CN114236543B - A bistatic forward-looking SAR trajectory design method for maneuvering platforms - Google Patents

Abstract

The invention discloses a method for designing a bistatic foresight SAR track of a maneuvering platform, which utilizes GEO-SAR as an irradiation source and a high-speed maneuvering platform as a receiving station to realize foresight imaging, comprehensively considers the flight performance and the imaging performance of the platform, models the actual situation, establishes a proper optimization function and a constraint condition, models the track design problem into a multi-constraint binocular optimization problem, discretizes state variables and control variables, and solves the optimization problem by adopting a multi-objective differential evolution algorithm, thereby realizing the bistatic foresight SAR track design of the high-speed maneuvering platform, calculating a flight path which meets the imaging indexes and can accurately reach a target landing point, and realizing the foresight imaging performance optimization of the high-speed maneuvering platform.

Description

A bistatic forward-looking SAR trajectory design method for maneuvering platforms

technical field

The invention belongs to the technical field of radar, and relates to a method for designing a bibase forward-looking SAR trajectory of a mobile platform based on GEO-SAR satellite irradiation.

Background technique

Synthetic Aperture Radar (SAR) is an all-weather, all-weather high-resolution imaging system. By transmitting a linear frequency modulation (LFM) signal with a large time and wide bandwidth, the distance direction of the echo signal is matched and filtered to obtain a pulse compression signal. In this way, high resolution in the range direction is obtained, and high resolution in the azimuth direction is achieved by using synthetic aperture technology. The imaging quality is not affected by weather conditions (clouds, light), etc., and it has the characteristics of detecting and locating long-distance targets. Typical applications of SAR include disaster monitoring, resource exploration, geological surveying and mapping, etc.

Compared with single-base SAR, bi-base SAR has the following advantages: 1. It has good concealment, bi-base SAR transmits and receives separately, and the receiving platform does not transmit signals, so it is not easy to be detected; 2. It can detect targets from different angles, while traditional single-base SAR Radar can only pick up the backscatter properties of a target. With the double-base configuration, the target can be detected from multiple angles such as front view, side view, and rear view. Especially the ability to detect stealth targets is greatly enhanced; 3. The system is flexible, and the transceiver platforms can be configured arbitrarily according to needs, and they are independent of each other.

Geosynchronous Orbit Synthetic Aperture Radar (GEO-SAR) is a geosynchronous orbit synthetic aperture radar satellite, which operates in a geosynchronous orbit with a certain inclination, and its operation period is the same as the earth's rotation period. With larger mapping bandwidth and shorter revisit period, it can be widely used in disaster monitoring and tectonic imaging. Bistatic GEO-SAR can also improve imaging performance conveniently and efficiently by adjusting the flight parameters of the receiver.

Recently, the research on bistatic SAR has been heating up, and good progress has been made in imaging algorithms, synchronization methods and experiments. Literature "Meng Ziqiang, Li Yachao, Wang Zongfu, Wu Chunfeng, Xing Mengdao, Bao Zheng. Ballistic Design Method for the Subduction Section of Missile-borne Bistatic Forward-Looking SAR[J]. Systems Engineering and Electronic Technology, 2015,37(04):768-774" A trajectory design method for missile-borne bistatic forward-looking SAR is proposed. However, this method only considers the trajectory design of the transmitter. At the same time, the analysis of imaging performance in this design method mainly considers the resolution in the range direction. Other indicators such as signal-to-noise ratio The included angle between the resolution and the resolution was not analyzed; the literature "Resolution calculation and analysis in bistatic SAR with geostationary illuminator," IEEE Geosci. The spatial resolution is analyzed in the case of spherical surface and large equivalent angle, but the analysis method and characteristics of spatial resolution cannot be directly applied to GEO bistatic SAR with non-zero inclination angle.

Contents of the invention

In order to solve the above-mentioned problems existing in the prior art, the present invention proposes a bistatic forward-looking SAR trajectory design method for a mobile platform.

