A Patrol Route Design for Inclined Geosynchronous Orbit Satellites in Space Traffic Management

Abstract

Conducting surveys and the timely acquisition of satellite status, especially for high-value geostationary orbit (GEO) targets, is of great significance for space traffic management. This article proposes an approach for patrolling inclined geosynchronous orbit (IGSO) targets based on crossing points and spiral rings. The method involves six steps: (1) calculate the crossing position and crossing time of the IGSO targets; (2) design a spiral trajectory that satisfies the desired patrol time; (3) divide IGSO targets into regions using a dichotomy approach; (4) calculate the bidirectional longitude drift rate within each region; (5) determine the starting position of patrol for each region; and (6) determine the transfer trajectory for each region. By selecting a class of IGSO satellites as the target set, the proposed approach is analyzed and validated in detail. The results show that the patrol orbit can effectively achieve patrol all of IGSO targets, with a period of no more than 40 days and less than 13.5 kg fuel consumption. The total fuel consumption of a single patrol cycle in all regions does not exceed 91.82 kg.

1. Introduction

Space activities, including not only commercial mega constellation launches and on-orbit services, but also space exploration missions by more and more countries and international organizations, are increasing day by day, making the space environment more complex. The importance of space traffic management (STM) is becoming increasingly prominent. Space traffic management refers to the planning, supervision, coordination, and emergency response activities carried out by relevant management agencies through technical, legal, and other means to ensure the safe, efficient, and orderly launch, orbit operation, re-entry, and other activities of satellites. Space traffic management is also used to maintain the space environment and prevent unsafe factors such as electromagnetic interference, physical collisions, and environmental damage [1,2,3,4]. Although the legal issues of space traffic management are still under discussion and studied by scholars around the world, research of its technological issues, especially in future scenarios, could provide support for subsequent space traffic management and also contribute to the research of legal issues.

Space traffic management revolves around three core goals: safety, orderliness, and sustainability. Safety is the primary consideration, ensuring the normal operation of satellites and related equipment and preventing the occurrence of various accidents and incidents. As an important component of space traffic management, the space situational awareness (SSA) system is primarily responsible for monitoring and tracking of space targets, ensuring the safety and sustainable use of the space environment.

In the vast expanse of space, the GEO zone is a key area for space traffic management. It is generally considered that the GEO zone is a circular orbit with inclination of 0° and a radius of 42,164 km from the Earth’s center. In a strict sense, there is only one GEO in space. In order to effectively use and protect GEO, the Inter-Agency Space Debris Coordination Committee (IADC) has artificially delineated a GEO zone as shown in Figure 1 [5]. It can be seen that the GEO zone’s upper and lower height is 35,786 km ± 200 km, and the north-south range is latitude ±15°. In the zone, the inner ±75 km zone is the GEO working zone, while the rest is GEO maneuvering zone. Satellites in the GEO zone are widely used in navigation, communication, early warning systems, and other fields due the unique orbital advantage, such as the sub-satellite point trajectory and global coverage [6].

The prerequisite for space traffic management in the GEO zone is to obtain the status information of GEO targets quickly and accurately. However, due to the long distance and poor tracking accuracy, the ground-based space situation awareness system for GEO satellite inspection is limited. Space-based observations, which can achieve close range, long-term, and multi-angle tracking and observation of space targets, become the only means to obtain detailed information on GEO satellites. Due to the vastness of GEO, the GEO SSA typically adopts a method of traverse patrol, which is similar to traffic patrol, called space traffic patrol. Space traffic patrol means to go around a zone or an area at regular times to check that it is safe and that there is no trouble at close range by using one or more satellite. Thus, we believe that the future space traffic management system should include three levels. The first level is mainly cataloging and orbit anomaly patrols based on ground-based equipment. The second layer is a long-distance range patrol, which can further confirm whether there are any abnormal postures or behaviors, and this is the focus of our study. The third layer is the close-in detailed investigation, which means that based on the previous two layers, the spacecraft approaches the target and carries out On-orbit service.

According to the UCS database, as of May 2023, there are 590 controllable GEO satellites [7]. In terms of satellite types, GEO satellites can be divided into three categories: military, civilian/government, and commercial. The number of commercial satellites is largest, accounting for about 57%, and the numbers of military and civilian/government are equivalent, accounting for 21% and 22%, respectively. When considering that the perigee and apogee are both in the working zone, the number of qualified satellites is 495, accounting for about 84%. When considering the distribution of orbital inclination, it can be seen that most GEO satellites have a small inclination angle, the number of satellite inclinations exceeding 0.1° is 179, accounting for more than 30%. Those satellites belong to a different orbital plane due to their inclination, which poses a huge challenge to space traffic patrol. In this article, satellites with an inclination angle exceeding 0.1° are defined as IGSO.

Essentially, the space traffic patrol problem belongs to the problem of multiple targets rendezvous. There are a lot of domestic and international researchers who study this topic, especially in terms of on-orbit services. These articles can be divided into four categories: one to multiple (OTM), multiple to multiple (MTM), peer to peer (PTP), and mixed.

OTM means that one service satellite refuels multiple targets. In OTM, most scholars consider the service order as the main variable for optimization, with the minimum consumption as the optimization objective. Those scholars transform it into a traveling salesman problem (TSP) or a mixed integer nonlinear programming (MILP) problem. Ref. [8] studied the OTM service order problem with the service satellite and targets in same circular orbit. Ref. [9] studied the route problem of servicing GEO targets with small inclination, while same question was addressed in Ref. [10]. In addition, a spiral cruise orbit for GEO satellites was designed using relative orbit equations in Refs. [11,12]. Genetic algorithms (GA), particle swarm optimization (PSO), and other optimization algorithms are also used to solve the problem [13,14,15,16,17].

MTM means that multiple service satellites refuel multiple targets. The MTM problem is much more complex than OTM. Many factors, such as the initial position and traverse cost, will affect the path planning. Moreover, considering the limited capacity of service satellites, the location and number of fuel stations affect the final result. For MTM, different scenarios are set, and different models are established. Ref. [18] assumes that the service satellite can return to the fuel station for refueling and continue the service. Taking the service sequence and the time of return to station as variables, the service plan with lowest consumption is optimized. By converting the problem into a location-route problem (LRP), a service plan is given in Ref. [19], and similar research is reported in Refs. [20,21,22,23].

