CN105445738A - GEO satellite-machine double-base SAR receiving station flight parameter design method based on genetic algorithm - Google Patents

Abstract

The invention discloses a GEO satellite-machine double-base SAR receiving station flight parameter design method based on a genetic algorithm and mainly aims at solving a problem of solving a GEO satellite-machine double-base SAR airborne receiving station flight parameter under a given imaging index condition. The method is characterized in that the genetic algorithm is used to solve an imaging performance equation set so that an airborne receiving station flight parameter satisfying a given spatial resolution and an imaging signal to noise ratio is acquired. A realization process comprises the following steps of (1) selecting an appropriate GEO-SAR satellite as an irradiation source; (2) establishing an imaging scene coordinate system; (3) determining a function relation of imaging performance and establishing the imaging performance equation set; (4) converting the imaging performance equation set into a multi-target optimization problem; (5) using a rapid non-dominated sorting genetic algorithm to solve the optimization problem; (6) acquiring the airborne receiving station flight parameter and realizing given imaging performance.

Description

GEO satellite-aircraft bistatic SAR receiving station flight parameter design method based on genetic algorithm

Technical Field

The invention belongs to the technical field of radars, and particularly relates to a flight task design of a GEO double-base SAR receiving station meeting the requirement on imaging performance.

Background

Synthetic Aperture Radar (SAR) is a full-time and all-weather high-resolution imaging system, and obtains distance high resolution by transmitting large time-width product linear frequency modulation signals and obtaining pulse compression signals through matched filtering during receiving, and the azimuth high resolution is realized by utilizing a synthetic aperture technology. The imaging quality is not influenced by weather conditions (cloud cover, illumination) and the like, and the method has the characteristic of detecting and positioning a long-distance target.

The geosynchronous orbit synthetic aperture radar (GEO-SAR) has larger mapping bandwidth and shorter revisit period compared with LEO-SAR, so that the geosynchronous orbit synthetic aperture radar (GEO-SAR) can be widely applied to disaster monitoring and earth structure imaging. For the single-basis GEO-SAR, a long synthetic aperture time is required to obtain good azimuthal resolution, but a long synthetic aperture time can introduce a large atmospheric phase delay. Compared with the single-base GEO-SAR, the GEO satellite-aircraft bistatic SAR can conveniently and efficiently improve the imaging performance by adjusting the flight parameters of the receiving station.

The GEO satellite-airborne bistatic SAR uses GEO satellites as illumination sources, and has a large illumination bandwidth and a shorter revisit period compared with LEO satellites. A GEO satellite-machine dual-base configuration system based on an approximate zero-inclination-angle GEO satellite is provided in the document Bistatic geosynchronous SARforlandand geographic information and service analysis (InProc.10THEUSAR, June2014, pp.1-4), low-spatial resolution imaging in an L wave band and a KU wave band can be realized, and system parameters are analyzed according to preset imaging performance. Meanwhile, in the document "wireless communication based on wireless communication and wireless communication," a two-base configuration system in which GEO is used as an irradiation source and LEO or unmanned aerial vehicle is used as a receiving station is proposed, but the system belongs to a single-base fixed two-base SAR. The spatial resolution is analyzed in the literature "resolution calculation and analysis of stationary sars with geometrical analysis of the column," ieee geosci. remotesens. lett., vol.10, No.1, pp.194-198, jan2013 "taking into account the surface of the ellipsoid and the large equivalent angle. But the analysis method of the spatial resolution and the characteristics of the spatial resolution cannot be directly applied to the GEO satellite-based bistatic SAR with a non-zero dip angle.

Disclosure of Invention

The invention aims to design a method for solving flight parameters of a rapid non-dominated sorting genetic algorithm aiming at the defects in the background art, and solve the problem of design of a receiving station flight task caused by a special configuration of a GEO satellite-borne SAR.

The invention provides a flight parameter design method of a GEO satellite-aircraft bistatic SAR receiving station based on a genetic algorithm, which specifically comprises the following steps:

step S1: selecting a proper GEO-SAR satellite as a radiation source

Firstly, calculating the space position coordinate and the speed of a GEO-SAR satellite, and calculating the coordinate of an aiming point through the attitude of the satellite and the beam pointing direction, wherein the position of the satellite in an inertial coordinate system can be expressed as:

$R_s=\left[\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right]=A_{or}\left[\begin{array}{c}{r}_s\\ 0\\ 0\end{array}\right]---\left(1\right)$

wherein

$\begin{array}{c}{A}_{or}=\left[\begin{array}{ccc}\cos \mathrm{\Ω}& -\sin \mathrm{\Ω}& 0\\ \sin \mathrm{\Ω}& \cos \mathrm{\Ω}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos i& -\sin i\\ 0& \sin i& \cos i\end{array}\right]\\ \left[\begin{array}{ccc}\cos \mathrm{\ω}& -\sin \mathrm{\ω}& 0\\ \sin \mathrm{\ω}& \cos \mathrm{\ω}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos f& -\sin f& 0\\ \sin f& \cos f& 0\\ 0& 0& 1\end{array}\right]\end{array}---\left(2\right)$

r_s＝[a(1-e²)]/(1+ecosf)(3)

Omega is the ascension of the ascending intersection point, i is the inclination angle of the track, omega is the argument of the perigee, f is the true perigee angle, a is the major semi-axis of the track, and e is the eccentricity of the track. GEO satellite velocity may be expressed as:

$V_s=\sqrt{u/\[a(1-e^2)\]}\·A_{or}\·\left[\begin{array}{c}e\sin f\\ 1+e\cos f\\ 0\end{array}\right]---\left(4\right)$

where μ is the gravitational constant. The position vector of the beam aiming point in the inertial coordinate system can be expressed as:

