Non-cooperative target initial orbit determination method based on spaceborne angle-only short-arc observation

Abstract

This invention discloses a method for initial orbit determination of a non-cooperative target under space-based angle-only extremely short arc observation, belonging to the technical field of initial orbit determination. This invention addresses the failure of existing initial orbit determination methods under extremely short arc conditions. The method includes: using the unit normal vector of the target orbital plane as the variable to be optimized; obtaining the relationship between the slant distance and the unit normal vector based on the geometric relationship between the target position vector, the slant distance between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector; transforming the solution of the unit normal vector into the solution of the X-axis and Y-axis components of the unit normal vector based on the normalization characteristics of the unit normal vector, and determining the constraints of the solution; and employing a heuristic algorithm to solve for the X-axis and Y-axis components of the unit normal vector, thereby determining the unit normal vector and achieving initial orbit determination. This invention improves the accuracy and robustness of initial orbit determination.

Description

Technical Field

[0001] This invention relates to a method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observation, belonging to the technical field of initial orbit determination. Background Technology

[0002] Existing methods for determining the initial orbit of spatial targets using only angle measurement can be broadly classified into three categories: traditional initial orbit determination methods, distance-based initial orbit determination methods, and neural network-based initial orbit determination methods.

[0003] Traditional initial orbit determination methods: These methods, dating back to the 18th century, derive a set of positive definite equations about the initial orbital state of the target through theoretical derivation of orbital dynamics. Solving these equations yields the initial orbit. These methods typically employ long-arc observations, requiring high geometric properties of the observed arc segment. When dealing with extremely short arc observation segments, these methods generally fail, exhibiting divergent or convergent results. The reason for this is that under extremely short arc conditions, the coefficient matrix elements of the equations are highly similar, leading to singularity in the matrix. Therefore, the algorithm fails.

[0004] Initial trajectory determination methods based on distance search: These methods generally use distance (or distance-velocity) as the decision variable and employ heuristic algorithms to optimize the variable, thereby obtaining the initial trajectory information of the target. Since the optimization variable is generally distance information, which has a large solution space and complex feasible region distribution in Cartesian coordinates, these algorithms generally have high requirements for prior information about the target. Furthermore, under noisy conditions, the observability of extremely short arcs further decreases, and the success rate further declines.

[0005] Neural network-based initial trajectory determination methods: These methods use neural networks to calculate the orbital elements of the target. The general algorithm process involves first selecting the network structure, generating training and validation sets, and then training the network until convergence. The performance of this method is dependent on the training data, limiting its applicability. Furthermore, the generalization and interpretability of neural networks remain key factors restricting their application.

[0006] In space situational awareness systems, conducting sky surveys using space-based observation platforms to acquire and catalog the status of space targets is a crucial means of enhancing space situational awareness capabilities. However, due to the limited field of view of optical payloads and the extremely high relative speed between the space-based observation platform and the target, most observation arcs last only tens of seconds; these arcs are commonly referred to as extremely short arcs. Traditional methods generally fail in the case of extremely short arcs. Furthermore, the presence of observation noise further increases the difficulty of providing reliable solutions. Summary of the Invention

[0007] To address the problem that existing initial orbit determination methods fail under extremely short arc conditions, this invention provides an initial orbit determination method for non-cooperative targets under space-based angle-only extremely short arc observation.

[0008] The present invention provides a method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation, comprising:

[0009] Using the unit normal vector of the target orbital plane as the variable to be optimized, the relationship between the slant distance and the unit normal vector is obtained based on the geometric relationship between the target position vector, the slant distance between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector.

[0010] Based on the normalization property of the unit normal vector, the solution of the unit normal vector is transformed into the solution of the X-axis component and the Y-axis component of the unit normal vector, and the constraint conditions for the solution are determined.

[0011] A heuristic algorithm is used to solve for the X-axis component and Y-axis component of the unit normal vector, thereby determining the unit normal vector and achieving initial trajectory determination of the target.

