Acta Astronautica 66 (2010) 220 -- 229

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Acta Astronautica
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Satellite scheduling considering maximum observation coverage time and

minimum orbital transfer fuel cost
Kai-Jian Zhu, Jun-Feng Li, He-Xi Baoyin∗
School of Aerospace, Tsinghua University, Beijing 100084, China

A R T I C L E I N F O A B S T R A C T

Article history: In case of an emergency like the Wenchuan earthquake, it is impossible to observe a
Received 17 January 2009 given target on earth by immediately launching new satellites. There is an urgent need
Received in revised form for efficient satellite scheduling within a limited time period, so we must find a way to
23 May 2009
reasonably utilize the existing satellites to rapidly image the affected area during a short
Accepted 28 May 2009
time period. Generally, the main consideration in orbit design is satellite coverage with
Available online 3 August 2009
the subsatellite nadir point as a standard of reference. Two factors must be taken into
Keywords: consideration simultaneously in orbit design, i.e., the maximum observation coverage time
Satellite scheduling and the minimum orbital transfer fuel cost. The local time of visiting the given observation
Primer vector sites must satisfy the solar radiation requirement. When calculating the operational orbit
Orbit transfer elements as optimal parameters to be evaluated, we obtain the minimum objective function
Optimization by comparing the results derived from the primer vector theory with those derived from
the Hohmann transfer because the operational orbit for observing the disaster area with
impulse maneuvers is considered in this paper. The primer vector theory is utilized to
optimize the transfer trajectory with three impulses and the Hohmann transfer is utilized
for coplanar and small inclination of non-coplanar cases. Finally, we applied this method in
a simulation of the rescue mission at Wenchuan city. The results of optimizing orbit design
with a hybrid PSO and DE algorithm show that the primer vector and Hohmann transfer
theory proved to be effective methods for multi-object orbit optimization.
© 2009 Elsevier Ltd. All rights reserved.

1. Introduction partial coverage properties than those with unrepeated

ground tracks. Moreover, only the repetition periods of
On May 12, 2008, an earthquake measuring 8.0 on the the repeated orbits are required in the analysis. The or-
Richter scale occurred at Wenchuan, Sichuan, China, lo- bits of space remote sensing platforms usually control the
cated at about (N31◦ , E103◦ ). In this emergency, a great ground resolution, area coverage, and the frequency of cov-
deal of observation information and pictures about the dis- erage parameters. The orbit altitude affects the resolution
aster area were badly needed. Sufficient space surveillance and the swath width. The higher the spacecraft moves,
ability would ensure the successful accomplishment of the the wider the affected area covered and correspondingly the
rescue and aid mission. Generally, circular orbits with re- lower the resolution. Although lower orbit altitudes enable
peated ground tracks are selected in the orbit design of a a spacecraft to get higher resolution, orbit perturbations
surveillance mission. These orbits proved to have better due to atmospheric drag should not be overlooked. Ground
surveillance can be looked on as the observation of multiple
discrete points (locations) on the surface of the earth. These
∗ Corresponding author. kinds of missions require that the satellite visits all the given
E-mail address: baoyin@tsinghua.edu.cn (H.-X. Baoyin). sites within a given period. A site is said to be visited if the

0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.actaastro.2009.05.029
 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229 221

