3-Dimensional Continuous-Thrust Trajectory Optimization for Electric Solar Wind Sails using Bezier Curves

P oyec o Fin de Ca e a
Ingenie ía de Telecomunicación
Fo ma o de Publicación de la Escuela Técnica
Supe io de Ingenie ía
Au o : F. Ja ie Payán Some
Tu o : Juan José Mu illo Fuen es
Dep. Teo ía de la Señal y Comunicaciones
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2013
T abajo Fin de G ado
G ado en Ingenie ía Ae oespacial
3-Dimensional Con inuous-Th us T ajec o y
Op imiza ion o Elec ic Sola Wind Sails
using Bezie Cu es
Au o : Miguel Ga cía U eña
Tu o : Guille mo Pacheco Ramos
Dp o. Ingenie ía Ae oespacial y Mecánica de Fluidos
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2025
T abajo Fin de G ado
G ado en Ingenie ía Ae oespacial
3-Dimensional Con inuous-Th us T ajec o y
Op imiza ion o Elec ic Sola Wind Sails using
Bezie Cu es
Au o :
Miguel Ga cía U eña
Tu o :
Guille mo Pacheco Ramos
P o eso Ayudan e Doc o
Dp o. Ingenie ía Ae oespacial y Mecánica de Fluidos
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2025
T abajo Fin de G ado:
3-Dimensional Con inuous-Th us T ajec o y Op imiza ion o
Elec ic Sola Wind Sails using Bezie Cu es
Au o : Miguel Ga cía U eña
Tu o : Guille mo Pacheco Ramos
El ibunal nomb ado pa a juzga el abajo a iba indicado, compues o po los siguien es p o eso es:
P esiden e:
Vocal/es:
Sec e a io:
acue dan o o ga le la cali icación de:
El Sec e a io del T ibunal
Fecha:

Ag adecimien os
E
n los úl imos 4 años de mi ida no solo me he o mado académicamen e sino que ambién
pe sonalmen e. No hace mucho ponía po p ime a ez un pie en es a escuela de ingenie os
lleno de ilusión y ganas po deja mi huella de alguna o ma u o a, no a dé en da me cuen a
que odo iba a se muy dis in o de ese momen o en adelan e. Me gus a ía dedica es e p oyec o a
odas esas pe sonas que me han acompañado en mi camino du an e es a e apa, especialmen e a las
inc eíbles pe sonas que me lle o de es a ca e a, desde el p incipio has a aho a Blasco, Agus ín,
Ál a o, Emilio, Tachi, Lucía, Cale o, Collado y Fe nando, sin ellos nada hab ía sido lo mismo,
ayudándonos en esas noches in e minables an es de un examen o simplemen e un “aguan a que
podemos”, since amen e no hab ía podido llega a donde es oy aho a sin ellos, muy ag adecido
po que compa an mi “locu a”.
Quie o hace especial mención a mi compañe a de camino du an e la mi ad del ayec o, muchas
g acias Ca men, po eco da me quién soy y que puedo hace cuando no eo o a al e na i a o
simplemen e eco da me que soy capaz de log a lo que me p oponga, ambién po da me el ejemplo
de siemp e elegi lo que e hace eliz y no lo que se end ía que elegi . También quie o ag adece
eno memen e a mi amilia, sin ellos simplemen e nada hab ía sido como ha sido, g acias po
ayuda me en odos los aspec os de mi ida y siemp e apoya me sin impo a la decisión que oma a
ue a más ace ada o menos, g acias a mi he mana Elena, po se el ejemplo más ce cano de que
siemp e hay que segui hacia adelan e sin impo a los con a iempos.
Finalmen e, quie o exp esa mi eno me g a i ud a mi u o , Guille mo, jun os hemos lle ado
a cabo es e p oyec o a eces muy complacien e, o as eces un poco us an e, pe o siemp e
guiándome y ayudándome en los momen os que lo he necesi ado, ojalá odo el mundo pusie a las
ganas y la pasión como él.
Espe o que es e p oyec o cumpla con las expec a i as del lec o , pues, las mías las ha supe ado
con c eces, y que se haya quedado una pequeña pa e de mí e lejada en él, siemp e con Volun ad,
Ag adecimien o, Ganas, Ilusión, Amo y Sueños.
Miguel Ga cía U eña
Se illa, 2025
I
Resumen
Tí ulo
Op imización de T ayec o ias In e plane a ias pa a Velas Eléc icas Sola es con Cu as de Bezie
Resumen del abajo
En es e T abajo Fin de G ado se p esen a un mé odo de op imización de ayec o ias de ans e encia
in e plane a ia empleando una ela sola eléc ica (E-sail). Pa iendo de un modelo dinámico que
incluye las ue zas del Sol, se o mula el p oblema de con ol óp imo como una P og amación No
Lineal (NLP). También, la ayec o ia con inua se disc e iza median e cu as de Bézie , lo que
pe mi e ans o ma la op imización en un p oblema de dimensión ini a y educi signi ica i amen e
el iempo de cómpu o op imizando una se ia de coe icien es geomé icos del desa ollo polinómico.
Los esul ados demues an la e icacia de la ap oximación con Bézie en é minos de p ecisión y
cos e compu acional, así como la ac ibilidad de las soluciones ob enidas pa a dis in as en anas de
lanzamien o.
Palab as cla e
Vela sola eléc ica (E-sail), P og amación no lineal (NLP), Cu as de Bézie , Ó bi a in e plane a ia,
Op imización de ayec o ias, P oblema de con ol óp imo (OCP), Misiones de endez ouz.
Conclusiones
El obje i o es demos a que las cu as de Bezie ec ean con p ecisión las ayec o ias de elas
sola es eléc icas mien as educen los iempos de cómpu o en e diez y cien eces en e a la
ansc ipción di ec a; se alidan ans e encias óp imas a Ma e, Júpi e y a ios as e oides, e iden-
ciando obus ez an o pa a ó bi as plane a ias como pa a ayec o ias muy excén icas; el análisis de
la acele ación ca ac e ís ica con i ma que los mayo es aho os de iempo de uelo se alcanzan en el
égimen de bajo empuje, apo ando una egla p ác ica pa a dimensiona la ela; el ba ido de o den
polinómico iden i ica g ados 8–12 como equilib io ideal en e exac i ud y cos e compu acional;
las ayec o ias sua es ob enidas acili an la implemen ación de los con olado es LQR y MPC
p opues os como abajo u u o pa a mi iga la a iabilidad del ien o sola ; la me odología se
in eg a con las e emé ides calculadas con SPICE y admi e eop imización en uelo g acias a su baja
dimensionalidad; en conjun o, es e p oyec o o ece una he amien a ápida y iable pa a el diseño
p elimina de misiones pa a Velas Eléc icas Sola es y sien a las bases pa a u u as ex ensiones con
asis encias g a i a o ias y con ol p edic i o.
III
XNo a ion
RTN Radial-T ans e se-No mal
YSa u n
σSail “clocking” angle.
SPICE Spacec a Plane Ins umen C-ma ix E en s Toolki .
Final ime o he ans e .
τNo malized ime pa ame e .
τcCylind ical e e ence ame
˜aρ,˜aθ,˜azDimensionless p opulsion accele a ion componen s.
˜
ρ,˜
θ,˜zDimensionless posi ion coo dina es.
˜ i
Co esponding dimensionless eloci y componen s o
(˜
ρ,˜
θ,˜z).
uΦUni ec o in he φcoo dina e
uρUni ec o in he ρcoo dina e
uzUni ec o in he zcoo dina e
ZU anus
UT Time Uni .
UV Veloci y Uni .
ρ, θ, zVeloci y componen s in ρ,θ,zdi ec ions.
ϕAzimu hal angle in cylind ical coo dina es.
♀Venus
x ec o o op imiza ion a iables.
ucon ol ec o (κ,αn,σ).

