Data collection by satellites during and after a natural disaster is of great significance. In this work, a reconfigurable satellite constellation is designed for disaster monitoring, and satellites in the constellation are made to fly directly overhead of the disaster site through orbital transfer. By analyzing the space geometry relations between satellite orbit and an arbitrary disaster site, a mathematical model for orbital transfer and overhead monitoring is established. Due to the unpredictability of disasters, target sites evenly spaced on the Earth are considered as all possible disaster scenarios, and the optimal reconfigurable constellation is designed with the intention to minimize total velocity increment, maximum and mean reconfiguration time, and standard deviation of reconfiguration times for all target sites. To deal with this multiobjective optimization, a physical programming method together with a genetic algorithm is employed. Numerical results are obtained through the optimization, and different observation modes of the reconfigurable constellation are analyzed by a specific case. Superiority of our design is demonstrated by comparing with the existing literature, and excellent observation performance of the reconfigurable constellation is demonstrated.
A global or regional observation satellite constellation is used for some emergency or significant missions such as disaster monitoring, and usually, the observation pattern of a specific constellation is restricted by both on-board instruments and geometric configuration of satellites [
The satellite constellation reconfiguration is defined as “a deliberate change of the relative arrangements of satellites in a constellation by addition or subtraction of satellites and orbital maneuvering in order to achieve desired changes in coverage or capacity” [
In recent years, plenty of research about constellation reconfiguration has been carried out. De Weck et al. [
As for regional or specific target observation, repeating ground track orbit is a usual choice, which has been proved to have better partial coverage properties than the unrepeated one [
Previous investigations mentioned above considered plenty of approaches to optimize the reconfiguration process and also the use of repeating orbit for partial coverage. However, most of them were carried out with a known initial constellation, a known final constellation, or a set of known observation locations, and few of them considered the reconfiguration for monitoring the unpredictable disaster site as well as the design of a reconfigurable initial constellation. Besides, most of the previous researches took fuel consumption as the optimization target, and few of them considered the optimization of reconfiguration time or tradeoff between them. In addition, none of the current constellations with the task of observing the Earth can achieve active maneuvers to collect data for the natural disaster or other significant events [
In this paper, we aim to design a ReCon, which can achieve timely and adequate observation for an arbitrary and unpredictable disaster site through orbital maneuvers. Satellites in the constellation flying directly over the disaster area for data collection are the purpose of the reconfiguration. Due to the unpredictability of disaster, target sites evenly spaced on the Earth are considered as all possible disaster scenarios. For the optimal ReCon, several optimization objectives are considered, which are (1) minimizing fuel consumption, which is equivalent to minimizing the total velocity increment during reconfiguration; (2) minimizing maximum and mean reconfiguration time to ensure timely observation for the target site; and (3) minimizing standard deviation of reconfiguration times to ensure similar reconfiguration ability for all target sites. To deal with this multiobjective optimization, a physical programming method together with a genetic algorithm is employed. The preferred design metrics in physical programming are considered according to the specific requirement or preference of the designer, which makes the design much more feasible, practical, and reasonable. Furthermore, GOM and ROM of the ReCon are analyzed to demonstrate the superior observation performance for normal global scan as well as some emergency disaster monitoring. As a whole, the main contributions of our work can be summarized as follows: firstly, a mathematical model for orbital transfer and overhead monitoring is established by analyzing the space geometry relations between satellite orbit and an arbitrary disaster site. Secondly, a ReCon for unpredictable disaster monitoring is designed, and several optimization objectives are considered to optimize the reconfiguration process as well as to ensure that the ReCon has similar reconfiguration ability to observe any possible disaster site. Thirdly, two observation modes are presented in the ReCon, and normal global scan as well as emergency disaster monitoring with excellent observation performance can be achieved, respectively.
The rest of this paper is organized as follows: The detailed problem formulation process is described in Section
Problem formulation is presented in this section. To simplify the formulation process, as well as to make the model easier to establish and understand, some reasonable and necessary assumptions ought to be made. Firstly, maneuvers for orbital transfer are impulsive, and this is reasonably accurate during the initial mission planning [
The aim of our work is to design a ReCon for disaster monitoring. There are two observation modes for the ReCon, which are GOM for normal operations and ROM for contingent responses [
Change rates
Also, for the nodal period of initial orbit, the corresponding angular velocity
A ground track is considered as repeating or periodic if it repeats at a fixed time interval allowing a cyclic observation of the Earth [
For the target repeating ground track orbit, assume that change rates of the RAAN, the argument of perigee, and mean anomaly are
Also, radius
For the nodal period of orbit, the corresponding angular velocity
In this work, the repetition period of one nodal day is considered to ensure the high frequency of observation, which also means that
It should be mentioned that
The transfer time
Due to the unpredictability of disaster, target sites around the Earth’s surface in step of 1 degree are set as possible scenarios, and all satellites in the constellation are considered to monitor these target sites through orbital maneuvers. To illustrate the model mathematically, one arbitrary target site on the ground and an orbit with one satellite are considered as the basic scenario. The main task of this subsection is to calculate the specific transition time of each satellite monitoring each target site in the basic scenario and then to form the whole constellation reconfiguration process. It should be noted that transition time refers to the time from the mission beginning to the satellite finishing the transfer, while reconfiguration time represents the maximum transition time of all satellites in the constellation.
