Rotoft Tjeto Tking b Supevise NL Contol A. Douin, O. Lengeke, A.B. Rmos. Mo-Co Abstt The pupose of this ommunition is to pesent new nonline ontol stutue fo tjeto tking tking epliitl into ount tutos stution. Hee tjeto tking b fou oto ift is onsiee. Afte intouing the flight nmis equtions fo the fou oto ift, tjeto tking ontol stutue bse on two le non line invese ppoh is opte supevision le is intoue to tke into ount the possible tutos stution. ne Tems Rotoft, nonline invese ontol, stution supevision, tjeto tking.. NTRODUCTON n the lst es lge inteest hs isen fo the fou oto onept sine it ppes to pesent simultneousl hoveing, oienttion tjeto tking pbilities of inteest in mn ptil pplitions []. The flight mehnis of fou oto ift e highl non line iffeent ontol ppohes integl LQR tehniques, integl sliing moe ontol [] hve been onsiee little suess to hieve not onl utonomous hoveing oienttion, but lso tjeto tking n this ppe, some simplifing ssumptions e opte the flight nmis equtions fo fou oto ift fie pith bles, o otoft, e onsiee. One impotnt limittion to pefom utomti guine fo otoft is elte the one w effet of otos its stution levels. Then the pupose of this stu is to intoue supevision le in non line invese ontol stutue to impove mneuvebilit tjeto tking effetiveness b this lss of otoft. This ppoh hs been le onsiee in the se of ift tjeto tking b iffeent uthos [3,, 5]. t ppes tht the flight nmis of the onsiee otoft pesent two level input ffine stutue whih is me ppent when new set of equivlent inputs is efine. This llows the evelopment of non line invese ontol ppoh two sles, one evote to ttitue ontol one evote to oienttion tjeto tking. Howeve this is one in genel out A. Douin is MAAA Lboto, Automtion Reseh Goup t ENAC, Toulouse, ne, ntoine.ouin@en.f. O. Lengueke is Mehtoni Roboti Lboto, Univesi Autonom e Bumng-UNAB, Bumng, Colombi, olengeke@unb.eu.o A.B. Rmos is nstituto e Ciênis Ets, UNE, tjubá, Bil,, mos@unifei.eu.b. Mo-Co is MAAA Lboto, Automtion Reseh Goup t ENAC, Toulouse, ne, Tel.+33567358, f. +335673, feli.mo@en.f onsieing tutos stutions when these ou, tjeto tking pbilit n be lgel ffete.. ROTORCRAT LGHT DYNAMCS The onsiee sstem is shown in figue whee otos one thee e lokwise while otos two fou e ounte lokwise. n ppeni the nmis of the otos e biefl hteie. The min simplifing ssumptions opte espet to flight nmis in this stu e igi oss stutue, no win, negligible eonmi ontibutions esulting fom tnsltionl spee, no goun effet s well s negligible i ensit effets ve smll oto esponse s. t is then possible to wite simplifie otoft flight equtions [6]. The moment equtions n be witten s: p& + k q q& 3 + k p & k + 3 whee p, q, e the omponents of the bo ngul p 3 ig. ou oto ift spee, k, k, being the moments of ineti in bo-is m the totl mss of the otoft. The Eule equtions e given b: & p + tn sin q + tn os & os q sin ψ& sin / os q + os / os whee,, ψ e espetivel the pith, bnk heing ngles. q
The eletion equtions witten ietl in the lol Eth efeene sstem e suh s: / mos ψ sin os + sin ψ sin / msin ψ sin os os ψ sin g + / M os os whee, e the ente of gvit oointes whee : + + + 3 3 the onstints: i m i {,,3, } 5 i. NL CONTROL APPROACH OR TRAJECTORY TRACKNG Hee we e inteeste in ontolling the fou oto ift so tht its ente of gvit follows given pth given heing ψ while ttitue ngles emin smll. Mn potentil pplitions equie not onl the ente of gvit of the evie to follow given tjeto but lso the otoft to pesent given oienttion. A. Attitue Contol om equtions it ppes tht the effetiveness of the oto tutos is muh lge espet to the oll pith is thn espet to the w is. Then it is onsiee hee tht ttitue ontol is involve ontolling the ngles. n equtions the effet of oto foes ppes s iffeenes so, we efine new ttitue inputs u u s: u 6. 