1 In oduc ion
"I ha e lo ed he s a s so passiona ely ha I do no ea
he nigh ."
Miguel
F
om he beginning o ime, he sky has been he objec o eno mous admi a ion, om he i s
ci ilisa ions o he p esen day, om he i s b igh poin s obse ed by mankind o he mos
exo ic e en s in he uni e se. All his has made mankind ask mo e and mo e ques ions, making he
cu iosi y o know and mas e he laws o he uni e se and i s cons i uen pa s, g ow. All his has
led us o ake a look a ou nea es neighbo s. Howe e , he e has always been an ine i able ba ie ,
ime.
1.1 Mo i a ion
In e s ella a el is a subjec ha has always ascina ed eade s, he possibili y o a elling o
new wo lds and becoming explo e s o he mode n age; howe e , as good as i sounds, he idea o
in e s ella a el ades away when he ma hema ics o how o achie e i is analysed. I we look a
ou "home", he Milky Way Galaxy, which has a diame e o 105,700 ligh -yea s (abou
10 ·1017
kilome es).
Conside ing ha he g ea es dis ance e e eached by a p obe is abou 24,000 million kilome e s,
eached by he Voyage 1, which has been on an escape ajec o y om he sola sys em o 30 yea s.
As can be seen, we s ill ha e a long way o go o unde s and whe e we li e.
I we ocus on a less ambi ious aspec , he closes plane a y sys em o he Ea h, "Alpha Cen au i",
a a dis ance o 4.25 ligh -yea s, (ligh akes 4.25 ligh -yea s o a el ha dis ance a a speed o
1, 079, 252, 849.6 km/h). To make i e en clea e , we ha e ha he human-made objec ha has
eached he highes speed is he Pa ke p obe, hanks o a g a i y-assis ed maneu e using he Sun,
which was 692.017 km/h. A his speed, i is es ima ed ha he ime o a i al a "Alpha Cen au i"
would be 8000 yea s, which sugges s ha i is no easible oday. As has jus been demons a ed, he
a el o o he plane a y sys ems is highly imp obable, so we mus wai some hund eds o yea s ill
we become explo e s again. Howe e , we can ocus on wha is nea us, he Sola Sys em. [12]
Since he beginning o he space ace, he e ha e been nume ous in e plane a y missions o
isi he plane s o he Sola Sys em and unde s and hem be e . Fo example, he "Vene a"
capsules aimed a Venus, ei he o pho og aph he plane and o iden i y he main componen s o he
a mosphe e in case habi abili y was possible ( he conclusions we e nega i e), he "Ma ine " p obes
(1964-1971), which we e mainly used by NASA o he a i al and descen o Ma s. Jupi e has
1
2Chap e 1. In oduc ion
been he mos isi ed plane wi h abou 7 lybys wi h "Pionee 10" (1973) and "Pionee 11" (1974),
"Voyage 1" (1979) and "Voyage 2" (1979), "New Ho izons" (2007) and "Juno" (2016-p esen ).[
21
]
The ex e io plane s, Nep une and U anus, ha e only been isi ed once by he Voyage p obes on
an escape ajec o y om he sola sys em, aking ad an age o hei passage by hese plane s o
pe o m g a i y assis ance and gain mo e speed. All hese missions ha e used adi ional p opulsion
sys ems, which ake ad an age o he ac ion- eac ion e ec o gene a e h us by using uel, bu e en
so, in o de o educe ligh imes, hey ha e had o use g a i y assis maneu e s, which p o ide
ex a h us by aking ad an age o he speed a which he plane s o bi a ound he Sun. Al hough
hese maneu e s sa e uel and ime, he main p oblem wi h hem is ha he plane s ha e o be in
a speci ic posi ion wi h espec o he Ea h, so he e is a wai ing ime, which de ines he launch
windows. This ime inc eases eno mously i mo e han one plane is in ol ed, which is no mal,
as he ene gy equi emen s o deep space missions a e e y high. Fo example, he plane a y
con igu a ion ollowed by ‘’Voyage 2‘’, passing Jupi e , Sa u n, U anus and Nep une, is epea ed
e e y 174 yea s, which is a e y long ime. [19]
Figu e 1.1 Vogaye 1 P obe P opulsion Sys em, showing RTG [15].
A possible solu ion o his p oblem is ha ins ead o gene a ing a la ge inc ease in speed o e a
‘sho ’ pe iod o ime, a con inuous p opulsion wi h a small h us , bu which o e a longe pe iod
o ime can each high speeds. This is why he Low-Th us Con inuous P opulsion Sys em was
de eloped. Wi h hese sys ems i would be possible o educe launch wai ing imes o e en combine
hem wi h g a i a ional maneu e s, bu hey ha e a majo limi a ion: he uel hey use uns ou .
This is whe e Low-Th us P opellan less Con inuous P opulsion Sys ems come in. [14]
The main idea behind his is o use he sun as he p ima y sou ce o ene gy p oduc ion, specially,
using he sola wind. The di e en echnologies ha ha e been s udied a e Elec ic, Magne ic and
Pho onic sails.
To demons a e he easibili y and pe o mance o E-sail, some missions ha e been de eloped,
such as ESTCube-1 de eloped by he Uni e si y o Ta u, which es ed his echnology in a LEO
o bi , as a i s app oxima ion, since he ela i e speed o he plasma is 7 km/s which is much
slowe han he sola wind (300-800 km/s) bu he e is a highe concen a ion, so i can be used as
a p elimina y phase be o e a mission ou side he Ea h’s magne osphe e o ully demons a e he
pe o mance o his sys em. Ano he mission is Ligh Sail 2, which is able o change i s o bi only
using sola ene gy , and he Ika os mission, wi h a simila objec i e.[3]
1.2 S a e o he A 3
In his p ojec ocus will be on Sola Elec ic Sails, as hey a e a iable solu ion o all he p oblems
desc ibed abo e.
1.2 S a e o he A
The E-sail concep was p oposed by Pekka Janhunen in 2004 [
8
]. He was a Finnish space physicis ,
as obiologis and in en o . Be o e conside ing E-sails, Pekka Janhunen was analyzing he magne ic
sola wind sail, analyzed by Dana And ews and Robe Zub in in 1988. The Eu opean Space Agency
(ESA) had gi en a p ojec o Janhunen’s eam o analyze whe he he magne ic sail was some hing
ha he Agency should in es iga e u he . Janhunen wasn’ a space echnologis bu a leading space
plasma physicis ins ead, which was enough o analyze he sola wind sail concep .[24]
Figu e 1.2 Elec ic Sola Wind Sail concep [18].
A e ha , he ESA s udy concluded ha he magne ic sail wo ks in heo y, bu i s implemen a ion
would need a kilome es-long(1-20 km), supe conduc ing cable, which would equi e ei he some
ligh weigh and eliable sun-shield design o a nea - oom- empe a u e supe conduc ing ma e ial.
Howe e , deep imme sion in o sola wind plasma physics and in e plane a y p opulsion p o ided
Janhunen wi h an idea. I was based on eplacing he hea y elec omagne s wi h hin wi es ca ying
an elec ic cha ge.
The p inciple o ope a ion is simple, he p opulsion in E-sails is based on he conse a ion
o linea momen um caused by he epulsion o he sola wind plasma, which is in e cep ed by
elec ically cha ged wi es. The elec ic ield gene a ed by he wi es a ec s he ajec o y o he
p o ons in he sola wind, educing he componen o hei momen um in his di ec ion. This loss
o momen um by he p o ons is ans e ed o he wi es, which expe ience a Coulomb o ce ( he
cha ged wi e unde goes a o ce due o he plasma). This o ce is hen ans e ed o he p obe as
h us . To main ain he high ol age bias on each e he , equi es emi ing collec ed elec ons back
in o he deep space media ia an elec on gun on he spacec a .[11]
The ope a ion di e s om o he p e iously conside ed me hods, such as pho onic and magne ic
sails, which a e also based on he conse a ion o linea momen um bu use di e en mechanisms.
Pho onic sails ely on he momen um ans e om pho ons in he sola adia ion, while magne ic
sails use magne ic ields o de lec he ajec o y o incoming p o ons.
The main ad an age p oposed by E-sails lies in hei h us beha io , which dec eases as
1/
,
whe e is he dis ance o he Sun. In con as , he h us o pho onic sails dec eases as 1/ 2.
4Chap e 1. In oduc ion
This means ha o a mission planing o dis ances a away om he Sun, E-sails migh be mo e
app op ia e spacec a since hey ha e a smoo h g adien whe eas he o he dec eased quickly.
The p ima y eason o his di e ence is ha he dynamic p essu e o he sola wind dec eases wi h
1/ 2
, bu he e ec i e a ea o he sail—p opo ional o he en elope su ounding he wi es—inc eases
wi h
, ollowing he same end as he Debye leng h o he plasma. (The Debye leng h is he dis ance
o e which he elec ic ield o a cha ge is shielded,so he pa icle can’ go h ough he wi es and
p oduce h us by bouncing.) As a esul , he p oduc o hese ac o s leads o a h us dependence
o 1/ .
As p e iously demons a ed, his sys em p oposes c ucial elemen s o u ilizing he sola wind,
whose in luence ex ends h oughou he en i e sola sys em. I can be used in uncon en ional
missions ha would be impossible wi h adi ional p opulsion sys ems, such as pola o bi s o sola
obse a ion o non-Keple ian o bi s, which equi e con inuous h us co ec ions o be main ained
a ound he Lag ange poin s.
I we ocus on he e olu ion o he E-sails, he i s design was based on a squa e shape design
whose cha ac e is ics was 10-100
m2
su ace, nea 2010 and 2018, bu las endencies, in con as ,
lean owa ds a ci cula shape ins ead, wi h his geome y i will be mo e easy o deploy and main ain
he s uc u e s abili y by inducing a cen i ugal o ce so he wi es a e main ained s e ched- his
is no possible in a squa e-shape geome y- and also, inc easing he e ec i e a ea o he sail o
p oduce mo e h us (1000-10000 m2) and dec ease he densi y o he sail 1−2.5g/m2.
Al hough op imizing E-sails o highe pe o mance is concei able, hei eliance on he a iable
sola wind can jeopa dize c i ical mission phases. To elimina e his dependency, a g ound-based lase
can con inuously illumina e he sail, enabling sus ained h us ega dless o sola wind condi ions.
By beaming ene gy om Ea h o he spacec a , he e ec i e sail a ea could be inc eased o be ween
10 000 and 1 000 000 m2while educing sail a eal densi y o as li le as 0.1 g/m2d ama ically
imp o ing h us gene a ion and mission eliabili y [15].
Figu e 1.3 E olu ion o he E-sails h oughou he ime [10].
1.3 Aim o he p ojec 5
1.3 Aim o he p ojec
As p e iously men ioned, Elec ic sails (E-sails) s ill ha e signi ican po en ial o ad ancemen
be o e hey can eliably each dis an poin s in space. The main goal o his p ojec , howe e , is no
o in es iga e he physical mechanisms behind he h us gene a ed by E-sails, no hei ope a ional
beha io . Ins ead, he emphasis is placed on enhancing he mission planning s age. Speci ically, his
p ojec aims o s eamline he p elimina y design phase o space missions by employing sophis ica ed
ajec o y op imiza ion echniques. Inco po a ing Bezie cu es ep esen s a p omising me hod,
which no only imp o es compu a ional e iciency bu also yields p ecise ajec o y solu ions and
educes o e all compu a ional ime equi ed o op imal ans e pa hs. Ul ima ely, his app oach
will signi ican ly educe design cos s associa ed wi h space missions, enabling mo e e icien and
cos e ec i e mission planning.
•
NLP p oblem planning and solu ion: To begin, we es ablish a comp ehensi e h us model
o he E-sail, de ailing he expec ed h us le els and he key con ol pa ame e s go e ning
spacec a maneu e ing. Building upon his ounda ion, we o mula e he unde lying nonlinea
p og amming (NLP) p oblem o op imize he ans e ajec o y. We hen sol e he NLP
ia di ec ansc ip ion me hods, disc e izing he con ol and s a e a iables o ans o m
he con inuous- ime op imiza ion in o a ini e-dimensional nonlinea p og am. Using an
op imiza ion sol e , we ob ain ime op imal ajec o ies ha sa is y all mission cons ain s.
Finally, we p esen he op imized ans e solu ions o missions o Ma s, Jupi e and se e al
as e oids, highligh ing hei easibili y and pe o mance me ics.
•
OCP solu ion using Bezie cu es: Se e al e inemen s o he baseline p oblem a e in o-
duced in his sec ion, ollowed by an analysis ha shows he Bezie cu e o mula ion can
educe he o e all compu a ional ime by an o de o magni ude. Th oughou he p ojec
he spacec a ’s a i ude is ea ed as p esc ibed. Al hough ac i e esea ch is unde way on
spin plane con ol using ol age modula ion and o he means ha couple di ec ly in o he
sail’s dynamics we model he a i ude subsys em as a black box, assuming ha he desi ed
o ien a ion can be eached ins an aneously a any poin along he ajec o y.
Fu he mo e, he p esen analysis p oceeds unde he ollowing assump ions:
–
Only he g a i a ional a ac ion o he Sun is conside ed. We ha e neglec ed he o he
plane ’s o ces.
–
The launch da e can be changed wi h eedom, always being as close as possible o he
eal launch windows.
1.4 S uc u e o he p ojec
The layou o his p ojec is ske ched in he pa ag aphs ha ollow. Each pa ag aph o e s an
expanded synopsis o a single chap e , cla i ying i s speci ic aims, he me hods applied, and he way
i s esul s do e ail wi h he o e a ching in es iga ion. In his manne , he eade can quickly g asp
how he s udy de elops om ounda ional heo y h ough me hodological inno a ion o he inal
assessmen and app ecia e how he indi idual pieces combine o o m a cohe en whole.
Chap e 1: In oduc ion. In his opening chap e lays he ounda ion o he p ojec by i s
explaining he unde lying mo i a ion ha d i es he esea ch. I hen su eys he cu en s a e o he
a , highligh ing key ad ances and open challenges in he E-sails ield. Building on his con ex , he
chap e clea ly s a es he p ojec objec i es, which se e as guiding miles ones o he ensuing wo k.

