Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 503168, 35 pages
doi:10.1155/2009/503168
Research Article
Optimization of Low-Thrust Limited-Power
Trajectories in a Noncentral Gravity
Field—Transfers between Orbits with
Small Eccentricities
Sandro da Silva Fernandes
Departamento de Matemática, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos,
SP, Brazil
Correspondence should be addressed to Sandro da Silva Fernandes, sandro@ief.ita.br
Received 12 October 2008; Revised 14 February 2009; Accepted 15 March 2009
Recommended by Dane Quinn
Numerical and first-order analytical results are presented for optimal low-thrust limited-power
trajectories in a gravity field that includes the second zonal harmonic J2 in the gravitational
potential. Only transfers between orbits with small eccentricities are considered. The optimization
problem is formulated as a Mayer problem of optimal control with Cartesian elements—position
and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle,
successive canonical transformations are performed and a suitable set of orbital elements is
introduced. Hori method—a perturbation technique based on Lie series—is applied in solving
the canonical system of differential equations that governs the optimal trajectories. First-order
analytical solutions are presented for transfers between close orbits, and a numerical solution is
obtained for transfers between arbitrary orbits by solving the two-point boundary value problem
described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a
shooting method. A comparison between analytical and numerical results is presented for some
maneuvers.
Copyright q 2009 Sandro da Silva Fernandes. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper presents numerical and analytical results for optimal low-thrust limited-power
transfers between orbits with small eccentricities in a noncentral gravity field that includes
the second zonal harmonic J2 in the gravitational potential. This study has been motivated by
the renewed interest in the use of low-thrust propulsion systems in space missions verified
in the last two decades due to the development and the successes of two space mission
powered by ionic propulsion: Deep Space One and SMART 1. Several researchers have
obtained numerical and sometimes analytical solutions for a number of specific initial orbits
and specific thrust profiles in central or noncentral gravity field 1–26.
2
Mathematical Problems in Engineering
It is assumed that the thrust direction is free and the thrust magnitude is unbounded
5, 27, that is, there exist no constraints on control variables. It is supposed that the orbital
changes caused by the thrust and the perturbations due to the oblateness of the central body
have the same order of magnitude. The optimization problem is formulated as a Mayer
problem of optimal control with position and velocity vectors—Cartesian elements—as
state variables. After applying the Pontryagin Maximum Principle 28, successive Mathieu
transformations are performed and a suitable set of orbital elements is introduced. Hori
method—a perturbation technique based on Lie series 29—is applied in solving the
canonical system of differential equations that governs the optimal trajectories. For transfers
between close quasicircular orbits, new set of nonsingular orbital elements is introduced and
the new Hamiltonian and the generating function, determined through the algorithm of Hori
method, are transformed to these new elements through Mathieu transformations. First-order
analytical solutions are obtained for transfers between close quasicircular orbits, considering
maneuvers between near-equatorial orbits or nonequatorial orbits. These solutions are
given in terms of systems of algebraic equations involving the imposed variations of
the orbital elements, the duration of the maneuvers, the effects of the oblateness of the
central body, and the initial values of the adjoint variables. The study of these transfers is
particularly interesting because the orbits found in practice often have a small eccentricity,
and the problem of slight modifications corrections of these orbits is frequently met 5.
For transfers between arbitrary orbits, a slightly different set of nonsingular elements is
introduced, and the two-point boundary value problem described by averaged maximum
Hamiltonian, expressed in nonsingular elements, is solved through a shooting method. A
comparison between analytical and numerical results is presented for some maneuvers.
2. Optimal Space Trajectories
A low-thrust limited-power propulsion system, or LP system, is characterized by low-thrust
acceleration level and high specific impulse 5. The ratio between the maximum thrust
acceleration and the gravity acceleration on the ground, γmax /g0 , is between 10−4 and 10−2 .
For such system, the fuel consumption is described by the variable J defined as
J
1
2
t
γ 2 dt,
2.1
t0
where γ is the magnitude of the thrust acceleration vector γ , used as control variable. The
consumption variable J is a monotonic decreasing function of the mass m of space vehicle:
J Pmax
1
1
−
,
m m0
2.2
where Pmax is the maximum power, and m0 is the initial mass. The minimization of the final
value of the fuel consumption Jf is equivalent to the maximization of mf .
The general optimization problem concerned with low-thrust limited-power transfers
no rendezvous will be formulated as a Mayer problem of optimal control by using Cartesian
elements as state variables. Consider the motion of a space vehicle M powered by a limitedpower engine in a general gravity field. At time t, the state of the vehicle is defined by the
Mathematical Problems in Engineering
3
Mtf rt
Γ
Mt0 O0
g
Of
Figure 1: Geometry of the transfer problem.
position vector rt, the velocity vector vt, and the consumption variable J. The geometry
of the transfer problem is illustrated in Figure 1. The control γ is unconstrained, that is, the
thrust direction is free, and the thrust magnitude is unbounded.
The optimization problem is formulated as follows: it is proposed to transfer the space
vehicle M from the initial state r0 , v0 , 0 at the initial time t0 0 to the final state rf , vf , Jf at the specified final time tf , such that the final consumption variable Jf is a minimum. The
state equations are
dr
v,
dt
dv
gr γ ,
dt
dJ 1 2
γ ,
dt
2
2.3
where gr is the gravity acceleration. For transfers between orbits with small eccentricities,
the initial and final conditions will be specified in terms of singular orbital elements
introduced in next sections. It is also assumed that tf − t0 and the position of the vehicle
in the initial orbit are specified.
According to the Pontryagin Maximum Principle 28, the optimal thrust acceleration
γ ∗ must be selected from the admissible controls such that the Hamiltonian function H
reaches its maximum. The Hamiltonian function is formed using 2.3,
1
H pr •v pv •gr γ pJ γ 2 ,
2
2.4
where pr , pv , and pJ are the adjoint variables, and dot denotes the dot product. Since the
optimization problem is unconstrained, γ ∗ is given by
γ∗ −
pv
.
pJ
2.5
The optimal thrust acceleration γ ∗ is modulated 5, and the optimal trajectories are governed
by the maximum Hamiltonian function H ∗ , obtained from 2.4 and 2.5:
H ∗ pr •v pv •gr −
pv 2
.
2pJ
2.6
4
Mathematical Problems in Engineering
The consumption variable J is ignorable, and pJ is a first integral. From the
transversality conditions, pJ tf −1; thus,
pJ t −1.
2.7
Equation 2.6 reduces to
H pr •v pv •gr pv 2
.
2
2.8
The consumption variable J is determined by simple quadrature.
For noncentral field, the gravity acceleration is given by 30
μ
gr − 3 r r
∂V2
∂r
T
,
2.9
with the disturbing function V2 defined by
V2 −μJ2
ae 2
r3
P2 sin ϕ ,
2.10
where μ is the gravitational parameter, ae is the mean equatorial radius of the central body,
J2 is the coefficient for the second zonal harmonic of the potential, ϕ is the latitude, and P2
is the Legendre polynomial of order 2. ∂V2 /∂r is a row vector of partial derivatives, and
superscript T denotes its transpose.
Using 2.8 and 2.9, the maximum Hamiltonian function can be put in the following
form:
H ∗ H0 HJ2 Hγ∗ ,
2.11
where
μ
r,
r3
∂V2 T
,
HJ2 pv •
∂r
H0 pr •v − pv •
Hγ∗ 2.12
pv 2
.