The technical solution of the present invention is: a method for designing a dual-base forward-looking SAR trajectory of a mobile platform, which specifically includes the following steps:

Step S1. Flight trajectory modeling,

分别代表x轴，y轴，z轴三个方向的加速度，假设同一段飞行轨迹中，高速运动平台沿三个方向的加速度都是恒定不变的，通过设计每段飞行轨迹的加速度，再对加速度进行积分，可以得到总的飞行轨迹。In order to plan the flight path of the high-speed motion platform more conveniently, the flight trajectory is divided into N segments on average according to time, and the time of each segment is t. The acceleration of the high-speed motion platform is recorded in the ground coordinate system as

Represent the acceleration in the three directions of x-axis, y-axis and z-axis respectively. Assuming that in the same flight trajectory, the acceleration of the high-speed motion platform along the three directions is constant, by designing the acceleration of each flight trajectory, and then Acceleration is integrated to obtain the total flight trajectory.

Step S2. Establishing an optimization function,

According to the mission objectives, four trajectory performance indicators, control energy, flight time, resolution unit area, and non-imaging time, are comprehensively selected to establish an optimization function.

(1) Control energy: due to the volume limitation of the flight platform, the fuel carried is limited, so the control energy of the flight trajectory is an important index to describe the trajectory performance.

The optimization function governing the energy is established as:

Wherein, i represents the segment number of the flight trajectory, for example: when i=1, it represents the first segment of the flight trajectory. The control energy is the sum of the squares of the accelerations in the three directions in each flight trajectory, and the purpose of optimization is to minimize the control energy of the flight trajectory.

(2) Flight time: Because the longer the flight time, the longer the platform is exposed to, thus increasing the risk of the platform, the optimization function for controlling the flight time is established as:

f ₂ = kNt (2)

Among them, k is the weight coefficient, which is used to ensure that the optimization function of flight time and control energy is in the same order of magnitude, and the purpose of optimization is to minimize the flight time of the platform.

(3) Resolution unit area: The high-speed motion platform is equipped with a synthetic aperture radar receiver to passively receive signals from GEO-SAR satellites. According to mission requirements, bistatic SAR imaging is required for the target area. Each segment of the flight trajectory divided by time is divided into M segments according to the time, and the imaging resolution performance of each sub-segment flight trajectory is calculated.

The optimization function of the resolution unit area is recorded as:

为每一子段的分辨单元面积，分辨单元面积越小，雷达成像性能就越好，从而能准确的识别目标，ρ_gr为距离分辨率，ρ_az为方位分辨率，α是分辨方向夹角。in,

is the resolution unit area of each sub-segment, the smaller the resolution unit area, the better the radar imaging performance, so that the target can be accurately identified, ρ _gr is the range resolution, ρ _az is the azimuth resolution, α is the resolution angle .

Among them, the distance resolution ρ _gr is:

where, c is the speed of light, B _r is the signal bandwidth, H ^⊥ is the ground projection matrix can be expressed as:

是P_G的转置。Among them, I is the unit matrix, _PG is the normal unit vector of the coordinate system of the imaging area,

is the transpose of _PG .

u _TA (t ₀ ) is the unit vector from the target to the transmitting station at time t ₀ , u _RA (t ₀ ) is the unit vector from the target to the receiving station at time t ₀ ;

The azimuth resolution ρ _az is expressed as:

Among them, λ is the carrier wavelength, T _a is the synthetic aperture time, ω _TA (t) is the angular velocity of the transmitting station, and ω _RA (t) is the angular velocity of the receiving station.

The resolution angle α can be expressed as:

α＝cos ^-1 (Ξ Θ) (7)

Among them, Θ represents the unit vector of the distance resolution direction, and Ξ represents the unit vector of the azimuth resolution direction.

