CN106508003B - The spacecraft relevant path forecasting procedure of multimetering data fitting parameter - Google Patents

Abstract

The invention belongs to dynamics and control of spacecraft technical field, discloses a kind of spacecraft relevant path forecasting procedure of multimetering data fitting parameter.The method by being converted to regard to 12 constant coefficients and key player on a team, the homogeneous equation group of cosine tradition based on dynamic law C_W equations, and homogeneous equation group procession formula is converted, construction least square fitting equation, using the accumulation of multimetering data, solve 12 constant coefficients, recycle 12 constant coefficients for solving gained as equation coefficient, relevant path forecast is carried out according to the time.Using said method, can significantly overcome the impact that motion model error and measurement initial value error are forecast to relevant path, improve spacecraft relevant path forecast precision.

Description

The spacecraft relevant path forecasting procedure of multimetering data fitting parameter

Technical field

The invention belongs to dynamics and control of spacecraft technical field, it is related to a kind of spacecraft relative movement orbit forecast Method.

Background technology

Spacecraft Equation of Relative Motion with Small (Clohessy-Wiltshire equations, abbreviation C_W equations) is description flight The one group of differential equation mutually moved between device, is that distance is much smaller than orbit radius between circular orbit and aircraft in spacecraft orbit Hypothesis under have analytic solutions.Because the equation is one group of reduced equation, can simply and quickly it describe between aircraft Relative motion, is the Classical Equation of relative motion relation between current description spacecraft.But in actual applications, due to C_W Equation model is simple, there are two deficiencies：One is that forecast precision is relatively low, is not suitable for forecasting for a long time；Two be that equation holds It is vulnerable to the influence of measurement initial value error, causes prediction error larger.

The content of the invention

The present invention is not enough for C_W equation models forecast precision, it is proposed that a kind of sky of multimetering data fitting parameter Between aircraft relevant path forecasting procedure.

A kind of spacecraft relevant path forecasting procedure of multimetering data fitting parameter of the present invention, including with Lower step：

Homogeneous equation group is constructed first with C_W equations, it is as follows：

X (t)=a₁cos(ωt)+b₁sin(ωt)+c₁+d₁

Y (t)=a₂cos(ωt)+b₂sin(ωt)+c₂t+d₂ (1)

Z (t)=a₃cos(ωt)+b₃sin(ωt)+c₃+d₃

In above formula, ω is that aircraft moves angular speed, and t is aircraft run duration, and x (t), y (t), z (t) are phase To the position under coordinate system, a₁、b₁、c₁、d₁、a₂、b₂、c₂、d₂、a₃、b₃、c₃、d₃For coefficient to be solved, Wherein d₁=0, d₃=0；

Formula (1) is arranged, obtained：

Then coefficient, structural matrix are solved using least square method fitting：

Z=A (t) × H+ ε (5)

In above formula, correspondence (2) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in x directions,

In above formula, correspondence (3) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in y directions,

In above formula, correspondence (4) formula, then：

H-matrix is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in z directions,

According to the principle of least square, can solve coefficient matrix is：

H=(A^TA)^-1A^TZ

By the use of the coefficient matrix H obtained by solution as the coefficient of formula (1), relevant path forecast is carried out according to the time.

The present invention is changed by entering line translation and parameter to C_W equations, and tradition is relative according to initial motion state forecast The pattern of motion, is changed into by way of accumulating multimetering data and solving coefficient, so that it is pre- to overcome C_W equations The problem of precision is low is reported, and is significantly improved due to the larger caused model rapid divergence problem of its measurement error, is greatly carried High spacecraft relevant path forecast precision.

Brief description of the drawings

Fig. 1 spacecrafts relevant path forecast coordinate system defines (RTN coordinate systems)；

Spacecraft relevant path actual value and forecast result of Fig. 2 a based on C_W equations；

Spacecraft relevant path prediction error figures of Fig. 2 b based on C_W equations；

Fig. 3 a use the spacecraft relevant path actual value and forecast result of multimetering data fitting parameter；

Fig. 3 b use the spacecraft relevant path prediction error figure of multimetering data fitting parameter.

