CN114253291A - Spacecraft formation guidance method and system based on linear pseudo-spectral model predictive control - Google Patents

Spacecraft formation guidance method and system based on linear pseudo-spectral model predictive control Download PDF

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CN114253291A
CN114253291A CN202111536437.3A CN202111536437A CN114253291A CN 114253291 A CN114253291 A CN 114253291A CN 202111536437 A CN202111536437 A CN 202111536437A CN 114253291 A CN114253291 A CN 114253291A
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杨良
陈万春
王冲冲
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Beihang University
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Abstract

The invention provides a spacecraft formation guidance method and a spacecraft formation guidance system based on linear pseudo-spectral model predictive control, which comprises the following steps: determining a predicted orbit of a main spacecraft of the target spacecraft formation and a terminal state of an accompanying spacecraft based on the initial state of the target spacecraft formation and the formation configuration relation; determining a relative motion disturbance equation of the formation of the target spacecraft based on the predicted orbit and the terminal state; discretizing a relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation of + terminal state correction; combining the linear relation and the quadratic performance index to construct an augmentation performance index; and solving the augmentation performance index based on the optimal control problem solving method, and deducing to obtain a control variable analytical solution accompanying the spacecraft. The invention solves the technical problem of low solving efficiency of the optimal control problem of spacecraft formation in the prior art.

Description

Spacecraft formation guidance method and system based on linear pseudo-spectral model predictive control
Technical Field
The invention relates to the technical field of spacecraft formation guidance, in particular to a spacecraft formation guidance method and system based on linear pseudo-spectrum model predictive control.
Background
The purpose of formation flight of spacecraft is to perform tasks in a coordinated manner with a set of spacecraft. Compared with a single complex aircraft, the method has potential advantages in cost reduction, flexibility enhancement, observation baseline improvement and better viability and reliability. Therefore, formation flying of spacecraft is one of the effective techniques in space exploration, and has become one of the hot spots of research in recent years. Designing robust and reliable guidance, navigation and control (GNC) technology is critical to the spacecraft formation mission. The optimal formation reconstruction guidance is a key aspect for ensuring a safe operating environment and maximizing the return of scientific tasks. Scharf investigated existing optimal reconstruction guidance algorithms for spacecraft formation flight formation maintenance and reconstruction. These algorithms can be broadly divided into two categories depending on the type of thrust used: pulse control and continuous low thrust control. It is known that since the small thrust control uses an electric propulsion system, it has advantages of accurate thrust output, less propellant consumption, and the like, as compared with the pulse control. Therefore, in the past few years, intensive research has been conducted on optimal reconstructed guidance for continuous control.
Sabol and Burns studied several satellite formation flight designs and their evolution over time according to the well-known Hill equation, and quantitatively analyzed the effects of different orbital elements on formation flight. No and Lee (2009) propose an analytical solution for multi-spacecraft formation maintenance, in which power series and trigonometric functions are used to represent relative orbital motion. Richards (2012) uses Mixed Integer Linear Programming (MILP) to design optimal trajectories for spacecraft fuel consumption minimization, where constraints on avoiding obstacles and plume collisions are involved. Campbell (2012) proposes an algorithm for quickly finding the shortest time and minimum fuel maneuver for a satellite from an initial stable formation to a final stable formation on a circular orbit. In the calculation process, the Hamilton-Jacobian-Bellman optimality is fully utilized.
With the development of numerical techniques and computational science, a general optimal control problem can be solved by a pseudo-spectrum method. The pseudo-spectrum method has higher precision and efficiency, and is widely applied to maintenance and reconstruction of the formation task. Huntington and Rao use a gaussian pseudo-spectral approach to solve the problem of spacecraft minimum fuel reconstruction with certain geometric constraints. The result shows that the Gaussian pseudospectrum method has good performance in both calculation precision and efficiency. Aoude and How propose a two-phase path planning method to provide the best reconstruction maneuver. In the method, a fast search random tree (RRT) method provides a good starting point for a Gaussian pseudo-spectrum method, thereby further improving the calculation efficiency. Wu also optimizes the low thrust orbits of the satellite formation by applying a Legendre pseudospectral method, which comprises a nonlinear relative satellite kinematic model and a J2 effect. However, the pseudo-spectrum method is to convert the optimal control problem into a nonlinear programming problem, and the solving efficiency is low.
Disclosure of Invention
In view of the above, the present invention aims to provide a spacecraft formation guidance method and system based on linear pseudo-spectrum model predictive control, so as to alleviate the technical problem of low solution efficiency of the optimal control problem for spacecraft formation in the prior art.