The ascending pass of a satellite is first considered for the monitoring mission. The basic scenario is presented in Figure
Monitoring during ascending pass.
Suppose that the longitude and latitude of the target site
In the inertial system, target site
After a specific time, target site
For point
Therefore, the defined “minimum distance 1”
Here, we take
As for the descending pass, similar analysis can be carried out. As shown in Figure
Monitoring during descending pass.
Let
Then, in the spherical right triangle
Therefore, the defined “minimum distance 2”
Similarly, we take
Once
As illustrated in Figure
Orbital transfer.
At
Then, the phase difference
As shown in Figure
The difference between
For the case that
Therefore, the total transition time
For the case that
So far, time limits
In conclusion, for the basic scenario that one satellite overhead monitors one arbitrary target site during ascending pass, the calculation process can be summarized as the following seven steps.
Input target site
Calculate minimum distance
Calculate time limit
Calculate total transition time
Judge whether
Calculate total velocity increment
Output
Similarly, analysis for descending pass can be carried out, and the mathematical model for one satellite monitoring one target can be concluded as Figure
Mathematical model for one satellite monitoring one target.
As a result, when considering
Constellation design is generally a complicated optimization problem with plenty of design variables; especially when there are several design objectives, the computational process can be very time-consuming [
Firstly, radius
Secondly, radius
Thirdly, the repetition period of the target orbit is one nodal day, and there is only one satellite in each orbit plane. The number of satellites
Therefore, once the number of satellites
Based on the analysis above, our design can be concluded as designing the optimal altitude of the initial orbit and phase differences between each orbit plane, and the purpose is to minimize the fuel consumption, maximum reconfiguration time, mean transition time of all satellites for all targets, the standard deviation of the reconfiguration time for all targets, and maximum waiting days for all target sites.
Before formulating the optimization problem, symbols of design variables and design metrics need to be defined. Assume that there are
Also, the maximum waiting days of the constellation for the target site
In conclusion, the optimization problem for ReCon design can be formulated as
A physical programming method is an efficient way to deal with multiobjective problems, of which the main idea is to convert the problem into a single-objective problem by using preference functions that capture the designer’s preferences [
Preference functions are defined with the intention to reflect the preferences of the designer, and each preference function corresponds to a design metric. Assume that there are
Generally, preferences in physical programming are divided into the following four main classes: Class 1, smaller is better; Class 2, larger is better; Class 3, value is better; and Class 4, range is better. Each class is divided into soft and hard types as well, such as Class 1S and Class 1H, where the suffix letters S and H represent soft type and hard type, respectively. For the soft one, value of the preference function is gradually changed in the feasible region, yet for the hard one, value of the preference function in the feasible region stays the same, which is set to be 0. As a whole, preference functions can be concluded to be eight basic types, as listed in Figure
Preference function classification for physical programming.
Furthermore, for soft types, design metrics can be divided into several regions. Here, we take Class 1S as an example, and the rest of soft types can be defined in the similar pattern. Regions for Class 1S can be listed as follows:
where
Preference function regions for Class 1S.
Preference functions are formulated based on the six interval divisions of the design metrics. By designing the value and the slope of preference function at each region end point, and fitting the value of preference function in each region, the specific preference function can be obtained.
Class 1S is illustrated as an example. Let
To calculate the value
To calculate the slope
Then, the slope of the first region can be expressed as
Slopes for the adjacent region can be expressed as
Therefore, the value of preference function in each region can be obtained, which is illustrated as follows.
When
When
As a result, the preference function of Class 1S is formulated. Other types of the preference functions can be formulated in a similar way.
Finally, the aggregate objective function is formed, and the multiobjective function is consequently converted to a single-objective function, which can be expressed as
In this paper,
In this section, simulation based on the proposed model is carried out. Initial conditions and preferences of design metrics in physical programming are set up by referring to the literature [
Initial conditions for the simulation are presented as follows. The inclination is selected as 90° for the global monitoring, and the number of satellites
Parameters of the final repeating ground track orbit.
Parameter | Value | Unit |
---|---|---|
1436.07 | Minute | |
15 | — | |
95.74 | Minute | |
1 | — | |
1436.07 | Minute | |
547.883 | km |
As for design variables and design metrics, specific ranges are determined according to literature [
Based on the boundaries of the initial altitude, the differences of angular velocity of the initial and target orbits are determined, and the transfer ability can be expressed as equation (
Details of the region end points for Class 1S.
Objective | Unit | Highly desirable | Desirable | Tolerable | Undesirable | Highly undesirable |
---|---|---|---|---|---|---|
m/s | 17 | 20 | 25 | 32 | 40 | |
Hour | 100 | 125 | 150 | 175 | 200 | |
Hour | 24 | 36 | 48 | 60 | 72 | |
Hour | 0 | 12 | 24 | 36 | 48 |
A genetic algorithm is employed for the optimization. Parameters used for the genetic algorithm are listed in Table
Parameters of the genetic algorithm.
