An Optimal Single-Pulse Interception Method for Time-Free Spacecraft

Abstract

The present invention discloses an optimal single-pulse interception method for a time-free spacecraft, which belongs to the field of aerospace technology. The implementation method of the present invention is as follows: establish a geocentric equatorial inertial coordinate system with the center of the earth as the origin, and establish the dynamic equations and interception condition constraints of the interceptor and the target on the basis of the coordinate system; derive the first-order necessary conditions for the optimal single-pulse interception, and establish the optimal interception equation group; use the high-order Taylor expansion method to simplify the optimal interception equation group to obtain an analytical nonlinear equation group; use the Newton iteration method to solve the nonlinear equation group to obtain the optimal pulse, that is, under the condition of freedom at the interception time, the interception of the target in orbit is achieved by solving the optimal pulse of the interceptor. The present invention has the following advantages: (1) strong robustness and high reliability; (2) optimal maneuvering pulse; (3) no strict restrictions and constraints on the orbital height of the interceptor; (4) semi-analytical solution, efficient solution and interception.

Description

Technical Field

[0001] The present invention relates to an optimal single-pulse interception method for a time-free spacecraft, in particular to an optimal single-pulse interception method for a spacecraft in a near-earth circular orbit under time-free conditions, and belongs to the field of aerospace technology. Background Art

[0002] With the development of aerospace technology in various countries, the exploration of unknown space and maintaining space technology advantages have become strategic goals of major countries. Space attack and defense is becoming a research hotspot involving the space field. In the field of space attack and defense, an important issue is the problem of spacecraft interception. The problem of spacecraft interception is a typical space maneuvering problem. When the interceptor engages with an exoatmospheric target, it should maneuver to approach the target so that its position vector is equal at the moment of interception. According to the maneuvering mode, the interception problem can be divided into two types of problems: pulse interception and limited thrust interception. For the limited thrust interception problem, in order to simplify the analysis, when the duration of continuous thrust is much shorter than the interval between thrusts or the non-maneuvering time of the spacecraft, the continuous thrust can be approximated by pulse thrust. Therefore, it is very necessary to study the problem of pulse interception. In addition, according to the number of pulses applied, pulse interception can be divided into single pulse and multi-pulse interception. Although the energy considered in multi-pulse interception is usually less than that in single pulse interception, considering the efficient combat requirements of spacecraft, compared with multi-pulse interception, single pulse interception can reduce the subsequent orbital support cost required for combat and improve reliability. In addition, compared to a fixed interception moment, the optimal single pulse problem considering a free interception moment will reduce the pulse size, thereby reducing fuel consumption, allowing the spacecraft to carry more payloads and more efficiently damage the target. However, the interception moment becomes unknown, which adds a degree of freedom in the solution and may reduce the efficiency of the solution. Therefore, the time-free optimal single pulse interception method for spacecraft proposed in this patent can not only complete the solution of the time-free optimal single pulse interception problem, but also obtain a semi-analytical solution through simplification, which significantly improves the calculation efficiency and can meet more spacecraft interception mission scenarios.

[0003] In the prior art of the developed single-pulse optimal pulse interception method for spacecraft [1] (see: Meng Shaofei, Liu Xinxue, Fu Dan, et al. Iterative algorithm for single-pulse most energy-saving interception trajectory [J]. Systems Engineering and Electronics, 2016, 38(12): 2821-2826.), the problem of single-pulse single-circle most energy-saving interception under the condition that the interception starting point is fixed but the interception end point is not fixed in orbit is considered. From the perspective of the changing law of the transfer velocity increment, a numerical iterative algorithm for the energy-saving interception trajectory is designed. This not only improves the speed and reliability of the algorithm, but also avoids falling into the problem of local optimality. However, multiple layers of iteration will increase the computational complexity and solution time.

