CN114253291B - Spacecraft formation guidance method and system based on linear pseudo spectrum model predictive control - Google Patents
Spacecraft formation guidance method and system based on linear pseudo spectrum model predictive control Download PDFInfo
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Abstract
The invention provides a spacecraft formation guidance method and a spacecraft formation guidance system based on linear pseudo-spectrum model predictive control, wherein the spacecraft formation guidance method and the spacecraft formation guidance system comprise 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 target spacecraft formation based on the predicted orbit and the terminal state; based on a Gaussian pseudo-spectrum method, discretizing a relative motion disturbance equation to obtain a linear relation of +terminal state correction; combining the linear relation and the quadratic performance index to construct an amplified performance index; and solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analysis solution accompanying the spacecraft. The method and the device solve the technical problem of low solving efficiency of the optimal control problem of spacecraft formation in the prior art.
Description
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 spacecraft formation flight is to perform tasks in a coordinated manner with a group of spacecraft. It has potential advantages over a single complex aircraft in terms of reduced cost, increased flexibility, improved observation baseline, and better viability and reliability. Therefore, spacecraft formation flight is one of the effective technologies in space exploration, and has become one of the hot spots of research in recent years. Design robust and reliable guidance, navigation and control (GNC) techniques are critical to spacecraft formation flight missions. Among other things, optimal formation reconfiguration guidance is a key aspect to ensure a safe operating environment and maximize return on scientific tasks. Scharf investigated the existing optimal reconfiguration guidance algorithm for spacecraft formation flying formation maintenance and reconfiguration. These algorithms can be broadly divided into two categories depending on the type of thrust used: pulse control and continuous low thrust control. As is well known, since the low thrust control uses an electric propulsion system, it has advantages of accurate thrust output, less consumption of propellant, and the like, as compared with the pulse control. Therefore, in the last few years, intensive research has been conducted on the optimal reconfiguration 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 impact of different orbital elements on formation flight. No and Lee (2009) propose an analytical solution for multi-spacecraft formation maintenance, where power series and trigonometric functions are employed to represent relative orbital motion. Richards (2012) uses Mixed Integer Linear Programming (MILP) to design the optimal trajectory for the minimum fuel consumption of the spacecraft, where constraints related to avoidance obstacles and plume collisions are involved. Campbell (2012) proposes an algorithm for quickly finding the shortest time and minimum fuel maneuvers for satellites on circular orbits from an initial stable formation to a final stable formation. Hamilton-Jacobian-Bellman optimality is fully utilized in the calculation process.
With the development of numerical technology and computational science, general optimal control problems can be solved by pseudo-spectroscopy. The pseudo spectrum method has higher precision and efficiency, and is widely applied to the maintenance and reconstruction of formation tasks. Huntington and Rao use gaussian pseudo-spectrum methods to solve the problem of minimum fuel reconstruction for spacecraft with certain geometric constraints. The result shows that the Gaussian pseudo-spectrum method has good performance in terms of calculation accuracy and efficiency. Aoude and How propose a two-stage path planning method to provide the best reconstruction maneuver. In the method, the rapid search random tree (RRT) method provides a good start for the Gaussian pseudo-spectrum method, so that the calculation efficiency is further improved. Wu also optimizes the low thrust orbit of satellite formation using the Legendre pseudospectrum method, which includes a nonlinear relative satellite kinematic model and J2 effects. However, the pseudo-spectrum method converts the optimal control problem into a nonlinear programming problem, and has low solving efficiency.
Disclosure of Invention
In view of the above, the invention aims to provide a spacecraft formation guidance method and a spacecraft formation guidance system based on linear pseudo-spectrum model predictive control, so as to solve the technical problem of low efficiency of solving the optimal control problem of spacecraft formation in the prior art.
In a first aspect, an embodiment of the present invention provides a spacecraft formation guidance method based on linear pseudo-spectrum model prediction control, including: determining a predicted orbit of a main spacecraft and a terminal state of an accompanying spacecraft of a target spacecraft formation based on an initial state of the target spacecraft formation and a formation configuration relation; 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 the relative motion between the primary spacecraft and the accompanying spacecraft; 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 relation calculation expression among initial change, terminal change and control variable of the accompanying spacecraft; combining the linear relation and the quadratic performance index to construct an amplified performance index; and solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analysis solution of the accompanying spacecraft.
