A real-time trajectory planning method for constellation formation reconstruction

Abstract

The present invention discloses a real-time trajectory planning method for star cluster formation reconstruction, which belongs to the field of star cluster formation reconstruction trajectory planning. The present invention uses the terminal non-convex configuration constraint to enable the star cluster to form an circling configuration that is conducive to the coordinated observation of the circling center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target, establishing the most fuel-saving performance index represented by the acceleration integral term, and saving more fuel. The present invention converts the terminal non-convex configuration constraint of the star cluster formation reconstruction into a unique star cluster formation non-convex equality constraint through variable substitution, relaxes the star cluster formation non-convex equality constraint into a star cluster formation concave inequality, and reduces the constraint dimension. The present invention improves the robustness of the star cluster formation reconstruction trajectory planning for different initial working conditions by accurately penalizing the relaxed variables into the objective function. The present invention realizes the convexity of the star cluster formation reconstruction trajectory planning problem by linearizing the concave inequality constraint, and significantly improves the convergence of the iterative optimization of the star cluster formation reconstruction trajectory.

Description

Technical Field

[0001] The present invention relates to a real-time trajectory planning method for star cluster formation reconstruction, and in particular to a real-time trajectory planning method suitable for star cluster formation reconstruction under terminal non-convex configuration constraints, belonging to the field of star cluster formation reconstruction trajectory planning. Background Art

[0002] Star clusters, due to their small size, maneuverability, and ability to carry a variety of payloads, have been used in many space science exploration and observation missions, such as observing the interaction between the solar and Earth magnetic fields, high-precision modeling of atmospheric density and plasma, and constructing asteroid gravitational fields. When implementing these missions, star clusters must frequently and complexly reconfigure their formations to eliminate the effects of spatial perturbations or execute new sub-tasks. A key issue in star cluster formation reconfiguration is trajectory planning. This involves generating a trajectory from an initial formation configuration to a final formation configuration, taking into account the dynamic characteristics of each member. Due to the large number of members in a star cluster and the potentially complex final configuration, solving this trajectory planning problem in real time and with high reliability becomes crucial. Failure to reliably generate formation reconfiguration trajectories in real time can lead to mission failure, resulting in unnecessary resource loss and waste. Therefore, research on real-time trajectory planning methods for star cluster formation reconfiguration is of great significance. In addition, according to the constraint form of the terminal formation configuration, the above trajectory planning problem can be divided into two types of star cluster formation reconstruction trajectory planning problems with terminal linear configuration and terminal non-convex configuration. Compared with the terminal linear configuration, the star cluster formation reconstruction trajectory planning problem with terminal non-convex configuration constraints is suitable for more complex and diverse flight missions. However, the non-convexity of the terminal configuration constraint increases the complexity of the solution space of the star cluster formation reconstruction trajectory planning problem, which may reduce the efficiency and reliability of problem solving. Therefore, the real-time trajectory planning method for star cluster formation reconstruction proposed in this patent can not only effectively solve the star cluster formation reconstruction trajectory planning problem with terminal non-convex configuration constraints, but also improve the planning robustness by performing concave decomposition and precise penalty on the terminal non-convex configuration constraints, and further improve the computational efficiency by using the sequential convex optimization framework, so that the developed method is suitable for real-time trajectory planning of star clusters.

[0003] Among the previously developed real-time trajectory planning methods for swarm formation reconfiguration, the prior art [1] (see: C. Lippe, S. D'Amico, Optimal spacecraft swarm reconfiguration through chief orbit refinement, Acta Astronautica 183 (2021) 162–175.) considered orbit refinement and convex programming methods for the main spacecraft, significantly reducing the computational cost of swarm formation reconfiguration and extending the mission life. However, it did not consider the terminal non-convex configuration constraints, which limited the application of this method.

[0004] Therefore, the real-time trajectory planning problem of constellation formation reconstruction needs to consider the terminal non-convex configuration constraints, and an efficient and reliable real-time trajectory planning algorithm is urgently needed to be developed. Summary of the Invention

[0005] Aiming at the problem of star cluster trajectory planning with terminal non-convex configuration constraints, the main purpose of the present invention is to provide a real-time trajectory planning method for star cluster formation reconstruction. By equivalent transformation and relaxation penalty processing of the terminal non-convex constraints, the real-time solution of the star cluster formation reconstruction trajectory planning problem is realized based on the sequential convex programming framework, and the real-time trajectory planning for star cluster formation reconstruction is realized, which has the following advantages: (1) the star cluster formation reconstruction process is the most fuel-efficient; (2) the initial feasibility of the iterative algorithm can be guaranteed; (3) the iterative algorithm has good convergence and high reliability; (4) the sequential convex programming solution has high efficiency in star cluster trajectory planning.

[0006] The purpose of the present invention is achieved through the following technical solutions:

