Comparison of LPM, GPM and RPM for Optimization of Low-Thrust Earth-Mars Rendezvous Trajectories

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1 Proceedngs o 4 IEEE Chnese Gudance, avgaon and Conrol Conerence Augus 8-, 4 Yana, Chna Comparson o LPM, GPM and RPM or Opmzaon o Low-Thrus Earh-Mars Rendezvous Trajecores Yuka Wang, Xuqang Jang and Shuang L Absrac The perormance comparson s perormed among Legendre (LPM), Gauss (GPM), and Radau (RPM) pseudospecral mehods, whch are ulzed o opmze he low-hrus Earh-Mars rendezvous rajecores. In order o provde a ar comparson condon, a shape-based approach s mplemened o provde equvalen nal guesses or all he nex accurae opmzaon, and he SQP algorhm s used o solve he resulng nonlnear programmng problem or each o he dscrezaon. The behavor and perormance o he hree pseudospecral mehods are compared on he msson o Earh-Mars Rendezvous n 3. I. ITRODUCTIO Low-hrus engnes, whch have hgher specc mpulse and lower uel consumpon, are provdng new opporunes n msson desgn. Because he hrus amplude s very small, wll ake long me or he engne o accelerae o a requred speed. Due o he long connues hrus duraon as well as a varey o dynamc consrans; s a ormdable ask o desgn such a low-hrus rajecory. In hs paper, a wo-sep approach ha combnes global opmzaon wh local opmzaon s appled o desgn such a low-hrus rajecory. Shape-based mehod s a wdely used global opmzaon mehod n low-hrus rajecory desgn, ncludng exponenal snusod developed by Peropoulos [,] and a 6 h nverse polynomal developed by Wall [3,4]. The exponenal snusod shape-based mehod has been mplemened as soware named TOUR-LTGA [5]. Some researcher clamed ha he 6 h nverse polynomal shape-based mehod provdes sgncan compuaonal savng over exponenal snusod shape-based mehod. Bu has no ye been proven n praccal mssons. In general, he ndrec mehods and he drec mehods [6] are wo ypes o numercal mehods ha are used or he opmzaon o a low-hrus rajecory. In he ndrec mehods, we rsly need o oban he necessary condons or opmaly, where he calculus-o-varaons echnque s used. And hen, s urn o solve he resulng boundary value problems: wo-pon boundary value problems (TPBVP, or shor), or mul-pon boundary value problems (MPBVP, or shor). The prmary advanages o ndrec mehods are hgh accuracy and as convergence. However, here are some Resrach suppored by he aonal aural Scence Foundaon o Chna (Gran o and 68457). Yuka Wang s wh College o Asronaucs, anjng Unversy o Aeronaucs and Asronaucs, anjng 6, Chna. Yongsheng Zhu s wh Shangha Engneerng Cener or Mcrosaelles, Shangha 3, Chna. Xuqang Jang s wh College o Asronaucs, anjng Unversy o Aeronaucs and Asronaucs, anjng 6, Chna. Shuang L s wh College o Asronaucs, anjng Unversy o Aeronaucs and Asronaucs, anjng 6, Chna (phone and ax: 86(5)848985; e-mal: lshuang@nuaa.edu.cn). drawbacks as ollows. Frs, s non-rval o derve he analyc expressons or he rs-order necessary condons, especally when complex consrans such as gravy asssed and perurbaon are consdered. Second, he resulng TPBVP/MPBVP can be dcul o solve because o sensvy problems. Thrd, or problems wh pah nequales s necessary o guess he sequence o consraned and unconsraned subarcs beore eraon can begn. By comparson, he drec mehod s deren. An analyc