CN113221314A - Modeling method for radar echo signal disturbed by angular motion of spinning tail projectile - Google Patents

Abstract

The invention relates to a method for modeling the initial disturbance of radar echo signals from the angular motion of the spin tail projectile, which is characterized in that the specific steps are as follows: establishing a dynamic equation of the spin tail projectile; Solve; determine the angular motion mode of the spin tail projectile; determine the definition of the relevant coordinate system; calculate the distance from the projectile scattering point to the radar; radar echo Doppler signal; it analyzes the projectile axis and speed of the spin tail projectile in the process of space motion The change mechanism of the angular motion between the vectors is then transformed into the projection transformation of the radar measurement information to this movement, which provides a theoretical basis for using the radar measurement information to identify the motion characteristics, aerodynamic characteristics and structural characteristics of the target.

Description

A Modeling Method for Disturbing Radar Echo Signals from Angular Motion of Spin-tail Projectile

technical field

The invention relates to a modeling method for the start of the angular motion of the spin tail projectile to disturb the radar echo signal. The invention relates to the modeling and simulation technology of Pule signal, which belongs to the field of radar echo signal modeling and simulation.

Background technique

The flight motion of the projectile in the air includes the motion of the center of mass and the rotation around the center. The core of the projectile motion model of the spin tail is constructed according to the kinematics and dynamics equations of the motion of the center of mass of the projectile and the rotation around the center. The rotation of the projectile around the center determines the flight stability of the projectile. There are two main methods to achieve the flight stability of the projectile: rotation stabilization and tail stabilization.

In the field of radar measurement, the motion of the center of mass of the projectile belongs to translation, including the radial velocity, radial acceleration or high-order acceleration and other motion components, and is the main component of the Doppler effect of radar measurement of moving targets (Gao Xudong. The principle and application of projectile flight [ M]. Beijing: Beijing Institute of Technology Press, 2018.). The projectile moves around the center, including motion components such as spin, cone spin, tumbling, vibration, etc., which form a small modulation on the Doppler signal of the radar target, which is often regarded as "noise" or "residual" when processing ballistic time domain data cull.

At the beginning of this century, V.C.Chen used microwave radar to measure and analyze the radial micro-motion (Micro-motion) effect of moving targets other than the translation of the center of mass, and defined it as the post-Micro-Doppler (Chen V.C.Analysis ofradar micro -Doppler with time-frequency transforms[J].Proceedings of the 10th IEEE workshop on Statistical Signal and Array Processing,USA, 2000.), in the field of radar target detection and recognition, it has become a research hotspot in academia and engineering at home and abroad (William Z L,Alan AG.Overview of the lincoln laboratory ballistic missile defense program[J].Lincoln Laboratory Journal,2002,13(1):932.)(Liu Yongxiang, Li Xiang, Zhuang Zhaowen. Progress in Radar Target Recognition Technology in Missile Defense System [J]. Systems Engineering and Electronic Technology, 2006, 28(8):1188-1193.).

There are a lot of research results on the micro-motion identification of ground targets, air targets and high-altitude targets. The use of micro-Doppler features can realize the classification and precise identification of ground moving targets such as tanks and armored vehicles (Barbaross S. Doppler-rate filtering for detecting moving targets with synthetic aperture radars[A].Proceedings of the SPIE on Millimeter Wave and Synthetic Aperture Radar[C].Orlando,USA:SPIE Press,1989.140-147.) (Huang Jian, Li Xin, Huang Xiaotao, etc. Based on micro-Doppler features Tank Target Parameter Estimation and Identification [J]. Journal of Electronics and Information, 2010, 32(5): 1050-1055.); Millimeter-wave radar can be used to analyze the helicopter rotor micro-Doppler spectrum and judge the helicopter movement state ( Nalecz M, Andrianik R R, Wojtkiewicz A. Micro-Doppler analysis of signal received by FMCWradar[A].Proceedings of International Radar Symposium[C].Dresden,Germany:IEEEPress,2003.231-235.) (Chen Peng, Hao Shiqi, Hu Yihua, etc. .Micro-Doppler characteristic analysis of moving helicopter rotor[J].Infrared and Laser Engineering, 2015,44(1):118-121.); especially in missile defense systems, micro-motion feature information has become the key to target recognition. It is an important auxiliary means, and the research results of the fretting characteristics of ballistic missiles can provide a basis for effectively identifying the threat target in the middle of the ballistic trajectory (Yang Guiling, Wang Zhaoying, Zhang Yu, etc. The characteristics of ballistic missile fretting characteristic parameters with radar echo signal-to-noise ratio change characteristics [J] .Information and Communication, 2017, 175(7): 10-12.)(Wang Weilin, Chen Lei, Lei Yongjun. Research on the fretting characteristics of the mid-course decoy of ballistic missiles [J]. Systems Engineering and Electronic Technology, 2016, 38(3): 487-492.).

However, none of the above studies involve the simulation and modeling of the radar echo signal of the projectile in the straight line segment of the trajectory which is disturbed by the initial disturbance.

SUMMARY OF THE INVENTION

The object of the present invention is to provide a modeling method for the initial disturbance of the radar echo signal of the angular motion of the spin tail projectile, which is in the field of conventional weapons test and identification, based on the dynamic and kinematic models of the projectile fretting as the premise, focusing on Pay attention to the signal level mapping relationship between the angular motion of the projectile caused by the initial disturbance in the ballistic straight section and the measuring radar, and establish the quantitative relationship between the micro-Doppler and projectile parameters and the fretting echo model, which can be used for interpreting the radar signal and accurately identifying the target. The motion parameters provide a reference; based on the dynamic equation of the spin-tail projectile characterized by the second-order variable coefficient differential equation of the projectile's return angle of attack, the various effects of the spin-tail projectile under the influence of the initial disturbance on the straight line segment of the trajectory are analyzed and studied. This kind of micro-motion is used to establish the quantitative relationship between micro-Doppler and projectile parameters and the micro-motion echo model, and improve the radar echo signal modeling system.