In a first aspect, an embodiment of the present invention provides an aeronautical spacecraft formation guidance method based on linear pseudo-spectrum model predictive control, including: determining a predicted orbit of a main spacecraft of a target spacecraft formation and a terminal state of an accompanying spacecraft based on an initial state and formation configuration relation of the target spacecraft formation; determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion disturbance equation is a linear equation characterizing relative motion between the primary spacecraft and the companion spacecraft; discretizing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation formula of terminal state correction; the linear relation is a linear relation calculation expression among the initial change, the terminal change and the control variable of the accompanying spacecraft; combining the linear relation and the quadratic performance index to construct an augmentation performance index; and solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analytical solution of the adjoint spacecraft.
Further, the method further comprises: processing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method and a preset multiplier to obtain initial co-modal estimation of the adjoint spacecraft; the preset multiplier is a Lagrange multiplier related to terminal state constraint of the accompanying spacecraft; determining initial control variables for the companion spacecraft based on the initial co-modal estimate.
Further, determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state, including: constructing an orbit dynamics equation of the adjoint spacecraft based on the terminal state; the orbit dynamics equation comprises a target disturbance term; the target disturbance term is an acceleration-related disturbance term caused by J2 perturbation of the earth; and performing Taylor series expansion on the orbit dynamics equation around the predicted orbit, and neglecting high-order terms to obtain a relative motion disturbance equation of the target spacecraft formation.
Further, solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analytic solution of the adjoint spacecraft, wherein the solving method comprises the following steps: and solving the augmentation performance index based on the KKT condition to obtain a control variable analytical solution of the adjoint spacecraft.
Further, discretizing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation of terminal state correction, comprising the following steps: transferring the time domain of the relative motion disturbance equation to a preset time interval; the preset time interval is a support point selection time interval of the Lagrange interpolation polynomial; the support points of the Lagrangian interpolation polynomial comprise-1 and the root of the Legendre polynomial; converting the relative motion disturbance equation after transferring to the preset time interval into a target algebraic equation based on the Lagrange interpolation polynomial; calculating the final state change of the adjoint spacecraft based on the target algebraic equation and the Gaussian discrete orthogonal rule; the final state change comprises a final change in position and velocity; and determining a linear relation of the terminal state correction based on the final state change of the accompanying spacecraft.
Further, the method further comprises: and taking the initial control variable of the adjoint spacecraft as the initial control input of the adjoint spacecraft, taking the control variable analysis solution of the adjoint spacecraft as the control input in the adjoint spacecraft guidance process, and carrying out reconstruction guidance on the target spacecraft formation.
In a second aspect, an embodiment of the present invention further provides a spacecraft formation guidance system based on linear pseudo-spectrum model predictive control, including: the device comprises a first determining module, a second determining module, a first calculating module, a second calculating module and a solving module; the first determining module is used for determining the predicted orbit of a main spacecraft of a target spacecraft formation and the terminal state of an accompanying spacecraft based on the initial state and formation configuration relation of the target spacecraft formation; the second determination module is used for determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion disturbance equation is a linear equation characterizing relative motion between the primary spacecraft and the companion spacecraft; the first calculation module is used for discretizing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation of terminal state correction; the linear relation is a linear calculation expression among the initial change, the terminal change and the control variable of the accompanying spacecraft; the second calculation module is used for combining the linear relation and the quadratic performance index to construct an augmentation performance index; and the solving module is used for solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analytic solution of the adjoint spacecraft.
Further, the system further comprises: a boundary control module to: processing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method and a preset multiplier to obtain initial co-modal estimation of the adjoint spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraints of the companion spacecraft; determining initial control variables for the companion spacecraft based on the initial co-modal estimate.
In a third aspect, an embodiment of the present invention further provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, where the processor implements the steps of the method according to the first aspect when executing the computer program.
In a fourth aspect, the present invention further provides a computer-readable medium having a non-volatile program code executable by a processor, where the program code causes the processor to execute the method described in the first aspect.
The invention provides a spacecraft formation guidance method and system based on linear pseudo-spectral model predictive control. The method provided by the embodiment of the invention does not need ballistic integration, and the Gaussian integration with the highest algebraic precision is used, so that the method has the characteristics of high calculation speed, high solving precision and the like, and the technical problem of low solving efficiency of the optimal control problem of spacecraft formation in the prior art is solved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.
Fig. 1 is a flowchart of a spacecraft formation guidance method based on linear pseudo-spectrum model predictive control according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of the flight motions of a primary spacecraft and a companion spacecraft provided in accordance with an embodiment of the present invention;
fig. 3 is a schematic diagram of a real-time optimal control architecture for formation reconfiguration of a master-slave spacecraft according to an embodiment of the present invention;
fig. 4 is a schematic diagram of a spacecraft formation guidance system based on linear pseudo-spectrum model predictive control according to an embodiment of the present invention;
fig. 5 is a schematic diagram of another spacecraft formation guidance system based on linear pseudo-spectral model predictive control according to an embodiment of the present invention.
Detailed Description
The technical solutions of the present invention will be described clearly and completely with reference to the accompanying drawings, and it should be understood that the described embodiments are some, but not all embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, are within the scope of the present invention.
