A low-cost multi-satellite co-location method for GEO orbits

Abstract

The present invention provides a low-cost multi-satellite co-location method for GEO orbit, which belongs to the field of satellite orbit technology. The method includes setting the target number of co-located satellites and solving the theoretical minimum separation distance; taking the less movement of the thrust vector adjustment mechanism as the optimization goal, solving the optimization problem model of the co-located satellite maneuver plan; according to the optimization problem model, verifying the minimum separation distance of each pair of satellites, and completing the multi-satellite co-location based on the theoretical minimum separation distance and the target number of co-located satellites. The present invention solves the low-cost multi-satellite co-location problem that takes into account the less movement of the thrust vector adjustment mechanism, the minimum fuel consumption and the larger number of satellite co-location.

Description

Technical Field

[0001] The present invention belongs to the technical field of satellite orbits, and in particular relates to a low-cost multi-satellite co-location method for GEO orbits. Background Art

[0002] In order to improve the utilization rate of geostationary orbit resources, a method is proposed for multiple satellites to share the same orbital position in the geostationary orbit (GEO). Multi-satellite colocation refers to placing multiple satellites at a fixed point in the geostationary orbit to form a satellite group (or constellation) and share an orbital position. The distribution of these satellites in orbit must be maintained at a specified position. Practice has proved that multi-satellite colocation does have good application value and significant advantages, prompting people to actively explore and develop this technology. However, the frequency isolation and spatial isolation of multiple satellites in the GEO orbit must be well solved, that is, it must be ensured that the colocation system is not subject to radio interference (anti-interference), space collision, and obstruction; at the same time, since the satellites in the multi-satellite colocation are relatively close, in order to meet the frequency isolation and space isolation requirements, the maneuvering control actions required by the satellite are bound to be more frequent and more costly. Therefore, how to balance the contradiction between multi-satellite colocation and low cost is the core issue of multi-satellite colocation technology research.

[0003] In order to ensure that the satellites in the co-located satellite cluster do not drift out of the orbit holding window and avoid collisions, the co-located satellites need to maintain longitude, eccentricity and inclination. The maintenance method is based on the orbit holding method of a single geostationary orbit satellite, which is an extension of the orbit holding method of a single geostationary orbit satellite, and pursues the optimization of the overall orbit holding of the co-located satellite cluster. With the increase in the number of co-located satellites, the concept of "master-slave satellite" orbit determination is proposed to meet the requirements of satellites for orbit position determination accuracy. This method concentrates the measurement and control work of the entire co-located satellite cluster on a "master satellite", and the ground station mainly tracks and measures the orbit of the master satellite. There are inter-satellite links between the co-located satellite clusters. The master satellite is responsible for relative measurement of other satellites (slave satellites) in the satellite cluster and issues position holding control instructions. Other satellites use the master satellite as a reference for relative position holding. However, this method requires the satellite to carry laser measurement equipment, which is costly. Therefore, this method has not been put into practical use and remains in the theoretical research stage.

[0004] The GEO distributed constellation consists of a group of satellites distributed in a unified and adjacent geostationary orbit. It uses networking coordination, co-orbit control and other technologies to integrate satellite resources that are adjacent and independently distributed in space. The satellites work together to improve the capabilities of space communication transmission and coverage, which can effectively solve the problems currently faced by communication satellites, such as scarce orbital positions, scarce spectrum resources, and limited payload of single satellites, and achieve the effect that the whole is greater than the sum of the parts. However, during the orbit control process, the service life of the satellite thrust vector adjustment mechanism is limited. If the action exceeds a certain number of times, it may be damaged and fail, thereby greatly shortening the on-orbit service life of the GEO satellite. In order to meet the engineering needs of low-cost multi-satellite co-location in GEO orbit, it is necessary to combine the dynamic characteristics of GEO orbit. When designing the multi-satellite co-location method for distributed constellation formation, it is required that the satellites in the constellation should maintain a close distance and ensure that the satellites cannot collide with each other; the satellites should have a close distance, a small relative angular velocity, a small opening angle and an angle change range, so as to achieve as many co-located satellites as possible. Therefore, it is urgent to propose a GEO orbit multi-satellite co-location method for the constellation in response to the demand for low-cost multi-satellite co-location with less movement of the satellite thrust vector adjustment mechanism and the lowest fuel consumption. Under the condition that the satellite thrust vector adjustment mechanism is less moved and the fuel consumption is the lowest, the number of satellites co-located is maximized to achieve low-cost multi-satellite co-location. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the present invention provides a low-cost multi-satellite co-location method for GEO orbit, which solves the problem of low-cost multi-satellite co-location with consideration of less movement of the thrust vector adjustment mechanism, minimum fuel consumption and a larger number of satellites co-located.

[0006] In order to achieve the above objectives, the technical solution adopted by the present invention is: a low-cost multi-satellite co-location method for GEO orbit, comprising the following steps:

[0007] S1. Set the number of target co-located satellites and solve the theoretical minimum separation distance;

[0008] S2. Taking the thrust vector control mechanism as the optimization target, solve the optimization problem model of the co-located satellite maneuver plan;

[0009] S3. According to the optimization problem model, the minimum separation distance of each pair of satellites is verified, and based on the theoretical minimum separation distance and the target number of co-located satellites, multi-satellite co-location is completed.

[0010] The beneficial effects of the present invention are as follows: the present invention solves the theoretical minimum separation distance and verifies the minimum separation distance of each pair of satellites. When the satellite thrust vector adjustment mechanism is less moved, the number of satellites co-located is maximized, thereby realizing low-cost multi-satellite co-location. The problems currently faced by communication satellites, such as scarce orbital positions, scarce spectrum resources, and limited single-satellite payload, can be effectively solved. The whole is greater than the sum of its parts, the orbital resources of the geostationary synchronous orbit are fully utilized, and the number of times the satellite thrust vector adjustment mechanism is used is reduced, thereby ultimately prolonging the service life of the thrust vector adjustment mechanism during actual operation of the satellite.

[0011] Furthermore, the S1 comprises the following steps:

[0012] S101, constructing an electric thrust and torque calculation model;

[0013] S102, constructing a satellite dynamics model including a thrust vector adjustment mechanism angle and electric thrust according to the electric thrust and torque calculation model;

[0014] S103, setting the number of target co-located satellites;

[0015] S104. According to the number of target co-located satellites, the theoretical minimum separation distance is solved by calculating the relative eccentricity and the tilt vector.