2
H0 is the undisturbed Hamiltonian function, HJ2 and Hγ∗ are disturbing functions concerning
the oblateness of the central body and the optimal thrust acceleration, respectively. HJ2 and
Hγ∗ are assumed to be of the same order of magnitude.
Mathematical Problems in Engineering
5
3. Transformation from Cartesian Elements to a Set of
Orbital Elements
Consider the canonical system of differential equations governed by the undisturbed
Hamiltonian H0 :
dr
v,
dt
μ
dv
− 3 r,
dt
r
μ
dpr
3 pv − 3pv •er er ,
dt
r
dpv
−pr ,
dt
3.1
where er is the unit vector pointing radially outward of the moving frame of reference
Figure 2. The general solution of the state equations is well known in Astrodynamics
31, and the general solution of the adjoint equations is obtained through properties of
generalized canonical systems 32, as described in the appendix. Thus,
a 1 − e2
er ,
r
1 e cos f
μ
v
e sin fer 1 e cos f es ,
a1 − e2 r sin f
1 − e3 cos E
a
2
er
pω −
pM
pr 2 2apa 1 − e cos E pe √
a
e
r
1 − e2
√
e cos f
1 − e2 cos f
sin f
pe −
pM es
pω a
ae
ae1 − e2 pΩ
1
a
− pω cotI sin ω
sin E pI cos ω √
r
sin I
a 1 − e2
pΩ
a
− pω cotI cos ω ew ,
cos E pI sin ω −
1 − e2
r
sin I
1 − e2 cos f
1
2
2ae sin fpa 1 − e sin f pe −
pω
pv √
e
na 1 − e2
⎫
3/2 ⎬
1 − e2
2e
cos f −
pM er
⎭
e
1 e cos f
a 2
pa 1 − e2 cos f cos E pe
2a 1 − e
r
1 − e2 sin f
1
2
1
pω − 1 − e pM es
e
1 e cos f
r pΩ
r
cos ω f pI sin ω f
− pω cot I
ew ,
a
a
sin I
3.2
3.3
3.4
3.5
6
Mathematical Problems in Engineering
es
Z
er
ew
M
r
ωf
Y
Ω
Line of nodes
X
Figure 2: Frames of reference.
where es and ew are unit vectors along circumferential and normal directions of the moving
frame of reference, respectively; a is the semimajor axis, e is the eccentricity, I is the inclination
of orbital plane, Ω is the longitude of the ascending node, ω is the argument
of pericenter, f is
the true anomaly, E is the eccentric anomaly, M is the mean anomaly, n μ/a3 is the mean
motion, and r/a, r/a sin f, and so forth are functions of the elliptic motion which can
be expressed explicitly in terms of the eccentricity and the mean anomaly through Lagrange
series 31. The true, eccentric and mean anomalies are related through the equations:
f
tan 2
1e
E
tan ,
1−e
2
3.6
M E − e sin E.
The unit vectors er , es , and ew of the moving frame of reference are written in the fixed frame
of reference as
er cos Ω cos ω f − sin Ω sin ω f cos I i
sin Ω cos ω f cos Ω sin ω f cos I j sin ω f sin Ik,
es − cos Ω sin ω f sin Ω cos ω f cos I i
− sin Ω sin ω f cos Ω cos ω f cos I j cos ω f sin Ik,
3.7
ew sin Ω sin Ii − cos Ω sin Ij cos Ik.
Equations 3.2–3.5 define a Mathieu transformation between the Cartesian elements
r, v, pr , pv and the orbital ones a, e, I, Ω, ω, M, pa , pe , pI , pΩ , pω , pM . Since the Hamiltonian
function is invariant with respect to this canonical transformation, it follows, from 2.11
through 3.5, that
H ∗ H0 HJ2 Hγ ∗ ,
3.8
Mathematical Problems in Engineering
where
H0 npM ,
√
∂V
1 − e2 ∂V2 2 ∂V2
2
−
HJ2 pa 1 − e2
pe
na ∂M
∂ω
∂M
na2 e
∂V2
∂V2
1
−
cos I
pI
√
∂Ω
∂ω
na2 1 − e2 sin I
√
∂V2
e cot I ∂V2
1 − e2 ∂V2
1
pΩ −
pω
√
∂e
na2 e
1 − e2 ∂I
na2 1 − e2 sin I ∂I
∂V2
1 − e2 ∂V2
1
−2
−
pM ,
na
∂a
ae
∂e
2
1
1
2
2aep
pe
Hγ∗ 2 2
1
−
e
1
−
cos
2f
a
2n a 1 − e2 2
1 − e2
2
pe p ω
2 1 − e sin 2f −apa pω −
2e
3/2
1 − e2
−2e
2
cos f apa pM pe p M
sin f
4 1−e
1 e cos f
2e
2
1 − e2 1 cos 2f pω 2
2
2e
5/2 2 1 − e2
−2e
cos f cos fpω pM
−
1 e cos f
e2
3 2
2 a 2
1 − e2
−2e
2
2
2
cos f pM 4a 1 − e
pa 2
1 e cos f
r
e2
2 a cos E cos f pa pe
4a 1 − e2
r
2
1 − e2 cos E cos f2 pe 2
2 4a 1 − e2
a
1
sin f 1 e
r
1 e cos f
1/2
× p a pω − 1 − e 2
pa pM
2
2 1 − e2 1
cos E cos f 1 e
1 e cos f
× sin f pe pω − 1 − e2 pe pM
7
8
Mathematical Problems in Engineering
2
1 − e2
1
2
1
sin f pω − 1 − e pM
e
1 e cos f
2 pΩ
1 r 2
2
pI − pω cot I
2 a
sin I
2 pΩ
1 r 2
2
− pω cot I
cos 2 ω f pI −
2 a
sin I
r 2
pΩ
− pω cot I
sin 2 ω f pI
,
a
sin I
3.9
with the disturbing function V2 given by
V2
μ ae 2
J2
a a
3 3 a 3 2
1 3 2
a
− sin I sin I cos 2 ω f .
r
2 4
4 r
3.10
The new Hamiltonian function H ∗ , defined by 3.8–3.10, describes the optimal lowthrust limited-power trajectories in a noncentral gravity field which includes the perturbative
effects of the second zonal harmonic of the gravitational potential.
4. Averaged Maximum Hamiltonian for Optimal Transfers
In order to eliminate the short periodic terms from the maximum Hamiltonian function H ∗ ,
Hori method 29 is applied. It is assumed that H0 is of zero-order and Hγ∗ is of the first-order
in a small parameter associated to the magnitude of the thrust acceleration, and HJ2 has the
same order of Hγ∗ .
Consider an infinitesimal canonical transformation:
, pω
, pM
a, e, I, Ω, ω, M, pa , pe , pI , pΩ , pω , pM −→ a , e , I , Ω , ω , M , pa , pe , pI , pΩ
.
4.1
The new variables are designated by the prime. According to the algorithm of Hori method
29, at order 0,
.
F0 n pM
4.2
F0 denotes the new undisturbed Hamiltonian. Now, consider the canonical system described
by F0 :
da
0,
dt
dΩ
0,
dt
de
0,
dt
dω
0,
dt
dI 0,
dt
dM
n ,
dt
Mathematical Problems in Engineering
9
dpa
3 n p ,
dt
2 a M
dpΩ
0,
dt
dpI
0,
dt
dpe
0,
dt
dpM
0,
dt
dpω
0,
dt
4.3
general solution of which is given by
a a0 ,
Ω Ω0 ,
pa pa 0 e e0 ,
ω ω0 ,
M M0 n t − t0 ,
3 n t − t0 pM ,
2
a
pΩ
,
pΩ
0
I I0 ,
pe pe 0 ,
pω
pω
,
0
pI pI 0 ,
4.4
pM
pM
.
