A pulse orbit game strategy based on reachable domain coverage

Abstract

A kind of impulse orbit game strategy based on reachable domain coverage, the present application relates to impulse orbit game strategy.The purpose of the present application is to solve the problem that existing artificial intelligence-based method cannot obtain the fuel condition for interceptor to prevent target from escaping successfully in theory.Process is as follows:one, after giving the nominal orbit parameters of interceptor and target at initial time, the fuel-optimal impulse time and interception time of target without maneuvering are found, and the optimal interceptor impulse vector is calculated;Two, the pulse reachable domain envelope of target at interception time is calculated;Three, the minimum pursuit impulse amplitude required for interceptor to cover target pulse reachable domain at t * f time is obtained;Four, the optimal solution that meets reachable domain coverage constraint is obtained by not optimizing the first impulse of interceptor and optimizing the first impulse of interceptor respectively;Five, the minimum pursuit impulse amplitude of interceptor that meets reachable domain coverage constraint is obtained and is corrected.The present application is used in aerospace field.

Description

TECHNICAL FIELD

[0001] The present application relates to a pulse orbit game strategy based on reachable set coverage. BACKGROUND

[0002] The orbit interception problem is a typical problem in the field of spaceflight. In this problem, a suitable transfer orbit is usually designed to make the interceptor and the target reach the same position at the same time. However, most of the current researches are carried out for non-maneuverable targets. In these researches, the interceptor only needs to perform the corresponding pulse maneuver according to the optimization result to successfully intercept the target. If the target has a certain maneuvering capability, it can maneuver to escape when it finds that the interceptor poses a threat to it, and the above researches are no longer applicable. At this time, the orbit interception problem is also called the orbit game problem. For the orbit game problem, the bilateral optimal control method based on differential game theory is usually used to solve it. It is worth noting that the research on the orbit game problem based on differential game theory is mostly carried out under the assumption of continuous thrust model. Considering that pulse thrust is still a major form of maneuvering for current spacecraft, it is necessary to study the orbit game problem under pulse thrust, especially the fuel condition for the interceptor to prevent the target from escaping successfully cannot be obtained theoretically by the existing artificial intelligence-based method. SUMMARY

[0003] The purpose of the present application is to solve the problem that the existing artificial intelligence-based method cannot obtain the fuel condition for the interceptor to prevent the target from escaping successfully theoretically, and to propose a pulse orbit game strategy based on reachable set coverage.

[0004] The specific process of a pulse orbit game strategy based on reachable set coverage is as follows:

[0005] Step one, after the initial time t0 and the nominal orbit parameters of the interceptor and the target at the initial time t0 are given, a two-dimensional search algorithm is used to find the fuel-optimal pulse time of the target without maneuvering and the interception time and the optimal interceptor pulse vector

[0006] Step two, given that the target performs an escape pulse maneuver with an amplitude of ΔV Tmax at the time , the pulse reachable set envelope of the target at the interception time is calculated based on the two-body state transition matrix;

[0007] Step three, given the time t2 of the second pulse maneuver of the interceptor, the minimum pursuit pulse amplitude required for the interceptor to cover the target pulse reachable set at the time is obtained;

[0008] Step four, constructing two fuel optimization problems;

[0009] Based on steps one, two and three, constructing optimization problem one:

[0010] The first pulse of the interceptor is not optimized, and the optimal solution satisfying the reachable domain coverage constraint is obtained, including the optimal interceptor pursuit pulse time The optimal interceptor interception time And the minimum pursuit pulse amplitude of the interceptor satisfying the reachable domain coverage constraint

[0011] Based on steps two and three, constructing optimization problem two:

[0012] The first pulse of the interceptor is optimized, and the optimal solution satisfying the reachable domain coverage constraint is obtained, including the optimal interceptor first pulse time The optimal interceptor pursuit pulse time The optimal interceptor interception time The optimal interceptor first pulse aiming point And the minimum pursuit pulse amplitude of the interceptor satisfying the reachable domain coverage constraint

[0013] Step five, the minimum pursuit pulse amplitude of the interceptor satisfying the reachable domain coverage constraint obtained in step four is modified.

[0014] The beneficial effects of the present application are:

[0015] The present application designs a pulse orbit game strategy based on reachable domain coverage. In the strategy, an ellipsoid is used to approximate the reachable domain envelope of the target, and then the necessary conditions required for the interceptor to cover the target reachable domain envelope are derived. Then, while satisfying the reachable domain coverage constraint, the fuel performance requirements of the interceptor are further reduced through numerical optimization. For the orbit game scene between two spacecrafts using pulse thrust, the strategy proposed in the present application can provide an effective solution.

[0016] The present application proposes a pulse orbit game strategy based on reachable domain coverage. In the present application, the ellipsoid envelope of the target's reachable domain at a given time is analytically solved based on the two-body state transition matrix. For a given pursuit pulse time and interception time, the minimum pursuit pulse amplitude that satisfies the reachable domain coverage constraint is obtained by solving a nonlinear equation system. On this basis, two fuel optimization problems under the reachable domain coverage constraint are constructed according to whether the first pulse of the interceptor is optimized. Finally, the optimization results are corrected based on the Kepler equation and Lambert solution, improving the coverage performance of the interceptor on the target's reachable domain. The pulse orbit game problem between two spacecrafts can be solved by using the strategy proposed in the present application, and the fuel condition required for the interceptor to capture the target can be theoretically given.

[0017] The existing method is based on artificial intelligence, and the advantage of the method based on reachable domain coverage in the present application is that if the fuel satisfies the reachable domain coverage condition, the target can be prevented from escaping successfully. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1 The flowchart of the present application is shown in the figure;

[0019] Figure 2 The orbit game scene based on reachable domain coverage is shown in the figure;

[0020] Figure 3 The comparison of the target reachable domain envelope calculation results of the analytical method and the shooting method is shown in the figure;

[0021] Figure 4 The contour plot of the interceptor pursuit pulse amplitude is shown in the figure, Contours is the contour line, Delta V2 represents the interceptor pursuit pulse amplitude, alpha is the azimuth angle of vector y, beta is the pitch angle of vector y, and rad is radian;

[0022] Figure 5 The reachable domain coverage relationship obtained by optimizing P1 is shown in the figure;

[0023] Figure 6 The reachable domain coverage relationship obtained by optimizing P2 is shown in the figure. DETAILED DESCRIPTION

[0024] Specific implementation method one: the specific process of the pulse orbit game strategy based on reachable domain coverage in the present embodiment is as follows:

[0025] Step one: after the initial time t0 and the nominal orbit parameters of the interceptor and the target at the initial time t0 are given, the fuel-optimal pulse time of the target without maneuvering is found by using a two-dimensional search algorithm and the interception time and the optimal interceptor pulse vector is calculated.

