A computational approach to reduce the revisit time using a genetic algorithm

Abstract

The genetic algorithms (GAs) have been recently used in many design problems including an orbit design and trajectory optimization problems due to their global search capability and the robust characteristics. Furthermore, the GAs do not require convenient analytical representations of the problem any more. For this reason, the GAs can provide good solutions to the orbital dynamics design problem, especially a local coverage problem of Low Earth Orbit (LEO), which is difficult to formulate and handle the problem since an orbit propagation and local coverage analysis should be performed at the same time. This paper presents the possibility to using a genetic algorithm as a computational approach to finding the temporary target orbit in order to reduce the average revisit time of current mission orbit over a particular target site during the specified days. Through a comprehensive simulation study, the possibility of using a genetic algorithm to apply this kind of a temporary reconnaissance mission using a single LEO spacecraft is successfully demonstrated.

Full text

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International Conference on Control, Automation and Systems 2007
Oct. 17-20, 2007 in COEX, Seoul, Korea
A Computational Approach to Reduce the Revisit Time
Using a Genetic Algorithm
Hae-Dong Kim1 , Ok-Chul Jung1 and Hyochoong Bang2
1 Satellite Mission Operations Department, Korea Aerospace Research Institute, Daejeon, Korea
(Tel : +82-42-860-2812; E-mail: haedkim, ocjung@kari.re.kr)
2 Division of Aerospace Engineering, KAIST, Daejeon, Korea
(Tel : +82-42-869-3722; E-mail: hcbang@fdcl.kaist.ac.kr)
1. INTRODUCTION

The genetic algorithms (GAs) have been used and
won a great popularity in many design problems
including orbital dynamics design due to their global
search capability and the robust characteristics.
Moreover, the GAs do not require analytical
representations of the problem and even objective
function’s derivatives any more [1]. For this reason, the
GAs can provide good solutions to the orbital dynamics
design problems for which one aims to find not
necessarily an optimal solution, but at least a reasonably
good one.
The applications of GAs to aerospace problems can
be found in literature. Kim and Spcencer [2] introduced
the GA to solve a fuel-optimal spacecraft rendezvous
problem. Crain et al. [3] developed a hybrid
optimization approach combining the global search
properties of genetic algorithms with the local search
characteristics of recursive quadratic programming for
the interplanetary flyby mission optimization. Rauwolf
and Coverstone-Carrol [4] investigated the effectiveness
of a genetic algorithm to design near-optimal low-thrust
trajectories. Regarding the satellite constellation design
problem, the GA was attempted to minimize revisit time
and consider zonal coverage by Lang [5] and Ely et al.
[6], respectively. Recently, Abdelkhalik and Mortari [7,
8] applied the genetic algorithm to orbit transfer and
orbit design problem to find out the natural orbit for
ground surveillance.
Although GAs have demonstrated better global
convergence ability than classical algorithms through
above earlier works, several disadvantages for applying
GAs to solve those kind of problems such as slow
convergence speed and computational burden are
reported.
This paper presents the possibility of using a genetic
algorithm as a computational approach to design the
target orbit in order to reduce the Average Revisit Time
(ART) of current mission orbit over a particular target
site during the specified days. Through a comprehensive
simulation study, the possibility of using a genetic
algorithm to apply this kind of a temporary
reconnaissance mission using a single LEO spacecraft is
successfully demonstrated. In this study, J2 perturbation
is only considered for orbit propagation and a simple
GA algorithm is applied.

2. Problem Statement

Traditionally, the performances of remote sensing
satellite in LEO can be represented in terms of the
revisit time (average, maximum, minimum), percentage
area of coverage, access duration, resolution, and the
probability of detection, namely coverage characteristics
of the spacecraft. Those kinds of characteristics should
be taken into account for the orbit design. Moreover,
most remote sensing spacecraft in LEO has an optical
system, which means that the swath width of the optical
system is proportional to the orbit height. Consequently,
low-altitude orbits may result in lack of coverage
capability. This problem can be solved by means of
making higher orbits or increasing the number of
spacecraft like a constellation. However, the former
leads to resolution loss, the latter is expensive.
Regarding the mission type of remote sensing
satellite, there are two types, surveillance and
reconnaissance. Surveillance is supervision over the
whole earth or an area with a size far exceeding the
swath width. Duration of a surveillance cycle cannot be