[0007] The present invention discloses a real-time trajectory planning method for star cluster formation reconstruction. The method takes the virtual center of the terminal non-convex configuration of the star cluster formation reconstruction as the origin to establish a relative motion coordinate system for the reconstruction of the star cluster formation. The construction of the relative motion coordinate system facilitates the convexification of the star cluster dynamics equation. On the basis of the relative motion coordinate system for the reconstruction of the star cluster formation, the dynamics equation constraint, acceleration magnitude constraint and terminal non-convex configuration constraint of the star cluster are established. The terminal non-convex configuration constraint includes terminal non-convex position constraint and terminal non-convex velocity constraint. The terminal non-convex configuration constraint enables the star cluster to form an circling configuration that is conducive to collaborative observation of the circling center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target. The acceleration integral term is used to represent the fuel consumption in the optimization index, so as to minimize the fuel consumption in the process of the star cluster formation reconstruction and increase the diversity of subsequent tasks. The dynamics equation constraint, acceleration magnitude constraint and terminal non-convex configuration constraint of the star cluster are used as constraint conditions, and the acceleration integral term is used to represent the fuel consumption as the optimization index. A star cluster formation reconstruction trajectory planning problem is established; the terminal non-convex configuration constraint of the star cluster formation reconstruction is converted into a unique star cluster formation non-convex equality constraint through variable substitution equivalence, and the non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a star cluster formation concave inequality constraint, which is more suitable for processing large-scale terminal configuration constraints in the star cluster formation and significantly improves the efficiency of star cluster formation reconstruction trajectory planning; at the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of the star cluster formation reconstruction and improve the robustness of the star cluster formation reconstruction trajectory planning for different initial working conditions; the star cluster formation concave inequality constraint is linearized to obtain the convex form of the star cluster formation reconstruction trajectory planning problem; the convex form of the star cluster formation reconstruction trajectory planning problem is iteratively solved until convergence to obtain the star cluster formation reconstruction trajectory; the linearization of the star cluster formation concave inequality constraint significantly improves the convergence of the iterative solution, thereby enabling the star cluster to realize real-time trajectory planning of star cluster formation reconstruction under the condition of limited onboard resources. Under the condition of limited onboard resources, the application of the present invention can obtain the trajectory of constellation formation reconstruction in real time. The constellation will be guided to the optimized terminal configuration along the most fuel-saving trajectory and complete the coordinated observation of the target.

[0008] The present invention discloses a real-time trajectory planning method for constellation formation reconstruction, comprising the following steps:

[0009] Step 1: Taking the virtual center of the terminal non-convex configuration of the constellation formation reconstruction as the origin, establish the relative motion coordinate system of the constellation formation reconstruction. The construction of the relative motion coordinate system facilitates the convexification of the constellation dynamics equation in step 2.

[0010] The virtual center of the terminal non-convex configuration of the constellation formation reconstruction is selected as the coordinate origin o to establish the constellation formation reconstruction relative motion coordinate system. The x-axis points from the center of the Earth to the virtual center point, and the y-axis is along the velocity direction of the virtual center point. The z-axis forms a right-handed rectangular coordinate system with the x- and y-axes, thus realizing the construction of the constellation formation reconstruction relative motion coordinate system.

[0011] Step 2: On the basis of the relative motion coordinate system of the star cluster formation reconstruction, the dynamic equation constraints, acceleration magnitude constraints and terminal non-convex configuration constraints of the star cluster are established. The terminal non-convex configuration constraints include terminal non-convex position constraints and terminal non-convex velocity constraints. Through the terminal non-convex configuration constraints, the star cluster forms an orbiting configuration that is conducive to collaborative observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target; in the optimization index, the acceleration integral term is used to represent the fuel consumption, so as to minimize the fuel consumption in the star cluster formation reconstruction process and increase the diversity of subsequent tasks; with the dynamic equation constraints, acceleration magnitude constraints and terminal non-convex configuration constraints of the star cluster as constraints, and the acceleration integral term representing the fuel consumption as the optimization index, the star cluster formation reconstruction trajectory planning problem is constructed.

[0012] Step 2.1: Based on the constellation formation's reconstructed relative motion coordinate system, establish the constellation's dynamic equation constraints, acceleration magnitude constraints, and terminal non-convex configuration constraints. The terminal non-convex configuration constraints include terminal non-convex position constraints and terminal non-convex velocity constraints. Through the terminal non-convex configuration constraints, the constellation forms an orbiting configuration at the terminal moment that is conducive to coordinated observation of the orbiting center, thereby improving the constellation formation's observation efficiency of the target.

[0013] The dynamic equation of the star cluster in the relative motion coordinate system of the star cluster formation reconstruction is in convex form and can be expressed as:

[0014]

[0015] Where d = 1, 2, ..., D represents the number of the members in the cluster, and D represents the total number of members in the cluster; r d =[x d ,y d ,z d ] T and v d =[v d,x ,v d,y ,v d,z ] T Represents the position vector and velocity vector of member d, respectively, and a d =[a d,x ,a d,y ,a d,z ] T Represents the acceleration vector of member d. d,0 and vd,0 Respectively represent the position and velocity vector of member d at the initial time t0, which are known values. The expressions of matrices A1 and A2 are as follows:

[0016]

[0017] Where n is the orbital angular velocity of the coordinate origin o (virtual center). The acceleration of the star cluster should satisfy:

[0018] ||a d ||≤a max (3)

[0019] Among them, a max is the upper limit of the allowed acceleration. The terminal non-convex configuration constraint of the star cluster formation is as follows:

[0020]

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] where t f Reconstruct the terminal time for the constellation as a known value. and Respectively represent the semi-major axis, phase angle, ascending node longitude, and z-maximum value of the non-convex orbital configuration orbit. Formulas (4)-(6) are the terminal non-convex position constraints of the star cluster formation, while formulas (7)-(9) are the terminal non-convex velocity constraints of the star cluster formation. Through the terminal non-convex configuration constraint, the star cluster forms an orbital configuration that is conducive to coordinated observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target. In addition, the z-maximum value Need to meet:

[0027]

[0028] in, is the upper limit of the maximum value in the z direction. At the terminal moment t f , the phase angle between any two adjacent members d1 and d2 and Need to meet:

[0029]

[0030] Where d1 = 1, 2, ..., D-1 and d2 = d1 + 1. To ensure that all members enter the same orbit, the right ascension of the ascending node and the maximum value in the z direction of each member need to satisfy:

[0031]

[0032] Step 2.2: Use the acceleration integral term to represent fuel consumption in the optimization index to minimize the fuel consumption during the constellation formation reconstruction process and increase the diversity of subsequent tasks;

[0033] The fuel consumed by the constellation during formation reconstruction can be equivalently expressed by the velocity increment, which can be further characterized by the acceleration integral term. The specific calculation formula is as follows:

[0034]

[0035] Where ΔV represents the velocity increment required by the constellation during formation reconfiguration, which is equal to the sum of the integral terms of the acceleration of each member in the constellation. The optimization metric of minimizing fuel consumption during formation reconfiguration is considered to increase the diversity of subsequent missions.

