(2011) 34 (3) ISSN

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1 Ceriotti, Matteo and McInne, Colin () Generation of optial trajectorie for Earth hybrid pole itter. Journal of Guidance, Control and Dynaic, 34 (3). pp ISSN , hi verion i available at Strathprint i deigned to allow uer to acce the reearch output of the Univerity of Strathclyde. Unle otherwie explicitly tated on the anucript, Copyright and Moral Right for the paper on thi ite are retained by the individual author and/or other copyright owner. Pleae check the anucript for detail of any other licence that ay have been applied. You ay not engage in further ditribution of the aterial for any profitaking activitie or any coercial gain. You ay freely ditribute both the url ( and the content of thi paper for reearch or private tudy, educational, or not-for-profit purpoe without prior periion or charge. Any correpondence concerning thi ervice hould be ent to the Strathprint adinitrator: trathprint@trath.ac.uk he Strathprint intitutional repoitory ( i a digital archive of Univerity of Strathclyde reearch output. It ha been developed to dieinate open acce reearch output, expoe data about thoe output, and enable the anageent and peritent acce to Strathclyde' intellectual output.

2 Generation of Optial rajectorie for Earth Hybrid Pole-Sitter Matteo Ceriotti * and Colin R. McInne Univerity of Strathclyde, Glagow G XJ, United Kingdo Abtract A pole-itter orbit i a cloed path that i contantly above one of the Earth pole, by ean of continuou low thrut. hi work propoe to hybridize olar ail propulion and olar electric propulion (SEP) on the ae pacecraft, to enable uch a pole-itter orbit. Locally-optial control law are found with a ei-analytical invere ethod, tarting fro a trajectory that atifie the pole-itter condition in the Sun-Earth circular retricted three-body proble. hee olution are ubequently ued a firt gue to find optial orbit, uing a direct ethod baed on peudopectral trancription. he orbital dynaic of both the pure SEP cae and the hybrid cae are invetigated and copared. It i found that the hybrid pacecraft allow aving on propellant a fraction. Finally, i it hown that for ufficiently long iion, a hybrid pole-itter, baed on id-ter technology, enable a conitent reduction in the launch a for a given payload, with repect to a pure SEP pacecraft. a Acceleration, / A Area, Noenclature f d g, h i Ditance fro the Earth, AU otal thrut Coefficient for non-ideal ail Generic tep I p Specific ipule, g Standard gravity acceleration on Earth urface, 9.8 J Cot function * Reearch Fellow, Advanced Space Concept Laboratory, Departent of Mechanical Engineering, AIAA Meber Director, Advanced Space Concept Laboratory, Departent of Mechanical Engineering, AIAA Meber Page of 35

3 Ma, kg gibal hruter gibal a, kg pl Payload a, kg prop Propellant a, kg rad Radiator a, kg SEP SEP thruter a, kg tank Propellant tank a, kg n thruter Nuber of thruter ˆ ˆn P r r Direction of olar ail acceleration Solar ail noral unit vector Power Poition vector Reflectivity coefficient rˆ, ˆ, ˆ Unit vector defining axe of reference frae (B) t ie, d t iion Duration of the iion ˆt û u U v e Solar ail tangent unit vector (in the plane of Sun vector) SEP thrut vector, N Generic unit vector Control vector Effective potential Exhaut velocity, / v w W x xˆ, yˆ, z ˆ Velocity vector Weight Energy flux denity of the Sun at AU, State vector 367 W Unit vector defining axe of reference frae (A) Cone angle, deg Page of 35

4 Lightne nuber Clock angle, deg eq Obliquity of equator on the ecliptic (3.5 deg) t ie increent Efficiency Angle between olar ail acceleration and Sun vector Dienionle a of the Earth, in Sun-Earth yte Sail loading, kg/ 3 Critical ail loading (.53 kg for Sun-Earth yte) Angular velocity, rad/ Subcript Referred to beginning of the iion Referred to the larger priary a (Sun) Referred to the aller priary a (Earth) a f gue ax SEP F Referred to total acceleration Referred to final tie or end of the iion Firt gue Maxiu Referred to the olar ail Referred to SEP engine Referred to SEP thrut vector Referred to the thin fil olar cell Supercript Optial B A, Expreed in reference frae (A), (B) ˆ Unit vector Differentiation with repect to tie Page 3 of 35

5 Introduction Non-Keplerian orbit (NKO) are trajectorie obtained in a perturbed and/or controlled two-body Newtonian gravitational field. A pacecraft can achieve a NKO by ean of a continuou control acceleration, which force the natural dynaic of the yte []. A particular kind of NKO i the pole-itter, in which a pacecraft i contantly above one of the Earth pole, i.e. lying on the Earth polar axi []. he pole-itter can provide a platfor for continuou, real-tie, ediu-reolution obervation of the Earth pole, with a full heipheric view. Currently, obervation of the Earth pole i perfored with one pacecraft or a contellation of pacecraft in highly-inclined low or ediu Earth orbit [3], and the heipheric view of the pole i recontructed through copoite iage that are ade of everal iage taken at different tie. hee polar iage copoite are ued to generate atopheric otion vector (AMV) and to identify tor yte. According to Lazzara [4], thee two application aong other would benefit fro a true pole-itter pacecraft. For exaple, both at the North Pole and South Pole, an interval of latitude exit in which AMV are not available, due to a gap in iaging between geotationary and polar orbiting atellite. Converely, the poibility of eeing the Polar Region with potential high patial and teporal reolution would iprove hort ter forecating and undertanding of atopheric phenoena. Another ignificant benefit of the pole-itter orbit would be for polar telecounication. It i well known that line-of-ight telecounication to conventional pacecraft in geotationary orbit i not poible at high latitude. Although the ditance of the pacecraft fro the Earth could preclude highbandwidth telecounication, a pole-itter will alway have a pole in ight, providing a continuou flow of data with, for exaple, cientific South Pole tation [5]. he ae pacecraft could thu accoplih both real-tie obervation and telecounication tak over the Polar Region. hi will be a key iue in the future, a change to the arctic ice pack open navigation channel for hipping. he concept of having a pacecraft in an orbit that allow direct-link telecounication and viibility of one of the Earth pole ha been propoed [, 5, 6], but only partly invetigated. hi paper tudie optial pole-itter orbit in oe detail and propoe to hybridize olar ail propulion and olar electric propulion (SEP) on the ae pacecraft, to enable a near-ter pole-itter iion. Hybridizing thee two propulion yte i a recent idea [7], neverthele reearch i flourihing in thi field, invetigating it potential for novel, intereting application: artificial equilibria above L in the Sun-Earth yte for Earth obervation [8], optial interplanetary tranfer to Venu and Mar [9, ], and diplaced periodic orbit in the Earth-Moon yte []. Finally, JAXA wa recently ucceful in deploying the ail of the firt hybrid olar ail deontrator, IKAROS []. he reaon for thi interet i due to the fact that in the hybrid yte, at the cot of increaed pacecraft and iion deign coplexity, the two propulion yte copleent each other, cancelling their reciprocal diadvantage and liitation. In principle, SEP can provide thrut in Page 4 of 35