6Chap e 1. In oduc ion
Finally, an o e iew o he documen ’s s uc u e is p o ided, ou lining how each subsequen chap e
con ibu es o ul illing hese objec i es and leading he eade logically om p oblem o mula ion
o inal conclusions.
Chap e 2: Backg ound in O bi al Mechanics. This chap e de i es he spacec a ’s undamen al
equa ions o mo ion and o mula es a con inuous h us model o elec ic sails (E-sails), no ing key
p ac ical limi a ions. I concludes by es ablishing he physical con ex equi ed o he ajec o y
analysis ha ollow.
Chap e 3: Op imal Mission Planning. This chap e cas s he mission as an op imal con ol
p oblem, ansla es i in o a nonlinea p og amming (NLP) o ma h ough di ec ansc ip ion, and
p esen s ime and p opellan op imal ans e ajec o ies o Ma s, Jupi e , and a ep esen a i e se
o as e oids.
Chap e 4: Bezie Cu e Based Planning. A e ou lining he ma hema ical de ini ion o
Bezie cu es, his chap e e o mula es he ans e p oblem wi hin ha pa ame ic amewo k and
benchma ks he esul ing ajec o ies agains he di ec me hod, compa ing compu a ional speed
and solu ion easibili y.
Chap e 5: Conclusions and Fu u e Wo k. The closing chap e consolida es he p ojec ’s
key esul s, e alua es he o e all e iciency o he p oposed me hods, and sugges s di ec ions o
ex ending he app oach o u u e mission scena ios.
All o ou p ocedu es a e based on he p oblem o mula ion desc ibed in [
18
], specially he Bezie
p oblem o mula ion.
2 Backg ound in o bi al mechanics
I
n his chap e a look a he basics o o bi al mechanics is aken, de eloping a con inuous h us
model ha shapes well wi h he mo ion o he E-sails h oughou in e plane a y space. The aim
o his chap e is o p opose a apid shape based me hod whe e he concep o Bezie cu e is used
o e icien ly design he h ee dimensional in e plane a y ajec o y o a spacec a p opelled by
E-sails.
2.1 Equa ion o mo ion
2.1.1 Deduc ion o he Equa ion o Mo ion
Fo his deduc ion, he Law o Uni e sal G a i a ion disco e ed by Si Isaac New on (1642-1727)
is used. This law desc ibes he g a i a ional e ec ha a massi e body expe iences when i is
nea ano he massi e body. When mo e plane s a e in ol ed, no analy ical solu ion exis s o hese
equa ions, so nume ical esul s a e equi ed; howe e , o he p oblem being sol ed, he wo body
model is su icien . The Law o Uni e sal G a i a ion s a es ha he g a i a ional o ce exe ed by
wo massi e bodies is di ec ly p opo ional o he p oduc o hei masses and in e sely p opo ional
o he squa e o he dis ance be ween hei cen es o mass. The equa ion is p esen ed in ec o
o m, so he a ac i e ec o o ce always lies along he line ha connec s hei cen es o mass.
These a e:
F12 =Gm1m2
2
R2−R1
|R2−R1|(2.1)
F21 =Gm1m2
2
R1−R2
|R1−R2|(2.2)
7
8Chap e 2. Backg ound in o bi al mechanics
Figu e 2.1 Desc ip ion o he g a i a ional Fo ce be ween wo bodies.
whe e G is he Uni e sal G a i a ional Cons an , G = 6.67430
×10−11 m3kg−1s−2
,
Fi j
is he
o ce in he body
i
exe ed by he body
j
, as i can be seen in Figu e 2.1, and
R1
,
R2
a e he posi ion
ec o s o each body in an ine ial ame o e e ence, as i can be seen in Figu e 2.1. In wha
ollows, he ec o deno es he di ec ion om mass 1 o mass 2:
=R2−R1, =| |(2.3)
A e ha , New on’s second law is in oduced o de i e he di e en ial equa ion o mo ion.
F=m·a(2.4)
Using (2.1) and (2.2) we ha e:
m1·¨
R1=F12 =G·m1·m2
2·
(2.5)
m2·¨
R2=F21 =−G·m1·m2
2·
(2.6)
Di iding by m1and m2yields he equa ion in e ms o ¨
R1and ¨
R2.
¨
R1=G·m2
2·
(2.7)
¨
R2=−G·m1
2·
(2.8)
Recalling he de ini ion o in (2.3)
¨
R2−¨
R1=¨
=−G·m1+m2
2·
(2.9)
This equa ion has an analy ical solu ion in he case whe e only wo bodies a e in ol ed; in ha
case, he cen e o g a i y o he sys em always mo es wi h cons an speed, so o con enience, he
de ini ion o ou ine ial e e ence sys em is made a his poin .
To simpli y he p oblem, ce ain hypo heses can be made. When one body has a signi ican ly
g ea e mass han he o he , he cen e o mass o he sys em lies e y close o he massi e body, so
his poin can be assumed o coincide wi h he cen e o he massi e body. Unde his simpli ica ion,
R1is cons an and he equa ion o mo ion o he second mass is:
2.1 Equa ion o mo ion 9
¨
=−µ
2·
(2.10)
whe e
µ=Gm1
(s anda d g a i a ional pa ame e ). Because
m1≫m2
, i ollows ha
m1+m2≈
m1
. Fo example, in he Sun-Ea h sys em he g a i a ional pa ame e has he alue
µ⊙≈1.327 ×
1020 m3s−2.
2.1.2 The 3D-p oblem
Fo an in e plane a y mission o ano he plane , i is possible o assume some hypo heses o simpli y
he p oblem. Fo ins ance, i he plane s a e o bi ing in he eclip ic plane ( he plane whe e Ea h
o bi s a ound he Sun), a wo-dimensional, co-plana model can be used ha i s well wi h eali y.
Also, using a ci cula model o he o bi s is a good app oxima ion because he eccen ici y o
he plane s is no big. Howe e , i a mission is o be de eloped o o he bodies ha a e o bi ing
a ound he Sun (like as e oids), hese simpli ica ions may no be as good as hey can be in a i s
app oach. A compa ison be ween he eccen ici y
e
and he inclina ion
i
o some plane s and some
as e oids in he Sola Sys em is p esen ed:
Table 2.1
Compa ison be ween eccen ici ies and inclina ions o plane s in he sola sys em espec
o he Eclip ic plane.
Plane '♀♂X Y Z [
i(º) 7.01 3.39 1.85 1.31 2.49 0.77 1.77
e 0.2056 0.0068 0.0934 0.0484 0.0541 0.0472 0.0086
Table 2.2 Compa ison be ween eccen ici ies and inclina ions o as e oids in he Sola Sys em .
As e oid Didymos Apophis Bennu 3200 Phae hon Dionysus
i(º) 3.41 3.33 6.035 22.2 13.9
e 0.38 0.19107 0.2037 0.89 0.542
Thus, i is e iden ha he app oxima ion is no sui able o as e oids i he p e iously men ioned
simpli ica ions a e applied, as i can be seen in Table 2.2, so inclina ions and eccen ici ies mus be
included in he calcula ions.
This p ojec ocuses on Ma s and Jupi e , as well as on Dionysus and Didymos; howe e , he
s udy can be ex ended o o he bodies wi hou di icul y.
To analyse he h ee dimensional p oblem, i is necessa y o de ine a launch da e ha alls wi hin
he plane ’s synodic pe iod. Wi h his in o ma ion, he o bi al elemen s o bo h Ea h and he
a ge plane (o as e oid) can be p edic ed and he op imal ans e ajec o y calcula ed. Since he
o bi al elemen s a y o e ime, hey a e ob ained om epheme ids, which allows hese esul s o
be inco po a ed seamlessly in o he p ojec .
16 Chap e 2. Backg ound in o bi al mechanics
2.2.2 Th us Cons ain
This sec ion conside s he p ac ical limi a ions o he E-sail, ocusing on he sail pi ch angle
αn
, he
cha ac e is ic accele a ion, he eloci ies a ainable by he spacec a , and he maximum payload
ha he p obe can ca y on an in e plane a y mission. Fi s , he sail pi ch angle is analysed, since
i is one o he mos c i ical pa ame e s aken in o accoun in he op imisa ion p ocedu e (as is
κ
,
discussed la e ). Cu en simula ion esul s and s udies place his angle in he ange
αnmax ∈[60◦,70◦]
.
Using he p opulsi e-accele a ion ec o desc ibed in (2.18), a ma hema ical exp ession can be
ob ained ha ela es
αnmax
o he h ee componen s o his ec o a e e y poin o he ajec o y,
he eby de e mining he easible ange o accele a ion componen s o he mission. A ypical
Radial–T ans e se–No mal (RTN) e e ence ame is in oduced, whe e
ˆ
iR
is he adial uni ec o ,
ˆ
iT
he ans e se uni ec o , and
ˆ
iN
he no mal uni ec o (wi h he o igin a he spacec a ’s cen e
o mass). This ame di e s om he body axes because he RTN ame is ied o he spacec a ’s
o bi o a ing slowly as posi ion and eloci y change whe eas he body ame is igidly ixed o he
spinning ehicle and u ns a i s spin a e; in essence, RTN speci ies he desi ed h us di ec ion in
o bi al space, while he body ame dic a es how he c a mus o ien i sel o p oduce ha h us .
Figu e 2.6 De ini ion o he Radial-T ans e se-No mal e e ence ame [26].
The p opulsi e accele a ion ec o can be w i en as a unc ion o he sail pi ch angle
αn
, by
p ojec ing in he h ee axes ha we ha e de ined be o e. This is:
aR=ac ⊕
2 (1+cos2αn)(2.31)
aT=ac ⊕
2 sin αncos αncos σ(2.32)
aN=ac ⊕
2 sin αncos αnsin σ(2.33)
He e,
⊕
deno es he Sun Ea h dis ance (1 AU) and
he heliocen ic dis ance o he spacec a .
The equa ion he e o e yields he componen s o he accele a ion ec o ac oss he ange o
αn
, bu
i is mo e con enien ly exp essed in he ollowing dimensionless o m:
˜aR= ⊕
2 (1+cos2αn)(2.34)
˜aT= ⊕
2 sin αncos αncos σ(2.35)

2.2 Con inuous Th us Model 17
˜aN= ⊕
2 sin αncos αnsin σ(2.36)
In he i s case, he uncons ained scena io is analysed, whe e
αn
spans
0◦
–
90◦
. This a ia ion
enables he desc ip ion o he “ o ce bubble”, which ep esen s he easible alues o he accele a ion-
ec o componen s a a speci ic poin along he ajec o y:
Figu e 2.7 Rep esen a ion o he accele a ion cons ain in e ms o αn∈[−π
2,π
2]and σ∈[0,2π].
In his case
= ⊕
is adop ed, bu he p ocedu e can be gene alised s aigh o wa dly. Taking
in o accoun he p e iously men ioned cons ain s on αn, he esul ing o ce bubble is:
Figu e 2.8 Rep esen a ion o he accele a ion cons ain in e ms o αn∈[0,65]and σ∈[0,2π].
As s a ed abo e, he es ic ions in
αn
a ec he componen s o he accele a ion ec o , p o iding
an easy geome ical way o e i y whe he he con ol-pa ame e esul s ob ained in he op imisa ion
18 Chap e 2. Backg ound in o bi al mechanics
Table 2.4 Spacec a mass budge o a cha ac e is ic accele a ion ac=1 mm/s2.
Payload (kg) 100 200 500 1000
Numbe o e he s N 12 16 24 34
Te he s Leng h L (km) 4.02 5.77 9.27 12.9
p oblem a e consis en wi h he p opulsion model and disca ding hose ha canno sa is y hese
es ic ions.[22]
In addi ion, he limi a ions on he h us ha he E-sail can p oduce a a gi en dis ance om he
Sun a e analysed; a sa u a ion ol age
V
exis s, app oxima ely de ined by he condi ion ha
eV
su passes he kine ic ene gy o he sola -wind p o ons (abou
1keV
), so, o a gi en heliocen ic
dis ance and he a e age numbe o
eV
ha pass h ough he E-sail, he maximum h us a ainable
a ha dis ance can be de e mined.[9]
The cha ac e is ic accele a ion is a undamen al pa ame e o he E-sail; hus, an analysis is made
in o de o ge an exp ession in unc ion o he E-sail sys em pa ame e s. De ining he pa ame e
as:
= VV0− 0(2.37)
whe e
0
=24.16
nNm−1
and
=24.16nNm−1kV−1
. The spacec a ’s cha ac e is ic accele a ion
is he e o e:
ac= N L
m(2.38)
In his equa ion, he cha ac e is ic accele a ion is exp essed in e ms o he design pa ame e s:
N
(numbe o e he s), L( e he leng h), V0(nominal ol age), and m(payload mass).
This exp ession p o ides a ma hema ical ela ion be ween he cha ac e is ic accele a ion and he
payload budge , enabling he maximum payload o a easible and ealis ic E-sail o be es ima ed o
each alue o ac:
In his able a clea o e iew o he mass budge o an E-sail is ob ained. I can be obse ed ha
inc easing he payload equi es mo e e he s and g ea e e he leng h. Al hough cu en E-sails
a e p ima ily based on CubeSa s, whe e mass budge is no a signi ican conce n, his ac o will
become inc easingly ele an o u u e in e plane a y and in e s ella missions. Such missions will
ca y payloads o oughly 30–1000 kg and will demand cha ac e is ic accele a ions o up o abou
3ms−2. The e o e, i is impo an o accoun o his aspec in he p esen p ojec .[9]
3 Op imal mission planning
I
n his chap e , he o mula ion and esolu ion o he op imal con ol p oblem a e add essed,
inco po a ing all he cons ain s ha mus be sa is ied along he ajec o y. The goal is o
de e mine he op imal con ol pa ame e s as unc ions o ime o a con inuous-p opulsion model
in an in e plane a y ajec o y, minimising he ans e ime. To achie e his objec i e, Bezie
unc ions a e in oduced as a shape-based algo i hm designed o minimal compu a ional ime.
Va ious in e plane a y ans e s a e conside ed, including Ea h–Ma s, Ea h–Jupi e , and as e oid
ajec o ies a ound he Sun, such as Dionysus and Didymos. The p oblem is o mula ed as a
nonlinea -p og amming (NLP) con ol p oblem, exp essing he s a e a iables in e ms o Bézie
unc ions ha depend on he op imisa ion pa ame e
and p o ide a polynomial app oxima ion o
he ajec o y poin s. An op imisa ion ou ine is subsequen ly applied o e ine hese coe icien s,
yielding he minimum ans e -o bi ime wi h he lowes compu a ional cos .
3.1 Op imal Con ol P oblem Fo mula ion
This sec ion add esses he op imal con ol p oblem be ween wo bodies, de ailing he p oblem
o mula ion and i s solu ion by means o a Runge–Ku a p ocedu e.
3.1.1 P oblem hypo heses
In he esolu ion o he OCP ha has been de eloped in his sec ion, he hypo heses and simpli ica ion
which we e dec ibe in he sec ion Sec ion 2.1.2 and 2.2.3, a e aken in o accoun .
3.1.2 The Op imal Con ol P oblem
Op imal con ol heo y is a ma hema ical op imiza ion me hod o inding a con ol law o minimize
ce ain objec i e unc ion while simul aneously being subjec o a se o cons ain s. Gi en a se o
n i s -o de di e en ial equa ions desc ibing gene ic dynamics:
˙
x= (x,u, )(3.1)
he m con ol unc ions
u
( ),
∈
[
i,
] mus be de e mined such ha he ollowing pe o mance
index
J=ϕ(x( ), )+Z
i
L(x,u, )d (3.2)
is minimized and q inal bounda y condi ions
ψ(x( ),u( ), ) = 0(3.3)
19
20 Chap e 3. Op imal mission planning
a e sa is ied. The solu ion o his p oblem is de i ed by he calculus o a ia ions and i s comple e
in es iga ion is beyond he pu poses o his epo . Thus, only he de i a ion o he Eule -Lag ange
equa ions will be b ie ly ecalled. In oducing wo kinds o Lag ange mul iplie s he q-dimensional
cons an ec o
ν
o he inal bounda y cons ain s and he n-dimensional a iable ec o o adjoin
o cos a e a iable o he dynamics he a gumen pe o mance index is de ined as
J=ϕ(x( ), )+νTψ(x( ),u( ), )+Z
i
[L(x,u, )+λT((x,u, )−˙
x)]d (3.4)
I is impo an o obse e ha he dynamics (3.1) a e included in he pe o mance index (3.2) in
he same ashion as a cons ain , so he op imal solu ion mus bo h minimise he objec i e unc ion
and sa is y he dynamics. This iewpoin o e s an al e na i e o ha o dynamical sys em heo y.
Al hough an Eule –Lag ange app oach could easibly sol e he op imisa ion p oblem, he p esen
wo k adop s a di ec ansc ip ion me hod ha con e s he dynamics in o a nonlinea p og amming
(NLP) p oblem, which is hen sol ed wi h a ou h o de Runge–Ku a scheme. When he cos
unc ion
J
consis s solely o he i s e m
ϕ
, he o mula ion is said o be in Maye o m, as is he
case he e, and he p oblem is subjec o he s a e equa ions:
˙
x( ) = (x( ),u( ), )(3.5)
and also o a se ies o pa h cons ain s
h(x( ),u( ), )≤0(3.6)
and bounda y condi ions
g(x( 0),u( 0),x( ),u( )) = 0(3.7)
In his o mula ion he s a e ec o is
x= [ρ,θ,z, ρ, θ, z]T
and he con ol ec o is
u=
[κ,αn,σ]T. Fo a minimum ime ajec o y he cos unc ion educes o he inal ime i sel :
J( ) = (3.8)
and he dynamics equa ions a e:
˙
ρ= ρ(3.9)
˙
θ= θ
ρ(3.10)
˙z= z(3.11)
˙ ρ= 2
θ
ρ−µ⊙ρ
(ρ2+z2)3/2+κac ⊕
2(ρ2+z2)(ρcos2αn+zsin αncos αncos σ+ρ)(3.12)
˙ θ=− ρ θ
ρ+κac ⊕
2(ρ2+z2)cos αnsin αnsin σpρ2+z2(3.13)
˙ z=−µ⊙z
(ρ2+z2)3/2+κac ⊕
2(ρ2+z2)(zcos2αn−ρsin αncos αncos σ+z)(3.14)
Th ee di e en pa h cons ain s a e de ined: he alues o he con ol pa ame e s, he dynamics
equa ions and he bounda y es ic ion in he depa u e and in he a i al. These a e de ined as
ollows:
3.1 Op imal Con ol P oblem Fo mula ion 21
0≤κ≤1(3.15)
−π
2≤αn≤π
2(3.16)
0≤σ≤2π(3.17)
ρ( 0) = ρ0
θ( 0) = θ0
z( 0) = z0
ρ( 0) = ρ0
θ( 0) = θ0
z( 0) = z0
And he bounda y cons ain in he a i al:
ρ( ) = ρ
z( ) = z
ρ( ) = ρ
θ( ) = θ
z( ) = z
A a i al he alue o
θ
is no used because i is unknown. To ob ain he alues essen ial o he
NLP p oblem, a high p ecision epheme is calcula o , SPICE, de eloped by NASA, is employed,
as discussed in (2.1.1). Thus, a launch da e o he E-sail mus be de ined because, when he
ellip ical and h ee dimensional p oblem is conside ed, o a ional symme y is los and a bi a y
pa ame e s canno be assigned; he posi ion o Ea h a he launch da e mus he e o e be de e mined.
To simpli y he p oblem, wo addi ional obse a ions mus be made: As onomical Uni s (AU)
a e adop ed o handle he equa ions con enien ly, and he sola g a i a ional pa ame e is se o
µ⊙=1AUUV2. Hence:
1AU =149.598 ·106km
1UV =29.7847 km/s
1UT =58.1324 days
1UA =5.9301 mm/s2
Wi h Uni s o Time (UT) and Uni s o Accele a ion (UA)
Chap e 2.2.3 de ines he “Fo ce Bubble,” he geome ical zone wi hin which he modulus o he
accele a ion ec o mus lie; on his basis, a cha ac e is ic accele a ion ha sa is ies he go e ning
equa ions can be de i ed. Fo he p esen p ojec he alue
ac=1mms−2
is adop ed, which
co esponds o ac=0.16863UA.
In addi ion, wi hou loss o gene ali y, he inal ime is assumed o sa is y
>0
, meaning ha he
p oblem conside s only he in e al a e he launch da e and excludes any solu ions ha p ecede i .
Wi h all o hese conside a ions in place, he inal OCP o mula ion is s a ed as ollows:

22 Chap e 3. Op imal mission planning
min
κ,αn,σ
subjec o:
˙
ρ= ρ
˙
θ= θ
ρ
˙z= z
˙ ρ= 2
θ
ρ−µ⊙ρ
(ρ2+z2)3/2+κac ⊕
2(ρ2+z2)(ρcos2αn+zsin αncos αncos σ+ρ)
˙ θ=− ρ θ
ρ+κac ⊕
2(ρ2+z2)cos αnsin αnsin σpρ2+z2
˙ z=−µ⊙z
(ρ2+z2)3/2+κac ⊕
2(ρ2+z2)(zcos2αn−ρsin αncos αncos σ+z)
ρ( 0) = ρ0
θ( 0) = θ0
z( 0) = z0
ρ( 0) = ρ0
θ( 0) = θ0
z( 0) = z0
ρ( ) = ρ
z( ) = z
ρ( ) = ρ
θ( ) = θ
z( ) = z
0≤κ≤1
−π
2≤αn≤π
2
0≤σ≤2π
∈[0, ], >0
Whe e ac=0.16863 UA,µ⊙=1AU UV2, ⊕=1AU
3.1 Op imal Con ol P oblem Fo mula ion 23
3.1.3 The Nonlinea P og amming P oblem
Essen ially, any nume ical me hod o sol ing he ajec o y op imisa ion p oblem inco po a es an
i e a i e p ocedu e wi h a ini e se o unknowns. The nex sec ion shows how an op imal-con ol
p oblem can be ans o med in o a nonlinea p og amming (NLP) p oblem; unlike an op imal con ol
o mula ion, an NLP p oblem in ol es no dynamics. Suppose ha he
n
dimensional a iable
x
mus be chosen o sol e
min
xF(x)
subjec o he m equali y cons ain s
c(x) = 0(3.18)
whe e m ≤n. The Lag angian o his p oblem is
L(x,λ) = F(x)−λTc(x),(3.19)
which is a scala unc ion o he n a iables
x
and he m Lag ange mul iplie s
λ
. The necessa y
condi ions o a poin (x∗,λ∗) o be a cons ained op imum equi e sol ing he ollowing sys em
∇xL(x,λ) = g(x)−GT(x)λ=0,(3.20)
∇λL(x,λ) = −c(x) = 0(3.21)
whe e g=
∇xF
and
G
a e he g adien s o he objec i e unc ion F(x) and he Jacobian o he equali y
cons ain ec o c(x), espec i ely. The las sys em o equa ions can be sol ed ia New on’s me hod
o ind he (n+m) a iables (
x∗,λ∗
) o simila nume ic me hods. Gi en a gene ic ini ial guess (
x,λ
),
i s co ec ions (
∆x,∆λ
) o cons uc he new solu ion (
x
+
∆x
,
λ+∆λ
) a e gi en by sol ing he linea
sys em
HL-GT
G0∆x
∆λ=−g
−c(3.22)
also e e ed as Ka ush-Kuhn-Tucke sys em; Ka ush-Kuhn-Tucke condi ions he e m
HL
is he
Hessian o (3.6) in x, namely
HL=∇2
xF−
m
∑
i=1
λi∇2
xci(3.23)
I is impo an o obse e ha an equi alen way o de ine he sea ch di ec ion
∆x
is o minimize he
quad a ic o m
1
2∆xTHL∆x+gT∆x(3.24)
subjec o he linea cons ain
G∆x=−c(3.25)
This is he eason why his p oblem is also e e ed o as a quad a ic p og amming (QP) p oblem.
The NLP p oblem o mula ed abo e can be gene alized o he case ha occu s when inequali y
cons ain s a e imposed; he mcons ain s a e o he o m
c(x)≥0(3.26)
Cons ains ha a e s ic ly sa is ied, i.e.
ci
(x)> 0, a e called inac i e; he emaining ac i e se o
cons ain s a e on hei bounds, i.e.
ci
(x)=0. I he ac i e se o cons ain s is known, he inac i e
cons ain s a e igno ed and he p oblem is simply sol ed using he me hod o an equali y cons ained
p oblem discussed abo e. In summa y, he gene al NLP p oblem equi es inding he
n
ec o s o
24 Chap e 3. Op imal mission planning
sol e
min
xF(x)(3.27)
subjec o he m cons ain s
cL≤c(x)≤cU(3.28)
and bounds
xL≤x≤xU(3.29)
In his o mula ion, equali y cons ain s can be imposed by se ing cj,L=cj,U[25]
3.1.4 Di ec T ansc ip ion
Op imal ajec o y design is a con inuous op imal con ol p oblem ha can be sol ed wi h he
Eule Lag ange equa ions; his p ocedu e is e med he indi ec me hod. An al e na i e philosophy
ansla es he con inuous op imal con ol p oblem in o a nonlinea p og amming (NLP) p oblem,
sol ing o a ini e se o a iables a p ocedu e known as di ec ansc ip ion, o he di ec me hod. In
a di ec app oach he solu ion o he op imal con ol p oblem is s ic ly connec ed o he nume ical
in eg a ion o he di e en ial equa ions: he se o equa ions go e ning E-sail mo ion is ansc ibed
in o a ini e se o equali y cons ain s, and i he NLP solu ion sa is ies hese cons ain s, he o iginal
op imal con ol p oblem is sol ed wi hin he nume ical accu acy o he scheme employed (he e, a
Runge–Ku a me hod). A ini e se o a iables is i s equi ed; he e o e, he p oblem is disc e ised.
De ining he se o unknowns as
x= [x1,x2,x3,x4,...]T
, he gene ic NLP p oblem is o mula ed as
an op imisa ion in which a scala objec i e unc ion F(x)is minimised.
Fo ge ing hese esul s, a se o inequali y cons ain s and bounds a e de ined, ep esen ed as:
subjec o (cL≤c(x)≤cU
xL≤x≤xU
Equali y pa h cons ain s and a iable bounds a e inco po a ed by imposing uppe and lowe
limi s wi hin he op imisa ion p oblem, hus ensu ing ha he sys em dynamics sa is y he p esc ibed
equa ions. Wo king in a disc e e ime domain equi es he de ini ion o a ime mesh on which he
dynamical equa ions a e e alua ed as pa h cons ain s, namely:
=[ 0, 1, 2,..., N−1, ]T, whe e k+1= k+h,
The pa ame e
h
, designa ed he s ep size o he ime mesh, is essen ial because an inapp op ia e
choice can p eclude any iable solu ion; a alue scaled o he p oblem,
h=
(N−1)
, is he e o e
adop ed, whe e
N
is he numbe o nodes. An uni o m ime mesh is selec ed a his s age[
9
], since
no ap io i in o ma ion abou he solu ion exis s; his assump ion is deemed easonable o an ini ial
app oxima ion and can be e isi ed o ede ine h a e p elimina y esul s ha e been analysed[
25
].
Once a ini e se o ime ins an s is de ined, he unknown coe icien s also become ini e: each node
ca ies six s a e a iables plus h ee con ol pa ame e s, gi ing a o al o
N(6+3)+1
op imisa ion
a iables, he inal e m ep esen ing he las componen o he op imisa ion ec o .
nT= (6+3)·N+1(3.30)
The “1” co esponds o he ime a iable, which cons i u es he objec i e o be minimised. In he
p esen o mula ion a mesh is adop ed ha adap s o each case; using a mos 300 poin s yields
a sa is ac o y comp omise be ween p ecision and compu a ional speed when sol ing he OCP.
Once he mesh is es ablished and he p oblem a iables a e de ined, he dynamics equa ions a e
inco po a ed: a ou h o de Runge–Ku a scheme is employed o his pu pose, as discussed nex .
3.1 Op imal Con ol P oblem Fo mula ion 25
The RK4 me hod is a nume ical echnique o sol ing o dina y di e en ial equa ions o he o m
x′( ) = ( ,x( )),x( 0) = x0.
I app oxima es he solu ion
x( )
by compu ing in e media e slopes and hen combining hem o
upda e he alue o x.
The algo i hm is desc ibed as ollows:
Gi en a s ep size h, he me hod compu es:
k1= ( n,xn),
k2=  n+h
2,xn+h
2k1,
k3=  n+h
2,xn+h
2k2,
k4= ( n+h,xn+hk3).
Then, he nex alue is gi en by:
xn+1=xn+h
6(k1+2k2+2k3+k4).
And his exp ession is he non-linea cons ain ha includes he dynamics o he p oblem, and
mus be e i ied by he op imal con ol s a e ec o . A block diag am is made o unde s anding he
dependence o he p ocedu e wi h k.
The ollowing diag am illus a es he e alua ion poin s in one s ep o he RK4 me hod:
( n,xn)k1= ( n,xn)
k2=  n+h
2,xn+h
2k1
k3=  n+h
2,xn+h
2k2
k4= ( n+h,xn+hk3)xn+1
k1
h
2
h
2
h
2
h
2
Figu e 3.1 Single-s ep scheme o he ou h o de Runge Ku a me hod .
I should also be no ed ha he alues o he con ol pa ame e s in he ec o
uk
emain cons an
h oughou each in e al, and, in addi ion o he p e iously s a ed equa ions, inequali y pa h con-
s ain s a e imposed o en o ce he minimum and maximum bounds on
κ
,
αn
, and
σ
wi hin e e y
in e al; hese cons ain s a e exp essed as:
0≤κk≤1
−π
2≤αnk ≤π
2
0≤σk≤2π
Wi h hese elemen s in place, he ull p oblem can now be exp essed in an NLP o mula ion:
32 Chap e 3. Op imal mission planning
sail p opulsion.
Unlike impulsi e maneu e s, which ely on sho , high ene gy bu ns, sola sails gene a e a
cons an bu e y low accele a ion o e long pe iods o ime. In his case, wi h a cha ac e is ic
accele a ion o
ac=1mm/s2
, he spacec a g adually spi als ou om Ea h’s o bi o each Ma s.
This esul s in longe ans e du a ions bu signi ican ly educes he need o p opellan , enabling
highly e icien missions.
F om a physical s andpoin , he ime is alid and expec ed. The ajec o y makes op imal use o
he con inuous sola adia ion p essu e, and he longe du a ion allows he sail o achie e he equi ed
heliocen ic dis ance and phase angle alignmen wi h Ma s using only he gen le, cumula i e push
om sunligh .
When ocusing on he con ol pa ame e s, a bang–bang beha io is obse ed, as an icipa ed; his
con ol s a egy applies maximal co ec ions a each poin o ensu e e icien spacec a a i al a
he a ge , in ag eemen wi h he beha io seen in he o he pa ame e s.

