A method for generating optimal low-thrust transfer trajectory based on switching function characteristics

Abstract

The application discloses a kind of optimal minimum-thrust transfer trajectory generation methods based on switching function characteristics, belong to aerospace field.The application implementation method is: in the geocentric inertial coordinate system, construct average dynamics model to describe the state of spacecraft, randomly generate the starting state of spacecraft of fuel consumption optimal transfer trajectory;By eliminating the control variable yaw angle and yaw switching angle of the switching function of YSS, construct the shutdown section transfer time expression composed of state and co-state, deduce the value range of the starting co-state of spacecraft when YSS switching function crosses 0 line upwards, randomly give starting co-state variable within the boundary range of above-solved co-state variable;Calculate the shutdown section transfer time under current starting state-co-state, judge whether transfer time constraint is satisfied;With starting state and starting co-state variable as integral starting point, simultaneously forward and reverse integration, generate complete optimal transfer trajectory, that is, complete the rapid generation of optimal minimum-thrust transfer trajectory.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for generating optimal low-thrust transfer trajectories based on switch function characteristics, in particular to a method for quickly generating optimal trajectories of small-thrust transfer between circular orbits under the control of natural precession and yaw switch, which is suitable for generating large-scale optimal low-thrust transfer trajectory data sets and is used for training neural networks to quickly evaluate fuel consumption for complex space missions, and belongs to the field of aerospace. BACKGROUND

[0002] In the low earth orbit region, complex multi-target space missions such as large-scale satellite constellation deployment and reconstruction and active debris removal tasks are actively carried out. These tasks usually involve multiple orbit maneuvers between near-circular orbits, and the overall speed increment is large. Small-thrust technology is a promising choice for these tasks due to its cost-effectiveness, and compared with traditional high-thrust systems, it can achieve the same speed change with less propellant consumption.

[0003] Although many methods can well solve the optimization problem of single low-thrust transfer trajectory, for complex multi-target space missions such as multi-debris access, in addition to the top-level rendezvous sequence planning, there are also bottom-level millions of single-satellite inter-orbit transfer optimization. Therefore, in complex space missions, fast fuel consumption evaluation is very necessary, because the traditional optimization method relying on a large amount of calculation load cannot meet the efficiency requirements of global search.

[0004] In recent years, machine learning methods such as deep neural networks have been used to quickly evaluate the fuel consumption of low-thrust transfer. A large-scale high-quality optimal transfer trajectory data set is the basis for high-quality network training. For the method of generating optimal transfer trajectory data set, it can be started from the perspective of directly solving the optimal transfer problem, and a large number of optimal transfer problems are directly solved using indirect method to obtain a sufficient amount of data. This method is easy to implement and the calculation idea is clear, but its convergence rate is low, and the calculation time of a single transfer trajectory is as long as about 60s, which is difficult to construct a data set of sufficient scale.

[0005] For the calculation of the optimal transfer trajectory between specific circular orbits, the first technology [1] (see Low-Thrust Transfer Between Circular Orbits Using Natural Precession and Yaw Switch Steering[J] Journal of Guidance Control and Dynamics, Wen, C., Zhang, C., Cheng, Y., and Qiao, D., 2021) proposes a small-thrust transfer strategy under the yaw switch steering (YSS) of natural precession and yaw switch steering. The method improves the solving efficiency under the premise of ensuring the optimality, and generates a single transfer trajectory in 6s, but the direct generation of the data set success rate is only about 40%

[0006] However, the optimal trajectory constructed by the optimal sample reverse generation method is highly concentrated around the nominal trajectory in space, so that the constructed data set is too concentrated in the spatial distribution, and is only suitable for specific transfer problems and does not have universality. SUMMARY

[0007] In order to solve the problem of slow calculation speed of the minimum fuel consumption orbit transfer, the purpose of the present application is to provide an optimal small-thrust transfer trajectory generation method based on the characteristics of the switch function, which constructs an average dynamics model to describe the spacecraft state in the geocentric inertial coordinate system, and randomly generates the initial state of the spacecraft of the fuel-optimal transfer trajectory;By eliminating the control variables of the yaw angle and the yaw switching angle of the YSS switch function, an expression composed of the state and the co-state is constructed, the value range of the initial co-state of the spacecraft when the YSS switch function crosses the 0 line upwards is derived, and the initial co-state variable is randomly given within the above-solved co-state variable boundary range;Calculate the transfer time of the shutdown section under the current initial state-co-state, and judge whether the transfer time constraint is satisfied;Take the initial state and the initial co-state variable as the integral starting point to generate a complete optimal transfer trajectory, that is, to complete the fast generation of the optimal small-thrust transfer trajectory.