PTP assumes that each service satellite and target have the ability to maneuver. PTP is actually an optimal matching problem, and the model is established depending on the constraints of the scene conditions. All models vary slightly. To make all targets fuel abundant with the lowest cost, the PTP problem was transformed into an optimal matching problem model and solved using auction algorithms. Ref. [24] studied the PTP traverse mode assuming that the targets are in a circular orbit with an uneven fuel distribution. The PTP refueling model added the fuel quantity threshold as a new parameter, where targets with a larger quantity were defined as fuel abundant, and those with a smaller quantity were defined as fuel deficient [25]. Furthermore, some scholars assumed that each service satellite can return to the original position of an satellite after rendezvous; transformed it into a three-index assignment problem, also called a E-PTP (egalitarian peer to peer) problem; and solved it using search algorithms [26]. Some others solved the problem by converting it into an optimal matching problem (OMP) [27] or a mixed mode [28,29].

Mixed traverse is relatively complex, and generally two or more categories are used to achieve the goal of reducing fuel consumption or more effective traverse. Refs. [30,31] simplify the MTM and PTP problems to achieve equilibrium through weighting, resulting in a relatively ideal state for fuel consumption and a fuel gap between targets. Ref. [32] selected some service satellites to only provide fuel for some targets in the constellation. After these targets receive fuel, they transform into service satellites to provide fuel for other targets. Through enumeration and PSO comprehensive algorithm comparisons, it was verified that the hybrid traverse strategy is more cost-effective than other strategies.

Although there are many literature studies on this topic, most of them focus on the coplanar rendezvous, and there is relatively little research on the non-coplanar rendezvous problem. Bai et al. used the dual target rendezvous orbit cluster to obtain the rendezvous orbit approaching multiple non-coplanar targets by searching and calculating the closest distance to other targets [33,34]. Zhang et al. used Lambert’s theorem to determine the service satellite orbits that were close to two targets, then improved the orbit using least squares method and genetic algorithm, and obtained the orbit that approaches three non-coplanar targets [35,36,37]. Ref. [38] demonstrated that a service satellite orbit that can rendezvous with multiple satellites in the Walker constellation without orbit maneuver. Based on this, the author summarized the rendezvous orbit constraints for the maximum number of constellation satellites and then used phase modulation to change the initial phase of the service satellite to rendezvous with all satellites in the orbit plane. The strategy is verified that fuel consumption is less compared to Lambert’s theorem. Ma N. designed the traversal orbit for GPS satellites using a single pulse coplanar orbit maneuver. After traversing one orbit plane, the maneuver is carried out at the intersection with next one, and the same operation is repeated until the traversal of all satellites [39].

The above articles mainly focus on the Walker constellation, but the IGSO orbit is not a regular part of the Walker constellation. To solve the rendezvous multiple targets problem on a general orbital plane, Zhang et al. proposed the crossing point method, and achieving the rendezvous orbit based on the method [40,41].

According to orbital dynamics, each satellite is in a plane passing through the center of the Earth. That is, the orbital planes of any two satellites will inevitably intersect, forming two intersection points, called crossing points, as shown in Figure 2.

Select any orbit plane as the reference plane, and the target orbit passes through the reference plane and intersects at points A and A’, where point A north of the equatorial plane is called the north crossing point, and point A’ south of the equatorial plane is called the south crossing point. The two points have a phase difference of 180° within the reference plane.

In this article, we propose a patrol route design approach based on the gradient longitude drift rate and crossing points to patrol IGSO targets. Section 2 introduces the approach in detail. The proposed approach is illustrated in Section 3 using numerical experiments. Additionally, a case study is illustrated in Section 4. Finally, the conclusion and potential future research directions are provided in Section 5.

2. Materials and Methods

For safety, the patrol satellite should not frequently pass through the GEO working zone and should not affect the normal satellites in working zone. Thus, the patrol satellite should be located in GEO maneuver zone defined by the IADC. It can be seen from the definition that the GEO maneuver zone is divided into two independent parts, above or below GEO. Therefore, the patrol orbit zone should be located in a part of the maneuver zone. When the satellite is in the maneuver zone, it will drift relative to GEO. Taking the standard GEO as the reference coordinate system, the patrol route relative to the GEO is shown in Figure 3.

It can be seen from the figure that when the satellite is located below GEO, its trajectory relative to GEO is wave-shaped (elliptical orbit) or linear (circular orbit). The sub-satellite point longitude becomes larger and drifts from west to east. Conversely, when located above the GEO, its relative trajectory is also wave-shaped or linear, and the sub-satellite point longitude becomes smaller and drifts from east to west.

By utilizing the relative drift, it is possible to patrol the GEO targets. The longitude drift rate D is expressed as follows:

$$D=-0.0128\left(a_s-a_c\right)$$

(1)

where a_c is the radius of GEO. Therefore, by designing the drift rate reasonably, it is possible to achieve a natural patrol of all targets in the GEO zone. As shown in Figure 4, assume that a GEO zone contains six targets. The patrol satellite first drifts from west to east, observing targets 2 and 6 at apogee. Then, implementing an orbital maneuver at the zone boundary, it turns around and drifts from east to west, observing targets 1, 3, 4, and 5 at perigee. Thereby, the satellite achieves the patrol of the entire zone’s targets. Afterwards, through orbital maneuvers, the satellite will turn around again and enter the east-drift orbit, conducting periodic patrol of targets in the zone.

The above strategy is very effective when the sub-satellite points of targets are stationary, but IGSO targets are up and down relative to the equatorial plane. Patrolling these targets not only requires consideration of relative position, but also relative time, which is a more complex spatiotemporal problem.

If we use the strategy above to implement the patrol of IGSO targets, there will be a situation where the patrol satellite nears the crossing point, but the target does not reach the crossing point. If the patrol satellite could hold on near the crossing point for a period of time, waiting while the target crosses the reference plane, then the problem can be smoothly transformed into the stationary problem. According to the space special orbit design theory [41], when the height difference of a_c and a_s meets certain conditions, the patrol satellite can achieve a holding-on spiral ring near the target as shown in Figure 5.

When the target crosses the reference plane, the patrol satellite happens to be on the spiral ring. By designing the spiral ring reasonably, the cycle of the spiral ring can cover the crossing time within a day. This transforms the patrol problem of IGSO targets into a problem of geostationary targets. Using the gradient longitude drift rate method shown in Figure 4, the patrol satellite can achieve the patrol of IGSO targets.

This article proposes a gradient longitude drift rate patrol configuration approach based on crossing points and spiral rings to patrol IGSO targets and includes following steps:

Step 1: Calculate the crossing position and crossing time of the IGSO targets;

Step 2: Design a spiral trajectory that satisfies desired patrol time;

Step 3: Divide IGSO targets into regions based on a dichotomy approach;

Step 4: Calculate the bidirectional longitude drift rate within each region;

Step 5: Determine the starting position of patrol for each region;

Step 6: Determine the transfer orbit for each region.

