A zero-range line-of-sight multiple-maneuver penetration method

Abstract

The application discloses a zero-range line direction multi-time maneuver penetration method, which comprises the following steps: taking the maneuver times, maneuver time and maneuver direction of a penetration warhead as optimization design variables, taking the zero-control miss distance, shift point deviation and fuel consumption as optimization indexes, and establishing a penetration trajectory optimization problem description model; designing the initialization stage, search stage and inertia weight of a PSO algorithm based on the optimization design variables of the maneuver times and maneuver time and the dimension uncertainty problem; converting the penetration trajectory optimization problem description model to a non-orthogonal coordinate system, deriving a free flight trajectory analytical solution, and combining the analytical method with the numerical method to quickly solve the zero-range line direction on the basis; and using the designed PSO algorithm to take the solved zero-range line direction as the maneuver direction to optimize the penetration trajectory of the multi-time maneuver. The application reduces the dimension of the optimization design variables, improves the penetration trajectory optimization efficiency, and can obtain the expected effect in a short time.

Description

Technical Field

[0001] The technical field of this invention is missile penetration, and in particular, it relates to a method for multiple maneuvering penetration along the zero-range line. Background Technology

[0002] Relevant data indicates that missile defense systems have achieved a considerably high success rate in intercepting traditional ballistic missiles without mid-course maneuvering capabilities. In this context, maneuvering during the mid-course phase to evade missile defense interception is crucial for improving the survivability of ballistic missiles in the mid-course phase. To enable missiles to achieve their mission of penetrating enemy interception, their trajectory design needs to be optimized. The essence of the ballistic missile mid-course penetration trajectory optimization problem is an optimal control problem. The basic approach to solving this type of problem is to design appropriate performance indicators and constraints, and then use optimization algorithms to solve for the control variables. However, existing mainstream trajectory optimization methods rely on continuous and differentiable system models and performance indicators. The discontinuous nature of the system model caused by the multiple mid-course maneuvering requirements of ballistic missiles makes mainstream trajectory optimization methods difficult to apply.

[0003] Theoretically, applying a velocity increment along the zero-range line to a missile does not affect its final impact point. If this characteristic is applied to maneuver penetration trajectory optimization, using the zero-range line as the maneuver direction and only the timing of the maneuver as the optimization design variable, a zero-range line maneuver penetration trajectory optimization method can theoretically reduce the number of trajectory optimization design variables, improve the efficiency of penetration trajectory optimization, and reduce fuel consumption when the trajectory returns to the original target point after maneuver penetration. Currently, methods for solving the zero-range line can be divided into analytical and numerical methods. Analytical methods for solving the zero-range line direction are simple and computationally efficient, but they do not consider the effects of Earth's rotation and gravitational perturbations. Furthermore, the analytical formula is mainly derived based on first-order Taylor expansions, omitting higher-order derivative terms, resulting in truncation errors. Therefore, its accuracy is not high and cannot meet the accuracy requirements of ballistic missiles. Numerical methods are based on the idea of ​​ballistic iteration. They calculate the zero-range line direction by numerically integrating the trajectory of a ballistic missile maneuvering in different directions. However, this method requires a large number of numerical integration ballistic calculations, which is time-consuming and inefficient, making it difficult to perform online calculations on the missile. Summary of the Invention

[0004] The purpose of this invention is to provide a method for multiple maneuvering penetration in the zero-range direction, which avoids the requirement of continuous and differentiable system models and performance indicators in traditional ballistic optimization methods. At the same time, by using the zero-range direction as the maneuvering penetration direction, the dimensionality of optimization design variables is reduced, the efficiency of penetration trajectory optimization is improved, and the desired effect can be obtained in a shorter time.

[0005] The technical solution to achieve the objective of this invention is: a method for multiple maneuvering penetration in the zero-range direction, comprising the following steps:

[0006] Step 1, establish a description model for the ballistic missile penetration trajectory optimization problem: Based on the dynamics and kinematics models of ballistic missiles and interceptor missiles, the number of maneuvers, timing of maneuvers, and direction of maneuvers of the penetrating warhead are used as optimization design variables, and the magnitude of zero-control miss distance, shift handover point deviation, and fuel consumption are used as optimization indicators to establish a description model for the ballistic missile penetration trajectory optimization problem.

[0007] Step 2, PSO Algorithm Design: Based on the multiple optimization design variables such as the number of maneuvers and maneuver time in Step 1, and the problem of uncertain dimensions, the initialization phase, search phase, and inertia weight of the PSO algorithm are designed.

[0008] Step 3, rapid solution for zero-range line direction; transform the description model of the penetration ballistic optimization problem into a non-orthogonal coordinate system, derive the analytical solution of the free flight trajectory, and on this basis, combine the analytical method and the numerical method to rapidly solve the zero-range line direction;

[0009] Step 4, optimization of multiple maneuver penetrations in the zero-range direction: Using the PSO algorithm designed in Step 2, the zero-range direction obtained in Step 3 is used as the maneuver direction to optimize the trajectory of multiple maneuver penetrations and obtain the optimal multiple maneuver penetration method.

[0010] Compared with existing technologies, the advantages of this invention are as follows: This invention has better performance index convergence speed and optimization efficiency, enabling successful penetration of the penetrating warhead. At the same time, the position deviation of the handover point caused by maneuvering penetration is small, which will not put ballistic correction pressure on the terminal guidance precision strike in the reentry phase. It can effectively ensure strike accuracy while penetrating the ballistic trajectory. Comparing the rapid solution of ballistic trajectory based on multiple maneuvering penetrations in the zero-range direction proposed in this invention with the solution of ballistic trajectory without using the zero-range direction as the maneuvering penetration direction, simulation results show that the rapid solution of ballistic trajectory based on multiple maneuvering penetrations in the zero-range direction proposed in this invention has better performance index convergence speed and optimization efficiency. Attached Figure Description

[0011] Figure 1 This is a geometric diagram of the non-orthogonal absolute coordinate system in the implementation method.

[0012] Figure 2 This is a schematic diagram of the geometric relationship of the elliptical orbit in the implementation method.

[0013] Figure 3 This is a graph showing the variation of longitudinal deviation (m) with azimuth angle in the implementation method.

[0014] Figure 4 This is a graph showing the variation of the transverse deviation (m / s) with the azimuth angle in the implementation method.

[0015] Figure 5 This is a ballistic simulation curve diagram used in the implementation method. Figure 5 (a) in the diagram is the full trajectory diagram. Figure 5 (b) in the image is a magnified view of the ballistic trajectory.

[0016] Figure 6 This is a graph showing the change of latitude over time in the implementation method.