6 alot any direction (a long a the exhaut flow doe not interfere with other pacecraft coponent), in particular providing an acceleration coponent toward the Sun, that the ail cannot generate. Siilarly, the hybrid pacecraft can be een a an SEP pacecraft, in which an auxiliary olar ail provide part of the acceleration, enabling a aving of propellant a, and lowering the deand on the electric thruter, poibly with oe interval in which it could be turned off. In thi work, an extenion of both the iple, contant altitude pure-sep pole-itter orbit propoed by Driver [], and the hybrid artificial equilibria propoed by Baig and McInne [8] i propoed. With repect to the forer, thi work extend Driver analyi by conidering hybrid propulion and developing a ethod for deigning true optial poleitter orbit. In the latter work [8], the pacecraft i tationary at an artificial equilibriu point above the Earth pole at the uer or winter oltice only. herefore, the full heipheric view of the pole only occur at one particular tie in the year, while at other epoch the polar region i partially out of ight. Intead, the new, continuou pole-itter orbit preented in thi paper are a practical realization of the olar ail pole-itter orbit propoed qualitatively by Forward [6]. Hybrid propulion require new failie of optial orbit with repect to the ue of claic SEP, a the preence of the olar ail generate a continuou acceleration without propellant conuption, and o introduce additional control variable, a well a attitude contraint. rajectory deign for hybrid low-thrut propulion wa addreed for co-planar, circle-to-circle interplanetary tranfer in the two-body proble [9, ]. Under thee auption, the author find the optial teering law uing an indirect approach. However, thi technique cannot be applied to the pole-itter proble, ainly becaue of the path contraint that force the pacecraft to be above the pole at any tie. herefore, a novel ethod for iniu-fuel pole-itter orbit in the circular retricted three-body proble (CR3BP) i propoed in thi paper. After a brief decription of the hybrid pacecraft dynaic (ection I), ection II preent a technique for generating quai-optial periodic orbit, baed on an invere control ethod, that iniize the thrut pointwie. hee orbit are ubequently ued a a firt gue for olving the optial control proble nuerically, and the reulting optial olution are preented for different ize of ail and iion requireent. Finally, ection III briefly copare, with a preliinary a budget, the a of a pure SEP pacecraft and the hybrid pacecraft in different cenario, for a given payload. I. Equation of Motion he circular retricted three-body proble decribe the otion of an infiniteial a (the pacecraft) under the gravitational attraction of two large ae (priarie), which are orbiting around their coon center of a in circular otion. he odel neglect the influence of the pacecraft on the otion of the priarie. A i coon, a Page 5 of 35

7 rotating reference frae i conidered, in which the origin i at the center of a of the yte, the ˆx axi i collinear with the two priarie, pointing toward the aller a, the ẑ axi i aligned with the angular velocity of the yte, and the ŷ axi coplete the right-handed Carteian reference frae (ee Fig. ). hi reference frae will be referred to a (A). Note that thi frae i not inertial. In the following, all the vector are expreed in thi frae, unle pecified by a different upercript. herefore, the upercript (A) will be oitted when not trictly needed. he ae of the two priarie are denoted and (with ), and z ˆ the angular velocity of the yte. he equation that decribe the otion of the pacecraft in thi yte are: r r r a () U where r i the poition vector, a i the acceleration due to external, non-gravitational force (i.e. thrut), and the effective potential i U r r x y r. he a ratio i defined a. In the following, the yte of equation will be ued in their canonical non-dienional for, that i auing and the unit of ditance being the eparation of the two priarie. With thee auption, the poition along the ˆx axi of i, and the poition of i. In thi work, the priary i the Sun, and the priary i the Earth: for thee bodie, he two vector r r and r r are the poition of the pacecraft with repect to the two priarie and repectively ( Fig. ). For a hybrid propulion pacecraft, the acceleration a i ade of two contribution: one due to the olar radiation preure on the pacecraft ail, a ; the econd i provided by the thrut of the olar electric propulion (SEP) yte, a. he two fraction of the total acceleration vector will be decribed in the following ubection. ẑ eq t Winter oltice r ŷ r r ˆx (Earth) (Sun) Fig. Reference frae (A) of the retricted three body proble, and apparent preceion of the Earth polar axi due to rotation of reference frae. Page 6 of 35