[0026] Step two, given the target, at the time of the escape maneuver with amplitude ΔV Tmax , based on the two-body state transition matrix to calculate the target at the time of interception pulse reachable domain envelope;

[0027] Step three, given the time t2 of the second pulse maneuver (i.e. pursuit pulse maneuver) of the interceptor, obtain the minimum pursuit pulse amplitude required for the interceptor to cover the target pulse reachable domain at time;

[0028] Step four, build two fuel optimization problems;

[0029] Based on steps one, two, three, build optimization problem one:

[0030] The first pulse of the interceptor is not optimized, and the optimal solution that satisfies the reachable domain coverage constraint is obtained, including the optimal interceptor pursuit pulse time optimal interceptor interception time and the minimum pursuit pulse amplitude of the interceptor that satisfies the reachable domain coverage constraint

[0031] Based on steps two, three, build optimization problem two:

[0032] Optimize the first pulse of the interceptor, and the optimal solution that satisfies the reachable domain coverage constraint is obtained, including the optimal interceptor first pulse time optimal interceptor pursuit pulse time optimal interceptor interception time optimal interceptor first pulse aiming point and the minimum pursuit pulse amplitude of the interceptor that satisfies the reachable domain coverage constraint

[0033] Further reduce the fuel performance requirements of the interceptor (minimize the fuel consumed by the interceptor);

[0034] Step five, correct the minimum pursuit pulse amplitude of the interceptor that satisfies the reachable domain coverage constraint obtained in step four , and further improve the coverage performance of the interceptor on the target reachable domain (the coverage performance specifically refers to the percentage of successful interception times in Table 3).

[0035] Step five only corrects in the optimization results of step four, because step four is based on the state transition matrix optimization, and step five is a correction of the optimization results under the two-body nonlinear model; other results obtained in step four are not corrected.

[0036] The dynamic model used in the algorithm is a two-body model, which is expressed as:

[0037]

[0038]

[0039] where μ represents the gravitational constant of the earth, r and v represent the position vector and velocity vector of the spacecraft (target and interceptor) in the J2000 earth-centered inertial system, respectively, where |r| represents the magnitude of the corresponding position vector; where represents the first derivative of r; where represents the first derivative of v;

[0040] Specific embodiment two: the difference between this embodiment and the specific embodiment one is that the initial time t0 and the nominal orbit parameters (nominal position vector and nominal velocity vector) of the interceptor and the target at the initial time t0 are given in step one, and a two-dimensional search algorithm is used to find the optimal fuel pulse time of the target without maneuvering and the interception time and the optimal interceptor pulse vector The specific process is as follows:

[0041] The search range of the pulse time [t 1min , t 1max ] and the search range of the interception time [t f min , t f max ] are given;

[0042] In the J2000 earth-centered inertial system, the nominal position vector and the nominal velocity vector of the interceptor at the initial time are denoted as r C0 and v C0 , and the nominal position vector and the nominal velocity vector of the target at the initial time are denoted as r T0 and v T0 ;

[0043] Given any pulse time t1 and interception time t f , the position vector r1 and the velocity vector v1 of the interceptor at the pulse time t1 are solved according to the Kepler equation; the position vector r f of the target at the interception time t f is solved according to the Kepler equation;

[0044] The pulse vector Δv1 applied by the interceptor at the pulse time t1 is obtained by solving the Lambert problem, which is expressed as Δv1=Lambert(r1, r f , t f -t1)-v1;

[0045] The magnitude of the pulse applied by the interceptor at pulse time t1, ΔV1, is expressed as ΔV1=||Δv1||;

[0046] Since Δv1 is determined by the pulse time t1 and the interception time t f Therefore, ΔV1 is determined by both the pulse time t1 and the interception time t. f A bivariate function, i.e., ΔV1=q(t1,t) f );

[0047] ΔV1 represents the magnitude of the pulse applied by the interceptor at pulse time t1; q() represents the relationship between ΔV1 and t1 and t f A binary functional relationship;

[0048] When the target is stationary, a two-dimensional search algorithm is used to find the optimal fuel pulse moment. and interception time

[0049] The two-dimensional search algorithm is a genetic algorithm;

[0050] The interceptor at the optimal fuel pulse moment is solved based on the Kepler equation. Position vector r1 * and velocity vector

[0051] The target's optimal interception time is determined using Kepler's equations. position vector

[0052] Based on the optimal pulse timing for fuel Interception time Interceptor at the optimal fuel pulse moment Position vector r1 * The interceptor at the optimal fuel pulse moment velocity vector And the target at the optimal fuel interception time position vector The optimal interceptor pulse vector is obtained based on the Lambert algorithm.

[0053] The other steps and parameters are the same as in Specific Implementation Method 1.

[0054] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that, in step two, the target device is given in... The amplitude at time ΔV T max The escape pulse maneuver is used to calculate the target's state transition at the moment of interception based on the two-body state transition matrix. the pulse impulse envelope of the target at time

[0055] In step two, the interceptor applies a pulse impulse at time When the target enters the intercept orbit, it immediately perceives the threat of the interceptor and executes an escape pulse maneuver at time T max with a magnitude of ΔV Tf . Considering that most interception tasks have high requirements for the interception time, i.e., the task needs to be completed in a short time, and the spacecraft (including the interceptor and the target) carries limited fuel during the game phase, a linearized method can be used to solve the pulse reachable envelope of the target at time . The specific process is as follows:

[0056] Suppose that, in the case of no maneuver, the nominal position vector of the target at time is

[0057] When the target applies an escape pulse maneuver at time , the position vector of the target at time changes;

[0058] Near the nominal orbit (corresponding to the orbit of the target without applying an escape pulse), the linearized relationship between the position vector change amount Δr Tf and the escape pulse maneuver Δv T applied by the target at time is described as

[0059] Δr Tf = Φ T12 Δv T (1)

[0060] where Δv T is the escape pulse maneuver applied by the target at time ;

