METHOD FOR CONSTRUCTING A FREE TRAJECTORY OF A BALLISTIC MISSILE AT A SPECIFIED LAUNCH ANGLE

Abstract

A method for constructing a free trajectory of a ballistic missile at a specified launch angle includes: setting of an initial state of iterations: based on geodetic coordinates of a launch point and a target point, a launch epoch and the specified launch angle, assuming that the earth does not rotate and a flight time is zero; generating a new flight time in a two-body force model by using known quantities and obtained auxiliary quantities; taking a difference between flight times obtained before and after the iterations as a condition for judging convergence; outputting designed parameters of the ballistic missile after the convergence is reached, or performing a differential correction including the J2 perturbation to improve the precision of the trajectory constructed, and taking a position error of the target point as a convergence condition of the differential correction.

Description

TECHNICAL FIELD

The present invention pertains to the technical field of ballistic missile trajectory construction in celestial mechanics, and more particularly, relates to a method for constructing a free trajectory of a ballistic missile at a specified launch angle.

BACKGROUND

The construction of trajectories for ballistic missiles is a prerequisite for ballistic research and data design. A complete ballistic trajectory includes a powered flight phase, a free flight phase and a reentry phase. The construction of a complete ballistic trajectory is typically accomplished based on specific performance parameters of the ballistic missile, involving some indefinite technical parameters and physical conditions that are complicated and time-consuming to compute, which cannot meet the requirements of general ballistic trajectory research and application. The free flight phase constitutes most of the flight time in the ballistic trajectory and involves a simple force analysis that mainly relates to the gravity of the earth exerted at the center of mass of the missile, in which phase the associated dynamic equations of motion of the missile are easy to solve and analyze. Hence, the development of a fast and effective method for constructing the free flight phase of a ballistic trajectory has important implications for providing a general trajectory simulation environment in the laboratory and aiding the design of ballistic data.

A trajectory constructed by taking the powered flight phase or the reentry phase or both of them as an extension of the free flight phase is referred to as a free trajectory. Article 1 (Bai Hefeng, Wu Ruilin, method for constructing trajectory of TBM [J], Modern Defense Technology, 1998(1):39-43.) proposes a method for constructing a free trajectory considering earth's rotation based on a minimum-energy trajectory theory, which is referred to as Method 1 hereinafter. Article 2 and Article 3 published later adopt the same construction method as Method 1. Specifically, Article 2 (Gu Tiejun, Liu Jian, Nie Cheng, Generation of TBM's Trajectory in the Simulation of Anti-TBM Battle [J], Modern Defense Technology, 2001(4).) corrects the reentry phase of an elliptical ballistic trajectory. Article 3 (Zhang Feng, Tian Kangsheng, Study on Construction of Trajectory for Ballistic Missile Early Warning Simulation System [J], Fire Control & Command Control, 2012, 37(3):94-98.) validates a constructed trajectory via Satellite Tool Kit (STK) software. The latter two methods also focus on the construction of free trajectories based on the minimum-energy theory.

In practice, the ballistic construction and data design must consider a combination of factors, typically focusing on not only the minimum energy as a crucial parameter, but also on other factors, such as the penetration effect of the missile on the impact velocity and the ground track of the trajectory to avoid certain deployment or sensitive areas. To comprehensively consider all factors, it is highly desirable to provide a more universal method for trajectory construction to produce alternative trajectories that are to be optimized and utilized according to specific needs in practice. Obviously, Method 1 does not meet such needs. Moreover, Method 1 has poor construction precision as it assumes that the earth is a homogeneous sphere and both the launch point and the target point are located on the surface of the sphere such that the missile is only subjected to the central gravity of the earth, and the launch point and the target point have equal geocentric radius vector moduli. Such a simplified model will cause a deviation in the impact point of the missile in terms of geometry and dynamics. This deviation increases with the range and the modulus difference between the true geocentric radius vectors of the launch point and the target point, even reaching up to tens of kilometers with regard to an intercontinental missile. In a case where the deviation of the impact point of the missile is required to be controlled on the order of hundreds of meters, consideration must be given to the oblateness of the earth while taking into account the major zonal harmonics perturbation, that is J₂ perturbation, of the earth's gravitational field in the dynamic model. Method 1, however, neither considers the effect of the perturbation nor the unequal geocentric radius vector moduli of the launch point and the target point, and thus fails to meet the requirements for high-precision trajectory construction.

SUMMARY

An objective of the present invention is to provide a method for constructing a free trajectory of a ballistic missile at a specified launch angle. The method is capable of constructing a positive-flying trajectory and a negative-flying trajectory at a specified launch angle in a two-body force model and a dynamic model including the J2 perturbation of the earth's gravitational field, respectively, to derive all trajectories from a launch point to a target point by traversing the launch angle, thereby providing a wealth of alternative trajectories for general ballistic research, data design, and various application scenarios.

To achieve the objective of the present invention, the present invention adopts the following technical solutions. A method for constructing a free trajectory of a ballistic missile at a specified launch angle includes the following steps:

step 1: data preprocessing: sequentially computing a body-fixed rectangular coordinate vector {right arrow over (R)}_A of a launch point A and a body-fixed rectangular coordinate vector {right arrow over (R)}_B of a target point B, a difference Δφ between a geodetic latitude and a geocentric latitude of the launch point A, a horizontal angle ϕ between an AB direction and a north-pole direction of the launch point A in a body-fixed coordinate system, and a difference ε between a ratio of a modulus of the geocentric radius vector of the launch point A to a modulus of the geocentric radius vector of the target point B and 1;

step 2: setting of an initial state of iterations: assuming that the earth does not rotate and a flight time T is zero;

step 3: computing, in a two-body force model, a launch velocity vector {right arrow over ({dot over (r)})}_A of a positive-flying trajectory or a negative-flying trajectory in a True Equator Mean Equinox of Epoch (TEMEE) coordinate system and a launch velocity vector {right arrow over ({dot over (R)})}_A of the positive-flying trajectory or the negative-flying trajectory in the body-fixed coordinate system, a first type of non-singular orbital elements σ of the ballistic missile at a launch epoch, and a plurality of other variables to generate a new flight time T*; and

step 4: letting T=T*, repeating step 3 to iteratively compute the launch velocity vector {right arrow over ({dot over (r)})}_A and the launch velocity vector {right arrow over ({dot over (R)})}_A of the ballistic missile, the first type of non-singular orbital elements of the ballistic missile at the launch epoch, a half-range angle and the flight time until |T−T*| is less than a set threshold S_t to complete the iterations to finally obtain the launch velocity vector {right arrow over ({dot over (R)})}_A and the flight time T of the ballistic missile in the two-body force model, and outputting designed parameters v_p, v_r, Ã, T, σ.

Compared with the prior art, the present invention has the following significant advantages. Taking the rotation of the earth into account, the present invention combines the three-dimensional (3D) geodetic coordinates of the launch point and the target point to precisely construct a free trajectory from the launch point to the target point based on a constraint on the launch angle. The advantages of the present invention include: (1) When the launch angle, the launch epoch, and the geodetic coordinates of the launch point and the target point are specified, the method is capable of constructing a free trajectory from the launch point to the target point by using a two-body force model when the rotation of the earth is considered. (2) When the launch angle, the launch epoch, and the geodetic coordinates of the launch point and the target point are specified, the method is capable of constructing a free trajectory from the launch point to the target point under the rotation of the earth by using a dynamic model that takes into account a combination of the gravity at the center of the earth and the J₂ perturbation. (3) When the launch angle, the launch epoch and the geodetic coordinates of the launch point and the target point are specified, the method is capable of constructing not only a positive-flying trajectory but also a negative-flying trajectory from the target point to the launch point based on the premise of a rational launch angle. (4) Theoretically, the method is capable of constructing all free trajectories from the launch point to the target point by traversing the launch angle and specifying the positive-flying or negative-flying direction of the trajectories, thereby providing a wealth of complete alternative trajectories for various ballistic research, data design and application. (5) In the method, the launch point and the target point can be flexibly selected, where the launch point may be set as a conventional launch point or an orbit injection point, and the target point may be similarly set as a reentry point. (6) The method eliminates singularities in mathematics without special requirements on the coordinates of the launch point, such that the method is still applicable to the case where the launch point is selected to be located at the north or south pole or adjacent regions. (7) The method introduces special computation techniques. For example, the method employs the Bulirsch-Stoer (BS) rational polynomial extrapolation integration technique to compute the trajectories when the effect of the J₂ perturbation of the earth is taken into account, such that the method is suitable for the computation of highly-eccentric orbits, thereby ensuring the reliability and efficiency of the computed result in extreme cases. (8) The method is capable of constructing all free trajectories from the launch point to the target point by means of traversing, including not only the classic minimum-energy trajectory (with the smallest launch velocity in inertial space) but also the practical minimum-energy trajectory (with the smallest launch velocity in the body-fixed coordinate system). (9) The method is capable of outputting various designed parameters of the missile for a comprehensive and precise characterization of the free trajectory constructed, including not only the magnitude (relative to the inertial space and the ground) and direction (the azimuth of the velocity in the horizontal plane at the launch point) of the velocity at the launch point, and the flight time of the missile, but also six orbital elements of the missile relative to the TEMEE coordinate system at the launch epoch.

The present invention will be further explained in detail below with reference to the drawings.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates a positive-flying ballistic missile.

FIG. 2 illustrates a negative-flying ballistic missile.

FIG. 3 illustrates the launch angle h which is the intersection angle between the missile's velocity vector and the horizontal plane of the launch point A, where the horizontal plane may be replaced by the tangent plane of the reference ellipsoid in practice.

FIG. 4 illustrates a flowchart of the proposed method for constructing a free trajectory at a specified angle.

FIG. 5 illustrates the definition of the pseudo body-fixed coordinate at launch epoch t₀.

FIG. 6 illustrates the definition of the TEMEE coordinate system at epoch t, in which γ and Ŷ indicate the true equinox and the mean equinox, respectively.

FIG. 7 illustrates the definition of the conventional horizontal coordinate system XYZ and the target-pointing horizontal coordinate system X*Y*Z*.

FIG. 8 illustrates a sketch of the celestial sphere and the triangle formed by points P, Z_A, and Z_B.

FIG. 9 illustrates spherical triangles KAA′ and KBB′ formed by the earth equator, missile trajectory and meridians.

FIG. 10A and FIG. 10B illustrate spherical triangles formed by the velocity vector, zenith and geocentric radius vector of the launch point A in the Northern hemisphere and in the Southern hemisphere, respectively.

FIG. 11 illustrates the launch velocity vector, the zenith vector and the geocentric radius vector of the launch point A.

FIG. 12 illustrates the positive-flying free trajectories (left) and the negative-flying free trajectories (right) constructed by traversing the launch angle when using the two-body force model according to Embodiment 1.

FIG. 13 a graph illustrating the function between the launch angle and the launch velocity of the positive-flying trajectories constructed by traversing based on the two-body force model.

FIG. 14 a graph illustrating the function between the launch angle and the launch velocity of the negative-flying trajectories constructed by traversing based on the two-body force model.

FIG. 15 illustrates the ground track of the minimum-energy missile trajectory constructed by the two-body force model according to Embodiment 1.

FIG. 16 illustrates the positive-flying free trajectories constructed by traversing the launch angle when using the two-body force model according to Embodiment 2.

DETAILED DESCRIPTION OF THE EMBODIMENTS

According to the present invention, referring to FIG. 4, a method for constructing a free trajectory of a ballistic missile at a specified launch angle specifically includes the following steps:

Step 1: data preprocessing: a body-fixed rectangular coordinate vector {right arrow over (R)}_A of the launch point A and a body-fixed rectangular coordinate vector {right arrow over (R)}_B of the target point B, a difference Δ_φ between the geodetic latitude and the geocentric latitude of the launch point A, a horizontal angle ϕ between the AB direction and the north-pole direction of the launch point (when the point A is not at or above the north or south pole) and a difference ε between a ratio of the modulus of the geocentric radius vector of the point A to the modulus of the geocentric radius vector of the point B and 1 are sequentially computed by specific steps as follows.

1. The geodetic coordinates of the points A and B are transformed from spherical coordinates (L, B, H) to three-dimensional (3D) rectangular coordinates (X, Y, Z) according to equation (1), and the 3D rectangular coordinates are expressed by {right arrow over (R)}_A and {right arrow over (R)}_B as follows:

$\begin{array}{cc}\{\begin{array}{c}X=\left(N+H\right)\cos B\cos L\\ Y=\left(N+H\right)\cos B\sin L\\ Z=\left[N\left(1-e_c^2\right)+H\right]\sin B\end{array};& \mathrm{equation}\left(1\right)\end{array}$

wherein, N is the radius of curvature of the prime vertical of a point, and N=α_e/√{square root over (1−e_c²(sin B)²)}.