The technical scheme of the present invention is realized in the following way: a modeling method for the start of the angular motion of the spin tail projectile to disturb the radar echo signal, which is characterized in that the specific steps are as follows:

Step 1: Establish the dynamic equation of the spin tail projectile;

According to the ballistic theory, the dynamic equation of the spin-tail projectile can be expressed by the second-order variable coefficient differential equation of the complex angle of attack motion

Δ is the angle of attack of the projectile flying in the air, which can generally be represented by a complex number;

其中，k_zz为赤道阻尼力矩，b_y为升力，b_x为阻力，g为本地重力加速度，θ弹丸攻角的标量值，v为弹丸的飞行速度；

Among them, k _zz is the equatorial damping moment, b _y is the lift force, b _x is the drag force, g is the local gravitational acceleration, θ is the scalar value of the angle of attack of the projectile, and v is the flight speed of the projectile;

M=k _z , k _z is the static moment, which determines the frequency of attitude movement;

C为极转动惯量，

为弹丸的自转速度，A为赤道转动惯量，v为飞行速度的比值；

C is the polar moment of inertia,

is the rotation speed of the projectile, A is the equatorial moment of inertia, and v is the ratio of the flight speed;

b_y为升力，k_y为马格努斯力矩；

b _y is the lift force, k _y is the Magnus moment;

为弹道弯曲引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile angle of attack movement caused by the curvature of the trajectory;

为由弹丸结构不对称引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile angle of attack motion caused by the projectile structure asymmetry;

为由弹丸气动不对称引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile's angle of attack motion caused by the aerodynamic asymmetry of the projectile;

为由垂直气流引起的对弹丸攻角的扰动项。

is the perturbation term to the projectile angle of attack caused by the vertical airflow.

Δ₀为弹丸的复攻角初值，

表示Δ₀对时间t的一阶导数，v₀为弹丸初速。方程及其起始条件可反映出弹丸由各种因素引起的攻角运动。其中，H代表阻尼运动，主要取决于赤道阻尼力矩k_zz、升力b_y和阻力b_x等；M主要与静力矩k_z有关，姿态运动频率主要取决于此项；T主要与升力b_y及马格努斯力矩k_y有关，它影响飞行稳定性；P为弹丸的自转速度和飞行速度的比值；||Δ|| is the angle between the space projectile axis and the velocity vector, and Δ′ and Δ″ are the first and second derivatives of the complex attack angle to the ballistic arc length s. When the initial condition is s=0, Δ=Δ ₀ ,

Δ ₀ is the initial value of the re-attack angle of the projectile,

Represents the first derivative of Δ ₀ with respect to time t, and v ₀ is the initial velocity of the projectile. The equation and its initial conditions can reflect the angle of attack motion of the projectile caused by various factors. Among them, H represents the damping motion, which mainly depends on the equatorial damping moment k _zz , the lift b _y and the resistance b _x , etc.; M is mainly related to the static moment k _z , and the attitude motion frequency mainly depends on this; T is mainly related to the lift b _y and Magnus moment _ky is related, it affects flight stability; P is the ratio of projectile's rotation speed and flight speed;

Step 2: Solve the homogeneous differential equation of the projectile complex angle of attack;

Aiming at the angular motion characteristic of the projectile that is only disturbed by the initial disturbance in the straight line segment, the homogeneous differential equation of the projectile re-attack angle is solved;

Δ″+(H-iP)Δ′-(M+iPT)Δ=0

The characteristic root of this equation with respect to the complex angle of attack Δ is

l _1,2 =λ _1,2 +iω _1,2

In the formula, λ ₁ , λ ₂ are called damping exponents, ω ₁ , ω ₂ are called the modal frequencies to the ballistic arc length s; then the angle of attack solution is

常数k₁,k₂是实数，

为初始相位，由起始条件确定；则攻角的解又可以写成In the formula, C ₁ , C ₂ are undetermined coefficients, which are complex numbers and can generally be written as

The constants k ₁ , k ₂ are real numbers,

is the initial phase, determined by the initial conditions; then the solution of the angle of attack can be written as

其中，k₁,k₂和

由初始条件确定；In the formula,

where k ₁ , k ₂ and

determined by initial conditions;

表示模为1，幅角为Φ_1,2的一个向量，当幅角以角频率ω_1,2改变时，此单位模复数的矢端将在复数平面上画出一个圆；The two complex numbers on the right side of the above formula are modal vectors, and K ₁ and K ₂ are called modal amplitudes. According to the vector notation of complex numbers,

Represents a vector whose modulus is 1 and its argument is Φ _1,2 . When the argument changes with the angular frequency ω _1,2 , the vector end of this unit modulus complex number will draw a circle on the complex plane;

Step 3: Determine the angular motion mode of the spin tail projectile;

When Δ=Δ ₀ and Δ′=Δ′ ₀ are combined with the solution of the angle of attack when the ballistic arc length s=0, the undetermined coefficients C ₁ and C ₂ can be solved as

Therefore, it is determined that the radar echo signal caused by the initial disturbance is modeled by the fast and slow two-circle motion modes;

Step 4: Determine the definition of the relevant coordinate system;

