ 一10 一15 一2 一1.5 一1 一〇5 0 05 1 1.5 2 Roll allgleγ(deg) (b)Driving fbrces due to the q−axis current fbr the S−directional drive塩1.=1A, Fig.6ユ2・11: Driving fbrces fbr roll angle! at yaw and pitch anglesα=β=Odeg “’hen the armature cun’ents for the .i一dii’ectional drive are supplied、 185 ・r=・N’、 Mover 7’、>0 Displaced in 7 イ!−axis cur「ent 夢ぷe∫〉° Stator Positive torque gelleratioll (a)Generated torques Tr due to the d−aXis current for the x−dii’eetional drive. .・’・= .X、 Mover 71, =O O S DiSPIaced inγ 〔 t∫−aXIS current ㊤, for.v−drive t >0 1・ 「’‘・ ・㌧ 「 0(No magnetization) Stator No↑orque generation (b)Generated torques T, due to the q−axis current for the x−directional drive. T>0 ’1’;A’、 Mover Stator Positive torqしle generation (c)Generated torques Z, due to the d・−axis current fbr the v−directional drive. .N’=x、 Mover τ>0 Stator Po9. ilive torque generaUol1 (d)Generated torques Tv due to the q−axis current for the〕・−directional drive. Fig.6.1.2・12: Scllematic Ntie“’s of generation of torques 7r.. 186 As the analysis results above show, the driving fbrcesノ『t, F,., F., Tr, T,.,乙can be expressed丘o皿the d−and q−axis currentsん.,ん,1,t)、,ソ, as fbllows: ε, F: ノ‘tr KI.−F(d,β,γ) l、ir ......,.’.....㊨............,......................,......,.”.......(6.1.2−7) 6×4皿atri z、・ 1‘{ト・ Ig), T, where Kl.・1・is a 6 x 4 matrix and all elements of the matrix nonhnearly depend on the yaw angleα, pitch angleβ and roll angle/In this study, the pitch and roll displacements of the mover are assumed to be very sma11(β定Odeg and 7=Odeg) because of small air gap(less than l mm)between the mover and stator, and in the range, all elements of Ki.i・ almost linearly depend on the pitch and roll displacements. Further皿ore, if the yaw displaeements are assu皿ed also to be very srnall(αzO deg), all elements of Ki.v・almost linearly depend on the yaw displacements, and the system・ constant matrix Ki・・7・is expressed apProximately as follows: κノィ. E、・ 0 迭 0 O Ki.〔. O 1‘tr l,1x ______...______.__.(6.1.2−8) K’ il, fl KIr・ Jd,, ll、・ −Kl《・ K〃・cr J,t,・ z. κ〃17 κ〃,β 7i,. whereκ,ぞ,κ1c, and Kyアare constant(in this analysis, fbr a O,5・mm ahr gap, Ki,c・零17mN, κア1’ ・12mN・mm, and Kl7, =4.5 mN・mm). Equation(6.1.2−8)indicates that the driving f()rces due to the d−axis currents Jd, and泓、. are equal because of the symmetry of the actuator. Therefbre, even迂the two currents Idv andん,. are controlled, only 1・DOF driving fbrces can be controUed in the range withinαre O deg,β≒Odeg、 andγたOdeg. Therefbre, controUing the fbur arnlature currents in the吻一f已ame controls the 3・DOF motions of the mover(fbr instance, x−, y−andご一皿otions, or x−,」一, andα一motions). In order to realize both 3・DOF motion controls on a plane and magnetic suspension, the planar actuator needs to be redesigned. 187 6.1.3.Conceptual De8ign of Fundamental St】ra(加re In order to suspend the mover, su8pension fbrces that balance the fbrce of gravity need to be generated. E quation(6.1.2・3)indicate8 that negative d−aXis currents(ldu,侮< 0)generate suspension fbrces(尾>0). Figure 6.1.3・1shows schematic views of when the d−axts currents are supplied. Negative d−aXis curTents to actively contro1 leVitation fbrces(Fz>0)always generate re8to血g torques against theβ一and r− displacements. The restoring torque8 stabilize theβ一and r−motions of the mover. Equation (6.1.2・3)also shows that the g−axis currents Igx, Iay generate the translational fbrces Fx,ちon a plane without vertical fbrces」Fz. Therefbre, the d−and g− a)dS C㎜ents伝, lg。, ldy, lgy: > independently control the translationa1]1orces Fx,ち, Fz > stabiUze the pitch and roU motions. 且owever, the d‘−axis currents utiized to control the suspension fbrces」巳, generate yaw−directional torques proportional to the yaw angleα, that is, they generate instable yaw motions. Therefbre, in order to realize both 3・DOF motion controls on a plane and magnetic suspension, a stabilization mechanism for the yaw motions is needed. Then, we can consider the fbUow血g two methods toward addition of the stal)ilization mechanism;redesign of structures of the permanent・magnet mover or stationary armature conductors. Fab亘cating the permanent・magnet mover is d避icult in bonding each permanent・magnet component. On the other hand, the armature conductors can be且exibly and easily manu£actured by means of mult且ayered printed ch℃uits. In this study, the armature conductors are redesigned to offer stable yaw motion with less interference to the translationa1, pitch, and roll motions. The torques acting on the mover depend on the relative yaw, pitch, and roU distances between the mover and the armature conductors, but relative pitch and roll distances should be a lways nearly equa1 to O deg in order to maintain a sma皿air gap. The torques also depend on pitch lengths of the armature conductOrs, which determine an alowable max血um width of those as shown in Fig.6.1.3・2. The width of the armature conductors also determines an aUowable maximum current of those, and so design of the armature conductors includmg Pitch lengths as a parameter tends to become comphcate. In this study, new armature conductors with different relative distances in the yaw direction丘om the armature conductors fbr the x−and.y−(lirectional drives are introduced to control the yaw motion as shown in Fig.6.1.3−3. 188 含・・V…一・ NSNSNSN Mover Mtlll NsNsNsNsNsN Stator Negative d−axis Current(/dく0) (a)Generation of the levitation fbrces F.. 竺Ω血gforqαe Mover (displaced inβor/) NsNsNsNsNsN Stator Negative d−axis Current(ld<0) (b)Generation of the restoring torques万, and Tr. Propulsion force M°ve「 Stator sNsNsNsNsN q−axis Current Iq (c)Generation of the propulsion forces Er. and F,.. Fig.6.1.3・1: Conceptual design of a magnetically leVitated planar actuator, 189 」 一 ... .. Fig.6.1.3・2: Allowable maximum width of the armature conductors determined by pitch length of those. ‘Tilt r””NL...7 ,’/ 、、 Fig.6.1.3・3: New introduced armature eonductors tilted in the yaw dii’ection. 190 Figures 6.1.2・3 and 6.1.2・4 mdicate8 that the d−−axis current generates translational fbrces Fz and torques Tz, and the 4−axis current generates translational fbrcesノ㌦, Fy and torques Tx,7}when the pitch and roU position8 are not displaced(β=γ=Odeg). So, at least fbur kinds of the g−axi8 currents, that is, fbur pah!s of polyphase currents are needed to actively contro16・DOF motions. Furthermore, Figs.6.1.2・3 and 6.1.2・4 indicate that the d−and g−a】ds currents gellerate o】〔且y torques without tran81ational fbrces when the relative yaw distance is 24・7deg or 45 deg・As mentioned in Subsection 6.1.2, a magnitude of torque 7三resulting from the mover tiled by 24.7 deg i81arger than, that by 45 deg. Therefore in this study, the armature conductors are tilted by 24.7 deg tn the yaw direction丘om the armature conductors for the x−directional drive, I term this arrangement ”armature conductors fbr theα一directional drive.”When the yaw angle of the moverα=Odeg, the d−aXis currents fbr theα一directional drive Ida: > generate only torques T, > without vertical fbrces凡. Therefbre, the d−−axis currents Idα can separate the generation of the vertical fbrces 尺and torques Tz, and stabiize the yaw motion. To date, the 4−and q−−axis currents are generated by three・phase currents, but they can be also be generated by two・phase currents. In this study, a magneticaUy levitated planar actuator with three pairs of two・phase armature conductOrs is organized as shown in Fig.6.1.3・4. Tables 6.1.3・1and 6.1.3・2show the specifications of the m血iaturized permanent・magnet mover and a triple・layered printed circUit board mo皿t血g armature conductors, respectively. 191 y Top view tt グ ’ パ.,、.后〆〃:万 ./’ s,. β i.…川・.・.、.1.・・2〃拶 憂§難lll繧 z x α γ ’ Mover (1{albach arra)..) 賎 , ら .・ら ら ら 辮1彗R。∋。や StatOr e日:{8} ぐ■ 1.● ,$ 鵬1’ (3−la)’erprinted circuit) 医.コ1. 、、 ロカ s h ド ごロ ノ1鍵:1 Two conductors アf。,)・.d,ive T、vo conductors 1 fbr x−drive 二し :.、÷:∼.. 鞍壷鍵._.一 ’、・∼.∼ ’ Two conductors i fbr a−drive ‘みasis li .1 . Mover er τ/2 S−ax】s 膓 1/ ノ∫ tt StatOr @∫ ” Thickness:0.lmm Side view z α ア Insulating layer(Thickness:0.2 mm)! (a)Fundamental structure. (b)]N{anufactured stator and mover. Fig.6.1.3・4: Magnetically levitated planar actuator. 192 Table 6.1.3・1: Specifications of miniaturized permanent・magnet mover. Mate亘al NdFeB(Shin・Etsu Chemical Co., Ltd.) Residua1且ux density Br 1.35−1.41T Overall dimension ll mm×11 mm×2mm PM component 2mm×2mm×2mm, or 2 mm×1mm×2mm Tota1 mass 1.89 Table 6.1.3・2: Speci丘cations of triple・layered printed ch℃uit board. Number of conductor layers 3 PitCh of meander pattern,τ 2.1mm Number of turns of meander pattern 16 Width of conductors 0.8mm Thickness of conductors 30 一 35 pm Thickness of insulating layer O.1,0rO2mm Resistance of each conductor 1.0Ω 193 6.2.Dynamic BehaVior of Mover The mover has 3・DOF translational and rotational motions because there is no mechanical suspension mechanism. When the physical quantities of the mover motion are repre8ented, it is extremely important what coordinates are respected. The translational motions are often represented w誌h respec乞to乞he stationary coordinate, and the rotational motion8 are often represented w比h respec乞to the mover coor《㎞ate. This section introduces an equation fbr the 6−DO亙]【notions ef the mover that desc亘bes the dynamic behaVior. ・ 6.2.1.Mass and lnertia Tensor The mass M and inertia tensor J.’ef the m《rver are determined by mass・de】窪sity a丑{垂 dimensions. The mass M was measured us垣g an eleetronic sca le(LIBROR, EB・・320鐙,. Shimadzu Corp.)that has a O.1−g resolutio江.「駈e scale垣《五ca給《l that. mass・擁:=・1.8:gi. which agreed with the theore昼cal value calcUlatedi丘o]【n mass de丑s迦ρ=7↓6③×103三 kg!m3 and volumeγ=224 mm3. The hle垣a teDsorみ.’Wtt}1 respecも:挺樋e・mover coor(linate axes xh、ym.7m with an頭9in at O’, correspen砲g to the ce疏e宝・of;搬ss・o£;』・・ mover shown in Fig.6.2.1−1, can be represe亘ted as a 3 x 3 mat亘x a…s島避Ows・; J.