8 A. Solar Sail Solar ailing i the exploitation of olar radiation preure produced by photon reflecting on a urface of the pacecraft (the ail) to produce a force, and thu an acceleration, without uing any propellant. In thi work, a nonperfect olar ail force odel, which take into account pecular reflection and aborption of the photon, will be ued. Scattered reflection and eiion by re-radiation are neglected. A detailed decription of thi ail odel i found in Ref. [8]; here an overview i preented for ake of copletene. According to Ref. [3], and including the auption entioned before, an approxiate expreion for the olar ail acceleration due to the Sun, acting on a ailcraft of a and ail area A, can be odeled a: a co ˆ in ˆ g h co n t () r he acceleration i expreed through it two coponent: one noral to the ail, along ˆn, and one perpendicular to the ail, in the plane containing the Sun vector ˆr, along ˆt. he angle between the direction of the Sun vector ˆr and the noral to the olar ail ˆn i known a the cone angle, and it hold that co nr. ˆ ˆ he two coefficient g and h are function of the optical propertie of the ail urface. Ideally a olar ail i ade of a light, thin fil of highly reflective aterial, ince axiizing it reflectivity axiize the agnitude of the force. However, in the hybrid pacecraft, it cannot be aued the ail aterial i unifor. In fact, part of the ail i covered with thin fil olar cell (FSC), which provide the neceary electrical power for the SEP engine. he FSC have different optical propertie than the ret of the ail (it reflectivity i lower, a part of the light i aborbed and converted into olar power). An area A of the ail i covered with highly reflective aterial, with reflectivity coefficient r, while the reaining urface A F A A i covered with FSC with reflectivity coefficient F r. Under thee auption, AF g r rf r A AF hr rf r A (3) he paraeter i the yte lightne nuber at beginning of life, defined a A (4) where i the a of the pacecraft at an initial reference tie t, and the critical ail loading.53 kg i y 3 a contant for the Sun-Earth te. If the unit vector ˆ i defined in the direction of a, Eq. () can be rewritten a Page 7 of 35

9 a a g h ˆ (5) ˆ co in co r he direction of ˆ with repect to the Sun-line i identified by the angle (ee Fig. ). It i poible to find a relationhip between and [8], uch that the acceleration Eq. (5) can be expreed a a function of the cone angle only: tan g h tan g htan (6) An intrinic liitation of olar ailing i that the thrut force can only point away fro the Sun. hi i equivalent to tating that the noral vector ˆn cannot be directed toward the Sun, which tranlate into the attitude contraint: (7) o uniquely define the direction of ˆ in pace, it i ueful to define a new reference frae, naed (B), depending on the poition of the pacecraft, defined a: rˆ ˆ zˆ rˆ ( B): zˆ rˆ ˆ ˆ ˆ r A vector defined in the reference frae (B) can be expreed in frae (A) conidering the tranforation: A ˆ ˆ ˆ u r u B (8) where ˆ ˆ ˆ r i the rotation atrix fro (A) to (B). A dicued, the cone angle define the pitch attitude of the olar ail with repect to the Sun direction ˆr. he attitude of the ail around thi vector i deterined through the o called clock angle : thi i the angle, eaured around ˆr, of the coponent of ˆn perpendicular to ˆr, tarting fro the direction of ˆ (ee Fig. ). he Carteian coordinate of the unit vector ˆn and ˆ, expreed a a function of their cone and clock angle, can be defined in frae (B) by: co B n ˆ in in ; in co co B ˆ in in (9) in co Page 8 of 35

10 ˆ rˆ ˆ ˆ zˆ rˆ zˆ rˆ ˆn ˆ ˆr Sail ˆr ˆr Fig. Frae (B), ail noral (n) and ail acceleration () cone and clock angle. A reflectivity of r.9, r.4 for the ail and the FSC repectively [ 7] will be aued. Alo, the FSC F occupie 5% of the total area of the ail, hence AF A.5. hi i etiated to be a conervative value, baed on previou tudie [8]. he actual area of the thin fil olar cell can be deterined once the requireent on the thrut are defined, by izing the power ubyte. he analyi will be perfored for a et of value of the lightne nuber, ranging fro (pure SEP) to., that repreent a future etiation of ail perforance. A value of.5 can be aued for a near-ter yte [4]. B. SEP hrut he thruter i aued to be teerable and can provide an adjutable thrut force. he direction of the thrut vector can be defined through the unit vector u, i.e. pacecraft acceleration i therefore: ˆ ˆ u. he relation between the thrut and the reulting a ˆ ˆ au u () Analogouly to ˆn, the unit vector u can alo be expreed in ter of it cone and clock angle,. hee angle are defined in the ae way a for the olar ail; therefore, in frae (B): ˆ co B u ˆ in in () in co In the cae of the olar ail, the paraeterization through the cone and clock angle i ueful due to the relationhip between the cone angle to the agnitude of the force. In the cae of the SEP thrut, the choice i dictated by having the ae paraeterization for the two part of the acceleration. A pecific ipule of Ip 3 i conervatively aued, baed on current ion engine technology (exiting NSAR/DS [ 5] or EADS/Atriu RI-X [6]) that can provide level of thrut uitable for the pacecraft and Page 9 of 35

11 iion under conideration. Higher value of pecific ipule can be achieved with current technology, for exaple the FEEP thruter can provide up to,, but the thrut i liited to N [7]. II. Pole-Sitter Orbit Deign A pole-itter pacecraft, during the operation phae of it iion, i contantly aligned with the polar axi of the Earth. If preceion of the equinoxe and nutation are neglected, the polar axi of the Earth doe not change it direction while the Earth i orbiting the Sun. herefore, in the rotating reference frae (A), the polar axi rotate with a otion of apparent preceion. It angular velocity i the oppoite of that of frae (A), or. herefore the polar axi pan a full conical urface in frae (A) every year (ee again Fig. ). he cone half angle i the tilt of the axi relative to the ecliptic, i.e. 3.5 deg. herefore, the pole-itter hall follow the Earth polar axi, and decribe a -year-periodic eq orbit. he equation of otion of the hybrid propulion pacecraft, in reference frae (A), can be found tarting fro Eq. () and ubtituting the two acceleration due to the ail (Eq. (5)) and SEP (Eq. ()). An additional equation i required that relate the a conuption with the thrut force. By introducing the tate vector differential equation can be written a a firt order yte: r v A A x U a ˆ ˆ v v au v e x r v, the () e Ipg where i the rate of change of pacecraft a and v i the exhaut velocity. he dynaic of the pacecraft Eq. () are to be contrained to follow the apparent preceion of the polar axi, and hence aintain the pole-itter condition. It i aued that the initial tie t coincide with the winter oltice, and therefore at any generic intant of tie t the pole-itter pacecraft i on the cone at poition: ineq cot ineqin d t co d t r t d t t (3) eq where d t i the ditance fro the center of the Earth, and i in general a continuou function of tie. Without lo of generality, due to the yetry of the proble in the ˆx - ŷ plane, the North Pole cae i conidered in thi work. Note that in a pure SEP pacecraft, the ter a ˆ A in the yte in Eq. () i null, and thu the coupling between the equation of the a flow and the equation for rv, i through the acceleration a, that i defined a a function of Page of 35