Combined with the ballistic theory and the definition of the coordinate system related to the measurement radar, the spatial geometric relationship of the angular motion of the projectile linear segment can be established. Taking the radar tail tracking as an example, four coordinate systems oriented according to the right-hand rule are introduced, including the radar coordinate system Q-UVW , the gun position coordinate system O ₁ -xyz, the reference coordinate system O-XYZ and the projectile axis coordinate system O-ξηζ, the gun position coordinate system takes the muzzle center as the origin O ₁ , and the horizontal axis O ₁ x is the shooting surface and the gun muzzle horizontal plane The line of intersection of the radar is positive, and the vertical axis O ₁ y is in the shooting plane and is perpendicular to the horizontal axis O ₁ z; the radar coordinate system takes the site center as the origin Q, and the horizontal axis QU points to true north in the horizontal plane direction, QV is the direction of the vertical line and is perpendicular to the horizontal axis QW; the reference coordinate system takes the center of mass of the projectile as the origin O, which is always parallel to the radar coordinate system; the origin of the projectile axis coordinate system is on the center of mass of the projectile, and the Oξ axis moves forward along the projectile axis is positive, the On-axis is positive upwards perpendicular to the projectile axis, and the oζ-axis is determined by the right-hand rule;

Step 5: Calculate the distance from the projectile scattering point to the radar;

The angular motion of the projectile is decomposed into the rotation of the polar axis of the projectile, the nutation of the polar axis around the gyro-momentum moment vector (fast circular motion) and the precession of the gyro-momentum moment vector around the velocity vector line (slow circular motion) motion with three degrees of freedom overlay. Starting from the superposition of three motions of projectile spin, nutation and precession, the radar echo signal model of the projectile target's strong scattering point is established;

决定，即本地坐标系 O-xyz分别围绕z轴顺时针旋转

围绕x轴顺时针旋转θ_e，再围绕z 轴顺时针旋转φ_e，就变换为参考坐标系O-XYZ。其表达式为In electromagnetic scattering, the curvature discontinuity is the strong scattering center of the target. For the spin tail projectile, the projectile vertex, the cone-column junction and the tail scattering point are the strong scattering center of the target. A representative scattering point is used to model the radar echo signal. In the elastic axis coordinate system, let the spin angular velocity be ω′ _s =(ω _sξ ,ω _sη ,ω _sζ ) ^T , the nutation angular velocity be ω′ _u =(ω _uξ ,ω _uη ,ω _uζ ) ^T , the precession The angular velocity is ω′ _c =(ω _cξ ,ω _cη ,ω _cζ ) ^T , the angular velocity scalars Ω _s =||ω′ _s ||, Ω _u =||ω′ _u || and Ω _c =||ω′ _c ||, the unit angular velocity of each rotational angular velocity in the reference coordinate system is ω _s =R _init ·ω′ _s /Ω _s , ω _u =R _init ·ω′ _u /Ω _u and ω _c =R _init ·ω ′ _c /Ω _c . R _init by initial Euler angles

Determined, that is, the local coordinate system O-xyz rotates clockwise around the z-axis respectively

Rotate θ _e clockwise around the x-axis, and then rotate φ _e clockwise around the z-axis to transform into a reference coordinate system O-XYZ. Its expression is

R_spin为自旋旋转矩阵，r₀为发射前散射点P在雷达坐标系下的初始矢量，与发射的射角和射向有关；然后弹轴Oξ绕陀螺力矩矢量线OG轴做章动运动，则P坐标变为

R_nut为章动旋转矩阵；最后，陀螺力矩矢量线OG轴绕速度矢量线V₀进动，则P坐标变为

R_con为进动旋转矩阵。则弹体上P点在t时刻到雷达的距离为After the above analysis, the angular motion of the projectile can be described as: at time t, the scattering point P on the projectile first spins around the projectile axis Oξ, then the coordinate of P becomes

R _spin is the spin rotation matrix, r ₀ is the initial vector of the scattering point P in the radar coordinate system before the launch, which is related to the launch angle and launch direction; then the projectile axis Oξ performs nutation motion around the gyro moment vector line OG axis , then the P coordinate becomes

R _nut is the nutation rotation matrix; finally, the gyro moment vector line OG axis precesses around the velocity vector line V ₀ , then the P coordinate becomes

R _con is the precession rotation matrix. Then the distance from point P on the projectile to the radar at time t is

In the formula, according to the Euler-Rodrigues rotation formula around the vector axis, each rotation matrix is

和

分别是ω_s、ω_n和ω_c所对应的斜对称矩阵；in,

and

are the oblique symmetric matrices corresponding to ω _s , ω _n and ω _c respectively;

Step 6: Radar echo Doppler signal;

Assume that the radar transmits a single carrier frequency continuous wave signal, and the transmitted signal is in the form of

s=exp(j2πf ₀ t)

Among them, f ₀ is the carrier frequency. Then the echo of the scattering point P is s _P =σexp(j2πf ₀ (t-τ)), where σ is the scattering coefficient, τ is the time delay of the scattering point _P , and τ=2RP (t)/c is satisfied;

Taking the transmitted signal formula as a reference signal, and coherently processing the target echo, the coherent phase signal is obtained as

where the phase term Φ(t)=4πf ₀ R _P (t)/c. Taking the derivation of the phase term with respect to time t, the Doppler frequency of the echo is obtained as

为

的单位向量，当目标位于雷达远场时，n≈R₀/||R₀||，为雷达视线方向 LOS的单位向量；in,

for

The unit vector of , when the target is located in the far field of the radar, n≈R ₀ /||R ₀ ||, which is the unit vector of the radar line-of-sight direction LOS;

From this, the Doppler frequency caused by the translation of the target can be obtained:

The projectile scattering point is caused by the angular motion of the projectile at the micro-Doppler frequency:

It can be seen from the above formula that the expression of the micro-Doppler frequency of the projectile radar echo caused by the angular motion of the projectile axis in space is relatively complicated. Through analysis, it can be seen that the variation of radar micro-Doppler frequency with time is periodic, the motion period T _top of the top scattering point A is the least common multiple of the nutation period T _s and the precession period T _c ; the motion period of the tail scattering point P T _bom is the least common multiple of the spin period T _s , the precession period T _c and the _nutation period Tu , that is, the following relationship holds:

T _top =k ₁ T _c = _{k 2} Tu ,T _bom =k ₁ T _c =k _{2 Tu} =k ₃ T _s k ₁ ,k ₂ ,k ₃ ∈N

Among them, N is the set of natural numbers.

The positive effect of the invention is that the radar echo characteristics of the micro-moving target reflect the fine characteristics of the target, such as structural characteristics, electromagnetic scattering characteristics and motion characteristics. The change mechanism of the angular motion is then transformed into the projection transformation of the radar measurement information to this movement, which provides a theoretical basis for using the radar measurement information to identify the motion characteristics, aerodynamic characteristics and structural characteristics of the target. Aiming at the situation that the spin tail projectile is in the straight line segment of the trajectory, the projectile angular motion is affected by the initial disturbance, the quantitative relationship between the micro-Doppler and projectile parameters is established, and the radar micro-Doppler echo signal construction method is proposed, which fills the gap in the The research on the space motion of the spin-tail projectile affected by the initial disturbance in the straight line segment is lacking, and the radar micro-Doppler signal modeling system is improved to provide the research on the micro-Doppler effect of the spin-tail projectile on the radar echo signal. theoretical support.

Description of drawings

1 is a time-frequency diagram of a radar micro-Doppler echo signal generated by the initial disturbance of the angular motion of the spin tail projectile generated by modeling and simulation of the present invention.

FIG. 2 is a time-frequency diagram of a micro-Doppler signal obtained by performing Gabor transform on the micro-Doppler signal in FIG. 1 according to the present invention.

FIG. 3 is a projectile angle of attack curve calculated by the present invention through simulation data.

Fig. 4 is the Doppler time-frequency diagram obtained by the actual measurement of the projectile of the present invention.

FIG. 5 is a time-frequency diagram of the matching simulation of the time-frequency signal of the measured projectile according to the priori conditions such as the actual size of the projectile, the projectile shape coefficient, and the flight speed of the present invention.

Detailed ways

Below in conjunction with accompanying drawing and example, the present invention will be further described: a kind of modeling method of the initial disturbance radar echo signal of spin tail projectile angular movement, with radar-related parameters, the translation parameters of the spin tail projectile in the air and the microscopic The dynamic parameters and the scattering characteristic parameters of the projectile are used as the main input. After determining the relationship between the radar and the target, the change of the angular motion between the projectile axis and the velocity vector during the space movement of the projectile is converted into radar measurement information. The projection transformation of the spin tail projectile angular motion to the radar's micro-Doppler signal model is established. The specific implementation steps are as follows:

Step 1: Establish the dynamic equation of the spin tail projectile;

According to the ballistic theory, the dynamic equation of the spin-tail projectile can be expressed by the second-order variable coefficient differential equation of the complex angle of attack motion

Δ is the angle of attack of the projectile flying in the air, which can generally be represented by a complex number;

其中，k_zz为赤道阻尼力矩，b_y为升力，b_x为阻力，g为本地重力加速度，θ弹丸攻角的标量值，v为弹丸的飞行速度；

Among them, k _zz is the equatorial damping moment, b _y is the lift force, b _x is the drag force, g is the local gravitational acceleration, θ is the scalar value of the angle of attack of the projectile, and v is the flight speed of the projectile;

M=k _z , k _z is the static moment, which determines the frequency of attitude movement;

C为极转动惯量，

为弹丸的自转速度，A为赤道转动惯量，v为飞行速度的比值；

C is the polar moment of inertia,

is the rotation speed of the projectile, A is the equatorial moment of inertia, and v is the ratio of the flight speed;

b_y为升力，k_y为马格努斯力矩；

b _y is the lift force, k _y is the Magnus moment;

为弹道弯曲引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile angle of attack movement caused by the curvature of the trajectory;

为由弹丸结构不对称引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile angle of attack motion caused by the projectile structure asymmetry;

为由弹丸气动不对称引起的对弹丸攻角运动的扰动项；

is the disturbance term to the projectile's angle of attack motion caused by the aerodynamic asymmetry of the projectile;

为由垂直气流引起的对弹丸攻角的扰动项。

is the perturbation term to the projectile angle of attack caused by the vertical airflow.

Δ₀为弹丸的复攻角初值，

表示Δ₀对时间t的一阶导数，v₀为弹丸初速。方程及其起始条件可反映出弹丸由各种因素引起的攻角运动。其中，H代表阻尼运动，主要取决于赤道阻尼力矩k_zz、升力b_y和阻力b_x等；M主要与静力矩k_z有关，姿态运动频率主要取决于此项；T主要与升力b_y及马格努斯力矩k_y有关，它影响飞行稳定性；P为弹丸的自转速度和飞行速度的比值。||Δ|| is the angle between the space projectile axis and the velocity vector, and Δ′ and Δ″ are the first and second derivatives of the complex attack angle to the ballistic arc length s. When the initial condition is s=0, Δ=Δ ₀ ,

Δ ₀ is the initial value of the re-attack angle of the projectile,

Represents the first derivative of Δ ₀ with respect to time t, and v ₀ is the initial velocity of the projectile. The equation and its initial conditions can reflect the angle of attack motion of the projectile caused by various factors. Among them, H represents the damping motion, which mainly depends on the equatorial damping moment k _zz , the lift b _y and the resistance b _x , etc.; M is mainly related to the static moment k _z , and the attitude motion frequency mainly depends on this; T is mainly related to the lift b _y and Magnus moment _ky is related, it affects flight stability; P is the ratio of projectile's rotation speed and flight speed.