nadir point of the satellite passes through a given specified ent transfer cases in which the spacecraft performed one
neighborhood of that target. The neighborhood size is deter- or more impulsive maneuvers to achieve the rendezvous.
mined by the mission scenario and work characteristics of Propellant constraints were included, and the problem was
sensors. Recently some researchers have investigated some solved by using an unconstrained parameter optimization
effective methods for orbit design. Abdelkhalik [1] adopted algorithm. Coverstone-Carroll and Prussing [9] addressed a
the Genetic Algorithm (GA) to solve several orbital prob- similar problem of cooperative rendezvous for spacecraft
lems that are characterized by many local minima, such as equipped with variable-thrust (power-limited) propulsion
space surveillance of a few specified sites. Two types of con- systems. Both the Clohessy-Wiltshire linear force-field
straint conditions are considered, i.e., the maximum resolu- model and the inverse-square central force-field model
tion of each observation point with a given imaging sensor were used. The optimal control problem was solved by
and the maximum observation time in the total flight time. means of a direct numerical approach that can discretize the
Satellite constellations in circular orbits are widely used in states and the controls. For a general multiple-spacecraft
all kinds of applications and [2–4] many constellation or- orbit-transfer problem, several spacecrafts move on dif-
bit design schemes have been investigated for local and ferent orbits in the beginning and finally are transferred
global coverage of the Earth's surface. Pontani [5] mainly re- to the same orbit, which can be different from any of the
searched constellation configurations that can maximize the original starting orbits. Herein, the problem is calculated as
total duration of visibility of a specific target and satisfy var- placing a few spacecrafts at different locations on the same
ious operational requirements, and proposed a new method final orbit. The initial orbits are the existing satellite orbits
for optimizing the constellation configurations of low-earth- and the final orbit is designed to meet the requirements
orbit (LEO) satellites for local observation. This new method of surveillance. Clearly, for the inverse-square force field, a
should satisfy four requirements, which are to minimize the two-impulse transfer between circular orbits would yield
maximum gap (MGap), maximize the maximum coverage the Hohmann transfer as the optimal two-impulse transfer
(MCov), minimize the MGap with a lower bound on the for one spacecraft [10–12]. It is quite necessary for us to
MCov, and maximize the MCov with an upper bound on the consider the multiple-spacecraft orbit transfer that satisfies
MGap. Visibility functions are used to define a correlation the constraint condition of the final space state. The nec-
function via Fourier transform which allows finding analyt- essary conditions for an optimal transfer of one spacecraft
ical solutions with high accuracy. Ulybyshev [6] proposed a between two orbits that can be assumed to be a parking or-
new approximate method for analyzing the coverage area bit and a final operational orbit have been well documented
of real time communication systems based on LEO satellite in the literature. The transfer of multiple spacecrafts to the
constellations in circular orbits. The novel geometric ana- final orbit is an extension of the single-spacecraft trans-
lytic method was used to process the statistical parameters fer problem as a classical Lambert problem with added
of the coverage area of LEO satellite constellations and the constraints that result from the nature of the departure ma-
algorithm for obtaining service areas. Ulybyshev [7] also ex- neuver and the final orbit-insertion maneuver(s) required
tended the simple coverage to a more complex scenario as- of all spacecrafts for satisfying the spacing constraint. Re-
sociated with full or partial visibility of a geographic region cently some literature processed the orbit transfer problem
by a satellite from the constellation and proposed a new through the primer vector theory which is defined as the
method for satellite constellation design that searches for adjoint to the velocity vector in the variational Hamilto-
the solution only in the two-dimensional space applications nian formulation [13]. If any of Lawden's conditions are
for combined maps of the satellite constellation and cover- unsatisfied, the transfer trajectory is not optimal. We can
age requirements. Basically, orbit design must include, as a use the primer vector history to obtain information on how
first step, an accurate coverage analysis for fixing an opti- to improve its velocity cost. Lawden solved a fixed-time
mal set of parameters, such as the number of observed sites rendezvous in his initial work and his theory was further
and their spatial distribution, according to the required op- extended as a milestone by Lion and Handelsman, and later
erational purposes. On the other hand, the motion relative by Jezewski and Rozendal to solve the N-impulse optimal
to the Earth's surface creates more difficulties because good transfer problem [14,15]. A more detailed derivation of the
numerical methods for calculating and analyzing the charac- primer vector theory can be found in [16]. As a prerequisite
teristics of the orbit coverage are needed in order to obtain of the satellite scheduling problem, we look at a way to
the generalized analytical solutions more easily. solve the single spacecraft transfer problem that minimizes
In addition to orbit design, satellite scheduling must the mission cost and is subject to the mission constraints.
also consider the orbital transfer, from the initial orbit to Without loss of generality, this paper considers the sun-
the operational orbit during a limited time with minimum synchronous orbit design and analyzes the influence of orbit
fuel cost, which is very important. Several researchers have elements on coverage properties. Then the search space of
addressed the optimization of trajectory transfer of multi- the solution can be reduced according to the former anal-
ple spacecrafts. This problem is approximately considered ysis results. The solar radiation condition is also processed
as a cooperative rendezvous in which both spacecraft take as a constraint in orbit design in order to ensure the imag-
an active role in order to further reduce propellant con- ing quality. As the second stage of satellite scheduling, or-
sumption considering the common active and passive cases. bit transfer is very significant for the mission design. In the
Mirfakhraie and Conway [8] examined time-fixed impulsive optimization of orbit elements, we must calculate the fuel
rendezvous between active spacecrafts in a central force cost of the orbit transfer that is optimized by comparing
field. Differential cost gradients were developed in differ- the bi-elliptic transfer with the Hohmann transfer. In order
222 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229

to illustrate this method, we use the Wenchuan earthquake

rescue mission as an example and obtain the optimal solu-
tion by using a hybrid optimization algorithm of the particle
swarm optimization (PSO) and the differential evolution (DE)
algorithm.