12 the control through a. However, a long a the SEP yte can provide the neceary thrut (i.e. the thrut i not aturated), then the yte in Eq. () can be caled in a, without affecting the evolution of the other tate. herefore, the optial olution in ter of r t, vt can be found uing a unitary initial a. he actual evolution of the a over tie for a given pacecraft i found by ultiplying the a fraction by the initial a of the pacecraft. hi ean that the initial a i only a caling factor for the ae equation, and doe not affect the otion of the pacecraft. Furtherore, if a periodic orbit i conidered, and until the propulion yte i able to provide the neceary peak thrut, the propellant a fraction for each orbit period doe not depend on the initial a. hi i not the cae for the hybrid pacecraft: in fact, the pacecraft a i coupled with the equation of otion through the ter a, which depend explicitly on the a, and the yte cannot be caled. In fact, while with SEP the agnitude of the thrut can be varied to copenate for the a reduction, the agnitude of the thrut generated by the ail cannot be controlled independently of it direction. hi ean that the acceleration provided by the ail, for a given cone angle, i higher a the a decreae, and the propellant a fraction needed for each orbit period i different depending on the actual value of the initial a in that period. In thi work, iniu fuel pole-itter orbit will be deigned, by olving an optial control proble. hi require a firt gue that i accurate enough to allow the optiizer to converge quickly and oothly to a locally optial olution. he firt gue will be generated uing an invere ethod, a decribed in the following ubection. A. Invere Method An invere ethod i propoed in thi work to generate feaible, ub-optial pole-itter orbit. Invere ethod have been ued in different area of control deign. Although the definition of invere ethod i quite broad, and ubtantially different approache are found in literature, the coon background to all of the i that the control hitory i found by inverting the equation of otion, once the evolution of the tate ha been pre-aigned. Invere ethod have been ued for aircraft flight control (ee Ref. [8] and other work referenced therein), and for controlling robotic ar [9]. In thi work it i hown that the pacecraft control can be obtained along a pre-aigned orbit, by iniizing the SEP thrut pointwie.. hrut Vector Optiization In thi ubection the following proble i addreed: aigned a kineatic law for the pacecraft otion through the poition r t, a function of tie, find the control u t at each intant of tie that enable that orbit. At each generic intant of tie t, the total acceleration needed a can be coputed fro Eq. (), in fact all the other ter are Page of 35

13 deterined when r t (and it tie derivative) i given. he acceleration can alo be expreed in odulu, cone and clock angle in the following way. he odulu i iply: a r r U (4) he cone and clock angle a and a can be found rewriting â in frae (B): co a in in rˆ ˆ ˆ aˆ aˆ B ina co a a a A (5) Inverting Eq. (5) it i found that: co a a aˆ B r B B aˆ aˆ tan, in a (6) in a in which the four-quadrant invere tangent function i ued. Finally, the agnitude of the thrut force needed (which can be provided by both propulion yte) i iply f a. Note that if no olar ail i ued, then the thrut hall be provided by the SEP yte copletely, and therefore the control are iply f,, a. If at any tie f exceed the axiu SEP thrut available, then the a elected orbit cannot be followed with a pure SEP yte: either another orbit i to be choen, or an additional propulion yte (e.g. ail) i to be ued. It can alo be noted that the a of the pacecraft i not contant throughout the orbit, but i a function of tie, and can be coputed knowing the previou control hitory. SEP-only olution are not optial for the hybrid pacecraft, in ter of propellant a conuption, ince they do not exploit the olar ail. A different approach can be ued: ince the kineatic i given, and o i the total acceleration, the attitude of the olar ail can be found, uch that the SEP thrut i iniu at each intant of tie. he agnitude of the SEP thrut can be expreed a a function of the ail cone and clock angle: a, aa ˆ in which a i known through Eq. (), and the ter a ˆ can be coputed uing Eq. (5), (6), and (9), and depend on the ail cone and clock angle, and the pacecraft a. herefore, at each point in the trajectory, and for a given a, the ail angle, can be deterined by olving the NLP proble: arg in, a (7) Page of 35

14 If the a a a function of tie i known, then for each point on a given trajectory, the control can be coputed by olving the proble defined by Eq. (7). It wa deotrated analytically in [8] that the iniu-thrut olution i uch that, in the general cae: and ince a i known through (6), the proble can be reduced to: a (8) a a arg in, (9) he iniization of Eq. (9) i olved nuerically, uing an SQP ethod [] ipleented in the MALAB function fincon. o prevent the poibility of converging to a local iniu, 4 different tarting point are ued, pread in the interval,. Note that the nuerical olution of the proble in Eq. (9) iplie that a cloed for olution of the invere ethod i not available. Once, are found, the other control can be eaily coputed: the odulu of the SEP thrut i iply given by a; it direction can be found conidering the SEP acceleration vector a a, ˆ that can be expreed in a frae (B) and then the angle can be found through Eq. (6) applied to a. he ethod that i ued here to find the acceleration at each pecific point r t can be extended to find entire orbit. hi will be the ubject of the following ubection.. Extenion to Periodic Orbit Here one-year periodic pole-itter orbit are ought. he initial a of the pacecraft, at tie t, the winter oltice, i aued to be kg. Note that thi a in general i not the launch a, but the a reaining after perforing the tranfer and injection into the pole-itter orbit. hi part of the iion i not a ubject of thi paper. he control at ucceive intant of tie, together with the a conuption, can be approxiated by dicretizing the trajectory into everal point, each of which i paced by a tie interval t. At each point, control can be etiated through Eq. (7). Once the thrut i known, it can be aued that thi level of thrut reain alot contant for an interval t. herefore the differential equation for the a in yte Eq. intant t i to the following intant ti t ti : () can be integrated fro the i th tie t () i i v e Algorith i ued to copute the entire one-year periodic orbit. he final tie in nondienional tie. t f coincide with Page 3 of 35