Step 2: Solve the homogeneous differential equation of the projectile complex angle of attack;

Aiming at the angular motion characteristics of the projectile only subject to initial disturbance in the straight line segment, the homogeneous differential equation of the projectile re-attack angle is solved.

Δ″+(H-iP)Δ′-(M+iPT)Δ=0

The characteristic root of this equation with respect to the complex angle of attack Δ is

l _1,2 =λ _1,2 +iω _1,2

In the formula, λ ₁ , λ ₂ are called damping exponents, and ω ₁ , ω ₂ are called the modal frequencies to the ballistic arc length s.

Then the angle of attack is solved as

常数k₁,k₂是实数，

为初始相位，由起始条件确定。则攻角的解又可以写成In the formula, C ₁ , C ₂ are undetermined coefficients, which are complex numbers and can generally be written as

The constants k ₁ , k ₂ are real numbers,

is the initial phase, determined by the initial conditions. Then the solution for the angle of attack can be written as

其中，k₁,k₂和

由初始条件确定。In the formula,

where k ₁ , k ₂ and

determined by initial conditions.

表示模为1，幅角为Φ_1,2的一个向量。当幅角以角频率ω_1,2改变时，此单位模复数的矢端将在复数平面上画出一个圆。The two complex numbers on the right side of the above formula are modal vectors, and K ₁ and K ₂ are called modal amplitudes. According to the vector notation of complex numbers,

Represents a vector with a modulus of 1 and an argument of Φ _1,2 . When the argument changes at the angular frequency ω _1,2 , the vector end of this unit modulo complex number will draw a circle in the complex plane.

Step 3: Determine the angular motion mode of the spin tail projectile;

When Δ=Δ ₀ and Δ′=Δ′ ₀ are combined with the solution of the angle of attack when the ballistic arc length s=0, the undetermined coefficients C ₁ and C ₂ can be solved as

Therefore, it is determined that the radar echo signal caused by the initial disturbance is modeled by the fast and slow two-circle motion modes.

Step 4: Determine the definition of the relevant coordinate system;

Combined with the ballistic theory and the definition of the coordinate system related to the measurement radar, the spatial geometric relationship of the angular motion of the straight line segment of the projectile can be established. Take the radar tail tracking as an example, as shown in Figure 1.

Four coordinate systems oriented according to the right-hand rule are introduced, including the radar coordinate system Q-UVW, the gun position coordinate system O ₁ -xyz, the reference coordinate system O-XYZ and the projectile axis coordinate system O-ξηζ. The gun position coordinate system takes the center of the muzzle as the origin O ₁ , the horizontal axis O ₁ x is the intersection of the shooting surface and the horizontal plane of the muzzle, the forward shooting direction is positive, and the vertical axis O ₁ y is in the shooting surface and is in line with the horizontal axis O ₁ The z-phase is vertical; the radar coordinate system takes the site center as the origin Q, the horizontal axis QU points to the true north direction in the horizontal plane, and QV is the direction of the plumb line and is perpendicular to the horizontal axis QW; the reference coordinate system takes the projectile center of mass as the origin O , which is always parallel to the radar coordinate system; the origin of the projectile axis coordinate system is on the projectile center of mass, the Oξ axis is positive forward along the projectile axis, the Oη axis is positive upward perpendicular to the projectile axis, and the oζ axis is determined by the right-hand rule.

Step 5: Calculate the distance from the projectile scattering point to the radar;

The angular motion of the projectile is decomposed into the rotation of the polar axis of the projectile, the nutation of the polar axis around the gyro-momentum moment vector (fast circular motion) and the precession of the gyro-momentum moment vector around the velocity vector line (slow circular motion) motion with three degrees of freedom overlay. Starting from the superposition of the projectile spin, nutation and precession, the radar echo signal model of the projectile target's strong scattering point is established.

决定，即本地坐标系 O-xyz分别围绕z轴顺时针旋转

围绕x轴顺时针旋转θ_e，再围绕z 轴顺时针旋转φ_e，就变换为参考坐标系O-XYZ。其表达式为In electromagnetic scattering, the curvature discontinuity is the strong scattering center of the target. For the spin tail projectile, the projectile vertex, the cone-column junction and the tail scattering point are the strong scattering center of the target. A representative scattering point is used to model the radar echo signal. In the elastic axis coordinate system, let the spin angular velocity be ω′ _s =(ω _sξ ,ω _sη ,ω _sζ ) ^T , the nutation angular velocity be ω′ _u =(ω _uξ ,ω _uη ,ω _uζ ) ^T , the precession The angular velocity is ω′ _c =(ω _cξ ,ω _cη ,ω _cζ ) ^T , the angular velocity scalars Ω _s =||ω′ _s ||, Ω _u =||ω′ _u || and Ω _c =||ω′ _c ||, the unit angular velocity of each rotational angular velocity in the reference coordinate system is ω _s =R _init ·ω′ _s /Ω _s , ω _u =R _init ·ω′ _u /Ω _u and ω _c =R _init ·ω ′ _c /Ω _c . R _init by initial Euler angles