Abstract: The genetic algorithms (GAs) have been recently used in many design problems including an orbit
design and trajectory optimization problems due to their global search capability and the robust
characteristics. Furthermore, the GAs do not require convenient analytical representations of the problem any
more. For this reason, the GAs can provide good solutions to the orbital dynamics design problem, especially
a local coverage problem of Low Earth Orbit (LEO), which is difficult to formulate and handle the problem
since an orbit propagation and local coverage analysis should be performed at the same time. This paper
presents the possibility to using a genetic algorithm as a computational approach to finding the temporary
target orbit in order to reduce the average revisit time of current mission orbit over a particular target site
during the specified days. Through a comprehensive simulation study, the possibility of using a genetic
algorithm to apply this kind of a temporary reconnaissance mission using a single LEO spacecraft is
successfully demonstrated.

Keywords: Average Revisit Time, Genetic Algorithms, Orbit Design, Low Earth Orbit

184
978-89-950038-6-2-98560/07/$15 ⓒICROS

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decreased by orbital maneuvering, while reconnaissance
is an observation of discrete sites of interest and the
spacecraft is able to observe all designated targets
within a specified time span with an unlimited
maneuvering capability [4].
In this paper, temporary reconnaissance mission using
a single LEO spacecraft
accomplishing a global Earth observation mission is
considered. This means that the optimization problem
addressed in this paper is to reduce (or minimize) the
current average revisit time of a single LEO spacecraft
over a particular target site during the specified days.
“Revisit time” typically means an interval of time
during which a specified point on the ground is not
visible to the satellite. The ART records the average of
all the revisit times for a given point whereas the worst
and the best revisits time report respectively the longer
revisit time and the shorter revisit time for a given point
during the specified time span.
However, remote sensing satellite in LEO the altitude
is usually 500 to 900 km, while the requirement for a
daily repeat (D=1) of a set of 14, 15, 16 tracks (N=)
from equations (1) and (2) may be deduced that the
corresponding orbit altitudes are only 894, 567, and 275
km [9]. This means that the current mission orbit should
be maneuvered using a fuel if its orbit is not flying on
above 3 different altitudes. Consequently, the fuel to
need to move the spacecraft from current orbit to 1-day
exact repeat ground track orbit (1-Day Exact RGT orbit)
might be too expensive and even impossible due to the
fuel budget.
In addition, in order to reduce or minimize the current
ART, we should know the current ART with respect to
sensor characteristics such as a field of view, off-nadir
viewing capability, and other constraints. A lot of
research [10-16] has been done on optimizing
constellations for continuous coverage and local
coverage of the Earth to minimize revisit time. However,
no analytical approach is available to find the best
constellation for partial global access, even though the
continuous global and zonal access problems have been
treated via approximate analytical approach [17, 18].
However, it might be difficult to resolve the solution
to the coverage problem using a single LEO spacecraft
over a particular target during the specified time span
because the orbit should be propagated and the sensor
characteristics should be analyzed with respect to the
current orbit at the same time.
In this paper, we attempt to apply the GA algorithm
as a computational approach to find the effective way to
reduce the average revisit time of current mission orbit
over a particular target site during the specified days, for
which orbit might be away from the 1-day Exact RGT
orbit. Through a comprehensive simulation study, we
found the temporary target orbit which increases the
number of imaging chances and reduces the ART of
current mission orbit at the same time with an
acceptable fuel cost. For this, firstly, semi-major axis (a)
is set to be a design variable to get the temporary target
orbit and then an inclination (i) and Right Ascension
which is currently
and Ascending Node (RAAN) as well as the altitude is
selected as design variables. The circular orbit is
assumed, i.e. zero for this study. The current orbit
parameters in osculating elements are a =7063.265188
km, i=98.132297 deg., w (Argument of Perigee) = 0
deg., RAAN = 81.654027 deg., and M (Mean Anomaly)
= 0 deg. The current RGT with respect to above orbit is
28 days. Regarding the sensor characteristics, the
off-nadir viewing capability is limited up to ± 20
degrees due to the GSD (Ground Sample Distance)
deterioration and an optical system which is only
available during the daytime (i.e. electro optical
imaging system) is assumed. The useful swath of a
sensor is 15 km. The target site is located at 126.9352 E,
37.5424 N in KOREA. Consequently, the current ART
and the number of imaging chance of given orbit over
the target site during 30 days are 2.9984 days and 9
times, respectively.