Further, the method further comprises: based on a Gaussian pseudo-spectrum method and a preset multiplier, processing the relative motion disturbance equation to obtain initial cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraints of the accompanying spacecraft; based on the initial collaborative estimate, an initial control variable for the companion spacecraft is determined.
Further, determining a relative motion disturbance equation for the target spacecraft formation based on the predicted orbit and the terminal state, comprising: constructing an orbit dynamics equation of the accompanying spacecraft based on the terminal state; the orbit dynamics equation comprises a target disturbance term; the target disturbance term is a disturbance term related to acceleration caused by the perturbation of the J2 term of the earth; and carrying out Taylor series expansion on the orbit dynamics equation around the predicted orbit, and ignoring higher-order terms to obtain a relative motion disturbance equation of the target spacecraft formation.
Further, based on an optimal control problem solving method, solving the augmentation performance index, deriving to obtain a control variable analysis solution of the accompanying spacecraft, including: and solving the augmentation performance index based on KKT conditions to obtain the control variable analysis solution of the accompanying spacecraft.
Further, based on a gaussian pseudo-spectrum method, discretizing the relative motion disturbance equation to obtain a linear relation for terminal state correction, including: transferring the time domain of the relative motion disturbance equation to a preset time interval; the preset time interval is a time interval selected by a supporting point of the Lagrangian interpolation polynomial; the support points of the Lagrangian interpolation polynomial comprise the roots of the-1 and Legendre polynomials; converting the relative motion disturbance equation after being transferred to the preset time interval into a target algebraic equation based on the Lagrangian interpolation polynomial; calculating the final state change of the accompanying spacecraft based on the target algebraic equation and Gaussian discrete orthometric rule; the final state change includes a final change in position and speed; 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 accompanying spacecraft as initial control input of the accompanying spacecraft, and taking the control variable analysis solution of the accompanying spacecraft as control input in the accompanying spacecraft guidance process to reconstruct and guide 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 prediction control, including: the system 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 a predicted orbit of a main spacecraft formed by the target spacecraft and a terminal state of an accompanying spacecraft based on an initial state and a formation configuration relation of the target spacecraft formed; the second determining 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 the relative motion between the primary spacecraft and the accompanying 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 initial change, terminal change and 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 augmented performance index; the solving module is used for solving the augmentation performance index based on an optimal control problem solving method, and deducing and obtaining the control variable analysis solution of the accompanying spacecraft.
Further, the system further comprises: a boundary control module for: based on a Gaussian pseudo-spectrum method and a preset multiplier, processing the relative motion disturbance equation to obtain initial cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraints of the accompanying spacecraft; based on the initial collaborative estimate, an initial control variable for the companion spacecraft is determined.
In a third aspect, an embodiment of the present invention further provides an electronic device, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor, where the processor executes the computer program to implement the steps of the method described in the first aspect.
In a fourth aspect, embodiments of the present invention also provide a computer readable medium having non-volatile program code executable by a processor, the program code causing the processor to perform the method of the first aspect.
The invention provides a spacecraft formation guidance method and a spacecraft formation guidance system based on linear pseudo-spectrum model predictive control, which are characterized in that a reference orbit of each spacecraft 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 under the discrete condition of a Gaussian pseudo-spectrum method, an analysis optimal correction solution which meets the correction of terminal position and speed and has optimal performance indexes is deduced, so that an optimal guidance law which meets the configuration requirements is obtained. Because the method provided by the embodiment of the invention does not need ballistic integration, the Gaussian integration with highest algebraic precision is used, 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 that are needed in the description of the embodiments or the prior art will be briefly described, and it is obvious that the drawings in the description below are some embodiments of the present invention, and other drawings can be obtained according to the drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a spacecraft formation guidance method based on linear pseudo-spectrum model predictive control provided by an embodiment of the invention;
FIG. 2 is a schematic diagram of a main spacecraft and a flight motion of an accompanying spacecraft according to an embodiment of the present invention;
fig. 3 is a schematic diagram of a real-time optimal control architecture for master-slave spacecraft formation reconstruction 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 provided by an embodiment of the invention;
fig. 5 is a schematic diagram of another spacecraft formation guidance system based on linear pseudo-spectrum model predictive control according to an embodiment of the invention.
