CN114861123A - GLONASS satellite coordinate fitting method based on Hermite interpolation method - Google Patents

Abstract

The invention relates to the technical field of satellite navigation, in particular to a GLONASS satellite coordinate fitting method based on the Hermite interpolation method. A new interpolation function is constructed by knowing the function value and the derivative value on the interpolation node, so that the interpolation function is the same as the original function. The degree of closeness is better, which greatly reduces the error between the approximate satellite coordinates and the real satellite coordinates, can better match the real satellite motion model, and effectively improves the computing efficiency and shortens the integration time.

Description

GLONASS satellite coordinate fitting method based on Hermite interpolation method

technical field

The invention relates to the technical field of satellite navigation, in particular to a GLONASS satellite coordinate fitting method based on the Hermite interpolation method.

Background technique

As an important device for testing receiver performance indicators, satellite signal simulators should not only be able to simulate and generate satellite navigation signals anytime and anywhere, but more importantly, have the ability to run indefinitely. In order to achieve continuous generation of satellite navigation signals, it is necessary to study the recursive algorithm of satellite positions.

The GLONASS navigation system uses the PZ-90 geodetic coordinate system and adopts frequency division multiple access technology, that is, each satellite uses a different operating frequency to transmit signals. Since the broadcast ephemeris parameters broadcast by the GLONASS satellite are not the same as the GPS ephemeris, it cannot be directly calculated by the ephemeris parameters like GPS.

At present, the commonly used method for calculating the precise position of GLONASS satellites is the Runge-kutta orbit integration method, which uses an integration method with a fixed integration step to calculate the satellite position of the GLONASS satellite at the next moment. When the integration step is too long, the calculation time will be greatly prolonged, and the calculated GLONASS satellite position has a larger error compared with the real situation.

SUMMARY OF THE INVENTION

The object of the present invention is to provide a GLONASS satellite coordinate fitting method based on the Hermite interpolation method, aiming to solve the problem that when the integration step is too long, the Runge-kutta orbit integration method will cause the calculation time to be greatly prolonged, and the calculated GLONASS satellite The problem that the position has a large error compared with the real situation.

To achieve the above object, the present invention provides a GLONASS satellite coordinate fitting method based on the Hermite interpolation method, comprising the following steps:

According to the GLONASS satellite ephemeris, establish the satellite orbital motion differential equation;

According to the satellite motion state information at the reference time, integrate the orbital motion differential equation to obtain the satellite motion state information at the next moment at the reference time;

Taking the satellite motion state information of the reference moment and the satellite motion state information of the next moment as two interpolation nodes of the Hermite interpolation method, and deriving the Hermite interpolation method to obtain a third-order interpolation formula;

The position of the satellite at any moment is approximated by the third-order interpolation formula.

Wherein, the satellite motion state information at the reference time includes satellite position, satellite velocity, and sun-moon perturbation acceleration.

Wherein, the integral of the orbital motion differential equation includes:

The orbital motion differential equation is integrated using a fourth-order Runge-kutta orbital integration algorithm.

Compared with the prior art, the GLONASS satellite coordinate fitting method based on the Hermite interpolation method of the present invention has the beneficial effect of: constructing a new interpolation function by knowing the function value and the derivative value on the interpolation node, so that the interpolation The function and the original function have a better degree of closeness, which greatly reduces the error between the approximate satellite coordinates and the real satellite coordinates, can better match the real satellite motion model, and effectively improves the computing efficiency and shortens the integration time.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to these drawings without creative efforts.

FIG. 1 is a schematic diagram of steps of a GLONASS satellite coordinate fitting method based on the Hermite interpolation method according to an embodiment of the present invention.

FIG. 2 is a flow chart of GLONASS orbit integral calculation based on Hermite interpolation method according to an embodiment of the present invention.

Detailed ways

Please refer to FIG. 1 and FIG. 2. FIG. 1 is a schematic diagram of steps of a method for fitting GLONASS satellite coordinates based on the Hermite interpolation method provided by an embodiment of the present invention. Specifically, as shown in Figure 1, the GLONASS satellite coordinate fitting method based on the Hermite interpolation method may include the following steps:

S101 , establishing an orbital motion differential equation of the satellite according to the GLONASS satellite ephemeris.

S102. Integrate the orbital motion differential equation according to the satellite motion state information at the reference time to obtain satellite motion state information at the next moment at the reference time.

S103. Use the satellite motion state information at the reference time and the satellite motion state information at the next time as two interpolation nodes of the Hermite interpolation method, and derive the Hermite interpolation method to obtain a third-order interpolation formula.

S104, approximately obtain the position of the satellite at any time by using the third-order interpolation formula.

Specifically, in the GLONASS satellite ephemeris, the position of the GLONASS satellite is given in the form of the motion vector of the satellite in the PZ-90 coordinate system. The main contents of the GLONASS satellite ephemeris are shown in the following table:

illustrate symbol unit reference time t_b s satellite clock offset τ_n(t_b),γ_n(t_b) s satellite location x_n(t_b),y_n(t_b),z<sub>n</sub >(t_b) m satellite speed x'_n(t_b),y'_n(t_b),z'_n(t_b) m/s sun-moon perturbation acceleration x”_n(t_b),y”_n(t_b),z”_n(t_b) m/s²

Table 1 Main content of GLONASS ephemeris

Since the GLONASS satellite ephemeris is updated every half an hour, the valid interval for ephemeris fitting is 30 minutes. The force analysis of the GLONASS satellite includes the universal gravitation of the earth, the aspherical gravitational perturbation of the earth, the gravitational perturbation of the sun and the moon, the gravitational perturbation of other planets, the perturbation of solar radiation pressure, the gravitational perturbation of the earth's tides, etc. of the force. Due to the short integration interval of fitting, in order to simplify the model of satellite orbital motion, it can be considered that the satellite is only affected by the earth's gravitational force, the earth's aspherical gravitational force and the sun-moon gravitational force in a short period of time. For the non-spherical gravitational force of the Earth, the high-order perturbation term coefficients have little influence, so they are ignored here, and only the harmonic coefficients below the second order are considered. As for the sun-moon perturbation item, the satellite ephemeris has been given, and within the allowable error range, the sun-moon perturbation acceleration can be considered as a fixed value. Based on the above, we can establish the orbital motion equation of the GLONASS satellite in the PZ-90 geocentric and geofixed coordinate system.

Assuming the t _b reference time, the three-dimensional coordinates of the GLONASS satellite in the PZ-90 coordinate system are x(t _b ), y(t _b ), z(t _b ), the earth's gravitational constant is GM, and the vector radius from the satellite to the earth's center of mass The length is r.

Radial length:

The radial acceleration obtained by the satellite due to the gravity of the center of the earth is:

The acceleration of the satellite caused by the non-spherical gravity of the earth is:

The acceleration of the satellite caused by the non-spherical gravity of the earth is:

为地球重力位二阶带谐系数，值为-0.00108263。日月引力摄动加速度为x'_n(t_b),y'_n(t_b),z'_n(t_b)，假设地球自转的角速度为ω，考虑地球自转产生的加速度的影响。可得GLONASS卫星的轨道运动微分方程如下：Among them, a _e is the radius of the Earth's equator, which is 6378136m,

is the second-order harmonic coefficient of the Earth's gravity potential, with a value of -0.00108263. The sun-moon gravitational perturbation acceleration is x' _n (t _b ), y' _n (t _b ), z' _n (t _b ), assuming that the angular velocity of the earth's rotation is ω, and the influence of the acceleration caused by the earth's rotation is considered. The differential equation of orbital motion of the available GLONASS satellite is as follows:

After establishing the orbital motion differential equation of the GLONASS satellite, according to the satellite position x _n (t _b ), y _n (t _b ), z _n (t _b ) at the reference time t _b , the satellite velocity x' _n (t _b ), y' _n (t _b ), z' _n (t _b ), the sun-moon perturbation acceleration x” _n (t _b ), y” _n (t _b ), z” _n (t _b ), the integration step is selected as h, the fourth-order Runge-kutta orbital integration algorithm is used to perform a numerical integration to obtain the satellite motion state at time t _b +h. The integral formula of the Runge-kutta orbital integration method is as follows:

Calculate the acceleration of the satellite at the starting time through the sun-moon perturbation acceleration, and then integrate the acceleration twice, and then combine the speed of the satellite at the starting time, the motion position of the satellite at the time t _b +h can be obtained as follows:

After several integrations, the satellite coordinates of the GLONASS satellite at time t _k can be obtained.

After the satellite motion states of t _b and t _k are obtained by the Runge-kutta orbit integration algorithm, the Hermite interpolation algorithm is used. Because the orbital equation function f(t) has a continuous derivative on [t _b , t _k ], and there are n+1 mutual dissimilarities ∈ [t _b , t _k ], then all the n+1 dissimilar points have There is only one 2n+1 Hermite interpolation polynomial.

where the nth degree polynomial l _i (t) is:

When there are only two end-to-end interpolation points, the interpolation polynomial can be obtained by substituting:

From the above interpolation polynomial, the motion state of the GLONASS satellite at any time can be calculated.

The invention adopts the Hermite interpolation method to construct a new interpolation function by knowing the function value and the derivative value on the interpolation node, so that the interpolation function and the original function have a better degree of closeness, and greatly reduce the approximate satellite coordinates and the real satellite coordinates. The error of the coordinates can be better matched with the real satellite motion model, and the calculation efficiency is effectively improved and the integration time is shortened.

The above disclosure is only a preferred embodiment of the present invention, and of course, it cannot limit the scope of rights of the present invention. Those of ordinary skill in the art can understand that all or part of the process of implementing the above embodiment can be implemented according to the present invention. The equivalent changes made by the claims still belong to the scope covered by the invention.

Claims (3)

1. a GLONASS satellite coordinate fitting method based on Hermite interpolation method, is characterized in that, comprises the steps: According to the GLONASS satellite ephemeris, establish the satellite orbital motion differential equation; According to the satellite motion state information at the reference time, integrate the orbital motion differential equation to obtain the satellite motion state information at the next moment at the reference time; Taking the satellite motion state information of the reference moment and the satellite motion state information of the next moment as two interpolation nodes of the Hermite interpolation method, and deriving the Hermite interpolation method to obtain a third-order interpolation formula; The position of the satellite at any moment is approximated by the third-order interpolation formula. 2. the GLONASS satellite coordinate fitting method based on Hermite interpolation method as claimed in claim 1, is characterized in that, The satellite motion state information at the reference time includes satellite position, satellite velocity and sun-moon perturbation acceleration. 3. the GLONASS satellite coordinate fitting method based on Hermite interpolation method as claimed in claim 2, is characterized in that, described to described orbital motion differential equation integral, comprises: The orbital motion differential equation is integrated using a fourth-order Runge-kutta orbital integration algorithm.

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