[0012] According to the method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation of the present invention, the geometric relationships between the target position vector, the slant range between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector are as follows:

[0013] ,

[0014] In the formula For the target position vector, This represents the slant distance between the space-based platform and the target location. Let be the unit vector of the target's azimuth. This is the position vector of the space-based platform.

[0015] The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to the present invention, the method for obtaining the relationship between the slant range and the unit normal vector is as follows:

[0016] Let the unit normal vector and the target position vector be... Dot product:

[0017] ,

[0018] In the formula It is the unit normal vector;

[0019] Then the slope distance for:

[0020] .

[0021] The initial orbit determination method for non-cooperative targets under space-based angular-only extremely short arc observation according to the present invention is obtained based on the normalization property of the unit normal vector:

[0022] ,

[0023] In the formula The x-axis component of the unit normal vector. The Y-axis component of the unit normal vector. The Z-axis component of the unit normal vector;

[0024] ,

[0025] Therefore, the unit normal vector The solution is transformed into the X-axis component of the unit normal vector. and the unit normal vector Y-axis component Solve for it.

[0026] According to the method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observations of the present invention, based on the X-axis component of the unit normal vector and the unit normal vector Y-axis component The sum of the squares is not greater than 1, thus obtaining the X-axis component of the unit normal vector. and the unit normal vector Y-axis component The solution space lies within the unit circle.

[0027] According to the method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observations of the present invention, the constraint conditions for solving circular or elliptical orbit targets are as follows:

[0028] ,

[0029] In the formula It is the semi-nominal diameter. for The target position vector at time t. for The target position vector at time t. for The target position vector at time t. This is the eccentricity rate.

[0030] The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to the present invention includes a heuristic algorithm for solving the X-axis component and Y-axis component of the unit normal vector, comprising:

[0031] Define the population size NP, the maximum number of iterations G, the variable dimension 2, the scaling factor S, and the crossover probability CR for the heuristic algorithm.

[0032] Represent the population as :

[0033] ,

[0034] individual For the variables to be optimized:

[0035] , ;

[0036] In the formula This is the i-th solution for the x-axis component of the unit normal vector. This is the i-th solution of the Y-axis component of the unit normal vector;

[0037] In each iteration of the solution process, the current population is evaluated based on the scaling factor S and the crossover probability CR. Mutation and crossover are performed to obtain the population for the current iteration. Candidate update values ​​are selected, and individuals with excellent performance are chosen through an evaluation function to determine the population for the current iteration. Update value;

[0038] During the iterative solution process, when the population When the evaluation function of all variables to be optimized for updating values ​​is 0 or the maximum number of iterations G is reached, the current population is... The updated value serves as the final solution, yielding the X-axis and Y-axis components of the unit normal vector.

[0039] The method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observation according to the present invention includes the following mutation operations:

[0040] Before generation [G / 2], the current population... middle The mutation operation is as follows:

[0041] ,

[0042] In the formula For the mutated individual, and Let them be three distinct integers in the range [1, NP].

[0043] In subsequent evolution, the current population middle The mutation operation is as follows:

[0044] ,

[0045] In the formula For the current population The optimal individual in the process.

[0046] According to the method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observations of the present invention, the crossover operation adopts a binary crossover:

[0047] ,

[0048] In the formula This represents the j-th component of the i-th crossover individual, where j equals 1 or 2; After crossover operation , After crossover operation ; It is a random number uniformly distributed in [0,1]. After mutation operation , After mutation operation ; correspond , correspond ; It is a number randomly selected from 1 and 2.

[0049] According to the method for initial orbit determination of non-cooperative targets under space-based angular-only extremely short arc observations of the present invention, the method for selecting individuals using evaluation functions includes:

[0050] ,

[0051] In the formula ; for The result of the g-th iteration;

[0052] In the formula For the evaluation function, the input is the unit normal vector of the orbital plane. :

[0053] ,

[0054] In the formula Let be the target's azimuth unit vector at time m, calculated recursively. Let M be the target azimuth unit vector at the m-th moment of actual observation, and M be the number of observations in the extremely short arc.