[0016] The beneficial effect of the above further solution is that the present invention obtains the theoretical minimum separation distance by solving it, which provides a clear indicator requirement for judging whether the multi-star co-location is successful.

[0017] Furthermore, the satellite dynamics model is expressed as follows:

[0018]

[0019]

[0020] in, represents the satellite's inertia matrix, represents the first derivative of the angular velocity vector, represents the first derivative of the angular momentum of the momentum wheel, represents the first derivative of the quaternion, represents the angular velocity vector, represents the angular momentum of the momentum wheel, represents the torque of the electric thrust acting on the satellite, represents the disturbance torque acting on the satellite, represents the total external torque acting on the satellite, represents the quaternion transformation matrix, represents a quaternion, , and Respectively represent the satellite x , y and z The acceleration vector in the direction.

[0021] The beneficial effect of the above further scheme is that: taking the all-electric propulsion satellite equipped with a deployable thrust vector adjustment mechanism as the research object, the satellite attitude dynamics and kinematics equations are established.

[0022] Furthermore, the expression of the electric thrust and torque calculation model is as follows:

[0023]

[0024]

[0025]

[0026] in, represents the torque of the electric thrust of a single electric thruster acting on the satellite, , and Respectively In the satellite body coordinate system x , y and z Projections in all directions, represents the torque of the electric thrust acting on the satellite, represents transpose, represents the satellite electric thrust in this system, represents the thrust application point, represents the satellite centroid coordinates, , and They represent the three-axis coordinates of the thrust action point, represents the thrust vector corresponding to the first electric thruster, represents the thrust vector corresponding to the second electric thruster, represents the torque of the electric thrust of the first electric thruster acting on the satellite, It represents the torque exerted on the satellite by the electric thrust of the second electric thruster.

[0027] The beneficial effect of the above further scheme is that according to the above process, the relationship between the thrust vector adjustment mechanism angle and the electric thrust and the electric thrust torque can be obtained. Substituting the electric thrust and torque into the attitude orbit equation, the influence of the electric thrust and the thrust vector adjustment mechanism angle of a single satellite on the satellite attitude and orbit control can be evolved, thereby characterizing the satellite maneuvering process.

[0028] Furthermore, the expression of the theoretical minimum separation distance is as follows:

[0029]

[0030]

[0031]

[0032]

[0033]

[0034] in, represents the theoretical minimum separation distance, represents the relative eccentricity, Relative eccentricity in the set Inside, Relative inclination, Indicates the relative inclination in the set Inside, represents the right ascension of the Earth's stationary position, represents the magnitude of the relative eccentricity, represents the magnitude of the relative inclination, represents the phase angle between relative eccentricity and tilt, represents an angle used to describe the phase relationship between relative eccentricity and tilt, Indicates the radius of the relative eccentricity vector control window.

[0035] The beneficial effect of the above further scheme is: an optimization equation for solving the theoretical minimum separation distance is given, and the minimum safe separation distance between satellites is ensured through constraints, ensuring the safe separation and effective control of satellites in orbit.

[0036] Furthermore, the S2 comprises the following steps:

[0037] S201, solving the relative state according to the state of the following satellite;

[0038] S202, defining a state error according to the relative state;

[0039] S203, determining a state constraint function according to the state error;

[0040] S204, defining a diagonal scaling matrix for the constraint function;

[0041] S205, based on the diagonal scaling matrix and the satellite dynamics model, constructing an optimization target with less movement of the thrust vector adjustment mechanism;

[0042] S206. Based on the optimization objective, solve the optimization problem model of the co-located satellite maneuver plan.

[0043] The beneficial effect of the above further scheme is that the thrust vector and the thrust vector adjustment mechanism angle are scaled and maintained between 0 and 1 so as to be maintained within a uniform range during the optimization process, and an optimization problem model for solving the co-located satellite maneuver plan is established.

[0044] Furthermore, the expression of the optimization problem model is as follows:

[0045]

[0046]

[0047] in, represents the weight factor, represents the change in the thrust vector adjustment mechanism angle after scaling, express The weight factor of represents the scaled thrust vector, denotes the elements of the slack variable vector, j represents the index in the constraint, M represents the length of the slack variable vector, represents the slack variable vector, represents the number of constraints, represents the propulsion force scaling matrix, Indicates The thrust vector of each satellite contains the thrust values ​​of each thruster. , , and Represents the maximum thrust provided by the thrusters of the 1st satellite, 2nd satellite, 3rd satellite and 4th satellite respectively.

[0048] The above further scheme has the beneficial effect that an optimization problem model is established to minimize fuel consumption and thrust vector control mechanism movement while keeping the relative state within the convex control window. The optimization results will give the thrust vector (after scaling) and thrust vector control mechanism rotation angle of each follower satellite, as well as the corresponding slack variables for handling constraints, and these results can be used to guide the actual satellite maneuvering operation.

[0049] Furthermore, the S3 comprises the following steps:

[0050] S301. Based on the optimization problem model, and according to the satellite maneuver plan results with the combined optimization objectives of less movement of the thrust vector control mechanism and minimum thruster fuel consumption, the minimum separation distance of each pair of satellites in the radial-normal plane within one year is verified;

[0051] S302. Determine whether the minimum separation distance of each pair of satellites is greater than the theoretical minimum separation distance. If so, set a target number of co-located satellites to achieve multi-satellite co-location. Otherwise, the optimization goal of reducing the movement of the thrust vector control mechanism cannot be achieved, and the number of co-located satellites is reduced, and return to S1.

[0052] The beneficial effect of the above further scheme is: through the above design, the maximum number of co-located satellites that meets the optimization goals of less movement of the thrust vector regulation mechanism and minimum fuel consumption is determined, solving the low-cost multi-satellite co-location problem that takes into account less movement of the thrust vector regulation mechanism, minimum fuel consumption and a larger number of co-located satellites. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 The figure is a flow chart of the method of the present invention.

[0054] Figure 2 Schematic diagram of the installation of the thrust vector adjustment mechanism on the satellite in this embodiment.

[0055] Figure 3 It is a top view of the thrust vector adjustment mechanism model in this embodiment.