Parameter | Value |
---|---|
Population size | 50 |
Generations | 100 |
Crossover fraction | 0.8 |
Migration fraction | 0.2 |
Function tolerance |
Based on the introduced physical programming method and genetic algorithm, optimization is carried out, and results are given as follows.
Change of the best and mean fitness value during the optimization are presented in Figure
Best and mean fitness in each generation.
Values of design variables are listed in Table
Values of design variables.
Variable | |||||
---|---|---|---|---|---|
Value | 597.766 | 216.153 | 72.357 | 287.738 | 144.232 |
Unit | km | deg | deg | deg | deg |
From the results presented in Table
Furthermore, if we consider the walker-
Values of design metrics are given in Table
Values of design metrics.
Variable | |||||
---|---|---|---|---|---|
Value | 27.172 | 150.010 | 50.661 | 22.562 | 6 |
Unit | m/s | Hour | Hour | Hour | Day |
From the results presented in Table
Also, the maximum reconfiguration time is about 150 hours, while the mean transition time is only 50 hours. That is to say, for any target site on the ground, the whole constellation can finish the reconfiguration in 6 days, and the average transition time for each satellite is about 2 days. Therefore, satellites in the constellation can achieve the orbital transfer gradually within 6 days for any target on the global, and the target site can be overhead monitored during the whole reconfiguration process. Besides, the standard deviation of reconfiguration times for all target sites is minimized to about 22.6 hours, which is much less than the maximum reconfiguration time, and similar reconfiguration ability for all target sites can be achieved.
In addition, a repeating ground track orbit is chosen to be the target orbit of all satellites in our design, which means the disaster site can always be directly overhead monitored by the whole constellation after the reconfiguration. Once the observation mission ends, the constellation can convert back to the GOM in an inverse process.
In this subsection, a practical case is studied to demonstrate the superiority and effectiveness of the designed ReCon. A monitoring mission for a forest fire in Liangshan, Sichuan, China, is introduced. The GOM and ROM of the Recon, as well as the BLASTOFF proposed in literature [
The mission starting time is 2020.3.30.10:00:00 UTC, and the geographic latitude and longitude of the target site are 28.531°N and 101.241°E. For the ReCon, RAAN and AOL of the anchor satellite at the starting time are both set to 0, and orbital altitude and phase differences of the rest satellites are provided in Table
Observation sequences of different constellations for the target site are presented in Figure
Observation sequences of different constellations.
The total number of observations varies with the time from the mission beginning for each constellation, and the specific change trend can be concluded based on the observation sequences, as illustrated in Figure
Total observation numbers of different constellations.
From Figures
In the foregoing case, each satellite in the ROM monitors the target site during the ascending pass, and the orbit of each satellite is a repeating orbit with the same repeating period. Therefore, all satellites in the ROM are flying along the same repeating ground track, and this constellation is also the so-called Flower constellation [
Configurations of different constellations.
GOM
ROM1
ROM2
Based on the presented configurations, ground tracks and average observation numbers per day of the GOM, the ROM1, and the ROM2 can be analyzed, as shown in Figure
Ground tracks and average observation numbers per day of different constellations.
GOM
ROM1
ROM2
To further analyze different properties of the GOM and the ROM, the low-resolution mode of the GF-1 camera is considered, of which the swath width is 800 km. Similarly, average observation numbers per day of the GOM, the ROM1, and the ROM2 are calculated, and results are given in Figure
Average observation numbers per day of different constellations.
GOM
ROM1
ROM2
In addition, to verify the accuracy of the proposed mathematical model, distances between the target site and the satellite ground tracks in the ROM1 and the ROM2 are presented in Figure
Enlarged views of the ground tracks of the ROM1 and the ROM2.
ROM1
ROM2
In this paper, a reconfigurable constellation, which can achieve timely and adequate observation for an unpredictable disaster site through orbital maneuvers, is successfully designed. By analyzing the space geometry relations between an arbitrary satellite orbit and disaster site, a mathematical model for orbital transfer and overhead monitoring is established. Aiming at minimizing both total velocity increment and reconfiguration time, optimization of the initial constellation and the reconfiguration process are combined and carried out, and a physical programming method along with a genetic algorithm is employed for the multiobjective optimization problem. Preferences in physical programming are set by referring to literature [
As for the results of the optimization, the optimal reconfigurable constellation is extremely close to the walker-
Furthermore, it should be mentioned that the preferences in our model are designed according to the designer, and if less velocity increment or less reconfiguration time is required, it is easy to adjust our model and obtain the preferred optimal results. Future work would be to monitor 2 or more disaster sites, as well as to monitor the moving disaster site through constellation reconfiguration. This research can provide basic idea and method for the preliminary design of reconfigurable constellation for disaster monitoring.
The data used to support the findings of this study are included within the article.
The authors declare that there is no conflict of interest regarding the publication of this paper.
The authors are grateful to Prof. Li’s advice on how to best improve this paper. This work was supported by the Hunan Provincial Natural Science Foundation of China (No.2020JJ4657) and the National Natural Science Foundation of China (No.11372345).