[0017] Figure 7 This is a graph showing the change of longitude over time in the implementation method.

[0018] Figure 8 This is a graph showing the change in height over time in the implementation method.

[0019] Figure 9 This is a vector diagram of the maneuvering direction unit in the implementation method.

[0020] Figure 10 This is a graph showing the changes in performance index values ​​during the implementation of the method.

[0021] Figure 11 This is a graph showing the changes in the performance index values ​​of ballistic optimization during 500 trials in the implementation method.

[0022] Figure 12 This is a graph showing the statistical results of the shift handover point deviation in the implementation method.

[0023] Figure 13 This is a graph showing the statistical results of zero-control miss distance in the implementation method.

[0024] Figure 14 This is a graph showing the total maneuver time statistics in the implementation method.

[0025] Figure 15 This is a graph showing the final performance index values ​​in the implementation method.

[0026] Figure 16 This is a distribution diagram of the final performance index values ​​and zero-control miss distance in the implementation method.

[0027] Figure 17 This is a distribution diagram showing the deviation between the final performance index value and the handover point in the implementation method.

[0028] Figure 18 This is a distribution diagram of the final performance index values ​​and total maneuver time in the implementation method.

[0029] Figure 19 This is a graph showing the changes in performance index values ​​during the implementation of the method.

[0030] Figure 20This is a flowchart illustrating the rapid solution of ballistic trajectories for multiple maneuvers and penetrations based on the zero-range direction in the implementation method. Detailed Implementation

[0031] This invention provides a solution for solving the trajectory of missiles with multiple maneuvers for penetration. First, it introduces the PSO optimization algorithm to address the problem that traditional trajectory optimization algorithms rely on continuous and differentiable system models and performance indicators. The PSO algorithm optimizes the number of maneuvers and the time of the missile's penetration. Then, by using the zero-range line direction as the penetration direction, it reduces the design variables for penetration trajectory optimization, improves the efficiency of maneuvering penetration trajectory optimization, and achieves accurate and rapid solution for maneuvering penetration trajectories.

[0032] Based on the dynamics and kinematics model of a ballistic missile penetration warhead, this paper mathematically describes the trajectory optimization problem by using the number of maneuvers, maneuver time, and maneuver direction of the penetration warhead as optimization design variables, and zero-control miss distance, shift point deviation, and fuel consumption as optimization indicators. Addressing the discontinuity problem in the missile dynamics system caused by the flexible on / off liquid orbital control engine of the penetration warhead, the paper optimizes the penetration trajectory based on the PSO algorithm. Furthermore, considering the multiple optimization design variables (number of maneuvers, maneuver time, and maneuver direction) and their uncertain dimensions, corresponding improvements are made to the initialization phase, search phase, and inertial weight design of the PSO algorithm.

[0033] By combining analytical and numerical methods, a semi-analytical zero-range solution method is constructed, improving computational efficiency while meeting accuracy requirements. First, a non-orthogonal absolute coordinate system is established. The Earth's gravitational force, including the J2 term, is decomposed within this non-orthogonal coordinate system, deriving the missile's differential equations of motion in this system. From this, an approximate analytical solution for the free-flight trajectory is further derived. Then, based on the analytical solution of the trajectory, a fast method for solving the zero-range line is proposed. This method iteratively solves for the partial derivatives of the longitudinal and transverse strokes with respect to the velocity increment using the analytical equations of the trajectory, thereby obtaining the zero-range line direction vector. This improves computational accuracy while maintaining computational speed.

[0034] Therefore, this invention mainly addresses the problem of solving the ballistic missile penetration trajectory problem involving multiple maneuvers, proposing a rapid solution method based on multiple maneuvers along the zero-range line. Traditional penetration trajectory optimization requires the system model and performance indicators to be continuous and differentiable. However, using intelligent optimization algorithms to optimize the number of maneuvers, maneuver time, and maneuver direction results in too many optimization design variables, leading to low optimization efficiency. Compared to the above methods, this invention proposes using the zero-range line as the maneuver penetration direction and optimizing the number of maneuvers and maneuver time using an improved PSO algorithm. This avoids the requirement of continuous and differentiable system model and performance indicators in traditional trajectory optimization methods. Furthermore, using the zero-range line as the maneuver penetration direction reduces the dimensionality of optimization design variables, improves the efficiency of penetration trajectory optimization, and can achieve the desired results in a shorter time.

[0035] Based on the above ideas, this implementation proposes a method for multiple maneuvering penetration along the zero-range line, combined with... Figure 20 Specifically, it includes:

[0036] Step 1: Based on the dynamics and kinematics models of ballistic missiles and interceptor missiles, the number of maneuvers, timing of maneuvers, and direction of maneuvers of the penetrating warhead are used as optimization design variables, and the magnitude of zero-control miss distance, shift handover point deviation, and fuel consumption are used as optimization indicators to complete the mathematical description of the ballistic optimization problem.

[0037] Step 2, Improved design of PSO algorithm: Optimization of penetration trajectory based on PSO algorithm: For the penetration trajectory optimization problem description model established in Step 1, the idea of ​​physical planning method is adopted. The preference function is used to reflect the designer's preference for the target weights, and the multi-objective optimization problem is transformed into a single-objective optimization problem.

[0038] Step 3, rapid solution for zero-range line direction: Considering the gravitational perturbation effect caused by the Earth's J2 term, a non-orthogonal absolute coordinate system is established. The Earth's gravity, including the J2 term, is decomposed in the non-orthogonal coordinate system. The differential equation of motion of the missile in the non-orthogonal coordinate system is established, and the approximate analytical solution of the free flight trajectory is derived. Based on the analytical solution of the trajectory, a rapid solution for the zero-range line is proposed. It is only necessary to calculate the longitudinal and transverse deviations caused by the velocity increments in the three directions through the analytical solution of the trajectory to obtain the partial derivatives of the longitudinal and transverse deviations with respect to the velocity increments, and then obtain the zero-range line direction vector.

[0039] Step 4, Optimization of multiple maneuvers for penetration in the zero-range direction: In view of the limitations of the basic PSO algorithm, and guided by the requirement of multi-dimensional uncertainty of the number of maneuvers and maneuver time in the penetration trajectory optimization problem, the PSO algorithm was improved in a targeted manner, and the penetration trajectory optimization was solved by using the zero-range line as the maneuver trajectory change direction.

[0040] Furthermore, the establishment of the description model for the penetration ballistic optimization problem specifically includes:

[0041] 1: Establish an optimization design model

[0042] To achieve multiple mid-course maneuvers and penetrations, it is first necessary to define the mathematical model of the penetration warhead.