[0055] The beneficial effects of this invention are as follows: This invention addresses the initial orbit determination problem under extremely short arc observation conditions. By combining it with a heuristic algorithm, it innovatively proposes the orbital plane normal vector as an optimization variable, which improves the weak observability under extremely short arc conditions and enhances the accuracy and robustness of the initial orbit determination.

[0056] Compared with traditional methods, the method of this invention is no longer limited to three observations within the arc segment, but makes full use of all observation data, thus enabling the calculation of more reliable orbit information. Compared with the initial orbit determination method based on distance search, the method of this invention transforms the search space from an open Euclidean space to a closed unit circle, which fundamentally alleviates the weak observability under the condition of extremely short arcs, thus having higher accuracy and stronger robustness. Compared with the initial orbit determination method based on neural networks, the method of this invention is not limited to a specific training set, and the specific algorithm is derived from orbit dynamics, which has strong applicability and interpretability.

[0057] The core improvement of this invention lies in selecting the target orbital plane normal vector as the optimization variable, abandoning the traditional approach of searching in Cartesian space or orbital element space, and transforming the high-dimensional, unbounded search problem into a search for the normal vector. This variable directly reflects the geometric attitude of the orbit, reducing the coupling between variables. Utilizing the normalization property of the normal vector, the originally open and unbounded search space is strictly constrained to a unit circle, greatly narrowing the search domain of feasible solutions. This geometric constraint acts as a "regularization," fundamentally eliminating a large number of physically meaningless solutions (such as hyperbolas and orbits that do not satisfy physical laws), thereby mathematically alleviating the inherent ill-conditioned nature of extremely short arc scenarios. The initial state can be randomly selected within the unit circle, eliminating the dependence on high-precision prior information and solving the problem of divergence caused by large initial value deviations in traditional methods.

[0058] The method of this invention improves the weak observability in extremely short arc scenarios: by using the orbital plane normal vector as the optimization variable, the search space is transformed from an open Cartesian distance space or orbital element space into a closed normalized unit circle. The introduction of constraints further limits the feasible domain of the solution space and discards candidate solutions that do not meet the preset conditions or physical laws.

[0059] The algorithm's dependence on prior information is reduced: Optimization approaches based on distance / orbital elements often require significant prior information; otherwise, the algorithm is prone to getting trapped in local optima or failing to converge. The optimization variables in this invention can be randomly selected within the unit circle, resulting in significantly higher coverage of the solution space than other methods, thus significantly reducing the algorithm's dependence on prior information.

[0060] Reducing the correlation of optimization variables: Optimization variables based on distance / orbital elements often exhibit high coupling, meaning that a set of optimization variables is not always independent. For example, the slant distance and relative velocity at two different times are highly correlated, while the relative velocity information is unknown under angle-only conditions, causing difficulties in algorithm initialization. The optimization variables proposed in this invention are unit vectors, which can be randomly selected within the feasible region. Different variables are not coupled and are only the components of the unit normal vector in different directions, thus significantly reducing the correlation between variables. Attached Figure Description

[0061] Figure 1 This is a schematic diagram of the observation configuration of the space-based platform;

[0062] Figure 2 This is a flowchart of the optimization process for a heuristic algorithm;

[0063] Figure 3 This is a schematic diagram of the feasible region constructed according to the constraints in the first scenario (Sce.1) during simulation analysis.

[0064] Figure 4 This is a schematic diagram of the feasible region constructed according to the constraints in the second scenario (Sce.2) during simulation analysis.

[0065] Figure 5 This is a schematic diagram of the feasible region constructed according to the constraints in the third scenario (Sce.3) during simulation analysis.