‘」㌶J. Jm’=J♪。’」ガJ♪. ................_...右.Q.輪...⑰.飢_◆..。...。.繊◆_.....◆..........}見.軌.}}}__..〈6こ2二玉ヨi) J.’」4J= where the diagona1 elements 」. ’,み’, aロ、d X』’』are the. m.o敬e,n.ts・o銑he・返le頭al abou±磁e. xm−,ルー, and z〃7−axes passing throuきh the cen搬・o£磁ass of血e血◎ve鵬主espect担e雄;. a盈《美 the o任《五agonal elements・ん’,」)実’,4零’,み’;ム.㌧銀dふ♪a苦e竈e騨od迫cts of the血e斑a・ These elements can be de丘鵬d as the章遊o“厄.殴琴e寧惑ion;: 」〆=1ρ硯2δバr)・rk)ばγ...___一一___._。。。_繊漣 whereρis mass densi観P=[司 聾 縫τIs a…袈s垣Q穎,、 vecto壬曇壁旦.疋磯a越顛.¢鰻鎗奪w姪鮭 ・espect t・the m・ver℃oo叉・撫蜘㈱s獅癒弓麺鴫;.左・=・1,,2違殴磁.煕磁s磯』・ Position vector r, and 4R is KTQ駐e¢ke置《1¢1ちa・無垣e琴毒.i塾亀en§o書痴.邑o£ぴ璽eetaD・膠a鍵頚越:§m5. which has u頑。顛ma領撫siも顯、嘘垣e綱磯垣壌・¢Ω壁撫撫撫題藤,頑轟、癒・ origin at O cξEロbeξepξese旦垣《董as・趨且◎ΨS; 襲挺 吉〃‘。2+1。2)・ ・ @・ 誕2+1.2)・ 」・L o o ⊥M ............._....._.._......(6.2.1・3) (1.2+1.・) 12 where lx, ly, and lz are the len gths of the edges of the prism as shown in Fig.6.2.1・2. Next, /of the same prism with respect to the we can easily calculate the inertia tensor Jo coordinate axes x“p7, with the origin at O, pata皿e1 tO the coordinate x.Or.z. with each other as fbnOWS: b2+c2 Jo, ’=Jo’+M −ba −ca 一ab −ac c2+a2 _bc ◆...__.._.............◆....◆......................(6.2.1−4) −一・cb a2+b2 c]ア◆ where dt=[a b ls the displacement vector from the origin O to the origin O,. The inertia tensor Jm’of the mover with respect to the coordinate axes x3iv。tt、 Js’can be calculated丘om Eqs.(6.2.1・3)and(6.2.1・4)as fb皿ows: 0.1828 0 0 Jst= ・10−7kg・m・.___◆_._.____._____.(62.1−5) 0 0.1828 0 0 0 0.3543 As we can see, the inertia tensor/1’is a diagonal matrix. The diagonal elements of the inertia tensor/1’and the coordinate axes U、z. are referred to the princip a1 moments of mertia and the principal axes, respectively. Once the principal moments and their axes of the mover are known, the inertia tensor Jm, with respect to any other axes passing through the center of mass, can be fbund by a shnnarity transfbrmation de丘ned by the Euler angles relating the two coordmates. If the transfbrmation matrix is given as R, the inertia tensor Jm’can be represented as fbUows: Jm’=RJ5’R7...◆....◆..._.....◆..........◆............◆........................_....._...................(6.2.1・6) The transformation matrix R from the stationary・coordinate axes石γ冨, to the mover・ coordinate axes xh,yh,z. shown in Fig.6.2.1−1is given as fblows: cos(π/4) −sin(π/4) O R=・in(π/4)…(π/4)0 ...◆...........................◆...................._....◆.............(6.2.1−7) 0 0 Therefore, we can ca lcUlate the inertia tensor Jm ’ of the mover with respect to the mover・coordmate axes x∂,㎡m as fb皿ows: 0.1828 0 0 Jm’= 0 0.1828 0 ・10−7kg・m・._.__.____.__...____(6.2.1−8) 0 0 0.3543 195 ㍗1η 〃1 Fig,6.2.1・1: Mover with mover・coordinate axes x“∂・“ご“, and stationary℃oordinate axeS芯):、ご.v・ )’t ・rf Fig.6.2.1−2: Rectangular prism with two mutuaUy−parallel coordinate axes. 196 62.2.EUIer Angle and Arigular Velocity In order to define the 3・DOF rotational orientation of the mover, the Euler angle needs to be de丘ned[GolO1, TajO6]. In this study, Euler angleφ=[α β /71 is de丘ned 丘℃mα,βandγas orderly counterclockwise rotations around the stationary二,一,)・.s−and ぷ一axes passing through the center of mass of the mover, respectively, as shown in Fig. 6.2.2・1.At f丘st, an immediate coordinate xlγ}zl is defined to be rotated f士om the stationary coordinate U宕, byαaround the二、−axis. Then, an皿mediate coordinate xぴz二2 is defined to be rotated丘om the coordinate x1γ向byβaround the y.−axis. Finally, the mover coordinate x“」,“浮,,、 is defined to be rotated丘om the coordinate x2yユニ2 byγaround the.v.,.−axis. Next, the orientation of the mover coordinate x“D,,,ξ“, with respect to the stationary coordinate x.O’・、z,, R。“、, is introduced宜om the Euler angleφWhen a body is rotated counterclockwise by Vi areund an arbitraエy vectorλ=[λ1 λ2 ゐ]㌧the rotation matrix Rv, can be represented as follows: R,、ニE…9‘t・(?・IMI+7L,〃,+占〃、)輌・・a・11』(1−…V)__.…__….___(6.2.2−1) where E is a 3×3 unit matrix and Mt(∫=1,20r 3)is an infinitesimal rotation generator, which can be represented by the following equations: 0 ui: 0 E・ 0 βfl= 0 ....,..’..,_......._...._...__、..,...,.............㊨....._..................㊨㊨......㊨(6.2.2・2) llii]…・闇一・一・・一・6…2−・・ 0 0 0 −l 0 , 〃勺= l O Zl z∫ α zづ ・⇒ c−・i’.[〉 Ys β Xs zηf γ Xnl X] Stationary coordinate Fig.6.22−1: )/1il ・x’.s’ Definition of Euler angleφ=[α 197 Mover coordinate β〕]v’. At丘rst, a rotation matrix’ to rotate counterclockwise by a aro皿d the z.−axis, R。1 can be calculated from Eq8.(6.2.2’1)一(6.2.2−3). Because the unit vector of the zs−axis with respect t《)the 8tationary coordinate x跳zぷis represented as 11ぶ=[0 0 1]「, the rotation matriX R∫i can be represented as follows: R・1= Then, the unit vector of the ys−aXis with respect to the coordinate xOγlzl,彪is represented as follows: 0 4sニR。1−1 Slnα COSα 0 _........................◆..............◆....◆..................._............(6◆2.2−5) 0 Therefore, a rotation matrix tO rotate counterclockwise by fi around the ys−axis, R 12 can be calcUlated as follows: R12= cosβ+sin 2α・(1−cosβ) c・sα・sinα・(1−c・sβ) …α・・inα・(1−・・sβ) …β+…2α・(1−…β) 一sinα・s口1β __(6.2.2・6) sin a ・ sin 6 COS fi −COSα・sinβ COSα・sinβ Fina皿y, the unit vector of the xs−axis with respect to th is represented as follows: COSα・COSβ λms=Rl 2−I Rs1−10=−sinα・cosβ ___..___.__._.._____._.___(6.2.2・7) O sinβ Therefore, a rotation matrix to rotate co皿terclockwise by r aro皿d theひaxis, R2nt can be calculated as fbUows: R、m= [R、mlR、m、R、m、]...__.__.__.__..____.____.__..____.(6.2.2・8) …r+…2α・C・S2β・(i−…r) R2ml= ・inβ・・in・1−…α・・inα…S2β・(1− COS 7) .............._._....◆◆.......(6.22・9) ・inα・C・・β・・in・1+…α・…β・・iψ(1−・・Sγ) 一・inβ・・in・7−…α・・inα・c・・’β・(1−…γ) R2m2= …r+・in2α…S2β・(1−COS 7) __.._....._............(6.2.2−10) …α・…fi…i・r−・inα・c・・,B・・inβ・(1−…γ) 一・inα・…β・・inγ+…α・…β・・inβ・(1−COS 1) R2m3= 一、。、α・C・・,B・・in・r・一一・inα・…β・・inβ・(1−…γ) ....._._.............._(6.2.2・11) …γ+・in2β・(1−…r) The rotation matrix of the mover coordinate x例ymz功with respect to the stationary coordinate xW。, R。m, can be calcUlated from the rotation matrices Rsi, R i 2, R2m as follows: 198 R、m = Rs’IRI2R2m COSα・COSβ 一Sinα・COSβ Sinβ =Sinα・COsγ+cOSα・Sinβ・Sin l COSα・cosγ一Sinα・sinβ・s㎞γ 一cosβ・Sinγ Sinα・Sinγ一COSα・Sinβ・CoSγcOSα・Sinγ+Sinα・Sinβ・cOSγCOSβ・COSγ ............_......._.....__(6.22・12) The rotation matrix of the stationary coordinate xSy、zs with respect to the mover coord丘1ate )chJlh,zm, R“,s, can be calculated as fbUows: R朋。=R。m−1=R.m「 COSα・COSβSinα・COSγ+COSα・Sinβ・SinγSinα・Sinγ一COSα・Sinβ・COSγ =−Sinα・cOSβcOSα・COSγ一Sinα・Sinβ・SinγCOSα・Sinγ+sinα・Sinβ・COSγ Sinβ 一COSβ・Sin 1 COSβ・cOSγ ◆...◆._.....◆◆..◆◆.....◆.......◆..(6.2.2・13) We can convert positions with respect to the mover coordinate x〃品into those with respect to the stationary coordinate x躍, as fbUows丘om Eq.(6.2.2−13). The angular velocity of the mover with respect to the mover coordinate x〃㎡加, as shown in Fig.6.2.2・2, cz)sm ’=[∂,’toy’tDz’]τcan. be calculated as fblows: ω、m』R、m’IR12−1ω。1’+R、m−1ω12’+ω、m’ =Rωφω誓 ’’’”°’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’”°’’”◆”◆◆’◆◆°°(』 Sin a・Sinγ一COSα・Sin fi・COSγ S桓α・cOsγ+COSα・S㎞β・Sinγ COSα・cOSβ Rωφ=cOSα・Sinγ+S㎞α・Sinβ・COSγ coSα・COSγ一S01α・S㎞β・S㎞γ 一Sinα・COSβ COSβ・COSγ 一COSβ・Sinγ Sinβ ......_.◆◆_◆◆..............◆◆◆.◆(6.2.2−15) where∂,1’,ω12’, andσ㌦’are angular velocities of the mover about the z。−axis with respect to the immediate coordinate xlylzl, the ys−axis with respect to the immediate coordinate x2y2乏, and the xs−axis with respect to the mover coor{五nate㌔y房m, respectively. The angular velocities tv。i ’,ω12’, and∂2m’can be calculated from the unit vectors Zl、,ん,ん、 a nd EUIer angleφ=[α β IT as follows: ω、ll=み。芸・ω121=ろ、誓・ω・加・=・1,m・書一・………・…………・……・・……・・…(6・2・2−・6) Then, we can calcUlate the differential of the EUIer angle(4/dt)丘om E qs.(6.2.2・14) and(6.2.2・15)as fbUows: 誓=R。φ⑧一1ω一1…・…・…・・……………・………・・………・…………・・……・………………(6・2・2−・7) 199 1 一l Rdi,. c・・(2α) ラ コ Slnα・Smγ一COSα・Sinβ・COSr cosα’Smγ+sinα・sinβ・cosγ x Slnα・cosγ+cosα・Smβ・sinγ COSβ・COSγ COSα・COSβ シ COSα’COSγ一sinα・sinfi・sinγ 一COSβ・sinγ 一Smα・COSβ sinβ .__._._._,,_.___..(6.2.2−18) Equation(62.2・18)in〔licates thatthe matrix 1∼、41 cannot be defined, and therefbre the Euler angleφcannot be uniquely determined丘om this equation when the Euler ・ng1・α=±45,・r±135 d・g・Th…ti・nt・ti・n・f th・m・ver i・・ft・n・al1・d…i・gUl。。 posture.”However, in this study, it is assumed tha七the mover is driven in the range within the Euler angleα≒Odeg. Therefbre, a singular posture cannot occur, and the differential of the Euler angle(dφ/dt)can be calculated fi・oM Eqs.(6.22・17)alld (6.2.2−18). z,η A:. (」)α Ym Xm Fig.6.2.2・2: Angular velocity di,。、’=[co.r ひ、、 tv,]Tand Euユer angleφ=[α β 7〕 T. 200 6.2.3.Equation of Motion The equation of the motion of the mover can be represented by the translational fbrces acting on the mover F、m=1万 ち 」Fz]「and tOrques around the mover center O・ Ts加’=[Tx’万刎「a8制・WS・ dり ”元加=塩+Fg……・・…・……・…・……・…・・…一…………・・……………・…・…….…….(6.2.3・1) み,14 R加’=T。m・−tD。m・・(」.’・p。胡う……………….…….…….……...……………….……(6.蹄2) where巳坊輌v, vz】Tand Fg=[00−Mg]Tar・v,1。city。fth。 m。ver and th。 f。,ce。f gravity acting on the mover, respectively」 Equations(6.2.3・1)and(6.2.3・2)represent 3・DOF translational and rotational motion equations of the mover, respectively. Al variables in the translational and rotational motion equations are represented with respect to the stationary coordinate xSy,z。 and mover coordinate xhJl..7励respectively. The position r,m and EUIer angleφof the mover can be represented by the velocity vs〃2 and angular velocity m,〃!, respective1)ちas fbUOWS: dr。m =ッsm_.____...______..__.__...◆...............................◆◆.....◆.......◆.(6.2.3−3) fl=Rωφ⑯一1ω、m・・………………………・……………・…・.