0
The subscript 0 denotes the constants of integration.
Introducing the general solution defined above into the equation of order 1 of the
algorithm of Hori method, it reduces to
∂S1
HJ2 Hγ ∗ − F1 .
∂t
4.5
According to 29, the mean value of HJ2 Hγ ∗ must be calculated, and, S1 is obtained through
integration of the remaining part. F1 and S1 are given by the following equations:
a 2 −2 3
1
e
F1 − n J2 1 − 5 cos2 I pω
1 − e2
cos I pΩ
2
a
2
⎧
5 − 4e2 2
a ⎨ 2 2 5 2
2
4a pa 1 − e pe pω
2μ ⎩
2
2e2
pΩ
3 2
5 2
5e2 sin 2ω 1 e
p
− cot I pω
e cos 2ω 2
2
21 − e2 21 − e2 I sin I pI2
1
21 − e2 pΩ
− cot I pω
sin I 2 3
1 e2
2
⎫
⎬
5
− e2 cos 2ω
,
⎭
2
10
Mathematical Problems in Engineering
3 a 2
−3/2
3 2 a 3
a
3 2 e
2
a pa
sin I
− 1−e
cos 2 ω f
S1 J2 1 − sin I
a
2
r
2
r
e
1
1
3 sin2 I cos
2
ω
cos
2ω
−
f
f
3f
−
cos
2ω
2 e 1− e2 2
2
3
3 −3/2
1 − e2
1 3 2 a
−
sin
I
− 1 − e2
e
2 4
r
3 2 a 3
pe
cos 2 ω f
sin I
4
r
e
1
3 sin 2I 1
pI
cos 2 ω f cos 2ω f cos 2ω 3f
4 1 − e2 2 2
2
3
1
3 cos I − f − M e sin f sin 2 ω f 2
2 1 − e2 2
1
e
pΩ
sin 2ω f sin 2ω 3f
2
3
3 5 cos2 I − 1 f − M e sin f 2
4 1 − e2 2 a a
1 3 cos2 I − 1
2
1−e 1 sin f 4 e 1 − e2 r
r
2
a a 3 sin2 I 2
1−e −
− 1 sin 2ω f 8 e 1 − e2 r
r
2
a 1
a 2
sin 2ω 3f
1−e r
r
3
3 3 − 5 cos2 I sin 2 ω f 2
8 1 − e2 1
e sin 2ω f sin 2ω 3f
pω
3
√
1 a5
8 1 − e2
2 2
2
8e sin E a pa 8 1 − e sin E a pa pe −
cos E pa pω
2 μ3
e
5
3
1 2
− e sin E sin 2E − e sin 3E pe2
1−e
4
4
12
√
1 − e2 5 1
1 2
e
3
−
e
e
cos
2E
cos
E
−
cos
3E
pe pω
e
2
2
6
Mathematical Problems in Engineering
⎡
2 ⎤
−1
pΩ
2
2
⎣pI ⎦
1−e
− pω cot I
sin I 11
−1
3 3
3 2
1 3
× −e e
sin E e sin 2E − e sin 3E 1 − e2
8
8
24
⎤
⎡
2
p
p
Ω
Ω
× ⎣pI2 cos 2ω 2pI
− pω
cot I sin 2ω −
− pω
cot I cos 2ω ⎦
sin I sin I 5 5 3
1 1 2
1 1 3
× − e e
e
sin E sin 2E − e e
sin 3E
4
8
4 8
12
24
−1/2
pΩ
2
2
1−e
−pI sin 2ω 2pI
− pω cot I cos 2ω
sin I ⎤
2
pΩ
− pω
cot I sin 2ω ⎦
sin I 5 1 1 1 2
× e cos E −
e
cos 2E e cos 3E
4
4 4
12
2 pω
3 1 2
1 5 3
.
2
e −e
sin E − e
sin 2E e sin 3E
4
4 2
12
e
4.6
Terms factored by pM
have been omitted in equations above, since only transfers no rendezvous are considered 2.
It should be noted that the averaged maximum Hamiltonian and the generating
function become singular for circular and/or equatorial orbits. In order to avoid these
singularities, suitable sets of nonsingular elements will be introduced in the next sections.
5. Optimal Transfers between Close Quasicircular Orbits
In this section, approximate first-order analytical solutions will be obtained for transfers
between close quasicircular orbits.
5.1. Transfers between Close Quasicircular Nonequatorial Orbits
Consider the Mathieu transformation 33 defined by
pa pa ,
a a ,
h e cos ω ,
k e sin ω ,
5.1
I I ,
Ω Ω ,
M ω ,
pω − pM
pω − pM
sin ω ,
cos ω ,
ph pe cos ω −
pk pe sin ω e
e
pI pI ,
pΩ
pΩ
,
p pM
.
5.2
12
Mathematical Problems in Engineering
The new set of canonical variables, designated by the double prime, is nonsingular for circular
nonequatorial orbits.
Introducing 5.2 into 4.6 and using the expansions of the elliptic motion in terms of
eccentricity and mean anomaly 31, one finds that the new averaged maximum Hamiltonian
and the generating function are written up to the zeroth order in eccentricity as
2
a 2
1
pΩ
3
a
5 2
e
2 2
2
2
,
4a pa p pk pI cos IpΩ F1 − nJ2
2
a
2μ
2 h
2
sin2 I
3 ae 2
7
1 2
2
S1 J 2
apa sin I cos 2 ph cos sin I −5 cos cos 3
2
a
4
3
7
1
pk sin sin2 I −7 sin sin 3
4
3
1
1
pI sin 2I cos 2 pΩ cos I sin 2
4
2
1
pΩ
1 a5
pI cos 2
8apa ph sin − pk cos − 3ph pk 2 μ3
2
sin I
2
pΩ
1 2
2
2
3 ph − pk p I −
sin 2 .
4
sin2 I
5.3
Double prime is omitted to simplify the notation.
