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

Path Planning Method for GEO-UAV Bistatic SAR Based on Differential Evolution

technical field

The invention belongs to the technical field of radar, and in particular relates to a UAV receiving station flight path planning method for GEO-UAV bi-base SAR.

Background technique

Synthetic Aperture Radar (SAR) is an all-weather, all-weather high-resolution imaging system. It transmits a large time-width product chirp signal and receives a pulse compression signal through matched filtering to obtain high resolution in the range direction. Using synthetic aperture technology to achieve high resolution in azimuth, the imaging quality is not affected by weather conditions (clouds, light), etc., and has the characteristics of detecting and locating long-distance targets.

Compared with LEO SAR, Geosynchronous Orbit Synthetic Aperture Radar (GEO-SAR) has a larger mapping bandwidth and shorter revisit period, which makes it widely used in disaster monitoring, tectonic imaging and other fields. Using GEO-SAR satellites as radiation sources, airborne or UAV receiving stations can passively receive echoes from target scenes to achieve high-resolution bistatic SAR imaging.

In the GEO-UAV bistatic SAR, the flight path of the UAV receiving station can be pre-set to make it autonomously complete the flight mission in the three-dimensional complex terrain and the bistatic SAR imaging task. The path planning of GEO-UAV bistatic SAR is to find a group of paths in the three-dimensional terrain that can achieve the optimal imaging performance of bistatic SAR and ensure flight safety and short flight time. Modeling two-dimensional UAV path planning in the literature "Route planning for unmannedaerial vehicle (UAV) on these increasing hybrid differential evolution and quantum-behaved particles warm optimization (IEEE Trans. It is a single-objective optimization problem, and a path planning method based on differential evolution of particle swarm optimization (DEQPSO) is proposed. At the same time, in the literature "Three-dimensionalofflinepathplanningforUAVsusingmultiobjectiveevolutionaryalgorithms(inProc.Congr.Evol.Comput.Singapore, 2007, pp.3195–3202)", the three-dimensional UAV path planning problem is modeled as a dual-objective optimization problem, considering the path length and path threat respectively. , and solve it with the multi-objective genetic algorithm NSGAII. However, the above methods do not consider the imaging performance of UAV as the receiving station of bistatic SAR, so they cannot be applied to the path planning problem of GEO-UAV bistatic SAR.

Contents of the invention

The purpose of the present invention is to design a GEO-UAV bi-base SAR path planning method based on the multi-objective differential evolution algorithm for the defects in the background technology, so as to solve the optimal path design problem of the UAV receiving station.

The specific technical scheme of the present invention is: provide a kind of GEO-UAV dual base SAR path planning method based on differential evolution, specifically comprise the following steps:

Step 1: Generate 3D Terrain

Based on the geographic location of the imaging scene, the background 3D terrain of the UAV path is generated through a digital map. In addition, the simulated terrain can be obtained through the numerical simulation of the following formula

$\begin{array}{c}\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&CenterDot;</span>\sqrt{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&CenterDot;</span>\sqrt{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

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

Step 2: Path Modeling of UAV Receiving Station

The path of the UAV receiving station is modeled as a set of control points of a spline curve. Assuming that the number of control points is N _c , the starting point and the ending point of the path are denoted as P _start and P _end respectively, and the imaging points passed by the path of the UAV receiving station are P _im . Except for the above three determined control points, the remaining N _c -3 control points are free control points. The sequence of control points of a spline curve can be expressed as

S _ctrl ＝(P _start ,P ₁ ,...,P _mid ,P _im ,P _mid+1 ,...P _n ,P _end )(2)

Wherein, P _mid and P _mid+1 are adjacent control points of the imaging point P _im , and mid=(N _c -3)/2. And the three-dimensional coordinates of P _im are solved by P _mid and P _mid+1 :

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Through the path modeling of the above UAV receiving station, the UAV path of the spline curve generated by S _ctrl moves from the required starting point P _start to P _end , and passes through the imaging point P _im .

