Satellite constellation method to achieve desired revisit performance for multiple targets

Abstract. We propose a method for optimizing satellite constellations and a series of process strategies to achieve the desired revisit performance for each target in the area of interest. Using a repeat ground track orbit of periodic characteristics, a tailored regional-satellite-constellation method is developed to achieve the desired revisit performance for each target. The study consists of four parts representing four programs: finding orbit elements through optimization techniques, producing an access matrix between targets and satellites, deriving the number of satellites for each target for the desired revisit performance, and deploying satellite groups by minimizing the orbital plane. Then, the superiority of this optimization method is verified through numerical comparisons and contour distributions for revisit performance compared with the existing conventional Walker method.


Introduction
With the advent of new spaces, there has been an increase in the diversity of missions using multiple low-cost group satellites. 1 These trends can be broadly divided into two types of missions.It is a field of space Internet consisting of a mega constellation for satellite communication and surveillance and reconnaissance for ground observation.Among these fields, the focus of this study was to improve the revisit performance of the region of interest using a large number of microsatellite groups.A region of interest can be defined as an area where ground observation is often required owing to the high density of military threats or frequent occurrence of national disasters.The revisit performance of such a region is based on how often the area is observed.In practice, the degree of observation is related to the efficiency of the mission designer deploying a satellite group with limited system cost, and it is important to select a satellite constellation method.
Satellite constellation methods are broadly classified into three types: Walker, street of coverage (SOC), and repeat ground track (RGT).3][4][5][6][7] A common characteristic of these studies was the symmetrical and regular nature of the constellations for continuous global coverage.Ballard, 8 Lang, 9 and Lang et al. 10 conducted an intensive study of the Walker constellation to find an optimal solution that provides continuous global coverage with a minimum number of satellites.2][13][14][15] The SOC constellation started with the distribution of orbits evenly spaced on the equatorial plane and expanded to the study of finding the optimal orbit for the zone of interest. 16,17In practical applications, the properties to be considered for mission operation in addition to the coverage performance and system cost of the satellite are diverse and complex, including coverage time, life cycle, space debris, launch, and station keeping.][20] In literature surveys, satellite constellation is largely divided into continuous and periodic coverage problems, and continuous coverage such as Walker and SOC is conceptually distinct from periodic coverage. 21Continuous coverage does not consider the rotation of the Earth for coverage issues.Moreover, it can be calculated in a unit sphere without any error in the nature of the coverage.However, periodic coverage directly affects the coverage characteristics owing to the rotation of the Earth, which is therefore a factor to consider when solving the periodic coverage problem.The complexity of the periodic coverage problem has been emphasized in many studies. 22,23The RGT orbit is representative of periodic coverage and provides a consistent observational image of any target on the Earth's surface during a repeat cycle.In addition, the coverage performance problem is solved by the relative dynamics between the satellite and the observed target on Earth's surface.
2][33] Particularly, Li et al. 31 presented an optimal design model that visits a single target twice a day in the ascending and descending stages during one revisit cycle with the aim of addressing operationally responsive space missions.The results provide an analytical method for designing a responsive orbit, which is useful for mission designers.Lee 32 presented a closed-form solution for the design of a circular RGT orbit through a geometric approach and proved that the RGT dynamics are like the different epicycle systems of Ptolemy's model.In addition, the revisit performance of the satellite group dramatically increased based on the relationship between the self-intersection point of the RGT orbit and the specific target.He and Li 33 built a database of RGT orbits through grid search and numerical methods and employed the pruning method to select the minimum orbits that cover the most discrete points.Based on this, a systematic constellation design methodology achieves a rapid revisit frequency and uniform coverage with a minimum number of satellites and maximum revisit time constraints.The commonality of the above studies is the focus on the target (single or multiple)-based RGT mission design method.
This study is a further development of those conducted on literature surveys.It finds the orbital element and the number of satellites for each target by applying a target-tailored optimization method, which minimizes the revisit time and obtains the required operational capability for the revisit performance of the target.This study is organized into four parts: finding the optimal orbital elements for each target, developing a matrix of the number of accesses for each target, finding the optimal number of satellites for each target, and minimizing the orbital plane to obtain the RGT constellation.Each part represents a program that consists of an interworking system.