$\left[\begin{array}{c}{x}_p\\ y_p\\ z_p\end{array}\right]=A_{or}{A}_{re}{A}_{ea}{\[-r,0,0\]}^T+\left[\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right]---\left(5\right)$

wherein,

$A_{re}=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{cos\θ}}_y& {\mathrm{sin\θ}}_y\\ 0& -{\mathrm{sin\θ}}_y& {\mathrm{cos\θ}}_y\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\θ}}_p& {\mathrm{sin\θ}}_p& 0\\ -{\mathrm{sin\θ}}_p& {\mathrm{cos\θ}}_p& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\θ}}_r& 0& -{\mathrm{sin\θ}}_r\\ 0& 1& 0\\ {\mathrm{sin\θ}}_r& 0& {\mathrm{cos\θ}}_r\end{array}\right]---\left(6\right)$

$A_{ea}=\left[\begin{array}{ccc}cos\mathrm{\γ}& 0& -ksin\mathrm{\γ}\\ 0& 1& 0\\ k\sin \mathrm{\γ}& 0& cos\mathrm{\γ}\end{array}\right]---\left(7\right)$

wherein, theta_yIs yaw angle, θ_pTo a pitch angle, θ_rIs a roll angle. Gamma is the view angle under the antenna, k is 1 for the right view of the antenna, and k is-1 for the left view of the antenna.

The longitude and latitude of the aiming point may be expressed as:

$\begin{array}{c}{\mathrm{\&theta;}}_{lat}=\arctan \&lsqb;\frac{z_p}{\sqrt{x_p^2+y_p^2}}\&rsqb;\\ {\mathrm{\&theta;}}_{long}=\left\{\begin{array}{cc}\arctan \&lsqb;y_p/x_p\&rsqb;,& X_t>0,Y_t>0\\ \arctan \&lsqb;y_p/x_p\&rsqb;+2\mathrm{\&pi;},& X_t>0,Y_t<0\\ \arctan \&lsqb;y_p/x_p\&rsqb;+\mathrm{\&pi;},& X_t<0\end{array}\right.\end{array}---\left(8\right)$

the radius of the earth at the center point can be expressed as:

$R_a=\sqrt{R_e^2{R}_p^2/\{{\&lsqb;R_p{\mathrm{cos\&theta;}}_{lat}\&rsqb;}^2+{\&lsqb;R_e{\mathrm{sin\&theta;}}_{lat}\&rsqb;}^2\}}---\left(9\right)$

the velocity of the beam center point is:

$V_p=\left[\begin{array}{c}-{\mathrm{\&omega;}}_e{R}_{a}cos{\mathrm{\&theta;}}_{long}sin{\mathrm{\&theta;}}_{lat}\\ {\mathrm{\&omega;}}_e{R}_a{\mathrm{cos\&theta;}}_{long}{\mathrm{cos\&theta;}}_{lat}\\ 0\end{array}\right]---\left(10\right)$

wherein, ω is_eFor rotational angular velocity of the earth

Step S2: establishing a suitable imaging scene coordinate system

And taking the beam center point as an origin, taking the direction of the geocentric pointing to the beam center point as a Z axis, and establishing an imaging scene coordinate system by the X axis in a plane formed by the Z axis of the beam center point inertial coordinate system.

The satellite position and velocity can be expressed as:

$\begin{array}{c}{R}_T^T\left(t\right)=F\&CenterDot;R_s-{\&lsqb;0,0,R_a\&rsqb;}^T\\ V_T^T=F\&CenterDot;\left(V_s\right(t)-V_p(t\left)\right)\end{array}---\left(11\right)$

wherein,

$F=\left[\begin{array}{ccc}{\mathrm{sin\&theta;}}_{lat}& 0& -{\mathrm{cos\&theta;}}_{lat}\\ 0& 1& 0\\ {\mathrm{cos\&theta;}}_{lat}& 0& {\mathrm{sin\&theta;}}_{lat}\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\&theta;}}_{long}& {\mathrm{sin\&theta;}}_{lat}& 0\\ -{\mathrm{sin\&theta;}}_{lat}& {\mathrm{cos\&theta;}}_{lat}& 0\\ 0& 0& 1\end{array}\right]---\left(12\right)$

θ_latlatitude of the center point, θ_longThe longitude of the center point.

Step S3: establishing a functional relationship of an imaging index

For a shift-invariant bistatic forward-looking SAR, the bistatic configuration will determine the imaging performance given the system parameters. Distance resolution

${\mathrm{\&rho;}}_{gr}=\frac{0.886c}{B_r\left|\right|H^{\&perp;}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T\left|\right|}---\left(13\right)$

Where c is the speed of light, B_rIs the signal bandwidth, H^⊥It is the ground projection matrix that can be expressed as:

$H^{\&perp;}=I-P_G\&CenterDot;P_G^T---\left(14\right)$

i is the identity matrix, P_GIs the normal unit vector of the imaging region coordinate system,is P_GThe transposing of (1).

u_TA(t₀) Is at t₀Unit vector of time target to transmitting station

$u_{TA}\left(t_0\right)=\frac{P_A-R_T\left(t_0\right)}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(15\right)$

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

$u_{RA}\left(t_0\right)=\frac{P_A-R_R\left(t_0\right)}{\left|\right|P_A-R_R\left(t_0\right)\left|\right|}---\left(16\right)$

P_AIs the target point position, R_T(t₀) The position of the receiving station, i.e. the position of the satellite.

$\begin{array}{c}{(P_A-R_R(t_0\left)\right)}^T\\ =H_R{\mathrm{tan\&theta;}}_R{M}_1\&CenterDot;\frac{H^{\&perp;}{(P_A-R_T(t_0\left)\right)}^T}{\left|\right|H^{\&perp;}{(P_A-R_T(t_0\left)\right)}^T\left|\right|}+{\&lsqb;0,0,H_R\&rsqb;}^T\end{array}---\left(17\right)$

Wherein H_RTo the height of the receiving station, theta_RIs the receiving station angle of incidence. M₁Is a rotation matrix

$M_1=\left[\begin{array}{ccc}cos\mathrm{\&phi;}& sin\mathrm{\&phi;}& 0\\ -sin\mathrm{\&phi;}& cos\mathrm{\&phi;}& 0\\ 0& 0& 1\end{array}\right]---\left(18\right)$

Phi is the bistatic projection angle, the azimuth resolution:

${\mathrm{\&rho;}}_{az}=\frac{0.886\mathrm{\&lambda;}}{{\&Integral;}_{t_0-T_a/2}^{t_0+T_a/2}\left|\right|H^{\&perp;}\left({\mathrm{\&omega;}}_{TA}\right(t)+{\mathrm{\&omega;}}_{RA}\left(t\right)\left)\right||dt}---\left(19\right)$

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.