τ
D
ττ
τ
⎝⎠

1/3
2
a
μ
π
⎜⎟
⎝⎠
⎝⎠

Where τ ,
ES
τ
are the satellite’s nodal period,
Earth’s rotation period, and the orbital period of Earth
round the Sun, respectively. μ is the gravitational
parameter of the Earth and a is the semi-major axis.

3. Genetic Algorithms

The Genetic Algorithms (GAs) are a stochastic global
search method based on the Darwinian concept of
natural selection. Although GAs were invented by John
Holland in the 1960s and 1970s, they have successfully
become a useful tool for optimization and design
problems since David E. Goldberg presented the theory
of genetic algorithms, giving an unambiguous, concise
definition in the late 1980s. Finally, after Dave Davis
successfully showed the utility of a properly conceived
genetic algorithm in advanced problem solving in 1991,
interesting and use of genetic algorithms has grown
substantially and applied into a wide variety of
problems [16].
Typically, a genetic algorithm begins with an initial
population of individuals. An individual can be
translated as an each solution and each set of solution is
can be referred to as a generation or population. The
selection of the initial population is generally random
and spread throughout the search space. Every
individual in a generation is evaluated though the
“fitness” which is a measurement of the performance
and most fit individuals are then used to assign the next
generation of individuals in order to enhance the fitness.
The next generation is then produced by probabilistic
operators such as selection (reproduction), crossover,
and mutation. “Selection” is the process of picking the
1
E
E
ES
N
−=
⎛
⎜
⎞
⎟
(1)
2
τ
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
=×
(2)
E τ ,
185

Page 3

best individual to be transferred to the next generation.
“Crossover” is a recombination operator, creating
offspring with the characteristics of each parent.
“Mutation” is the operation of changing the
characteristics of the individual without reproduction
with another by means of altering the bits at random.
The GA continues to transform generation after
generation using those kinds of operators until the
stopping criteria defined by fitness function is met. In
this study, 5 different kinds of a “Selection” method are
applied to investigate the convergence rate. The fitness
function as shown in Eq. (3) is to minimize the average
revisit time during the specified time span, 30 days in
this simulation study. Throughout the whole simulation,
the population size is 10 and the “uniform” is selected
as a creation function. The “Gaussian” and “Scattered”
are selected as a Mutation method and Crossover
method, respectively. Regarding the stopping criteria,
the number of both of “Generation” and “Stall
Generation” is 50. The Matlab Genetic Toolbox® [19] is
used for this study.
However, these selected parametric numbers and
different methods relevant to each operator should be
carefully tuned in order to obtain the best final solution,
even though the GA can provide a good initial solution
and a reasonable global solution typically.

f = (ART/86400) – 1 (3)

4. Computational Approach

In order to propagate the orbit and evaluate the ART,
The COTS (Commercial Off-the-Shelf) orbit analysis
package and coverage
COVERAGE® [20] were integrated with the GA
module. The concept of the whole algorithm follows the
flowchart in Fig. 1. The orbit is firstly propagated
including J2 perturbation only during the time span (30
days) by the initial conditions generated by the GA
module and then, the fitness function is evaluated. If
stopping criteria is not satisfied with a fitness value, the
selected parents reproduce the new generation via
crossover and mutation. This procedure is repeated until
the number of stall generation (n=50) is reached or no
improvement from generation to generation is achieved.


tool, i.e. STK®
and


Fig. 1 The concept of the whole algorithm
The case studies to solve the problem given in this
study are divided into two parts, one is using a single
design variable (semi-major axis, a) and another is
using 3 design variables (semi-major axis a, inclination
i, and RAAN). For former cases, the “Selection” and the
“range of the initial population” are varied in order to
investigate the effect of themselves on the convergence
and effectiveness. Moreover, to overcome the rapid
convergence with an inadequate fitness value in case of
the range of initial population of 0~300, the method of
Mutation and Crossover were changed into “Adaptive
feasible” and “Heuristic” instead of “Gaussian” and
“Scattered”, respectively.
For latter cases, we selected “Uniform” as a method
of the “Selection” due to the convergence speed, which
was already proven via former case study. However, the
ranges of the initial population of a, i, and RAAN are
0~10, 0~1, and 0~50, respectively. This is because
broad initial values may find out the 1-day exact RGT
orbit that needs too expensive fuel cost. For all cases,
Elitism is used as a reproduction operator. The
conditions of case studies simulated in this paper are
summarized in Table 1.