[0043]

[0044] Where g is the gravitational acceleration experienced by the missile, G I With G B Here, are the direction cosine matrices between the launch inertial coordinate system and the launch coordinate system, and between the missile body coordinate system and the launch coordinate system, respectively; m is the current mass of the missile; R is the aerodynamic force acting on the missile (since the mid-course trajectory is outside the atmosphere, the aerodynamic force is 0); T is the engine thrust; and V... I Let r be the velocity vector of the missile in the launch inertial coordinate system. I Let I be the geocentric radius vector of the missile in the launch inertial coordinate system. sp ρ is the specific impulse of the engine propellant, and g0 is the gravitational acceleration at sea level.

[0045] Through mathematical modeling, the flight state variables of the penetrating warhead can be clearly defined, including three velocity components, three position components, and mass in the launch inertial coordinate system, which can be expressed as x = [V I ,r I [m] T Its state variables can be obtained by solving a system of differential equations.

[0046] 2: Optimization of target analysis and performance index design

[0047] In actual mid-course multiple-maneuver penetration trajectory optimization, the most important aspect is to deviate as much as possible from the original trajectory, increasing the zero-control miss distance to ensure successful penetration. However, on the other hand, excessive ballistic maneuvering, while ensuring successful penetration, may cause excessive deviation in the handover point position. If this deviation exceeds the maximum correction capability of the reentry trajectory, the missile will be unable to complete its mission of striking the target. In addition, it is desirable to minimize the fuel consumption of the penetration warhead to ensure that after successfully evading the interceptor, the warhead still has sufficient fuel to deal with potential subsequent interception threats. Therefore, based on the above comprehensive considerations, the following optimization objectives are proposed: ① Maximum zero-control miss distance; ② Minimum handover point position deviation; ③ Minimum maneuvering fuel consumption. Therefore, its performance indicators are designed as follows:

[0048]

[0049] Where w1, w2, and w3 are the weight coefficients for each objective; R miss0Zero-control miss distance; d r denoted as the landing position deviation; p1(), p2(), and p3() are the zero-control miss distance, total engine operating time, and landing position deviation preference functions, respectively.

[0050] 3: Optimize design variables

[0051] For the ballistic optimization problem of multiple maneuvers of a penetration warhead using a liquid-fueled orbital control engine with flexible switching, the optimization design variables include: maneuver timing (time of each engine start and stop) and maneuver direction, which can be expressed in the following form:

[0052]

[0053] Where i∈{1,2,…,n}; n is the total number of maneuvers (number of times the engine is started and stopped); This refers to the start time of each maneuver (engine start-up time); This refers to the end time of each maneuver (engine shutdown time); u i (t) represents the direction vector of each maneuver (unit vector u in the inertial coordinate system). i (t)=[u x (t),u y (t),u z (t)] T ).

[0054] Since the penetration warhead carries a limited amount of fuel, each maneuver will not be too long. To reduce the complexity of the optimization design variables and improve optimization efficiency, assume the maneuver direction vector u... i (t) is a static optimization variable whose value does not change with time during maneuvers, i.e.

[0055]

[0056] 4: Process state constraints

[0057] Since the midcourse trajectory of a ballistic missile is outside the atmosphere, aerodynamic forces during flight are negligible. Therefore, constraints related to dynamic pressure, angle of attack, sideslip angle, overload, and heat flux do not need to be considered during flight. While the low-thrust liquid engine used in the penetration warhead can be repeatedly started and stopped, it has a limited lifespan. Therefore, in multiple maneuvering penetration trajectory optimizations, there exists a maximum total number of cycles, n. max The number of operations meets the constraints.

[0058] n≤n max (52)

[0059] Meanwhile, due to the limited fuel capacity of the penetration warhead and the delay in the engine thrust build-up process, there is a constraint on the total engine operating time T. total_max and the minimum time constraint ΔT for a single operation min ,Right now

[0060]

[0061] 5: Terminal state constraints

[0062] In practical engineering applications, flight altitude is generally used as the criterion for determining the midcourse and reentry phases of a ballistic missile's trajectory, and the location of these phases is called the reentry handover point. This invention, based on practical engineering applications, optimizes the terminal altitude constraint H of the midcourse multiple-maneuver penetration trajectory. f The reentry handover point height H was determined. Target ,Right now

[0063] H f =H Target (54)

[0064] Furthermore, combined Figure 2 The improved PSO algorithm design specifically includes:

[0065] 1: Improved design during the initialization phase

[0066] Since the optimization design variables include maneuver timing (engine on / off time) and maneuver direction, during the initialization phase, the number of maneuvers performed by each individual in the entire population is first uniformly generated according to the following formula.

[0067]

[0068] Where i∈{1,2,…,SN}; n max The maximum number of maneuvers is given; ceil() is the floor function, which returns the nearest integer greater than or equal to the given element.

[0069] After initializing the number of maneuvers, the start time of each maneuver is determined using the following formula. and the speed of start time adjustment initialization.

[0070]

[0071]

[0072]

[0073] Where, j∈{1,2,…,n} i};t s，min t s,maxThese are the lower and upper bounds of the maneuver start time, respectively; v max (t s ), v min (t s ) represent the upper and lower bounds of the adjustment speed for the start time of the maneuver; ΔT min `min()` is the shortest time for a single engine operation; `sort()` is an ascending sort function that sorts the engine start times from low to high; `unifrnd(a,b)` is a random number generator that generates a random number between (a,b). `lim_min(x,Δx)` min Let be the minimum interval constraint function for elements, satisfying the minimum time constraint for a single engine operation. Its formula is:

[0074]

[0075] Where x is the input vector and y is the output value after the minimum interval of the restricted elements.

[0076] After completing the initialization of the maneuver start time, the engine operating time is adjusted using the following formula: v(ΔT) i,j ) and the end time of each maneuver Perform initialization.

[0077] v(ΔT i,j ) = v min (ΔT)+(v max (ΔT)-v min (ΔT))unifrnd(0,1) (60)

[0078]

[0079] ΔT i =randfixedsum(n i ,T total_max, ΔT min ,ΔT max (62)

[0080]

[0081] Among them, v max (ΔT), v min (ΔT) represents the upper and lower limits of the engine operating time adjustment speed, respectively; T is the total engine operating time. total_max ΔT is the engine's maximum total operating time. maxThe maximum operating time of a single engine cycle; `randfixedsum(a,b,c,d)` is a function that generates a random array whose sum is a set value, where `a` is the number of array elements, `b` is the sum of the array elements, and `c` and `d` are the ranges of values ​​for the elements in the random array; ΔT i This is an array representing the duration of each maneuver.