Satellite's Surveillance and Reconnaissance Cycle
Over the past few decades, the security environment in the Asia-Pacific region has changed dynamically owing to a variety of circumstances.This security situation is changing the character of traditional regional threats into a global, diverse, and unspecified feature.To respond effectively to these changes, establishing a command and control system that enables prompt operation against threats that may occur at any place and time is necessary.The use of the observe, orient, decide, act (OODA) loop, which is widely known, not only for military purposes but also for companies, can satisfy security needs.The OODA loop is a concept developed by Colonel John Boyd, an F-16 pilot of the U.S. Air Force, as part of an effective strategy to fight and defeat enemies in various flight-operation environments.The purpose of this study is to strengthen the surveillance and reconnaissance capability of satellite groups through the OODA loop for the rapid power projection of space assets against arbitrary threats.
The OODA loop comprises an inner and an outer loop.The inner loop has an observe → orient → decide → observe cyclic structure, and the outer loop follows an observe → orient → decide → act → observe cycle.This cycle is repeated at least once or more times owing to the uncertainty of the threat.In particular, with the inner loop, there are few delay factors and incomplete information, but quick decision support is required by operating a fast loop.However, the outer loop has various delay factors.Delay factors affect the action-observation phase, including the time it takes for the commander to act after deciding, and the time it takes for the projected force to be applied.Therefore, the OODA loop can be optimized using technologies, such as artificial intelligence (AI) or big data, and simple procedures, such as information collection, analysis, and targeting of the inner loop with less influence from delay factors.This study optimized the decision cycle for timely information collection by interlocking the inner loop with the average revisit time of the satellite groups.
As shown in Fig. 1, the surveillance and reconnaissance cycle of the satellite group for the target in the area of interest takes ∼5 to 10 min to acquire satellite images and ∼5 min to receive and process the satellite images at the ground control station.The next step is the reading/identification of the target, which plays an important role in the decision cycle and typically takes hours to days.Recently, efforts have been made to shorten the reading/identification step to a few minutes using AI or big data.As a final step, it takes the commander ∼15 min to make a decision.Consequently, a minimum of 30 þ α minutes, where the α value is the threshold for statistical significance, is required to interlock the satellite's surveillance and reconnaissance cycle with the command and control systems.

Optimization Problem Using Genetic Algorithm
This study established the following two optimization problems that must be solved to interlock the desired revisit performance of the satellite group with the decision cycle of the OODA loop.
1. determine the elements of the optimal circular RGT orbit for each target with the minimum average revisit time 2. determine the optimal number of satellites required for each target to achieve the desired average revisit time.
The optimization technique for solving the above two problems is the applied genetic algorithm: Problem 1 uses a multi-objective optimization problem (MOOP), and problem 2 uses a singleobjective optimization problem (SOOP).To apply the MOOP to problem 1, the average revisit time and number of accesses are defined as the revisit performance, and the optimized orbit element is subsequently determined.The reason for including the number of accesses in revisit performance is that the higher the mission altitude of the satellite, the longer the access time of the satellite sensor; therefore, even if the average revisit time is good, the value of the number of accesses is not the optimal solution.In addition, MOOP can be applied to various optimization techniques using the number of accesses as a variable.In this study, the MOOP selects the Pareto front, which is the optimal solution to the revisit times and number of accesses in any environment, to maximize the revisit performance for the target.In the SOOP of problem 2, to find the optimal number of satellites, the total number of satellites required to achieve the desired revisit time is the learning target of the genetic algorithm.Typically, genetic algorithms do not always succeed in optimal solution search because a heuristic search is performed based on a random population.An optimal solution does not exist if appropriate options are not selected, or different results appear each time it is executed.In addition, when the total number of satellites is used as a single objective function, various optimal solutions are obtained because the number of objective function variables is equal to the number of targets.To solve this problem, this study proposes a robust optimal solution by grouping targets with similar locations and properties and using them as an objective function.The two problems presented above used a genetic algorithm built-in function in MATLAB.To find the optimal orbital elements of each target, the MOOP is performed by linking MATLAB's "gamultiob" built-in function with the Systems Tool Kit (STK).To find the optimal number of satellites for the target, the SOOP uses the "ga" built-in function.

Optimization Programs
In this study, four programs were developed to optimize the revisit performance of a satellite constellation.
The above four development programs are interworking systems based on MATLAB-STK.