3.2 Simula ions and esul s 33
3.2.2 Ea h To Jupi e ajec o y
The ans e ence be ween Ea h and Jupi e is analyzed. In addi ion, his case is sligh ly mo e
complex han be o e since we ha e o cope wi h bigge dis ances and need o be mo e accu a e.
Consequen ly, he simula ion ime has inc eased no ably, which makes sense wi h he scale o he
p oblem. Howe e , we ound a good and ealis ic solu ion.
•Op imal Time ≈60.58705 UT ≈9.643 yea s.
•Launch Da e
:
2031 JANUARY 01 00 : 00 : 00 UT
. I is conside ed o be a good op imal
da e because i gi es a easonable ime o he ans e ence and se e al simula ions has been
done and he esul s show ha he op imal da e is close o his mon h.
•Numbe o poin s
: N=275. In his case we needed mo e p ecision han be o e o his
eason he numbe o nodes ha e inc eased.
•Compu a ion ime:13074.17 s≈3.632 h.
Figu e 3.7 Ea h o Jupi e T ans e ence wi h a cha ac e is ics accele a ion o ac=1 mm/s2.
34 Chap e 3. Op imal mission planning
Figu e 3.8
Con ol pa ame e s h oughou he ajec o y om Ea h o Jupi e wi h
ac=1 mm/s2
.
Figu e 3.9
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Jupi e wi h
ac=1 mm/s2.
3.2 Simula ions and esul s 35
Figu e 3.10
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Jupi e wi h
ac=1 mm/s2.
Figu e 3.11 T ajec o y in 2-D be ween Ea h and Jupi e compa ed wi h he ini ial guess.
36 Chap e 3. Op imal mission planning
In his case, i is ob ious ha due o he posi ion o Jupi e ( a es han Ma s) i will be necessa y
o ob ain mo e ene gy o each he o bi , his is why we no ice a change in he shape o he o bi
and we see mo e u ns a ound he Sun be o e each o Jupi e wi h a endez- ous. This beha iou ,
can be also seen in he posi ion and eloci y a iables, in which when he spacec a is close o he
Sun he eloci ies each a maximum alue. Focusing on he compa ison be ween op imal ans e
and ini ial guess i can be said ha in he i s s ages o he o bi ei he ma ch each o he (so we can
assume ha he ini ial guess is good) bu in he inal ones he di e ence be ween guess and inal
o bi is high no ably, showing he easibili y o he me hod.
3.2 Simula ions and esul s 37
3.2.3 Ea h To Dionysus ajec o y
The ans e be ween Ea h and Dionysus is analyzed using a mesh o 250 poin s o achie e highe
p ecision on his mo e eccen ic and inclined o bi . Dionysus, an Apollo ype nea Ea h as e oid
(NEA) composed p ima ily o silica es, o e s an oppo uni y o gain new insigh s in o he o ma ion
o Ea h and he sola sys em. The esul s ob ained a e:
•Op imal Time ≈21.95278 UT, his is almos 1277 days.
•Launch Da e
:
2030 JANUARY 01 00 : 00 : 00 UT
. This da e was chosen a e analyzing
se e al op ions and de e mining ha he op imal ans e da e lay close o i . The as e oid’s
synodical pe iod is app oxima ely 1.38yea s, which de ines he launch window, so, wi h an
app op ia e da e (such as he one selec ed), i is possible o each Dionysus.
•Numbe o poin s: N=250.
•Compu a ion ime:12150.39 s≈3.37 h.
Figu e 3.12 Ea h o Dionysus T ans e ence wi h a cha ac e is ics accele a ion o ac=1 mm/s2.

38 Chap e 3. Op imal mission planning
Figu e 3.13
Con ol pa ame e s h oughou he ajec o y om Ea h o Dionysus wi h
ac=
1 mm/s2.
Figu e 3.14
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Dionysus wi h
ac=1 mm/s2.
3.2 Simula ions and esul s 39
Figu e 3.15
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Dionysus wi h
ac=1 mm/s2.
Figu e 3.16 T ajec o y in 2-D be ween Ea h and Dionysus compa ed wi h he ini ial guess.
40 Chap e 3. Op imal mission planning
In his sec ion, he compa ison be ween he inal op imal ajec o y and he ini ial guess has
been added, enabling a clea e unde s anding o he esul s. Focusing on he g aphical ou pu s,
he ajec o y and eloci y p o iles o he Dionysus mission exhibi physically cohe en beha io ,
e lec ing he g adual, con inuous changes cha ac e is ic o low h us p opulsion. Ra he han he
ab up jumps ypical o impulsi e bu ns, he spacec a ’s pa h ansi ions smoo hly om Ea h
depa u e h ough ans e adjus men s o inal eloci y ma ching wi h he as e oid. This beha io
ma ked by modes con inuous accele a ion, oscilla o y co ec ions, and p og essi e educ ion in
ela i e eloci y aligns wi h op imized low h us ajec o ies ha espec o bi al mechanics and
e icien uel managemen . Mo eo e , he highes eloci ies occu a he pe ihelion poin s, consis en
wi h inc eased sola p essu e and g a i a ional in luence close o he Sun.
3.2 Simula ions and esul s 41
3.2.4 Ea h To Didymos ajec o y
The Ea h o Didymos ans e is analysed using a mesh o 275 poin s, o e ing a sui able balance
be ween p ecision and compu a ional speed. The inclusion o his small body is jus i ied by NASA’s
DART (Double As e oid Redi ec ion Tes ) mission o he Didymos-Dimo phos sys em, which
demons a ed as e oid de lec ion echniques by impac ing Dimo phos o al e i s o bi a ound
Didymos. The esul s ob ained o his ans e a e as ollows:
•Op imal Time ≈15.09409 UT, his is almos 877 days.
•Launch Da e
:
2031 JANUARY 01 00 : 00 : 00 UT
. This da e is di e en om he p e ious
since i is one o he possibles op imal da e o he mission.
•Numbe o poin s
: N=275. Fo his case, he numbe o mesh poin s was inc eased o
enhance solu ion accu acy and enable a sensi i i y analysis o he ans e ime; consequen ly,
he compu a ional ime inc eased no ably.
•Compu a ion ime:20279.74s≈5.63 h.
Figu e 3.17
Compa ison be ween Ea h o Didymos T ans e ence wi h a cha ac e is ics accele a ion
o ac=1 mm/s2.
48 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
By inc easing he numbe o con ol poin s, a quad a ic Bezie cu e can be made, which
ma hema ical o m is:
B(τ) = (1−τ)[(1−τ)P0+τP1]+τ[ (1−τ)P1+τP2]whe e 0 ≤τ≤1(4.2)
Which can be in e p e ed as he linea in e polan o he co esponding poin s on he linea Bezie
cu es om P0 o P1and om P1 o P2, his is:
Figu e 4.2 Bezie cu e o o de 2.
As can be deduced, inc easing he numbe o poin s imp o es he app oxima ion o he o iginal
cu e. When he numbe o con ol poin s is aised o eigh , he esul ing i appea s as ollows:
Figu e 4.3 Bezie cu e o o de 8.
The Bezie cu e shapes he con ol poin s in each case; he main pu pose is o de e mine he
op imal con ol poin s ha sa is y he cons ain s along he spacec a ’s ajec o y, om which he
op imal ans e pa h is compu ed.

4.1 De ini ion o Bezie Cu es 49
4.1.1 Polynomial o m
The Bezie cu es ha e been de ined as a linea combina ion o bo h he pa ame e
τ
and he con ol
poin s; bu hey can be gene alized as ollows:
B(τ) =
n
∑
i=0n
i(1−τ)n−iτiPi
= (1−τ)nP0+n
1(1−τ)n−1τP1+n
n−1(1−τ)τn−1Pn−1+τnPn,0⩽τ⩽1
(4.3)
Whe e n
ia e he binomial coe icien s.
When he exp ession is adap ed o he p esen p oblem, he s a e a iables can be exp essed in
e ms o Be ns ein polynomials and he con ol poin s.[5]
Fo he o mula ion he de ini ion o he spacec a dimensionless cylind ical componen s and
ime a e p esen ed:
[˜
ρ,θ,˜z]and τ≜
,wi h τ∈[0,1].(4.4)
Using his shape-based app oach o he op imal ans e ajec o y design; he componen s o
he spacec a (dimensionless) posi ion ec o
˜
ρ,θ,˜z
a e expanded in he domain o
τ
using Bezie
unc ions o o de n ∈N≥3, gi en by
i(τ) =
n
∑
j=0
Bj(τ)Pi,jwi h i = [ ˜
ρ,θ,˜z](4.5)
whe e
Pi,j
a e he unknown geome ic coe icien s (i.e, he so-called con ol poin s), and
Bj(τ)
a e he Be ns ein basis polynomials o deg ee n, de ined as [1]:
Bj(τ) = n!τj(1−τ)n−j
j!(n−j)!wi h j ∈[1,2,...,n](4.6)
In gene al, a di e en alue o n can be chosen o each spacec a s a e ( ha is
nρ,nθ,nz
. Howe e ,
o he sake o simplici y, in a i s app oach, i is conside ed he same numbe o each s a e a iable.
Taking in o accoun equa ions (3.36) and (3.37), he i s and he second
τ−
de i a i es o he
i h−
coo dina e app oxima ion can be w i en, in a compac o m, as
i′(τ) = ˜ i=
n
∑
j=0
B′j(τ)Pi,jwi h i = [ ˜
ρ,θ,˜z](4.7)
and
i′′(τ) = ˜ i=
n
∑
j=0
B′′
j(τ)Pi,jwi h i = [ ˜
ρ,θ,˜z](4.8)
whe e he de i a i e o he Be ns ein polynomial is:
B′j(τ) = 










−n(1−τ)n−1i j=0
n!τj−1(1−τ)n−j
(j−1)!(n−j)!−n!τj(1−τ)n−j−1
j!(n−j−1)!i j∈[1,n−1]
nτn−1i j=n
(4.9)
and
50 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
B′′
j(τ) =

































n(n−1)(1−τ)n−2i j=0
n(n−1)(n−2)τ(1−τ)n−3−2n(n−1)(1−τ)n−2i j=1
n!τj−2(1−τ)n−j
(j−2)!(n−j)!−2n!τj−1(1−τ)n−j−1
(j−1)!(n−j−1)!
+n!τj(1−τ)n−j−2
j!(n−j−2)!i j∈[2,n−2]
n(n−1)(n−2)τn−3(1−τ)−2n(n−1)τn−2i j=n−1
n(n−1)τn−2i j=n
(4.10)
In pa icula , he bounda y alues o
Bj
and
B′j
a e ob ained by subs i u ing
τ=0
(ini ial ime
ins an ), o τ=1( inal ime) in equa ions (3.40) and (3.41), we ge :
Bj(0) = 