[0056] Figure 4 It is the front view of the thrust vector adjustment mechanism model in this embodiment. DETAILED DESCRIPTION

[0057] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0058] Example

[0059] like Figure 1 As shown, the present invention provides a low-cost multi-satellite co-location method for GEO orbit, and its implementation method is as follows:

[0060] S1. Set the number of target co-located satellites and solve the theoretical minimum separation distance. The implementation method is as follows:

[0061] S101, constructing an electric thrust and torque calculation model;

[0062] S102, constructing a satellite dynamics model including a thrust vector adjustment mechanism angle and electric thrust according to the electric thrust and torque calculation model;

[0063] S103, setting the number of target co-located satellites;

[0064] S104, according to the number of target co-located satellites, by calculating the relative eccentricity and the tilt vector, solving the theoretical minimum separation distance;

[0065] In this embodiment, the relationship between the thrust vector adjustment mechanism angle and the electric thrust and electric thrust torque is obtained through S101, wherein the thrust vector adjustment mechanism angle can calculate the conversion matrix from the electric thrust coordinate system to the satellite body coordinate system, the elements of the first three rows of the third column of the conversion matrix are inverted, that is, the unit vector of the thrust direction (the thrust direction is opposite to the thruster), and the first three rows of the fourth column are the real electric thrust action points in the whole satellite body coordinate system, which can be used to calculate the electric thrust torque and the resultant electric thrust. Substituting the electric thrust and torque into the attitude orbit equation of S102, a satellite dynamics model including the thrust vector adjustment mechanism angle and electric thrust can be obtained, that is, a satellite dynamics model configured with the thrust vector adjustment mechanism is constructed. After the satellite dynamics model is constructed, it can be applied to the subsequent S205 and S206, and an optimization problem model with less movement of the thrust vector adjustment mechanism and less fuel consumption as the combined optimization objectives is established, and the optimization problem is solved to obtain the co-located satellite orbit maneuvering plan.

[0066] S2. Taking the thrust vector control mechanism as the optimization target, the optimization problem model of the co-located satellite maneuver plan is solved. The implementation method is as follows:

[0067] S201, solving the relative state according to the state of the following satellite;

[0068] S202, defining a state error according to the relative state;

[0069] S203, determining a state constraint function according to the state error;

[0070] S204, defining a diagonal scaling matrix for the constraint function;

[0071] S205, based on the diagonal scaling matrix and the satellite dynamics model, constructing an optimization target with less movement of the thrust vector adjustment mechanism;

[0072] S206. Solving the optimization problem model of the co-located satellite maneuver plan based on the optimization objective;

[0073] S3. According to the optimization problem model, the minimum separation distance of each pair of satellites is simulated, and based on the theoretical minimum separation distance and the target number of co-located satellites, multi-satellite co-location is completed. The implementation method is as follows:

[0074] S301. Based on the optimization problem model, and according to the satellite maneuver plan results with the combined optimization objectives of less movement of the thrust vector control mechanism and minimum thruster fuel consumption, the minimum separation distance of each pair of satellites in the radial-normal plane within one year is verified;

[0075] S302. Determine whether the minimum separation distance of each pair of satellites is greater than the theoretical minimum separation distance. If so, set a target number of co-located satellites to achieve multi-satellite co-location. Otherwise, the optimization goal of reducing the movement of the thrust vector control mechanism cannot be achieved, and the number of co-located satellites is reduced, and return to S1.

[0076] In this embodiment, the satellite dynamics model configured with the thrust vector adjustment mechanism is established as follows:

[0077] The satellite attitude dynamics and kinematics equations are as follows:

[0078]

[0079] Quaternions are used to describe attitudes without any strangeness and with high efficiency:

[0080]

[0081] in, represents the satellite's inertia matrix, represents the first derivative of the angular velocity vector, represents the first derivative of the angular momentum of the momentum wheel, represents the first derivative of the quaternion, represents the angular velocity vector, represents the angular momentum of the momentum wheel, represents the torque of the electric thrust acting on the satellite, represents the disturbance torque acting on the satellite, represents the total external torque acting on the satellite, Represents a quaternion transformation matrix.

[0082] The Gaussian perturbation equation using the classical orbital principle is as follows:

[0083]

[0084] in, Indicates time, represents the Earth's gravitational constant, represents the semi-major axis of the orbit, is the orbital eccentricity, represents the orbital inclination, represents the right ascension of the ascending node, represents the argument of perigee, represents the true anomaly angle, represents the mean anomaly angle, represents the semi-diameter of the track, represents the orbital angular momentum, represents the satellite's geocentric distance, Indicates the satellite speed. represents the external acceleration vector except gravitational acceleration, , and They represent the radial, lateral (the forward direction perpendicular to the center of the Earth in the orbital plane) and normal components of the control acceleration of the satellite. For a fully electric propulsion satellite, the external acceleration includes the acceleration generated by the thrust of the electric thruster and the perturbation acceleration vector:

[0085]

[0086] in, represents the external acceleration vector except gravitational acceleration, It represents the satellite's own system to the orbit RTN coordinate system (the R axis is along the direction from the center of the earth to the center of mass of the satellite, the T axis is perpendicular to the X axis in the orbital plane and points to the direction of the satellite's forward movement, and the N axis is determined according to the right-hand rule). , and Represent external forces respectively. represents the satellite electric thrust in this system, It represents the orbital perturbation force of the satellite in its own system.

[0087] For electric propulsion satellites, electric thrust is all the forces except interference. In order to simulate the motion characteristics of all-electric propulsion satellites, it is necessary to establish an accurate model of electric thrust and torque. The electric thrust model design method is given later.

[0088] In this embodiment, the electric thrust and torque calculation model is established as follows:

[0089] The thrust vector adjustment mechanism studied adopts a four-joint structure, which can adjust the direction and point of action of electric thrust in three-dimensional space. The thrust vector adjustment mechanism consists of four joint drive shafts, a root base plate, a thruster support plate, and a thrust rod. The layout on the satellite is as follows: Figure 2 shown.

[0090] Figure 2 The coordinate system shown in the figure is the satellite body coordinate system. The state of the vector control mechanism in the figure is not deployed, and the electric thrust point is at the center of the circle on the plane of the electric thruster. The four joint angles of each thrust vector control mechanism are , and , in the compressed state, the angles of the four joints are , the status of the organization is Figure 2 As shown, the four joint angles returned by the rotation transformer are The Z axis of the joint coordinate system coincides with the joint axis. The rotation polarity of the joint is defined based on the joint axis using the right-hand screw rule, such as Figure 3 and Figure 4 shown.