OOP
The purpose of this program is to determine the optimal RGT orbital elements with the minimum average revisit time for each target.The satellite orbit is typically described by six orbit elements E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 4 ; 3 7 0 fa; e; i; Ω; ω; M 0 g: In Eq. ( 1), the semi-major axis a and eccentricity e determine the size and shape of the orbit.The ascending node Ω, inclination i, and argument of perigee ω represent the orientation of the orbital plane, and the initial mean anomaly M 0 is the position of the satellite at time t 0 .Because this study considers a circular RGT orbit, e and ω are excluded when determining the OOP for each target.The ðΩ; MÞ phasing mechanism is used to design the orbit such that the group of satellites assigned to each target flies the same RGT.Consequently, the orbit elements to be searched are a; i, and Ω [32].

Orbit frequency γ
To form an RGT orbit, an integer number of orbit periods and an integer number of days must be equal.Thus, the RGT orbit can be expressed as the orbital frequency γ, which is the ratio of the rotational angular velocity of the Earth to that of the satellite E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 4 ; 1 8 9 where n, _ M, _ ω, ω È , and _ Ω denote the mean motion of the satellite, change in the mean motion, change in the perigee, rotational velocity of the Earth, and change in the ascending node, respectively.Each element of Eq. ( 2) under Earth's J 2 perturbation is defined as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 a ; 1 1 4 ; 1 2 8 where p ¼ að1 − e 2 Þ, the Earth's radius R e and J 2 ¼ 0.00108263.
The orbital frequency γ is the real number expressed as a fraction of the nodal day N d and the number of revolutions of the satellite R s as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 7 ; 4 3 1 where N d and R s are positive integer numbers prime to each other.The range of an arbitrary real number γ is the range of the altitude of the satellite; for a γ range of 14.5 to 15.5, the altitude of the circular orbit considering J 2 is ∼340 to 660 km.In determining the optimal solution, there are many possible altitudes for a given range of γ, and the complexity is related to the mathematics of the decimal point of γ.The following equation represents the R s of the RGT orbits that can be selected from a given range of γ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 7 ; 3 2 3 If only one decimal place is considered for γ, the number of R s that can be taken for each N d is as shown in Table 1.
The utilization of Table 1 minimizes the computational effort by searching for optimal orbit elements using R s values for each N d as input variables in the genetic algorithm and finds the optimal RGT orbit in a given γ range.

Inclination i and ascending node Ω
It is inefficient to search the entire range of 0 to 90 deg for the inclination optimized for the target.For a low Earth orbit observation satellite, Fu et al. 34 showed the best revisit performance for the target was when it had an inclination 3 to 5 deg higher than the target's latitude.Therefore, this study expands the search range to find the optimal inclination in the range of the target latitude þ10 deg, which is approximately twice the value presented in Ref. 34; however, this does not pose an issue with finding the optimal solution by performing a number of simulations.
The longitude of the ascending node Ω uses the ground track separation interval of the equator.A successful RGT on any nodal day was repeated at 2π γ•N d intervals, which represents an angular separation between the equatorial ground tracks.Therefore, the search range for Ω is 0 to 2π γ•N d .

Access Matrix
In the previous section (on OOP), we determined the orbital elements of the RGT that minimize the revisit performance for each target.A satellite deployed in an optimal RGT orbit for each target periodically accesses its own and other targets.Consider multiple targets in the region of interest, and let the serial numbers of the targets be j ¼ 1;2; 3; : : : ; m.The number of accesses of the satellite deployed as the optimal orbit element for the j th target is called the apparent access ðA j Þ and the number of times accessed for the remaining targets other than the j th target is called the relative access ðR jk ; j ≠ kÞ.This correlation can be expressed as an AM, as presented in Table 2.
The advantage of AM is that it is useful for expressing the standard form of the optimization problem and effective for analyzing the number of accesses per target.The characteristic of AM is that the diagonal line is a principal element, implying that for A j of each target with the largest number of accesses, the relationship between A j ≥ R jk is established, and it is always greater than zero.

