Transferts Terre-Lune en poussée faible par contrôle feedback. La mission Smart-1
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Transferts Terre-Lune en poussée faible par contrôle feedback. La mission Smart-1. Alex Bombrun — Jonathan Chetboun — Jean-Baptiste Pomet. N° 5955 1er août 2006. SN 0249-6399. apport de recherche. ISRN INRIA/RR--5955--FR+ENG. Thème NUM.
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(156) Ç7¾ . . 2.5 2 1.5 1. dx/dt. 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1. −0.5. 0. 0.5. 1. 1.5. x. )D. µ = 0.012153,. C=3. θTY~je~}0xzml³º³ºÙz|m ¼ P@~x|xlm8´¸@¹°j³ºm@»T³º±°TY}jm x|
(157) z|ml³º³¹¹zm8yV"|zÓÌ@mJe³ºm zmlk_xpx~jºÌ
(158) }\z³º_yvw}@k_º\~jm7Ü. kF k ≤ Fmax. YÑ. m ˙ =−. Fmax. mJxzÛ×Pm"z|Ùz|m^ÛÐmlze³7k7@x|xm)j~ ÎcD\Ð. kF k ve. Vm x{z³'Ìw¹zmVxxm89¯ {mJ"zº@}1ymJx£Y YÉ m[xvyxziJkDm[}j@}l@}\z|½@³ºRT\xxiJjm["º}e¹}\zJ£@³ºmJx9mJkDºil|mJx[θ\~jmO³c¯ Y}8}jz|m m"z \~jF"Y|mJxP@}ymJ}\z~yÏ7@|@k_i"z|mJx[ymp³c¯ Y°Ùz|mVÐ^Éy mpj|@°j³ºilk_m²ym C mJj³¹mJl@}\zc,|½@e³º,Peml~y,zhxl¯,Jhlº|m }x_"mVx_}j@~Ì@ml³º³ºmJx_"w@yY}j}jlmVx'@~yÏj\~jml³º³¹mVxD@}Í
(159) {@~jzm³º³º@}j£YÙz|~ym L mlz7@Ѭ³º ´µ@"m F mVx{z ve. x. y. x. y. .
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(161) "ȾŠ¸ÀlĹ¾ Æ'ÃÁ
(162) ¿>ÆĹ¾Å"¾|¾YÀYà 7ǸÁ¿TÇ
(163) Æ . VØ. 0.4 0.3 0.2. y. 0.1 0 −0.1 −0.2 −0.3 −0.4 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. x. 9. Êe|°j¹zm8~jz@~ym8³º_ ~j}jm8P@~j. jJ"Yk_T\xJm²j@}x³ºm mJTiJm k_@°jº³ºm c˙ = e˙ x = e˙ y =. ÓÌYmJ'Ü. ðSRTUTT. . . h˙ x. =. h˙ y. =. L˙ =. QSW. C=3. ô [Ü 0 2. c2 FS µZ c A ey (sin LFQ + FS − Y FW ) µ Z Z c B ex (− cos LFQ + FS + Y FW ) µ Z Z c X cos LFW µ 2Z c X sin LFW µ 2Z µ2 2 c Y Z + FW c3 µZ. Z = 1 + ex cos L + ey sin L A = ex + (1 + Z) cos L B = ey + (1 + Z) sin L X = 1 + h2x + h2y Y = hx sin L − hy cos L. Îc@@Ð. ÎcYÐ.
