CN109828594B - Electromagnetic spacecraft configuration reconstruction method with low fuel consumption and stable process - Google Patents
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Abstract
The invention belongs to the technical field of electromagnetic spacecrafts, and particularly relates to a configuration reconstruction method of an electromagnetic spacecraft with low fuel consumption and stable process. The method mainly comprises the following steps: (S1) calculating an initial relative equilibrium state configuration and a target relative equilibrium state configuration of the electromagnetic spacecraft formation system, respectively, (S2) designing a time interval as a connection period of the expected initial relative equilibrium unstable manifold and the target relative equilibrium stable manifold; (S3) respectively selecting unstable manifolds in the initial relative equilibrium state configuration global manifolds and a certain number of discrete state points on the target relative equilibrium state configuration global manifolds relative to the stable manifolds in the equilibrium state configuration global manifolds, establishing a scalar optimization objective function related to the deviation between the state points, designing pulse speed increment and electromagnetic magnetic moment to enable the scalar optimization objective function to approach 0, and realizing the reconstruction of the two spacecrafts. The invention introduces electromagnetic force control in configuration reconstruction design, fully utilizes the advantage of not consuming fuel and has better stability.
Description
Technical Field
The invention belongs to the technical field of electromagnetic spacecrafts, and particularly relates to a configuration reconstruction method of an electromagnetic spacecraft with low fuel consumption and stable process.
Background
With the enhancement of the capability of the micro-nano spacecraft to perform the on-orbit task, the requirement of realizing the single large spacecraft task or the new concept on-orbit task through the micro-nano spacecraft cluster gradually becomes a typical mode. The flexibility of the configuration is realized through the micro-nano spacecraft clustering and reconstruction, and the on-orbit task capability, the precision, the robustness and the upgrading capability can be obviously improved. At present, aiming at spacecraft cluster configuration reconstruction under the action of conventional inertial thrust, a plurality of researches are carried out, and some optimization design methods are provided; the introduction of the satellite-borne controllable electromagnetic device enables the micro-nano spacecraft control dynamics to show new characteristics, including no fuel consumption, existence of relative equilibrium state and stability, satisfaction of dynamics conservation and the like.
The prior art is mainly based on a reconstruction planning method, the intrinsic characteristics of a control object (particularly a micro-nano electromagnetic spacecraft) are less considered, and a designed reconstruction optimization scheme has the defects of insufficient mining potential, suboptimum and the like, and comprises the following steps: (1) the self characteristics of the electromagnetic spacecraft are not fully utilized, and the configuration reconstruction consumes a large amount of fuel; (2) the characteristics of the configuration reconstruction process are not sufficient, and dynamic unstable characteristics may exist in the process.
Disclosure of Invention
According to the method, the relative equilibrium state characteristic of the electromagnetic cluster spacecraft is fully utilized from the control dynamics characteristic of the micro-nano electromagnetic spacecraft, and the configuration reconfiguration requirements of various requirements such as long term, multiple times, expandability and the like can be met by obtaining the electromagnetic spacecraft configuration reconfiguration method meeting the low fuel consumption and the process stability based on the electromagnetic control dynamics invariant manifold design and the unstable-stable manifold switching optimization control design. The specific technical scheme of the invention is as follows:
a low-fuel-consumption and stable-process electromagnetic spacecraft configuration reconstruction method is applied to an electromagnetic spacecraft formation system consisting of double electromagnetic spacecrafts, and mainly comprises the following steps:
(S1) respectively calculating the initial relative equilibrium state configuration and the target relative equilibrium state configuration of the electromagnetic spacecraft formation system, wherein the specific process is as follows:
(S11) taking the centroid CM of the electromagnetic spacecraft formation system as an origin oCMEstablishing a reference coordinate system o for dynamic modelingCMxyz，oCMThe x-axis is along the CM ground center distance direction, oCMy-axis along track speed direction, oCMz axis and oCMx-axis, oCMThe y axis forms a right-hand rectangular coordinate system based on a coordinate system oCMxyz, establishing a translational motion kinetic equation of each spacecraft relative to the CM;
(S12) solving the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction of the electromagnetic spacecraft formation system according to the translational motion kinetic equation of each spacecraft;
(S13) obtaining a relative equilibrium state X of the system according to the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction*Then according to the relative equilibrium state X of the system*Linearizing the translation motion dynamic model of the relative CM to obtain the relative equilibrium state X of the electromagnetic spacecraft formation in the system*The linearized model of (1);
(S14) mixing X*Substituting designed manifold initial state into system relative equilibrium state X*Obtaining a global manifold by integrating the linearized model, wherein the global manifold comprises a stable manifold and an unstable manifold;
(S2) designing a time interval, which is a connection period of the desired initial relatively equilibrium unstable manifold and the target relatively equilibrium stable manifold;
(S3) respectively selecting unstable manifolds in the initial relative equilibrium state configuration global manifolds and a certain number of discrete state points on the target relative equilibrium state configuration global manifolds, establishing a scalar optimization objective function related to the deviation between the state points, designing pulse speed increment and electromagnetic magnetic moment to enable the scalar optimization objective function to approach 0, and realizing the reconstruction of two spacecrafts from an initial distance L1 to an expected distance L2.