[0082] After completing the initialization assignment of maneuver timing, the maneuver direction is selected using the zero-range line direction. The fitness value is obtained by calculating the performance index according to the formula, and the initialization work of the entire PSO algorithm is completed.

[0083] 2: Improved design during the search phase

[0084] In particle swarm optimization (PSO), the most crucial part is updating particle positions, and its search efficiency directly impacts the algorithm's performance. Since the optimization design variables for multiple maneuver penetration ballistics problems include maneuver timing and direction, during the search phase, the dimensionality of the solutions differs for particles with different maneuver counts, preventing information exchange between them. Therefore, particle velocity and position updates can only rely on local and global optima of the same dimensionality.

[0085] During particle update, the maneuver start time is first updated based on the local and global optimal solutions of the same dimension.

[0086]

[0087]

[0088] in, and These represent the start time of the maneuver in the local optimal solution and the global optimal solution, respectively.

[0089] Subsequently, similar to updating the maneuver start time, the engine operating time was updated.

[0090]

[0091] ΔT i,j,new =ΔT i,j +Δv(ΔT i,j,new (67)

[0092] in, and These represent the engine operating time in the local optimal solution and the global optimal solution, respectively.

[0093] After the start time and working time of the maneuver are updated, constraints are imposed on the start time range, start time interval, total working time, etc., and the end time of the maneuver is further updated.

[0094]

[0095]

[0096] ΔT i,j,new =bound_protection(ΔT) min ,ΔT max ,ΔT i,j,new (70)

[0097]

[0098]

[0099]

[0100] Wherein, `bound_protection(LB,UB,x)` is the bounds protection function, which implements the bounds-out wraparound function, and its formula is:

[0101]

[0102] Where LB is the lower boundary; UB is the upper boundary; x is the protected input value; and y is the protected output value.

[0103] After updating the maneuver end time, the maneuver direction is chosen using the zero-range line direction. The fitness value is obtained by calculating the performance index according to the formula. If a new solution has a better performance index, the new solution replaces the old optimal solution. The local and global optimal solutions for particles of different dimensions (i.e., different maneuver counts) are updated accordingly for use in the next iteration.

[0104] 3: Improved design of inertia weight

[0105] In early practical applications of the PSO algorithm, it was found that particle movement speed was often too fast, easily exceeding the optimal solution of the objective function. Therefore, the concept of inertia weight w was introduced into the PSO algorithm to reduce particle movement speed. However, in practice, it was found that when the inertia weight w was set large, the particle movement speed was fast, resulting in strong global search capability but weak local search capability; when the inertia weight w was set small, the particle movement speed was slow, resulting in weak global search capability but strong local search capability. However, in actual problem solving, it was found that due to the different characteristics of various optimization problems, the constant value design strategy could not adequately meet the requirements of speed and robustness.

[0106] To achieve a balance between global and local search in the penetration trajectory optimization problem and accelerate convergence to the optimal solution, this invention designs the inertia weight w as an exponentially decreasing function. The exponentially changing inertia weight coefficient w yields better global optimization capability in the initial stage of optimization and better local optimization capability in the later stage of optimization, i.e.

[0107] w = (w0 - w) ∞ )e -k·i +w ∞ (75)

[0108] Where w0 is the initial inertia weight, w ∞ is the final inertia weight, and k is the exponential convergence coefficient.

[0109] Furthermore, the rapid solution for the zero-range direction specifically includes:

[0110] 1: Definition of a non-orthogonal absolute coordinate system

[0111] Assume the missile's current position is K, and the geocentric radius vector r of the missile in the launch inertial frame is known. K With velocity vector V K The missile's free flight motion is decomposed into two component motions: the motion within the initial orbital plane and the motion along ω. e The direction of motion. Therefore, a non-orthogonal absolute coordinate system O is defined. π -x π y π z π The origin of the coordinate system is the missile's center of mass, and y π Axis and geocentric radius r K The directions are the same, x π The axis lies in the initial orbital plane and intersects with the y-axis. π The axis points perpendicularly to the direction of missile motion, z π The axis passes through point K and ω e Parallel and opposite in direction, such as Figure 8 As shown. Let KD be the circular arc KD that intersects the initial orbital plane π with the Earth's surface. π The axis is tangent to the arc. Let KN be the great circle arc passing through point K, and let the angle between KN and KD be denoted as . but It is the dihedral angle between the initial orbital plane and the meridional plane.

[0112] The vector r at any point P on the missile's trajectory can be expressed as component r1 and component Z. π The vector sum, i.e.

[0113] r = r1 + Z π (76)

[0114] Where r1 can represent the component of the missile's geocentric radius vector in the π plane, with its endpoint being P1; Z π For the point passing through point P1 and z π A vector parallel to the axis, whose endpoint P is the actual position of the missile.

[0115] Furthermore, the angles related to r and r1 are defined as follows: the geocentric latitude of r is φ, the geocentric latitude of r1 is φ1, and δ is denoted as δ. φ =φ1-φ; Define r K The geocentric angle to r1 is β, see Figure 1 .

[0116] 2: Establishment of the differential equations of motion

[0117] (1) Decomposition of Earth's gravitational force g

[0118] The formula for Earth's gravitational field can be expressed as:

[0119]

[0120] in

[0121]

[0122] (2) Initial orbital plane

[0123] According to Newton's second law and the law of angular momentum, we can obtain

[0124]

[0125] C = |(r K ×V K )|=r K V K cosθ K (80)

[0126] Where C is a constant representing the orbital angular momentum, and θ K The local speed inclination angle.

[0127] To solve the system of equations, replace the variable t with the variable β, and let... Then there is

[0128]

[0129] (3)z π Equations of motion in the axial direction

[0130] According to Newton's second law, z π The equation of motion in the axial direction is

[0131]

[0132] To solve the system of equations, let Then there is

[0133]

[0134] The combined formula yields

[0135]

[0136] (4)g r1 With g ω Transform into explicit functions of η, ζ, and β

[0137] Based on the formula, we can first obtain

[0138]

[0139]

[0140] Depend on Figure 1 Medium triangle PP1O E Applying the Law of Cosines, we have

[0141]

[0142] Divide both sides of the equation by r1 2 get

[0143]

[0144] Take the reciprocal of the expression, since J2 is 10. -3 The order of magnitude, as defined by ζ, is also 10. -3 The order of magnitude can be further approximated by the square root approximation formula as follows:

[0145]

[0146] Note: Approximate formula for square root extraction

[0147]

[0148] In addition, by Figure 1 The geometric relationship between sinφ1 and sinφ can be obtained.

[0149] r1 sinφ11=r sinφ+Z π (91)

[0150] Therefore, from equations and , we can obtain...