[0061] Δr Tf is the position vector change amount of the actual position r Tf of the target after applying the escape pulse maneuver at time relative to the nominal position ,

[0062] is the two-body state transition matrix of the target from time to time , and Φ T is written as

[0063]

[0064] where denotes the set of all 6x6 real matrices, Φ T11 denotes the target the pair of position vectors at time the matrix of partial derivatives of position vectors at time T12 denotes the target the pair of position vectors at time the matrix of partial derivatives of velocity vectors at time T21 denotes the target the pair of velocity vectors at time the matrix of partial derivatives of position vectors at time T22 denotes the target the pair of velocity vectors at time the matrix of partial derivatives of velocity vectors at time

[0065] When the target applies an escape pulse of amplitude at time T max then all possible escape pulse vectors lie on a spherical surface, denoted as

[0066]

[0067] where

[0068]

[0069] where diag(·) denotes the function that generates a diagonal matrix, Λ T denotes the escape pulse vector spherical surface matrix, · denotes the multiplication sign, and the superscript T denotes the transpose;

[0070] The pulse reachable region envelope of the target at time can then be obtained by performing a linear transformation on the pulse vector spherical surface denoted by equation (3), denoted as

[0071]

[0072] where

[0073]

[0074] where A T denotes the reachable region ellipsoidal matrix;

[0075] Equation (5) shows that, in the linear case, the pulse reachable region envelope of the target at time can be approximated by an ellipsoidal equation; thus, based on the two-body state transition matrix, the pulse reachable region ellipsoidal envelope of the target at time is obtained (equation (5)).

[0076] Other steps and parameters are the same as in specific implementation method one or two.

[0077] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that, in step three, the time t2 at which the interceptor applies the second pulse maneuver (i.e., the pursuit pulse maneuver) is given, and the interceptor's position at... The minimum chasing pulse amplitude required to cover the reachable domain of the target pulse at any given moment; the specific process is as follows:

[0078] Step 3.1: Based on the target obtained in Step 2... The pulse at a given moment can reach the envelope of the target. The reachable ellipsoid matrix A at time t is T It is symmetric and positive definite, therefore for the reachable ellipsoid matrix A T Perform Choleskey decomposition, represented as

[0079]

[0080] In the formula, H T Let A represent the ellipsoid matrix of the reachable region. T The matrix obtained by Choleskey decomposition;

[0081] Step 3.2, Define vector y as...

[0082] y = H T Δr Tf (8)

[0083] Equation (5) can then be rewritten as

[0084] y Τ y = 1 (9)

[0085] Equation (9) shows that the vector y is constrained on a unit sphere; any point taken on the unit sphere is represented as

[0086] y=[cosβcosα,cosβsinα,sinβ] Τ (10)

[0087] Vector y corresponds to a point in three-dimensional space;

[0088] In the formula, α∈[0,2π) is the azimuth angle of vector y, and β∈[-π / 2,π / 2] is the pitch angle of vector y;

[0089] Step 3: Based on an arbitrary point chosen on the unit sphere, the point corresponding to the selected point on the envelope of the target pulse reachability domain is obtained.

[0090]

[0091] Two-body state transition matrix Φ based on interceptor C The interceptor's chasing pulse vector is obtained as follows:

[0092]

[0093] in For the interceptor from t2 to The two-body state transition matrix of the nominal orbit at time. It is matrix Φ C Chinese elements Let be the set of all 3-order square matrices in the real number field;

[0094] Φ C Written as

[0095]

[0096] Where Φ C11 Interceptor The partial derivative matrix of the position vector at time t1 with respect to the position vector at time t2, Φ C12 Interceptor The partial derivative matrix of the position vector at time t2 with respect to the velocity vector at time t2, Φ C21 Interceptor The partial derivative matrix of the velocity vector at time t2 with respect to the position vector at time t2, Φ C22 Interceptor The partial derivative matrix of the velocity vector at time t1 with respect to the velocity vector at time t2;

[0097] Steps three and four: Given the chasing pulse time t2 and the interception time... At that time, the interceptor chasing pulse amplitude depends only on the variables α and β, described as follows:

[0098] ΔV2=g(α,β) (13)

[0099] Where ΔV2=||Δv2|| represents the amplitude of the interceptor chasing pulse; g() represents the bivariate functional relationship between ΔV2 and α and β, and |||| represents the modulus sign of the vector;

[0100] Formula (13) is obtained by going from formula (10) to formula (12);

[0101] Step 35: The minimum pursuit pulse amplitude required for the interceptor to cover the target's maneuverable reach domain satisfies the following condition, expressed as follows:

[0102]

[0103] Step 36: Solve the nonlinear equation system (14) using the Newton-Raphson iteration method to find the root ΔV2, and select the maximum value of ΔV2 as the interceptor. The minimum pursuit impulse amplitude required for the target pulse to cover the reachable set.

[0104] It should be noted that there can be multiple solutions to the nonlinear equations (14), and all solutions to equation (14) need to be obtained and compared to select the solution with the maximum amplitude.

[0105] The other steps and parameters are the same as one of the first to third embodiments.

[0106] The fifth embodiment is different from one of the first to fourth embodiments in that in the equation (14)

[0107]

[0108] wherein

[0109]

[0110]

[0111]

[0112] The other steps and parameters are the same as one of the first to fourth embodiments.

[0113] The sixth embodiment is different from one of the first to fifth embodiments in that in the step four, two fuel optimization problems are constructed

[0114] Based on the steps one, two and three, an optimization problem one is constructed

[0115] The first pulse of the interceptor is not optimized, and the optimal solution satisfying the reachable set coverage constraint is obtained, including the optimal interceptor pursuit impulse time The optimal interceptor interception time And the minimum pursuit impulse amplitude of the interceptor satisfying the reachable set coverage constraint

[0116] Based on the steps two and three, an optimization problem two is constructed

[0117] The first pulse of the interceptor is optimized, and the optimal solution satisfying the reachable set coverage constraint is obtained, including the optimal interceptor first pulse time The optimal interceptor pursuit impulse time The optimal interceptor interception time The optimal interceptor first pulse aiming point And the minimum pursuit impulse amplitude of the interceptor satisfying the reachable set coverage constraint

[0118] Further reduce the fuel performance requirement of the interceptor (minimize the fuel consumed by the interceptor);