[0036] Step 2.3: Using the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration as constraints, and the acceleration integral term representing fuel consumption as the optimization indicator, construct a constellation formation reconstruction trajectory planning problem.

[0037] With the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration, and the acceleration integral term representing fuel consumption as the optimization metric, the following trajectory planning problem for constellation formation reconstruction is constructed:

[0038]

[0039] st formula (1)-(12)(15)

[0040] Step 3: The terminal non-convex configuration constraint of the star cluster formation reconstruction in step 2 is converted into a unique star cluster formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a star cluster formation concave inequality constraint, which is more suitable for processing large-scale terminal configuration constraints in star cluster formations and significantly improves the efficiency of star cluster formation reconstruction trajectory planning. At the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of star cluster formation reconstruction and improve the robustness of star cluster formation reconstruction trajectory planning for different initial working conditions.

[0041] Step 3.1: The terminal non-convex configuration constraint of the constellation formation reconstruction in step 2 is converted into a unique constellation formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a constellation formation concave inequality constraint. This reduces the constraint dimension and is more suitable for handling large-scale terminal configuration constraints in constellation formations, significantly improving the efficiency of constellation formation reconstruction trajectory planning.

[0042] The terminal non-convex configuration constraints of the constellation formation reconstruction shown in equations (4)-(12) are uniformly written as:

[0043] g(y)=0 (16)

[0044] l(y)≤0 (17) In which, g in formula (16) represents the terminal equality constraint of the constellation formation reconstruction in formulas (4)-(9) and (11)-(12), and l in formula (17) represents the terminal inequality constraint of the constellation formation reconstruction in (10), and y is given by r d , v d , and According to the convex-concave decomposition theory, the terminal equality constraint (16) and the terminal inequality constraint (17) of the constellation formation reconstruction are further equivalently written as:

[0045] g1(y)-g2(y)=0 (18)

[0046] l1(y)-l2(y)≤0 (19)

[0047] Among them, g1, g2, l1, and l2 are all convex functions. In addition, the following variable substitutions for constellation formation reconstruction are introduced:

[0048] g2(y)-α=0 (20)

[0049] l2(y)-β=0 (21)

[0050] Where α and β are new optimization variables introduced. Then, equations (18)-(19) are replaced by equations (20)-(21) and the following equations:

[0051] g1(y)-α=0 (22)

[0052] l1(y)-β≤0 (23)

[0053] Although g1, g2, and l2 are all convex functions, equations (20)-(22) are still non-convex because their feasible sets are non-convex. However, equation (23) is convex. For convenience, define:

[0054]

[0055]

[0056] Using definitions (24) and (25), equations (20)-(22) are unified into the following form:

[0057] h(y)-λ=0 (26)

[0058] in is convex. So far, the non-convexity in the terminal non-convex constraint of the constellation formation reconstruction has been concentrated into the s non-convex equations in equation (26). In order to further concentrate the non-convexity, the equivalent change of equation (26) is as follows:

[0059] h(y)-λ≤0 (27)

[0060]

[0061] Where, Equation (27) represents s convex inequality constraints, and Equation (28) represents the only non-convex equality constraint for the star cluster formation.

[0062] Furthermore, we introduce slack variables The non-convex equality constraint H(y)-Γ(λ)=0 for the star cluster formation is relaxed along the concave direction to the following form:

[0063] H(y)-Γ(λ)≤0 (29)

[0064]

[0065]

[0066] Equations (29) and (30) are convex constraints, while Equation (31) is a concave inequality constraint for the star cluster formation. By converting multiple non-convex configuration constraints of the star cluster formation terminal into a unique concave inequality constraint for the star cluster formation, the constraint dimension is reduced, making it more suitable for processing large-scale terminal configuration constraints in the star cluster formation, and significantly improving the efficiency of the star cluster formation reconstruction trajectory planning:

[0067] Step 3.2: At the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of the constellation formation reconstruction and improve the robustness of the constellation formation reconstruction trajectory planning for different initial working conditions.

[0068] The slack variable Accurately penalizing the objective function to ensure the initial feasibility of constellation formation reconstruction, the optimization index (14) becomes:

[0069]

[0070] Among them, c p>0 is a sufficiently large penalty coefficient. The new optimization index (32) improves the robustness of the constellation formation reconstruction trajectory planning for different initial working conditions.

[0071] Step 4: Linearize the concave inequality constraint of the star cluster formation obtained in step 3 to obtain the convex form of the star cluster formation reconstruction trajectory planning problem; iteratively solve the convex form of the star cluster formation reconstruction trajectory planning problem until convergence to obtain the star cluster formation reconstruction trajectory; the linearization of the concave inequality constraint of the star cluster formation significantly improves the convergence of the iterative solution, thereby enabling the star cluster to realize real-time trajectory planning of star cluster formation reconstruction under the condition of limited onboard resources.