expresson, ha s or he necessary condons, s no requred n he drec mehods, as well as he nal guesses o he adjon varables. The essence o drec approach s o conver he opmal conrol problem no a parameer opmzaon problem whch could be solved by usng an exsng nonlnear programmng algorhm. The drec mehods can manly be dvded no collocaon mehods [7], derenal ncluson mehods [8], pseudospecral mehods [9,] and so on. The pseudospecral mehods arose rom specral mehods. And, he lud dynamcs problems are used o be solved by specral mehods. Over he las decade, he pseudospecral mehods have been wdely used or solvng he opmal conrol problems. Two key properes o pseudospecral mehods are he ses o collocaon pons and dscrezaon pons. We dscreze he conrol and collocae he derenal equaons o mee he dynamcs a he collocaon pons. Smlarly, we dscreze he sae a he dscrezaon pons, whch are enorced no he nonlnear program. Based on he ses o collocaon pons he pseudospecral mehods can be classed as Legendre Pseudospecral Mehod (LPM) [,], Gauss Pseudospecral Mehod (GPM) [3,4,,5], and Radau Pseudospecral Mehod (RPM) [6]. In hs paper, he perormance, ncludng he accuracy and compuaonal me usng he hree deren pseudospecral mehods are compared wh respec o low-hrus Earh-Mars rendezvous rajecores. II. SPACECRAFT MODEL A. Equaons o moon The ollowng equaons o moon are consdered r = v () Where, μ T v = r+ u () 3 r m T m = (3) gi r : Poson vecor o spacecra sp /4/$3. 4 IEEE 46

2 v : Velocy vecor o spacecra m : Mass o spacecra μ : Gravaonal parameer o he aracng body u : Thrus acceleraon drecon un vecor T : Thrus magnude g : Earh sea-level gravaonal acceleraron o I sp he aracng body : Specc mpulse There are many low-hrus magnude models. In hs paper, he solar sal model s used as ollow. η P T = (4) gi r sp Where, η s he hrus ecency, P s he engne npu power a AU. The values oη, P and I sp can be consan or varable wh upper and lower boundares. B. Opmal conrol problem saemen We ransorm he low hrus rajecory opmzaon problem o an opmal conrol problem. Take he Earh-Mars rendezvous msson as an example, s opmal conrol problem saemen ollows: Mnmze he cos unconal: J = m (5) Subjec o he dynamc consrans: x=(x,u) (6) Boundary condons: And pah consrans: Φ = r rlp = Φ = v vlp = Φ 3 = r rap = Φ = v v = 4 Ap (7) C = u = (8) Where, m s he nal mass o spacecra, r and v are he nal poson vecor and velocy vecor o spacecra respecvely, r and v are he nal poson vecor and velocy vecor o spacecra respecvely, r LP and v LP are he poson vecor and velocy vecor o Earh a deparure respecvely, r AP and v AP are he poson vecor and velocy vecor o Mars a arrval respecvely, x = [, r v, m] s he sae varables, u s he conrol varables. As a resul, he low-hrus rajecory opmzaon problem o Earh-Mars rendezvous s convered no an opmal conrol problem wh decson vecor P = [, rv, m, u,, ].Where s he deparure dae, and s he arrval dae. III. GLOBAL OPTIMIZATIO Global opmzaon or prelmnary desgn plays a crucal role n desgn o low-hrus rajecores, whch may provde a as nal guess or he nex accurae opmzaon. In hs paper, an analyc, shape-based mehod developed by Peropoulos and Izzo s appled o desgn he prelmnary low-hrus rajecores. Accordng o he research [,, 7], hrus arcs are presened by exponenal