3 u n the heing position nmis, the effets of oto foes moments ppe s sums, so we efine new guine inputs v v s: v + v + 6. 3 t is suppose tht u u n be me to v signifintl while v v n emin onstnt. Then the ttitue nmis n be ewitten une the ffine input fom: X f X, V + g X U 7. Y ', 7. X ' p, q,,, U u, u V v, v 8 ' ' Then, tking pofit of non line invese ontol theo, it ppes tht ll the ttitue ngles hve eltive egees equl to one tht thee is no intenl nmis while the output equtions n be ewitten s: Y & M Y U + N X V + P X 9 os M Y tg sin k sin k sin N X k tg os k tg os P X [ P, P ]' whee: P k os p qsin + os p + tg qsin + tg os. P kq + k p tg sin + q tg sin / t + tg os / t. Then, while ±π /, the ttitue nmis given b 9 e invetible. Then it ppes oppotune to opt s ptil ontol objetive to ssign to the ttitue ngles seon oe line nmis tows thei uent efeene vlues: whee Y & & 3,,, e espetivel mping ntul fequen pmetes while e efeene vlues fo the ttitue ngles. Then the esulting non line invese ttitue ontol lw is U M Y N X V + P X & B. Guine Contol Lw Consieing tht the ttitue nmis e stble muh fste thn the guine nmis, the guine equtions n be ppoimte b the ontol ffine fom: k os ψ v v os / mos ψ sin os + sin ψ sin v + v 5 / msin ψ sin os os ψ sin v + v g + / m os os v + v Hee lso, the outputs of the guine nmis pesent eltive egees equl to while the intenl nmis, whih e onene the ttitue ngles e suppose le stbilie. Then, onsieing tht seon oe line nmis e lso of inteest fo the guine vibles, we n efine esie eletions b: whee Y ψ ψ& ψ ψ ψ ψ ψ & & & ψ,,, ψ,,, 6, e espetivel mping ntul fequen pmetes while ψ,, e efeene vlues fo the ttitue ngles. Of ouse, mn othe shemes n be popose to efine esie eletions t the guine level. One esie eletions e me vilble, the invesion of the guine nmis bings nol the solution:
os v m & + + + g & ψ k os 7. os v m & + + + g + & ψ k os 7. tg & osψ + sinψ /& g 8. + & sinψ osψ sin 8. + + + g ψ &,,, & NL guine ontol, NL ttitue ontol of unfesible efeene vlues fo the inputs b moifing, s less s possible, the uent ontol objetives. Aoing to 5, 6 7, the ontol signls shoul be suh s: m ui m i,. v i m i,. Conitions. implies fo the esie ttitue eletions to stisf the following onitions: onition: & & m. & os N P. + m & os.3 m N + P + m & & tg m 3. & m + N + P tg N + P 3. & + N + P tg N + P 3.3 m m / v,v,, ψ,,, p, q,, ig. Popose ontol stutue Then, etuning to the epession of the ttitue ontol lw, it hppens tht the ente of gvit eletion tems ompenste eh othes the lw beomes simpl: U M Y N X + P X & 9 N N X N Attitue nmis Guine nmis u,u, Attitue ontol loop, ψ&, &, &, &,,, Y sin os os ψ os os sin os os The whole popose ontol stutue is given in the bove figue. V. LGHT CONTROL SUPERVSON Guine ontol loop Sine the bove ontol ppoh oes not onsie epliitl the input level onstints, we intoue hee supevision le whose funtion is to voi the genetion Then, efeene vlues fo instnt ttitue ngles eletions n be obtine fom the solution of the following line quti optimition poblem: α β α + β m. & β & m. & α tg β &.3 &, & Obseve tht the solution of this poblem is equl to if it is fesible espet to onstints..3, othewise the solution will be on the boe of the onve fesible set. Then if α β e solution of this poblem, u u e u α M Y N X V + P X 5 u β n the se of v v eltions. onsieing the epessions of the bove ppoh les to the onsietion of n intite non onve optimition poblem. A iffeent ppoh is popose hee. Let λ be suh s: & λ, λ, + g λ & + g 6 then oing to 8. 8.: 7 esible efeene vlues fo & &, & &, & & ψ& & n be obtine fom the solution of the following line quti optimition poblem: λ λ, μ + η μ ψ 8.