2. The geocentric latitudes φ_A and φ_B of the points A and B are computed according to equation (2):

$\begin{array}{cc}\phi ={\sin }^{-1}\frac{Z}{\sqrt{X^2+Y^2+Z^2}}.& \mathrm{equation}\left(2\right)\end{array}$

3. The difference Δ_φ between the geodetic latitude and the geocentric latitude of the point A is computed according to equation (3):

$\begin{array}{cc}\mathrm{\Delta \phi }=B_A-{\phi }_A.& \mathrm{equation}\left(3\right)\end{array}$

4. The horizontal angle ϕ between the AB direction and the north-pole direction of the launch point in the body-fixed coordinate system is computed according to a system of equations (4) and (5):

$\begin{array}{cc}\{\begin{array}{c}\sin \varphi =\frac{\cos B_B\sin \left(L_A-L_B\right)}{\sin q}\\ \cos \varphi =\frac{\sin B_B\cos B_A-\cos B_B\sin B_A\cos \left(L_A-L_B\right)}{\sin q}\end{array};& \mathrm{equation}\left(4\right)\end{array}$

wherein, q is the zenith angle of the point B from the zenith of the point A and is computed by:

$\begin{array}{cc}\{\begin{array}{c}\cos q=\sin B_B\sin B_A+\cos B_B\cos B_A\cos \left(L_A-L_B\right)\\ \sin q=\sqrt{1-{\cos }^{2}q}\end{array}q\in \left(0,\pi \right);& \mathrm{equation}\left(5\right)\end{array}$

in a particular case where the point A is located at or above the north or south pole, equations (4) and (5) are still satisfied, and are simplified as:

$\{\begin{array}{cc}\varphi =\pi -\left(L_A-L_B\right)& A\mathrm{is}\mathrm{at}\mathrm{or}\mathrm{above}\mathrm{the}\mathrm{north}\mathrm{pole}\\ \varphi =L_A-L_B& A\mathrm{is}\mathrm{at}\mathrm{or}\mathrm{above}\mathrm{the}\mathrm{south}\mathrm{pole}\end{array}\forall L_A\in \left[0,360°\right).$

5. The difference ε between a ratio of the modulus of the geocentric radius vector of the point A to the modulus of the geocentric radius vector of the point B and 1 is computed according to equation (6):

$\begin{array}{cc}\varepsilon =\frac{{\overrightarrow{R}}_B}{{\overrightarrow{R}}_A}-1.& \mathrm{equation}\left(6\right)\end{array}$

Step 2: setting of the initial state of iterations: assuming that the earth does not rotate and the flight time T is zero.

If the earth does not rotate, the modulus v_p of the launch velocity vector of the missile in the body-fixed coordinate system is equal to the modulus v_r of the launch velocity vector in the TEMEE coordinate system, that is,

$\frac{v_p}{v_r}=1.$

Meanwhile, the launch azimuth A* in the target-pointing horizontal coordinate system is zero.

Step 3: the launch velocity vector {right arrow over ({dot over (r)})}_A of the positive-flying trajectory or the negative-flying trajectory in the TEMEE coordinate system and the launch velocity vector {right arrow over ({dot over (R)})}_A of the positive-flying trajectory or the negative-flying trajectory in the body-fixed coordinate system, the first type of non-singular orbital elements a of the missile at the launch epoch, the flight time, and a plurality of other variables are computed in the two-body force model to generate a new flight time T* specifically by the following steps.

1. The coordinate vectors {right arrow over (r)}_A and {right arrow over (r)}_B of the points A and B in the TEMEE coordinate system are computed in combination with the flight time T according to equation (7):

$\begin{array}{cc}\{\begin{array}{c}{\overrightarrow{r}}_A=M\left(t_0\right){\overrightarrow{R}}_A\\ {\overrightarrow{r}}_B=M\left(t_0+T\right){\overrightarrow{R}}_B\end{array};& \mathrm{equation}\left(7\right)\end{array}$

wherein, t₀ is the launch epoch, and M is a time-dependent transformation matrix from the body-fixed coordinate system to the TEMEE coordinate system. In a first iteration, the flight time T is zero, and {right arrow over (r)}_B=M(t₀){right arrow over (R)}_B is satisfied.

2. The half-range angle is computed according to equations (8) and (9) as follows:

a half intersection angle is expressed by:

$\begin{array}{cc}\delta =\frac{1}{2}\times {\cos }^{-1}\frac{{\overrightarrow{r}}_A·{\overrightarrow{r}}_B}{{\overrightarrow{r}}_A{\overrightarrow{r}}_B};& \mathrm{equation}\left(8\right)\end{array}$

then the half-range angle is computed by:

$\begin{array}{cc}\beta =\{\begin{array}{ccc}\delta & \beta \in \left(0,\frac{\pi }{2}\right)& \mathrm{positive}-\mathrm{flying}\mathrm{trajectory}\\ \pi -\delta & \beta \in \left(\frac{\pi }{2},\pi \right)& \mathrm{negative}-\mathrm{flying}\mathrm{trajectory}\end{array}.& \mathrm{equation}\left(9\right)\end{array}$

3. An inclination i and a right ascension Ω of an ascending node of an elliptical trajectory/orbit are computed according to equation (10):

$\begin{array}{cc}\{\begin{array}{cc}\left[\begin{array}{c}\sin i\sin \Omega \\ -\sin i\cos \Omega \\ \cos i\end{array}\right]=\frac{{\overrightarrow{r}}_A\times {\overrightarrow{r}}_B}{{\overrightarrow{r}}_A\times {\overrightarrow{r}}_B}& \mathrm{positive}-\mathrm{flying}\mathrm{trajectory}\\ \left[\begin{array}{c}\sin i\sin \Omega \\ -\sin i\cos \Omega \\ \cos i\end{array}\right]=\frac{{\overrightarrow{r}}_B\times {\overrightarrow{r}}_A}{{\overrightarrow{r}}_B\times {\overrightarrow{r}}_A}& \mathrm{negative}-\mathrm{flying}\mathrm{trajectory}\end{array}i\in \left(0,\pi \right)\Omega \in \left[0,2\pi \right).& \mathrm{equation}\left(10\right)\end{array}$

4. True arguments of latitude μ_A and μ_B of the points A and B on the elliptical trajectory/orbit are computed according to equation (11):

$\begin{array}{cc}\{\begin{array}{c}\sin u_A=\frac{\sin {\phi }_A}{\sin i}\\ \cos u_A=\frac{\sin {\phi }_B-\sin {\phi }_A\cos 2\beta }{\sin i\sin 2\beta }\\ u_B=u_A+2\beta \end{array}.& \mathrm{equation}\left(11\right)\end{array}$

5. The cosine of an angle α between the launch velocity vector and the geocentric radius vector of the launch point in the body-fixed coordinate system is computed according to equation (12):

$\begin{array}{cc}\cos \alpha =\cos \mathrm{\Delta \phi sin}h-\sin \mathrm{\Delta \phi cos}h\cos \left(A^{*}-\varphi \right);& \mathrm{equation}\left(12\right)\end{array}$

wherein, h is the launch angle, and A*=0 in the first iteration.

6. The tangent of an angle θ between the launch velocity vector and the geocentric radius vector of the launch point in the TEMEE coordinate system is computed according to equation (13):

$\begin{array}{cc}\{\begin{array}{c}\cos \theta =\frac{v_P}{v_r}\cos \alpha \\ \sin \theta =\sqrt{1-{\cos }^2\theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\end{array};& \mathrm{equation}\left(13\right)\end{array}$

wherein,

$\frac{v_P}{v_r}=1$

in the first iteration.

7. A bias Δω of the argument of perigee is introduced and computed according to equation (15), and then the argument of perigee ω is computed according to equation (14).

According to the nature of the elliptical trajectory/orbit of a ballistic missile, the apogee of the trajectory is located in an outer space of the earth and between the points A and B. When the geocentric radius vector moduli of the points A and B are equal, the argument of apogee is equal to the mean value of the true arguments of latitude of the points A and B and then the mean value minus π obtains the argument of perigee ω. The argument of perigee ω computed in this way, however, generally has a deviation due to unequal geocentric distances of the point A and the point B. The bias Δω of ω is introduced to obtain:

$\begin{array}{cc}\omega ={\omega }_0+\mathrm{\Delta \omega }\omega \in \left[0,2\pi \right);& \mathrm{equation}\left(14\right)\end{array}$

wherein,

${\omega }_0=\frac{u_A+u_B}{2}-\pi ,$

wherein μ_A and μ_B have been computed according to equation (11); Δω is computed according to equation (15):

$\begin{array}{cc}\tan \Delta \omega =\frac{\varepsilon }{2\left(1+\varepsilon \right)\cot \theta -\varepsilon \cot \beta }\mathrm{\Delta \omega }\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]& \mathrm{equation}\left(15\right)\end{array}$

accordingly, true anomalies of the points A and B on the elliptical trajectory/orbit are expressed by Δω as follows:

$\begin{array}{cc}\{\begin{array}{c}{f}_A=\pi -\beta -\Delta \omega \\ f_B=\pi +\beta -\Delta \omega \end{array}.& \mathrm{equation}\left(16\right)\end{array}$

8. An eccentricity e of the elliptical trajectory/orbit is computed according to equation (17):

$\begin{array}{cc}e=\frac{\cot \theta }{\sin \left(\beta +\Delta \omega \right)+\cot \theta \cos \left(\beta +\Delta \omega \right)}.& \mathrm{equation}\left(17\right)\end{array}$

9. The rationality of the specified launch angle h is judged as follows. If any one of the following conditions is satisfied, then the specified launch angle is judged to be irrational such that a rationally designed trajectory cannot be obtained, the current construction procedure of the trajectory is ended, and a launch angle is re-specified.

For the positive-flying trajectory:

(1) If e≥1, then the specified launch angle is judged to be excessively large; and

(2) If cot θ≤0 or |tan Δω|≥tan β, then the specified launch angle is judged to be excessively small.

For the negative-flying trajectory:

$\begin{array}{cc}\mathrm{If}e\ge 1\mathrm{or}\cot \theta \ge -\tan \beta +\frac{\varepsilon }{2\left(1+\varepsilon \right)}\left(\tan \beta +\cot \beta \right),& \left(1\right)\end{array}$

then the specified launch angle is judged to be excessively large; and

$\begin{array}{cc}\mathrm{if}\cot \theta \le \max \left\{\frac{\varepsilon }{2\left(1+\varepsilon \right)}\cot \beta ,0\right\},& \left(2\right)\end{array}$

then the specified launch angle is judged to be excessively small.

10. The true anomalies f_A and f_B of the points A and B on the elliptical trajectory/orbit are computed according to equation (16).

11. The orbital elements σ of the missile at the launch epoch are computed according to equations (18) to (21).

σ is a set of the first type of non-singular orbital elements, wherein the inclination i and the right ascension Ω of the ascending node of the orbit are computed according to equation (10), and a semi-major axis α and ξ, η, λ are computed as follows:

$\begin{array}{cc}a=\frac{{\overrightarrow{r}}_A\left[1-e\cos \left(\beta +\mathrm{\Delta \omega }\right)\right]}{1-e^2};& \mathrm{equation}\left(18\right)\\ \{\begin{array}{c}\zeta =e\cos \omega \\ \eta =-e\sin \omega \\ \lambda =\omega +M_A\end{array};& \mathrm{equation}\left(19\right)\end{array}$

wherein M_A is computed as follows:

$\begin{array}{cc}\{\begin{array}{c}{E}_A=f_A-2{\tan }^{-1}\left(\frac{e\sin f_A}{1+\sqrt{1-e^2}+e\cos f_A}\right)\\ E_B=f_B-2{\tan }^{-1}\left(\frac{e\sin f_B}{1+\sqrt{1-e^2}+e\cos f_B}\right)\end{array};& \mathrm{equation}\left(20\right)\\ \{\begin{array}{c}{M}_A=E_A-e\sin E_A\\ M_B=E_B-e\sin E_B\end{array}.& \mathrm{equation}\left(21\right)\end{array}$

12. The flight time T* of the missile is computed according to equation (22):

$\begin{array}{cc}\{\begin{array}{c}{T}^{*}=\frac{M_B-M_A}{n}\\ n=\sqrt{\mu /a^3}\end{array}.& \mathrm{equation}\left(22\right)\end{array}$

13. The launch velocity v_r of the missile in the TEMEE coordinate system and the launch velocity v_p of the missile in the body-fixed coordinate system are computed according to equations (23) to (25) as follows:

the launch velocity vectors {right arrow over ({dot over (r)})}_A and {right arrow over ({dot over (R)})}_A of the missile in the TEMEE coordinate system and the body-fixed coordinate system satisfy:

$\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{\overrightarrow{r}}}_A=\sqrt{\frac{\mu }{\rho }}\left[\left(-\sin u_A+\eta \right)\overrightarrow{P}+\left(\cos u_A+\xi \right)\overrightarrow{Q}\right]\\ \overrightarrow{P}=\left[\begin{array}{c}\cos \Omega \\ \sin \Omega \\ 0\end{array}\right]\\ \overrightarrow{Q}=\left[\begin{array}{c}-\sin \Omega \cos i\\ \cos \Omega \cos i\\ \sin i\end{array}\right]\\ p=a\left(1-e^2\right)\end{array};& \mathrm{equation}\left(23\right)\\ {\stackrel{.}{\overrightarrow{R}}}_A=M\left(t_0\right){\stackrel{.}{\overrightarrow{r}}}_A-{\stackrel{.}{\theta }}_g\left[\begin{array}{c}-Y_A\\ X_A\\ 0\end{array}\right];& \mathrm{equation}\left(24\right)\\ \mathrm{then}:& \\ \{\begin{array}{c}{v}_p={\stackrel{.}{\overrightarrow{R}}}_A\\ v_r={\stackrel{.}{\overrightarrow{r}}}_A\end{array};& \mathrm{equation}\left(25\right)\end{array}$

wherein, {dot over (θ)}_g is a change rate of the Greenwich sidereal hour angle in the TEMEE coordinate system and is 360°.985612288/day.

14. The launch azimuth A* in the target-pointing horizontal coordinate system is computed according to equations (26) and (27):

the launch velocity vector in the target-pointing horizontal coordinate system is denoted as {right arrow over ({dot over (R)})}_A*(X*,Y*,Z*), which is derived by performing a coordinate rotation on {right arrow over ({dot over (R)})}_A as follows:

$\begin{array}{cc}{\stackrel{.}{\overrightarrow{R}}}_{A^{*}}=\mathrm{QW}{\stackrel{.}{\overrightarrow{R}}}_A;& \mathrm{equation}\left(27\right)\\ \left[\begin{array}{c}{X}^{*}\\ Y^{*}\\ Z^{*}\end{array}\right]={\stackrel{.}{\overrightarrow{R}}}_{A^{*}}\left[\begin{array}{c}\cos h\cos A^{*}\\ -\cos h\sin A^{*}\\ \sin h\end{array}\right];& \mathrm{equation}\left(28\right)\end{array}$

wherein, W is a rotation matrix from the body-fixed coordinate system to the conventional horizontal coordinate system, and Q is a rotation matrix from the conventional horizontal coordinate system to the target-pointing horizontal coordinate system.