OOS
An RGT orbit has the characteristic of repeating the same trajectory for a specific period.Therefore, the revisit performance of the target exhibits a linear characteristic with respect to the number of satellites.AM indicates the number of accesses to one satellite for each target.If the revisit performance for each target is defined as χ j , the χ j of one satellite for each target is represented as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 4 ; 3 3 9 for j ¼ 1;2; 3; : : : ; m.In a more extended manner, if the number of satellites flying an identical RGT for each target is S j , the revisit performance for the satellite group for each target is expressed as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 4 ; 2 6 2 where R jk is the relative access of the k satellite to the j target.Equation ( 7) is a standard form suitable for use as a constraint in the optimization problem to achieve the target revisit performance for each target.Let f be the objective function of the optimization problem and S be the decision variable.The generalized optimization standard form for m targets is as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 a ; 1 1 4 ; E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 b ; 1 1 4 ; 1 0 7 R jk • S k ≤ χ j ; j ¼ 1;2; 3; : : : ; m; where Eq. (8a) is minimized subject to Eq. (8b).Furthermore, χ j denotes the desired revisit performance for each target.In this study, it was assumed that all targets had the same desired revisit performance, χ.Thus, assuming that λ t is the desired average revisit time for each target, the number of accesses for each target is calculated as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 1 1 7 ; 7 1 2 χ ¼ 86;400 λ t .( 9) In applying Eq. ( 8) to the optimization problem of a genetic algorithm, there are several considerations.First, several optimal solutions exist because a random-distribution-based heuristic search is applied to determine the optimal number of satellites for each target.This makes it difficult for the mission designer to select an optimal solution for each target.Second, as many orbital planes as the total number of satellites are required for the deployment of the satellite group to the entire target, which incurs a significant launch cost.To overcome this problem, in this study, the optimal solution was determined by grouping in consideration of the characteristics of the target, and an objective function was used to minimize the total number of satellites and orbital planes: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 1 1 7 ; 5 6 8 where G k;max is the maximum number of satellites for each target group, which will be described in detail later, and is equal to the number of orbital planes in the group.The constraint is the same as that in Eq. (8b).In summary, the OSS program implements the optimization problem in a standard form based on AM and derives the optimal solution by utilizing the relationship between the total number of satellites and orbital plane minimization.

RGT
In this section, the satellite group assigned to each target derived from the OOS program flies the same RGT.To achieve this, the following periodic conditions must be satisfied.
E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 7 ; 3 9 8 where P t is the orbital period of the satellite.Subsequently, the following relationship from Eq. ( 11) is given E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 7 ; 3 3 6 M 0 ¼ −γΩ: Using Eq. ( 12), we obtain the satellite phasing rule such that N s satellites have the same RGT E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 a ; 1 1 7 ; 3 0 0 where Ω 1 and M 10 are the orbital elements of the first satellite assigned to the target and Θ is the spacing of the orbital planes.Θ is the equally spaced distribution of N s satellites and is defined as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 1 1 7 ; 2 2 3 The satellite phasing rules in Eq. ( 13) have the same concept as the phasing rules in the Flower Constellations that use independent integer parameters. 35,36This study deployed each assigned satellite group for m targets, and the phasing mechanism for each target was as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 a ; 1 1 7 ; 1 5 1 (15b) for j ¼ 1;2; 3; : : : ; m, where Ω j is the optimal ascending node for each target obtained through OOP.Note that because Ω j is different for each target, the spacing of the ascending nodes is not constant, even if the orbital plane is deployed with a constant Θ.However, the initial mean anomalies were equally spaced.The total satellite N s constellation according to Eq. ( 15) requires as many orbital planes as the number of satellites because each satellite has an ascending node, which incurs a significant launch cost to construct a satellite group.Therefore, it is necessary to minimize the orbital plane of the satellite group.To design a mission to fly the same RGT to one target, the satellite group must have the same orbital frequency γ and inclination i.To exploit this characteristic, targets with similar locations and properties are grouped together.Then, in the OOP program, the optimal orbit element for each group is determined, and the nodal day of γ is used to minimize the orbital plane.The N d of the optimal solution γ obtained by OOP indicates that the RGT orbit is formed after the Earth has rotated N d times.In other words, there is an initial condition to have the same RGT as the N d number on one orbital plane, which is the initial mean anomaly.
Consider m targets and l target group within the region of interest.
E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 a ; 1 1 4 ; 5 8 0 ; t e m p : i n t r a l i n k -; e 0 1 6 b ; 1 1 4 ; 5 4 5 where G l ≤ T m .Moreover, γ for each group obtained from the OOP program is defined as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 1 1 4 ; 5 2 7 where R s and N d are primes of each other.The optimal number of satellites G s for each group obtained from the OSS program was as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 8 ; 1 1 4 ; 4 6 6 From Eqs. ( 17) and ( 18), the number that can minimize the orbital plane for each group is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 9 ; 1 1 4 ; 4 1 7 where P k is an integer.Therefore, the orbital plane minimization number for the entire target group is as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 0 ; 1 1 4 ; 3 5 7 Another way to minimize the orbital plane is to use the difference in the optimal Ω for each target.If the N d value of γ is smaller than the number of orbital planes desired by the mission designer, γ and i for each group obtained above are the same, and only Ω j for each target is searched again through OOP.
Figure 2 shows two different RGTs with the same γ and i designed using the ðΩ; Here, #2 RGT can also be designed using the phasing mechanism of ðΩ 1−2 ; M 0 10−2 Þ; ðΩ 2−2 ; M 0 20−2 Þ; ðΩ 3−2 ; M 0 30−2 Þ.Consequently, it is possible to minimize the orbital plane by adjusting M 0 0 for #2 RGT to the orbital plane for #1 RGT.Based on this characteristic, the minimum possible number of orbital planes is as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 1 ; 1 1 4 ; 1 7 5 Then, the satellite phasing rule is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 2 a ; 1 1 4 ; 1 2 0 where Θ ¼ 2π P total .

