A PSO-RPM-based off-line three-dimensional trajectory planning method for multi-cruise missile cooperative penetration

Abstract

The application discloses a kind of multi-cruise missile cooperative penetration off-line three-dimensional trajectory planning methods based on PSO-RPM, it is related to missile flight path planning technical field, comprising: according to the first flight path constraint model of first waypoint constraint, voyage constraint, flight path angle constraint and first threat area constraint, based on particle swarm algorithm, according to particle velocity and position update model, the voyage cost function of first flight path constraint model is solved, the final flight path set of each cruise missile is obtained, and the cooperative speed of each cruise missile is found;According to the second flight path constraint model of cooperative constraint, boundary constraint, control variable and flight path angle constraint, and second threat area constraint, combined with cruise missile dynamics model and second performance index function, the optimal control problem model of three-dimensional trajectory planning is constructed, and the cooperative penetration off-line three-dimensional trajectory planning strategy of cruise missile is obtained based on pseudospectral method solution.The application can efficiently and accurately plan safe trajectory.

Description

Technical Field

[0001] This invention relates to the field of missile flight path planning technology, and more specifically, to a method for offline three-dimensional trajectory planning of multi-cruise missile cooperative penetration based on a two-order PSO-RPM strategy. Background Technology

[0002] During coordinated missile penetration, missiles face threats from radar and anti-aircraft missiles, and are constrained by their own capabilities, such as trajectory inclination, deflection, range, and flight dynamics. Coordinated trajectory planning is one of the means to improve the probability of successful penetration. The hp-Ruda pseudospectral method is a commonly used method in trajectory planning, often solving the nonlinear optimization problem transformed by the pseudospectral method through sequential quadratic programming. Due to the sensitivity of the initial value of the sequential quadratic programming algorithm, directly solving it may result in long solution time or no solution. Therefore, the selection of the initial value often affects the success or failure of the algorithm. The problem of coordinated penetration of multiple cruise missiles is more complex, with nonlinear motion models, coupled relative motion relationships between missiles, and the need to consider dynamics and obstacle avoidance constraints. The feasible region of the optimization problem may be a very narrow non-convex region. Therefore, obtaining a feasible or even suboptimal initial value is particularly crucial. Summary of the Invention

[0003] The present invention provides a method for offline three-dimensional trajectory planning of multi-cruise missile cooperative penetration based on a two-order PSO-RPM strategy, which can solve the above-mentioned problems.

[0004] To solve the above problems, the technical solution adopted by the present invention is as follows:

[0005] This invention provides a multi-cruise missile cooperative penetration offline three-dimensional trajectory planning method based on PSO-RPM, characterized by the following steps:

[0006] S1. Treat cruise missiles as particles and construct particle velocity and position update models. Construct a first trajectory constraint model based on midway point constraints, range constraints, trajectory angle constraints, and first threat zone constraints. Use the range cost function as the first performance index function. Based on the particle swarm algorithm, solve the first performance index function according to the particle velocity and position update models and the first trajectory constraint model to obtain the candidate trajectory set for each cruise missile. For each cruise missile, its candidate trajectory set includes its optimal candidate trajectory and several suboptimal candidate trajectories.

[0007] S2. Traverse the candidate trajectory set of each cruise missile, find the final trajectory of each cruise missile as the initial trajectory, and make the coordination time t of each cruise missile not equal to 0. Based on the final trajectory and coordination time of each cruise missile, obtain the coordination speed of each cruise missile as the initial speed.

[0008] S3. Construct a second trajectory constraint model based on the cooperative constraints, boundary constraints, control variables and trajectory angle constraints, as well as the second threat zone constraints. Construct a three-dimensional trajectory planning optimal control problem model based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function.

[0009] S4. After interpolating the initial trajectory, use it as the initial value guess. Use the initial velocity as the velocity of the optimal control problem. Solve the optimal control problem model of three-dimensional trajectory planning based on the pseudospectral method through the sequential quadratic programming algorithm to obtain the cruise missile cooperative penetration offline three-dimensional trajectory planning strategy.