1i j=0
0i j∈[1,n]
(4.11)
Bj(1) = 


0i j∈[0,n−1]
1i j=n
(4.12)
B′j(0) = 










−ni j=0
ni j=1
0i j∈[2,n]
(4.13)
B′j(1) = 










0i j∈[0,n−2]
−ni j=n−1
ni j=n
(4.14)
Using equa ions (3.36) and (3.38), he bounda y alue o he gene ic dimensionless
i h−
coo di-
na e app oxima ion and i s i s τ−de i a i e a e gi en by
i(0) = Pi,0,i(1) = Pi,n,˜ i(0) = nPi,1−Pi,0,
˜ i(1) = nPi,n−Pi,n−1(4.15)
wi h i=
[˜
ρ,θ,˜z]
. The wel e geome ic coe icien s
[Pi,0,Pi,1,Pi,n,Pi,n−1]
can be ob ained, as a
unc ion o he gi en componen s o he spacec a s a e ec o a he ini ial and he inal ime
, by
combining
P˜
ρ,0=ρ0
⊕
,Pθ,0=θ0,P˜z,0=z0
⊕
(4.16)
4.2 P oblem o mula ion including Bezie Cu es 51
P˜
ρ,1= ρ0
⊕
+ρ0
⊕
,Pθ,1= θ0
n ⊕
+θ0,P˜z,1= z0
⊕
+z0
⊕
(4.17)
P˜
ρ,n−1=ρ
⊕− ρ
n ⊕
,Pθ,n−1=θ − θ
n ⊕
,P˜z,n−1=z
⊕− z
n ⊕
,(4.18)
P˜
ρ,n=ρ
⊕
,Pθ,n=θ ,P˜z,n=z
⊕
.(4.19)
As desc ibed abo e, he Bezie cu es me hod has been de ined o he p esen p oblem and can
now be o mula ed o ob ain he op imal ans e ; since he a i al and depa u e condi ions a e
ixed by he Bezie coe icien s, only he ee coe icien s along he ajec o y equi e op imisa ion,
esul ing in 3(n−3)+1unknown pa ame e s.[6]
4.2 P oblem o mula ion including Bezie Cu es
Fo a gi en mission scena io, he p oblem is o ind he minimum ime ans e ajec o y such
ha he cons ain s o he accele a ion ec o a e all sa is ied. To ha end, he componen s o he
p opulsi e accele a ion ec o
{ax0,ay0,az0}
a e exp essed as unc ions o he Bezie cu e-based
app oxima ion gi en by equa ions (4.4) and (4.3) using he ollowing p ocedu e.
Fo a gi en ligh ime
and a se o
N
unknown geome ic coe icien s
{Pi,2,...,Pi,n−2}
, wi h
n>3and
i∈{˜
ρ,˜
θ,˜z}
he dimensionless componen s o he p opulsi e accele a ion
{˜aρ,˜aθ,˜az}
can be ob ained om he
equa ions o mo ion by mo ing he g a i a ional and ine ial e ms om he igh -hand side o he
le -hand side.
The componen s o he p opulsi e accele a ion
{aρ,aθ,az}
a e hen calcula ed om equa ions
(2.23), whe eas he componen s
{ax0,ay0,az0}
may be w i en, acco ding o equa ions (2.14), (2.22)
and (4.26), as


ax0
ay0
az0

=

cosφ0−sinφ
0 1 0
sinφ0 cosφ


aρ
aθ
az

(4.20)


ax0
ay0
az0

=








aρ˜z−az˜
ρ
p˜
ρ2+˜z2
aθ
aρ˜
ρ+az˜z
p˜
ρ2+˜z2








,(58)
whe e φis he o bi al inclina ion angle.
In o de o ob ain he componen s o he dimensionless accele a ion ec o , i can be w i en he
dynamical equa ions in a dimensionless o m using
⊕
and he
ha de ine he pa ame e
τ
. Doing
his, we ob ain he ollowing exp essions:
τ≜
,τ∈[0,1](4.21)
˜
ρ′=˜ ˜
ρ≜ ˜
ρ(4.22)
52 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
θ′=˜ θ
˜
ρ≜ θ(4.23)
˜z′=˜ ˜z≜ ˜z(4.24)
˜ ′˜
ρ=˜ 2
θ
˜
ρ−˜
µ⊙˜
ρ
(˜
ρ2+˜z2)3/2+κ˜ac
2˜
ρcos2αn+˜zsinαncosαncosσ+˜
ρ≜ ˜ ˜
ρ(4.25)
˜ ′
θ=−˜ ˜
ρ˜ θ
˜
ρ+κ˜ac
2cosαnsinαnsinσp˜
ρ2+˜z2≜ ˜ θ(4.26)
˜ ′˜z=−˜
µ⊙˜z
(˜
ρ2+˜z2)3/2+κ˜ac
2˜zcos2αn−˜
ρsinαncosαncosσ+˜z≜ ˜ ˜z(4.27)
Whe e he a iables a e de ined by:
˜
ρ≜ρ
⊕
,˜z≜z
⊕
,˜
µ≜µ⊙ 2
3
⊕
,˜ac≜ac 2
⊕
,
˜ ˜
ρ≜ ρ
⊕
,˜ θ≜ θ
⊕
,˜ ˜z≜ z
⊕
.
(4.28)
Wi h his o mula ion, he dimensionless componen s o he spacec a ’s accele a ion ec o a e
compu ed di ec ly om he Bezie coe icien s; by en o cing he accele a ion cons ain s a he
accele a ion le el since accele a ion is exp essed in he same polynomial basis he need o in eg a e
he dynamics o eco e eloci ies and posi ions is ob ia ed, and he accele a ion ec o cons ain s
a e imposed con inuously along he en i e ajec o y.
Focusing on his exp essions we can easily iden i y he componen s ha we a e looking o :
˜a˜
ρ=κ˜ac
2(˜
ρcos2αn+˜zsinαncosαncosσ+˜
ρ)(4.29)
˜a˜
θ=κ˜ac
2cosαnsinαnsinσp˜
ρ2+˜z2(4.30)
˜a˜z=k˜ac
2˜zcos2αn−˜
ρsinαncosαncosσ+˜z(4.31)
By subs i u ing hese exp essions in o equa ions (4.24)–(4.26) and ans e ing he g a i a ional
e m o he le hand side, he dimensionless accele a ion componen s can be w i en explici ly as
unc ions o he Bezie coe icien s
˜ ′
θ+˜ ˜
ρ˜ θ
˜
ρ=˜a˜
ρ(4.32)
˜ ′
θ+˜ ˜
ρ˜ θ
˜
ρ=˜a˜
θ(4.33)
˜ ′˜z+˜
µ⊙˜z
(˜
ρ2+˜z2)3/2=˜a˜z(4.34)
Since he dimensional componen s ha e been exp essed in e ms o he dimensionless ones, he
analy ical ela ions be ween hem ollow by using he no malized ime pa ame e
τ= /
, yielding
he equi alences:
4.2 P oblem o mula ion including Bezie Cu es 53
d2
d 2=1
2
d2
dτ2(4.35)
aτ(τ) = 2
⊕
a( )⇐⇒ a( ) = ⊕
2
aτ(τ)(4.36)
So, i can be ob ained he ela ionship be ween dimensional and dimensionless componen s:
aρ= ⊕
2
˜aρ(4.37)
aθ= ⊕
2
˜aθ(4.38)
az= ⊕
2
˜az(4.39)
Wi h hese esul s, he dimensional accele a ion ec o componen s a e ob ained in e ms o he
Bezie coe icien s no ing ha he dimensionless componen s a e compu ed ia he Bezie expansion
and he ca esian ame componen s [ax0,ay0,az0]can hen be calcula ed as ollows:


ax0
ay0
az0

=

cosφ0−sinφ
0 1 0
sinφ0 cosφ


aρ
aθ
az

=








aρ˜z−az˜
ρ
p˜
ρ2+˜z2
aθ
aρ˜
ρ+az˜z
p˜
ρ2+˜z2








.(4.40)
Wi h hese exp essions, he Ca esian componen s o he accele a ion a e de ined by he Bézie
coe icien s, enabling us o o mula e he accele a ion cons ain ha he spacec a mus sa is y
con inuously along i s ajec o y.
4.2.1 Accele a ion Cons ain s
In Chap e 2, he cons ain was o mula ed di ec ly om he sys em dynamics. Now ha he
di e en ial equa ions ha e been emo ed om he cons ain se , hose cons ain s mus be eph ased
in e ms o he accele a ion ec o , which i sel is a unc ion o he Bezie coe icien s. In so doing,
he ajec o y es ic ions a e imposed di ec ly on he coe icien s being op imized. To ha end, he
ans e se uni ec o is in oduced:
ˆ
≜ˆ
׈
n׈
sinαn
wi h αn=0(4.41)
The p opulsi e accele a ion ec o lies on he plane spanned by
ˆ
and
ˆ
since i may be w i en in
e ms o adial and ans e se componen s:
a=ac ⊕
˜a ˆ
+˜a ˆ
,(4.42)
whe e
˜a =κcos2αn+1
2,˜a =κsinαncosαn
2.(4.43)

54 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
Making a compa ison be ween equa ions (4.37) and (2.20) e eals ha :
a ≜˜a ac ⊕
≡az0(4.44)
a ≜˜a ac ⊕
=qa2
x0+a2
y0(4.45)
And combining equa ions (4.37) i may be e i ied ha :
˜a2
+˜a −3
4κ2=κ
42(4.46)
As
κ
a ies o e he in e al
[0,1]
, he ci cles de ined by equa ion (4.40) enclose a egion
S
in
which he dimensionless h us accele a ion mus lie. Because he sail pi ch angle
αn∈[−π
2,π
2]
(so
ha he no mal
ˆn
always poin s an i-Sun), he angen om he o igin o any o hese ci cles sa is ies
˜a =2√2 ˜a ,
ega dless o
κ
. The e o e, he lowe bounda y o
S
is p ecisely his line o slope
2√2
, and one may
cha ac e ize
S=(˜a ,˜a )˜a ⩾2√2 ˜a ,(˜a ,˜a )lies on o below he ci cle o some κ∈[0,1].
Figu e 4.4
Dimensionless P opulsi e accele a ion componen s and admissible egion o he p opul-
si e accele a ion.
Wi h he aid o he geome y in he igu e, he egion can be di ided in o wo cons ain s in e ms
o ˜a ; hus, he dimensionless cons ain s o be sa is ied by he accele a ion ec o a e:
S=˜a ,˜a : ˜a ⩽