Embodiment

Relevant path is described under relative coordinate system, and coordinate system definition is as shown in Figure 1.Wherein, the origin of coordinates is following the trail of winged Row device barycenter, the earth's core is defined as x directions with following the trail of aircraft barycenter line direction, follows the trail of spacecraft orbit face normal direction (R × V directions) is defined as z directions, and right hand theorem determines y directions.

The spacecraft relevant path forecasting procedure of multimetering data fitting parameter described in the present embodiment, by by classics C_W equation initial motions state transformation is constant, and expands into the trigonometric function equation that constant is coefficient, to having converted Into trigonometric function equation reconfigure, as on 12 constant coefficients and key player on a team, the homogeneous equation group of cosine；Then By the conversion of homogeneous equation group procession formula, least square fitting equation is constructed, using the accumulation of multimetering data, 12 constant coefficients are solved, by the use of 12 constant coefficients obtained by solution as equation coefficient, relative rail is carried out according to the time Mark is forecast.Detailed process is as follows：

(1) conversion of C_W equations

It is the hypothesis that circular orbit and interstellar distance are much smaller than orbit radius ignoring active control and Perturbation Effect, track Under, by the linear differential equation system that C_W equation simplifications are constant coefficient, analytic solutions are：

In above formula,For initial position, initial velocity, ω is that aircraft moves angular speed, T is aircraft run duration, and x (t), y (t), z (t) are the position under relative coordinate system；

Conversion process is carried out to above formula, it is known that：

X (t)=Acos (ω t+ φ)+B

Y (t)=2Acos (ω t+ φ)+C (2)

Z (t)=Dcos (ω t- θ)

Wherein：

(2) constructing variable equation

Formula (2) is deployed：

X (t)=Acos (ω t+ φ)+B=Acos (φ) cos (ω t)-Asin (φ) sin (ω t)+B

Y (t)=2Acos (ω t+ φ)+C=2Acos (φ) cos (ω t) -2Asin (φ) sin (ω t)+C (3)

Z (t)=Dcos (ω t- θ)=Dcos (θ) cos (ω t)-Dsin (θ) sin (ω t)

Wherein, φ, θ are constant, and C is time t 1 function, and (3) formula can transform to：

X (t)=a₁cos(ωt)+b₁sin(ωt)+c₁+d₁

Y (t)=a₂cos(ωt)+b₂sin(ωt)+c₂t+d₂ (4)

Z (t)=a₃cos(ωt)+b₃sin(ωt)+c₃+d₃

In above formula, a₁、b₁、c₁、d₁、a₂、b₂、c₂、d₂、a₃、b₃、c₃、d₃For coefficient to be solved, wherein d₁=0, d₃=0.

(3) fitting solves parameter

Formula (4) is arranged, obtained：

Then coefficient, structural matrix are solved using least square method fitting：

Z=A (t) × H+ ε (8)

In above formula, correspondence (5) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in x directions,

In above formula, correspondence (6) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in y directions,

In above formula, correspondence (7) formula, then：

H-matrix is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in z directions,

According to the principle of least square, can solve coefficient matrix is：

H=(A^TA)^-1A^TZ

(4) forecast based on parameter fitting relevant path

By the use of the coefficient matrix H obtained by solution as the coefficient of formula (4), relevant path forecast is carried out according to the time.

The relevant path forecast result using tradition C_W equations and the present embodiment methods described is contrasted below：

By taking orbit altitude about 1100km circular orbit satellite as an example (orbital tracking a=7474305m, e=0.00045), Relevant path is carried out using traditional C_W equations to give the correct time in advance, using 1 group of result of Cross-Link measurement as initial value, forecast precision is： 1 orbital period x direction forecast precision is about 100m, and y directions forecast precision is about 200m, and z directions forecast precision is about For 4m.Predicted value is contrasted as shown in Fig. 2 a with true value, shown in forecast precision as Fig. 2 b.

By taking orbit altitude about 1100km circular orbit satellite as an example (orbital tracking a=7474305m, e=0.00045), Relevant path is carried out using the present embodiment methods described to give the correct time in advance, is joined with 600 groups of (10 minutes) results of Cross-Link measurement Number is fitted, and forecast precision is：1 orbital period x direction forecast precision is about 1m, and y directions forecast precision is about 25m, Z directions forecast precision is about 5m.Predicted value is contrasted as shown in Fig. 3 a with true value, shown in forecast precision as Fig. 3 b.

