Dynamic optimal trajectory and inertial factor graph integrated navigation method under elliptic geometric constraint

Abstract

The present application aims at the autonomous navigation demand of low-cost high-rotation ammunition, and provides a factor graph combination navigation method of dynamically optimizing trajectory and inertia under elliptic geometric constraint. The method takes ideal trajectory as a reference, constructs an elliptic geometric constraint relationship, generates multiple candidate trajectories meeting the geometric constraint condition around the ideal trajectory, and establishes a candidate trajectory set. The similarity between the measured inertia data (gyroscope and accelerometer) and the inertia data inversely calculated from each candidate trajectory is evaluated, and the trajectory sequence meeting the threshold requirement is dynamically optimized. The optimized trajectory and the inertia pre-integration result are used as constraint factors to design the factor graph combination navigation scheme of the optimized trajectory and inertia fusion, so as to realize the high-precision autonomous positioning of high-rotation ammunition in the satellite denial scene. The present application solves the technical problems of rapid error accumulation and significant decrease of navigation accuracy in the pure inertia navigation scheme under the satellite denial condition.

Description

Technical Field

[0001] This invention belongs to the field of high-spin munition navigation under satellite denial conditions, specifically involving a factor graph combination navigation method for dynamically optimizing trajectories and inertia under elliptical geometric constraints. Background Technology

[0002] In complex electromagnetic warfare environments, global navigation satellite systems are susceptible to interference or even complete failure. Furthermore, under satellite denial conditions, purely inertial navigation schemes suffer from rapid error accumulation and decreased navigation accuracy. Therefore, research into autonomous navigation technology for high-spin munitions is of great significance. Especially in the absence of external information support, achieving high-precision, autonomous, and robust navigation capabilities is one of the key directions for current guidance technology development.

[0003] However, most existing autonomous navigation methods rely on external observation information or multi-sensor collaboration, or use data-driven error compensation. While these methods have achieved good results on conventional platforms, they still have significant limitations in high-spinning munition environments with high initial velocities (≥800m / s), high rotation (≥20r / s), and high overloads (≥10000g). Therefore, it is necessary to further explore the munition's own motion mechanism and introduce more targeted dynamic constraints to assist inertial navigation. Currently, Lyu et al. introduced a dynamic model to assist inertial navigation updates, significantly reducing errors under satellite signal interruption conditions, indicating that dynamic constraints play an important role in the absence of external observations. However, this method is currently only applied to vehicle navigation, and there is limited research on its application in high-spinning munition navigation. Therefore, it is necessary to propose a high-spinning munition navigation method that considers dynamic constraints. Summary of the Invention

[0004] To address the need for autonomous navigation of low-cost high-spin munitions, this invention proposes a factor graph-based navigation method that dynamically optimizes the trajectory and inertia under elliptical geometric constraints. This method solves the technical challenges of rapid error accumulation and significant decrease in navigation accuracy inherent in pure inertial navigation schemes under satellite denial conditions, thereby improving the positioning accuracy of high-spin munitions.

[0005] The technical solution adopted in this invention is:

[0006] A factor graph-based navigation method for dynamically optimizing trajectory and inertia under elliptical geometric constraints includes the following steps:

[0007] Based on the ideal trajectory, an elliptical geometric constraint equation is constructed, and multiple candidate trajectories that satisfy the geometric constraint conditions are generated around the ideal trajectory.

[0008] The similarity between the measured output data of the inertial device and the inertial data of each candidate ballistic inversion inertial device and the ideal ballistic inertial data is evaluated, and the ballistic sequence that meets the preset threshold requirements is dynamically selected from the candidate ballistics and the ideal ballistics.

[0009] Using the optimized ballistic sequence and inertial pre-integration results as constraint factors, a factor graph combined navigation scheme that integrates the optimized ballistic sequence and inertial data is designed to achieve autonomous positioning of high-spin munitions in satellite denial scenarios.