….………….….…….…………(6.2.3・4) Equations(6.2.3・1)一(6.2.3・4)can represent dynamic behaviors of the mover with 6 DOF. 201 6.3. Planar Motion Control with Stable Magnetic Levitation This section di8cu8ses 8ix・current controls to stably leVitate the mover and actively control the xrナ, z−, andα一motions. There are two important things fbr the motion controls: 〉 to generate independent translational forces」Fx, Fy, and F. with stable torques in theγ一andβ一directions. 〉 to generate torques in the a−direction with less interference to translational fbrces Fx,1『}, and Fz. This section丘rst presents driving fbrces resulting丘om three pairs of two・phase armature current8, and then the driVing force℃ontrol system. 6.3.1.1}ran 81ational Motion Control In this study, three pairs of two・phase c㎜ents杉=1ん 勾79=x,ア, orα), as shown in Fig.6.3.1・1, are assumed to be supplied to the three pairs of two・phase armature conductors as shown jn the fbUowmg equations: ・1ノー一・ノ…ψv)….….………………...…….….……….………….……..…………….…....(6.3.1・1) ・、ノ=・ノ・inψv)…..….…………….….._…………….….…….………._………….….…(6.3.1−2) Figure 6.3.1・2 shows phasor diagrams fbr the relation between the dq−frame and α’ タ’一丘ame. The currents lix and lly generate the opposite・phase magnetic丘eld to that resUlting from the permanent・magnet mover when the mover position in the x−and .y−directions(x,.y)=(xs, Ys)and the Euler angleφ=(0,0,0). Theα’−axis are a五gned to the oPPosite side of the current 1りaxis, and theβ’−axis leads theα’−axis by 90 deg. The current llαgenerates a magnetic field that is tilted byψ=−24.7 deg around theα一 direction丘om that caused by current lix. Bearing this in mind, the armature currents in the∂dq−frame々・and loj・can be represented by the currenbS lif and 12j as follows: [1:]=[:翻 譲㌻6][;1:]………◆…・………・……・……・……・……・・……(6・3・・−3) [il:]=[:馴 鵡㌻劉囲………・……………・…・・………・……・…・……・(6・3・・−4) [ill]{:翻 識劉6:]一…・…・……・…………・・……・………・……(6・3・・−5) αs=xs cosψ_ys sin q..____..,.__.____..____..__..______..._(6.3.1−6) 202 fi =axi g. ▲|brx−drive d−axis . . 1 . . . . tbr a−drive ll。,謂:v,▲1 ;r㌃芸i,c 442∴二;ivc lI ” ノ (y−axlS tbr a−dri、ノL’ fi=axis ギ lbr 1,−drive ヒ_レα’−asis ib「ipdri、’e ・t ‘ 1..1 1 1 1 1 ‖刀 #・#・iH:i鵬オ]・1 氈E−ax】s 一 ゜1・P ・ 一 . 一 2τ1,.11.it・.ハ1・a4・.s ・一・ ・> ibrJ一drive d−−asis !..F {br、一drive ’ ノ./ ん㌧., 〈・.:・忠,2・, .〆:え アβ 貰墜懇 .1 ノ.”...’ 王ド xα tt 1‖ x 11 α Gt 一㌘←ヨ← γ s’ つ 「) Fig.6.3.1・1: dg−frame and a’fiLframe for the x−一, y−, alldα一directional {irives. 与 β’−axis fbrノーdrive ∫、・∫、Z・ゲ . び=エ.y. orα) S−ax1S う 品rゾーdrive Magneticneld ’ f@’ 、 due to mover 、、 、 膓, @ ‘トaxis n)rノーdrive \泓 免 π 1 丁ノ・ 1bl’∫−d1 β, Fig.6.3.1−2: 、 ソ 一ax Jl/ Phasor diagram showing relation between dq−frame andα’βL☆ame. 203 These pairs of d・−and・−axis currents generate the translational fbrcesノ『、“, and torques TvnJ’as fbllowsl E ∫‘tr E F: ∫,lx κ(r、・,tt ) ip) ∫d). ..,..................,...,.....”........,㊨㊨....㊨㊨.....㊨㊨....㊨...”(6.3.1−7) 71,’ 6×6 rnatrix 1ぴ Idα T、・ T,1 ttα where K is a 6 x 6 matrix, and all elements of K depend on the mover position,w、 and Euler angleφWhere Euler angleφ=0, K can be approximated as shown in Fig.6.3.1−3, and therefore 3−DOF translational forces万, F,., and F, can be independently controlled by two・phase currents i. and i」.. In this study」references of the translational fbrces E,nt’= [4.’ F、.’ 1『』’]’1 are deterロ1ined丘om the mover positions,、,,、=Lx’y :・]T and position references rsm’=[x’」’ z’ nフ’by three PID controls. F、’.’・砿一r,n,)一・’芸・………………’一…・…………・…………・…・…・…………(・・…一・) where Pl.・=[Pノ.1・Pl.・yノ)ノ.L一]and D∫,ニ[Di−:, Dハ. Di.L.]are proportional and differentia1 ホ コ parameters, respectively. ln this study, refbrences of the armature currentsみand it. are ロ calculated丘o皿those of the translationa1 forces E,.“, as fbUows: Drivjng forces 4・ 乙 Cur「ents Translational force control 0 0000 000 00 0 0 00 Tx「 「)・’ 3x4 3 x 2 ∫dU Idl・ 14y 人31.人「ささくo (fa κ(へ。.φ) [ ㌫,]一[笥[;:1:] ldα matrix mat「ix T:1 [〉[;・]・[剖[;:d /q.v (∫、凶,<o→1・:・ω 亀,。,quec。,,., 6 × 6 matrix ∫、tv 匡i]一開[;:1::畷 cl・v ∫tiT. tt:’ Fig.6.3.1・3: Control method for driving forces. 204 闇=[Kn K12K31 K32]]蠕;2]・………………………………一……・……・…・………・・…(6・3…9) [1::]{雛:]’i「乏12]・………一・…………・・……一一………………………(一) Supplying the armature currentsみandんequal to the refbrencesピand↓’generates the translational fbrces Fsm equal to the refbrences 1㌃胡゜. 6.3.2.Torque Chatracteristics and Rotational Motion Control The armature currents i. and iy generate not only the translational fbrces Fsm, but also the torques T。m’. Therefbre, it is extremely important to investigate how the torques T。m’resUlting from the armature currents ix and iy in且uence the rotational motions of the mover. When the Euler angleφ霜0, the torques 7…’, Ty’, and 7二’are do血nant on the Euler angleα,βand 7, respectively. Next I perfbrmed a numerical analysis of the torque characteristics due to the armature currents fbr the x−directional drive when rotational motions with more than 2 DOF occur in the range within−2 deg<α,βandγ <2deg. Figure 6.3.2・1 shows the system constants K61(=7三’/、ldu)and」K62(=7∵/Iqx), which are dominant on the a−motion, for the Euler angleα. The system constant K61 is independent on the Euler anglesβand%and the system constantκ62 is almost independent on the Euler anglesαandμFigure 6.3.2・2 shows the system constants Ks i (=Ty’/ ldU)andκ52(=7ジ/Igx), which are dominant on theβ一motion, f()r the Euler angleβ The system constant Ks i is hldependent on the Euler angle%and the differential(∂Ks l/ ∂βis independent on the Euler anglesαand 7. The system constant Ks2 is almost independent on the Euler anglesα,」6, and Z Figure 6.3.2・3 shows the system constants K41(=T}’/ldU)and陥2(=Tx’/Igx), which are dominant on theγ一motion, fbr the Euler angle 1. The system constantκ41 is independent on the Euler anglesαandβand the system constant K42 is almost independent on the Euler anglesβand万 205 15 10 ⊇ 5 盲 ξ z 0 日 ) 5 一5 (β,7) (0,2),(2,2) 一10 一15 一2 一15 −1 −0.5 0 05 1 1.5 2 Euler angleα(deg) (a)K61 (=r.J/娠)at(βr)=(0,0),(2,0),(O,2), and(2,2). 15 (0,2) (2,2) 10 ? ㌫_一一k.__._ 5 ) 芦 ZT 0 ∈ ) s 一5 (β、1) (0,0) (2, O) 一10 一15 一2−1.5−1−0500.51152 Euler angleα(deg) (b)K62(=T.ソlvx)at(βr)=(0,0),(2,0),(0,2), and(2,2), Fig.6.3.2−1: Analysis result of torque万’due to the armature currents fbr the x−directional drive fbr the Euler angleα. 206 15 10 (2,0),(2,2) ⊇ 5 ) 巨 子 5 0 7, 一5 (α,7) (0,0),(0,2) 一10 一15 一2 −15 −1 −05 0 05 1 15 2 Euler angleβ(deg) (a)κ5i(=ny:11dr)at(α,カ=(O,0),(2,0),(O,2), and(2,2). 15 10 A < \ 5 (O,O) F z 写 E ) 0 (2,2) (α,/)=(2,0) (O,2) 1 誤 ・L. t 一5 t 1 1 一10 t ’ ” , ● ■ . 一 ・ 桓 一 ・ ・ ρ ・ ● 、 一15 一2 −L5 −1 −05 0 05 1 15 2 Euler aiigleβ(deg) (b)κ52(=71.’/ly.v)at(α,]・)=(O,0),(2,0),(0,2), and(2,2). Fig.6.3.2・2: Analysis resuユt of torque Tv’due to the armature currents f(〕r the x−directional drive f(〕r the Euler angleβ. 207 15 10 ? (α,β) (0,0),(0,2) 5 ) 芦 ZTE 0 ) (α,β) (2,0),(2,2) ≒ 一5 一10 一15 一2 −15 −1 −05 0 05 1 15 2 Euler angleγ(deg) (a)κ4i(=7∵!idv)at(α,β)=(0,0),(2,0),(0,2), and(2,2). 15 10 ? 5 (2,2). (2,0). ) a Z で 0 宍・一キ・一・Se ・−lie ・ 一“.・−t.・.一指 日 ) 今 一5 (α,β)(o,o) (O,2) 一10 一工5 一2 −15 −1 −05 0 05 1 1.5 2 Euler angle 1 (deg) (b)Ka2(=Z, V lgr)at(α,β1=(0.0),(2,0),(O,2), and(2,2). Fig.6.3.2・3: Analysis result of torque T,’due to the armature currents fbr the x−directional drive for the Euler angle那 208 From these results, when rotational motions with more than 2 DOF occur, K is alm・・t・in・agreem・nt・with・K・・T・in・Eq・(6・12・3)・Theref・re, n・gativ・φaxis current砿レ that・・nt・・1 the su・p・n・i・n f・・ces・F。 9・n・・ate・tabl・・restO・ing t・・qu・・T,’,・T.’.且・w・ver, th・q−axis c㎜・nt・that・・nt・・1 th・t・an・lati・nal・f・rce・Fx,ち9・nerate・tO・que8互万, T. ’, which are not stable restoring torques. So next I perfbrmed a numerical analy8is of the torque characteristics due tO the armature cur rentS for the a−−clirectiona1 drive. Figure 6.3.2−4 shows the匂)rques due to the armature condu由rs for the o←《血ectional drive at(βフ)=(0,0). When the Euler angles(βフ)=(0,0), the d−aXis current ldαgenerates only the torque Tz’and the g−axis current Igαgenerates only the torques Ty’, Tx’. Therefbre, the torque81ジand Tx’cannot be i皿dependently controned by the armature eurrents for the a−・directiona1 drive. Figure 6.3.2・5 shows the torques from the armature conductors for the a・−clirectiona1 曲・at(B・P=(2・2)・Th・d−・and・q−axi・・curr・nt・g・n・・ate・Tz’, Ty’, Tr・, but the t・rqu・TY is much less than the torques万’and 7>’. Therefore in this study, the torques T.’and Tx’ are controlled by the two armature currents for the a・−directiona1 drive. When the Euler angle ¢N O and angUlar velocity a}.,’ ft O, a linearized equation of the rotational motion can be obtained from Eqs.(6.2.3・2)and(6.2.3・4)as fbUows: 雲=R・φ〔肱L卿励1ω一》一薯〕.…………..….……………(_) 駕R。φω一1 T。mt=TE=[TaηTr]τ In this study,7ご, which is the reference of TE, is determined by a PD control from つ the Euler angleαand the refbrenceαas fbUows: T・’=P・・(a’一α)−D環・……・……………・…・・……・……・……・………・………・・……(6・3・2−2) where PTa and DTa are proportional and(i遜erential parameters, respectively. Then, the references 7ジand 7膓are determmed to be zero because of the supPression of theβ一and γ一motions. The torque references T.’and T.’can be calculated from the refbrence 7言by Eq.(6.3.2・1). Then, the refbrences of the armature currents fbr theα一dhごectional drive Idα’ ≠獅п@lgα’can be calcUlated for the torque references Tr’硲and 7ゾas fbnows: 闇=[El:瓢::]一[;]〕・……………・…………………・・…………………(6・3・2−3) where Txa’and 7三。’are torques due to the armature currents ix and ii,, and can be represented as fbUows: 209 1dU [TiZ]=[E::E:::1:2 19x ....................._......◆_........_......_._.....(6.3.2・4) ldy lgy ■ Supplying the armature currents iα equal to the references iα generates TE nearly equal to Ti, and controls the rotationa1 translationa1 motion8. 210 motion8 with less interference to the 30 §2° Tv〃、〆tX) (β,γ)(o,o) …’° i° 三’1° K4s( ㎏一20 〃dtt) .」・. ・十..’. 