For the convenience of understanding, the symbols and their definitions used in this article are shown in

Table A1. List of symbols.

D	Longitude drift rate
$\left(a_s,e_s,i_s,{\Omega }_s,{\omega }_s,{\tau }_s\right)$	Patrol satellite orbital element
ac	Semi-major axis of GEO
$\left(a_t,e_t,i_t,{\Omega }_t,{\omega }_t,{\tau }_t\right)$	Target orbital element
$t_{cross}$	Time of the crossing point
$r_{cross}$	Geocentric distance of the crossing point
${\varphi }_{cross}$	Ascending intersection angle of the crossing point
ucross	Argument of latitude of the crossing point
μ	The gravitational constant of the Earth
E	Eccentric anomaly
Tt	Target’s orbital period
Te	The period of rotation of the Earth
Ωcross	Right ascension of the ascending node of the crossing point
vturn	Velocity of the turning point
rturn	Position of the turning point
vaim	Velocity of the aiming point
raim	Position of the aiming point
Tring	The period of the spiral ring
νturn	Energy parameter at the turning point
Tins	Time of the earliest aiming point
λi (i = 1, 2, …, n)	Targets’ sub-satellite point longitude
Δλi (i = 1, 2, …, n)	Difference in longitude between adjacent targets
Δλmax	Largest longitude difference gap between targets
Δλsec-max	Second largest longitude difference gap between targets
Dmin	Min longitude drift rate
Dmax	Max longitude drift rate
Ti (i = I, II, …)	Periods of the zones
T0	The given limitation period
Zi (i = 1, 2, …, N)	Target set in zone i
Mi (i = 1, 2, …, N)	Number of targets in zone i
DE	Eastward longitude drift rate of the patrol satellite
DW	Westward longitude drift rate of the patrol satellite
QE	Targets with the same remainder of DE
QW	Targets with the same remainder of DW
Δq	Region thresholds for the target set
TE	Initial time of the eastward drift
TW	Initial time of the westward drift
λE	Initial position of the eastward drift
λW	Initial position of the westward drift
aE	Semi-major axis of the eastward drift orbit
aW	Semi-major axis of the westward drift orbit
aT	Semi-major axis of the transfer orbit
raE	Position of the eastward drift orbit apogee
rpE	Position of the eastward drift orbit perigee
raW	Position of the westward drift orbit apogee
rpW	Position of the westward drift orbit perigee
vaE	Velocity of the eastward drift orbit apogee
vpE	Velocity of the eastward drift orbit perigee
vaW	Velocity of the westward drift orbit apogee
vpW	Velocity of the westward drift orbit perigee
v1	Velocity of the first maneuver
v2	Velocity of the second maneuver
rp	Position of the patrol orbit perigee
ra	Position of the patrol orbit apogee
vp	Velocity of the patrol orbit perigee
va	Velocity of the patrol orbit apogee

.

2.1. Calculate the Crossing Position and Time of the IGSO Targets

Assuming a IGSO target crosses the orbital plane, the target orbital element is $(a_t,e_t,i_t,{\Omega }_t,{\omega }_t,{\tau }_t)$. Set the orbital plane of patrol satellite as the reference plane, which can be represented by $(i_s,{\Omega }_s)$. The coordinates of crossing point in reference plane can be represented as $(t_{cross},r_{cross},{\varphi }_{cross})$, where $t_{cross}$ is the moment when target crosses through the plane, $r_{cross}$ is the geocentric distance of the crossing point, and ${\varphi }_{cross}$ is the ascending intersection angle of the crossing point. Thus, the following is obtained:

$$\begin{array}{l}\cos {\varphi }_{cross}=\cos ({\Omega }_t-{\Omega }_s)\cos u_{cross}+\sin u_{cross}\sin ({\Omega }_t-{\Omega }_s)\cos (180°-i_t)\\ \sin {\varphi }_{cross}={\displaystyle \frac{\sin u_{cross}\sin i_t}{\sin i_s}}={\displaystyle \frac{\sin ({\Omega }_t-{\Omega }_s)\sin i_t}{\sin (i_t-i_s)}}\end{array}$$

(2)

Among them, $u_{cross}$ is the argument of latitude of the target, which can be obtained as follows:

$$tg{\varphi }_{cross}={\displaystyle \frac{\sin ({\Omega }_t-{\Omega }_S)}{\cos ({\Omega }_t-{\Omega }_s)\cos i_s+\sin i_s\cot (180°-i_t)}}$$

(3)

The crossing time can be obtained from the Kepler equation:

$$t_{cross}=(E_{cross}-e_t\sin E_{cross})\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}+{\tau }_t$$

(4)

$$tg{\displaystyle \frac{E_{cross}}{2}}=\sqrt{{\displaystyle \frac{1-e_t}{1+e_t}}}tg{\displaystyle \frac{(u_{cross}-{\omega }_t)}{2}}$$

(5)

Due to the periodic motion, the target periodically crosses the reference plane, with a time interval of orbital period T_t, and this is expressed as follows:

$$t_{cross}^m=t_{cross}+m{T}_t\hspace{1em}\hspace{1em}\hspace{1em}m=0,1,2,\cdots$$

(6)

In this article, the patrol satellite operates within the equatorial plane; thus, the following is obtained:

$$\begin{array}{l}{i}_s=0\\ {\Omega }_s=0\end{array}$$

(7)

Then, the crossing point becomes the ascending and descending nodes of the target, and the crossing point can be represented by $(t,\Omega )$. Thus, Formulas (5)~(7) derive the crossing time as follows:

$$t_{cross}^m=(E_{cross}-e_t\sin E_{cross})\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}+{\tau }_t+2\pi m\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}\hspace{1em}\hspace{1em}\hspace{1em}m=0,1,2,\cdots$$

(8)

Considering that most IGSO targets are in near circular orbit, the following are noted:

$$\begin{array}{l}E\approx M\approx f\\ e\approx 0\end{array}$$

(9)

Formula (8) can be simplified as follows:

$$\begin{array}{ll}{\displaystyle \frac{d{t}_{cross}^m}{dt}}& ={\displaystyle \frac{d(E_{cross}+2\pi m)}{dt}}\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}+(E_{cross}+2\pi m){\displaystyle \frac{d\sqrt{\frac{a_t^3}{\mu }}}{dt}}\\ & ={\displaystyle \frac{d{E}_{cross}}{dt}}\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}+{\displaystyle \frac{3}{2}}(E_{cross}+2\pi m)\sqrt{{\displaystyle \frac{a_t}{\mu }}}{\displaystyle \frac{da}{dt}}\\ & ={\displaystyle \frac{d\lambda }{dt}}\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}+{\displaystyle \frac{3}{2}}{\displaystyle \frac{(E_{cross}+2\pi m)}{a_t}}\sqrt{{\displaystyle \frac{a_t^3}{\mu }}}{\displaystyle \frac{da}{dt}}\approx -{\displaystyle \frac{3}{2a_c}}\left(a_t-a_c\right)\end{array}$$