[0119] The specific process is as follows:

[0120] 41. Based on steps one, two, and three, construct optimization problem one, defined as P1, described as follows:

[0121] The interceptor's first pulse is not optimized, meaning the interceptor is still in... Execute at all times The pulse maneuver; therefore, it can only be achieved by optimizing the interceptor's pursuit pulse time t2 and interception time t. f Reduce the interceptor's fuel consumption; the specific process is as follows:

[0122] In fact, for a given pursuit pulse time t2 and interception time t f The minimum chasing pulse amplitude required by the interceptor to satisfy the reachability domain coverage constraint can be obtained through the process in step three. However, since there are multiple solutions to equation (14), all solutions need to be obtained and compared to obtain the solution with the maximum amplitude. If the method in step three is used in the optimization process, it will lead to low computational efficiency of the optimization algorithm. Therefore, in step four, the chasing pulse amplitude of the interceptor is also considered as an optimization variable;

[0123] 411. At a certain interception moment t f The set of reachable regions of the target is denoted as

[0124]

[0125] In the formula, Indicates r Tf Must meet

[0126] Given the chasing pulse amplitude ΔV2 of the interceptor at time t2, then the interceptor at t f The set of reachable regions at time t is described as follows

[0127]

[0128] In the formula, r Cf After the interceptor applies the chasing pulse at time t2, at t f The actual location at that moment; When the interceptor does not apply a chasing pulse at time t2, at t f Nominal position of time; A C Let be the reachable region ellipsoid matrix of the interceptor, denoted as

[0129]

[0130] In the formula, Λ C Represents the sphere matrix of the chasing pulse vector;

[0131] Four one two, based on four one one, by optimizing the interceptor pursuit pulse moment t2, pursuit pulse amplitude ΔV2 and the interception moment t f , reduce the fuel consumption of the interceptor; the specific process is:

[0132] The constraint conditions are as follows:

[0133] (1) Fuel constraint: the interceptor must have a certain amount of fuel left for performing the pursuit pulse maneuver, that is,

[0134] ΔV2> 0 (22)

[0135] (2) Pursuit pulse time constraint: the pursuit pulse time needs to be after the first pulse time and before the terminal interception time, that is,

[0136]

[0137] (3) Terminal interception time constraint: the terminal interception time cannot exceed the upper limit of time t fmax set by the interception task, that is,

[0138] t f ≤t f max (24)

[0139] (4) Reachable domain coverage constraint: at the terminal interception moment, the reachable domain set of the interceptor needs to completely cover the reachable domain set of the target to prevent the target from escaping, which is expressed as

[0140]

[0141] In the above optimization problem, the reachable domain coverage constraint is expressed as the inclusion relationship between two sets. Since it is necessary to quickly judge the coverage between two reachable domains in the optimization process, the present application adopts the following method to transform the reachable domain coverage constraint:

[0142] 1) Sample the target reachable domain ellipsoid envelope to obtain a set of sampling points describing the target reachable domain ellipsoid envelope; specifically:

[0143] First, sample in the unit sphere represented by equation (9). A point in the unit sphere can still be represented by equation (10); therefore, only α and β need to be sampled. If the sampling step size is set to τ, the total number of sampling points can be written as

[0144] N T =N α ·N β (26)

[0145] Where N T represents the total number of sampling points, N α represents the number of sampling in the α direction, and Nβ This indicates the number of samples in the β direction;

[0146] N α and N β The expression is as follows:

[0147]

[0148] Where ceil(·) and floor(·) represent the rounding up and rounding down functions, respectively;

[0149] Then, for each sampling point within the unit sphere, the transformation is performed using equation (11) to obtain the point on the ellipsoidal envelope of the target pulse reachability domain corresponding to each sampling point; finally, the set of sampling points describing the ellipsoidal envelope of the target reachability domain is obtained, denoted as

[0150]

[0151] Where, r T,j This represents the position vector of the j-th sampling point on the envelope of the reachable domain ellipsoid;

[0152] 2) For each sampling point on the ellipsoidal envelope of the target reachability region, it must be located within the reachability region of the interceptor, denoted as...

[0153]

[0154] Further written as

[0155]

[0156] Thus, the reachable domain coverage constraints based on equations (26) to (29) have been transformed into the constraints described by equation (30), which can quickly determine the coverage relationship between the reachable domains of the interceptor and the target during the optimization process;

[0157] In summary, the optimization problem P1 can be described as follows:

[0158]

[0159] Where J represents the fuel optimization index;

[0160] For equation (31), the interior-point method is used for optimization to obtain the optimal solution that satisfies the reachability domain coverage constraint, including the optimal chasing pulse time of the interceptor. Optimal interceptor interception time And the minimum chasing pulse amplitude that the interceptor satisfies the reachability domain coverage constraint.

[0161] 42. Optimization problem two, based on steps two and three, is defined as P2 and described as follows:

[0162] Based on the target's escape maneuver capability, the first impulse of interceptor is involved in the optimization process;

[0163] If the first impulse time and the aim point of interceptor are t1 and The pursuit impulse vector Δv1 is solved by Lambert algorithm, and the pursuit impulse amplitude ΔV1 = ||Δv1|| is obtained based on the pursuit impulse vector Δv1;

[0164] The target also perceives the threat of interceptor at t1, and immediately executes the escape impulse maneuver; therefore, the reachable set of target at t f The reachable set of target at t f is still expressed by equation (19); for given interceptor pursuit impulse time t2 and pursuit impulse amplitude ΔV2, the reachable set of interceptor at t f is expressed by equation (20); then the reachable set coverage constraint is finally converted into the form of equation (30);

[0165] So far, the first impulse time t1, the first impulse aim point of interceptor, the pursuit impulse time t2, the pursuit impulse amplitude ΔV2 and the terminal interception time t f of interceptor can be optimized to meet the reachable set coverage constraint equation (30), and further reduce the fuel demand of interceptor;

[0166] The specific process is as follows:

[0167] On the basis of considering the constraint conditions in optimization problem one, the following constraints need to be further considered:

[0168] (1) First impulse time constraint: the first impulse time needs to be after the initial time and before the pursuit impulse time, that is

[0169] t0≤t1≤t2 (32)

[0170] (2) First impulse aim point constraint: the aim point of first impulse needs to be selected within the target impulse reachable set, that is

[0171]

[0172] In summary, the optimization problem P2 can be described as

[0173]

[0174] For equation (34), the interior point method can also be used for optimization to obtain the optimal solution that meets the reachable set coverage constraint, including the optimal interceptor first impulse time the optimal interceptor pursuit impulse time the optimal interceptor interception time First impulse aiming point of optimal interceptor Minimum pursuit impulse magnitude satisfying interceptor reachable set coverage constraint

[0175] Other steps and parameters are the same as one of the first to fifth embodiments.

