A spacecraft impulse maneuver optimization method considering collision avoidance

Abstract

A method for optimizing spacecraft pulse maneuvers considering collision avoidance. The present invention relates to a method for optimizing spacecraft pulse maneuvers considering collision avoidance. The present invention belongs to the field of satellite orbit control. The purpose of the present invention is to solve the problem that the existing method for solving the optimal collision avoidance pulse time for spacecraft fuel is relatively long. The process is as follows: Step 1, based on the reachable domain envelope model, construct an optimization model for a single pulse collision avoidance problem with a fixed pulse time; Step 2, based on the optimization model of Step 1, use the Newton iteration method to solve the minimum pulse of the spacecraft at a given collision avoidance moment; Step 3, perform a one-dimensional search on the collision avoidance moment in Step 2 to obtain the optimal collision avoidance time, and obtain the minimum pulse with free collision avoidance time based on the optimal collision avoidance time.

Description

Technical Field

[0001] The present invention relates to a spacecraft pulse maneuver optimization method considering collision avoidance and belongs to the field of satellite orbit control. Background Art

[0002] With the continuous advancement of space technology, the number of man-made objects in the space environment is increasing, and the amount of space debris in orbit is also increasing. By 2020, more than 20,000 objects larger than 5 cm in diameter had been recorded in orbit. Furthermore, mega-constellation projects, including SpaceX's Starlink, are contributing to an increasingly crowded space environment. This increasing amount of space debris poses a serious threat to spacecraft in orbit. Furthermore, spacecraft in orbit also face the risk of collision with other non-cooperative spacecraft. Therefore, a method for rapidly solving spacecraft impulse maneuvers is needed to achieve collision avoidance.

[0003] For the collision avoidance problem, the current pulse-based collision avoidance strategy is mainly based on optimization methods, and the collision avoidance constraints are processed into safety distance constraints and collision probability constraints. For the risk of sudden collisions with a shorter time, planning the collision avoidance pulse requires a certain amount of time. Summary of the Invention

[0004] The purpose of the present invention is to solve the problem that the existing method for solving the collision avoidance pulse time of the optimal spacecraft fuel is long, and to propose a spacecraft pulse maneuver optimization method taking collision avoidance into consideration.

[0005] A spacecraft impulse maneuver optimization method considering collision avoidance is as follows:

[0006] Step 1: Based on the reachable domain envelope model, an optimization model for the single pulse collision avoidance problem with fixed pulse time is constructed;

[0007] Step 2: Based on the optimization model of step 1, use the Newton iteration method to solve the minimum pulse of the spacecraft at a given collision avoidance time;

[0008] Step 3: Perform a one-dimensional search on the collision avoidance time in step 2 to obtain the optimal collision avoidance time. Based on optimal collision avoidance time Get the minimum pulse of collision avoidance time freedom.

[0009] The beneficial effects of the present invention are:

[0010] The present invention uses the reachable domain to realize the rapid calculation of the collision avoidance pulse of the spacecraft. The reachable domain is the set of all reachable points under orbital dynamics. At present, the analysis of the reachable domain has been used in various scenarios such as uncertainty propagation, collision probability calculation, and inter-spacecraft game. After obtaining the reachable domain information of the incoming target, the spacecraft can change its orbit to avoid the target threat. The collision avoidance method based on the reachable domain has a unique advantage, that is, the geometric relationship between the spacecraft and the reachable domain can be clearly obtained, and then it can be intuitively judged whether the spacecraft can truly achieve the collision avoidance task of the non-cooperative target.

[0011] This paper proposes a method for rapidly calculating the minimum fuel pulse required for spacecraft collision avoidance, taking collision avoidance into account. This method uses an ellipsoidal approximation to describe the reachable envelope of a non-cooperative target under measurement uncertainty. This method then constructs a framework for optimizing the minimum fuel pulse required for spacecraft collision avoidance. A semi-analytical method is then used to calculate the required minimum fuel pulse solution for spacecraft collision avoidance. This proposed strategy provides an effective solution for spacecraft using pulsed thrust and non-cooperative targets subject to measurement uncertainty.

[0012] The present invention proposes a method for rapidly calculating the minimum fuel collision avoidance pulse for a spacecraft, taking collision avoidance into account. Based on a two-body state transfer matrix, the present invention analytically solves the ellipsoidal envelope of the reachable domain caused by uncertainty at a given moment for a non-cooperative target. Furthermore, for a given pulse moment, by solving a two-dimensional nonlinear equation, the spacecraft satisfies the collision avoidance constraint, and the minimum pulse collision avoidance problem is further simplified to a one-dimensional search problem. The strategy proposed in the present invention is used to solve the pulse collision avoidance problem between the spacecraft and the non-cooperative target, and the minimum fuel pulse solution required for the spacecraft to achieve collision avoidance can be quickly obtained.

