Nomenclature
a = semimajor axis [km]
dr a = radius direction component of thrust [km/s2]
d a ƒΖ = argument direction component of thrust
[km/s2]
dh a = normal direction of the orbit plane
component of thrust [km/s2]
T a = thrust acceleration magnitude [km/s2]
e = eccentricity [-]
f = true anomaly [rad]
h = angular momentum [km2/s]
i = inclination [rad]
M = mean anomaly [rad]
n = mean motion [rad/s]
p = semi-latus rectum [km]
r = orbital radius [km]
p r = altitude of perigee [km]
ƒΆ = longitude of ascending node[rad]
w = argument of peri-apsis [rad]
I. Introduction
Now, the moon is recognized as an important
destination for space science and exploration. In this
research, the trajectories between the Earth and the
moon are discussed.
To explore the moon efficiently, it is necessary to
transport more payloads to the moon in a period of
time. Now, to transport a spacecraft from the Earth
to the moon, a large expendable booster is used. By
reusing the booster, the above purpose can be
achieved. That is because transportation amount is
increased and production costs are reduced. In this
transportation system, it is difficult to use low-thrust
propulsion for orbital transfer to the moon. That is
because it takes a long time by using it. The flight
time it takes before main mission should be short.
The lunar transportation system is roughly
composed of the payload module (spacecraft,
supply and so on) and the propulsion module
(ROTV) as shown in Fig.1. The payload module
(included man) is separated from the lunar
transportation system in LTO and is reached LLO
by its single injection. The ROTV (not included
man) is returned to initial LEO through ETO. The
concept of the ROTV is shown in Fig.2.
This paper aims at improving the payload ratio of
the ROTV. This is equal to the minimization of fuel
consumption. In this research, we focus on the
optimization problem of the Earth return trajectories
with small fuel consumption. In ETO, we discuss
multi-impulse flight using electrical propulsion and
aero-assisted flight using aero-braking. The
optimum trajectory control law for the ROTV is
shown. The analytical suboptimum control strategy
is also shown and the effect of the strategy is
investigated. It is important to obtain the
suboptimum control law analytically, because it
takes a long time to obtain the optimum results
numerically.
In this paper, we discuss the lunar transfer
trajectory in section II, the Earth transfer trajectory
in section III, and the conclusion is described in
section IV.
Payload
ROTV [Propulsion module]
Fig.1 Image of Lunar Transportation System
‡A Rendezvous Docking [LEO]
Moon
‡C Lunar Orbit Injection [LLO]
‡D ETO
‡E Return [LEO]
‡B LTO
‡@ Payload Launch [LEO]
Fig.2 Concept of ROTV
II. Lunar Transfer Trajectory (Lunar
Free Return Trajectory)
A lunar free return trajectory is used as the lunar
transfer trajectory for this lunar transportation
system, because safety for the payload module is
insured by using the trajectory. It is one of the most
important things in LTO to insure safety. A lunar
free return trajectory have been studied and used
since the Apollo era [1]. The trajectory has
beneficial character istics for this lunar
transportation system. One of the characteristics is
that the lunar transportation system is automatically
returned to the Earth by only a single injection
maneuver at the Earth departure even if the lunar
orbit injection is not performed. The return flight
time is also short. The fuel consumption needed in
the total operation is saved by using this trajectory,
because the fuel consumption of this trajectory is
nearly equal to that of Hohmann trajectory.
A lunar free return trajectory was obtained by
solving restricted four body problem. The result is
shown in Fig.3.
( )
E js/c j (1)
s/c s/c3s/c j j js/c3 j3
= - ƒΚ - ƒΚ -
r r r
j =M,S
ƒ°
? ?
? ?
? ?
? ?
???? @@ @
@@@@@@@@@@@
r r
r r
Fig.3 Lunar Free Return Trajectory
in inertial coordinate system
III. Earth Transfer Trajectory
A. Numerical Control Laws using Low-thrust
Propulsion (Optimum trajectories)
Now, the optimum control problem is to find the
control history which minimizes fuel consumption.
In this paper, the problem is calculated by using the
method called Direct Collocation with Nonlinear
Programming (DCNLP).
The equation of motion of ROTV in polar
coordinate system is expressed as
{ [( ) ] }
[ ( ) ]
[( ) ( ) ]
(
2
dr dƒΖ
dr dƒΖ
dh
dh
dr dƒΖ dh
dr dƒΖ
da=2a esinfa +pa
dt h r
de = 1 psinfa + p+r cosf +re a
dt h
di=rcosƒΖa
dt h
dƒΆ=rsinƒΖa
dt hsini
dw=1 -psinfa + p+r sinfa -rsinƒΖcosia
dt he hsini
dM = n+ b pcosf - 2re a - p+r sinfa
dt ahe
? ? ?
? ? ?
? ? ???????????????
@
@
2)
The control variables are control acceleration
expressed in polar coordinate system, , , ( ) dr d dh a a a ƒΖ .
(3)
dr T out in
a a e e i i
dƒΖ T out in T
a a e e i i
dh T in
a acos sin
a = acos cos =a w +w +w
w +w +w
a asin
ƒΣ ƒΣ
ƒΣ ƒΣ
ƒΣ
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
u u u
u u u
Here, the steering angle in ƒΣ and out ƒΣ are defined in
Fig.4.
R
in ƒΣ
S
W
Thrust
out ƒΣ
Earth
Fig.4 Definition of angle of thrust
The vectors a u , e u and i u are the unit direction
vectors for each basic steering law and the
values a w , e w and i w are weights for each law. The
control angles for each unit vector are expressed as
follows.