The first embodiment is as follows:
fig. 1 is a flowchart of an aircraft formation guidance method based on linear pseudo-spectrum model predictive control according to an embodiment of the present invention. As shown in fig. 1, the method specifically includes the following steps:
and S102, determining the predicted orbit of the main spacecraft of the target spacecraft formation and the terminal state of the companion spacecraft based on the initial state and formation configuration relation of the target spacecraft formation.
Specifically, the predicted orbit of a main spacecraft of a target spacecraft formation is determined based on the initial state of the target spacecraft formation, and the terminal state of an accompanying spacecraft of the target spacecraft formation is determined by further combining the formation configuration relationship ten years later.
Step S104, determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion perturbation equation is a linear equation characterizing the relative motion between the primary and the companion spacecraft.
Step S106, discretizing a relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation formula for correcting the terminal state; the linear relationship is a linear computational expression between initial changes, terminal changes, and control variables that accompany the spacecraft.
And step S108, combining the linear relation and the quadratic performance index to construct an augmentation performance index.
And S110, solving the augmentation performance index based on the optimal control problem solving method, and deducing to obtain a control variable analytic solution accompanying the spacecraft.
Optionally, the embodiment of the invention solves the augmentation performance index based on the KKT condition to obtain a control variable analytic solution associated with the spacecraft.
The invention provides a spacecraft formation guidance method based on linear pseudo-spectral model predictive control, which is characterized in that a reference orbit of each aircraft is obtained through analysis of an initial state of spacecraft formation, a quasi-linear method is adopted for linear processing to obtain a relative motion disturbance equation, and finally, an analytic optimal correction solution which meets terminal position and speed correction and has optimal performance indexes is deduced under a Gaussian pseudo-spectral method discrete condition to obtain an optimal guidance law which meets configuration requirements. The method provided by the embodiment of the invention does not need ballistic integration, and the Gaussian integration with the highest algebraic precision is used, so that the method has the characteristics of high calculation speed, high solving precision and the like, and the technical problem of low solving efficiency of the optimal control problem of spacecraft formation in the prior art is solved.
In embodiments of the invention, the predicted orbit of the primary spacecraft may be determined by an orbit element, and the orbit element may be solved from an initial state. In particular, for a classical two-body system, a particular track may be defined by six track elements, as follows:
(e a iΩωM) (1)
wherein e is the eccentricity of the track, a is the semi-major axis of the track, and these two elements define the shape and size of the ellipse; i is the inclination of the orbit and Ω is the ascension of the intersection, these two elements defining the direction of the orbital plane in which the ellipse lies. ω is the argument of the perigee for determining the direction of the archwines in the plane of the orbit. M is the mean anomaly which defines the position of the spacecraft on the orbit at any instant. Although the differential equation of the spacecraft with respect to the earth's motion is non-linear, the equation has an analytical solution. Thus, once the initial position and velocity vectors of the spacecraft on the orbit are determined, the corresponding orbit elements can be determined in an analytical form.
Specifically, since the angular momentum vector h and the eccentricity vector e are both constant vectors and can be expressed as a function of the position and velocity vectors, they can be first calculated from the initial position and velocity vectors:
h＝r×v (2)
Figure BDA0003412735470000071
it should be noted that the magnitude of the eccentricity vector is the eccentricity of the track. The radius p is then a function of the angular momentum vector and the gravitational constant, and can be given by:
Figure BDA0003412735470000081
if p is determined, the semi-principal axis can be given by the following equation:
Figure BDA0003412735470000082
the orbital inclination i can be calculated from the orbital angular momentum vector:
Figure BDA0003412735470000083
now, a vector n is defined, which is perpendicular to the plane determined by the orbital angular momentum vector and the north pole axis (K):
Figure BDA0003412735470000084
definition u is the sum of the true perigee angle and the perigee argument. Then, according to the vector projection theorem, the ascension Ω at the intersection point, the argument ω at the perigee, and the parameter u:
Figure BDA0003412735470000085
Figure BDA0003412735470000086
Figure BDA0003412735470000087
the true paraxial angle, i.e. the angle between the two vectors r and e, can then be calculated:
θ0＝u0-ω (11)
the off-center angle E may be expressed as:
Figure BDA0003412735470000088
finally, the mean anomaly can be obtained:
M(t0)＝E0-e sin E0 (13)
the six orbital elements that determine the elliptical orbit are calculated from the initial position and velocity of the spacecraft, so its position at any time can be solved analytically.
The embodiment of the invention is concerned with the master-slave aircraft formation widely used in the flight of the spacecraft formation. Specifically, fig. 2 is a schematic view of the flying motion of a main spacecraft and an associated spacecraft according to an embodiment of the present invention. As shown in fig. 2, the formation consists of N +1 spacecraft, the equation of motion of which is assumed to be a classical thrust vector controlled two-body problem.