[0010] In a preferred embodiment of the present invention, the particle velocity and position update model in step S1 is as follows:

[0011]

[0012] Where i is the particle number, x i =(x i，x x i，y x i，z ), representing the missile's three-dimensional position coordinates, and the velocity vector of the i-th particle is v. i =(v i，x v i，y v i，z After t iterations, the position vector x of the i-th particle is... i The best historical position is w represents the inertia weight coefficient, (c1, c2) represent the particle's learning factor and acceleration factor, respectively, r1 and r2 are both random numbers between (0, 1) and follow a uniform distribution, M represents the population size, and t max represents the maximum number of iterations for the particle swarm, and t represents the current number of iterations for the particle swarm.

[0013] In a preferred embodiment of the present invention, the linear decreasing formula for the inertia weight coefficient w is:

[0014]

[0015] Among them, the maximum weight w max The minimum weight w is 0.95. min It is 0.4.

[0016] When the particle velocity exceeds the velocity range, the velocity update formula is:

[0017]

[0018] When the particle position exceeds the boundary of the solution space, the position update formula is:

[0019]

[0020] Where: v represents the particle's current velocity, x represents the particle's current position, and xp U x represents the upper bound of the particle's position boundary value. L The lower bound of the boundary value representing the particle's position.

[0021] In a preferred embodiment of the present invention, the range constraint in step S1 is:

[0022]

[0023] l min ≤l i

[0024] Among them, l max It is the longest allowed voyage, l min It is the shortest permitted flight distance.

[0025] In a preferred embodiment of the present invention, the track angle constraint in step S1 is:

[0026]

[0027]

[0028] Where, ψ V,max For the maximum permissible track deviation angle, θ max This is the maximum permissible track inclination angle, and the horizontal projection of the i-th track segment is denoted as a. i =(x i -x i-1 , z i -z i-1 ) T .

[0029] In a preferred embodiment of the present invention, it is assumed that there are m cylindrical threat sources, and the center point of the j-th threat source is (x j,threat y j,threat ), with a radius of R i,threat The height is H j,threat In each flight path segment, select q points, and denote the k-th point as P. k =(p k,x p k,y p k,z If the trajectory angle constraint in step S1 is..., then the trajectory angle constraint in step S1 is...

[0030]

[0031] p k,z ≥H j,threat , k = 1, 2, ..., q.

[0032] 7. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 6, characterized in that n trajectory segments are provided, and the length of the i-th trajectory segment is l. i Then the first performance index function is

[0033]

[0034] In a preferred embodiment of the present invention, the trajectory planning problem in the particle swarm optimization algorithm can be transformed into the following optimal control problem:

[0035] min J=f(x)

[0036] stg i (x)≤0, i=1,2…M

[0037] Where J represents the fitness value, f(x) represents the fitness evaluation function, and g i (x) represents the i-th inequality constraint, and M represents the number of inequality constraints;

[0038] To handle constraints, a constraint violation function is introduced:

[0039]

[0040] The criteria for evaluating the quality of particles are as follows:

[0041] (1) When particle x i and particle x j When the constraints are satisfied, if f(x) i )≤f(x j If ) holds true, then particle x i If it is better, then particle x j Excellent;

[0042] (2) When particle x i and particle x j If f does not satisfy the constraints vc (x i )≤f vc (x j If ) holds true, then particle x i If it is better, then particle x j Excellent;

[0043] (3) When particle x i Satisfying the constraints, particle x j If the constraints are not satisfied, then f(x) j )≤f(x i And f vc (x j If )≤ε, then particle x j If it is better, then particle xi For optimal performance, ε is the predetermined accuracy requirement.