[0091] When the electric thruster is used to control the satellite, the angles of each joint are different, and the projection and action points of the electric thrust on the three axes of the satellite are different, which will have different effects on the satellite's state and orbit. Therefore, it is necessary to establish a dynamic model of the thrust vector adjustment mechanism to deduce the relationship between the electric thrust, electric thrust torque and the joint angles of the mechanism, so as to provide a basis for designing the control method of the thrust vector adjustment mechanism, the satellite attitude and orbit control method, and verifying the control system. The DH parameters of the thrust vector adjustment mechanism are shown in Table 1. Table 1 is the DH parameter table of the thrust vector adjustment mechanism.

[0092] Table 1

[0093]

[0094] The two thrust vector adjustment mechanisms are exactly the same. Taking one of them as an example, the DH connecting rod model of the thrust vector adjustment mechanism is established, as shown in Figure 3 and Figure 4 shown. Figure 3 and Figure 4 The coordinate systems {1}~{4} in the figure are the coordinate systems established based on joints 1~4 (the coordinate system {1} is the coordinate system in the figure The coordinate system {e} is the electric thrust coordinate system, with its origin at the electric thrust action point, fixedly connected to the coordinate system {4}, and the Ze axis direction is the opposite direction of the electric thrust direction.

[0095] Use front The parameter method obtains the Parameters, the coordinate system can be directly calculated Relative to the coordinate system The transformation matrix is ​​as follows:

[0096]

[0097] Electric thrust coordinate system The origin is located at the center of the thruster nozzle, defined as Figure 3 As shown in the figure, when the joints of the mechanism rotate, the transformation relationship of this coordinate system relative to the coordinate system of the end of the robot arm {4} is a fixed value:

[0098]

[0099] in, Represents the electric thrust coordinate system The origin is located at the center of the thruster nozzle, defined as Figure 3 As shown in the figure, when the joints of the mechanism rotate, the coordinate system is relative to the coordinate system at the end of the robot arm. The conversion relationship is a fixed value. For a specific thrust vector adjustment mechanism, , It is a fixed value and can be obtained by direct length measurement, and the joint angle can be measured in real time by a rotary transformer. Indicates the offset in the DH parameter, which indicates the offset distance in the Z-axis direction compared to the previous joint. Specifically: is the offset of the first joint. is the offset of the second joint. is the offset of the third joint. is the offset of the fourth joint. is the link length in the DH parameter, which represents the distance between adjacent joints. Specifically: is the length of the link between the first and second joints. is the length of the link between the second and third joints. is the length of the link between the third and fourth joints.

[0100] The coordinates of the origin of the mechanism base coordinate system in the satellite system are , which is a fixed coordinate system on the entire satellite and does not change with the rotation of the mechanism joints. The axis direction is along the arm direction, The axis direction is along the axis direction of joint 1, and the mechanism arm and satellite The angle between the axes is According to the definition of the mechanism base coordinate system and the whole satellite coordinate system, the transformation matrix between the mechanism base coordinate system and the whole satellite coordinate system can be obtained:

[0101]

[0102] in, Represents the transformation matrix between the mechanism base coordinate system and the entire satellite coordinate system, It indicates that the X-axis direction is along the arm direction, the Z-axis direction is along the axis direction of joint 1, and the angle between the mechanism arm and the satellite X-axis. , , Represents the mechanism base coordinate system The origin is in the coordinates of the satellite's own system.

[0103] Electric thrust coordinate system Satellite based system The transformation matrix between them is as follows:

[0104]

[0105] in, , , Represents the transformation matrix between the coordinate systems of each joint of the mechanism, represents the transformation matrix of the mechanism coordinate system 1 relative to the mechanism base coordinate system, It represents the transformation matrix of the coordinate system established at the actual thrust action point of the mechanism electric thruster relative to the mechanism coordinate system 4. Homogeneous transformation matrix In the figure, the inverse of the first three rows of the third column is the thrust direction unit vector (the thrust direction is opposite to the thruster direction), and the first three rows of the fourth column are the actual electric thrust action points in the entire satellite body coordinate system.

[0106] for Figure 2 The two thrust vector adjustment mechanisms in the calculation are different according to the installation angle and installation position. Also different. The unit vector of the electric thrust direction calculated by the above method is represented by Indicates the thrust application point. Assume that the thrust generated by the electric thruster is nominally , then the projection of the electric thrust generated by a single electric thruster on each axis in this system is:

[0107]

[0108] in, Indicates the magnitude of the electric thrust generated by a single electric thruster. , , represents the component of the electric thrust on each axis, , , Represents the calculated electric thrust direction unit vector.

[0109] use represents the coordinates of the satellite's center of mass, then the electric thrust torque is calculated as follows:

[0110]

[0111] in, represents the torque of the electric thrust of a single electric thruster acting on the satellite, , , express In the satellite body coordinate system x , y and z The projection in the direction, represents transpose, is the representation of satellite electric thrust in this system, represents the thrust application point, represents the satellite centroid coordinates, , , They respectively represent the three-axis coordinates of the thrust application point.

[0112] For the case of two thrust vector adjustment mechanisms, the corresponding force and torque of each electric thruster are calculated according to the above method, and the resultant force and torque acting on the satellite can be obtained by adding them together:

[0113]

[0114] Among them, for the case where two thrust vector adjustment mechanisms are configured, the corresponding force and torque of each electric thruster are calculated according to the above method, and the resultant force and torque acting on the satellite can be obtained by adding them together. represents the thrust vector corresponding to the first electric thruster, represents the thrust vector corresponding to the second electric thruster, represents the torque of the electric thrust of the first electric thruster acting on the satellite, It represents the torque exerted on the satellite by the electric thrust of the second electric thruster.

[0115] According to the above process, the relationship between the thrust vector adjustment mechanism angle and the electric thrust and electric thrust torque can be obtained. By substituting the electric thrust and torque into the attitude and orbit equation, the influence of the electric thrust on the attitude and orbit control can be simulated.