Table 1 The conditions of case studies
Number of
Variables
CASE
Parametric
Operator
Description
(Method)
Roulette
Stochastic
Uniform
Tournament
Reminder
Uniform
0 ~ 50
0 ~ 100
0 ~ 500
Proportional
Adaptive feasible
Heuristic
a : 0 ~ 10
i : 0 ~ 1
RAAN : 0~50
1
2
3
4
5
6
7
8
9
Selection
Range of Initial
Population
Fitness Scale
Mutation
Crossover
1
(a)
10
3
(a, i, RAAN)
11
Range of Initial
Population

5. Results

Cases 1~5 in Fig. 2 show the results for a case study
of “Selection” operator. Best fitness value is achieved in
case of selecting “Tournament” method, while
“Uniform” method shows a rapid convergence of the
GA. As shown in Table 2, the number of imaging
chance during 30-days is increasing from 9 to 12 for all
cases 1~5, even though the current ART value of given
orbit (=2.9984 days) is varied from 2.3057 days to
1.9984 days case by case. This is because the ART
measures the average of all the revisit times for a given
point including the longer and shorter revisit time. Thus,
“Uniform” method of “Selection” operator is applied to
the remaining cases 6~11 due to the rapid convergence
speed and efficiency. Consequently, the result of design
variable, “semi-major axis” is 7065.6 km on average for
cases 1~5, which deviation from the current altitude is
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only about 2.4 km.
Cases 6~8 presents the effect of the range of the
initial population on the GA performance. When the
initial population value is increased up to 100, which
means the altitude can be changed up to 100km from the
current altitude, the ART can be reduced to 1.76 days
and the number of imaging chance is increased up to 15
times. The altitude deviation is 34 km. The increment of
the imaging chance during 30-days is about 67% with
respect to the current imaging chance. Assume chemical
propulsion is used for the orbit maneuver in these cases.
If the thruster specific impulse is 220s and the
spacecraft mass is 800 kg, the amount of fuel needed for
orbit maneuver in this case is about 6.65 kg. However,
the fuel cost of this case is relatively small compared to
the 1-day exact RGT case (i.e., 894 km or 567 km).
When the initial population value is increased up to
500, the resulting semi-major axis is 7266.45 km. This
responding altitude (888.31 km) is much close to the
1-day exact RGT case (894km). From the cases 6~8, it
is clear that we should take into account the fuel cost in
selecting the range of the initial population or
incorporate with fuel consumption constraints as a
penalty function.
However, the GA stops just after iterating 4 times
when the range of the initial population is 0~300. To
avoid the stall cases 9 and 10 are applied. The ART is
1.30173 days and resulting semi-major axis is 7263.205
km when the fitness scale is changed from “Rank” to
“Proportional”. When the “Reproduction, Mutation,
Crossover” are changed from default methods (already
mentioned above) to “Elitism, Adaptive feasible,
Heuristic”, the results are almost same with the case 8 in
terms of the ART, semi-major axis, and the number of
imaging chances, respectively. The reason of stall in
case 9 and 10 with default methods is under
investigation.
Finally, Case 11 presents the result when the
inclination and RAAN are added to the design value as
well as a semi-major axis. The default parametric
operators are used but the range of the initial population
is 10, 1, and 50 for a semi-major axis, inclination,
RAAN, respectively. In this case, the ART is close to 1
day and the number of imaging chance is increased by
2.3 times, even though the altitude deviation from the
current altitude is 4.389km. However, the fuel
consumption in the change of the inclination and RAAN
is about 1194 kg for the current chemical propulsion.
This fuel cost will not be allowed because the satellite
used in this problem was assumed not to be a
reconnaissance satellite originally. In other words, the
inclination and RAAN as well as semi-major axis are
also major orbital parameter to affect the coverage of
spacecraft and the fuel cost will be particularly
expensive in out of plane change. Consequently, we
should take into account the fuel cost in selecting the
range of the initial population of the inclination and
RAAN.
Regarding the ground track during 30 days, the
original ground track of current orbit is shown in Fig. 3.
The repeat cycle of current orbit is 28 days, which
means that the spacecraft has a little chance (9 times) to
take an image over a target site (Seoul) with sensor
characteristics and operational constraints as described
above. Fig. 4 shows Case 4 that achieves the ART less
than 2 days with an altitude deviation of 2.4 km only.
The ground track shifts westward in a non-sequential
manner. Fig. 5 shows Case 6 that achieves the number
of imaging chance more than 14 times with an altitude
deviation of 34 km. Case 8 that achieves the number of
imaging chance more than 28 times is shown in Fig. 6.
The ground track shifts sequentially westward from the
east boundary day by day. Fig. 7 shows Case 11, which
the design variables are 3 (i.e., a, i, RAAN). In this case,
the ground track is so close to that of 5-day exact RGT
orbit.