Program Flow Chart
The developed programs consist of classes 1 and 2, and Fig. 3 shows the algorithm of the developed programs.The class 1 algorithm inputs the latitude and longitude of the target and mission period for the RGT orbit and finds the optimal orbit elements ði; Ω; γÞ for each target by interworking with STK in OOP.The AM program implements the number of accesses to one satellite for each target as a matrix based on the solution obtained from the OOP, and the OSS finds the optimal number of satellites for each target to achieve the desired revisit performance.The RGT  program linked with STK distributes satellite groups equally spaced using the satellite phasing rule.
In class 2-1, targets with similar characteristics are grouped together to minimize the orbital plane, and the optimal solution is found using the same procedure as in class 1.If the denominator of the optimal solution of γ is small, and it is difficult to minimize the orbital plane considered by the mission designer, the class 2-2 program uses the same i and γ values of class 2-1 and finds Ω for each target through the OOP.The orbital plane minimization of class 2-2 was achieved using Eqs.( 21) and ( 22).

Numerical Simulations
This section presents the simulation results of the four developed programs in achieving the revisit performance of the 30 þ α minutes of the decision cycle of the OODA loop described in the problem statement.For the area of interest for numerical simulation, nine commonly known ballistic missile operation area (BMOA) targets located in North Korea on the Korean peninsula were selected in consideration of the suitability of the research purposes and target distribution.

OOP
The OOP program aims to find orbit elements with the minimum average revisit time of one satellite for each target by applying a genetic algorithm technique based on MATLAB-STK. Figure 4(a) shows nine targets in BMOA.The actual target is an area of ∼50 km × 50 km; however, in this study, it was considered as a point target.The search range for finding the optimal orbit element for each target of one satellite was mentioned in the previous section.The orbit frequency γ is in the range of 14.5 to 15.5, which is the surveillance and reconnaissance altitude of interest in this study, that is, in the range of 340 to 660 km.The incidence angle of the synthetic aperture radar payload is set at 15 to 35 deg considering the technology at the microsatellite level.
Figure 4(b) shows the orbital elements searched by the OOP program for nine targets.The optimal solution is 15, which has the smallest nodal day in the given range of γ and has the simplest geometric RGT structure.The mission altitude varies depending on the location of the target but is ∼490 to 500 km.The optimal solutions for the ascending node is 6 to 9 deg, and the inclination is 3 to 3.5 deg for the target latitude.
Figure 5(a) shows the number of accesses to the BMOA #4 target obtained through the OOP, and it is clearly confirmed that there are four accesses a day.The target was located at the center of the diamond mesh structure of the RGT trajectory.As shown in Fig. 5(b), all nine targets are located at the center of the diamond shape, and the value of the apparent access (A j ) is four.