${\mathrm{\&omega;}}_{TA}\left(t\right)=\frac{\&lsqb;I-u_{TA}^T\left(t_0\right)u_{TA}\left(t_0\right)\&rsqb;V_T^T}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(20\right)$

${\mathrm{\&omega;}}_{RA}\left(t\right)=\frac{\&lsqb;I-u_{RA}^T\left(t_0\right)u_{RA}\left(t_0\right)\&rsqb;V_R^T}{\left|\right|P_A-R_R\left(t_0\right)\left|\right|}---\left(21\right)$

Wherein,which is a transpose of the velocity vector of the transmitting station,is a transpose of the velocity vector of the transmitting station.

$V_R^T=V_R{M}_2\frac{H^{\&perp;}{V}_T^T\left(t_0\right)}{\left|\right|H^{\&perp;}{V}_T^T\left(t_0\right)\left|\right|}---\left(22\right)$

If the velocity of the receiving station is parallel to the x-y plane, M₂Can be expressed as:

$M_2=\left[\begin{array}{ccc}cos\mathrm{\&psi;}& sin\mathrm{\&psi;}& 0\\ -sin\mathrm{\&psi;}& cos\mathrm{\&psi;}& 0\\ 0& 0& 1\end{array}\right]---\left(23\right)$

psi is a double-base flight direction included angle;

direction angle resolution:

α＝cos^-1(Ξ·Θ)(24)

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

$\mathrm{\&Theta;}=\frac{H^{\&perp;}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T}{\left|\right|u_{TA}\left(t_0\right)+u_{RA}\left(t_0\right)\left|\right|}---\left(25\right)$

$\mathrm{\&Xi;}=\frac{H^{\&perp;}{\left({\mathrm{\&omega;}}_{TA}\right(t_0)+{\mathrm{\&omega;}}_{RA}(t_0\left)\right)}^T}{\left|\right|{\mathrm{\&omega;}}_{TA}\left(t_0\right)+{\mathrm{\&omega;}}_{RA}\left(t_0\right)\left|\right|}---\left(26\right)$

Signal-to-noise ratio:

$SNR=\frac{P_t{G}_t{G}_r{\mathrm{\&sigma;}}_0{\mathrm{\&rho;}}_{az}{\mathrm{\&rho;}}_{gr}{\mathrm{\&lambda;}}^2{T}_a{D}_c}{{\left(4\mathrm{\&pi;}\right)}^3{R}_T^2{R}_R^2{L}_T{\mathrm{kT}}_0{F}_n}---\left(27\right)$

wherein, P_tFor transmitting the peak power of the signal, G_tFor transmitting antenna gain, G_rFor receiving antenna gain, σ₀For standardization of radar cross-section, p_azFor azimuthal resolution, p_grTo distance resolution, D_cIs the duty ratio, L_TK is Boltzmann constant, T, for propagation loss₀As noise temperature, F_nIs the receiving station noise figure.

Step S4: modelling as a system of non-linear equations

Imaging index azimuthal resolution ρ_azDistance resolution ρ_grAngle of resolution α, SNR is θ_RPhi, psi.

For a given imaging index ρ_grD，ρ_azD，α_D，SNR_DThe task design can be modeled as a system of non-linear equations:

F(x)＝0(28)

wherein,

$F\left(x\right)=\left[\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\\ f_3\left(x\right)\\ f_4\left(x\right)\end{array}\right]---\left(29\right)$

f₁(x)＝ρ_gr(θ_R,φ,ψ)-ρ_grD

f₂(x)＝ρ_az(θ_R,φ,ψ)-ρ_azD

f₃(x)＝α(θ_R,φ,ψ)-α_D(30)

f₄(x)＝SNR(θ_R,φ,ψ)-SNR_D

x＝(θ_R,φ,ψ)^Tis a decision vector consisting of three decision variables:

0＝(0,0,0)^T(32)

step S5: conversion to Multiobjective Optimization Problem (MOP):

to solve the above system of nonlinear equations and obtain multiple solutions simultaneously, we convert the system of nonlinear equations into a multi-objective optimization problem consisting of two objective functions. The multi-objective optimization problem of the system of linear equations may become:

$\left\{\begin{array}{c}{\mathrm{minF}}_1\left(x\right)={\mathrm{\&theta;}}_R+{\mathrm{\&Sigma;}}_{i=1}^4\left|f_i\left(x\right)\right|\\ \min F_2\left(x\right)=1-{\mathrm{\&theta;}}_R+4max\left(\right|f_1\left(x\right)|,\mathrm{...},|f_4\left(x\right)\left|\right)\end{array}\right.---\left(33\right)$

step S6: non-dominated genetic algorithm to solve MOP

S6.1 initializing parent data

Let G equal to 0, G be genetic algebra, G_maxIs the maximum algebra.

Randomly generating an initial population P_GThere are NBody

Calculating an initial population P_GThe fitness value of each individual, i.e., the two objective function values of equation (33) above.

S6.2 if G ∈ [1, G_max]Step S6.3 is executed, otherwise go to step S7;

s6.3 contest selection

Binary race selection: in population P_GOptionally, the two individuals compare the objective function values and select the objective function value to be smaller. Is carried out N times from the original P_GN individuals are selected.

S6.4 Cross mutation

Simulated binary crossover operator (SBX):

if the number P ∈ [0,1 ] is randomly generated]，P＜P_c，P_cAnd if the probability is the cross probability, pairing each decision variable of the N individuals. Assume one pair of them isAndwherein G is an algebra, and a number u is randomly generated_i∈[0,1]And i is the ith decision variable.

