CN102998690B - Attitude angle direct resolving method based on global position system (GPS) carrier wave double-difference equation - Google Patents

Abstract

The invention provides an attitude angle direct resolving method based on a global position system (GPS) carrier wave double-difference equation. The attitude angle direct resolving method mainly comprises building a carrier wave double-difference equation, guiding carrier attitude information into the carrier wave double-difference equation and solving an attitude angle by utilizing nonlinear least squares. The attitude angle direct resolving method can effectively reduce or eliminate short base line public errors, achieves high-precision resolving, can directly resolve the attitude angle by leading the attitude angle information into the double-difference equation, greatly reduces intermediate evaluated errors, evaluates the attitude angle by utilizing the nonlinear least squares method, improves system resolving speed, finishes real-time resolving of the attitude information and is more suitable for real-time carrier attitude measuring.

Description

A kind of attitude angle direct solving method based on the two eikonal equations of gps carrier

Technical field

The present invention relates to a kind of real-time attitude measurement method of large-scale carrier, what particularly relate to is a kind of attitude angle direct solving method based on the two eikonal equations of gps carrier.

Background technology

In order to meet the demand of different application platforms to attitude information, existing a large amount of attitude measurement equipment comes out at present, such as the star sensor of measuring for space attitude of carrier, horizon tracking device, sun sensor, magnetometer etc.; Be used to land or underwater carrier that magnetic compass, the gyro-magnetic compass in course are provided; Be used to various aircraft, land transportation equipment, boats and ships, the device of diving, space carrier that inertia device of attitude information etc. is provided.And this wherein, star sensor is subject to the impact of weather, landform or other objective factor, can not provide in real time attitude or orientation values; Inertial navigation set complex structure, expensive, when working time inertial device error when long can accumulate in time and causes the reduction of surveying appearance precision.

Use GPS positioning signal to carry out attitude of carrier measurement, only need to utilize receiver cheaply can provide the attitude information of degree of precision, with this, replace the tradition involving great expense and survey appearance equipment, can also complete location and the time service of carrier simultaneously, and be subject to the impact of environment little, can carry out for a long time high precision and survey appearance task.

The measurement of GPS attitude of carrier is to utilize to be arranged on the relative position between receiving antenna on carrier fixed position, and coordinate transformation relation is determined attitude angle.Concrete methods of realizing is, utilize the high-accuracy position system of installing on carrier as base station, and at three (or many) receiving antennas of carrier fixed position installation, the relative position that uses gps signal to measure between antenna and base station is obtained baseline at WGS-84(World Geodetic System-84) be lower coordinate, according to known baseline carrier system, descend coordinate to determine that attitude rotation matrix finally obtains attitude angle.

At present, GPS attitude algorithm algorithm is mainly that attitude angle is obtained in the processing based on to positioning result, and conventional mathematics method of estimation is applied in attitude algorithm, by calculation result, is divided into and asks attitude matrix and ask Eulerian angle two classes.

The method that solves attitude matrix is determined attitude angle, to utilize positioning equation to set up the positioning equation group of multi-satellite and a plurality of epoch, by solving overdetermined equation, complete the estimation of baseline coordinate, utilize the coordinate transformation relation of baseline in different coordinates to estimate attitude matrix, thereby determine attitude angle.In these class methods, need to use mathematics estimation means to realize determining of baseline calculating coordinate and attitude angle.Conventional method comprises:

(1) the coordinate relation in Department of Geography and carrier system according to baseline, the TRIAD algorithm that utilizes the orthogonality of attitude matrix to carry out optimal estimation;

(2) calculation method based on Wahba problem: as QUEST(QUaternion ESTimator) algorithm, FOAM(FastOptimal Attitude Matrix) algorithm, Euler-q algorithm, SVD(Singular Value Decomposition) algorithm etc.;

(3) utilize the least square direct estimation method of many epoch of baseline calculating coordinate;

(4) utilize the lower baseline coordinate of pseudorange or carrier wave observed quantity and carrier system, set up the method that observation equation is estimated attitude matrix.