For transfers between close orbits, F1 and S1 can be linearized around a suitable
reference orbit, and an approximate first-order analytical solution can be determined. This
solution is given by
Δx Ap0 B,
5.4
where Δx Δα Δh Δk ΔI ΔΩ T , p0 is the 5 × 1 vector of initial value of the adjoint
variables, A is a 5 × 5 symmetric matrix concerning the optimal thrust acceleration and B
is a 5 × 1 vector containing the perturbative effects of the oblateness of the central body. The
variables α a/a and pα apa are introduced to make the adjoint and the state vectors
dimensionless, and the variables Ω Ω sin I and pΩ pΩ / sin I are introduced to simplify
the matrix A. The adjoint vector is constant. The matrix A and the vector B are given by
⎡
⎤
aαα aαh aαk 0
0
⎢
⎥
⎢ahα ahh ahk 0
0 ⎥
⎢
⎥
⎢
⎥
⎥,
a
A⎢
a
a
0
0
⎢ kα kh kk
⎥
⎢
⎥
⎢ 0
0
0 aII aIΩ ⎥
⎣
⎦
0
0
0 aΩ I aΩ Ω
5.5
Mathematical Problems in Engineering
13
⎡
bα
⎤
⎢ ⎥
⎢ bh ⎥
⎢ ⎥
⎢ ⎥
⎥
B⎢
⎢ bk ⎥,
⎢ ⎥
⎢ bI ⎥
⎣ ⎦
bΩ
where
aαα
)
&
' 5 ) f
( a ))
4
,
μ3 ))
0
aαh ahα
) f
&
)
' 5
)
(a
) ,
4
sin
)
μ3
)
0
aαk akα
) f
&
)
' 5
)
(a
−4
cos )) ,
μ3
)
0
ahh
)
&
) f
' 5
)
(a 5
3
sin 2 )) ,
3
4
μ 2
)
0
ahk akh
) f
&
)
'
)
3 ( a5
) ,
−
cos
2
)
3
4 μ
)
0
akk
)
&
) f
' 5
)
(a 5
3
− sin 2 )) ,
3
4
μ 2
)
0
)
&
) f
' 5
)
(a 1
1
) ,
aII sin
2
)
4
μ3 2
)
0
aIΩ aΩ I
) f
&
)
'
)
1 ( a5
−
cos 2 )) ,
4 μ3
)
0
aΩ Ω
)
&
) f
' 5
)
(a 1
1
− sin 2 )) ,
3
4
μ 2
)
0
) f
)
bα ε sin2 I cos 2 ) ,
0
5.6
14
Mathematical Problems in Engineering
) f
)
1
7
bh ε cos sin2 I −5 cos cos 3 )) ,
4
3
0
) f
)
1
7
bk ε sin sin2 I −7 sin sin 3 )) ,
4
3
0
bI ) f
ε
)
sin 2I cos 2 ) ,
4
0
) f
)
ε
1
bΩ − sin 2I − sin 2 )) ,
2
2
0
5.7
with f 0 ntf − t0 , t0 is the initial time, tf is the final time, and ε 3/2J2 ae /a2 . The
overbar denotes the orbital elements of the reference orbit.
The analytical solution defined by 5.4–5.7 is in agreement with the ones obtained
through different approaches in 6, 10. In 6 the optimization problem is formulated with
Gauss equations in nonsingular orbital elements as state equations, and in 10 Hori method
is applied after the transformation of variables. Equations 5.4–5.7 represent a complete
first-order analytical solution for optimal low-thrust limited-power transfers between close
quasicircular nonequatorial orbits in a gravity field that includes the second zonal harmonic
J2 in the gravitational potential. These equations contain arbitrary constants of integration
that must be determined to satisfy the two-point boundary value problem of going from
the initial orbit at the time t0 0 to the final orbit at the specified final time tf T . Since
they are linear in these constants, the boundary value problem can be solved by simple
techniques.
An approximate expression for the optimal thrust acceleration γ ∗ can be obtained from
3.5 and 5.2, by using the expansions of the elliptic motion, and it is given, up to the zeroth
order in eccentricity, by
γ∗ 1 *
ph sin − pk cos er 2 pα ph cos pk sin es
na
+
pI cos pΩ sin ew .
5.8
According to Section 2, the optimal fuel consumption variable J is determined by
simple quadrature of the last differential equation in 2.3, with γ ∗ given by 5.8. An
approximate expression for J is given by
ΔJ 1,
aαα pα2 2aαh pα ph 2aαk pα pk ahh ph2 2ahk ph pk akk pk2
2
2
aII pI2 2ah Ω pI pΩ aΩ Ω pΩ
.
5.9
For transfers involving a large number of revolutions, 5.4–5.7 can be greatly
simplified by neglecting the short periodic terms in comparison to the secular ones, and,
Mathematical Problems in Engineering
15
the system of algebraic can be solved analytically. Thus,
1
pα 4
pΩ 2
μ3
a5
μ3
μ3
Δh
Δ
a5
μ3
Δα
Δ
a5
2
pk 5
a5
2
ph 5
pI 2
μ3
Δk
Δ
a5
ΔI
,
,
5.10
Δ
ΔΩ sin I
,
Δ
,
ε
sin 2I .
2
Introducing these equations into 5.8 and 5.9, γ ∗ and J are obtained explicitly in terms of
the imposed variations of the orbital elements, the perturbative effects due to the oblateness
of the central body and the duration of the maneuver:
Δk
Δk
1 Δα 4 Δh
2 Δh
sin −
cos er cos sin es
γ 2
2 Δ 5 Δ
a 5 Δ
Δ
Δ
ΔI
ΔΩ
cos ε cos I sin I sin ew ,
2
Δ
Δ
∗
ΔJ
μ
1
4
Δ
5.11
2
1 Δα 2 4 Δh2 Δk2
ΔΩ
ΔI 2
2
4
ε cos I sin I
.
2
5
a5 2 Δ
Δ
Δ
Δ
5.12
μ3
5.2. Transfers between Close Quasicircular Near-Equatorial Orbits
Consider the Mathieu transformation 33 defined by
a a ,
ξ e cosω Ω ,
η e sinω Ω ,
I
I
cos Ω ,
sin Ω ,
Q sin
λ M ω Ω ,
P sin
2
2
5.13
16
Mathematical Problems in Engineering
pa pa ,
pξ pe cosω Ω −
pη
pe
sinω Ω pω
− pM
e
pω
− pM
e
sinω Ω ,
cosω Ω ,
pP pI
sin Ω
2 cos Ω
− pΩ
,
− pω
cosI /2
sinI /2
pQ
pI
cos Ω
2 sin Ω
p
,
−
p
ω
Ω
cosI /2
sinI /2
5.14
pλ pM
.
The new set of canonical variables, designated by the double prime, is nonsingular for circular
equatorial orbits.
Introducing 5.14 into 4.6 and using the expansions of the elliptic motion in terms of
eccentricity and mean anomaly 31, one finds that the new averaged maximum Hamiltonian
and the generating function are written up to the zeroth order in eccentricity and first-order
in inclination as
F1 a 2 1
a
3
5 2
e
2
nJ2
4a2 pa2 pξ pη2 pP2 pQ
,
QpP − P pQ 2
a
2μ
2
8
S1 3 ae 2
1
1
J2
pξ cos λ pη sin λ QpP P pQ sin 2λ P pP − QpQ cos 2λ
2
a
2
2
1
2
1
a5
3 2
2
2
2
p − pη p − pQ sin 2λ
8apa pξ sin λ − pη cos λ 4 ξ
16 P
μ3
5.15
5.16
3
1
− pξ pη pP pQ cos 2λ .
2
8
Double prime is omitted to simplify the notation.