[0013] The present invention can meet the need for rapid planning of minimum collision avoidance pulses for spacecraft under the risk of sudden collision in space. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 Flow chart of the method of the present invention;

[0015] Figure 2 This is a schematic diagram of the maneuvering trajectory for spacecraft collision avoidance based on the reachable domain;

[0016] Figure 3 Schematic diagram of the changes in the orbits of spacecraft and non-cooperative targets and the reachable areas of non-cooperative targets;

[0017] Figure 4 Schematic diagram of the spacecraft's collision avoidance maneuver trajectory; (a) Schematic diagram of the positional relationship between the spacecraft and the reachable region of the non-cooperative target; (b) Schematic diagram of the cross-section of the reachable region when the spacecraft reaches the boundary of the reachable region of the non-cooperative target;

[0018] Figure 5 This is the distance diagram between the spacecraft and the boundary of the reachable region under single pulse obtained by the method of the present invention. DETAILED DESCRIPTION

[0019] Specific implementation method 1: Combination Figure 1 This embodiment describes a method for optimizing spacecraft pulse maneuvers that considers collision avoidance. The specific process is as follows:

[0020] Step 1: Based on the reachable domain envelope model, an optimization model for the single pulse collision avoidance problem with fixed pulse time is constructed (21);

[0021] The process of obtaining the reachable domain envelope model (6) is as follows:

[0022] The dynamic models used are all two-body models, expressed as:

[0023]

[0024]

[0025] Where μ represents the Earth's gravitational constant, r and v represent the position vector and velocity vector of the J2000 geocentric inertial spacecraft (target and interceptor), respectively, ||·|| represents the modulus of a vector or matrix, and ||r|| represents the magnitude of the position vector of the J2000 geocentric inertial spacecraft. represents the first derivative of r; represents the first-order derivative of v;

[0026] At the initial time t0, the position and velocity vectors of the spacecraft are r s0 and v s0 , the measured values of the position and velocity vectors of the non-cooperative target are and

[0027] At the initial time t0, the true value of the initial position and velocity of the non-cooperative target r c0 and v c0 Respectively expressed as

[0028]

[0029] Among them, δr c0 Denotes the error between the initial position measurement value and the true value of the non-cooperative target, δv c0 represents the error between the initial velocity measurement value and the true value of the non-cooperative target;

[0030] Assume that the error satisfies the zero-mean Gaussian distribution and the covariance matrix is

[0031]

[0032] Among them, I 3×3 represents the third-order identity matrix, σ r and σ v denote the standard deviation of position and velocity, respectively; E(·) denotes taking the mean of the values in the brackets; the superscript T denotes taking the transpose;

[0033] like Figure 2 This is a schematic diagram of the maneuvering trajectory for spacecraft collision avoidance based on the reachable domain;

[0034] The position vector r and the velocity vector v are uniformly described as a six-dimensional state vector x = [r T ,v T ] T In this invention, the non-cooperative target is defined by the initial measurement state The state obtained by recursion according to the Kepler equation is the nominal state, a certain t f The nominal state of the non-cooperative target at time It can be obtained according to the Kepler equation;

[0035] State measurement error at the initial time t0 Under this condition, the non-cooperative target is at time t f The true state x cf The nominal state obtained by recursion based on the initial measurement value and Kepler equation different;

[0036] For non-cooperative goals at t f The state is measured at the initial moment The nominal state obtained by recursion according to the Kepler equation;

[0037] True state x cf I don't know. It is obtained recursively, so the first step to establish the reachable domain is to describe the range in which the true value may appear;

[0038] By using the linearized relationship, time t f Nominal state With the real state x cf The deviation vector Δx cf for

[0039]

[0040] in, For non-cooperative goals, from time t0 to t f The two-body state transfer matrix at time ;

[0041] The two-body state transfer matrix is expressed as

[0042]

[0043] in, represents the set of all 6-order square matrices in the real number field, Φ c11 represents the non-cooperative goal t f The partial derivative matrix of the position vector at time t0 with respect to the position vector at time t0, Φ c12 represents the non-cooperative goal t f The partial derivative matrix of the position vector at time t0 with respect to the velocity vector at time t0, Φ c21 represents the non-cooperative goal t f The partial derivative matrix of the velocity vector at time t0 with respect to the position vector at time t0, Φ c22 represents the non-cooperative goal t f The partial derivative matrix of the velocity vector at time t0 with respect to the velocity vector at time t0;

[0044] The non-cooperative target is f The covariance matrix at time Expressed as

[0045]

[0046] Among them, P c0 is the covariance matrix at the initial moment,

[0047]

[0048] Among them, 0 3×3 Represents a 3-order square matrix with all elements being 0;

[0049] In this paper, we only focus on the reachable domain of non-cooperative target positions. The boundary equation of the reachable domain ellipsoid of non-cooperative target is:

[0050]

[0051] Among them, r cf For non-cooperative goals at time t f The true position vector of For non-cooperative goals at t f At this moment, according to the initial measurement state The nominal position vector obtained by recursion with the Kepler equation, A c11 The error ellipse equation for the non-cooperative target is the reachable domain position ellipsoid matrix.