[0004] The single pulse interception problem of a time-free spacecraft is essentially a nonlinear and non-convex problem, so there is no analytical solution. If the first-order necessary conditions for the optimal single pulse interception problem can be derived and the semi-analytical solution of the above problem can be obtained by appropriate simplification, the speed of numerical iteration can be improved, thereby significantly expanding the application scenarios of this problem. Summary of the invention

[0005] The technical problem to be solved by the optimal single-pulse interception method for a spacecraft with time freedom disclosed in the present invention is to intercept an on-orbit target by solving the optimal pulse of the interceptor under the condition of freedom at the interception moment; the method has the following advantages: (1) strong robustness and high reliability; (2) optimal maneuvering pulse; (3) no strict restrictions and constraints on the orbital altitude of the interceptor; and (4) semi-analytical solution with efficient solution.

[0006] The objective of the present invention is achieved through the following technical solutions:

[0007] The present invention discloses a method for optimal single-pulse interception of a time-free spacecraft, which is a semi-analytical method. First, a geocentric equatorial inertial coordinate system with the center of the earth as the origin is established, and the dynamic equations and interception condition constraints of the interceptor and the target are established on the basis of the coordinate system; then, the first-order necessary conditions for the optimal single-pulse interception are derived, and the optimal interception equation group is established; the optimal interception equation group is simplified using a high-order Taylor expansion method to obtain an analytical nonlinear equation group. Finally, the nonlinear equation group is solved using the Newton iteration method to obtain the optimal pulse, that is, under the condition of freedom at the interception time, the interception of the target in orbit is achieved by solving the optimal pulse of the interceptor.

[0008] The present invention discloses an optimal single pulse interception method for a time-free spacecraft, establishes the dynamic equations of the interceptor and the target, derives the first-order necessary conditions for the optimal single pulse interception problem, establishes the optimal interception equation group, and uses the high-order Taylor expansion method to obtain an analytical nonlinear equation group. The Newton iteration method is used to solve the equation group, which has high solving efficiency and obvious advantages.

[0009] The present invention discloses an optimal single-pulse interception method for a time-free spacecraft, comprising the following steps:

[0010] Step 1: Establish a geocentric equatorial inertial coordinate system.

[0011] The center of the earth is selected as the coordinate origin O to establish the geocentric equatorial inertial coordinate system. The xy plane coincides with the earth's equatorial plane. The x-axis points to the vernal equinox, and the z-axis is along the earth's rotation axis (i.e., perpendicular to the equatorial plane) and points to the North Pole. The y-axis, the x-axis, and the z-axis form a right-handed rectangular coordinate system, thus completing the establishment of the geocentric equatorial inertial coordinate system.

[0012] Step 2: Establish the dynamic equations of the interceptor and the target and the interception condition constraints in the geocentric equatorial inertial coordinate system.

[0013] Step 2.1: Establish the dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system;

[0014] The dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system are expressed as:

[0015]

[0016] where μ represents the Earth's gravitational constant, subscript p represents the interceptor, subscript e represents the target, and t represents the flight time. p (t) = [x p (t)y p (t)z p (t)] T and v p (t) = [v x,p (t)v y,p (t)v z,p (t)] T They represent the position coordinates and velocity coordinates of the interceptor in the Earth-centered equatorial inertial coordinate system oxyz, r p (t)=||r p (t)|| is the modulus of the interceptor position vector. Similarly, r e (t) = [x e (t)y e (t)z e (t)] T and v e (t) = [v x,e (t)v y,e (t)v z,e (t)] T They represent the position coordinates and velocity coordinates of the target in the geocentric equatorial inertial coordinate system oxyz, r e (t)=||r e (t)|| is the modulus of the target position vector. t0 represents the initial time, which can usually be considered as t0 = 0, r p0 and v p0 represents the position and velocity of the interceptor at the initial moment, and r e0 and v e0 Represents the position and velocity of the target at the initial moment. Δv p Represents the initial velocity increment of the interceptor, which is an unknown quantity to be determined.