[0116] In this embodiment, the theoretical minimum separation distance is solved as follows:

[0117] This embodiment proposes a mapping between orbital element differences and coordinates in the radial, tangential, and normal reference frames attached to the leading satellite. The relative orbital element is obtained by subtracting the leader element from the follower element and is represented by the symbol. The absolute orbital element in the following equation refers to the leader satellite. This first-order mapping is introduced as part of the relative motion model:

[0118]

[0119] in, , and Represent the satellite coordinates, represents the right ascension of the geosynchronous orbital position [radians], represents the mean orbital motion [radians per second], represents the relative mean orbital motion [rad / s], represents the first element in the relative eccentricity vector, represents the perigee angle of mean longitude, represents the second element in the relative eccentricity vector, represents the first element in the relative inclination vector, Represents the second element in the relative inclination vector.

[0120] This linear mapping relates the relative orbital elements to the relative Cartesian positions in the radial, tangential, normal (RTN) reference frame for near-circular, near-equatorial orbits. This mapping still depends on the state of the pilot satellite. If necessary, this can be done by Replace with ,Will Replace with ,Will Replace with to remove this dependency and apply to satellites located within a geosynchronous orbit slot (i.e., if the mapping is valid between any two satellites within a slot, it is also valid between a satellite and the center of its slot).

[0121] The error analysis of linear mapping was performed for the remaining analyzed cases, and it was found that the relative error was defined as:

[0122]

[0123] in, is the error between the approximate linear mapping and the exact nonlinear mapping, which is less than . Express the relative eccentricity and inclination vector in polar coordinates:

[0124]

[0125] in, represents the relative eccentricity vector, represents the magnitude of the relative eccentricity vector, represents the angle relative to the eccentricity vector, represents the relative inclination vector, represents the magnitude of the relative inclination vector, represents the right ascension of the ascending node [rad], represents the first element in the relative eccentricity vector, represents the perigee angle of mean longitude, represents the second element in the relative eccentricity vector, represents the first element in the relative inclination vector, Represents the second element in the relative inclination vector.

[0126] Use respectively and To express the relative eccentricity and the magnitude of the inclination vector, use and To express its phase. Further definition is the angle between the satellite position and the relative eccentricity vector, is the angle between the relative eccentricity and the inclination vector, so the relative separation in the RTN coordinate system can be written as:

[0127]

[0128] in, and They represent the angle between the satellite position and the relative eccentricity vector and the angle between the relative eccentricity and the inclination vector, respectively. Indicates relative mean longitude.

[0129] These equations directly show the relationship between the relative eccentricity and the phase of the inclination vector and the corresponding relative motion, can be obtained directly from the relative eccentricity and the inclination vector:

[0130]

[0131] in, represents the angle between the relative eccentricity and the inclination vector, represents the relative eccentricity vector, represents the relative inclination vector, represents the magnitude of the relative eccentricity vector, Represents the relative inclination vector size.

[0132] Due to the large uncertainty in the tangential direction, it is usually only dependent on the separation distance in the radial-normal plane. From the above, it can be observed that in this plane, the parallel relative eccentricity and inclination vectors (i.e. or ) results in the maximum separation distance.

[0133]

[0134] in, Indicates the maximum separation distance.

[0135] Since in reality the satellite cannot always be controlled to maintain , the present invention studies the minimum separation distance in the radial-normal plane as a function of the relative eccentricity and the inclination vector. Assuming that the difference in the semi-major axis is small ( ),have:

[0136]

[0137] in, represents the minimum separation distance on the radial-normal plane, Indicates satellite position Axis coordinates, Indicates satellite position Axis coordinates.

[0138] for For a specific relative eccentricity and inclination vector configuration, the minimum separation distance in the radial-normal plane can be obtained by We can find the derivative of the minimized equation by taking the following equation:

[0139]

[0140] can be written as:

[0141]

[0142] , As shown below:

[0143]

[0144]

[0145] in, represents a constant describing the effect of the combination of relative eccentricity and inclination on the minimum separation distance, The formula describes how changes in relative eccentricity and inclination affect the minimum separation distance between satellites. The combined effects of eccentricity and inclination are taken into account, as well as the phase relationship between them (represented by the cosine term). This is critical to ensuring safe separation of satellites in orbit. Represents an angle that describes the phase relationship between relative eccentricity and tilt. By calculating the phase relationship between relative eccentricity and tilt, The formula for provides an angle that is important in determining the relative motion and separation distance between satellites. This phase relationship affects the relative positions of the satellites in orbit, and thus the minimum separation distance between them.

[0146] The minimum value is found by solving the following equation:

[0147]

[0148] in, Indicates derivative.

[0149] This will be occurs at , so there are four solutions in the domain of interest, two of which correspond to the maximum separation distances and two of which correspond to the minimum separation distances. The symmetry of the relative motion requires that only one of the solutions needs to be checked to find the minimum distance. If you choose Can be verified :

[0150]

[0151] Therefore, by Substituting into the minimization equation and taking the square root obtains the minimum:

[0152]

[0153] in, represents the minimum separation distance, represents the right ascension of the geostationary position [rad], Represents the relative eccentricity vector size [-], represents an angle used to describe the phase relationship between relative eccentricity and tilt, represents the magnitude of the relative tilt vector [rad], It represents the angle between the relative eccentricity and the inclination vector [rad].

[0154] This equation can be used to find the minimum separation distance as a function of the relative eccentricity and the inclination vector.

[0155] Set up a safety isolation strategy and use a leader / follower architecture to control a set of conjugate satellites. The designated leader satellite is controlled using a desired method, assuming the predicted leader satellite state trajectory Can be accessed by each follower satellite (note that in order to obtain the "predicted" state, the current state is propagated via the latest maneuver plan). Follower satellites are controlled relative to this pilot satellite track, and are usually assumed to follow this track within a predefined control window.

[0156] The typical coordination strategy for conjugate geosynchronous satellites is the eccentricity-inclination vector separation strategy. This strategy relies entirely on the radial-normal plane to ensure safe separation. The key idea of ​​this strategy is to control the relative eccentricity and inclination vectors so that the radial and normal motions are approximately phase difference, so the normal separation is maximum when the radial separation is zero, and vice versa. As mentioned above, when the relative eccentricity and the inclination vector are parallel (i.e. ), this will happen.

[0157] Using this strategy, and defining the nominal relative eccentricity and inclination vector between the follower satellite and the pilot satellite, and The relative eccentricity and inclination vector are then controlled within tolerance windows that are relative to the nominal The vector is centered on The norm is bounded:

[0158]

[0159] in, and denote the nominal relative eccentricity and inclination vector, respectively, and represents the corresponding limits, Indicates the setting description relative eccentricity vector control window, Indicates the setting description relative tilt vector control window, represents the relative eccentricity vector, Represents the relative inclination vector.