os m + + + g λ μ m k os os m + + + g λ + μ m k os 8. 8.3 ψ-ψ.6...8 λ whee η is hee onstnt. Let μ be the solution of the bove poblem, then the ontol inputs n be tken s: os v m λ & + + & + g 9. μ k os os v m λ & + + & + g + μ 9. k os Then,,, 3, stisf onition 5. u + / u + / 3. v 3 u v v / v / 3. V. CASE STUDES u Hee we onsiee two ses: one whee the objetive is to hove t n initil position of oointes,, while quiing new oienttion ψ, one whee the otoft is tking the helioïl tjeto of equtions: ρ osν t ρ sinν t δ + γ t 3 ψ ν t + π / whee ρ is onstnt ius γ is onstnt pth ngle. A. Heing ontol t hove n this se we get the guine ontol lws: v m g ψ& & v m g + ψ& & 3 k k the following efeene vlues fo the ttitue ngles: 33 Hee the heing eletion is & ψ ψ ψ ψ ψ ψ 3 Stting fom n hoiontl ttitue,, ttitue inputs u u given b eltion emin equl to eo. Then, figues 3 ispl some simultion esults:.6...5.5.5 3 3.5.5 5 ig. Heing esponse uing hove B. Tjeto tking se n this se we get the guine ontol lws: v v m ρ ν + g 38 Hee the pemnent efeene vlues fo the ttitue ngles e suh s: 39. ρ ν sin 39. ρ ν + g the esie guine oienttion eletions e ρ ν os ν ρν sin ν, ψ Attitue inputs e given b eltion whee now: M / os tg tg / / N X ' [ ] n figues 5 to 7 simultion esults e isple whee t initil the otoft is hoveing: t.5.5.5 -.5 - -.5 - -.5 -.5 - -.5 - -.5.5.5.5 ig. Evolution of otoft hoiontl tk 5, 3 t....8 5.6,.5.5.5 3 3.5.5 5 ig 3. Hove ontol inputs...5.5.5 3 3.5.5 5 ig 5. Evolution of otoft ltitue
.5.5.5 3 3.5.5 5 lim t + A.7, Omeg /s.5 8 6.5...3..5.6.7.8.9 3 5 omeg /s ig 7. Rotoft tjeto tking inputs 5...3..5.6.7.8.9. V. CONCLUSON n this ommunition nonline invese ontol tehnique pplie to otoft tjeto tking hs been onsiee. This ppoh les to the esign of two level ontol stutue bse on nltil lws. Howeve the possibilit of tutos stution hs le to the esign of supevision le whose objetive is to moif efeenes vlues fo the nonline invese ontol lws so tht the tking pefomne is mintine s muh s possible. The pplibilit of the popose ppoh ppe eptble sine the ompleit of the esulting optimition poblems to be solve online ppe to be the low. Then the popose ppoh shoul enlge the fiel of pplitions fo otoft. This ppoh oul be pte to the supevision of tutos stution othe utonomous ift. APPENDX The oto engine nmis e hteie b the eltion between the input voltge V the ngul te. A possible moel of oto nmis is & K Q + KV / τ A. τ, whee τ, K Q K e given positive pmetes whee the voltge input is suh s: V V m A. negligible esponse fo the voltge geneto. The step esponse V onstn of the oto is solution of the sl Riti eqution: & K Q + KV / τ V A.3 τ. A ptiul solution of the ssoite iffeentil eqution is suh s: + KV KQτ A. τ KQ n the genel se, the solution of A.3 n be witten s A.5 t / τ ' + e t / τ ' + K Qτ ' e τ ' τ / + K K τ V V Q A.6 ig 8. Two emples of oto step esponse t ppes fom figue 8 tht the nmis of the oto m be lose to those of fist oe line sstem onstnt τ, but s n be seen in A.6, this vlue is funtion of V. f the esie nmis fo the output e suh s: & A.8 T whee T is ve smll onstnt V n be hosen suh s: τ τ + + τ K K T T The oto foes e then i i while the oto moments e Q A.9 f i to A. M k i to A. i i whee f k e positive onstnt pmetes. REERENCES [] G. Hoffmnn, D.G. Rjnn, S.L. Wsle, D. Dostl, J.S. Jng, C.J. Tomlin, The Stfo Tetsbe of Autonomous Rotoft fo Multi-Agent Contol, 3 Digitl Avionis Sstems Confeene, Slt Lke Cit, UT, Novembe. [] H. K. Khlil, Nonline Sstems, Pentie Hll, 3 E.,. [3] S.N. Singh A.A. Sh, Nonline eouple ontol snthesis fo mneuving ift, Poeeings of the 978 EEE onfeene on Deision Contol, Pistw, NJ, 978. [] R. Ghosh C.J. Tomlin., Nonline nvese Dnmi Contol fo Moel-bse light, Poeeing of AAA-GNC,. [5] R. Asep, T.J. Shen,. Mo-Co, An pplition of the nonline invese tehnique to flight-pth supevision ontol, Poeeings of the 9th ntentionl Confeene of Sstems Engineeing, Ls Vegs,NV, 993. [6] B. Etkin, L. R. Rei, Dnmis of light-stbilit Contol. John Wile & Sons. New Yok, NY, 996. [7] W.C. Lu,. Mo-Co, light Mehnis Diffeentil ltness. Dinon, Poeeings of Dnmis Contol Confeene, lh Soltei, Bil, pp. 83-839,. [8] A.Douin, A.B. Rmos. Mo-Co, Rotoft Tjeto Tking b Nonline nvese Contol, Dinon, São Pulo, 7. [9] S.A. Beltn Meno, O. Lengeke,H. Gonle Auñ. Mo- Co, Contol PD e ltu e un quioto, 3 ntentionl Mehtoni Confeene, Bumng, Colombi, Otobe.