[0072] At a given reference profile y [k-1] On the other hand, the concave inequality constraint of the star cluster formation obtained in step 3 is linearized to the following form:

[0073]

[0074] Where k represents the kth iteration. Furthermore, the convex form of the constellation formation reconstruction trajectory planning problem is obtained as follows:

[0075]

[0076] st formulas (1)-(3), (23), (27), (29), (30), (33) (35) are iteratively solved to the convex form of the constellation formation reconstruction trajectory planning problem until convergence, and the constellation formation reconstruction trajectory is obtained. The linearization of the constellation formation concave inequality constraint significantly improves the convergence of the iterative solution, thereby enabling the constellation to achieve real-time trajectory planning for constellation formation reconstruction under the condition of limited onboard resources. In order to ensure the reliability of the solution, as an optimal method, the primal-dual interior point solution method is used to iteratively solve the convex problem until convergence to obtain the constellation formation reconstruction trajectory.

[0077] The method also includes step five: under the condition of limited onboard resources, the real-time trajectory planning method for constellation formation reconstruction in steps one to four is applied to quickly and reliably obtain the trajectory of constellation formation reconstruction. The constellation will be guided to the optimized terminal configuration along the most fuel-efficient trajectory and complete the coordinated observation of the target.

[0078] Beneficial effects:

[0079] 1. The present invention discloses a real-time trajectory planning method for constellation formation reconstruction. By using the terminal non-convex configuration constraint, the constellation forms an orbiting configuration at the terminal moment that is conducive to coordinated observation of the orbiting center, thereby improving the observation efficiency of the constellation formation on the target and establishing the most fuel-efficient performance index represented by the acceleration integral term, thereby further saving fuel.

[0080] 2. The present invention discloses a real-time trajectory planning method for constellation formation reconstruction. By replacing variables, the terminal non-convex configuration constraint of constellation formation reconstruction is converted into a unique constellation formation non-convex equality constraint, and then the constellation formation non-convex equality constraint is relaxed into a constellation formation concave inequality, thereby reducing the constraint dimension. The method is more suitable for processing large-scale terminal configuration constraints in constellation formations, and significantly improves the efficiency of trajectory planning for constellation formation reconstruction.

[0081] 3. The present invention discloses a real-time trajectory planning method for constellation formation reconstruction, which ensures the initial feasibility of constellation formation reconstruction by accurately penalizing slack variables into the objective function and improves the robustness of constellation formation reconstruction trajectory planning for different initial working conditions.

[0082] 4. The present invention discloses a real-time trajectory planning method for constellation formation reconstruction. By linearizing the concave inequality constraint, the trajectory planning problem of constellation formation reconstruction is convexified, which greatly improves the convergence of the iterative optimization trajectory of constellation formation reconstruction and increases the reliability of the solution of the present invention.

[0083] 5. The present invention discloses a real-time trajectory planning method for constellation formation reconstruction, which significantly improves the problem-solving efficiency through a sequential convex programming framework and increases the real-time performance of the optimization solution of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 is a flow chart of a real-time trajectory planning method for constellation formation reconstruction according to the present invention;

[0085] Figure 2 is a schematic diagram of the relative motion coordinate system in step 1 of the present invention;

[0086] Figure 3 is the flight trajectory of the constellation reconstruction process in this embodiment;

[0087] Figure 4 is a curve showing the acceleration of the star cluster changing with time in this embodiment. DETAILED DESCRIPTION

[0088] In order to better illustrate the purpose and advantages of the present invention, the present invention is explained in detail below by analyzing the implementation of a real-time trajectory planning method for constellation formation reconstruction.

[0089] Example 1:

[0090] like Figure 1 As shown, this embodiment discloses a real-time trajectory planning method for constellation formation reconstruction, and the specific implementation method is as follows:

[0091] Step 1: Taking the virtual center of the terminal non-convex configuration of the constellation formation reconstruction as the origin, establish the relative motion coordinate system of the constellation formation reconstruction. The construction of the relative motion coordinate system facilitates the convexification of the constellation dynamics equation in step 2.

[0092] The virtual center of the terminal non-convex configuration of the constellation formation reconstruction is selected as the coordinate origin o to establish the relative motion coordinate system of the constellation formation reconstruction, as follows: Figure 2 As shown in the figure, the x-axis points from the center of the Earth to the virtual center point, and the y-axis points in the direction of the virtual center point's velocity. The z-axis, along with the x- and y-axes, forms a right-handed rectangular coordinate system, which enables the reconstruction of the relative motion coordinate system for the constellation formation.

[0093] Step 2: On the basis of the relative motion coordinate system of the star cluster formation reconstruction, the dynamic equation constraints, acceleration magnitude constraints and terminal non-convex configuration constraints of the star cluster are established. The terminal non-convex configuration constraints include terminal non-convex position constraints and terminal non-convex velocity constraints. Through the terminal non-convex configuration constraints, the star cluster forms an orbiting configuration that is conducive to collaborative observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target; in the optimization index, the acceleration integral term is used to represent the fuel consumption, so as to minimize the fuel consumption in the star cluster formation reconstruction process and increase the diversity of subsequent tasks; with the dynamic equation constraints, acceleration magnitude constraints and terminal non-convex configuration constraints of the star cluster as constraints, and the acceleration integral term representing the fuel consumption as the optimization index, the star cluster formation reconstruction trajectory planning problem is constructed.