7ーノ/∫、,の κr,) 一30 一2 一15 一1 −05 0 0.5 ユ 15 2 Euler angleα(deg) (a)陥5(=Tx’/ldα),κ55(=T).’ノIda), and K65(=7∵/ノLta)at(6,1)=(O,0), 30 κ4〔、( t92° /f、i,‘) ◇ …1° κ56(T、㌦/1、i a) ’− 魔?f−x←一一)←・_)← i° ’■’口 ¥ 十… 一や.・.一÷唱 き一1° 1、力よ) (β,γ)(0,0) 》−20 一30 一2 一15 一1 −O.5 0 0.5 1 1.5 2 Euler angleα(deg) (b)Kj6(=7J,’11、iα),ムロ6(=T,.ソJ,ノロ), and K66(=r,ソiga)at(β1]’)=(0,0). Fig.6.3.2−4: Analysis result of the torques丘om the armature conductors f()r the α一directional drive fヒ)r the Euler angleαat(β2)=(0,0). 211 30 (β,γ) (2,2) 書2° …1° i° 三一1° κ T,〃、1。) ’三)←’−x−一一k−一一)←一一. jke−..→k−・_*一._ ◇ ・十’ K4S(Tx〃tia) 一+”・’+’ ’十” t< −20 一÷“ ・十一 κ 7層,〃、1川 一30 一2−15−1−0.500511.52 Euler angleα (deg) (a)K4s(=7r.’!Jda),κ55(=7}ソlda), andκと5(ニT,’/ノliロ)at(fi,))=(2,2). 30 κ4・(ア.・ソノ、∫“) 宕20 ξ 乏i° …。 κ56( //、J,x) ’一 h←’一・・一一一 )e・一…一._,、一 三 き一1° ”・・ {一・・一一L・… ↓一・一+一一一+・一・→一一・一+一一・一 κ,、。(γ二.∫、の 葛.20 (β, (2,2) 一30 一2−1.5−1−0.500.511.52 Euler angleα(deg) (b)K46(=TV〆1qa), klsG←TI.’/Jen), and K66(=7三ソ1,∼a)at(a ].)=(2,2). Fig.6.3.2・5: A1ユalysis result of the torques f士om the armature conductors fol’the a−directional drive fbr the Euler angleαat(fi,7)=(2,2). 212 6.4.Numerical Ana]y8i80f Mover Motion This section present8 the analytical condition8 of the 6・DOF motions of the mover and the analysis resUlt8. 6.4.1.Ana lytical Model and Condition 8 Motion characteristics with 6 DOF can be obtained by solving Eqs.(6.2.3・1)一 (62・3・4)using the Runge・Kutta method. In order to numerica皿y solve the equations, it is necessary to calculate the driving fbrces Fsm and Tsm’at each t皿e step. The calculation at each time step consists of an integration of Lorentz fbrce acting on the line segments as shown in Eqs.(3.3.1・3)and(3.3.1・4), and so reqUi res a lot of computation time. The flux density B acting on the armature conductors greatly depends on the mover position rsm and Euler angleφThere」e()re, the driving fbrces I『』m and Tsm’are fUnctions of the mover position rsm and Euler angleφIn this study, the system・constant matriX K was calcUlated and the data table of K was made before the motion analysis. Then, the system℃onstant matrix K is calculated丘om the mover position rsm and Euler angleφ1)y inte叩01ating it with the data table at each tilne step. Figure 6.4.1・1 shows a flow chart of the motion analysis. The analysis conditions are shown as fbnows: 〉 time step dt=0.2 ms 〉 control period tc=2ms 〉 initial position ri=O 〉 mitia1 EUIer angleφ・=0. When the z−position is zero, the mover is assumed to be on the stator. The proportional and differentia1 parameters are determined so that the settling times in the x−, v−,2−, andα一motions are less than 1 s. In this analysis, to investigate the planar motion control and magnetic levitation, the fo皿owing two position references are given: (1) Magnetic suspension at spec迅c positions: In this analysis, the position refbrences are given as fblows:the mover position輪寧 =[000.15]τand Euler angl・α窃=Od・9.・Theref・・e, the・1arge・q−axis currenS伝and Iny・to generate the translational forces Fx and Fy are unnecessary. In this condition, the magnetic levitation of the mover is easy to be stabilized because there are small torques 213 T」.’and Tx’, which are not restoring torques. (II) Planar motion control with magnetic suspension: In this analysis, in order to verify the compatibility ofboth the 3−DOF planar motion control and magnetic suspension, the position references are given as follows: ン x’=2cos(nt)mm ン」・’=2sin(nt)皿m ン∼=0.15mm オ > Euler angleα =Odeg. In this analysis, the q−axis currents 4∫x andん,. used to generate the translational forces丹and 4, in且uence the magnetic suspension characteristics, and this influence was investigated. Pllvsical mod〔ll (T輌me step:dt) Cしtl’1’c」Ilt colltl、olle1・ (S・mpling tim・・r。) r.、.φ Calcuiation of reference fbl℃es Data table of Calculation of K(r..φ) K(r..φ) F F τ.’咋 Calculation of cun℃ntS CaIculatiOn driving forces F F T.t, T Motion equation ”.φ L . _ ・ 一 → 一 一 一 ・ 一 Figure 6.4.1・1: . 一 ・ 一 . 一 一 一 一 一 . 一 . 一 ・ − H”一 . − Flow chart of 6・DOF motion analvsis. w 214 6.4.2.Numerica 1 Arialysis ResUlts Numerical analysis of the mover motions under the previously mentioned conditions (1)and(II)in Subsection 6.4.1 were perfbrmed. These analysi8 resUltS are shown as follows under each of the above conditions: (1) Magnetic suspension at specfic positions: Figure 6.42・1 shows the analysis result of the mover motions under analysis condition(1). Figure 6.4.2・1 indicates that the mover call be positioned at these reference positions in the x−,γ一, z−, andα」directions with less suppressedβ一and 7・−di8placements. Theref()re, the mover can be magnetically suspended with stability. Figure 6.4.2・2 shows the analysis resUlt of the armature cur rents皿der analysis condition(1). The d−axis currents Idu and Idy used to generate the suspension fbrces are absolutely less than O.36 A and O.45 A, respectively. The q−axis currents Iqx and Iay used to generate the translational fbrces Fx and Fy are absolutely less than 3 mA, therefbre, high・resolution current controls are necessary to control the mover motions. The armature currents fbr theα一directional cirive are absolutely less than O.04 A. (II) Planar motion control with magnetic suspension: Figure 6.4.2・3 shows the analysis result of the mover motions under analysis condition(II). Figure 6.4.2・3 indicates that the mover can track the reference positions in the x−and y−clirections, and be positioned in the z−and α一clirections with suppression of theβ一and r一displacements. Therefore, mover motions can be cのtrolled with stable magnetic leVitation. Figure 6.4.2・4 shows the analysis resUlt of the armature currents under analysis condition(II). The q一axis currents伝and 4γare absolutely less than 7mA, but slightly larger than those in analysis(1)・The q−axis currents伝 and Ig・used to control the translational forces Fx and Fy also generate simultaneously the torques写and万’, respectively. Therefore, displacement of the EUIer angles 6 and 1 under analysis condition(II)is larger than that in analysis condition(1)due to the greater q−axis currents Igx and Iq, for the planar motions. 215 Therefore, 1 proposed a planar actuator with a magnetically levitated mover capable of large planar motions over the stator, and demonstrated both 3・DOF planar motion and magnetic levitation controls by applying three pairs(minimum number)of two・ phase armature currents control by numerical analysis of the 6・DOF rnotion. 0」8 0.16 P 巨0・14 二 〇.12 1‘ 二,0」 ㌶ 0.08 ≡ ≧0.06 む 呂0.04 ■ x=v=O = ぎ0,02 1’ 0 .1’ −0.02 00.511.522,533.5 4 Timじr〔s) (a)1[E’anSlational motions x,」:,ご, 0、5 0.4 ハ 夢0.3 三 コ α =0 k 〇2 ミミ0.1 ..c’.\’. . .. .:.’.一.r、二r㌃㍉㌘〉ヤー さ O 、,} ’. o E“ , 、、 一” 1 t . 瓦一〇,1 ) . 巨 」 −0.2 ヱ ,宗一〇,3 ↓t. − .0.4 −0.5 0 0.5 1 L5 2 2.5 3 G.5 4 Time t(S) (b)Rotational motionsα,β7. Fig.6.4.2・1: AnalyticaUy・obtained mover motions under analysis condition(1). 216 0 1・0.355 /−O.36− −0.05 i,1 ・0,1 −O.365 ’ <.O.15 ) 一〇,44 ∫、4」. Es−0.2 −0.445 ミー0.25 −0,45 ξ一〇.3 2 ヒ 2.5 3 3.5 4 3.5 4 δ一〇・35 /.t” −O.4 −045 1,t」. −O.5 0 0.5 1 2 2、5 3 Time t(s) (a)dLaxis currents Idr andレused to generate sus pension forces F.. 0.04 ∫、1、冶 0.03 ハ < ) 0.02 よ −0.Ol g 、tr ミ 1二、ご 1、∼ .1.・.三㌧...,..∴ 1 ㌔.㌧ \ 、 tm . 」. .、 ・ . . . @ O ;.−0.01 ト 1 ..1轄 \/\、 ∵∼’・∵一⇔一’ コ べ ぬ 1 .、. 唐堰@4 L , ’ 、、 ’ ミ §・0,02 ヒ = U−0.03 −0.04 . . 0 0.5 | 1.5 2 2.5 3 3.5 4 Time(s) (b)d− and q−axis currents伝,んlda, and lva used to control planar motions. 2 y L5 § .戸 1 K \ 0.5 ⊆. 0 言 8 日 \ 一〇.5 旦 昏 一1 ’δ ・1.5 ψ 一2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t(s) (a)Planar motions x, y. 0.18 0,16 0.14 言 巨0.12 シ .1 0.1 芭 巨0・08 皇0・06 言・… 0.02 0 −0.02 O O.S 1 1.5 2 2.5 3 3.5 4 Time t(s) (b)Vertical motionご. 0.S O,4 命 弓 0.3 ) 0.2 く 0.1 べ ti 0 ・一. u. ラ・一:‘一く5’一ン㌻『一ン、「’.・’一一一一 r㌔←一二∨r.一’・T・‘∼:一一t・−s:・_・・..L一 v る一〇.1 臣 −02 旨 言 −O.3 二 ・0.4 ・O.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t〔s) (c)R・tati・nal m・ti・nsα,βγ Fig.6.4.2−3: Analytically−obtained mover motions under analysis condition(II). 218 0 −O.05 −0.l n を0.15 さ’−O.2 ざ一〇.25 §−o.3 ヒ δ一〇・35 −0.4 −0.45 −0.5 0 1 0.5 1.5 2 2,5 35 3 4 Timc川s) (a)d−aXis currents J、lt and ldv. used to generate the suspension forces F.. 0.06 3 ベ 0,05 ハ fvn ‘1.’輌 ξ0・04 : 、o.03 『. F’『・’1 ,1’ ミ ミ0・02 .、 ,” .、 1・. ’・[一、 ,、 ,’ ,1 ノ. 1’・ … , . 1 −「.” L「er ・・1.’、 ノ l ll ミo・Ol タ 〉 き 0 5 」, 、 ’ ..:. q∼「 . ・ .,烏 .1, 、 べ s 三.べ 、. /、ノ、 》 .\.、 ’ u−0.01 ∼.... 一〇.02 o 0.5 】 】.5 2 2.5 3 3.5 4 Time ’ 〔s) (b)φand g−axis currents伝, i,v、,んα, andム,αused to control planar motions. Fig.6.4.2・41 .Analytically・obtained armature currents under analysis condition(II)、 219 6.5.Summary of Chapter 6 This chapter pre8ents a fbasibi五ty veri五cation of a planar actuator with both 3・DOF planar motions and magnetic suspension of the mover in order to further improve perfbrmance. Then, based on a皿merical analysis of the 6・DOF ciriving forces, a planar actuator haVing a mover positioned above a plane and magnetically leVitated by only siX currents and the six・current・control alg磁thm were conceptuaUy designed. Furthermore, I vahdated the designed planar actuator by numerical analysis of the 6−DOF motions. The results obtained in this thesis illdicate the possibility of the reahzatioll of a high・perfbrmance MD OF planar actuator: 〉 decoupled 3・DOF motion contro1 and magnetic leVitation on a plane. 〉 wide movable area by a smaU number(six)of armature conductors. 〉 extendible movable area regardless of the number of armature conductors. 〉 smaU m皿imeter・sized mover. 〉 no problematic wiring to adversely affect drive performance. As the next step, it is necessary to design an experimental system fbr the verification of the 6−DOF motion characteristics and conduct experimental tests. 