Further, the linearized model of the two electromagnetic spacecraft formation at X in the step (S13) is:
wherein δ X (τ) ═ X (τ) -X*(τ), X (τ) represents the system state at time τ, X*(τ) represents the relative equilibrium state of the system at time τ; i is3×3Representing a 3 x 3 identity matrix, the B matrix has different forms depending on the radial, normal and tangential relative equilibrium modes, as follows:
the radial direction is opposite to the equilibrium state,
in the normal direction, the state of the relative equilibrium,
tangential relative equilibrium state
In the formula (II)
And P ═ mu1zμ2z-μ1yμ2y；
Wherein L is the distance between two electromagnetic spacecrafts, M1＝m2/(m1+m2)，m1And m2Mass of the spacecraft 1,2, nCMIs the centroid CM orbital motion angular velocity, mu0For vacuum permeability (μ)1x,μ1y,μ1z)、(μ2x,μ2y,μ2z) Electromagnetic magnetic moment mu of two spacecrafts respectively1、μ2At oCMProjection components of the xyz coordinate system.
Further, the specific process in the step (S14) is as follows:
let the stable eigenvector of matrix A be VsUnstable eigenvector is VuThe steady flow form is WsUnstable manifold is WuThen, then
The initial state of the stable manifold is
The initial state of the unstable manifold is
Wherein epsilon represents a control parameter, the initial state
Substituting the linearized model into t ∈ [0 ∞ ] and t ∈ (- ∞ 0)]Integrating to obtain a stable manifold; will be in the initial state
Substituting the linearized model into t ∈ [0 ∞ ] and t ∈ (- ∞ 0)]And integrating to obtain the unstable manifold.
Further, the step (S3) includes the steps of:
selecting a time point tau on the initial relative equilibrium unstable manifoldu，0≤τu≤0.5τm，τmSelecting a time point tau on a target relative equilibrium stable manifold for a set configuration reconstruction times，0≤τs≤0.5τm。
The time interval [0,0.5 taum]Equally dividing the flow into N parts, and respectively calculating the stable manifold at each time point and the discrete state W corresponding to the unstable manifolds(τs)∈Ws、Wu(τu)∈WuFurther, furtherCalculating the deviation gamma (tau) of the corresponding stateu,τs)＝||r(τs)-r(τu)1+1000nCM||v(τs)-v(τu)1Memory for recording
Indicating the time corresponding to the minimum deviation value between the states;
the control scheme is as follows: applying pulse speed increment at 2N +1 time nodes, and applying constant electromagnetic magnetic moment control at 2N stages; the constraint conditions are as follows:
wherein Δ vρDenotes the ρ -th pulse velocity increment, ρ being an integer, and ρ being 1,2, …,2N +1, μkDenotes the kth electromagnetic moment, k being an integer, k being 1,2, …, 2N; u represents a control amount, Δ vmax、μmaxRespectively, capability thresholds of corresponding devices;
designing a comprehensive objective function J as follows:
performing Optimization calculation on the comprehensive objective function by adopting Particle Swarm Optimization (PSO for short), and obtaining the objective function meeting the task requirement
The control quantity U with the smallest sum is applied with the pulse speed increment.
In order to better understand the technical solution of the present invention, the related principles and derivation processes are further explained below.
