@@@@@@@@@@@Nomenclature
ƒÊ@@Fgravity constant of the Earth
¢@@:equatorial radius of the Earth
‚î@@:second-order zonal harmonic coefficient of the
@@@@@Earth gravitational potential (‚î@@1.0826~
@@@@@10|3)
‚’@@: position vector of spacecraft in inertial system
17@@: velocity vector of spacecraft in inertial system
‚@@: semimajor axis of (chief) reference orbit
‚ƒ@@@: eccentricity of ‚’‚…’Ñrence orbit
i@@@: inclination of reference orbit
ƒÖ@@:argument of perigee of reference orbit
‚Ž@@: right ascension of reference orbit
M@@: mean anomaly of chief spacecraft
’@@:eccentric anomaly of chief spacecraft
ƒÊ@@:true anomaly of chief spacecraft
ƒÆ@@@:true latitude of chief spacecraft (ƒÆƒÊ\ƒÖ)
91 C 92 : components of eccentricity vector {qi = e COSU!B
@@@@@92‚ƒ sin ƒÖ)
‚Ž@@: mean orbit angular rate
Subscripts ‚ƒ and ‚„ express variables of a chief spacecraft
and those of ‚ deputy spacecraft, respectively¡
@@@@@@@@@@@1. Introduction
@It is well known that the ‚îterm of the Earth gravity
potential has effects on the formation flight of multiple
spacecraft^. In the case of ‚ circular reference orbit, ini-
tial conditions in the Hill coordinates to cancel the ‚î
effects have already been derived^. Even in the case of
an elliptic ‚’‚…’Ñrence orbit, initial conditions to cancel the
‚îeffects have been derived^ but the initial velocities
are not expressed in the time derivatives of Hill coor-
dinates, and thus, they are not necessarily suitable for
the deputy spacecraft initial conditions. In this paper,
the initial velocity of the deputy spacecraft to cancel
the‚îeffects are derived in the time derivative of Hill
coordinates in the case of an elliptic reference orbit.
@@2. Transition matrix in Hill coordinates
2¡1 Hill coordinates
@‚` spacecraft which moves on ‚ reference orbit is called
a chief spacecraft, and ‚ spacecraft which makes ‚ for-
mation flight with respect to the chief spacecraft is called
‚ deputy spacecraft. A local orthogonal coordinates
with the origin at the center of mass of the chief space-
craft is selected so that the X-axis is aligned with the di-
rection from the Earth center of mass to the chief space-
craft, and the z-axis is aligned with the normal (the‚’‚-
tation ‚–‚…‚ƒt‚‚’)of the chief spacecraft orbital plane. This
local coordinates is called the Hill coordinates. Set the
position vector of the chief spacecraft in the inertial CO-
ordinates as Vc and its second derivative of time in the
inertial coordinates as —ç.‚s‚ˆ‚… equations of motion of
the chief spacecraft become as follows:
‘Ñ7à\‚k‚iƒm
where /jj is the force by the ‚îterm exerted on the chief
spacecraft. Let the components ofç‚i‚Q expressed in the
Hill coordinates be Ú‚i@™{@fB£‚×These components
are expressed as follows by the use of the orbit elements
of the chief spacecraft:
2.2 Transformation of state vector
@Set the position vector ‚’ expressed in the Hill coordi-
nates as
and the state vector ‚ of the spacecraft as
‚Ú‚i@‚’!‚i@‚s‚š@‚s‚˜
‚ç@”ªv‚s@@@@@i6j
spacecraft with respect
The state vector of the deputy
to the chief spacecraft, Sp, is given as follows:
@@@@@@@@@@@Sp = Pd-pB
Set the orbit elements of the spacecraft as
‚X@ƒj@ƒŒ@ƒÆ@i@91@92@n£‚V¥
‚“@@ƒŒ‚ƒ@‚‰@‚Ž@ƒÖ@–¦
and the deviation of the orbit elements between the
deputy spacecraft and the chief spacecraft as ƒÂ9 and
ƒÂ‚“:
@@@@@@@SX, VyƒjPy and ‚u‚š@@‚u‚š. By using the sub-
script O to express the variables at time to, Px, ‚!/ and Pz
are calculated by ƒÓpit,to) from PxO,P!yO and PzO- In the
case of neglecting the effects of J2C the following equa-
tions should be satisfied to cancel the time-dependent
terms ofƒ‹‚‚Ž‚„I:
This equation is known as the formation condition of an
elliptic reference orbit^ Let us express ño as follows by
using this equation:
‚ê‚ƒ can be expressed as 10110‚—s:
@@@‚ê‚ƒ‡D‡D(1
where¢a is calculated as-'-
In order to obtain the time evolution of (5s, we make the
following approximation based on the above equation.