Step 4: let T=T*, step 3 is repeated to iteratively compute the launch velocity vector of the missile, the orbital elements of the missile at the launch epoch, the half-range angle, the flight time and a plurality of other variables until |T−T*| is less than a set threshold S_t to complete the iterations to finally obtain the launch velocity vector {right arrow over ({dot over (r)})}_A and the flight time T of the missile in the two-body force model, and designed parameters v_p, v_r, Ã, T, σ of the missile are output. The specific steps thereof include:

1. It is judged whether |T−T*|<S_t is satisfied; if yes, let T=T*, and proceeding to the next step; otherwise, let T=T*, and returning to step 3.

2. A declination à of the launch velocity vector in the target-pointing horizontal coordinate system relative to the target point in the horizontal plane is computed according to equation (28):

$\begin{array}{cc}\{\begin{array}{cc}\tilde{A}=A^{*}& A^{*}\le \pi \\ \tilde{A}=A^{*}-2\pi & A^{*}>\pi \end{array}.& \mathrm{equation}\left(28\right)\end{array}$

3. The designed parameters v_p, v_r, Ã, T, σ of the missile are output.

Through steps 1 to 4, the designed parameters of the missile from the launch point to the target point are derived based on the two-body force model to support various simulation applications and trajectory construction that requires a lower precision. In a case where the trajectory construction requires higher precision and the J₂ perturbation of the earth's gravitational field needs to be considered, the results obtained in the two-body force model are taken as initial values to establish a constraint equation with a constant launch angle and a differential equation based on a position error propagation of the target point for a differential correction. The specific implementation thereof includes steps 5 and 6.

Step 5: the launch velocity vector {right arrow over ({dot over (r)})}_A and the flight time T obtained by the two-body force model are taken as reference solutions {right arrow over ({dot over (r)})}_A⁰ and T⁰ for the differential correction; if a distance |Δ{right arrow over (R)}_BB*| between the target point B and a target point B* obtained by a perturbation extrapolation based on the reference solutions is less than a given threshold S_R, then the correction of the designed parameters of the missile is ended, and corrected parameters v_p, v_r, Ã, T, σ are output; otherwise, proceeding to step 6. In the iterative process, the thresholds are small quantities, where S_t may be 10 μs, and S_R may be 1 cm.

Step 5 includes the following sub-steps:

1. The target point obtained by the perturbation extrapolation based on the reference solutions {right arrow over ({dot over (r)})}_A⁰ and T⁰ is denoted as B*, and partial derivative matrices

$\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\overrightarrow{r}}_A},\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}$

and a position vector {right arrow over (r)}_B* of B* in the TEMEE coordinate system are computed by a numerical integration.

To adapt to computing highly-eccentric trajectories, the numerical integration is performed by a Gragg-Bulirsch-Stoer first-order integrator (Article 4: Bulirsch R, Stoer J. Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods [J]. Numerische Mathematik, 1966, 8 (1):1-13.), where the differential equations to be integrated are:

$\begin{array}{cc}\{\begin{array}{c}\frac{d\overrightarrow{r}}{\mathrm{dt}}=\overrightarrow{v}\\ \frac{d\overrightarrow{v}}{dt}=\overrightarrow{F}\left(\overrightarrow{r}\right)\\ \frac{d}{\mathrm{dt}}\left(\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\right)=\frac{\partial \overrightarrow{v}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\\ \frac{d}{\mathrm{dt}}\left(\frac{\partial \overrightarrow{v}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\right)=\frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}·\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\end{array};& \mathrm{equation}\left(29\right)\end{array}$

derive initial values as:

$\begin{array}{cc}\{\begin{array}{c}\overrightarrow{r}\left(t_0\right)={\overrightarrow{r}}_A\\ \overrightarrow{v}\left(t_0\right)={\stackrel{.}{\overrightarrow{r}}}_A^0\\ \frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}{|}_{t=t_0}=0\\ \frac{\partial \overrightarrow{v}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}{|}_{t=t_0}=I\end{array};& \mathrm{equation}\left(30\right)\end{array}$

integrate from t=t₀ to t=t₀+T⁰ to obtain:

$\begin{array}{cc}\{\begin{array}{c}{\overrightarrow{r}}_{B^{*}}=\overrightarrow{r}{|}_{t=t_0+T^0}\\ \frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}=\overrightarrow{v}{|}_{t=t_0+T^0}\\ \frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}=\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}{|}_{t=t_0+T^0}\end{array};& \mathrm{equation}\left(31\right)\end{array}$

the force function {right arrow over (F)}({right arrow over (r)}) only including the J2 perturbation and its partial derivative matrix

$\frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}$

in equation (29) are expressed as:

$\begin{array}{cc}\{\begin{array}{c}\overrightarrow{F}={\left(\nabla U\right)}_r=M\left(t\right)·{\left(\nabla U\right)}_R\\ \frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}={\left({\nabla }^{2}U\right)}_r=M\left(t\right)·{\left({\nabla }^{2}U\right)}_R·M^T\left(t\right)\end{array};& \mathrm{equation}\left(32\right)\end{array}$

wherein, t is an integration time; subscripts r and R represent a gradient or tensor of U in the TEMEE coordinate system and the body-fixed coordinate system, respectively; U represents a gravitational potential function of the earth in the body-fixed coordinate system and includes a central gravitational potential U₀ and a J₂ gravitational potential U₁:

$\begin{array}{cc}U=U_0+U_1.& \mathrm{equation}\left(33\right)\end{array}$

Accordingly, the gradient and tensor of the gravitational potential function U of the earth are obtained as follows:

$\begin{array}{cc}\{\begin{array}{c}{\left(\nabla U\right)}_R={\left(\nabla U_0\right)}_R+{\left(\nabla U_1\right)}_R\\ {\left({\nabla }^{2}U\right)}_R={\left({\nabla }^2{U}_0\right)}_R+{\left({\nabla }^2{U}_1\right)}_R\end{array};\mathrm{wherein},& \mathrm{equation}\left(34\right)\\ \{\begin{array}{c}{\left(\nabla U_0\right)}_R=-\frac{\mu }{r^3}{\left(X,Y,Z\right)}^T\\ {\left({\nabla }^2{U}_0\right)}_R=\frac{\mu }{r^3}\left[\begin{array}{ccc}-1+3\frac{X^2}{r^2}& \frac{3\mathrm{XY}}{r^2}& \frac{3\mathrm{XZ}}{r^2}\\ \frac{3\mathrm{XY}}{r^2}& -1+3\frac{Y^2}{r^2}& \frac{3\mathrm{YZ}}{r^2}\\ \frac{3\mathrm{XZ}}{r^2}& \frac{3\mathrm{YZ}}{r^2}& 1+3\frac{Z^2}{r^2}\end{array}\right]\\ {\left(\nabla U_1\right)}_R={\left(\frac{\partial U_1}{\partial X},\frac{\partial U_1}{\partial Y},\frac{\partial U_1}{\partial Z}\right)}^T\\ {\left({\nabla }^2{U}_1\right)}_R=\left[\begin{array}{ccc}\frac{{\partial }^2{U}_1}{\partial X^2}& \frac{{\partial }^2{U}_1}{\partial X\partial Y}& \frac{{\partial }^2{U}_1}{\partial X\partial Z}\\ \frac{{\partial }^2{U}_1}{\partial X\partial X}& \frac{{\partial }^2{U}_1}{\partial Y^2}& \frac{{\partial }^2{U}_1}{\partial Y\partial Z}\\ \frac{{\partial }^2{U}_1}{\partial Z\partial X}& \frac{{\partial }^2{U}_1}{\partial Z\partial Y}& \frac{{\partial }^2{U}_1}{\partial Z^2}\end{array}\right]\end{array};& \mathrm{equation}\left(35\right)\end{array}$

wherein, (X, Y, Z)^T is a 3D rectangular coordinate vector of the missile in the body-fixed coordinate system, and r=R=√{square root over (X²+Y²+Z²)}; with reference to Article 6 (Balmino G, Barriot J P, Valès N. Non-singular formulation of the gravity vector and gravity gradient tensor in spherical harmonics [J]. Manuscr Geod, 1990, 15.), the elements composing the above matrices or vectors are expressed by:

$\begin{array}{cc}\{\begin{array}{c}\mathrm{let}{\theta }_1=\frac{X}{r};{\theta }_2=\frac{Y}{r};{\theta }_3=\frac{Z}{r}=\zeta \\ {\stackrel{\_}{P}}_2\left(\zeta \right)=\sqrt{5}\times \frac{\left(3{\zeta }^2-1\right)}{2};{\stackrel{\_}{P}}_2^{!}\left(\zeta \right)=3\sqrt{5}\zeta ;{\stackrel{\_}{P}}_2^{″}\left(\zeta \right)=3\sqrt{5}\\ D_r=-3{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2\left(\zeta \right);D_3={{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\overline{P}}_2^{\prime }\left(\zeta \right)\\ S_{rr}=12{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2\left(\zeta \right);S_{r3}=-3{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2^{\prime }\left(\zeta \right);S_{33}={{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2^{″}\left(\zeta \right)\\ \tilde{\varphi }=D_r-D_3{\theta }_3\\ \frac{\partial U_1}{\partial X}=\frac{\mu }{r^2}\tilde{\varphi }{\theta }_1;\frac{\partial U_1}{\partial Y}=\frac{\mu }{r^2}\tilde{\varphi }{\theta }_2;\frac{\partial U_1}{\partial Z}=\frac{\mu }{r^2}\left(D_3+\overline{\varphi }{\theta }_3\right)\\ \frac{{\partial }^2{U}_1}{\partial X^2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1^2\right\}\\ \frac{{\partial }^2{U}_1}{\partial X\partial Y}=\frac{\mu }{r^3}\left[S_{\mathrm{rr}}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1{\theta }_2\\ \frac{{\partial }^2{U}_1}{\partial X\partial Z}=\frac{\mu }{r^3}\left\{\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_1+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1{\theta }_3\right\}\\ \frac{{\partial }^2{U}_1}{\partial Y2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_2^2\right\}\\ \frac{{\partial }^2{U}_1}{\partial Y\partial Z}=\frac{\mu }{r^3}\left\{\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_2+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_2{\theta }_3\right\}\\ \frac{{\partial }^2{U}_1}{\partial Z^2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+S_{33}+2\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_3+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_3^2\right\}\end{array};& \mathrm{equation}\left(36\right)\end{array}$

wherein, C₂₀ is a normalized spherical harmonic coefficient corresponding to the J₂ perturbation in an expansion of a spherical harmonic series of the earth's gravitational potential.

2. A position vector difference Δ{right arrow over (R)}_BB* between the target point B and the target point B* obtained by the perturbation extrapolation based on the reference solutions and {right arrow over ({dot over (r)})}_A⁰ and T⁰ in the body-fixed coordinate system are computed according to equations (37) and (38):

$\begin{array}{cc}\Delta {\overrightarrow{R}}_{{\mathrm{BB}}^{*}}={\overrightarrow{R}}_B-{\overrightarrow{R}}_{B^{*}}=\left[\begin{array}{c}{X}_B\\ Y_B\\ Z_B\end{array}\right]-\left[\begin{array}{c}{X}_{B^{*}}\\ Y_{B^{*}}\\ Z_{B^{*}}\end{array}\right];& \mathrm{equation}\left(37\right)\end{array}$

wherein, a coordinate vector {right arrow over (R)}_B*({right arrow over (X)}_B*, {right arrow over (Y)}_B*, {right arrow over (Z)}_B*) of B* in the body-fixed coordinate system is obtained from {right arrow over (r)}_B* through a coordinate transformation:

$\begin{array}{cc}{\overrightarrow{R}}_{B^{*}}=M^T\left(t_0+T^0\right){\overrightarrow{r}}_{B^{*}}.& \mathrm{equation}\left(38\right)\end{array}$

3. It is judged whether Δ{right arrow over (R)}_BB* is less than the given threshold S_R; if yes, the designed parameters v_p, v_r, Ã, T, σ of the missile are recomputed and output based on the reference solutions and corresponding equations in steps 3 and 4, and the correction is ended; otherwise, proceeding to step 6 to perform a differential correction in order to estimate a launch velocity vector increment and a flight time increment.

Step 6: the constraint equation with the constant launch angle and a differential equation based on a position error propagation of the target point are established; the launch velocity vector increment Δ{right arrow over ({dot over (r)})}_A and the flight time increment ΔT are computed, and corrected solutions are denoted as and {circumflex over (T)}; the corrected solutions are taken as reference solutions for a next differential correction, that is, let: {right arrow over ({dot over (r)})}_A⁰=, T⁰={circumflex over (T)}, and repeat the sub-steps of step 5.