For each mission, the accompanying spacecraft needs to finely control its relative position and eventually achieve a specific geometry. At the same time, some tasks also require the final relative speed to be constrained to perfectly guarantee the subsequent formation flight. In other words, if the master-slave spacecraft formation and reference orbits (predicted orbits) are known, the final constraints on the position and velocity of each companion spacecraft are determined:
δxi(tf)＝δxif，δvi(tf)＝δvif (19)
wherein i represents the ith spacecraft, tfIs the terminal time determined by the lead spacecraft and the mission.
Optionally, step S104 further includes the steps of:
step S1041, constructing an orbit dynamics equation accompanying the spacecraft based on the terminal state; the orbit dynamics equation comprises a target disturbance term; the target disturbance term is an acceleration-related disturbance term caused by J2 perturbation of the earth;
and step S1042, performing Taylor series expansion on the orbit dynamics equation around the predicted orbit, and neglecting a higher-order term to obtain a relative motion disturbance equation of the target spacecraft formation.
In the embodiment of the invention, the thrust is assumed to be much smaller than the gravity of the earth, so that the obtained disturbance solution has high precision and can be used for developing a formation reconstruction guidance algorithm. Specifically, the orbital dynamics problem can be expressed as follows using ordinary differential equations (i.e., orbital dynamics equations associated with spacecraft):
Figure BDA0003412735470000091
wherein the content of the first and second substances,
Figure BDA0003412735470000092
is a position vector in the geocentric coordinate system,
Figure BDA0003412735470000093
Figure BDA0003412735470000094
is a velocity vector in the geocentric coordinate system, u ═ ux uy uz) Is the thrust acceleration vector in the geocentric coordinate system, af(r) is a non-linear term related to the position vector only, aJ2(r) is a perturbation function, specifically an acceleration-dependent perturbation term caused by perturbation of the earth by the J2 term. The specific expressions of these two terms are:
Figure BDA0003412735470000101
where μ is the gravitational constant of the central body. It should be noted that the embodiment of the present invention does not consider the disturbance acceleration caused by the non-spherical gravity and the atmospheric resistance except for the item J2. The reason is that embodiments of the present invention focus on the formation reconstruction guidance problem in a short time, and therefore it is reasonable to ignore the effects of these disturbance accelerations. Next, under the same initial conditions, the orbital dynamics equations are expanded around the ideal orbit in a taylor series. The disturbance equation is used for describing the relative motion between the main spacecraft and the accompanying spacecraft, and the following relative motion disturbance equation can be obtained by neglecting a high-order term:
Figure BDA0003412735470000102
wherein the content of the first and second substances,
Figure BDA0003412735470000103
optionally, step S106 further includes the following specific steps:
step S1061, transferring the time domain of the relative motion disturbance equation to a preset time interval; the preset time interval is a support point selection time interval of the Lagrange interpolation polynomial; the support points of the Lagrangian interpolation polynomial include-1 and the root of the Legendre polynomial;
step S1062, based on the Lagrange interpolation polynomial, converting the relative motion disturbance equation after the transfer to the preset time interval into a target algebraic equation;
step S1063, calculating the final state change of the accompanying spacecraft based on a target algebraic equation and a Gaussian discrete orthogonal rule; the final state change includes a final change in position and velocity;
step S1064, determining a linear relational expression for correcting the terminal state based on the final state change of the accompanying spacecraft.