[0176] Seventh embodiment: the difference between this embodiment and one of the first to sixth embodiments is that the minimum pursuit impulse magnitude satisfying the reachable set coverage constraint of the interceptor obtained in step four in the step five is modified to further improve the coverage performance of the interceptor on the target reachable set; the specific process is as follows:

[0177] The escape impulse maneuver vector exerted by the target at the time of or can be denoted as

[0178]

[0179] wherein denotes the azimuth angle of the escape impulse maneuver vector exerted by the target at the time of or , and γ ∈ [-π / 2, π / 2] denotes the elevation angle of the escape impulse maneuver vector exerted by the target at the time of or ; ΔV T max denotes the maximum value of the escape impulse magnitude;

[0180] The escape impulse maneuver vector Δv T exerted by the target is processed based on the Kepler equation, and the actual position vector r Tf of the target at the optimal interception time is obtained;

[0181] If the interceptor wants to reach the position vector r Tf , the pursuit impulse vector exerted by the interceptor at the time of can be obtained by solving a standard Lambert problem, and is denoted as

[0182]

[0183] wherein Δv2 denotes the pursuit impulse vector exerted by the interceptor at the time of , r C2 denotes the position vector of the interceptor at the time of , and v C2 denotes the velocity vector of the interceptor before the pursuit impulse is exerted at the time of ; ​​​

[0184] The interceptor pursuit impulse amplitude ΔV2=||Δv2||;

[0185] The interceptor pursuit impulse amplitude ΔV2is a two-dimensional function of and γ (when other parameters are given, Δv2is related to r Tf and Δv Tf , and Δv T is determined by T and γ. Therefore, ΔV2is a two-dimensional function of and γ.); expressed as

[0186]

[0187] In the formula, f() represents the binary function relationship between ΔV2and and γ;

[0188] The condition that the minimum pursuit impulse amplitude required by the interceptor satisfies is expressed as

[0189]

[0190] The Newton iteration method is used to solve the root ΔV2of the nonlinear equation group formula (38), and the maximum value of ΔV2is selected as the minimum pursuit impulse amplitude required by the interceptor to cover the target impulse reachable domain at the time .

[0191] It should be noted that the nonlinear equation group formula (38) may have multiple solutions, and all solutions of formula (38) need to be obtained and compared to select the solution with the maximum amplitude.

[0192] The other steps and parameters are the same as one of the first to sixth embodiments.

[0193] Embodiment eight: Different from one of the first to seventh embodiments, in the formula (38)

[0194]

[0195] Among them is obtained through formula (16);

[0196] ​This represents the partial derivative of the impulse output with respect to the terminal position input in the Lambert problem (see Zhang G, Zhou D, Mortari D, et al. Covariance analysis of Lambert's problem via Lagrange's transfer-time formulation[J]. Aerospace science and technology, 2018, 77: 765-773.).

[0197] And r Tf about The partial derivatives of γ are written as

[0198]

[0199] The other steps and parameters are the same as those in any of the specific implementation methods one to seven.

[0200] Example

[0201] Let the nominal position velocity vector of the interceptor at the initial moment be:

[0202]

[0203] The nominal position and velocity vector of the target at the initial moment is:

[0204]

[0205] First, the search range for the pulse time is set to t1∈[0,5000]s, and the search range for the interception time is t f ∈[t1,10000]s. Then, a genetic algorithm is used to perform a two-dimensional search to obtain the optimal pulse time under the condition of no target maneuvering. The optimal interception time is The pulse vector that the interceptor needs to apply at the optimal pulse time is: Its pulse amplitude is ΔV1 * = 0.4109 km / s. Considering the interceptor at... Apply the first pulse at the right time It entered the interception trajectory. Simultaneously, the target device... The ability to escape can be achieved by applying a pulse maneuver in any direction at any time, with the maximum amplitude of the maneuver pulse set to ΔV. T max =100m / s.

[0206] Then, the method in step two is used to calculate the target device at... The reachable region ellipsoidal envelope at time t is calculated and compared with the results of the target shooting method, such as... Figure 3The envelope of the target reachable region calculated by the shooting method is shown in Fig. 4. It can be seen that the envelope of the target reachable region calculated by the shooting method is basically consistent with that calculated by the analytical method.

[0207] If the pursuit impulse time of the interceptor is given as The interception time is still The minimum pursuit impulse amplitude required for the interceptor to cover the target reachable region is solved by the method in step three. First, the contour plot of the function represented by equation (13) is drawn with α, β as independent variables and ΔV2 as function value, as shown in Fig. 5. Figure 4 .

[0208] Figure 4 It can be seen from Fig. 5 that the function represented by equation (13) has four extreme points. The initial guess is determined according to the contour plot, and then the Newton iteration method is used to solve the nonlinear equation group corresponding to equation (14). Finally, by comparing all the solutions, the solution with the maximum amplitude is 0.2012 km / s. That is, if the interceptor wants to cover the target reachable region at , the minimum pulse amplitude required at t2 is at least 0.2012 km / s.

[0209] The maximum duration of the interception task is set as t f max = 10000 s. The two fuel optimization problems described in step four are solved. First, for optimization problem P1, the interior point method is used for optimization, and the optimal pursuit impulse time is obtained as The optimal interception time is The minimum pursuit impulse amplitude that satisfies the reachable region coverage constraint is The total fuel consumption is 0.5437 km / s, which is 0.0684 km / s less than the unoptimized result in step three. The reachable region coverage relationship is shown in Fig. 6. Figure 5 .

[0210] For optimization problem P2, the interior point method is still used for optimization, and the optimal first impulse time is finally obtained as The optimal pursuit impulse time is The optimal interception time is The optimal first impulse aiming point is The minimum pursuit impulse amplitude that satisfies the reachable region coverage constraint is Then, the Lambert algorithm is used to obtain the first impulse amplitude of the interceptor as The total fuel consumption of the interceptor is 0.5156 km / s, which is 0.0281 km / s less than the optimization result of P1. The reachable region coverage relationship is shown in Fig. 7. Figure 6 .