Step 3: Model path planning as a multi-objective optimization problem

First, the UAV path is discretized, and the discrete points of the path are expressed as N _dis is the number of discrete points, then the path distance can be calculated by the following formula

in, is the length of the i-th discrete path. In addition, the UAV receiving station needs to keep a certain safe distance from the ground. Assuming that the minimum safe distance is r _safe , then the threat value of the terrain to the UAV path is as follows

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; g</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Among them, N _g is the number of terrain grid points. r _i,j is the distance between the i-th path discrete point and the j-th terrain grid point.

In addition, the path of the UAV must meet two conditions. First, the path cannot collide with the terrain; second, the turning angle of the UAV path cannot exceed the actual maximum turning angle θ _max . Note that the number of points colliding with the terrain in the discrete points of the path is N _cons1 , and the number of discrete points whose rotation angle exceeds θ _max is N _cons2 . Then N _cons1 =0 and N _cons2 =0 are required.

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

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; l</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; z</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}\mathrm{<span\; class="notranslate">\; r</span>}}}{\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

where ρ _gr is the range resolution:

${\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 0.886</span>}\mathrm{<span\; class="notranslate">\; c</span>}}{{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; h</span>}}^{<span\; class="notranslate">\; \&perp;</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

where, c is the speed of light, t ₀ is the imaging center moment, B _r is the signal bandwidth, H ^⊥ is the ground projection matrix can be expressed as:

${\mathrm{<span\; class="notranslate">\; h</span>}}^{<span\; class="notranslate">\; \&perp;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; G</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

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 , which can be obtained through satellite-ground coordinate conversion.

u _RA (t ₀ ) is the unit vector from the target to the receiving station at time t ₀ :

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}}{<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Among them, P _A is the position of the target point, and P _im is the position of the receiving station.

Azimuth resolution:

${\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; z</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 0.886</span>}\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}{{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; h</span>}}^{<span\; class="notranslate">\; \&perp;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ||</span>\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

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:

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}}{<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}}{<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

Among them, R _T (t ₀ ) is the position coordinate of the transmitting station at the moment of imaging center, is the transpose of the velocity vector of the transmitting station, is the transpose of the velocity vector of the transmitting station.

Resolve direction angles:

α＝cos ^-1 (Ξ Θ)(13)

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

$\mathrm{<span\; class="notranslate">\; \&Theta;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}^{<span\; class="notranslate">\; \&perp;</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}}{<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}^{<span\; class="notranslate">\; \&perp;</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}}{<span\; class="notranslate">\; ||</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; R</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ||</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

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

$\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}& {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}& {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; l</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

stN _consi =0,i=1,2

Among them, w ₁ and w ₂ are the weighting coefficients of path length function and terrain threat function respectively.

Step 4: Using multi-objective differential evolution algorithm to solve

4.1 Initialize iteration parameters

Initialize the iteration parameters of the multi-objective differential evolution algorithm, including the population size N, the maximum number of iterations G _max , the scaling factor F and the crossover rate C _r , and randomly generate the initial population X _G , G=0, including N individuals.

4.2 Crossover mutation

For each individual x _i,G ,i=1,2,...,N in the G-th generation population X _G , generate a new individual v _i,G :

${\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

in, and are three individuals randomly selected from X _G.

After obtaining N new individuals v _{i, G} , i=1, 2, ..., N, perform a mutation operation to obtain a test group U _G . Each individual u _i,G in the test population U _G can 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, G} can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}& \mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; rand</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}}& \mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; e</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

Through the above cross-mutation operation, the test population U _G corresponding to the G generation population X _G is obtained, and X _G and U _G are combined to obtain the population R _G =X _G ∪ U _G .