[0052] The error ellipse equation of the non-cooperative target can be reached by the domain position ellipsoid matrix A c11 Calculated by the following formula:

[0053]

[0054] Among them, k 2 is the state error vector The Mahalanobis distance is defined by the user based on the desired probability threshold;

[0055] is the covariance matrix P c (t f ,t0), i=1,2; j=1,2;

[0056] represents the set of all 3-order square matrices in the real number domain;

[0057]

[0058] Step 2: Based on the optimization model of step 1, the Newton iteration method is used to solve the minimum pulse of the spacecraft at a given collision avoidance time (31);

[0059] Step 3: Perform a one-dimensional search (such as the golden section algorithm) on the collision avoidance time in step 2 to obtain the optimal collision avoidance time. Based on optimal collision avoidance time Get the minimum pulse with free collision avoidance time (step 2 is the optimal pulse under given collision avoidance time conditions, but the collision avoidance time is free, so we need to search for an optimal collision avoidance time in one dimension to get the minimum pulse with free collision avoidance time).

[0060] Specific embodiment 2: This embodiment differs from specific embodiment 1 in that: in step 1, based on the reachable domain envelope model, an optimization model (21) for the single pulse collision avoidance problem with a fixed pulse time is constructed;

[0061] The specific process is:

[0062] Step 1: Calculate the position vector of the spacecraft under the action of the pulse; the specific process is:

[0063] The time t under the pulse f Spacecraft position vector r sf The linear relationship is described as:

[0064]

[0065] in, is the time t in the original orbit f Spacecraft position vector, Δv s1 The first collision avoidance pulse for a spacecraft, The time from spacecraft t0 to t f The two-body state transfer matrix at time ;

[0066] Transfer matrix Φ s (t f ,t0) is expressed as

[0067]

[0068] Among them, Φ s11 represents the spacecraft t f The partial derivative matrix of the position vector at time t0 with respect to the position vector at time t0, Φ s12 represents the spacecraft t f The partial derivative matrix of the position vector at time t0 with respect to the velocity vector at time t0, Φ s21 represents the spacecraft t f The partial derivative matrix of the velocity vector at time t0 with respect to the position vector at time t0, Φ s22 represents the spacecraft t f The partial derivative matrix of the velocity vector at time t0 with respect to the velocity vector at time t0;

[0069] Step 1 and 2: Based on step 1, analyze the single pulse collision avoidance problem with fixed pulse time and give the constraint expression (12) that needs to be satisfied for collision avoidance; the specific process is as follows:

[0070] Set time t f Spacecraft position vector r sf Substituting into the ellipsoid boundary equation (6) of the reachable region of the non-cooperative target, we can obtain the distance between the spacecraft and the non-cooperative target at the pulse Δv s1 Under the action of t f The collision detection function F(Δv s1 ,t f );

[0071]

[0072] When F ≥ 0, the spacecraft is outside the ellipsoid or at the boundary of the reachable area of non-cooperative spacecraft, and it is considered that there is no collision risk at this time;

[0073] When F < 0, the spacecraft is located in the ellipsoid of the reachable domain of non-cooperative spacecraft. It is considered that there is a collision risk at this time. It is necessary to be outside the ellipsoid of the reachable domain of non-cooperative spacecraft at all times during the mission time period. Therefore, the collision avoidance constraint is

[0074]

[0075] Among them, t0 represents the initial time, t end Indicates the end time;

[0076] Step 13: Simplify the constraint expression (12) of step 12 to obtain the new collision avoidance constraint (17);

[0077] The specific process is:

[0078] The minimum value of F is greater than or equal to 0, and the constraints in formula (12) are naturally satisfied. Let the time corresponding to the minimum value of F be t m , define t m To avoid collision,

[0079]

[0080] in

[0081]

[0082] Among them, F(Δv s1 ,t m ) represents the distance between the spacecraft and the non-cooperative target during the pulse Δv s1 Under the action of t m The collision detection function at the moment, Represents the function F(Δv s1 ,t) first-order partial derivative with respect to time t; t represents time;

[0083] Initial measurement state based on non-cooperative target Using the Kepler equation, we can get t m The nominal position vector at time and the nominal velocity vector

[0084] According to the initial state of the spacecraft Using the Kepler equation, we can get t m The position vector of the spacecraft in the original orbit at the moment With velocity vector

[0085]

[0086] The time derivative of the state transfer matrix is expressed as

[0087]

[0088] For any given pulse Δv s1 , t m Calculated by formula (13), the collision avoidance constraint (12) is also expressed as,

[0089] W(Δv s1 )=F(Δv s1 ,t m )≥0 (17)