[0008] The purpose of the present application is realized by the following technical solutions.

[0009] The optimal small-thrust transfer trajectory generation method based on the characteristics of the switch function disclosed by the present application comprises the following steps:

[0010] Step 1: Construct an average orbit dynamics equation to describe the state of the spacecraft, and randomly generate the integral initial spacecraft state X of the optimal transfer trajectory s ;

[0011] The spacecraft state X is represented by the velocity V, the orbit inclination I and the ascending node right ascension Ω, i.e., X = [V, I, Ω];

[0012] According to the YSS strategy, the control variable thrust size f, f ∈ [0, f max ], the yaw angle β, β ∈ [-π, π] and the latitude amplitude angle θ are introduced, and the average orbit dynamics equation is obtained by using the Gauss variation equation:

[0013]

[0014] In formula (1),

[0015]

[0016] Where, the equatorial radius J2 perturbation coefficient J2 = 1.0825267 × 10 -3 , the gravitational parameter μ = 3.986004418 × 10 14 m 3 / s 2

[0017] The integral starting state of the optimal transfer trajectory is X s = [V s ,I s ,Ω s ], and the construction expression of X s is:

[0018]

[0019] Where, I s is the integral starting orbit inclination, Ω s is the integral starting ascending node right ascension, Ω s0 is a specific integral starting ascending node right ascension value; the subscript min represents the minimum value, the subscript max represents the maximum value, and rand(0, 1) represents a random sample in the range of (0, 1) subject to uniform distribution; the starting velocity V s is replaced by the starting orbit height h s for more intuitive representation of the position of the spacecraft relative to the ground; the conversion relationship between the velocity and the orbit height h is:

[0020]

[0021] Step two: analyze the condition when the switch function in the YSS strategy crosses the 0 line upward, derive the value range of the corresponding co-state variable, and generate the integral starting co-state variable of the optimal transfer trajectory;

[0022] The state X of the spacecraft and the co-state λ = [λ V ,λ I ,λΩ The switch function S is expressed as:

[0023]

[0024] where λ V is the velocity co-state variable, and the expression of the intermediate variable K is:

[0025]

[0026] where λ I is the orbit inclination co-state variable, and λ Ω is the ascending node right ascension co-state variable;

[0027] The derivative of the switch function is:

[0028]

[0029] When the switch function first crosses zero, the initial co-state variable must satisfy S = 0 and is expressed as:

[0030]

[0031] The integral initial velocity co-state variable must satisfy the constraint equation (9) and is expressed as:

[0032]

[0033] where, and The signs of and are randomly given in data generation; since the initial spacecraft state X is known, the values of are randomly generated within the allowed range of equation (9);

[0034] The integral initial velocity co-state variable must satisfy the constraint equation (10):

[0035]

[0036] When and are randomly given within the boundary constraint range, the integral initial orbit inclination co-state variable is directly obtained by equation (11):

[0037]

[0038] The complete initial co-state variable is thus obtained

[0039] Step three: calculate the transfer time Δt in the current integral initial state and the integral initial covariant c , judge whether the transfer time constraint is satisfied; if satisfied, proceed to step four, otherwise return to step one iteration.

[0040] Δt c The expression is:

[0041]

[0042] Judge whether the shutdown segment time is within the maximum allowed transfer time t fmax , that is, Δt c <t fmax . If satisfied, proceed to step four, otherwise return to step one iteration.