Calculate dispersion factor β_qi

${\mathrm{\&beta;}}_{qi}=\left\{\begin{array}{cc}{\left(2u_i\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& u_i\&le;0.5\\ {\left(\frac{1}{2(1-u_i)}\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& u_i>0.5\end{array}\right.---\left(34\right)$

Wherein, η_cIs a cross factor.

The offspring individuals are obtained by the following formula:

$\begin{array}{c}{x}_i^{(1,G+1)}=0.5\&lsqb;(1+{\mathrm{\&beta;}}_{qi})x_i^{(1,G)}+(1-{\mathrm{\&beta;}}_{qi})x_i^{(2,G)}\&rsqb;\\ x_i^{(2,G+1)}=0.5\&lsqb;(1-{\mathrm{\&beta;}}_{qi})x_i^{(1,G)}+(1+{\mathrm{\&beta;}}_{qi})x_i^{(2,G)}\&rsqb;\end{array}---\left(35\right)$

polynomial mutation operator (PLM):

if the number P ∈ [0,1 ] is randomly generated]，P＜P_m，p_m1/D, D being the number of decision variables, for each of N individualsAnd carrying out mutation operation on the decision variables. The progeny individuals are generated by the following formula:

$y_i^{(1,G+1)}=x_i^{(1,G+1)}+\mathrm{\&delta;}(x_i^u-x_i^l)---\left(36\right)$

wherein,andupper and lower bounds for the ith decision variable, respectively. Given by the formula, r_iFor randomly generating a number r_i∈[0,1]

$\mathrm{\&delta;}=\left\{\begin{array}{cc}{\&lsqb;2r_i+(1-2r_i){(1-{\mathrm{\&delta;}}_1)}^{{\mathrm{\&eta;}}_m+1}\&rsqb;}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}-1& r_i\&le;0.5\\ 1-{\&lsqb;2(1-r_i)+2(r_i-0.5){(1-{\mathrm{\&delta;}}_2)}^{{\mathrm{\&eta;}}_m+1}\&rsqb;}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}& r_i>0.5\end{array}\right.---\left(37\right)$

η therein_mIs the index of variation of the polynomial,₁and₂given by:

$\begin{array}{c}{\mathrm{\&delta;}}_1=\frac{x_i^{(1,G+1)}-x_i^l}{x_i^u-x_i^l}\\ {\mathrm{\&delta;}}_2=\frac{x_i^u-x_i^{(1,G+1)}}{x_i^u-x_i^l}\end{array}---\left(38\right)$

the filial generation population generated by selection, crossing and mutation is marked as Q_G。

Parent child merge

Calculating the progeny population Q according to equation (33)_GThe objective function value of each individual.

Merging parents and offspring to generate a new population: h_G＝P_G∪Q_G。

S6.5 non-dominant ranking

Using a fast non-dominant selection algorithm pair H_GThe 2N individuals in the set are subjected to non-dominated sorting.

For minimizing multi-objective problems, n objective components f_iVector f (x) comprising (i-1, …, n), (f ═ x) and (f ═ y)₁(x),f₁(x),…,f_n(x) Given any two decision variables x)_u,x_v∈ U, wherein U is the value range of the decision variable.

If and only if, forAll have f_i(x_u)＜f_i(x_v) Then x_uDominating x_v。

If and only if, forAll have f_i(x_u)≤f_i(x_v) And at least one j ∈ {1, …, n } is present, such that f_j(x_u)＜f_j(x_v) X is then_uWeak domination x_v。

If and only if, forLet f_i(x_u)＜f_i(x_v) And at the same time,let f_j(x_u)＞f_j(x_v) X is then_uAnd x_vAre not mutually exclusive.

For H_GEach individual i is provided with the following two parameters n_iAnd S_i,n_iTo govern the number of solution individuals of an individual i in the population, S_iIs the set of solution individuals governed by individual i.

First, all n in the population are found_iIndividuals of 0, store them in the current set F₁。

Then F for the current set₁Each individual j in (a) looks at the set of individuals S it governs_jWill aggregate S_jN of each individual k in_kSubtracting l, i.e. the number of solution individuals dominating the individual k, minus 1 (since the individual j dominating the individual k has been stored in the current set F₁) If n is_k-1 ═ 0 then stores the individual k in another set F₂。

Finally, F is mixed₁As a first-level non-dominant individual set, and endowing the individuals in the set with the same non-dominant order i_rankThen continue to pair F₂Making the above-mentioned hierarchical operations and assigning corresponding non-dominant ordersUntil all individuals are ranked.

S6.6 Congestion calculation

The congestion distance calculation principle is as follows:

(1) arranging the solution sets in the congestion degree calculation set F according to an ascending order;

(2) the crowd distance of the first and last individuals is set to infinity;

(3) the congestion distance of the ith individual is set to the sum of normalized values of the differences between all the objective function values of the (i + 1) th individual and the (i-1) th individual.

Can be expressed as:

$dis\left(i\right)=\underset{m=1}{\overset{M}{\&Sigma;}}\frac{F_m(i+1)-F_m(i-1)}{\max \left(F_m\right)-\min \left(F_m\right)}---\left(39\right)$

wherein F_m(i-1) is the mth objective function value of the ith individual, and M is the number of objective functions.

And finally, sorting the set F in a descending order according to the congestion distance.

Because both the child and parent individuals are contained in H_GIn (3), the non-dominating set F after non-dominating sorting₁The individual contained in (A) is H_GAmong them, F is first₁Will be put into a new parent population P_G+1In (1). If F₁Is less than N, P continues_G+1Middle filling next-level non-dominating set F₂Until addition of F_nWhen the size of the population exceeds N, for F_nThe individuals in the Chinese tree are sorted according to the crowding degree, and the top N-P is taken_G+1I individuals, make P_G+1The number of individuals reaches N.

S6.7G ═ G +1, return to step S6.2

Step S7: output optimization solution

To P_G+1Apply non-dominant ordering and label the set with the highest dominant rank as F₁

From F according to constraints of the environment₁To select the optimal solution S_op。

Step S8: the obtained optimal solution S_opThe given imaging index is realized as a flight parameter of the airborne receiving station.