(164) 4D. j¾ ×ÀÁ
(165) Ây¿- eÁ
(166) Ç7À"Æ^Ây¿ 7
(167) Á
(168) Ç7¾ . . @~·³¹mRY°j³¹iJk_mOy6m C ml³¹mJ·"@}\z|½Y³¹Y»^}j@~x·~yzº³¹x@}x9³ºmJx·¹}\zJ£@³ºmJxFj|mlk_ºil|mJxFP@~j"@}Tx{z|~jº|m ~}jm ´µ@}"zº@}-ym·v\j~}j
(169) Ì V \~j·xz|@°j¹³ºxm £@³º@°T³ºmlk_ml}\z~j}jm @|°j¹zm8"º°j³ºm7Ü ÎcY¢Ð c V = (e − e ) + k (e − e ) + k ( − 1) + k (h − h ) + k (h − h ) c ÝYkDk_me³¹mVxO´µY}^z|¹Y}x[ym²·v\j~j}
(170) Ì x@}\z[ml³º³ºmJx¶ck_álk_mVx[ymVx[º}\zl£Y|@³¹mVxRj|mlk_ºil|mJx[j~±k_@~y¶ ÌYmlk_ml}\zV»y¹³·mlÏyºxzm8P@~jhT@"~}jm89¯ mJ}\z|m²VmJ³¹³ºmJx~j}´µmlmJj°@à±|@°~x{z|m8x{z°jº³¹x|}\zJ»j@¯ mVx{z ¼ yºm8\~j Pml|k_m"z 9¯ ÓÌY@º V˙ ≤ 0 ÉFo}ymJxe@°y{mV^z|Ù´¸x²ym)³7z|hjiJxm_9¯ p³¹mlÏ "Yk)°j|~j}0mVx{z ymDkDmlzz|m'ml}0JÌ\¹¶ jml}lmD³ºm_´¸¹z)\~·¯ ~j}´µmlmJj°@àÖjm'z{vwPm±·v\j~j}
(171) Ì1TmJ~yzá"z|m7Pml´µ@|k7}\z²P@~8³¹m7j|@°³¹iJkDm7ym z||@}x´µmlz·¯ Y°j¹zm mJ},z|mlk_xk_º}j¹k'~jk-É k. 2. k. x. 2. xc. 1. y. 2. yc. 2. 2. 3. x. 2. xc. 4. y. yc. c. k. k. . . ! #" %$ &' (*)+,-. . . ! & !). Zzºejm)k7º}\zml}T}\zJ»T@}Öl@}xºjil|m³¹m'j|@°j³ºilk_m|mJxz|mlº}\zeymJxpz|@xpl@|xpl@}\z|½@³º@ÉFÊe} º}\z|yy~j¹z ³ºmJx V\~
(172) z|¹Y}x8y~jY°j³ºilk_m7"@}\z|½Y³¹Dml}mJj|ml}@}Yz ³ºmJx J\~zº@}x8j~ j|@°³¹iJkDm7x|}x l@}\z|½@³ºm-θØYвmlz)ml}v
(173) {@~jz|}\z8~j}jm±Ylll³ºl
(174) zº@} TmJz|~j|°
(175) z|"mY»·x~j)³º°Yxm±ymVx8J\~zº@}x8j~ Y°j³¹iJk_m jm C mJj³ºml"Y}\z|½@³ºDÎcØ@Ð²Ü x ¨ =. 2y˙. x+µ ρ31 y (1 − µ) 3 ρ1. + x − (1 − µ). y¨ = −2x˙ +. y. −. − µ −. x−1+µ ρ32 y µ 3 ρ2. Îc IYÐ. + ux + uy. mDl@}\z|½@³ºm |mlj|Jxml}\z|m³>¯ @J"J³¹J|zº@}Ö\~jR"Y|mJxP@} ¼ ³º¦P@~x|xJm'y~k_z|ml~j m"z mVx{zmlÏwºk_ }(ux³ºmJ,xulw)@y@}j}lmJx y~-|mlPil|mpz|@~j|}@}YzVÉ x. . y. & ,- &' & ) . (* ) / !. " %$ ! ! " . & . . m,P@º}YzDym¦£@}£@m θml}\z|m,³0gml|m7mlz'³0·~}jmVÐmJxz'~j}ªP@º}\z'9¯ J\~jº³¹º°j|m Î>j}x)³¹m |mlPil|m8z|@~j|}}\z^Ðy~Öxvyx{z|ilk_Lm'x|}xp"@}\z|½Y³¹mYÉFÊe}1j|Jxml}\zm)ºl~j}m)"z|~ymD@x|xm )@J@yJkD\~jm)ym ³)xz|@°jº³¹xzº@}ym²lmeP@º}\z ·¯ VY~¹³º¹°mDÜw³ºDx{z°jº³ººx|
