Lunar Lander Landing Site Decision in Low-Fuel Case

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1 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 Lun Lnde Lndng Ste Deson n Low-Fuel Cse Mn-Guk Seo Seong-Mn Hong nd Mn-Je Thk KAIST Deptment of Aeospe Engneeng Dejeon Koe Abstt. Ths ppe dels wth lun lndng ste deson lgothms fo the se when the lun lnde s not ble to eh the ognl lndng ste. A new lndng ste whh mnmzes the fuel onsumpton of the lun ove s detemned fom lndng ste nddtes loted wthn the mmum ehble downnge of the lun lnde. A pth plnnng lgothm s ntodued to detemne the new lndng ste whh hs the shotest pth to the ognl lndng ste mong the nddtes. Numel smultons e onduted to vef the pefomne of the poposed lgothm. Intoduton In lun lndng msson the lnde m need new lndng ste f the lndng on the ognl lndng ste s not possble due to vous dstubnes dung the opeton. A good nddte fo the new lndng ste s n e fom whh the ove s ble to tvel to the ognl lndng ste whee the msson should tke ple wth mnmum eneg onsumpton. A new lndng ste deson lgothm s desgned wth full nfomton of obstles to stsf ths equement. The fst step of the lgothm s to obtn the sfe ehble egon of lun lnde b solvng the lndng tjeto optmzton poblem fo mmum downnge. The new lndng ste s hosen mong ll the lndng ste nddtes s the one whose pth to the ognl lndng ste s detemned to be the shotest one b pplng pth plnnng lgothm. Lndng gudne s n mpotnt ssue n the plnet eploton feld. The ommon objetve of plnet lndng gudne lgothms of pevous studes s to pesel lnd on the pedetemned lndng ste wth mnmum fuel onsumpton[-5]. Ths mples tht those lgothms e not sutble to fgue out the mmum downnge of the lun lnde unde fuel lmts. In ths ppe the optml lndng gudne poblem s efomulted s downnge mmzton poblem. The dnms of the lun lnde s smpl modeled n - dmensonl lol Ctesn oodnte sstem. The optml soluton s obtned usng Guss Pseudospetl Optmzton Solve (GPOPS). The oute-most ontou ponts of the ehble lndng egon whh e new lndng ste nddtes e detemned fom the mmum ehble downnge nd the obstle nfomton. Pth plnnng lgothms wth obstle vodne studed n mn pevous tles[6-3] do not fous on gunteeng shotest pth. Ths ppe suggests new pth plnnng lgothm espell fo the lun ove to mnmze pth length. The obstles on the lun sufe e ssumed to be gven s les nd polgons. The esultng pth mp onssts of stght lne segments nd ul s. The shotest pth sttng fom the lndng ste s obtned b pplng Djkst s lgothm[4]. The lndng ste nddte wth the mnmum pth length s hosen s the new lndng ste. Ths ppe s ognzed s follows. The lndng tjeto optmzton fo downnge mmzton s hndled n Seton. The pth plnnng lgothm fo lun lnde s ntodued n Seton 3. Smulton esults e gven n Seton 4 fo the pefomne nlss of the lgothm. The ovell onluson of ths ppe s ddessed n Seton 5. Lndng Tjeto Optmzton fo Downnge Mmzton In the se when lun lnde nnot eh the ognl lndng ste whee eploton mssons e supposed to be pefomed lndng ste deson lgothm s needed to fnd new lndng ste. One equement of the new lndng ste s tht the lun lnde should be ble to lnd on t sfel wth lmted mount of fuel of the thuste nd the tttude ontol sstem. Ths mples tht the lndng ste nddtes should be the ponts whee the lun lnde n mke lndng nd not on obstle egons. The lun ove hs to tvel to the ognl lndng ste to pefom ts mssons fte the lun lnde ehes the new lndng ste. In ode to mnmze the eneg nd the tme onsumpton the pont whh hs the shotest pth to the ognl lndng ste s hosen s the new lndng ste mong nddte ponts. As mentoned pevousl the ponts n the ehble egon fo lun lnde on the lun sufe e nddtes of the new lndng ste. Sne the ognl lndng ste s The Authos publshed b EDP Senes. Ths s n open ess tle dstbuted unde the tems of the Cetve Commons Attbuton Lense 4. (