The invention has the beneficial effects that: the method realizes the design of the GEO satellite-aircraft bistatic SAR flight task by applying a non-dominated sorting genetic algorithm. Firstly, selecting a proper GEO-SAR satellite as an irradiation source, then converting position parameters of the satellite into a representation under an imaging scene coordinate system, then constructing a functional relation of imaging performance, then modeling the design of a GEO satellite-machine bistatic SAR flight task into a nonlinear equation set, converting the nonlinear equation set into a multi-objective optimization problem, and finally solving through a genetic algorithm of non-dominated sorting so as to realize a given imaging index.

The method has the advantages that the method solves the optimization problem by using the powerful global search function of the genetic algorithm, has high efficiency, and avoids the dilemma of getting into the local optimal solution when the general iterative algorithm solves the optimization problem and the solution deviation caused by unstable numerical value; the flight mission designed by the method can enable the GEO satellite-aircraft bistatic SAR to realize a given imaging index, so that the GEO satellite-aircraft 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 graphical representation of an imaging metric of 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 index for measuring SAR imaging performance has azimuth resolution rho_azDistance resolution ρ_grResolution direction angle α, signal-to-noise ratio snr, generally requires that the resolution be as small as possible, that the resolution direction angle (i.e., the angle between the distance and azimuth directions) be perpendicular, and that the signal-to-noise ratio be as large as possible.

The method comprises the following specific steps:

the method comprises the following steps: and analyzing the space geometric model of the satellite to obtain the position and the speed of the GEO-SAR satellite, and solving the position and the speed of the central point of the wave beam according to the position of the satellite.

The parameters of the GEO-SAR system are shown in table 1.

TABLE 1

Step two: and D, establishing a coordinate system of the imaging scene according to the beam center point obtained in the step one, wherein the coordinate system is established by taking the beam center point as an origin point and taking the direction of the geocentric pointing to the center point as a Z axis. And converting the position and speed vectors of the satellite into a representation under the coordinate system of the imaging scene.

Step three: determining the azimuth resolution rho of the imaging index according to the parameters of the transmitting station obtained in the step two_azDistance resolution ρ_grResolution azimuth α, SNR and receiver station angle of incidence θ_RThe system parameters are shown in table 2, and the functional relation between the biradical projection angle phi and the biradical flight direction included angle psi is shown in table 2.

TABLE 2

Step four: and establishing a nonlinear equation set according to the functional relation obtained in the step three and the given imaging index.

Step five: and 4, converting the nonlinear equation system obtained in the step four into an optimization problem consisting of two objective functions.

Step six: and (4) solving the multi-objective optimization problem obtained in the step five by using a rapid non-dominated sorting genetic algorithm, wherein the flow chart of the algorithm is shown in figure 1. The parameters of the genetic algorithm are shown in table 3.

TABLE 3

Step seven: and selecting a solution meeting the imaging index from the optimal solution solved by the rapid non-dominated sorting genetic algorithm.

Optimal solution set S_opAs shown in table 4.

TABLE 4

Step eight: will optimize the solution S_opThe given imaging index is achieved as a flight parameter of the GEO bistatic SAR, as shown in fig. 2.

Claims (1)

1. A GEO satellite-aircraft bistatic SAR receiving station flight parameter design method based on a genetic algorithm specifically comprises the following steps:

step S1: selecting a proper GEO-SAR satellite as a radiation source

Firstly, calculating the space position coordinate and the speed of a GEO-SAR satellite, and calculating the coordinate of an aiming point through the attitude of the satellite and the beam pointing direction, wherein the position of the satellite in an inertial coordinate system can be expressed as:

$R_s=\left[\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right]=A_{or}\left[\begin{array}{c}{r}_s\\ 0\\ 0\end{array}\right]---\left(1\right)$

wherein,

$\begin{array}{c}{A}_{or}=\left[\begin{array}{ccc}\cos \mathrm{\&Omega;}& -\sin \mathrm{\&Omega;}& 0\\ \sin \mathrm{\&Omega;}& \cos \mathrm{\&Omega;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos i& -\sin i\\ 0& \sin i& \cos i\end{array}\right]\\ \left[\begin{array}{ccc}\cos \mathrm{\&omega;}& -\sin \mathrm{\&omega;}& 0\\ \sin \mathrm{\&omega;}& \cos \mathrm{\&omega;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos f& -\sin f& 0\\ \sin f& \cos f& 0\\ 0& 0& 1\end{array}\right]\end{array}---\left(2\right)$

r_s＝[a(1-e²)]/(1+ecosf)(3)

omega is the ascension of the intersection point, i is the inclination angle of the track, omega is the argument of the perigee, f is the true perigee angle, a is the major semi-axis of the track, and e is the eccentricity of the track;

GEO satellite velocity is expressed as:

$V_s\left(t\right)=\sqrt{u/\&lsqb;a(1-e^2)\&rsqb;}\&CenterDot;A_{or}\&CenterDot;\left[\begin{array}{c}e\sin f\\ 1+e\cos f\\ 0\end{array}\right]---\left(4\right)$

where μ is the gravitational constant;

the position vector of the aiming point of the beam in the inertial coordinate system is expressed as:

$\left[\begin{array}{c}{x}_p\\ y_p\\ z_p\end{array}\right]=A_{or}{A}_{re}{A}_{ea}{\&lsqb;-r,0,0\&rsqb;}^T+\left[\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right]---\left(5\right)$

wherein,

$A_{re}=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{cos\&theta;}}_y& {\mathrm{sin\&theta;}}_y\\ 0& -{\mathrm{sin\&theta;}}_y& {\mathrm{cos\&theta;}}_y\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\&theta;}}_p& {\mathrm{sin\&theta;}}_p& 0\\ -{\mathrm{sin\&theta;}}_p& {\mathrm{cos\&theta;}}_p& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\&theta;}}_r& 0& -{\mathrm{sin\&theta;}}_r\\ 0& 1& 0\\ {\mathrm{sin\&theta;}}_r& 0& {\mathrm{cos\&theta;}}_r\end{array}\right]---\left(6\right)$