[0043] Step four: take the initial spacecraft state X s =[a s ,I s ,Ω s ] obtained in step one and the initial covariant obtained in step two as the integral starting point, and integrate Δt1 in reverse and Δt c +Δt2 in forward, to generate the complete optimal transfer trajectory; Δt1 is the first segment startup time; Δt2 is the second segment startup time;

[0044] Ensure that the overall transfer time Δt1+Δt c +Δt2 is within the maximum allowed transfer time (0,t fmax ];

[0045] The maximum thrust acceleration f max :

[0046] f max =f mmax +(f mmax -f mmin )×rand(0,1) (m / s 2 ) (13)

[0047] Where f mmax is the maximum allowed value of the maximum thrust acceleration, f mmin is the minimum allowed value of the maximum thrust acceleration;

[0048] To ensure that the fuel consumption ΔV is within the range specified by the maximum allowed value ΔV max , Δt1 and Δt2 are randomly given according to equation (14):

[0049]

[0050] According to the YSS strategy, the optimal transfer trajectory co-state variable satisfies formula (15):

[0051]

[0052] The starting co-state is integrated according to formula (15), and the co-state variable at any time is obtained, and the corresponding optimal control is:

[0053]

[0054] Where, f * , θ * and β * are the optimal thrust size, yaw angle and latitude amplitude angle respectively.

[0055] The optimal control is brought into the dynamic equation of formula (1), so that the starting spacecraft state X s =[a s , I s , Ω s ] and the starting co-state are taken as the integration starting point, and are simultaneously integrated in reverse Δt1 and forward Δt c +Δt2, that is, the optimal transfer trajectory is generated.

[0056] Advantages:

[0057] 1. The optimal small-thrust transfer trajectory generation method based on the switching function characteristics disclosed in the application determines the co-state variable boundary by using the switching function characteristics of the YSS strategy, directly integrates to obtain the optimal transfer trajectory, avoids the problem of guessing the initial value of the co-state variable in the traditional method of directly solving the optimal transfer problem, significantly improves the data generation efficiency, and can be used for large-scale data set generation.

[0058] 2. The optimal small-thrust transfer trajectory generation method based on the switching function characteristics disclosed in the application takes random state and random co-state satisfying the boundary requirement as the starting point of orbit integration, simultaneously integrates forward and reversely to obtain the complete optimal transfer trajectory, the trajectory data generated by using the method is not constrained by the nominal trajectory, the data set has a relatively uniform spatial distribution characteristic, and can be used for fuel consumption prediction in a larger space range such as low orbit. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 is a flowchart of the optimal small-thrust transfer trajectory generation method based on the switching function characteristics disclosed in the application.

[0060] Figure 2 is the switching function of the optimal transfer trajectory generated in the example of the application. DETAILED DESCRIPTION

[0061] For better illustrating the purposes and advantages of the present application, the summary of the application is further explained below in combination with the drawings and examples.

[0062] Embodiment 1:

[0063] In order to verify the feasibility of the method, the present application is now given the value range of transfer time, fuel consumption, orbital altitude, orbital inclination, initial ascending node right ascension, maximum thrust acceleration for low-orbit transfer task, and the specific parameters are shown in the following table:

[0064] Table 1 Parameter boundaries in data generation

[0065]

[0066] Among them, according to the ascending node right ascension property in YSS strategy, in order to simplify the calculation, the initial ascending node right ascension is set to a constant value here.

[0067] As shown in Figure 1 , the optimal small-thrust transfer trajectory generation method based on the characteristics of the switching function disclosed in the present embodiment has the following specific implementation steps:

[0068] Step 1: Construct the average orbit dynamics equation to describe the spacecraft state, and randomly generate the integral initial spacecraft state X of the optimal transfer trajectory s ;

[0069] The spacecraft state X is represented by the velocity V, the orbital inclination I and the ascending node right ascension Ω as X = [V, I, Ω];

[0070] According to the YSS strategy, the control amount thrust size f, f ∈ [0, f max ], yaw angle β, β ∈ [-π, π] and latitude amplitude angle θ are introduced, and the average orbit dynamics equation is obtained by using the Gauss variation equation:

[0071]

[0072] In formula (17),

[0073]

[0074] Among them, the equatorial radius J2 perturbation coefficient J2 = 1.0825267 × 10 -3 , gravitational parameter μ = 3.986004418 × 10 14 m 3 / s 2

[0075] The integral initial state of the optimal transfer trajectory is X s = [V s ,I s ,Ωs ], X s The configuration expression of S is:

[0076]