[0160] The key question is: what is the minimum separation distance of the relative eccentricity and inclination vectors within these (convex) tolerance windows? From the equations presented previously, it can be inferred that, as The increase and , The control window is defined for the relative orbital elements between the follower satellite and the pilot satellite. If the vector is controlled within these windows, there can be a large gap between the two follower satellites. and smaller , Note that in these terms, the worst case occurs between F1 and F2, or equivalently between F2 and F3, assuming that with F2 fixed, the tolerance window for F3 is widened to and To find an exact bound on the minimum separation distance, a non-convex optimization problem can be solved:

[0161]

[0162] St:

[0163]

[0164] set up and , and assume that for all and ,have (This is a reasonable assumption, since will result in ), the analytical solution to the optimization problem can be found by substituting as follows: If we set and for all and (this is a very reasonable assumption as it leads to) and assume, we get the analytical solution to the optimization problem:

[0165]

[0166] in, represents the radius of the relative eccentricity vector control window. Substituting it into the minimum value equation Please note that you need to substitute to find the minimum separation distance between two follower satellites. Therefore, by choosing a set of (convex) control windows of the relative eccentricity and inclination vectors and maintaining the relative orbital elements within these windows, a certain minimum separation distance can be guaranteed, which can be obtained by solving the equation Or the equation under more stringent conditions and This approach provides an alternative way to deal with the non-convex constraint of maintaining a minimum separation distance between satellites. An important advantage of this approach is that the minimum separation distance constraint can be handled by defining two different convex constraints. Since these constraints are defined with respect to the (relative) orbital elements, the satellites follow a very natural orbital motion (i.e. the (relative) orbital elements are the integration constants in the Kepler two-body problem). The only disturbances that need to be compensated come from the differential orbital perturbations, the control acceleration of the pilot satellite and the disturbances caused by sensor and actuator errors.

[0167] In this embodiment, the relative state, state error and constraints are solved as follows:

[0168] The relative state is obtained by subtracting the pilot satellite state from the follower satellite state. Sampling is performed at discrete time points to match the discretized follower satellite states, thereby obtaining , the relative state is formed as follows:

[0169]

[0170] in, Indicates the difference between the follower satellite state and the leader satellite state, Indicates The state of following the satellite, Indicates the status of the leading satellite, represents the state transfer matrix, describing the change of satellite state over time, Indicates The mass of the following satellite, represents the input matrix, describing the influence of control input on satellite state, Indicates The thrust configuration matrix of each satellite maps the thrust to the corresponding acceleration. Indicates The thrust vector of the satellite, Indicates The disturbance acceleration of each satellite includes the influence of all dominant disturbances.

[0171] in, Indicates i Follow the satellite, Indicates Follow satellites. Note that the term for each satellite in the fleet is is also different because it depends on all the main perturbations, including the solar radiation pressure, which in turn depends on the characteristics of the specific satellite. In fact, the differential perturbations other than the solar radiation pressure are very similar for different satellites and can be cancelled when forming the relative state. However, the invention decided to keep these perturbations because it allows the absolute state to be evaluated and possibly constrained. Now let For the The expected relative state of the following satellite, the state error is defined as:

[0172]

[0173] in, Indicates the difference between the relative state and the desired state, Indicates the difference between the follower satellite state and the leader satellite state, Indicates the relative state that you want to follow the satellite to achieve. represents the state transfer matrix, describing the change of satellite state over time, Indicates Follow satellites at discrete times status, Indicates The mass of the satellite, represents the input matrix, describing the influence of control input on satellite state, Indicates The thrust configuration matrix of each satellite maps the thrust to the corresponding acceleration. Indicates The thrust vector of the satellite, Indicates The disturbance acceleration of the satellites includes the influence of all dominant disturbances, Indicates the status of the leader satellite.

[0174] The key task of the configuration control algorithm is to keep the convex function of this error below a certain bound:

[0175]

[0176] in, represents the state constraint function, Indicates the upper limit of the status range.

[0177] Or an affine function of the upper and lower bound errors.

[0178]

[0179] in, represents the state constraint function, Indicates the lower limit of the status range, Indicates the upper limit of the state range, "lb" and "ub" refer to the lower limit and upper limit respectively.

[0180] In this embodiment, the scaling of state and control variables is as follows:

[0181] In the formulation of the optimization problem, it is good practice to scale the optimization variables and constraints so that they vary in a more uniform range. Appropriate scaling can enhance robustness and improve the convergence of the optimization problem. To achieve the goal of optimizing the thrust vector control mechanism with less movement, the optimization variables are set to the angular changes of the four corners of the two thrust vector control mechanisms during attitude and orbit control. , and To minimize the movement of the thrust vector control mechanism, the angle change should be as small as possible. The vector optimization variables are scaled so that they are in the range This can be achieved by the following scaling law:

[0182]

[0183] in, Indicates The change in the thrust vector control mechanism angle after satellite scaling, represents the zoom scale, Indicates The change in the original thrust vector control mechanism angle of each satellite, , and Respectively represent the maximum angle changes of the four corners during attitude and orbit control. A similar method is also applicable to constraint conditions. In this work, the present invention only considers the constraints with symmetric boundaries (i.e. is an affine constraint on , or a convex constraint that is greater than or equal to zero and has only an upper bound. If express No. elements, then through Scale the constraint so that the scaled constraint In the interval or The internal changes correspond to and ,in, Indicates The original constraint vector of a satellite represents the constraint conditions of the satellite at a specific time point. Indicates Satellite The upper limit value of a constraint indicates the maximum value allowed by the constraint. Indicates The scaling constraint vector of the satellite represents the constraint value after scaling. Similar to the scaling of the thrust vector, the present invention defines a diagonal scaling matrix for the constraint:

[0184]

[0185] in, Indicates The scaling constraint vector for each satellite, Indicates The purpose of the formula is to scale the constraint vector so that its value is in a uniform range (usually [0, 1] or [-1,1]) for easy processing in the optimization problem. By scaling, the optimization algorithm can handle the constraints more effectively and improve the stability and convergence of the solution.

[0186] The scaled variables will be used in the formulation of the optimization problem.