[0090] Where W(Δv s1 ) indicates that F is in the pulse of Δv s1 When , the minimum value is taken;

[0091] Step 14: Use the Lagrange multiplier method to transform the inequality constraint expression (17) in step 13 into an equality constraint; the specific process is:

[0092] The optimization index is expressed as:

[0093]

[0094] Using the Lagrange multiplier method and introducing the Lagrange constant λ, the Lagrange equation H is

[0095] H=Δv s1 +λW(Δv s1 ) (19)

[0096] satisfy

[0097]

[0098] If λ=0, there is Δv s1 =0 3×1 , 0 3×1 represents a 3D column vector with all elements equal to 0, which is not realistic. Therefore, λ≠0, that is, when the fuel is optimal, W(Δv s1 )=0, at this time, the inequality constraint is transformed into an equality constraint, that is, F(Δv s1 ,t m )=0;||·|| represents taking the modulus of a vector or matrix;

[0099] Step 15: Based on the equality constraints of step 14, the final collision avoidance pulse optimization model is given; the specific process is as follows:

[0100] The collision avoidance pulse optimization model is

[0101]

[0102] Other steps and parameters are the same as those in the first embodiment.

[0103] Specific embodiment 3: This embodiment differs from specific embodiment 1 or 2 in that: in step 2, based on the optimization model of step 1, the Newton iteration method (29) is used to solve the given collision avoidance time t m Minimum pulse of the lower spacecraft;

[0104] The specific process is:

[0105] Step 21: Simplify the given collision avoidance time t m The optimal pulse optimization model;

[0106] The specific process is:

[0107] Formula (11) can be written as

[0108]

[0109] Among them, c is the three-dimensional column vector defined by the simplified function F, Ψ is the matrix The third-order square matrix composed of eigenvalues, the acquisition process of the two is as follows

[0110] c=ρ T Δv s1 +y0

[0111]

[0112] Ψ=diag(ψ1,ψ2,ψ3)

[0113] Among them, y0 is a three-dimensional column vector defined by the simplified variable c, and ψ1, ψ2, and ψ3 are all matrices , diag(·) is the function that generates the diagonal matrix, each column of the matrix ρ is the eigenvector corresponding to the eigenvalue of the matrix Θ, with ||ρ||=1, ||·|| represents the modulus of the vector or matrix;

[0114] Expressing Equation (14) as the equation of vector c is:

[0115]

[0116] Among them, B and D are coefficient matrices, and the acquisition process is as follows:

[0117]

[0118] The purpose of this section is to find the spacecraft that satisfies constraints (12) and (13) under the premise that the distance between the spacecraft and the boundary of the reachable region of the non-cooperative target is the shortest at a given moment, and let ||Δv s1 ||Minimum Δv s1 ;

[0119] Since ||ρ||=1 and at t m Given that y0 is a constant vector, given the collision avoidance time t m The optimal pulse optimization can be equivalent to finding c that satisfies the constraints in equations (22) and (23). * , where when c=c * When ||c-y0|| is the smallest, it is not difficult to see that formula (22) is a standard ellipsoid about the variable c. Therefore, the variable c is expressed in the form of an angle, that is,

[0120] c=[ε1 cosθsinφ,ε2sinθsinφ,ε3cosφ] T (twenty four)

[0121] Among them, ε1, ε2 and ε3 are the lengths of the three semi-axes of the ellipsoid, θ and φ are angles, satisfying θ∈(0,360°], φ∈[0,180°];

[0122] So given the collision avoidance time t m The optimal pulse optimization model is

[0123]

[0124] Step 2: Using the Lagrange equation, transform (25) into a system of two-variable equations;

[0125] The specific process is:

[0126] Using the Lagrange multiplier method, the Lagrange multiplier λ is introduced t , Lagrange equation H t for

[0127] H t =||c(θ,φ)-y0|| 2 +λ t (c T Bc+Dc) (26)

[0128] The optimal solution satisfies

[0129]

[0130] The two derivatives are expressed as

[0131]

[0132] Eliminate the Lagrange constant λ using the two sub-formulas of formula (27) t , we get a binary equation system S(θ,φ)

[0133]

[0134] Among them, k θ and k φ is the coefficient of the binary equation system S(θ,φ), and the acquisition process is

[0135]

[0136] Step 23: Use Newton iteration to solve the binary equations in step 22 and get the given collision avoidance time t m The minimum pulse of the spacecraft at

[0137] The specific process is:

[0138] Formula (28) is a set of two-variable equations about angles, which can be solved by Newton iteration.

