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,
[email protected]) 2 Division of Aerospace Engineering, KAIST, Daejeon, Korea (Tel : +82-42-869-3722; E-mail:
[email protected])
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 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.
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
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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
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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 which is currently 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
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.
⎛ ⎝
Nτ ⎜ 1 −
τ ⎞ ⎟ = Dτ τ ⎠ E
(1)
E
ES
1/ 3
2 ⎛ ⎛ τ ⎞ ⎞ a = ⎜ μ ×⎜ ⎟ ⎟⎟ ⎜ 2 π ⎝ ⎠ ⎠ ⎝
(2)
Where τ , τ E , τ ES 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
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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
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
(3)
CASE
Parametric Operator
Fitness Scale Mutation Crossover
Roulette Stochastic Uniform Tournament Reminder Uniform 0 ~ 50 0 ~ 100 0 ~ 500 Proportional Adaptive feasible Heuristic
Range of Initial Population
a : 0 ~ 10 i:0~1 RAAN : 0~50
1
4. Computational Approach
2
In order to propagate the orbit and evaluate the ART, The COTS (Commercial Off-the-Shelf) orbit analysis package and coverage tool, i.e. STK® and 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.
1 (a)
3 4 5 6 7 8 9 10
3 (a, i, RAAN)
11
Description (Method)
Selection
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
Fig. 1 The concept of the whole algorithm
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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.
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.
Table 2 Simulation Results Case
Fitness Value
ART (day)
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
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
Delta Value (km, deg.) +2.1081 +2.4249 +2.4148 +2.4165 +2.3728 +34.043 +36.257 +203.187 +199.939 +204.112
a = 7067.6549 i = 99.412646 RAAN=187.12
+4.3897 +1.2803 +105.46
No. of Imaging Chances 12 12 12 12 12 16 15 29 21 29 20
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, sensor characteristics, and sensor constraints of the resulting orbit over a single target site during the specified days at the same time.
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Best: 0.99839 Mean: 0.99839 Best: 0.99845 Mean: 1.6353 Best: 1.3057 Mean: 1.7503
2.8
2.6
Best fitness
2.6
Best fitness Best fitness
Mean fitness
Mean fitness
2.4
Mean fitness
2.6 2.4
2.4
2.2
2.2
2
1.8
Fitness value
2 Fitness value
Fitness value
2.2
1.8 1.6
2 1.8 1.6
1.4
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1.6
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1
1
0
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5
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25 30 Generation
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CASE 3 Best: 0.76278 Mean: 2.2164 3
Best: 1.3057 Mean: 1.3057
Best fitness
2.1
Best: 0.99841 Mean: 1.0984 Best fitness
Mean fitness
2
2.5
Mean fitness
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1.9
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2.6
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CASE 6
Best: -0.0028281 Mean: 0.67182 Best: 0.30173 Mean: 2.3789
3.5
Best: 0.87287 Mean: 0.87287
2.5
2.6 Best fitness
3
Mean fitness
2.4
1.8 1.6
Fitness value
2
Fitness value
2 Fitness value
2
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1 0.8
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20
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CASE 9
Best: 0.070342 Mean: 0.22105 2.5
Best: -0.034942 Mean: 0.12641
Best fitness
2.5
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2
2
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-0.5