[0187] In this embodiment, the optimization problem expression is as follows:

[0188] An optimization-based approach is used to determine the tracking satellite’s hold maneuver. i The coordination control problem of the present invention specifies the following optimization problem:

[0189]

[0190] In this embodiment, an optimization goal is added , represents the scaled thrust vector, which is used to solve the maneuver plan:

[0191]

[0192] in, represents the weight factor, represents the change in the thrust vector adjustment mechanism angle after scaling, express The weight factor of denotes the elements of the slack variable vector, j represents the index in the constraint, M represents the length of the slack variable vector, represents the slack variable vector, represents the number of constraints, represents the propulsion force scaling matrix, Indicates The thrust vector of each satellite contains the thrust values ​​of each thruster. , , and Respectively represent the maximum thrust provided by the thrusters of the first satellite, the second satellite, the third satellite, and the fourth satellite, represents the scaled thrust vector, ranging between [0, 1]. The purpose of this formula is to normalize the thrust vector so that its value is between 0 and 1, so that it is easier to handle and compare during the optimization process.

[0193] use It indicates the change of the thrust vector adjustment mechanism angle after scaling. If the total change of the four angles is small, it is considered that the thrust vector adjustment mechanism moves less. Represents the scaled thrust vector, which means that the thrust consumption is as small as possible. Therefore, the optimization goal of this equation is to minimize the movement of the thrust vector control mechanism and the thrust consumption. The result of the solution is the satellite co-location maneuver plan obtained by solving the optimization problem. Specifically, the optimization problem aims to minimize the propellant consumption and the movement of the thrust vector control mechanism, while keeping the relative state within the convex control window. The optimization result will give the thrust vector (after scaling) and the rotation angle of the thrust vector control mechanism of each follower satellite, as well as the corresponding slack variables, which are used to deal with the constraints. These results can be used to guide the actual maneuvering operation of the satellite. The specific process of implementing the maneuvering plan is as follows: According to the optimization results, determine the thrust magnitude and direction of each follower satellite at different time points and the change in the angle of the thrust vector control mechanism. Implement the maneuvering plan: Convert the optimized propulsion plan into actual thruster operation, which is realized by simple on / off control, and adjust the change in the angle of the thrust vector control mechanism. During the implementation process, monitor the status of the satellite to ensure that the satellite remains in the predetermined relative position and orbit. In this way, the relative motion of multiple satellites in geostationary orbit can be effectively controlled to ensure that they are safely co-located in the same orbital slot.

[0194] The first term, thrust vectoring mechanism angle, represents the rotation operation of the thrust vectoring mechanism, and the second term, thrust, represents the actual pushing operation. The goal of the optimization problem is to minimize the thrust vector and thrust vectoring mechanism rotation, which means that the thrust vector and thrust vectoring mechanism rotation in the optimization result will indicate the magnitude and direction of the thrust and thrust vectoring mechanism angle that each follower satellite needs to apply at different time points. This thrust vector and thrust vectoring mechanism angle are scaled to keep them in a uniform range during the optimization process. The final thrust force value will be used to guide the satellite's thruster operation, which is achieved through simple on / off control, and the thrust vectoring mechanism angle change is used to guide the thrust vectoring mechanism operation. Together, the two realize the co-located satellite maneuver plan with the optimization goals of less thrust vectoring mechanism movement and less propellant consumption.

[0195] In this problem, the optimization variables include:

[0196] Scaled thrust vector adjustment mechanism angle change ; represents the scaled thrust vector; the slack variable vector , whose length is ,Right now The number of constraints.

[0197] In this embodiment, the cost function is described as follows:

[0198] The cost function contains three items. The first item is the change in the angle of the thrust vector control mechanism. -norm. This term represents the minimization of the change in the rotation angle to achieve the optimization goal of less movement of the thrust vector control mechanism. The second term is the thrust vector of the electric propulsion satellite -norm. This term represents the minimization of the thrust vector to achieve the optimization goal of minimizing fuel consumption. The third term is a dead zone linear penalty function, which can be interpreted as follows: whenever the slack variable (Absolute value) less than constraint function Each time a limit is violated, it is penalized by a cost function. This implementation allows the optimizer to choose a desired control window that violates the state variable at a certain cost. The advantage of this implementation is that the optimization problem does not become unsolvable when the constraints cannot be satisfied. (in ), which adds the relative weights between the three terms in the cost function. The weights of the slack variables can be further adjusted to distinguish their relative importance.

[0199] In this embodiment, the constraints are as follows:

[0200] Three types of state constraints are used in this work. The first two types of constraints are about the eccentricity and tilt vector errors. and of -norm bounds. To get a concrete example, let Represents discrete time points The Relative state error. For time point The eccentricity vector error -norm constraints, including scaling and slack variables, as follows: The present invention uses three types of state constraints. The first two types of constraints are eccentricity and tilt vector errors and -norm. To get a concrete example, let denote the relative state error in discrete time. The eccentricity vector error -norm constraints in time, including scale and slack variables:

[0201]

[0202] in, Indicates Satellite The upper limit value of a constraint indicates the maximum value allowed by the constraint. Indicates Satellites at discrete times The second relative state error represents the difference between the relative state and the expected state. Indicates Satellites at discrete times The third relative state error represents the difference between the relative state and the expected state. Indicates the allowed range. The purpose of the formula is to Satellites at discrete times The 2 norm of the relative state error is constrained to ensure that it is within the allowed range. If the 2 norm of the relative state error exceeds the upper limit of the constraint, it will be penalized by a linear penalty function.

[0203] And about the tilt vector error:

[0204]

[0205] in, Indicates Satellite The upper limit value of a constraint indicates the maximum value allowed by the constraint. Indicates Satellites at discrete times The fourth relative state error represents the difference between the relative state and the expected state. Indicates Satellites at discrete times The fifth relative state error represents the difference between the relative state and the expected state. Indicates the allowed range of settings.

[0206] The third type of constraint is the longitude error The box constraint is implemented as follows:

[0207]

[0208] in, Indicates Satellites at discrete times The sixth relative state error represents the difference between the relative state and the expected state.

[0209] For control variables The constraint is the angle limit of the four corners of the satellite thrust vector control mechanism. After scaling, the control variable Each element in is constrained to be between 0 and 1.

[0210] In this embodiment, the orbital maneuver is implemented as follows:

[0211] Before solving the optimization problem, a propulsion scheme needs to be given. Assume that the thruster has only one operating point and is an on / off type thruster. norm is penalized, and the resulting maneuver plans are always sparse (i.e., they contain only a small number of non-zero elements). In implementing the maneuver plan, the present invention uses the following logic: if subsequent time intervals of a single thruster have non-zero thrust, then these time intervals are merged into a single burn, centered around the weighted average of the individual thrusts, with a total impulse equal to the sum of the individual elements.

