CN109375648B - Elliptical orbit satellite formation configuration initialization method under multi-constraint condition - Google Patents

Elliptical orbit satellite formation configuration initialization method under multi-constraint condition Download PDF

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CN109375648B
CN109375648B CN201811492401.8A CN201811492401A CN109375648B CN 109375648 B CN109375648 B CN 109375648B CN 201811492401 A CN201811492401 A CN 201811492401A CN 109375648 B CN109375648 B CN 109375648B
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翟光
杨少伟
张景瑞
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Beijing Institute of Technology BIT
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Abstract

The invention discloses an initialization method for formation configuration of an elliptic orbit satellite under a multi-constraint condition, relates to an initialization method for formation configuration of a satellite, and belongs to the field of guidance and control of spacecrafts. The implementation method of the invention comprises the following steps: through classical orbit dynamics and coordinate transformation, deducing to obtain an analytic solution of a true-near-point angular domain T-H equation, analyzing dynamics constraint, periodically surrounding flight constraint, establishing monitoring camera view field constraint and thrust amplitude constraint, establishing a multi-constraint optimal control model of formation, solving through a Gaussian pseudo-spectrum method, obtaining a coordination value at any moment, realizing real-time monitoring on spacecraft structural integrity, and reducing fuel consumption and maximum amplitude of a propulsion system by increasing eccentricity and initialization time. The invention can realize the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition, and realize the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.

Description

Elliptical orbit satellite formation configuration initialization method under multi-constraint condition
Technical Field
The invention relates to a satellite formation configuration initialization method, in particular to an optimal initialization method for building an elliptical orbit satellite formation under a multi-constraint condition, and belongs to the field of spacecraft guidance and control.
Background
In recent years, satellite formation technology has attracted attention for its wide future application. The satellite cluster formation flying technology based on the small satellites is generally concerned and accepted by relevant organizations in the fields of military affairs, industry, commerce, scientific research and the like due to the strong technical advantages and wide application prospects of the technology. The two-star formation is the most basic form of satellite formation, and the formation is composed of a master star and slave stars, and the slave stars surround the master star to perform on-orbit monitoring tasks, such as detecting whether the appearance structure of the master star and a solar panel are normal or not, so that the cost is reduced, and the risk of the astronauts for moving outside the cabin is reduced. Meanwhile, the fly-around monitoring is also an important helpful and key technology for realizing on-orbit services such as fuel filling, on-orbit maintenance, material supply, space rendezvous and the like for the target spacecraft. Therefore, the research on the design of the orbiting configuration of the satellite formation and the configuration initialization control is of great significance.
The configuration and control of the formation of the satellite are based on relative dynamics, and a circular orbit relative motion model is given by a C-W equation. When the orbit of the target spacecraft is an elliptical orbit, the C-W equation can introduce obvious errors due to the existence of the eccentricity of the orbit, and the T-H equation is adopted to describe the linear relative motion of the target spacecraft. However, since the T-H equation has obvious nonlinear characteristics in the time domain, the solution of the relative motion is difficult, and therefore the invention linearizes the T-H model through true paraxial angular domain transformation and deduces an analytic solution.
The control in the formation satellite release process generally adopts a thruster with limited amplitude, and in order to ensure the safety in the release process, a visible light camera is generally required to be used for monitoring, so the engineering constraint conditions need to be considered when the formation is initialized. The solution of the trajectory optimization problem can be essentially divided into an indirect method, a direct method and a mixed method. The indirect method strictly meets the first-order requirement of optimality in theory, but in the actual calculation process, due to the high sensitivity to initial guessing of the cooperative state, the convergence radius is small, and the solution is difficult, so the application is limited to a certain extent. Compared with an indirect method, the direct method can effectively avoid the solution of the cooperative state, and can effectively reduce the calculated amount by properly selecting the scale of the discrete nodes.
Disclosure of Invention
The invention discloses an initialization method for formation configuration of an elliptic orbit satellite under multi-constraint conditions, which aims to solve the technical problems that: and initializing the formation configuration of the elliptic orbit satellite under the multi-constraint condition, and realizing effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.
The purpose of the invention is realized by the following technical scheme.
The invention discloses an initialization method for formation configuration of elliptic orbit satellites under multiple constraint conditions, which comprises the following steps:
the method comprises the following steps: and deducing to obtain the analytic solution of the true-near-point angular domain T-H equation through classical orbit dynamics and coordinate transformation.
Step 1.1: respectively establishing an inertia coordinate system and a track coordinate system;
and establishing a fixed inertial coordinate system with the earth as a center, wherein the X axis points to the spring equinox, the Z axis is consistent with the rotation axis direction of the earth, and the Y axis is determined according to the right-hand rule. And establishing an orbit coordinate system which takes the main star as a mass center and rotates relative to the origin, wherein the x axis of the orbit coordinate system points to the center of the main star from the center of the earth, the z axis is vertical to the orbit plane, and the y axis is determined according to the right-hand rule.