Determined, that is, the local coordinate system O-xyz rotates clockwise around the z-axis respectively

Rotate θ _e clockwise around the x-axis, and then rotate φ _e clockwise around the z-axis to transform into a reference coordinate system O-XYZ. Its expression is

R_spin为自旋旋转矩阵；然后弹轴Oξ绕陀螺力矩矢量线OG轴做章动运动，则P坐标变为

R_nut为章动旋转矩阵；最后，陀螺力矩矢量线OG轴绕速度矢量线V₀进动，则P坐标变为

R_con为进动旋转矩阵。则弹体上P点在t时刻到雷达的距离为After the above analysis, the angular motion of the projectile can be described as: at time t, the scattering point P on the projectile first spins around the projectile axis Oξ, then the coordinate of P becomes

R _spin is the spin rotation matrix; then the elastic axis Oξ performs nutation motion around the gyro moment vector line OG axis, then the P coordinate becomes

R _nut is the nutation rotation matrix; finally, the gyro moment vector line OG axis precesses around the velocity vector line V ₀ , then the P coordinate becomes

R _con is the precession rotation matrix. Then the distance from point P on the projectile to the radar at time t is

In the formula, according to the Euler-Rodrigues rotation formula around the vector axis, each rotation matrix is

和

分别是ω_s、ω_n和ω_c所对应的斜对称矩阵。in,

and

are the oblique symmetric matrices corresponding to ω _s , ω _n and ω _c , respectively.

Step 6: Radar echo Doppler signal;

Assume that the radar transmits a single carrier frequency continuous wave signal, and the transmitted signal is in the form of

s=exp(j2πf ₀ t)

Among them, f ₀ is the carrier frequency. Then the echo of the scattering point P is s _P =σexp(j2πf ₀ (t-τ)), where σ is the scattering coefficient, τ is the time delay of the scattering point _P , and τ=2RP (t)/c is satisfied.

Taking the transmitted signal formula as a reference signal, and coherently processing the target echo, the coherent phase signal is obtained as

where the phase term Φ(t)=4πf ₀ R _P (t)/c. Taking the derivation of the phase term with respect to time t, the Doppler frequency of the echo is obtained as

为

的单位向量。当目标位于雷达远场时，n≈R₀/||R₀||，为雷达视线方向 LOS的单位向量。in,

for

unit vector of . When the target is located in the far field of the radar, n≈R ₀ /||R ₀ ||, which is the unit vector of the radar line-of-sight direction LOS.

From this, the Doppler frequency caused by the translation of the target can be obtained:

The projectile scattering point is caused by the angular motion of the projectile at the micro-Doppler frequency:

It can be seen from the above formula that the expression of the micro-Doppler frequency of the projectile radar echo caused by the angular motion of the projectile axis in space is relatively complicated. Through analysis, it can be seen that the variation of radar micro-Doppler frequency with time is periodic, the motion period T _top of the top scattering point A is the least common multiple of the nutation period T _s and the precession period T _c ; the motion period of the tail scattering point P T _bom is the least common multiple of the spin period T _s , the precession period T _c and the _nutation period Tu , that is, the following relationship holds:

T _top =k ₁ T _c = _{k 2} Tu ,T _bom =k ₁ T _c =k _{2 Tu} =k ₃ T _s k ₁ ,k ₂ ,k ₃ ∈N

Among them, N is the set of natural numbers.

First, set the radar parameters, including the position, system, frequency, sampling frequency, sampling time, line of sight vector and window function of the radar; then, set various motion parameters of the spin tail projectile, including translation speed, spin Frequency, nutation frequency, precession frequency, nutation angle, precession angle, etc.; finally, set the number of projectile strong scattering points, the position on the projectile, etc. There are no special requirements for the setting of parameters, only general indicators are set.

Radar main parameter settings:

1) Single carrier frequency continuous wave radar;

2) carrier frequency f ₀ =10GHz;

3) sampling interval f _s =20000Hz;

4) Take the radar position as the coordinate origin;

5) Tail-tracking.

Projectile main parameter settings:

1) Translation velocity vector (200, 200, 200) m/s;

2) Spin frequency ω _s =300πrad/s;

3) Precession frequency ω _c =4πrad/s;

4) Nutation frequency ω _s = 20πrad/s;

5) Nutation angle φ=1.2;

6) Precession angle ψ=8.2.

Spin tail projectile scattering characteristic settings:

1) Select the tip scattering point and 2 tail scattering points for calculation;

2) The position of the projectile axis coordinate system of the scattering point, the projectile top scattering point P1 is (0.5, 0, 0) m, the tail scattering point P2 is (-0.3, 0.06, 0) m, and the tail scattering point P3 is (-0.3, - 0.06,0)m;

Secondly, determine the radar coordinate system, the missile coordinate system and the reference coordinate system, calculate the transformation relationship between each coordinate system through the Euler rotation transformation matrix, and calculate the rotation matrix that rotates around the vector axis.

According to the Euler-Rodrigues rotation formula around the vector axis, each rotation matrix is

Thirdly, by calculating the distance expression of each scattering point relative to the continuous wave radar under the superposition of various motions such as translation, spin, nutation and precession of the projectile, and separate the Doppler signal born to the radar signal.

Finally, the Doppler signals generated by translation and micro-motion are separated and time-frequency analysis is performed,

The translational Doppler signal is:

Micro-Doppler Signal:

The modeling and simulation results are shown in Figure 1. The horizontal axis is time, and the simulation time length is 4 seconds; the vertical axis is the frequency amplitude of micro-Doppler; the solid curve with smaller amplitude in the figure is the micro-Doppler signal generated by the tip scattering point; the larger amplitude And the overlapping curves are the Doppler signals generated by two different tail scatter points. The two vertical lines show the period of the time-frequency signal, which is exactly the same as the theoretical calculated value.