[0212] In this embodiment, according to the satellite relative eccentricity control, relative inclination control, and relative average accuracy control results, the minimum separation distance of each pair of satellites in one year is simulated, which is specifically:

[0213] Analyze each pair of satellites (total Combination of n The minimum separation distance of each pair of satellites in the radial-normal plane is calculated during the entire simulation process of one year. If the minimum separation distance of each pair of satellites is greater than the theoretical separation distance, the proposed number of co-located satellites can be achieved. Otherwise, the optimization goal of reducing the movement of the thrust vector control mechanism cannot be achieved, and the number of co-located satellites needs to be reduced and run again.

Claims

1. A low-cost multi-satellite co-location method for GEO orbit, characterized in that: The following steps are involved: S1. Set the number of target co-located satellites and solve the theoretical minimum separation distance; S2. Taking the thrust vector control mechanism as the optimization target, solve the optimization problem model of the co-located satellite maneuver plan; The expression of the optimization problem model is as follows: if Among them, α represents the weight factor, represents the change in the thrust vector adjustment mechanism angle after scaling, and β represents The weight factor of represents the scaled thrust vector, s j represents the element of the slack variable vector, j represents the index in the constraint, M represents the length of the slack variable vector, s represents the slack variable vector, represents the number of constraints, T i represents the propulsion force scaling matrix, T i represents the propulsion vector of the i-th satellite, including the thrust values ​​of each thruster, and Respectively represent the maximum thrust provided by the thrusters of the first satellite, the second satellite, the third satellite, and the fourth satellite; S3. According to the optimization problem model, the minimum separation distance of each pair of satellites is verified, and based on the theoretical minimum separation distance and the target number of co-located satellites, multi-satellite co-location on the GEO orbit is completed.

2. The low-cost multi-satellite co-location method for GEO orbit according to claim 1, characterized in that: The S1 comprises the following steps: S101, constructing an electric thrust and torque calculation model; S102, constructing a satellite dynamics model including a thrust vector adjustment mechanism angle and electric thrust according to the electric thrust and torque calculation model; S103, setting the number of target co-located satellites; S104. According to the number of target co-located satellites, the theoretical minimum separation distance is solved by calculating the relative eccentricity and the tilt vector.

3. The low-cost multi-satellite co-location method for GEO orbit according to claim 2, characterized in that: The satellite dynamics model is expressed as follows: Where I represents the satellite's inertia matrix, represents the first derivative of the angular velocity vector, represents the first derivative of the angular momentum of the momentum wheel, represents the first derivative of the quaternion, represents the angular velocity vector, H represents the angular momentum of the momentum wheel, M e It represents the torque of the electric thrust acting on the satellite, M d is the disturbance torque acting on the satellite, M e +M d represents the total external torque acting on the satellite, represents the quaternion transformation matrix, q represents the quaternion, ω x ,ω y and ω z Represent the acceleration vector of the satellite in the x, y and z directions respectively.

4. The low-cost multi-satellite co-location method for GEO orbit according to claim 2, characterized in that: The expression of the electric thrust and torque calculation model is as follows: M e (i)=[M ex M ey M ez ] T =F e ×(L e -L s ) L e =[L ex ,L ey ,L ez ] F e =F e (1)+F e (2),M e =M e (1)+M e (2) Among them, M e (i) represents the torque of the electric thruster acting on the satellite, M ex 、M ey and M ez Respectively represent M e The projection in the x, y and z directions of the satellite body coordinate system, M e represents the torque of the electric thrust acting on the satellite, T represents the transpose, and F e It represents the satellite electric thrust in this system, L e Indicates the thrust application point, L s represents the satellite center of mass coordinates, L ex , L ey and L ez They represent the three-axis coordinates of the thrust action point, F e (1) represents the thrust vector corresponding to the first electric thruster, F e (2) represents the thrust vector corresponding to the second electric thruster, M e (1) represents the torque of the electric thrust of the first electric thruster acting on the satellite, M e (2) represents the torque exerted on the satellite by the electric thrust of the second electric thruster.

5. The low-cost multi-satellite co-location method for GEO orbit according to claim 2, characterized in that: The expression of the theoretical minimum separation distance is as follows: minimizeρ xz,min St: Δe∈S e Δi∈S i c2=atan2(δi 2 sin2δω,-δe 2 +δi 2 cos2δω) Among them, ρ xz,min represents the theoretical minimum separation distance, Δe represents the relative eccentricity, Δe∈S e Relative eccentricity in the set S e In the example, Δi represents the relative inclination, Δi∈S i Indicates the relative inclination in the set S i In it, a represents the right ascension of the geostationary position, δe represents the magnitude of the relative eccentricity, δi represents the magnitude of the relative inclination, δω represents the phase angle between the relative eccentricity and the inclination, c2 represents an angle used to describe the phase relationship between the relative eccentricity and the inclination, and r e Indicates the radius of the relative eccentricity vector control window.

6. The low-cost multi-satellite co-location method for GEO orbit according to claim 2, characterized in that: The S2 comprises the following steps: S201, solving the relative state according to the state of the following satellite; S202, defining a state error according to the relative state; S203, determining a state constraint function according to the state error; S204, defining a diagonal scaling matrix for the constraint function; S205, based on the diagonal scaling matrix and the satellite dynamics model, constructing an optimization target with less movement of the thrust vector adjustment mechanism; S206. Based on the optimization objective, solve the optimization problem model of the co-located satellite maneuver plan.

7. The low-cost multi-satellite co-location method for GEO orbit according to claim 6, characterized in that: The S3 comprises the following steps: S301. Based on the optimization problem model, and according to the satellite maneuver plan results with the combined optimization objectives of less movement of the thrust vector control mechanism and minimum thruster fuel consumption, the minimum separation distance of each pair of satellites in the radial-normal plane within one year is verified; S302. Determine whether the minimum separation distance of each pair of satellites is greater than the theoretical minimum separation distance. If so, determine the target number of co-located satellites to achieve multi-satellite co-location in the GEO orbit. Otherwise, the optimization goal of reducing the movement of the thrust vector control mechanism cannot be achieved, and the number of co-located satellites is reduced, and return to S1.

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