Detailed Description
The following description of the embodiments of the present invention will be made apparent and fully in view of the accompanying drawings, in which some, but not all embodiments of the invention are shown. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Embodiment one:
fig. 1 is a flowchart of a spacecraft formation guidance method based on linear pseudo-spectrum model predictive control, which is provided by an embodiment of the invention. As shown in fig. 1, the method specifically includes the following steps:
step S102, based on the initial state and the formation configuration relation of the target spacecraft formation, determining the predicted orbit of the main spacecraft and the terminal state of the accompanying spacecraft of the target spacecraft formation.
Specifically, firstly, based on the initial state of target spacecraft formation, the prediction orbit of the main spacecraft of the target spacecraft formation is determined, and the terminal state of the accompanying spacecraft of the target spacecraft formation is determined by further combining the formation configuration relation after ten years.
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 disturbance equation is a linear equation that characterizes the relative motion between the main spacecraft and the accompanying spacecraft.
Step S106, discretizing a relative motion disturbance equation based on a Gaussian pseudo-spectrum method to obtain a linear relation of terminal state correction; the linear relationship is a linear computational expression between the initial variation, the terminal variation, and the control variable accompanying the spacecraft.
Step S108, combining the linear relation and the quadratic performance index to construct the amplified performance index.
And step S110, solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analysis solution of the accompanying spacecraft.
Optionally, the embodiment of the invention solves the augmentation performance index based on the KKT condition to obtain a control variable resolution solution for the accompanying spacecraft.
The invention provides a spacecraft formation guidance method based on linear pseudo-spectrum model predictive control, which is characterized in that a reference orbit of each spacecraft 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 under the discrete condition of a Gaussian pseudo-spectrum method, an analysis optimal correction solution which meets the terminal position and speed correction and has optimal performance indexes is deduced to obtain an optimal guidance law which meets the configuration requirements. Because the method provided by the embodiment of the invention does not need ballistic integration, the Gaussian integration with highest algebraic precision is used, 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 an embodiment of the present invention, the predicted orbit of the main spacecraft may be determined by orbit elements, which may be solved by an initial state. In particular, for classical two-body systems, one particular trajectory may be determined by six trajectory elements, as follows:
(e a iΩω M) (1)
where e is the eccentricity of the track, a is the semi-major axis of the track, the two elements defining the shape and size of the ellipse; i is the inclination of the orbit and Ω is the right ascent point, and these two elements define the direction of the orbit plane in which the ellipse lies. ω is the perigee argument for determining the direction of the arching in the orbit plane. M is the angle of the closest point, which defines the position of the spacecraft on orbit at any instant in time. Although the differential equation of spacecraft with respect to earth motion is nonlinear, the equation has an analytical solution. Thus, once the initial position and velocity vector of the spacecraft on orbit is determined, the corresponding orbit element can be determined in analytical form.
In particular, since the angular momentum vector h and the eccentricity vector e are both constant vectors and can be expressed as a function of position and velocity vectors, they can be calculated first from the initial position and velocity vectors:
h＝r×v (2)
it should be noted that the magnitude of the eccentricity vector is the eccentricity of the track. The semi-diameter p is then a function of the angular momentum vector and the gravitational constant, and can be given as:
if p is determined, the semi-principal axis can be found by the following formula:
the orbital tilt i can be calculated from the orbital angular momentum vector:
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):
definition u is the sum of the true near point angle and the near point argument. Then, according to the vector projection theorem, the ascent point right angle Ω, the near-spot argument ω, and the parameter u:
the true near point angle, i.e. the angle between the two vectors r and e, can then be calculated:
θ 0 ＝u 0 -ω (11)
the near point angle E can be expressed as:
finally, the closest point angle can be obtained:
M(t 0 )＝E 0 -esinE 0 (13)
the six orbital elements that determine the elliptical orbit are calculated from the initial position and velocity of the spacecraft, and therefore their position at any time can be resolved in an analytical way.
The embodiment of the invention aims to consider master-slave spacecraft formation widely used in spacecraft formation flight. Specifically, fig. 2 is a schematic diagram of a main spacecraft and a flight motion of an accompanying spacecraft according to an embodiment of the present invention. As shown in fig. 2, the formation consists of n+1 spacecraft, whose equation of motion is assumed to be a two-body problem of classical thrust vector control.