Step 1.2: obtaining an analytic solution according to classical orbit dynamics;
defining the relative position vector rho ═ r of the master star and the slave stard-rcWherein r isd∈R3And rc∈R3Respectively, the position vectors of the slave and master stars. The relative dynamics between the master and slave stars are expressed in the inertial reference frame as:
Figure BDA0001895738600000021
where μ represents a constant earth gravity coefficient and u ∈ R3Representing the actively controlled acceleration produced by a propeller fixed to the tracker. From the inertial reference system to the orbital coordinate system, equation (1) is approximately expressed as:
Figure BDA0001895738600000022
where ω is the orbital angular velocity of the main star and the subscript o denotes the vector described in the rotating orbital coordinate system. Substituting formula (1) into formula (2), the relative motion is expressed as
Figure BDA0001895738600000031
Wherein
Figure BDA0001895738600000032
Where f is the true proximal angle of the principal star. Finally, substituting equations (1) and (2) into equation (3) yields:
Figure BDA0001895738600000033
Figure BDA0001895738600000034
Figure BDA0001895738600000035
wherein
Figure BDA0001895738600000036
If the active control is omitted, equations (5) to (7) are the T-H equations. The T-H equation describes the relative dynamics of free flight orbits of the two spacecrafts, and the inner and outer motions of the orbital plane are decoupled. Due to the non-linearity of the T-H equation, it is difficult to find an analytical solution of the T-H equation in the time domain. Therefore, the T-H equation is rewritten by the following transformation:
Figure BDA0001895738600000037
taking the first and second derivatives of f,
Figure BDA0001895738600000038
Figure BDA0001895738600000039
the following operators are defined:
Figure BDA00018957386000000310
Figure BDA0001895738600000041
finally, substituting equations (9) to (13) into equations (5) to (7), the relative kinetics are re-expressed as:
Figure BDA0001895738600000042
Figure BDA0001895738600000043
Figure BDA0001895738600000044
obviously, the equations (14) - (15) are linearized and solved under the given initial conditions:
Figure BDA0001895738600000045
Figure BDA0001895738600000046
Figure BDA0001895738600000047
wherein
Figure BDA0001895738600000048
h and p are the angular momentum of the primary star and the positive semi-focal distance, respectively. c. Ci( i 1, 2.., 6) is the initial state and initial of the main starThe constant coefficient determined by the true paraxial point angle is calculated according to the following formula:
Figure BDA0001895738600000049
Figure BDA00018957386000000410
Figure BDA00018957386000000411
Figure BDA00018957386000000412
Figure BDA00018957386000000413
Figure BDA0001895738600000051
analytic solution of T-H equation into
Figure BDA0001895738600000052
Namely, the derivation is realized to obtain the analytic solution of the true near point angular domain T-H equation.
Step two: and deducing a periodic fly-around constraint condition, and establishing a monitoring camera view field constraint and a thrust amplitude constraint.
Step 2.1: deriving a periodic fly-around constraint in a true anomaly angle domain;
since the expressions (21) to (26) are all sinusoidal, for a given true paraxial angle f, if c4(f) Is equal to 0, i.e
Figure BDA0001895738600000053
It can be ensured that a periodic relative trajectory is obtained. Further, the following formulae (9), (10) and (27):
Figure BDA0001895738600000054
equation (28) is a prerequisite for a periodic trajectory in an arbitrary elliptical orbit, and if equation (27) is satisfied, the corresponding periodic relative motion is expressed as
Figure BDA0001895738600000055
Figure BDA0001895738600000056
Figure BDA0001895738600000057
However, since equation (27) cannot guarantee a periodic fly-around trajectory, equation (30) is rewritten as follows:
Figure BDA0001895738600000058
then, combining the formula (32) and the formula (29), the in-plane relative trajectory is represented as
Figure BDA0001895738600000059
When the in-plane trajectory flies around, the three-dimensional trajectory can be ensured to fly around. In formula (33), when c1When 0, the orbit is reduced to an ellipse centered at (0,0), and the half axis thereof changes with time. The periodic fly-around constraint is described as:
Figure BDA0001895738600000061
where k is a coefficient selected according to the orbit around, k may be set to zero if it is desired to follow the relative trajectory in the plane from the star.
Step 2.2: and establishing a monitoring camera view field constraint.
To ensure initialisation of formation configurationThe FOV geometry of the surveillance camerA consists of two cones, point Ot is the intersection of the vertices of the two cones, representing the optical center of the lens, l is the central axis of the geometry, representing the optical axis, rectangular ABCD is the base of the lower cone, representing the CCD array, and α and β are the FOV angles, α and β are determined by the size of the CCD and the focal length of the lens
Figure BDA0001895738600000062
A non-rotating coordinate system fixed at the optical center of the lens. Therefore, the distance from the point Ot to the centroid of the main star is extremely small, and the distance from Ot to the centroid of the main star is ignored, and therefore, the monitoring coordinate system coincides with the body coordinate system of the main star.
The position vector considering the points is:
Figure BDA0001895738600000063
Figure BDA0001895738600000064
the plane OA 'D' may then be expressed as:
Figure BDA0001895738600000065
where the subscript b denotes the vector described in the body coordinate system. Similarly, the equations for the other three planes are expressed as:
Figure BDA0001895738600000071
thus, the monitoring constraint is expressed as:
Figure BDA0001895738600000072
further, assume that the attitude of the primary star is (θ φ ψ) in the orbital coordinate systemT. Where θ, φ and ψ are the pitch, yaw and roll angles, respectively, and then the vectors in the body coordinate system are converted to the orbital coordinate system as shown in equation (40):
Figure BDA0001895738600000073
wherein L isob(θ, φ, ψ) is a transformation matrix from the body coordinate system to the orbit coordinate system. The formula (40) is substituted for the formula (39) to obtain the monitoring constraint in the orbit coordinate system
Figure BDA0001895738600000074
Step 2.3: establishing an amplitude constraint for the electric propulsion;
since the electric propulsion is fixed to the slave, the attitude motion of the slave causes the thrust direction to change with respect to the orbital coordinate system, making the configuration initialization significantly more complex, so the configuration initialization is simplified by the attitude of the slave which should be kept constant with respect to the orbital coordinate system, and the thrust acceleration is always along the axis of the orbital coordinate system, so the electric propulsion magnitude constraint is expressed as:
Figure BDA0001895738600000081
wherein
Figure BDA0001895738600000082
Representing the maximum magnitude of thrust in each direction.
Step three: and establishing a multi-constraint optimal control model for formation based on the true near point angular domain T-H equation analytic solution obtained by derivation in the step one and the monitoring camera view field constraint and thrust amplitude constraint established in the step two.
Step 3.1: establishing a basic optimal control model without considering the constraint condition;
the general form of nonlinear optimal control is expressed as: in the time interval t0,tf]In the method, a state control function for minimizing a performance index functional is found
Figure BDA0001895738600000083
Simultaneous states are constrained by system dynamics equations
Figure BDA0001895738600000084
Boundary constraint
Ψ(x(t0),x(tf))＝0 (45)
And existing state and path constraints controlling mixing
C(x(t),u(t))≥0 (46)
Wherein E is called a terminal performance index, also called a Mayer type performance index; the integral term containing F is called the process performance indicator, also known as the Lagrange-type performance indicator. Defining an augmented Hamilton function as
L(x(t),u(t),λ,μ)＝H(x(t),u(t),λ)+μTC(x(t),u(t)) (47)
Where λ is the co-state variable and H is the Hamilton function
H(x(t),u(t),λ)＝λTf(x(t),u(t))+F(x(t),u(t)) (48)
And mu satisfies the complementary condition
μTC(x(t),u(t))＝0 (49)
The optimal control and state is obtained by solving the following equations:
Figure BDA0001895738600000091
where the superscript denotes the optimal solution that satisfies the aforementioned constraints.
Step 3.2: establishing a multi-constraint optimal control model based on the monitoring camera view field constraint and the thrust amplitude constraint established in the step two;
angular velocity and angular acceleration taking into account true paraxial angle
Figure BDA0001895738600000092
Figure BDA0001895738600000093
Wherein the content of the first and second substances,
Figure BDA0001895738600000094
representing the average track angular velocity, while a and e represent the semi-major axis and eccentricity of the track, respectively. Then, by substituting equation (52) for equations (5) and (6), the augmented kinetic equation is restated as:
Figure BDA0001895738600000095
wherein
Figure BDA0001895738600000096
Shows augmentation kinetics, and
Figure BDA0001895738600000097
representing the augmented state vector. The expression in said equation (34) is considered as a nonlinear state constraint on the augmented dynamics, enabling the problem to be solved by using a direct method.
Once the formation configuration has started, the slaves are released, the initial state of the configuration being determined by the release position and speed, i.e.
Figure BDA0001895738600000101
The final state of the formation configuration is constrained by the periodic fly-around:
Figure BDA0001895738600000102
wherein c isi(i ═ 1, 2.., 6) calculated according to equations (21) to (26), the slave star must be located in the FOV of the master star CCD camera during configuration, i.e. according to equation (41), the relative states should satisfy the following constraints:
Figure BDA0001895738600000103
the thrust on each shaft should also satisfy the following constraints:
Figure BDA0001895738600000104
the mathematical description of the multi-constraint optimal control problem is therefore:
Figure BDA0001895738600000105
equation (58) is subject to the dynamic constraints of equation (53), the initial state constraints of equation (54), the period fly-around constraints of equation (55), the FOV constraints of equation (56), and the thrust magnitude constraints of equation (57).
Step four: and solving the multi-constraint optimal control model to obtain a solution of the original optimal control problem and then obtain a collaborative value at any moment, namely realizing the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition and realizing the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.
And step four, the method for solving and establishing the multi-constraint optimal control model of the formation comprises a direct method, an indirect method and a mixed method.
The Gaussian pseudo-spectrum method is a direct method, and is consistent with a first-order necessary condition in an optimal control theory in the solving process; on the other hand, according to the principle of co-state mapping, the gaussian pseudo-spectrum method can obtain co-state information the same as the optimal control theory, therefore, the method for solving and establishing the multi-constraint optimal control model for formation described in the fourth step is preferably a gaussian pseudo-spectrum method for solving, and when the gaussian pseudo-spectrum method is used for solving and establishing the multi-constraint optimal control model for formation, the concrete implementation method of the fourth step comprises the following steps:
step 4.1: solving by a Gaussian pseudo-spectrum method;
the basic thought of the Gaussian pseudospectrum method is to disperse unknown state time history and control time history on a series of Gaussian points, then to respectively construct Lagrange interpolation polynomial by using the dispersed state and control to approximate to the real state and control time history, to convert the dynamic differential equation constraint into a series of algebraic constraints by means of simple mathematical knowledge, to finally convert the optimal control problem into a parameter optimization problem constrained by a series of algebraic constraints, and to solve by using other numerical methods such as a sequential quadratic programming method.
The Gaussian point is zero point of Legendre polynomial and is unevenly distributed in the interval [ -1,1]Therefore, it is necessary to match the real time interval t ∈ [ t ]0,tf]And (3) carrying out conversion:
Figure BDA0001895738600000111
substitution of t by T as an independent variable after conversion, t0,tfBecomes a system parameter. Accordingly, the original time domain optimal control problem is expressed in the τ domain as:
Figure BDA0001895738600000112
using the kth legendre-gaussian point and the state at the initial node, derive a lagrange interpolation polynomial with order K + 1:
Figure BDA0001895738600000113
wherein
Figure BDA0001895738600000121
Lagrange interpolation can ensure that the approximation at the interpolation node is exactly the same as the actual state, while the values at other points are only approximately equal. Similarly, using control of the Kth Legendre-Gaussian point, the controlled Lagrangian interpolation polynomial is derived as
Figure BDA0001895738600000122
Wherein
Figure BDA0001895738600000123
Differentiating the formula (61) in the equation to obtain
Figure BDA0001895738600000124
Wherein the differential matrix is determined as follows:
Figure BDA0001895738600000125
wherein K is 1,2, 1, K, and i is 0,1K(τ) Legendre polynomials of order K, i.e.
Figure BDA0001895738600000126
Substituting the formula (67) into the formula (66), and algebraicizing the kinetic constraint
Figure BDA0001895738600000127
The terminal value should be obtained by Gauss quadrature method
Figure BDA0001895738600000131
Wherein ω iskIs a Gaussian weight, calculated as
Figure BDA0001895738600000132
ωkAnd DkiDetermined only by the number of LG points K and prior to the optimization algorithmAnd (3) off-line calculation and solving, wherein the continuous performance index functional of the formula (60) is obtained by solving by Gaussian integration:
Figure BDA0001895738600000133
therefore, the initial optimal control problem is transformed into a nonlinear programming problem by the gaussian pseudo-spectrum method, which is described as follows:
Figure BDA0001895738600000134
solving the nonlinear programming problem to obtain the solution of the original optimal control problem and obtain the covariant value at any moment, namely realizing the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition and realizing the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.
Has the advantages that:
the invention discloses an initialization method for formation configuration of an elliptic orbit satellite under multi-constraint conditions, which is characterized in that a true-anomaly angular domain T-H equation analytic solution is obtained through classical orbit dynamics and coordinate transformation by deduction, dynamics constraint is analyzed, periodic fly-around constraint is carried out, and monitoring camera view field constraint and thrust amplitude constraint are established, so that a multi-constraint optimal control model for formation is established and solved through a Gaussian pseudo-spectrum method to obtain a coordination value at any moment, the real-time monitoring on the structural integrity of a spacecraft is realized, and the fuel consumption and the maximum amplitude of a propulsion system are reduced by increasing the eccentricity and the initialization time.
Description of the drawings:
FIG. 1 is a flow chart of an initialization method for formation configuration of elliptic orbit satellites under multiple constraints according to the present invention;
FIG. 2 is a schematic diagram of a coordinate system for a two-star formation according to the present invention;
FIG. 3 is a view of the FOV geometry of step two of the present invention;
FIG. 4 is a schematic view of FOV constraints in step two of the present invention;
FIG. 5 is a release trajectory from the star and a fly-around trajectory of the present invention;
FIG. 6 is a projection of the release trajectory from the star and the orbit around the fly in the x-y plane of the present invention;
FIG. 7 is a projection of the release trajectory from the star and the orbit around the flight in the y-z plane of the present invention;
FIG. 8 is a time history of the relative position of the satellites according to the present invention;
FIG. 9 is a time history of relative velocity from the stars according to the present invention;
FIG. 10 is a time history of the present invention from the star to the optimal thrust acceleration;
Detailed Description
To better illustrate the objects and advantages of the present invention, the following detailed description of the embodiments of the present invention is provided in conjunction with the accompanying drawings. In the whole simulation process, all orbit perturbations are ignored, meanwhile, the orbit attitude of the main satellite is considered to be unchanged in the whole process, the auxiliary satellite is fixed on the main satellite before release, the formation initialization process is started by an ejection device, and then the auxiliary satellite enters a preset relative orbit flying around. And (3) simulating to convert the original optimal control model into a series of nonlinear programming problems (NLPs) by using a GPOPS-II tool box in a Matlab environment, and then finishing final solution by using SNOPT software.
Example 1:
in the method for initializing formation configuration of an elliptic orbit satellite under multiple constraint conditions disclosed by the embodiment, the reference orbit parameters are as follows: the semi-major axis a is 8378km, the inclination angle i is 15 °, the ascension angle Ω is 45 ° and the perigee angular distance ω is 30 °. At the beginning of the control, an initial true paraxial angle f is assumed0The parameter k within the period fly-around constraint is 0.5, and the FOV range of the surveillance camera is 60 ° × 60 °. The secondary satellite is a micro satellite with the mass of 10 kilograms and is provided with a three-dimensional electric propulsion device, and the maximum amplitude of the thrust in each direction is 100 mN. The time taken for release (denoted t)f) Set to 1500 seconds, the orbit eccentricity e is 0.05 from the initial state of the star.
The method comprises the following steps: and deducing to obtain the analytic solution of the true-near-point angular domain T-H equation through classical orbit dynamics and coordinate transformation.
As shown in FIG. 2, a fixed inertial frame centered on the earth is established with the X-axis pointingAt spring equinox, the Z axis is consistent with the rotation axis direction of the earth, and the Y axis is determined according to the right-hand rule. And establishing an orbit coordinate system which takes the main star as a mass center and rotates relative to the origin, wherein the x axis of the orbit coordinate system points to the center of the main star from the center of the earth, the z axis is vertical to the orbit plane, and the y axis is determined according to the right-hand rule. According to classical orbital dynamics, the data such as the semi-major axis a being 8378km, the inclination angle i being 15 °, the ascension point omega being 45 ° and the perigee angular distance omega being 30 ° are substituted into the equations (17) (18) (19) to obtain an analytic solution
Figure BDA0001895738600000151
Step two: and deducing a periodic fly-around constraint condition, and establishing a monitoring camera view field constraint and a thrust amplitude constraint.
Step 2.1: deriving a periodic fly-around constraint in a true anomaly angle domain;
step 2.2: establishing a monitoring camera view field constraint;
as shown in FIGS. 3 and 4, the FOV geometry of the surveillance camerA is made up of two cones, point Ot is the intersection of the vertices of the two cones, representing the optical center of the lens, l is the central axis of the geometry, representing the optical axis, rectangle ABCD is the bottom of the lower cone, representing the CCD array, and α and β are the FOV angles, α and β are determined by the size of the CCD and the focal length of the lens
Figure BDA0001895738600000152
A non-rotating coordinate system fixed at the optical center of the lens. Therefore, the distance from the point Ot to the centroid of the main star is extremely small, and the distance from Ot to the centroid of the main star is ignored, and therefore, the monitoring coordinate system coincides with the body coordinate system of the main star.
Step 2.3: establishing an amplitude constraint for the electric propulsion;
step three: and establishing a multi-constraint optimal control model for formation based on the true near point angular domain T-H equation analytic solution obtained by derivation in the step one and the monitoring camera view field constraint and thrust amplitude constraint established in the step two.
Step 3.1: establishing a basic optimal control model without considering the constraint condition;
step 3.2: establishing a multi-constraint optimal control model based on the monitoring camera view field constraint and the thrust amplitude constraint established in the step two;
step four: and solving the multi-constraint optimal control model to obtain a solution of the original optimal control problem and then obtain a collaborative value at any moment, namely realizing the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition and realizing the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.
Step 4.1: solving by a Gaussian pseudo-spectrum method;
will be true of the angle f0The parameter k within the period fly-around constraint is 0.5, and the FOV range of the surveillance camera is 60 ° × 60 °. The secondary satellite is a micro satellite with the mass of 10 kilograms and is provided with a three-dimensional electric propulsion device, and the maximum amplitude of the thrust in each direction is 100 mN. The time taken for release (denoted t)f) The initial state of the star is set to 1500 seconds, and constraint conditions such as the orbital eccentricity e being 0.05 are substituted into yes (72), and the optimal solution J being 0.0067 is obtained. Fig. 5 shows the configuration and periodic fly-around trajectory of the slave star in the orbital coordinate system, with the quadrangular pyramid in fig. 5 representing the FOV of the camera.
Fig. 4 shows that the optimal release trajectory is always within the quadrangular pyramid until the slave stars enter the periodic orbit around, which enables real-time monitoring of the slave stars to ensure the safety of the formation configuration, and the trajectory is also smooth, with no oscillations observed during release. After 1500 seconds of release has expired, the slave star moves continuously by following the desired periodic fly-around trajectory, which also demonstrates the correctness of the constraint in equation (55) when the periodic fly-around relative motion in the elliptical orbit is initialized.
FIGS. 6 and 7 show the projection of the trajectory in the x-y and z-y planes: since the eccentricity of the reference track is small, the closed projections in the two planes approximate an ellipse, which is consistent with the conclusion drawn by equation (33). The time history from the relative position and velocity of the star is shown in figures 8 and 9, respectively, and during release the relative position and velocity smoothly change from an initial value to a desired final value with peak values of relative velocity of 0.85m/s (x direction), 1.5m/s (negative y direction) and 0.47m/s (z direction), respectively.
As shown in fig. 10, the optimum thrust acceleration also smoothly changes from a positive value to a negative value throughout the release process, and the maximum magnitude of the thrust acceleration occurs on the x-axis of 1500s, but the thrust acceleration does not exceed 30% of the maximum magnitude in any of the three directions.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (3)

1. An initialization method for formation configuration of elliptic orbit satellites under multiple constraint conditions is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: deducing to obtain an analytic solution of a true-near-point angular domain T-H equation through classical orbit dynamics and coordinate transformation;
step two: deducing a periodic fly-around constraint condition, and establishing a monitoring camera view field constraint and a thrust amplitude constraint;
step three: establishing a multi-constraint optimal control model for formation based on the true near point angular domain T-H equation analytic solution obtained by derivation in the first step and the monitoring camera view field constraint and thrust amplitude constraint established in the second step;
step four: solving the multi-constraint optimal control model to obtain a solution of an original optimal control problem and then obtain a collaborative value at any moment, namely realizing the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition, and realizing the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite;
the specific implementation method of the step one is that,
step 1.1: respectively establishing an inertia coordinate system and a track coordinate system;
establishing a fixed inertial coordinate system with the earth as a center, wherein an X axis points to spring equinox, a Z axis is consistent with the direction of a rotating shaft of the earth, and a Y axis is determined according to a right-hand rule; establishing an orbit coordinate system which takes the main satellite as a mass center and rotates relative to an origin, wherein the x axis of the orbit coordinate system points to the center of the main satellite from the center of the earth, the z axis is vertical to an orbit plane, and the y axis is determined according to a right-hand rule;
step 1.2: obtaining an analytic solution according to classical orbit dynamics;
defining the relative position vector rho ═ r of the master star and the slave stard-rcWherein r isd∈R3And rc∈R3Position vectors of the slave star and the master star, respectively; the relative dynamics between the master and slave stars are expressed in the inertial reference frame as:
Figure FDA0002354807000000011
where μ represents a constant earth gravity coefficient and u ∈ R3Representing the actively controlled acceleration generated by a propeller fixed to the tracker; from the inertial reference system to the orbital coordinate system, equation (1) is approximately expressed as:
Figure FDA0002354807000000012
where ω is the orbital angular velocity of the primary satellite and the subscript o denotes the vector described in the rotating orbital coordinate system; substituting formula (1) into formula (2), the relative motion is expressed as
Figure FDA0002354807000000021
Wherein
Figure FDA0002354807000000022
Wherein f is the true proximal angle of the principal star; finally, substituting equations (1) and (2) into equation (3) yields:
Figure FDA0002354807000000023
Figure FDA0002354807000000024
Figure FDA0002354807000000025
wherein
Figure FDA0002354807000000026
Neglecting active control, equations (5) to (7) are T-H equations; the T-H equation describes the relative dynamics of free flight orbits of the two spacecrafts, and the internal and external motion of the orbital plane is decoupled; due to the nonlinearity of the T-H equation, an analytic solution of the T-H equation is difficult to find in a time domain; therefore, the T-H equation is rewritten by the following transformation:
Figure FDA0002354807000000027
taking the first and second derivatives of f,
Figure FDA0002354807000000028
Figure FDA0002354807000000029
the following operators are defined:
Figure FDA00023548070000000210
Figure FDA00023548070000000211
finally, substituting equations (9) to (13) into equations (5) to (7), the relative kinetics are re-expressed as:
Figure FDA0002354807000000031
Figure FDA0002354807000000032
Figure FDA0002354807000000033
obviously, the equations (14) - (15) are linearized and solved under the given initial conditions:
Figure FDA0002354807000000034
Figure FDA0002354807000000035
Figure FDA0002354807000000036
wherein
Figure FDA0002354807000000037
h and p are the angular momentum of the main star and the positive semi-focal distance respectively; c. Ci(i ═ 1, 2.., 6) is a constant coefficient determined by the initial state of the main star and the initial true proxel angle, calculated as:
Figure FDA0002354807000000038
Figure FDA0002354807000000039
Figure FDA00023548070000000310
Figure FDA00023548070000000311
Figure FDA00023548070000000312
Figure FDA00023548070000000313
analytic solution of T-H equation into
Figure FDA00023548070000000314
Namely, deducing to obtain an analytic solution of a true near point angular domain T-H equation;
the concrete implementation method of the second step is that,
step 2.1: deriving a periodic fly-around constraint in a true anomaly angle domain;
since the expressions (21) to (26) are all sinusoidal, for a given true paraxial angle f, c4(f) Is equal to 0, i.e
Figure FDA0002354807000000041
Periodic relative trajectories can be ensured; further, the following formulae (9), (10) and (27):
Figure FDA0002354807000000042
equation (28) is a prerequisite for a periodic trajectory in an arbitrary elliptical orbit, and if equation (27) is satisfied, the corresponding periodic relative motion is expressed as
Figure FDA0002354807000000043
Figure FDA0002354807000000044
Figure FDA0002354807000000045
However, since equation (27) cannot guarantee a periodic fly-around trajectory, equation (30) is rewritten as follows:
Figure FDA0002354807000000046
then, combining the formula (32) and the formula (29), the in-plane relative trajectory is represented as
Figure FDA0002354807000000047
When the in-plane track flies around, the three-dimensional track can be ensured to fly around; in formula (33), when c1When the orbit is 0, the orbit is reduced to an ellipse with (0,0) as the center, and the half axis of the ellipse changes with time; the periodic fly-around constraint is described as:
Figure FDA0002354807000000048
where k is a coefficient selected according to the orbit around, k may be set to zero if it is desired to follow the relative trajectory in the plane from the star;
step 2.2: establishing a monitoring camera view field constraint;
to ensure the safety of the initialization of the formation configuration, the master satellite is equipped with a visible camera to monitor the release process of the slave satellite, the slave satellite is located in the field of view of the camera until the slave satellite forms a periodic orbit around the fly, the FOV geometry of the surveillance camera is composed of two cones, the point Ot is the intersection of the vertices of the two cones, which represents the optical center of the lens, l is the central axis of the geometry, which represents the optical axis, the rectangle ABCD is the bottom of the lower cone, which represents the CCD array, and α and β are the FOV angles, the FOV angles α and β are determined by the size of the CCD and the focal length of the lens, the range of the FOV is determined by the size ofThe cone O-A 'B' C 'D' represents, but has an infinite depth; the extent of the field of view is represented by a cone, also with infinite depth, and the surveillance coordinate system is represented as
Figure FDA0002354807000000051
A non-rotating coordinate system fixed on the optical center of the lens; therefore, the distance from the point Ot to the centroid of the primary star is extremely small, and the distance from Ot to the centroid of the primary star is ignored, so that the monitoring coordinate system is consistent with the body coordinate system of the primary star;
the position vector considering the points is:
Figure FDA0002354807000000052
Figure FDA0002354807000000053
the plane OA 'D' may then be expressed as:
Figure FDA0002354807000000054
wherein the subscript b denotes a vector described in the body coordinate system; the equations for the other three planes are expressed as:
Figure FDA0002354807000000055
thus, the monitoring constraint is expressed as:
Figure FDA0002354807000000061
further, assume that the attitude of the primary star is (θ φ ψ) in the orbital coordinate systemT(ii) a Where θ, φ and ψ are the pitch, yaw and roll angles, respectively, and then the vectors in the body coordinate system are converted to the orbital coordinate system as shown in equation (40):
Figure FDA0002354807000000062
wherein L isob(θ, φ, ψ) is a transformation matrix from the body coordinate system to the orbit coordinate system; the formula (40) is substituted for the formula (39) to obtain the monitoring constraint in the orbit coordinate system
Figure FDA0002354807000000063
Step 2.3: establishing an amplitude constraint for the electric propulsion;
configuration initialization is simplified by keeping the attitude of the satellites constant relative to the orbital coordinate system, and thrust acceleration is always along the axis of the orbital coordinate system, so the electric propulsion magnitude constraint is expressed as:
Figure FDA0002354807000000064
wherein
Figure FDA0002354807000000065
Representing the maximum magnitude of thrust in each direction;
the third step is realized by the concrete method that,
step 3.1: establishing a basic optimal control model without considering the constraint condition;
the general form of nonlinear optimal control is expressed as: in the time interval t0，tf]In the method, a state control function for minimizing a performance index functional is found
Figure FDA0002354807000000071
Simultaneous states are constrained by system dynamics equations
Figure FDA0002354807000000072
Boundary constraint
Ψ(x(t0)，x(tf))＝0 (45)
And existing state and path constraints controlling mixing
C(x(t)，u(t))≥0 (46)
Wherein E is called a terminal performance index, also called a Mayer type performance index; the integral term containing F is called a process performance index, also called a Lagrange-type performance index; defining an augmented Hamilton function as
L(x(t)，u(t)，λ，μ)＝H(x(t)，u(t)，λ)+μTC(x(t)，u(t)) (47)
Where λ is the co-state variable and H is the Hami1ton function
H(x(t)，u(t)，λ)＝λTf(x(t)，u(t))+F(x(t)，u(t)) (48)
And mu satisfies the complementary condition
μTC(x(t)，u(t))＝0 (49)
The optimal control and state is obtained by solving the following equations:
Figure FDA0002354807000000073
wherein superscript denotes an optimal solution satisfying the aforementioned constraint condition;
step 3.2: establishing a multi-constraint optimal control model based on the monitoring camera view field constraint and the thrust amplitude constraint established in the step two;
angular velocity and angular acceleration taking into account true paraxial angle
Figure FDA0002354807000000081
Figure FDA0002354807000000082
Wherein the content of the first and second substances,
Figure FDA0002354807000000083
represents the average track angular velocity, while a and e represent the semi-major axis and eccentricity of the track, respectively; then, by substituting equation (52) for equations (5) and (6), the augmented kinetic equation is restated as:
Figure FDA0002354807000000084
wherein
Figure FDA0002354807000000085
Shows augmentation kinetics, and
Figure FDA0002354807000000086
represents an augmented state vector; the expression in said formula (34) is considered as a nonlinear state constraint on the augmented dynamics, enabling the problem to be solved by using a direct method;
once the formation configuration has started, the slaves are released, the initial state of the configuration being determined by the release position and speed, i.e.
Figure FDA0002354807000000087
The final state of the formation configuration is constrained by the periodic fly-around:
Figure FDA0002354807000000091
wherein c isi(i ═ 1, 2.., 6) calculated according to equations (21) to (26), the slave star must be located in the FOV of the master star CCD camera during configuration, i.e. according to equation (41), the relative states should satisfy the following constraints:
Figure FDA0002354807000000092
the thrust on each shaft should also satisfy the following constraints:
Figure FDA0002354807000000093
the mathematical description of the multi-constraint optimal control problem is therefore:
Figure FDA0002354807000000094
equation (58) is subject to the dynamic constraints of equation (53), the initial state constraints of equation (54), the period fly-around constraints of equation (55), the FOV constraints of equation (56), and the thrust magnitude constraints of equation (57).
2. The method for initializing formation configuration of elliptic orbit satellites under multiple constraints as claimed in claim 1, wherein: and step four, the method for solving and establishing the multi-constraint optimal control model of the formation comprises a direct method, an indirect method and a mixed method.
3. The method for initializing formation configuration of elliptic orbit satellites under multiple constraints as claimed in claim 1, wherein: the Gaussian pseudo-spectral method is a direct method, when the Gaussian pseudo-spectral method is used for solving and establishing the multi-constraint optimal control model of formation, the concrete implementation method of the step four comprises the following steps,
step 4.1: solving by a Gaussian pseudo-spectrum method;
the basic thought of the Gaussian pseudospectrum method is that unknown state time history and control time history are dispersed on a series of Gaussian points, Lagrange interpolation polynomial is respectively constructed by using the dispersed state and control to approximate to real state and control time history, the constraint of a kinetic differential equation is converted into a series of algebraic constraints, the optimal control problem is finally converted into a parameter optimization problem constrained by the series of algebraic constraints, and other numerical methods such as a sequence quadratic programming method are utilized to solve the problem;
the Gaussian point is zero point of Legendre polynomial and is unevenly distributed in the interval [ -1,1]Therefore, it is necessary to match the real time interval t ∈ [ t ]0，tf]And (3) carrying out conversion:
Figure FDA0002354807000000101
substitution of t by T as an independent variable after conversion, t0，tfBecoming system parameters; accordingly, the original time domain optimal control problem is expressed in the τ domain as:
Figure FDA0002354807000000102
using the kth legendre-gaussian point and the state at the initial node, derive a lagrange interpolation polynomial with order K + 1:
Figure FDA0002354807000000103
wherein
Figure FDA0002354807000000104
Lagrange interpolation can ensure that the approximation at the interpolated node is exactly the same as the actual state, while the values at other points are only approximately equal; using the control of the Kth Legendre-Gaussian point, the controlled Lagrange interpolation polynomial is derived as
Figure FDA0002354807000000105
Wherein
Figure FDA0002354807000000111
Differentiating the formula (61) in the equation to obtain
Figure FDA0002354807000000112
Wherein the differential matrix is determined as follows:
Figure FDA0002354807000000113
wherein K is 1,2, 1, K, and i is 0,1K(τ) Legendre polynomials of order K, i.e.
Figure FDA0002354807000000114
Substituting the formula (67) into the formula (66), and algebraicizing the kinetic constraint
Figure FDA0002354807000000115
The terminal value should be obtained by Gauss quadrature method
Figure FDA0002354807000000116
Wherein ω iskIs a Gaussian weight, calculated as
Figure FDA0002354807000000117
ωkAnd DkiDetermined only by the number of LG points K and solved off-line prior to the optimization algorithm, the continuous performance indicator functional of equation (60) is solved by gaussian integration:
Figure FDA0002354807000000118
therefore, by the gaussian pseudo-spectrum method, the initial optimal control problem is transformed into a nonlinear programming problem, which is described as follows:
Figure FDA0002354807000000121
solving the nonlinear programming problem to obtain the solution of the original optimal control problem and obtain the covariant value at any moment, namely realizing the initialization of the formation configuration of the elliptic orbit satellite under the multi-constraint condition and realizing the effective optimal control under the multi-constraint condition through the initialization of the formation configuration of the satellite.
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Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP0978957A1 (en) * 1998-08-06 2000-02-09 Alcatel Method for allocating links between a set of zones and a set of satellites
US6418312B1 (en) * 1999-08-30 2002-07-09 Motorola, Inc. Management of perferred communications periods in a satellite communication system
CN103412573A (en) * 2013-07-22 2013-11-27 西北工业大学 Elliptical orbit spacecraft relative position regressing control method based on cascade connection equation
CN104252548A (en) * 2013-06-27 2014-12-31 上海新跃仪表厂 Method of designing injection target point of Mars probe with optimal fuel
CN107391813A (en) * 2017-07-03 2017-11-24 北京航空航天大学 A kind of energetic optimum moon based on the moon high ladder ground transfer orbit design method
CN107526368A (en) * 2017-09-12 2017-12-29 北京理工大学 A kind of multiple-pulse ring moon satellites formation initial method for considering error
CN107589756A (en) * 2017-09-12 2018-01-16 北京理工大学 A kind of Benyue satellites formation initial method
CN108438255A (en) * 2018-03-14 2018-08-24 上海航天控制技术研究所 Satellite is diversion Formation Configuration initial method under a kind of engineering constraints
CN108875174A (en) * 2018-06-06 2018-11-23 北京航空航天大学 A kind of constant quasi-periodic orbit based on multistage shooting method determines method

Patent Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP0978957A1 (en) * 1998-08-06 2000-02-09 Alcatel Method for allocating links between a set of zones and a set of satellites
US6418312B1 (en) * 1999-08-30 2002-07-09 Motorola, Inc. Management of perferred communications periods in a satellite communication system
CN104252548A (en) * 2013-06-27 2014-12-31 上海新跃仪表厂 Method of designing injection target point of Mars probe with optimal fuel
CN103412573A (en) * 2013-07-22 2013-11-27 西北工业大学 Elliptical orbit spacecraft relative position regressing control method based on cascade connection equation
CN107391813A (en) * 2017-07-03 2017-11-24 北京航空航天大学 A kind of energetic optimum moon based on the moon high ladder ground transfer orbit design method
CN107526368A (en) * 2017-09-12 2017-12-29 北京理工大学 A kind of multiple-pulse ring moon satellites formation initial method for considering error
CN107589756A (en) * 2017-09-12 2018-01-16 北京理工大学 A kind of Benyue satellites formation initial method
CN108438255A (en) * 2018-03-14 2018-08-24 上海航天控制技术研究所 Satellite is diversion Formation Configuration initial method under a kind of engineering constraints
CN108875174A (en) * 2018-06-06 2018-11-23 北京航空航天大学 A kind of constant quasi-periodic orbit based on multistage shooting method determines method

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits;Gokhan Inalhan;《JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS_2002》;20020228;第25卷(第1期);第48-59页 *
Relative Motion Between Elliptic Orbits: Generalized Boundedness Conditions and Optimal Formationkeeping;Pini Gurfil;《JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS_2005》;20050831;第28卷(第4期);第761-767页 *
日地Halo轨道的多约束转移轨道分层微分修正设计;张景瑞;《宇航学报》;20151031;第36卷(第10期);第1114-1124页 *
椭圆轨道目标的绕飞初始化问题研究;王炳程;《中国优秀硕士学位论文全文数据库-工程科技Ⅱ辑》;20150215(第2期);第C031-541页 *

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