Step 6 includes the following sub-steps:

1. A system of linear differential equations of the launch velocity vector increment Δ{right arrow over ({dot over (r)})}_A and the flight time increment ΔT are established according to equations (39) to (41):

$\begin{array}{cc}\left[\begin{array}{cc}G& 0\\ C& D\end{array}\right]\left[\begin{array}{c}\Delta {\stackrel{.}{\overrightarrow{r}}}_A\\ \Delta T\end{array}\right]=\left[\begin{array}{c}0\\ \Delta {\overrightarrow{R}}_{{\mathrm{BB}}^{*}}\end{array}\right];& \mathrm{equation}\left(39\right)\end{array}$

wherein, Δ{right arrow over ({dot over (r)})}_A(Δ{right arrow over ({dot over (x)})}_A, Δ{right arrow over ({dot over (y)})}_A, Δ{right arrow over (ż)}_A) and Δ{right arrow over (R)}_BB*(Δ{right arrow over (x)}_BB*, Δ{right arrow over (y)}_BB*, Δ{right arrow over (z)}_BB*) are expressed in terms of components as:

$\begin{array}{cc}\left[\begin{array}{cc}G& 0\\ C& D\end{array}\right]\left[\begin{array}{c}\Delta {\stackrel{.}{\overrightarrow{x}}}_A\\ \Delta {\stackrel{.}{\overrightarrow{y}}}_A\\ \Delta {\stackrel{.}{\overrightarrow{z}}}_A\\ \Delta T\end{array}\right]=\left[\begin{array}{c}0\\ \Delta {\overrightarrow{x}}_{{\mathrm{BB}}^{*}}\\ \Delta {\overrightarrow{y}}_{{\mathrm{BB}}^{*}}\\ \Delta {\overrightarrow{z}}_{{\mathrm{BB}}^{*}}\end{array}\right];& \mathrm{equation}\left(40\right)\end{array}$

wherein, G is a 1×3 matrix, C is a 3×3 matrix, D is a 3×1 matrix, which are expressed as:

$\begin{array}{cc}\{\begin{array}{c}G=\left[-\sin h\cos A^{*}\sin h\sin A^{*}\cos h\right]{\mathrm{QWM}}^T\left(t_0\right)\\ C=M^T\left(t_0+T^0\right)\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\\ D=M^T\left(t_0+T^0\right)\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}+{\stackrel{.}{\theta }}_g\left[\begin{array}{c}{Y}_{B^{*}}\\ -X_{B^{*}}\\ 0\end{array}\right]\end{array};& \mathrm{equation}\left(41\right)\end{array}$

wherein,

$\frac{\partial {\overrightarrow{r}}_B}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\mathrm{and}\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}$

are computed by the numerical integration in step 5. The matrix G generates the constraint equation with the constant launch angle.

2. The system of linear differential equations are solved to obtain a set of unique solutions of Δ{right arrow over ({dot over (r)})}_A and ΔT;

the system of linear equations (40) includes four equations and four unknown quantities so that a set of unique solutions of Δ{right arrow over ({dot over (r)})}_A and ΔT are obtained, and then the corrected solutions and {circumflex over (T)} are obtained according to equation (42):

$\begin{array}{cc}\{\begin{array}{c}={\stackrel{.}{\overrightarrow{r}}}_A^0+\Delta {\stackrel{.}{\overrightarrow{r}}}_A\\ \hat{T}=T^0+\Delta T\end{array}.& \mathrm{equation}\left(42\right)\end{array}$

3. The corrected solutions are taken as the reference solutions for the next differential correction, that is, let {right arrow over ({dot over (r)})}_A⁰=, T⁰={circumflex over (T)}, and repeat the sub-steps of step 5.

To facilitate the understanding of the technical solutions of the present invention, the technical solutions are described clearly and completely in conjunction with the embodiments below. The embodiments apply the variables and symbols defined in the technical solutions and derive corresponding results by assigning specific numerical values to the variables. The definitions of the coordinate systems and the transformation methods are first introduced below, and then the technical solutions are described in detail in conjunction with the embodiments.

I. Coordinate Systems and Transformation Matrices

1. Body-Fixed Coordinate System

The body-fixed coordinate system is an earth-fixed coordinate system that rotates with the earth. The body-fixed coordinate system can conveniently describe the spatial position of a point on the earth's surface, and is divided into a conventional body-fixed coordinate system and a pseudo body-fixed coordinate system according to different directions to which the Z-axis points. The difference (caused by polar motion) between the conventional body-fixed coordinate system and the pseudo body-fixed coordinate system has a minimal effect on the coordinates of the ground point. FIG. 5 illustrates the definition of the pseudo body-fixed coordinate system, in which the Z-axis points to the instantaneous pole, and the reference plane is an instantaneous equatorial plane. In the conventional body-fixed coordinate system, the Z-axis points to a conventional pole, and the reference plane is a plane orthogonal to a line connecting the center of the earth and the conventional pole. In consideration of the research background of the present invention, the difference between the conventional pole and the instantaneous pole can be ignored so that the conventional body-fixed coordinate system and the pseudo body-fixed coordinate system are collectively referred to as the body-fixed coordinate system (or geodetic coordinate system) in two forms of spherical coordinates and 3D rectangular coordinates.

2. TEMEE Coordinate System

As shown in FIG. 6, the TEMEE coordinate system is a hybrid geocentric coordinate system, in which the reference plane is an instantaneous true equatorial plane, and the X-axis is located in the reference plane and points to the mean equinox of the epoch (epoch J2000.0 herein). The mean equinox is, in fact, an imaginary point on the instantaneous true equator, and is located μ+Δμ east of the true equinox, wherein μ is the general precession of right ascension, and Δμ is the nutation of the right ascension, which are computed with reference to Article 5 (Wu Lianda. Orbits and Detection of Artificial Satellites and Space Debris, China Science and Technology Press, 2011.). With the complementary advantages of the inertial system and the instantaneous true equatorial geocentric system, the TEMEE coordinate system not only avoids causing changes in the potential function of the earth's gravitational field, but also involves minimal additional perturbation that is negligible in solving problems requiring general precision. Hence, the TEMEE coordinate system is the preferred coordinate system for the analysis method of artificial satellites.

3. Conventional Horizontal Coordinate System and Target-Pointing Horizontal Coordinate System

As shown in FIG. 7, the conventional horizontal coordinate system takes the launch point of the missile as the origin and the horizontal plane passing through the origin as the reference plane, where the X-axis points to the north pole in the reference plane. In the target-pointing horizontal coordinate system, the X*-axis points to the target point B in the reference plane. The launch azimuth A* in the present invention is defined in the target-pointing horizontal coordinate system. In consideration of the research background of the present invention, the difference between the geoid and the reference ellipsoid can be ignored so that the tangent plane of the reference ellipsoid is used instead of the horizontal plane.

4. Transformation Matrix M from the Pseudo Body-Fixed Coordinate System to the TEMEE Coordinate System

The difference between the pseudo body-fixed coordinate system and the TEMEE coordinate system (epoch J2000.0) is only the Greenwich sidereal hour angle θ_g in the X-axis, so that the coordinate transformation matrix M is expressed as:

$M=R_Z\left[-{\theta }_g\right]=\left[\begin{array}{ccc}\cos \left(-{\theta }_g\right)& \sin \left(-{\theta }_g\right)& 0\\ -\sin \left(-{\theta }_g\right)& \cos \left(-{\theta }_g\right)& 0\\ 0& 0& 1\end{array}\right];$

wherein, θ_g is computed by:

${\theta }_g=280°·460618375+360^{\circ }·9856122882d$

d=MJD(UT1)−51544.5 is a Julian day number from 12^h UT₁ on Jan. 1, 2000 to the launch epoch t of the missile.

The transformation matrix from the TEMEE coordinate system to the pseudo body-fixed coordinate system is denoted as M^T in consideration of its orthogonality.

5. Transformation Matrix W from the Body-Fixed Coordinate System to the Conventional Horizontal Coordinate System

In a case where the difference between the origins of the body-fixed coordinate system and the conventional horizontal coordinate system is not considered, a transformation between the two coordinate systems is accomplished by two rotations using a transformation matrix that is a function of the geodetic latitude and longitude of the ground point. Assuming that the geodetic latitude and longitude of the ground point are L and B, respectively, then the transformation matrix W from the body-fixed coordinate system to the conventional horizontal coordinate system is:

$W=R_Y\left[B-\pi {/}_2\right]·R_Z\left[L-\pi \right]=\left[\begin{array}{ccc}-B\sin B\cos L& -\sin B\sin L& \cos B\\ \sin L& -\cos L& 0\\ \cos B\cos L& \cos B\sin L& \sin B\end{array}\right];$

Similarly, the transformation matrix from the conventional horizontal coordinate system to the body-fixed coordinate system is denoted as W^T.

5. Transformation Matrix Q from the Conventional Horizontal Coordinate System to the Target-Pointing Horizontal Coordinate System

The conventional horizontal coordinate system and the target-pointing horizontal coordinate system have the same origin and reference plane, and only differ in the direction to which the X-axis points. In the conventional horizontal coordinate system, if the azimuth of the target point B is ϕ, then the classic horizontal coordinates are rotated counterclockwise around the Z-axis by ϕ to be transformed into the target-pointing horizontal coordinates. Accordingly, the transformation matrix Q is expressed by:

$Q=R_Z\left[\varphi \right]=\left[\begin{array}{ccc}\cos \varphi & \sin \varphi & 0\\ -\sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right];$

Similarly, the transformation matrix from the target-pointing horizontal coordinate system to the conventional horizontal coordinate system is denoted as Q^T.

II. Implementation of the Present Invention in Conjunction with the Embodiments

Embodiment 1: All positive-flying and negative-flying free trajectories from the launch point A to the target point B are constructed based on randomly given launch epoch to of a missile and spherical coordinates of the two points (as shown in Table 1) by traversing using a two-body force model and a dynamic model including the J2 perturbation of the earth's gravitational field, respectively.

TABLE 1
Embodiment 1
(Launch epoch: 13:45:18.732 on Aug. 24, 2012 Beijing time;
launch angle h: traversed)
	Geodetic longitude	Geodetic latitude	Geodetic height
Point	L (°)	B (°)	H (m)
A	86.384	−60.756	14.00
B	116.403	60.905	49.00

Embodiment 2: All positive-flying and negative-flying free trajectories from the launch point A (at the north pole) to the target point B are constructed based on randomly given launch epoch to of a missile and spherical coordinates of the point B (as shown in Table 2) by traversing using a two-body force model and a dynamic model including the J2 perturbation of the earth's gravitational field, respectively.

TABLE 2
Embodiment 2
(Launch epoch: 13:55:25.6 on May 18, 2012 Beijing time;
launch angle h: traversed)
	Geodetic longitude	Geodetic latitude	Geodetic height
Point	L (°)	B (°)	H (m)
A	∀LA ∈ [0, 360)	90.000	1000.00
B	240.500	−45.905	0.00

In addition to the above known parameters, the present invention also involves the following geophysical constants:

Equatorial radius of the reference ellipsoid: α_e=6378136 m.

Geocentric gravitational constant: μ=0.39860043770442×10¹⁵ m³/s².

Oblateness of the earth: f=1/298.25781.

Eccentricity of the meridian of the earth: e_c²=2f−f²=0.0066946.

Among the above parameters, the launch epoch and the geodetic coordinates of the launch point are known, and the coordinates of the launch point in the inertial space are obtained by rotating the coordinate system. Due to the rotation of the earth, the position of the target point in the inertial space changes at all times. In this regard, the flight time is a key variable for the constructed trajectory to hit the target point with a changing position, and the flight time is computed based on the position of the target point in the inertial space. In the present invention, two variables of the flight time and inertial coordinates of the target point are iteratively computed simultaneously by specific steps as follows:

Step 1: data preprocessing: the 3D rectangular coordinate vector {right arrow over (R)}_A of the launch point A and the 3D rectangular coordinate vector {right arrow over (R)}_B of the target point B, the difference Δ_φ between the geodetic latitude and the geocentric latitude of the launch point A, the horizontal angle ϕ between the AB direction and the north-pole direction of the launch point in the body-fixed coordinate system and the difference ε between a ratio of the modulus of the geocentric radius vector of the point A to the modulus of the geocentric radius vector of the point B and 1 are sequentially computed.

Step 1 includes the following sub-steps:

1. The geodetic coordinates of the points A and B are transformed from spherical coordinates (L, B, H) to 3D rectangular coordinates (X, Y, Z) according to equation (1), and the 3D rectangular coordinates are expressed by {right arrow over (R)}_A and {right arrow over (R)}_B as:

$\begin{array}{cc}\{\begin{array}{c}X=\left(N+H\right)\cos B\cos L\\ Y=\left(N+H\right)\cos B\sin L\\ Z=\left[N\left(1-e_c^2\right)+H\right]\sin B\end{array};& \mathrm{equation}\left(1\right)\end{array}$

wherein, N is the radius of curvature of the prime vertical of a point, and N=α_e/√{square root over (1−e_c²(sin B)²)}.

In Embodiment 2, the point A is located at the north pole, and thus its geodetic longitude is indefinite. In the present invention, the geodetic longitude of the point A is specified as an arbitrary value between 0° and 360°, such that the construction of trajectories is carried out smoothly without affecting the result.