2 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 supposed to be outsde of the ehble egon n ths ppe ll the pths fom the nddtes should oss the oute-most ontou of the ehble lndng egon. Ths mens tht the lndng ste nddtes e lmted to the ponts on the oute-most ontou. The lndng tjeto optmzton fo downnge mmzton s utlzed to fgue out the ontou. Sevel ssumptons e onsdeed to defne lndng tjeto optmzton poblem. ) The moton of the lun lnde s desbed n netl Ctesn oodnte sstem wth ts ogn on the lun sufe whh s ssumed to be flt plne. ) The lttude s s noml to the lun sufe nd the othe es e on the sufe. 3) The effet due to the otton of the moon s negleted nd the unfom gvttonl eleton s onsdeed. 4) The ntl velot omponent noml to the lttude s nd the tttude ngle te e. 5) The mjo s of the lun lnde the s pllel to the mn thuste deton s pllel to the lttude s t the ntl tme. 6) The otton wth espet to n s noml to the mjo s s possble b the ombnton of the tttude ontol sstem. Unde those ssumptons ) ~ 6) the tjeto optmzton poblem s defned n the -dmensonl netl Ctesn oodnte sstem nd the oute-most ontou s defned s le enteed t the ognl lndng ste wth the mmum downnge s the dus. The effet of lttude ontol thustes on the lun lnde s velot hnge nd the tttude ontol s onduted b tttude ontol sstem equpped on the lnde. The ontol nputs of the lun lnde e defned s mn thuste nd tttude thuste n ths model The dnm model of the lun lnde s defned fom ll the ssumptons gven bove. V h Vh TD os V m TD sn V h g () m TD Ttt m Isp g q Ttt LT q I zz whee nd h e downnge nd lttude of lun lnde. V nd V h denote velot omponents n eh s deton. Mss moment of net moment m of the tttude thuste tttude ngle nd tttude ngle te of lun lnde e smbolzed s m I zz L T nd q. The tttude ngle s defned s the ngle between the mjo s nd the downnge s. I sp s spef mpulse of lun lnde engne whh s used t the mn thuste nd tttude ontol thuste nd g s the gvttonl eleton on the lun sufe. The ontol nput vbles e mn thust T D nd tttude ontol thust T tt. h O V h g T D Fgue. Coodnte Sstem nd Stte Vble Defntons. The objetve funton s defned to mmze the mgntude of. mn () TD Ttt The ntl ondtons of stte vbles e gven fom the ssumptons ddessed befoe. t ht h Vt Vht (3) mt m t9 qt The lttude of the lun lnde s bounded to be lge thn to potet t fom shng on the lun sufe. The sensos lke mes on the lun lnde mesue the nfomton of the lun sufe fo sfe lndng nd hzd vodne. The tttude ngle nd ts te e lmted to onsde the opeton pefomne of those sensos. ht f Vtf Vhtf (4) tf 9 qtf h mn m (5) qmn q qm The mn thust s lmted b the pefomne of the tuto. The tttude ontol thuste s lso lmted unde tuto pefomne but t n be negtve vlue onsdeng both deton of the thuste. TD T mn D TD m (6) Ttt T mn tt Tttm The estton on the mount of fuel s epessed s mss lmt of the lun lnde. mmn m mm (7) The soluton of the optmzton poblem defned n ()-(7) s obtned b pplng pope optmzton lgothms. GPOPS s the optmzton tool utlzed n ths ppe. Ths tool s bsed on the vble-ode Gussn qudtue method nd dels wth ontnuoustme optml ontol poblems. The oute-most ontou of the ehble egon s deded fom the optml soluton V V