For transfers between close orbits, F1 and S1 can be linearized around a suitable
reference orbit, and an approximate first-order analytical solution can be determined. This
solution is given by
Δx Cp0 D,
5.17
where Δx Δα Δξ Δη ΔP ΔQT , p0 is the 5×1 vector of initial value of the adjoint variables,
C is a 5×5 matrix concerning the optimal thrust acceleration, and D is a 5×1 vector containing
the perturbative effects of the oblateness of the central body. The matrix C and the vector D
Mathematical Problems in Engineering
17
are given by
⎡
⎤
cαα cαξ cαη 0
0
⎢
⎥
⎢ cξα cξξ cξη 0
0 ⎥
⎢
⎥
⎢
⎥
⎥,
C⎢
c
c
c
0
0
⎢ ηα ηξ ηη
⎥
⎢
⎥
⎢ 0 0 0 cP P cP Q ⎥
⎣
⎦
0 0 0 cQP cQQ
⎡
0
5.18
⎤
⎢ ⎥
⎢ dξ ⎥
⎢ ⎥
⎢ ⎥
⎥
D⎢
⎢ dη ⎥,
⎢ ⎥
⎢ dP ⎥
⎣ ⎦
dQ
5.19
where
cαα
)
&
' 5 )λ f
( a ))
4
λ ,
μ3 ))
cαξ cξα
λ0
cαη cηα
λ0
)λ f
&
)
' 5
)
(a
) ,
−4
cos
λ
)
3
μ
)
λ0
cξη cηξ
)λ f
&
)
'
)
3 ( a5
−
cos 2λ)) ,
4 μ3
)
λ0
cP P
cP Q
cQP
)λ f
&
)
' 5
)
(a
) ,
4
sin
λ
)
3
μ
)
)
&
)λf
' 5
)
(a 5
3
cξξ λ sin 2λ )) ,
3
4
μ 2
)
λ0
cηη
)
&
)λ f
' 5
)
(a 5
3
λ − sin 2λ )) ,
3
2
4
μ
)
λ0
&
' 1
1 ( a5
1
sin
sin
−
,
Δλ
cos
εΔλ
−
ελ
−
ελ
ελ
ελ
2
2
f
0
0
f
8 μ3
2
2
&
' 1
1 ( a5
1
,
cos
cos
sin
εΔλ
−
−
ελ
−
ελ
Δλ
ελ
ελ
2
2
f
0
0
f
8 μ3
2
2
&
' 1
1 ( a5
1
cos
cos
,
sin
εΔλ
−
−
ελ
−
ελ
−Δλ
ελ
ελ
2
2
f
0
0
f
8 μ3
2
2
18
Mathematical Problems in Engineering
cQQ
&
' 1
1 ( a5
1
,
sin
sin
cos
εΔλ
−
−
ελ
−
ελ
Δλ
ελ
ελ
2
2
f
0
0
f
8 μ3
2
2
)λ f
2
)
ae
3
)
dξ J2
cos λ) ,
)
2
a
λ0
dP P0
)λf
2
)
ae
3
)
dη J 2
sin λ) ,
)
2
a
λ0
1
1
cos εΔλ − ε cos ελf − 2 ελ0 ε cos ελ0 − 2 ελf − 1
2
2
1
1
Q0 sin εΔλ ε sin ελf − 2 ελ0 − ε sin ελ0 − 2 ελf ,
2
2
dQ Q0
1
1
cos εΔλ ε cos ελf − 2 ελ0 − ε cos ελ0 − 2 ελf − 1
2
2
1
1
P0 − sin εΔλ ε sin ελf − 2 ελ0 − ε sin ελ0 − 2 ελf .
2
2
5.20
The adjoint variables are
pa pa0 ,
pξ pξ 0 ,
pη pη 0 ,
5.21
pP pP0 cos εΔλ pQ0 sin εΔλ,
pQ −pP0 sin εΔλ pQ0 cos εΔλ,
where Δλ λf −λ0 ntf −t0 . The overbar denotes the orbital elements of the reference orbit.
Equations 5.20 do not contain higher order terms in ε as appear implicitly in the solution
presented in 7.
The analytical solution defined by 5.17–5.21 is in agreement with the ones obtained
through different approaches in 7, 10. In 7 the optimization problem is formulated with
Gauss equations in nonsingular orbital elements as state equations, and in 10 Hori method
is applied after the transformation of variables. Equations 5.17–5.21 represent a complete
first-order analytical solution for optimal low-thrust limited-power transfers between close
quasicircular near-equatorial orbits in a gravity field that includes the second zonal harmonic
J2 in the gravitational potential. These equations contain arbitrary constants of integration
that must be determined to satisfy the two-point boundary value problem of going from
the initial orbit at the time t0 0 to the final orbit at the specified final time tf T . Since
they are linear in these constants, the boundary value problem can be solved by simple
techniques.
Mathematical Problems in Engineering
19
As described in Section 5.1, approximate expressions can be obtained for the optimal
thrust acceleration γ ∗ and the fuel consumption J. These expressions are given by
1 1
pξ sin λ − pη cos λ er 2 pα pξ cos λ pη sin λ es pP cos λ pQ sin λ ew ,
na
2
,
1
ΔJ cαα pα2 2cαξ pα pξ 2cαη pα pη cξξ pξ2 2cξη pξ pη cηη pη2
2
2
.
cP P pP2 cP Q cQP pP pQ cQQ pQ
γ∗ 5.22
For transfers involving a large number of revolutions, 5.17–5.21 can be greatly
simplified by neglecting the short periodic terms in comparison to the secular ones, and,
the system of algebraic can be solved analytically. Thus,
1
pα 4
pP pQ μ3
μ3
a5
Δα
Δλ
a5
a5
2
pξ 5
2
pη 5
μ3
Δξ
Δλ
Δη
Δλ
,
,
,
5.23
μ3 8 ,
ΔP
cos
εΔλ
−
ΔQ
sin
εΔλ
−
P
Q
P
0
0
0
a5 Δλ
μ3 8 a5 Δλ
Q0 ΔQ cos εΔλ P0 ΔP sin εΔλ − Q0 .
The subscript denoting the constants is omitted. Introducing these equations into 5.22, γ ∗
and J are obtained explicitly in terms of the imposed variations of the orbital elements,
the perturbative effects due to the oblateness of the central body and the duration of the
maneuver:
γ∗ Δη
Δη
1 Δα 4 Δξ
2 Δξ
sin
cos
cos
sin
λ
−
λ
e
λ
λ
es
r
2 Δλ 5 Δλ
a2 5 Δλ
Δλ
Δλ
μ
4 Pf cos λ θf − P0 cos λ θ0 Qf sin λ θf − Q0 sin λ θ0 ew ,
Δλ
20
Mathematical Problems in Engineering
ΔJ
1
Δλ 4
1 Δα 2 4 Δξ2 Δη2
2
5
a5 2 Δλ
Δλ
εΔλ
ΔP 2 ΔQ2
4
16 cos εΔλ
sin
2
2
2
Δλ
Δλ
εΔλ
εΔλ
− Q0 cos
× Pf P0 sin
2
2
εΔλ
εΔλ
Q0 sin
,
Qf P0 cos
2
2
μ3
5.24
with θ0 εnt − t0 and θf εnt − tf .
5.3. General Remarks on Optimal Transfers between Close
Quasicircular Orbits
Considering the analytical solutions described above, general remarks about the effects of the
oblateness of the central body on optimal transfers between close quasicircular orbits can be
stated.
1 Matrices A and C, defined by 5.5 and 5.18, respectively, can be decomposed into
two square matrices 3 × 3 and 2 × 2. This fact shows that there exists an uncoupling
between the in-plane modifications and the rotation of the orbital plane. Similar
results were obtained by Edelbaum 1 and Marec 3 for transfers in a Newtonian
central field.
2 There is a normal component of the optimal thrust acceleration that counteracts
the perturbative effects due to the oblateness of the central body. This normal
component is proportional to ε.
3 In general, the maneuvers in a noncentral gravity field are more expensive than
the maneuvers in Newtonian central field, taking into account that the perturbative
effects due to the oblateness of the central body must be counteracted.
4 Fuel can be saved for long-time maneuvers which involves modification of the
longitude of the ascending node if the terminal orbits are direct 0◦ < I < 90◦ and ΔΩ < 0, and if the terminal orbits are retrograde 90◦ < I < 180◦ and ΔΩ > 0.
5 The extra consumption needed to counteract the perturbative effects due to the
oblateness of the central body reaches its maximum for maneuvers between orbits
with 45◦ or 135◦ of inclination and its minimum null extra consumption for
maneuvers between equatorial or polar orbits the gravity field is symmetric in
these cases.