(176) z|¹Y}±}·¯ mJxz 'j|¹YF\~jme³ºyl@³¹mY»Y°YxJmex~j ³¹m ³ºº}jJ@xY»wm"z³>¯ @},´¸¹zymJxx¹k'~j³
(177) zº@}TxYѦ³c¯ @},}m xm8xY~"ºm²@x 9¯ @°yz|ml}jºym Ú Tmlz¹z|x{Û "Y}\z|½@³ºmJxJ» @¯ mVx{z¶ ¼ ¶×j¹|m7Y~ ¯ mJ³¹³ºmJx'}jm,l@}lml|}jml}\zYx³ºm¦lYx9¯ ~j}fx|
(178) z|ml³º³¹¹zm ¼ TY~xxlm±´¸º°j³¹m θx|~y´°jºml}ªx~j }x'~j}ªPm"z|Ùz_Ì@Yºx¹}T£@m¦ym @Ѫ³ºmJxD"Y}Yz|½Y³¹mVx³º¹}Jº|mJx)x@}\z)}T
(179) z~mJ³¹³ºmlk_ml}\zDTmlz¹z|xÐ^ÉÝm"zzm xmJ"zº@}mJxz@¹³º³¹mJ~jxyJl@}jL2}jmV^z|lm8ym8³Dx~j¹zmy~-jP@zJÉ twYÙz (a, b) ~j}TY¹}\zp9¯ J\~jº³º¹°j|m@ÉFÊe}¦V"|Ùz8Ü Îc\Ð x = a + ξ, y = b + η YÑ ξ m"z η x@}\z ³¹mVx"wY|yY}j}jJmJxmJ³ºzºÌ@mJx @~¦P@º}\z·¯ VY~¹³º¹°mYÉ·m P@º}Yzym£Y|@}j£@memJxzlml³º~j YÑ1³>¯
(180) zz@"zº@}1ym³±gml|m8mlz²ym³± ~j}jm'xm"Yk_TmJ}xmJ}\zJÉP Y~j³ºm)P@º}\zeym)£@}£@m \~jm }@~xl@}xyl|@}xJ»w@}1 a = 0 m"z b = 0.836892 ÉjÊe}¦´¸@Ùz~}0ylÌYml³º@jPmlk_ml}\zym Ω ~-Ì@YLºxº}£Ym jm (a, b) Ü Î¸ØYYÐ 1 1 Ω = Ω(a, b) + Ω (a, b)ξ + Ω (a, b)η + Ω (a, b)ξ + Ω (a, b)ξη + Ω (a, b)η + O(3) 2. 2. 2. x. y. 2. xx. 2. xy. 2. yy. .
(181) jÆ ¿µÅ"¾"ƾlÆ^ƾ Ây¿P¾'¾"¿ jÁÂ
(182) "ȾŠ¸ÀlĹ¾ Æ'ÃÁ
(183) ¿>ÆĹ¾Å"¾|¾YÀYà 7ǸÁ¿TÇ
(184) Æ . Ó. mJx J\~zº@}xj~k_Y~jÌ@mJkDmJ}\z ³ºº}jJ@xJmJx x@}\zy@}DÜ ξ¨ − 2η˙ η¨ + 2ξ˙. = Ωxx (a, b)ξ = Ωxy (a, b)ξ. \~j9Pml~Ì@ml}\zpxm kDmlzz|mx@~x ³'´µ@|k_mDÜ. + Ωxy (a, b)η + Ωyy (a, b)η. + ux + uy. + O(2) + O(2). θØ\@Ð. X˙ = AX + Bu. ÓÌYmJ'Ü X = (ξ. η. ξ˙ η) ˙ t. . m"z. 0 0 1 0 0 0 A= Ωxx (a, b) Ωxy (a, b) 0 Ωxy (a, b) Ωyy (a, b) −2. @~e³ºm'J@xpj~ xvwxziJk_m'gml|ml¶× ~j}jm@»@Ñ ³Dk7
(185) z|"m A xl¯ Jl¹z8Ü. u = (ux 0 1 2 0. µ = 0.012153. . m"z. λ1 λ2 λ3 λ4. = = = =. Î>Ø@ØYÐ. uy ) t . 0 0 B= 1 0. 0 0 0 1. »ym"z²TY~je³ºm)P@º}\z²ymD @£@}j£Ym. 0 0 1 0 0 0 A= 11.29391 0 0 0 −4.14696 −2. mJx Ì
(186) ³ºml~jx j|@j|mJx ym8³Dk7
(187) z|"m A xY}\z Ü. Î>ØjVÐ. 0 1 2 0. Î>Ø D\Ð L2. ». Î>ØY@Ð. 2.931837 −2.931837 2.334248i −2.334248i. Î>Ø@YÐ. 'Ì
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(189) z|"m ymzmJ³¹³ºm xYz|m\~jm[³ºmJxÌ
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