[0017] Step 2.2: Establish interception condition constraints.

[0018] Consider that the interceptor and the target meet the interception condition at the terminal moment, that is, at the interception time t f The interceptor and target have the same position vector

[0019] r p (t f )=r e (t f ) (2)

[0020] Interception time t f is the variable that needs to be optimized.

[0021] Step 3: Derive the first-order necessary conditions for optimal single-pulse interception and establish the optimal interception equations.

[0022] Step 3.1: Derive the first-order necessary conditions for optimal single-shot interception;

[0023] Using the dynamics in formula (1) to recursively obtain:

[0024]

[0025] in The terminal position of the interceptor is the interception time t f With the velocity increment Δv p The function of The terminal position of the target is only the interception time t f There is no analytical expression for either function. Note is a set of real numbers. Then using the intercept condition in the constraint of equation (2), we get:

[0026]

[0027] Formula (4) shows that the velocity increment Δv p and interception time t f Satisfy the equality constraint. In fact, when the interception time t is given f , the intercept problem is a Lambert problem, which can be solved directly using the Lambert problem, that is, there is a velocity increment Δv p and interception time t f Corresponding; let v p1 =Δv p +v p0 , then the condition satisfied by the optimal speed increment is:

[0028]

[0029] right Continue the derivation, first take the variation of both ends of equation (4), and get:

[0030]

[0031] Further, we get:

[0032]

[0033] The two ends of formula (7) are multiplied by Δv p , the necessary condition for the optimal speed increment is:

[0034]

[0035] Step 3.2: Establish the optimal interception equations;

[0036] Combining equation (4) and equation (8), we get the optimal pulse interception equations as follows:

[0037]

[0038] Among them, the unknown quantity is The number of equations is 4, and the equations are closed. Solving the above optimal intercept equations can obtain the optimal speed increment.

[0039] Step 4: Simplify the optimal pulse interception equations to obtain the analytical nonlinear equations.

[0040] At the initial time t0=0, the position vector r of the interceptor p (t) is subjected to Taylor expansion, and we get:

[0041]

[0042] Take the first five terms of the sum of the series in formula (10):

[0043]

[0044] Formula (11) can be expressed in matrix form:

[0045] r p (t) = Φ p τ p (12)

[0046] Among them, Φ p and τ p It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows:

[0047]

[0048] τ p =[1 tt 2 t 3 t 4 ] T (14)

[0049]

[0050] At the interception time t f When , we can get from formula (12):

[0051] r p (t f )=Φ p τ pf (16)

[0052] Among them, τ pf The expression is as follows:

[0053] τ pf =[1 t f t f 2 t f 3 t f 4 ] T (17)

[0054] According to formula (8), two partial derivative calculation formulas are derived:

[0055]

[0056]

[0057] Among them, λ p , Π p , Λ pi ,i=1,2,..,5 are intermediate variables introduced to simplify the expression. The expression is as follows:

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066] Similarly, according to equations (10)-(23), the analytical approximate expression of the terminal position of the target is derived. At the initial time t0=0, the position vector r of the target ise (t) is subjected to Taylor expansion, and we get:

[0067]

[0068] Take the first five terms of the series sum shown in formula (24):

[0069]

[0070] Formula (25) can be expressed in matrix form:

[0071] r e (t) = Φ e τ e (26)

[0072] Among them, Φ e and τ e It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows:

[0073]

[0074] τ e =[1 tt 2 t 3 t 4 ] T (28)

[0075]

[0076] At the interception time t f When , we can get from formula (26):

[0077] r e (t f )=Φ e τ ef (30)

[0078] Among them, τ ef The expression is as follows:

[0079] τ ef =[1 t f t f 2 t f 3 t f 4 ] T (31)

[0080] Similarly, the calculation formula of the partial derivative of the terminal position with respect to the interception time is derived:

[0081]

[0082] Among them, λ e It is an intermediate variable introduced to simplify the expression. Its expression is as follows:

[0083]

[0084] Using equations (18), (19) and (32), the necessary condition in equation (8) is expressed as:

[0085]

[0086] in

[0087]

[0088] In summary, the simplified optimal impulse nonlinear equations are as follows:

[0089]

[0090] Furthermore, (36) can be written as a general nonlinear system of equations:

[0091] F(x)=0 (37)

[0092] in The form of the above system of equations is completely analytical.