3 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 s the le on the lun sufe enteed t the ntl poston on the sufe wth the mmum downnge s dus. Assumng tht the nfomton of obstles e gven the lndng ste nddtes e defned s the ponts dvdng ths ontou nto etn smll nd equl ntevls nd not loted on the obstles. 3 Pth Plnnng Algothm fo Lun Rove As ddessed befoe the new lndng ste s the one mong nddtes wth the mnmum pth length to the ognl lndng ste. The pth plnnng lgothm between two ponts s equed fo ths poess. The desgned lgothm should not onl ensue the mnmum pth length but lso guntee the sfet of lnde b plnnng the pth not to oss the obstles on the lun sufe. The pth plnnng lgothm ntodued n ths ppe stts fom onstutng pth nddtes fom lndng ste nddte to the ognl lndng ste. The obstles on the lun sufe e ssumed to be gven s geometl fgues s stted below. Eh of the ognl lndng ste nd lndng ste nddte s ble to be epessed s pont on the lun sufe. The tes e ble to be ppomted s les[5]. The ul obstle enteed t wth the dus of s epessed s C n ths ppe. The nfomton of othe obstles whh e not ound lke oks e ble to be deteted fom the senso mesuements[6]. Eh of those obstles s ble to be ppomted s polgon whh s fgue onssts of stght segments suoundng t[7]. In the se when the obstles e les the mnmum pth s onssts of some of followng pth segment nddtes: tngent lnes fom two end ponts to the les ommon tngent lnes of two bt seleted les nd ul onnetng ponts of ontt on eh le[8]. When the obstles e polgons pth segment nddtes e s follows: stght segments between two end ponts nd the vetees of the polgons stght segments between two vetees fom dffeent polgons nd the sdes of the polgons[9]. Sne the obstles on the lun sufe nludes both les nd polgons the goup of pth segment nddtes onssts of ll the sots of them stted bove nd the tngent lnes fom the vetees of the polgons to the ul obstles. The stght lne onnetng two ponts nd e gven s follows. PPbPP PP (8) whee pp bpp (9) pp The pth segment s defned b (8) nd two end ponts. nd C fom the pont p p outsde of the le e obtned s below. CPCP () whee The two tngent lnes to the le CP f f g CP CP CP p CP p p () fcp p p gcp p p The ponts of ontt on the tngent lnes t t obtned s () nd () e lulted s f f t g t t t CP () t CPt CP whee f t CP CP (3) g t CP CP The equtons of pth segment nddtes e gven s nd () nd the end ponts of them e p p t t. The ommon tngent lnes between two les C nd C e gven s follows. b (4) whee b f f f f f f f f f f f f f f f f (5) b f f f (6) nd The pont of ontt on C t t one on C t t e lulted s t t t t (7) The mthemtl sgns n font of the sme vble e equl. Ths esults n fou ommon tngent lnes eept fo se. Two tngent lnes obtned when the mthemtl sgns n font of nd of (5) e dffeent fom eh othe nd the ponts of ontt on them e possble n ths se. The equtons of pth segment nddtes e 3

4 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 epessed s (4). t t t t e the end ponts of them. The pth nddtes should be desgned not to pss though n of obstles fo sfet of the lun ove. Ths mples tht the pth segment nddtes whose pt s nluded n n of the obstles should be eluded. The eptblt test s desgned fo n bt pth segment nddte denoted b two end ponts nd nd the equton of stght lne nd b. The fst poess of ths test dels wth ul obstles. The dstne between n bt pth segment nddte nd the ente of n bt ul C s lulted s below. obstle b d (8) b It s obvous tht the pth segment nddte does not pss though the ul obstle n d se. Two nteseton ponts of b C nd LC e lulted when d LC n ode to fgue out the fesblt of the segment. whee LC LC f f h g h LC LC b b LC LC LC LC f bb LC LC (9) g () b h LC b The pth segment nddte s eluded when n of two LC stsfes LC sne ths mens tht C. Ths some pt of t s loted nsde of poess s ppled to ll the possble ombntons of