[0044] In a preferred embodiment of the present invention, it is set that Let be the final trajectory of the i-th cruise missile, then its cooperative velocity is . Where t is the coordination time, t = min{t1∩t2∩...t} N},

[0045] The i-th cruise missile arrives at the target within the time range t. i for:

[0046]

[0047] Among them, L i V represents the set track length of the i-th cruise missile. min V represents the minimum flight speed of the i-th cruise missile. max This represents the maximum flight speed of the i-th cruise missile.

[0048] In a preferred embodiment of the present invention, step S3 specifically includes:

[0049] S31. The dynamic model of the cruise missile is constructed as follows:

[0050]

[0051] Where N represents the number of cruise missiles, i represents the i-th cruise missile, g represents the gravitational acceleration, and m i V represents the mass of the i-th cruise missile. i Let x represent the velocity of the i-th cruise missile. i y i , z i (n) represents the position of the i-th cruise missile in the ground coordinate system. xi n yi n zi ) represents the triaxial overload component of the i-th cruise missile, (θ) i , ψ Vi ) represents the inclination angle and deflection angle of the i-th cruise missile;

[0052] Define state variables and control variables as follows:

[0053]

[0054]

[0055] in, The cruise missile dynamics model can then be rewritten in the following state-space form:

[0056]

[0057] S32. Construct a second trajectory constraint model based on collaborative constraints, boundary constraints, control variables, trajectory angle constraints, and the second threat zone constraint:

[0058]

[0059]

[0060] Among them, R i，safe R represents the minimum safe distance for the i-th cruise missile. link Indicates the maximum permissible communication range for cruise missiles. Let t represent the initial state of the i-th cruise missile. f Indicates the time when they arrive at the target simultaneously. This indicates the terminal state of a cruise missile.

[0061] (n xi min n xi max ), (n yi min n yi nax ), (n zi min n zi max ) represent the minimum and maximum allowable overload components along the x, y, and z axes of the i-th cruise missile, respectively, and (θ) i min θ i max ), (ψ Vi min , ψ Vi max ) represent the minimum and maximum allowable trajectory inclination and trajectory deflection angles of a cruise missile, respectively;

[0062] S33. Construct the second performance index function:

[0063]

[0064] S34. Based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function, a three-dimensional trajectory planning optimal control problem model is constructed:

[0065] min J = t f -t0

[0066]

[0067] Compared with the prior art, the beneficial effects of the present invention are:

[0068] This paper addresses the optimal trajectory planning problem for multi-cruise missile cooperative penetration, considering factors such as bypassing threat areas, passing through midway points, satisfying maximum range, avoiding collisions between cruise missiles, and meeting dynamic and practical performance constraints. A particle swarm optimization (PSO) algorithm is used to obtain initial trajectories and velocities without considering collision avoidance and dynamic constraints. The initial values ​​obtained by the PSO algorithm are then used as initial value guesses for the hp-Ruda pseudospectral method. Solving this problem using sequential quadratic programming reduces solution time and the possibility of no solution. Furthermore, by considering all constraints, the paper obtains the actual feasible trajectories for each cruise missile, which better reflects the actual battlefield environment.

[0069] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, embodiments of the present invention are described below in detail with reference to the accompanying drawings. Attached Figure Description

[0070] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0071] Figure 1 This is a flowchart of the PSO-RPM two-stage strategy multi-cruise missile cooperative penetration process of the present invention;

[0072] Figure 2 This is a flowchart of the particle swarm algorithm in this invention. Detailed Implementation

[0073] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0074] Please refer to Figure 1 This invention provides an offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on a two-order PSO-RPM strategy. It considers the optimal trajectory planning problem for multi-cruise missile cooperative penetration, including bypassing threat areas, passing through midway points, satisfying maximum range, avoiding collisions between cruise missiles, and satisfying dynamic and practical performance constraints. The initial trajectory and velocity are obtained using a particle swarm optimization (PSO) algorithm without considering collision avoidance and dynamic constraints. The initial values ​​obtained by the PSO algorithm are used as initial value guesses for the hp-Ruda pseudospectral method. Considering all constraints, the actual feasible trajectories of each cruise missile are obtained. The specific method of this invention is as follows:

[0075] S1. Treat cruise missiles as particles and construct particle velocity and position update models. Construct a first track constraint model based on midway point constraints, range constraints, track angle constraints, and first threat zone constraints. Use the range cost function as the first performance index function. Based on the particle swarm optimization (PSO) algorithm, solve the first performance index function according to the particle velocity and position update models and the first track constraint model to obtain the candidate track set for each cruise missile. For each cruise missile, its candidate track set includes its optimal candidate track and several suboptimal candidate tracks.

[0076] The specific steps are as follows:

[0077] 1. Treat cruise missiles as particles and construct a particle velocity and position update model.

[0078] In the three-dimensional search space, assume the three-dimensional position vector of the i-th particle is x. i =(x i，x x i,y x i，z ), representing the missile's three-dimensional position coordinates, and the velocity vector of the i-th particle is v. i =(v i，x v i,y v i，z After t iterations, the position vector x of the i-th particle is... i The best historical position is The optimal global position for the entire population is The formulas for updating the velocity and position of the i-th particle in iteration t+1 are:

[0079]

[0080] Where: w represents the inertia weight coefficient, (c1, c2) represent the particle's learning factor and acceleration factor, respectively, r1 and r2 are random numbers between (0, 1) and follow a uniform distribution, M represents the population size, and t max represents the maximum number of iterations for the particle swarm, and t represents the current number of iterations for the particle swarm.

[0081] The formula for the linear decrease of inertia weight is:

[0082]

[0083] Where: the maximum weight w is selected based on experience. max The minimum weight w is 0.95. min It is 0.4.

[0084] When the particle velocity exceeds the velocity range, the velocity update formula is:

[0085]

[0086] When the particle position exceeds the boundary of the solution space, the position update formula is:

[0087]

[0088] Where: v represents the particle's current velocity, x represents the particle's current position, and xp U x represents the upper bound of the particle's position boundary value. L The lower bound of the boundary value representing the particle's position.

[0089] 2. Construct the first trajectory constraint model based on midway point constraints, range constraints, trajectory angle constraints, and first threat zone constraints.

[0090] (1) Midway point constraint

[0091] To satisfy the midpoint constraint of the trajectory, a parallel particle swarm optimization (PSO) algorithm is employed. Assume the initial position of the cruise missile is... Midway point is The end position is The PSO algorithm selects the starting and ending points as: initial position -> intermediate point, and intermediate point -> final position, respectively. Finally, the first and second track segments are connected to obtain the final track that satisfies the intermediate point constraint.

[0092] (2) Range constraints

[0093] Let l max It is the longest allowed voyage, l min If it is the shortest allowed flight distance, then the flight distance constraint is:

[0094]

[0095] l min ≤l i (6)

[0096] (3) Track angle constraint

[0097] Let ψ V,max For the maximum permissible track deviation angle, θ max This is the maximum permissible track inclination angle, and the horizontal projection of the i-th track segment is denoted as a. i =(x i -x i-1 , z i -z i-1 ) T The track angle constraint is then:

[0098]

[0099]

[0100] (4) First Threat Zone Constraints

[0101] Suppose there are m cylindrical threat sources, and the center point of the j-th threat source is (x j,threat y j,threat ), with a radius of R j,threat The height is H j,threat In each flight path segment, select q points, and denote the k-th point as P. k =(p k,x p k,y p k,z The threat zone constraint is then:

[0102]

[0103] p k,z ≥H j,threat , k=1,2…q (10)

[0104] Equations (5)-(10) are transformed into inequality constraints of the form g(x)≤0, as shown below:

[0105]

[0106] 3. First performance index function

[0107] In particle swarm optimization (PSO), a cost function is needed to evaluate the quality of particles. In this invention, the performance metric in the PSO algorithm is the range cost function. Assuming there are n tracks, and the length of the i-th track segment is l... i Then the first performance index function is:

[0108]

[0109] 4. Particle Evaluation Criteria

[0110] After the above processing, the trajectory planning problem can be transformed into the following optimal control problem:

[0111]

[0112] Where: J represents the fitness value, f(x) represents the fitness evaluation function, i.e., the range cost model, g i (x) represents the i-th inequality constraint, and M represents the number of inequality constraints.