[0010] Furthermore, the elliptic geometric constraint equations include:

[0011] The equation of the ellipse centered at point A and parallel to the OXY plane, where point A is the position of the ideal trajectory at the current moment;

[0012] The geometric constraint equations of the line containing the major axis of the ellipse;

[0013] The geometric constraint equations of the line containing the minor axis of the ellipse;

[0014] The geometric constraint equations for the straight line passing through the center of the ellipse at a 45° angle;

[0015] The geometric constraint equations for a straight line passing through the center of the ellipse at a direction of 135°.

[0016] Furthermore, the geometric constraint equations for the line containing the major axis of the ellipse are:

[0017] ;

[0018] Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the line containing the major axis on the x-axis and y-axis, respectively. For the major half-axis, It is the short half-axis.

[0019] Furthermore, the geometric constraint equations for the line containing the minor axis of the ellipse are:

[0020] ;

[0021] Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the line containing the minor axis on the x-axis and y-axis, respectively. For the major half-axis, It is the short half-axis.

[0022] Furthermore, the geometric constraint equations for the straight line passing through the center of the ellipse at a 45° angle are:

[0023] ;

[0024] Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the x-axis and y-axis along a straight line passing through the center of the ellipse at a 45° angle. For the major half-axis, It is the short half-axis.

[0025] Furthermore, the geometric constraint equations for the line passing through the center of the ellipse at a 135° angle are as follows:

[0026] ;

[0027] Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the x-axis and y-axis along a straight line passing through the center of the ellipse at a 135° angle. For the major half-axis, It is the short half-axis.

[0028] Furthermore, the inertial data of the candidate ballistic inversion inertial devices include:

[0029] The output of the gyroscope in the candidate ballistic inversion inertial device is:

[0030] ;

[0031] in, , , Let x represent the pitch angle, yaw angle, and roll angle of the candidate trajectory at time k, respectively. Sampling time, , These are the angular velocities in the x and z directions, respectively; the angular velocity in the y direction is consistent with the ideal trajectory. The yaw angle and pitch angle are:

[0032] ;

[0033] in, , , Representing the velocities of the candidate trajectory at time k, respectively. Velocity components along the x, y, and z axes , Let be the position coordinates of the i-th candidate trajectory at time k, i = 1, 2, 3, ..., N, where N is the number of candidate trajectories generated around the ideal trajectory that satisfy the geometric constraints.

[0034] The roll angle is consistent with the ideal trajectory;

[0035] The output of the candidate ballistic inversion inertial device accelerometer is:

[0036] ;

[0037] in, Let be the attitude transformation matrix from the navigation coordinate system to the vehicle coordinate system at time k. This is the projection of the Earth's rotational angular velocity onto the navigation coordinate system. This is the projection of the angular velocity of the navigation system relative to the Earth system onto the navigation coordinate system. This is the projection of the gravitational acceleration vector onto the navigation coordinate system.

[0038] Furthermore, the similarity between the measured inertial device output data and the inertial data of each candidate ballistic inversion inertial device and the ideal ballistic inertial data is evaluated. Ballistic sequences that meet preset threshold requirements are dynamically selected from the candidate and ideal trajectories. Specifically, this includes:

[0039] Using the inertial data of the measured trajectory from inertial devices as the matching benchmark, a local sliding window similarity measure is performed on the inertial data of each candidate trajectory and the ideal trajectory. The average matching error, angular velocity error, and specific force error of each candidate trajectory and the ideal trajectory within the sliding window are calculated, and then fused through a weighted method to form a comprehensive error index, namely, similarity.

[0040] ;

[0041] In the formula, Let be the angular velocity of the m-th trajectory in the x and z directions at time t. Let be the measured angular velocity vectors of the ballistic trajectory in the x and z directions at time t; Let be the triaxial specific force of the m-th trajectory at time t. The triaxial specific force of the measured trajectory at time t; These are the weighting coefficients. Let be the angular velocity Euclidean distance of the m-th trajectory at time t after the moving average. Let be the Euclidean distance of the m-th trajectory at time t after a moving average; m = 1, 2, 3, ..., N+1, where N+1 represents the total number of ideal and candidate trajectories. It is a 2-norm. This is a comprehensive error index;

[0042] After normalizing the comprehensive error index, it is compared with the preset threshold, and the optimal ballistic sequence that meets the preset threshold condition is obtained at each time step.