pth segment nddte nd ul obstle. The seond poess whh s ppled to the pth segment nddtes not eluded n the fst poess hndles polgon obstles. In the se when some pt of pth segment nddte s ld nsde of polgon obstle t should nteset wth one o moe sdes of ths polgon. The nteseton pont of n stght lne whh bt pth segment s nluded n nd stght lne pssng though two vetees P P nd P P s epessed s follows. LP LP P P P P P P P P P P b LP b () The pth segment nddte s eluded when ths nteseton pont s ld on t LP nd the sde of polgon obstle wth those vetees s the end ponts P LP P t the sme tme. Ths poess s ppled to ll the bt ombntons of pth segment nddte nd sde of polgon obstle. The sfe pth segment nddtes e obtned fom the eptblt test. The onstuton of pth nddtes s ompleted b onnetng the end ponts of the sfe pth segment nddtes whh e lled s nodes n the lte pt of ths ppe on obstles. The nodes on polgon obstles e obvousl the vetees. Ths mples tht ll the sdes of eve polgon obstles should be nluded n the goup of pth segment nddtes. The pth segment nddtes on ul obstles e obtned s the ul s between two neb nodes on the sme ul obstle. When pt of ul obstle s ovelpped b nothe ul obstle the pth segment nddtes on t nd ontnng the ovelpped pt of ts ontou should be eluded. Two ul obstles C nd C e defned to be ovelpped when the followng ondton s stsfed. () The nteseton ponts of those two les e obtned s below. whee f f h g h b b b f b b g h b b Two ul s on (3) (4) C lled A nd B onnetng two nteseton ponts gven n () est; one of them s loted nsde of C nd the othe one s not. The ovelpped s fgued out fom the ente of A ARC A ARC nd the A ente of B ARC ARC. B B 4

5 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 whee ARCA ARCA ARCB ARCB b b b b b b b b If ARCA ARCA (5) (6) the pth segment nddtes ontnng A should be eluded. The ones ontnng B e eluded when ARC B ARC B needed to be ppled to. The sme poess s C fo the vefton of n uneptble ul. The optml pth fom lndng ste nddte to the ognl lndng ste s detemned b pplng Djkst s lgothm[4]. The vsblt djen mt s onstuted fom the lengths of the pth segment nddtes nd the oodntes of the nodes. The optml pth plnnng lgothm eplned n ths seton s ppled to ll the lndng ste nddtes obtned n Seton. Ths povdes the lengths of the pth sttng fom ll the lndng ste nddtes. The one wth the mnmum pth length s deded s the optml lndng ste. 4 Smulton Smultons e pefomed to vef the pefomne of the poposed lndng ste deson lgothm. The lun lnde s ssumed to be pled stton on the pont z. km (7) h The ognl lndng ste on the lun sufe OL OL s defned s below. OL 68 km OL 5 (8) The mp of obstles s gven s shown n the followng fgue. (km) Intl Poston Ognl Lndng Ste (km) Fgue. Coodnte Sstem nd Stte Vble Defntons. 4.. Lndng Tjeto Optmzton fo Downnge Mmzton The lndng tjeto optmzton s pefomed fst to fgue out the mmum downnge of the lun lnde. The ntl ondtons of the lun lnde e deded fom (3) nd (7). t ht km Vt Vht mt 4kg t9 qt (9) The temnl onstnts e gven n (4). The lmts of tttude ngle nd ts te e defned s below. mn 7 m (3) qmn / se qm / se The ontol nputs e gven s follows. TD kn T mn D kn m (3) Ttt 5 N T 5 mn tt N mn The mmum nd the mnmum mss of the lun lnde e spefed to dede the mount of vlble fuel dung lndng. mmn kg mm 4kg (3) The moment of net nd moment m e lso hosen s Izz 5 kgm LT.m (33) The optml lndng tjeto s obtned wth GPOPS s follows. 5

6 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT Alttude (m).5.5 Mss Vton (kg) V (m/s) DownRnge (m) () Lun Lnde Tjeto Tme (se) (b) Mss Tme (se) () Velot Component n Downnge Deton 5 V h (m/s) Thet (deg) 9 8 Pth te (Deg/se) Tme (se) Tme (se) Tme (se) (d) Velot Component n Alttude Deton (e) Atttude Angle (f) Atttude Angle Rte 5 Thust (N) Atttude Thuste (N) Tme (se) (g) Mn Thust Tme (se) (h) Atttude Angle Rte Commnd Fgue 3. Lndng Tjeto Optmzton fo Mmum Downnge. The mmum downnge m s.km. As shown n Fg. 3-() to (f) stte vbles stsf the temnl onstnts fo soft lndng. The ontol nputs the tttude ngle nd ts te e bounded b the own lmts s gven n Fg. 3-(e) to (h). All of the vlble fuel s onsumed s ddessed n Fg. 3-(b). geen dotted lnes e the fesble pth segment nddtes obtned wth the optml lndng ste nd the ed les on the obstles e nodes. The sold mgent lne epesents the optml pth nd ts length s 9.447km. 4.. Optml Lndng Ste Deson wth Pth Plnnng Algothm The oute-most ontou s defned s the le enteed t the ntl poston on the lun sufe wth the dus of m fgued out n the pevous step. ponts on ths ontou dvdng t n equl ntevls e hosen s the lndng ste nddtes. The lotons nd shpes of obstles e gven s dwn n Fg.. The esult of optml lndng ste deson lgothm s obtned s Fg. 4. Fg. 4 shows tht the pth segment nddtes nd nodes e geneted fom the gven obstle mp. The n ponts ld on the oute-most ontou of the ehble lndng egon e the fesble nddtes of new lndng ste nd the ed pont loted on s the optml lndng ste. The (km) (km) Fgue 4. Optml Pth nd Lndng Ste fo Lun Rove. 6

7 MATEC Web of Confeenes (6) DOI:.5/ mteonf/65493 MIMT 6 5 Conluson The lndng ste deson lgothm fo lun lnde s poposed n ths ppe. The oute-most ontou of sfe ehble egon s defned b solvng lndng tjeto optmzton poblem. Ths poblem s defned to fgue out the mmum downnge wth the temnl onstnts nd the eneg lmts n onsdeton. GPOPS s utlzed to fgue out the soluton. The lgothm to fnd the shotest pth between lndng ste nddte nd n ognl lndng ste s desgned to selet the most sutble lndng ste fo lun ove eneg onsumpton mnmzton. The pth segment nddtes nd nodes e geneted fom gven obstle mp b pplng the theoes of geomet. The shotest pth s deved b pplng Djkst s lgothm. Intentonl Confeene on Robot Intellgene Tehnolog nd Applton (3) 8. D. S. Km K. Yu Y. Cho D. Km C. Yp Computtonl Sene nd Its Appltons-ISA 4 (4) 9. G. H. Moon Mste s Thess Koe Advned Insttute of Sene nd Tehnolog Dejeon Republ of Koe (4) Aknowledgment Ths wok ws suppoted b the Ntonl Reseh Foundton of Koe(NRF) Gnt funded b the Koen Govenment(MSIP)(No. 4MA3A3A334589). Refeenes. A. R. Klumpp Automt (974). R. R. Sost J. R. Re AIAA Gudne Nvgton nd Contol Confeene (5) 3. F. Njson nd K. D. Mese AIAA Gudne Nvgton nd Contol Confeene (5) 4. B. Aç kmeşe S. R. Ploen Jounl of Gudne Contol nd Dnms 3 5 (7) 5. L. Blkmoe B. Açıkmeşe D. P. Shf Jounl of Gudne Contol nd Dnms 33 4 () 6. B. Slno L. Svo L. Vlln G. Oolo Robots: Modelng Plnnng nd Contol (Spnge Sene & Busness Med 9) 7. V. J. Lumelsk A. A. Stepnov IEEE Tnstons on Automt Contol AC-3 (986) 8. V. J. Lumelsk A. A. Stepnov Algothm - 4 (987) 9. K. Fuzmu H. Smet IEEE Tnstons on Robots nd Automton 5 (989). E. Gt M. G. Slk D. P. Mlle R. J. Fsb IEEE Intentonl Confeene on Robots nd Automton (99). M. Geke Poeedngs of the Amen Contol Confeene 4 (999). N. A. Melho R. Smmons IEEE Intentonl Confeene on Robots nd Automton (7) 3. M. Tokh Fuzz Sets nd Sstems 59 (8) 4. E. W. Djkst Numeshe Mthemtk (959) 5. B. J. Jeon B. G. Pk M. J. Thk 3 As-Pf Intentonl Smposum on Aeospe Tehnolog (3) 6. F. Junhu C. Pngun C. Huto Sstems nd Contol n Aeonuts nd Astonuts (ISSCAA) 3d Intentonl Smposum on (). 7. K. S. Km G. H. Moon J. W. Km J. W. Jeong H. L. Cho H. C. Shm M. J. Thk The nd 7