[0066] Figure 6 This is a statistical diagram illustrating the number of successes under noise-free conditions;

[0067] Figure 7 This is a statistical diagram illustrating the number of successes under noisy conditions;

[0068] Figure 8 This is a schematic diagram of the root mean square error of position under noisy conditions;

[0069] Figure 9 This is a schematic diagram of the root mean square error of velocity under noise conditions. Detailed Implementation

[0070] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0071] Specific Implementation Method 1: Combination Figure 1 As shown, this invention provides a method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation, including:

[0072] Using the unit normal vector of the target orbital plane as the variable to be optimized, the relationship between the slant distance and the unit normal vector is obtained based on the geometric relationship between the target position vector, the slant distance between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector.

[0073] Based on the normalization property of the unit normal vector, the solution of the unit normal vector is transformed into the solution of the X-axis component and the Y-axis component of the unit normal vector, and the constraint conditions for the solution are determined.

[0074] A heuristic algorithm is used to solve for the X-axis component and Y-axis component of the unit normal vector, thereby determining the unit normal vector and achieving initial trajectory determination of the target.

[0075] Furthermore, when observing targets using a space-based platform, the observation geometry is as follows: Figure 1 As shown.

[0076] The geometric relationships between the target position vector, the slant range between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector are as follows:

[0077] ,

[0078] In the formula For the target position vector, This represents the slant distance between the space-based platform and the target location. Let be the unit vector of the target's azimuth. This is the position vector of the space-based platform.

[0079] Differentiating the position vector with respect to time yields the velocity vector:

[0080] ,

[0081] In the formula The velocity vector representing the target. Represents the rate of change of interstellar distances. Represents the rate of change of the target's azimuth vector. The velocity vector represents the observation platform.

[0082] The core purpose of initial trajectory determination is to provide information about unknown targets, especially trajectory information, namely the target's position vector and velocity vector, or the equivalent orbital six-axis number.

[0083] In this embodiment, the determination of position and velocity vectors is transformed into the solution of the unit normal vector N of the target orbital plane.

[0084] The method for obtaining the relationship between the slope distance and the unit normal vector is as follows:

[0085] Let the unit normal vector and the target position vector be... Dot product:

[0086] ,

[0087] In the formula It is the unit normal vector;

[0088] Then the slope distance for:

[0089] .

[0090] It can be seen that obtaining the target orbital plane normal vector is equivalent to obtaining the slant range information. The unit vector representing the target's azimuth... Since the target's position vector can be obtained through observation, obtaining the normal vector is equivalent to obtaining the target's position vector at each observation time. Knowing the target's position vector at two specific moments, the target's velocity vector can be obtained by solving the Lambert problem, thus yielding complete target trajectory information.

[0091] Based on the normalization property of the unit normal vector, we obtain:

[0092] ,

[0093] In the formula The x-axis component of the unit normal vector. The Y-axis component of the unit normal vector. The Z-axis component of the unit normal vector;

[0094] ,

[0095] Therefore, the unit normal vector The solution is transformed into the X-axis component of the unit normal vector. and the unit normal vector Y-axis component Solve for it.

[0096] Furthermore, based on the X-axis component of the unit normal vector and the unit normal vector Y-axis component The sum of the squares is not greater than 1, thus obtaining the X-axis component of the unit normal vector. and the unit normal vector Y-axis component The solution space lies within the unit circle.

[0097] For targets in circular or elliptical orbits, the constraints for solving are:

[0098] ,

[0099] In the formula It is the semi-nominal diameter. for The target position vector at time t. for The target position vector at time t. for The target position vector at time t. Let be the eccentricity. The problem then becomes finding a set of solutions representing the true state of the target normal vector within a constrained unit circle. This implementation uses the differential evolution algorithm from a heuristic approach for the search and solution; the specific process is described below.