[0211] The fuel consumption of the optimization results of the above two types of optimization problems is shown in Table 1:

[0212] Fuel consumption of optimal solution of optimization problem P1 and P2

[0213]

[0214]

[0215] From the above results, it can be found that the total fuel required by the optimal solution of P2 is better than that of P1, which is due to the optimization of the first pulse of the interceptor in P2.

[0216] Since the above processes are all based on the results of the two-body state transition matrix, in order to obtain a more accurate solution under the two-body nonlinear model, the optimal pursuit pulse solution obtained in step four is modified by using the method in step five. The modified results of the solution of P1 are And the modified results of the solution of P2 are Further, in order to compare the coverage performance of the interceptor on the target pulse reachable domain before and after modification, the Monte-Carlo shooting method is used for verification. Considering that the interceptor and the target both use the two-body dynamics model, at the moment of the first pulse of the interceptor, the target simultaneously performs an escape maneuver with an arbitrary direction and an amplitude of ΔV T max = 100 m / s; and then, the Lambert algorithm is used to solve the pursuit pulse amplitude ΔV2 required by the interceptor to intercept the target. In the case before modification, if the interceptor is considered to successfully intercept the target; otherwise, it is considered that the target successfully escapes. The same judgment is made for the case after modification. 100000 times of Monte-Carlo simulation calculation are performed, and the percentage of successful interception times before and after modification is counted respectively, and the results are shown in Table 2. At the same time, Table 2 also shows the total fuel consumption before and after modification.

[0217] Table 2 Comparison results of total fuel consumption and coverage performance before and after modification

[0218]

[0219] The above results show that compared with the results before modification, although the fuel requirement of the interceptor is slightly increased after modification, the probability of successful interception is also increased to 100%, which indicates that the modification method effectively improves the reachable domain coverage performance.

[0220] The present application also has other various embodiments, and those skilled in the art can make various corresponding changes and modifications according to the present application without departing from the spirit and essence of the present application, but these corresponding changes and modifications should all belong to the protection scope of the claims attached to the present application.

Claims

1. A pulse orbit game strategy based on reachable domain coverage, characterized in that: The specific process is as follows: Step one, given the initial time t0 and the nominal orbit parameters of interceptor and target at initial time t0, the two-dimensional search algorithm is used to find the optimal pulse time of target without maneuver and the interception time and the optimal interceptor pulse vector is calculated Step 2: Given the target device in The amplitude at time ΔV Tmax The escape pulse maneuver is used to calculate the target's state transition at the moment of interception based on the two-body state transition matrix. The pulse reachable domain envelope; Step three, given the time t2 at which the second impulse maneuver of the interceptor is applied, obtain the minimum pursuit impulse magnitude required at time t2 for the interceptor to cover the target impulse reachable set at time t2. Step three, given the time t2 at which the second impulse maneuver of the interceptor is applied, obtain the minimum pursuit impulse magnitude required at time t2 for the interceptor to cover the target impulse reachable set at time t2. Step four, constructing two fuel optimization problems; Based on steps one, two and three, an optimization problem one is constructed: The first pulse of the interceptor is not optimized, and an optimal solution satisfying the reachable set coverage constraint is obtained, including an optimal interceptor pursuit pulse time An optimal interceptor interception time And a minimum pursuit pulse amplitude of the interceptor satisfying the reachable set coverage constraint Based on steps two and three, an optimization problem two is constructed: Optimization for the first pulse of interceptor, to obtain the optimal solution satisfying the reachable set coverage constraint, including the optimal interceptor first pulse time Optimal interceptor pursuit pulse time Optimal interceptor interception time Optimal interceptor first pulse aiming point And the minimum pursuit pulse amplitude of the interceptor satisfying the reachable set coverage constraint Step five, the minimum pursuit pulse amplitude of the interceptor obtained in step four satisfying the reachable domain coverage constraint Amendments are made.

2. The strategy based on the reachable domain coverage of the impulsive orbital game according to claim 1, characterized in that: The step one gives the initial time t0 and the nominal orbit parameters of the interceptor and the target at the initial time t0, and adopts a two-dimensional search algorithm to find the optimal pulse time of the target without maneuver and the interception time and calculates the optimal interceptor pulse vector The specific process is: Given the pulse time search range [t 1min ,t 1max ] and the intercept time search range [t fmin ,t fmax ]; In the J2000 Earth-centered inertial system, the nominal position vector and the nominal velocity vector of the interceptor at the initial time are denoted as r C0 and v C0 , respectively, and the nominal position vector and the nominal velocity vector of the target at the initial time are denoted as r T0 and v T0 , respectively. Given any one impulse time t1 and intercept time t f , solve for the position vector r1 and velocity vector v1 of the interceptor at impulse time t1 according to Kepler's equation; solve for the position vector r f of the target at intercept time t f according to Kepler's equation; The impulse vector Δv1 applied by the interceptor at the impulse time t1 is obtained by solving the Lambert problem and is expressed as Δv1 = Lambert(r1, r f ,t f -t1)-v1; The pulse amplitude ΔV1 exerted by the interceptor at the pulse moment t1 is represented as ΔV1=||Δv1||; Since Δv1is determined by the impulse time t1and the intercept time t f , Δv1is also a function of the impulse time t1and the intercept time t f , i.e. Δv1= q(t1, t f ). AV1 represents the pulse amplitude value applied by the interceptor at the pulse time t1; q() represents the binary function relationship between AV1 and t1 and t f . In the case of a target without mobility, the optimal moment of fuel injection is found by a two-dimensional search algorithm and the moment of interception The two-dimensional search algorithm is a genetic algorithm; Solving the position vector of the interceptor at the optimal time of fuel impulse according to Kepler's equation and the velocity vector at the optimal time of fuel impulse according to Kepler's equation Solving the position vector of a target at the fuel-optimal intercept time according to Kepler's equation ​ Based on the optimal pulse timing for fuel Interception time Interceptor at the optimal fuel pulse moment position vector Interceptor at the optimal fuel pulse moment velocity vector And the target at the optimal fuel interception time position vector The optimal interceptor pulse vector is obtained based on the Lambert algorithm.