(10)

It can be seen that the change in crossing time is closely related to the semi-major axis difference between the target and GEO. The difference between two consecutive crossing times is expressed as follows:

$$\begin{array}{ll}\Delta t& =-{\displaystyle \frac{3}{2a_c}}\left(a_t-a_c\right)2\pi \sqrt{{\displaystyle \frac{a_t^3}{\mu }}}\\ & =2\pi \sqrt{{\displaystyle \frac{a_c^3}{\mu }}}\left(-{\displaystyle \frac{3}{2a_c}}\left(a_t-a_c\right)\sqrt{{\displaystyle \frac{a_t^3}{a_c^3}}}\right)\\ & =-{\displaystyle \frac{3}{2a_c}}\left(a_t-a_c\right)T_e\end{array}$$

(11)

If the semi-major axis difference is 100 km, it can be obtained that the daily variation of crossing time is about 306.5 s. Considering that the semi-major axis change is only a few kilometers under the perturbation, it can be considered that the crossing time of the IGSO target in this article remains basically unchanged. The Ω_cross will also experience precession drift due to the perturbation influence, where the Ω_cross decreases by about 0.0089° per day. However, under controlled conditions, the Ω_cross will remain within ±0.1° of its sub-point longitude. Therefore, in this paper, we consider that the crossing points of selected targets remain unchanged.

2.2. Design a Spiral Trajectory That Satisfies the Desired Patrol Time

As shown in Figure 5, the intersection point is called the aiming point, and the point where the trajectory turns from west to east or from east to west is called the turning point. The satellite that passes through the same aiming point twice is in a spiral loop, which is a closed teardrop trajectory. The turning point is the point where the velocity of the patrol satellite is equal to the geostationary satellite. Using the vitality formula, the turning point is expressed as follows:

$$v_{turn}^2={\displaystyle \frac{2\mu }{r_{turn}}}-{\displaystyle \frac{\mu }{a_s}}={\displaystyle \frac{\mu }{a_c}}$$

(12)

Among them, v_turn is the velocity of the turning point, and r_turn is the position of the turning point. The turning point position is expressed as follows:

$$r_{turn}={\displaystyle \frac{2a_s{a}_c}{a_c+a_s}}$$

(13)

According to the displacement difference between aiming point and turning point, it is approximately equal to half of the difference between the aiming point and the satellite. Thus, the position of the aiming point is expressed as follows:

$$r_{aim}=\left\{\begin{array}{cc}2r_{turn}+75-a_c& r_{turn}<a_c\\ 2r_{turn}-75-a_c& r_{turn}>a_c\end{array}\right.$$

(14)

The period of the spiral ring is the time interval where the satellite continuously passing through the same aiming point, which is approximately twice the time between turning points. Similarly, the following can be obtained:

$$T_{ring}\approx 4\sqrt{{\displaystyle \frac{a_s^3}{\mu }}}\left(\arccos \left({\displaystyle \frac{1-{\nu }_{turn}}{e_s}}\right)+e_s\sin \left(\arccos \left({\displaystyle \frac{1-{\nu }_{turn}}{e_s}}\right)\right)\right)$$

(15)

where ν_turn is the energy parameter at the turning point, which is expressed as follows:

$${\nu }_{turn}={\displaystyle \frac{v_{turn}^2}{\frac{\mu }{r_{turn}}}}={\displaystyle \frac{\frac{\mu }{a_c}}{\frac{\mu }{r_{turn}}}}={\displaystyle \frac{r_{turn}}{a_c}}={\displaystyle \frac{2a_s}{a_c+a_s}}$$

(16)

Assuming that the crossing times within a day is {t₁, t₂, …, t_n} and the earliest aiming point is T_ins, the following are obtained:

$$\begin{array}{l}{T}_{ring}\ge \max \left\{t_i\right\}-\min \left\{t_i\right\}\\ T_{ins}=\min \left\{t_i\right\}\end{array}$$

(17)

2.3. Divide IGSO Targets into Regions Using the Dichotomy Approach

It can be seen that when the longitude drift rate is constant, the patrol period of targets depends on the maximum longitude difference. On the other hand, if the distribution of longitude differences between targets within the zone is uneven, there will be a situation where the targets are concentrated on both sides, which causes a great waste of time. Therefore, when designing the patrol configuration, it is necessary to consider dividing all targets reasonably to ensure that the longitude differences within each zone are consistent, so that the patrol periods of each zone are not significantly different. Thus, periodic patrol of all targets is achieved.

Without losing generality, sort n IGSO targets in ascending order according to their sub-satellite point longitude, and record the longitude as λ_i (i = 1, 2, …, n), and λ₁ < λ₂ <…< λ_n. The difference gaps in longitude between adjacent targets is recorded as Δλ_i (i = 1, 2, …, n), then the following is obtained:

$$\begin{array}{l}\begin{array}{cc}\Delta {\lambda }_i={\lambda }_{i+1}-{\lambda }_i& i=1\cdots n-1\end{array}\\ \Delta {\lambda }_n={\lambda }_1-{\lambda }_n+360\end{array}$$

(18)

When the distribution of IGSO targets is uneven, there will inevitably be several large longitude gaps. If the divided area covers the gaps, there will be a significant increase in the patrol period due to a single target. Therefore, when dividing zones, it is necessary to avoid these huge gaps as much as possible.

This paper presents a zone division method based on the dichotomy approach, which can quickly perform zone division. Set the maximum and second largest longitude difference gap between targets as follows:

$$\begin{array}{l}\begin{array}{cc}\Delta {\lambda }_{\max }=\max \left\{\Delta {\lambda }_i\right\}& i=1\cdots n\end{array}\\ \begin{array}{cc}\Delta {\lambda }_{\sec -\max }=\max \left\{〈\Delta {\lambda }_i|\Delta {\lambda }_{\max }〉\right\}& i=1\cdots n\end{array}\end{array}$$

(19)

Due to the characteristics of a circular distribution, these two longitude differences divide all targets into two zones, as shown in Figure 6.