2. Orbit coverage analysis

In this section, we start a typical coverage analysis from

examination of known facts related to coverage geometry
and assume that the Earth is a round body and the orbit is
approximately circular in shape. Fig. 1 shows typical satellite
coverage for an observer on the Earth. The satellite is located
at the orbital altitude h, and the projection of the subsatellite
point onto the Earth's surface determines a coverage circle
with an angle of . The well-known relation among the cov-
erage angle , the orbit altitude h, and the elevation angle e
is given by
 
Re
 = cos−1 cos e − e (1)
Re + h
Fig. 2. Difference of varying magnitudes of i.
where Re represents the Earth's radius. Note that the size of
the coverage circle is only dependent on the satellite orbit
altitude. With the orbit ephemeris, we can get the longitude
and latitude of the sub-point as shown by

 =  + tan−1 (tan u cos i)

 = sin−1 (sin u sin i)
u=w+f
 =  − [G0 + e (t − t0 )]
= (2)

Fig. 3. Difference of varying magnitudes of .

where  and  are the right ascension and declination of

the satellite in the Earth-centered inertial coordinate sys-
tem, while u, f, , i, G0 and e are the argument of latitude,
the argument of perigee, the true anomaly, the right ascen-
sion of ascending node (RAAN), the inclination of the orbit,
the Greenwich sidereal time and the Earth angle velocity re-
spectively. We can see that the subsatellite point is only de-
pendent on the last four orbit elements, which is described
by = [a, e, i, , w, f ]. Figs. 2–4 show the difference of differ-
ent magnitudes of the orbit elements.
In Fig. 2, the coverage of the latitude of the subsatellite
point is only between the 60◦ of north latitude and the 60◦
of south latitude, which can be deduced from Eq. (2) con-
sidering the orbit inclination is 60◦ .
In Fig. 3, the adjoint of different loci of the subsatellite
Fig. 1. Observer-to-satellite geometry. point is an interval of about 40◦ according to the difference of
 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229 223

Fig. 4. Difference of varying magnitudes of u.

Fig. 5. Longitude and latitude error of two-body and J2 model.

RAAN (40◦ ) which can be deduced from Eq. (2). The interval and the argument of latitude:
between the sub-point tracks during two adjacent periods
u̇ = ẇ + ḟ
on the same orbit is about 24.2◦ , which is shown by  2  
3nJ2 Re
5 2
ẇ = − sin i − 2
˙)
 = TN (e −  (3) 2(1 − e2 )2 a2
  2  
3nJ2 Re 3
where TN is the node period and ḟ = n 1 − e2 − sin2 i − 1 . (7)
2(1 − e ) a
2 2
 7/2 We can see that the difference of the longitude and latitude
˙ = −9.97 Re
 cos i. (4) of the two-body model from the J2 model in Fig. 5 is diver-
a
gent along with time.
˙ is degrees per day and we can see from Another important factor in the orbit design is the solar
The unit of 
radiation, which determines the imaging quality. In the case
Eq. (4) that the orbit plane rotates from the east to the west
of the sun-synchronous orbit, the solar radiation condition
if i < 90 and the orbit plane rotates from the west to the east
on the same latitude (ascending arc or descending arc) is the
if i > 90. The sun-synchronous orbit can be determined by
˙ = 0.9856. same, so that the local mean solar time of the subsatellite
combining a and i, which means satisfying 
point on the same latitude is the same. In order to extend
In Fig. 4, u is the different adjoint locus of 120◦ , which
the observation time and shorten the interval of the Earth's
moves eastward and repeats if u = 360. The difference of
shadow, we adopted 6:00 a.m. or 18:00 p.m. as the local
longitude, dL, can be calculated by
mean solar time at which the spacecraft could take images
u on the ascending arc and the descending arc.
˙)
dL = (e −  (5) It is well known that the local solar time of the subsatel-
n
lite point is determined only by the difference between the
where solar right ascension h and the satellite's right ascension s :
 T = 12 − (h − s )/15. (8)
ue
n= (ue = 398600.5 km3 /s2 ). (6)
a3 According to the sine theorem of a spherical triangle shown
in Fig. 6, we can get
We consider the sun-synchronous orbit in orbit design
where the semi-major axis and inclination are multiple con- sin u = sin /sin i
straints. In the approximate circle orbit, dL is determined by sin( − s ) = − cos i sin u/cos ,
u. Because the difference of RAAN will make the longitude
cos( − s ) = cos u/cos (9)
excursion shown in Fig. 3, we must synthesize the effect of
RAAN and the argument of altitude in the orbit design. It is where represents the declination of the satellite.
well known that perturbations (like atmospheric drag, solar
radiation pressure and third body effects) can be regarded 3. Orbit transfer optimization
as negligible factors in the sun-synchronous orbit design.
However, the J2 perturbation of Earth oblateness must be The main concern in orbit transfer is the fuel cost, as fuel
considered in the integration model which affects the RAAN is the key parameter of spacecraft lifetime. Multiple satel-
224 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229

3. During the transfer, p ⱕ 1. In the cruise phase, p < 1, and

p = 1 when an impulse is performed.
4. As a consequence of these conditions, for impulses that
are not at the initial or final times, ṗ = 0.