[0100] Combination Figure 2 As shown, the core of heuristic algorithms is to mimic biological evolution in nature, that is, individuals with high fitness have a greater probability of producing offspring, thereby improving the adaptability of the entire population and continuously evolving until the stopping condition is met or the maximum number of iterations is reached.

[0101] Methods for solving the X-axis and Y-axis components of the unit normal vector using heuristic algorithms include:

[0102] Define the population size NP, the maximum number of iterations G, the variable dimension 2, the scaling factor S, and the crossover probability CR for the heuristic algorithm.

[0103] Represent the population as It consists of NP individuals:

[0104] ,

[0105] individual It includes two variables to be optimized:

[0106] , ;

[0107] In the formula This is the i-th solution for the x-axis component of the unit normal vector. This is the i-th solution of the Y-axis component of the unit normal vector;

[0108] In each iteration of the solution process, the current population is evaluated based on the scaling factor S and the crossover probability CR. Mutation and crossover are performed to obtain the population for the current iteration. Candidate update values ​​are selected, and individuals with excellent performance are chosen through an evaluation function to determine the population for the current iteration. Update value;

[0109] During the iterative solution process, when the population When the evaluation function of all variables to be optimized for updating values ​​is 0 or the maximum number of iterations G is reached, the current population is... The updated value serves as the final solution, yielding the X-axis and Y-axis components of the unit normal vector.

[0110] Mutation operations include:

[0111] Before generation [G / 2], the current population... middle The mutation operation is as follows:

[0112] ,

[0113] In the formula For the mutated individual, and Let them be three distinct integers in the range [1, NP].

[0114] In subsequent evolution, the current population middle The mutation operation is as follows:

[0115] ,

[0116] In the formula For the current population The optimal individual in the process.

[0117] This implementation selects two different mutation strategies. Before generation [G / 2], a random mutation strategy is used. This strategy has good coverage of the solution space and can conduct a broad search in the early stages of evolution, avoiding early entrapment in local optima. In subsequent evolution, an elite strategy with fine-grained solution capabilities is used. This strategy focuses on the current best individual, which is beneficial to improving the accuracy of the final solution.

[0118] The crossover operation mimics the crossing over of chromosomes in nature and is performed after the mutation operation is completed. In this embodiment, the crossover operation uses a binary crossover:

[0119] ,

[0120] In the formula This represents the j-th component of the i-th crossover individual, where j equals 1 or 2; After crossover operation , After crossover operation ; It is a random number uniformly distributed in [0,1]. After mutation operation , After mutation operation ; correspond , correspond ; It is a number randomly selected from 1 and 2. After mutation and crossover, if the individual is superior to the original individual, it is retained in the next generation; otherwise, it is eliminated.

[0121] In this embodiment, the method for selecting individuals using the evaluation function includes:

[0122] ,

[0123] In the formula ; for The result of the g-th iteration;

[0124] In the formula For the evaluation function, the input is the unit normal vector of the orbital plane. :

[0125] ,

[0126] In the formula Let be the target's azimuth unit vector at time m, calculated recursively. Let M be the target azimuth unit vector at the m-th moment of actual observation, and M be the number of observations in the extremely short arc.

[0127] The evaluation function calculation process is as follows: After selecting a set of solutions within the feasible region of the unit circle, the slant distances at two specific moments are calculated, and the target's position vectors at those two moments are obtained. The target's velocity vectors at the corresponding moments are obtained by solving the Lambert problem. When both the position and velocity vectors are known, the trajectory information is unique and definite, and the target's position information at subsequent moments can be recursively derived, thereby obtaining the target's azimuth vector information. If the candidate solution is exactly the true state, the recursively derived azimuth vector should be consistent with the observed azimuth vector, and the evaluation function value... The value should be 0. Therefore, intuitively, the closer the evaluation function value is to 0, the closer the candidate solution is to the true state. After iteratively solving the problem using a heuristic algorithm, the individual in the final population whose evaluation function value is closest to 0 can be considered the final solution.