The direct method estimating of attitude angle is generally to utilize the relation at known aerial position and rotation attitude angle, and substep solves attitude angle.Comprise: make a baseline be arranged on the major axes orientation of carrier, first obtain crab angle and the angle of pitch, the rotation relationship of recycling second baseline is obtained two antennas of roll angle and surveyed appearance method; Utilize two antennas to survey appearance formula and determine crab angle and the angle of pitch, many antennas that other antenna is obtained to roll angle through twice rotation are surveyed appearance method.

Generally, in GPS survey appearance, there are the following problems:

1. during utilization mathematics method of estimation is estimated attitude matrix, directly do not obtain attitude angle result, the information acquisition that need to carry out many epoch completes the calculating of final carriage angle, and this point has been introduced further evaluated error on the one hand, has also affected on the other hand the real-time of attitude measurement.

2. use direct measuring method to carry out attitude angle, not needing baseline carrier is coordinate, also need not calculate attitude matrix, but the carrier coordinate z axle component of all antenna baselines is zero, the matrix that coordinate forms is full rank not, can cause attitude result unreliable, and precision is far away from mathematics method of estimation.

Summary of the invention

The object of the present invention is to provide and a kind ofly can improve estimation procedure precision, and realize a kind of attitude angle direct solving method based on the two eikonal equations of gps carrier of Real-time Determination of Attitude requirement.

The object of the present invention is achieved like this:

A kind of attitude angle direct solving method based on the two eikonal equations of gps carrier comprises the following steps:

(1) adopt three GPS receivers to measure antenna collection gps satellite signal and obtain carrier wave observation information

using the carrier wave observation information of No. 0 antenna in place, right angle as base station signal;

(2) choose public satellites in view i, i=1,2, M, and using the satellite of elevation angle maximum in satellites in view as with reference to satellite (being numbered 1), receiver obtains after triantennary carrier wave observation information, take antenna for base station observation information as benchmark, respectively the carrier signal of the relatively same visible satellite of each antenna is done to poor single poor carrier phase observed quantity of obtaining to base station signal

n=0,1,2, then corresponding different antennae, does poor two poor carrier wave observed quantities that obtain by the poor carrier information of each satellite list to the poor carrier wave observed quantity of list of reference satellite

(3) utilize the relational expression of carrier wave observed quantity shown in formula (1) and attitude angle, set up solving of attitude equation:

Wherein,

represent the two poor carrier wave observed quantity of n antenna; λ represents carrier wavelength, unit: m;

represent two poor integer ambiguities;

represent that satellite i is to the sight line vector of reference; R represents that geography is tied to the attitude matrix of carrier system;

represent the geographical attitude transition matrix that is tied to the earth's core system; Y, r, p represents three attitude angle of carrier, is respectively course angle, roll angle, the angle of pitch, i.e. equation unknown number; b _nrepresenting baseline vector under carrier coordinate system, is known quantity, unit: m.

(4) attitude measurement system Real-time Obtaining list carrier signal epoch is carried out two poor processing, by step (1)-(4), calculates in real time the attitude information of carrier.

The present invention can also comprise:

The described attitude angle direct solving method based on the two eikonal equations of gps carrier, is characterized in that: described antenna adopts right angle orthogonal formula layout, i.e. form right angle triangle projective planum, and mounting distance is more than 10m.

The described attitude angle direct solving method based on the two eikonal equations of gps carrier, is characterized in that: described solving of attitude equation calculation process is:

1) because formula (1) equal sign the right integer ambiguity is known, therefore equation (1) is expressed as:

y _t＝f(x)

Wherein,

represent that integer ambiguity solves rear epoch of the two poor carrier wave observed quantity of t constantly; $f\left(x\right)={\mathrm{\&lambda;}}^{-1}\left[\begin{array}{c}-{({\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^2-I_0^1)}^T\\ -{({\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^3-I_0^1)}^T\\ ...\\ -{(I_0^M-I_0^1)}^T\end{array}\right]A_R^E{R}^T\left(x{b}_n\right)$ The function that x is independent variable is take in expression, x=[y r p] ^t.