[0151]

[0152] Then, by the cosine theorem for spherical triangles, we can obtain...

[0153]

[0154] make but

[0155] sinφ1=E(β) (94)

[0156] (5) Approximate equations of motion

[0157] Substitute the expression into the expression and ignore ζ. 2 Higher-order minor terms can be obtained

[0158] sinφ≈(1+ζE(β))(E(β)-ζ)=E(β)-ζ+ζE 2 (β)-ζ 2 E(β) (95)

[0159] Ignore ζ 2 Higher-order minor terms can be obtained

[0160] sin 2 φ=E 2 (β)+2ζE 3 (β)-2ζE(β) (96)

[0161] Substitute equations 1, 2, 3, and 4 into equations 5 and 6, and ignore ζ. 2 , And higher-order minor terms such as ζJ2, which can be summarized as follows:

[0162]

[0163]

[0164] in

[0165]

[0166] Substituting equations 1 and 2 into equations 3 and 4, we obtain the set of differential equations of motion for a free-flight trajectory in a non-orthogonal coordinate system:

[0167]

[0168] 3: Solutions to the differential equations of motion

[0169] Since ζ and J2 are both 10 in the system of differential equations of motion -3 Since it is on a small order of magnitude, it can be approximated as...

[0170]

[0171] (1) Solution of elliptical orbit

[0172] Elliptical orbit equations

[0173]

[0174] Given the initial condition: β = 0, The solution to the equation can be found as follows:

[0175]

[0176] in

[0177] ξ=ξ00+β (104)

[0178]

[0179]

[0180] In the above formula, ξ is the geocentric angle from the apogee to point P1, with clockwise rotation being positive; ξ0 is the geocentric angle from the apogee to point K, with clockwise rotation being positive. The geometric relationship between ξ, ξ0, and β is as follows: Figure 2 As shown.

[0181] (2) Solution to equation ζ

[0182] To solve this equation, we first transform E(β) into a function of ξ.

[0183] E(β)=E s sinξ+E C cosξ=F(ξ) (107)

[0184] in

[0185]

[0186] Therefore, the equation concerning ζ in the formula can be rewritten as follows:

[0187]

[0188] The solution to the system of equations can be found as follows:

[0189]

[0190] in

[0191]

[0192] (3) Solution of equation δη

[0193] Substituting equations 1, 2, and 3 into the third equation of the system of equations, we can find the solution.

[0194]

[0195] in

[0196]

[0197]

[0198] (4) Solution of flight time t

[0199] Let T be the flight time from the initial point K to any point P. From the fifth equation of the system of equations, we know...

[0200]

[0201] Since η = η0 + δη, it can be approximated as

[0202]

[0203] Substituting the expressions into the equation, we can find the answer.

[0204] T=Γ(ξ)-Γ(ξ0) (117)

[0205] In the formula

[0206]

[0207] in

[0208]

[0209]

[0210] 4: Determination of ballistic parameters

[0211] This patent utilizes the geocentric radius r in the inertial frame. K The velocity vector V in the inertial frame K and the unit Earth rotation angular velocity vector in the inertial frame. Find the given flight time T C Under the condition, T C The ballistic parameters at any given moment.

[0212] The equation shows that the flight time T is an explicit function of ξ, but finding the corresponding flight time T = T C The corresponding ξ does not have a corresponding explicit functional relationship, so iterative solution is required. The specific solution method is as follows:

[0213] (1) For a given r K V K and Using the formula for calculating angular momentum and the transformation matrix, the orbital angular momentum C and the velocity in the north-sky-east coordinate system are obtained.

[0214] (2) After obtaining p, e, and ξ0 from equation (1) to equation (2), calculate ξ using the following equation (3). [1] :

[0215]

[0216] (3) Starting from i=1, the iteration begins as follows:

[0217]

[0218] Wherein, Φ(ξ) [i] ), a 1η With b 1η Calculated by the formula, Γ(ξ) [i] ) and Γ(ξ0) are calculated by the formula.

[0219] (4) Regarding the set accuracy requirement Δ1, when |Δξ is satisfied [i] When | < Δ1, terminate the iteration. Take ξ. C =ξ [i+1] For T C The Earth's core at any given moment.

[0220] When T is solved C The geocentric angle ξ at a given moment C After that, ξ C Substituting into equations 1, 2, 3, and 4, we can find η0(ξ). C ), ζ(ξ C ), δη(ξ C ), η(ξ) C ), T(ξ) C After obtaining relevant parameters such as ), the ballistic parameters are solved using the following formulas:

[0221] (1) Geocentric latitude φ1 of point P1

[0222] φ1=arcsin(F(ξ)) (123)

[0223] Where F(ξ) is the geocentric angle transformation function;

[0224] (2) Geocentric latitude φ of point P

[0225] φ=arcsin((sin(φ1)-ζ)(1+ζsin(φ1))) (124)

[0226] (3) The geocentric distance r of point P P

[0227]

[0228] η = 1 / |r1| is the derivative of the magnitude of the geocentric radius vector r in the π plane;

[0229] (4) Absolute longitude difference between point K and point P

[0230]

[0231] In the formula, the dihedral angle The velocity components at point K in the northeast-east coordinate system were calculated.

[0232]

[0233] in,

[0234] (5) The relative longitude difference λ between point K and point P KP

[0235]

[0236] (6) Longitude λ of point P p

[0237] λ P =λ K +λ KP (129)

[0238] (7) The velocity vector V of point P in the inertial frame. P

[0239]

[0240] In the formula

[0241]

[0242]

[0243] (8) The geocentric radius r in the inertial frame at point P P

[0244]

[0245] In the formula

[0246]

[0247]

[0248] r P Here, B0 is the geocentric vector of the missile's actual position, L0 is the latitude of the missile's initial launch point, and a is the longitude of the missile's initial launch point. e For the Earth's semi-major axis, The square of the first eccentricity of the Earth;

[0249] (5) Solving for the zero-range line direction vector

[0250] When a velocity increment is applied to the missile along the zero-range line, theoretically it does not affect the missile's final impact point; that is, the velocity increment along the zero-range line satisfies:

[0251]

[0252] Where L is the longitudinal stroke; H is the transverse stroke; V is the velocity vector; and δV is the applied velocity increment.