2. The geocentric latitudes φ_A and φ_B of the points A and B are computed according to equation (2):

$\begin{array}{cc}\phi ={\sin }^{-1}\frac{z}{\sqrt{X^2+Y^2+Z^2}}.& \mathrm{equation}\left(2\right)\end{array}$

3. The difference Δ₁₀₀ between the geodetic latitude and the geocentric latitude of the point A is computed according to equation (3):

$\begin{array}{cc}\mathrm{\Delta \phi }=B_A-{\phi }_A.& \mathrm{equation}\left(3\right)\end{array}$

4. The horizontal angle ϕ between the AB direction and the north-pole direction of the launch point in the body-fixed coordinate system is computed according to equations (4) and (5):

$\begin{array}{cc}\{\begin{array}{c}\sin \varphi =\frac{\cos B_B\sin \left(L_A-L_8\right)}{\sin q}\\ \cos \varphi =\frac{\sin B_B\cos B_A-\cos B_B\sin B_A\cos \left(L_A-L_B\right)}{\sin q}\end{array};& \mathrm{equation}\left(4\right)\end{array}$

wherein, q is the zenith angle of the point B from the zenith of the point A, and is computed by:

$\begin{array}{cc}\{\begin{array}{c}\cos q=\sin B_B\sin B_A+\cos B_B\cos B_A\cos \left(L_A-L_B\right)\\ \sin q=\sqrt{1-{\cos }^{2}q}\end{array}q\in \left(0,\pi \right).& \mathrm{equation}\left(5\right)\end{array}$

In a particular case where the launch point is located at or above the north or south pole, equations (4) and (5) are still satisfied, and are simplified as:

$\{\begin{array}{cc}\varphi =\pi -\left(L_A-L_B\right)& A\mathrm{is}\mathrm{at}\mathrm{or}\mathrm{above}\mathrm{the}\mathrm{north}\mathrm{pole}\\ \varphi =L_A-L_B& A\mathrm{is}\mathrm{at}\mathrm{or}\mathrm{above}\mathrm{the}\mathrm{south}\mathrm{pole}\end{array}\forall L_A\in \left[0,360°\right).$

Equations (4) and (5) are derived from the formulas of spherical trigonometry applied to the spherical triangle O-PZ_AZ_B in FIG. 8. O is the center of the earth. P, Z_A, Z_B and γ are the projections of the north pole, the zenith of the launch point, the zenith of the target point and the equinox on the celestial sphere, respectively. The great circle defined by is an extension of the equator on the celestial sphere, where E is the intersection point of the great circle passing P and Z_B and the equatorial plane, and F is the intersection point of the great circle passing P and Z_A and the equatorial plane. The great circle defined by is perpendicular to OZ_A; and the great circle defined by is perpendicular to OZ_B. The difference between the geoid and the reference ellipsoid is ignored, such that angular distance ∠POZ_B represents the complementary angle of the geodetic latitude of the point B and is equal to

$\left(\frac{\pi }{2}-B_B\right).$

Angular distance ∠POZ_A represents the complementary angle of the geodetic latitude of the point A and is equal to

$\left(\frac{\pi }{2}-B_A\right).$

Spherical angle Z_BPZ_A represents an angle between meridians passing through the points A and B, and is equal to (L_A−L_B). Spherical angle ∠Z_APZ_B is the zenith angle of the point B from the zenith of the point A and is denoted as q. ∠Z_BZ_AP is denoted as ϕ to be computed. According to the law of sines of spherical trigonometry and the five-element formulas, the following two equations are derived:

$\{\begin{array}{c}\frac{\sin \varphi }{\sin \left(\frac{\pi }{2}-B_B\right)}=\frac{\sin \left(L_A-L_B\right)}{\sin q}\\ \sin q\cos \varphi =\cos \left(\frac{\pi }{2}-B_B\right)\sin \left(\frac{\pi }{2}-B_A\right)-\\ \sin \left(\frac{\pi }{2}-B_B\right)\cos \left(\frac{\pi }{2}-B_A\right)\cos \left(L_A-L_B\right)\end{array},$

simplified as the following equations for computing ϕ:

$\{\begin{array}{c}\sin \varphi =\frac{\cos B_B\sin \left(L_A-L_B\right)}{\sin q}\\ \cos \varphi =\frac{\sin B_8\cos B_A-\cos B_B\sin B_A\cos \left(L_A-L_B\right)}{\sin q}\end{array}.$

According to the law of cosines of spherical trigonometry, q is expressed by:

$\cos q=\cos \left(\frac{\pi }{2}-B_B\right)\cos \left(\frac{\pi }{2}-B_A\right)+\sin \left(\frac{\pi }{2}-B_B\right)\sin \left(\frac{\pi }{2}-B_A\right)\cos \left(L_A-L_B\right),$

rewritten as:

$\{\begin{array}{c}\cos q={\sin B}_B{\sin B}_A+{\cos B}_B{\cos B}_A\cos \left(L_A-L_B\right)\\ \sin q=\sqrt{1-{\cos }^{2}q}\end{array}q\in \left(0,\pi \right);$

wherein the value of q is limited within (0,π). The two points A and B must not coincide or be located at the two ends of the diameter of the earth for the following reason. If the target point and the launch point coincide, then it does not meet the actual need for constructing a trajectory. If the target point and the launch point are located at the two ends of the diameter of the earth, then there exist countless elliptical trajectories passing through the two points at the specified launch angle, such that it is impossible to derive a unique set of designed parameters of the missile.

5. The difference ε between a ratio of the modulus of the geocentric radius vector of the point A to the modulus of the geocentric radius vector of the point B and 1 is computed according to equation (6):

$\begin{array}{cc}\varepsilon =\frac{{\overrightarrow{R}}_B}{{\overrightarrow{R}}_A}-1.& \mathrm{equation}\left(6\right)\end{array}$

The moduli of the geocentric radius vectors of the points A and B are typically not equal, and the geodetic height difference between the points A and B is much smaller than the radius of the earth. Thus, ε is typically a small non-zero quantity.

Step 2: setting of the initial state of iterations: assuming that the earth does not rotate and the flight time T is zero.

If the earth does not rotate, the launch velocity v_p of the missile in the body-fixed coordinate system is equal to the launch velocity v_r in the TEMEE coordinate system, that is,

$\frac{v_p}{v_r}=1.$

Meanwhile, the launch azimuth A* in the target-pointing horizontal coordinate system is zero.

Step 3: the launch velocity vector {right arrow over ({dot over (r)})}_A of the positive-flying trajectory or the negative-flying trajectory in the TEMEE coordinate system and the launch velocity vector {right arrow over ({dot over (R)})}_A of the positive-flying trajectory or the negative-flying trajectory in the body-fixed coordinate system, the orbital elements σ of the missile at the launch epoch, the flight time and a plurality of other variables are computed in the two-body force model to obtain a new flight time T*.

Step 3 includes the following sub-steps:

1. The coordinate vectors {right arrow over (r)}_A and {right arrow over (r)}_B of the points A and B in the TEMEE coordinate system are computed in combination with the flight time T according to equation (7):

$\begin{array}{cc}\{\begin{array}{c}{\overrightarrow{r}}_A=M\left(t_0\right){\overrightarrow{R}}_A\\ {\overrightarrow{r}}_B=M\left(t_0+T\right){\overrightarrow{R}}_B\end{array};& \mathrm{equation}\left(7\right)\end{array}$

wherein, t₀ is the launch epoch, and M is a time-dependent transformation matrix transformation matrix from the body-fixed coordinate system to the TEMEE coordinate system. In the first iteration, the flight time T is zero, and {right arrow over (r)}_B=M(t₀){right arrow over (R)}_B is satisfied.

2. The half-range angle β is computed according to equations (8) and (9):

a half intersection angle is expressed by:

$\begin{array}{cc}\delta =\frac{1}{2}\times {\cos }^{-1}\frac{{\overrightarrow{r}}_A·{\overrightarrow{r}}_B}{{\overrightarrow{r}}_A{\overrightarrow{r}}_B};& \mathrm{equation}\left(8\right)\end{array}$

then the half-range angle is computed by:

$\begin{array}{cc}\beta =\{\begin{array}{ccc}\delta & \beta \in \left(0,\frac{\pi }{2}\right)& \mathrm{positive}-\mathrm{flying}\mathrm{trajectory}\\ \pi -\delta & \beta \in \left(\frac{\pi }{2},\pi \right)& \mathrm{negative}-\mathrm{flying}\mathrm{trajectory}\end{array};& \mathrm{equation}\left(9\right)\end{array}$

3. The inclination i and the right ascension Ω of the ascending node of the elliptical trajectory/orbit are computed according to equation (10):

$\begin{array}{cc}\{\begin{array}{cc}\left[\begin{array}{c}\sin i\mathrm{sin\Omega }\\ -\sin i\mathrm{cos\Omega }\\ \cos i\end{array}\right]=\frac{{\overrightarrow{r}}_A·{\overrightarrow{r}}_B}{{\overrightarrow{r}}_A\times {\overrightarrow{r}}_B}& \mathrm{positive}-\mathrm{flying}\mathrm{trajectory}\\ \left[\begin{array}{c}\sin i\mathrm{sin\Omega }\\ -\sin i\mathrm{cos\Omega }\\ \cos i\end{array}\right]=\frac{{\overrightarrow{r}}_B·{\overrightarrow{r}}_A}{{\overrightarrow{r}}_B\times {\overrightarrow{r}}_A}& \mathrm{negative}-\mathrm{flying}\mathrm{trajectory}\end{array}i\in \left(0,\pi \right)\Omega \in \left[0,2\pi \right).& \mathrm{equation}\left(10\right)\end{array}$

4. The true arguments of latitude μ_A and μ_B of the points A and B on the elliptical trajectory/orbit are computed according to equation (11):

$\begin{array}{cc}\{\begin{array}{c}{\sin u}_A=\frac{{\mathrm{sin\phi }}_A}{\sin i}\\ {\cos u}_A=\frac{{\mathrm{sin\phi }}_B-{\mathrm{sin\phi }}_A\mathrm{cos2\beta }}{\sin i\mathrm{sin2\beta }};\\ u_B=u_A+2\beta \end{array}& \mathrm{equation}\left(11\right)\end{array}$

Equation (11) is derived from the law of sines applied to the spherical triangles in FIG. 9. In the spherical triangle O-KAA′, spherical angle ∠AKA′ is equal to the orbital inclination i of the trajectory; angular distance ∠AOA′ is equal to the geocentric latitude φ_A of the point A; angular distance ∠AOK is equal to the true argument of latitude μ_A of the point A; and sin μ_A is expressed according to the law of sines as follows:

${\sin u}_A=\frac{{\mathrm{sin\phi }}_A}{\sin i}.$

Similarly, in the spherical triangle O-KBB′,

${\sin u}_B=\frac{{\mathrm{sin\phi }}_B}{\sin i}.$

The half-range angle β is known and μ_B=μ_A+2β is satisfied, which is substituted into the above equations to obtain:

${\sin u}_B=\sin \left(u_A+2\beta \right)={\sin u}_A\mathrm{cos2\beta }+{\cos u}_A\mathrm{sin2\beta }.$

The expressions of sin μ_A and sin μ_B are substituted to obtain:

$\frac{{\mathrm{sin\phi }}_B}{\sin i}=\frac{{\mathrm{sin\phi }}_A}{\sin i}\mathrm{cos2\beta }+{\cos u}_A\mathrm{sin2\beta },$

simplified as:

${\cos u}_A=\frac{{\mathrm{sin\phi }}_B-{\mathrm{sin\phi }}_A\mathrm{cos2\beta }}{\sin i\mathrm{sin2\beta }}.$

5. The cosine of the angle α between the launch velocity vector and the geocentric radius vector of the launch point in the body-fixed coordinate system is computed according to equation (12):

$\begin{array}{cc}\mathrm{cos\alpha }=\mathrm{cos\Delta \phi sin}h-\mathrm{sin\Delta \phi cos}h\cos \left(A^{*}-\varphi \right);& \mathrm{equation}\left(12\right)\end{array}$

wherein, h is the launch angle, and A*=0 in the first iteration.

Equation (12) is derived from the law of cosines applied to the spherical triangles in FIG. 10A and FIG. 10B. The spherical triangle in FIG. 10A and FIG. 10B is formed by the projections of the velocity vector, radius vector and zenith of the point A on the celestial sphere, where the spatial directions of the velocity, the zenith and the radius vector of the point A are shown in FIG. 11. In the spherical triangle A-ZRV, angular distance ∠ZAR represents the difference Δ₁₀₀ between the geodetic latitude and the geocentric latitude of the point A. According to the definition of the launch angle h, angular distance ∠ZAV is the complementary angle of h, namely

$\tilde{\alpha }=\frac{\pi }{2}-h.$

∠PZV is the azimuth of the launch velocity vector and denoted as Θ, which is equal to (A*−ϕ). Angular distance ∠VAR is the angle α between the launch velocity vector and the geocentric radius vector of the launch point in the body-fixed coordinate system and is to be computed. According to the law of cosines, the angle is expressed by:

$\cos \alpha =\cos \mathrm{\Delta \phi cos}\tilde{\alpha }+\sin \mathrm{\Delta \phi sin}\tilde{\alpha }\cos \left(\pi -\theta \right);$

{tilde over (α)} and Θ are substituted into the above equation to obtain:

$\cos \alpha =\cos \mathrm{\Delta \phi sinh}-\sin \mathrm{\Delta \phi cos}h\cos \left(A^{*}-\varphi \right).$

6. The tangent of the angle θ between the launch velocity vector and the geocentric radius vector of the launch point in the TEMEE coordinate system is computed according to equation (13):

$\begin{array}{cc}\{\begin{array}{c}\cos \theta =\frac{v_P}{v_r}\cos \alpha \\ \sin \theta =\sqrt{1-{\cos }^2\theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\end{array}& \mathrm{equation}\left(13\right)\end{array}$

Due to the rotation of the earth, the launch velocity vector in the TEMEE coordinate system is a resultant of the launch velocity vector in the body-fixed coordinate system and the rotational velocity vector of the earth. Since the rotational velocity vector of the earth is perpendicular to the geocentric radius vector, the launch velocity vectors in the two coordinate systems have equal components in the direction of the geocentric radius vector. Accordingly:

${\stackrel{.}{\overrightarrow{r}}}_A·\frac{{\overrightarrow{r}}_A}{{\overrightarrow{r}}_A}={\stackrel{.}{\overrightarrow{R}}}_A·\frac{{\overrightarrow{R}}_A}{{\overrightarrow{R}}_A},$

simplified as:

${\stackrel{.}{\overrightarrow{r}}}_A\cos \theta ={\stackrel{.}{\overrightarrow{R}}}_A\cos \alpha .$

Because |{right arrow over ({dot over (r)})}_A|=v_r; |{right arrow over ({dot over (R)})}_A|=v_p, the above equation can be rewritten as:

$v_r\cos \theta =v_P\cos \alpha ;$

wherein, in the first iteration,

$\frac{v_P}{v_r}=1.$

7. A bias Δω of the argument of perigee is introduced and computed according to equation (15), and then the argument of perigee ω is computed according to equation (14).