By solving this equation, we obtain ƒÂs{t) as follows:
S‹Ð)’p(ƒŠo)ƒÂ‚“(Zo)C’p(ƒŠo)‚…‚˜‚(•D‚¯‚×o)) (28)
Although it is not easy to obtain ƒÓ‚“ in usual,ƒÓ‚“ can
be obtained relatively easily, because the upper triangle
part of A, are zeros except for the (6,6) component. Let
the (6,6) component of A, be as66, and ‹©‚U‚U becomes
The time exponential function in ^,{t,to)is ‚…‚˜‚(‚‚“66(Z|
to)) only. If we make the following approximation:
@@@@‚…‚˜‚(‹©66(fˆêZo))—í‚P{‹©66(Zˆêto),
non-zero components ofƒÓs{t,to) can be expressed as
follows where ƒÓ‚“‚èand ‚ express the (—çj)‚ƒ‚‚‚‚‚Ž‚…‚Žt
ofƒÓ‚“ and ‚i‚Qàß‚ŽB{t - to)/a'^, respectively:
2.‚U Time evolution of 5p
@Using the above procedure, the state transition matrix
between Sp{‚èand Sp{to) can be obtained. Let the state
transition matrix be
‚¿(‚èo)ƒÉ”Ï—â—âƒÊZ)’p(ƒŠo)AB,ito)A^Hito) (31)
Note that the components of AHq, A—‘CƒÓs; -^sq and
AgH are functions of the orbit elements of the chief
spacecraft:@Uc, ec, ic, We and Uc- Therefore, the orbit
elements at time to are used to obtain ƒÓsit, to), Asgito)
and Agnito) and those at time ‚y are used to obtain
Anqit) and Ag,,{‚è. However, as the time variations of
the mean orbit elements ttc, ec and ic can be neglected
under the influences of J2C the values of these elements
at time to can be used as those at time t. The true
‚‚Ž‚‚‚‚Œy‹©is‚•sed to designate time as well as Z. The
mean orbit element of the argument of perigee ‹©is var-
ied as in (22) and it is approximated as
@@@sin(wc(i)) ~ sin(‚S(Zo))\‚ƒ‚s(‚S(Zo))ƒÀ@@@(33)
@@@‚ƒ‚s(ƒÖB(Z))—í‚ƒ‚s(ƒÖB(Zo))|s‚‰‚Ž‚nB(Zo))ƒÀ@@@(34)
@Set the time at the perigee as the initial time to and
the true anomaly of the chief spacecraft at time ‚y as
ƒÊ‚ƒ. Set the each component of 5p at time Z in the Hill
coordinates as
where Vxƒji>X, VyƒjPy and ‚u‚š@@‚u‚š. By using the sub-
script O to express the variables at time to, Px, ‚!/ and Pz
are calculated by ƒÓpit,to) from PxO,P!yO and PzO- In the
case of neglecting the effects of J2C the following equa-
tions should be satisfied to cancel the time-dependent
terms ofƒ‹‚‚Ž‚„I:
This equation is known as the formation condition of an
elliptic reference orbit^ Let us express ño as follows by
using this equation:
Then p^ and py are expressed as follows:
@As described later, there are time-dependent terms
inƒ‹CI‚‚Ž‚„™{by the effects of ‚i‚Q.@Some of these
time-dependent terms can be cancelled by the time-
dependent terms caused by ¢ño.@This procedure is
described in the next section.