3. The strategy of the impulse orbital game based on the reachable domain coverage according to claim 2, characterized in that: In step two, the target device is given in The amplitude at time ΔV Tmax The escape pulse maneuver is used to calculate the target's state transition at the moment of interception based on the two-body state transition matrix. The pulse reachable domain envelope; the specific process is as follows: Assuming the target is in a non-motorized situation, the target's nominal position vector at time instant t is where x(t) is the target's actual position vector at time instant t. When the target vehicle is at a moment of time, an escape maneuver is applied, which causes a change in the position vector of the target vehicle at a moment of time; Position vector change Δr Tf And the targeter is in The escape impulse maneuver Δv T The linearized relationship is described as Δr Tf = Φ T12 Δv T (1) where Δv T is the escape impulse maneuver applied by the target at the time instant t. Δr Tf the actual position r at the time of the application of the escape pulse maneuver to the target Tf from the nominal position the position vector change amount, from the target to the reference frame to the two-body state transition matrix, Φ, of the nominal orbit at the instant T is written as wherein denotes the set of all 6x6 matrices in the real numbers, Φ T11 denotes the target time position vector pair time position vector pair T12 denotes the target time position vector pair time position vector pair T21 denotes the target time position vector pair time position vector pair T22 denotes the target time position vector pair time position vector pair When the targeter is at the magnitude of the escape pulse applied at that instant takes the maximum value AV Tmax then all possible escape pulse vectors lie on a circle, denoted as Wherein where diag( ) denotes a function that generates a diagonal matrix, Λ T represents an escape pulse vector circle spherical matrix, • denotes a multiplication sign, and a superscript T denotes a transpose; The target device is in The pulse reachable region envelope at time t can be obtained by linearly transforming the pulse vector circle sphere represented by equation (3) and is represented as In the formula In the formula, A T denotes the reachable domain ellipsoid matrix.

4. The strategy of the impulse orbit game based on the reachable domain coverage according to claim 3, characterized in that: The time t2 at which the second impulse maneuver is applied to the interceptor in step three is determined such that the interceptor reaches the minimum pursuit impulse magnitude required to cover the target impulse reachable domain at the time. The specific process is as follows: Step three i. Based on the target object obtained in step two the pulse reachable region envelope of the target object at the moment, the reachable region ellipsoid matrix A T is symmetric positive definite, so the Choleskey decomposition is performed on the reachable region ellipsoid matrix A T and expressed as In the formula, H T denotes a matrix obtained by Choleskey decomposition on the ellipsoid matrix A T of the reachable domain Step three two, the vector y is defined as y = H T Δr Tf (8) Then formula (5) can be rewritten as y T y = 1 (9) Formula (9) indicates that the vector y is constrained on a unit sphere; any point on the unit sphere is represented as y = [cos β cos α, cos β sin α, sin β] T (10) In the formula, α∈[0, 2π) is the azimuth angle of the vector y, and β∈[-π / 2, π / 2] is the pitch angle of the vector y; Step three three, based on any point on the unit sphere, the point corresponding to the selected point on the target pulse reachable domain envelope is obtained as Intercept-based two-body state transition matrix Φ C The pursuit impulse vector of the interceptor is obtained as where is the interceptor's state at time t2 is the two-body state transition matrix from t2 to is the matrix Φ C is the element in the matrix Φ is the set of all 3x3 matrices in the real field. Φ C written as where Φ C11 represents the interceptor the partial derivative matrix of the position vector at time t2 with respect to the position vector at time t2, Φ C12 represents the interceptor the partial derivative matrix of the velocity vector at time t2 with respect to the position vector at time t2, Φ C21 represents the interceptor the partial derivative matrix of the velocity vector at time t2 with respect to the velocity vector at time t2, Φ C22 represents the interceptor the partial derivative matrix of the velocity vector at time t2 with respect to the velocity vector at time t2, Φ Step three Four, when given the chase pulse time t2 and the intercept time the interceptor chase pulse amplitude is only related to the variables a and b, described as ΔV2=g(α, β) (13) Wherein ΔV2=||Δv2|| represents the pursuit pulse amplitude of the interceptor; g() represents the binary function relationship between ΔV2 and α and β, and |||| represents the modulus symbol of the vector; Step three five, the minimum pursuit pulse amplitude required for the interceptor to cover the target maneuver reachable domain satisfies the following condition, represented as Step three six, the Newton iteration method is used to solve the root AV2 of the nonlinear equation set (14), and the maximum value of AV2 is selected as the minimum pursuit pulse amplitude required for the interceptor to cover the target pulse reachable domain at the moment.