220 Chapter 7 Conclusions This chapter concludes this thesis and suggests fUture work. 221 7.Conclusions This chapter pre8ents the accomp五8hments and technical contributions of this thesis as conclusions, and a180 makes suggestions for future work. 7.1.Conclu8ions In this study, I designed planar actuators that have a small mover capable of travelmg over a wide movable area on a plane, and which is driven by a 8mall number of armature conductors. These planar actuators form spatiaUy superimposed magnetic ch℃Uits for the M D OF motion controls. Magnetic circUits are the most irmovative of all planar actuators and enable the extensions of the movable area regardless of the number of armature conductors.且owever, there is a disadvantage to magnetic cir℃uits that needs to be solved, which is that realizing decoupled controls among the driVing forces in each degree of fヒeedom is difEicUlt. The most important assertion, and technical contribution of this thesis is the design of the planar actuators so as to achieve independently control more・degree・of’freedom mover motions by using spatia皿y superimposed magnetic ci1℃uits. Chapter l presented an血troduction to and apphcations fbr MDOF drive systems. Multiple moving・part actuators, consisting of multiple 1・DOF actuators, have been most utilized in MD OF面ve systems.且owever, there are severa 1 disadvantages with multiple moving・part actuators that make it diflicUlt to improve the accuracy and response of the mover drive. In order to solve these disadvantages, single moVing・part actuators, capable of clirect drive with MDOF, have been studied. Chapter l then introduced important element technologies, including magnetic materials and cir℃uits, position sensing, and suspension and guide mechanisms. With this in mind, the purpose and technical cont亘butions of this study were deta iled. Finally, the structure of this thesis was outhned. Chapter 2 presented classi丘cation of M])OF drive systems and remarks about their features and technical issues. MDOF drive systems can be classi丘ed by the number of moVing parts, form of driVing forces, and drive p血ciple. Synchronous planar actuators, with which this study deals, have especially good controllabMty of the driVing forces in planar actuators. With these technical details in mind, I then summarized the speci丘cations of synchronous planar actuators that had been developed. In synchronous planar actuators, planar actuators with a permanent−magnet mover realize 222 sophisticated motion controls, but have in8ufEiciently wide movable area皿1ess the planar actuator8 have a large number of armature conductors. The planar actuator that Iproposed in・this study is aimed at achieving comp atibility of both sophisticated motion controls and a wide movable area u8ing j ust a sma皿number of armature conductors. In Chapter 2, I clari丘ed the orientation of my proposed planar actuator in relation to preVious planar actuators. Chapter 3 presented the fundamental conceptual design of my proposed planar actuator, which aims to resolve the technical i8sues of preVious planar actuators. The drive principle of the planar actuator i8 based on two・orthogonal五near・synchronous motors. The planar actuator fbrm sp atiany sup erimposed magnetic circUits corresponding t《)the magnetic ch℃uits of the two・orthogonal linear’synchronous motOrs. There are two polyphase armature conductors, a nd exciting these armature conductOrs generates two・directional multipole magnetic field over the stators. Therefbre, increasing the length of al1 the armature conductors easily expands the movable area. Based on the numerical analysis results of the driv士ng fbrces, I designed a decoupled control algoritim for 2・DOF translational and 1・DOF rotational motions. Chapter 4 presented a design for a n exp e血lenta 1 system for an investigation into the d rive characteristics of the p1anar actuator. 1 implemented a control a lgoritim into aDSP connected to AD/DA converter boards, and designed a 3・DOF positiol1・sensing system using three laser・displacement sensors, as wen as a suspension mechanism fbr the mover using ball bearings. Then, specifications of these experimenta1 apparatuses were presented. Chapter 5 presented an eXperimenta 1 verification of the 3−DOF motion controls of the mover on a plane, and the resUlbS of the experiment. From these eXperimental results, I successfUlly demonstrated that 3−DOF motions could be independently controlled by two pairs of three・phase currents. The movable area in the translational motions can be in丘nitely extended, and the rotationa1 motions is in the range withi n the yaw angle=±26 deg. Furthermore, the driVing forces are periodic With a 90−deg period in the yaw direction, and the mover can travel in multiple 90・deg steps in the yaw direction. Therefbre, the planar actuator has a wider movable area than preVious planar actuators, a lthough it only has two polyphase armature conductors. Chapter 6 presented a feasibility verification of the magnetic suspension of a mover capable of 3・DOF planar motions in order tO e五minate friction forces between the mover and ba皿bearings, aimed at incremental improvement of the drive performance. Based on a nume亘cal analysis of the 6・DOF driving forces, I designed a planar actuator that has spatiaUy superimposed magnetic chrcUits fbrmed by only six currents and a 223 permanent°magnet mover・so that the mover motion8 coUld be independently controlled mthe 3’DOF translation8 and 1・DOF rotations above a plane. The(hive characteristics were validated by a numerical analysis of the 6・DOF motio】ms. This thesis demonstrated the following signi丘cant accomplishments of a novel study: 〉 experimental verification of the design and control of a long・stroke 3・DOF planar actuator. 〉 numerical verification of design and control of a planar actuator with a stably and magnetica皿y leVitated mover capable of 3・DOF planar motions. 7.2.Future Work This section discusses future works aimed at incremental improvements in the perfbrmance of the planar actuator as fbnows: 〉 Improvements tO the drive system: ◇ realization of decoupled 6・DOF motion controls by redesigning the mover or stator structuヱe. ◇ improvements to the spec迅cations of the controUer boards(input/output range resolution, samp㎞g time, and so on), that would improve drive characteristics such as positioning precision and response. ◇ investigation of a movable area out of plane. ◇・ consideration of p ayloads nlo皿ted on the mover. 〉 Improvements to the position・sen、sing system; 十 rea五zation of 6・DOF position・sensing system, prefbrably integrated with the mover or staiX)r. 十c輌ation of sensor si…画s agぬst the expe血ental enVironment such as temperature and thermal expansion. 224 In conclu8ion・this thesi8 presents high・performance M D OF planar actuators with a permanent°magnet mover capable of traveling over a wide movable area on a plane, with just a small皿mber of 8tationary armature conductors. The combination of the mover and stator can generate spatiaUy superimposed magnetic fields fbr the MDOF drive, and therefore increasing the length of the armature conductors can easily expand the movable area re gardless of the皿mber of armature conductors. A plana r actuator was conceptua皿y designed and fab亘cated. The fab亘cated planar aCtuator can independently control the 3・DOF motions of the mover. Furthermore, in order to eliminate deterioration of the drive characteristics due to friction fbrces, the planar actuator was redesigned so that the mover could be stably levitated and the 3・DOF motions on a plane could be controlled. Then, the mover motion characte亘stics were successfUlly veri丘ed by means of a numerical analysis. Next, a sma U fabrication size was realized by integrating the perma nent・magnet array and armature conductors for the MD OF drive. The planar actuator has the first mi皿imeter・sized mover and woUld provide a significant starting poin.t when used with sina皿 electromechanical components in an M DOF drive. 225 Appendices A. Fabrication of the Smalle8t且albach Pemlanent・Magnet Mover B. structure 6f Manufa6tured P血ted circuit B。ard C. ⑪・FP。8i』S舳g Ut正zmg La8er.Di8Plaeement Sen80r8 226 A・Fabrication of the Smale8t且albach Permanent・Magnet Mover In this stud訂鋤亘・ated・th・・maUe・t 2・D且alba・h p・・man・nt・magnet・array,・whi。h measure just l l mm×11 mm×2mm. The permanent−magnet array consists of one group of 16 permanent magnets and one of 24 permanent magnets, which measure 2 mm×2mm×2mm・and 2 mm×2mm×1mm, respectively Mr. K()ji Myata and Mr. Y両iDoi・Shin’Et8u Chemical Co・, Ltd. kindly provided these permanent magnets for this 8tudy. In the且albach permanent−magnet array, adj acent permaneIlt magnets are mutuaUy・ubjected・t・repUl・i・n・f・rce・. The・ef・re,1・fabri・at・d th・p・㎜an・nt・magn。t array l)y l〕onding the pemmanent magnets using these excellent adhesives;Araldite standard (Epoxy ad hesive)and LOCTITE 326 LVUV (Ultraviolet cure adhesive) combined with LOCTITE 7649(Primer). First,1 fabricated the permanent・magnet array on a 2・mm iron plate, mounting a square・rUler’shaped 1.2−mm h℃n plate in order to丘x the permanent magnets using the iron plate during bonding between the permanent magnets. For the bond between the permanent magnets,1 used LOCTITE, which l)onds quickly(less than one minute), and has a relatively high shear strength(18.5 N/mm2), bonding only the lateral sides of the permanent magnets. So in other words, I fabricated a且albach permanent・magnet array using only LO CTITE.