@3. Initial Conditions considering‚îeffects
3.1 Formation condition
@When the chief spacecraft and the deputy spacecraft
are in the formation flight, the formation is gradually
changed by the ‚îeffects. We consider here the initial
conditions to minimize the formation change due to the
‚îeffects. Hereafter, by regarding Sp as a function of
the true anomaly of the chief spacecraft Vc, we set 5p at
the perigee as
The following two cases are considered to make the for-
mation flight under the ‚îeffects:
@Case 1@:@Pyf =P!yO@atƒÊ‚Ãƒj2Ntt( N: integer)
@Case 2@:@{hŽl\Žµ}‚O
Case ‚Œ means that the deputy spacecraft returns to the
initial position in the Hill coordinates after N orbiting,
where N is assumed in the range ‚‚†ƒÀá1. Case ‚Q
means that the deviation of the orbit elements Af\ƒÖof
the deputy spacecraft from those of the chief spacecraft
does not change in time.
3.‚Q Case l
@Set the time at •ƒj2NTr as ‚è. When Vyo is set as
(37)C‚‹…C‚‚ªandp‚Ãare obtained in the following form
with the inclusion of ‚îeffects:
@@@‚‹…@@PxO@@@@@@@@@@@@@@@@@@(43)
@@@Iç@@Io\‚‚ª‚Q|3(1\F¿(‚Æ-to)¢‚èo¢)
@@@‚‰D@@‚o‚šO\PzJ2@@@@@@@@@@@@@@@(45)
where Pyj2 ‚‚Ž‚„™{‚i‚Q are terms caused by the ‚îeffects.
As shown in (43)‚¯he‚îeffects do not appear in ƒ‹çin
the first order. Therefore, ¢ño is determined as follows
from the condition ‚‚†IçƒjIo:
@@@@@@¢‚èoƒj3(1\‚ÆŒ³(y¼Íßo)@@@@@@(46)
In this case, Pzj2 becomes as follows:
Although this PzJ2 cannot be cancelled by ¢‚èoC the ‚„-
fects of PzJ2 is relatively small as compared with those
OiPyJ2, because the initial condition ‚‚†Io(in the mov-
ing direction) does not affect on‚o‚š-‚iO and the effects of
the initial condition ‚‚†‘o(in the out-of-plane direc-
tion) are cancelled by the small phase difference in the
vicinity ofƒÊ‚ƒƒj2NTr as shown in (40).‚ay calculating
¢ño from (46), VyO becomes as follows where the chief
orbit elements are the initial values:
3.‚R Case ‚Q
@As the condition of Case 2 is obtained from the de-
viation of the orbit elements, it is evaluated as follows
from the time evolution of Ss in (27):
@@@@Žçƒ€(ZO)•DŸŽo)ðÜ(ZO)‰¾(ZO)(49)
When Vyo is given in (37) in the initial condition of
Spito)‚¯he term
@@@@@@@@@@@@@@@@@@@@@@@@@is‚‚‚Œ‚i‚ H 111!‚lj
s‚’@‚•!‚o‚i‚‚‚Éj‚‚g‹©‚•‚!‚Œ‚‚vSndaCI IE‹©‚‚‚m‰³@K—Ÿ‚Š
@@@@@@@@@@¥pgssgjddns OS IB s! ‚•‚“·s‚‚„ ‚‚`!‚v‚Œ‚‚Œ‚b
JO ‚•‚!’@FIEƒ©9T0UAi ab‰c)’@‚•!pJOOO TTTJJ 9b‹©uoT:jTSod
‚Œ’jl!‹©sl! pUB ‚Œw—z)‚„‚b’@‚•‚!l!sod lJRI7)a7)E‚„sS‚Ž‚„‚‚o
3m. u99Ai:j9q gDugjgnip aln!Ils!‚i‚Œ‚b112nobIl ‚” ‚sE‚É)
UT@‹‚Œw—z)‚„‚b’@s‚’@‚•!pJOOO TTTJJ 9b‹©‚•‚!l!sod ‚Œ’jl
-!‚•!‚bol s‚•‚i‚Ž‚e)J :jsora|'B lJRI7)a7)E‚„sS‚Ž‚„‚‚o ‚‚Œ‚Œ‚®l asE‚É)
UT ‹s‚•‚!l!P‚•o7)I’jl!‹©‚bJo ‚•‚!‚™)ajLlo7)Aq pgssgjddns gq
weo luÝuodraoo n 10 ^JPP all‚®sllns‚‚i ‚s‚b‚‚‚i‚i ‚i‚Œ‚‚e)s!