$A_{ea}=\left[\begin{array}{ccc}cos\mathrm{\&gamma;}& 0& -ksin\mathrm{\&gamma;}\\ 0& 1& 0\\ k\sin \mathrm{\&gamma;}& 0& cos\mathrm{\&gamma;}\end{array}\right]---\left(7\right)$

wherein, theta_yIs yaw angle, θ_pTo a pitch angle, θ_rThe angle is a roll angle, gamma is an antenna downward viewing angle, k is 1 for an antenna right view, and k is-1 for an antenna left view;

the longitude and latitude of the aiming point may be expressed as:

$\begin{array}{c}{\mathrm{\&theta;}}_{lat}=\arctan \&lsqb;\frac{z_p}{\sqrt{x_p^2+y_p^2}}\&rsqb;\\ {\mathrm{\&theta;}}_{long}=\left\{\begin{array}{cc}\arctan \&lsqb;y_p/x_p\&rsqb;,& X_t>0_t,Y_t>0\\ \arctan \&lsqb;y_p/x_p\&rsqb;+2\mathrm{\&pi;},& X_t>0_t,Y_t<0\\ \arctan \&lsqb;y_p/x_p\&rsqb;+\mathrm{\&pi;},& X_t<0\end{array}\right.\end{array}---\left(8\right)$

the radius of the earth at the center point is expressed as:

$R_a=\sqrt{R_e^2{R}_p^2/\{{\&lsqb;R_p{\mathrm{cos\&theta;}}_{lat}\&rsqb;}^2+{\&lsqb;R_e{\mathrm{sin\&theta;}}_{lat}\&rsqb;}^2\}}---\left(9\right)$

the velocity of the beam center point is:

$V_p=\left[\begin{array}{c}-{\mathrm{\&omega;}}_e{R}_a{\mathrm{cos\&theta;}}_{long}{\mathrm{sin\&theta;}}_{lat}\\ {\mathrm{\&omega;}}_e{R}_a{\mathrm{cos\&theta;}}_{long}{\mathrm{cos\&theta;}}_{lat}\\ 0\end{array}\right]---\left(10\right)$

wherein, ω is_eThe rotational angular velocity of the earth;

step S2: establishing a suitable imaging scene coordinate system

Taking the beam center point as an origin, taking the direction of the geocentric pointing to the beam center point as a Z axis, and establishing an imaging scene coordinate system by the X axis in a plane formed by the Z axis of the beam center point inertial coordinate system;

the satellite position and velocity can be expressed as:

$\begin{array}{c}{R}_T^T\left(t\right)=F\&CenterDot;R_s-{\&lsqb;0,0,R_a\&rsqb;}^T\\ V_T^T=F\&CenterDot;\left(V_s\right(t)-V_p)\end{array}---\left(11\right)$

wherein,

$F=\left[\begin{array}{ccc}{\mathrm{sin\&theta;}}_{lat}& 0& -{\mathrm{cos\&theta;}}_{lat}\\ 0& 1& 0\\ {\mathrm{cos\&theta;}}_{lat}& 0& {\mathrm{sin\&theta;}}_{lat}\end{array}\right]\left[\begin{array}{ccc}{\mathrm{cos\&theta;}}_{long}& {\mathrm{sin\&theta;}}_{lat}& 0\\ -{\mathrm{sin\&theta;}}_{lat}& {\mathrm{cos\&theta;}}_{lat}& 0\\ 0& 0& 1\end{array}\right]---\left(12\right)$

wherein, theta_latLatitude of the center point, θ_longLongitude as the center point;

step S3: establishing a functional relationship of an imaging index

For shift-invariant bistatic forward-looking SAR, range resolution:

${\mathrm{\&rho;}}_{gr}=\frac{0.886c}{B_r\left|\right|H^{\&perp;}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T\left|\right|}---\left(13\right)$

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

$H^{\&perp;}=I-P_G\&CenterDot;P_G^T---\left(14\right)$

i is the identity matrix, P_GIs the normal unit vector of the imaging region coordinate system,is thatThe transposing of (1).

u_TA(t₀) Is at t₀Unit vector of time target to transmitting station:

$u_{TA}\left(t_0\right)=\frac{P_A-R_T\left(t_0\right)}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(15\right)$

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

$u_{RA}\left(t_0\right)=\frac{P_A-R_R\left(t_0\right)}{\left|\right|P_A-R_R\left(t_0\right)\left|\right|}---\left(16\right)$

P_Ais the target point position, R_T(t₀) Is the position of the receiving station, i.e. the position of the satellite;

$\begin{array}{c}{(P_A-R_R(t_0\left)\right)}^T\\ =H_R{\mathrm{tan\&theta;}}_R{M}_1\&CenterDot;\frac{H^{\&perp;}{(P_A-R_T(t_0\left)\right)}^T}{\left|\right|H^{\&perp;}{(P_A-R_T(t_0\left)\right)}^T\left|\right|}+{\&lsqb;0,0,H_R\&rsqb;}^T\end{array}---\left(17\right)$

wherein H_RTo the height of the receiving station, theta_RFor the receiving station angle of incidence, M₁For the rotation matrix:

$M_1=\left[\begin{array}{ccc}cos\mathrm{\&phi;}& sin\mathrm{\&phi;}& 0\\ -sin\mathrm{\&phi;}& cos\mathrm{\&phi;}& 0\\ 0& 0& 1\end{array}\right]---\left(18\right)$

wherein phi is a biradical projection angle;

azimuth resolution:

${\mathrm{\&rho;}}_{az}=\frac{0.886\mathrm{\&lambda;}}{{\&Integral;}_{t_0-T/2}^{t_0+T/2}\left|\right|H^{\&perp;}\left({\mathrm{\&omega;}}_{TA}\right(t)+{\mathrm{\&omega;}}_{RA}(t\left)\right)\left|\right|dt}---\left(19\right)$

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:

${\mathrm{\&omega;}}_{TA}\left(t\right)=\frac{\&lsqb;I-u_{TA}^T\left(t_0\right)u_{TA}\left(t_0\right)\&rsqb;V_T^T}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(20\right)$

${\mathrm{\&omega;}}_{RA}\left(t\right)=\frac{\&lsqb;I-u_{RA}^T\left(t_0\right)u_{RA}\left(t_0\right)\&rsqb;V_R^T}{\left|\right|P_A-R_R\left(t_0\right)\left|\right|}---\left(21\right)$

wherein,which is a transpose of the velocity vector of the transmitting station,is a transpose of the velocity vector of the transmitting station.