snusods. Exponenal snusods are geomerc curves ha are gven n polar coordnaes as r k e ksn( kθ + φ) = (9) Where, k, k, k and φ are consans. Based on he work o Peropoulos, Izzo proposed low-hrus lamber problem and solved numercally. Accordng o he algorhm developed by Izzo [7], le s consder he Earh-Mars case and use JPL ephemers DE45 o evaluae he posons and veloces o he planes. To maxmze he nal mass o spacecra, we consder he ollowng objecve uncon: mn m p J = () m Where, m p s he propellan mass, m s he nal mass o spacecra. We selec he ollowng bounds or he decson varables: Launch me [3 //, 3 / / 3], me o lgh Δ [, 5], k [., ], revoluons [, 3] ; meanwhle, we consder boh o he hyperbolc speed o he deparure hyperbola and arrval hyperbola wh respec o cener plane s no exceed.5 km / s. Derenal Evoluon echnque [8] wll be called o ry locang he global opmum o () and he resuls are summarzed n Table I. Fg. vsualzes he opmum Earh-Mars ranser orb Fgure. Earh-Mars ranser orb

3 TABLE I. GLOBAL OPTIMAL FOR THE EARTH-MARS CASE k 3//5 5//.53 IV. PSEUDOSPECTRAL METHODS Whou loss o generaly, consder he ollowng opmal conrol problem n Mayer orm. Mnmze he cos unconal: J =Φ( y( ),, y, ) () Subjec o dynamc consrans: Boundary condons: Pah consrans: dy ( ( ), ( ), ) d = y u () φ ( y( ),, y( ), ) = (3) Cy ( ( ), u( ), ) (4) Where, y ( ) and u ( ) are sae varables and conrol varables respecvely. The pseudospecral mehods, smply, are a class o drec collocaon. The sae and conrol are parameerzed by usng global polynomals a a se o dscrezaon pons. As a resul, ransorms he opmal conrol problem o he nonlnear programmng problem. And, we oban he derenal-algebrac equaons by enorcng he equaon o moons and pah consrans a a se o collocaon pons. GPM dscrezaon pons collocaon pons hree ses o pons are dened on he doman[,], bu der sgncanly. The LG pons nclude neher o he endpons. Whle he LGL pons nclude boh o he endpons, and he LGR pons nclude only one o he endpons. oe ha, he me nerval can be ransormed rom [,] o he me nerval [, ] va he ransormaon (5). + = τ + (5) A. LPM Consder he LGL pons, ( τ, τ,..., τ ), whereτ =, τ =. Le L ( τ ), ( =,..., ), be he Lagrange polynomals o degree. τ τ L ( τ ) =, ( =,..., ) (6) τ τ j j The sae, y ( τ ), s approxmaed by L ( τ ) as = y( τ) Y( τ) Y L ( τ) (7) Where, Y = Y ( τ ). ex, he dervave o he sae s approxmaed by derenang he approxmaon o (6) wh respec oτ, = y ( τ) Y( τ) YL ( τ) (8) The ollowng derenal-algebrac are ormed by equang he rgh hand sde o he sae dynamc consrans () o he sae approxmaon n (8) a he LGL pons: YL ( τk) = ( Yk, Uk, τk;, ), ( k =,..., ) (9) DkY = ( Yk, Uk, τk ;, ), Dk = L ( τk ) () Le LGL Y be dened as RPM LPM Fgure. The dscrezaon pons and collocaon pons o hree pseudospecal mehods There are hree ypes o collocaon pons or he pseudospecral mehods: Legendre-Gauss (LG) pons, Legendre-Gauss-Lobao (LGL) pons and Legendre-Gauss-Radau (LGR) pons (as shown n Fg.). All Y LGL Y = () Y Then, he opmal conrol problem ()~(4) s ranscrbed no he ollowng nonlnear programmng problem (LP): mn J =Φ( Y( τ ), τ, Y ( τ ), τ ) () s.. LGL DkY ( Yk, Uk, τk;, ) =, ( k=,..., ) φ( Y( τ), τ, Y( τ), τ) =, CY ( k, Uk, τk;, ), ( k=,..., ) (3) Where he LP varables are ( Y,..., Y ), ( U,..., U ), and. 463