sv ‹‰³‚sE‚É)Jo(09)Jo ‚•‚™¹)‚‚i‚i‚‚V)‚bb!M. s:j|ns9J ‚•‚!’@I
-‚Ž‚!s 3m. SMOUS f‹—Ÿj pUB ‚Œ ‚sE‚É)Jo(8F)Jo ‚•‚™¹)‚ƒG107)
‚bb!M s:jms9J ‚•‚!’@‚Œ‚Ž‚!s 3u% sMous g gjnSijj ‹UMOUS
3xe s‚•‚!l!P‚•o7)I’jl!‹©‚bJo ‚•‚™¹)‚ƒG107)‚bb!M. s:j|ns9J
‚•‚!’@‚Œ‚Ž‚!s 3J_% es‚™)aJJa‚bJo 93U9S9jd ‚b‚•ƒhl‚˜‚‚m
@@@@@@@@@@@@‚“‚•‚!1!pUOD ‚Œ‚…!1!‚•! io ‚•‚!‚ÝxJo½‰³¥ƒ„
@@@Ks‚™)aJJa‚bAq Paq‚i‚Ž‚Œs!P Xn'Bnp'BjS s! ’¬ml7)a7)Eds
S‚Ž‚„‚‚o ‚bJoS‚‚‚o‹½‚•‚!%'ewLio} ‚besgjnSii as‚b‹©
uMoqs sy@‹s‚™)aJJa‚bb!M. gSBO 3m. SM.0b‰³@K—Ÿ‚Š
‚‚Œ!UM es‚‚‚i‚i‚‚b—}‚b!M. ‚s‹è)‚bs‚l‚b‚Œ‚‚Œ‚Ž—Ÿ˜¦
@@@@@•«ˆê‚h
@@@[‚Ž¨]009\‚Çƒj‹å@'oOS =‚±j‚h@‚b‚‚O‚XÝ‚’‚Â
@@@@@@@@@@‚n‚i‚te[‚Ž¨]‚h^@–èKO•«
@@@@@@@sE%3S gjB gra!I I’jl!UT 3m. fe UBJogoEds jaiqo
‚bJo s:ju9ra9p ‚Œ!(Flo 3m. uguM‰³P‚•E I ‹s—Ÿ‚Š ‚•! UMOUS
3xe s:j|ns9J ‚•‚!’@‚Œ‚Ž‚!Sàsa7)‚•‚Œq‚i‚Ž‚Œs!PA‚•E—}‚b!‚l‚•‚!l
-ora ‚•EFlalda)I JO gsBO 3J_% ‹©Pa‚š!jesi s! ‚•‚!%'ewLio}
j'Bmojp-j'egu E %'evfl OS %3S 3xe s‚•‚!l!pUOO TE!l!‚•!‚±3S9qH
@@@Ž¬@@@@@Mj@@@@@‚É‚Æ“yJL/‚Ö£
@@@@@@@@@@@‚nŠÌ‚Î@‚èo/ideO‚‚i‚„
@@:SMOTTOI SB s‚’@‚•!pJOOO TTTJJ 3m. UT UBJOgOE‚„sS‚Ž‚„‚‚o
‚bJo s‚•‚!l!P‚•o7)I’jl!‹©‚b—}s‚‚l‚Š!(Flo j!^dnp we uo
sJ)ƒ©Ora ^J'BJD93'Bds J9!UD 3m. gjguM asŒÞ)‚b‚•o 2‚•!sn7)‚‚Š
@@@@@@@@‚Œ!qJo‚±)!ld!||3 UB io ‚“‚•‚!1!pUOD ‚Œ‚…!1!"I l¥ƒ„
Œá‚n|T){iO\’w@@Žq‚n|I}ƒ‘“y!/Mˆó
’@‰³‚•!‚ªu‚è‚•!sS‰³‚ÇH9{‚è‰³‚•!‚ªu‚è‚•!sS‰³‚ÇHg£@P
@@@@@@@@@@oád@Ó‚nPT)’U“yJL/ˆ¢Ž™@|
@@@@@@@@@@@@‚Š‚Š‚Œ!‚ªr‚è‰³‚t!SS‰³‚ÇHg
@@@@@@@@@@sgraoogq llnsajl‹©X ¥p9A!¥igp s! o‚è
ð)’jŽM‚‚‚i‚ià)‚•lld làè!q‚i‚‹©l‹©s—}‹|‚‹©12‚•!s•Þ‚c‚‚o
0% spuodsgjjoo 0 anlEƒ©‚ƒ©‚(˜Ê‹©12‹©‚e‚v‹(6F)‚i‚n M.OJ
b‚˜!s‹©I P‚•E M.OJ ‚‚i‚t‹©12‹©PPEAqPa’@nlEƒ©‚‹©‚•‹è)
Fig. 4 Deputy Motion in Hill Coordinates (with Vyo
correction in Case 2)
@@@@@@@@@@@5. Conclusions
@In this paper, we have analyzed the ‚îeffects on the