The longitude coverage ranges of the two zones is shown below:

$$\begin{array}{l}\mathrm{I}:{\lambda }_{\max +1} ~ {\lambda }_{\sec -\max }\\ \mathrm{I}\mathrm{I}:{\lambda }_{\max +1} ~ {\lambda }_{\sec -\max }\end{array}$$

(20)

Set the range of longitude drift rate to the following:

$$D\subseteq \left\{D_{\min } ~ D_{\max }\right\}$$

(21)

Then, the maximum patrol period of the two zones can be calculated as follows:

$$\begin{array}{l}{T}_I={\displaystyle \frac{{\lambda }_{\sec -\max }+360-{\lambda }_{\max +1}}{D_{\min }}}\\ T_{II}={\displaystyle \frac{{\lambda }_{\max }-{\lambda }_{\sec -\max }}{D_{\min }}}\end{array}$$

(22)

Compare the sizes of T_I and T_II, select the zone with a larger period, select the maximum longitude difference gap within the zone, and perform secondary division, as shown in Figure 7.

Compare the periods of the three zones T_I, T_II, and T_III and also select the zones with larger periods for segmentation using the maximum longitude difference gap until all zones meet the patrol period T₀ (T₀ is the given limitation period), as shown in Figure 8.

The process steps for dividing the entire zone are shown in Figure 9.

2.4. Calculate the Bidirectional Longitude Drift Rate Within Each Region

After dividing the entire target into multiple zones, it is necessary to determine the east/west longitude drift rate within each zone. Assuming n targets are divided into N zones, denoted as Z_i (i = 1, 2, …, N), each zone contains M_i targets represented by their sub-satellite point longitude and denoted as λ_ij (i = 1, 2, …, N, j = 1, 2, …, M_i). Thus, the following is obtained:

$$\begin{array}{l}{Z}_i=\left\{\begin{array}{cccc}{\lambda }_{i1}& {\lambda }_{i2}& \cdots & {\lambda }_{i{M}_i}\end{array}\right\}\\ {\displaystyle \sum _{i=1}^N{M}_i}=n\end{array}$$

(23)

It can be seen from Figure 4 that if target k and target l can be observed in one patrol, their longitude difference needs to be an integral multiple of the drift rate, which is expressed as follows:

$$\begin{array}{cc}\left|{\lambda }_k-{\lambda }_l\right|=pD& p\in N\end{array}$$

(24)

The above equation also indicates that the remainder of D for target k and target l is the same, which indicates the following:

$$\mathrm{mod}\left(\begin{array}{cc}{\lambda }_k& D\end{array}\right)=\mathrm{mod}\left(\begin{array}{cc}{\lambda }_l& D\end{array}\right)$$

(25)

Mod represents the remainder of two numbers. Assuming that the eastward longitude drift rate of the patrol satellite is D_E and the westward is D_W, it can be inferred from the above analysis that the conditions for achieving patrol of targets can be transformed into the existence of D_E and D_W. The eastward target set Q_E refers the targets with D_E, and the westward target set Q_W refers the targets with D_W. When these meet, the union of the Q_E and Q_W can include all targets as follows:

$$Z=Q_E\cup Q_W$$

(26)

Taking zone N as an example for analysis, the sub-satellite point longitude of the targets is λ_j (j = 1, 2, …, M_N). Considering that the GEO satellite works in a certain longitude box, keep the sub-satellite point to one decimal, denoted as λ_j^* (j = 1, 2, …, M_N). Simultaneously, simplify the longitude drift rate to one decimal as follows:

$$D\subseteq \left\{\begin{array}{cccc}{D}_1& D_2& \cdots & D_d\end{array}\right\}$$

(27)

The ordered expression is provided as follows:

$$q_{D_i}=\left\{\mathrm{mod}\left(\begin{array}{cc}{\lambda }_j^{\ast }& D_i\end{array}\right)\right\}$$

(28)

Take Q corresponding to the mode of the above set, denoted as Q_Di, then the following is obtained:

$$Q_{D_i}=\left\{\left.{\lambda }_j\right|\mathrm{mod}\left(\begin{array}{cc}{\lambda }_j^{\ast }& D_i\end{array}\right)=MODE\left(q_{D_i}\right)\right\}$$

(29)

where MODE() means the most frequently occurring numerical value in a set of data. Join the above target sets until patrol of all targets is achieved as follows:

$$Z_N=Q_{D_k}\cup Q_{D_l}$$

(30)

Considering that the orbital periods for patrol satellite are not significantly different, the drift rate affects the number of patrol cycles under certain longitude difference conditions. The larger the drift rate is, the more longitudes drift each day, the fewer cycles required for patrol, and the shorter the corresponding patrol cycle. On the contrary, the smaller the drift rate is, the longer the patrol period. Therefore, patrol satellites with high longitude drift rates should be selected first. When satellites with high longitude drift rates do not meet the requirements for patrol, they choose to use smaller longitude drift rates for patrol.

In addition, according to the constraint analysis, there is redundancy in the observation distance. By adjusting the observation distance, a certain sub-satellite point longitude threshold can be brought to the target’s sub-satellite point longitude, as shown in Figure 10.

Set the point longitude threshold to Δq. The same remainder condition is transformed into the following:

$$q_{D_i}=\left\{\begin{array}{cc}\begin{array}{cc}{\lambda }_i\in Z& {\lambda }_j\in Z\end{array}& \left|\mathrm{mod}\left(\begin{array}{cc}{\lambda }_i^{\ast }& D_i\end{array}\right)-\mathrm{mod}\left(\begin{array}{cc}{\lambda }_j^{\ast }& D_i\end{array}\right)\right|\le \Delta q\end{array}\right\}$$

(31)

Correspondingly, the target set corresponding to the mode becomes the target set corresponding to the same set of residues with the most elements as follows:

$$Q_D=\left\{q_D\left|num\left(q_D\right)\right.=\max \left(num\left(q_{D_i}\right)\right)\right\}$$

(32)

Take the intersection of the target set Q generated by each longitude drift rate mentioned above, and find two sets of longitude drift rates for the east-west longitude drift of the patrol satellite.

2.5. Determine the Starting Position of Patrol for Each Region

Assuming that the set of eastward drift targets and westward drift targets are Q_E and Q_W, respectively, the initial positions λ_E and λ_W are expressed as follows.

$$\begin{array}{l}{\lambda }_E=\min \left(Q_E\right)-p_E{D}_E\\ {\lambda }_W=\max \left(Q_W\right)+p_W{D}_W\end{array}$$

(33)

Here, p_E and p_W meet the following:

$$\begin{array}{l}\min \left(Q_E\right)-p_E{D}_E\le \min \left(Z_i\right)\le \min \left(Q_E\right)-p_E{D}_E+D_E\\ \max \left(Q_W\right)+p_W{D}_W-D_W\le \max \left(Z_i\right)\le \max \left(Q_W\right)+p_W{D}_W\end{array}$$

(34)

At this point, the initial time T_E of the eastward drift is the 0 o’clock of λ_E local time, and the initial time T_W of the westward drift is the 12 o’clock of λ_W local time.