According to the nature of the elliptical trajectory/orbit, the apogee of the trajectory is located in an outer space of the earth and between the points A and B. When the geocentric radius vector moduli of the points A and B are equal, the argument of apogee is equal to the mean value of the true arguments of latitude of the points A and B, and the mean value minus π obtains the argument of perigee ω. The argument of perigee ω computed in this way, however, generally has a deviation due to unequal geocentric distances of the point A and the point B. The bias Δω of ω is introduced to obtain:

$\begin{array}{cc}\omega ={\omega }_0+\mathrm{\Delta \omega }\omega \in \left[0,2\pi \right);& \mathrm{equation}\left(14\right)\end{array}$

wherein,

${\omega }_0=\frac{u_A+u_B}{2}-\pi ,$

wherein μ_A and μ_B have been computed according to equation (11); Δω is computed according to equation (15):

$\begin{array}{cc}\tan \Delta \omega =\frac{\varepsilon }{2\left(1+\varepsilon \right)\cot \theta -\varepsilon \cot \beta }\mathrm{\Delta \omega }\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right];& \mathrm{equation}\left(15\right)\end{array}$

accordingly, the true anomalies of the points A and B on the elliptical trajectory/orbit are expressed by Δω as follows:

$\begin{array}{cc}\{\begin{array}{c}{f}_A=\pi -\beta -\Delta \omega \\ f_B=\pi +\beta -\Delta \omega \end{array}.& \mathrm{equation}\left(16\right)\end{array}$

Equation (15) is derived as follows. A polar coordinate system is established on an orbital plane, then a polar coordinate equation of an elliptic curve is expressed as:

$r=\frac{p}{1+e\cos f};$

wherein, p is a semi-laths rectum. The polar radii of the points A and B are:

$\{\begin{array}{c}{r}_A=\frac{p}{1+e\cos f_A}\\ r_B=\frac{p}{1+e\cos f_B}\end{array};$

r_A=|{right arrow over (R)}_A| and r_B=|{right arrow over (R)}_B| are substituted into equation (6) to obtain:

$\varepsilon =\frac{1+e\cos f_A}{1+e\cos f_B}-1,$

rewritten as:

$1+e\cos f_A=\left(1+\varepsilon \right)\left(1+e\cos f_B\right).$

The expressions of f_A and f_B in equation (16) are substituted to obtain:

$\sin \mathrm{\Delta \omega }=\frac{\varepsilon \left(1-e\cos \left(\beta -\Delta \omega \right)\right)}{2e\sin \beta }.$

Equation (17) for computing e is substituted into the expression of sin Δω to obtain the most compact form expressed as equation (15).

8. The eccentricity e of the elliptical trajectory/orbit is computed according to equation (17).

$\begin{array}{cc}e=\frac{\cot \theta }{\sin \left(\beta +\Delta \omega \right)+\cot \theta \cos \left(\beta +\Delta \omega \right)};& \mathrm{equation}\left(17\right)\end{array}$

wherein, θ is the angle between the launch velocity vector of the missile and the geocentric radius vector in the TEMEE coordinate system, and is determined jointly by the eccentricity of the elliptical orbit and the true anomaly of the launch point. The relationship between the three is expressed as:

$\tan \theta =\frac{1+e\cos f_A}{e\sin f_A},$

rewritten as:

$e=\frac{\cot \theta }{\sin f_A-\cot \theta \cos f_A},$

f_A=π−β−Δω is substituted into the above equation to obtain:

$e=\frac{\cot \theta }{\sin \left(\beta +\Delta \omega \right)+\cot \theta \cos \left(\beta +\Delta \omega \right)}.$

9. The rationality of the specified launch angle h is judged as follows. If any one of the following conditions is satisfied, then the specified launch angle is judged to be irrational such that a rationally designed trajectory cannot be obtained, the current construction procedure of the trajectory is ended, and a launch angle is re-specified.

For the positive-flying trajectory:

(1) If e≥1, then the specified launch angle is judged to be excessively large; and

(2) If cot θ≤0 or |tan Δω|≥tan β, then the specified launch angle is judged to be excessively small.

For the negative-flying trajectory:

$\begin{array}{cc}\mathrm{If}e\ge 1\mathrm{or}\cot \theta \ge -\tan \beta +\frac{\varepsilon }{2\left(1+\varepsilon \right)}\left(\tan \beta +\cot \beta \right),& \left(1\right)\end{array}$

then the specified launch angle is judged to be excessively large; and

$\begin{array}{cc}\mathrm{If}\cot \theta \le \max \left\{\frac{\varepsilon }{2\left(1+\varepsilon \right)}\cot \beta ,0\right\},& \left(2\right)\end{array}$

then the specified launch angle is judged to be excessively small.

10. The true anomalies f_A and f_B of the points A and B on the elliptical trajectory/orbit are computed according to equation (16).

11. The orbital elements a of the missile at the launch epoch are computed according to equations (18) to (21).

σ is a set of the first type of non-singular orbital elements, wherein the inclination i and the right ascension Ω of the ascending node of the orbit are computed according to equation (10), and a semi-major axis α and ξ, η, λ are computed as follows:

$\begin{array}{cc}a=\frac{\left|{\overrightarrow{r}}_A\right|\left[1-e\cos \left(\beta +\Delta \omega \right)\right]}{1-e^2};& \mathrm{equation}\left(18\right)\\ \{\begin{array}{c}\xi =e\cos \omega \\ \eta =-e\sin \omega \\ \lambda =\omega +M_A\end{array};& \mathrm{equation}\left(19\right)\end{array}$

wherein M_A is computed as follows:

$\begin{array}{cc}\{\begin{array}{c}{E}_A=f_A-2{\tan }^{-1}\left(\frac{e\sin f_A}{1+\sqrt{1-e^2}+e\cos f_A}\right)\\ E_B=f_B-2{\tan }^{-1}\left(\frac{e\sin f_B}{1+\sqrt{1-e^2}+e\cos f_B}\right)\end{array};& \mathrm{equation}\left(20\right)\\ \{\begin{array}{c}{M}_A=E_A-e\sin E_A\\ M_B=E_B-e\sin E_B\end{array}.& \mathrm{equation}\left(21\right)\end{array}$

12. The flight time T* of the missile is computed according to equation (22):

$\begin{array}{cc}\{\begin{array}{c}{T}^{*}=\frac{M_B-M_A}{n}\\ n=\sqrt{\mu /a^3}\end{array}.& \mathrm{equation}\left(22\right)\end{array}$

13. The launch velocity v_r of the missile in the TEMEE coordinate system and the launch velocity v_p of the missile in the body-fixed coordinate system are computed according to equations (23) to (25), where the launch velocity vectors {right arrow over ({dot over (r)})}_A and {right arrow over ({dot over (R)})}_A of the missile in the TEMEE coordinate system and the body-fixed coordinate system satisfy:

$\begin{array}{cc}\{\begin{array}{c}{\stackrel{\stackrel{.}{\to }}{r}}_A=\sqrt{\frac{\mu }{p}}\left[\left(-\sin u_A+\eta \right)\overrightarrow{P}+\left(\cos u_A+\xi \right)\overrightarrow{Q}\right]\\ \overrightarrow{P}=\left[\begin{array}{c}\cos \Omega \\ \sin \Omega \\ 0\end{array}\right]\\ \overrightarrow{Q}=\left[\begin{array}{c}-\sin \Omega \cos i\\ \cos \Omega \cos i\\ \sin i\end{array}\right]\\ p=a\left(1-e^2\right)\end{array};& \mathrm{equation}\left(23\right)\\ {\stackrel{\stackrel{.}{\to }}{R}}_A=M\left(t_0\right){\stackrel{\stackrel{.}{\to }}{r}}_A-{\stackrel{.}{\theta }}_g\left[\begin{array}{c}-Y_A\\ X_A\\ 0\end{array}\right];\mathrm{then}\text{:}& \mathrm{equation}\left(24\right)\\ \{\begin{array}{c}{v}_p={\stackrel{\stackrel{.}{\to }}{R}}_A\\ v_r={\stackrel{\stackrel{.}{\to }}{r}}_A\end{array};& \mathrm{equation}\left(25\right)\end{array}$

wherein, {dot over (θ)}_g is a change rate of the Greenwich sidereal hour angle in the TEMEE coordinate system and is 360°.985612288/day.

14. The launch azimuth A* in the target-pointing horizontal coordinate system is computed according to equations (26) and (27):

the launch velocity vector in the target-pointing horizontal coordinate system is denoted as {right arrow over ({dot over (R)})}_A*(X*, Y*, Z*), and which is derived by performing a coordinate rotation on {right arrow over ({dot over (R)})}_A as follows:

$\begin{array}{cc}{\stackrel{\stackrel{.}{\to }}{R}}_{A^{*}}=\mathrm{QW}{\stackrel{\stackrel{.}{\to }}{R}}_A;& \mathrm{equation}\left(26\right)\\ \left[\begin{array}{c}{X}^{*}\\ Y^{*}\\ Z^{*}\end{array}\right]={\stackrel{\stackrel{.}{\to }}{R}}_{A^{*}}\left[\begin{array}{c}\cos h\cos A^{*}\\ -\cos h\sin A^{*}\\ \sin h\end{array}\right];& \mathrm{equation}\left(27\right)\end{array}$

wherein, W is a rotation matrix from the body-fixed coordinate system to the conventional horizontal coordinate system, and Q is a rotation matrix from the conventional horizontal coordinate system to the target-pointing horizontal coordinate system.

Step 4: let T=T*, step 3 is repeated to iteratively compute the launch velocity vector of the missile, the orbital elements of the missile at the launch epoch, the half-range angle, the flight time and a plurality of other variables until |T−T*| is less than the set threshold S_t to complete the iterations to finally obtain the launch velocity vector {right arrow over ({dot over (r)})}_A and the flight time T of the missile in the two-body force model.

Step 4 includes the following sub-steps:

1. It is judged whether |T−T*|<S_t is satisfied; if yes, let T=T*, and proceeding to the next step; otherwise, let T=T*, and returning to step 3.

2. The declination à of the launch velocity vector in the target-pointing horizontal coordinate system relative to the target point in the horizontal plane is computed according to equation (28):

$\begin{array}{cc}\{\begin{array}{cc}\tilde{A}=A^{*}& A^{*}\le \pi \\ \tilde{A}=A^{*}-2\pi & A^{*}>\pi \end{array}.& \mathrm{equation}\left(28\right)\end{array}$

3. The designed parameters v_p, v_r, Ã, T, σ of the missile obtained based on the two-body force model are output.

Step 5: the launch velocity vector {right arrow over ({dot over (r)})}_A and the flight time T obtained by the two-body force model are taken as reference solutions {right arrow over ({dot over (r)})}_A⁰ and T⁰ for the differential correction; if the distance |Δ{right arrow over (R)}_BB*| between the target point B and the target point B* obtained by a perturbation extrapolation based on the reference solutions is less than the given threshold S_R, then the correction of the designed parameters of the missile is ended, and corrected parameters v_p, v_r, Ã, T, σ are output; otherwise, proceeding to step 6.

Step 5 includes the following sub-steps:

1. The target point obtained by the perturbation extrapolation based on the reference solutions {right arrow over ({dot over (r)})}_A⁰ and T⁰ is denoted as B*, and partial derivative matrices

$\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\overrightarrow{r}}_A},\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}$

and a position vector {right arrow over (r)}_B* of B* in the TEMEE coordinate system are computed by a numerical integration.

To adapt to computing highly-eccentric trajectories, the numerical integration is performed by a Gragg-Bulirsch-Stoer first-order integrator, where the differential equations to be integrated are:

$\begin{array}{cc}\{\begin{array}{c}\frac{d\overrightarrow{r}}{\mathrm{dt}}=\overrightarrow{v}\\ \frac{d\overrightarrow{v}}{dt}=\overrightarrow{F}\left(\overrightarrow{r}\right)\\ \frac{d}{dt}\left(\frac{\partial \tilde{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\right)=\frac{\partial \overrightarrow{v}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\\ \frac{d}{dt}\left(\frac{\partial \overrightarrow{v}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\right)=\frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}·\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\end{array};& \mathrm{equation}\left(29\right)\end{array}$

derive initial values as:

$\begin{array}{cc}\{\begin{array}{c}\overrightarrow{r}\left(t_0\right)={\overrightarrow{r}}_A\\ \overrightarrow{v}\left(t_0\right)={\stackrel{.}{\overrightarrow{r}}}_A^0\\ {\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}}_{t=t_0}=0\\ {\frac{\partial \overrightarrow{\nu }}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}}_{t=t_0}=I\end{array};& \mathrm{equation}\left(30\right)\end{array}$

integrate from t=t₀ to t=t₀+T⁰ to obtain:

$\begin{array}{cc}\{\begin{array}{c}{{\overrightarrow{r}}_{B^{*}}=\overrightarrow{r}}_{t=t_0+T^0}\\ {\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}=\overrightarrow{v}}_{t=t_0+T^0}\\ {\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}=\frac{\partial \overrightarrow{r}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}}_{t=t_0+T^0}\end{array};& \mathrm{equation}\left(31\right)\end{array}$

the force function {right arrow over (F)}({right arrow over (r)}) only including the J2 perturbation and its partial derivative matrix

$\frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}$

in equation (29) are expressed as:

$\begin{array}{cc}\{\begin{array}{c}\overrightarrow{F}={\left(\nabla U\right)}_r=M\left(t\right)·{\left(\nabla U\right)}_R\\ \frac{\partial \overrightarrow{F}}{\partial \overrightarrow{r}}={\left({\nabla }^{2}U\right)}_r=M\left(t\right)·{\left({\nabla }^{2}U\right)}_R·M^T\left(t\right)\end{array};& \mathrm{equation}\left(32\right)\end{array}$

wherein, t is an integration time; subscripts r and R represent a gradient or tensor of U in the TEMEE coordinate system and the body-fixed coordinate system, respectively; U represents a gravitational potential function of the earth in the body-fixed coordinate system and includes a central gravitational potential U₀ and a J₂ gravitational potential U₁:

$\begin{array}{cc}U=U_0+U_1.& \mathrm{equation}\left(33\right)\end{array}$

Accordingly, the gradient and tensor of the gravitational potential function U of the earth are obtained as follows:

$\begin{array}{cc}\{\begin{array}{c}{\left(\nabla U\right)}_R={\left(\nabla U_0\right)}_R+{\left(\nabla U_1\right)}_R\\ {\left({\nabla }^{2}U\right)}_R={\left({\nabla }^2{U}_0\right)}_R+{\left({\nabla }^2{U}_1\right)}_R\end{array};\mathrm{wherein},& \mathrm{equation}\left(34\right)\\ \{\begin{array}{c}{\left(\nabla U_0\right)}_R=-\frac{\mu }{r^3}{\left(X,Y,Z\right)}^T\\ {\left({\nabla }^2{U}_0\right)}_R=\frac{\mu }{r^3}\left[\begin{array}{ccc}-1+3\frac{X^2}{r^2}& \frac{3\mathrm{XY}}{r^2}& \frac{3\mathrm{XZ}}{r^2}\\ \frac{3\mathrm{XY}}{r^2}& -1+3\frac{Y^2}{r^2}& \frac{3YZ}{r^2}\\ \frac{3\mathrm{XZ}}{r^2}& \frac{3YZ}{r^2}& 1+3\frac{Z^2}{r^2}\end{array}\right]\\ {\left(\nabla U_1\right)}_R={\left(\frac{\partial U_1}{\partial X},\frac{\partial U_1}{\partial Y},\frac{\partial U_1}{\partial Z}\right)}^T\\ {\left({\nabla }^2{U}_1\right)}_R=\left[\begin{array}{ccc}\frac{{\partial }^2{U}_1}{\partial X^2}& \frac{{\partial }^2{U}_1}{\partial X\partial Y}& \frac{{\partial }^2{U}_1}{\partial X\partial Z}\\ \frac{{\partial }^2{U}_1}{\partial Y\partial X}& \frac{{\partial }^2{U}_1}{\partial Y^2}& \frac{{\partial }^2{U}_1}{\partial Y\partial Z}\\ \frac{{\partial }^2{U}_1}{\partial Z\partial X}& \frac{{\partial }^2{U}_1}{\partial Z\partial Y}& \frac{{\partial }^2{U}_1}{\partial Z^2}\end{array}\right]\end{array};& \mathrm{equation}\left(35\right)\end{array}$

wherein, (X, Y, Z)^T is a 3D rectangular coordinate vector of the missile in the body-fixed coordinate system, and r=R=√{square root over (X²+Y²+Z²)}. With reference to Article 6 (Balmino G, Barriot J P, Valès N. Non-singular formulation of the gravity vector and gravity gradient tensor in spherical harmonics [J]. Manuscr Geod, 1990, 15.), the elements composing the above matrices or vectors are expressed by:

$\begin{array}{cc}\{\begin{array}{c}\mathrm{let}{\theta }_1=\frac{X}{r};{\theta }_2=\frac{Y}{r};{\theta }_3=\frac{Z}{r}=\zeta \\ {\stackrel{\_}{P}}_2\left(\zeta \right)=\sqrt{5}\times \frac{\left(3{\zeta }^2-1\right)}{2};{\stackrel{\_}{P}}_2^{\prime }\left(\zeta \right)=3\sqrt{5}\zeta ;{\stackrel{\_}{P}}_2^{\mathrm{\prime \prime }}\left(\zeta \right)=3\sqrt{5}\\ D_r=-3{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2\left(\zeta \right);D_3={{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2^{\prime }\left(\zeta \right)\\ S_{rr}=12{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2\left(\zeta \right);S_{r3}=-3{{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2^{\prime }\left(\zeta \right);S_{33}={{\stackrel{\_}{C}}_{20}\left(\frac{a_e}{r}\right)}^2{\stackrel{\_}{P}}_2^{\mathrm{\prime \prime }}\left(\zeta \right)\\ \tilde{\varphi }=D_r-D_3{\theta }_3\\ \frac{\partial U_1}{\partial X}=\frac{\mu }{r^2}\tilde{\varphi }{\theta }_1;\frac{\partial U_1}{\partial Y}=\frac{\mu }{r^2}\tilde{\varphi }{\theta }_2;\frac{\partial U_1}{\partial Z}=\frac{\mu }{r^2}\left(D_3+\tilde{\varphi }{\theta }_3\right)\\ \frac{{\partial }^2{U}_1}{\partial X^2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1^2\right\}\\ \frac{{\partial }^2{U}_1}{\partial X\partial Y}=\frac{\mu }{r^3}\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1{\theta }_2\\ \frac{{\partial }^2{U}_1}{\partial X\partial Z}=\frac{\mu }{r^3}\left\{\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_1+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_1{\theta }_3\right\}\\ \frac{{\partial }^2{U}_1}{\partial Y^2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_2^2\right\}\\ \frac{{\partial }^2{U}_1}{\partial Y\partial Z}=\frac{\mu }{r^3}\left\{\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_2+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_2{\theta }_3\right\}\\ \frac{{\partial }^2{U}_1}{\partial Z^2}=\frac{\mu }{r^3}\left\{\tilde{\varphi }+S_{33}+2\left(S_{r3}-D_3-S_{33}{\theta }_3\right){\theta }_3+\left[S_{rr}+2\left(D_3-S_{r3}\right){\theta }_3-\tilde{\varphi }+S_{33}{\theta }_3^2\right]{\theta }_3^2\right\}\end{array};& \mathrm{equation}\left(36\right)\end{array}$

wherein, C₂₀ is a normalized spherical harmonic coefficient corresponding to the J₂ perturbation in an expansion of a spherical harmonic series of the earth's gravitational potential.

2. The position vector difference Δ{right arrow over (R)}_BB* between the target point B and the target point B* obtained by the perturbation extrapolation based on the reference solutions {right arrow over ({dot over (r)})}_A⁰ and T⁰ in the body-fixed coordinate system is computed according to equations (37) and (38):

$\begin{array}{cc}\Delta {\overrightarrow{R}}_{{\mathrm{BB}}^{*}}={\overrightarrow{R}}_B-{\overrightarrow{R}}_{B^{*}}=\left[\begin{array}{c}{X}_B\\ Y_B\\ Z_B\end{array}\right]-\left[\begin{array}{c}{X}_{B^{*}}\\ Y_{B^{*}}\\ Z_{B^{*}}\end{array}\right];& \mathrm{equation}\left(37\right)\end{array}$

wherein, the coordinate vector {right arrow over (R)}_B*({right arrow over (X)}_B*, {right arrow over (Y)}_B*, {right arrow over (Z)}_B*) of B* in the body-fixed coordinate system is obtained from {right arrow over (r)}_B* through a coordinate transformation:

$\begin{array}{cc}{\overrightarrow{R}}_{B^{*}}=M^T\left(t_0+T^0\right){\overrightarrow{r}}_{B^{*}}.& \mathrm{equation}\left(38\right)\end{array}$

3. It is judged whether Δ{right arrow over (R)}_BB* is less than the given threshold S_R; if yes, the designed parameters v_p, v_r, Ã, T, σ of the missile are recomputed and output based on the reference solutions and corresponding equations in steps 3 and 4, and the correction is ended; otherwise, proceeding to step 6 to perform a differential correction in order to estimate a launch velocity vector increment and a flight time increment.

Step 6: the constraint equation with the constant launch angle and a differential equation based on a position error propagation of the target point are established; the launch velocity vector increment Δ{right arrow over ({dot over (r)})}_A and the flight time increment ΔT are computed, and corrected solutions are denoted as and {circumflex over (T)}; the corrected solutions are taken as reference solutions for a next differential correction, that is, let: {right arrow over ({dot over (r)})}_A⁰=, T⁰={circumflex over (T)}, and repeat the sub-steps of step 5.

Step 6 includes the following sub-steps:

1. A system of linear differential equations of the launch velocity vector increment Δ{right arrow over ({dot over (r)})}_A and the flight time increment ΔT are established according to equations (39) to (41):

$\begin{array}{cc}\left[\begin{array}{cc}G& 0\\ C& D\end{array}\right]\left[\begin{array}{c}\Delta {\stackrel{.}{\overrightarrow{r}}}_A\\ \Delta T\end{array}\right]=\left[\begin{array}{c}0\\ \Delta {\overrightarrow{R}}_{BB}*\end{array}\right];& \mathrm{equation}\left(39\right)\end{array}$

wherein, Δ{right arrow over ({dot over (r)})}_A(Δ{right arrow over ({dot over (x)})}_A, Δ{right arrow over ({dot over (y)})}_A, Δ{right arrow over (ż)}_A) and Δ{right arrow over (R)}_BB*(Δ{right arrow over (x)}_BB*, Δ{right arrow over (y)}_BB*, Δ{right arrow over (z)}_BB*) are expressed in terms of components as:

$\begin{array}{cc}\left[\begin{array}{cc}G& 0\\ C& D\end{array}\right]\left[\begin{array}{c}\Delta {\stackrel{.}{\overrightarrow{x}}}_A\\ \Delta {\stackrel{.}{\overrightarrow{y}}}_A\\ \Delta {\stackrel{.}{\overrightarrow{z}}}_A\\ \Delta T\end{array}\right]=\left[\begin{array}{c}0\\ \Delta {\overrightarrow{x}}_{{\mathrm{BB}}^{*}}\\ \Delta {\overrightarrow{y}}_{{\mathrm{BB}}^{*}}\\ \Delta {\overrightarrow{z}}_{{\mathrm{BB}}^{*}}\end{array}\right];& \mathrm{equation}\left(40\right)\end{array}$

wherein, G is a 1×3 matrix, C is a 3×3 matrix, D is a 3×1 matrix, which are expressed as:

$\begin{array}{cc}\{\begin{array}{c}G=\left[-\sin h\cos A^{*}\sin h\sin A^{*}\cos h\right]QW{M}^T\left(t_0\right)\\ C=M^T\left(t_0+T^0\right)\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\stackrel{.}{\overrightarrow{r}}}_{A^{*}}}\\ D=M^T\left(t_0+T^0\right)\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}+{\stackrel{.}{\theta }}_g\left[\begin{array}{c}{Y}_{B^{*}}\\ -X_{B^{*}}\\ 0\end{array}\right]\end{array};& \mathrm{equation}\left(41\right)\end{array}$

wherein,

$\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial {\stackrel{.}{\overrightarrow{r}}}_A}\mathrm{and}\frac{\partial {\overrightarrow{r}}_{B^{*}}}{\partial T}$

are computed by the numerical integration in step 5. The matrix G generates the constraint equation with the constant launch angle.

2. The system of linear differential equations are solved to obtain a set of unique solutions of Δ{right arrow over ({dot over (r)})}_A and ΔT;

the system of linear equations (40) includes four equations and four unknown quantities so that a set of unique solutions of Δ{right arrow over ({dot over (r)})}_A and ΔT are obtained, and then the corrected solutions and {circumflex over (T)} are obtained according to equation (42):

$\begin{array}{cc}\{\begin{array}{c}={\stackrel{.}{\overrightarrow{r}}}_A^0+\Delta {\stackrel{.}{\overrightarrow{r}}}_A\\ \hat{T}=T^0+\Delta T\end{array}.& \mathrm{equation}\left(42\right)\end{array}$

3. The corrected solutions are taken as the reference solutions for the next differential correction, that is, let {right arrow over ({dot over (r)})}_A⁰=, T⁰={circumflex over (T)}, and repeat the sub-steps of step 5.

III. Results of Constructing the Trajectory

According to the method for constructing the free trajectory, when the geodetic coordinates of the launch point and the target point and the launch angle are given, the launch velocity vector and flight time of the trajectory can be derived by the two-body force model and the dynamic model including the J2 perturbation, respectively. Conversely, when the coordinates of the launch point and the launch velocity vector are known, an integration is performed over the length of the flight time by the corresponding force model. The integral of the geocentric radius vector should coincide with the coordinates of the target point, and the integral coordinates in the integration range forms the trajectory of the missile.

According to the known data of Embodiment 1, the launch angle is traversed from 1° by an increment of 1° to construct 46 positive-flying trajectories (at launch angle in the range of 1-46°) and 23 negative-flying trajectories (at launch angle in the range of 1-23°). If the launch angle falls outside the above ranges, then it is impossible to obtain a rationally designed trajectory. FIG. 12 illustrates all positive-flying free trajectories (left) and negative-flying free trajectories (right) constructed by the two-body force model. Different gray values represent different launch angles. The impact points of the trajectories are all converged at the target point B, so are the impact points of highly-eccentric orbits (the eccentricity of an elliptical orbit at a launch angle of 46° is 0.77), which verifies the correctness of the method for constructing the trajectory of the present invention. Each successfully constructed trajectory corresponds to a set of output parameters, including v_p, v_r, Ã, T, σ. For example, when the specified launch angle is 10°, the output parameters of the positive-flying and negative-flying trajectories constructed according to Embodiment 1 are shown in Table 3. The comparison of the data in Table 3 indicates that when the launch angle is constant, the launch velocity of the negative-flying trajectory is much higher than that of the positive-flying trajectory, which is matched with the fact that a longer-range trajectory requires more launch energy. Therefore, the minimum-energy trajectory from the launch point to the target point must be a shorter-range positive-flying trajectory. By traversing the launch angle, the method for constructing the trajectory of the present invention is capable of determining not only a classic minimum-energy trajectory (with the smallest launch velocity in the inertial space) but also a practical minimum-energy trajectory (with the smallest launch velocity in the body-fixed coordinate system) among numerous successfully constructed positive-flying trajectories. FIGS. 13 and 14 respectively illustrate the functions between the launch angle and the launch velocity of the positive-flying and negative-flying trajectories constructed by means of traversing according to Embodiment 1, in which the positive-flying trajectory with the smallest launch velocity corresponding to a launch angle of 14° can be identified visibly, while the launch velocity of the negative-flying trajectory is a monotonically increasing function of the launch angle without a minimum value. When the primary concern for trajectory selection is not the minimum energy but the velocity at the impact point or the ground track, the ground track needs to be superimposed on the map data for optimization, as shown in FIG. 15, wherein the velocity at the impact point is directly proportional to the launch angle. In either case, the method for constructing the trajectory of the present invention can provide a wealth of alternative trajectories.