[0139]

[0140] Where m is the number of Newton iterations in formula (29),

[0141] θ (m+1) is the value of the angle θ for the m+1th iteration;

[0142] φ (m+1) is the value of the angle φ for the m+1th iteration;

[0143] θ (m) is the value of the angle θ for the mth iteration;

[0144] φ (m) is the value of the angle φ for the mth iteration;

[0145] S (m) θ is the angle iteration for the mth time (m) and φ (m) The corresponding function value;

[0146] J H is the Jacobian matrix of the binary equation system S(θ,φ) in equation (28);

[0147]

[0148] Where S1 represents the

[0149] S2 represents c in formula (28) T Bc+Dc=0;

[0150] Indicates the mth iteration θ (m) and φ (m) The value of the corresponding Jacobian matrix;

[0151] Given a set of initial angle values θ in equation (29) (0) and Then, the optimal solution c is solved iteratively through formula (29) * The corresponding angle θ * and φ * , satisfying θ * ∈(0,360°],φ * ∈[0,180°]; at this time, the collision avoidance time is given as t m The minimum pulse of the spacecraft at for

[0152]

[0153] Other steps and parameters are the same as those in the first or second embodiment.

[0154] Specific embodiment 4: This embodiment differs from any one of the specific embodiments 1 to 3 in that: the Jacobian matrix in steps 2 and 3

[0155]

[0156]

[0157] Among them, Equations (32) to (35) represent the four components of the Jacobian matrix in Equation (30), where the second-order partial derivatives with respect to c are as follows

[0158]

[0159] Then we can use the expression of Jacobian matrix in equation (30).

[0160] The other steps and parameters are the same as those in the first to third embodiments.

[0161] Specific embodiment 5: The difference between this embodiment and any one of the specific embodiments 1 to 4 is that the initial value θ of a set of angles in formula (29) in steps 2 and 3 is (0) and By coordinates get;

[0162] The coordinates The distance coordinate y0 on the standard ellipsoid about c defined by equation (22) is [y 01 ,y 02 ,y 03 ] T The coordinates of the nearest point;

[0163] The specific process is:

[0164] 1) Distance coordinate y0 on the ellipsoid = [y 01 ,y 02 ,y 03 ] T The coordinates of the nearest point satisfy

[0165]

[0166] in, for The x-axis component of for The y-axis component of for The z-axis component of

[0167] y 01 is the x-axis component of y0, y 02 is the y-axis component of y0, y 03 is the z-axis component of y0;

[0168] The above coordinates all fall into the coordinate system of the ellipsoid. The coordinate system of the ellipsoid takes the center of the ellipsoid as the origin, and the three main axes of the ellipsoid are the x, y, and z axes;

[0169] κ is a parameter variable;

[0170] 2) κ satisfies

[0171]

[0172] This is a one-dimensional equation with respect to the parameter variable κ, where κ∈(-min(ε1,ε2,ε3) 2 ,∞);

[0173] The root of U(κ)=0 is solved by Newton iteration, the process is:

[0174]

[0175] Where l is the number of Newton iterations in equation (38), κ (l+1) is the value of the parameter variable κ for the l+1th iteration, κ (l) is the value of the parameter variable κ for the first iteration, U(κ (l) ) is the lth iteration κ (l) The corresponding function value;

[0176] U′(κ) is the first-order derivative of the function U with respect to κ;

[0177]

[0178] U′(κ (l) ) is the lth iteration κ (l) The value of the corresponding first-order derivative of the function;

[0179] The initial value of the Newton iteration problem is κ (0) =-min(ε1,ε2,ε3) 2 ;

[0180] 3) Solve equations (38) and (39) to obtain κ and substitute it into equation (36) to obtain Then, using formula (40), we can get the initial value of the angle in formula (29):

[0181]

[0182] The initial value of the iteration of formula (29) is given by considering only the first constraint condition in the optimization problem (21). Formula (22) defines a standard ellipsoid about c. The standard ellipsoid is in a coordinate system with the center of the ellipsoid as the origin and the three main axes of the ellipsoid as the x, y, and z axes. Find c that satisfies formula (22) and minimizes ||c-y0|| as The calculation process is transformed into finding the coordinates of the point on the ellipsoid closest to the end point of the vector y0 starting from the origin, using Calculate the initial value θ in equation (29) (0) and

[0183] The other steps and parameters are the same as those in the first to fourth embodiments.

[0184] The following examples are used to verify the beneficial effects of the present invention:

[0185] Example 1:

[0186] Assume that the initial orbit parameters of the spacecraft and the initial measured orbit elements of the non-cooperative target are as shown in Table 1, and the position measurement errors of the non-cooperative target are The speed measurement error is When constructing the reachable domain, k=3 is taken, and the probability that the true value falls within the error ellipsoid is 97.07%. The results were run on a CoreTM i7 2.60GHz and MATLAB 2018b running Windows 10.

[0187] The minimum value of the function F on the original orbit of the spacecraft occurs at time At this point, F = -0.9206, and the spacecraft is located inside the reachable region of the non-cooperative target. At this point, the distance between the spacecraft and the non-cooperative target is 0.4654m, and the boundary of the reachable region is 1.1860m.