The linear pseudo-spectral model predictive control is proposed to iteratively provide control improvements to effectively solve the non-linear control problem. When implementing linear pseudo-spectral model predictive control, one of the most important steps is to derive analytical correction equations to eliminate final position and velocity errors. This is accomplished by discrete linearized equations of relative motion perturbation using a pseudo-spectral approach. In general, there are three common pseudospectral methods: gaussian pseudo-spectrometry, laguo pseudo-spectrometry and legendre pseudo-spectrometry. Previous research shows that the Gaussian pseudo-spectrum method not only has higher discrete precision than other methods, but also theoretically proves the convergence of the algorithm. Therefore, the embodiment of the invention adopts a Gaussian pseudo-spectrum method. The specific discretization and derivation process is as follows:
the first step in implementing gaussian pseudo-spectroscopy is to transfer the time domain to the time interval [ -1,1] (i.e., the preset time interval mentioned above), since the support points of the lagrange interpolation polynomial are chosen as the orthogonal points located in the time interval [ -1,1], which is done by the following mapping function:
Figure BDA0003412735470000111
definition, dT ═ tf-t0) And/2, used in the following derivation. Substituting equation (20) into equation (16) and τ as an argument, one can obtain:
Figure BDA0003412735470000112
lagrangian interpolating polynomials of degree N +1 are defined LN (τ), with support points of-1 and the root of Legendre (Legendre) polynomials (abbreviated as LG points):
Figure BDA0003412735470000113
it is clear that LN (. tau.) has the following properties:
Figure BDA0003412735470000114
and:
xN(τl)＝x(τl)； (24)
the derivation of equation (22) yields, at the LG point:
Figure BDA0003412735470000115
where D is a differential approximation matrix of N (N +1) times, derived from the derivative of each Lagrangian polynomial at the LG point. The elements of D are represented as:
Figure BDA0003412735470000121
δ x can be considered as a state error vector, i.e.:
Figure BDA0003412735470000122
by substituting equation (27) into equation (21), the set of linear kinetic equations is converted into a set of algebraic equations (i.e., target algebraic equations):
Figure BDA0003412735470000123
wherein the content of the first and second substances,
δR＝[δr1 … δrn]T，δV＝[δv1 … δvn]T，U＝[u1 … un]T；
Figure BDA0003412735470000124
Figure BDA0003412735470000125
is defined as:
Figure BDA0003412735470000126
wherein k is 1, 2. Generally, only a few LG points are needed to obtain high accuracy, and therefore, in the following numerical simulation, no more than 8 LG points are needed. The differential approximation matrix is decomposed into two parts, and the equation (28) is modified as follows:
D1δr0+D2：nδR＝dTδV；
Figure BDA0003412735470000127
wherein D is1Is part of D associated with an initial change of state, D2：nIs another part of D that is related to the change of state of the LG point. The position and velocity variation of all LG points can be expressed as:
Figure BDA0003412735470000128
Figure BDA0003412735470000129
Figure BDA0003412735470000139
the differential approximation matrix has the following characteristics:
D1＝-D2：n1 (33)
substituting equation (33) into equation (32), the change in the position and velocity vector of the LG point can be expressed as:
Figure BDA0003412735470000131
Figure BDA0003412735470000132
since the support points do not include the final point, the final state change is calculated using the gaussian orthogonality rule:
Figure BDA0003412735470000133
wherein, ω isiIs the weight of the gaussian integral:
Figure BDA0003412735470000134
wherein the content of the first and second substances,
Figure BDA0003412735470000135
is the derivative of an N degree Legendre polynomial. The resulting change in position and velocity is therefore expressed as:
Figure BDA0003412735470000136
Figure BDA0003412735470000137
wherein W is [ omega ]1 … ωn]，
Figure BDA0003412735470000138
Substituting equation (34) into equation (37), the final change in position and velocity can be expressed as a function of the initial position, initial velocity, and control of the LG point.
Figure BDA0003412735470000141
Figure BDA0003412735470000142
Defining some new variables Krr、Krv、KvrAnd KvvThe terminal change in position and velocity has a simple linear relationship, as follows:
Figure BDA0003412735470000143
wherein the content of the first and second substances,
Figure BDA0003412735470000144
Figure BDA0003412735470000145
Figure BDA0003412735470000146
Figure BDA0003412735470000147
Figure BDA0003412735470000148
Figure BDA0003412735470000149
Figure BDA00034127354700001410
Figure BDA00034127354700001411
it is clear that this formula reveals the relationship between the initial change of state, the terminal change and the control variable. For simplicity of derivation, equation (39) may be rewritten as a general expression (i.e., a linear relation of terminal state correction):
δxf＝Kxδx0+KuU+Kc (40)
wherein the content of the first and second substances,
Figure BDA0003412735470000151
it should be noted that the controlled quantity is discretized at the LG point. Kx is a 6 by 6 matrix and Ku is a 6 by 6 x n matrix. Obviously, if the final state is determined, there will be thousands of solutions because the number of control variables is much greater than the constraints of the final state. It is an aim of embodiments of the present invention to provide optimal control of accompanying spacecraft to achieve a particular position in a formation, and therefore it is necessary to provide a performance index, as follows:
Figure BDA0003412735470000152
where R is a control weight matrix. Likewise, a gaussian integral formula is applied to (41) to obtain.
Figure BDA0003412735470000153
Wherein the content of the first and second substances,
Figure BDA0003412735470000154
the augmented performance index is obtained by correlating equation (40) with the performance index:
Figure BDA0003412735470000155
where v is the lagrangian multiplier associated with the terminal state constraint. It is clear that since the performance index is quadratic and the equality constraint is linear for the control variables, an analytical solution can be derived from the KKT condition. The corresponding KKT conditions are:
Figure BDA0003412735470000156
the optimal solution for the control and lagrange multipliers (i.e., the control variables that accompany the spacecraft) can be expressed analytically as:
Figure BDA0003412735470000157
in an embodiment of the invention, the method further comprises solving for optimal control of the boundary accompanying the spacecraft. Specifically, the method comprises the following steps:
based on a Gaussian pseudo-spectrum method and a preset multiplier, processing a relative motion disturbance equation to obtain initial co-modal estimation accompanied with the spacecraft; presetting a multiplier as a Lagrange multiplier related to terminal state constraint of the accompanying spacecraft;
based on the initial co-modal estimation, initial control variables accompanying the spacecraft are determined.