6 As the transfer time increases, the extra fuel consumption increases.
7 The extra fuel consumption is greater for transfers between low orbits.
Mathematical Problems in Engineering
21
Table 1: Maneuver of modification of all orbital elements.
ae /a0
T tf − t0
100
150
200
250
—
0.950
0.850
0.750
0.650
0.550
0.450
0.350
0.250
0.150
Maneuver of modification of all orbital elements
Initial orbit
Final orbit
Orbital elements
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
1.000
0.0025
30.00◦
30.00◦
60.00◦
1.150
0.0050
35.0◦
37.5◦
62.50◦
a
e
I
Ω
ω
6. Optimal Long-Time Transfers between Arbitrary Orbits
In this section the two-point boundary-value problem for long-time transfers between
arbitrary orbits is formulated. In Section 7, this boundary-problem problem will be solved
numerically through a shooting method, and the solution will be compared to the analytical
ones in the case of orbits with small eccentricities.
Since the averaged maximum Hamiltonian, given by 4.6, becomes singular for
circular and/or equatorial orbits, a set of nonsingular elements is introduced.
Consider the Mathieu transformation 33 defined by
a a ,
ξ e cosω Ω ,
η e sinω Ω ,
I
cos Ω ,
P sin
2 I
sin Ω ,
Q sin
2
pa pa ,
p ω − pM
sinω Ω ,
pξ pe cosω Ω −
e
p ω − pM
cosω Ω ,
pη pe sinω Ω e
pP pI
sin Ω
2 cos Ω
− pΩ
,
− pω
cosI /2
sinI /2
pQ
pI
cos Ω
2 sin Ω
pΩ
.
− pω
cosI /2
sinI /2
6.1
6.2
This transformation of variables is the same one given by 5.12, excepting the fast-phase,
which is unnecessary short-periodic terms have been eliminated. Double prime designates
the new variables.
22
Mathematical Problems in Engineering
Introducing 6.2 into 4.6, F1 is written as
a 2 −2 ,
3
e
1 − ξ 2 − η2
1 − 2 P 2 Q2
F1 − nJ2
ξpη − ηpξ P pQ − QpP
2
a
2 1
1 − 5 1 − 2 P 2 Q2
ξpη − ηpξ
2
5 5 2
5 5 2
a
4a2 pa2 pξ2
− ξ − 2η2 pη2
− η − 2ξ2 − ξηpξ pη
2μ
2 2
2 2
−1
1
3
3
1 − ξ 2 − η2
1 ξ 2 η2
2
2
2
1
−1
2 1 1 2
2
pP pQ
1 − P 2 − Q2
− P pP QpQ ×
4
4
4
2
× QpP − P pQ 4 ξpη − ηpξ P pQ − QpP
2 4 P 2 Q2 ξpη − ηpξ
−1 5
1 − P 2 − Q2
ξ2 − η2 P 2 − Q2 4ξηP Q
8
2 2
× 2 ξpη −ηpξ P pQ −QpP 1 − P 2 − Q2 pP2 pQ
−
5
5 2
2
− ξηpP pQ ξ − η2 pP2 − pQ
2
8
2 2 5
.
ξpη −ηpξ − ξ −η P pQ QpP 2ξη P pP −QpQ
2
6.3
Double prime is omitted to simplify the notation. Note that 5.15 is a simplification of 6.3
for quasicircular near-equatorial orbits.
The two-point boundary-value problem for long-time transfers between arbitrary
orbits is formulated by the following system of canonical differential equations:
dx
dt
∂F1
∂p
T
,
dp
∂F1 T
−
,
dt
∂x
6.4
where x a ξ η P QT and p pa pξ pη pP pQ T , and the boundary-conditions
at0 a0 ,
a tf af ,
ξt0 ξ0 ,
ξ tf ξf ,
ηt0 η0 ,
η tf ηf ,
P t0 P0 ,
P t f Pf ,
Qt0 Q0 ,
Q tf Qf ,
6.5
with initial time t0 0 and final time tf T . The explicit form of state and adjoint equations
is derived by using MAPLE Software 9, and they are not presented in text because they are
extensive.
Mathematical Problems in Engineering
23
Table 2: Small and moderate amplitude maneuvers.
ae /a0
0.975
T tf − t0
Orbital elements of initial orbit
100
a 1.0
1.05
1.10
150
e 0.0025
1.15
1.20
200
I 30.00◦
1.25
1.30
250
◦
1.35
1.40
◦
1.45
1.50
Ω 30.00
ω 60.00
—
ρ yf /y0
Maneuver of modification of all
orbital elements
0.00024
0.0002
0.00016
T 100
J
0.00012
T 150
T 200
8E − 005
T 250
4E − 005
0
0.2
0.4
0.6
0.8
1
ae /a0
Figure 3: Consumption for maneuver of modification of all orbital elements.
7. Results
In this section, numerical and analytical results are compared for maneuvers described in
Tables 1 and 2. Earth is the central body, such that J2 1.0826 × 10−3 . In Table 1, orbital
elements of initial and final orbits are defined for a maneuver of modification of all orbital
elements considering different values of transfer durations T tf − t0 and ratios ae /a0 . In
Table 2, orbital elements of initial orbit, values of transfer duration and ratio ρ yf /y0 ,
where y denotes the orbital element that is changed in the maneuver, are defined for small
orbit corrections and moderate amplitude maneuvers of modification of semimajor axis
and inclination of the orbital plane. Results are presented in Figures 3 through 8 using
canonical units. In these units, a0 1.0 and μ 1.0. For example, in metric units, taking
ae 6378.2 km, ρ 0.950 corresponds to a0 6713.89 km, and T 100 time units
corresponds to 24.20 hours 1.008 day, and ρ 0.150 corresponds to a0 42521.34 km, and
T 250 time units corresponds to 964.45 hours 40.185 days. Thus, maneuvers described in
the tables below have duration of 1 to 74 days, approximately.
On the other hand, considering
∗
that the value of |γ | is approximately given by 2ΔJ/T short periodic terms are neglected,
|γ ∗ | has values between 1.0 × 10−4 and 4.0 × 10−3 canonical units for all results presented in
Figures 3–5. These values of magnitude of the thrust acceleration are typical for LP systems
5, 27.
24
Mathematical Problems in Engineering
Maneuver of modification of semimajor
axis in central field
0.0002
Maneuver of modification of semimajor
axis in noncentral field — ae /a0 0.15
0.0002
T 100
T 100
0.00016
0.00016
T 150
0.00012
T 150
0.00012
J
J
T 200
8E − 005
T 250
4E − 005
T 200
8E − 005
T 250
4E − 005
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1
1.1
1.2
ρ
1.4
1.5
1.6
ρ
a
b
Maneuver of modification of semimajor
axis in noncentral field — ae /a0 0.55
0.0002
1.3
Maneuver of modification of semimajor
axis in noncentral field — ae /a0 0.975
0.0002
T 100
T 100
0.00016
0.00016
T 150
T 150
0.00012
J
0.00012
8E − 005
8E − 005
T 250
4E − 005
T 200
J
T 200
T 250
4E − 005
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1
1.1
1.2
ρ
c
1.3
1.4
1.5
1.6
ρ
d
Figure 4: Consumption for maneuver of modification of semimajor axis.