[0188] The collision avoidance mission is completed within one cycle of the spacecraft, t end =6307.12s. To address the issue of not returning to the original orbital phase after collision avoidance, the simulation results of a single-pulse semi-analytical method are presented, in which a pulse is applied at the start of the mission.

[0189] Table 1 Spacecraft initial orbit parameters and non-cooperative target initial measurement orbit parameters

[0190]

[0191] The spacecraft applies a pulse to avoid collision at the initial moment, that is, t1 = 0, and the minimum avoidance pulse corresponding to the minimum avoidance pulse is obtained. The collision avoidance pulse is The pulse amplitude is The calculation of the above results takes 0.1581s. The orbit of the spacecraft and the non-cooperative target before and after the maneuver and the reachable area of the non-cooperative target are as follows: Figure 3 As shown, the relationship between the orbit of the spacecraft before and after the maneuver and the position of the reachable area of the non-cooperative target is as follows: Figure 4 shown.

[0192] Figure 3 The image shows the changes of the original orbit of the spacecraft, the orbit after maneuvering, the observed orbit of the non-cooperative target and its reachable domain over time (the reachable domain is magnified 100 times). Figure 4 (a) shows the position of the spacecraft in the original orbit and the orbit after maneuvering at different times, as well as the reachable area of the non-cooperative target. Figure 4 (b) shows the cross section of the non-cooperative target reachable region when the spacecraft reaches the boundary of the non-cooperative target reachable region after the maneuver. Figure 4 As can be seen from (b) in the figure, for the original orbit without pulse application, the spacecraft is is in the reachable domain of the non-cooperative target at the moment. After the pulse is applied, the spacecraft is The moment is exactly at the boundary of the reachable region of the non-cooperative target. Figure 4 As can be seen in (a), Spacecraft in collision avoidance orbits are outside the reachable domain.

[0193] The distance between the spacecraft and the boundary of the reachable domain changes with time as follows Figure 5 shown. Figure 5 In the spacecraft Time Satisfaction At this time, the spacecraft is outside the reachable domain of the non-cooperative target, and its distance to the boundary of the reachable domain is 0.007m. This distance is not 0 due to the error caused by the two-body state transfer matrix.

[0194] For the minimum pulse problem of collision avoidance alone, the simulation results of the proposed method are compared with those of a single-pulse numerical optimization method and a multi-pulse numerical optimization method in which the optimization variables are all pulse times and pulse timings are free. The average fuel consumption, percentage increase in fuel consumption, average computation time, and percentage reduction in computation time of the above methods are shown in Table 2. The percentage increase in fuel consumption represents the percentage increase in fuel consumption of the single-pulse semi-analytical method with a fixed pulse time compared to the optimization method with free pulse timings, while the percentage reduction in computation time represents the percentage reduction in computation time of the single-pulse semi-analytical method with a fixed pulse time compared to the optimization method with free pulse timings.

[0195] It can be seen that, firstly, the fuel consumption of the method proposed in the present invention is almost the same as that of the single-pulse, double-pulse and triple-pulse methods based on numerical optimization. Therefore, collision avoidance can be performed using only a single pulse, and the method proposed in this patent can be approximately considered to be optimal. Secondly, the method proposed in this patent can significantly shorten the calculation time compared with the numerical optimization method. Its calculation time is shortened by 79.12%, 96.12% and 99.09% compared with the single-pulse, double-pulse and triple-pulse numerical optimization methods, respectively. This shows that the proposed method can effectively shorten the problem-solving time while ensuring collision avoidance and low fuel consumption.

[0196] Table 2 Fuel consumption and calculation time of the method of the present invention and the numerical optimization method

[0197]

[0198] The present invention may have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art may make various corresponding changes and modifications based on the present invention, but these corresponding changes and modifications should all fall within the scope of protection of the claims attached to the present invention.