$V_R^T=V_R{M}_2\frac{H^{\&perp;}{V}_T^T\left(t_0\right)}{\left|\right|H^{\&perp;}{V}_T^T\left(t_0\right)\left|\right|}---\left(22\right)$

If the velocity of the receiving station is parallel to the x-y plane, M₂Expressed as:

$M_2=\left[\begin{array}{ccc}cos\mathrm{\&psi;}& sin\mathrm{\&psi;}& 0\\ -sin\mathrm{\&psi;}& cos\mathrm{\&psi;}& 0\\ 0& 0& 1\end{array}\right]---\left(23\right)$

wherein psi is a double-base flight direction included angle;

direction angle resolution:

α＝cos^-1(Ξ·Θ)(24)

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

$\mathrm{\&Theta;}=\frac{H^{\&perp;}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T}{\left|\right|u_{TA}\left(t_0\right)+u_{RA}\left(t_0\right)\left|\right|}---\left(25\right)$

$\mathrm{\&Xi;}=\frac{H^{\&perp;}{\left({\mathrm{\&omega;}}_{TA}\right(t_0)+{\mathrm{\&omega;}}_{RA}(t_0\left)\right)}^T}{\left|\right|{\mathrm{\&omega;}}_{TA}\left(t_0\right)+{\mathrm{\&omega;}}_{RA}\left(t_0\right)\left|\right|}---\left(26\right)$

signal-to-noise ratio:

$SNR=\frac{P_t{G}_t{G}_r{\mathrm{\&sigma;}}_0{\mathrm{\&rho;}}_{az}{\mathrm{\&rho;}}_{gr}{\mathrm{\&lambda;}}^2{T}_a{D}_c}{{\left(4\mathrm{\&pi;}\right)}^3{R}_T^2{R}_R^2{L}_T{\mathrm{kT}}_0{F}_n}---\left(27\right)$

wherein R is_TFor transmitting station distance, R_RFor receiving station distance, T_aFor synthetic aperture time, P_tFor transmitting the peak power of the signal, G_tFor transmitting antenna gain, G_rFor receiving antenna gain, σ₀For standardization of radar cross-section, p_azFor azimuthal resolution, p_grTo distance resolution, D_cIs the duty ratio, L_TK is Boltzmann constant, T, for propagation loss₀As noise temperature, F_nIs the receiving station noise figure;

step S4: modelling as a system of non-linear equations

For the imaging index ρ_az、ρ_grα, SNR is θ_RPhi, psi, where p_azAzimuthal resolution, p_grDistance resolution, α resolution direction angle, signal-to-noise ratio, SNR;

for a given imaging index ρ_grD，ρ_azD，α_D,SNR_DThe task design can be modeled as a system of non-linear equations:

F(x)＝0(28)

wherein,

$F\left(x\right)=\left[\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\\ f_3\left(x\right)\\ f_4\left(x\right)\end{array}\right]---\left(29\right)$

$\begin{array}{c}{f}_1\left(x\right)={\mathrm{\&rho;}}_{gr}({\mathrm{\&theta;}}_R,\mathrm{\&phi;},\mathrm{\&psi;})-{\mathrm{\&rho;}}_{grD}\\ f_2\left(x\right)={\mathrm{\&rho;}}_{az}({\mathrm{\&theta;}}_R,\mathrm{\&phi;},\mathrm{\&psi;})-{\mathrm{\&rho;}}_{azD}\\ f_3\left(x\right)=\mathrm{\&alpha;}({\mathrm{\&theta;}}_R,\mathrm{\&phi;},\mathrm{\&psi;})-{\mathrm{\&alpha;}}_D\\ f_4\left(x\right)=SNR({\mathrm{\&theta;}}_R,\mathrm{\&phi;},\mathrm{\&psi;})-{\mathrm{SNR}}_D\end{array}---\left(30\right)$

x＝(θ_R,φ,ψ)^Tis a decision vector consisting of three decision variables:

0＝(0,0,0)^T(32)

step S5: conversion to Multiobjective Optimization Problem (MOP):

to solve the system of nonlinear equations (28) and obtain multiple solutions simultaneously, the system of nonlinear equations is converted into a multi-objective optimization problem consisting of two objective functions, the multi-objective optimization problem of the system of linear equations may become:

$\left\{\begin{array}{c}{\mathrm{minF}}_1\left(x\right)={\mathrm{\&theta;}}_R+{\mathrm{\&Sigma;}}_{i=1}^4\left|f_i\left(x\right)\right|\\ {\mathrm{minF}}_2\left(x\right)=1-{\mathrm{\&theta;}}_R+4max\left(\right|f_1\left(x\right)|,\mathrm{...},|f_4\left(x\right)\left|\right)\end{array}\right.---\left(33\right)$

step S6: non-dominated genetic algorithm to solve MOP

S6.1 initializing parent data

Let G equal to 0, G be genetic algebra, G_maxIs the maximum algebra.