Figure 3 shows the consumption J as function of ratio ae /a0 and values of T for the
maneuver of modification of all orbital elements described in Table 1. Figures 4 and 5 show
the consumption J as function of ratio ρ yf /y0 and T for the small and moderate amplitude
maneuvers described in Table 2, considering that the maneuvers are performed in central or
noncentral gravity field. In these figures, solid line represents analytical solution, and dashed
line represents numerical solution. Analytical solutions, including short periodic terms, are
obtained by solving the linear system of algebraic equations given by 5.4–5.7 transfer
between nonequatorial orbits with the semimajor axis and the inclination of the reference
orbit defined by a a0 af /2 and I I0 If /2, respectively. In order to include the
short periodic terms, it is assumed that the initial position of the vehicle is the pericenter of
Mathematical Problems in Engineering
0.0008
25
Maneuver of modification of inclination
in central field
Maneuver of modification of inclination
in noncentral field — ae /a0 0.15
0.0008
T 100
T 100
0.0006
0.0006
T 150
T 150
J
0.0004
J
T 200
T 200
T 250
0.0002
T 250
0.0002
0.0004
0
0
1
1.1
1.2
1.3
1.4
1.5
1
1.6
1.1
1.2
a
0.0008
1.3
1.4
1.6
b
Maneuver of modification of inclination
in noncentral field — ae /a0 0.55
Maneuver of modification of inclination
in noncentral field — ae /a0 0.975
0.0008
T 100
T 100
0.0006
0.0006
T 150
T 150
J
1.5
ρ
ρ
0.0004
T 200
J
T 200
T 250
0.0002
0.0004
T 250
0.0002
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1
1.1
1.2
ρ
c
1.3
1.4
1.5
1.6
ρ
d
Figure 5: Consumption for maneuver of modification of inclination.
the initial orbit. Variables I and Ω are transformed into nonsingular variables P and Q to
compare the analytical results with the numerical results. Numerical solutions are obtained
by solving the two-point boundary value problem described by 6.4 and 6.5 through a
shooting method 34.
Considering the maneuvers described in Tables 1 and 2, Figures 3–5 show that
analytical and numerical solutions yield quite similar values of the consumption J in the
following cases: i maneuvers of modification of all orbital elements with ratio ae /a0 < 0.550;
ii maneuvers of modification of semimajor axis with ratios ρ < 1.25 and ae /a0 < 0.550; iii
maneuvers of modification of inclination with ratios ρ < 1.50 and ae /a0 < 0.550, for all values
of T . For maneuvers with very high ratio ae /a0 , analytical and numerical solutions only yield
similar results for the same maneuvers described above for T 100 time units.
26
Mathematical Problems in Engineering
a
1.16
0.02
0.04
1.12
0.01
0.02
ξ
1.08
η
0
−0.01
1.04
−0.02
−0.02
1
0
20
40
60
−0.04
0
80 100
20
40
60
0
80 100
20
a
b
0.24
0.2
0.236
0.18
0.232
Q 0.16
0.228
0.14
40
60
80 100
Time
Time
Time
P
0
c
0.00025
0.0002
0.00015
J
0.0001
5E − 005
0.12
0.224
0
20
40
60
80 100
0
0
20
40
Time
60
0
80 100
20
d
40
60
80 100
Time
Time
e
f
1.2
0.01
1.16
0.01
1.12
a
ξ
η
0
0
1.08
1.04
−0.01
−0.01
1
0
50 100 150 200 250
0
50 100 150 200 250
Time
g
P
0.2
0.236
0.18
0.232
Q 0.16
0.228
0.14
0.224
Time
i
0.0002
0.00016
J
0.00012
8E − 005
4E − 005
0.12
50 100 150 200 250
0
0
50 100 150 200 250
Time
j
50 100 150 200 250
h
0.24
0
0
Time
0
50 100 150 200 250
Time
k
Time
l
Figure 6: Time history of state variables for maneuver of modification of all orbital elements with ae /a0 0.950—T 100 and 250.
Mathematical Problems in Engineering
a
27
1.16
0.02
0.02
1.12
0.01
0.01
ξ
1.08
η
0
−0.01
1.04
−0.01
−0.02
1
0
20
40
60
−0.02
0
80 100
20
40
60
80 100
0
a
0.2
0.236
0.18
0.232
Q 0.16
0.228
0.14
40
60
80 100
0.0002
0.00016
0.00012
J
8E − 005
4E − 005
0
0
80 100
60
c
0.12
0.224
20
40
Time
b
0.24
0
20
Time
Time
P
0
20
40
60
0
80 100
20
d
40
60
80 100
Time
Time
Time
e
f
1.16
0.008
1.12
a
0.008
0.004
1.08
0
ξ
1.04
0.004
η
0
−0.004
−0.004
−0.008
−0.008
1
0
0
50 100 150 200 250
50 100 150 200 250
g
P
0
Time
h
i
0.24
0.2
0.0001
0.236
0.18
8E − 005
0.232
Q 0.16
0.228
0.14
J
50 100 150 200 250
2E − 005
0
0
50 100 150 200 250
Time
j
6E − 005
4E − 005
0.12
0.224
0
50 100 150 200 250
Time
Time
0
50 100 150 200 250
Time
k
Time
l
Figure 7: Time history of state variables for maneuver of modification of all orbital elements with ae /a0 0.550—T 100 and 250.
28
Mathematical Problems in Engineering
a
1.16
0.02
0.02
1.12
0.01
0.01
ξ
1.08
η
0
−0.01
1.04
−0.01
−0.02
1
0
20
40
60
−0.02
0
80 100
20
40
a
0
0.00016
0.236
0.18
0.00012
0.232
Q 0.16
0.228
0.14
J
60
0
20
40
60
0
80 100
d
0.02
0.02
1.12
0.01
0.01
0
η
−0.02
0
50 100 150 200 250
50 100 150 200 250
Time
0.236
0.18
0.232
Q 0.16
0.228
0.14
6E − 005
4E − 005
J
2E − 005
0
0
50 100 150 200 250
0
50 100 150 200 250
Q
Time
j
Time
i
0.12
50 100 150 200 250
50 100 150 200 250
h
0.2
0
0
Time
0.24
0.224
0
−0.02
0
g
80 100
−0.01
−0.01
1
60
f
1.16
ξ
40
Time
e
1.04
20
Time
Time
1.08
80 100
4E − 005
0
80 100
60
8E − 005
0.12
40
40
c
0.2
20
20
Time
0.24
0
P
80 100
b
0.224
a
60
Time
Time
P
0
k
Time
l
Figure 8: Time history of state variables for maneuver of modification of all orbital elements with ae /a0 0.150—T 100 and 250.
Mathematical Problems in Engineering
29
Z
Line of apsis
I
e
h
ω
Y
N
Ω
X
Line of nodes
Figure 9: Vectors e, h, and N.
Figures 6–8 represent the time history of state variables nonsingular orbital elements
for maneuvers of modification of all orbital elements Table 1 considering three values of
ratios ae /a0 —0.150, 0.550, and 0.950—and two values of transfer duration T —100 and 250
time units. According to these figures, the analytical solution yields a good representation of
the numerical solution for all state variables, excepting the semimajor axis for the maneuver
with ae /a0 0.950 and T 250. The numerical solution shows that the semimajor axis is
strongly perturbed by the oblateness of earth, and its time evolution cannot be described by
the linear approximation given by the analytical solution. On the other hand, the analytical
solutions shows that the amplitude of the short periodic terms decreases as the transfer
duration increases such that these terms can be omitted for very long-time transfers.