[0077] where I s is the integral initial orbit inclination, Ω s is the integral initial right ascension of the ascending node, Ω s0 is the specific integral initial right ascension of the ascending node; the subscript min represents the minimum value, the subscript max represents the maximum value, and rand(0, 1) represents a random sample in the range of (0, 1) subject to uniform distribution; the initial velocity V s is replaced by the initial orbit height h s for a more intuitive representation of the position of the spacecraft relative to the ground; the conversion relationship between the velocity and the orbit height h is:

[0078]

[0079] At this time, the randomly generated initial orbit height h s = 456.0750 km, i.e. V s = 7637.0335 m / s, the initial orbit inclination I s = 85.3057°, and the initial right ascension of the ascending node Ω s = 0°.

[0080] Step two: analyze the conditions when the switching function in the YSS strategy crosses the 0 line upward, derive the value range of the corresponding co-state variables, and generate the integral initial co-state variables of the optimal transfer trajectory;

[0081] In the YSS strategy, the switching function expression is:

[0082]

[0083] By eliminating the control variables of the YSS switching function, the yaw angle β and the yaw switching angle θ, the switching function S expression composed of the spacecraft state X and the co-state λ = [λ V , λ I , λ Ω ] is constructed as:

[0084]

[0085] where λ V is the velocity co-state variable, and the expression of K is:

[0086]

[0087] where λ I is the orbit inclination co-state variable, and λ Ωis the mean anomaly at epoch;

[0088] derivative of the switching function is:

[0089]

[0090] the initial co-state variable at the first zero crossing of the switching function must satisfy S = 0 and is given by:

[0091]

[0092] the initial co-state variable of the integral must satisfy constraint (26) and is given by:

[0093]

[0094] where and the sign of and are randomly given in the data generation; since the initial spacecraft state X is known, is randomly generated within the allowed range of equation (9); here and the sign of and i.e., both are positive. Since I s = 85.3057°, which satisfies satisfies the second condition in equation (26), i.e., the initial co-state of the velocity is randomly generated here

[0095] the initial co-state variable of the integral must satisfy constraint expression (10):

[0096]

[0097] Since according to equation (27), we have the initial co-state of the ascending node is randomly generated here

[0098] When and are randomly given within the boundary constraint range, can be directly obtained and is given by:

[0099]

[0100] is calculated to be Thus the complete initial covariant can be obtained

[0101] Step three: calculate the transfer time Δt of the shutdown phase under the current integral initial state and integral initial covariant c , to determine whether the transfer time constraint is satisfied;

[0102] Δt c The expression is:

[0103]

[0104] Determine whether the shutdown phase time is within the maximum allowed transfer time t fmax , that is, Δt c < t fmax . If it is satisfied, proceed to step four, otherwise return to step one and iterate. The calculation can obtain Δt c = 5.6628 day, which satisfies the time constraint.

[0105] Step four: take the initial spacecraft state X s = [a s , I s , Ω s ] obtained in step one and the initial covariant obtained in step two as the integral starting point, and integrate Δt1 in reverse and Δt c + Δt2 in forward, to generate the complete optimal transfer trajectory; Δt1 is the first segment on time; Δt2 is the second segment on time;

[0106] Ensure that the overall transfer time Δt1 + Δt c + Δt2 is within the maximum allowed transfer time (0, t fmax ];

[0107] The maximum thrust acceleration f max :

[0108] f max = f mmax + (f mmax - f mmin ) × rand(0, 1) (m / s 2 ) (30)

[0109] Where f mmax is the maximum allowed value of the maximum thrust acceleration, f mmin is the minimum allowed value of the maximum thrust acceleration; Here f max = 0.0042 m / s 2 is randomly generated.

[0110] To ensure that the fuel consumption ΔV is within the maximum allowed value ΔV maxThe Δt1 and Δt2 are randomly given by formula (31) within the prescribed range:

[0111]

[0112] Here Δt1 = 0.0639 day, Δt2 = 1.3805 day;

[0113] According to the YSS strategy, the optimal transfer trajectory co-state variable satisfies formula (15):

[0114]

[0115] The initial co-state is integrated by formula (15), and the co-state variable at any time is obtained, and the corresponding optimal control is:

[0116]

[0117] Where, f * , θ * and β * are the optimal thrust size, yaw angle and latitude amplitude angle respectively.