[0093] Step 5: Obtain the optimal pulse through Newton iteration method, that is, under the condition of freedom at the interception moment, intercept the target in orbit by solving the optimal pulse of the interceptor.

[0094] Given an initial value x (0) , iteratively update equation (38) until the convergence condition shown in equation (39) is satisfied

[0095] x (k+1) =x (k) -F′(x (k) ) -1 F(x (k) ),k=0,1,... (38)

[0096] ||x (k+1) -x (k) ||≤δ (39)

[0097] Where F′(x (k) ) -1 is the Jacobian matrix of the vector function F(x), and δ is the threshold set by the user. Finally, the optimal pulse Δv p and the corresponding interception time t f , that is, under the condition of freedom at the interception moment, the interception of the target in orbit is achieved by solving the optimal pulse of the interceptor.

[0098] Beneficial effects:

[0099] 1. The present invention discloses an optimal single-pulse interception method for a spacecraft with time freedom. Under the condition of freedom at the interception time, the interceptor can successfully intercept the target by using a single pulse maneuver, and has no strict requirements on its own orbital altitude, which has obvious advantages.

[0100] 2. The present invention discloses an optimal single-pulse interception method for a time-free spacecraft, which can solve the single-pulse interception problem of a time-free spacecraft. Compared with the interception problem with fixed time, it has higher flexibility, richer application scenarios and a wider range of applicability.

[0101] 3. The present invention discloses a method for optimal single-pulse interception of a time-free spacecraft, which derives the necessary conditions for the optimal single-pulse interception problem, thereby obtaining a solution that satisfies the optimality conditions. Under high-precision approximation, an approximate optimal solution to the original problem can be obtained, which is more fuel-efficient than the traditional method of giving a given interception time.

[0102] 4. The present invention discloses an optimal single-pulse interception method for a time-free spacecraft, which solves the optimal pulse nonlinear equations by Newton's iteration method. It is essentially a semi-analytical solution with high solving efficiency.

[0103] 5. The optimal single-pulse interception method for a time-free spacecraft disclosed in the present invention has strong robustness and high reliability because it applies a pulse at the initial moment and the optimal pulse interception equation group has universal applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 is the pulse increment Δv in step 3 of the present invention p and interception time t f Schematic diagram of the relationship;

[0105] Figure 2 It is a flow chart of an optimal single pulse interception method of a time-free space vehicle of the present invention;

[0106] Figure 3 is the flight trajectory of the interceptor and the target in this embodiment;

[0107] Figure 4 It is the curve of the position vector of the interceptor and the target changing with time in this embodiment. DETAILED DESCRIPTION

[0108] In order to better illustrate the purpose and advantages of the present invention, the present invention is explained in detail below by means of simulation analysis of an optimal single-pulse interception method for a time-free space vehicle.

[0109] Embodiment 1:

[0110] like Figure 2 As shown, the present embodiment discloses a time-free optimal single-pulse interception method for a spacecraft, comprising the following steps:

[0111] Step 1: Establish a geocentric equatorial inertial coordinate system.

[0112] The center of the earth is selected as the coordinate origin O to establish the geocentric equatorial inertial coordinate system. The xy plane coincides with the earth's equatorial plane. The x-axis points to the vernal equinox, and the z-axis is along the earth's rotation axis (i.e., perpendicular to the equatorial plane) and points to the North Pole. The y-axis, the x-axis, and the z-axis form a right-handed rectangular coordinate system, thus completing the establishment of the geocentric equatorial inertial coordinate system.