且owever, the adhesive strength was not high enough, and the bonded permanent・magnet array often became unglued when the electromagnetic forces for the MD OF drive acted upon the permanent・magnet array. Next, in order tO strengthen the adhesion, I coated the且a lbach permanent・magnet array, bonded with LO CTITE, With Ara ldite, which bonds slowly(more than 12 hours) but has greater shear strength. Araldite is Viscous, and keeping a flat coating using Airaldite is (lifiicUlt. So, after the Araldite hardened completely, 1 removed the unwanted Ara ldite using sandpaper tO flatten the surface of the permanent・magnet anray. Figure A・1 shows the fabrication procedure fbr the smallest 2・D且albach permanent’magnet array. 227 ’一一 @一 ≡ 一一 一 一 一 一 , ← 一 一 一 一 一 一 一 一 一 一 一 一 一一 一“∼ 一 一 ’“ 一一 一 ^香一 一 一 ←・、 一 一’ 一 一 一 一 一 一 一 一 一r −一 ,、 t 、 1 I 1,2イnm≡ti】畝 I I iron副ate I I I I I l I I I I I I I 1 Bonding surface I l 2.O−mm−thick J I iron Plate l 1 I 与 l I I I t t I t I I I t2・mm・thiCl《 1 1 i刷幽佃 l I l I I l I l I d I t I I I 1 I I I 1 I 1 合 I 1 1 I 1 1 1 1 1 1 1 1 1 t Bonding the pemanent−magnet components with ultraviolet cure adhesive l L ’ 、 一 一 一 “ 一 一 一 一 一 一 一 r − − r − 一 一 ≡ ≡ 一 , ← 一 _ 一 一 r − 一 ^ 一 一 “ 一 〔 _ _ _ 一 _ 一 ← , 一 一 一 , 梧 一 一 一 一 一 _ _ _ _ 一 〔 ノ 母 ”一 一 一 一 一 − 一 ^ ≡ 一 “ 〔 一 一 一 一 一 一 一 一 一 一 一 一 一 ’ , 一 “ 一 一 一 一 一 一 一 一 一 一 一 一 一 一 一 一 ← ← 一 一 一 一 一 一 一 一 一 一 一 一 s ’ 、 I 戸 I I I 1 I l lEpoxy I l Coating the permanent−magnet erray with epoxy adhesive 、 ’ 一 一 A − 一 ’ A − w − 一 一 ≡ ← r ’ r − A − 一 一 一 一 一 P − r − 一 一 一 一 A − r ≡ 〔 ≡ 一 一 一 一 一 一 一 一 一 一 一 合 梧 一 一 一 一 一 _ ≡ 一 ’ 与 、 、 ’ ’ , , $ 1 1 1 I l l : l I I I I d I ‘ l 1 I ‘ I 1 1 1 1 1 , 1 Remov泊g the hardened extra epoxy adhesive with sandpaper 1 1 1 ’ 、 、 一 _ _ 一 一 _ 一 _ 一 ← “ 一 一 一 一 一 一 一 一 一 一 ’ ∼ A P − − P − , − P A − ≒ 一 一 一 一 一 一 一 A − 一 一 一 一 一 一 一 一 一 ∼ − 一 _ 一 一 ’ Fig. A・1: Fabrication procedure fbr the smaUest 2・D Halbach permanent’ magnet array. 228 B.Structure of ManUfacture d Printed CircUit Board A・m・nti・n・d in Chapter 3, in the exp・rim・nt・・n 3・DOF m・ti・n・・nt,。1。n a plan,, ad・ubl・’layered頭nt・d・ir・uit b・ard wa・ut血zed in・rder t・9・n・・ate a multip。1。 magneti・丘eld that ha・arbit・卿amphtude and pha・e in th・x−andγ一血e,ti。n、. Th。 P血t・d・i「・uit b・a・d・・n・i・t・・f tw・35・μm・thi・k・・ndu・t・r layers and a 100・μm・thi,k in・Ulating・layer・andwi・h・d b・tween th・tw・c・ndu・t・・layers. In・a・h・・ndu,t。r・layer, 0・8’mm’wide st可・・f・・PPer釦m鯉e菰gn・d at 1.76・mm(・・rre・p・nding t。。n。・t}血d。f the pitch length of the 3・DOF planar actuatOr)mtervals. Three・phase conductors fbr the x−and・y−directi・nal血ve8訂・th・n伽m・d by inserting the extemal・ir・Uit・・sh。wn by da・h・d Hnes in Fig・B’1・Th・丘卿sh・w・h・w・x・iting tw・pairs・f three・phase conductors generates a multipole magnetic field above the centered 90 mm×90 mm area of the printed circUit board. The intervals between the strips of copper film near the end of each strip are longer than those near the center in order to secure areas wide ・n・ugh t…lder, and・2・5’mm−diam・ter land・a・e aligned at 3.5−mm interval・. Figure B・2shows the manufactured double・1ayered prmted ch℃uit board. In Chapter 6, a triple・1ayered printed circUit board was designed in order to generate a multipole magnetic丘eld that has arbitrary amplitude and phase in the x−,アー, and xct−directions shown in Fig.6.3.1・1.Across・section View of the triple・layered p血ted c血cuit board i8 shown m Fig. B−3. The total thickness of the p血ted c迂cuit board is O.425mm. The丘rst, second, and third conductor layers have two・phase armature conductors for the x−, y−, andα一clirection,a1 drives as shown丘1 Figs. B・4, B・5, and B・6, respectively. The丘rst and thrd conduct《)r layers consist of 18・μm・thick eopper film and 12・μm・thick through・hole plating, and the second conductor layer consists of 35’pm’thick coPPer丘hn・The width of aU the conductors is O.8 mm. In the p血ted chrcUit board・there are a lot of O・3・mm・diameter through holes, including 12・pm・thick through’hole plating in order to fbrm mutua皿y insulated three pairs of two・phase printed circUits. There are 15・μm・thick solder・resist layers and 5・mm・diameter lands with 1・4’mm’diameter through holes on the top and bottom surfaces. Figure B・7 shows the manUfactured triple・1ayered printed circUit board. 229 i’↑’警・1ぷ’i警・1・旨・il:’書・}・・÷.;:・1”i亨・…・号.il:?÷・il’÷.ll:’警・、峯.ピ’i警・、、ち.1::”iぶ・、 1・’ …” A:、.・… ld:‘・+ ’’’” h u.「’ 呼’”}… 呼’r” 寧・ ;−ぐ”l s::.t…’1 .v ’”t’ 、’ 煤DF’’” lI.1− ’’’” P) ∫學一 v・)一’:㌦一 v、,t+・・、:,,、. Fig. B−1: Structure of the double・1ayered printed c辻cuit board. The solid五nes represent the copper丘lm and the dashed lines represent external circuits. 230 1 1 き Fig. B・2: Photograph of the manufactured double・layered printed chrcuit board. 231 Total thickness: 0.425mm 15−pm solder resist イ8−L↓mCu刊m ロ +12−FtM PTH First conductor layer 35−um Cu川m . 100一μmprepreg Second conductor layer 18一μmCu film Third conductor|ayer +12一μmPTH PTH:Through−hole plating Through hole Fig. B・3: Cross・section view of triple−layered printed c血cuit board. 232 ∈ E oco 80mm Fig. B・4: Structure ofthe first conductor layer. Red and pink lines represent the two・phase armature conductors fbr the.r−directional drive;dark and light green lines represent the two−phase armature conductors{br the〕一directional drivel and dark and Ught blue lines represent the two・phase ar皿ature conductors fol−the a−directional drive. 233 ∈ E 8 80mm Fig. B・5: Structure of the second conductor layer. Red and pink lines represent the two−phase armature conductors fol’the.v−directional drivel dark and light green lines represent the two・phase ai’mature conductors for the.1一dii’ectional drive;and dark and light blue lines represent the two・phase armatLuごe conductors for theα一clii’ectional drive, 234 ● ● ↑・、α ↑・iα ● ● ● E E ↑・。 O oo ● ● ↑・1.,. 80mm Fig. B−6: Structure of the third conductor layer. Dark and light blue li皿es represent the two・phase armature conductors for theα一directiena1 drive. 235 (a)Top View. (b)Bottom view, Fig. B・7: Photographs of the manufactured triple−layered pri nted circuit board. 236 C.6・DOF Po8ition Sensing Utilizing Laser・Di8placement Sensor8 In order to suspend the mover without mechanical contact, it is extremely important to detect the 6’DOF positions of the mover. In this study, a position・sensing method utiizing six laser・displacement sensors was investigated to precisely detect the position of the extremely 8mall mover, the dimension of which are approximately l l mm×11 mm×2mm. The 8ix la8er・displacement sensors are arranged as shown in Fig. C・1.As mentione d in Chapter 4, we mea8ure the distance from the sensor head to the surface of an object using the sensors and the principle of laser triangulation. The sensors output a voltage proportiona1 to the magnitude of the displacementS from reference distance D. Sensors 1, 2,and 3 irradiate d ifferent lateral sides of the mover, and Sensors 4,5, and 6 irradiate the same top surface of the mover. In this study, a sensor coor(li nate xδiizi is de丘ne d to be tited by−45 deg around the z,−axis to the stationary coordinate xSyszs. Six laser・displacement sensors are aligned so that path of the laser beam丘om Sensor i with respect to the sensor coordmate xdil∫, rl,(i=1,2,3,4,5,0r 6)can be represented as fbUows; m−D−9・・]ア輌・】・ rn= _.........................◆.............._.....................(C・1) mx23_D_LO2 2]ア却 1 0]T.............◆.............◆....◆...........◆..◆................(C・2) rl・= m一x23 D+∠0 2 2]τ+剛 一1 0r.◆................◆..◆.........._..........................(C・3) rl・一 db・c・・仇司ア+N・[・in e4・一…e4】・………・・(C−4) rl・= m一輌一Xl・−3・ ͡儂・ ]「+N・ [・ ・in e・ −c・・es]・……・…・…・(C−5) rl・ o¥・5−−db・C・・弓 m x;・{+巻・db・ ・in e・ +;]T+N・[・一・in・e6−…e・ ]r……・……・・(σ6) rl・= where w,1, and h are width,1ength, an・d height of the mover, respectivel跳勘andアグ(i,ノ:1, 2,3,4,5,0r 6)are relative positions between Sensors 1,2,3,4,5,0r 6, andハli is a positive number. Laser beams from Sensors 4,5, and 6 are tilted byθ4, es, and e6 to the z axis, respectively. When the mover position with respect to the stationary coordmate x躍、r、m=[0 0 0]アand the Euler angleφ=[0 0 0]「, distances between the sensor head and measurement point in Sensors 4,5, and 6 are∂db4, db5, and∂db6, respectively. 237 The orientation of the mover can be calculated relatively easily by using a new Euler angleφ=[al β ri]Tthat is de丘ned by counterclockwise丘rst al−rotation around the zr axi8, secondβ一rotation around the yraXis, and third 7t−rotation around the xraxis. The Euler angle〃can be repre8ented by the amount of a di8placement in the measurement points of Sensor 5, and 6(△S5, and△S6, respectively, as shown in Fig. C・2) as fbllows: k一△S5 COS e5+△S6 COS e6△Ss S㎞θ5+△S6 sin e6+Y56〕・…・………・……・……………・・…・・…・……………・(σ7) γ・…−1 Next, the Euler angleβcan be represented byβ’, which is a tilt angle of the mover to the x1一γ1 plane about theγr−axis, as fbUows: 旋mβr=tanβ㍑・cosγ∫.........................◆......◆◆........_.◆◆.............................................(C−8) Then, the tilt angle〃can be represented by the Euler angle乃and displacement of the measurement points威,△S5,△∫6 as fbUows: 〃=伍n−1 kas・C°se4一器…Vk,γ・}〕…………・……・…………・・一……(◇9) Therefbre, the EUIer angle lt can be obtained by Eqs.(C・8)and(C・9)as shown in the fblowmg equation: 輌一1 kaS・c°se4一器㌶γ’◆aS・s’n es・c・・乃〕……・………・………・…(◇・・) Next, m order to obtain the Euler angleα∼, output signals of Sensors 2 and 3 are necessary. The yr−d血ectional positions of the points measured by Sensors 2 and 3,}乞and }∼as shown in Fig. C・3, can be calcUlated from the output signals. The y1−directional distance between the two measurement points (Y3−}’2) depends on only the EUIer angle φ=【alβ 乃]「and can be represented by the Euler angle as fbUows: ろ一ち=yL1−.YL2_____..._.__.___.__.___._._____._.._..___(C・11) アLl=ocosγ,+asin ll tan rl’_.__....______._.._____.____.__..._(C・12) yL2=x23 tanα∫’.__._◆______...___.__._.______..._____.._._(C・13) whereαジand戊〆eXPress tilt angles about the :・raXis in xi−yt plane and about the xraxis in cross・section B−B’, respectively, and can be represented as fbUows: kSin a∫ Sin fil COSγ1+COSα, Sin 71Sin al Sin /71 sin h 一 cosα1 coSγ1〕・・…………・◆…………・……・…・…・…………(C・4) γ’1=tm−1 k、。