CN110321598B - Spacecraft relative motion analysis solving method under J2 perturbation condition - Google Patents
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Abstract
The invention discloses a method for analyzing and solving relative motion of a spacecraft under a J2 perturbation condition, which can carry out a space orbital evolution task of relative motion aiming at spacecraft formation/clustering under an earth non-spherical J2 gravity perturbation condition, and can realize the analytic evolution of the relative orbital motion state of the spacecraft formation/clustering by constructing a relative motion state transition matrix. The evolution of absolute orbit is carried out by using a J2 perturbation absolute orbit kinetic equation of members of a spacecraft formation/cluster, a relative motion differential equation taking a relative orbit state as a variable is obtained by carrying out coordinate transformation on absolute orbit difference among the members, and a relative motion state transition matrix and an analytic solution are obtained by solving the differential equation by a Taylor series expansion method.
Description
Technical Field
The invention belongs to the technical field of spacecraft formation relative motion dynamics modeling, and particularly relates to a spacecraft relative motion analytic solving method under the condition of J2 perturbation.
Background
With the development of aerospace technology, modern aerospace missions tend to be diversified and complicated more and more, and functions and applications which are difficult to realize by a single spacecraft are realized by formation flying of a plurality of spacecrafts, so that the method is an important development trend in the field of aerospace engineering. The premise of realizing the specific task of spacecraft formation flight is to maintain a certain geometric configuration, and the premise of carrying out configuration maintenance control is to model the relative motion dynamics of formation.
Currently, the existing spacecraft relative motion dynamics models are various, and various models are often limited and limited by the use range, orbit eccentricity, the type of modeling disturbance force and the calculation workload required by integration among the spacecrafts, but can only be applied to the problem of characteristics. It is often difficult to directly determine the quality of a dynamic model, and only in a specific task scene is it appropriate to see whether the model is suitable.
The earliest model of relative motion dynamics was proposed by Clhessy and Wiltshire to establish the C-W equation for the space station or "Rendezvous" phase in the process of orbital assembly of large satellites. The model takes time as an independent variable, describes the change of the relative position and the speed of two spacecrafts, and is suitable for the relative motion of a near-circular orbit under the condition of a short distance. Subsequently, for the non-linear problem of relative motion on an arbitrary elliptical orbit, tschauner and Hempel linearize the relative motion, and derive a linearized model with a true anomaly as an argument and a state transition matrix thereof.
Both the CW equation and the TH equation are linearization models based on the two-body problem, the perturbation effect is not considered, and other scholars respectively try to add atmospheric resistance perturbation, earth aspheric perturbation, march-Sandy gravitational perturbation and the like into a dynamics model on the basis. Since the spacecraft absolute orbit element has a slowly time-varying nature under perturbation influence, some authors introduce another state parameter describing the relative motion of the spacecraft, called the Relative Orbital Element (ROE), which is a linear or non-linear combination of six orbital elements of two spacecraft. The model developed by Gim and Alfriend considers the J2 perturbation of the earth oblateness and the orbital eccentricity, and obtains a state transition matrix based on ROE. Rao Yanrui and the like consider the influence of perturbation forces such as earth non-spherical attraction, atmospheric resistance, three-body attraction and the like, and add corresponding perturbation items to the relative motion model.
In orbit control and maneuver of spacecraft formation flight, the description of the relative orbit number is not intuitive enough, and the relative position and the relative speed still need to be converted into real-time. A group of nonlinear differential equations is deduced by Kechichian, dynamics of a tracked spacecraft relative to a target spacecraft reference coordinate system is described, large ellipse orbit eccentricity and the influence of J2 perturbation of the earth are considered, the equations are further simplified by subsequent Threon and the like, a numerical integration result is obtained, and a relative motion analytic solution under the condition of J2 perturbation is not obtained in related research results.
Therefore, in the prior art, orbital evolution is performed on relative motion of a spacecraft formation by performing kinetic analytic modeling under J2 perturbation by taking a relative orbital element as a state or numerical integration evolution is performed on relative motion under J2 perturbation by taking a relative position and a relative speed as a state, and a J2 perturbation relative motion analytic solution method taking the relative position and the relative speed as a state is lacked.
Disclosure of Invention
In view of the defects of the prior art, the invention aims to provide a method for analyzing and solving the relative motion of a spacecraft under the condition of J2 perturbation, which can realize the evolution of the relative motion orbit of a spacecraft formation under the condition of J2 perturbation by using a state transition matrix obtained by the method without increasing the calculation amount too much.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the invention relates to a method for resolving and analyzing relative motion of a spacecraft under a J2 perturbation condition, which comprises the following steps of:
1) Establishing absolute orbit kinetic equations containing J2 perturbation of the two spacecrafts under an inertial coordinate system respectively;
2) Differentiating the absolute orbit dynamics equation under the J2 perturbation of the two spacecrafts to obtain a perturbation relative motion dynamics equation of the spacecrafts under an inertial system;
3) Establishing a coordinate transformation matrix from an inertial coordinate system i to a track coordinate system l (LVLH);
4) Performing coordinate transformation on the spacecraft shooting relative motion kinetic equation under the inertial system by using the established coordinate transformation matrix to obtain a shooting relative motion kinetic equation under the orbit system;
5) Taking the relative position and speed under the track system as state quantities, and arranging the established shot relative motion kinetic equation under the track system into a vector linear differential equation form;
6) And solving a vector linear differential equation to obtain a state transition matrix.
Further, the orbit dynamics equations of the two spacecrafts under the inertial coordinate system established under the condition of considering the earth non-spherical J2 perturbation in the step 1) are as follows:
wherein r is t And v t Respectively the relative position and velocity, r, of the spacecraft 1 in the inertial system c And v c Relative position and velocity, g (r), respectively, under the inertial system of the spacecraft 2 t ) And g (r) c ) The acceleration of the earth's gravity of the spacecraft 1 and 2, respectively:
where μ is the gravitational constant, R eq Is the mean radius of the earth, n = [0,0,1 =] T Is a unit vector of the rotation axis direction of the earth under the inertia system,
and &>
Are respectively r t And r c Unit vector of, J 2 =0.00108263 is the earth's aspherical perturbation coefficient, | | | | | represents modulo arithmetic.
Further, the spacecraft shooting relative motion kinetic equation under the inertial system obtained in the step 2) is as follows:
wherein r and v are the relative position and velocity between the two spacecrafts under the inertial system, respectively.
Further, the transformation matrix from the inertial coordinate system to the orbital coordinate system (LVLH) established in step 3) is as follows:
wherein,
denotes a coordinate transformation matrix from i system to l system, and T denotes a matrix transposition operation.
Further, the shot relative motion kinetic equation under the orbital system established in the step 4) is as follows:
wherein the superscript l denotes the projection in the orbital coordinate system l, Ω × Is a cross-product matrix of the orbital angular velocity omega of the spacecraft 2,
denotes that g (r) is in r c The partial derivative of (c);
wherein,
I 3×3 representing a 3 x 3 identity matrix.
Further, in the step 5), the relative position and speed under the orbital system are used as state quantities, that is, x (t) = [ r = [ r ] ] l ,v l ] T The equation of the relative motion dynamics of the object under the orbital system is arranged into a linear differential equation in the form as follows:
where F (t) is a system matrix of the form:
further, the step 6) specifically includes: equation (8) is solved according to the linear system theory, and the result is as follows:
x(t)＝Φ(t)x(0) (10)
wherein, x (0) = [ r ] 0 l ,v 0 l ] T Given initial relative position and speed, Φ (t) is the state transition matrix corresponding to time t, and the specific form is as follows:
Φ(t)＝e t·F(t) (11)。
where e is a natural constant.
The invention has the beneficial effects that:
the method can carry out the space orbital evolution task of relative motion on the spacecraft formation/cluster under the earth aspheric J2 gravity perturbation condition, and can realize the analytic evolution of the relative orbital motion state of the spacecraft formation/cluster by constructing a relative motion state transition matrix. The evolution of absolute orbit is carried out by using a J2 perturbation absolute orbit kinetic equation of members of a spacecraft formation/cluster, a relative motion differential equation taking a relative orbit state as a variable is obtained by carrying out coordinate transformation on absolute orbit difference among the members, and a relative motion state transition matrix and an analytic solution are obtained by solving the differential equation by a Taylor series expansion method.
Drawings
FIG. 1 is a schematic of the modeling method of the present invention;
FIG. 2a is a graph of X-axis direction relative position estimation error;
FIG. 2b is a graph of Y-axis direction relative position estimation error;
FIG. 2c is a graph of Z-axis direction relative position estimation error;
FIG. 3a is a graph of relative velocity estimation error in the X-axis direction;
FIG. 3b is a graph of Y-axis direction relative velocity estimation error;
FIG. 3c is a graph of Z-axis direction relative velocity estimation error.
Detailed Description
In order to facilitate understanding of those skilled in the art, the present invention will be further described with reference to the following examples and drawings, which are not intended to limit the present invention.
The invention discloses a method for resolving and resolving relative motion of a spacecraft under a J2 perturbation condition, and aims to solve the problems that in the prior art, orbital evolution is carried out on relative motion of a spacecraft formation through dynamic resolution modeling under the J2 perturbation by taking a relative orbital element as a state, or numerical integral evolution is carried out on relative motion under the J2 perturbation by taking a relative position and a relative speed as states, and a J2 perturbation relative motion resolving and resolving method taking the relative position and the relative speed as states is lacked. The method obtains a relative motion differential equation with a relative orbit state as a variable by carrying out coordinate transformation on absolute orbit difference between the spacecrafts, solves the differential equation by a Taylor series expansion method to obtain a relative motion state transition matrix and an analytic solution, and can be suitable for orbital analytic evolution of relative motion flight tasks of spacecraft formation/clustering.
Referring to fig. 1, the following is specifically explained:
1. the orbit dynamics equations of the two spacecrafts in the inertial coordinate system under the condition of considering the aspheric J2 perturbation of the earth are as follows:
wherein r is t And v t Respectively the relative position and velocity, r, of the spacecraft 1 in the inertial system c And v c Relative position and velocity, g (r), respectively, under the inertial system of the spacecraft 2 t ) And g (r) c ) The acceleration of the earth's gravity of the spacecraft 1 and 2, respectively.
2. And (3) carrying out difference on the absolute orbit kinetic equation under the J2 perturbation of the two spacecrafts to obtain the shot relative motion kinetic equation of the spacecrafts under the inertial system:
wherein r and v are the relative position and velocity between the two spacecrafts under the inertial system, respectively.
3. The established inertial to orbital coordinate system (LVLH) coordinate transformation matrix is as follows:
wherein,
denotes a coordinate transformation matrix from i system to l system, and T denotes a matrix transposition operation.
4. The established shooting relative motion kinetic equation under the orbital system is as follows:
wherein the superscript l denotes the projection, Ω, under the orbital coordinate system l × Is a cross-product matrix of the orbital angular velocity omega of the spacecraft 2,
denotes that g (r) is at r c The partial derivative of (c).
5. Taking the relative position and speed under the track system as state quantities, i.e. x (t) = [ r l ,v l ] T The equation of the relative motion dynamics of the object under the orbital system is arranged into a linear differential equation in the form as follows:
where F (t) is the system matrix.
6. Equation (5) is solved according to the linear system theory, and the result is as follows:
x(t)＝Φ(t)x(0) (6)
wherein, x (0) = [ r = 0 l ,v 0 l ] T Given initial relative position and speed, Φ (t) is the state transition matrix corresponding to time t, and the specific form is as follows: Φ (t) = e t·F(t) Where e is a natural constant.
Examples of the process of the invention: an example validation of the present invention is described in connection with fig. 2a to 3c, setting the following calculation conditions and technical parameters:
1) The initial orbit parameters of the satellite A in the inertial coordinate system are as follows:
[5023.5585km,5023.5585km,0km,-1.8109km/s,1.8109km/s,7.0411km/s]；
2) The initial orbit parameters of the satellite B in the inertial coordinate system are as follows:
[5023.4575km,5023.6791km,0.4700km,-1.8108km/s,1.8104km/s,7.0413km/s]；
3) Simulation time 27000 seconds;
4) The state update period is 10 seconds;
based on the relative motion analysis method and the set calculation conditions and technical parameters, matHEMATICS software is adopted for simulation verification. As shown in fig. 2a to fig. 3c, the evolution error curves of the relative position and the relative velocity between two spacecrafts are shown respectively, and the curves in the drawings show that the evolution errors of the three-axis relative position are all within 1m, the relative velocity is all within 0.001m/s, and the relative orbital evolution precision is very high.
Therefore, the method can realize the analytic modeling and solving of the J2 perturbation relative orbital motion under a Cartesian coordinate system and realize the accurate orbital evolution of the relative motion of the spacecraft.
While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.
Claims (7)
1. A method for resolving analysis of relative motion of a spacecraft under J2 perturbation conditions is characterized by comprising the following steps:
step 1) establishing absolute orbit kinetic equations containing J2 perturbation of two spacecrafts in an inertial coordinate system respectively;
step 2) carrying out difference on the absolute orbit dynamics equations under the J2 perturbation of the two spacecrafts to obtain the relative perturbation motion dynamics equation of the spacecrafts under the inertial system;
step 3) establishing a coordinate transformation matrix from an inertial coordinate system i to a track coordinate system l;
step 4) carrying out coordinate transformation on the spacecraft shooting relative motion kinetic equation under the inertial system by utilizing the established coordinate transformation matrix to obtain the shooting relative motion kinetic equation under the orbit system;
step 5) taking the relative position and speed under the track system as state quantities, and arranging the established shot relative motion kinetic equation under the track system into a vector linear differential equation form;
and 6) solving a vector linear differential equation to obtain a state transition matrix.
2. The method for resolving the relative motion of the spacecraft under the J2 perturbation condition according to the claim 1, wherein the orbital dynamics equations of the two spacecrafts under the inertial coordinate system under the consideration of the earth non-spherical J2 perturbation condition in the step 1) are as follows:
wherein r is t And v t Respectively the relative position and velocity, r, of the spacecraft 1 in the inertial system c And v c Relative position and velocity, g (r), respectively, under the inertial system of the spacecraft 2 t ) And g (r) c ) The acceleration of the earth's gravity of the spacecraft 1 and 2, respectively:
wherein, muIs the constant of gravity of the earth, R eq Is the mean radius of the earth, n = [0,0,1 =] T Is a unit vector of the rotation axis direction of the earth under the inertia system,
and &>
Are respectively r t And r c Unit vector of, J 2 =0.00108263 is the earth's aspherical perturbation coefficient, | | | | | represents modulo arithmetic.
3. The method for resolving the spacecraft relative motion under the J2 perturbation condition according to claim 2, wherein the spacecraft photographic relative motion kinetic equation under the inertial system obtained in the step 2) is as follows:
wherein r and v are the relative position and velocity between the two spacecrafts under the inertial system, respectively.
4. The method for resolving the analysis of the relative motion of the spacecraft under the condition of J2 perturbation according to the claim 3, wherein the coordinate transformation matrix from the inertial coordinate system to the orbit coordinate system established in the step 3) is as follows:
wherein,
denotes a coordinate transformation matrix from i system to l system, and T denotes a matrix transposition operation.
5. The method for analytically solving the relative motion of the spacecraft under the condition of J2 perturbation according to claim 4, wherein the equations of the perturbed relative motion in the orbital system established in the step 4) are as follows:
wherein the superscript l denotes the projection, Ω, under the orbital coordinate system l × Is a cross-product matrix of the orbital angular velocity omega of the spacecraft 2,
denotes that g (r) is in r c The partial derivative of (c);
wherein,
I 3×3 representing a 3 x 3 identity matrix.
6. The method for resolving the spacecraft relative motion under the J2 perturbation condition according to the claim 5, wherein the relative position and speed under the orbital system in the step 5) are state quantities, namely x (t) = [ r = l ,v l ] T The equation of the shot relative motion dynamics under the orbital system is arranged into a linear differential equation in the form as follows:
where F (t) is a system matrix of the form:
7. the method for resolving the spacecraft relative motion under the J2 perturbation condition according to claim 6, wherein the step 6) specifically comprises: equation (8) is solved according to the linear system theory, and the result is as follows:
x(t)＝Φ(t)x(0) (10)
wherein,
given initial relative position and velocity, Φ (t) is the state transition matrix corresponding to time t, the specific form is as follows:
Φ(t)＝e t·F(t) (11)
where e is a natural constant.
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