15 Algorith Invere ethod for coputing control. : Set t t; 3: Do r t, r t, r t fro kineatic, 4: Given find total acceleration uing (4) and (6) 5: Find, with (8) and olving proble (9) 5: Find other control,, 6: Update a with (): t ve 7: Update tie: t t t 8: While t tf Firtly, the analyi with a pole-itter orbit that keep a contant ditance fro the Earth pole i conidered (Fig. 3 (a)). he kineatic of thi trajectory i iply the one in Eq. (3), with dt cont d. t t t eq d ˆx d eq d ˆx Fig. 3 a) b) Shape-baed pole-itter orbit: (a) Contant ditance fro the Earth; (b) ilted orbit. Uing Algorith, the locally-optial hybrid olution for the contant-ditance pole-itter i obtained. he trajectory of the pacecraft over one year i plotted in Fig. 4, uperipoed on the cone decribed by the Earth polar axi. he bold arrow are proportional to the total local acceleration a fro Eq. (), that need to be counterbalanced by the total thrut, for aintaining the orbit. he other vector field refer to the reference cae.5. he gray arrow repreent the acceleration of the olar ail a. heir direction i lightly tilted with repect to ˆn (not plotted), due to the non-ideal ail that wa conidered. Finally, the black, non-bold arrow repreent the SEP thrut. It can be noted that the gravitational acceleration i otly directed toward z ˆ, therefore the reulting thrut hall be in the oppoite direction. he olar ail thrut ha a coponent in zˆ, but it i accopanied by a ignificant coponent along xˆ, i.e. facing away fro the Sun. he SEP i therefore providing the iing coponent in coponent. z ˆ, but alo counteracting the reidual ŷ Page 4 of 35

16 Fig. 5 how the agnitude of the acceleration provided by the ail (a) and SEP (b), for different value of : the olid line refer to, i.e. a pure SEP yte with no ail, the long-dahed line to.5 and the hort-dahed line to.. he bold circled line i the agnitude of the gravitational force. In the cae of pure SEP, the SEP acceleration i obviouly the ae a the gravitational acceleration. In the hybrid cae, clearly the SEP acceleration i lower for higher value of. Additionally, fro the figure it i poible to ee that higher acceleration i required fro SEP around the uer oltice, roughly in the interval t.3,.7 year. he axiu thrut required i the rather high value of 7 N for the pure SEP cae, which goe down to 69 N and 46 N for increaing value of. he pacecraft a a function of tie i plotted in Fig. 6. Fro the figure, it i viible the a gain of the hybrid propulion cae with repect to the pure SEP pacecraft. hi plot alo highlight that the pacecraft a flow rate (and hence the SEP thrut) i alot contant: thi i due to the fact that the doinant gravitational ter i due to the Earth (which i contant at contant ditance), while the Sun and other force only caue light perturbation. Fig. 4 rajectory and thrut vector for =.5 over one year on a flat orbit d =. AU. Page 5 of 35

17 x -4 x -4 Acceleration, / = =.5 =. Acceleration, / = =.5 =. a gc a gc a) t, y b) t, y Fig. 5 Acceleration of the ail (a) and SEP (b) for different value of, over one year on a flat orbit d =. AU = =.5 =., kg Fig t, y Ma trend for different value of, over one year on a flat orbit d =. AU. A paraetric analyi can be perfored for orbit aintaining a contant ditance fro the Earth, a a function of the ditance itelf. Fig. 7 illutrate the propellant a fraction of the pacecraft in a contant-ditance pole-itter orbit, for one year, a a function of the ditance of the orbit, for three value of. hee orbit becoe extreely expenive when the ditance fro the Earth decreae, a the Earth gravitational attraction becoe predoinant with repect to the other force of the CR3BP. hi reult atche what wa found in []. It i alo intereting to note that the propellant conuption ha a iniu for ditance of about.7 AU,.75 AU and.8 AU, for the three value of. hee are therefore the optial ditance for a contant-ditance pole-itter orbit. Page 6 of 35

18 prop / = =.5 = d, AU Fig. 7 Propellant a fraction needed for one year on a flat, circular orbit, a a function of the ditance d fro Earth (initial pacecraft a = kg). Finally, Fig. 8 how the propellant a fraction needed for a pacecraft of kg to aintain an orbit at d.7 AU fro the Earth. hi graph i relevant becaue it highlight that a conitent gain in propellant a i obtained by adding a all ail to the pure SEP pacecraft: in fact, the lope of the curve i highet for all value of. A the lightne nuber increae toward very high value, the gain in propellant a for a given increae of becoe le. However, thi graph jutifie the invetigation of the hybrid pacecraft, een a a pure SEP yte with a all auxiliary ail. he trend of thi graph i the ae for different value of d. he effect of the ail on the total a budget of the pacecraft will be dicued later. Note that even if a higher ail produce a higher acceleration a, it direction i till contrained and related to the cone angle, and in general it i not aligned to the total acceleration vector a (ee vector in Fig. 4). herefore, a higher ail acceleration i in general accopanied by a greater acceleration coponent in an unwanted direction, which hall be copenated by uing the thruter. Page 7 of 35