4.3 Non-dominated sorting and next generation group selection

Sort the 2N individuals in R _G using a non-dominated selection algorithm under constraints. For any given two individuals x _i,G and x _{j,G in the merged group R G .} If x _{i, G} satisfy the constraints but x _{j, G} do not, then x _{i, G} dominates x _{j, G} ; if none of them satisfy the constraints, and the constraints of x _{i, G} are smaller than x _{j, G} , x _i,G dominates x _j,G . If x _{i, G} and x _{j, G} all satisfy the constraints, then compare their objective function values, that is, F ₁ and F ₂ in formula (16), if the objective function values of x _{i, G} are smaller than x _{j, G} , then x _i,G dominates x _j,G . The "constraints" here can be regarded as prior art in this field, and will not be described in detail.

Compare the dominance relationship between each individual in R _G , sort the 2N individuals according to the dominance relationship in the group, and select the N individuals with the highest dominance level to form the next generation group X _G+1 .

4.4 Judging the loop termination condition

Update the number of iterations G=G+1, if the number of iterations G=G _max , go to step 5; if the number of iterations G<G _max , go back to step 4.2.

Step 5: Generate the optimal path for the UAV receiving station

The final generation population obtained by the iterations in step 4 That is, the optimal solution of the multi-objective optimization problem (16). For each optimal solution The UAV receiving station path can be generated, and GEO-UAV bistatic SAR imaging can be realized at the imaging point P _im .

Beneficial effects of the present invention: the method of the present 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; then establishes a multi-objective optimization problem for UAV path planning, and uses multi-objective difference The evolutionary algorithm is used to solve the problem, and multiple groups of UAV optimal paths can be obtained. Specifically, the method of the present invention utilizes the multi-objective differential evolution algorithm for GEO-UAV bistatic SAR path planning, first generates a simulated three-dimensional complex terrain through a digital map or simulation, and then models the flight path of the UAV receiving station as a spline curve A set of control point sequences; secondly, the GEO-UAV bistatic SAR path planning is modeled as a multi-objective optimization problem under constraints, and the multi-objective differential evolution algorithm is used to solve the optimization problem to obtain the optimal The optimal path can meet different UAV 3D terrain navigation and bistatic SAR imaging performance requirements. The method of the present invention utilizes the advantages of strong global search ability and good stability of the differential evolution algorithm, and obtains multiple groups of UAV receiving station paths at the same time, so that the GEO-UAV dual-base SAR system can be widely used in earth remote sensing, resource exploration, geological surveying and mapping.

Description of drawings

Fig. 1 is a schematic flow chart of the method 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

Embodiments of the present invention will be further described below in conjunction with the accompanying drawings.

The present invention mainly uses the method of simulation experiment to verify, and all steps and conclusions are verified by simulation in Matlab2013.

Step 1: Generate 3D simulated terrain according to formula (1). In order to simulate the real terrain height and surface, the following terrain parameters are selected: a, b, e, g=1, c, d, f=1.8. The two-dimensional grid spacing of the terrain is 0.2km, and the total length is 20km. Then the terrain height information matrix is expressed as Z(x,y).

Step 2: Path modeling of UAV receiving station. According to formula (2), the UAV receiving station path is modeled as a set of control point coordinates, x=((x ₁ ,y ₁ ,z ₁ ),(x ₂ ,y ₂ ,z ₂ ),...,(x _n ,y _n ,z _n )), in the simulation analysis, set the number n of UAV path control points to 9, where the first point is the starting point of the path, and its coordinates are set to (4,4,0.3) km, the optimal control point is the end point of the path, and its coordinates are set to (18, 17, 1.4) km. The GEO-UAV bibase SAR parameters are shown in Table 1.

Table 1

Step 3: Model the path planning problem as a multi-objective optimization problem. For a set of control points x, first generate a continuous path spline. Discretize the path spline curve into _Ndis discrete points, _Ndis =1000; when calculating the path length, first calculate the straight-line distance between every two adjacent discrete path points, and then sum all the straight-line distances; in addition, r _safe is taken as 200m; finally, a dual-objective optimization model under constraints is established according to formula (16), where the path length and terrain threat weighting coefficient of the first objective function F ₁ (x) are set as w ₁ =w ₂ = 0.5.