formation flight of an elliptic reference orbit, and have
derived the initial velocity in the moving direction of the
deputy spacecraft to cancel the ‚îeffects. Here we have
used two methods; one is to derive the state transition
matrix in the Hill coordinates and to obtain the initial
velocity to cancel the position change after N orbiting
based on the transition matrix, and another one is to de-
rive the initial velocity which makes the time change of
the difference of the orbit elements between two space-
craft zero. The effectiveness of the initial velocities to
cancel the ‚îeffects are verified by the numerical simu-
lations.
@@@@@@@@@@@@References
1. S. Schaub and J. L. Junkins: Analytical Mechanics
@@of Space Systems, AIAA Educational Series, pp.
@@489-543, 593-628, 693-696, 2003
2. T. Shima, ‚j. Yamada, I. Jikuya and ‚r. Yoshikawa:
@@Trajectory Planning of ‚ Spacecraft based on the
@@Modified Hill's Equation, TransactiƒÆns of the J’@a71
@@SƒÆ‚„‚„!/‰Ây‚Úechanical En¼Þ‰Áeers, ‚u‚O‚P. 72C ‚m‚O. 722,
@@pp. 3364-3371, 2006 (‚‰‚Ž Japanese)
3. P. Sengupta, ‚q. Sharma and ‚r. R. Vadali:@Peri-
@@odic Relative Motion Near ‚ Keplerian Elliptic Or-
@@bit with Nonlinear Differential Gravity, u
@@Guidanc‚ç‚bƒÆEƒ¡ƒÆ4 and p!mamics, ‚u‚O‚P. 29C ‚m‚O. 5C
@@pp. 1110-1121, 2006
4. D.Gim and ‚j. T. Alfriend: State Transition Matrix
@@of Relative Motion for the Perturbed Noncircular
@@Reference Orbit, ‚iƒÆurnal of Guidanc‚ç‚bƒÆntrol, and
@@D!‚‰‚‰‚Ž‚‚‚‰‚ƒ‚“C Vol. 26C ‚m‚O. 6C pp. 956-971, 2003
5. K. Yamada, ‚s. Shima and S. ‚x‚shika‚—‚:@Space-
@@craft Formation Flying in Eccentric Orbits, Hƒ¡ƒÆ-
@@‚ƒ‚…‚…‚„‚‰‚Ž‚‘‚“ of 16t‚ˆ ‚v‚‚’k‚“‚ˆ‚‚ ‚‚Ž ‚i‚`‚w‚` ‚`‚“t‚’‚‚„‚‰‚Š‚s‚‰‚‚s‚Ž-
@@ic‚“ ‚‚Ž‚„ ‚e‚Œ‚‰‚‡‚ˆt ‚l‚…‚ƒ‚ˆ‚‚Ž‚‰‚ƒ‚“C pp. 22-27, 2006