˜a
2√2,˜a ∈[0,2
3],
1
16 −˜a −3
42,˜a ∈[2
3,1].
(4.47)
4.2 P oblem o mula ion including Bezie Cu es 55
Finally, o a gi en adius
and cha ac e is ic accele a ion
ac
, he p opulsi e accele a ion compo-
nen s {ax0,ay0,az0}mus sa is y:
qa2
x0+a2
y0⩽az0
2√2,az0
ac( ⊕/ )∈[0,2
3],(4.48)
qa2
x0+a2
y0⩽ac
⊕
s1
16 −az0
ac( ⊕/ )−3
42,az0
ac( ⊕/ )∈[2
3,1].(4.49)
Finally, he es ic ions on he accele a ion ec o componen s ha mus be sa is ied by he
spacec a a each poin ha e been ob ained; o en o ce hese cons ain s,
m=150
poin s a e chosen
along he ajec o y.[6]
4.2.2 Fixed Coe icien o he ans e ence
In he op imiza ion, we employ MATLAB’s
mincon
o de e mine he ee Bezie con ol poin s
namely he
3(n−3)
unknown pa ame e s while he wel e bounda y coe icien s emain ixed by
he depa u e and a i al condi ions. By combining bo h ixed and op imized coe icien s, we
econs uc he comple e in e plane a y ajec o y. The alues o he ixed coe icien s we e de ined
p e iously in equa ions (4.16)-(4.19)
These coe icien s a e en i ely dic a ed by he depa u e and a i al bounda y condi ions. To
s eamline hei use in ou op imiza ion p ocedu e, we assemble hem in o a single ec o wi h he
ollowing s uc u e:
P=Pρ,0Pρ,1··· Pρ,n−1Pρ,nPθ,0Pθ,1··· Pθ,n−1Pθ,nPz,0Pz,1··· Pz,n−1Pz,nT(4.50)
By s uc u ing he ixed bounda y coe icien s his way, we a oid ecompu ing he depa u e and
a i al condi ions a each i e a ion. Ins ead, we impose hem once as cons ain s and le he op imize
sol e only o he emaining ee pa ame e s, signi ican ly imp o ing compu a ional e iciency.
56 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
4.2.3 Ini ial Guess
The basic idea o ini ialize he unknown coe icien s in he NLP sol e is o p o ided an app oxima ion
o he spacec a coo dina es [
ρ,θ,z
] a he m disc e iza ion poin s. The unknown coe icien s can
hem be calcula ed by i ing he unc ions o Bezie cu es h oughou his se o disc e e poin s.
No e ha he o al ligh ime
is a a iable ha mus be op imized. Howe e , a guess alue o
he ligh ime
app
may be ob ained as he a io o he no m o he ec o ial di e ence be ween
he angula momen um o a ge and pa king o bi s o he E-sail o que induced by h us . This
exp ession is:
app =qµ⊙p0+p −2µ⊙√p0p cos∆i
κapp ac ⊕cosαn1sinαn1
(4.51)
whe e
p0/p
a e he semila us ec um o he pa king o bi and he a ge o bi espec i ely, also,
∆i
is he a ia ion in o bi al inclina ion,
κapp ≜1
3
is he es ima ed a e age h us coe icien s, and
αn1
≜
54.7 deg is he sail pi ch angle ha maximizes he h us cone angle. To ge a be e imp ession
abou his app oxima ion, in he case o Ma s, he ollowing esul s a e gi en:
p0=1AU
p =1.5104 AU
ac=0.1686 UA
app =8.7427 UT
(4.52)
Compa ing his i s app oxima ion wi h he op imal esul ob ained o he Ea h–Ma s di ec a-
jec o y,
op =8.191067UT
, demons a es ha i cons i u es a good s a ing poin o he op imisa ion
p ocess.
Fi s , a e an ini ial es ima e o he ans e ime has been compu ed, a p elimina y guess o he
ee Bezie coe icien s is gene a ed by i ing a hi d-o de Bezie cu e o he bounda y condi ions
hus de e mining all coe icien s and e alua ing his cu e a he
m
disc e iza ion poin s; hese
alues se e as ini ial es ima es o he ee coe icien s, which a e hen supplied o he op imize o
i e a i e e inemen . I is necessa y o calcula e he ixed coe icien s:
Pρ,0=ρi,Pρ,1=ρi+ app ρi
3,Pρ,2=ρ − app ρ
3,Pρ,3=ρ ,
Pθ,0=θi,Pθ,1=θi+ app θi
3,Pθ,2=θ − app θ
3,Pθ,3=θ ,
Pz,0=zi,Pz,1=zi+ app zi
3,Pz,2=z − app zi
3,Pz,3=z .
(4.53)
No e ha he alue o
θ
needs o be upda ed in each
τk
s ep, due o his, alue changes h ough he
ajec o y. This is:
θ k+1=θ k+2nπn=1,2... (4.54)
This is specially essen ial wi h he ini ial guess gene a ion, since i his de ail is no inco po-
a ed, a w ong esul s could be ob ained. Wi h he hi d o de Bezie cu e, he app oxima ion o
ρapp,θapp,zapp can be w i en as:
ρapp(τ) = (1−τ)3Pρ,0+3τ(1−τ)2Pρ,1+3τ2(1−τ)Pρ,2+τ3Pρ,3,
θapp(τ) = (1−τ)3Pθ,0+3τ(1−τ)2Pθ,1+3τ2(1−τ)Pθ,2+τ3Pθ,3,
zapp(τ) = (1−τ)3Pz,0+3τ(1−τ)2Pz,1+3τ2(1−τ)Pz,2+τ3Pz,3.
(4.55)
4.2 P oblem o mula ion including Bezie Cu es 57
Consequen ly, he disc e e app oxima ion da a alues o
ρapp,θapp and zapp
can be ob ained by
e alua ing equa ions a he m disc e iza ion poin s, o his wo k m=150. Then an ini ial guess
o he unknown pa ame e s in bezie unc ions can be ob ained om equa ion (4.4). Besides, a
low diag am is p esen e in o de o ge a be e comp ehension abou how o implemen hese las
sec ions oge he .
Ini ial Guess P ocess
Inpu da a
x0= (ρi,˙
ρi,θi,˙
θi,zi,˙zi),
x = (ρ ,˙
ρ ,θ ,˙
θ ,z ,˙z ),
ac, ⊕,κapp,n
1. Es ima e app
app =√µ⊕(p0+p )−2µ⊕√p0p cos∆i
κapp ac ⊕cosαn1sinαn1
2. Fixed coe icien s
Cubic Bézie (
n=3
). Fo
each i∈{ρ,θ,z}:
Pi,0=i0
Pi,1=i0+ app ˙
i0
3
Pi,2=i − app ˙
i
3
Pi,3=i
3. E alua e app oxima-
ion
Compu e
{ρapp(τk),θapp(τk),zapp(τk)}
a
k=1,...,m
disc e iza ion
nodes.
4. Ini ial Sol e
Sol e o he
3(n−3)
ee
coe icien s
Pi,2...n−2by imposing
n
∑
j=0
Bj(τk)Pi,j=
iapp(τk) (k=1,...,m).
5. Assemble ec o
{ ,app,Pi,0...,n}
eady o mincon.
End o Guessing P ocess
The igu e below ga he s he comple e se o ini ial guesses ha will be used as s a ing poin s
o each celes ial body conside ed in his wo k. E e y en y in his able has been p oduced
wi h he p ocedu e de ailed ea lie : he cubic Bezie amewo k is ixed, bounda y coe icien s
a e assigned, and he emaining deg ees o eedom a e sol ed by en o cing he app oxima ion
condi ions, p esen ing he guesses side-by-side se es a dual pu pose.
Fi s , i highligh s he in e nal consis ency o he me hod, showing how a single algo i hm
yields cohe en esul s ac oss a wide spec um o ini ial condi ions. Second, i o e s a quali a i e
benchma k be o e he ec o s a e handed o he
mincon
op imize o u he e inemen . All
symbols, uni s, and sign con en ions ma ch hose in oduced in he p eceding sec ions, ensu ing
aceabili y om he heo e ical de elopmen o he g aphical alida ion.
64 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
4.3.2 Ea h o Jupi e T ajec o y wi h Bezie cu es
The solu ion o Ea h o Jupi e is analyzed. I is easy o app ecia e ha he compu a ion ime has
dec eased and, al hough, he ole ance o cons ain s has been inc eased o add ess he accele a ion
ec o cons ain s he esul s a e highly ealis ic. The s ep and cons ain ole ances ha a e being
used ha e a alue o TOL 10−3.
•Op imal Time ≈55.02569 UT which is ≈8.76795 yea s.
•Launch Da e
:
2031 JANUARY 01 00 : 00 : 00 UT
. Close o he launch window o Jupi e
in 2031
•Bezie cu e o de : n=8.
•Compu a ion ime:74.8118s
Figu e 4.14 Ea h o Jupi e T ans e ence using Bezie Cu es.

4.3 Simula ions and esul s using Bezie cu es 65
Figu e 4.15
Con ol pa ame e s h oughou he ajec o y om Ea h o Jupi e wi h a Bezie cu e
o o de 8.
Figu e 4.16
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Jupi e wi h a
Bezie cu e o o de 8.
66 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
Figu e 4.17
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Jupi e wi h a
Bezie cu e o o de 8.
Figu e 4.18 T ajec o y in 2-D be ween Ea h and Jupi e compa ed wi h he ini ial guess.
4.3 Simula ions and esul s using Bezie cu es 67
To ob ain a physically easible solu ion and o he con e gence o he p oblem, he ole ance
was inc eased. Al hough his lowe ole ance may make he esul s less eliable, and hey can be
unexpec ed om he solu ion ha was hough, hey s ill p oduced a physically plausible ajec o y
whose inal ans e ime ma ches ha o he NLP solu ion. The u u e wo k mus ocus on enhancing
his.
No e ha o his solu ion, he inal op imal cu e has a b oaden o m han in he NPL p oblem,
bu a con inuous e olu ion o ei he , s a e a iables and eloci ies shows ha his ajec o y is
physically possible. Mo eo e , he key ac o o in oducing his me hod (as i has been men ioned
be o e) is he elimina ion o ough changes in he accele a ion ec o , ob aining a so e shape o
he h us pa ame e κ, which a oid nume ical p oblem, specially i de i a ion is necessa y.
Mo ing on he bidimensional ep esen a ion, he ini ial guess cu e di e s a li le bi mo e han in
he p e ious one, his migh be a signal o imp o ing he accu acy o he ini ial guess me hod, when
u he dis ances a e in ol ed. I is likely ha hus i has been necessa y o dec ease he ole ance
cons ain s.
The loops ha has been added a e sligh ly supe io (n=3) his due o he posi ion o Jupi e , abou
5.2 AU, so he spacec a will need mo e o each he objec i e.[4.2.3]
68 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
4.3.3 Ea h o Dionysus T ajec o y wi h Bezie cu es
In his sec ion he ans e ence be ween Ea h and Dyonisus is p esen ed, in o de o ge hese esul s
he cons ain ole ance has been inc eased (TOL
10−4
), since mee ing he accele a ion ec o
cons ain has been mo e di icul his ime, howe e he esul s ha a e ob ained a e physically
possibles. In addi ion, a cha ac e is ic accele a ion o
ac=1 mm/s2
is used. As i has been saw
be o e, he educ ion on he compu a ional ime is qui e signi ican , so we can p o e ha he me hod
wo ks. The esul s a e:
•Op imal Time ≈20.06444 UT which is 1167.75 days.
•Launch Da e:2031 JANUARY 01 00 : 00 : 00 UT.
•Bezie cu e o de : n=8.
•Compu a ion ime:202.4325s.
Figu e 4.19 Ea h o Dionysus T ans e ence using Bezie Cu es.
4.3 Simula ions and esul s using Bezie cu es 69
Figu e 4.20
Con ol pa ame e s h oughou he ajec o y om Ea h o Dionysus wi h a Bezie
cu e o o de 8.
Figu e 4.21
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Dionysus wi h
a Bezie cu e o o de 8.

70 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
Figu e 4.22
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Dionysus wi h
a Bezie cu e o o de 8.
Figu e 4.23 T ajec o y in 2-D be ween Ea h and Dionysus compa ed wi h he ini ial guess.
4.3 Simula ions and esul s using Bezie cu es 71
The eloci y p o iles emain ee o spikes, ensu ing ha he sola -sail endu es only gen le,
con inuous h us ing; his is e lec ed in he con ol-pa ame e plo s, whe e he “pi ch” angle
αn
swi ches only once om a high- h us egime nea depa u e o a shallow, coas -op imized a i ude,
while he clocking angle
σ
and cu a u e
κ
desc ibe single, well shaped lobes a he han oscilla o y
bang-bang beha io . In he 2-D p ojec ion a be e es ima ion o he ini ial guess migh be done in
o o pu o wa d a mo e e icien op imiza ion p ocess, mo eo e , a couple o manual loops ha e
been added o ge ing a good ini ial guess.[4.2.3]
By compa ison, he classical di ec - ansc ip ion app oach (N = 250 RK4 nodes) eaches Dyonisus
in abou 1 277 days nea ly 110 days longe while demanding o e
12150s(≈3.4h)
o CPU ime.
Thus, he Bezie me hod no only ims ligh ime by 9 % bu also slashes compu a ion ime om
hou s o jus
≈200s
, all wi hou comp omising he physical ealism o dis ances, eloci ies, o
con ol demands.
72 Chap e 4. Op imal Mission Planning wi h Bezie Cu es
4.3.4 Ea h o Didymos T ajec o y wi h Bezie cu es
Finally he endez ous ajec o y be ween Ea h and Didymos is p esen ed. This ime a sligh ly
in e io op imal ime has been ob ained bu despi e ha , he esul is comple ely alid in a ea ly
s age design mission. Howe e in his case, a solu ion wi h a high alue o ole ance (TOL
10−6
)
has been ound, as an imp o emen om he p e ious analysis. The esul s a e:
•Op imal Time ≈11.65088 UT which is 678.03 days.
•Launch Da e:2031 JANUARY 01 00 : 00 : 00 UT.
•Bezie cu e o de : n=8.
•Compu a ion ime:259.07384s
Figu e 4.24 Ea h o Didymos T ans e ence using Bezie Cu es.
4.3 Simula ions and esul s using Bezie cu es 73
Figu e 4.25
Con ol pa ame e s h oughou he ajec o y om Ea h o Didymos wi h a Bezie
cu e o o de 8.
Figu e 4.26
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Didymos wi h
a Bezie cu e o o de 8.
80 Chap e 5. Conclusions and u u e wo k
e sus 877 days and 5.6 h o he con en ional sol e which shows he high e ec i i y o he
polynomial me hod.
Mo eo e ,a dedica ed s udy on Dionysus showed ha aising he Bezie deg ee om
n=10
o
n=40
sho ens
by
17 %
bu inc eases un ime nine old; diminishing e u ns appea beyond
n≈20
, which cap u es
97 %
o he a ainable sa ing o only
55 %
o he maximum CPU cos .
This iden i ies n∈[10,20]as a swee spo o p elimina y E-sail design.
Table 4.2 demons a ed ha p opulsi e upg ades ha e hei g ea es le e age in he low h us
egime: aising
ac
om 0.5 o 0.6
mm/s2
sha ed almos 400 days o he Ea h o Ma s c uise,
whe eas he same 0.1
mm/s2
inc emen abo e
1 mm/s2
yielded a me e 13 day imp o emen . Wi h
his app oach also a op imal ange o he achas been analyzed.
The smoo hness and explici accele a ions con inui y o Bezie ajec o ies make hem ideal
e e ence p o iles o he LQR and Model P edic i e Con ol schemes ha can be added in u u e
p ojec s. Because he polynomial solu ion deli e s analy ic s a e de i a i es, he guidance laws can
be uned wi hou nume ical di e en ia ion, educing onboa d compu a ional load and imp o ing
obus ness o sola wind luc ua ions.
All ajec o ies we e c oss checked agains NASA HORIZONS epheme ids; g aphical o e lays
con i med ha he op imised pa hs emain wi hin a ew hund ed hs o an AU o he published o bi al
planes. Benchma k compa isons (Appendix A) u he showed excellen ag eemen wi h ligh
imes epo ed in ecen E-sail s udies.
In summa y, his hesis es ablishes a ep oducible, apid con e gence pipeline o E-sail mission
ea ly design due o:
•
Reduces op imisa ion un imes om hou s o seconds wi hou sac i icing oo much physical
accu acy.
•P o ides smoo h, con olle - iendly s a e and con ol his o ies.
•Quan i ies key design ades in ol ing cha ac e is ic accele a ion and polynomial o de .
•Demons a es e sa ili y ac oss plane a y and high-eccen ici y as e oid a ge s.
Taken oge he , hese achie emen s ma k a signi ican s ep owa ds in eg a ing ad anced shape
based echniques in o ou ine in e plane a y mission planning, pa ing he way o cos e ec i e
explo a ion o he Sola Sys em wi h p opellan less con inuous h us .
All o hese s udies ha e heigh ened ou awa eness o he nume ous op imal ajec o ies wi hin
he Sola Sys em and, mo eo e , ha e unde sco ed how c ucial i is o accoun o mission ime.
5.2 Fu u e Wo k
This hesis has delibe a ely elied on a se o simpli ying assump ions o ende he E-sail op i-
misa ion p oblem ac able. A he same ime, Op imal Con ol Theo y is e ol ing apidly and is
being ex ended o e e mo e sophis ica ed ae ospace sys ems. Consequen ly, a wide spec um o
enhancemen s can be en isioned o he me hodology de eloped he e. The esul s p esen ed o e a
clea baseline o wha can be achie ed wi h cu en shape based and di ec ansc ip ion echniques
o Elec ic Sola Wind Sail mission design; he p ospec i e e inemen s ou lined below should
he e o e be iewed ei he as means o sha pen he p esen solu ions o as adap a ions ha enable
he amewo k o ackle new a ge s and ope a ional scena ios. The mos p omising a enues o
imp o emen a e summa ised in he ollowing lis .
•
Imp o e Con e genceThe u u e wo k should be o gua an ee con e gence in e e y es ed
scena io wi hou dec easing he ole ance wi h su icien accu acy. Achie ing his goal en ails
ou lining conc e e esea ch di ec ions o ins ance, e examining he ini ial i e a e o imp o e