[0128] Application scenarios of this invention:

[0129] 1. Initial Orbit Acquisition of Non-Cooperative Targets by Space-Based Platforms: Space-based surveillance satellites and target satellites often have high relative velocities, resulting in extremely short target dwell times within the field of view (often only tens of seconds). Traditional methods cannot handle such extremely short arc observations. This invention focuses on initial orbit determination of targets under extremely short arc observations, offering higher accuracy and robustness compared to existing methods, and is suitable for initial orbit acquisition of non-cooperative targets.

[0130] 2. Space Target Maneuver Detection and Rapid Recovery: When a non-cooperative target maneuvers (changes orbit), the original orbital elements become invalid, potentially leading to target loss. In this case, it is necessary to quickly re-determine the orbit using the first short arc segment captured. The orbit after the maneuver is unknown, and the method of this invention allows for cold start without precise initial value guessing. Since it does not require accumulating long arc segment data, the approximate parameters of the new orbit can be determined based solely on a small segment of data after the orbit change, providing a foundation for subsequent precise orbit determination.

[0131] 3. Discovery of Uncataloged Debris and New Targets: Monitoring satellite disintegration or collision events generates a large amount of uncataloged debris. This debris lacks historical orbital data, and sensors often only capture discontinuous short segments. The method of this invention is suitable for the initial rapid cataloging of debris clouds from collision / disintegration events, providing an initial state for accurate tracking and improving cataloging efficiency.

[0132] 4. Multi-Arc Segment Correlation Preprocessing: In space-based surveillance systems, multiple observations of the same target often exist, resulting in multiple discontinuous extremely short arc observation data. Fully utilizing these multi-segment extremely short arc observation data can significantly improve the geometric ill-conditioned nature of a single extremely short arc, yielding more accurate orbital information. However, due to the large number of observed arc segments, arc segment correlation is essential for precise orbit determination. The correlation approach involves using the method of this invention to calculate the initial orbital elements corresponding to each arc segment, and determining whether the arc segments are correlated based on the relationship between the orbital elements of different arc segments. Therefore, the information provided by the method of this invention can serve as the basis for arc segment correlation, providing preprocessing for multi-arc segment correlation.

[0133] The method of this invention introduces the orbital plane normal vector as a new optimization variable. Compared with the variable based on distance / orbital elements, the new variable has the characteristic of normalization, which fundamentally improves the weak observability in the case of extremely short arcs, greatly reduces the dependence on prior information, and improves the accuracy and robustness of initial orbit determination for extremely short arcs.

[0134] Due to the normalization characteristics of the normal vector, the search range can be limited to the unit circle. Adding constraints such as semi-circle and eccentricity can further limit the feasible region and further compress the solution range.

[0135] This invention transforms the initial trajectory determination problem into an optimization variable problem that satisfies specific constraints, and ultimately uses a differential evolution algorithm to search for the optimization variables. Furthermore, considering the optimization process, the optimization is performed in two stages: the first stage focuses on the coverage of the solution space, considering global optimization to avoid the algorithm getting trapped in local optima too early; the second stage focuses on fine-grained search, aiming to improve the accuracy of the final solution.

[0136] Simulation analysis:

[0137] The simulation scenario settings are shown in Table 1, where `obser` represents the primary star. Three simulation scenarios were selected, corresponding to near-circular, small eccentricity, and medium eccentricity targets, respectively. During the simulation, the observation duration was set to 60 seconds, and the observation frequency to 1 Hz. The feasible region was constructed according to the constraints as follows: Figures 3 to 5 As shown, the actual state is within the feasible region.

[0138] Table 1

[0139]

[0140] Figures 3 to 5 In the diagram, sky blue represents feasible regions, and yellowish-brown represents prohibited regions. It represents the actual state.