2.6. Determine the Transfer Orbit for Each Region

Assuming that the patrol satellite does not consume fuel during the westward and eastward drifts (ignoring the pulses applied during orbit maintenance), the classic double pulse orbit transfer method is used for the turning process. Eastward drift occurs first and then westward, as shown in Figure 11.

From the previous section, it can be seen that when patrol satellite turns around from east drift to west drift, it should maneuver from the apogee of the east drift orbit to the perigee of the west drift orbit. As previously analyzed, the patrol position of the eastward drift orbit is apogee, while the position of the westward drift orbit is perigee. The period is approximately 24 h. The above transfer orbit allows the patrol satellite to carry out normal missions after turning around. The specific transfer process is shown in Figure 12.

According to Formula (1), the semi-major axes of the east and west drift orbits are expressed as follows:

$$\begin{array}{l}{a}_E=a_c-{\displaystyle \frac{D_E}{0.0128}}\\ a_W=a_c+{\displaystyle \frac{D_W}{0.0128}}\end{array}$$

(35)

The perigee of the transfer orbit is the eastward drift orbit apogee r_aE, and the apogee of the transfer orbit is the westward drift orbit perigee r_pW. Thus, its semi-major axis is expressed as follows:

$$a_T={\displaystyle \frac{r_{aE}+r_{pW}}{2}}$$

(36)

Based on orbital dynamic knowledge, it is easy to obtain the velocity of the eastward drift orbit apogee v_aE; the velocity of westward drift orbit apogee v_pW; and v₁ and v₂ of the transfer orbit apogee, respectively, as follows:

$$\begin{array}{l}{v}_{aE}=\sqrt{{\displaystyle \frac{\mu }{a_E}}{\displaystyle \frac{r_{pE}}{r_{aE}}}}\\ v_{pW}=\sqrt{{\displaystyle \frac{\mu }{a_W}}{\displaystyle \frac{r_{aW}}{r_{pW}}}}\\ v_1=\sqrt{{\displaystyle \frac{\mu }{a_T}}{\displaystyle \frac{r_{pW}}{r_{aE}}}}\\ v_2=\sqrt{{\displaystyle \frac{\mu }{a_T}}{\displaystyle \frac{r_{aE}}{r_{pW}}}}\end{array}$$

(37)

Therefore, the velocity increment required for the transition is expressed below:

$$\Delta v_{EW}=v_1-v_{aE}+v_{pW}-v_2$$

(38)

Similarly, when the patrol satellite transitions from a west to east drift, it shifts from the perigee of the westward drift orbit to the apogee of the eastward drift orbit, as shown in Figure 13.

It can be seen that the required velocity increment is expressed as follows:

$$\Delta v_{WE}=v_1-v_{aE}+v_{pW}-v_2$$

(39)

Thus, the velocity increment of patrol of the targets in the zone once is obtained as follows:

$$\Delta V=\Delta v_{EW}+\Delta v_{WE}$$

(40)

According to the engine principle, the relationship between the velocity increment and fuel consumption is expressed as follows:

$$\Delta m=m_0\left(1-e^{-{\displaystyle \frac{\Delta V}{V_e}}}\right)$$

(41)

Here, m₀ is the mass of the patrol satellite, and V_e is the engine equivalent exhaust velocity, which is expressed as follows:

$$V_e=I_{sp}\cdot g_0$$

(42)

I_sp is the engine specific impulse, and g₀ is the ground gravity acceleration, which is equal to 9.8 m/s².

3. Results

Satellites can be divided into three categories: military, civilian/government, and commercial. According to the UCS database, there are a total of 48 IGSO military targets. In our paper, the analysis and design of patrol orbit configurations will be carried out for these targets. It is assumed that the mass of the patrol satellite is 1000 kg and the engine specific impulse is 300 s. Based on the above steps, this section provides a specific analysis of the patrol for IGSO targets.

3.1. Calculate the Crossing Position and Crossing Time of the IGSO Targets

The crossing time of IGSO targets and the results are shown in

Table 1. The time of crossing the target.

NORAD Number	Time (UTCG)	NORAD Number	Time (UTCG)	NORAD Number	Time (UTCG)
23467	17:03:00	27168	15:57:00	41724	14:42:00
41937	22:45:00	26694	12:00:00	41940	10:06:00
43162	15:21:00	39120	13:54:00	38352	17:49:20
25019	13:01:20	43700	15:37:20	28158	11:58:20
38254	23:37:20	22787	14:06:00	44337	19:45:10
37481	11:03:00	28542	17:54:00	44204	11:29:00
27711	8:03:00	25639	16:36:00	37256	15:19:20
26715	13:27:00	38091	15:46:20	37804	11:01:20
22988	12:27:00	26880	8:31:20	37234	13:07:20
26695	8:42:00	39375	19:03:00	37377	21:06:00
33055	8:42:00	28117	12:54:00	44231	5:23:00
20253	22:37:20	25967	19:06:00	41586	0:25:20
38466	15:51:00	39234	5:48:00	43683	21:04:20
20776	13:12:00	29155	4:00:00	36287	9:07:20
44481	8:03:30	38953	15:19:20	37210	19:52:20
24737	15:46:20	42949	21:27:00	25258	17:54:00

.

3.2. Design a Spiral Trajectory That Satisfies Desired Patrol Time

The closer the observation distance is, the more detailed characteristic information can be obtained. Therefore, when the distance between the patrol satellite and GEO is the smallest, the observation effect is the best. That is, the observation position should be selected as the apogee (east drift orbit) or perigee (west drift orbit) of the patrol satellite. It can be seen that the conditions for forming a spiral ring are that the apogee velocity is less than the target orbit, while the perigee velocity is greater than the target orbit. This is noted as follows:

$$\begin{array}{c}{v}_a\le v_c\\ v_p\ge v_c\end{array}$$

(43)

Among them, v_a and v_p are the perigee and apogee velocities of the patrol orbit, respectively.