[0113] To handle constraints, a constraint violation function is introduced:

[0114]

[0115] The criteria for evaluating the quality of particles are as follows:

[0116] (1) When particle x i and particle x jWhen the constraints are satisfied, if f(x) i )≤f(x j If ) holds true, then particle x i If it is better, then particle x j Excellent;

[0117] (2) When particle x i and particle x j If f does not satisfy the constraints vc (x i )≤f vc (x j If ) holds true, then particle x i If it is better, then particle x j Excellent;

[0118] (3) When particle x i Satisfying the constraints, particle x j If the constraints are not satisfied, then f(x) j )≤f(x i And f vc (x j If )≤ε, then particle x j If it is better, then particle x i The optimal value is ε, where ε is the predetermined accuracy requirement.

[0119] Using the PSO algorithm described above, a set of candidate tracks for each cruise missile is obtained. For each cruise missile, its set of candidate tracks includes its optimal candidate track and several suboptimal candidate tracks. The particle swarm algorithm flow of this invention is as follows: Figure 2 As shown.

[0120] S2. Traverse the candidate trajectory set of each cruise missile, find the final trajectory of each cruise missile as the initial trajectory, and make the coordination time t of each cruise missile not equal to 0. Based on the final trajectory and coordination time of each cruise missile, obtain the coordination speed of each cruise missile as the initial speed.

[0121] Assume the time range for the i-th cruise missile to reach the target is t. i for:

[0122]

[0123] Among them, L i V represents the set track length of the i-th cruise missile. min V represents the minimum flight speed of the i-th cruise missile. max This represents the maximum flight speed of the i-th cruise missile.

[0124] To achieve simultaneous arrival of multiple cruise missiles at the target, the coordination time is defined as t = min{t1∩t2∩...t}. N}

[0125] Starting from the optimal trajectory, iterate through the alternative trajectories of each cruise missile until t≠0.

[0126] Assumption Let be the final trajectory of the i-th cruise missile, then its cooperative velocity is . This speed will be used in modeling the trajectory planning problem using the pseudospectral method. Before using the pseudospectral method to solve the cruise missile trajectory, first... Interpolation is performed on the initial discrete points of the pseudospectral method to obtain the initial guess values ​​for the three-dimensional position information.

[0127] S3. Construct a second trajectory constraint model based on the cooperative constraints, boundary constraints, control variables and trajectory angle constraints, as well as the second threat zone constraint. Construct a three-dimensional trajectory planning optimal control problem model based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function.

[0128] 1. Constructing a cruise missile dynamics model

[0129] To simplify the problem, we assume the cruise missile is a point mass and ignore its attitude changes. Secondly, based on the mission context, we assume the cruise missile flies at a constant velocity, different types of threats are uniformly represented as cylinders, and communication between cruise missiles is good, without delay or interference. Therefore, the cruise missile dynamics model based on overload is:

[0130]

[0131] Where: N represents the number of cruise missiles, i represents the i-th cruise missile, g represents the gravitational acceleration, and m i V represents the mass of the i-th cruise missile. i Let x represent the velocity of the i-th cruise missile. i y i , z i (n) represents the position of the i-th cruise missile in the ground coordinate system. xi n yi n zi ) represents the triaxial overload component of the i-th cruise missile, (θ) i , ψ Vi ) represents the inclination angle and deflection angle of the i-th cruise missile.

[0132] Define state variables and control variables as follows:

[0133]

[0134] in, The cruise missile dynamics model (15) is then rewritten in the following state-space form:

[0135]

[0136] 2. Construct a second trajectory constraint model based on collaborative constraints, boundary constraints, control variables and trajectory angle constraints, as well as second threat zone constraints.

[0137] (1) Coordination constraints

[0138] Each cruise missile requires collision avoidance during flight, and the communication distance between cruise missiles is limited. Therefore, the cooperative constraints are:

[0139]

[0140] Among them, R i，safe R represents the minimum safe distance for the i-th cruise missile. link This indicates the maximum permissible communication range for cruise missiles.

[0141] (2) Boundary constraints

[0142] The boundary constraints of the cruise missile trajectory optimization problem include initial state constraints and terminal state constraints, as shown below:

[0143]

[0144]

[0145] in: Let t represent the initial state of the i-th cruise missile. f Indicates the time when they arrive at the target simultaneously. This indicates the terminal state of a cruise missile.

[0146] (3) Control variables and trajectory angle constraints

[0147] The control variables for cruise missiles are constrained as follows:

[0148]

[0149] Where: (n ximin n ximax ), (n yimin n yimax ), (n zimin n zimax ) represent the minimum and maximum allowable overload components along the x, y, and z axes of the i-th cruise missile, respectively, and (θ) imin θ imax ), (ψ Vimin , ψ Vimax ) represent the minimum and maximum allowable trajectory inclination and trajectory deflection angles of a cruise missile, respectively.

[0150] (4) Second Threat Zone Constraints

[0151] Suppose there are m cylindrical threat sources, and the center point of the j-th threat source is (xj,threat y j,threat ), radius R j,threat Height is H j,threat The second threat constraint is as follows:

[0152]

[0153] Equations (19) and (20) can be expressed as equality constraints E(x(t0), t0, x(t)). f ), t f Equations (18), (21), and (22) represent inequality constraints C(x(t), u(t), t) ≤ 0, thus yielding the second trajectory constraint model, as shown in the following equation:

[0154]

[0155]

[0156] 3. Second performance index function

[0157] To ensure that multiple cruise missiles arrive at the target point simultaneously, the simultaneous arrival and penetration time of the cruise missile formation is selected as the performance index, i.e., the second performance index function is:

[0158]

[0159] 4. A three-dimensional trajectory planning optimal control problem model is constructed based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function.

[0160]

[0161] S4. After interpolating the initial trajectory obtained in step S2, use it as the initial value guess for the hp-Ruda pseudospectral method. Use the initial velocity obtained in step S2 as the velocity of the optimal control problem. Solve the optimal control problem model of the three-dimensional trajectory planning based on the hp-Ruda pseudospectral method through the sequential quadratic programming algorithm to obtain the cruise missile cooperative penetration offline three-dimensional trajectory planning strategy.

[0162] The specific steps are as follows:

[0163] First, construct the optimal control problem. Substitute the initial velocity obtained in step S2 into equation (15) to obtain the original optimization problem under various constraints, as shown in equation (24).

[0164] Then, the optimal control problem is transformed into a finite-dimensional nonlinear programming problem using the pseudospectral method and solved, as follows:

[0165] Step 1: Time Domain Transformation

[0166] First, define the time interval [t0, t... f Discretize into K subintervals [t] k-1 , t k ], where k = 1, 2, ..., K.

[0167] Subintervals according to rules Perform a transformation so that t∈[t k-1 , t k Transformed into:

[0168] τ∈[-1,1].

[0169] Step 2: Discretization of state and control variables

[0170] Select N in the kth sub-region k Each point N k L-order Lagrange interpolation polynomial representation i (τ), i = 1, ..., N k +1, then we get:

[0171]

[0172] State variable x (k) The expression for (τ) is:

[0173]

[0174] control variable u (k) The expression for (τ) is:

[0175]

[0176] Step 3: Transformation of dynamic differential equations into algebraic equations

[0177] By differentiating the approximate state variables obtained in the second step, we obtain...

[0178]

[0179] Complete N k The transformation from the dynamic differential equations to algebraic equation constraints at LGR points can be obtained as follows:

[0180]

[0181] in:

[0182] Step 4: Discrete Equality Constraints and Inequality Constraints

[0183]

[0184] In summary, the hp-Radau pseudospectral method can discretize the three-dimensional trajectory planning optimal control problem model (24) into a nonlinear programming problem as shown in equation (31). After interpolating the initial trajectory obtained in step S2, it is used as the initial value guess for the sequential quadratic programming method, and then the discretized problem can be solved by the sequential quadratic programming method.

[0185]

[0186] Step 5: Update the judgment criteria and update method

[0187] Suppose that L additional LGR collocation points are generated in the k-th segment using the Barycentric Lagrange interpolation method. The approximate error of the dynamic system within the k-th interval is calculated by the following formula.

[0188]

[0189] Assume the maximum accuracy error is ε max If the following formula is satisfied in the k-th segment, it means that the approximate error of the collocation points in the k-th segment of the dynamic system is satisfied; otherwise, the collocation points are redistributed.

[0190]

[0191] in: Representing vectors respectively The maximum value of the elements in the middle.

[0192] Before refining the mesh, it is necessary to determine in advance whether to increase the number of collocation points or the polynomial order. The curvature function of the k-th segment of the curve is:

[0193]

[0194] make This represents the maximum value of the curvature function of the k-th segment of the curve at the sampling point. This represents the average value of the curvature function of the k-th segment of the curve at the sampling points.

[0195]

[0196] If r k ≤r max When the number of discrete points is increased, the number of discrete points becomes:

[0197]

[0198] in: This indicates the number of points in the k-th segment after the update. This represents the number of collocation points before and after the k-th segment, ceil() represents the floor function, and X represents an arbitrary constant.

[0199] If r k >r max Let's further divide the k-th segment. Then the number of segments after the re-division is:

[0200]

[0201] Where: Y represents an arbitrary constant.

[0202] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A multi-cruise missile cooperative penetration offline three-dimensional trajectory planning method based on PSO-RPM, characterized in that, Includes the following steps: S1. Treat cruise missiles as particles and construct particle velocity and position update models. Construct a first trajectory constraint model based on midway point constraints, range constraints, trajectory angle constraints, and first threat zone constraints. Use the range cost function as the first performance index function. Based on the particle swarm algorithm, solve the first performance index function according to the particle velocity and position update models and the first trajectory constraint model to obtain the candidate trajectory set for each cruise missile. For each cruise missile, its candidate trajectory set includes its optimal candidate trajectory and several suboptimal candidate trajectories. S2. Iterate through the candidate trajectory set of each cruise missile, find the final trajectory of each cruise missile as the initial trajectory, and make the coordination time of each cruise missile... The coordinated velocity of each cruise missile is obtained based on its final trajectory and coordination time, and is used as the initial velocity. S3. Based on the cooperative constraints, boundary constraints, control variables, trajectory angle constraints, and second threat zone constraints, construct a second trajectory constraint model. Then, based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function, construct a three-dimensional trajectory planning optimal control problem model. The second trajectory constraint model is as follows: in, Indicates the first The triaxial overload component of a cruise missile Indicates the first The trajectory inclination and trajectory deflection of a cruise missile. Indicates the first Minimum safe distance for a cruise missile Indicates the maximum permissible communication range for cruise missiles. Indicates the first The initial state of a cruise missile. Indicates the time when they arrive at the target simultaneously. This indicates the terminal state of a cruise missile. They represent the first Minimum and maximum allowable overload components along the x, y, and z axes for a cruise missile; These represent the minimum and maximum allowable trajectory inclination and trajectory deflection angles for cruise missiles, respectively. S4. After interpolating the initial trajectory, use it as the initial value guess. Use the initial velocity as the velocity of the optimal control problem. Solve the optimal control problem model of three-dimensional trajectory planning based on the pseudospectral method through the sequential quadratic programming algorithm to obtain the cruise missile cooperative penetration offline three-dimensional trajectory planning strategy.

2. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 1, characterized in that, The particle velocity and position update model in step S1 is as follows: in, For particle serial numbers, , representing the three-dimensional position coordinates of the missile, the first The velocity vectors of the particles are ,go through After the nth iteration, the th Particle position vectors The best historical position is , Indicates the inertia weighting coefficient. These represent the learning factor and acceleration factor of the particle, respectively. and All Random numbers between and following a uniform distribution Indicates population size, This represents the maximum number of iterations in the particle swarm optimization. This indicates the current iteration number of the particle swarm.

3. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 2, characterized in that, Inertia weight coefficient The linear descent formula is: Among them, the maximum weight for minimum weight for , When the particle velocity exceeds the velocity range, the velocity update formula is: When the particle position exceeds the boundary of the solution space, the position update formula is: in: Indicates the particle's current velocity. Indicates the current position of the particle. The upper bound of the boundary value representing the particle's position. The lower bound of the boundary value representing the particle's position.

4. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 3, characterized in that, The range constraint in step S1 is in, It is the longest permitted flight distance. It is the shortest allowed flight distance. For the first Segment length of the flight path.

5. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 4, characterized in that, The track angle constraint in step S1 is in, The maximum permissible track deviation, It is the maximum allowable track inclination angle, the first The horizontal projection of the segment track is denoted as .

6. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 5, characterized in that, Assume there is The first cylindrical threat source, the... The central point of the threat source is , radius is The height is Take in each track segment The point, the first Each point is recorded as Then the threat zone constraint in step S1 is 。 7. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 6, characterized in that, It has For a given flight path, the first performance index function is: 。 8. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 7, characterized in that, In the particle swarm optimization algorithm, the trajectory planning problem is transformed into the following optimal control problem: in, Represents the fitness value. This represents the fitness evaluation function. Indicates the first Inequality constraints Indicates the number of inequality constraints; To handle constraints, a constraint violation function is introduced: The criteria for evaluating the quality of particles are as follows: (1) When the particle and particles When the constraints are satisfied, if If true, then the particle For better, otherwise particles Excellent; (2) When the particle and particles If the constraint is not satisfied, If true, then the particle For better, otherwise particles Excellent; (3) When the particle Satisfying constraints, particles If the constraint is not satisfied, and Then the particle For better, otherwise particles Of which, This is the predetermined accuracy requirement.

9. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 8, characterized in that, set up For the first The final trajectory of the cruise missiles, their coordinated speed is Where t is the coordination time, , No. Time range of cruise missiles reaching the target for: in, Indicates the first The length of the trajectory of each cruise missile is set. Indicates the first The minimum flight speed of a cruise missile, Indicates the first The maximum flight speed of a cruise missile.

10. The offline three-dimensional trajectory planning method for multi-cruise missile cooperative penetration based on PSO-RPM according to claim 9, characterized in that, Step S3 specifically includes: S31. The dynamic model of the cruise missile is constructed as follows: in, Indicates the number of cruise missiles. Indicates the first cruise missiles, Represents gravitational acceleration. Indicates the first The mass of a cruise missile Indicates the first The speed of a cruise missile Indicates the first The position of a cruise missile in a ground coordinate system; Define state variables and control variables as follows: in, The cruise missile dynamics model can then be rewritten in the following state-space form: ； S32. Construct the second trajectory constraint model; S33. Construct the second performance index function: ； S34. Based on the cruise missile dynamics model, the second trajectory constraint model, and the second performance index function, a three-dimensional trajectory planning optimal control problem model is constructed: 。

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