This can be written as three equations for the components

along the orbit in the radial, transverse and normal directions
[17].

ps = k3 e sin f − k5 cos f + k6 [2(1 − e cos E)

3e sin f
− (E − e sin E)] (10)
1 − e2
 
sin E
pt = k3 (1 + e cos f ) + k4 (1 − e cos E) + k5 sin f + 
1 − e2
3(1 + e cos f )
− k6  (E − e sin E) (11)
1 − e2

pw = k1 sin E + k2 (cos E − e) (12)

where ki (i = 1, 2, . . . , 6) are constants to be determined and

E is the eccentric anomaly. We can get the time derivatives
Fig. 6. Orbit plane geometry.
of these components:
  2
dp
lites are scheduled on the operational orbit from the given = − k3 (1 + e cos f )2 − k4 (1 + e cos f )
satellite resource. The rigorous satellite phase on the oper- dt s h3 ha
2
ational orbit is not considered. The satellites are distributed
− k5 sin f (1 + e cos f )
on the same orbit in sequence so that the revisiting time h3
 
can be decreased. Generally, the problem is considered as a e 2 (1 + e cos f )2
+ k6 − sin f + 3 M  (13)
constrained orbit transfer problem for multiple spacecrafts. ha h3 1 − e2
The satellite transfers from the initial orbit to the final or-
bit occur during a certain time period and must satisfy the   2
dp e
end-point constraint. The “fixed-time” problem represents = k4 sin f +k5 3 (e+ cos f )−k6 (1+ cos f ) (14)
dt t ha h ha
a special case of transfer optimization with additional con-
 
straints on the initial and final time of the transfer. dp (e + cos f ) sin f
Generally, the coplanar orbit transfer between the = k1  − k2 (15)
dt w ha 1 − e2 ha
circular orbits considers the Hohmann transfer as the most
optimal transfer, because the bi-elliptic and bi-parabolic where M is the mean anomaly and h is the angular momen-
transfer take a lot of transfer time when the ratio of fi- tum. Since the primer vector is a first-order theory based on
nal radius to initial radius is greater than 11.94. For the local variables, it will converge on local optimal neighboring
non-coplanar case, the Hohmann transfer is still the opti- trajectories of the reference trajectory. Therefore, the opti-
mal transfer when the total inclination change is less than mal solution is highly dependent on the reference trajectory.
60.185◦ . Otherwise, the bi-elliptical transfer will minimize It also depends on other design parameters discussed later
the fuel consumption if the transfer time is within the ac- in this paper. Once the primer vector history is computed,
ceptable time limit. When the transfer time increases, the we can determine the optimality of the trajectory using the
fuel cost decreases. four necessary Lawden conditions.
This paper adopts the Hohmann transfer and bi-elliptic Without loss of generality, this paper makes an assump-
transfer simultaneously in the satellite scheduling. The bi- tion that the second impulse is located outside of the final
elliptic transfer can be used in the primer vector theory, orbit, where f = E = M = and the first and the third impulse
which focuses on the optimization of velocity increment. The occur at f =E=M =0. The inclined Hohmann transfer is more
primer vector theory takes its roots in the calculus of several economical in some cases and the inclined bi-elliptic transfer
variables and requires a set of first order necessary condi- is more economical in other cases in terms of the inclination
tions, so the trajectory has to be locally optimal. The neces- shown in Figs. 7–12. The radius of the initial orbit is set as
sary conditions, first derived by Lawden [13], are expressed one and that of the final orbit is greater than it. The velocity
in terms of the primer vector. increments shown in these figures are all non-dimensional.
We can see that the effect of the bi-elliptic transfer and that
1. The primer vector and its first derivative must be contin- of the bi-parabolic transfer are almost the same, since the
uous during the entire history. two transfers need to change the inclination at the second
2. The thrust impulses are applied in the direction of the impulse, which is generated at a large distance. As the ra-
primer vector when p = 1. tio of the final radius to the initial radius increases, the fuel
 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229 225

Fig. 7. Ratio of final radius to initial radius R = 1. Fig. 9. Ratio of final radius to initial radius R = 3.

Fig. 10. Inclination change = 20◦ .