Step 4: Use the multi-objective differential evolution algorithm to solve the optimization problem in (16). First initialize the parameters of the differential algorithm, as shown in Table 2.

Table 2

The initial N individuals are randomly generated to form the initial population X ₀ , and the cross-mutation operation is performed according to formulas (17) and (18) to generate a test population, which is combined with the initial population to obtain a merged population R ₀ containing 2N individuals.

Perform non-dominated sorting on each individual in R ₀ , rank the individual with the highest dominance level at the front, and the individual with the lowest dominance level at the end, and select N individuals with high dominance level to form the next generation group X ₁ . Add 1 to the number of iterations G, and judge whether it exceeds the maximum number of iterations G _max , if not, return to step 4.2 to continue cross-mutation, merging and non-dominated sorting, update individuals in the group until the number of iterations reaches G _max , and jump out Loop to get the optimal solution matrix Figure 1 shows the flow of the multi-objective differential evolution algorithm.

Step 5: Select the optimal solution matrix Individuals in , generate the path of the UAV receiving station, and measure the navigation performance of its UAV as well as the GEO-UAV bistatic SAR imaging performance. Figure 2, Figure 3 and Figure 4 are three Individually generated UAV receiving station paths in , with different path lengths, terrain threats, and bistatic SAR imaging performance. ( _a ), (b) and (c) in Fig. 5 are the contour maps of the imaging results of the target point PA by the UAV receiving station in the three paths respectively.

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:

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

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:

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

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:

$S_{cell}\left(x\right)=\frac{{\mathrm{\&rho;}}_{az}\&CenterDot;{\mathrm{\&rho;}}_{gr}}{\sin \mathrm{\&alpha;}}---\left(6\right)$

where ρ is_grFor distance resolution:

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

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:

$H^{\&perp;}=I-P_G\&CenterDot;P_G^T---\left(8\right)$

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:

${\mathrm{\&rho;}}_{az}=\frac{0.886\mathrm{\&lambda;}}{{\&Integral;}_{t_0-T_a/2}^{t_0+T_a/2}\left|\right|H^{\&perp;}\left({\mathrm{\&omega;}}_{TA}\right(t)+{\mathrm{\&omega;}}_{RA}(t\left)\right)\left|\right|dt}---\left(10\right)$

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:

${\mathrm{\&omega;}}_{TA}\left(t\right)=\frac{\&lsqb;I-u_{TA}^T\left(t_0\right)u_{TA}\left(t_0\right)\&rsqb;V_T^T}{\left|\right|P_A-R_T\left(t_0\right)\left|\right|}---\left(11\right)$

${\mathrm{\&omega;}}_{RA}\left(t\right)=\frac{\&lsqb;I-u_{RA}^T\left(t_0\right)u_{RA}\left(t_0\right)\&rsqb;V_R^T}{\left|\right|P_A-P_{im}\left|\right|}---\left(12\right)$

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:

$\mathrm{\&Theta;}=\frac{H^{\&perp;}{\left(u_{TA}\right(t_0)+u_{RA}(t_0\left)\right)}^T}{\left|\right|u_{TA}\left(t_0\right)+u_{RA}\left(t_0\right)\left|\right|}---\left(14\right)$

$\mathrm{\&Xi;}=\frac{H^{\&perp;}{\left({\mathrm{\&omega;}}_{TA}\right(t_0)+{\mathrm{\&omega;}}_{RA}(t_0\left)\right)}^T}{\left|\right|{\mathrm{\&omega;}}_{TA}\left(t_0\right)+{\mathrm{\&omega;}}_{RA}\left(t_0\right)\left|\right|}---\left(15\right)$

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

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

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:

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

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.