5.2 Fu u e Wo k 81
i s quali y, expe imen ing wi h al e na i e op imise s, such as CasaDi, and assessing hei
impac on bo h solu ion p ecision and compu a ion ime. These s eps will p o ide a obus
pa h owa d eliably mee ing he desi ed con e gence c i e ia in u u e s udies.
•
Adap i e mesh e inemen . Replace he cu en uni o m g id wi h an adap i e ime–mesh
ha alloca es ex a nodes only whe e he dynamics a e s i o swi ching occu s, p ese ing
accu acy while educing he NLP size and CPU ime.
•
Ad anced model p edic i e guidance. In eg a e eceding and sh inking ho izon MPC
a ian s comple e wi h ho izon adap a ion and cons ain igh ening o o ecas sola -wind
luc ua ions and gene a e con ol ac ions ha keep he E-sail wi hin i s h us “ o ce bubble”
e en unde unce ain y.
•
Concu en launch window op imisa ion. T ea he depa u e da e as an op imisa ion
a iable, allowing he sol e o pick bo h launch epoch and h us his o y ha join ly minimise
o al ime o ligh , a he han ixing he launch da e.
•
Me a heu is ic ini ial guess gene a ion. Use popula ion based algo i hms gene ic, di e en-
ial e olu ion o pa icle swa m o deli e di e se, high quali y s a ing poin s o he NLP,
enla ging he basin o con e gence and e ealing al e na i e ajec o y amilies.
•
G a i y assis ajec o ies. Ex end he dynamics o include hi d body pe u ba ions and
design con inuous h us ans e s ha exploi plane a y lybys, aiming o sho e c uise imes
and lowe cha ac e is ic accele a ion equi emen s.
•
Robus con ol agains sola wind a iabili y. Replace de e minis ic h us bounds wi h a
s ochas ic model o dynamic p essu e and syn hesise obus LQR/MPC con olle s ha mee
e minal cons ain s wi h a chosen con idence le el.
•
Adap i e Bezie segmen a ion. Le he Bezie deg ee and he numbe o pa ches a y along
he pa h, cap u ing sha p manoeu es wi hou global o e pa ame e isa ion and keeping he
decision- ec o compac .
•
Indi ec me hod benchma king. Implemen a Pon yagin based shoo ing sol e o c oss
alida e di ec - ansc ip ion and Bezie esul s, helping de ec possible sub-op imal local
minima and quan i y op imali y gaps.
•
Onboa d eop imisa ion capabili y. Exploi he low-dimensional Bezie pa ame e se o
enable eal ime, mid cou se e a ge ing whene e epheme ids a e upda ed o sail deg ada ion
es ima es change.
Appendix A
Code Valida ion
In his sec ion, he esul s ob ained wi h he MATLAB code de eloped he ein a e compa ed wi h
he solu ions commonly epo ed in he li e a u e; his compa ison alida es he code o he p esen
scena io.
A.1 G aphical Valida ion
Fi s ly, he plo p oduced by he de eloped code o he plane a y and as e oid o bi s is compa ed
wi h he ac ual o bi ob ained om he NASA Ho izons Sys em.
A.1.1 Ea h o Ma s
Figu e A.1 Compa ison be ween o bi al planes o Ea h and Ma s wi h NASA Ho izons Sys em .
Figu e A.2 Compa ison be ween o bi al planes o Ea h and Ma s in he NLP p oblem .
83
84 Chap e A. Code Valida ion
Fo his case, i is known ha Ea h and Ma s o bi in he eclip ic plane, which has an inclina ion
o abou
i=23.5◦
. In he i s image (A.1), bo h plane s a e seen in he eclip ic plane as a e e ence,
so i can be con i med ha hey o bi wi hin he same plane. In he second image, i can be obse ed
ha , unless hey a e o a ed, bo h bodies o bi oge he , he eby e i ying he g aphical solu ion.
A.1.2 Ea h o Jupi e
Figu e A.3
Compa ison be ween o bi al planes o Ea h and Jupi e wi h NASA Ho izons Sys em.
Figu e A.4 Compa ison be ween o bi al planes o Ea h and Jupi e in he NLP p oblem.
Bo h o bi s closely esemble ha depic ed in he op image; consequen ly, he scena io unde
conside a ion accu a ely ep esen s he o mula ed p oblem. As in he p e ious case, bo h plane a y
o bi s lie wi hin he eclip ic plane, so no signi ican incon enience is expec ed o a spacec a o
pe o m a endez ous wi h Jupi e .
A.1 G aphical Valida ion 85
A.1.3 Ea h o Dionysus
Figu e A.5
Compa ison be ween o bi al planes o Ea h and Dionysus wi h NASA Ho izons Sys em.
Figu e A.6 Compa ison be ween o bi al planes o Ea h and Ma s in he NLP p oblem.
As in he p e ious case, he o bi s appea o be qui e simila , and he high eccen ici y o Dionysus
is ep oduced wi h accu acy. The ela i e inclina ion be ween Ea h and Dionysus is likewise
consis en in bo h cases. Addi ionally, a poin a which he o bi s app oach each o he e y closely
can be obse ed, and his ea u e is p esen in he ep esen a ion. The e o e, i can be concluded
ha he model p o ides a good app oxima ion o he eal o bi al con igu a ion.

86 Chap e A. Code Valida ion
A.1.4 Ea h o Didymos
Figu e A.7
Compa ison be ween o bi al planes o Ea h and Didymos wi h NASA Ho izons Sys em.
Figu e A.8 Compa ison be ween o bi al planes o Ea h and Didymos in he NLP p oblem.
Bo h igu es show he o bi al planes o Ea h and Didymos a ound he Sun. The model (Figu e
A.8) closely ma ches he NASA Ho izons e e ence (Figu e A.7), accu a ely cap u ing he inclina ion
be ween he o bi s and he ellip ical shape o Didymos’ ajec o y. The spa ial p opo ions a e
consis en . Despi e s ylis ic di e ences, he physical ep esen a ion in ou model emains ai h ul
o he eal o bi al mechanics. This con i ms ha ajec o y da a and o bi al elemen s a e well
implemen ed and ealis ic in his new case.
A.2 Nume ical Valida ion 87
A.2 Nume ical Valida ion
To es he alidi y o he nume ical esul s, he ans e o bi is compu ed o he cases desc ibed in
Chap e 3 and compa ed wi h o he h ee dimensional models, p o iding an o e all alida ion o
he ob ained esul s. A cha ac e is ic accele a ion o 1mm/s2is employed.
A.2.1 Ea h o Ma s
In his sec ion, wo models a e compa ed o demons a e he accu acy o he de eloped code.
Al hough sligh di e ences exis be ween he models, hese a e explained he ein. Ano he model,
namely [
18
], is employed o compa ison. The co esponding alues a e p esen ed in he nex able.
Fo consis ency, a depa u e da e o 1 Feb ua y 2029 is selec ed, acknowledging ha his da e is no
op imal.
Table A.1
Fligh Times compa a i e be ween di e en models o a Ea h o Ma s ajec o y wi h
ac=1 mm/s2wi h a depa u e da e on 1s Feb ua y 2029.
S udy This P ojec 3 Dimensional model om [18]
Fligh Time [Days] 650 637
Bo h esul s a e qui e simila , di e ing by only 13 days; he e o e, he code can be conside ed
alid o he analysis, and con idence may be placed in he esolu ion me hod, as i yields ealis ic
esul s.
A.2.2 Ea h o Dionysus
As in he p e ious e i ica ion, he model is compa ed o he Ea h o Dionysus ans e ; a launch
da e o 20 Ma ch 2024 is selec ed o he analysis, and he ollowing esul s a e ob ained.
Table A.2
Fligh Times compa a i e be ween di e en models o a Ea h o Dionysus ajec o y
wi h ac=1 mm/s2wi h a depa u e da e on 20 h Ma ch 2024.
S udy This P ojec 3 Dimensional model om [18]
Fligh Time [Days] 1362 1078
In his case, esul s di e om hose ob ained in he p e ious example because a educed numbe
o poin s was employed. Compa ison wi h he analysis p esen ed in 3.6, whe e a inal ime o
app oxima ely
=1200days
was ob ained, shows a close ag eemen wi h he e e ence model.
Ne e heless, he esul s a e physically plausible, so con idence can be placed in he p esen analysis.
Lis o Figu es
1.1 Vogaye 1 P obe P opulsion Sys em, showing RTG [15] 2
1.2 Elec ic Sola Wind Sail concep [18] 3
1.3 E olu ion o he E-sails h oughou he ime [10] 4
2.1 Desc ip ion o he g a i a ional Fo ce be ween wo bodies 8
2.2 Desc ip ion o he cylind ical e e ence ame τc11
2.3 De ini ion o he posi ion and no mal ec o s o he E-sail o he p opulsi e accele a ion ec o 13
2.4 De ini ion o he e e ence ame τ014
2.5 Rep esen a ion o he a io a/ac o di e en alues o he con ol pa ame e κ14
2.6 De ini ion o he Radial-T ans e se-No mal e e ence ame [26] 16
2.7 Rep esen a ion o he accele a ion cons ain in e ms o αn∈[−π
2,π
2]and σ∈[0,2π]17
2.8 Rep esen a ion o he accele a ion cons ain in e ms o αn∈[0,65]and σ∈[0,2π]17
3.1 Single-s ep scheme o he ou h o de Runge Ku a me hod 25
3.2 Ea h o Ma s T ans e ence wi h a cha ac e is ics accele a ion o ac=1 mm/s229
3.3 Con ol pa ame e s h oughou he ajec o y om Ea h o Ma s wi h ac=1 mm/s230
3.4
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Ma s wi h
ac=1 mm/s230
3.5
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Ma s wi h
ac=1 mm/s231
3.6 2-D ans e ence in compa ison o he ini ial guess 31
3.7 Ea h o Jupi e T ans e ence wi h a cha ac e is ics accele a ion o ac=1 mm/s233
3.8 Con ol pa ame e s h oughou he ajec o y om Ea h o Jupi e wi h ac=1 mm/s234
3.9
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Jupi e wi h
ac=1 mm/s234
3.10
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Jupi e wi h
ac=1 mm/s235
3.11 T ajec o y in 2-D be ween Ea h and Jupi e compa ed wi h he ini ial guess 35
3.12 Ea h o Dionysus T ans e ence wi h a cha ac e is ics accele a ion o ac=1 mm/s237
3.13 Con ol pa ame e s h oughou he ajec o y om Ea h o Dionysus wi h ac=1 mm/s238
3.14
E olu ion o he s a es a iables h oughou he ajec o y om Ea h o Dionysus wi h
ac=1 mm/s238
3.15
E olu ion o he s a es eloci ies h oughou he ajec o y om Ea h o Dionysus wi h
ac=1 mm/s239
3.16 T ajec o y in 2-D be ween Ea h and Dionysus compa ed wi h he ini ial guess 39
3.17
Compa ison be ween Ea h o Didymos T ans e ence wi h a cha ac e is ics accele a ion o
ac=1 mm/s241
3.18 Con ol pa ame e s h oughou he ajec o y om Ea h o Didymos wi h ac=1 mm/s242
89
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