AM
The AM program is a matrix of the number of accesses of one satellite for each target based on the optimal solution of the OOP.Table 3 shows the AM for nine BMOA targets, which is a 9 × 9 matrix.The diagonal of AM is A j , all of which are four, which means that the number of accesses of the satellites deployed with optimal orbital elements for each target is four per day.The AM rows represent the number of accesses per satellite to the target, and the columns indicate the number of accesses per target to the satellite.This access pattern was repeated continuously based on the nodal day.

OSS
The OSS program aims to determine the optimal number of satellites required to achieve revisit performance for each target based on the AM values obtained in the previous section.The objective function for the optimization problem is set such that the combination of the total number of satellites for the nine targets is minimized The constraint λ t ¼ 48 was applied to achieve an average revisit time of 30 min for each target.
The objective function in Eq. ( 23) consists of nine variables.In addition, because the genetic algorithm applied in this study is a heuristic search based on a random population, the optimal combination of the number of satellites in the OSS has several solutions, as shown in Table 4. Table 4 shows 10 different optimal solution combinations out of 152 non-overlapping optimal

RGT
The RGT program deploys satellite constellations for each target based on the solution for the optimal number of satellites obtained in the previous section.Satellite constellations were equally spaced according to the satellite phasing rule.To verify the performance of the developed program, the numerical value of the average revisit time of the satellite group and the revisit performance distribution through the contour were compared with the commonly used Walker method.
Figure 7 shows the results of comparing the RGT class 1 to the Walker method in terms of average revisit time.The RGT constellation selects the first optimal solution (7 0 1 1 7 5 0 3) from the combination in Table 4.For the Walker method, the same sensor swath, orbit altitude of ∼490 km, and satellite group 43:28/7/1 were set under mission conditions such as the RGT specifications.As shown in Fig. 7, the RGT achieved an average revisit time of 30 min for all nine targets.By contrast, for the Walker method, targets #5, #6, and #8 had a revisit time of 30 min, but other targets existed outside the desired revisit performance.As shown in Fig. 8, the contour distribution of the revisit performance clearly shows lateral characteristics in the Walker method, while the RGT is customized to the target and distributed.
Table 5 lists the specifications and results of the class 2 program.The class 1 program requires 28 orbital planes equal to the total number of satellites to construct a constellation because each target has Ω.As emphasized, this incurs a significant launch cost for the constellation.To solve this problem, this study performs orbital plane minimization using a class 2 algorithm.The class 2-1 algorithm uses the denominator γ to minimize the orbital plane.Because the denominator of the optimal solution of γ is 16 and 2 for each respective group, the entire satellite group can be deployed in 11 orbital planes using Eq.(20).The class 2-2 algorithm utilizes the maximum number of satellites for each group among the number of satellites obtained from the OSS, and by applying Eq. ( 21), it is possible to deploy the entire satellite group in the eight orbital planes.Even if the orbital plane is minimized, there is no challenge in achieving the revisit performance, and Figs. 9 and 10 show the results of class 2-2 compared with Walker.
In summary, the merit of this study is that it achieves the desired revisit performance by deploying tailored satellite groups for each target regardless of the target distribution.Observing the contour of the revisit performance, the parts with good revisit performance were concentrated around the target location.By contrast, for the Walker method, the lateral characteristics of the revisit performance are clearly shown, and the target located in the upper part of the satellite's orbit is good; however, the target in the lower part is limited.In addition, because the width of the lateral band of the contour is thin, it has weak characteristics for widely distributed targets.

Conclusions
This study proposes four programs to achieve the desired revisit performance for each target in the area of interest.The program provides robust and reliable results that satisfy the required operational performance of the satellite group, regardless of the location and size of the target distribution.This study makes the following general and practical contributions to mission designers in a real environment.In the optimization search process, we learn the RGT orbit design method and check the analytical results of the RGT orbit for the number of mutual accesses between the targets.In addition, the required operational performance is set by assigning the same or weighting the desired revisit performance for each target or target group.
In addition, the launch cost can be reduced by applying various methods to minimize the orbital plane of the satellite group.These achievements contribute to the success of the planned mission by ensuring the surveillance and reconnaissance capability of the target in the area of interest.

Fig. 3
Fig. 3 Flow chart of the overall program.

Table 1
Searchable R s with one decimal place considered for γ.

Table 5
Specifications of class 2 program.