For each mission, the accompanying spacecraft needs to carefully control its relative position and ultimately achieve a specific geometry. At the same time, some tasks also require constraints on the final relative speed to perfectly guarantee the subsequent formation flights. In other words, if master-slave spacecraft formation and reference orbit (predicted orbit) are known, then the final constraints on the position and velocity of each accompanying spacecraft are determined:
δx i (t f )＝δx if ,δv i (t f )＝δv if (19)
wherein i represents the ith spacecraft, t f Is the end time determined by the lead spacecraft and 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 includes a target disturbance term; the target disturbance term is a disturbance term related to acceleration caused by the perturbation of the J2 term of the earth;
and step S1042, carrying out Taylor series expansion on the orbit dynamics equation around the predicted orbit, and neglecting higher-order terms to obtain a relative motion disturbance equation of the target spacecraft formation.
In the embodiment of the invention, the thrust force is assumed to be much smaller than the earth gravity, and the disturbance solution obtained by the method has high precision and can be used for developing a formation reconstruction guidance algorithm. In particular, the orbit dynamics problem can be expressed by the ordinary differential equation (i.e., the orbit dynamics equation accompanying the spacecraft) as follows:
wherein r= (x y z) ∈ 3 Is a position vector in the geocentric coordinate system, v= (v) x v y v z )∈ 3 Is a velocity vector in the geocentric coordinate system, u= (u) x u y u z ) Is a thrust acceleration vector in a geocentric coordinate system, a f (r) is a nonlinear term related only to the position vector, a J2 (r) is a perturbation function, specifically an acceleration-dependent perturbation term caused by the J2 term perturbation of the earth. The specific expressions of the two terms are:
where μ is the gravitational constant of the central body. It should be noted that the embodiments of the present invention do not consider the disturbance acceleration caused by non-spherical gravity and atmospheric resistance other than the J2 term. The reason is that the embodiments of the present invention focus on the problem of formation reconfiguration guidance in a short time, and therefore it is reasonable to ignore the effects of these disturbance accelerations. Next, under the same initial conditions, the orbit dynamics equation expands in taylor series around the ideal orbit. 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 higher order term:
wherein,
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 time interval selected by the support points of the Lagrangian interpolation polynomial; the support points of the Lagrangian interpolation polynomial include the roots of the-1 and Legendre polynomials;
step S1062, converting the relative motion disturbance equation after transferring to the preset time interval into a target algebraic equation based on the Lagrangian interpolation polynomial;
step S1063, calculating the final state change of the accompanying spacecraft based on the target algebraic equation and Gaussian discrete orthogonal rule; the final state change includes a final change in position and speed;
in step S1064, a linear relation for terminal state correction is determined based on the final state change of the accompanying spacecraft.
The linear pseudo-spectrum model predictive control is proposed to iteratively provide control improvements to effectively solve the nonlinear control problem. In implementing linear pseudo-spectrum model predictive control, one of the most important steps is deriving analytical correction formulas to eliminate final position and velocity errors. This is accomplished by discrete linearization of the relative motion disturbance equation using pseudo-spectral methods. In general, there are three common pseudospectral methods: gaussian pseudoscopy, laduo pseudoscopy and Legend pseudoscopy. Previous studies have shown that the gaussian pseudospectrum method not only has higher discrete accuracy than other methods, but also demonstrates the convergence of the algorithm theoretically. 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 shift the time domain to time interval [ -1,1] (i.e., the preset time interval described above), because the support point for the Lagrangian interpolation polynomial is selected to lie at the intersection of time interval [ -1,1], this is done by the following mapping function:
definition, dt= (t f -t 0 ) And/2, used in the following derivation. Substituting equation (20) into equation (16), and taking τ as an argument, one can obtain:
a lagrangian interpolation polynomial with LN (τ) of n+1 degrees is defined, with its support points being the root (abbreviated LG point) of the-1 and Legendre (Legendre) polynomials:
obviously, LN (τ) has the following characteristics:
and:
the derivative of equation (22) is available, at the LG point:
where D is a differential approximation matrix of N× (N+1), derived from the derivative of each Lagrangian polynomial at the LG point. The elements of D are expressed as:
δx may be considered as a state error vector, namely:
by substituting equation (27) into equation (21), the set of linear dynamics equations is converted into a set of algebraic equations (i.e., target algebraic equations):
wherein,
δR＝[δr 1 … δr n ] T ，δV＝[δv 1 … δv n ] T ，U＝[u 1 … u n ] T ；
is defined as:
where k=1, 2,..n. In general, very high accuracy can be obtained with very few LG points, so that in the latter numerical simulations, LG points do not exceed 8. The differential approximation matrix is decomposed into two parts, and the deformation for equation (28) is as follows:
D 1 δr 0 +D 2：n δR＝dTδV；
wherein D is 1 Is a part of D related to the initial change of state, D 2：n Is another part of D related to the state change of the LG point. The position and speed changes of all LG points can be expressed as:
the differential approximation matrix has the following characteristics:
D 1 ＝-D 2:n 1 (33)
substituting formula (33) into formula (32), the change in position and velocity vector of the LG point can be expressed as:
since the support points do not include the final points, the gaussian orthogonal rule is used to calculate the final state change:
wherein omega i Is the weight of the gaussian integral:
wherein,is the derivative of the N degree Legendre polynomial. Thus, the final change in position and velocity is expressed as:
wherein w= [ ω ] 1 …ω n ]，
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.
Defining new variables K rr 、K rv 、K vr And K vv There is a simple linear relationship between the terminal changes in position and velocity, as follows:
wherein,
it is apparent that this formula reveals the relationship between the initial change in state, the terminal change and the control variable. For simplicity of derivation, equation (39) may be rewritten as a general expression (i.e., linear relation for terminal state correction):
δx f ＝K x δx 0 +K u U+K c (40)
wherein,
it should be noted that the control amount is discretized at the LG point. Kx is a 6 by 6 matrix and Ku is a 6 by 6 xn matrix. Obviously, if the final state is determined, there are thousands of solutions, since the number of control variables is far greater than the constraints of the final state. It is an aim of embodiments of the present invention to provide optimal control for an accompanying spacecraft to achieve a specific position of formation, and therefore it is necessary to provide a performance index as follows:
wherein R is a control weight matrix. Likewise, a gaussian integral formula is applied in (41).
Wherein,
the enhanced performance index is obtained by correlating equation (40) with the performance index:
where v is the lagrangian multiplier associated with the terminal state constraint. It is apparent that since the performance index is quadratic and the equation constraint is linear for the control variable, the analytical solution can be derived from the KKT condition. The corresponding KKT conditions are:
the optimal solution of the control and lagrangian multipliers (i.e., the control variables that accompany the spacecraft) can be analytically expressed as:
in an embodiment of the invention, the method further comprises solving the boundary optimal control of the accompanying 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 cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraint of the accompanying spacecraft;
based on the initial cooperative estimation, an initial control variable accompanying the spacecraft is determined.
In the embodiment of the invention, in order to solve the boundary optimal control, an theorem 1 is introduced as follows:
theorem 1: for linear optimal control problems with quadratic performance metrics, if gaussian pseudospectral dispersion is used, initial and terminal co-state estimates can be derived from the KKT multiplier pi associated with terminal constraints.
Wherein lambda is 0 ，λ f Is the initial and terminal coordination, matrix M λ ,M u The definition is given in the following demonstration.