4 There are hree key properes o he Legendre pseudospecral mehod as ollow. Fs, he dscrezaon pons and he collocaon pons are same n he Legendre pseudospecral mehod. Second, he degree o he Lagrange polynomal, ha s used o approxmae he sae, s, and s one less han he number o dscrezaon pons. Thrd, he dynamc consrans are collocaed a LGL pons. B. GPM Consder he LG pons, ( τ, τ,..., τ ), whereτ >, and τ <+. Dene oher wo new pons τ = andτ + =.ex, le L ( τ ) be he Lagrange polynomal o degree gven by τ τ τ τ j j L ( τ ) =, ( =,..., ) (4) The sae, y ( τ ), s approxmaed by L ( τ ) as = = y( τ) Y( τ) Y L ( τ) (5) I s essenal o noe ha τ = s no a LG pon bu s used n sae approxmaon. ex, derenae he (5) wh respec o τ and gve a resul: = = y ( τ) Y( τ) YL ( τ) (6) The ollowng derenal-algebrac are ormed by equang he rgh hand sde o he sae dynamc consrans () o he sae approxmaon n (6) a he LG pons: YL ( τk) = ( Yk, Uk, τk;, ), ( k =,..., ) (7) DkY = ( Yk, Uk, τk ;, ), Dk = L ( τk ) (8) I s noed here ha he sae a he nal pon s no ncluded n he ormulaon, whch s approxmaed by he ollowng LG quadraure: Y( τ ) = Y( τ ) + w Y ( τ ) + k k k= = Yτ + wk Yk Uk τk k= ( ) (,, ;, ) (9) Where w k, k, are he LG quadraure weghs. LG Dene Y as Y LG Y = (3) Y Then, he opmal conrol problem ()~(4) s ransormed no he ollowng nonlnear programmng problem (LP): mn J =Φ( Y( τ ), τ, Y ( τ ), τ ) (3) + + LG DY k Y ( k, Uk, τ k;, ) =, ( k =,..., ) Y( τ+ ) Y( τ) wk( Yk, Uk, τk;, ) =, k= (3) φ( Y( τ), τ, Y( τ+ ), τ+ ) =, CY ( k, Uk, τ k;, ), ( k =,..., ) Where he LP varables are ( Y,..., Y ), ( U,..., U ), and. I s noed ha he nal conrol U and nal conrol U + are no obaned n he soluon o LP. There are hree key properes o he Gauss pseudospecral mehod as ollow. Frs, he dscrezaon pons are he LG pons plus he nal ponτ and he nal ponτ +. Second, he degree o Lagrange polynomal used o approxmae he sae s. Thrd, he dynamc consrans are only collocaed a he collocaon pons. C. RPM Consder LGR pons, ( τ, τ,..., τ ), Whereτ = andτ <. Dene a new ponτ + =, and le L () τ be he Lagrange polynomal o degree gven by + τ τ L ( τ ) =, ( =,..., + ) (33) τ τ j j The sae, y ( τ ), s approxmaed by L () τ as = + y() τ Y() τ Y L () τ (34) I s mporan o noe ha τ + = s no a LGR pon bu s used n sae approxmaon. ex, an approxmaon o he dervave o he sae y () τ s gven by derenang he (33) wh respec oτ, = + y () τ Y() τ YL () τ (35) The ollowng derenal-algebrac are ormed by equang he rgh hand sde o he sae dynamc consrans (3-) o he sae approxmaon n (3-5) a he LGR pons: + YL ( τk) = ( Yk, Uk, τ;, ), ( k=,..., ) (36) + DkY = ( Yk, Uk, τ;, ), Dk = L ( τk ) (37) LGR Dene Y as Y LGR Y = (38) Y + Then, he opmal conrol problem ()~(4) s ranscrbed 464

5 no he ollowng nonlnear programmng problem (LP): mn J =Φ( Y( τ), τ, Y ( τ+ ), τ+ ) (39) s.. LGR DY k ( Yk, Uk, τ;, ) =, ( k=,..., ) φ( Y( τ), τ, Y( τ ), τ ) = + + (4) CY ( k, Uk, τ;, ), ( k=,..., ) Where he LP varables are ( Y,..., Y ), ( U,..., U ), and. I s mporan o noe ha he nal conrol U s obaned n he soluon o he LP, whle he nal conrol U s al o solve. + There are hree key properes o he Radau pseudospecral mehod as ollow. Frs, he dscrezaon pons are LGR pons plus he nal pon τ +. Second, he degree o Lagrange polynomal used o approxmae he sae