[0043] Furthermore, using the optimized ballistic sequence and inertial pre-integration results as constraint factors, a factor graph-based combined navigation scheme integrating the optimized ballistic sequence and inertial data is designed, specifically including:

[0044] The position, velocity, and inertial pre-integration of the preferred ballistic sequence are selected as constraint factors to construct a factor graph. The plug-and-play nature of the factor graph is fully utilized. Then, the Gauss-Newton method is used to optimize the factor graph. Finally, the factor graph is marginalized to obtain the factor graph combined navigation scheme.

[0045] A factor graph-based navigation system for dynamically optimizing ballistics and inertia under elliptical geometric constraints includes:

[0046] The candidate trajectory set construction unit, based on the ideal trajectory, constructs an elliptical geometric constraint equation and generates multiple candidate trajectories that satisfy the geometric constraint conditions around the ideal trajectory;

[0047] The candidate trajectory dynamic optimization unit is used to evaluate the similarity between the output data of the measured inertial device and the inertial data of each candidate trajectory inversion inertial device and the ideal trajectory inertial data, and dynamically optimize the trajectory sequence that meets the preset threshold requirements from the candidate trajectories and the ideal trajectory.

[0048] The autonomous positioning unit uses the optimized ballistic sequence and inertial pre-integration results as constraint factors to design a factor graph combination navigation scheme that integrates the optimized ballistic sequence and inertial data, thereby achieving autonomous positioning of high-spin munitions in satellite denial scenarios.

[0049] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention generates multiple candidate trajectories through elliptical geometric constraints, breaking through the adaptability limitations of traditional fixed trajectory models; it dynamically optimizes the trajectory sequence based on inertial data similarity, realizing accurate matching between constraint information and actual flight state; it designs a factor graph navigation architecture that fuses optimized trajectories with inertial pre-integration, effectively suppressing the error accumulation of pure inertial navigation in satellite denial scenarios, and significantly improving the autonomous navigation and positioning accuracy of high-dynamic high-spin munitions. Attached Figure Description

[0050] Figure 1 This is a diagram of the factor graph combination navigation framework for dynamically optimizing ballistics and inertia under elliptical geometric constraints proposed in this invention.

[0051] Figure 2 This is a schematic diagram of a ballistic dynamic optimization method based on elliptical geometric constraints.

[0052] Figure 3 A schematic diagram of the construction of a factor map for optimizing ballistics and inertia. Detailed Implementation

[0053] The implementation of the present invention will now be described in detail with reference to the accompanying drawings.

[0054] Combination Figure 1 A factor graph-based navigation method for dynamically optimizing trajectories and inertia under elliptical geometric constraints, comprising:

[0055] Based on the ideal trajectory, an elliptical geometric constraint relationship is constructed, and multiple candidate trajectories that satisfy the geometric constraint conditions are generated around the ideal trajectory to form a candidate trajectory set.

[0056] The similarity between the output data of the measured inertial devices (gyroscopes, accelerometers) and the inertial data inverted from each candidate trajectory is evaluated, and the trajectory sequence that meets the preset threshold requirements is dynamically selected.

[0057] Using the optimized trajectory and inertial pre-integration results as constraint factors, a factor graph combined navigation scheme integrating the optimized trajectory and inertial data is designed to achieve high-precision autonomous positioning of high-spin munitions in satellite denial scenarios. The method of this invention will be described in detail below.

[0058] Figure 2 The image shows eight candidate trajectories generated under elliptical geometric constraints. Using the ideal trajectory as a reference, an elliptical geometric constraint is constructed centered at the coordinate point of each time step. This ellipse is parallel to the OXY plane, meaning the generated candidate trajectories are at the same height as the ideal trajectory. Assuming the distance between the measured trajectory and the ideal trajectory diverges as a function of time, the shape of the ellipse dynamically changes over time, exhibiting a simple linear function relationship as shown below:

[0059] (1)

[0060] In the formula, Let k be the semi-major axis of the ellipse at time k. The semi-major axis of the ellipse corresponding to the landing point. Let k be the minor semi-axis of the ellipse at time k. Let be the minor semi-axis of the ellipse corresponding to the impact point, and T be the flight time of the ideal trajectory. Since the range generally varies more than the sideslip, the axis corresponding to the range is set as the major axis when constructing the ellipse.

[0061] Taking a certain time k as an example, a schematic diagram of obtaining candidate trajectories using the elliptic geometric constraint concept is attached. Figure 2 Using the ideal trajectory as a reference, candidate trajectories are generated in its vicinity. Point A is the current position of the ideal trajectory, with coordinates A(x1, y1, z1). Geometric constraints include: 1) an ellipse centered at point A, parallel to the OXY plane, with its major semi-axis being... The minor semi-axis is ; 2) Constraints on the line containing the major axis of the ellipse; 3) Constraints on the line containing the minor axis of the ellipse; 4) Constraints on the line passing through the center of the ellipse at a 45° angle; 5) Constraints on the line passing through the center of the ellipse at a 135° angle. These geometric constraints constitute the coordinate points of the candidate trajectories, generating 8 candidate trajectories.

[0062] With point A as the center of the ellipse, parallel to the OXY plane, the semi-major axis of the ellipse at time k is... The minor semi-axis is Then the equation of the ellipse is:

[0063] (2)

[0064] The line containing the major axis of the ellipse is:

[0065] (3)

[0066] Given that the altitude is consistent with the ideal trajectory, combining equations (2) and (7), the geometric constraint equations of the ellipse and the line containing its major axis are:

[0067] (4)

[0068] Solving equation (4) yields the coordinates of the intersection point of the constructed ellipse and the line containing the major axis. Since the height is limited to z = z1, the three-dimensional coordinates of the intersection point are:

[0069] (5)

[0070] The geometric constraint equations for the ellipse and the line containing its minor axis are:

[0071] (6)

[0072] Solving equation (6) yields the coordinates of the intersection point of the constructed ellipse and the line containing the minor axis. Since the height is limited to z = z1, the three-dimensional coordinates of the intersection point are:

[0073] (7)

[0074] The geometric constraint equations between the ellipse and the line passing through the center of the ellipse at a 45° angle are:

[0075] (8)

[0076] Solving equation (8) yields the coordinates of the intersection point of the constructed ellipse and the straight line passing through the center of the ellipse at a 45° angle. Since the height is limited to z = z1, the three-dimensional coordinates of the intersection point are:

[0077] (9)

[0078] The geometric constraint equations between the ellipse and the line passing through the center of the ellipse at a 135° angle are:

[0079] (10)

[0080] Solving equation (10) yields the coordinates of the intersection point of the constructed ellipse and the line passing through the center of the ellipse at a 135° angle. Since the height is limited to z = z1, the three-dimensional coordinates of the intersection point are:

[0081] (11)

[0082] Based on equations (4) to (11), eight candidate trajectories can be obtained. The coordinates of these eight trajectories P1 to P8 are as follows: , , , , , , , .

[0083] After obtaining 8 candidate trajectories, the velocity can be calculated as follows:

[0084] (12)

[0085] In the formula, Sampling time, Let be the position coordinates of the i-th trajectory at time k, where i = 1, 2, 3, 4, 5, 6, 7, 8.