Claims

1. A method for optimizing spacecraft impulse maneuvers with consideration of collision avoidance, characterized by: The specific process of the method is: Step 1: Based on the reachable domain envelope model, an optimization model for the single pulse collision avoidance problem with fixed pulse time is constructed; Step 2: Based on the optimization model of step 1, use the Newton iteration method to solve the minimum pulse of the spacecraft at a given collision avoidance time; Step 3: Perform a one-dimensional search on the collision avoidance time in step 2 to obtain the optimal collision avoidance time. Based on optimal collision avoidance time Get the minimum pulse of collision avoidance time freedom; In the step 1, based on the reachable domain envelope model, an optimization model for the single pulse collision avoidance problem with a fixed pulse time is constructed; The specific process is: Step 1: Calculate the position vector of the spacecraft under the action of the pulse; the specific process is: The time t under the pulse f Spacecraft position vector r sf The linear relationship is described as: in, is the time t in the original orbit f Spacecraft position vector, Δv s1 The first collision avoidance pulse for a spacecraft, The time from spacecraft t0 to t f The two-body state transfer matrix at time ; Represents the set of all 6-order square matrices in the real number domain; Transfer matrix Φ s (t f ,t0) is expressed as Among them, Φ s11 represents the spacecraft t f The partial derivative matrix of the position vector at time t0 with respect to the position vector at time t0, Φ s12 represents the spacecraft t f The partial derivative matrix of the position vector at time t0 with respect to the velocity vector at time t0, Φ s21 represents the spacecraft t f The partial derivative matrix of the velocity vector at time t0 with respect to the position vector at time t0, Φ s22 represents the spacecraft t f The partial derivative matrix of the velocity vector at time t0 with respect to the velocity vector at time t0; Step 1 and 2: Based on step 1, analyze the single pulse collision avoidance problem with fixed pulse time and give the constraint expression (12) that needs to be satisfied for collision avoidance; the specific process is as follows: Set time t f Spacecraft position vector r sf Substitute the ellipsoid boundary equation of the reachable region of the non-cooperative target into In the pulse Δv, the spacecraft and the non-cooperative target are obtained. s1 Under the action of t f The collision detection function F(Δv s1 ,t f ); When F ≥ 0, the spacecraft is outside the ellipsoid or at the boundary of the reachable area of non-cooperative spacecraft, and it is considered that there is no collision risk at this time; When F < 0, the spacecraft is located in the ellipsoid of the reachable domain of non-cooperative spacecraft. It is considered that there is a collision risk at this time. It is necessary to be outside the ellipsoid of the reachable domain of non-cooperative spacecraft at all times during the mission time period. Therefore, the collision avoidance constraint is Among them, t0 represents the initial time, t end Indicates the end time; r cf For non-cooperative goals at time t f The true position vector of For non-cooperative goals at t f At this moment, according to the initial measurement state The nominal position vector obtained by recursion with the Kepler equation; and are the measured values of the position and velocity vectors of the non-cooperative target, respectively; The superscript T means to find the transpose; A c11 is the error ellipse equation of the non-cooperative target reachable domain position ellipsoid matrix; it is calculated by the following formula: Among them, k 2 is the state error vector Mahalanobis distance; is the covariance matrix P c (t f ,t0), i=1,2; j=1,2; represents the set of all 3-order square matrices in the real number domain; Step 13: Simplify the constraint expression (12) of step 12 to obtain the new collision avoidance constraint (17); The specific process is: The minimum value of F is greater than or equal to 0, and the constraints in formula (12) are naturally satisfied. Let the time corresponding to the minimum value of F be t m , define t m To avoid collision, in Among them, F(Δv s1 ,t m ) represents the distance between the spacecraft and the non-cooperative target during the pulse Δv s1 Under the action of t m The collision detection function at the moment, Represents the function F(Δv s1 ,t) first-order partial derivative with respect to time t; t represents time; Initial measurement state based on non-cooperative target Using the Kepler equation to find t m The nominal position vector at time and the nominal velocity vector According to the initial state of the spacecraft Using the Kepler equation to find t m The position vector of the spacecraft in the original orbit at the moment With velocity vector r s0 and v s0 are the position and velocity vectors of the spacecraft at the initial time t0; The time derivative of the state transfer matrix is expressed as For any given pulse Δv s1 , t m Calculated by formula (13), the collision avoidance constraint (12) is also expressed as, W(Δv s1 )=F(Δv s1 ,t m )≥0(17) Where W(Δv s1 ) indicates that F is in the pulse of Δv s1 When , the minimum value is taken; Step 14: Use the Lagrange multiplier method to transform the inequality constraint expression (17) in step 13 into an equality constraint; the specific process is: The optimization index is expressed as: Using the Lagrange multiplier method and introducing the Lagrange constant λ, the Lagrange equation H is H=Δv s1 +λW(Δv s1 ) (19) satisfy If λ=0, there is Δv s1 =0 3×1 , 0 3×1 represents a 3D column vector with all elements equal to 0, which is not realistic. Therefore, λ≠0, that is, when the fuel is optimal, W(Δv s1 )=0, at this time, the inequality constraint is