[0141] Simulation settings: In the differential evolution algorithm, NP=100, G=200, S=0.9, CR=0.9, observation noise is Gaussian noise, and the standard deviation is... One hundred Monte Carlo simulations were performed. Four heuristic algorithms were compared: the N-method proposed in this invention, the existing DR method, the existing AR method, and the existing aemM method. The core difference lies in the selection of optimization variables. After solving, the candidate solutions were uniformly transformed into position and velocity vectors. If the solution satisfies the following relationship with the true state, the solution is considered successful; otherwise, it fails:

[0142] ,

[0143] In the formula Represents a position or velocity vector. It represents the true state of the corresponding vector. Figure 6 and Figure 7 The success rate curves of each algorithm are shown under noise-free and noisy conditions, respectively. The overall success rate is the minimum of the success rates in the position and velocity vectors. Figure 8 and Figure 9 The root mean square errors of the position and velocity vectors obtained by different algorithms under noisy conditions are shown respectively.

[0144] analyze Figure 6 It can be seen that, under noise-free conditions, the N method proposed in this invention can converge stably in all three scenarios; the AR method is the next best, with a slightly lower success rate than the N method; the DR method has a success rate of about 70%, indicating that the algorithm itself has limitations, namely, it cannot achieve stable convergence under noise-free conditions; the aeM method has a success rate of about 90% under near-circular conditions, but as the eccentricity increases, the success rate of the algorithm drops rapidly to about 10% when it reaches 0.1, and the algorithm completely fails when it reaches 0.4, unable to provide effective orbital information.

[0145] contrast Figure 6 and Figure 7It can be seen that under noisy conditions, the success rate of the N method is still significantly better than other methods. It can still converge stably under near-circular conditions and the success rate is about 70% under elliptical conditions. Moreover, there is no significant difference between the two scenarios with eccentricity of 0.1 and 0.4. The DR method is the second best. Compared with the performance under no noise, the success rate drops significantly, about 70% under near-circular conditions and about 40% under elliptical conditions. The success rate is second only to the N method. The AR method's performance drops rapidly under noisy conditions. The success rate is about 20% under near-circular conditions and almost fails under elliptical scenarios, indicating that the algorithm is very sensitive to noise and has the worst robustness. The aeM method shows strong robustness under near-circular conditions, with the success rate maintained at around 85%. However, due to the characteristics of the algorithm itself, the success rate does not change significantly under elliptical scenarios and still fails at eccentricity of 0.4.

[0146] Figure 8 and Figure 9 The results show the root mean square errors (RMS) of position and velocity vectors. The N method yields approximately 11 km for position and 0.02 km / s for velocity in near-circular conditions, and approximately 72 km and 0.12 km / s for velocity in elliptical conditions. When the eccentricity is less than 0.1, the aeM method achieves high accuracy in solving the position vector (approximately 55 km), but its velocity RMS error is significantly higher than the N method, especially in elliptical scenarios, reaching 0.5 km / s and 1.5 km / s. Therefore, the failure of the aeM method in elliptical scenarios is mainly due to the failure in velocity vector solving. The DR method's RMS for position ranges from 218 km to 525 km, and its RMS for velocity ranges from 0.4 km to 0.79 km. The AR method's RMS for position ranges from 340 km to 750 km, and its RMS for velocity ranges from 0.68 km / s to 1.4 km / s. Figure 8 and Figure 9 In the diagram, sce1, sce2, and sce3 correspond to Sce.1, Sce.2, and Sce.3, respectively. Comparative simulations show that the N-method proposed in this invention significantly outperforms other methods in both accuracy and robustness, demonstrating greater engineering applicability.

[0147] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for initial orbit determination of a non-cooperative target under space-based angle-only short-arc observation, characterized in that include, Using the unit normal vector of the target orbital plane as the variable to be optimized, the relationship between the slant distance and the unit normal vector is obtained based on the geometric relationship between the target position vector, the slant distance between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector. Based on the normalization property of the unit normal vector, the solution of the unit normal vector is transformed into the solution of the X-axis component and the Y-axis component of the unit normal vector, and the constraint conditions for the solution are determined. A heuristic algorithm is used to solve for the X-axis component and Y-axis component of the unit normal vector, thereby determining the unit normal vector and achieving initial trajectory determination of the target.