(4) Imaging unavailable time: Since the final section of the flight trajectory of the high-speed platform needs to fly in the direction of the target, the imaging performance of the final section of the flight deteriorates sharply, resulting in imaging failure. Let the unimaging time of the flight trajectory be t _{un_image} , so the unimaging time will be controlled The optimization function is written as:

f ₄ =t _{un_image} (10)

It is necessary to minimize the non-imaging time of the end segment to optimize the imaging performance of the whole segment.

In summary, considering the four objective functions of control energy, flight time, resolution unit area, and non-imaging time, it is modeled as a dual-objective optimization function, aiming at the kinetic energy and energy constraints of the flying platform, while considering the minimum control energy and the shortest flight time Create optimization function 1:

Aiming at the imaging performance of the entire flight path of the flight platform, the optimization function 2 is established by considering the minimum resolution unit area and the shortest imaging time:

F ₂ =f ₃ +f ₄ (12)

Taking the above two optimization functions into consideration, the flight path design is carried out.

Step S3. Determine the constraints,

After the optimization function is established, the constraint conditions should be determined according to the task requirements. The main considerations are three constraints: the terminal position of the flying platform, the maneuverability of the flying platform, and the angle of sight.

(1) Flying platform terminal position constraints: According to the mission requirements, the flying platform must ensure that it lands in the target area after flying with a certain trajectory. In order to make the flying platform land at the specified position, the constraints are established:

||[R _dx (Nt)-R _dx ,R _dy (Nt)-R _dy ,R _dz (Nt)-R _dz ]||＝0 (13)

Among them, the position vector of the flight platform at time t [R _dx (t), R _dy (t), R _dz (t)], [R _dx (Nt), R _dy (Nt), R _dz (Nt)] Represents the position vector of the flying platform at landing time Nt, || || means to take the 2-norm operation, that is, to take the length of the vector. The target position vector is [R _tx , R _ty , R _tz ], then finally it can be obtained by ||[R _dx (Nt)-R _dx , R _dy (Nt)-R _dy , R _dz (Nt)-R _dz ]|| Determine whether the terminal constraint is met before the aircraft lands, that is, when the constraint condition (13) is satisfied, the landing position of the flying platform coincides with the target position.

(2) Mobility constraints of the flying platform: Considering the strength of the flying platform, the shape of the fuel and the air resistance during the flight, the acceleration that the flying platform can provide is limited, and the excessive acceleration will affect the stability of the flying platform. Severe cases may lead to stall or disintegration.

Constrain the acceleration (a _x , a y , a z ) in the three directions of x, _y , and _z , as shown in formula (14).

a_{x_max},a_{y_max},a_{z_max}分别为飞行平台沿x，y，z三个方向加速度的最大值。in,

a _{x_max} , a _{y_max} , and a _{z_max} are the maximum acceleration values of the flying platform along the x, y, and z directions respectively.

(3) Sight angle constraint: Since the flight platform needs to image the target area to find the mission point to land during the flight, when the sight angle of the aircraft exceeds the field of view, the target point cannot be observed by the flight platform, which will affect the flight The platform target point specifies the location.

The view angle constraint is set to:

|σ(t)|≤σ _max (1)

Among them, σ(t) represents the line-of-sight angle of the flight platform at time t, and σ _max is the maximum beam pointing angle of the radar antenna of the flight platform.

In summary, the established optimization function and constraints are as follows:

Step S4. Use the multi-constraint differential evolution algorithm (DE) to find the optimal path of the flight platform.

The specific instructions are as follows:

The differential evolution algorithm first generates a random initial population, and generates a new individual by summing the vector difference of any two individuals in the population with the third individual, and then compares the new individual with the corresponding individual in the contemporary population, if the new If the fitness of the individual is better than the fitness of the current individual, the new individual will replace the old individual in the next generation, otherwise the old individual will be kept. Through continuous evolution, good individuals are retained, low-quality individuals are eliminated, and the search is guided to approach the optimal solution.

The main control parameters of DE algorithm include: population size (NP), scaling factor (F) and crossover probability (CR). NP mainly reflects the amount of information in the population, the crossover probability (CR) represents the degree of information exchange between each generation of variants, and the scaling factor (F) is the factor that has the greatest impact on the performance of the algorithm, mainly affecting the overall performance of the algorithm Optimizing ability.

The specific sub-steps of the differential evolution algorithm are as follows:

Step S41. According to different scenarios, select an appropriate population size (NP), maximum number of iterations (GM), crossover probability (CR) and scaling factor (F).

如式(17-18)所示Step S42. Let X _i,G =(x _1,i,G ,x _2,i,G ,x _3,i,G ,…,x _D,i,G ) represent the i-th individual in the G-th generation , D is the dimension of the optimization problem, x _1,i,G ,x _2,i,G ,x _3,i,G ,…,x _D,i,G represent the value of each search variable of the individual, x _{j, i, 0} shall be randomly generated within the specified maximum value x _{j, max} and minimum value x _{j, min} , where,

As shown in formula (17-18)

Randomly generate the initial population:

x _j,i,0 = x _j,min +rand _ij [0,1]×(x _j,max -x _j,min ) (18)

Among them, rand _ij [0,1] represents a uniformly distributed random number between 0 and 1.

Step S43. mutating through formula (19) to form a new intermediate individual,

是随机从种群中选取的三个互不相同的向量。where F is the scaling factor for scaling,

are three different vectors randomly selected from the population.

Step S44. After generating the intermediate individual vector, start to generate trajectory vector u _i,G =[u _1,i,G ,u _2,i,G ,…,u _D,i,G ] through crossover operation, the crossover operation can increase individual In the most commonly used binomial crossover operator, as long as the randomly generated number between 0 and 1 is less than or equal to the value Cr, it will be exchanged with the current individual x _i,G , as shown in formula (20) Show:

where j _rand ∈ [1,2,…,D] is an index randomly selected to ensure that at least one component of u _i,G is selected from v _i,G .

Step S45. Finally, calculate the optimization function and constraint conditions of the offspring population obtained after mutation and crossover, and select each group of parent individuals and offspring individuals according to the feasibility criterion:

1) If both the parent individual and the child individual are infeasible solutions, then select the individual with a smaller constraint violation;

2) If one of the parent individual and the child individual is a feasible solution and the other is an infeasible solution, then choose a feasible solution

3) If both the parent individual and the child individual are feasible solutions, then select the individual with the smallest sum F ₁ + F ₂ of the objective function as the next generation individual, and select each group of parent and child individuals to obtain the next generation population, When the termination condition is reached or the evolution algebra reaches the maximum, the evolution is terminated, and the best individual is output as the optimal solution to obtain the optimal path.

Beneficial effects of the present invention: the method of the present invention utilizes GEO-SAR as the irradiation source, and the high-speed maneuvering platform as the receiving station to realize forward-looking imaging, comprehensively considers the flight performance and imaging performance of the platform, models the actual situation, and establishes a suitable optimization function and constraint conditions, the trajectory design problem is modeled as a multi-constrained dual-objective optimization problem, and then the state variables and control variables are discretized, and the multi-objective differential evolution algorithm is used to solve the optimization problem, thus realizing the bi-basic forward-looking SAR trajectory design can calculate a flight path that satisfies the imaging index and can accurately reach the target landing point, so as to realize the optimization of the forward-looking imaging performance of the high-speed motion platform.

Description of drawings

Fig. 1 is a schematic diagram of the geometry used in the embodiment of the present invention.

Fig. 2 is a flowchart of a method provided by an embodiment of the present invention.

Fig. 3 is a schematic diagram of the three-dimensional flight trajectory of the bistatic forward-looking SAR of the high-speed maneuvering platform generated by the embodiment of the present invention.

Fig. 4 is a schematic diagram of the flight trajectory of the bistatic forward-looking SAR XY plane of the high-speed maneuvering platform generated by the embodiment of the present invention.

Fig. 5 is a schematic diagram of the flight trajectory of the bistatic forward-looking SAR ZY plane generated by the embodiment of the present invention.

Fig. 6 is a schematic diagram of the flight trajectory of the bistatic forward-looking SAR ZX plane generated by the embodiment of the present invention.

Fig. 7 is a schematic diagram of the distance resolution of the optimal flight trajectory at the end stage of the high-speed aircraft according to the embodiment of the present invention.

Fig. 8 is a schematic diagram of the azimuth resolution of the optimal flight trajectory at the end stage of the high-speed aircraft according to the embodiment of the present invention.

Detailed ways

The present invention will be further described in detail below in combination with specific embodiments.

The geometric structure diagram of the embodiment of the present invention is as shown in Figure 1, and the GEO-SAR system parameter that the specific implementation mode adopts is as shown in Table 1, and the satellite selects the geosynchronous orbit satellite 36,000 kilometers away from the earth's surface, in the present embodiment Think of it as stationary relative to the target, the incident angle of the satellite is the angle between the star-eye line and the Z axis)

Table 1

The specific process is shown in Figure 2, including the following steps:

Step S1: Divide the flight trajectory into N segments on average according to time, as shown in Figure 2, the time of each segment is t, and the acceleration of the high-speed motion platform is recorded as [a _x , a _y , a _z ] in the ground coordinate system, Represent the acceleration in the three directions of x-axis, y-axis and z-axis respectively.

Step S2: According to the task objective, comprehensively select the four factors of the minimum control energy, the shortest flight time, the smallest resolution unit area, and the shortest imaging failure time to establish an optimization function.

Step S3: After establishing the optimization function, the constraint conditions should be determined according to the task requirements. The main considerations are three constraints: the terminal position of the flying platform, the maneuverability of the flying platform, and the angle of sight.

Step S4: Use the differential evolution algorithm (DE) to find the optimal path of the flying platform. Specific steps are as follows:

Step S41. Set appropriate population size (NP), scaling factor (F) and crossover probability (CR).

如式(23-24)所示Step S42. Let X _i,G =(x _1,i,G ,x _2,i,G ,x _3,i,G ,…,x _D,i,G ) represent the i-th individual in the G-th generation , D is the dimension of the optimization problem, x _j,i,0 should be randomly generated within the range of the specified maximum value x _j,max and minimum value x _j,min , where

As shown in formula (23-24)

Randomly generate the initial population:

x _j,i,0 = x _j,min +rand _ij [0,1]×(x _j,max -x _j,min ) (24)

Among them, rand _ij [0,1] represents a uniformly distributed random number between 0 and 1.

Step S43: mutating through formula (25) to form a new intermediate individual,

是随机从种群中选取的三个互不相同的向量。where F is the scaling factor for scaling,

are three different vectors randomly selected from the population.

Step S44. Generate trajectory vector u _i,G =[u _1,i,G ,u _2,i,G ,...,u _D,i,G ] through crossover, if the randomly generated number between 0 and 1 is less than Or equal to the value Cr, it will be exchanged with the current individual x _{i, G} , as shown in formula (26):

where j _rand ∈ [1,2,…,D] is an index randomly selected to ensure that at least one component of u _i,G is selected from v _i,G .

Step S45. The optimization function and constraint conditions of the offspring population obtained after mutation and crossover are selected according to the feasibility criterion for each group of parent and offspring individuals:

1) If both the parent and child individuals are infeasible solutions, select the individual with a smaller constraint violation;

2) If one of the parent and child is a feasible solution and the other is an infeasible solution, then choose a feasible solution;

3) If both the parent and offspring are feasible solutions, select the individual with the smallest objective function sum F ₁ +F ₂ as the next generation individual, and select each group of parent and offspring individuals to obtain the next generation population.

When the termination condition is reached or the evolution algebra reaches the maximum, the evolution is terminated, and the best individual is output as the optimal solution to obtain the optimal path.

The optimal path is shown in Figure 3. In order to further display the details of the optimal path, the cross-sectional views of the X-Y, Y-Z, and X-Z planes of the optimal path are drawn respectively, as shown in Figure 4, Figure 5, and Figure 6; Figure 7, Figure 8 are the range resolution and azimuth resolution curves under the optimal path, respectively. It can be seen that the imaging performance under this path meets the task index.

It can be seen from the specific embodiments of the present invention that the present invention can realize the planning of the flight trajectory of the bistatic forward-looking SAR of the high-speed maneuvering platform, by modeling the flight trajectory, establishing optimization functions and constraint conditions, and modeling the problem as a multi-constraint For the dual-objective optimization problem, the discretization method is used to discretize the state variables and control variables, and finally the differential evolution algorithm is used to obtain the optimal solution. The method of the invention solves the problem of bibase forward-looking SAR flight trajectory planning of the high-speed maneuvering platform, can be used in the flight trajectory design of the high-speed maneuvering platform, and can be used in fields such as earth remote sensing, autonomous landing, and autonomous navigation.

Claims (2)

1. A dual-base forward-looking SAR trajectory design method for a mobile platform, specifically comprising the steps of: Step S1. Flight trajectory modeling, Divide the flight trajectory into N segments on average according to time, and the time of each segment is t. The acceleration of the high-speed motion platform is recorded in the ground coordinate system as

Represent the acceleration in the three directions of x-axis, y-axis and z-axis respectively; Step S2. Establishing an optimization function, According to the mission objectives, four trajectory performance indicators, including control energy, flight time, resolution unit area, and non-imaging time, are selected to establish an optimization function: (1) Control energy: the optimization function of control energy is established as:

Among them, i represents the segment number of the flight trajectory, and the control energy is the sum of the squares of the accelerations in the three directions in each segment of the flight trajectory; (2) Time-of-flight: The optimization function for controlling the time-of-flight is established as: f ₂ = kNt Among them, k is the weight coefficient, which is used to ensure that the optimization function of flight time and control energy is in the same order of magnitude; (3) Resolution unit area: divide each segment of the flight trajectory into M segments according to time, and calculate the imaging resolution performance of each sub-segment flight trajectory, The optimization function of the resolution unit area is recorded as:

in,

is the resolution unit area of each sub-section, ρ _gr is the distance resolution, ρ _az is the azimuth resolution, and α is the angle between resolution directions; The range resolution ρ _gr is:

where, c is the speed of light, B _r is the signal bandwidth, H ^⊥ is the ground projection matrix can be expressed as:

Among them, I is the unit matrix, _PG is the normal unit vector of the coordinate system of the imaging area,

is the transpose of _PG ; u _TA (t ₀ ) is the unit vector from the target to the transmitting station at time t ₀ , u _RA (t ₀ ) is the unit vector from the target to the receiving station at time t ₀ ; The azimuth resolution ρ _az is expressed as:

Wherein, λ is the carrier wavelength, T _a is the synthetic aperture time, ω _TA (t) is the angular velocity of the transmitting station, and ω _RA (t) is the angular velocity of the receiving station; The resolution angle α is expressed as: α＝cos ^-1 (Ξ·Θ) Wherein, Θ represents the unit vector of the distance resolution direction, Ξ represents the unit vector of the azimuth resolution direction,

(4) Unimaging time: Let the flight path unimaging time be t _{un_image} , and the optimization function controlling the unimaging time is recorded as: f ₄ =t _{un_image} Considering the four objective functions of control energy, flight time, resolution unit area, and non-imaging time, it is modeled as a dual-objective optimization function, aiming at the kinetic energy and energy constraints of the flight platform, while considering the minimum control energy and the shortest flight time to establish an optimization function. :

Aiming at the imaging performance of the entire flight path of the flight platform, the optimization function 2 is established by considering the minimum resolution unit area and the shortest imaging time: F ₂ =f ₃ +f ₄ Step S3. Determine the constraints, Determine the three constraints of the terminal position of the flying platform, the maneuverability of the flying platform, and the angle of sight; (1) Flying platform terminal position constraint: In order to make the flying platform fall at the specified position, the constraint conditions are established: ||[R _dx (Nt)-R _dx ,R _dy (Nt)-R _dy ,R _dz (Nt)-R _dz ]||＝0 Among them, it is specified that the position vector of the flying platform at time t is [R _dx (t), R _dy (t), R _dz (t)], [R _dx (Nt), R _dy (Nt), R _dz (Nt) ] _{represents the} position vector _of the flight platform at landing time Nt _, || )-R _dx ,R _dy (Nt)-R _dy ,R _dz (Nt)-R _dz ]||Determine whether the aircraft meets the terminal constraints before landing; (2) Maneuverability constraints of the flying platform: constrain the accelerations (a _x , a y , a z ) in the three directions of x, _y , and _z :

in,

a _{x_max} , a _{y_max} , and a _{z_max} are the maximum acceleration values of the flying platform along the x, y, and z directions respectively; (3) Sight angle constraint: the sight angle constraint is set to: |σ(t)|≤σ _max , where σ(t) represents the sight angle of the flight platform at time t, and σ _max is the maximum beam pointing angle of the radar antenna of the flight platform ; The established optimization function and constraints are as follows:

s.t. ||[R _dx (Nt)-R _dx ,R _dy (Nt)-R _dy ,R _dz (Nt)-R _dz ]||＝0

|σ(t)|≤σ _max Step S4. Use the multi-constraint differential evolution algorithm to find the optimal path of the flight platform. 2. a kind of mobile platform bibase forward-looking SAR trajectory design method according to claim 1, the concrete steps of step S4 are: Step S41. According to different scenarios, select the population size (NP), the maximum number of iterations (GM), the crossover probability (CR) and the scaling factor (F); Step S42. Let X _i,G =(x _1,i,G ,x _2,i,G ,x _3,i,G ,…,x _D,i,G ) represent the i-th individual in the G-th generation , D is the dimension of the optimization problem, x _1,i,G ,x _2,i,G ,x _3,i,G ,…,x _D,i,G represent the value of each search variable of the individual, x _{j, i, 0} shall be randomly generated within the specified maximum value x _{j, max} and minimum value x _{j, min} , where,

Randomly generate the initial population: x _j,i,0 ＝x _j,min +rand _ij [0,1]×(x _j,max -x _j,min ) Among them, rand _ij [0,1] represents a uniformly distributed random number between 0 and 1; Step S43. Pass

Mutate to form a new intermediate individual, where F is the scale factor of scaling,

are three different vectors randomly selected from the population; Step S44. Generate trajectory vector u _i,G =[u _1,i,G ,u _2,i,G ,…,u _D,i,G ] by intersection, as long as the randomly generated numbers between 0 and 1 are less than Or equal to the value Cr, it will be exchanged with the current individual x _i,G :

where j _rand ∈ [1,2,…,D] is an index randomly selected, ensuring that at least one component of u _i,G is selected from v _i,G ; Step S45. Calculate the optimization function and constraint conditions of the offspring population obtained after mutation and crossover respectively, and select each group of parent and offspring individuals according to the feasibility criterion: If both the parent and child individuals are infeasible solutions, select the individual with a smaller constraint violation; If one of the parent and child is a feasible solution and the other is an infeasible solution, then choose a feasible solution; If the parent and child are both feasible solutions, select the individual with the smallest objective function sum F ₁ + F ₂ as the next generation individual, and select each group of parent and child individuals to obtain the next generation population, when the termination condition is reached Or terminate the evolution when the evolution algebra reaches the maximum, and output the best individual as the optimal solution to obtain the optimal path.

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