s. Thrd, he dynamc consrans are only collocaed a he LGR collocaon pons. V. STIMULATIO RESULTS Earh-Mars rendezvous msson n 3 s presened here. We assume ha, he spacecra nal mass s kg, and he low-hrus specc mpulse s 3 seconds. The npu power and ecency o he engne s assumed o be 6.5 kw and.65 respecvely. Summarzed as: Inpu power: P = 6.5 kw Specc mpulse: IsP = 3 s Engne ecency: η =.65 Inal mass: m = kg All compuaon were compleed on he PC o Penum(R) Dual-core CPU Each mehod ranscrbes he low-hrus opmzaon problem no an LP ha consss o 4 nodes, and he SQP algorhm s appled o solve he LP. The resuls are summarzed on Table II. I can be seen rom Table II: All he hree pseudospecral mehods can converge o he ermnal consrans wh he nal values provded n secon II, and he resuls are accuracy hghly. The opmal resuls o launch dae, arrval dae, and lgh me are close n hree pseudospecral mehods, whle he number o eraon and compuaonal me s deren rom each oher. I wll ake.78 seconds or he LPM o opmze he Earh-Mars rendezvous low-hrus rajecory, whch s much longer han he RPM seconds and he GPM seconds. Alhough he values o nal mass are close o each oher, he RPM ge he maxmum nal mass whch s one o he mos key eaures consdered n space mssons. Through he above comparson, we can conclude ha he RPM s he ases and opmal among he hree pseudospecral mehods. Thus on hs coun cases, he RPM s superor o he oher wo mehods. I should be noed ha alhough he compuaon me or he LPM s longer han he oher wo approaches, he nal and nal conrol s obaned n he soluon whle he RPM can only oban he nal conrol and he GPM can oban neher he nal conrol nor he nal conrol. Lasly, o gve an nuve comparson, Fg.3~5 shows he ranser orb, he me hsory o conrol and velocy respecvely by usng he LPM; Fg.6~8 shows he ranser orb, he me hsory o conrol and velocy respecvely by usng he GPM; Fg.9~shows he ranser orb, he me hsory o conrol and velocy respecvely by usng he RPM. TABLE II. COMPARISO OF THE THREE DIFFERET PSEUDOSPECTRAL METHODS Parameers LPM GPM RPM Deparure dae (YY/MM/DD) 3//. 5 3// // Arrval dae (YY/MM/DD) 5// // //3.447 Flgh me (day) Arrval hyperbolc excess velocy.8e e e-8 (km/s) Spacecra nal mass (kg) umber o eraon Compuaon me (s) y [au] Earh-Mars Rendezvous Orb x [au] Fgure 3. The Earh-Mars ranser orb(lpm) 465

6 conrol hsory conrol hsory ux uy uz me o lgh [day] - ux uy uz me o lgh [day] Fgure 4. The me hsory o conrol(lpm) Fgure 7. The me hsory o conrol(gpm) 3 3 velocy hsory [km/s] - - velocy hsory [km/s] vx vy vz me o lgh [day] Fgure 5. The me hsory o veloces(lpm) -3 vx vy vz me o lgh [day] Fgure 8. The me hsory o veloces(gpm) Earh-Mars Rendezvous Orb Earh-Mars Rendezvous Orb.5.5 y [au] y [au] x [au] Fgure 6. The Earh-Mars ranser orb(gpm) x [au] Fgure 9. The Earh-Mars ranser orb(rpm) 466