、α、c。、慧蓋きnβ、inγ、〕…………・……・………一・・…・……………・・(σ・5) α・・=㎞一1 The Euler angleα∼can be calculated from Eqs.(C・11)一(C−15)and represented as fbUOWS; 238 (X23 C。、_、inβ.、inγ、_、。、γ、)、〕.…一_.(C−、6) α’=・in・−1 kx23C。,/7,一(ろ一ろ)、m働.、in、r∫ 幅)C°Sγ’〕 一一1 Next, in order iX)obtain the mover positions, a normal vector of each surface〃励。∫and aposition vector of each sur£ace center’功。∫(i=1,2,3,4,5,0r 6)with respect to the sensor coordinate xδrl, are introduced as shown in Fig. C・4. In Fig. C・4,0and O’express origins at the sensor and mover coordinates, respectively, and Oi’ expresses center of surface i(∫=1,2,3,4,5,0r 6). When the mover is not displaced丘om the base position, the normal vector nms,,o and position vector rms,,o can be represented as fblows: nm。1,。=[io o]T, nms、,。=[−i O o】τ, nm。3,。=[01 o]T, nms、,。=Eo−1 o]T, nm。5,。=[Oo Ir, rm。1,。=[w/2 rm。3,。=[o oo]T, 〃2 rm。5,。=[O o nm,6,。=[Oo o]T, h/2]T, __.___._____._.____._......(C・17) −1】τ rm。2,。=[−w/200]T, rms4,0=[O −〃2 0]T, ._____.._...............__._.....(C・18) rm。6,。=[O o−h/2]「 in。i can be ca lcUlated by the normal vector The normal vectoM胡、, and position vector’ nms輌,o and position vectOr r.∫i,o(∫=1,2, 3,4,5,0r 6)as fbnows: 〃“∬,=R〃”nmsゴ,o……・……………・…・・……・・…………・……………….・・_.___.__.___(C・19) rms,=Rlm rmsi,o………・………・…・……・・………・………・…………・…….・・_____._..__(C−20) where R,. eXpresses the orientation of the mover with respect to the laser coordinate xδtzi and ca n be represented by the Euler angleφas fb皿ows: COSα1 COS Pl 一sinα, COS 6i sin防 Rlm= Sinα, COS乃+COSα, S㎞防Smγ, COSα, COS 71− sin ai sin /ili sin 7∫ 一 COS /ili sin h Sinα, Sin rl−COSα1 Sin Pl COS r, COSα∫Sin ll+SinαノSin 」6i COS 71 COS /ill cos lt ........_....._._..._........._(C・21) Aposition vector of an arbitrary point on a surface i with respect to the sensor coord nate xOlzl, rlsi(i=1,2,3,4,5,0r 6)satisfy the following equation: nmsゴT・(rl.ゴーrlm)=0.__.____◆___._.__.............◆◆..◆....................___._.(C−22) where rlm expresses the mover position with respect to the laser coordinate xOi・i. The mover position rt. can be calcUlated firom the EUIer angleφ by E qs.(C・17)一(C・22)with respect to the three Surfaces 1,3(or 2),6(or 4,0r 5). 239 .㌦必 一Vm’A’ D、 i}二 iB’ A 〔 L’L”四゜’四曽「’「 d’「’層” 一・… A’ P・・一・………×…………・… c「iンα Cross’section at A−A’ iB /、 SCils‘,r.、 〔 乃’ B B’ A’4s ご二 Cross’section at B−B’ (a)Case in which the mover is not displaced加m the base position, yz )・sご「s !7, x・・ e・ ←1 iB’ }: i 〔 …⊇ ↓つ B 鋤 BI /i x: 一一一…一言。s,.se,、i。na、B.BI (b)Case in which the mover is displaced丘om the base position. Fig. C 1: Position relation among the six laser bean〕s and mover. 240 〔 le‘lsor 5 SeTisnr 6 Disρlaced Disρlaced 6, A l Y56→ A「 B B’ 一’一’一’一’−Ntt〔〕i§〒)1’a5ed Not displaced (a)Measurement point of Sensor 4. (b)Measurement points of Sensors 5 and 6. Fig. C−2: Definition of displacements in the measurement points of Sensors 4,5, and 6丘om the base positions(△S4,△ぷ5, and昼). ._ bQ− Xt t.1 二1 iB }二 (a)Cross・section view in the x,っヤplane. (b)Cross−section view at B−B’. Fig. C3: Displacements in the measul’ement points of Sensors 4,5, and 6丘om the base positions(△S4,△S5, and△S6). 241 Mover ’o. 0.’ ’一・一一’一’一一一 it〃nyr, lt,ns.1 n“n.vt ’i,n,C, .1’ x/ (a)Case in which the皿over is not displaced f壬om the base position. Mover ll、。,r,.o il.,.、4.1) ’t秩g.1.。P .1’ A’ P (b)Case in which the皿over is displaced丘o皿the base position、 Fig. C・4: De丘nition of the normal vector of each surface tt,,1。t and the position vector ofeach surface center r、n、、 with respect to the sensor coordinate xO,ノニノ. 242 Next, the mover position r加and Euler angleφwith respect to the laser coordinate xδ1β’are transformed with re8pect to the stationary coordinate xSivsl。, because the control system of the mover po8ition r。n, and Euler angleφwith respect to the stationary coordmate x泥ぷwere designed in Chapter 6. The la8er coordinate xO/P, are tilted by−45 deg around the zs−axis丘om the stationary coordinate xSyszs. Therefbre, the mover po8ition rsm with respect to the stationary coordinate xSysz、 can be represented by that rlm the laser coordinate xt]J tzt as fbllows: rsm=Rsi rim..___...._.._____......_._._.._.__..___.__..___.___..(C・23) where R。∫expresse8 a rotation matrix that generates a−45 deg counterclockwise rotation around the z.−axis and can be represented as fb皿ows: …(45°)・in(45・) Rs,= −sin(45°) cos(45°) _.................................◆.....◆..............................◆.(C・24) 0 0 The orientation Rノ胡, def吐1ed by the Euler angleφ=[α∼ β li]T based on rotations aro皿d the z , yrr, and xt−axes as shown in Eq.(C・21), can also be represented by the Euler angleφ=[α β ク]τbased on rotations aroulld the 2。一,必「, and xs−axes. Vectors of the必一and x.−axes with respect to the laser coor(linate xO/itt,,んand友, are represented as fbUows: _1/万 一1 ・11y=Rsl 1/」 .......................◆◆....舎.........◆◆..◆◆.......◆.................◆..............(C−25) 0 1!E 一1 ・llx= Rsl =1/,万 ........◆........◆◆............................_......._._........_........_(C・26) 0 From Eqs.(6.2.2・1)一(6.2.2・3)and(C−25)一(C・26), the orientation Rlm can also be represented by utilizing the EUIer angle 一 as follows; Rlm=[Rlml Rlm2 Rlm3】..______...____._._._◆.._____._._____(C・27) where」Ri. i,Rtm2, and Rim3 can be represented as follows: ・inα・←・inβ・Sin・7+…β一…γ)+…α・(一・inβ・・inγ+…β+…γ) Rlml 1・m灘1欝㌶驚ご:ぽご㌶sγ) ._.___......___........_◆(C・28) 243 ・inα・6inβ・・i・γ一…β一…r)+…α・←・inβ・・inr+…P−…γ) 1 Rlm2 ・inα・←・inβ・・i・γ一…β+…γ)+…α・(・in B・・in・1+…β+…γ) 2 孤㎞α価β・…r+・in・1)+…α・(一・inβ・…γ+・inγ)) ....◆..._..◆...................◆.....◆.(C・29) sinβ+COS fi・sinγ 1 Rlm・= sinβ一COS fi・sinγ .......◆..........._............_...._..............◆.................(C・30) E…β・C・・γ From Eqs.(C・21)and(C・27)一(C・30), the Euler angleφ can be represented by the (lifllerent EUIer angle th as follows: 一1〔 〕……・………(c−3・) α=、in(・in a・+…α’蹄防・in・7’+…防)+(・in al 一…α・)…γ・ 2cosβ β=・in−1 ksin讐輌 ・〕………………・………………………………一・……・・……(c・32) k万s三網・…・…………一…・◆………・…・……・…・…・…………・……・……(C−33) γ=・in・−1 As mentioned above, the 6・DOF mover position can be detected by using the six laserdisplacement sensors. Figure C−5 shows the calculation procedure of the 6・DOF position丘om the output signals of the six laser・displacement sensors. Next, I fabricated the position−sensing system shown. in Fig. C・6, and then mvestigated the characteristics. The spec靴丘cations of the fabricated position・sensing system are shown as fbUows: 〉 Sensors 1,2, and 3:LK・080[KeyO 1] 〉 Sensors 4,5, an,d 6:LKGO80[KeyO2】 〉 tilted angles of laser beams fヒom Sensors 4,5, and 6 to zraxis:e4=25 deg, es= 15deg, and e6=15 deg 〉 distances between sensor head and measurement point in Sensors 4,5, and 6: db4=70 mm, db5=68 mm, and{必6=68 mm. The resUlts show there are important problems to be resolved;the detected positions include errors caused by d皿ension and placement errors of each piece of experimental apparatuses, property variations of the sensors and power ampli丘ers due to temperature variations, electrical noise, and so on. Furthermore, these errors can induce identi丘cation errors in the system・constallt matrix K in the motion・control algorithm, and deteriorate the motion’control characteristics. Therefore, calibrating the position sensing system is an extremely important issue. 244 Sensor 1 Position M“.」・∫,”,二ん,、 ジEuler angk渇 >Euler angle∠31 Euler angle ai >EUIer angle rl ジEuler aロgleβ >Sensor 1 Rotation matrix R,t アSensor 3(or 2) >Sensor 6(or 4,0r 5) 》Sensor 2 >Sen80r 3 Position艮w,」:、力r三,,, >Position Xtm > PositionYlm Euler an β >Eule >Sens >Sens 91e乃 >Posiぱon:,m Euler angleαβ >EUler angle at >E司er angleβ >Euler angle ri メ. S_rド・.:.・.. Q1..・・∵.一.......:、・・.−L’ Fig. C・5: Calculation procedure fbr the 6−DOF position丘om the output signals of the six laserdisplacement sensors. 245 (a)Tbp view. (b)Side view. (c)]Mover and stator. 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Viandenput,”Magnetically Levitated Planar Actuator with Movmg Magnets,,’The hコt已rna tional Eleetrie/lfaehines虚thゴves Confere2コcθ (IEmOC’07), Vol.1, pp.272・278, Antalya, Turkey, May 2007. [VianO7b] C.M.M. van Lierop,」.W. Jansen, E. Lomonova, A.A.且. Damen, P.P.J. van den Bosch, and A.」.A. Vandenput,’℃ommutation of a MagneticaUy Levitated Planar Actuator with Moving・Magnets,”The Internati’αロa1 SymipositM2 Q刀Linear刀n’陀●血血dロ8勾・APP必’α?tions a刀必2007?, OS6.2, Li皿e, France, September 2007. 【VianO7c】 A.Lebedev, D. Thakkar, D. Laro, E. Lomonova, and A.」.A. Viandenput, ”Contactless Linear Electromechanical Actuator: Experimental Verification of the lmproved Design,”7he ln te』ma施nal Symposium on・乙加θぼτ・Pti’ves for lndusti v Aρplica施刀s(Z、DZA.2007?, OS11.3, Ulle, France, September 2007. 259 [YanO4] T.Yano,”MUIti Dimensional Drive System,”The 14th lnternational Sympo殴tum on Po wer Electroni’es,1弘96励a1伽’ves, A utomation and Motion《微EZ2レ脇004λpp.457・462, Capri, Italy, June 2004. [YanO7] T.Yano, Y Kubota, T. Shikayama, and T, SuzUki,”Basic Characteristic80f a Multi・pole Spherical Synchronou8 Motor,” i2ternatibna l Symposj’tM20n Mj’αro’Nano Meehatrom’es and Human Seienee (/ZVail/S200 Z?, pp.383・388, Nagoya, Japan, November 2007. [YanO8a] T.Yano, Y Kubota, T. Shikayama, and T. Suzuki,”Development of a Spherical Synchronous Motor with ’1kAro Degrees of Freedom,”The 26Mh Symposj’tUZI on Eleetromagneties and Dynamies(sead20?,21B1−1, pp.133・138, Beppu, May 2008(in Japanese).. 矢野智昭,久保田義昭,鹿山透,鈴木健生,「2自由度球面同期モータの開 発」,第20回「電磁力関連のダイナミクス」シンポジウム,21B1・1, pp.133・138,別府,2008年5月. [YanO8b】 T.Yano,”Development of a且igh Torque Spherical Motor −Proposal of aHexahedron−Tetrahedron Based Spherical Stepping Motor−,” Journa l Of the Japan Soeiety Of Apρlied Eleetromagneties and MeehatU’cs, Vol.16, No2, PP.108・113, June 2008(in Japanese). 