1. Dynamic modeling and normalization design
Setting the mass centers (represented by a symbol 'CM') of two electromagnetic spacecraft formation systems to operate on a circular orbit, and selecting a dynamic modeling reference coordinate system as a Hill system o at the CMCMxyz (shown in fig. 2): oCMx is along the direction of CM ground center distance, oCMy in the track speed direction, oCMz and oCMx、oCMy constitutes a right-hand rectangular coordinate system. Based on oCMIn an xyz coordinate system, the translational kinematic equation of the spacecraft i (i ═ 1,2) relative to the CM is derived as:
in the formula, ri＝[xi yi zi]TDistance vector for spacecraft i relative to CM is at oCMProjection component, n, of an xyz coordinate systemCMFor CM orbital angular velocity, miIs the spacecraft i mass, uiIs the inertial thrust (characterized by pulse velocity increment) borne by the spacecraft i, Fi EMThe electromagnetic force applied to the spacecraft i is calculated by the following expression:
in the formula, mu0＝4π×10-7H/m is the vacuum permeability (mu)1,μ2) Electromagnetic magnetic moments, r, of spacecraft 1 and spacecraft 2, respectively21＝r1-r2Is a relative distance vector, r, pointing from the electromagnetic spacecraft 2 to the electromagnetic spacecraft 121＝||r21||2Symbol | | | non-conducting phosphor2Representing 2 norm.
Further, m is defined in terms of centroid CM1r1+m2r20 and r21＝r1-r2The definition can be given as follows:
by substituting the formula (3) into the formula (2), the relation between the electromagnetic force applied to the spacecraft and the relative CM distance vector can be obtained as follows:
in the formula, M1＝m2/(m1+m2)。
To eliminate n in formula (1)CMAnd
difference of magnitude, let τ be nCMt、
Substituted by formula (1) to obtain
2. Solving and reconstructing design of relative equilibrium state and corresponding manifold thereof
For two electromagnetic spacecrafts distributed along the radial direction, the normal direction and the tangential direction of the orbit, the radial direction is as follows: oCMx-direction, tangential: oCMy-direction, normal: oCMIn the z direction, if the distance between the two electromagnetic spacecrafts is set to be L, the corresponding relative equilibrium state condition is solved as follows:
radial relative equilibrium state
Normal relative equilibrium state
Tangential relative equilibrium state
In the above formula, (. mu.)1x,μ1y,μ1z)、(μ2x,μ2y,μ2z) Are respectively mu1、μ2At oCMProjection components of the xyz coordinate system.
For a general dynamics system, the manifold calculation given to a specific state is complex, and a manifold analytical solution is difficult to obtain; the invention adopts an approximation method to carry out numerical calculation of manifold. Based on m1r1+m2r20 and m1r′1+m2r′2Selecting the system state as X ═ X ═ 01 y1 z1 x′1 y′1 z′1]T；(x1,y1,z1) For spacecraft 1 at oCMLocation projection component of xyz coordinate system, (x'1,y′1,z′1) For spacecraft 1 at oCMThe velocity projection component of the xyz coordinate system.
The relative equilibrium state X can be obtained by the distance L between the two electromagnetic spacecrafts, the radial/normal/tangential distribution and the corresponding relative equilibrium state condition*. In a relatively equilibrium state X*Processing a linear equation (5) to obtain that two electromagnetic spacecraft formation positions are X*The linearized model of (1) is:
wherein δ X (τ) ═ X (τ) -X*(τ), from the radial, normal, and tangential relative equilibrium modes, the 3 × 3 matrix B has the form:
radial relative equilibrium state
Normal relative equilibrium state
Tangential relative equilibrium state
In the formula (II)
And P ═ mu1zμ2z-μ1yμ2y。
For the matrix a of equation (9), there are stationary (corresponding to eigenvalues whose real parts are negative), unstable (corresponding to eigenvalues whose real parts are positive) and central eigenvectors (corresponding to eigenvalues whose real parts are zero), respectively. Wherein the stable eigenvector VsAnd unstable eigenvector VuRespectively form a stable feature space EsAnd unstable feature space Eu. At X*Stable manifold of a non-linear system (characterized by equation (5)) in a certain small space (characterized by a multiplicative small quantity epsilon) in the vicinity
Unstable manifold
Respectively with a stable feature space EsUnstable feature space EuTangent. Thus, corresponding to the system relative equilibrium state X*Its global manifold Ws(or W)u) The calculation is as follows: will be in the initial state
Or
Substituted into formula (9), and further directed tot＝∞Andt＝-∞and integrating to obtain the approximate global manifold. Solving mode see FIG. 3, WsTo stabilize the manifold, WuIs an unstable manifold;
and
are each X*Selected to be tangential to the stable, unstable manifold. Ws-、Ws+Are each relative to X*The negative direction and the positive direction of the flow form are stabilized; wu-、Wu+Are each relative to X*Negative, positive unstable manifold.