( ( ) )
( ( ) )
(4)
-1
a
-1
e
i
= tan esinf
ecosf +1
= tan sinf(1+ecosf)
(2+ecosf)cosf +e
= 3ƒΞ 2 if cos w+ f >0
ƒΞ 2 if cos w+ f <0
ƒΣ
ƒΣ
ƒΣ
?????
@
@@@@@
@
@@
Here, these control angles are the optimum steering
angles which maximize the time rate of change of
each orbital element. The characteristics for each
basic steering law are shown in Fig.5-10. In Fig.5,
Fig.7 and Fig.9, the trajectories calculated by using
each control law are shown, and in Fig.6, Fig.8 and
Fig.10, the thrust directions for each control law are
shown in inertial coordinate system.
Fig.5 Trajectory of basic control law
for minimum a??
in inertial coordinate system
Fig.6 Thrust directions of basic control law
for minimum a??
in inertial coordinate system
Fig.7 Trajectory of basic control law
for minimum e??
in inertial coordinate system
Fig.8 Thrust directions of basic control law
for minimum e??
in inertial coordinate system
Fig.9 Trajectory of basic control law
for minimum ??i
in inertial coordinate system
Fig.10 Thrust directions of basic control law
for minimum ??i
in inertial coordinate system
As to the boundary conditions, the results of LFRT
are used as the initial values for each orbital element,
and the values for each element in the initial parking
orbit are used as the final values. The closest
altitude to the Earth is restricted, because the safety
must be insured. The trajectory optimized by using
DCNLP is shown in Fig.11.
Fig.11 Optimum trajectory
in inertial coordinate system
(Numerical result)
B. Analytical Control Laws using Low-thrust
Propulsion (Suboptimum trajectories)
It is important to obtain the suboptimum control
law analytically, because it takes a long time to
obtain the optimum results numerically. To control
the ROTV simplify, the method used four control
laws for each discrete arc is proposed.
In this method, out-of-plane control is performed
first, and then in-plane control is performed. That is
because the fuel consumption for out-of-plane
control is small in the initial high-elliptic orbit. The
basic control law for minimum i?? is utilized as
out-of-plane control. In in-plane control, it is one of
the most important things to control the altitude of
the perigee. The basic control law for minimum a??
is the optimum control for minimum fuel
consumption if there is not restriction. However, the
control law can not be continued using, because the
altitude of perigee goes down as time passes and
then reaches the minimum restricted. So, the
steering law which fixes the altitude of perigee is
useful, because the closest altitude to the Earth is
kept to the minimum restricted and the aero-braking
can be used effectively. The control angle is
expressed as follows.
(5)
p
-1
r const.
{e(cosf +1)+2}(1- e)(1- = tan cosf)
(ecosf +1)(1+e)sinf
ƒΣ @
In Fig.12, the thrust directions of the control law
for constant p r are shown in inertial coordinate
system.
Fig.12 Thrust directions of control law
for constant p r
in inertial coordinate system
The basic control law for minimum e?? is easy to
use, because the thrust direction is roughly fixed in
inertial coordinate system. The altitude of perigee
goes up as time passes by using the control law. So,
in-plane control is realized by using the above three
control laws properly.
The effect of the suboptimum control law is nearly
equal to that of the optimum control law. The
suboptimum control law is better than the optimum
control, because the aero-braking can be used
effectively.
C. Effect of Coasting and Aero-braking using
Earth Atmosphere
The effect of coasting is considered. Here, the
coasting area is defined in Fig.13, and the efficiency
of coasting for each eccentricity is shown in Fig.14.
Coasting
area
Propulsive
area
Earth
apogee perigee
Fig.12 Definition of coasting area
Fig.13 Efficiency of coasting
The effect of aero-braking is shown in Fig.15. The
flight time also goes down exponentially as the
closest altitude to the Earth goes down.
Fig.15 Effect of aero-braking
The coasting area and the closest altitude to the
Erath restricted must be selected properly according
to the flight time restricted.
IV. Conclusion
The optimum return trajectory of minimum fuel
consumption for the ROTV was calculated.
However, it takes a long time to obtain the optimum
results numerically. So, the suboptimum trajectory
control strategy was proposed. The control strategy
is mainly divided into the four phases. In the phase1,
the out-of-plane control is performed by control law
for minimum i??. In the phase2, the altitude of
perigee goes down to the closest altitude restricted
by using the control law for minimum a?? . In the
phase3, the altitude of perigee is kept by using the
control law for constant p r and the altitude of
apogee goes down drastically by using the
aero-braking effectively. In the phase4, the altitude
of perigee goes up to that of the initial LEO by
using the control law for the minimum e?? . The
suboptimum control strategy is very useful for the
ROTV judging from the total performance. When
the coasting area and the closest altitude to the Erath
restricted is selected properly according to the flight
time restricted, fuel consumption is saved more.
References
[1] Richard H. Battin, "An Introduction to the
Mathematics and Methods of Astrodynamics,
Revised Edition", AIAA Educational Series
[2] John T. Betts, "Survey of Numerical Methods
for Trajectories Optimization", Journal of
Spacecraft and Rockets, Vol.21, No.2, pp.
193-207, 1998
[3] C.R. Hargarves and S.W. Paris, "Direct
Trajectory Optimization Using Nonlinear
Programming and Collocation", Journal of
Guidance, Control, and Dynamics, Vol.10,
No.4, pp. 338-342, 1987
[4] C.A.Kluever and S.R.Oleson, "Direct
Approach for Computing Near-Optimal
Low-Thrust Earth-Orbit Transfers", Journal of
Spacecraft and Rockets, Vol.35, No.4, pp.
509-515, 1998