According to orbital dynamics, the velocities of the perigee and apogee can be expressed as follows:

$$\begin{array}{l}\begin{array}{c}{v}_a=\sqrt{{\displaystyle \frac{\mu }{r_p+r_a}}}\sqrt{{\displaystyle \frac{2r_p}{r_a}}}\\ v_p=\sqrt{{\displaystyle \frac{\mu }{r_p+r_a}}}\sqrt{{\displaystyle \frac{2r_a}{r_p}}}\end{array}\\ v_c=\sqrt{{\displaystyle \frac{\mu }{a_c}}}\end{array}$$

(44)

By substituting into Formula (19), we can obtain the following:

$${\displaystyle \frac{r_p\left(r_p+r_a\right)}{2r_a}}\le a_c\le {\displaystyle \frac{r_a\left(r_p+r_a\right)}{2r_p}}$$

(45)

As analyzed above, the edge of the GEO maneuvering area is ±75 km, and the observation position is taken as the perigee (west drift) or apogee (east drift), as shown in Figure 14.

$$\begin{array}{c}\begin{array}{cc}\mathrm{East} & {\displaystyle \frac{r_p\left(r_p+a_c-75\right)}{2\left(a_c-75\right)}}\le a_c\le {\displaystyle \frac{\left(a_c-75\right)\left(r_p+a_c-75\right)}{2r_p}}\end{array}\\ \begin{array}{cc}\mathrm{West} & {\displaystyle \frac{\left(a_c+75\right)\left(a_c+75+r_a\right)}{2r_a}}\le a_c\le {\displaystyle \frac{r_a\left(a_c+75+r_a\right)}{2\left(a_c+75\right)}}\end{array}\end{array}$$

(46)

By organizing the above equation, we can obtain the following:

$$\begin{array}{c}\begin{array}{cc}\mathrm{East} \mathrm{drift} & a_c-a_s\ge {\displaystyle \frac{150a_c}{a_c+75}}\end{array}\\ \begin{array}{cc}\mathrm{West} \mathrm{drift} & a_s-a_c\ge {\displaystyle \frac{150a_c}{a_c-75}}\end{array}\end{array}$$

(47)

By rearranging Formulas (45)–(47), the relationship among the aiming point position, spiral ring period, and longitude drift rate D can be obtained as follows:

$$r_{aim}=\left\{\begin{array}{cc}2\left(2a_c-{\displaystyle \frac{2a_c^2}{2a_c-\frac{D}{0.0128}}}\right)+75-a_c& r<a_c\\ 2\left(2a_c-{\displaystyle \frac{2a_c^2}{2a_c-\frac{D}{0.0128}}}\right)-75-a_c& r>a_c\end{array}\right.$$

(48)

$$T_{ring}=2\sqrt{{\displaystyle \frac{a_c^3}{\mu }}}\sqrt{{\left(1+{\displaystyle \frac{-\frac{D}{0.0128}}{a_c}}\right)}^3}\left(\begin{array}{l}\arccos \left({\displaystyle \frac{a_s\left(-\frac{D}{0.0128}\right)}{\left(a_c+a_s\right)\left(\frac{D}{0.0128}-75\right)}}\right)\\ +{\displaystyle \frac{\frac{D}{0.0128}-75}{a_s}}\sin \left(\arccos \left({\displaystyle \frac{a_s\left(\frac{D}{0.0128}\right)}{\left(a_c+a_s\right)\left(\frac{D}{0.0128}-75\right)}}\right)\right)\end{array}\right)$$

(49)

The aiming point position and spiral ring period with respect to the longitude drift rate are shown in Figure 15 and Figure 16.

3.3. Divide IGSO Targets into Regions Based on Dichotomy Approach

Use the dichotomy approach to divide the regions of the 48 IGSO targets. There are several significant longitude gaps in the distribution of the targets. If the divided region covers these gaps, there may be a significant increase in the patrol period due to a certain target. Therefore, when dividing regions, it is important to avoid these gaps as much as possible. The position and the longitude difference gap for IGSO targets are shown in

Table 2. IGSO target positions and their longitude differences.

Sub-satellite point longitude (°)	−177.1		−159.6		−159.0		−130.1		−120.0		−96.8		−90.0
Longitude gap (°)		17.5		0.6		28.9		10.1		23.2		6.8
Sub-satellite point longitude (°)	−90.0		−67.9		−38.9		−34.0		−17.8		−14.7		−10.1
Longitude gap (°)		22.1		29		4.9		16.2		3.1		4.6
Sub-satellite point longitude (°)	−10.1		−1.3		0.0		3.9		16.3		20.6		26.0
Longitude gap (°)		8.8		1.3		3.9		12.4		4.3		5.4
Sub-satellite point longitude (°)	26.0		28.8		29.0		35.6		59.0		69.5		70.0
Longitude gap (°)		2.8		0.2		6.6		23.4		10.5		0.5
Sub-satellite point longitude (°)	70.0		71.5		72.7		74.0		75.0		80.0		91.1
Longitude gap (°)		1.5		1.2		1.3		1.0		5.0		11.1
Sub-satellite point longitude (°)	91.1		92.0		93.1		98.0		103.8		106.0		113.9
Longitude gap (°)		0.9		1.1		4.9		5.8		2.2		7.9
Sub-satellite point longitude (°)	113.9		118.0		129.8		130.0		131.1		143.0		144.0
Longitude gap (°)		4.1		11.8		0.2		1.1		11.9		1.0
Sub-satellite point longitude (°)	144.0		144.5		160.0		172.3		−177.1
Longitude gap (°)		0.5		15.5		12.3		10.6

.

From Table 2, it can be seen that there are five longitude differences greater than 20°, but the first four notches are very concentrated, considering the actual situation such as period. Mark the longitude gaps greater than 10°, and control the patrol period of each region to be within 60 days. Based on the distribution characteristics of IGSO targets and the analysis of orbital longitude drift rate, the targets are divided into seven regions. The divisions are shown in

Table 3. IGSO target divisions.

Region	A	B	C	D	E	F	G
IGSO targets using the sub-satellite point longitude as a reference (°)	160.0	−130.1	−38.9	16.3	59.0	91.1	129.8
	172.3	−120.0	−34.0	20.6	69.5	92.0	130.0
	−177.1	−96.8	−17.8	26.0	70.0	93.1	131.1
	−159.6	−90.0	−14.7	28.8	71.5	98.0	143.0
	−159.0	−67.9	−10.1	29.0	72.7	103.8	144.0
			−1.3	35.6	74.0	106.0	144.5
			0.0		75.0	113.8
			3.9		80.0	118.0

.

3.4. Calculate the Bidirectional Longitude Drift Rate Within Each Region

As shown in Figure 4, as the longitude drift rate increases, the aiming point position relative to GEO increases linearly. As shown in Figure 16, the spiral ring period shows a rapid growth and then a slowing trend, eventually tending toward 15 h. If all of the crossing times correspond to an interval range of 0–12 h (according to the concept of crossing points, the crossing points always appear in pairs, and there must be a crossing point in this interval), and the patrol satellite adopts a strategy of regional control and a round-trip path. To achieve comprehensive coverage of crossing time, it requires that the spiral ring period cannot be less than 12 h. According to Figure 5, the longitude drift rate corresponding to a 12 h is 3.3°/day.

