CN112208796A - Gravity field mixing linearization method - Google Patents

Gravity field mixing linearization method Download PDF

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CN112208796A
CN112208796A CN202010940619.6A CN202010940619A CN112208796A CN 112208796 A CN112208796 A CN 112208796A CN 202010940619 A CN202010940619 A CN 202010940619A CN 112208796 A CN112208796 A CN 112208796A
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linearization method
linearization
expression
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CN112208796B (en
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张冉
王嘉炜
李惠峰
师鹏
程晓明
禹春梅
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Beihang University
Beijing Aerospace Automatic Control Research Institute
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B64G1/242—Orbits and trajectories
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/04—Constraint-based CAD
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention provides a gravity field mixing linearization method, which comprises the following steps: step one, acquiring a position track and relative variation thereof; step two, calculating an expression of a Taylor linearization method; step three, calculating an expression of a Jerveski linearization method; step four, calculating a mixing coefficient; step five, calculating an expression of the hybrid linearization method; through the steps, a novel gravity field linearization method is obtained, the effects of enhancing the convergence and the convergence speed of the trajectory planning method of the rocket power carrier are achieved, and the problems of difficult convergence and slow convergence of the trajectory planning method are solved. The method of the invention is scientific, has good manufacturability and has wide popularization and application value.

Description

Gravity field mixing linearization method
Technical Field
The invention relates to a mixed linearization method of a gravity field, which can be applied to a trajectory planning method of a rocket power carrier and belongs to the technical field of guidance, navigation and control.
Background
The problem of trajectory planning for rocket-powered vehicles is a type of optimal control problem. The trajectory planning method generally converts a trajectory planning problem into a parameter optimization problem to solve. The non-linearity of the parameter optimization problem determines the difficulty of solving the problem. In order to reduce the difficulty of solving the trajectory planning problem, the nonlinear motion equation of the rocket power carrier needs to be subjected to linearization processing. The gravitational field is the main nonlinear term in the nonlinear equation of motion of rocket-powered vehicles. Therefore, in planning the trajectory of the rocket-powered vehicle, it is generally necessary to linearize the gravitational field. Currently, there are two commonly used linearization methods.
The first linearization method is called taylor linearization method. The taylor linearization method uses taylor expansion of a gravity field at a certain position as an approximation. This method is highly accurate but very sensitive to initial guesses. Therefore, the taylor linearization method reduces the convergence of the trajectory planning method.
The second linearization method is called the Jervsky linearization method. The Jervsky linearization method uses the expansion of the gravitational field at a certain height as an approximation. This method is insensitive to initial guesses, but less accurate. Therefore, the Jervsky linearization method reduces the convergence speed of the trajectory planning method.
It can be seen that, by applying the existing gravity field linearization methods (taylor linearization method and yerzivski linearization method), one of the convergence or convergence speed of the trajectory planning method is reduced and cannot be obtained.
Disclosure of Invention
Objects of the invention
The invention aims to provide a gravity field mixing linearization method, namely a novel gravity field linearization method called as a mixing linearization method. The gravity field hybrid linearization method can be applied to a trajectory planning method of a rocket power carrier, and reduces the nonlinearity degree of a motion equation, thereby reducing the solving difficulty. There are two existing linearization methods: taylor linearization method and jezivski linearization method. The hybrid linearization method is a hybrid of the taylor linearization method and the yerzivski linearization method. The hybrid linearization method is applied to the trajectory planning method of the rocket power carrier, so that the advantages and disadvantages of the two linearization methods can be complemented, and the convergence speed of the trajectory planning method are further enhanced. The enhancement of convergence and convergence speed enhances the autonomy and reliability of the trajectory planning method of the rocket power carrier, and has important value in the realization of intelligent autonomous spacecrafts.
(II) technical scheme
The invention discloses a gravity field hybrid linearization method which can be applied to a trajectory planning method of a rocket power carrier, and comprises the following specific steps:
step one, acquiring position track and relative variation thereof
When the trajectory planning method needs to calculate an expression of the linearized gravity field, extracting a position trajectory from the trajectory planning method to serve as a linearized expansion point of the gravity field; meanwhile, calculating the relative variation of the position track (relative to the last iteration) for subsequent calculation;
step two, calculating an expression of the Taylor linearization method
Calculating an expression of a gravity field on a position track by applying a formula of a Taylor linearization method;
step three, calculating the expression of the Jervsky linearization method
Calculating an expression of the gravity field on the position track by applying a formula of a Jervsky linearization method;
step four, calculating a mixing coefficient
Calculating a blending coefficient to blend the taylor linearization method and the jezivski linearization method; the mixing coefficient is a number between 0 and 1; the invention provides an empirical formula, and the relative variation of the position track is applied to calculate the mixing coefficient;
step five, calculating mixed linearization method
And (3) mixing the Taylor linearization method and the Yeziwski linearization method by applying a mixing coefficient to obtain an expression of the mixed linearization method for the track planning method to call. The calculation steps of the hybrid linearization method are ended so far, and the trajectory planning method starts to perform other calculations. And when the trajectory planning method needs to update the expression of the mixed linearized gravity field, re-entering the step one.