[0113] Step 2: Establish the dynamic equations of the interceptor and the target and the interception condition constraints in the geocentric equatorial inertial coordinate system.

[0114] Step 2.1: Establish the dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system;

[0115] The dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system are expressed as:

[0116]

[0117] where μ represents the Earth's gravitational constant, subscript p represents the interceptor, subscript e represents the target, and t represents the flight time. p (t) = [x p (t)y p (t)z p (t)] T and v p (t) = [v x,p (t)v y,p (t)v z,p (t)] T They represent the position coordinates and velocity coordinates of the interceptor in the Earth-centered equatorial inertial coordinate system oxyz, r p (t)=||r p (t)|| is the modulus of the interceptor position vector. Similarly, r e (t) = [x e (t)y e (t)z e (t)] T and v e (t) = [v x,e (t)v y,e (t)v z,e (t)] T They represent the position coordinates and velocity coordinates of the target in the geocentric equatorial inertial coordinate system oxyz, r e(t)=||r e (t)|| is the modulus of the target position vector. t0 represents the initial time, which can usually be considered as t0 = 0, r p0 and v p0 represents the position and velocity of the interceptor at the initial moment, and r e0 and v e0 Represents the position and velocity of the target at the initial moment. Δv p Represents the initial velocity increment of the interceptor, which is an unknown quantity to be determined.

[0118] Step 2.2: Establish interception condition constraints.

[0119] Consider that the interceptor and the target meet the interception condition at the terminal moment, that is, at the interception time t f The interceptor and target have the same position vector

[0120] r p (t f )=r e (t f ) (41)

[0121] Interception time t f is the variable that needs to be optimized.

[0122] Step 3: Derive the first-order necessary conditions for optimal single-pulse interception and establish the optimal interception equations.

[0123] Step 3.1: Derive the first-order necessary conditions for optimal single-shot interception;

[0124] Using the dynamics in equation (1) to recursively obtain:

[0125]

[0126] in Indicates that the terminal position of the interceptor is the interception time t f With the velocity increment Δv p The function of The terminal position of the target is only the interception time t f There is no analytical expression for either function. Note is a set of real numbers. Then using the intercept condition in the constraint of equation (41), we can get:

[0127]

[0128] Formula (43) shows that the velocity increment Δv p and interception time t f Satisfy the equality constraint. In fact, when the interception time t is given f, the intercept problem is a Lambert problem, which can be solved directly using the Lambert problem, that is, there is a velocity increment Δv p and interception time t f Corresponding, such as Figure 1 As shown; let v p1 =Δv p +v p0 , then the condition satisfied by the optimal speed increment is:

[0129]

[0130] right Continuing the derivation, first take the variation of both ends of equation (43), and we can get:

[0131]

[0132] Further, we get:

[0133]

[0134] Multiply both ends of formula (46) by Δv p , the necessary condition for the optimal speed increment is:

[0135]

[0136] Step 3.2: Establish the optimal interception equations;

[0137] Combining equation (43) and equation (47), we can get the optimal pulse interception equations as follows:

[0138]

[0139] Among them, the unknown quantity is The number of equations is 4, and the equations are closed. Solving the above optimal intercept equations can obtain the optimal speed increment.

[0140] Step 4: Simplify the optimal pulse interception equations to obtain the analytical nonlinear equations

[0141] At the initial time t0=0, the position vector r of the interceptor p (t) is subjected to Taylor expansion, and we can obtain:

[0142]

[0143] Take the first five terms of the sum of the series in formula (49):

[0144]

[0145] Formula (50) can be expressed in matrix form:

[0146] r p (t) = Φ p τ p (51)

[0147] Among them, Φ p and τ p It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows:

[0148]

[0149] τ p =[1 tt 2 t 3 t 4 ] T (53)

[0150]

[0151] At the interception time t f When , we can get from the formula (51):

[0152] r p (t f )=Φ p τ pf (55)