Fig. 2 is a time-frequency diagram of a micro-Doppler signal obtained by performing Gabor transform on the micro-Doppler signal in Fig. 1; the horizontal axis is time, and the simulation time length is 4 seconds; the vertical axis is the frequency amplitude of the micro-Doppler The curve with the highest brightness in the center of the figure is the micro-Doppler signal generated by the tip scattering point; the curves with lower brightness and sinusoidal variation in the figure are the Doppler signals generated by two different tail scattering points.

At this point, the modeling of the radar echo signal of the start of the spin tail projectile angular motion is completed.

In order to verify the correctness of the model, the parameter extraction and comparison of the micro-Doppler signal and the matching verification of the model are carried out.

The projectile attack angle of the simulation data is calculated, and the time-frequency diagram of the measured projectile is matched and simulated to verify the correctness of the modeling.

Comparing the calculation results with the matching simulation, Figure 3 shows the projectile angle of attack curve calculated from the simulation data. The horizontal axis is time, and the simulation time length is 4 seconds; the vertical axis is the amplitude of the angle of attack of the projectile; its peak value is equal to the theoretical calculation value, and its variation law is consistent with the variation law of the projectile angle of attack in ballistics. Figure 4 is the Doppler time-frequency diagram obtained by the actual measurement of the projectile, and Figure 5 is the matching simulation time-frequency diagram of the measured projectile time-frequency signal according to the actual size of the projectile, projectile coefficient, flight speed and other prior conditions. By comparing Figure 4 and Figure 5, the correctness and effectiveness of the model are verified.

From the comparison of parameters and simulation verification, the radar echo signal model obtained by the method of the invention and generated by the initial disturbance of the angular motion of the spin tail projectile conforms to the ballistic theory, and the angle of attack between the projectile axis and the velocity vector of the projectile conforms to the projectile. The actual situation of angular motion changes verifies the correctness of the established model.

Claims (1)

1. the modeling method of a spin tail projectile angular motion initial disturbance radar echo signal, it is characterized in that concrete steps are as follows: Step 1: Establish the dynamic equation of the spin tail projectile; According to the ballistic theory, the dynamic equation of the spin-tail projectile can be expressed by the second-order variable coefficient differential equation of the complex angle of attack motion

Δ is the angle of attack of the projectile flying in the air, which can generally be represented by a complex number;

Among them, k _zz is the equatorial damping moment, b _y is the lift force, b _x is the drag force, g is the local gravitational acceleration, θ is the scalar value of the angle of attack of the projectile, and v is the flight speed of the projectile; M=k _z , k _z is the static moment, which determines the frequency of attitude movement;

C is the polar moment of inertia,

is the rotation speed of the projectile, A is the equatorial moment of inertia, and v is the ratio of the flight speed;

b _y is the lift force, k _y is the Magnus moment;

is the disturbance term to the projectile angle of attack movement caused by the curvature of the trajectory;

is the disturbance term to the projectile angle of attack motion caused by the projectile structure asymmetry;

is the disturbance term to the projectile's angle of attack motion caused by the aerodynamic asymmetry of the projectile;

is the disturbance term to the angle of attack of the projectile caused by the vertical airflow; ||Δ|| is the angle between the space projectile axis and the velocity vector, and Δ′ and Δ″ are the first and second derivatives of the complex attack angle to the ballistic arc length s. When the initial condition is s=0, Δ=Δ ₀ ,

Δ ₀ is the initial value of the re-attack angle of the projectile,

Represents the first derivative of Δ ₀ to time t, v ₀ is the initial velocity of the projectile; the equation and its initial conditions can reflect the angle of attack motion of the projectile caused by various factors. Among them, H represents the damping motion, which mainly depends on the equatorial damping moment k _zz , the lift b _y and the resistance b _x , etc.; M is mainly related to the static moment k _z , and the attitude motion frequency mainly depends on this; T is mainly related to the lift b _y and Magnus moment _ky is related, it affects flight stability; P is the ratio of projectile's rotation speed and flight speed; Step 2: Solve the homogeneous differential equation of the projectile complex angle of attack; Aiming at the angular motion characteristics of the projectile only subject to initial disturbance in the straight line segment, the homogeneous differential equation of the projectile re-attack angle is solved; Δ″+(H-iP)Δ′-(M+iPT)Δ=0 The characteristic root of this equation with respect to the complex angle of attack Δ is l _1,2 =λ _1,2 +iω _1,2 In the formula, λ ₁ , λ ₂ are called damping exponents, ω ₁ , ω ₂ are called the modal frequencies to the ballistic arc length s; then the angle of attack solution is

In the formula, C ₁ , C ₂ are undetermined coefficients, which are complex numbers and can generally be written as

The constants k ₁ , k ₂ are real numbers,

is the initial phase, determined by the initial conditions; then the solution of the angle of attack can be written as

In the formula,

where k ₁ , k ₂ and

determined by initial conditions; The two complex numbers on the right side of the above formula are modal vectors, K ₁ , K ₂ are called modal amplitudes; according to the vector representation of complex numbers,

Represents a vector whose modulus is 1 and its argument is Φ _1,2 . When the argument changes with the angular frequency ω _1,2 , the vector end of this unit modulus complex number will draw a circle on the complex plane; Step 3: Determine the angular motion mode of the spin tail projectile; When Δ=Δ ₀ and Δ′=Δ′ ₀ are combined with the solution of the angle of attack when the ballistic arc length s=0, the undetermined coefficients C ₁ and C ₂ can be solved as