On the other hand, in order to ensure that the patrol satellite can observe the target during the spiral loop, the position of the aiming point relative to GEO needs to meet the observation distance constraint condition (takes 250 km as an example). Similarly, as shown in Figure 5, the longitude drift rate corresponding to a relative distance of 250 km is 4.1°/day. In summary, the longitudinal drift rate of the patrol orbit for IGSO targets is expressed as follows:

$$\begin{array}{c}\begin{array}{cc}\mathrm{Eastward} \mathrm{drift} & 3.3°/\mathrm{day}\le D_E\le 4.1°/\mathrm{day}\end{array}\\ \begin{array}{cc}\mathrm{Westward} \mathrm{drift} & -4.1°/\mathrm{day}\le D_W\le -3.3°/\mathrm{day}\end{array}\end{array}$$

(50)

Also, consider setting region thresholds for the target set as follows:

$$\Delta q=0.2°$$

(51)

At this point, the corresponding aiming point position is 200 km. As shown in Figure 4, the corresponding longitude drift rate is 3.5°/day. Therefore, the set of longitude drift rates D as follows:

$$\begin{array}{c}\begin{array}{cc}\mathrm{Eastward} \mathrm{drift} & 3.3°/\mathrm{day}\le D_E\le 3.5°/\mathrm{day}\end{array}\\ \begin{array}{cc}\mathrm{Westward} \mathrm{drift} & -3.5°/\mathrm{day}\le D_W\le -3.3°/\mathrm{day}\end{array}\end{array}$$

(52)

The above calculation is applied to each region using the gradient longitude drift rate strategy, and the longitude drift rate and patrol period for each region were obtained as shown in

Table 4. Longitude drift rate and patrol period of regions.

Region	A	B	C	D	E	F	G
Westward drift rate (°/day)	3.4	3.4	3.4	3.4	3.5	3.4	3.3
Eastward drift rate (°/day)	3.5	3.4	3.5	3.3	3.5	3.4	3.3
Patrol period (day)	25	39	28	13	14	18	12

. From the table, it can be seen that the longest period does not exceed 40 days.

3.5. Determine the Starting Position of the Patrol for Each Region

The initial positions of the west drift and east drift are shown in

Table 5. The initial positions of each region.

Region	A	B	C	D	E	F	G
Initial longitude of eastward drift (°)	158.4	−130.2	−39.0	16.1	58.9	91.0	129.8
Initial longitude of westward drift (°)	−159.0	−66.0	6.8	35.6	82.0	120.6	147.6

.

3.6. Determine the Transfer Orbit for Each Region

The fuel consumption for one patrol cycle in each region can be obtained as shown in

Table 6. Fuel consumption for one patrol cycle.

Region	A	B	C	D	E	F	G
ΔV (km/s)	0.0393	0.0387	0.0393	0.0382	0.0399	0.0387	0.0376
Δm (kg)	13.28	13.08	13.28	12.91	13.48	13.08	12.71

. It can be seen that fuel consumption does not exceed 13.5 kg. The total fuel consumption of a single patrol cycle in all regions should not exceed 91.82 kg.

In summary, the IGSO patrol configuration is shown in Figure 17.

4. Case Study

Taking zone A as example, there are five targets in this region, with sub-satellite point longitudes of {−160.0°, 172.3°, −177.1°, −159.6°, −159.0°}, and the longitude drift rate used is expressed as follows:

$$\begin{array}{l}{D}_E=3.5°/\mathrm{day}\\ D_W=3.4°/\mathrm{day}\end{array}$$

(53)

The corresponding sets of east and west drift targets are expressed as follows:

$$\begin{array}{l}{Q}_E=\left\{172.3°-177.1°-159.6°\right\}\\ Q_W=\left\{\begin{array}{cc}160.0°& -159.0°\end{array}\right\}\end{array}$$

(54)

The corresponding initial positions for east and west drift are expressed as follows:

$$\begin{array}{l}{\lambda }_E=158.4°\\ {\lambda }_W=-159.0°\end{array}$$

(55)

The simulation results are shown in Figure 18.

It can be seen that the patrol satellite has achieved patrol of all targets. The patrol of various targets is shown in Figure 19.

During the patrol, the minimum observation distance and observation duration of the satellite for each target are shown in

Table 7. The minimum observation distance and duration for each target.

Target	1	2	3	4	5
Minimum patrol distance (km)	108.8	187.1	205.7	84.7	101.1
Patrol duration (s)	5820.6	1180.5	1585.7	1487.6	1352.9

.

From Table A1, it can be seen that for region A, the minimum observation distance of the patrol satellite for each target is less than 250 km, and the observation duration that meets the observation condition constraint is above 20 min, which can ensure the patrol of targets.

5. Conclusions

With the rapid development of mega constellations, the space environment is becoming increasingly harsh, and space traffic management is a future trend. Space traffic patrol is one of the important components in space traffic management. This involves going around a zone or an area at regular times to check that it is safe and that there is no trouble at close range using one or more satellites.

This paper focuses on the patrol of IGSO targets. A method based on the longitude drift rate and crossing points is proposed for the patrol IGSO targets and includes the following steps: (1) calculate the crossing position and crossing time of the IGSO targets; (2) design a spiral trajectory that satisfies the desired patrol time; (3) divide IGSO targets into regions using a dichotomy approach; (4) calculate the bidirectional longitude drift rate within each region; (5) determine the starting position of patrol for each region; and (6) determine the transfer trajectory for each region. Using the above methods, the STK tool was used to simulate a target in region A, and the simulation results verified that using the gradient longitude drift rate method based on crossing points and spiral rings can achieve patrol of IGSO targets. The patrol period should not exceed 40 days, and the fuel consumption is no more than 13.5 kg for a single zone patrol period. And, the total fuel consumption of a single patrol cycle in all regions should not exceed 91.82 kg. The approach proposed in this article not only can be applied in the design of GEO zone patrol route, but also be used to guide space traffic patrol, thereby enhancing space safety.

Potential future research directions include the following: (1) The optimal match between different types of patrol satellites and GEO targets should be determined; (2) The research in this article is only at the theoretical stage, and there will be many problems in practical engineering; (3) This article divided the areas and adopted a “single to many” patrol form for each area. It is worth further research and exploration to determine whether using a “many to many” approach is more efficient in regional simulation.

6. Patents

This work was supported by National Defense Science and Technology Excellence Youth Fund Project of China. All data, models, and code generated or used during the study appear in the submitted paper.