[0153] Among them, τ pf The expression is as follows:

[0154] τ pf =[1 t f t f 2 t f 3 t f 4 ] T (56)

[0155] According to formula (47), two partial derivative calculation formulas are derived:

[0156]

[0157]

[0158] Among them, λ p , Π p , Λ pi ,i=1,2,..,5 are intermediate variables introduced to simplify the expression. The expression is as follows:

[0159]

[0160]

[0161]

[0162]

[0163]

[0164]

[0165]

[0166]

[0167] Similarly, according to equations (49)-(62), the analytical approximate expression of the terminal position of the target is derived. At the initial time t0=0, the position vector r of the target is e (t) is subjected to Taylor expansion, and we can obtain:

[0168]

[0169] Take the first five terms of the series sum shown in equation (63):

[0170]

[0171] Express equation (64) in matrix form:

[0172] r e (t) = Φ e τ e (65)

[0173] Among them, Φ e and τ e It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows:

[0174]

[0175] τ e =[1 tt 2 t 3 t 4 ] T (67)

[0176]

[0177] At the interception time t f When , we can get from formula (65):

[0178] r e (t f )=Φ e τ ef (69)

[0179] Among them, τ ef The expression is as follows:

[0180] τ ef =[1 t f t f 2 t f 3 t f 4 ] T (70)

[0181] Similarly, the calculation formula of the partial derivative of the terminal position with respect to the interception time is derived:

[0182]

[0183] Among them, λ e It is an intermediate variable introduced to simplify the expression. Its expression is as follows:

[0184]

[0185] Using equations (57), (58) and (71), the necessary condition in equation (47) can be expressed as:

[0186]

[0187] in

[0188]

[0189] In summary, the simplified optimal impulse nonlinear equations are as follows:

[0190]

[0191] Furthermore, (75) can be written as a general nonlinear system of equations:

[0192] F(x)=0 (76)

[0193] in The form of the above system of equations is completely analytical.

[0194] Step 5: Use Newton iteration method to solve and obtain the optimal pulse.

[0195] Given an initial value x (0) , iteratively update equation (77) until the convergence condition shown in equation (78) is satisfied

[0196] x (k+1) =x (k) -F′(x (k) ) -1 F(x(k) ),k=0,1,... (77)

[0197] ||x (k+1) -x (k) ||≤δ (78)

[0198] Where F′(x (k) ) -1 is the Jacobian matrix of the vector function F(x), and δ is the threshold set by the user. Finally, the optimal pulse Δv p and the corresponding interception time t f .

[0199] In order to verify the feasibility of the method, the initial positions and velocities of the interceptor and the target are selected as shown in Table 1. Assume that the radius of the earth is 6378 km.

[0200] Table 1 Initial states of interceptor and target spacecraft

[0201] <![CDATA[Initial position vector r0 (m)]]> <![CDATA[Initial velocity vector v0 (m / s)]]> Interceptor <![CDATA[[-4752.6-4602.11467.1] T ×10 3 ]]> <![CDATA[[3.947-5.72-2.86] T ×10 3 ]]> Target <![CDATA[[-3268.1-6038.2-327.1] T ×10 3 ]]> <![CDATA[[-3.011.165.50] T ×10 3 ]]>

[0202] By using the Newton iteration method to solve the nonlinear equations in equation (76), the optimal pulse increment and the corresponding interception time can be obtained, thereby realizing the interception of the target by the interceptor. In order to verify the advantages of this method in terms of pulse optimality and computational efficiency during single pulse interception, a simulation comparison analysis is performed between this method and the fixed-time Lambert method.

[0203] Using the initial state information of the interceptor and the target in Table 1, directly solving the Lambert problem as a comparison method, given a flight time of 200s, the pulse increment is 601.2319m / s, while the optimal pulse solved by this method is 122.31m / s, which is only 20% of the comparison method. At the same time, the interception time of this method is 208.83s. The specific results can be seen in Table 2. This example shows the optimality of the pulse obtained by this method, and the solution time of this method is only 7 milliseconds, which reflects the computational efficiency of this method and has the potential to be applied in real-time application scenarios.