矢野智昭,「高トルク球面モータの開発一正六面体と正四面体に基づく球 面ステッヒ゜ングモータの提案一」,日本AEM学会誌, Vb1.16, No.2, pp.108・113,2008月6月. [Yan93】 T.Yano and M. Kaneko,”Basic Consideration of Actuators with MUIti Degrees of Freedom HaVing an ldentical Center of Rotation,” Journa l Of the Roboties Soeiety of Japan, Vb1.11, No.6, pp.107・114, June 1993 (in Japanese). 矢野智昭,金子真,「回転中心を同一とする多自由度アクチュエータの基 礎的検討」,日本ロボット学会誌,Vb1.11, No.6, pp.107・114,1993年6月. [XiaO7】 且.Li, C. Xia, P. Song, and T. Shi,”Magnetic Field Analysis of A Halbach Array PM Spherica1 Motor,” IEEE ln terna tiona 1 Conferenee 伽Automation and Logtstics, pp.2019・2023, Jinan, Chma, August 2007. 260 Publications Journal Paper8 [P1] YUeda and且. Ohsaki,“A Long・Stroke Planar Actuator with Multiple Degrees of Freedom by Minimum Number of Polyphase Currents,”Motion Control, I V・T脇CH Book, ISBN:978・953・7619・X・X, September 2009(to be submitted). [P2】 YUeda and H. Ohsaki,‘℃omp act Three・Degree・of’Freedom Planar Actuator with Only SiX Currents Cap able of DriVing over Large displacements in Yaw Direction,”IEEJ 7}ansaction on lndustry Apph’ea tionθ, Vb1.129, No.3, March 2009(in Japanese, to be published). 上里垂込,大崎博之,「2組の3相交流電流を適用してヨー方向に広い範囲で駆動 が可能な小形の3自由度平面アクチュエータ」,電気学会論文誌D,Vbl.129, No.3, 2009年3月(掲載予定). [P3] YUeda and且. Ohsaki,“Positioning of a Maglev Planar Actuator by ControUing Three Sets of Two−Phase Currents,”Jo urna 1 of thθJapan Soeiety Of Applied Eleetromagnetics and Meehantes, Vbl.17, No.1, March 2009(in Japanese, to be published). 上田靖人,大崎博之,「3組の2相電流制御による磁気支持平面アクチュエータの 位置決め」,日本AEM学会誌, Vbl.17, No.1,2009年3月(掲載予定). [P4] YUeda and且. Ohsa]d,“Six・Degree・of’Freedom Motion Analysis of a Pla nar Actuator with a Magnetically Levitated Mover by Six・Phase Current controls,”IEEE 7}ransaetion on Magneties, Vbl.44, No.11, Part 2, pp.4301− 4304,November 2008. 【P5] YUeda and且. Ohsaki,“A Planar Actuator with a Sma且Mover ’lllraveling over Large Yaw and T}ranslational Displacements,”IEEE 7}ransaetion on Magneties, Vb1.44, No.5, PP.609・616, May 2008. 261 International Conference Proceed ings [P6] YUeda and且. Ohsaki,“Armature Conductor Design of a Long・Stroke Planar Actuator with Multiple Degrees of Freedom,”The 7th lnterna施ηぼ1 SympoSi’ltM on Linear thゼves」for lndustria1 Appliea tions(fLDZA20092, Incheon, Korea, September 2009(to be 8ubmitted). [P7】 YUeda and且. Ohsaki,“Design and Control of a且igh・Perfbrmance Multi・Degree・of’Freedom Planar Actuator,”Symposi°醐Of Global COE a t 乙励’versl’ SアOf Tokyo oη5εo乙zrθう乙ife」EleetrotU’α%Tbkyo, Japan, January 2009. [P8] YUeda and且. Ohsald,“A PIanar Actuator with a MagneticaUy LeVitated Mover Cap able of Planar Motions by Only Six・Current Control,”The 9th Seoα1 M亙伽ぼ1乙Tllrzi’versity・∼面’verSl’tyαf Te幻りJoint Semihar oロ・酬θo垣皿 En8カleen’ng, Tbkyo, Japan, January 2009. ’ [P9] YUeda and且. Ohsald,“Six・Degree−o}Freedom Motion Analysis of a Planar Actuator with a Magneticany Levitated Mover by Six・Phase Cunrent Controls,”The 1加θma在伽a1互∼agne施s confereneθ(ln termag20082, GH・09, Madrid, Sp ain, May 2008. [P10] YUeda and且. Ohsaki,“Large「Yaw Motion Contro1 of a Planar Aetuator for Two・dimensional D亘ve,”The 6th ln tei1刀a亙伽a 1 Symposi’ILzzl on L加ear thゴves for lndustrialApp必已面oηθ(LDL4,200 Z?, OS9.1, Li皿e, France, September 2007. [P11] YUeda and且. Ohsaki,“Fundamental characteristics of a sma11 actuator with amagnetica皿y levitated mover,”The 4th Porver Conversi’on Ob唖τθ刀6θ (POC・Nagoya200”/?, pp.614・621, Nagoya, Japan, Ap姐2007. [P12】 YUeda and且. Ohsaki,“Two・(imensiona1 D亘ve with Yawing motion by a Sma皿Surface Motor,”Me 8th Seou〃Va ti伽a1乙麺゜versity’乙励’versity of 7bkyo e/bj’rn t Semhコar o刀Eleetriea1」動28カコeering, PP.79・82, Seou1, Korea, February 2007. [P13] YUeda and H. Ohsaki,“Application of Vector Con、trol to a Coreless Surface Motor based on a Permanent Magnet Type])inear Synchronous Motor,”The 2006 1n ternationa l Conference oη Eleetn;とral /lfaehiロes and Systems (ICEMS2006?, Nagasaki, Jap an, November 2006. [P14] 且.Ohsaki and Y Ueda,“Numerical Simulation of Mover Motion of a Surface Motor using Halbach Perman、ent Magnets,” The 18th lnternational Symposium伽P・wer・Eleetroni’〈rs,・Electrica1 Drives, Aut・mati伽and M・毎伽 (SPEEDAM72006?, pp.364・367, Taormina, Ita ly, May 2006. 262 [P15] YUeda and且. Ohsaki,“Two・dimensional Drive by a Coreless Surface MotOr using且albach Permanent Magnet Array,”The 7th乙励’versity of Tokyo’SeouZ 2V吾莇伽a1 乙麺’verSl’tアJoin t Se]ロ21iηaτon 1弛c伽頃」脇28カ1θering, PP.157・161, Tokyo, Japan, November 2005. [P16] YUeda and且. Oh8aki,“Positioning Characte血8tic80f a Coreles8 Surface Motor using且albach Permanent Magnet ArraピThe sth lnternational SymPOSI’un on L加θar thゴIzes for.lndustrial App】「ications d乙DL4.2005?, pp.270・273, Awaji, Japan, September 2005. [P17] YUeda, Y Kawamoto and且. Ohsaki,“Dynamic Characteristics of a Coreless Surface Motor using且albach Permanent Magnets,”The sth力6θm硫伽al Po rver Eleetroni’es(フ’onferenee(IPEO・2Vfiga ta20052, S4・1, N亘gata, JapaI1, Apri1 2005. Domestic Conference Proceedngs(in Japane8e) [P18] 上旦魅大崎博之,「6つの電機子導体を持つ磁気支持平面アクチュエータの平 面運動制御」,平成20年電気学会産業応用部門大会III, pp.135・136,高知,2007 年8月. Y. Ueda and H. Ohsaki,“Planar Motion Contro1 of a Maglev Planar Actuator with SiX Armature Conductors,” IEE 7 Ann ual Meetin8 0刀 Industry 4ρρ五cぼ施ηθIII, pp.135・136, Kochi, August 2008. [P19] 上田靖人,大崎博之,「光メモリ用の多自由度ドライブ装置の開発動向」,電気学 会交通・電気鉄道/リニアドライブ合同研究会,TER・08・19∫LD・08・19, PP.35・40, 鹿児島,2008年7月. YUeda and且. Ohsaki,“Survey of Development of Multi−degree・of’freedom Drive for Optica 1 Memories,” IEEJ Join t 7beh. Meeting on 71ransporta tion and Eleetrie Rai7rvay and Linear Diゴves, Kagoshima, TER・08−1911D・08・19, pp.35・40, July 2008. [P20] 上田靖人,大崎博之,「3組の2相電流制御による磁気支持平面アクチュエータの 位置決め」,第20回「電磁力関連のダイナミクス」シンポジウム,21B2・2, pp.165・170,別府,2008年5月. YUeda and且. Ohsaki,“Positioning of a Maglev Planar Actuator by Contro且ing Three Sets of Two・Phase Currents,” The 20肋Symposゴzun on Electromagneties and」Oyηamies(sea d20?,21B2・2, pp.165・170, Beppu, May 2008. 263 [P21] 」』拠大崎博之,「平面アクチュエータの3自由度回転姿勢に対する電磁力特 性」,電気学会全国大会,5・213,p.321,福岡,2008年3月. YUeda and H. Ohsaki,“Electromagnetic Force Characteristics of a Planar Actuator fbr Three・Degree・of’Freedom,”IEEJ Ann ua1 Meeting,5・213, p.321, Fukuoka, March 2008. [P22] 」;.!Mg}Hius,MA大崎博之,「ヨー方向に大変位できる平面アクチュエータの可動子の位 置検出」,電気学会リニアドライブ研究会,LD・07−34, pp.11・16,東京,2007年 10月. Y. Ueda and且. Ohsald,“Position Detection of a Mover of a Planar Actuator Cap able of ’lllraveling over Large Displacements inrYaw Direction,”IEE 7 Teeh. Meeting on、Linear t zゴves, LD・07・34, pp.11・16, Tokyo, October 2007. [P23】 上田靖人,大崎博之,「ヨー角に対する平面アクチュエータの位置決め特性」,平 成19年電気学会産業応用部門大会III, pp.137・138,大阪,2007年8月. YUeda and且. Ohsaki,“Positioning characteristics of a planar actuator for yaw angle,”IEE 7 Ann ua1 Meeting on lndustζ7 Appk’ea tions III, pp.137・138, Osaka, AugUst 2007. [P24] 上坦鐙ム,大崎博之,「小形平面モータの磁気支持力特性」,第19回「電磁力関 連のダイナミクス」シンポジウム,A312, pp.363・365,東京,2007年5月. Y. Ueda and H. Ohsaki,“Magnetic suspension force characteristics of a small Planar motOr,”The 19th Symp・訂㎜on Eleetromagnetics and Dymamics (sead192, A312, pp.363・365, Tokyo, May 2007. 【P25] 上田靖人,大崎博之,「小型多自由度アクチュエータの電磁力特性」,電気学会リ ニアドライブ研究会,LD・06・63, pp.79・84,東京,2006年10月. YUeda and且. Ohsaki,‘「Electromagnetic characteristics of a sman actuator for mUlti・degrees of丘eedom,”IEEJ 7Z?eh. Meeting on in’near Drives, LD・06・63, PP.79・84, Tbkyo, October 2006・ [P26] 上坦董ム,大崎博之,「永久磁石同期モータに基づく空心形サーフェスモータの電 磁力特性」,平成18年電気学会産業応用部門大会III, PP.155−158,名古屋, 2006年8月. YUeda and且. Ohsaki,“Electromagnetic Characteristics of a Coreless Surface Motor based on Permanent Magnet TYPe Synchronous MotOr,”IEEJ Ann ual Meθtiロg on lndustryAppk’ca tions III, pp.155・158, Nagoya, August 2006. 264 [P27] .k1911EiZS,大崎博之,「小型多自由度アクチュエータの駆動に関する検討」,電気 学会交通・電気鉄道/リニアドライブ合同研究会,TER・06・51/LD・06・29, pp.19・24,札幌,2006年7月. YUeda and且. Ohsaki,“lnvestigation about drive of a small actuator fbr mUlti・degree80f丘eedom,”IEEゾJoin t Teeh. Meeting on 1}ansρorta tion and Eleetric Ralway and Linear」thゴves, TER・06・51∫LD・06・29, pp.19・24, Sapporo, July 2006. [P28] 」;.1Ngws,MA大崎博之,「永久磁石リニア同期モータを原理とする空心形サーフェス モータの駆動特性に関する考察」,第18回「電磁力関連のダイナミクス」シンポ ジウム,A2PO1, pp.489−494,神戸,2006年5月. YUeda and且. Ohsaki,‘TDiscussion about Drive Contro1 of a Coreless Surface Motor based on Permanent Magnet TYpe Linear Synchronous Motor,”The 18th βジz互ρo匝LL囮on Eleetromagnetics a刀dl 1∼グηalzzτ已(sead18?, A2PO1, PP.489・494, Kobe, May 2006. [P29] 上田靖人,河本泰典,大崎博之,「ハルバッハ磁石を用いた空心形サーフェスモー タにおける可動子の回転運動抑制制御」,第17回「電磁力関連のダイナミクス」 シンポジウム,pp.249−252,高知,2005年6月. YUeda and且. Ohsaki,“Control of Mover Yawing Motion in a Coreless Surface Motor using且albach Permallent Magnet,“The 17th Symposium on Eleetromagneties and Dymamlbs(seadl Z?,2AMO6, pp.249・252, Kochi, June 2005. [P30】 河本泰典,上田靖人,大崎博之,「ハルバッハ磁石を用いた空心形サーフェスモー タの駆動特性」,電気学会リニアドライブ/半導体電力変換合同研究会, LD・04・98/SPC・04・170, pp.7・12,諏訪,2004年12月. YKawamoto,】͡alld且. Ohsaki,“Drive Characteristics of a Coreless TYpe Surface Motor Using且albach Permanent Magnets,”IEEJ Join t Tedh Meθting on Liηear Dn’ves and Semieonduetor」Power Converter, LD・04・98/ SPC・04・170, pp.7−12, Suwa, December 2004. [P31] 上田靖人,大崎博之,正田英介,「電磁吸引式磁気浮上車両の支持系へのフアジイ 制御の導入」,電気学会交通・電気鉄道/リニアドライブ合同研究会, TER・04・31/LD・04・52, pp.1−6,名古屋,2004年7月. YUeda,且. Ohsaki, and E. Masada,‘‘ApPlication of Fuzzy Control to Suspension System of Electromagnetic Suspension Type Magnetically LeVitated Vehicle,”IEEJ Joint 7Z?eh. Meeting on 7}ansportation and」Electrie Railrva 7 and Linear Drives, TER・04−31!LD−04−52, PP.1−6, Nagoya, July 2004・ 265 Other8 [P32] YUeda, H. Uesugi, M Nara, Y Fujli, and E. Ohkuma,“Campus Life is Changed!?,”The Journa1 oflEEJ, Vol.126, No.12, PP.775・778, December 2006 (in Japanese). 上田靖人,上杉春奈,奈良雅文,藤井康正,大熊栄一,「キャンパスライフが 変わる!?」,電気学会誌,Vb1.126, No.12, pp.775・778,2006年12月. [P33] YUeda,“Systematized technologie80f multi degrees of freedom motors (Section 7.3:Lens Drive f()r Optical Memories),”ZEEJ 7εφ皿bal Reρort, No.1140, pp.59・63, November 2008(in Japanese). .L[91WS,「多自由度モータのシステム化技術(7.3節:光メモリ用のレンズ駆動)」, 電気学会技術報告No.1140, pp.59・63,2008年11月. 266