In the embodiment of the present invention, in order to solve the boundary optimal control, a theorem 1 is introduced as follows:
theorem 1: for a linear optimal control problem with quadratic performance index, if gaussian pseudo-spectral dispersion is used, the initial and terminal covariant estimates can be derived from the KKT multiplier pi associated with the terminal constraints.
Figure BDA0003412735470000161
Wherein λ is0，λfIs the initial and terminal co-modal, matrix Mλ，MuThe definitions are made in the following proof.
And (3) proving that: consider the linear optimal control problem as follows. The performance index is a quadratic form of the control variable u:
Figure BDA0003412735470000162
the kinetic equation of a linear time-varying system is:
Figure BDA0003412735470000163
time interval of t e [ -1,1 [ ]]The initial and final states are fixed to x (-1) ═ x0，x(1)＝xf. Using gaussian pseudo-spectral dispersion one can obtain:
Dx＝Ax+Bu+C (50)
the state at the LG point can be represented as:
x＝-(D2：n-A)-1D1x0+(D2：n-A)-1Bu+(D2：n-A)-1C (51)
the terminal state can be obtained by gaussian integration:
Figure BDA0003412735470000164
substituting equation (51) into equation (52), the terminal state can be expressed as a function of the initial state and the control quantity at the LG point:
xf＝(I-WA(D2：n-A)-1D1)x0+W(A(D2：n-A)-1+I)Bu+ W(A(D2：n-A)-1+I)C (53)
for simplicity, equation (53) can be expressed in a simple manner:
xf＝Mxx0+Muu+Mc (54)
wherein the content of the first and second substances,
Mx＝I-WA(D2：n-A)-1D1
Mu＝W(A(D2：n-A)-1+I)B
Mc＝W(A(D2：n-A)-1+I)C
Figure BDA0003412735470000171
the augmentation performance indexes are as follows:
Figure BDA0003412735470000172
the KKT conditions are:
Figure BDA0003412735470000173
it is possible to obtain:
Figure BDA0003412735470000174
Figure BDA0003412735470000175
then define a new variable lambdafIt is expressed as:
Figure BDA0003412735470000176
wherein the content of the first and second substances,
Figure BDA0003412735470000177
Figure BDA0003412735470000178
Figure BDA0003412735470000181
thus, optimal control of the LG point can be expressed as a function λf：
Figure BDA0003412735470000182
Similarly, substituting equation (61) into equation (51), the state of the LG point can also be expressed as:
Figure BDA0003412735470000183
definition of
Figure BDA0003412735470000184
It is possible to obtain:
Figure BDA0003412735470000185
substituting λ into equation (61) and equation (62), the optimal control and state is represented as a linear function λ of the initial state sum:
u＝-R-1BTλ (64)
x＝-(D2：n-A)-1D1x0-(D2：n-A)-1BR-1BTλ+(D2：n-A)-1C (65)
the formula (65) is collated to obtain:
D2：nx+D1x0＝Ax-BR-1BTλ+C (66)
it is readily observed that equations (63), (64) and (66) constitute the first order requirement for the optimal control problem with discrete points LG:
Figure BDA0003412735470000186
thus, λ is the co-modal value of the optimal control problem at the LG point, and the variable λfIs the co-modal value of the terminal. Calculating an initial co-modal value from the terminal co-modal value and the co-modal value of the LG point using Gaussian integration:
Figure BDA0003412735470000187
substituting equations (59) and (63) into equation (68) yields:
Figure BDA0003412735470000188
theorem 1 shows that for the linear optimal control problem, if the gaussian pseudo-spectrum method is applied to the original optimal control problem, the initial co-state can be accurately estimated by the lagrangian multiplier associated with the terminal state constraint. This also means that a complete mapping between the KKT condition and the first order requirement is successfully derived. Compared with the collaborative mapping theorem, the method only needs the Lagrange multipliers related to terminal constraint, the number of the Lagrange multipliers is far less than that of the Lagrange multipliers related to state constraint, and the calculation efficiency is greatly improved. Finally, for a linear optimal control problem with quadratic performance indicators, the optimal control can be represented analytically as a function of the covariances. Therefore, it is not necessary to obtain the boundary control by numerically solving an additional nonlinear programming problem, and the initial optimal control can be calculated by an analytical expression as follows:
Figure BDA0003412735470000191
therefore, the initial control amount for the optimal convoy reconstruction guidance can be obtained by the formula (70), which is the guidance instruction acting on the accompanying aircraft at each guidance cycle.
Optionally, after determining the initial control variable and the control variable analytic solution accompanying the spacecraft, the method provided in the embodiment of the present invention further includes: and taking the initial control variable of the accompanying spacecraft as the initial control input of the accompanying spacecraft, taking the control variable analysis solution of the accompanying spacecraft as the control input in the process of making guidance of the accompanying spacecraft, and carrying out reconstruction guidance on the formation of the target spacecraft.