19 prop / Fig. 8. Propellant a fraction needed for one year on a flat, circular orbit (d =.7 AU), a a function of the ail lightne nuber (initial pacecraft a = kg). Different failie of pole-itter orbit, which reduce propellant a, can be obtained by relaxing the contraint of a contant ditance fro the Earth. It i poible to reduce the thrut and better exploit the capabilitie of the olar ail by tilting the orbit uch that the required acceleration i toward the Sun, and thu it can be counterbalanced ore efficiently by the olar ail. A different et of hape will now be conidered, in which the ditance fro the Earth at the winter oltice d and at the uer oltice d are introduced a paraeter ( Fig. 3 (b)). he ẑ coordinate i varied between thee two value with a inuoidal law: co t d t d d d () If for exaple d. AU and d.8 AU are elected, the orbit in Fig. 9 i obtained. Fro the vector field of the force, it can be een that during uer the alignent of the gravitational force i ore favorable. he plot of the acceleration a a function of tie (Fig. ) confir thi: while the ail provide ore or le a contant acceleration through the year, the SEP acceleration drop coniderably around uer. hi reult in ignificant aving in propellant a. he axiu thrut required i at the beginning of the orbit, when the pacecraft a i higher and the ditance fro the Earth i iniu, and range fro 43 N to 3 N for increaing value of. Note that thi lat value i lower than the one found for the contant ditance orbit, depite that the altitude at the initial point i the ae, and it i due to the different acceleration that the pacecraft require at that point. Fig. repreent the a a a function of tie. Over uer, the ail i exploited ot and thi i reflected in a lower propellant conuption. hi i to be copared with Fig. 6, where the fuel conuption wa alot contant. he a aving due to the tilting of the orbit i evident, although the average ditance of the pacecraft fro the Earth Page 8 of 35

20 during an orbit i higher. It i worth noting that the propellant a aving i not only due to the increaed ditance of the pacecraft during uer, but alo to the favorable alignent of the force. In fact, tilting the orbit in the other direction (i.e. d.8 AU and d. AU ) produce a uch wore reult in ter of propellant conuption. herefore it can be concluded that, in general, it i cheaper to oberve the North Pole fro a horter ditance in winter than in uer. Fig. 9 rajectory and thrut vector for =.5 over one year on a tilted orbit d =. AU, d =.8 AU. x -4 = =.5 =. a gc x -4 = =.5 =. a gc Acceleration, / Acceleration, / a) t, y b) t, y Fig. Acceleration of the ail (a) and SEP (b) for different value of, over one year on a tilted orbit d =. AU, d =.8 AU. otal acceleration i alo plotted on both graph. Page 9 of 35

21 , kg = =.5 = t, y Fig. Ma trend for different value of, over one year on a tilted orbit d =. AU, d =.8 AU. 3. Reark he olution generated with the propoed invere ethod are near-optial, in the ene that the thrut to aintain a given orbit i iniized at each intant of tie. herefore, for the pecific path conidered, the control profile that wa found i the one that allow iniu propellant conuption. If, for exaple, a payload require a contant ditance fro the Earth throughout the year, then the orbit i defined. Note that in the cae where the kineatic i fully deterined, it wa verified by everal nuerical experient that Algorith provide olution that are extreely iilar to thoe that can be found by olving an optial control proble, iniizing the propellant a, and keeping the orbit fixed. However, if there i no contraint or requireent on the orbit, but for exaple only a range of ditance i pecified (depending for exaple on the intruent requireent) then the ue a priori of a pecific orbit cannot be jutified. A dicued, different orbit could affect the aount of thrut needed at each intant of tie, and thu the propellant a over one year. Due to the contraint of the pole-itter pacecraft, all poible orbit can be decribed through the ditance fro the Earth a a function of tie during one year, dt. Aong thee, there hall be at leat one particular -year-periodic orbit, for any given value of, that iniize propellant a while aintaining the pacecraft above the pole. A een in Fig. 7, the propellant a fraction i not onotonically decreaing with the ditance fro the Earth, but it ha a iniu. Although the figure refer to the cae of flat orbit, the ae trend i een for tilted orbit. hi fact indicate that orbit do not becoe indefinitely cheap while increaing their ditance. herefore, it can be expected to find optial orbit without liiting the axiu ditance fro Earth. If the kineatic of the orbit are not known a priori (that i equivalent to ay that the ditance function dt i not given), then the ei-analytical procedure ued o far i not applicable, and the olution, that include deterining the optial trajectory, i found by olving an optial control proble. he optial control proble i one of finding the Page of 35

22 control hitory of a given dynaical yte, uch a to optiize a given cot function, while atifying the dynaic itelf and poibly other contraint, which can include initial/final condition. In the following ection, the olution will be found through a direct ethod. All direct ethod need to be initialized with an initial (or firt-gue) olution, which i cloe enough to the optial one to guarantee convergence; orbit and control hitorie found with the invere ethod will be ued for thi purpoe. B. Optial Orbit he proble i to find optial periodic orbit with repect to a given cot function, pecifically to iniize the propellant conuption, or axiize the final a. Given that a pole-itter orbit i fully deterined through the function d t, a poible approach i to obtain an equation for the dynaic of dt. If the poition vector i expreed through Eq. (3), then the three-coponent dynaic for r t in Eq. () can be tranfored into three calar equation, all involving,, d t d t d t. One of thee equation can be ued for decribing the dynaic of d, while the other two repreent differential contraint on the control, neceary for eeting the pole-itter condition (i.e. aintain the pacecraft above the pole). In thi work, rather than uing thi approach, the dynaic in Eq. () i ued, and then the pole-itter contraint are enforced explicitly, to enure that r t i copatible with Eq. (3). A the proble will be olved for one orbital period ( t ). he fixed initial tate ( t ) are:, f ry t t he firt condition, together with proper bound on r x, r z, guarantee that the tarting point i at the winter oltice. Other initial tate are free. herefore, the optiizer i allowed to ove the initial point on the xˆ y ˆ plane, pecifically raie or lower it, a well a the initial velocity. he periodicity of the olution i et requiring that the final tate, except the a, are the ae a the correponding initial tate: tf t, tf t r r v v Note that, depite the choice of the winter oltice a the initial point, ince the orbit i periodic, it doe not iply that the injection fro tranfer hould necearily happen in thi point. Since the pacecraft ha to tay along the polar axi of the Earth at each tie t t, two path contraint are introduced, uch that: tan, y,, x r r r tan, x, y, z eq r r t () Page of 35