[0086] Yaw angle of the candidate trajectory at time k Pitch angle for:

[0087] (13)

[0088] In the formula, , , They represent Based on Euler's equations, the angular velocity components along the x, y, and z axes can be obtained from the gyroscope output as follows:

[0089] (14)

[0090] in, , , Let x represent the pitch angle, yaw angle, and roll angle of the candidate trajectory at time k, respectively. For the sampling time, the angular velocity in the y-direction is consistent with the ideal trajectory; the roll angle and roll rate of the 8 candidate trajectories are consistent with the ideal trajectory.

[0091] Substituting equations (12) and (13) into the force ratio equation, we can obtain the accelerometer output result as follows:

[0092] (15)

[0093] In the formula, Let be the attitude transformation matrix from the navigation coordinate system to the vehicle coordinate system at time k. This is the projection of the Earth's rotational angular velocity onto the navigation coordinate system. This is the projection of the angular velocity of the navigation system relative to the Earth system onto the navigation coordinate system. The projection of the gravitational acceleration vector in the navigation coordinate system.

[0094] The results of the gyroscope and accelerometer outputs can be obtained from the trajectory according to equations (12) to (15).

[0095] A candidate trajectory inertial database is formed by inverting the ideal trajectory with the inertial data obtained from the generated candidate trajectories. During the trajectory optimization process, the inertial data of the measured trajectory is used as the matching benchmark. A local sliding window similarity measurement is performed on the inertial data of each candidate trajectory, and the differences between trajectories are quantified by calculating the Euclidean distance. For angular velocity, only the components in the x and z directions are selected, while for specific force, all three-axis components are selected. Considering the continuity and correlation of inertial data over time, a moving average of the obtained Euclidean distance is performed using a sliding window to obtain the overall matching error of the trajectory within a local time period. The average matching error, angular velocity error, and specific force error of each candidate trajectory within this window are calculated and fused using a weighted method to form a comprehensive error index.

[0096] (16)

[0097] In the formula, Let be the angular velocities of the m-th candidate trajectory in the x and z directions at time t. Let be the measured angular velocity vectors of the ballistic trajectory in the x and z directions at time t; Let m be the triaxial specific force of the m-th candidate trajectory at time t. The triaxial specific force of the measured trajectory at time t; These are weighting coefficients used to balance the influence of different inertial characteristics on the optimization results; Let be the angular velocity Euclidean distance of the m-th candidate trajectory at time t after the moving average. Let be the ratio of the candidate trajectory at time t to the Euclidean distance after moving average, where m = 1, 2, 3, 4, 5, 6, 7, 8, 9. The norm is 2. The candidate trajectories consist of nine trajectories, including the ideal trajectory, and the sequence of trajectories that meets the preset threshold requirements is dynamically selected.

[0098] In the optimization process, only candidate trajectories with normalized errors below a threshold are selected as valid trajectories at the current moment, with the threshold value ranging from (0,1). If no trajectory meets the threshold condition, the trajectory with the smallest error is selected as the candidate result.

[0099] When designing a factor graph-based navigation scheme that integrates optimized ballistics and inertial navigation, the position and velocity of the optimized ballistic sequence, along with inertial pre-integration, are selected as constraint factors to construct the factor graph. This fully utilizes the "plug-and-play" nature of factor graphs to constrain state nodes. Then, the Gauss-Newton method is used for factor graph optimization. Finally, factor graph marginalization is performed to ensure real-time performance and computational efficiency. See the appendix for details on the factor graph construction. Figure 3 It should be noted that inertial pre-integration is a well-known term in this field, and the specific process of factor graph optimization using the Gauss-Newton method and finally factor graph marginalization is also well-known in this field and will not be elaborated here.

[0100] This invention also provides a factor graph combined navigation system for dynamically optimizing ballistics and inertia under elliptical geometric constraints, comprising:

[0101] The candidate trajectory set construction unit, based on the ideal trajectory, constructs an elliptical geometric constraint equation and generates multiple candidate trajectories that satisfy the geometric constraint conditions around the ideal trajectory;

[0102] The candidate trajectory dynamic optimization unit is used to evaluate the similarity between the output data of the measured inertial device and the inertial data of each candidate trajectory inversion inertial device and the ideal trajectory inertial data, and dynamically optimize the trajectory sequence that meets the preset threshold requirements from the candidate trajectories and the ideal trajectory.