5. The strategy of the impulse orbit game based on the reachable domain coverage according to claim 4, characterized in that: In the formula (14) Wherein 6. The strategy of the impulse orbit game based on the reachable domain coverage according to claim 5, characterized in that: The two fuel optimization problems in step four are constructed; Based on steps one, two and three, an optimization problem one is constructed: The first pulse of the interceptor is not optimized, and an optimal solution satisfying the reachable set coverage constraint is obtained, including an optimal interceptor pursuit pulse time An optimal interceptor interception time And a minimum pursuit pulse amplitude of the interceptor satisfying the reachable set coverage constraint Based on steps two and three, an optimization problem two is constructed: Optimization for the first pulse of interceptor, to obtain the optimal solution satisfying the reachable set coverage constraint, including the optimal interceptor first pulse time Optimal interceptor pursuit pulse time Optimal interceptor interception time Optimal interceptor first pulse aiming point And the minimum pursuit pulse amplitude of the interceptor satisfying the reachable set coverage constraint The specific process is as follows: Four one, based on steps one, two and three, an optimization problem one is constructed, defined as P1, described as follows: The first pulse of the interceptor is not optimized, i.e. the interceptor still performs the pulse maneuver at the moment t2; only the pursuit pulse moment t f of the interceptor can be optimized to reduce the fuel consumption of the interceptor; the procedure is as follows: Four one one, at some interception time t f The set of reachable domains of the target is denoted by In the formulae, r represents r Tf Must meet If the pursuit pulse amplitude of the given interceptor at time t2 is ΔV2, then the reachable set of the interceptor at time t f is described as where r Cf is the actual position of the interceptor at time t f after the pursuit impulse is applied at time t2; is the nominal position of the interceptor at time t f after the pursuit impulse is not applied at time t2; A C is the attainable ellipsoid matrix of the interceptor, expressed as In the formula, Λ C denotes the pursuit impulse vector circle spherical matrix; Four one two, based on four one one, by optimizing the interceptor pursuit pulse moment t2, pursuit pulse amplitude ΔV2 and intercept time t f , reduce the fuel consumption of the interceptor; The specific process is: The constraint conditions are specifically as follows: (1) Fuel constraint: the interceptor must have a certain amount of fuel left for performing the pursuit pulse maneuver, that is ΔV2>0 (22) (2) Pursuit pulse time constraint: the pursuit pulse time needs to be after the first pulse time and before the terminal interception time, that is (3) Terminal interception time constraint: the terminal interception time cannot exceed the upper time limit t set by the interception task fmax i.e. t f ≤t fmax (24) (4) Reachable domain coverage constraint: at the terminal interception moment, the reachable domain set of the interceptor needs to completely cover the reachable domain set of the target, to prevent the target from escaping, represented as The following method is adopted to transform the reachable domain coverage constraint: 1) The target reachable domain ellipsoid envelope is sampled to obtain a sample point set describing the target reachable domain ellipsoid envelope; specifically: First, sampling is performed within the unit sphere represented by Equation (9), and a point within the unit sphere can still be represented by Equation (10); thus, only a and β need to be sampled; if the sampling step size is set to τ, the total number of sampling points can be written as N T = N α · N β (26) where N T represents the total number of sampling points, N α represents the number of samples in the a direction, N β represents the number of samples in the β direction; N α and N β The expression is as follows: Wherein ceil(·) and floor(·) represent the upward rounding and downward rounding functions respectively; Then, for each sample point in the unit sphere, formula (11) is used for transformation to obtain the point corresponding to each sample point on the target pulse reachable domain ellipsoid envelope; finally, the sample point set describing the target reachable domain ellipsoid envelope is obtained, denoted as where r T,j represents the position vector of the jth sampling point on the target reachable domain ellipsoid envelope; 2) Each sample point on the target reachable domain ellipsoid envelope needs to be located in the reachable domain of the interceptor, represented as Further written as At this point, based on formula (26) to formula (29), the reachable domain coverage constraint has been transformed into the constraint condition described in formula (30); Based on the above, the optimization problem P1 can be described as Wherein J represents the fuel optimization index; For equation (31), the interior point method is used to optimize the optimal solution that satisfies the reachable set coverage constraint, including the optimal pursuit impulse time of the interceptor Optimal interceptor interception time And the minimum pursuit pulse amplitude of the interceptor that satisfies the reachable set coverage constraint Four two, based on the optimization problem two constructed in steps two and three, defined as P2, described as follows: Based on the escape maneuvering capability of the target, the first pulse of the interceptor is included in the optimization process; If the first pulse time of the interceptor and the aim point are t1 and The pursuit impulse vector Δv1 is solved by using the Lambert algorithm, and the pursuit impulse amplitude ΔV1 = ||Δv1|| is obtained based on the pursuit impulse vector Δv1. The target also perceives the threat of the interceptor at time t1and immediately performs an escape impulse maneuver; therefore, the target's reachable set at time t f is still represented by equation (19); for a given interceptor pursuit impulse time t2and pursuit impulse magnitude AV2, the interceptor's reachable set at time t f is also represented by equation (20); then the reachable set coverage constraint is finally transformed into the form of equation (30); At this point, the first pulse time t1 of the interceptor, the first pulse aiming point The interceptor pursuit pulse time t2, the pursuit pulse amplitude ΔV2 and the terminal interception time t f Under the condition of satisfying the reachable domain coverage constraint (30), further reduce the fuel demand of the interceptor; The specific process is as follows: In addition to the constraint conditions considered in the optimization problem one, the following constraints also need to be further considered: (1) The first impulse time constraint: the first impulse time should be after the initial time and before the pursuit impulse time, i.e. t0≤t1≤t2(32) (2) The first impulse aiming point constraint: the aiming point of the first impulse should be selected within the target impulse reachable domain, i.e. In summary, the optimization problem P2 can be described as For equation (34), the interior point method is used to optimize to obtain the optimal solution satisfying the reachable set coverage constraint, including the optimal interceptor first pulse time Optimal interceptor pursuit pulse time Optimal interceptor interception time Optimal interceptor first pulse aiming point Minimum pursuit pulse amplitude satisfying the interceptor reachable set coverage constraint 7. The strategy of the impulse orbit game based on the reachable domain coverage according to claim 6, characterized in that: the minimum pursuit pulse amplitude of the interceptor obtained in step four satisfying the reachable set coverage constraint in step five correction is made; the specific process is as follows: The targeter is at or The escape impulse maneuver vector applied at the moment can be noted wherein denotes the azimuth angle of the escape pulse maneuver vector applied by the target at or denotes the elevation angle of the escape pulse maneuver vector applied by the target at or time; AV Tmax denotes the maximum value of the escape pulse amplitude; Applying an escape impulse maneuver vector Δv to the target based on the Kepler equation T Performing processing to obtain an actual position vector r of the target at the optimal intercept time Tf ; The interceptor, if it is to reach the position vector r Tf , the interceptor must apply a pursuit impulse vector at The pursuit impulse vector required at time instant can be obtained by solving a standard Lambert problem, denoted as where Δv2represents the pursuit impulse vector applied by the interceptor at time t2, r C2 represents the position vector of the interceptor at time t2, v C2 represents the velocity vector of the interceptor prior to application of the pursuit impulse at time t2. The interceptor pursuit impulse amplitude ΔV2=||Δv2||; The interceptor pursues the pulse amplitude ΔV2, which is a two-dimensional function of and γ; expressed as where f() represents a binary function relationship between ΔV2 and and γ. The condition that the minimum pursuit impulse amplitude required by the interceptor satisfies is The Newton iteration method is used to solve the root of the nonlinear equation set (38) for ΔV2, and the maximum value of ΔV2 is selected as the minimum pursuit pulse amplitude required for the interceptor to cover the target impulse reachable domain at the moment.

8. The strategy of the impulse orbit game based on the reachable domain coverage according to claim 7, characterized in that: In the formula (38) wherein obtained by formula (16); denotes the partial derivative of the impulse response with respect to the terminal position input in the Lambert problem; And r Tf With respect to The partial derivatives of φ and γ are written as

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