transformed into an equality constraint, that is, F(Δv s1 ,t m )=0;||·|| represents taking the modulus of a vector or matrix; Step 15: Based on the equality constraints of step 14, the final collision avoidance pulse optimization model is given; the specific process is as follows: The collision avoidance pulse optimization model is 2. The method for optimizing spacecraft impulse maneuvers with consideration of collision avoidance according to claim 1, characterized in that: In the step 2, based on the optimization model of the step 1, the Newton iteration method is used to solve the minimum pulse of the spacecraft at a given collision avoidance time; The specific process is: Step 21: Simplify the given collision avoidance time t m The optimal pulse optimization model; The specific process is: Formula (11) can be written as Among them, c is the three-dimensional column vector defined by the simplified function F, Ψ is the matrix The third-order square matrix composed of eigenvalues, the acquisition process of the two is as follows c=ρ T Δv s1 +y0 Ψ=diag(ψ1,ψ2,ψ3) Among them, y0 is a three-dimensional column vector defined by the simplified variable c, and ψ1, ψ2, and ψ3 are all matrices , diag(·) is the function that generates the diagonal matrix, each column of the matrix ρ is the eigenvector corresponding to the eigenvalue of the matrix Θ, with ||ρ||=1, ||·|| represents the modulus of the vector or matrix; Expressing Equation (14) as the equation of vector c is: Among them, B and D are coefficient matrices, and the acquisition process is as follows: Since ||ρ||=1 and at t m Given that y0 is a constant vector, given the collision avoidance time t m The optimal pulse optimization can be equivalent to finding c that satisfies the constraints in equations (22) and (23). * , where c=c * When ||c-y0|| is the smallest, equation (22) is a standard ellipsoid about the variable c. Therefore, the variable c is expressed in the form of an angle, that is, c=[ε1cosθsinφ,ε2sinθsinφ,ε3cosφ] T (24) Among them, ε1, ε2 and ε3 are the lengths of the three semi-axes of the ellipsoid, θ and φ are angles, satisfying θ∈(0,360°], φ∈[0,180°]; So given the collision avoidance time t m The optimal pulse optimization model is Step 2: Using the Lagrange equation, transform (25) into a system of two-variable equations; The specific process is: Using the Lagrange multiplier method, the Lagrange multiplier λ is introduced t , Lagrange equation H t for H t =||c(θ,φ)-y0|| 2 +λ t (c T Bc+Dc) (26) The optimal solution satisfies The two derivatives are expressed as Eliminate the Lagrange constant λ using the two equations in equation (27) t , we get a binary equation system S(θ,φ) Among them, k θ and k φ is the coefficient of the binary equation system S(θ,φ), and the acquisition process is Step 23: Use Newton iteration to solve the binary equations in step 22 and get the given collision avoidance time t m The minimum pulse of the spacecraft at The specific process is: Formula (28) is a set of two-variable equations about angles, which can be solved by Newton iteration. Where m is the number of Newton iterations in formula (29), θ (m+1) is the value of the angle θ for the m+1th iteration; φ (m+1) is the value of the angle φ for the m+1th iteration; θ (m) is the value of the angle θ for the mth iteration; φ (m) is the value of the angle φ for the mth iteration; S (m) θ is the angle iteration for the mth time (m) and φ (m) The corresponding function value; J H is the Jacobian matrix of the binary equation system S(θ,φ) in equation (28); Where S1 represents the S2 represents c in formula (28) T Bc+Dc=0; Indicates the mth iteration θ (m) and φ (m) The value of the corresponding Jacobian matrix; Given a set of initial angle values θ in equation (29) (0) and Then, the optimal solution c is solved iteratively through formula (29) * The corresponding angle θ * and φ * , satisfying θ * ∈(0,360°],φ * ∈[0,180°]; at this time, the collision avoidance time is given as t m The minimum pulse of the spacecraft at for 3. The method for optimizing spacecraft impulse maneuvers with consideration of collision avoidance according to claim 2, characterized in that: The Jacobian matrix of steps 2 and 3 Among them, Equations (32) to (35) represent the four components of the Jacobian matrix in Equation (30), where the second-order partial derivatives with respect to c are as follows 4. The method for optimizing spacecraft impulse maneuvers with consideration of collision avoidance according to claim 3, characterized in that: The initial value θ of a set of angles in equation (29) in steps 2 and 3 is (0) and By coordinates get; The coordinates The distance coordinate y0 on the standard ellipsoid about c defined by equation (22) is [y 01 ,y 02 ,y 03 ] T The coordinates of the nearest point; The specific process is: 1) satisfy in, for The x-axis component of for The y-axis component of for The z-axis component of y 01 is the x-axis component of y0, y 02 is the y-axis component of y0, y 03 is the z-axis component of y0; κ is a parameter variable; 2) κ satisfies Whereκ∈(-min(ε1,ε2,ε3) 2 ,∞); The roots of U(κ)×0 are solved by Newton iteration, the process is: Where l is the number of Newton iterations in equation (38), κ (l+1) is the value of the parameter variable κ for the l+1th iteration, κ (l) is the value of the parameter variable κ for the first iteration, U(κ (l) ) is the lth iteration κ (l) The corresponding function value; U′(κ) is the first-order derivative of the function U with respect to κ; U′(κ (l) ) is the time of the first iteration (l) The value of the corresponding first-order derivative of the function; The initial value of the Newton iteration problem is κ (0) =-min(ε1,ε2,ε3) 2 ; 3) Solve equations (38) and (39) to obtain κ and substitute it into equation (36) to obtain Then, using formula (40), we can get the initial value of the angle in formula (29):

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