As seen from the above comparison, spacecraft relevant path forecast precision is greatly improved using the present invention.

Although illustrative embodiment of the invention is described above, in order to the technical staff of this technology neck The present invention is understood, it should be apparent that the invention is not restricted to the scope of embodiment, to the common skill of the art For art personnel, as long as various change is in the spirit and scope of the present invention that appended claim is limited and is determined, this A little changes are it will be apparent that all utilize the innovation and creation of present inventive concept in the row of protection.

Claims (1)

1. a kind of spacecraft relevant path forecasting procedure of multimetering data fitting parameter, it is characterised in that Comprise the following steps：

Homogeneous equation group is constructed first with C_W equations, it is as follows：

$\begin{array}{c}x\left(t\right)=a_{1}cos\left(\mathrm{\ω}t\right)+b_1\sin \left(\mathrm{\ω}t\right)+c_1+d_1\\ y\left(t\right)=a_{2}cos\left(\mathrm{\ω}t\right)+b_2\sin \left(\mathrm{\ω}t\right)+c_{2}t+d_2\\ z\left(t\right)=a_{3}cos\left(\mathrm{\ω}t\right)+b_3\sin \left(\mathrm{\ω}t\right)+c_3+d_3\end{array}---\left(1\right)$

In above formula, ω is that aircraft moves angular speed, and t is aircraft run duration, and x (t), y (t), z (t) are phase To the position under coordinate system, a₁、b₁、c₁、d₁、a₂、b₂、c₂、d₂、a₃、b₃、c₃、d₃For coefficient to be solved, Wherein d₁=0, d₃=0；

Formula (1) is arranged, obtained：

${\left[\begin{array}{c}{x}_1\\ x_2\\ ...\\ x_i\end{array}\right]}_{i\×1}={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& 1& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& 1& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& 1& 1\end{array}\right]}_{i\×4}\·{\left[\begin{array}{c}{a}_1\\ b_1\\ c_1\\ d_1\end{array}\right]}_{4\×1}---\left(2\right)$

${\left[\begin{array}{c}{y}_1\\ y_2\\ ...\\ y_i\end{array}\right]}_{i\×1}={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& t& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& t& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& t& 1\end{array}\right]}_{i\×4}\·{\left[\begin{array}{c}{a}_2\\ b_2\\ c_2\\ d_2\end{array}\right]}_{4\×1}---\left(3\right)$

${\left[\begin{array}{c}{z}_1\\ z_2\\ ...\\ z_i\end{array}\right]}_{i\×1}={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& 1& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& 1& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& 1& 1\end{array}\right]}_{i\×4}\·{\left[\begin{array}{c}{a}_3\\ b_3\\ c_3\\ d_3\end{array}\right]}_{4\×1}---\left(4\right)$

Then coefficient, structural matrix are solved using least square method fitting：

Z=A (t) × H+ ε (5)

In above formula, correspondence (2) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

$A\left(t\right)={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& 1& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& 1& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& 1& 1\end{array}\right]}_{i\×4}$

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in x directions,

In above formula, correspondence (3) formula, then：

H is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

$A\left(t\right)={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& t& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& t& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& t& 1\end{array}\right]}_{i\×4}$

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in y directions,

In above formula, correspondence (4) formula, then：

H-matrix is the coefficient matrix of construction,

A (t) is the corresponding parameter matrixs of time t,

$A\left(t\right)={\left[\begin{array}{cccc}cos(\mathrm{\ω}\·t_1)& sin(\mathrm{\ω}\·t_1)& 1& 1\\ cos(\mathrm{\ω}\·t_2)& sin(\mathrm{\ω}\·t_2)& 1& 1\\ ...& ...& ...& ...\\ cos(\mathrm{\ω}\·t_i)& sin(\mathrm{\ω}\·t_i)& 1& 1\end{array}\right]}_{i\×4}$

Z is t of the aircraft under relative coordinate system_iThe moment corresponding actual relative measurement in z directions,

According to the principle of least square, can solve coefficient matrix is：

H=(A^TA)^-1A^TZ

By the use of the coefficient matrix H obtained by solution as the coefficient of formula (1), relevant path forecast is carried out according to the time.