Wherein, the expression of the taylor linearization method is calculated in the step two, which is specifically as follows:
acquiring a position track from a track planning method, calculating the gravity acceleration and the gravity tensor (namely the Jacobian matrix of the gravity acceleration to the position) on the position track, and calculating to obtain an expression of a gravity field under a Taylor linearization method by applying a formula of the Taylor linearization method;
the expression "calculating the Jersey-based linearization method" described in step three is embodied as follows:
acquiring a position track from a track planning method, and calculating to obtain an expression of a gravity field under the Yezewski linearization method by applying a formula of the Yezewski linearization method;
wherein, the "calculating the mixing coefficient" in step four is as follows:
the position track and the relative variation thereof are obtained from the track planning method, and the empirical formula provided by the invention is applied to calculate the mixing coefficient, wherein the mixing coefficient is close to 1 when the relative variation is large, and the mixing coefficient is close to 0 when the relative variation is small.
The "calculation mixing linearization method" in step five is specifically as follows:
and (4) applying the mixing coefficient calculated in the fourth step, and carrying out linear weighting on the expression of the Yezivski linearization method of the Taylor linearization method calculated in the second step and the third step to obtain the expression of the gravity field under the mixed linearization method.
Through the steps, a novel gravity field linearization method is obtained, the effects of enhancing the convergence and the convergence speed of the trajectory planning method of the rocket power carrier are achieved, and the problems of difficult convergence and slow convergence of the trajectory planning method are solved.
(III) the advantages and effects of the invention
The invention has the advantages and effects that:
(1) by mixing the Taylor linearization method and the Yezivski linearization method, the respective advantages of the two linearization methods can be fully exerted, and the convergence speed of the trajectory planning method are enhanced;
(2) by applying the empirical formula of the mixing coefficient provided by the invention, the mixing coefficient can be calculated based on the relative variation of the position track, so that the empirical formula of the mixing coefficient has stronger universality and can adapt to different track planning methods;
(3) the method of the invention is scientific, has good manufacturability and has wide popularization and application value.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is a comparison result of the hybrid taylor linearization method proposed by the present invention and the existing method.
FIG. 3 is a comparison result of the hybrid Taylor linearization method proposed by the present invention and the prior art method.
Detailed Description
The present invention is described in further detail below.
The nonlinear equation of motion of a rocket-powered vehicle is:
Figure BDA0002673524960000041
Figure BDA0002673524960000042
in the formula: r represents a position vector, V represents a velocity vector, g represents a gravitational acceleration vector, T represents a thrust vector, and m represents a rocket mass; the trajectory planning method requires discretization of the motion equations into several equality constraints. Nonlinear equation of motion constraint is formed after discretization, and the solving difficulty of the track planning problem is increased. The nonlinear equation of motion is linearized and then discretized to form linear equation constraint, and the solving difficulty is greatly reduced compared with the nonlinear equation constraint. In the nonlinear equation of motion of a rocket-powered vehicle, only the gravitational field is nonlinear. Therefore, a method for linearizing the gravity field is needed for the trajectory planning method.
The invention discloses a gravity field mixing linearization method, the flow chart of which is shown in figure 1, comprising the following steps:
step one, acquiring position track and relative variation thereof
And extracting a position track from the track planning method to be used as a linear expansion point of the gravity field. At the same time, the relative change in the position trajectory (relative to the last iteration) is calculated for subsequent calculations.