2) provide initial attitude angle

at initial attitude angle, place carries out single order Taylor series expansion to Equation f (x), order $\mathrm{\&Delta;x}={\left[\begin{array}{ccc}\mathrm{\&Delta;y}& \mathrm{\&Delta;r}& \mathrm{\&Delta;p}\end{array}\right]}^T=\begin{array}{}\end{array}{\left[\begin{array}{ccc}y-\hat{y}& r-\hat{r}& p-\hat{p}\end{array}\right]}^T$ For state variable, equation simplification is following form:

H·Δx＝Δy _t （2）

In formula, the i.e. estimated result x of state vector in the k time iteration _krelative measurement value

deviation; Observing matrix H is:

$H=\left[\begin{array}{ccc}& .& \\ & .& \\ & .& \\ ...& -{(I_0^i-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^1)}^T{A}_R^E\left(\frac{{\&PartialD;R}^T}{\&PartialD;x}\right)b_n& ...\\ & .& \\ & .& \\ & .& \end{array}\right]$

Wherein,

represent the partial derivative of attitude matrix to each attitude angle.

3) adopt least-squares estimation solution overdetermined equation (2), its solution is:

Δx＝(H ^TH) ^-1H ^TΔy _t （3）

4) set iteration cut-off threshold, estimate attitude angle final value x:

$x=\begin{array}{c}\left[\begin{array}{c}y\\ r\\ p\end{array}\right]\end{array}=\left[\begin{array}{c}\hat{y}+\mathrm{\&Delta;y}\\ \hat{r}+\mathrm{\&Delta;r}\\ \hat{p}+\mathrm{\&Delta;p}\end{array}\right]---\left(4\right)$

The principal feature of method of the present invention is as follows:

(1) gps carrier observed quantity can provide high-precision locating information, utilizes two poor modes to set up carrier wave equation and more can effectively eliminate the common error of surveying appearance baseline two ends, greatly improves attitude measurement accuracy;

(2) using attitude angle as unknown quantity, introduce in two difference measurements equations, reduced the pilot process of traditional survey attitude positioning method, can effectively reduce evaluated error, and can resolve by list information epoch, directly obtain attitude angle information;

(3) introducing of non-linear least square algorithm for estimating can realize the estimation of non-linear overdetermined equation, is resolved single epoch fast accomplished, can greatly improve the real-time of attitude of carrier information and estimate.

Beneficial effect of the present invention can be verified by following emulation:

1. solving of attitude simulating, verifying model is set up

The design of Attitude Simulation verification model utilizes known satellite position, by setting main antenna terrestrial coordinate, the lower coordinate of baseline carrier system and preset posture angle, can carry out the checking of Attitude Algorithm.

The design of verification system comprises following a few partial content:

(1) co-ordinates of satellite obtains

The actual almanac file of broadcasting of gps satellite that utilizes navcen.uscg.gov website to provide, calculates the real-time coordinate of all satellite in orbit, obtains the satellite trajectory of rough grade.Calculate at the elevation angle that completes all satellites according to setting customer location, obtains visible satellite positional information.

(2) carrier signal simulation

Set behind the position of main antenna and reference antenna, can show that according to predetermined attitude angle the ground of all antennas feels concerned about coordinate, satellite is known to antenna distance.By the carrier signal that adds certain measurement noise can realize each antenna, simulate, complete the foundation (system postulation integer ambiguity is known) of carrier wave observation equation.

(3) attitude of carrier result verification

By analog parameter, as Given information, according to different attitude algorithm methods, set up equation, solve attitude angle information, then contrast with preset posture angle, thus the checking of implementation algorithm.

The survey appearance array of setting up three antennas under carrier coordinate system, distribution situation as shown in Figure 3.

If reference antenna is at carrier rotation center, satellite is 5 ° by the elevation angle, and other simulated conditions are set as shown in table 1.

The setting of table 1 simulation parameter

2. solving of attitude method real-time checking

If be 0 during initial GPS, sample 100 epoch, observe algorithm static at carrier, by the rotation of fixed angles speed and in time sinusoidal real-time of rotating under three kinds of states resolve effect.Wherein certain list iteration situation epoch as shown in Figure 4, is found to reach stable through 4 loop iteration process attitude results.100 resolve epoch the result as shown in Figure 5, resolution error situation is as shown in Figure 6.