Randomly generating an initial population P_GThere are N individuals;

calculating an initial population P_GThe fitness value of each individual, namely the two objective function values of the above equation (33);

s6.2 if G ∈ [1, G_max]Step S6.3 is executed, otherwise go to step S7;

s6.3 contest selection

Binary race selection: in population P_GOptionally, the two individuals compare the objective function values and select the objective function value to be smaller. Is carried out N times from the original P_GSelecting N individuals;

s6.4 Cross mutation

Simulating a binary crossover operator:

if the number P ∈ [0,1 ] is randomly generated]，P＜P_c，P_cFor cross probability, each decision variable of N individuals is paired pairwise, and one pair is assumed asAndwherein G is an algebra, and a number u is randomly generated_i∈[0,1]I is the ith decision variable;

calculate dispersion factor β_qi：

${\mathrm{\&beta;}}_{qi}=\left\{\begin{array}{cc}{\left(2u_i\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& u_i\&le;0.5\\ {\left(\frac{1}{2(1-u_i)}\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& u_i>0.5\end{array}\right.---\left(34\right)$

Wherein, η_cIs a cross-over factor;

the offspring individuals are obtained by the following formula:

$\begin{array}{c}{x}_i^{(1,G+1)}=0.5\&lsqb;(1+{\mathrm{\&beta;}}_{qi})x_i^{(1,G)}+(1-{\mathrm{\&beta;}}_{qi})x_i^{(2,G)}\&rsqb;\\ x_i^{(2,G+1)}=0.5\&lsqb;(1-{\mathrm{\&beta;}}_{qi})x_i^{(1,G)}+(1+{\mathrm{\&beta;}}_{qi})x_i^{(2,G)}\&rsqb;\end{array}---\left(35\right)$

polynomial mutation operator:

if the number P ∈ [0,1 ] is randomly generated]，P＜P_m，p_mAnd D is the number of decision variables, performing mutation operation on each decision variable of the N individuals, and generating the filial generation individuals by the following formula:

$y_i^{(1,G+1)}=x_i^{(1,G+1)}+\mathrm{\&delta;}(x_i^u-x_i^l)---\left(36\right)$

wherein,andthe upper bound and the lower bound of the ith decision variable are respectively;

given by the formula, r_iFor randomly generating a number r_i∈[0,1]

$\mathrm{\&delta;}=\left\{\begin{array}{cc}{\&lsqb;2r_i+(1-2r_i){(1-{\mathrm{\&delta;}}_1)}^{{\mathrm{\&eta;}}_m+1}\&rsqb;}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}-1& r_i\&le;0.5\\ 1-{\&lsqb;2(1-r_i)+2(r_i-0.5){(1-{\mathrm{\&delta;}}_2)}^{{\mathrm{\&eta;}}_m+1}\&rsqb;}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}& r_i>0.5\end{array}\right.---\left(37\right)$

Wherein, η_mIs the index of variation of the polynomial,₁and₂given by:

$\begin{array}{c}{\mathrm{\&delta;}}_1=\frac{x_i^{(1,G+1)}-x_i^l}{x_i^u-x_i^l}\\ {\mathrm{\&delta;}}_2=\frac{x_i^u-x_i^{(1,G+1)}}{x_i^u-x_i^l}\end{array}---\left(38\right)$

the filial generation population generated by selection, crossing and mutation is marked as Q_G；

Merging parent and child:

calculating the progeny population Q according to equation (33)_GThe value of the objective function for each individual,

merging parents and offspring to generate a new population: h_G＝P_G∪Q_G；

S6.5 non-dominant ranking

Using a fast non-dominant selection algorithm pair H_GThe 2N individuals in the set are subjected to non-dominated sorting.

For minimizing multi-objective problems, n objective components f_iVector of (i ═ 1, …, n)

f(x)＝(f₁(x),f₁(x),…,f_d(x) Given any two decision variables x)_u,x_v∈ U, wherein U is the value range of the decision variable.

If and only if, forAll have f_i(x_u)＜f_i(x_v) Then x_uDominating x_v；

If and only if, forAll have f_i(x_u)≤f_i(x_v) And there is at least one j ∈ { 1.. multidot.n }, such that f_j(x_u)＜f_j(x_v) X is then_uWeak domination x_v；

If and only if, forLet f_i(x_u)＜f_i(x_v) And at the same time,let f_j(x_u)＞f_j(x_v) X is then_uAnd x_vDo not dominate each other;

for H_GEach individual i is provided with the following two parameters n_iAnd S_i,n_iTo govern the number of solution individuals of an individual i in the population, S_iA set of solution individuals governed by an individual i;

first find all n in the population_iIndividuals of 0, store them in the current set F₁；

Then F for the current set₁Each individual j in (a) looks at the set of individuals S it governs_j(ii) a Will gather S_jN of each individual k in_kSubtracting l, i.e. the number of solution entities dominating the individual k, minus 1, if n_k-1 ═ 0 then stores the individual k in another set F₂；

Finally F is put₁As a first-level non-dominant individual set, and endowing the individuals in the set with the same non-dominant order i_rankThen continue to pair F₂Performing the grading operation and assigning corresponding non-dominant orders until all individuals are graded;

s6.6 Congestion calculation

The congestion distance calculation principle is as follows:

(1) arranging the solution sets in the congestion degree calculation set F according to an ascending order;

(2) the crowd distance of the first and last individuals is set to infinity;

(3) the congestion distance of the ith individual is defined as the sum of normalized values of the differences between all objective function values of the (i + 1) th individual and the (i-1) th individual, and is expressed as:

$dis\left(i\right)=\underset{m=1}{\overset{M}{\&Sigma;}}\frac{F_m(i+1)-F_m(i-1)}{\max \left(F_m\right)-\min \left(F_m\right)}---\left(39\right)$

wherein, F_m(i-1) is the mth objective function value of the ith individual, and M is the number of objective functions;

finally, the set F is sorted in a descending order according to the crowding distance;

because both the child and parent individuals are contained in H_GIn (3), the non-dominating set F after non-dominating sorting₁The individual contained in (A) is H_GAmong them, F is first₁Will be put into a new parent population P_G+1Performing the following steps; if F₁Is less than N, P continues_G+1Middle filling next-level non-dominating set F₂Until addition of F_nWhen the size of the population exceeds N, for F_nThe individuals in the Chinese tree are sorted according to the crowding degree, and the top N-P is taken_G+1I individuals, make P_G+1The number of individuals reaches N;

S6.7G ═ G +1 returns to step S6.2;

step S7: output optimization solution

To P_G+1Apply non-dominant ordering and label the set with the highest dominant rank as F₁；

From F according to constraints of the environment₁To select the optimal solution S_op；

Step S8: the obtained optimal solution S_opThe given imaging index is realized as a flight parameter of the airborne receiving station.