And (3) proving: the linear optimal control problem is considered as follows. The performance index is a quadratic form of the control variable u:
the kinetic equation of the linear time-varying system is:
the time interval is t E < -1,1 [ - ]]The initial and final states are fixed to x (-1) =x 0 ,x(1)＝x f . Using gaussian pseudo-spectrum dispersion:
Dx＝Ax+Bu+C (50)
the state at the LG point can be expressed as:
x＝-(D 2：n -A) -1 D 1 x 0 +(D 2：n -A) -1 Bu+(D 2：n -A) -1 C (51)
the termination state can be obtained by gaussian integration:
substituting equation (51) into equation (52), the terminal state can be expressed as a function of the initial state and the control amount at the LG point:
x f ＝(I-WA(D 2：n -A) -1 D 1 )x 0 +W(A(D 2：n -A) -1 +I)Bu+W(A(D 2：n -A) -1 +I)C (53)
for simplicity, equation (53) can be expressed in a simple manner:
x f ＝M x x 0 +M u u+M c (54)
wherein,
M x ＝I-WA(D 2：n -A) -1 D 1
M u ＝W(A(D 2：n -A) -1 +I)B
M c ＝W(A(D 2：n -A) -1 +I)C
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the augmentation performance index is as follows:
the KKT conditions are:
the method can obtain the following steps:
then define a new variable lambda f It is expressed as:
wherein,
thus, optimal control of LG points can be expressed as a function lambda f ：
Similarly, substituting formula (61) into formula (51), the state of the LG point can also be expressed as:
definition of the definitionThe method can obtain the following steps:
substituting λ into equation (61) and equation (62), the optimal control and state is expressed as a linear function λ of the initial state sum:
u＝-R -1 B T λ (64)
x＝-(D 2：n -A) -1 D 1 x 0 -(D 2：n -A) -1 BR -1 B T λ+(D 2：n -A) -1 C(65)
the arrangement of formula (65) can be obtained:
D 2：n x+D 1 x 0 ＝Ax-BR -1 B T λ+C (66)
it can be easily seen that equations (63), (64) and (66) construct a first order requirement for the optimal control problem that is discrete at the LG point:
thus, λ is the synergistic value of the optimal control problem at the LG point, while the variable λ f Is the cooperative value of the terminal. Calculating an initial coordination value from the terminal coordination value and the coordination value of the LG point by using Gaussian integration:
substituting equations (59) and (63) into equation (68) yields:
theorem 1 shows that for a linear optimal control problem, if gaussian pseudo-spectroscopy 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 constraints. This also means that the complete mapping between the KKT condition and the first order requirement is successfully deduced. Compared with the collaborative mapping theorem, the method only needs Lagrangian multipliers related to terminal constraint, and the number of the Lagrangian multipliers is far smaller than that of the Lagrangian multipliers related to state constraint, so that the calculation efficiency is greatly improved. Finally, for linear optimal control problems with quadratic performance indicators, optimal control can be expressed as a function of the synergy by an analytical method. Therefore, it is not necessary to obtain the boundary control by numerically solving an additional nonlinear programming problem, and the optimal control at the beginning can be calculated by parsing the expression as follows:
thus, the initial control amount for optimal formation reconfiguration guidance can be obtained by equation (70), which is the guidance command that is applied to the accompanying aircraft at each guidance cycle.
Optionally, after determining the initial control variable and the control variable parsing solution of the accompanying spacecraft, the method provided by the embodiment of the invention further includes: and taking the initial control variable of the accompanying spacecraft as initial control input of the accompanying spacecraft, and taking the control variable analysis solution of the accompanying spacecraft as control input in the guidance process of the accompanying spacecraft to reconstruct and guide the target spacecraft.
Specifically, fig. 3 is a schematic diagram of a real-time optimal control architecture for formation reconstruction of a master-slave spacecraft according to an embodiment of the invention. As shown in fig. 3, the method proposed by the embodiment of the present invention is composed of three main parts: initializing, predicting the track of the main spacecraft and optimally configuring and reconstructing guidance along with the spacecraft. In the first part, the guidance period, mission objective and spacecraft formation flight are predetermined. It also provides an initial status for the primary and companion aircraft. In the second section, six orbit element analyses are used to predict the trajectory of the main aircraft. Also, for each companion aircraft, the position and speed of the terminal may be precisely determined according to the specific formation defined previously. In the third section, a relative motion disturbance equation is formulated around the predicted trajectory of the main aircraft. And the original optimal control problem under the system is converted into a group of linear algebraic equations by adopting linear pseudo-spectrum model predictive control, so that a series of analytic optimal control of the LG points is successfully deduced. Furthermore, the collaborative mapping theorem derived in 3.2 is used to calculate the initial optimal control for each accompanying spacecraft. Finally, this initial optimal control is used as a control input. In the next guidance cycle, the procedure of the second part is repeated, and by means of the proposed algorithm, a formed flight reconstruction trajectory with an optimal performance index will be achieved.