When the construction of the trajectories requires a precision of the order of kilometers or higher, the method for constructing the trajectory based on the two-body force model cannot meet such requirements. To this end, additional considerations must be given to the effects of perturbations such as the earth's non-central gravity, atmospheric drag and solar pressure. Having taken into account the J₂ perturbation of the earth's gravitational field, the method for precisely constructing the trajectories of the present invention only needs to append the perturbations such as the atmospheric drag and solar pressure into its framework when necessary. The ballistic parameters derived by the two-body force model, as shown in Table 3, are taken as reference solutions to perform a differential correction for the position error propagation of the target point. The J₂ perturbation is considered in the position propagation of the target point. The results of the differential correction are shown in Table 4, in which the symbol “↑” or “↓” represents an increase or decrease in the ballistic parameter after being corrected.

According to the known data, Embodiment 2 successfully constructs 54 positive-flying trajectories and 32 negative-flying trajectories by using the same traversal method as in Embodiment 1. FIG. 16 illustrates the 54 positive-flying trajectories. Since the launch point is located at the north pole, its longitude is indefinite and constitutes a singularity in Method 1. In this case, the method for constructing the trajectories of the present invention can still successfully construct and characterize the trajectories in the 3D rectangular coordinate system, as shown in FIG. 16, in which the launch points appear as discrete points, and the discrete points all have the latitude of 90° and thus are actually the same point in the spherical coordinate system. The above two embodiments both illustrate the feasibility and correctness of this method.

TABLE 3
Output parameters of positive-flying and negative-flying trajectories
constructed based on the two-body force model in Embodiment 1
(h = 10°; t0 = 56163.57313347)
Positive-flying	vp (m/s)	7610.599	σ	a (m)	6026011.677
	vr (m/s)	7691.680		i (°)	79.4004
	Ã (°)	3.346		Ω (°)	164.8504
	T (min)	35.45		ξ	−0.1828677
				η	0.0002563
				λ (°)	277.8481
Negative-flying	vp (m/s)	8520.429	σ	a (m)	7309475.919
	vr (m/s)	8412.881		i (°)	103.3056
	Ã (°)	−167.691		Ω (°)	350.2673
	T (min)	78.82		ξ	−0.2153108
				η	−0.0003193
				λ (°)	223.0456

TABLE 4
Output parameters of positive-flying and negative-flying trajectories
constructed when the J2 perturbation is considered in Embodiment 1
(h = 10°; t0 = 56163.57313347)
Positive-	vp	(m/s)	7605.442	σ	a	(m)	6018859.853
flying			↓ 5.157				↓ 7151.82
	vr	(m/s)	7686.569		i	(°)	79.3943
			↓ 5.111				↓ 0.0061
	Ã	(°)	3.358		Ω	(°)	164.8623
			↑ 0.012				↑ 0.0119
	T	(min)	35.44		ξ	−0.1832377
			↓ 0.01				↓ 0.00037
					η	−0.0009288
							↓ 0.00119
					λ	(°)	277.8532
							↑ 0.0051
Negative-	vp	(m/s)	8517.134	σ	a	(m)	7302264.995
flying			↓ 3.295				↓ 7210.92
	vr	(m/s)	8409.680		i	(°)	103.2933
			↓ 3.201				↓ 0.0123
	Ã	(°)	−167.718		Ω	(°)	350.2419
			↓ 0.027				↓ 0.0254
	T	(min)	78.95		ξ	−0.2148064
			↑ 0.13			↑ 0.000504
					η	−0.0010205
							↓ 0.0007
					λ	(°)	223.0432
							↓ 0.0024

IV. Definition of the Symbols and Variables Involved

	Symbol	Definition	Explanation
Known	t0	launch epoch
quantities	h	launch angle (or specified launch angle,	specified value
		whose value can be specified as any real	0 < h < 90°
		number within an allowable range)
	LA, BA, HA	geodetic longitude, latitude and height of	when the point is
		the launch point A	located at the north or
	LB, BB, HB	geodetic longitude, latitude and height of	south pole, the
		the target point B	longitude is assigned
			any value in the range
			of 0-360°
Constants	{right arrow over (R)}A(XA, YA, ZA)	rectangular coordinate vector of the point A
during		in the body-fixed coordinate system
iteration	{right arrow over (R)}B(XB, YB, ZB)	rectangular coordinate vector of the point B
		in the body-fixed coordinate system
	{right arrow over (r)}A(xA, yA, zA)	rectangular coordinate vector of the point A
		in the TEMEE coordinate system
	ε	difference between the ratio of the modulus
		of the geocentric radius vector of the point A
		to the modulus of the geocentric radius
		vector of the point B and 1
	q	zenith angle of the target point in the
		conventional horizontal coordinate system
		of the launch point
	ϕ	horizontal angle between the AB direction
		and the north-pole direction of the point A
		when the point A is not at or above the north
		or south pole
	Δφ	difference between the geodetic latitude and
		the geocentric latitude of the point A
Variables	T	flight time	with an initial value of
during			0
iteration	A*	launch azimuth in the target-pointing	with an initial value of
		horizontal coordinate system	0
	β	half-range angle
	{right arrow over (r)}B(xB, yB, zB)	rectangular coordinate vector of the point B
		in the TEMEE coordinate system
	A ({dot over (X)}A, {dot over (Y)}A, ŻA)	rectangular coordinate vector of the launch
		velocity in the body-fixed coordinate
		system
	A ({dot over (x)}A, {dot over (y)}A, żA)	rectangular coordinate vector of the launch
		velocity in the TEMEE coordinate system
	vp	modulus of the launch velocity vector in the	vp = |   A|
		body-fixed coordinate system
	vr	modulus of the launch velocity vector in the	vr = |   A|
		TEMEE coordinate system
	σ	first type of non-singular orbital elements at	6 components,
		the launch epoch	including
			a, i, Ω, ξ, η, λ
	ω	argument of perigee of the orbit
	Δω	bias of the argument of perigee
	fA fB	true anomalies of the points A and B on the
		elliptical trajectory/orbit
	uA uB	true arguments of latitude of the points A
		and B on the elliptical trajectory/orbit
	α	intersection angle between the launch
		velocity vector and the geocentric radius
		vector of the launch point in the body-fixed
		coordinate system
	θ	intersection angle between the launch
		velocity vector and the geocentric radius
		vector of the launch point in the TEMEE
		coordinate system
	A*(X*, Y*, Z*)	launch velocity vector in the target-pointing
		horizontal coordinate system
	Ã	declination of the launch velocity vector in	west declination:
		the target-pointing horizontal coordinate	negative; east
		system relative to the target point in the	declination: positive
		horizontal plane
Output	vp, vr, Ã, T, σ	the definition of each
quantities		symbol is given above

In addition, the present invention involves the following geophysical constants: the equatorial radius of the reference ellipsoid α_e=6378136 m, the geocentric gravitational constant μ=0.39860043770442×10¹⁵ m³/s², the oblateness of the earth f=1/298.25781, and the eccentricity of the meridian of the earth e_c²=2f−f²=0.0066946.

Claims

1. A method for constructing a free trajectory of a ballistic missile at a specified launch angle based on a two-body force model, comprising: step 1: data preprocessing: sequentially computing a body-fixed rectangular coordinate vector {right arrow over (R)}A of a launch point A and a body-fixed rectangular coordinate vector {right arrow over (R)}B of a target point B, a difference Δ100 between a geodetic latitude and a geocentric latitude of the launch point A, a horizontal angle ϕ between an AB direction and a north-pole direction of the launch point A in a body-fixed coordinate system, and a difference ε between a ratio of a modulus of geocentric radius vector of the launch point A to a modulus of a geocentric radius vector of the target point B and 1;step 2: setting of an initial state of iterations: assuming that the earth does not rotate and a flight time T is zero;step 3: computing, in the two-body force model, a launch velocity vector {right arrow over ({dot over (r)})}A of a positive-flying trajectory or a negative-flying trajectory in a True Equator Mean Equinox of Epoch (TEMEE) coordinate system and a launch velocity vector {right arrow over ({dot over (R)})}A of the positive-flying trajectory or the negative-flying trajectory in the body-fixed coordinate system, a first type of non-singular orbital elements σ of the ballistic missile at a launch epoch, and the flight time T to generate a new flight time T*; andstep 4: letting T=T*, repeating step 3 to iteratively compute the launch velocity vector {right arrow over ({dot over (r)})}A and the launch velocity vector {right arrow over ({dot over (R)})}A of the ballistic missile, the first type of non-singular orbital elements of the ballistic missile at the launch epoch, a half-range angle and the flight time until |T−T*| is less than a set threshold St to complete the iterations to finally obtain the launch velocity vector {right arrow over ({dot over (r)})}A and the flight time T of the ballistic missile in the two-body force model, and outputting designed parameters vp, vr, Ã, T, σ, wherein vp is a modulus of the launch velocity vector in the body-fixed coordinate system, vr is a modulus of the launch velocity vector in the TEMEE coordinate system, and à is a declination of a launch velocity vector in a target-pointing horizontal coordinate system relative to the target point B in a horizontal plane.

2. A method for constructing a free trajectory of a ballistic missile at a specified launch angle considering both a central gravity and a J2 perturbation of the earth based on a result of a two-body force model, comprising: step 1: data preprocessing: sequentially computing a body-fixed rectangular coordinate vector {right arrow over (R)}A of a launch point A and a body-fixed rectangular coordinate vector {right arrow over (R)}B of a target point B, a difference Δφ between a geodetic latitude and a geocentric latitude of the launch point A, a horizontal angle ϕ between an AB direction and a north-pole direction of the launch point A in a body-fixed coordinate system, and a difference ε between a ratio of a modulus of the geocentric radius vector of the launch point A to a modulus of the geocentric radius vector of the target point B and 1;step 2: setting of an initial state of iterations: assuming that the earth does not rotate and a flight time T is zero;step 3: computing, in the two-body force model, a launch velocity vector {right arrow over ({dot over (r)})}A of a positive-flying trajectory or a negative-flying trajectory in a True Equator Mean Equinox of Epoch (TEMEE) coordinate system and a launch velocity vector {right arrow over ({dot over (R)})}A of the positive-flying trajectory or the negative-flying trajectory in the body-fixed coordinate system, a first type of non-singular orbital elements σ of the ballistic missile at a launch epoch and the flight time T to generate a new flight time T*; andstep 4: letting T=T*, repeating step 3 to iteratively compute the launch velocity vector of the ballistic missile, the first type of non-singular orbital elements of the ballistic missile at the launch epoch, a half-range angle and the flight time until |T−T*| is less than a set threshold St to complete the iterations to finally obtain the launch velocity vector {right arrow over ({dot over (r)})}A and the flight time T of the ballistic missile in the two-body force model;step 5: taking the launch velocity vector {right arrow over ({dot over (r)})}A the flight time T obtained by the two-body force model as reference solutions {right arrow over ({dot over (r)})}A0 and T0 for a differential correction; if a distance |Δ{right arrow over (R)}BB*| between the target point B and a target point B* obtained by a perturbation extrapolation based on the reference solutions is less than a given threshold SR, ending a correction of designed parameters of the ballistic missile and outputting corrected parameters vp, vr, Ã, T, σ, wherein vp is a modulus of the launch velocity vector in the body-fixed coordinate system, vr is a modulus of the launch velocity vector in the TEMEE coordinate system, and à is a declination of a launch velocity vector in a target-pointing horizontal coordinate system relative to the target point B in a horizontal plane; otherwise, proceeding to step 6; andstep 6: establishing a constraint equation with a constant launch angle and a differential equation based on a position error propagation of the target point B; computing a launch velocity vector increment Δ{right arrow over ({dot over (r)})}A and a flight time increment ΔT, and denoting corrected solutions as and {circumflex over (T)}; taking the corrected solutions as reference solutions for a next differential correction by letting {right arrow over ({dot over (r)})}A0=, T0={circumflex over (T)} and repeating step 5.

3. The method according to claim 1, wherein step 1 specifically comprises: 3.1: transforming geodetic coordinates of the launch point A and the target point B from spherical coordinates (L, B, H) to 3D rectangular coordinates (X, Y, Z) according to equation (1), and expressing the 3D rectangular coordinates by {right arrow over (R)}A and {right arrow over (R)}B as;

4. The method according to claim 1, wherein step 3 specifically comprises: 4.1: computing coordinate vectors {right arrow over (r)}A and {right arrow over (r)}B of the launch point A and the target point B in the TEMEE coordinate system in combination with the flight time T according to equation (7):

5. The method according to claim 1, wherein step 4 specifically comprises: 5.1: judging whether |T−T*|<St is satisfied; if yes, letting T=T*, and proceeding to a next step; otherwise, letting T=T*, and returning to step 3;5.2: computing the declination à of the launch velocity vector in the target-pointing horizontal coordinate system relative to the target point B in the horizontal plane according to equation (28):

6. The method according to claim 2, wherein step 5 specifically comprises: 6.1: denoting the target point obtained by the perturbation extrapolation based on the reference solutions {right arrow over ({dot over (r)})}A0 and T0 as B*, and computing partial derivative matrices

7. The method according to claim 2, wherein step 6 specifically comprises: 7.1: establishing a system of linear differential equations of the launch velocity vector increment Δ{right arrow over ({dot over (r)})}A and the flight time increment ΔT according to equations (39) to (41):

8. The method according to claim 2, wherein step 1 specifically comprises: 3.1: transforming geodetic coordinates of the launch point A and the target point B from spherical coordinates (L, B, H) to 3D rectangular coordinates (X, Y, Z) according to equation (1), and expressing the 3D rectangular coordinates by {right arrow over (R)}A and {right arrow over (R)}B as;

9. The method according to claim 2, wherein step 3 specifically comprises: 4.1: computing coordinate vectors {right arrow over (r)}A and {right arrow over (r)}B of the launch point A and the target point B in the TEMEE coordinate system in combination with the flight time T according to equation (7):

10. The method according to claim 2, wherein step 4 specifically comprises: 5.1: judging whether |T−T*|<St is satisfied; if yes, letting T=T*, and proceeding to a next step; otherwise, letting T=T*, and returning to step 3;5.2: computing the declination à of the launch velocity vector in the target-pointing horizontal coordinate system relative to the target point B in the horizontal plane according to equation (28):