Position trajectory is expressed as a function r of time0The locus of positions in the last iteration is represented as a function of time
Figure BDA0002673524960000043
The calculation formula of the relative variation epsilon is as follows:
Figure BDA0002673524960000044
step two, calculating an expression of the Taylor linearization method
Calculating an expression of a gravity field on a position track by applying a formula of a Taylor linearization method; the non-linear gravitational field of the earth can be expressed as:
Figure BDA0002673524960000045
in the formula: g represents gravitational acceleration, and μ represents a gravitational constant of the earth;
at the position locus r0The expression of the taylor linearization method is:
Figure BDA0002673524960000051
in the formula: gTIs an expression of the taylor linearization method,
Figure BDA0002673524960000052
is shown at r0The tensor of gravity (i.e., the jacobian matrix of gravitational acceleration versus position).
Step three, calculating the expression of the Jervsky linearization method
Calculating an expression of the gravity field on the position track by applying a formula of a Jervsky linearization method;
at the position locus r0The expression for the Jervsky linearization method is:
Figure BDA0002673524960000053
in the formula: gJIs an expression of the yarrowia linearization method.
Step four, calculating a mixing coefficient
Calculating a blending coefficient to blend the taylor linearization method and the jezivski linearization method; the mixing coefficient is a number between 0 and 1; in the hybrid linearization method, the determination of the hybrid coefficient lambda is crucial; in order to enhance the convergence and the convergence speed of the trajectory planning method, the lambda is gradually transited from 1 to 0 along with the iteration;
the invention proposes the following empirical formula:
λ＝(1-e-∈)3 (7)
in the formula: e is the relative change calculated in step one; when epsilon is far more than zero, iteration is not converged, and then lambda is about 1; when the epsilon is close to zero, the iteration is close to convergence, and the lambda is about 0 at the moment; this empirical formula thus enables automatic selection of the mixing coefficients.
Step five, calculating an expression of a mixed linearization method
Applying a mixing coefficient to mix a Taylor linearization method and a Yezivski linearization method for a trajectory planning method to call;
the expression for the hybrid linearization method is:
gH(r)＝(1-λ)gT(r)+λgJ(r) (8)
in the formula: gHIs an expression of the hybrid linearization method;
it can be seen that the hybrid linearization method is a hybrid of the taylor linearization method and the jezivski linearization method, and λ is the mixing coefficient; when λ is 0, the hybrid linearization method is equivalent to the taylor linearization method; when λ ═ 1, the hybrid linearization method is equivalent to the yerzivski linearization method; when λ is between 0 and 1, the gravitational acceleration is a mixture of the two linearization method calculations, hence this method is referred to as the mixed linearization method.
Through the steps from one to five, an expression of the hybrid linearization method is calculated, and the method can be applied to a trajectory planning method. And if the iteration of the track planning method is not terminated, recalculating the expression of the hybrid linearization method from the first step for iteration.
Simulation case:
in the simulation case: setting the exhaust speed of the rocket to be 3400 m/s, the thrust of the rocket to be 600 kilo-newtons, the initial mass of the rocket to be 89 tons, and the final mass of the rocket to be 22 tons; setting an initial semi-major axis to be 3600 kilometers, an initial eccentricity to be 0.81, an initial orbit inclination angle to be 40 degrees, an initial ascent point right ascension to be 314.5 degrees, an initial perigee argument to be 334.5 degrees, and an initial true perigee angle to be 177.5 degrees; the goal of trajectory planning for rocket-powered vehicles is to maximize the termination semimajor axis, the termination eccentricity is 0, the termination orbit inclination is 42 degrees, and the termination ascent crossing right ascension is 313 degrees.
In the simulation case, a convex planning method is selected as a trajectory planning method of the rocket power carrier. The convex programming method can be applied to various gravity field linearization methods, including taylor linearization method, jezivski linearization method, and the hybrid linearization method proposed by the present invention. The simulation case is solved by applying the existing taylor linearization method and the jezivski linearization method, respectively, and the hybrid linearization method proposed by the present invention, and the results are shown in fig. 2 and fig. 3. When the Taylor linearization method is applied to solve the simulation case, the simulation case diverges in the 5 th iteration and cannot be planned out. Showing that the convergence of the Jervsky linearization method and the hybrid Taylor linearization method is superior to the Taylor linearization method. The convergence speed of the hybrid Taylor linearization method is superior to that of the Yeziwski linearization method, and higher trajectory precision can be obtained within fewer iteration times. It can be seen that in the same trajectory planning method (convex planning method), different gravity field linearization methods are applied, and the convergence property and the convergence speed of the trajectory planning method are different. The mixed linearization method provided by the invention can simultaneously enhance the convergence and the convergence speed of the trajectory planning method.
The simulation case verifies that the hybrid linearization method can fully exert the respective advantages of the hybrid Taylor linearization method and the Yeziwski linearization method, and further enhance the convergence and the convergence speed of the trajectory planning method.

Claims (1)

1. A gravity field hybrid linearization method is applied to a rocket power carrier, wherein:
the nonlinear equation of motion of a rocket-powered vehicle is:
Figure FDA0002673524950000011
Figure FDA0002673524950000012
in the formula: r represents a position vector, V represents a velocity vector, g represents a gravitational acceleration vector, T represents a thrust vector, and m represents a rocket mass; the trajectory planning method needs to discretize a motion equation into a plurality of equation constraints; forming nonlinear equality constraint after discretization of a nonlinear motion equation, and increasing the solving difficulty of a track planning problem; the nonlinear equation of motion is linearized and then discretized to form linear equation constraint, and the solving difficulty is greatly reduced compared with the nonlinear equation constraint; in the nonlinear equation of motion of the rocket-powered vehicle, only the gravitational field is nonlinear; therefore, the trajectory planning method requires a method for linearizing the gravity field;
the method is characterized in that: the method comprises the following steps:
step one, acquiring position track and relative variation thereof
Extracting a position track from a track planning method to be used as a linear expansion point of a gravity field; meanwhile, calculating the relative variation of the position track for subsequent calculation;
position trajectory is expressed as a function r of time0The locus of positions in the last iteration is represented as a function of time
Figure FDA0002673524950000013
The calculation formula of the relative variation epsilon is as follows:
Figure FDA0002673524950000014
step two, calculating an expression of the Taylor linearization method
Calculating an expression of a gravity field on a position track by applying a formula of a Taylor linearization method; the non-linear gravitational field of the earth is expressed as:
Figure FDA0002673524950000015
in the formula: g represents gravitational acceleration, and μ represents a gravitational constant of the earth;
at the position locus r0The expression of the taylor linearization method is:
Figure FDA0002673524950000016
in the formula: gTIs an expression of the taylor linearization method,
Figure FDA0002673524950000017
is shown at r0The tensor of gravity, i.e., the Jacobian matrix of gravitational acceleration versus position;
step three, calculating the expression of the Jervsky linearization method
Calculating an expression of the gravity field on the position track by applying a formula of a Jervsky linearization method;
at the position locus r0The expression for the Jervsky linearization method is:
Figure FDA0002673524950000021
in the formula: gJIs an expression of the Jerveski linearization method;
step four, calculating a mixing coefficient
Calculating a blending coefficient to blend the taylor linearization method and the jezivski linearization method; the mixing coefficient is a number between 0 and 1; in the hybrid linearization method, the determination of the hybrid coefficient lambda is crucial; in order to enhance the convergence and the convergence speed of the trajectory planning method, the lambda is gradually transited from 1 to 0 along with the iteration;
the following empirical formula is given:
λ＝(1-e-∈)3 (7)
in the formula: e is the relative change calculated in step one; when epsilon is far more than zero, iteration is not converged, and then lambda is about 1; when the epsilon is close to zero, the iteration is close to convergence, and the lambda is about 0 at the moment; therefore, the empirical formula can realize automatic selection of the mixing coefficient;
step five, calculating an expression of a mixed linearization method
Applying a mixing coefficient to mix a Taylor linearization method and a Yezivski linearization method for a trajectory planning method to call;
the expression for the hybrid linearization method is:
gH(r)＝(1-λ)gT(r)+λgJ(r) (8)
in the formula: gHIs an expression of the hybrid linearization method;
the hybrid linearization method is a hybrid of the taylor linearization method and the jezivski linearization method, and λ is a hybrid coefficient; when λ is 0, the hybrid linearization method is equivalent to the taylor linearization method; when λ ═ 1, the hybrid linearization method is equivalent to the yerzivski linearization method; when λ is between 0 and 1, the gravitational acceleration is a mixture of the two linearization method calculations, hence this method is referred to as the mixed linearization method.
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Patent Citations (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
TWI225622B (en) * 2003-10-24 2004-12-21 Sunplus Technology Co Ltd Method for detecting the sub-pixel motion for optic navigation device
CN105022403A (en) * 2015-05-11 2015-11-04 北京航天自动控制研究所 Method for determining the vertical locus control gain of gliding aircraft
CN108318856A (en) * 2018-02-02 2018-07-24 河南工学院 The target positioning of fast accurate and tracking under a kind of heterogeneous network
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CN110466804A (en) * 2019-08-30 2019-11-19 北京理工大学 The quick track optimizing method of rocket-powered decline landing mission
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