Therefore, the reconstruction of the relative equilibrium state of the two electromagnetic spacecraft formation can be carried out based on manifold, the characteristics of invariant manifold and electromagnetic control capability are utilized to the maximum extent, the use of inertial propulsion is effectively reduced, the specific design idea is shown in figure 4, and X is1 *Is in an initial relative equilibrium configuration, X2 *In a desired relative equilibrium configuration; reconfiguration along X1 *Starting from an unstable manifold, with or without intermediate inertial thrust action (pulsed velocity incremental application) to transition to X2 *The stable manifold of (2).
3. Pulse velocity increment and electromagnetic magnetic moment optimization design process
(1) Determining a desired moment of connection of a stable manifold to an unstable manifold
Selecting the time interval of 0 to taus≤0.5τmAnd 0. ltoreq. tauu≤0.5τmCalculating
Discrete state W corresponding to stable manifolds(τs)∈WsCalculating
Discrete state W corresponding to unstable manifoldu(τu)∈WuLet a discrete variable Γ (τ)u,τs)＝Ws(τs)-Wu(τu) The deviation between the selected states (or near-stray trajectories) is characterized.
To be [0,0.5 tau ]m]Equally dividing into N parts (in the embodiment, N is taken according to the actual situation of radial or normal formation design), and calculating gamma (tau) corresponding to each node by adopting a grid searching method as shown in FIG. 5u,τs) The value, Γ (τ)u,τs)＝||r(τs)-r(τu)||1+1000nCM||v(τs)-v(τu)||1，r(τs) Representing a steady manifold τsPosition of time, r (τ)u) Representing an unstable manifold τuPosition of time, v (τ)s) Representing a steady manifold τsVelocity at time v (τ)u) Representing an unstable manifold τuThe speed of the moment. | | non-woven hair1Representing a 1 norm.
By comparison, the time node corresponding to the minimum of gamma is obtained, so as to obtain
And (6) performing characterization.
(2) Optimizing objectives and constraints
The desired design target is Γ (τ)u,τs) Gradually moving to 0, the adopted control scheme is as follows: 2N +1 time points applying pulse velocity increment delta vρρ ═ 1,2, …,2N + 1; 2N stages apply a constant electromagnetic moment control mukK is 1,2, …, 2N. Then the corresponding optimal design variables and their constraints are:
in the formula,. DELTA.vmaxAnd mumaxT represents a transposed symbol to correspond to a capability threshold of the device.
Applying sums in consideration of pulse velocity increments
Minimum, and the design target Γ (τ) will be desiredu,τs) Introducing an optimization objective function as 0 to obtain a comprehensive objective function J as follows:
(3) pulse velocity increment and electromagnetic magnetic moment numerical optimization
Based on the established comprehensive objective function and the control quantity threshold value constraint condition, a Particle Swarm Optimization (PSO for short) is adopted for Optimization calculation to obtain a method satisfying the task requirement gamma (tau)u,τs) The control variable U with the smallest sum Δ V is applied to 0 and the pulse speed increment.