[0204] Table 2 Pulse increment and interception time of the two methods

[0205] <![CDATA[Interception moment t f (s)]]> <![CDATA[Pulse increment Δv p (m / s)]]> This method 208.83 122.31 Comparison Method 200 601.23

[0206] Furthermore, the flight trajectories of the interceptor and the target can be obtained by performing orbit recursion with the obtained optimal pulse. Figure 3 The flight trajectory of the interceptor and the target is shown, and finally the interceptor intercepts the target with the lowest pulse cost. Figure 4The curve of the position vector of the interceptor and the target changing with time is shown. At the end moment, the position vectors of the two coincide, further verifying the successful interception of the target by the interceptor.

[0207] The specific description above further illustrates in detail the purpose, technical solutions and beneficial effects of the invention. It should be understood that the above is only a specific embodiment of the present invention, which is used to explain the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. An optimal single-pulse interception method for a time-free spacecraft, characterized by: The following steps are included: Step 1: Establish the geocentric equatorial inertial coordinate system; The implementation method of step one is: The center of the earth is selected as the coordinate origin O to establish the geocentric equatorial inertial coordinate system. The xy plane coincides with the earth's equatorial plane. The x-axis points to the vernal equinox, and the z-axis is along the earth's rotation axis and points to the North Pole. The y-axis, the x-axis and the z-axis form a right-handed rectangular coordinate system, thus completing the establishment of the geocentric equatorial inertial coordinate system. Step 2: Establish the dynamic equations of the interceptor and the target and the interception condition constraints in the geocentric equatorial inertial coordinate system; The implementation method of step 2 is: Step 2.1: Establish the dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system; The dynamic equations of the interceptor and the target in the geocentric equatorial inertial coordinate system are expressed as: where μ represents the Earth's gravitational constant, subscript p represents the interceptor, subscript e represents the target, and t represents the flight time; r p (t) = [x p (t)y p (t)z p (t)] T and v p (t) = [v x,p (t)v y,p (t)v z,p (t)] T They represent the position coordinates and velocity coordinates of the interceptor in the Earth-centered equatorial inertial coordinate system oxyz, r p (t)=||r p (t)|| is the modulus of the interceptor position vector; similarly, r e (t) = [x e (t)y e (t)z e (t)] T and v e (t) = [v x,e (t)v y,e (t)v z,e (t)] T They represent the position coordinates and velocity coordinates of the target in the geocentric equatorial inertial coordinate system oxyz, r e (t)=||r e (t)|| is the modulus of the target position vector; t0 represents the initial time, which can usually be considered as t0 = 0, r p0 and v p0 represents the position and velocity of the interceptor at the initial moment, and r e0 and v e0 represents the position and velocity of the target at the initial moment; Δv p represents the initial velocity increment of the interceptor, which is an unknown quantity to be determined; Step 2.2: Establish interception condition constraints; Consider that the interceptor and the target meet the interception condition at the terminal moment, that is, at the interception time t f The interceptor and target have the same position vector r p (t f )=r e (t f ) (2) Interception time t f is the variable that needs to be optimized; Step 3: Derive the first-order necessary conditions for optimal single-shot pulse interception and establish the optimal interception equations; The implementation method of step three is: Step 3.1: Derive the first-order necessary conditions for optimal single-shot interception; Using the dynamics in formula (1) to recursively obtain: in The terminal position of the interceptor is the interception time t f With the velocity increment Δv p The function of The terminal position of the target is only the interception time t f There is no analytical expression for either function. Note is a set of real numbers; using the intercept condition in the constraint of formula (2), we get: Formula (4) shows that the velocity increment Δv p and interception time t f Satisfy the equality constraint; in fact, when the interception time t is given f , the intercept problem is a Lambert problem, which can be solved directly using the Lambert problem, that is, there is a velocity increment Δv p and interception time t f Corresponding; let v p1 =Δv p +v p0 , then the condition satisfied by the optimal speed increment is: right Continue the derivation, first take the variation of both ends of equation (4), and get: Further, we get: The two ends of formula (7) are multiplied by Δv p , the necessary condition for the optimal speed increment is: Step 3.2: Establish the optimal interception equations; Combining equation (4) and equation (8), we get the optimal pulse interception equations as follows: Among them, the unknown quantity is The number of equations is 4, and the equations are closed; solving the above optimal intercept equations will give the optimal speed increment; Step 4: Simplify the optimal pulse interception equations to obtain analytical nonlinear equations; Step 5: Obtain the optimal pulse through Newton iteration method, that is, under the condition of freedom at the interception moment, intercept the target in orbit by solving the optimal pulse of the interceptor.