Attitude result curve in analysis chart 6, under static condition, the direct solving of attitude methods and results precision based on the two eikonal equations of carrier wave can reach 10 ^-4the order of magnitude of degree, and all can effectively resolve in simulation time; When carrier rotates course angle temporal evolution with fixed angles speed, dynamically course angle precision reaches 10 equally ^-4number of degrees magnitude; When the carrier angle of pitch changes by sinusoidal rule in time, calculation accuracy has reached 10 ^-2number of degrees magnitude.

Accompanying drawing explanation

Fig. 1 is that method of the present invention is resolved process flow diagram;

Fig. 2 is carrier signal propagation characteristic diagram;

Fig. 3 is attitude algorithm phantom antenna distribution situation of the present invention;

Fig. 4 is list least square epoch iterative estimate conditional curve of the present invention;

Fig. 5 a is that carrier of the present invention is at stationary state solving of attitude result curve figure;

Fig. 5 b is that carrier of the present invention is at fixed angles speed rotation status solving of attitude result curve figure;

The sinusoidal rotation status solving of attitude of Fig. 5 c carrier of the present invention result curve figure;

Fig. 6 a is that carrier of the present invention is at stationary state solving of attitude error curve diagram;

Fig. 6 b is that carrier of the present invention is at fixed angles speed rotation status solving of attitude error curve diagram;

Fig. 6 c carrier of the present invention is at sinusoidal rotation status solving of attitude error curve diagram.

Embodiment

Below in conjunction with accompanying drawing, for example the present invention is described in detail:

1. attitude angle direct solution algorithm implementing procedure

(1) set up baseline vector and carrier wave observed quantity relation

GPS carrier 3 d pose is measured and generally to be adopted three receiving antennas to form two baseline vectors, so the attitude measurement system that this patent relates to consists of three antennas, and wherein No. 0 antenna is made as reference antenna, and 1, No. 2 antenna is from antenna.If three antennas are followed the tracks of M satellite simultaneously, for any satellite i wherein, can set up the carrier wave observation equation of receiver to satellite:

In formula, represent that antenna n receives the carrier wave observed quantity of i star; λ represents carrier wavelength, unit: m;

represent that antenna is to the actual range of satellite, unit: m; represent ionosphere delay, unit: m;

represent tropospheric delay, unit: m; δ t _nrepresent receiver clock correction, unit: s; δ t ⁱrepresent satellite clock correction, unit: s; represent integer ambiguity;

represent carrier wave measurement noise.

For every baseline, set up single eikonal equation to eliminate ionosphere time delay, troposphere time delay, and satellite clock correction equal error item.Because base length is much smaller than satellite, think that the satellite sight line vector of baseline two-end-point is identical, satellite-signal is propagated as shown in Figure 2.

According to single receiver, the carrier wave observation equation of satellite is set up to two stations to the poor observation equation of the list of same satellite i:

Further cancellation receiver clock correction, sets up single eikonal equation of another satellite j that triantennary observes simultaneously, and two single eikonal equations of simultaneous are made poor two poor carrier wave observation equations that obtain:

Baseline vector b _n0connect with two poor carrier wave observation equations.The sight line vector of satellite to three antenna is

principal and subordinate's antenna is to the poor geometric distance of list of satellite i

(unit: m), equal baseline vector main antenna to satellite i observed ray on the opposite number of projected length, that is:

$r_{n0}^i=-{\mathrm{<figure-callout\; id="0"\; label="b\; n"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">b</figure-callout>}}_n\&CenterDot;I_0^i---\left(4\right)$

Therefore, in two eikonal equations, the relation of two poor geometric distances and baseline can be expressed as:

$r_{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">n</figure-callout>}0}^{\mathrm{ij}}=-{\mathrm{<figure-callout\; id="0"\; label="b\; n"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">b</figure-callout>}}_n\&CenterDot;I_0^i+{\mathrm{<figure-callout\; id="0"\; label="b\; n"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">b</figure-callout>}}_n\&CenterDot;I_0^i=-(I_0^i-I_0^j)\&CenterDot;b_{n0}---\left(5\right)$

The two eikonal equations of carrier wave change into:

It is two poor that formula (6) has provided with baseline vector b _n0between relation.In formula,

being two poor carrier phase measurement values, is known quantity; b _n0three-dimensional baseline vector to be asked, unit: m;

being two poor integer ambiguities, is unknown integer.(2) set up solving of attitude model

In order to simplify, survey appearance algorithm process of solution, the evaluated error of avoiding pilot process to introduce, improves calculation accuracy, below directly found the relation of attitude angle and double difference, method is as follows:

If GPS attitude measurement system antenna and carrier connect firmly, antenna coordinate under carrier system is constant, and known, is respectively r _{0, B}, r _{1, B}, r _{2, B}.Forming two baseline vectors is b _n=r _n,B-r _{0, B}, n=1 wherein, 2.Under geographic coordinate system,

Utilize the transformational relation of coordinate system, by attitude matrix by baseline vector representation under carrier system, formula (6) is converted to formula (7):

In formula, R is the attitude matrix that geography is tied to carrier system,

it is the attitude transition matrix that geography is tied to the earth's core system.

After two poor integer ambiguities are determined, in equation, only having three attitude angle is unknown quantity, the relational expression of carrier wave double difference and Eulerian angle in utilization (7) formula, and setting up observation equation matrix can direct estimation go out attitude angle.

To every baseline, can set up M-1 two eikonal equation, equation matrix form is:

In formula (8), establishing No. 1 satellite is reference satellite.According to the relation of attitude angle and equation, by equation linearization, then by the method for least square, attitude angle is estimated, finally drawn stabilization result.

(3) least-squares estimation solving of attitude process

Three rotation attitude angles, as unknown number, are included in attitude matrix R, and attitude matrix has following form:

$R=\left(\begin{array}{ccc}\mathrm{CrCy}-\mathrm{SrSpSy}& \mathrm{CrSy}+\mathrm{SrSpCy}& -\mathrm{SrCp}\\ -\mathrm{CpSy}& \mathrm{CpCy}& \mathrm{Sp}\\ \mathrm{SrCy}+\mathrm{CrSpSy}& \mathrm{SrSy}-\mathrm{CrSpCy}& \mathrm{CrCp}\end{array}\right)---\left(9\right)$

In formula, S represents sin; C represents cos; Y, r, p are respectively carrier around the crab angle (Yaw) of local horizontal coordinates z axle rotation, around the roll angle (Roll) of y axle rotation, the angle of pitch (Pitch) rotating around x axle.Because model has nonlinear relationship for attitude angle, therefore adopt non-linear least square to estimate to determine attitude angle.

Think that formula (8) equal sign the right integer ambiguity is known, by the Representation Equation, be:

y _t＝f(x)

Wherein, represent that integer ambiguity solves rear epoch of the two poor carrier wave observed quantity of t constantly; $f\left(x\right)={\mathrm{\&lambda;}}^{-1}\left[\begin{array}{c}-{({\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^2-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^1)}^T\\ -{({\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^3-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^1)}^T\\ ...\\ -{(I_0^M-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_0^1)}^T\end{array}\right]A_R^E{R}^T\left(x\right)b_n$ The function that x is independent variable is take in expression, x=[y r p] ^t.

If initial attitude angle

at initial attitude angle, place carries out single order Taylor series expansion to Equation f (x), ignores higher order term.Order $\mathrm{\&Delta;x}={\left[\begin{array}{ccc}\mathrm{\&Delta;y}& \mathrm{\&Delta;r}& \mathrm{\&Delta;p}\end{array}\right]}^T=\begin{array}{}\end{array}{\left[\begin{array}{ccc}y-\hat{y}& r-\hat{r}& p-\hat{p}\end{array}\right]}^T$ For state variable, Nonlinear System of Equations can be similar to and be converted into following system of linear equations of expressing with matrix form:

H·Δx＝Δy _t （10）

In formula,

the i.e. estimated result x of state vector in the k time iteration _krelative measurement value

deviation.Observing matrix H is expressed as:

$H=\left[\begin{array}{ccc}& .& \\ & .& \\ & .& \\ ...& -{(I_0^i-I_0^1)}^T{A}_R^E\left(\frac{{\&PartialD;R}^T}{\&PartialD;x}\right)b_n& ...\\ & .& \\ & .& \\ & .& \end{array}\right]$