Specifically, fig. 3 is a schematic diagram of a real-time optimal control architecture for formation and reconstruction of a master-slave spacecraft, according to an embodiment of the present invention. As shown in fig. 3, the method proposed by the embodiment of the present invention is composed of three main parts: initialization, main spacecraft trajectory prediction and accompanying spacecraft optimal configuration reconstruction guidance. In the first part, the guidance period, mission objective and spacecraft formation flight are predetermined. It also provides initial conditions for the host aircraft and the companion aircraft. In the second part, the trajectory of the host aircraft is analytically predicted using six track elements. At the same time, the position and speed of the terminal can be precisely determined for each accompanying aircraft according to the specific formation defined previously. In the third section, a relative motion disturbance equation is formulated around the predicted trajectory of the host aircraft. And the linear pseudo-spectrum model prediction control is adopted to convert the original optimal control problem under the system into a group of linear algebraic equations, so that a series of analytic optimal controls of the LG points are successfully deduced. Furthermore, the co-modal mapping theorem derived in 3.2 is used to calculate the initial optimal control for each companion spacecraft. Finally, this initial optimal control is taken as a control input. In the next guidance period, the second part of the procedure is repeated, and the formation flight reconstruction trajectory with the optimal performance index is realized through the proposed algorithm.
The optimal spacecraft formation reconstruction guidance method based on the linear pseudo-spectrum model predictive control fully considers the influence of a J2 perturbation term and nonlinear dynamics, obtains a reference orbit of each aircraft through orbit equation analysis, adopts a quasi-linear method to carry out linearization processing to obtain an error propagation equation related to deviation, combines task configuration requirements, and deduces an analytic optimal correction solution which meets terminal position and speed correction and optimal performance indexes under a Gaussian pseudo-spectrum discrete condition. As the method does not need ballistic integration, adopts a Gaussian integral formula with the highest algebraic precision, and has the characteristics of high calculation speed, high solving precision and the like, and finally, the calculation example verifies that the method can obtain the optimal control instruction only by dozens of milliseconds, the position guidance precision is below 1 meter, the relative speed guidance precision is below 1 meter/second, and the configuration requirement is completely met.
Example two:
FIG. 4 is a schematic diagram of an aircraft formation guidance system based on linear pseudo-spectrum model predictive control according to an embodiment of the invention. As shown in fig. 4, the system includes: a first determination module 10, a second determination module 20, a first calculation module 30, a second calculation module 40 and a solving module 50.
Specifically, the first determining module 10 is configured to determine, based on an initial state of a formation of a target spacecraft and a formation configuration relationship, a predicted orbit of a main spacecraft of the formation of the target spacecraft and a terminal state of an associated spacecraft.
The second determination module 20 is configured to determine a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion perturbation equation is a linear equation characterizing the relative motion between the primary and the companion spacecraft.
The first calculation module 30 is configured to discretize the relative motion disturbance equation based on a gaussian pseudo-spectrum method to obtain a linear relation for terminal state correction; the linear relation is a linear relation calculation expression between initial variation, terminal variation and control variables accompanying the spacecraft.
And the second calculation module 40 is used for combining the linear relation and the quadratic performance index to construct an augmentation performance index.
And the solving module 50 is used for solving the augmentation performance indexes based on the optimal control problem solving method and deducing to obtain a control variable analytic solution accompanying the spacecraft.
The invention provides an optimal spacecraft formation reconstruction guidance system based on linear pseudo-spectral model predictive control, which is characterized in that a reference orbit of each aircraft is obtained through initial state analysis of spacecraft formation, linear processing is carried out by adopting a quasi-linear method to obtain a relative motion disturbance equation, and finally, an analytic optimal correction solution which meets terminal position and speed correction and has optimal performance indexes is deduced under a Gaussian pseudo-spectral method dispersion condition to obtain an optimal guidance law which meets configuration requirements. The method provided by the embodiment of the invention does not need ballistic integration, and the Gaussian integration with the highest algebraic precision is used, so that the method has the characteristics of high calculation speed, high solving precision and the like, and the technical problem of low solving efficiency of the optimal control problem of spacecraft formation in the prior art is solved.
Alternatively, fig. 5 is a schematic diagram of another spacecraft formation guidance system based on linear pseudo-spectral model predictive control provided according to an embodiment of the invention. As shown in fig. 5, the system further includes: a boundary control module 60 for:
based on a Gaussian pseudo-spectrum method and a preset multiplier, processing a relative motion disturbance equation to obtain initial co-modal estimation accompanied with the spacecraft; presetting a multiplier as a Lagrange multiplier related to terminal state constraint of the accompanying spacecraft; based on the initial co-modal estimation, initial control variables accompanying the spacecraft are determined.