7 .5 conrol hsory velocy hsory [km/s] ux uy uz me o lgh [day] Fgure. The me hsory o conrol(rpm) -3 vx vy vz me o lgh [day] Fgure. The me hsory o veloces(rpm) VI. COCLUSIO The perormance o Legendre (LPM), Gauss (GPM), and Radau (RPM) pseudospecral mehods s Comparavely analyzed. Smulaon resuls show ha he pseudospecral mehod has good convergence and hgh precson n low-hrus rajecory opmzaon problem. By comparng hese hree mehods appled o opmze Earh-Mars rendezvous msson n 3, we can easly conclude ha he RGM s superor o he oher wo mehods. ACKOWLEDGMET The work descrbed n hs paper was suppored by he aonal aural Scence Foundaon o Chna (Gran o. 6735), Innovaon Funded Projec o Shangha Aerospace Scence and Technology (Gran o. SAST3), and he Fundamenal Research Funds or he Cenral Unverses (Gran o. S494). The auhor ully apprecaes her nancal suppors. REFERECES [] A. E. Peropoulos, J. M. Longusk and. X. Vnh, Shape-based analyc represenaons o low-hrus rajecores or gravy-asss applcaons, n Proc. Asrodynamcs 999, Proceedngs o he AAS/AIAA Asrodynamcs Conerence, Grdwood, AK, UITED STATES, 6-9 Aug.999, pp [] A. E. Peropoulos and J. M. Longusk, Shape-based algorhm or auomaed desgn o low-hrus, gravy-asss rajecores, J. Spacecra Rockes, vol. 4, no. 5, pp , 4. [3] B. J. Wall and B. A. Conway, Shape-based approach o low-hrus rendezvous rajecory desgn, Journal o Gudance, Conrol, and Dynamcs, vol. 3, no., pp. 95-, 9. [4] B. J. Wall, Shape-based approxmaon mehod or low-hrus rajecory opmzaon, n Proc. AIAA/AAS Asrodynamcs Specals Conerence and Exhb, Hawa, 8-, Augues. 8, pp. -9. [5] A. E. Peropoulos and J. M. Longusk, Auomaed desgn o low-hrus gravy-asss rajecores, n Proc. AIAA/AAS Asrodynamcs Specals Conerence, Denver, CO, UITED STATES, 4-7 Aug., pp [6] J. T. Bes, Survey o numercal mehods or rajecory opmzaon, Journal o Gudance, Conrol, and Dynamcs, vol., no., pp. 93-7, 998. [7] C. R. Hargraves and S. W. Pars, Drec rajecory opmzaon usng nonlnear programmng and collocaon, n Proc. Asrodynamcs Conerence, Wllamsburg, UITED STATES, 8- Aug.986, pp. 3-. [8] H. Seywald, Trajecory opmzaon based on derenal ncluson, Journal o Gudance, Conrol, and Dynamcs, vol. 7, no. 3, pp , 994. [9] F. Fahroo and I. M. Ross, Pseudospecral mehods or nne-horzon opmal conrol problems, Journal o Gudance, Conrol, and Dynamcs, vol. 3, no. 4, pp , 8. [] D. Garg, W. W. Hager and A. V. Rao, Pseudospecral mehods or solvng nne-horzon opmal conrol problems, Auomaca, vol. 47, no. 4, pp ,. [] G. Elnagar, M. A. Kazem and M. Razzagh, The pseudospecral legendre mehod or dscrezng opmal conrol problems, IEEE Transacons on Auomac Conrol, vol. 4, no., pp , 995. [] L. H. Tu, J. P. Yuan and J. J. Luo, Opmal desgn o orbal ranser wh ne hrus based on Legendre pseudospecral mehod, Journal o Asronaucs, vol. 9, no. 4, pp , 8. [3] D. Benson, A gauss pseudospecral ranscrpon or opmal conrol, Ph.D. dsseraon, Deparmen o Aeronaucs and Asronaucs, Massachuses Ins. o Technology, Massachuses, 5. [4] G. T. Hunngon, Advancemen and analyss o a gauss pseudospecral ranscrpon or opmal conrol problems, Ph.D. dsseraon, Massachuses Ins. o Technology, Massachuses, 7. [5] H.B. Shang, P.Y. Cu and R. Xu, Fas Opmzaon o Inerplaneary Low-Thrus Transer Trajecory Based on Gauss Pseudospecral Mehod, Journal o Asronaucs, vol. 3, no. 4, pp. 5-,. [6] S. Kameswaran and L.T. Begler, Convergence Raes or Dynamc Opmzaon Usng Radau Collocaon, n Proc. SIAM Conerence on Opmzaon, Sockholm, Sweden, 5. [7] D. Izzo, Lamber's problem or exponenal snusods, Journal o Gudance, Conrol, and Dynamcs, vol. 9, no. 5, pp. 4-45, 6. [8] R. Sorn and K. Prce, Derenal evoluon a smple and ecen heursc or global opmzaon over connuous spaces, Journal o global opmzaon, vol., no. 4, pp ,