In formula,

represent the partial derivative of attitude matrix to each attitude angle, form is as follows:

$\frac{{\&PartialD;R}^T}{\&PartialD;y}=\left(\begin{array}{ccc}-\mathrm{CrSy}-\mathrm{CySpSr}& -\mathrm{CyCp}& -\mathrm{SrSy}+\mathrm{CySpCr}\\ \mathrm{CyCr}-\mathrm{SySpSr}& -\mathrm{SyCp}& \mathrm{<figure-callout\; id="0"\; label="CySr\; +\; SySpCr"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">CySr</figure-callout>}+\mathrm{SySpCr}\\ 0& 0& 0\end{array}\right)$

$\frac{{\&PartialD;R}^T}{\&PartialD;r}=\left(\begin{array}{ccc}-\mathrm{CySr}-\mathrm{<figure-callout\; id="0"\; label="SySpCr"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">SySpCr</figure-callout>}& 0& \mathrm{CyCr}-\mathrm{SySpSr}\\ -\mathrm{<figure-callout\; id="0"\; label="SySr\; +\; CySpCr"\; filenames="HDA00002467225800022.png,HDA00002467225800032.png"\; state="\{\{state\}\}">SySr</figure-callout>}+\mathrm{CySpCr}& 0& \mathrm{SyCr}+\mathrm{CySpSr}\\ -\mathrm{CpCr}& 0& -\mathrm{CpSr}\end{array}\right)$

$\frac{{\&PartialD;R}^T}{\&PartialD;p}=\left(\begin{array}{ccc}-\mathrm{SyCpSr}& \mathrm{SySp}& \mathrm{SyCpCr}\\ \mathrm{CyCpSr}& -\mathrm{CySp}& -\mathrm{CyCpCr}\\ \mathrm{SpSr}& \mathrm{Cp}& -\mathrm{SpCr}\end{array}\right)$

Least square solution is:

Δx＝(H ^TH) ^-1H ^TΔy _t （11）

By setting up above non-linear least square estimate equation, under the condition at given initial attitude angle, can complete the estimation of unknown parameter.That is, realized and utilized the two poor observation equation direct estimation of carrier wave to obtain attitude angle.

2. attitude angle direct solving method error condition is analyzed

If all satellite carrier observational errors are all mutually and separate, average is 0, two difference measurements value errors are

the two poor observation equations of simultaneous satellite carrier, carry out overdetermined equation by least square method and solve.The general type of overdetermined equation is:

Ax＝b （12）

, there is unique least square solution:

$A^H\mathrm{Ax}=A^{H}b\&DoubleRightArrow;x={\left(A^{H}A\right)}^{-1}{A}^{H}b$

In formula (12), the error of b is δ b, and the error of A is δ A, they on the impact of solution of equation all to square being directly proportional of the conditional number of A, the conditional number of overdetermined equation will be quadratic relationship and increase:

cond(A ^HA)＝[cond(A)] ² （13）

According to error propagation rule, the error of least square solution is

Due to based on the two eikonal equation direct solution attitude angle methods of carrier wave, only carry out a step least-squares estimation and completed asking for of attitude angle.From analyzing above, the evaluated error of vector x is:

cov[Δx]＝E[ΔxΔx ^T]＝(H ^TH) ^-1H ^TE[ΔyΔy ^T]H(H ^TH) ^-1 （14）

The variance of measuring error Δ y is

average is 0, and separate.Therefore, wherein I is unit matrix.

Definition dimensionless matrix

Therefore, pose estimation error and measuring error and measure matrix, base length is relevant.Wherein, in the situation that not considering other error components, the longer precision of baseline is higher; And from formula (15), observing matrix is only relevant with the geometric position of satellite, when how much distributions are better, measuring error is less on the estimation impact of attitude angle.