From the above description, the invention provides an optimal spacecraft formation reconstruction guidance method based on linear pseudo-spectrum model predictive control, which fully considers J2 perturbation items and nonlinear dynamics effects, obtains a reference orbit of each aircraft through orbit equation analysis, adopts a quasi-linear method to carry out linear processing, obtains an error propagation equation about deviation, derives an analysis optimal correction solution meeting terminal position and speed correction and having optimal performance indexes under the condition of Gaussian pseudo-spectrum discrete, and obtains an optimal guidance law meeting the configuration requirements by deriving the analysis optimal correction solution meeting terminal position and speed correction and having optimal performance indexes under the condition of Gaussian pseudo-spectrum discrete, and further combining the global collaborative analysis solution due to the fact that Legend-Gaussian nodes do not contain endpoint control, deriving optimal estimation of initial collaborative states, obtaining optimal solution at initial moments and forming closed loop iterative calculation conditions. The method does not need ballistic integration, adopts a Gaussian integration formula with highest algebraic precision, has the characteristics of high calculation speed, high solving precision and the like, and finally, the method can obtain an optimal control instruction only by tens of milliseconds through verification of calculation examples, has position guidance precision below 1 meter and relative speed guidance precision below 1 meter/second, and completely meets the configuration requirements.
Embodiment two:
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 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 solution module 50.
Specifically, the first determining module 10 is configured to determine, based on an initial state and a formation configuration relationship of a target spacecraft formation, a predicted orbit of a main spacecraft and a terminal state of an accompanying spacecraft of the target spacecraft formation.
A second determining module 20, 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 disturbance equation is a linear equation that characterizes the relative motion between the main spacecraft and the accompanying spacecraft.
The first calculation module 30 is configured to discretize a relative motion disturbance equation based on a gaussian pseudo-spectrum method, so as to obtain a linear relation for terminal state correction; the linear relationship is a linear relationship calculation expression between initial changes, terminal changes, and control variables accompanying the spacecraft.
A second calculation module 40 for combining the linear relationship and the quadratic performance index to construct an augmented performance index.
The solving module 50 is configured to solve the augmentation performance index based on an optimal control problem solving method, and derive a control variable resolution solution associated with the spacecraft.
The invention provides an optimal spacecraft formation reconstruction guidance system based on linear pseudo-spectrum model predictive control, which is characterized in that a reference orbit of each spacecraft 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 under the discrete condition of a Gaussian pseudo-spectrum method, an analysis optimal correction solution which meets the terminal position and speed correction and has optimal performance indexes is deduced, so that an optimal guidance law which meets the configuration requirements is obtained. Because the method provided by the embodiment of the invention does not need ballistic integration, the Gaussian integration with highest algebraic precision is used, 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 that provides power based on linear pseudo-spectrum model predictive control in accordance with 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 cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraint of the accompanying spacecraft; based on the initial cooperative estimation, an initial control variable accompanying the spacecraft is determined.
The embodiment of the invention also provides an electronic device, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor executes the computer program to realize the steps of the method in the first embodiment.
The present invention also provides a computer-readable medium having non-volatile program code executable by a processor, the program code causing the processor to perform the method of the first embodiment.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (6)
1. A spacecraft formation guidance method based on linear pseudo-spectrum model predictive control is characterized by comprising the following steps:
determining a predicted orbit of a main spacecraft and a terminal state of an accompanying spacecraft of a target spacecraft formation based on an initial state of the target spacecraft formation and a formation configuration relation;
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 the relative motion between the primary spacecraft and the accompanying spacecraft;
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 relation calculation expression among initial change, terminal change and control variable of the accompanying spacecraft;
combining the linear relation and the quadratic performance index to construct an amplified performance index;
solving the augmentation performance index based on an optimal control problem solving method, and deducing to obtain a control variable analysis solution of the accompanying spacecraft;
based on an optimal control problem solving method, solving the augmentation performance index, deducing and obtaining a control variable analysis solution of the accompanying spacecraft, wherein the method comprises the following steps:
solving the augmentation performance index based on KKT conditions to obtain a control variable resolution solution of the accompanying spacecraft;
the augmentation performance index J a The method comprises the following steps:
wherein dT= (t) f -t 0 )/2，t f Is the terminal time, t 0 Is the initial time; u= [ U ] 1 … u n ] T ，u＝(u x u y u z ) Is the thrust in the geocentric coordinate systemAn acceleration vector;ω i is the weight of the gaussian integral; />Is a conjugate matrix of the control weight matrix R; v is the lagrangian multiplier associated with the terminal state constraint; δx f Is the terminal change in position and velocity, δx 0 Is the initial change in position and velocity; k (K) x 、K u 、K c Is a defined variable;
the KKT conditions are as follows:
the method further comprises the steps of:
based on a Gaussian pseudo-spectrum method and a preset multiplier, processing the relative motion disturbance equation to obtain initial cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraints of the accompanying spacecraft;
determining an initial control variable for the accompanying spacecraft based on the initial cooperative estimation;
and taking the initial control variable of the accompanying spacecraft as initial control input of the accompanying spacecraft, and taking the control variable analysis solution of the accompanying spacecraft as control input in the accompanying spacecraft guidance process to reconstruct and guide the target spacecraft formation.