[0253] Then the unit vector in the direction of the zero range line is:

[0254]

[0255] From equations and , it can be seen that the key to solving for the zero-range line lies in the rapid and accurate calculation of the partial derivatives of the longitudinal and transverse strokes with respect to velocity. Furthermore, the partial derivatives of the longitudinal and transverse strokes with respect to velocity can be expressed as:

[0256]

[0257] Based on the missile's current state, apply velocity increments (±ΔV) respectively. x ±ΔV y ±ΔV z The deviation between the longitudinal and transverse travel (ΔL) can be obtained. x ,ΔL y ,ΔL z ,ΔH x ,ΔH y ,ΔH z Based on the concept of small deviation linearization, when the speed increment is sufficiently small, the partial derivatives of the longitudinal and transverse strokes can be approximated as follows:

[0258]

[0259] In the formula, the longitudinal deviation ΔL and the transverse deviation ΔH are:

[0260]

[0261] in, Let A be the average radius of the Earth; A and β be the azimuth and geocentric angle between the landing point and the current point, respectively, under the condition of no velocity increment; A - With β - These are the azimuth angle and the geocentric angle, respectively, between the landing point and the current point under the applied velocity increment condition. The formulas for calculating the azimuth angle and the geocentric angle are:

[0262] β=arccos(sinφK K sinφ P +cosφ K cosφ P cos(λ P -λ K (141)

[0263]

[0264] Using the ballistic analytical solution calculation method considering the J2 perturbation effect in the previous section, we calculate the longitudinal and transverse deviations caused by the velocity increments in the three directions. We then use the formula to calculate the partial derivatives of the longitudinal and transverse deviations and substitute them into the formula to obtain the zero-range direction vector.

[0265] Furthermore, the optimization of multiple maneuver penetrations along the zero-range line specifically includes:

[0266] Step 4.1: Solve for the zero-range line direction and use it as the direction of maneuver penetration in the mid-course of the ballistic trajectory;

[0267] Step 4.2: The improved PSO algorithm is used to optimize the design of the penetration trajectory.

[0268] Example

[0269] To verify the effectiveness of the zero-range line algorithm considering the J2 perturbation effect, the zero-range line direction was calculated along the x-axis of the launch inertial frame, the zero-range line direction without considering the effects of Earth's rotation and oblateness, and the zero-range line direction considering the J2 perturbation effect. A certain velocity increment was applied, and the impact on the impact point deviation was calculated. The zero-range line calculation method without considering the effects of Earth's rotation and oblateness is based on the analytical method in existing literature. The ballistic simulation conditions are shown in Table 1, and the calculation results are shown in Tables 2, 3, and 4.

[0270] Table 1 Ballistic Missile Simulation Parameters

[0271]

[0272] Table 2. Deflection of the landing point when velocity increments are applied along the x-axis of the inertial frame.

[0273]

[0274] Table 3. Landing point deviation when velocity increments are applied along the zero range line (analytical method without considering the effects of Earth's rotation and oblateness).

[0275]

[0276] Table 4. Landing point deviation when velocity increments are applied along the zero range line (This paper proposes a fast algorithm that considers the effects of Earth's rotation and oblateness).

[0277]

[0278]

[0279] As shown in Tables 2, 3, and 4, regardless of the direction in which the velocity increment is applied, the impact on the landing point deviation increases with the increase of the velocity increment. However, applying the velocity increment along the zero-range line can significantly reduce the impact on the landing point. Compared with the zero-range line calculation method that does not consider the influence of Earth's rotation and oblateness, the zero-range line calculation method proposed in this paper that considers the influence of Earth's rotation and oblateness can significantly reduce the landing point deviation. Specifically, the lateral deviation can be controlled within 1m, the longitudinal deviation can be controlled within the meter range within a velocity increment of 8m / s, and the longitudinal deviation can be controlled within 100m with a velocity increment of 20m / s, demonstrating high accuracy.

[0280] To further verify the applicability of the algorithm, simulations were conducted under different launch azimuth angles, applying a velocity increment of 10 m / s. The simulation results are as follows: Figure 3 and Figure 4 As shown.

[0281] from Figure 3 and Figure 4 Simulation results show that the maximum longitudinal deviation is approximately 16m, and the minimum is approximately 6m. The deviation is greatest at an azimuth angle of 90° and least at 270°, exhibiting a symmetrical variation pattern. The maximum transverse deviation is approximately 1.5m, and the minimum is approximately 0m. The deviation is least at approximately 20° and 160°, and least at 90° and 270°, also exhibiting a symmetrical variation pattern. Overall, the zero-range line solution algorithm has high accuracy, causes small impact point deviations, and has good adaptability under different launch azimuth angles. It can meet the accuracy requirements of ballistic missile mid-course penetration trajectory optimization design and lay the foundation for optimizing multiple maneuver penetration trajectories in the zero-range line direction.

[0282] Example of solving the trajectory of multiple maneuvers for penetration in the zero-range direction

[0283] (1) Simulation condition settings

[0284] This patent uses the zero-range line as the mid-course maneuvering penetration direction and employs an improved PSO algorithm to optimize the penetration trajectory. The initial position and velocity conditions for the penetration warhead and the interceptor warhead are detailed in Tables 1 and 5. The basic parameters, constraint settings, performance index parameter settings, satisfaction range and preference value settings, and algorithm parameter settings for the maneuvering warhead are detailed in Tables 6, 7, 8, 9, and 10.

[0285] Table 5 Simulation Parameters for Interceptor Missiles

[0286]

[0287]

[0288] Table 6 Basic Parameter Settings for Mobility Warheads

[0289]

[0290] The process of optimizing the penetration trajectory of a penetrating warhead involves constraints such as the number of engine operations, the shortest time, and the total operating time. The specific constraint settings are shown in Table 7.

[0291] Table 7 Constraint Settings

[0292]

[0293] The weighting coefficients in the ballistic optimization performance indicators, as well as the satisfaction range and preference value settings for multi-objective optimization, are shown in Tables 8 and 9.

[0294] Table 8 Performance Index Parameter Settings

[0295]

[0296] Table 9 Satisfaction Range and Preference Value Settings

[0297]

[0298] The relevant parameter settings for the improved PSO algorithm are shown in Table 10.

[0299] Table 10 Improved PSO Algorithm Parameter Settings

[0300]

[0301]

[0302] (2) Simulation Result Analysis

[0303] Using the above simulation conditions, optimized simulations were performed on multiple maneuver penetration trajectories. The simulation results are as follows: Figures 5-10 ,from Figures 5 to 10 The optimized trajectory results of multiple maneuvers along the zero-range line show that the maneuvering warhead, with an overload of 0.29g, continuously maneuvered multiple times along the zero-range line between 476s and 501s. This resulted in a zero-control miss distance of 14.6km when the maneuvering warhead intersected with the interceptor, far exceeding the interceptor's terminal guidance maneuver range, thus achieving successful penetration. Simultaneously, the shift point deviation caused by the maneuvering penetration was within 3m, which would not create trajectory correction pressure on the terminal guidance precision strike during the reentry phase, effectively ensuring strike accuracy while penetrating the defenses. Figure 10As can be seen from the performance index change curve, after the PSO algorithm completes particle initialization, its performance index value is already around 0.5, indicating that its solution has converged to near the optimal solution. After 50 iterations, its performance index has dropped to below 0.45. Its convergence speed is significantly improved compared to ballistic optimization without using the zero-range direction for penetration. Therefore, it verifies the effectiveness of using the zero-range line direction as the maneuver penetration direction to improve the optimization convergence speed.