23 he forer define the angular poition of the pacecraft around the Earth in the xˆ yˆ plane, while the latter contrain the ditance of the pacecraft eaured fro an axi paing fro the centre of the Earth and parallel to ẑ. Control are tranfored into Carteian coordinate, to prevent proble with abiguity and periodicity of angular variable, which can arie []. herefore, the thrut i decribed through it Carteian coponent along rˆ, ˆ, ˆ, repectively r,, he attitude of the olar ail i analogouly decribed through the coponent in the ae frae of the unit vector ˆn. It three coponent n, n, n r are ufficient to deterine the direction of ˆn, and it agnitude i not relevant. Even though a path contraint i enforced, to guarantee the uniquene of the olution: n n n. r he ail attitude contraint in Eq. (7) i enforced with the control bound n. Finally, bound are et on the r poition tate, uch that the pacecraft doe not exceed a axiu ditance d ax fro Earth. Different cot function will be ued and will be pecified in the following ubection: the overall ai i to iniize the propellant conuption. he optial control proble i olved nuerically uing a direct ethod baed on peudopectral trancription, ipleented in the tool PSOP. PSOP i coded in C++ by Becerra [] and i free and open ource. PSOP can deal with endpoint contraint, path contraint, and interior point contraint. Bound on the tate and control can be enforced, a well a interval for initial and final tate [3]. It ake ue of the ADOL-C library for the autoatic differentiation of objective, dynaic and contraint function. he NLP proble i olved through IPOP [4], an open ource C++ ipleentation of an interior point ethod for large cale proble. C. Reult he optiized olution preented here are for the three value of,.5,.. he reult ue 6 collocation point in the one year iion duration. It wa aeed that a higher nuber of point did not reult in any ignificant change in the tate or control hitory. he convergence of PSOP i eaier and fater uing a all nuber of point: therefore, all the following olution were found by iteratively increaing the nuber of point, fro to 6, and uing the optial olution found at one iteration a a firt gue for the following one. he next ubection will preent optial orbit that iniize the propellant conuption of the pacecraft over one period, with no contraint on the pacecraft ditance fro the Earth. In the one after, intead, a trade-off will be preented, quantifying the additional propellant needed to atify given contraint on the ditance. Page of 35

24 . Miniu Propellant Conuption In thi ection, propellant conuption will be iniized; therefore the cot function i iply: J t f that i, to axiize the final a after one period, or one year. Note that linear iniu-fuel proble, with contrained control agnitude, uually reult in a bang-off-bang control [5]. In thi cae, uch a control i not expected due to the path contraint on the tate: in general, a continuou control i needed to aintain a pole-itter orbit. Even if a liitation on the axiu agnitude of the thrut i taken into account, the SEP thrut law doe not preent an on-off tructure. However, it will be hown that the SEP yte could be witched off in oe arc, if the olar ail i ufficient to aintain the pole-itter condition. f In thi ection contraint are not enforced on the ditance of the pacecraft fro the Earth. he value d ax wa et to the rather high value of. AU. he ai i to find the cheapet orbit in ter of propellant conuption. Fig. (a) repreent the three different optial trajectorie, obtained with PSOP, for each value of ; Fig. (b) plot the ditance of the pacecraft for the Earth. he firt point to note i that all the optial orbit are naturally yetric with repect to the xz ˆˆ plane. Furtherore, the pure SEP olution would optially follow an orbit that i alo yetric with repect to an ideal plane paing through the Earth and perpendicular to ˆx. On the other hand, the optial orbit get cloer to the Earth in winter and farther in uer, a the lightne nuber of the olar ail increae. he ditance can even double fro winter to uer for a olar ail with a lightne nuber of x 6 4 = (pure SEP) =.5 =. 3.5 d, k 3.5 a) b) t, y Fig. Miniu propellant a pole-itter olution, for three different value of, during one year. (a) rajectorie. (b) Ditance fro the Earth (in k). he acceleration provided by the olar ail and the SEP yte i plotted in Fig. 3. Conidering the pure SEP cae in the analyi, two thrut region can be identified: one acro the uer oltice and one acro the winter oltice. Page 3 of 35

25 Due to the elected tie-frae, tarting at the winter oltice, the latter region i plit in the plot. In thee two region the SEP acceleration i alot contant. he two region are eparated by two hort arc, in the autun and pring equinoxe, in which the thrut becoe very low, and the pacecraft otion along thrut happen at the axiu ditance fro Earth (copare with Fig. (b)). ẑ i inverted: the intant with lowet If a olar ail i added (ee the cae.5 in Fig. 3), the SEP acceleration required in the two thrut arc decreae, and thrut region around the uer oltice becoe horter, while two non-thruted arc expand, increaing the tie when the thruter i off, and the olar ail i alot ufficient to aintain the pacecraft orbit. Eventually, for a uitably large olar ail (.), the thrut region around the uer oltice diappear, while the two ballitic arc erge: becoing only one, lating fro pring to autun approxiately. Hence the tilting of the orbit can be explained in the following way: having the pacecraft high in uer and a low in winter allow the orbit to exploit, for a coniderable aount of tie, the olar ail and the natural force only to drive the pacecraft back to the winter oltice point and cloe the orbit. he a a a function of tie i repreented in Fig. 4. In thi figure the ubtantial aving in propellant that i given by a relatively all ail i evident: for a -year orbit, the propellant a decreae fro 58 kg (for the pure SEP) to 97 kg (.5 ). Confiring the reult found before (ee Fig. 7), a reduction of propellant a i obtained by adding a relatively all ail. However, the aving in propellant a i le than proportional to the lightne nuber. Fig. 5 repreent the angle between the olar ail noral and the SEP thrut. he purpoe of thi plot i twofold. he firt i to how that, throughout the whole iion, the required thrut i never lying in the plane of the olar ail. hruting along the ail would poe evere proble due to exhaut gae ipinging on the ail itelf. he econd ai i to underline that the rotation of the thrut vector with repect to the ail i liited. If it i aued that the olar ail i fixed with the pacecraft, then it orientation can be changed by changing the attitude of the pacecraft. However, the SEP thruter cannot be fixed with repect to the pacecraft, a the thrut direction need to vary; intead, it hall be ounted on a gibal. he rotation needed i le than degree, in the wort cae (.), hence a iple echani can be ufficient to guarantee the required thruter pointing. Finally, able uarize oe feature of the three optial orbit (for each conidered value of ) that were deigned. Note that all the olution found iply the ue of the thruter. A pure ail iion can be een a a particular cae of the hybrid iion, in which no thrut i needed. Since the optial orbit were found iniizing the propellant a, it i expected that a no-thrut olution would have been found if it exited, at leat for the range of, ditance Page 4 of 35