Claims (1)

1. the attitude angle direct solving method based on the two eikonal equations of gps carrier, is characterized in that comprising the following steps:

(1) adopt three GPS receivers to measure antenna collection gps satellite signal and obtain carrier wave observation information

using the carrier wave observation information of No. 0 antenna in place, right angle as base station signal;

(2) choose public satellites in view i, i=1,2, M, and using the satellite of elevation angle maximum in satellites in view as with reference to satellite, receiver obtains after triantennary carrier wave observation information, take antenna for base station observation information as benchmark, respectively the carrier signal of the relatively same visible satellite of each antenna is done to poor single poor carrier phase observed quantity of obtaining to base station signal

n=0,1,2, then corresponding different antennae, does poor two poor carrier wave observed quantities that obtain by the poor carrier information of each satellite list to the poor carrier wave observed quantity of list of reference satellite

(3) utilize formula

shown in the relational expression of carrier wave observed quantity and attitude angle, set up solving of attitude equation, wherein,

represent the two poor carrier wave observed quantity of n antenna; λ represents carrier wavelength, unit: m;

represent two poor integer ambiguities;

represent that satellite i is to the sight line vector of reference antenna; R represents that geography is tied to the attitude matrix of carrier system;

represent the geographical attitude transition matrix that is tied to the earth's core system; Y, r, p represents three attitude angle of carrier, is respectively course angle, roll angle, the angle of pitch, i.e. equation unknown number; b _nrepresenting the baseline vector under carrier coordinate system, is known quantity, unit: m;

(4) attitude measurement system Real-time Obtaining list carrier signal epoch is carried out two poor processing, by step (1)-(4), calculates in real time the attitude information of carrier;

Described antenna adopts right angle orthogonal formula layout, i.e. form right angle triangle projective planum, and mounting distance is more than 10m;

Described solving of attitude equation calculation process is:

1) due to formula

equal sign the right integer ambiguity is known, therefore will

be expressed as:

y _t=f(x)

Wherein,

represent that integer ambiguity solves rear epoch of the two poor carrier wave observed quantity of t constantly; $f\left(x\right)={\mathrm{\&lambda;}}^{-1}\left[\begin{array}{c}{-(I_0^2-I_0^1)}^T\\ {-(I_0^3-I_0^1)}^T\\ ...\\ {-(I_0^M-I_0^1)}^T\end{array}\right]A_R^E{R}^T\left(x\right)b_n$ The function that x is independent variable is take in expression, x=[y r p] ^t;

2) provide initial attitude angle $x_0={\left[\begin{array}{ccc}\hat{y}& \hat{r}& \hat{p}\end{array}\right]}^T,$ At initial attitude angle, place carries out single order Taylor series expansion to Equation f (x), order $\mathrm{\&Delta;x}=\begin{array}{c}{\left[\begin{array}{ccc}\mathrm{\&Delta;y}& \mathrm{\&Delta;r}& \mathrm{\&Delta;p}\end{array}\right]}^T={\left[\begin{array}{ccc}y-\hat{y}& r-\hat{r}& p-\hat{p}\end{array}\right]}^T\end{array}$ For state variable, equation simplification is following form:

H·Δx=Δy _t

In formula,

the i.e. estimated result x of state vector in the k time iteration _krelative measurement value

deviation; Observing matrix H is:

$H=\left(\begin{array}{ccc}& \&CenterDot;& \\ & \&CenterDot;& \\ & \&CenterDot;& \\ ...& -{(I_0^i-I_0^1)}^T{A}_R^E\left(\frac{{\&PartialD;R}^T}{\&PartialD;x}\right)b_n& ...\\ & \&CenterDot;& \\ & \&CenterDot;& \\ & \&CenterDot;& \end{array}\right)$

Wherein,

represent the partial derivative of attitude matrix to each attitude angle;

3) adopt least-squares estimation solution overdetermined equation H Δ x=Δ y _t, its solution is:

Δx=(H ^TH) ^-1H ^TΔy _t

4) set iteration cut-off threshold, estimate attitude angle final value x:

$x=\left[\begin{array}{c}y\\ r\\ p\end{array}\right]=\begin{array}{c}\left[\begin{array}{c}\hat{y}+\mathrm{\&Delta;y}\\ \hat{r}+\mathrm{\&Delta;r}\\ \hat{p}+\mathrm{\&Delta;p}\end{array}\right].\end{array}$

Images