Table 2 Simulation Results
CaseFitness
Value
ART
(day)
Result of
Variables
(km, deg.)
7065.3733
7065.6901
7065.6799
7065.6817
7065.6379
7097.3082
7099.5226
7266.4518
7263.2046
7267.3774
a = 7067.6549
i = 99.412646
RAAN=187.12


Delta
Value
(km, deg.)
+2.1081
+2.4249
+2.4148
+2.4165
+2.3728
+34.043
+36.257
+203.187
+199.939
+204.112
+4.3897
+1.2803
+105.46
No. of
Imaging
Chances
12
12
12
12
12
16
15
29
21
29
20
1
2
3
4
5
6
7
8
9
10
11
1.3057
0.9985
0.9984
0.9984
1.3057
0.7627
0.8729
-0.003
0.3017
-0.035
0.0703
2.3057
1.9985
1.9984
1.9984
2.3057
1.7628
1.8729
0.9972
1.3017
0.9651
1.070
6. Conclusions

In this paper, we demonstrate the possibility of using
a genetic algorithm to search for a temporary target orbit
from the current mission orbit that will achieve a
reconnaissance mission over a particular target site
using a single LEO satellite during 30 days. The range
of the initial population of design values in this problem
is most important and should be taken into account for
the fuel cost. Through a comprehensive case study we
found the target orbit with an altitude of 719.168km,
which reduced the ART from 2.9984 days to 1.79 days
and also increased the number of imaging chance by
67% with a fuel cost of 6.65kg only. This result might
be acceptable compared with the 1-day exact RGT
solution in terms of fuel cost. Consequently, we were
able to successfully show that a GA can find a feasible
solution to this problem effectively as a computational
approach, while an analytical solution might not be
found easily because an orbit should be propagated and
analyzed the coverage
characteristics, and sensor constraints of the resulting
orbit over a single target site during the specified days at
the same time.

characteristics, sensor
187

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05 10 152025 303540 4550
1.4
1.6
1.8
2
2.2
2.4
2.6
Generation
Fitness value
Best: 1.3057 Mean: 1.7503


Best fitness
Mean fitness
051015202530354045 50
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Generation
Fitness value
Best: 0.99845 Mean: 1.6353


Best fitness
Mean fitness
05 1015 2025 3035 404550
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Generation
Fitness value
Best: 0.99839 Mean: 0.99839


Best fitness
Mean fitness
05 1015202530 35404550
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Generation
Fitness value
Best: 0.99841 Mean: 1.0984


Best fitness
Mean fitness
05 101520253035404550
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
Generation
Fitness value
Best: 1.3057 Mean: 1.3057


Best fitness
Mean fitness
05 101520253035404550
0.5
1
1.5
2
2.5
3
Generation
Fitness value
Best: 0.76278 Mean: 2.2164


Best fitness
Mean fitness

CASE 1 CASE 2 CASE 3
CASE 4 CASE 5 CASE 6
051015202530354045 50
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Generation
Fitness value
Best: 0.87287 Mean: 0.87287


Best fitness
Mean fitness
05 101520 2530 3540 4550
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Generation
Fitness value
Best: -0.0028281 Mean: 0.67182
05 1015 2025 3035404550
0
0.5
1
1.5
2
2.5
Generation
Fitness value
Best: 0.30173 Mean: 2.3789
051015202530 35404550
-0.5
0
0.5
1
1.5
2
2.5
Generation
Fitness value
Best: -0.034942 Mean: 0.12641
05101520253035 4045 50
0
0.5
1
1.5
2
2.5
Generation
Fitness value
Best: 0.070342 Mean: 0.22105


Best fitness
Mean fitness
CASE 7 CASE 8 CASE 9
CASE 10 CASE 11

Fig. 2 The concept of the whole algorithm

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