The beneficial effects obtained by adopting the invention are as follows: according to the method, the electromagnetic spacecraft configuration reconstruction method which meets the requirements of low fuel consumption and stable process is obtained by starting from the control dynamics characteristic of the micro-nano electromagnetic spacecraft, fully utilizing the relative equilibrium state characteristic of the electromagnetic cluster spacecraft and based on the design of the electromagnetic control dynamics invariant manifold and the design of the switching optimization control between the unstable manifold and the stable manifold. The invention has the advantages that: (1) electromagnetic force control is introduced into the reconstruction design of the micro-nano spacecraft configuration, so that the advantage of not consuming fuel is fully utilized; (2) obtaining a configuration reconstruction scheme of the micro-nano spacecraft with low fuel consumption through the optimization design of pulse velocity increment and electromagnetic magnetic moment; (3) the configuration reconstruction extends along the dynamic manifold, the sequence equilibrium state characteristic of electromagnetic control is fully utilized, and the stability is good; (4) the design scheme can meet the configuration reconfiguration requirements of the micro-nano spacecraft, such as long-term, multiple times, expandable tasks and the like.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a schematic diagram of a modeling reference system and a space configuration for formation of two electromagnetic spacecrafts;
FIG. 3 is a schematic diagram of a global manifold value solution for two electromagnetic spacecraft formations in a relatively balanced state;
FIG. 4 is a schematic diagram of a relative equilibrium state configuration reconstruction process of two electromagnetic spacecraft formation systems based on invariant manifold;
FIG. 5 is a graph illustrating the determination of expected stable manifold and unstable manifold connection times (τ) using a grid search methodu,τs) Schematic ofA drawing;
fig. 6 is a relative equilibrium global manifold at a radial direction L of 25m for two electromagnetic spacecrafts in an implementation, wherein (a) is a stable manifold and (b) is an unstable manifold;
fig. 7 is a relative equilibrium global manifold diagram in an implementation when the normal direction L of two electromagnetic spacecrafts is 25m, wherein (a) is a stable manifold diagram and (b) is an unstable manifold diagram;
FIG. 8 is a candidate manifold for radial reconstruction in an embodiment;
FIG. 9 is a radial optimal reconstruction trajectory and pulse velocity increment application node in an embodiment;
FIG. 10 is a radial control variable versus time graph wherein (a) is a plot of pulse velocity increment values and their application times; (b) the relationship graph of the electromagnetic magnetic moment control quantity and time is shown;
FIG. 11 is a candidate manifold for normal reconstruction in an embodiment;
FIG. 12 is a normal optimal reconstruction trajectory and pulse velocity increment application node in an embodiment;
FIG. 13 is a plot of normal control variable versus time, wherein (a) is a plot of pulse velocity increment values and their application times; (b) the electromagnetic magnetic moment control quantity is plotted against time.
Detailed Description
The invention is further illustrated by the following figures and examples.
FIG. 1 is a flow chart of the method of the present invention, a method for reconstructing a configuration of an electromagnetic spacecraft with low fuel consumption and stable process, which is applied to formation of dual electromagnetic spacecraft,
(S1) calculating an initial equilibrium configuration and a target equilibrium configuration of the electromagnetic spacecraft formation system, respectively, including the steps (S11) - (S14):
(S11) establishing a kinetic modeling reference coordinate system oCMxyz based on the coordinate system oCMxyz, establishing a translational motion kinetic equation of each spacecraft relative to the CM;
(S12) solving the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction of the electromagnetic spacecraft formation system according to the translational motion kinetic equation of each spacecraft;
(S13) obtaining a relative equilibrium state X of the system according to the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction*Then according to the relative equilibrium state X of the system*Establishing electromagnetic spacecraft formation in system relative equilibrium state X*The linearized model of (1);
(S14) substituting the initial state of the system into the relative equilibrium state X of the system*Obtaining a global manifold by integrating the linearized model, wherein the global manifold comprises a stable manifold and an unstable manifold;
(S2) designing a time interval, which is a desired stable manifold and unstable manifold connection period;
(S3) respectively selecting unstable manifolds in the initial equilibrium state configuration global manifolds, a certain number of discrete state points on the unstable manifolds in the target equilibrium state configuration global manifolds, establishing a scalar optimization objective function related to deviation between the state points, designing constraint conditions to enable the scalar optimization objective function to approach 0, and realizing reconstruction of two spacecrafts from an initial distance L1 to an expected distance L2.
The numerical simulation verification parameters of the present invention are shown in table 1.
TABLE 1 numerical simulation parameters
| Parameter(s) | Numerical value | Unit of |
| nCM | 7.26×10-5 | rad/s |
| m1 | 150 | kg |
| L | 25/50 | m |
| μ1x(radial relative equilibrium state) | 3587/20292 | Am2 |
| μ1z(Normal relative equilibrium state) | 2071/11716 | Am2 |
| Δvmax | 1 | mm/s |
| μmax | 30000 | Am2 |
| ε | 0.001 | - |
In which the electromagnetic magnetic moment (mu)1x，μ1z) The "/" data on both sides in the column correspond to values where the L data is both initially 25m apart and expected to be 50m apart.