Fig. 8. Ratio of final radius to initial radius R = 2.

fuel cost of the Hohmann transfer with that of the bi-elliptic

cost of the Hohmann transfer increases. However, the fuel transfer simultaneously because we must adjust the param-
costs of the other two transfers decrease because the final eters constantly in the optimization of orbit design in order
radius is close to the location of the second impulse. As the to search the solution space continuously. The initial tra-
inclination change increases, the fuel cost of the Hohmann jectory or first-guess is labeled as the reference trajectory.
transfer increases. But the fuel costs of the other two trans- The primer vector obeys the second order canonical form
fers do not change yet. In the last three figures, we note of the Euler–Lagrange equation and its state cannot be in-
that the three transfers will have equal costs at some ratio tegrated simultaneously as it is coupled with the spacecraft
when the inclination change is small. The Hohmann trans- state. Solving the primer vector history is equivalent to solv-
fer always has better performance than the other two trans- ing a two-point boundary value problem (TPBVP). This pa-
fers in the stage with a small ratio of final radius to initial per has fixed three impulses in the solution and divided the
radius. However, in the third figure, the Hohmann transfer whole transfer into two TPBVPs. We can move the location
velocity increment is greater than the other two despite the of the second impulse and every inclination change shown in
ratio. This means that the Hohmann transfer is suitable for Figs. 13 and 14.
the small inclination change and the influence of inclination The total fuel cost is:
change is greater than that of the ratio.

n
So we must minimize the inclination change in satel- vtotal = vi . (16)
lite scheduling. This scheduling optimization compares the k=1
226 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229

Fig. 13. Primer vector geometry at the first impulse.

Fig. 11. Inclination change = 40◦ .

Fig. 14. Primer vector geometry at the second impulse.

0 degrees, then ( 0 − ) is the Heaviside unit step func-

tion [(x) = 0 if x < 0, (x) = 1 if x > 0]. The problem is to
minimize the objective function, and equivalently, to mini-
mize the sum of the velocity maneuvers required by all the
spacecrafts if each spacecraft is identical and to maximize
the coverage time on condition that the solar altitude angle
must meet the requirement.

4. Scheduling frame
◦
Fig. 12. Inclination change = 60 .
The satellite scheduling problem that can satisfy all kinds
of constraints belongs to the global optimization finding the
global optimum of a given performance index in a large do-
If the fuel cost of the Hohmann transfer is less than that
main and typically characterized by the presence of a large
of the bi-elliptic transfer, n equals two; otherwise n equals
number of local optima. In this paper, we utilize the hybrid
three. The performance index
algorithm of PSO and DE which were both developed in the
J = −kTcoverage + vtotal 1990s and are now widely applied in various fields. In the
previous 30 iterations of every 50 iterations of the hybrid al-
t gorithm, PSO is first applied to extend the search ability. In
Tcoverage = ( 0 − ) cos  dt the latter 20 iterations of every 50 iterations, DE is applied
0
to converge rapidly. In the process of parameter optimiza-
cos  = cos cos( + e t)[cos u cos  − cos i sin u sin ] tion, the design variables are the five orbit elements, which
+ cos sin( + e t)[cos u sin  + cos i sin u cos ] are eccentricity, inclination, RAAN, argument of perigee and
+ sin sin u sin i (17) true anomaly. The semi-major axis is relative to the incli-
nation. The bounds of the variables are fixed according to
will consider the fuel cost of satellite scheduling and the the above analysis. Then we transform the orbit design and
subsatellite point coverage time where and  are the lati- scheduling problem into a parameter optimization problem.
tude and longitude of the target respectively. k is the weight In order to modify the local time of the subsatellite point,
that balances the two types of objective.  and  have been RAAN must be taken into account together with inclination
shown in Fig. 1. If the antenna angle of the satellite is set as during the orbit transfer. We can get the effective angle be-
 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229 227

Orbit parameter modification

Coverage time evaluation Fuel cost evaluation

Solar radiation condition Hohmann transfer Bi-elliptic transfer

Fig. 15. Frame for orbit parameter optimization.

tween the two orbit planes, which can be described using

angular momentum.
⎡ ⎤ ⎡ ⎤
hx h sin  sin i
⎣ hy ⎦ = ⎣ −h cos  sin i ⎦ (18)
hz h cos i

In the optimization in this study, the given observation

intervals are discretized and the state is integrated accord-
ing to time. The program first judges the angle between the
radius of the subsatellite point and the radius of the ob-
served target, and then determines whether the target can
be observed according to the Earth center angle if the first
judgment is true. In the present work, the optimal parame-
ters in the loop are searched using the hybrid algorithm and
the objective function shown in Fig. 15 is minimized.