2. The method of claim 1, wherein determining a relative motion disturbance equation for the target spacecraft formation based on the predicted orbit and the terminal state comprises:
constructing an orbit dynamics equation of the accompanying spacecraft based on the terminal state; the orbit dynamics equation comprises a target disturbance term; the target disturbance term is a disturbance term related to acceleration caused by the perturbation of the J2 term of the earth;
and carrying out Taylor series expansion on the orbit dynamics equation around the predicted orbit, and ignoring higher-order terms to obtain a relative motion disturbance equation of the target spacecraft formation.
3. The method of 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 time interval selected by a supporting point of the Lagrangian interpolation polynomial; the support points of the Lagrangian interpolation polynomial comprise the roots of the-1 and Legendre polynomials;
converting the relative motion disturbance equation after being transferred to the preset time interval into a target algebraic equation based on the Lagrangian interpolation polynomial;
calculating the final state change of the accompanying spacecraft based on the target algebraic equation and Gaussian discrete orthometric rule; the final state change includes a final change in position and speed;
and determining a linear relation of the terminal state correction based on the final state change of the accompanying spacecraft.
4. Spacecraft formation guidance system based on linear pseudo-spectrum model predictive control, characterized by comprising: the system comprises a first determining module, a second determining module, a first calculating module, a second calculating module and a solving module; wherein,
the first determining module is used for determining a predicted orbit of a main spacecraft formed by the target spacecraft and a terminal state of an accompanying spacecraft based on the initial state and the formation configuration relation of the target spacecraft;
the second determining 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 the relative motion between the primary spacecraft and the accompanying 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 initial change, terminal change and 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 augmented performance index;
the solving module is used for solving the augmentation performance index based on an optimal control problem solving method, and deducing and obtaining a control variable analysis solution of the accompanying spacecraft;
the solving module is specifically configured to:
solving the augmentation performance index based on KKT conditions to obtain a control variable resolution solution of the accompanying spacecraft;
the augmentation performance index J a The method comprises the following steps:
wherein dT= (t) f -t 0 )/2，t f Is the terminal time, t 0 Is the initial time; u= [ U ] 1 … u n ] T ，u＝(u x u y u z ) Is a thrust acceleration vector in the geocentric coordinate system;ω i is the weight of the gaussian integral; />Is a conjugate matrix of the control weight matrix R; v is the lagrangian multiplier associated with the terminal state constraint; δx f Is the terminal change in position and velocity,δx 0 is the initial change in position and velocity; k (K) x 、K u 、K c Is a defined variable;
the KKT conditions are as follows:
the system further comprises: a boundary control module for:
based on a Gaussian pseudo-spectrum method and a preset multiplier, processing the relative motion disturbance equation to obtain initial cooperative estimation of the accompanying spacecraft; the preset multiplier is a Lagrangian multiplier related to terminal state constraints of the accompanying spacecraft;
determining an initial control variable for the accompanying spacecraft based on the initial cooperative estimation;
and taking the initial control variable of the accompanying spacecraft as initial control input of the accompanying spacecraft, and taking the control variable analysis solution of the accompanying spacecraft as control input in the accompanying spacecraft guidance process to reconstruct and guide the target spacecraft formation.
5. An electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor implements the steps of the method of any of the preceding claims 1 to 3 when the computer program is executed.
6. A computer readable medium having non-volatile program code executable by a processor, the program code causing the processor to perform the method of any of claims 1-3.
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