2. The method according to claim 1, wherein the non-cooperative target initial orbit determination method under the condition of only angular observation of space-based is characterized in that, The geometric relationships between the target position vector, the slant range between the space-based platform and the target position, the target azimuth unit vector, and the space-based platform position vector are as follows: ， wherein is a target position vector, is an oblique distance between the space-based platform and the target position, is a target direction unit vector, is a space-based platform position vector.

3. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 2, characterized in that, The method for obtaining the relationship between the slope distance and the unit normal vector is as follows: Let the unit normal vector and the target position vector be... Dot product: ， In the formula It is the unit normal vector; Then the slope distance for: 。 4. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation as described in claim 3, is characterized in that... Based on the normalization property of the unit normal vector, we obtain: ， In the formula The x-axis component of the unit normal vector. The Y-axis component of the unit normal vector. The Z-axis component of the unit normal vector; ， Therefore, the unit normal vector The solution is transformed into the X-axis component of the unit normal vector. and the unit normal vector Y-axis component Solve for it.

5. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 4, characterized in that, Based on the X-axis component of the unit normal vector and the unit normal vector Y-axis component The sum of the squares is not greater than 1, thus obtaining the X-axis component of the unit normal vector. and the unit normal vector Y-axis component The solution space lies within the unit circle.

6. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 5, characterized in that, For targets in circular or elliptical orbits, the constraints for solving are: ， In the formula It is the semi-nominal diameter. for The target position vector at time t. for The target position vector at time t. for The target position vector at time t. This is the eccentricity rate.

7. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation as described in claim 6, is characterized in that... Methods for solving the X-axis and Y-axis components of the unit normal vector using heuristic algorithms include: Define the population size NP, the maximum number of iterations G, the variable dimension 2, the scaling factor S, and the crossover probability CR for the heuristic algorithm. Represent the population as : ， individual For the variables to be optimized: ， ； In the formula This is the i-th solution for the x-axis component of the unit normal vector. This is the i-th solution of the Y-axis component of the unit normal vector; In each iteration of the solution process, the current population is evaluated based on the scaling factor S and the crossover probability CR. Mutation and crossover are performed to obtain the population for the current iteration. Candidate update values ​​are selected, and individuals with excellent performance are chosen through an evaluation function to determine the population for the current iteration. Update value; During the iterative solution process, when the population When the evaluation function of all variables to be optimized for updating values ​​is 0 or the maximum number of iterations G is reached, the current population is... The updated value serves as the final solution, yielding the X-axis and Y-axis components of the unit normal vector.

8. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 7, characterized in that, Mutation operations include: Before generation [G / 2], the current population... middle The mutation operation is as follows: ， In the formula For the mutated individual, and Let them be three distinct integers in the range [1, NP]. In subsequent evolution, the current population middle The mutation operation is as follows: ， In the formula For the current population The optimal individual in the process.

9. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 8, characterized in that, Crossover operations use binary crossover: ， In the formula This represents the j-th component of the i-th crossover individual, where j equals 1 or 2; After crossover operation , After crossover operation ; It is a random number uniformly distributed in [0,1]. After mutation operation , After mutation operation ; correspond , correspond ; It is a number randomly selected from 1 and 2.

10. The method for initial orbit determination of a non-cooperative target under space-based angular-only extremely short arc observation according to claim 9, characterized in that, Methods for selecting individuals using evaluation functions include: ， In the formula ; for The result of the g-th iteration; In the formula For the evaluation function, the input is the unit normal vector of the orbital plane. : ， In the formula Let be the target's azimuth unit vector at time m, calculated recursively. Let M be the target azimuth unit vector at the m-th moment of actual observation, and M be the number of observations in the extremely short arc.

Images