[0094] Step 2.1: Based on the constellation formation's reconstructed relative motion coordinate system, establish the constellation's dynamic equation constraints, acceleration magnitude constraints, and terminal non-convex configuration constraints. The terminal non-convex configuration constraints include terminal non-convex position constraints and terminal non-convex velocity constraints. Through the terminal non-convex configuration constraints, the constellation forms an orbiting configuration at the terminal moment that is conducive to coordinated observation of the orbiting center, thereby improving the constellation formation's observation efficiency of the target.

[0095] The dynamic equation of the star cluster in the relative motion coordinate system of the star cluster formation reconstruction is in convex form and can be expressed as:

[0096]

[0097] Where d = 1, 2, ..., D represents the number of the members in the cluster, and D represents the total number of members in the cluster; r d =[x d ,y d ,z d ] T and v d =[v d,x ,v d,y ,v d,z ] TRepresents the position vector and velocity vector of member d, respectively, and a d =[a d,x ,a d,y ,a d,z ] T Represents the acceleration vector of member d. d,0 and v d,0 Respectively represent the position and velocity vector of member d at the initial time t0, which are known values. In this embodiment, the total number of members in the star cluster is D=5, r d,0 and v d,0 The values ​​are shown in the following table:

[0098] Table 1 r of each member d,0 and v d,0 Value

[0099] Member D <![CDATA[Initial position vector r d,0 (m)]]> <![CDATA[Initial velocity vector v d,0 (m / s)]]> Member 1 <![CDATA[[10.39,97.81,25.44] T ]]> <![CDATA[[0.36,1.52,1.16] T ×10 -3 ]]> Member 2 <![CDATA[[49.73,10.45,-7.26] T ]]> <![CDATA[[0.38,7.25,-2.12] T ×10 -3 ]]> Member 3 <![CDATA[[20.34,-91.35,-29.93] T ]]> <![CDATA[[-3.33,-2.97,-0.15] T ×10 -3 ]]> Member 4 <![CDATA[[-37.16,-66.91,-11.24] T ]]> <![CDATA[[2.44,5.42,2.03] T ×10 -3 ]]> Member 5 <![CDATA[[-43.30,50,22.98] T ]]> <![CDATA[[1.82,6.31,1.41] T ×10 -3 ]]>

[0100] The expressions of matrices A1 and A2 are as follows:

[0101]

[0102] Where n is the orbital angular velocity of the coordinate origin o (virtual center), which is n=7.27×10 - 5 rad / s. The acceleration of the star cluster should satisfy:

[0103] ||a d ||≤a max (38)

[0104] Among them, a max is the upper limit of the allowed acceleration, and its value is a max =0.05m / s 2 The terminal non-convex configuration constraints of the star cluster formation are as follows:

[0105]

[0106]

[0107]

[0108]

[0109]

[0110]

[0111] where t f The terminal time of constellation reconstruction is a known value, which is t f =2000s. and Respectively represent the semi-major axis, phase angle, ascending node longitude, and z-maximum value of the non-convex orbital configuration orbit. Formulas (39)-(41) are the terminal non-convex position constraints of the star cluster formation, while formulas (42)-(44) are the terminal non-convex velocity constraints of the star cluster formation. Through the terminal non-convex configuration constraint, the star cluster forms an orbital configuration that is conducive to coordinated observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target. In addition, the z-maximum value Need to meet:

[0112]

[0113] in, is the upper limit of the z-axis maximum value, and its value is At the terminal time t f , the phase angle between any two adjacent members d1 and d2 and Need to meet:

[0114]

[0115] Where d1 = 1, 2, ..., D-1 and d2 = d1 + 1. To ensure that all members enter the same orbit, the right ascension of the ascending node and the maximum value in the z direction of each member need to satisfy:

[0116]

[0117] Step 2.2: Use the acceleration integral term to represent fuel consumption in the optimization index to minimize the fuel consumption during the constellation formation reconstruction process and increase the diversity of subsequent tasks;

[0118] The fuel consumed by the constellation during formation reconstruction can be equivalently expressed by the velocity increment, which can be further characterized by the acceleration integral term. The specific calculation formula is as follows:

[0119]

[0120] Where ΔV represents the velocity increment required by the constellation during formation reconfiguration, which is equal to the sum of the integral terms of the acceleration of each member in the constellation. The optimization metric of minimizing fuel consumption during formation reconfiguration is considered to increase the diversity of subsequent missions.

[0121] Step 2.3: Using the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration as constraints, and the acceleration integral term representing fuel consumption as the optimization indicator, construct a constellation formation reconstruction trajectory planning problem.

[0122] Finally, with the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration, and the acceleration integral term representing fuel consumption as the optimization indicator, the following trajectory planning problem for constellation formation reconstruction is constructed:

[0123]

[0124] st formula (1)-(12)(50)

[0125] Step 3: The terminal non-convex configuration constraint of the star cluster formation reconstruction in step 2 is converted into a unique star cluster formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a star cluster formation concave inequality constraint, which is more suitable for processing large-scale terminal configuration constraints in star cluster formations and significantly improves the efficiency of star cluster formation reconstruction trajectory planning. At the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of star cluster formation reconstruction and improve the robustness of star cluster formation reconstruction trajectory planning for different initial working conditions.

[0126] Step 3.1: The terminal non-convex configuration constraint of the constellation formation reconstruction in step 2 is converted into a unique constellation formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a constellation formation concave inequality constraint. This reduces the constraint dimension and is more suitable for handling large-scale terminal configuration constraints in constellation formations, significantly improving the efficiency of constellation formation reconstruction trajectory planning.

[0127] The terminal non-convex configuration constraints of the constellation formation reconstruction shown in Equations (39)-(47) are uniformly written as:

[0128] g(y)=0 (51)

[0129] l(y)≤0 (52)

[0130] Where g in formula (51) represents the terminal equality constraint of the constellation formation reconstruction in formulas (39)-(44) and (46)-(47), and l in formula (52) represents the terminal inequality constraint of the constellation formation reconstruction in (45), and y is given by r d , v d , and According to the convex-concave decomposition theory, the terminal equality constraint (51) and the terminal inequality constraint (52) of the constellation formation reconstruction are further equivalently written as:

[0131] g1(y)-g2(y)=0 (53)

[0132] l1(y)-l2(y)≤0 (54)

[0133] Among them, g1, g2, l1, and l2 are all convex functions. In addition, the following variable substitutions for constellation formation reconstruction are introduced:

[0134] g2(y)-α=0 (55)

[0135] l2(y)-β=0 (56)

[0136] Where α and β are new optimization variables introduced. Then, equations (53)-(54) are replaced by equations (55)-(56) and the following equations:

[0137] g1(y)-α=0 (57)

[0138] l1(y)-β≤0 (58)

[0139] Although g1, g2, and l2 are all convex functions, equations (55)-(57) are still non-convex because their feasible sets are non-convex. However, equation (58) is convex. For convenience, define:

[0140]

[0141]

[0142] Using definitions (59) and (60), equations (55)-(57) are unified into the following form:

[0143] h(y)-λ=0 (61)

[0144] in is convex. So far, the non-convexity in the terminal non-convex constraint of the constellation formation reconstruction has been concentrated into the s non-convex equations in equation (61). In order to further concentrate the non-convexity, the equivalent change of equation (61) is as follows:

[0145] h(y)-λ≤0 (62)

[0146]

[0147] Where, Equation (62) represents s convex inequality constraints, and Equation (63) represents the only non-convex equality constraint for the star cluster formation.

[0148] Furthermore, we introduce slack variables The non-convex equality constraint H(y)-Γ(λ)=0 for the star cluster formation is relaxed along the concave direction to the following form:

[0149] H(y)-Γ(λ)≤0 (64)

[0150]

[0151]

[0152] Equations (64) and (65) are convex constraints, while Equation (66) is a concave inequality constraint for the star cluster formation. By converting multiple non-convex configuration constraints of the star cluster formation terminal into a unique concave inequality constraint for the star cluster formation, the constraint dimension is reduced, making it more suitable for processing large-scale terminal configuration constraints in the star cluster formation, and significantly improving the efficiency of the star cluster formation reconstruction trajectory planning:

[0153] Step 3.2: At the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of the constellation formation reconstruction and improve the robustness of the constellation formation reconstruction trajectory planning for different initial working conditions.

[0154] The slack variable Accurately penalizing the objective function to ensure the initial feasibility of constellation formation reconstruction, the optimization index (49) becomes:

[0155]

[0156] Among them, c p >0 is a sufficiently large penalty coefficient. The new optimization index (67) improves the robustness of the constellation formation reconstruction trajectory planning for different initial working conditions.

[0157] Step 4: Linearize the concave inequality constraint of the star cluster formation obtained in step 3 to obtain the convex form of the star cluster formation reconstruction trajectory planning problem; iteratively solve the convex form of the star cluster formation reconstruction trajectory planning problem until convergence to obtain the star cluster formation reconstruction trajectory; the linearization of the concave inequality constraint of the star cluster formation significantly improves the convergence of the iterative solution, thereby enabling the star cluster to realize real-time trajectory planning of star cluster formation reconstruction under the condition of limited onboard resources.

[0158] At a given reference profile y [k-1] On the other hand, the concave inequality constraint of the star cluster formation obtained in step 3 is linearized to the following form:

[0159]

[0160] Where k represents the kth iteration. Furthermore, the convex form of the constellation formation reconstruction trajectory planning problem is obtained as follows:

[0161]

[0162] st formulas (36)-(38), (58), (62), (64), (65), (68) (70) are iteratively solved to the convex form of the constellation formation reconstruction trajectory planning problem until convergence, and the constellation formation reconstruction trajectory is obtained. The linearization of the constellation formation concave inequality constraint significantly improves the convergence of the iterative solution, thereby enabling the constellation to achieve real-time trajectory planning for constellation formation reconstruction under the condition of limited onboard resources. In order to ensure the reliability of the solution, as an optimal method, the primal-dual interior point solution method is used to iteratively solve the convex problem until convergence to obtain the constellation formation reconstruction trajectory.

[0163] Step 5: Under the condition of limited onboard resources, the real-time trajectory planning method for constellation formation reconstruction in steps 1 to 4 is applied to quickly and reliably obtain the constellation formation reconstructed trajectory. The constellation will be guided to the ideal terminal configuration along the most fuel-efficient trajectory and complete the coordinated observation of the target.

[0164] To verify the advantages of this method in terms of initial feasibility, convergence, and computational efficiency, we used the initial state information in Table 1 and compared it with the classic convex-concave decomposition sequential convex optimization method. The specific results are shown in Table 2. This example demonstrates the significant advantages of this method in terms of initial feasibility, and its convergence is also far superior to the comparison method. Table 2 also demonstrates the computational efficiency and stability of this method, which is suitable for real-time application scenarios.

[0165] Table 2 Comparison of the results of the proposed method with existing methods

[0166]

[0167] also, Figure 3 The flight trajectory of the constellation formation reconstruction process is shown; Figure 4 The curve of the acceleration vector of the cluster changing with time is shown. It can be seen that the control curve of the cluster presents an "on-off" structure, indicating that the cluster has achieved the most fuel-efficient indicator requirements during the formation reconstruction process.

[0168] The above specific description further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above description is only a specific embodiment of the present invention, which is used to explain the present invention and is not used to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A real-time trajectory planning method for constellation formation reconstruction, characterized by: The following steps are included: Step 1: Taking the virtual center of the terminal non-convex configuration of the constellation formation reconstruction as the origin, establish the constellation formation reconstruction relative motion coordinate system, and construct the relative motion coordinate system to facilitate the convexification of the constellation dynamics equation in step 2; Step 2: On the basis of the relative motion coordinate system of the star cluster formation reconstruction, the dynamic equation constraint, acceleration magnitude constraint and terminal non-convex configuration constraint of the star cluster are established. The terminal non-convex configuration constraint includes a terminal non-convex position constraint and a terminal non-convex velocity constraint. Through the terminal non-convex configuration constraint, the star cluster forms an orbiting configuration that is conducive to collaborative observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target; in the optimization index, the acceleration integral term is used to represent the fuel consumption, so as to minimize the fuel consumption in the star cluster formation reconstruction process and increase the diversity of subsequent tasks; with the dynamic equation constraint, acceleration magnitude constraint and terminal non-convex configuration constraint of the star cluster as the constraint conditions, and the acceleration integral term representing the fuel consumption as the optimization index, the star cluster formation reconstruction trajectory planning problem is constructed; Step 3: The terminal non-convex configuration constraint of the constellation formation reconstruction in step 2 is converted into a unique constellation formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a constellation formation concave inequality constraint, which is more suitable for processing large-scale terminal configuration constraints in constellation formations and significantly improves the efficiency of constellation formation reconstruction trajectory planning. At the same time, the slack variables are accurately penalized into the objective function to ensure the initial feasibility of constellation formation reconstruction and improve the robustness of constellation formation reconstruction trajectory planning for different initial working conditions. Step 4: Linearize the concave inequality constraint of the star cluster formation obtained in step 3 to obtain the convex form of the star cluster formation reconstruction trajectory planning problem; Iteratively solve the convex form of the constellation formation reconstruction trajectory planning problem until convergence, and obtain the constellation formation reconstruction trajectory; The linearization of the concave inequality constraint of the star cluster formation significantly improves the convergence of the iterative solution, thereby enabling the star cluster to achieve real-time trajectory planning for star cluster formation reconstruction under the condition of limited onboard resources.

2. The real-time trajectory planning method for constellation formation reconstruction according to claim 1, wherein: The method also includes step five, in which, under the condition of limited onboard resources, the real-time trajectory planning method for constellation formation reconstruction in steps one to four is applied to quickly and reliably obtain the trajectory of constellation formation reconstruction. The constellation will be guided to the optimized terminal configuration along the most fuel-efficient trajectory and complete the coordinated observation of the target.

3. A real-time trajectory planning method for constellation formation reconstruction according to claim 1 or 2, characterized in that: The implementation method of step one is: The virtual center of the terminal non-convex configuration of the star cluster formation reconstruction is selected as the coordinate origin o to establish the relative motion coordinate system of the star cluster formation reconstruction; the x-axis points in the direction from the center of the earth to the virtual center point, and the y-axis is along the speed direction of the virtual center point; the z-axis and the x- and y-axes form a right-handed rectangular coordinate system, that is, the construction of the relative motion coordinate system of the star cluster formation reconstruction is realized.

4. The real-time trajectory planning method for constellation formation reconstruction according to claim 3, wherein: The implementation method of step 2 is: Step 2.1: Based on the constellation formation's reconstructed relative motion coordinate system, establish the constellation's dynamic equation constraints, acceleration magnitude constraints, and terminal non-convex configuration constraints. The terminal non-convex configuration constraints include terminal non-convex position constraints and terminal non-convex velocity constraints. Through the terminal non-convex configuration constraints, the constellation forms an orbiting configuration at the terminal moment that is conducive to coordinated observation of the orbiting center, thereby improving the constellation formation's observation efficiency of the target. The dynamic equation of the star cluster in the relative motion coordinate system of the star cluster formation reconstruction is in convex form and can be expressed as: Where d = 1, 2, ..., D represents the number of the members in the cluster, and D represents the total number of members in the cluster; r d =[x d ,y d ,z d ] T and v d =[v d,x ,v d,y ,v d,z ] T Represents the position vector and velocity vector of member d, respectively, and a d =[a d,x ,a d,y ,a d,z ] T represents the acceleration vector of member d; r d,0 and v d,0 Respectively represent the position and velocity vector of member d at the initial time t0, which are known values; the expressions of matrices A1 and A2 are as follows: Where n is the orbital angular velocity of the coordinate origin o (virtual center); the acceleration of the star cluster should satisfy: ||a d ||≤a max (3) Among them, a max is the upper limit of the allowed acceleration; the terminal non-convex configuration constraint of the star cluster formation is as follows: where t f Reconstruct the terminal time for the constellation as a known value; and They represent the semi-major axis, phase angle, ascending node longitude, and z-maximum value of the non-convex orbital configuration orbit respectively; formulas (4)-(6) are the terminal non-convex position constraints of the star cluster formation, and formulas (7)-(9) are the terminal non-convex velocity constraints of the star cluster formation; the terminal non-convex configuration constraint enables the star cluster to form an orbital configuration that is conducive to the coordinated observation of the orbiting center at the terminal moment, thereby improving the observation efficiency of the star cluster formation on the target; in addition, the z-maximum value Need to meet: in, is the upper limit of the maximum value in the z direction; at the terminal time t f , the phase angle between any two adjacent members d1 and d2 and Need to meet: Where d1 = 1, 2, ..., D-1 and d2 = d1 + 1. To ensure that all members enter the same orbit, the right ascension of the ascending node and the maximum value in the z direction of each member need to satisfy: Step 2.2: Use the acceleration integral term to represent fuel consumption in the optimization index to minimize the fuel consumption during the constellation formation reconstruction process and increase the diversity of subsequent tasks; The fuel consumed by the constellation during formation reconstruction can be equivalently expressed by the velocity increment, which is further characterized by the acceleration integral term. The specific calculation formula is as follows: Where ΔV represents the velocity increment required by the cluster during formation reconstruction, and the required velocity increment is equal to the sum of the integral terms of the acceleration of each member in the cluster. The optimization index of minimum fuel consumption is considered during the formation reconstruction process of the cluster to increase the diversity of subsequent tasks. Step 2.3: Using the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration as constraints, and the acceleration integral term representing fuel consumption as the optimization metric, construct a constellation formation reconstruction trajectory planning problem. With the constrains of the constellation's dynamic equations, acceleration magnitude, and terminal non-convex configuration, and the acceleration integral term representing fuel consumption as the optimization metric, the following trajectory planning problem for constellation formation reconstruction is constructed: st formula (1)-(12)(15).