5. Case analysis

When an earthquake happens, specific satellites to

Fig. 16. Coverage circle and belt (h = 700 km,  = 30◦ ).
observe the quake area cannot be launched as an immedi-
ate response, so the existing orbiting spacecrafts must be
used adequately. Here we provide a set of satellite including
15 targets, which is the approximate number of Chinese
orbiting satellites. These satellites, mainly used for space
exploration for science and technology, Earth observation
and weather forecasting, can be scheduled among the low
Earth orbits. The range of the semi-major axis of the given
15 satellites is from 6951 to 7182 km (Appendix A). All the
satellites are in the near circular orbit and almost all of
them are in sun-synchronous orbit. In order to save the fuel
cost of transfer, the operational orbit must be close to the
given orbit on condition that the designed orbit must meet
the mission requirements.
The disaster area is located at about 30.5–33◦ north and
103–106◦ east. Along the circular orbit at an altitude of
700 km, the coverage circle of the earth's surface can include
the whole region, because the antenna angle is 30◦ and the
coverage angle is about 3.7◦ , as shown in Fig. 16. The en-
tire disaster area can be surveilled in one coverage circle as
shown in Fig. 17, and we can look on the area as a point in
the orbit design.
From the expression (17), we can see that the optimiza-
tion objective function consists of the coverage time and
the fuel consumption for the orbit transfer. The weight k is Fig. 17. Distribution of the disaster area.
228 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229

Table 1 Table 5
Results of operational orbit and algorithm parameters (k = 1). Results of Hohmann transfer and bi-elliptic transfer (k = 10).

a (m) e i (deg.)  (deg.) w (deg.) M (deg.) Orbit number Transfer type Orbit 1 Orbit 2 Orbit 3

7133050.54 0.00012 98.40858 79.66662 59.77764 202.49820 1 Hohmann 2.2881 2.2580 2.2548
7179627.99 0.00060 98.60378 82.90337 85.10580 178.62048 bi-elliptic 2.2757 2.2458 2.2423
7091802.82 0.00043 98.23844 79.09009 34.53875 223.28280 2 Hohmann 13.8408 13.9954 13.9565
bi-elliptic 6.4879 6.4958 6.4955
3 Hohmann 14.5056 14.4424 14.4794
bi-elliptic 6.4585 6.4606 6.4626
4 Hohmann 5.9035 6.2492 6.1346
Table 2 bi-elliptic 5.4111 5.6405 5.5656
Results of Hohmann transfer and bi-elliptic transfer (k = 1). 5 Hohmann 3.9144 3.5925 3.7006
bi-elliptic 3.7973 3.5048 3.6023
Orbit number Transfer type Orbit 1 Orbit 2 Orbit 3 6 Hohmann 7.5906 7.9342 7.8250
1 Hohmann 2.2551 2.2747 2.2462 bi-elliptic 6.2156 6.2676 6.2525
bi-elliptic 2.2417 2.2629 2.2311 7 Hohmann 14.7441 14.7447 14.7604
2 Hohmann 14.0751 13.8841 14.1404 bi-elliptic 6.4316 6.4343 6.4362
bi-elliptic 6.5041 6.4878 6.5138 8 Hohmann 13.6441 0.5246 0.3897
3 Hohmann 14.4394 14.4756 14.4668 bi-elliptic 0.0483 0.5246 0.3897
bi-elliptic 6.4663 6.4572 6.4751 9 Hohmann 13.8992 14.0398 14.0069
4 Hohmann 6.3896 6.0159 6.4664 bi-elliptic 6.4467 6.4552 6.4552
bi-elliptic 5.7263 5.4887 5.7699 10 Hohmann 14.5250 14.4667 14.4873
5 Hohmann 3.4684 3.8070 3.4062 bi-elliptic 6.4115 6.4135 6.4125
bi-elliptic 3.3885 3.7012 3.3273 11 Hohmann 13.9274 14.0621 14.0123
6 Hohmann 8.0796 7.6998 8.1645 bi-elliptic 6.4394 6.4467 6.4431
bi-elliptic 6.2477 6.2308 6.2601 12 Hohmann 13.3959 13.5543 13.4932
7 Hohmann 14.7675 14.7350 14.8088 bi-elliptic 6.4213 6.4256 6.4227
bi-elliptic 6.4403 6.4305 6.4491 13 Hohmann 14.1057 14.2113 14.1669
8 Hohmann 0.6796 0.2639 0.7577 bi-elliptic 6.4457 6.4498 6.4460
bi-elliptic 0.6778 0.2620 0.7565 14 Hohmann 11.1437 10.8711 10.9555
9 Hohmann 14.1160 13.9370 14.1088 bi-elliptic 6.2956 6.2851 6.2937
bi-elliptic 6.4632 6.4476 6.4575 15 Hohmann 12.8287 12.6195 12.6811
10 Hohmann 14.412 14.4965 14.3579 bi-elliptic 6.3617 6.3541 6.3525
bi-elliptic 6.4079 6.4102 6.3992
11 Hohmann 14.0826 13.9651 14.0665
Coverage time 360.00 (s) 300.00 (s) 360.00 (s)
bi-elliptic 6.4429 6.4404 6.4354
12 Hohmann 13.589 13.4449 13.5791
bi-elliptic 6.4224 6.4219 6.4152
13 Hohmann 14.2253 14.1380 14.2060 Table 6
bi-elliptic 6.4492 6.4463 6.4415 Results of algorithm parameters and satellite number (k = 10).
14 Hohmann 10.7466 11.0609 10.6676
bi-elliptic 6.2768 6.4171 6.2665 Num-1 v1 Num-2 v2 Num-3 v3 NP F CR
15 Hohmann 12.5177 12.7680 12.4456
8 0.0483 1 2.2757 5 3.7973 50 0.8 0.8
bi-elliptic 6.3453 6.3629 6.3352
8 0.5246 1 2.2458 5 3.5048 100 0.8 0.7
8 0.3897 1 2.2423 5 3.6023 200 0.9 0.6
Coverage time 300.00 (s) 240.00 (s) 240.00 (s)

set as 30◦ . In order to estimate the coverage time of the sub-

Table 3
satellite, we set the time step as second-round and thus the
Results of algorithm parameters and satellite number (k = 1).
resulting coverage time is also second-round.
Num-1 v1 Num-2 v2 Num-3 v3 NP F CR Taking the solar irradiation and Earth shadow into ac-
8 0.6778 1 2.2417 5 3.3885 50 0.8 0.8 count, we set the local time of the subsatellite point as 6:00
8 0.2620 1 2.2629 5 3.7012 100 0.8 0.7 a.m. and 18:00 p.m. with an error of half an hour. Then the
8 0.7565 1 2.2311 5 3.3273 50 0.9 0.6
orbit must apply the appropriate RAAN.
In the second stage of the satellite scheduling mission,
Table 4 this paper simultaneously considers the Hohmann transfer
Results of operational orbit and algorithm parameters (k = 10). and the transfer of three impulses using the primer vector
theory, as three is the maximum number of impulses for op-
a (m) e i (deg.)  (deg.) w (deg.) M (deg.)
timal transfers between coplanar and noncoplanar circular
7174650.27 0.00011 98.58276 83.92074 106.60176 151.88940 orbits. During the optimization, the object function utilizes
7161144.15 0.00030 98.52592 80.85435 96.09912 163.76148
7152436.89 0.00067 98.48943 81.90154 149.14116 114.02604 the minimum fuel cost of the Hohmann transfer and the
bi-elliptic transfer. We can compare the velocity increment
of the two types of orbit transfer shown in Tables 2 and 5,
used to balance the coverage time and fuel cost. If we want which are consistent with Figs. 7–12.
to emphasize the coverage time, the weight should be in- The factor of the objective function is very important for
creased. From Tables 2 and 5, we can see that the coverage the optimization. Different factors result in different solu-
time increases when k is changed from 1 to 10. In this pa- tions as the object of optimization changes with the parame-
per, the coverage time is the interval between the moment ters. In this paper, we let the factors equal one or ten, which
when the observed target enters into the view of the radar means different tradeoffs of the fuel cost and the coverage
and the moment when it leaves with the view angle of radar time as shown in Tables 1–6. Tables 1 and 4 indicate the plan-
 K.-J. Zhu et al. / Acta Astronautica 66 (2010) 220 -- 229 229

ning orbits by using the hybrid optimal method and Tables References
2 and 5 show the velocity increment from given spacecraft
orbits in Appendix A to planning orbits where the unit is [1] O. Abdelkhalik, D. Mortari, Orbit design for ground surveillance using
genetic algorithms, Journal of Guidance, Control, and Dynamics 29
km/s. Tables 3 and 6 indicate the number of selected space- (5) (2006) 1231–1235.
craft orbits and the optimization parameters where NP is the [2] J.G. Walker, Some circular orbit patterns providing continuous whole
population size, CR is the crossover probability and F is the earth coverage, Journal of the British Interplanetary Society 24 (1971)
369–384.
mutation scaling factor. [3] L. Rider, Optimized polar orbit constellations for redundant earth
coverage, Journal of the Astronautical Sciences 33 (2) (1985)
147–161.
6. Conclusions [4] J.M. Hanson, A.N. Linden, Improved low-altitude constellation design
methods, Journal of Guidance, Control, and Dynamics 12 (2) (1989)
228–236.
This paper mainly considers the problems of orbit design [5] M. Pontani, P. Teofilatto, Satellite constellation for continuous and
and satellite scheduling for realizing optimal disaster res- early warning observation: a correlation-based approach, Journal of
Guidance, Control, and Dynamics 30 (4) (2007) 910–919.
cue and aid. This paper analyzes the characteristics of sun- [6] Y. Ulybyshev, Geometric analysis of low-earth-orbit satellite
synchronous orbit and demonstrates the effect of different communication systems: covering functions, Journal of Spacecraft
parameters in the orbit design. In the optimization of or- and Rockets 37 (3) (2000) 385–391.
[7] Y. Ulybyshev, Satellite constellation design for complex coverage,
bit elements, we obtain the optimal fuel cost by comparing Journal of Spacecraft and Rockets 45 (4) (2008) 843–849.
the primer vector theory with Hohmann transfer, and uti- [8] K. Mirfakhraie, B.A. Conway, Optimal cooperative time-fixed
lize the results to schedule multiple satellites. Provided the impulsive rendezvous, Journal of Guidance, Control, and Dynamics
17 (3) (1994) 607–613.
requirement of solar radiation is met, optimal parameters [9] V. Coverstone-Carroll, J.E. Prussing, Optimal cooperative power-
are determined by evaluating the performance of the ground limited rendezvous with propellant constraints, Journal of the
surveillance. The proposed fitness function is proved to be Astronautical Sciences 43 (3) (1995) 289–305.
[10] A. Miele, M. Ciarcia, J. Mathwig, Reflections on the Hohmann transfer,
effective. However, we could not verify that it is the best
Journal of Optimization Theory and Applications 123 (2) (2004)
formulation. The solution of the multiple levels and multiple 233–253.
objects optimization problem can be obtained by the hybrid [11] D.F. Lawden, Optimal transfers between coplanar elliptical orbits,
algorithm of PSO and DE, which is a novel idea for orbit de- Journal of Guidance, Control, and Dynamics 15 (3) (1991) 788–791.
[12] R.H. Battin, An Introduction to the Mathematics and Methods of
sign. The proposed hybrid algorithm is not sensitive to the Astrodynamics, revised edition, AIAA Reston, Virginia, 1999.
selection of the routine tuning parameters for the small di- [13] D.F. Lawden, Optimal Trajectories for Space Navigation, Butterworths,
mension optimization. This means that the method could London, 1963.
[14] P.M. Lion, M. Handelsman, Primer vector on fixed-time impulsive
be applied in the satellite constellation design and multiple trajectories, AIAA Journal 6 (1) (1968) 127–132.
satellite scheduling and satellite formation initialization. [15] D.J. Jezewski, H.L. Rozendaal, An efficient method for calculating
optimal free-space n-impulse trajectories, AIAA Journal 6 (11) (1968)
2160–2165.
[16] L.A. Hiday, Optimal transfers between libration-point orbits in
Acknowledgment the elliptical restricted three-body problem, Ph.D. Thesis, Purdue
University, 1992.
This work was supported by the National Natural Science [17] J.P. Gravier, C. Marchal, R.D. Culp, Optimal impulse transfers between
real planetary orbits, Journal of Optimization Theory and Applications
Foundation of China (nos. 10832004 and 10602027). 15 (5) (1975) 587–604.

Appendix A. Satellites set (Epoch 2008-5-12—18:00:00

Beijing time)

See Table A1

Table A1

a (m) e i (deg.)  (deg.) w (deg.) M (deg.)

6951243.66 0.00069655 81.1734 81.3068 227.7660 197.1606

6956768.52 0.00009949 82.4864 219.8569 261.4996 246.1430
6964759.09 0.00075607 97.6779 285.1372 117.7282 68.9045
6992639.75 0.00075122 82.5631 127.8716 163.0112 69.6364
7015822.33 0.00065468 82.5353 57.9830 187.5739 328.5504
7066634.97 0.00047777 98.0080 145.8512 159.4356 7.6336
7083522.23 0.00022845 98.1886 266.5466 5.2586 194.7408
7097154.50 0.00019476 98.7117 84.9228 352.8194 156.8457
7128694.42 0.00038771 98.5641 225.0141 194.4076 244.4087
7159845.33 0.00048555 98.3767 283.7099 66.7035 341.0212
7163961.50 0.00038912 97.3436 225.1685 205.5092 207.2123
7169195.30 0.00050084 98.5063 214.5738 30.4086 116.8738
7170636.97 0.00004070 98.2562 230.0121 172.9338 178.7962
7181222.13 0.00012167 96.5304 345.9248 254.1646 288.3165
7182403.54 0.00018589 98.3862 322.7639 63.4361 124.8821