CN102170658B - Geometric positioning improvement method under NLOS (non-line-of-sight) environment - Google Patents

Abstract

The invention relates to a geometric positioning improvement method under an NLOS (non-line-of-sight) environment, which adopts the Gauss-Newton iteration method with fast convergence, and combines with a grid searching method to optimize the iteration initial coordinate value. Compared with the existing geometric positioning method, the geometric positioning improvement method has the advantages that the complexity is moderated, the fast and stable convergence is provided, and the higher mobile station positioning accuracy can be obtained.

Description

Geometry location under a kind of NLOS environment is improved one's methods

Technical field

The geometry location that the present invention relates under a kind of NLOS environment is improved one's methods, and is applied to the Wireless Location in Cellular Network technical field.

Background technology

The geometry location method is that how much distribution relations according to one or more measurement parameters and base station and travelling carriage carry out location estimation to travelling carriage.

Be used for carrying out the parameter TOA of position calculation and the measure error of AOA mainly is comprised of two parts, namely systematic measurement error and NLOS propagate the error that produces.The systematic measurement error Gaussian distributed, along with the development of technology can reduce gradually, and the NLOS propagated error is existed by the meeting that affects of radio propagation environment all the time, and becomes the chief component of measure error.Propagating in numerous methods on the impact that mixes TOA/AOA method positioning accuracy for reducing NLOS, scattering model is wherein a kind of.Under the prerequisite of using the individual reflection model, existing geometry location method traditional TOA (Time Of Arrival)/AOA (Angle Of Arrival) method is arranged and improve one's methods, LOP (linear line of position) method and the HLOP that improves one's methods (hybrid LOP) method thereof etc., this several method utilizes the geometry site between travelling carriage, scattering object and the base station to estimate the position of travelling carriage, although method is simple, operand is little, positioning accuracy is but not high.Generally speaking, orientation problem to travelling carriage all is summed up as lsqnonlin, wherein the most basic a kind of method is Gaussian-Newton method, its feature is when having preferably the iteration initial value, has good convergence, if and selected relatively poor initial value, convergence will variation.

Summary of the invention

For avoiding above the deficiencies in the prior art, the geometry location that the present invention proposes under a kind of NLOS environment is improved one's methods, to solve the not high problem of positioning accuracy.The present invention uses the grid search method to optimize travelling carriage primary iteration coordinate figure, so that gauss-newton method obtains better convergence.

Purpose of the present invention is achieved through the following technical solutions:

Geometry location under a kind of NLOS environment is improved one's methods, and the method comprises parameter measurement and location of mobile station two stages of estimation, and concrete grammar is as follows:

1) the parameter measurement stage:

According to the base station signal parameter that moving table measuring obtains, time of arrival (toa)

With the direction of arrival degree

And the geometry site of travelling carriage, scattering object and base station has:

${\widetilde{\mathrm{\α}}}_i-\arctan \left(\frac{y_i-y_b^i}{x_i-x_b^i}\right)=0---\left(1\right)$

${\tilde{L}}_i-r_b^i-r_m^i={\tilde{L}}_i-\sqrt{{(x_i-x_b^i)}^2+{(y_i-y_b^i)}^2}-\sqrt{{(x_i-x_m)}^2+{(y_i-y_m)}^2}=0---\left(2\right)$

Wherein

With

Represent respectively the base station signal propagation distance and arrive angle, (x _i, y _i) and (x _m, y _m) represent respectively the position coordinates of base station, scattering object and travelling carriage, signal propagation distance

Can be by the measurement parameter signal propagation time

Multiply each other with light velocity C and to obtain:

${\tilde{L}}_i={\widetilde{\mathrm{\τ}}}_i\×C---\left(3\right)$

2) location of mobile station estimating stage: according to the base station signal parameter

And Adopt Gauss's Newton iteration method and in conjunction with grid search method Optimized Iterative initial coordinate values, resolve the position that obtains travelling carriage, specifically may further comprise the steps:

(1) dwindles feasible zone

Under the NLOS environment, because travelling carriage (x _m, y _m) and each base station

Between distance

By the distance of scattering object to travelling carriage

Add that scattering object is to the distance between the base station Consist of, namely

Then according to parameter With

The position of rough estimate travelling carriage:

$\left\{\begin{array}{c}{\hat{x}}_{\mathrm{ml}}^i=x_b^i+{\tilde{L}}_i\cos {\widetilde{\mathrm{\α}}}_i\\ {\hat{y}}_{m1}^i=y_b^i+{\tilde{L}}_i\sin {\widetilde{\mathrm{\α}}}_i\end{array}\right.---\left(4\right)$

$\left\{\begin{array}{c}{\hat{x}}_{m2}^i=x_b^i+({\tilde{L}}_i-R_d^i)\cos {\widetilde{\mathrm{\α}}}_i\\ {\hat{y}}_{m2}^i=y_b^i+({\tilde{L}}_i-R_d^i)\sin {\widetilde{\mathrm{\α}}}_i\end{array}\right.---\left(5\right)$

Constraints according to the proposition of GBSBCM model

Obtain the feasible zone of algorithm:

$0\≤x_m\≤(\sqrt{3}/2)R$ (6)

$0\≤y_m\≤\min \{\sqrt{3}{x}_m,-(\sqrt{3}/3)x_m+R\}$

$0\≤r_m^i=|x_i+{\mathrm{jy}}_i-(x_m+{\mathrm{jy}}_m)|\≤R_d^i,i=\mathrm{1,2,3}---\left(7\right)$

Wherein, R is radius of society, Be the maximum of scattering object distribution radius corresponding to each base station, i=1,2,3.

From the coordinate of above (4), (5), choose maximum, the min coordinates value of satisfied (6), (7), obtain a feasible zone [x who dwindles _Min, x _Max] * [y _Min, y _Max].

(2) grid search is determined the iteration initial coordinate values

In the feasible zone that this dwindles, carry out grid search, choose the grid point that satisfies constraints (7) and consist of candidate's point set CPS;

Coordinate figure to all candidate points among the CPS is averaging, the iteration initial value x that is optimized ₀

(3) adopt Gaussian-Newton method to estimate the position coordinates of travelling carriage

In theory, each base station signal arrives the propagation distance of travelling carriage and arrives angle and can be expressed as:

L(x)＝[L ₁(x)，L ₂(x)，L ₃(x)，α ₁(x)，α ₂(x)，α ₃(x)] (8)

X=[x wherein _m, y _m], $L_i\left(x\right)=\sqrt{{(x-x_b^i)}^2+{(y-y_b^i)}^2},$ ${\mathrm{\α}}_i\left(x\right)=\arctan \frac{y-y_b^i}{x-x_b^i}$

And the actual propagation distance of each base station signal with the arrival angle is:

$\tilde{L}=L\left(x\right)+n---\left(9\right)$

Wherein n is that NLOS propagates the error cause and obeys the systematic measurement error that average is zero Gaussian Profile;

Because the existence of error, (1), (2) always can not be met, and obtain thus target function:

$\mathrm{\ϵ}\left(x\right)={(\tilde{L}-L\left(x\right))}^T{\mathrm{\Σ}}_n^{-1}(\tilde{L}-L\left(x\right))---\left(10\right)$

∑ wherein _nCovariance matrix for noise n:

∑ _n＝E{nn ^T} (11)

The coordinate that then satisfies following formula namely can be used as the location estimation value of travelling carriage:

$\hat{x}=\underset{x}{\arg \min }\mathrm{\ϵ}\left(x\right)---\left(12\right)$

To (9) formula at iteration initial value x ₀The place carries out linearisation:

$L\left(x\right)=L\left(x_0\right)+\mathrm{\φ}\left(x\right){|}_{x=x_0}(x-x_0)---\left(13\right)$

Wherein

$\mathrm{\&phi;}\left(x\right)={{\&dtri;}_x}^T\&CircleTimes;L\left(x\right)=\left[\begin{array}{c}(x-x_1)/r_1,\left({y-y}_1\right)/r_1\\ \left({x-x}_2\right)/r_2,\left({y-y}_2\right)/r_2\\ (x-x_3)/r_3,\left({y-y}_3\right)/r_3\\ (y-y_1)/{r_1}^2,(x-x_1)/{r_1}^2\\ \left({y-y}_2\right)/{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000058134620000024.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2,\left({x-x}_2\right)/{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000058134620000024.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2\\ \left({y-y}_3\right)/{r_3}^2,(x-x_3)/{r_3}^2\end{array}\right],---\left(14\right)$

According to (10), (13) formula, following formula is carried out iterative:

$x^{(k+1)}=x^{\left(k\right)}+{\left({\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}\mathrm{\&phi;}\left(x^{\left(k\right)}\right)\right)}^{-1}$

$..{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))---\left(15\right)$

$=x^{\left(k\right)}+A^{\left(k\right),-1}\&CenterDot;{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))$

When twice iteration result's difference during less than an arbitrarily small positive number, iteration termination obtains final travelling carriage estimated coordinates

The invention has the advantages that:

Feasible zone, two optimization orders of grid search have been dwindled in the employing primary iteration coordinate figure of gauss-newton method obtains more stable, convergence fast, and moderate complexity can obtain higher mobile station positioning accuracy.

Description of drawings

Fig. 1 is the flow chart of concrete grammar of the present invention;

Fig. 2 is the GBSBCM model;

Fig. 3 is the geometrical relationship figure of travelling carriage, base station and scattering object.

Fig. 4 is the cell layout schematic diagram;

Fig. 5 (a) is that the average position error in different scattering object distribution maximum radius situations compares;

Fig. 5 (b) has shown the cumulative distribution function curve of each algorithm.

Embodiment

As shown in Figure 1, be the particular flow sheet of implementation method of the present invention.The improved mixing of the present invention TOA/AOA geometry location method realizes as follows:

(1) parameter measurement

The wireless signal parameter of moving table measuring base station, i.e. time of arrival (toa) (Time Of Arrival, TOA)

With direction of arrival degree (Angle Of Arrival, AOA)

And in conjunction with the geometry site of travelling carriage and base station, can extrapolate the position of travelling carriage.

In NLOS (non-line-of-sight, non line of sight) environment, owing to have obstacle or scattering object, there are larger error in the TOA that moving table measuring arrives and AOA parameter value.As shown in Figure 2, simulate the NLOS communication environments with individual reflection circle model (geometrically based single bounce macrocell circular model, GBSBCM) model.This model has utilized and an actual hypothesis that conforms to: in the macrocellular environment, antenna for base station is higher, and reverberation does not produce reflected signal near the base station.Scattering object centered by travelling carriage, R _dBe Gaussian Profile in the circle for radius.

As shown in Figure 3, the geometry site according to travelling carriage, scattering object and base station has:

${\widetilde{\mathrm{\&alpha;}}}_i-\arctan \left(\frac{y_i-y_b^i}{x_i-x_b^i}\right)=0---\left(1\right)$

${\tilde{L}}_i-r_b^i-r_m^i={\tilde{L}}_i-\sqrt{{(x_i-x_b^i)}^2+{(y_i-y_b^i)}^2}-\sqrt{{(x_i-x_m)}^2+{(y_i-y_m)}^2}=0---\left(2\right)$

Wherein

With

Represent respectively base station signal and arrive the propagation distance of travelling carriage and arrive angle,

(x _i, y _i) and (x _m, y _m) represent respectively the position coordinates of base station, scattering object and travelling carriage.The signal propagation distance

Can be by the measurement parameter signal propagation time Multiply each other with light velocity C and to obtain:

${\tilde{L}}_i={\widetilde{\mathrm{\&tau;}}}_i\&times;C---\left(3\right)$

(2) dwindle feasible zone

Under the NLOS environment, because travelling carriage (x _m, y _m) and each base station

Between distance

By the distance of scattering object to travelling carriage

Add that scattering object is to the distance between the base station

Consist of, namely

Then according to parameter

With

The position of rough estimate travelling carriage:

$\left\{\begin{array}{c}{\hat{x}}_{\mathrm{ml}}^i=x_b^i+{\tilde{L}}_i\cos {\widetilde{\mathrm{\&alpha;}}}_i\\ {\hat{y}}_{m1}^i=y_b^i+{\tilde{L}}_i\sin {\widetilde{\mathrm{\&alpha;}}}_i\end{array}\right.---\left(4\right)$

$\left\{\begin{array}{c}{\hat{x}}_{m2}^i=x_b^i+({\tilde{L}}_i-R_d^i)\cos {\widetilde{\mathrm{\&alpha;}}}_i\\ {\hat{y}}_{m2}^i=y_b^i+({\tilde{L}}_i-R_d^i)\sin {\widetilde{\mathrm{\&alpha;}}}_i\end{array}\right.---\left(5\right)$

And in cell layout schematic diagram as shown in Figure 4, the number of base stations that participates in the location is 3, and wherein base station 1 is serving BS, and travelling carriage is in the gray area of OABC encirclement.The constraints that proposes according to the GBSBCM model in addition

We obtain the feasible zone of algorithm:

$0\&le;x_m\&le;(\sqrt{3}/2)R$ (6)

$0\&le;y_m\&le;\min \{\sqrt{3}{x}_m,-(\sqrt{3}/3)x_m+R\}$

$0\&le;r_m^i=|x_i+{\mathrm{jy}}_i-(x_m+{\mathrm{jy}}_m)|\&le;R_d^i,i=\mathrm{1,2,3}---\left(7\right)$

Wherein, R is radius of society, Be scattering object distribution radius maximum corresponding to each base station, i=1,2,3.

From the coordinate of above (4), (5), choose maximum, the min coordinates value of satisfied (6), (7), obtain a feasible zone [x who dwindles _Min, x _Max] * [y _Min, y _Max].

(3) grid search is determined the iteration initial coordinate values

In the feasible zone that this dwindles, carry out grid search, choose the grid point that satisfies constraints (7) and consist of candidate point set CPS (Candidate Point Set).

Coordinate figure to all candidate points among the CPS is averaging, the iteration initial value x that is optimized ₀

(4) adopt the Gauss-Newton alternative manner to estimate the position coordinates of travelling carriage

In theory, each base station signal arrives the propagation distance of travelling carriage and arrives angle and can be expressed as:

L(x)＝[L ₁(x)，L ₂(x)，L ₃(x)，α ₁(x)，α ