CN105279581A - GEO-UAV Bi-SAR route planning method based on differential evolution - Google Patents

Abstract

The invention discloses a GEO (Geosynchronous orbit)-UAV (unmanned aerial vehicle) Bi-SAR (synthetic aperture radar) route planning method based on differential evolution. The GEO-UAV Bi-SAR route planning method based on differential evolution comprises 1) generating a three dimensional landform; 2) modeling a UAV accepting station route; 3) modeling the route planning as a multiobjective optimization problem for a constraint condition; 4) utilizing a multiobjective differential evolution algorithm to solve; and 5) obtaining the optimal solution, generating a UAV optimal path, and realizing autonomous navigation and Bi-SAR imaging of the UAV in the three dimensional complicated landform. The GEO-UAV Bi-SAR route planning method based on differential evolution models the UAV route planning problem which comprehensively considers the route length, the flight safety and the SAR imaging performance as a multiobjective optimization problem, and utilizes the improved differential evolution algorithm to solve and obtain a set of optimal UAV accepting station flight routes.

Description

GEO-UAV bistatic SAR path planning method based on differential evolution

Technical Field

The invention belongs to the technical field of radars, and particularly relates to a flight path planning method for a UAV receiving station of a GEO-UAV bistatic SAR.

Background

Synthetic Aperture Radar (SAR) is a full-time and all-weather high-resolution imaging system, and can obtain distance high resolution by transmitting large time-width product linear frequency modulation signal and receiving it, and can obtain pulse compression signal by means of matched filtering so as to obtain high resolution in direction of distance.

Geosynchronous orbit synthetic aperture radar (GEO-SAR) has larger mapping bandwidth and shorter revisit period compared with low orbit SAR, so that the geosynchronous orbit synthetic aperture radar can be widely applied to the fields of disaster monitoring, earth structure imaging and the like. By using the GEO-SAR satellite as a radiation source, an airborne or unmanned aerial vehicle receiving station can passively receive the echo of a target scene to realize high-resolution bistatic SAR imaging.

In the GEO-UAV bistatic SAR, a flight task and a bistatic SAR imaging task in three-dimensional complex terrain can be autonomously completed by presetting a flight path of a UAV receiving station. The GEO-UAV bistatic SAR path planning is to find a group of paths which can realize the optimal imaging performance of the bistatic SAR and ensure the flight safety and the shorter flight time in the three-dimensional terrain. In the literature, "routeplananning for planning and estimating a route of a UAV and a method of estimating a route of the UAV based on the UAV, wherein the UAV is modeled as a single-objective optimization problem, and a method of path planning based on a particle swarm differential evolution algorithm (DEQPSO) is proposed. Meanwhile, in the document "Three-dimensional orthogonal of inflatapthplanning for uavsusingmultiobjective evolution of goal, 2007, pp.3195-3202", the Three-dimensional UAV path planning problem is modeled as a binocular optimization problem, the influences of path length and path threat are respectively considered, and a multi-objective genetic algorithm nsgai is adopted for solving. However, the above methods do not consider the imaging performance of the UAV as a bistatic SAR receiving station, and therefore cannot be applied to the path planning problem of GEO-UAV bistatic SAR.

Disclosure of Invention

The invention aims to design a GEO-UAV bistatic SAR path planning method based on a multi-objective differential evolution algorithm aiming at the defects in the background art, and solve the problem of optimal path design of a UAV receiving station.

The specific technical scheme of the invention is as follows: the GEO-UAV bistatic SAR path planning method based on differential evolution is provided, and specifically comprises the following steps:

step 1: generating three-dimensional terrain

Generating a background three-dimensional terrain of the UAV path from the digital map according to the geographic location of the imaging scene. In addition, the simulated terrain can be obtained by numerical simulation of the following formula

$\begin{array}{c}z\left(x,y\right)=\sin \left(y+a\right)+b\·\sin \left(x\right)+c\·\cos \left(d\·\sqrt{y^2+x^2}\right)\\ +e\·\cos \left(y\right)+f\·\sin \left(f\·\sqrt{y^2+x^2}\right)+g\·\cos \left(y\right)\end{array}---\left(1\right)$

Wherein x and y are two-dimensional horizontal coordinates of the ground, z is the ground height, and a, b, c, d, e, f and g are first-order to seventh-order terrain parameters.

Step 2: UAV receiving station path modeling

Modeling the path of the UAV receiving station as a group of control points of a spline curve, and assuming that the number of the control points is N_cThe starting point and the ending point of the path are respectively marked as P_startAnd P_endThe imaging point of the UAV receiving station path is P_im. In addition to the three determined control points described above, the remaining N_c-3 control points are free control points. The sequence of control points of the spline curve can be represented as

S_ctrl＝(P_start,P₁,...,P_mid,P_im,P_mid+1,...P_n,P_end)(2)

Wherein, P_midAnd P_mid+1Is an imaging point P_imIs equal to (N)_c-3)/2. And P is_imIs represented by the three-dimensional coordinate P_midAnd P_mid+1Solving:

$P_{im}=\frac{P_{mid}+P_{mid+1}}{2}---\left(3\right)$

modeling by the UAV receiving station path, S_ctrlUAV path of generated spline curve from required starting point P_startMove to P_endAnd passes through the imaging point P_im。

And step 3: modeling path planning as a multi-objective optimization problem

The UAV path is first discretized and path discrete points are represented asN_disFor the number of discrete points, the path distance can be calculated by the following formula

Wherein,for the length of the ith discrete path, the UAV receiving station needs to be at a safe distance from the ground. Assume a minimum safe distance of r_safeThen the threat value of terrain to the UAV path is given by

$f_{threat}\left(x\right)=\underset{i=1}{\overset{N_{dis}}{\Σ}}\underset{j=1}{\overset{Ng}{\Σ}}{(r_{safe}/r_{i,j})}^2---\left(5\right)$

Wherein N is_gThe number of the terrain grid points. r is_i,jIs the distance between the ith path discrete point and the jth terrain grid point.

In addition, the path of the UAV satisfies two conditions. First, the path cannot collide with the terrain; second, the rotational angle of the UAV path cannot exceed the actual maximum rotational angle θ_max. Recording the number of points colliding with the terrain in the path discrete points as N_cons1Angle of rotation exceeding theta_maxThe number of discrete points of (2) is N_cons2. Then N is required_cons10 and N_cons2＝0。

On the other hand, aiming at the imaging performance of the GEO-UAV bistatic SAR, the area of a resolution unit is used as a measurement index. The area of the resolution cell can be expressed as

$S_{cell}=\frac{{\mathrm{\ρ}}_{az}\·{\mathrm{\ρ}}_{gr}}{\sin \mathrm{\α}}---\left(6\right)$

Where ρ is_grFor distance resolution:

${\mathrm{\ρ}}_{gr}=\frac{0.886c}{B_r||H^{\⊥}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T||}---\left(7\right)$

where c is the speed of light, t₀Is the imaging center time, B_rIs the signal bandwidth, H^⊥It is the ground projection matrix that can be expressed as:

$H^{\⊥}=I-P_G\·P_G^T---\left(8\right)$

wherein I is the identity matrix, P_GIs the normal unit vector of the imaging region coordinate system,is P_GThe transposing of (1). u. of_TA(t₀) Is at t₀The unit vector from the time target to the transmitting station can be obtained through satellite-ground coordinate conversion.

u_RA(t₀) Is at t₀Unit vector of time target to receiving station:

$u_{RA}\left(t_0\right)=\frac{P_A-P_{im}}{||P_A-P_{im}||}---\left(9\right)$

wherein, P_AIs the target point position, P_imIs the location of the receiving station.

Azimuth resolution:

${\mathrm{\ρ}}_{az}=\frac{0.886\mathrm{\λ}}{{\∫}_{t_0-T_a/2}^{t_0+T_a/2}||H^{\⊥}\left({\mathrm{\ω}}_{TA}\right(t)+{\mathrm{\ω}}_{RA}(t\left)\right)||dt}---\left(10\right)$

wherein λ is the carrier wavelength, T_aFor synthesizing the aperture time, omega_TA(t) is the angular velocity of the transmitting station, ω_RA(t) is the angular velocity of the receiving station:

${\mathrm{\ω}}_{TA}\left(t\right)=\frac{\[I-u_{TA}^T\left(t_0\right)u_{TA}\left(t_0\right)\]V_T^T}{||P_A-R_T\left(t_0\right)||}---\left(11\right)$

${\mathrm{\ω}}_{RA}\left(t\right)=\frac{\[I-u_{RA}^T\left(t_0\right)u_{RA}\left(t_0\right)\]V_R^T}{||P_A-P_{im}||}---\left(12\right)$

wherein R is_T(t₀) The position coordinates of the transmitting station at the moment of the imaging center,which is a transpose of the velocity vector of the transmitting station,is a transpose of the velocity vector of the transmitting station.

Resolving the included angle of the directions:

α＝cos^-1(Ξ·Θ)(13)

wherein Θ denotes a unit vector in the distance resolution direction, and xi denotes a unit vector in the azimuth resolution direction:

$\mathrm{\Θ}=\frac{H^{\⊥}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T}{||u_{TA}\left(t_0\right)+u_{RA}\left(t_0\right)||}---\left(14\right)$

$\mathrm{\Ξ}=\frac{H^{\⊥}{\left({\mathrm{\ω}}_{TA}\right(t_0)+{\mathrm{\ω}}_{RA}(t_0\left)\right)}^T}{||{\mathrm{\ω}}_{TA}\left(t_0\right)+{\mathrm{\ω}}_{RA}\left(t_0\right)||}---\left(15\right)$

therefore, the multi-objective optimization problem obtained by modeling the path planning problem can be expressed as:

$\left\{\begin{array}{cc}min& F_1\left(x\right)=w_1{f}_{dis}\left(x\right)+w_2{f}_{threat}\left(x\right)\\ min& F_2\left(x\right)=S_{cell}\left(x\right)\end{array}\right.---\left(16\right)$

s.t.N_consi＝0,i＝1,2

wherein, w₁And w₂Respectively, a path length function and a weighting factor for a terrain threat function.

And 4, step 4: solving by adopting multi-target differential evolution algorithm

4.1 initializing iteration parameters

Initializing iteration parameters of the multi-target differential evolution algorithm, including the group size N and the maximum iteration number G_maxScaling factor F and crossover rate C_rRandomly generating an initial population X_GAnd G ═ 0, comprising N individuals.

4.2 Cross mutation

For the G generation population X_GEach of the individuals x in_i,GI-1, 2, …, N, to generate a new individual v_i,G：

$v_{i,G}=x_{r_1^i,G}+F\·(x_{r_2^i,G}-x_{r_3^i,G})---\left(17\right)$

Wherein,andis X_GThree individuals randomly selected.

Obtaining N new individuals v_i,GI is 1,2, …, N, then carrying out mutation operation to obtain test population U_G. Test population U_GEach individual u in_i,GCan be expressed as u_i,G＝[u_1,i,G,u_2,i,G,...,u_D,i,G]Where D is the number of decision variables. Each decision variable u_j,i,GCan be derived from the following formula:

$u_{j,i,G}=\left\{\begin{array}{cc}{v}_{j,i,G}& if({\mathrm{rand}}_{i,j}\[0,1\]\≤Crorj=j_{rand})\\ x_{j,i,G}& otherwise\end{array}\right.---\left(18\right)$

through the cross mutation operation, the G-th generation group X is obtained_GTest population U of_GAnd X is_GAnd U_GCombining to obtain population R_G＝X_G∪U_G。

4.3 non-dominated sorting and Next Generation population selection

Using a pair of non-dominant selection algorithms under constraint conditions_G2N individuals were sorted. For merged population R_GGiven arbitrarily two individuals x_i,GAnd x_j,G. If x_i,GX satisfying the constraint condition_j,GIf the constraint is not satisfied, x_i,GDominating x_j,G(ii) a If none of the constraints is satisfied, and x_i,GIs less than x_j,G，x_i,GDominating x_j,G. If x_i,GAnd x_j,GAll satisfy the constraint condition, their objective function values, i.e. F in the formula (16), are compared₁And F₂If x_i,GHas an objective function value of less than x_j,GThen x_i,GDominating x_j,G. The "constraints" herein can be regarded as prior art in the field and will not be described in detail.

To R_GComparing the domination relations between each individual, sorting 2N individuals according to the domination relations in the population, and selecting the individuals with the highest domination level from the N individuals to form a next generation population X_G+1。

4.4 judging Loop termination conditions

Updating the iteration number G +1, if G is equal to G_maxIf yes, executing step 5; if the iteration number G is less than G_maxThen return to step 4.2.

And 5: generating an optimal path for a UAV receiving station

Final generation population by iteration in step 4I.e. the optimal solution of the multi-objective optimization problem (16). For each optimal solutionA UAV receiving station path may be generated and at an imaging point P_imAnd realizing GEO-UAV bistatic SAR imaging.

The invention has the beneficial effects that: the method of the invention generates a three-dimensional terrain according to a digital map or simulation, and then models the UAV path as a control point sequence of a spline function; and then establishing a multi-objective optimization problem of UAV path planning, and solving by adopting a multi-objective differential evolution algorithm to obtain a plurality of groups of UAV optimal paths. Specifically, the GEO-UAV bistatic SAR path planning is carried out by utilizing a multi-target differential evolution algorithm, firstly, a simulated three-dimensional complex terrain is generated through a digital map or simulation, and then, the flight path of the UAV receiving station is modeled into a group of control point sequences of a spline curve; secondly, modeling GEO-UAV bistatic SAR path planning as a multi-target optimization problem under a constraint condition, solving the optimization problem by adopting a multi-target differential evolution algorithm to obtain the optimal paths of a plurality of groups of UAV receiving stations, and meeting the requirements of different UAV three-dimensional terrain navigation and bistatic SAR imaging performance. The method of the invention utilizes the advantages of strong global search capability and good stability of the differential evolution algorithm, and simultaneously obtains a plurality of groups of UAV receiving station paths, so that the GEO-UAV bistatic SAR system can be widely applied to the fields of earth remote sensing, resource exploration, geological mapping and the like.

Drawings

FIG. 1 is a schematic flow diagram of the process of the present invention.

Fig. 2 is a schematic diagram of a UAV path simulation result 1 according to an embodiment of the present invention.

Fig. 3 is a schematic diagram of a UAV path simulation result 2 according to an embodiment of the present invention.

Fig. 4 is a schematic diagram of a UAV path simulation result 3 according to an embodiment of the present invention.

Fig. 5 is a schematic diagram of UAV path imaging results according to an embodiment of the present invention.

Detailed Description

The embodiments of the present invention will be further described with reference to the accompanying drawings.

The invention mainly adopts a simulation experiment method for verification, and all the steps and conclusions are verified in Matlab 2013.

The method comprises the following steps: and (3) generating a three-dimensional simulated terrain according to the formula (1). To simulate real terrain height and the surface, the following terrain parameters are selected: a, b, e, g is 1, c, d, f is 1.8. The two-dimensional grid spacing of the terrain is 0.2km, and the total length is 20 km. The terrain height information matrix is denoted as Z (x, y).

Step two: and modeling the path of the UAV receiving station. According to equation (2), the UAV receiver station path is modeled as a set of control point coordinates, x ═ x₁,y₁,z₁),(x₂,y₂,z₂),...,(x_n,y_n,z_n) In the simulation analysis, the number n of UAV path control points is set to 9, wherein the first point is a path starting point, the coordinates of the first point are set to (4,4,0.3) km, the most preferable control point is a path terminating point, the coordinates of the most preferable control point are set to (18,17,1.4) km, and the GEO-UAV bistatic SAR parameters are shown in table 1.

TABLE 1

Step three: the path planning problem is modeled as a multi-objective optimization problem, and for a set of control points x, a continuous path spline curve is generated first. Discretizing a path spline curve into N_disA discrete point, N_dis1000; when the path length is calculated, firstly, the straight-line distance between every two adjacent discrete path points is calculated, and then all the straight-line distances are summed; in addition, r_safeTaking the diameter as 200 m; finally, a dual-target optimization model under the constraint condition is established according to the formula (16), wherein the first target function F₁(x) Is set to w and the terrain threat weighting coefficient₁＝w₂＝0.5。

Step four: and (3) solving the optimization problem in the formula (16) by adopting a multi-target differential evolution algorithm. The parameters of the differential algorithm are first initialized as shown in table 2.

TABLE 2

Randomly generating initial N individuals to form an initial group X₀Performing cross mutation operation according to formulas (17) and (18) to generate a test population, and combining with the initial population to obtain a combined population R containing 2N individuals₀。

At R₀Middle pairEach individual is subjected to non-domination sorting, the individual with the highest domination level is ranked at the top, the individual with the lowest domination level is ranked at the last, and N individuals with high domination levels are selected to form a next generation group X₁. Adding 1 to the iteration number G and judging whether the iteration number G exceeds the highest iteration number G_maxIf not, returning to the step 4.2 to continue the cross mutation, combination and non-dominant sorting, and updating the individuals in the population until the iteration number reaches G_maxAnd jumping out of the loop to obtain an optimal solution matrixFIG. 1 is a flow chart of a multi-objective differential evolution algorithm.

Step five: selecting an optimal solution matrixGenerates a path for the UAV receiving station, and measures the navigation performance and GEO-UAV bistatic SAR imaging performance of its UAV. Three in FIG. 2, FIG. 3 and FIG. 4Have different path lengths, terrain threats, and bistatic SAR imaging performance. In FIG. 5, (a), (b) and (c) are the UAV receiving station to the target point P in three paths respectively_AA contour map of the imaging results of (1).

Claims (1)

1. A GEO-UAV bistatic SAR path planning method specifically comprises the following steps:

step 1: generating three-dimensional terrain

Generating a background three-dimensional terrain of the UAV path through a digital map according to the geographic position of an imaging scene, and specifically obtaining a simulated terrain through numerical simulation according to the following formula:

$\begin{array}{c}z\left(x,y\right)=\sin \left(y+a\right)+b\·\sin \left(x\right)+c\·\cos \left(d\·\sqrt{y^2+x^2}\right)\\ +e\·cos\left(y\right)+f\·\sin \left(f\·\sqrt{y^2+x^2}\right)+g\·\cos \left(y\right)\end{array}---\left(1\right)$

wherein x and y are two-dimensional horizontal direction coordinates of the ground, z is the height of the ground, and a, b, c, d, e, f and g are first-order to seventh-order topographic parameters;

step 2: UAV receiving station path modeling

Modeling the path of the UAV receiving station as a group of control points of a spline curve, and assuming that the number of the control points is N_cThe starting point and the ending point of the path are respectively marked as P_startAnd P_endThe imaging point of the UAV receiving station path is P_imN remains of_c-3 control points are free control points, and the sequence of control points of the spline curve can be expressed as:

S_ctrl＝(P_start,P₁,...,P_mid,P_im,P_mid+1,...P_n,P_end)(2)

wherein, P_midAnd P_mid+1Is an imaging point P_imIs equal to (N)_c-3)/2, and P_imIs represented by the three-dimensional coordinate P_midAnd P_mid+1Solving:

$P_{im}=\frac{P_{mid}+P_{mid+1}}{2}---\left(3\right)$

UAV receiving station path modeling by equations (2), (3), S_ctrlUAV path of generated spline curve from required starting point P_startMove to P_endAnd passes through the imaging point P_im；

And step 3: modeling path planning as multi-objective optimization

Discretizing the UAV path and representing path discrete points asN_disFor the number of discrete points, the path distance is calculated by:

wherein,for the length of the ith discrete path, the UAV receiving station needs to maintain a certain safety distance from the ground, assuming that the minimum safety distance is r_safeThen the threat value of terrain to UAV path is as follows:

$f_{threat}\left(x\right)=\underset{i=1}{\overset{N_{dis}}{\Σ}}\underset{j=1}{\overset{Ng}{\Σ}}{(r_{safe}/r_{i,j})}^2---\left(5\right)$

wherein N is_gFor the number of topographical grid points, r_i,jThe distance between the ith path discrete point and the jth terrain grid point;

the path of the UAV also satisfies two conditions: the path cannot collide with the terrain; the rotation angle of the UAV path cannot exceed the actual maximum rotation angle theta_max；

Recording the number of points colliding with the terrain in the path discrete points as N_cons1Angle of rotation exceeding theta_maxThe number of discrete points of (2) is N_cons2. Then N is required_cons10 and N_cons2＝0；

Aiming at the imaging performance of the GEO-UAV bistatic SAR, the area of a resolution unit is taken as a measurement index, and the area of the resolution unit is expressed as follows:

$S_{cell}\left(x\right)=\frac{{\mathrm{\ρ}}_{az}\·{\mathrm{\ρ}}_{gr}}{\sin \mathrm{\α}}---\left(6\right)$

where ρ is_grFor distance resolution:

${\mathrm{\ρ}}_{gr}=\frac{0.886c}{B_r\left|\right|H^{\⊥}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T\left|\right|}---\left(7\right)$

where c is the speed of light, t₀Is the imaging center time, B_rIs the signal bandwidth, H^⊥It is the ground projection matrix that can be expressed as:

$H^{\⊥}=I-P_G\·P_G^T---\left(8\right)$

wherein I is the identity matrix, P_GIs the normal unit vector of the imaging region coordinate system,is P_GTranspose of u_TA(t₀) Is at t₀The unit vector from the time target to the transmitting station is obtained through satellite-ground coordinate conversion;

u_RA(t₀) Is at t₀Unit vector of time target to receiving station:

$u_{RA}\left(t_0\right)=\frac{P_A-P_{im}}{\left|\right|P_A-P_{im}\left|\right|}---\left(9\right)$

wherein, P_AIs the target point position, P_imIs the location of the receiving station.

Azimuth resolution:

${\mathrm{\ρ}}_{az}=\frac{0.886\mathrm{\λ}}{{\∫}_{t_0-T_a/2}^{t_0+T_a/2}\left|\right|H^{\⊥}\left({\mathrm{\ω}}_{TA}\right(t)+{\mathrm{\ω}}_{RA}(t\left)\right)\left|\right|dt}---\left(10\right)$

wherein λ is the carrier wavelength, T_aFor synthesizing the aperture time, omega_TA(t) is the angular velocity of the transmitting station, ω_RA(t) is the angular velocity of the receiving station:

${\mathrm{\ω}}_{TA}\left(t\right)=\frac{\[I-u_{TA}^T\left(t_0\right)u_{TA}\left(t_0\right)\]V_T^T}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(11\right)$

${\mathrm{\ω}}_{RA}\left(t\right)=\frac{\[I-u_{RA}^T\left(t_0\right)u_{RA}\left(t_0\right)\]V_R^T}{\left|\right|P_A-P_{im}\left|\right|}---\left(12\right)$

wherein R is_T(t₀) The position coordinates of the transmitting station at the moment of the imaging center,which is a transpose of the velocity vector of the transmitting station,transpose for the transmitting station velocity vector;

resolving the included angle of the directions:

α＝cos^-1(Ξ·Θ)(13)

wherein Θ denotes a unit vector in the distance resolution direction, and xi denotes a unit vector in the azimuth resolution direction:

$\mathrm{\Θ}=\frac{H^{\⊥}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T}{\left|\right|u_{TA}\left(t_0\right)+u_{RA}\left(t_0\right)\left|\right|}---\left(14\right)$

$\mathrm{\Ξ}=\frac{H^{\⊥}{\left({\mathrm{\ω}}_{TA}\right(t_0)+{\mathrm{\ω}}_{RA}(t_0\left)\right)}^T}{\left|\right|{\mathrm{\ω}}_{TA}\left(t_0\right)+{\mathrm{\ω}}_{RA}\left(t_0\right)\left|\right|}---\left(15\right)$

the multi-objective optimization problem obtained by modeling the path planning problem is expressed as:

$\left\{\begin{array}{cc}min& F_1\left(x\right)=w_1{f}_{dis}\left(x\right)+w_2{f}_{threat}\left(x\right)\\ min& F_2\left(x\right)=S_{cell}\left(x\right)\end{array}\right.---\left(16\right)$

s.t.N_consi＝0,i＝1,2

wherein, w₁And w₂Weighting coefficients of a path length function and a terrain threat function respectively;

and 4, step 4: solving by adopting multi-target differential evolution algorithm

4.1 initializing iteration parameters

Initializing iterative parameters of a multi-target differential evolution algorithm, comprising: group size N, maximum number of iterations G_maxScaling factor F and crossover rate Cr; randomly generating an initial population X_GG ═ 0, comprising N individuals;

4.2 Cross mutation

For the G generation population X_GEach of the individuals x in_i,GI-1, 2, …, N, to generate a new individual v_i,G

$v_{i,G}=x_{r_1^i,G}+F\·\left(x_{r_2^i,G}-x_{r_3^i,G}\right)---\left(17\right)$

Wherein,andis X_GThree individuals randomly selected from the group;

obtaining N new individuals v_i,GI is 1,2, …, N, then carrying out mutation operation to obtain test population U_GTest population U_GEach individual u in_i,GCan be expressed as u_i,G＝[u_1,i,G,u_2,i,G,...,u_D,i,G]Wherein D is the number of decision variables;

each decision variable u_j,i,GIs derived from the following formula:

$u_{j,i,G}=\left\{\begin{array}{cc}{v}_{j,i,G}& if({\mathrm{rand}}_{i,j}\[0,1\]\≤Crorj=j_{rand})\\ x_{j,i,G}& otherwise\end{array}\right.---\left(18\right)$

wherein v is_j,i,GAnd x_j,i,GAre respectively provided withIs v is_i,GAnd x_i,GThe jth decision variable, rand_i,j[0,1]Is a random number between 0 and 1, j_randIs a random integer between 0 and D;

through cross mutation operation, the group X corresponding to the G generation is obtained_GTest population U of_GAnd X is_GAnd U_GCombining to obtain population R_G＝X_G∪U_G；

4.3 non-dominated sorting and Next Generation population selection

Using a pair of non-dominant selection algorithms under constraint conditions_G2N individuals in (A) are sorted, and the combined population R is subjected to sorting_GGiven arbitrarily two individuals x_i,GAnd x_j,G(ii) a If x_i,GX satisfying the constraint condition_j,GIf the constraint is not satisfied, x_i,GDominating x_j,G(ii) a If none of the constraints is satisfied, and x_i,GIs less than x_j,G，x_i,GDominating x_j,G(ii) a If x_i,GAnd x_j,GAll satisfy the constraint condition, their objective function values, i.e. F in the formula (16), are compared₁And F₂If x_i,GHas an objective function value of less than x_j,GThen x_i,GDominating x_j,G；

To R_GComparing the dominance relationship among each individual, and sequencing the 2N individuals according to the dominance relationship in the population; selecting N individuals with highest domination level to form a next generation group X_G+1；

4.4 judging Loop termination conditions

Updating the iteration number G +1, if G is equal to G_maxThen step 5 is executed, if the iteration number G is less than G_maxReturning to the step 4.2;

and 5: generating an optimal path for a UAV receiving station

Final generation population by iteration in step 4I.e. the optimal solution of the multi-objective optimization problem (16). For each optimal solutionA UAV receiving station path may be generated and at an imaging point P_imAnd realizing GEO-UAV bistatic SAR imaging.