5. The real-time trajectory planning method for constellation formation reconstruction according to claim 4, characterized in that: The implementation method of step three is: Step 3.1: The terminal non-convex configuration constraint of the constellation formation reconstruction in step 2 is converted into a unique constellation formation non-convex equality constraint through variable substitution. The non-convex equality constraint is relaxed along the concave direction by introducing slack variables to obtain a constellation formation concave inequality constraint. This reduces the constraint dimension and is more suitable for handling large-scale terminal configuration constraints in constellation formations, significantly improving the efficiency of constellation formation reconstruction trajectory planning. The terminal non-convex configuration constraints of the constellation formation reconstruction shown in equations (4)-(12) are uniformly written as: g(y)=0 (16) l(y)≤0 (17) Where g in formula (16) represents the terminal equality constraint of the constellation formation reconstruction in formulas (4)-(9) and (11)-(12), and l in formula (17) represents the terminal inequality constraint of the constellation formation reconstruction in (10), and y is given by r d , v d , and The independent variable of constituting the star cluster formation is: According to the convex-concave decomposition theory, the terminal equality constraint (16) and the terminal inequality constraint (17) of the star cluster formation reconstruction are further equivalently written as: g1(y)-g2(y)=0 (18) l1(y)-l2(y)≤0 (19) Among them, g1, g2, l1, and l2 are all convex functions; in addition, the following variable substitution for constellation formation reconstruction is introduced: g2(y)-α=0 (20) l2(y)-β＝0 (21) Where α and β are new optimization variables introduced; then, equations (18)-(19) are replaced by equations (20)-(21) and the following equations: g1(y)-α=0 (22) l1(y)-β≤0 (23) Although g1, g2, and l2 are all convex functions, equations (20)-(22) are still non-convex because their feasible sets are non-convex; while equation (23) is convex; for convenience, define: Using definitions (24) and (25), equations (20)-(22) are unified into the following form: h(y)-λ=0 (26) in is convex; so far, the non-convexity in the terminal non-convex constraint of the constellation formation reconstruction is concentrated into the s non-convex equations in equation (26); in order to further concentrate the non-convexity, the equivalent change of equation (26) is as follows: h(y)-λ≤0 (27) Where, equation (27) represents s convex inequality constraints; equation (28) represents the only non-convex equality constraint for the star cluster formation; Introducing slack variables The non-convex equality constraint H(y)-Γ(λ)=0 for the star cluster formation is relaxed along the concave direction to the following form: H(y)-Γ(λ)≤0 (29) Equations (29) and (30) are convex constraints, while Equation (31) is a concave inequality constraint for the star cluster formation. By converting multiple non-convex configuration constraints of star cluster formation terminals into a unique concave inequality constraint for the star cluster formation, the constraint dimension is reduced, making it more suitable for processing large-scale terminal configuration constraints in the star cluster formation, and significantly improving the efficiency of the star cluster formation reconstruction trajectory planning: Step 3.2: Accurately penalize the slack variables in the objective function to ensure the initial feasibility of constellation formation reconstruction and improve the robustness of constellation formation reconstruction trajectory planning for different initial working conditions. The slack variable Accurately penalizing the objective function to ensure the initial feasibility of constellation formation reconstruction, the optimization index (14) becomes: Among them, c p >0 is a sufficiently large penalty coefficient; the new optimization index (32) improves the robustness of the constellation formation reconstruction trajectory planning for different initial working conditions.

6. A real-time trajectory planning method for constellation formation reconstruction according to claim 5, characterized in that: In step four, At a given reference profile y [k-1] On the other hand, the concave inequality constraint of the star cluster formation obtained in step 3 is linearized to the following form: Where k represents the kth iteration. Furthermore, the convex form of the constellation formation reconstruction trajectory planning problem is obtained as follows: st formula (1)-(3), (23), (27), (29), (30), (33)(35) The convex form of the star cluster formation reconstruction trajectory planning problem is iteratively solved until convergence, and the star cluster formation reconstruction trajectory is obtained; the linearization of the concave inequality constraint of the star cluster formation significantly improves the convergence of the iterative solution, thereby enabling the star cluster to realize real-time trajectory planning of star cluster formation reconstruction under the condition of limited onboard resources.

7. The real-time trajectory planning method for constellation formation reconstruction according to claim 6, characterized in that: The convex problem is solved iteratively until convergence to obtain the constellation formation reconstruction trajectory. The iterative solution method adopts the primal-dual interior point solution method.

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