[0304] To further verify the feasibility of the algorithm, 500 maneuver penetration ballistic optimization solutions were performed. The simulation results are as follows: Figures 11-19 As shown. From Figure 11 It can be seen that the performance index converges to within 0.46 after about 50 iterations, and reaches between 0.445 and 0.45 after 150 iterations, indicating a significantly faster overall convergence speed. Figure 12 The statistical results show that the shift point deviation of most optimization results is within 3m, with a maximum of no more than 4m. Compared with the previous chapter, its distribution is more concentrated, and this level of deviation will not put any correction pressure on the terminal guidance trajectory during the reentry phase, nor will it affect the strike accuracy. From Figure 13 It can be seen that the zero-control miss distance between the warhead and the interceptor is mostly around 14.6 km, with a minimum of 14.3 km, far exceeding the maximum maneuvering distance of 3000 m during the terminal guidance phase of the interceptor, thus achieving effective penetration of the maneuvering warhead. From Figure 14 It can be seen that the total maneuver time is relatively concentrated on a time scale, mostly around 22.9s, with a minimum of 22.8s and a maximum of 23.6s. From Figure 15 It can be seen that the final performance index values ​​of the algorithm after 500 iterations are relatively concentrated, with most of them falling between 0.444 and 0.446. Figures 16-19 The relationship between the final performance index values ​​and the zero-control miss distance, shift handover point deviation, and total maneuver time shows that the distribution of the zero-control miss distance and total maneuver time in relation to the final performance index values ​​is relatively concentrated, with a small overall dispersion. However, there is a clear dividing line between the distribution of the shift handover point deviation and the final performance index. The minimum shift handover point deviation is around 3m. This is due to the method error of the zero-range rapid calculation method. Within a certain speed increment, there is a minimum shift handover point deviation limit, thus creating the dividing line between the distribution of the shift handover point deviation and the final performance index.

Claims

1. A method for multiple maneuvering penetration along the zero-range line, characterized in that, Includes the following steps: Step 1, establish a description model for the ballistic missile penetration trajectory optimization problem: Based on the dynamics and kinematics models of ballistic missiles and interceptor missiles, the number of maneuvers, timing of maneuvers, and direction of maneuvers of the penetrating warhead are used as optimization design variables, and the magnitude of zero-control miss distance, shift handover point deviation, and fuel consumption are used as optimization indicators to establish a description model for the ballistic missile penetration trajectory optimization problem. Step 2, PSO Algorithm Design: Based on the multiple optimization design variables such as the number of maneuvers and maneuver time in Step 1, and the problem of uncertain dimensions, the initialization phase, search phase, and inertia weight of the PSO algorithm are designed. Step 3, rapid solution for zero-range line direction; transform the description model of the penetration ballistic optimization problem into a non-orthogonal coordinate system, derive the analytical solution of the free flight trajectory, and on this basis, combine the analytical method and the numerical method to rapidly solve the zero-range line direction; Step 4, optimization of multiple maneuver penetrations in the zero-range direction: Using the PSO algorithm designed in Step 2, the zero-range direction obtained in Step 3 is used as the maneuver direction to optimize the trajectory of multiple maneuver penetrations and obtain the optimal multiple maneuver penetration method.

2. The method for multiple maneuvering penetration along the zero-range line according to claim 1, characterized in that, The specific steps in step 1 of establishing the description model for the penetration ballistic optimization problem include: Step 1.1, establish a mathematical model for penetrating warheads: Where g is the gravitational acceleration experienced by the missile, G I With G B Here, are the direction cosine matrices between the launch inertial coordinate system and the launch coordinate system, and between the missile body coordinate system and the launch coordinate system, respectively; m is the current mass of the missile; R is the aerodynamic force acting on the missile; T is the engine thrust; and V... I Let r be the velocity vector of the missile in the launch inertial coordinate system. I Let I be the geocentric radius vector of the missile in the launch inertial coordinate system. sp g is the specific impulse of the engine propellant, and g0 is the gravitational acceleration at sea level; Step 1.2, using the magnitude of the zero-control miss distance, the shift handover point deviation, and fuel consumption as optimization indicators, the performance indicators are designed as follows: Where w1, w2, and w3 are the weight coefficients for each objective; R miss0 The zero-control miss distance; i∈{1,2,…,n}, where n is the total number of maneuvers. The start time of each maneuver. d represents the end time of each maneuver. r represents the landing point position deviation; p1(), p2(), and p3() are the zero-control miss distance, total engine operating time, and landing point position deviation preference functions, respectively. Step 1.3, the optimization design variables for the number of maneuvers, timing of maneuvers, and direction of maneuvers of the penetrating warhead are expressed as: Among them, u i (t) represents the direction vector for each maneuver; Step 1.4, Design process state constraints There exists a maximum total number of operations n in multiple maneuver penetration ballistic optimizations. max The number of operations meets the constraints: n≤n max (4) There is a total engine operating time constraint T total_max and the minimum time constraint ΔT for a single operation min ,Right now Step 1.5, Design terminal state constraints The terminal altitude constraint H of the mid-course multiple maneuver penetration trajectory optimization f The reentry handover point height H was determined. Target ,Right now H f =H Target (6)。 3. The method for multiple maneuvering penetrations along the zero-range line according to claim 1, characterized in that, The initialization phase of designing the PSO algorithm in step 2 specifically includes: During the initialization phase, the number of maneuvers performed by each individual in the entire population is first uniformly generated according to the following formula; Where i∈{1,2,…,SN}; n max This represents the maximum number of maneuvers; ceil() is the floor function, which rounds up to the nearest integer greater than or equal to the given element. After initializing the number of maneuvers, the start time of each maneuver is determined using the following formula. and the speed of start time adjustment initialization; Where, j∈{1,2,…,n} i };t s，min t s,max These are the lower and upper bounds of the maneuver start time, respectively; v max (t s ), v min (t s ) represent the upper and lower bounds of the adjustment speed for the start time of the maneuver; ΔT min `min()` is the shortest time for a single engine operation; `sort()` is an ascending sort function that sorts the engine start times from low to high; `unifrnd(a,b)` is a random number generator function that generates a random number between (a,b); `lim_min(x,Δx)`... min Let be the minimum interval constraint function for elements, satisfying the minimum time constraint for a single engine operation. Its formula is: Where x is the input vector and y is the output value after the minimum interval of the restricted elements; After completing the initialization of the maneuver start time, the engine operating time is adjusted using the following formula: v(ΔT) i,j ) and the end time of each maneuver Perform initialization; v(ΔT i,j )=v min (ΔT)+(v max (ΔT)-v min (ΔT))unifrnd(0,1) (12) ΔT i =randfixedsum(n i ,T total_max ,ΔT min ,ΔT max ) (14) Among them, v max (ΔT), v min (ΔT) represents the upper and lower limits of the engine operating time adjustment speed, respectively; T is the total engine operating time. total_max ΔT is the engine's maximum total operating time. max The maximum operating time of a single engine cycle; `randfixedsum(a,b,c,d)` is a function that generates a random array whose sum is a set value, where `a` is the number of array elements, `b` is the sum of the array elements, and `c` and `d` are the ranges of values ​​for the elements in the random array; ΔT i This is an array representing the duration of each maneuver.