[0103] The autonomous positioning unit uses the optimized ballistic sequence and inertial pre-integration results as constraint factors to design a factor graph combination navigation scheme that integrates the optimized ballistic sequence and inertial data, thereby achieving autonomous positioning of high-spin munitions in satellite denial scenarios.

[0104] The present invention was verified by simulation experiments, and the experimental results are shown in Table 1.

[0105] Table 1 Comparison of Positioning Accuracy in Experimental Results

[0106] INS 7705.829 166.345 Ideal / INS 49.639 133.077 Optimal / INS 52.07 125.348 All / INS 66.827 76.511 DS / INS (This invention) 36.081 44.451

[0107] Experimental results show that the DS / INS (Dynamic Optimization of Trajectory and Inertia Combined Navigation Method) proposed in this invention, under elliptical geometric constraints, can effectively improve the positioning accuracy of high-spin munitions. Compared with traditional methods, the positioning accuracy of this invention can be improved by 27.3%, significantly reducing positioning errors. It can better adapt to the positioning requirements under high-spin dynamic conditions and has excellent engineering application value.

[0108] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A factor graph-based navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints, characterized in that, include: Based on the ideal trajectory, an elliptical geometric constraint equation is constructed, and multiple candidate trajectories that satisfy the geometric constraint conditions are generated around the ideal trajectory. The similarity between the measured output data of the inertial device and the inertial data of each candidate ballistic inversion inertial device and the ideal ballistic inertial data is evaluated, and the ballistic sequence that meets the preset threshold requirements is dynamically selected from the candidate ballistics and the ideal ballistics. Using the optimized ballistic sequence and inertial pre-integration results as constraint factors, a factor graph combined navigation scheme that integrates the optimized ballistic sequence and inertial fusion is designed to achieve autonomous positioning of high-spin munitions in satellite denial scenarios. The similarity between the measured inertial device output data and the inertial data of each candidate ballistic inversion inertial device and the ideal ballistic inertial data is evaluated. Ballistic sequences that meet preset threshold requirements are dynamically selected from the candidate and ideal trajectories. Specifically, this includes: Using the inertial data of the measured trajectory from inertial devices as the matching benchmark, a local sliding window similarity measure is performed on the inertial data of each candidate trajectory and the ideal trajectory. The average matching error, angular velocity error, and specific force error of each candidate trajectory and the ideal trajectory within the sliding window are calculated, and then fused through a weighted method to form a comprehensive error index, namely, similarity. ； In the formula, Let be the angular velocity of the m-th trajectory in the x and z directions at time t. Let be the measured angular velocity vectors of the ballistic trajectory in the x and z directions at time t; Let be the triaxial specific force of the m-th trajectory at time t. The triaxial specific force of the measured trajectory at time t; These are the weighting coefficients. Let be the angular velocity Euclidean distance of the m-th trajectory at time t after the moving average. Let be the Euclidean distance of the m-th trajectory at time t after a moving average; m = 1, 2, 3, ..., N+1, where N+1 represents the total number of ideal and candidate trajectories. It is a norm 2. This is a comprehensive error index; After normalizing the comprehensive error index, it is compared with the preset threshold, and the optimal ballistic sequence that meets the preset threshold condition is obtained at each time step.

2. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 1, characterized in that, The elliptic geometric constraint equations include: The equation of the ellipse centered at point A and parallel to the OXY plane, where point A is the position of the ideal trajectory at the current moment; The geometric constraint equations of the line containing the major axis of the ellipse; The geometric constraint equations of the line containing the minor axis of the ellipse; The geometric constraint equations for the straight line passing through the center of the ellipse at a 45° angle; The geometric constraint equations for a straight line passing through the center of the ellipse at a direction of 135°.

3. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 2, characterized in that, The geometric constraint equations for the line containing the major axis of the ellipse are: ； Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the line containing the major axis on the x-axis and y-axis, respectively. For the major half-axis, It is the short half-axis.

4. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 2, characterized in that, The geometric constraint equations for the line containing the minor axis of the ellipse are: ； Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the line containing the minor axis on the x-axis and y-axis, respectively. For the major half-axis, It is the short half-axis.

5. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 2, characterized in that, The geometric constraint equations for the line passing through the center of the ellipse at a 45° angle are: ； Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the x-axis and y-axis along a straight line passing through the center of the ellipse at a 45° angle. For the major half-axis, It is the short half-axis.

6. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 2, characterized in that, The geometric constraint equations for the line passing through the center of the ellipse at a direction of 135° are: ； Where x1 and y1 are the coordinates of point A on the x-axis and y-axis, respectively, and x and y are the coordinates of a point on the x-axis and y-axis along a straight line passing through the center of the ellipse at a 135° angle. For the major half-axis, It is the short half-axis.

7. The factor graph combination navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 1, characterized in that, The inertial data for candidate ballistic inversion inertial devices include: The output of the gyroscope in the candidate ballistic inversion inertial device is: ； in, , , Let x represent the pitch angle, yaw angle, and roll angle of the candidate trajectory at time k, respectively. Sampling time, , These are the angular velocities in the x and z directions, respectively; the angular velocity in the y direction is consistent with the ideal trajectory. The yaw angle and pitch angle are: ； in, , , Representing the velocities of the candidate trajectory at time k, respectively. Velocity components along the x, y, and z axes , Let be the position coordinates of the i-th candidate trajectory at time k, i = 1, 2, 3, ..., N, where N is the number of candidate trajectories generated around the ideal trajectory that satisfy the geometric constraints. The roll angle is consistent with the ideal trajectory; The output of the candidate ballistic inversion inertial device accelerometer is: ； in, Let be the attitude transformation matrix from the navigation coordinate system to the vehicle coordinate system at time k. This is the projection of the Earth's rotational angular velocity onto the navigation coordinate system. This is the projection of the angular velocity of the navigation system relative to the Earth system onto the navigation coordinate system. This is the projection of the gravitational acceleration vector onto the navigation coordinate system.

8. The factor graph combined navigation method for dynamically optimizing ballistics and inertia under elliptical geometric constraints according to claim 1, characterized in that, Using the optimized ballistic sequence and inertial pre-integration results as constraint factors, a factor graph-based combined navigation scheme integrating the optimized ballistic sequence and inertial fusion is designed, specifically including: The position, velocity, and inertial pre-integration of the preferred ballistic sequence are selected as constraint factors to construct a factor graph. The plug-and-play nature of the factor graph is fully utilized. Then, the Gauss-Newton method is used to optimize the factor graph. Finally, the factor graph is marginalized to obtain the factor graph combined navigation scheme.

9. A factor graph combined navigation system for dynamically optimizing ballistics and inertia under elliptical geometric constraints, implementing the factor graph combined navigation method of any one of claims 1-8, characterized in that, include: The candidate trajectory set construction unit, based on the ideal trajectory, constructs an elliptical geometric constraint equation and generates multiple candidate trajectories that satisfy the geometric constraint conditions around the ideal trajectory; The candidate trajectory dynamic optimization unit is used to evaluate the similarity between the output data of the measured inertial device and the inertial data of each candidate trajectory inversion inertial device and the ideal trajectory inertial data, and dynamically optimize the trajectory sequence that meets the preset threshold requirements from the candidate trajectories and the ideal trajectory. The autonomous positioning unit uses the optimized ballistic sequence and inertial pre-integration results as constraint factors to design a factor graph combination navigation scheme that integrates the optimized ballistic sequence and inertial data, thereby achieving autonomous positioning of high-spin munitions in satellite denial scenarios.

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