Therefore, it is determined that the radar echo signal caused by the initial disturbance is modeled by the fast and slow two-circle motion modes; Step 4: Determine the definition of the relevant coordinate system; Four coordinate systems oriented according to the right-hand rule are introduced, including the radar coordinate system Q-UVW, the gun position coordinate system O ₁ -xyz, the reference coordinate system O-XYZ and the projectile axis coordinate system O-ξηζ. The center of the mouth is the origin O ₁ , the horizontal axis O ₁ x is the intersection of the firing surface and the horizontal plane of the muzzle, the forward firing direction is positive, and the vertical axis O ₁ y is in the firing surface and perpendicular to the horizontal axis O ₁ z; the radar The coordinate system takes the site center as the origin Q, the horizontal axis QU points to the true north direction in the horizontal plane, QV is the direction of the plumb line and is perpendicular to the horizontal axis QW; the reference coordinate system takes the projectile center of mass as the origin O, which is always the same as the radar coordinate system. Parallel; the origin of the projectile axis coordinate system is on the center of mass of the projectile, the Oξ axis is positive forward along the projectile axis, the Oη axis is positive upwards perpendicular to the projectile axis, and the oζ axis is determined by the right-hand rule; Step 5: Calculate the distance from the projectile scattering point to the radar; The angular motion of the projectile is decomposed into the rotation of the polar axis of the projectile, the nutation of the polar axis around the gyro momentum moment vector (fast circular motion), and the precession of the gyro momentum moment vector around the velocity vector line (slow circular motion) motion with three degrees of freedom Superposition; starting from the superposition of three motions of projectile spin, nutation and precession, establish the radar echo signal model of the strong scattering point of the projectile target; In electromagnetic scattering, the curvature discontinuity is the strong scattering center of the target. For the spin-tail projectile, the projectile vertex, the cone-column junction and the tail scattering point are the strong scattering centers of the target. A representative scattering point is used to model the radar echo signal; in the projectile axis coordinate system, let the spin angular velocity be ω′ _s =(ω _sξ ,ω _sη ,ω _sζ ) ^T , and the nutation angular velocity be ω′ _u =(ω _uξ ,ω _uη ,ω _uζ ) ^T , the precession angular velocity is ω′ _c =(ω _cξ ,ω _cη ,ω _cζ ) ^T , the angular velocity scalar Ω _s =||ω′ _s ||, Ω _u =| |ω′ _u || and Ω _c =||ω′ _c ||, the unit angular velocity of each rotational angular velocity in the reference coordinate system is ω _s =R _init ·ω′ _s /Ω _s , ω _u =R _init ω′ _u /Ω _u and ω _c =R _init ω′ _c /Ω _c ; R _init is determined by the initial Euler angles

Determined, that is, the local coordinate system O-xyz rotates clockwise around the z-axis respectively

Rotate θ _e clockwise around the x-axis, and then rotate φ _e clockwise around the z-axis, it is transformed into the reference coordinate system O-XYZ; its expression is

After the above analysis, the angular motion of the projectile can be described as: at time t, the scattering point P on the projectile first spins around the projectile axis Oξ, then the coordinate of P becomes

R _spin is the spin rotation matrix,

is the initial vector of the scattering point P before the launch in the radar coordinate system, which is related to the launch angle and launch direction; then the projectile axis Oξ performs nutation motion around the gyro moment vector line OG axis, then the P coordinate becomes

R _nut is the nutation rotation matrix; finally, the gyro moment vector line OG axis precesses around the velocity vector line V ₀ , then the P coordinate becomes

R _con is the precession rotation matrix; then the distance from point P on the projectile to the radar at time t is

In the formula, according to the Euler-Rodrigues rotation formula around the vector axis, each rotation matrix is

in,

and

are the oblique symmetric matrices corresponding to ω _s , ω _n and ω _c respectively; Step 6: Radar echo Doppler signal; Assume that the radar transmits a single carrier frequency continuous wave signal, and the transmitted signal is in the form of s=exp(j2πf ₀ t) Among them, f ₀ is the carrier frequency; then the echo of the scattering point P is s _P =σexp(j2πf ₀ (t-τ)), where σ is the scattering coefficient, τ is the time delay of the scattering point P, and τ= 2R _P (t)/c; Taking the transmitted signal formula as a reference signal, and coherently processing the target echo, the coherent phase signal is obtained as

Among them, the phase term Φ(t)=4πf ₀ R _P (t)/c; taking the derivation of the phase term with respect to time t, the Doppler frequency of the echo is obtained as

in,

for

The unit vector of , when the target is located in the far field of the radar, n≈R ₀ /||R ₀ ||, which is the unit vector of the radar line-of-sight direction LOS; From this, the Doppler frequency caused by the translation of the target can be obtained:

The projectile scattering point is caused by the angular motion of the projectile at the micro-Doppler frequency:

It can be seen from the above formula that the expression of the micro-Doppler frequency of the projectile radar echo caused by the angular motion of the projectile axis in space is relatively complex; through analysis, it can be seen that the variation of the radar micro-Doppler frequency with time is periodic. , the motion period T _top of the top scattering point A is the least common multiple of the nutation period T _s and the precession period T _c ; the motion period T _bom of the tail scattering point P is the spin period T _s , the precession period T _c and the nutation period The least common multiple of T _u , that is, the following relationship holds: T _top =k ₁ T _c = _{k 2} Tu ,T _bom =k ₁ T _c =k _{2 Tu} =k ₃ T _s k ₁ ,k ₂ ,k ₃ ∈N Among them, N is the set of natural numbers.

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