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35
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CASE 10
0
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CASE 11
“Interplanetary Flyby Mission Optimization Using a Hybrid Global-Local Search Method”, Journal of Spacecraft and Rocket, Vol. 37, No. 4, July-Aug., 2000, pp.468-474. [4] Rauwolf, G. A., and Coverstone-Carroll, V. L., “Near-Optimal Low-Thrust Orbit Transfers Generated by a Genetic Algorithm”, Journal of Spacecraft and Rocket, Vol. 33, No. 6, Nov.-Dec., 1996, pp.859-862. [5] Lang, T. J., “A Parametric Examination of Satellite Constellations to Minimize Revisit Time for Low Earth Orbits Using a Genetic Algorithm”, AAS/AIAS Spacefligh Mechanics Meeting, AAS 01-345, 2001. [6] Ely, T. A., Crossley, W. A., and Williams, E. A., “Satellite Constellation Design for Zonal Coverage
Fig. 2 The concept of the whole algorithm
References [1] Ely, T. A., Crossely, W. A., and Williams, E. A., “Satellite Constellation Design for Zonal Coverage Using Genetic Algorithms”, The Journal of the Astronautical Sciences, Vol. 47, Nos. 3 and 4, July-Dec., 1999, pp.207-228. [2] Kim, Y. H., and Spencer, D. B., “Optimal Spacecraft Rendezvous Using Genetic Algorithms”, Journal of Spacecraft and Rockets, Vol. 39, No. 6, Nov., 2002, pp.859-865. [3] Crain, T., Bishop, R. H., Fowler, W., and Rock, K.,
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Using Genetic Algorithms”, The Journal of the Astronautical Sciences, Vol. 47, Nos. 3 and 4, July-Dec. 1999, pp.207-228. [7] Abdelkhalik, O., and Mortari, D., “Orbit Design for Ground Surveillance Using Genetic Algorithms”, Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, Sep.-Oct., 2006, pp.1231-1235. [8] Abdelkhalik, O., and Mortari, D., “N-Impulse Orbit Transfer Using Genetic Algorithms”, Journal of Spacecraft and Rocket, Vol. 44, No. 2, Mar.-Apr., 2007, pp.456-459. [9] Fortescue, P., Stark, J., and Swinerd, G., Spacecraft Systems Engineering, 3rd Edition, John Wiley & Sons Ltd, England, 2003. [10] Lang, T. J., “Streets of Coverage Constellations to Minimize Revisit Time in Low Earth Orbit”, AAS/AIAA Spacefligh Mechanics Meeting, AAS 05-154, 2005. [11] Gutenev. A., and Mozhaev, G.. V., “On The Orbital Structure of Elliptic Satellite Constellations Providing Minimum Revisit Time”, J. of Electrical and Electronics Engineering, Austrailia, Vol. 13, No. 4, 1993, pp.293-301. [12] Lang, T. J., “Walker Constellations to Minimize Revisit Time in Low Earth Orbit”, AAS/AIAS Spacefligh Mechanics Meeting, AAS-03-178, 2003. [13] Lang, T. J., “A Parametric Examination of Satellite Constellations to Minimize Revisit Time for Low Earth Orbits Using a Genetic Algorithm”, AAS/AIAS Spacefligh Mechanics Meeting, AAS 01-345, 2001. [14] Hayes, E. W., “A Method for Selecting Satellite Constellations to Minimize Revisit Time”, AIAA 26th Aerospace Sciences Meeting, AIAA 88-0162, 1988. [15] Middour, J. W., “Survey of Orbit Selection for Satellite Earth Surveillance”, AIAA Space Technology Conference, AIAA 99-4637, 1999. [16] Pegher, D. J., and Parish, J. A., “Optimizing Coverage and Revisit Time in Sparse Military Satellite Constellations: A Comparison of Traditional Approaches and Genetic Algorithms”, Master’s Thesis, Naval Postgraduate School, California, 2004. [17] Kantsiper B., and Weiss, S., “An Analytical Approach to Calculating Earth Coverage”, AAS/AIAS Spacefligh Mechanics Meeting, AAS 97-620, 1997. [18] Hayes, E. W., “Computation of Average Revisit Time for Earth Observing Satellites”, AIAA 26th Aerospace Sciences Meeting, AIAA 88-0574, 1988. [19] The MathWorks, Inc., Genetic Algorithm and Direct Search Toolbox, 2007. [20] http://www.agi.com
Fig. 3 Ground track for the current orbit
Fig. 4 Ground track for Case 4
Fig. 5 Ground track for Case 6
Fig. 6 Ground track for Case 8
Fig. 7 Ground track for Case 11
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