26 fro Earth and other technological paraeter conidered. No uch olution were found o that pure olar ail poleitter orbit are not believed to exit..5 x -4 = (pure SEP) =.5 =..5 x -4 = (pure SEP) =.5 =. Acceleration, /.5 Acceleration, / a) t, y b) t, d Fig. 3 Acceleration of the ail (a) and SEP (b) for three different value of, during one year Ma, kg = (pure SEP) =.5 = t, y Fig. 4 Ma a a function of tie, for three different value of, during one year = (pure SEP) =.5 =. Angle, deg t, y Fig. 5 Angle between the thrut vector u with repect to the olar ail noral n, for three different value of, during one year. Page 5 of 35

27 able Suary of characteritic of uncontrained optial orbit for three different value of ail lightne nuber. ie frae i one year; initial a i = kg. in d t, AU ax dt, AU t t f, kg prop ax t, N t Orbit Perforance rade-off In thi ubection the trade-off between an orbit with a particular feature, and the additional propellant a needed to aintain it i conidered. A poible, iportant requireent of the pole-itter pacecraft could be the axiu ditance fro the Earth. It ha been hown that fuel-optial orbit vary their ditance quite conitently, and thi can be an iue, for exaple, for guaranteeing a ufficient reolution for obervation of the Earth. herefore failie of optial orbit have been deigned, contraining the axiu ditance of the pacecraft, by gradually decreaing the value d ax. able preent the reult of the optial orbit for different value of d and.5. Note that for value of ax d ax lower than thoe conidered in the table, the orbit i ubtantially flat, and therefore the optial contrained olution coincide with the flat orbit found with the invere ethod, and the value of the final a after one year i the one repreented in Fig. 7. Note that the thrut peak value doe not change ignificantly a the orbit get cloer to the Earth, but the propellant needed i neverthele ore due to a higher value of the thrut required throughout the whole orbit. hi i due to the fact that the peak of thrut happen at the beginning of the orbit, at the winter oltice, and when the pacecraft a i highet. he ditance of the orbit fro Earth at the winter oltice i alot the ae a long a d ax.4 AU, and o i the thrut needed at that point. he failie of orbit that are obtained are plotted in Fig. 6. able Suary of optial orbit (iniu propellant a conuption over one year) obtained contraining the axiu ditance fro the Earth to d ax. Initial a i = kg, =.5. d ax, AU in dt, AU ax dt, AU t t f, kg prop ax t, N t Page 6 of 35

28 Fig. 6 A faily of optial orbit, contrained to everal value of d ax, for =.5. Another poible requireent for the pole-itter pacecraft payload could be to aintain a contant or quaicontant ditance fro the Earth. hi can be ueful, for exaple, for an intruent with a fixed focal length, in which the field of view cannot be varied, and thu a contant ditance fro the Earth i deirable uch that the intruent can be deigned for that ditance. It wa dicued already that if the orbit i flat, then the invere ethod can be ued to copute the optial control, and iniize the propellant a. In thi ection, however, a trade-off between the flatne of the orbit and the additional propellant a required will be perfored. It i aued that there i no particular requireent on the axiu ditance d ax. o achieve thi, in the cot function a weighed penalty that conider the oveent of the pacecraft along ẑ i introduced. If it i noted that a flat orbit i one in which the velocity along ẑ i null during the whole orbit, o a poible choice of the cot function could be: z w J f v dt. (3) By varying the weight w, a trade-off of propellant a againt a low average of the pacecraft velocity along ẑ i poible. Note that the cae w coincide with the optiization of the propellant a only. able 3 i a uary of the characteritic of the optial orbit found uing the cot function in Eq. (3), for everal value of the weight w. hee value have been deterined by trial and error, and conidering the relative order of agnitude of the two addend in the cot function (in non-dienional unit). A expected, the orbit flatten at the cot of oe additional propellant a. It wa alo found that it i ore expenive to flatten orbit when high value of are conidered. For value of w higher that thoe preented in the table, the orbit becoe ubtantially flat, tabilizing to a ditance of about.74 AU. A expected, thi value repreent the optial ditance for a flat pole-itter orbit, a found before, and coincide with the iniu of the curve plotted in Fig. 7. Page 7 of 35

29 able 3 Suary of optial orbit obtained iniizing a weighed u of propellant a and velocity in z. Different value of the weight w are conidered. Initial a i = kg, =.5. w in ax d t, AU f, kg prop ax t, N t d t, AU t t Fig. 7 A faily of optial orbit, for different value of the cot function weight w; =.5. III. Ma Budget Due to the additional coplexity of the hybrid pacecraft, it i intereting to ae whether thi yte allow a lower initial a for a given payload a pl. o thi ai, a preliinary a budget i preented here, for different iion cenario, conidering a pure SEP pacecraft and a hybrid one. For ake of coparion, the technological auption are baed on what wa choen in [8]. In that work, the author coputed the requireent for a pacecraft to be tationary in the Sun-Earth rotation frae, placed at.83 AU above the North Pole at the uer oltice (hence above the Lagrange point L ), a olar ail with.3, a payload a of kg, a SEP pecific ipule of 3, and a 5 year iion tiefrae. Here a iion in which d. 83 AU i conidered, therefore uing a faily of orbit like thoe in Fig. 6. he total pacecraft a can be plit a: prop tank thruter SEP gibal rad F pl n (4) where i the propellant a neceary for a given iion duration. he a of the tank i prop ax t iion tank [ 6]; the nuber of thruter i n, for redundancy and uch that a econd thruter can be ued. prop thruter after the perforance of the firt ha degraded (note that thi i a conervative choice: a higher payload fraction i Page 8 of 35