The embodiment of the present invention further provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and when the processor executes the computer program, the steps of the method in the first embodiment are implemented.
The embodiment of the invention also provides a computer readable medium with a non-volatile program code executable by a processor, wherein the program code causes the processor to execute the method in the first embodiment.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (10)

1. A spacecraft formation guidance method based on linear pseudo-spectral model predictive control is characterized by comprising the following steps:
determining a predicted orbit of a main spacecraft of a target spacecraft formation and a terminal state of an accompanying spacecraft based on an initial state and formation configuration relation of the target spacecraft formation;
determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion disturbance equation is a linear equation characterizing relative motion between the primary spacecraft and the companion spacecraft;
discretizing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation formula of terminal state correction; the linear relation is a linear relation calculation expression among the initial change, the terminal change and the control variable of the accompanying spacecraft;
combining the linear relation and the quadratic performance index to construct an augmentation performance index;
and solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analytical solution of the adjoint spacecraft.
2. The method of claim 1, further comprising:
processing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method and a preset multiplier to obtain initial co-modal estimation of the adjoint spacecraft; the preset multiplier is a Lagrange multiplier related to terminal state constraint of the accompanying spacecraft;
determining initial control variables for the companion spacecraft based on the initial co-modal estimate.
3. The method of claim 1, wherein determining a relative motion perturbation equation for the formation of the target spacecraft based on the predicted orbit and the terminal state comprises:
constructing an orbit dynamics equation of the adjoint spacecraft based on the terminal state; the orbit dynamics equation comprises a target disturbance term; the target disturbance term is an acceleration-related disturbance term caused by J2 perturbation of the earth;
and performing Taylor series expansion on the orbit dynamics equation around the predicted orbit, and neglecting high-order terms to obtain a relative motion disturbance equation of the target spacecraft formation.
4. The method of claim 1, wherein solving the augmented performance indicator based on an optimal control problem solution method to derive a control variable analytic solution for the companion spacecraft comprises:
and solving the augmentation performance index based on the KKT condition to obtain a control variable analytical solution of the adjoint spacecraft.
5. The method according to claim 1, wherein discretizing the relative motion disturbance equation based on a gaussian pseudo-spectrum method to obtain a linear relation for terminal state correction comprises:
transferring the time domain of the relative motion disturbance equation to a preset time interval; the preset time interval is a support point selection time interval of the Lagrange interpolation polynomial; the support points of the Lagrangian interpolation polynomial comprise-1 and the root of the Legendre polynomial;
converting the relative motion disturbance equation after the transfer to the preset time interval into a target algebraic equation based on the Lagrange interpolation polynomial;
calculating the final state change of the adjoint spacecraft based on the target algebraic equation and the Gaussian discrete orthogonal rule; the final state change comprises a final change in position and velocity;
and determining a linear relation of the terminal state correction based on the final state change of the accompanying spacecraft.
6. The method of claim 2, further comprising:
and taking the initial control variable of the adjoint spacecraft as the initial control input of the adjoint spacecraft, taking the control variable analysis solution of the adjoint spacecraft as the control input in the adjoint spacecraft guidance process, and carrying out reconstruction guidance on the target spacecraft formation.
7. A spacecraft formation guidance system based on linear pseudo-spectral model predictive control is characterized by comprising the following components: the device comprises a first determining module, a second determining module, a first calculating module, a second calculating module and a solving module; wherein the content of the first and second substances,
the first determination module is used for determining the predicted orbit of a main spacecraft of a target spacecraft formation and the terminal state of an accompanying spacecraft based on the initial state and formation configuration relation of the target spacecraft formation;
the second determination module is used for determining a relative motion disturbance equation of the target spacecraft formation based on the predicted orbit and the terminal state; the relative motion disturbance equation is a linear equation characterizing relative motion between the primary spacecraft and the companion spacecraft;
the first calculation module is used for discretizing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation of terminal state correction; the linear relation is a linear calculation expression among the initial change, the terminal change and the control variable of the accompanying spacecraft;
the second calculation module is used for combining the linear relation and the quadratic performance index to construct an augmentation performance index;
and the solving module is used for solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analytic solution of the adjoint spacecraft.
8. The system of claim 7, further comprising: a boundary control module to:
processing the relative motion disturbance equation based on a Gaussian pseudo-spectrum method and a preset multiplier to obtain initial co-modal estimation of the adjoint spacecraft; the preset multiplier is a Lagrange multiplier related to terminal state constraint of the accompanying spacecraft;
determining initial control variables for the companion spacecraft based on the initial co-modal estimate.
9. An electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the steps of the method of any of the preceding claims 1 to 6 are implemented when the computer program is executed by the processor.
10. A computer-readable medium having non-volatile program code executable by a processor, wherein the program code causes the processor to perform the method of any of claims 1-6.
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