4. The method for multiple maneuvering penetrations along the zero-range line according to claim 1, characterized in that, The search phase of designing the PSO algorithm in step 2 specifically includes: During the particle update process in the search phase, the maneuver start time is first updated based on the local and global optimal solutions of the same dimension: in, and These represent the start time of the maneuver in the local optimum and the global optimum, respectively. Subsequently, the engine operating time was updated: ΔT i,j,new =ΔT i,j +v(ΔT i,j,new ) (19) in, and These represent the engine operating times in the local optimal solution and the global optimal solution, respectively. After the start time and working time of the maneuver are updated, constraints are imposed on the start time range, start time interval, total working time, etc., and the end time of the maneuver is further updated. ΔT i,j,new =bound_protection(ΔT min ,ΔT max ,ΔT i,j,new ) (22) Wherein, `bound_protection(LB,UB,x)` is the bounds protection function, which implements the bounds-outbounds wraparound function, and its formula is: Where LB is the lower boundary; UB is the upper boundary; x is the protected input value; and y is the protected output value.

5. The method for multiple maneuvering penetrations along the zero-range line according to claim 1, characterized in that, The inertia weights designed in step 2 of the PSO algorithm specifically include: The inertia weight w is designed as an exponentially decreasing function, i.e. w=(w0-w ∞ )e -k·i +in ∞ (27) Where w0 is the initial inertia weight, w ∞ is the final inertia weight, and k is the exponential convergence coefficient.

6. The method for multiple maneuvering penetrations along the zero-range line according to claim 1, characterized in that, The rapid solution for the zero-range line direction in step 3 specifically includes: Step 3.1: Transform the description model of the penetration ballistic optimization problem into a non-orthogonal coordinate system, and derive the analytical solution of the free flight trajectory in the non-orthogonal coordinate system; Step 3.2, determine the velocity increment in the zero-range direction; Step 3.3: Calculate the partial derivatives of the longitudinal and transverse strokes with respect to velocity; Step 3.4: Solve for the zero-range direction vector.

7. The method for multiple maneuvering penetration along the zero-range line according to claim 1, characterized in that, The process of deriving the analytical solution for free-flight ballistics in a non-orthogonal coordinate system includes determining the given flight time T. C Under the condition, T C The ballistic parameters at any given time include: The vector r at any point P on the trajectory, r1 represents the component of the missile's geocentric radius vector r in the π plane, with endpoints P1 and Z. π For the point passing through point P1 and z π A vector parallel to the axis, whose endpoint P is the actual position of the missile, and the geocentric latitude φ1 of point P1 is: φ1=arcsin(F(ξ)) (28) Where ξ = ξ0 + β, ξ0 is the geocentric angle from the apogee to point K, clockwise is positive, and β is the geocentric radius r. K The geocentric angle with respect to component r1, F(ξ) is the geocentric angle transformation function; Geocentric latitude φ of point P: φ=arcsin((sin(φ1)-ζ)(1+ζsin(φ1))) (29) In the formula, ζ=Z π / |r1|, The distance from the center of point P to the Earth is r. P : In the formula, η=1 / |r1| is the derivative of the component modulus of the geocentric radius vector r in the π plane; Absolute difference in longitude from point K to point P In the formula, the dihedral angle The following was calculated using the velocity components at point K in the northeast-eastern coordinate system: in, The relative longitude difference λ between point K and point P KP : The longitude λ of point P p : l P =λ K +λ KP (34) The velocity vector V at point P in the inertial frame P : In the formula The geocentric radius r at point P in the inertial frame P : In the formula In the formula, r P Here, B0 is the geocentric vector of the missile's actual position, L0 is the latitude of the missile's initial launch point, and a is the longitude of the missile's initial launch point. e For the Earth's semi-major axis, It is the square of the first eccentricity of the Earth.

8. The method for multiple maneuvering penetration along the zero-range line according to claim 6, characterized in that, The velocity increment in the zero-range direction satisfies: Where L is the longitudinal stroke; H is the transverse stroke; V is the velocity vector; and δV is the applied velocity increment.

9. A method for multiple maneuvering penetration along the zero-range line according to claim 6, characterized in that, The calculation of the partial derivatives of the longitudinal and transverse strokes with respect to velocity includes... The partial derivatives of the longitudinal and transverse strokes with respect to velocity are expressed as follows: Based on the missile's current state, apply velocity increments ±ΔV respectively. x ±ΔV y ±ΔV z The deviation ΔL between the longitudinal and transverse strokes is obtained. x ,ΔL y ,ΔL z ,ΔH x ,ΔH y ,ΔH z Based on the idea of ​​small deviation linearization, the partial derivatives of the longitudinal and transverse axes are approximately expressed as: In the formula, the longitudinal deviation ΔL and the transverse deviation ΔH are: in, Let A be the average radius of the Earth; A and β be the azimuth and geocentric angle between the landing point and the current point, respectively, under the condition of no velocity increment; A - With β - These are the azimuth angle and the geocentric angle between the landing point and the current point, respectively, under the condition of applied velocity increment; the azimuth angle and the geocentric angle are: β=arccos(sinφ) K sinφ P +cosφ K cosφ P cos(λ P -l K )) (45) 。 10. A method for multiple maneuvering penetration along the zero-range line according to claim 6, characterized in that, The zero-range line direction vector is: Substituting the calculation result of equation (43) into equation (47) yields the zero-range direction vector.

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