8. Conclusion
In this paper an approach based on canonical transformations is presented for the problem of
optimal low-thrust limited-power maneuvers between close orbits with small eccentricities
in noncentral gravity which includes the perturbative effects of the second zonal harmonic of
the gravitational potential. Analytical first-order solutions are derived through this approach
for transfers between nonequatorial orbits and for transfers between near-equatorial orbits.
These analytical solutions have been compared to a numerical solution for long-time
transfers between arbitrary orbits described by averaged maximum Hamiltonian, expressed
in nonsingular orbital elements. The results show that the analytical solutions yield good
estimates of the fuel consumption for preliminary mission analysis considering maneuvers
with moderate amplitude, ratio ae /a0 and transfers duration. Numerical results also show
that the semimajor axis is strongly perturbed by the oblateness of earth, and its time evolution
cannot be described by the linear approximation given by the analytical solution.
Appendix
The general solution of the differential equations for the adjoint variables pr and pv is
obtained by computing the inverse of the Jacobian matrix of the point transformation between
30
Mathematical Problems in Engineering
the Cartesian elements and the orbital ones, defined by 3.2 and 3.3. This matrix is obtained
through the variations of the orbital elements induced by the variations in the Cartesian
elements, as described below.
Let us consider the inverse of the point transformation defined by 3.2 and 3.3:
a
r
,
2 − rv2 /μ
e2 1 −
cos I h2
,
μa
k•h
,
h
A.1
i•N
cos Ω ,
N
N•e
,
Ne
1
r
cos E 1−
,
e
a
cos ω where the eccentricity vector e, the angular momentum vector h, and the nodal vector N,
shown in Figure 9, are given, as function of the Cartesian elements r and v, by the following
equations:
e
1
μ
v2 −
μ
r − r•vv ,
r
A.2
h r × v,
A.3
N k × h.
A.4
Here the symbol × denotes the cross product. Note that the true anomaly f has been replaced
by eccentric anomaly E. These anomalies are related through the following equation:
f
tan 2
E
1e
tan .
1−e
2
A.5
Now, consider the variations in the Cartesian elements, δr and δv, given in the moving
frame of reference by
δr δξ er δη es δζ ew ,
δv δu er δv es δw ew .
A.6
Mathematical Problems in Engineering
31
The variations of the orbital elements—a, e, I, Ω, ω, and E—induced by the variations in the
Cartesian elements, δr and δv, are obtained straightforwardly from A.1 through A.6 and
are given by
2
a r•δr 2a2
v•δv,
δa 2
r
r
μ
δe −
δI δΩ δω h•δh
h2
δa,
μae
2μa2 e
h•δh
k•δh
csc I,
cot I −
2
h
h
A.7
N•δN
i•δN
csc Ω
cot Ω −
2
N
N
δe N•δN
1
cot ω −
N•δe e•δNcsc ω,
2
e
Ne
N
δE r•δr
r
1
− 2 δa cos E δe ,
e sin E ra
a
where the variations of the vectors e, h, and N are written as
μδe vs2 δξ − vs vr δη 2rvs δv er
−vs vr δξ vr2
μ
μ
2
δη − rvs δu − rvr δv es δζ − rvr δw ew ,
−
v −
r
r
δh −vs δζ er vr δζ − rδwes vs δξ − vr δη rδv ew ,
δN A.8
vs δξ − vr δη rδv sin I cos ω f − vr δζ − rδw cos I er
− vs δξ − vr δη rδv sin I sin ω f − vs cos I δζ es
vr δζ − rδw sin I sin ω f vs sin I cos ω f δζ ew .
Here vr and vs denote the radial and circumferential components of the velocity vector,
respectively see 3.3.
32
Mathematical Problems in Engineering
In the moving fame of reference, e, h, N, and the unit vectors i and k are written as
e
hvs
hvr
− 1 er −
es ,
μ
μ
h hew A.9
μa1 − e2 ew ,
A.10
N h sin I cos ω f er − h sin I sin ω f es ,
A.11
i cos Ω cos ω f − sin Ω sin ω f cos I er
− cos Ω sin ω f sin Ω cos ω f cos I es sin Ω sin Iew ,
k sin ω f sin Ier cos ω f sin Ies cos Iew .
A.12
A.13
From 3.2, 3.3, and A.2 through A.11, one gets the explicit form of the variations
of the orbital elements—a, e, I, Ω, ω, and E—induced by the variations in the Cartesian
elements, δr and δv. At this point, it is useful to replace the variation in the eccentric anomaly
E by the variation in the mean anomaly M, obtained from the well-known Kepler’s equation
M E − e sin E.
A.14
This variation is given by
δM r a
δE − sin Eδe.
A.15
Thus,
√
2
2e sin f
2 1 − e2 a
a
δξ √
δu δa 2
δv,
r
n
r
n 1 − e2
√
√
a 1 − e2
sin f
1 − e2
1 − e2 δη
sin
f
δu
cos
E
δξ
cos E cos f δv,
a
na
na
r2
r 1
sin E cos ω
1
cos ω f δw,
δζ cos E sin ω √
δI √
r
1 − e2
na 1 − e2 a
δe r 1
1
sin E sin ω
sin ω f δw,
δΩ δζ − cos E cos ω √
√
2
2
r sin I
a
1−e
na sin I 1 − e
Mathematical Problems in Engineering
δω 33
e cos f
sin f
sin E sin ω
cot I
δξ −
cos
E
cos
ω
−
δη
δζ
√
er
r
ae1 − e2 1 − e2
√
√
1
1 − e2
1 − e2
cos fδu sin f 1 −
δv
nae
nae
1 e cos f
cot I r sin ω f δw,
√
2
na 1 − e a
√
1 − e3 cos E
1 − e2
δM − √
cos fδη
sin fδξ ae
er 1 − e2
1 − e2
1 − e2
2e
1
cos f −
δu −
sin f 1 δv.
nae
1 e cos f
nae
1 e cos f
−
A.16
Equations A.16 can be put in the form
⎡
δa
⎤
⎡
δξ
⎤
⎥
⎢
⎢ ⎥
⎢ δe ⎥
⎢ δη ⎥
⎥
⎢
⎢ ⎥
⎥
⎢
⎢ ⎥
⎢ δI ⎥
⎢ δζ ⎥
⎥
⎥
⎢
−1 ⎢
⎥ Δ ⎢ ⎥,
⎢
⎢ δΩ ⎥
⎢ δu ⎥
⎥
⎢
⎢ ⎥
⎥
⎢
⎢ ⎥
⎢ δω ⎥
⎢ δv ⎥
⎦
⎣
⎣ ⎦
δM
A.17
δw
where the matrix Δ−1 is inverse Jacobian matrix of the point transformation between the
Cartesian elements and the orbital ones, defined by 3.2 and 3.3.
Following the properties of generalized canonical systems 32, the general solution of
the differential equations for the adjoint variables pr and pv is given by
⎡
pa
⎤
⎢ ⎥
⎢ pe ⎥
⎢ ⎥
⎢ ⎥
T ⎢ p I ⎥
pr
⎢ ⎥
−1
Δ
⎢ ⎥,
⎢ pΩ ⎥
pv
⎢ ⎥
⎢ ⎥
⎢ pω ⎥
⎣ ⎦
A.18
pM
with pr and pv expressed in the moving frame of reference. Equations 3.4 and 3.5 are
obtained straightforwardly from the above equation.
Acknowledgment
This research has been supported by CNPq under contract 305049/2006-2.
34
Mathematical Problems in Engineering
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