2. The optimal single-pulse interception method for a time-free spacecraft as claimed in claim 1, characterized in that: The implementation method of step 4 is: At the initial time t0=0, the position vector r of the interceptor p (t) is subjected to Taylor expansion, and we get: Take the first five terms of the sum of the series in formula (10): Formula (11) can be expressed in matrix form: r p (t)=Φ p t p (12) Among them, Φ p and τ p It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows: τ p =[1 t t 2 t 3 t 4 ] T (14) At the interception time t f When , we can get from formula (12): r p (t f )=Φ p t pf (16) Among them, τ pf The expression is as follows: τ pf =[1 t f t f 2 t f 3 t f 4 ] T (17) According to formula (8), two partial derivative calculation formulas are derived: Among them, λ p , Π p , Λ pi ,i=1,2,..,5 are intermediate variables introduced to simplify the expression. The expression is as follows: L p1 =[I 3×3 0 3×3 0 3×3 0 3×3 0 3×3 ] T Λ p2 =[0 3×3 t f I 3×3 0 3×3 0 3×3 0 3×3 ] T Similarly, according to equations (10)-(23), the analytical approximate expression of the terminal position of the target is derived. At the initial time t0=0, the position vector r of the target is e (t) is subjected to Taylor expansion, and we get: Take the first five terms of the series sum shown in formula (24): Express equation (25) in matrix form: r e (t)=Φ e t e (26) Among them, Φ e and τ e It is an intermediate variable introduced for the convenience of expression. The specific expression is as follows: τ e =[1 t t 2 t 3 t 4 ] T (28) At the interception time t f When , we can get from formula (26): r e (t f )=Φ e t ef (30) Among them, τ ef The expression is as follows: τ ef =[1 t f t f 2 t f 3 t f 4 ] T (31) Similarly, the calculation formula of the partial derivative of the terminal position with respect to the interception time is derived: Among them, λ e It is an intermediate variable introduced to simplify the expression. Its expression is as follows: Using equations (18), (19) and (32), the necessary condition in equation (8) is expressed as: in In summary, the simplified optimal impulse nonlinear equations are as follows: Furthermore, (36) is written as a nonlinear system of equations: F(x)=0 (37) in The form of the above system of equations is completely analytical.

3. The optimal single pulse interception method for a time-free spacecraft as claimed in claim 2, characterized in that: Step 5 is implemented as follows: Given an initial value x (0) , iteratively update equation (38) until the convergence condition shown in equation (39) is satisfied x (k+1) =x (k) -F′(x (k) ) -1 F(x (k) ),k=0,1,... (38) ||x (k+1) -x (k) ||≤δ (39) Where F′(x (k) ) -1 is the Jacobian matrix of the vector function F(x), and δ is the threshold set by the user; the optimal pulse Δv can be obtained in the end p and the corresponding interception time t f , that is, under the condition of freedom at the interception moment, the interception of the target in orbit is achieved by solving the optimal pulse of the interceptor.

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