[0118] The optimal control is brought into the dynamic equation of formula (1), and the initial spacecraft state X s = [a s , I s , Ω s ] and the initial are taken as the integration starting point, and Δt1 is simultaneously integrated in reverse and Δt c + Δt2 in positive direction, that is, the optimal transfer trajectory can be quickly generated.

[0119] Figure 2 The switching function of the generated optimal transfer trajectory is about 0.0138 s for 1000 above-mentioned trajectories;

[0120] The above specific description further details the purpose, technical scheme and beneficial effects of the application. It should be understood that the above description is only a specific embodiment of the application and is not used to limit the protection scope of the application. Any modification, equivalent replacement, improvement, etc. within the spirit and principle of the application should be included in the protection scope of the application.

Claims

1. A method for generating optimal low-thrust transfer trajectory based on switching function characteristics, characterized in that: comprising the steps of Step one: Construct the averaged orbital dynamics equations to describe the spacecraft state and randomly generate the integral initial spacecraft state of the optimal transfer trajectory ; The step one implementation method is, Spacecraft state By velocity , orbital inclination and longitude of ascending node are expressed as ; According to the YSS strategy, the control quantity thrust size is introduced , , the yaw angle , and the latitude amplitude , the average orbit dynamics equation is obtained by using the Gauss variation equation: Formula wherein, wherein the equatorial radius , J2 perturbation coefficient , gravitational parameter The initial state of the integral of the optimal transfer trajectory is , The construct expression is: where, is the integral initial orbit inclination, is the integral initial right ascension of the ascending node, is a specific integral initial right ascension of the ascending node value; subscript represents the minimum value, subscript represents the maximum value, represents a random sample subject to a uniform distribution in the range; initial velocity is replaced by the initial orbit altitude for a more intuitive representation of the position of the spacecraft from the ground; velocity and orbit altitude conversion relationship: Step two: analyze the condition of the switch function crossing the 0 line upward in the YSS policy, derive the value range of the corresponding co-state variable and generate the integral starting co-state variable of the optimal transfer trajectory; The step two implementation method is, The configuration is made from the spacecraft state and the co-state The switching function is composed of The expression is: wherein is the velocity co-state variable, the intermediate variable is given by wherein is the orbital inclination co-state variable, is the ascending node right ascension co-state variable; Derivative of a switching function is: switching function crosses zero for the first time, starting co-state variable must satisfy and is expressed as: Integral start speed covariate The constraint (9) must be satisfied, expressed as: wherein, and the symbol and are randomly given in the data generation; the values of are known, are randomly generated within the allowed range of the equation allowing Integral start speed covariate Constraint expression to be satisfied : When and are randomly given within the boundary constraint range, the integral initial orbit inclination co-state variable is directly obtained by ​ Thus obtaining the complete initial co-state variable ; Step three: calculate the transfer time of the shutdown phase under the current integral start state and the integral start co-state determine whether the transfer time constraint is satisfied; if satisfied, proceed to step four, otherwise return to step one iteration; In step three, The expression is: determining whether the shutdown segment time is within the maximum allowed transfer time within, i.e. ; if so, proceed to step four, otherwise return to step one for iteration; Step four: Start with the initial spacecraft state from step one and the initial co-state from step two Integrate backwards from the initial time Integrate forward from the initial time to generate the complete optimal transfer trajectory is the first segment on-time is the second segment on-time The step four implementation method is, to ensure overall transfer time within the maximum allowed transfer time ​​ Maximum thrust acceleration : wherein is the maximum thrust acceleration maximum allowable value, is the maximum thrust acceleration minimum allowable value; To ensure the fuel consumption of the maximum allowable value specified in the range, randomly given by equation (14) and : According to the YSS policy, the optimal transfer trajectory co-state variable satisfies the equation : The starting co-state is given by Integrating the above equation, we get the co-state variable at any time, and the corresponding optimal control is wherein, , and are the optimal thrust magnitude, yaw angle and latitude amplitude, respectively; Substituting optimal control Within the dynamic equations, starting from the initial spacecraft state and initial costate Starting from the integral, integrate simultaneously in reverse according to the dynamic equation. positive integral This means generating the optimal transfer trajectory.

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