(1) Global manifold
For the radial relative equilibrium state, T is integrated from the initial manifold (corresponding to the relative equilibrium state where two electromagnetic spacecrafts are 25m apart by L)CM(center of mass orbital period is TCM，TCM＝2π/nCM) Corresponding toThe global manifold is shown in fig. 6, in which the stable manifold is symmetrical to the unstable manifold, and the electromagnetic spacecraft 1 is symmetrical to the corresponding manifold of the electromagnetic spacecraft 2. In the figure, the steady manifold "+ branch" corresponds to "Ws+ ", steady manifold" -Branch "corresponds to" Ws- "; the unstable manifold "+ Branch" corresponds to "Wu+ ", unstable manifold" -branch "corresponds to the upper right superscript" Wu- ".
Similarly, for the normal relative equilibrium state, 3T is integrated from the initial manifold (corresponding to the relative equilibrium state where the distance L between two electromagnetic spacecrafts is 25 m)CMThe corresponding global manifold is shown in fig. 7. Comparing and analyzing the radial and normal global manifolds to know that: the radially opposed equilibrium manifolds are planar in configuration, while the normally opposed equilibrium manifolds are three-dimensional in configuration.
(2) Radial reconstruction
For the radial configuration reconstruction mode from the relative equilibrium state L of 25m to the relative equilibrium state L of 50m, according to the method of the present invention, a candidate "unstable manifold-stable manifold" switching mode is obtained as shown in fig. 8; get
The obtained optimal reconstructed trajectory and pulse velocity increment application node are shown in fig. 9, and the corresponding pulse velocity increment and electromagnetic magnetic moment are shown in fig. 10, and as can be seen from the analysis in fig. 10, the consumed pulse velocity increment is about 3.85mm/s in the radial direction "25 m → L50 m reconstructed" based on the invariant manifold design.
(3) Normal reconstruction
A candidate "unstable manifold-stable manifold" switching pattern is obtained from the theoretical design described above for the normal reconstruction pattern from the relative equilibrium state L of 25m to the relative equilibrium state L of 50m, as shown in fig. 11. Get
The obtained optimal reconstructed trajectory and pulse velocity increment application node are shown in fig. 12, and the corresponding pulse velocity increment and electromagnetic magnetic moment are shown in fig. 13, and as can be seen from the analysis of fig. 13, the consumed pulse velocity increment is about 5 in the normal "25 m → 50m reconstructed" based on the invariant manifold design.71mm/s。
The method comprehensively utilizes the internal control characteristics of the electromagnetic spacecraft, and enables the micro-nano spacecraft configuration reconstruction to extend along the electromagnetic control manifold as much as possible through the pulse speed increment and the electromagnetic magnetic moment control optimization design, so that the method has the remarkable advantages of less fuel consumption, internal stability and the like. The method passes numerical simulation verification (radial configuration reconstruction and normal configuration reconstruction), has ideal effect and is consistent with the design expectation.
Claims (4)
1. A configuration reconstruction method of an electromagnetic spacecraft with low fuel consumption and stable process is applied to an electromagnetic spacecraft formation system consisting of double electromagnetic spacecrafts and is characterized in that,
(S1) calculating an initial relative equilibrium configuration and a target relative equilibrium configuration of the electromagnetic spacecraft formation system, respectively, comprising the steps of:
(S11) taking the centroid CM of the electromagnetic spacecraft formation system as an origin oCMEstablishing a reference coordinate system o for dynamic modelingCMxyz，oCMThe x-axis is along the CM ground center distance direction, oCMy-axis along track speed direction, oCMz axis and oCMx-axis, oCMThe y axis forms a right-hand rectangular coordinate system based on a coordinate system oCMxyz, establishing a translational motion kinetic equation of each spacecraft relative to the CM;
(S12) solving the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction of the electromagnetic spacecraft formation system according to the translational motion kinetic equation of each spacecraft;
(S13) obtaining a relative equilibrium state X of the system according to the relative equilibrium state conditions of the radial direction, the normal direction and the tangential direction*Then according to the relative equilibrium state X of the system*Linearizing the translation motion dynamic model of the relative CM to obtain the relative equilibrium state X of the electromagnetic spacecraft formation in the system*The linearized model of (1);
(S14) mixing X*Substituting designed manifold initial state into system relative equilibrium state X*Obtaining a global manifold by integrating the linearized model, wherein the global manifold comprises a stable manifold and an unstable manifold;
(S2) designing a time interval, which is a connection period of the desired initial relatively equilibrium unstable manifold and the target relatively equilibrium stable manifold;
(S3) respectively selecting a certain number of discrete state points on an unstable manifold in the initial relative equilibrium state configuration global manifold and a stable manifold in the target relative equilibrium state configuration global manifold, establishing a scalar optimization objective function related to deviation between the state points, designing pulse speed increment and electromagnetic magnetic moment to enable the scalar optimization objective function to tend to 0, realizing reconstruction of two spacecrafts from an initial distance L1 to a desired distance L2, and enabling L1 and L2 to represent distance values.
2. The method for reconstructing a low-fuel-consumption and process-stable electromagnetic spacecraft configuration of claim 1, wherein in said step (S13), two electromagnetic spacecraft teams are located at X*The linearized model of (1) is:
wherein δ X (τ) ═ X (τ) -X*(τ), X (τ) represents the system state at time τ, X*(τ) represents the relative equilibrium state of the system at time τ; i is3×3Representing a 3 x 3 identity matrix, the B matrix has different forms depending on the radial, normal and tangential relative equilibrium modes, as follows:
the radial direction is opposite to the equilibrium state,
in the normal direction, the state of the relative equilibrium,
tangential relative equilibrium state
In the formula (II)
And P ═ mu1zμ2z-μ1yμ2y；
Wherein L is the distance between two electromagnetic spacecrafts, M1＝m2/(m1+m2)，m1,m2Mass of spacecraft 1, spacecraft 2, nCMIs the centroid CM orbital motion angular velocity, mu0For vacuum permeability (μ)1x,μ1y,μ1z)、(μ2x,μ2y,μ2z) Electromagnetic magnetic moment mu of two spacecrafts respectively1、μ2At oCMProjection components of the xyz coordinate system.
3. A low fuel consumption and process stable electromagnetic spacecraft configuration reconfiguration method according to claim 2, characterized in that said step (S14) is embodied by:
let the stable eigenvector of matrix A be VsUnstable eigenvector is VuThe steady flow form is WsUnstable manifold is WuThen, then
The initial state of the stable manifold is
The initial state of the unstable manifold is
WhereinEpsilon represents a control parameter, and
substituting the linearized model into t ∈ [0 ∞ ] and t ∈ (- ∞ 0)]Integrating to obtain a stable manifold; will be in the initial state
Substituting the linearized model into t ∈ [0 ∞ ] and t ∈ (- ∞ 0)]And integrating to obtain the unstable manifold.
4. A low fuel consumption and process stable electromagnetic spacecraft configuration reconfiguration method according to claim 2, wherein said step (S3) comprises the steps of:
selecting a time point tau on the initial relative equilibrium unstable manifoldu，0≤τu≤0.5τm，τmSelecting a time point tau on a target relative equilibrium stable manifold for a set configuration reconstruction times，0≤τs≤0.5τm；
The time interval [0,0.5 taum]Equally dividing the flow into N parts, and respectively calculating the stable manifold at each time point and the discrete state W corresponding to the unstable manifolds(τs)∈Ws、Wu(τu)∈WuFurther, the deviation r (tau) of the corresponding state is calculatedu,τs)＝||r(τs)-r(τu)||1+1000nCM||v(τs)-v(τu)||1Memory for recording
Indicates the time corresponding to the minimum value of deviation between states, where r (τ)s) Representing a steady manifold τsPosition of time, r (τ)u) Representing an unstable manifold τuPosition of time, v (τ)s) Representing a steady manifold τsVelocity at time v (τ)u) Representing an unstable manifold τuVelocity of time, | | | luminance1Represents a norm of 1;
the control scheme is as follows: applying pulse speed increment at 2N +1 time nodes, and applying constant electromagnetic magnetic moment control at 2N stages; the constraint conditions are as follows:
wherein Δ vρDenotes the ρ -th pulse velocity increment, ρ ═ 1,2, …,2N +1, μkRepresents the kth electromagnetic moment, k ═ 1,2, …, 2N; u represents a control amount, Δ vmax、μmaxRespectively, capability thresholds of corresponding devices;
designing a comprehensive objective function J as follows:
and (3) performing optimization calculation on the comprehensive objective function by adopting a particle swarm optimization algorithm to obtain a control quantity U which meets the task requirement gamma of 0 and has the minimum sum of pulse velocity increment application.
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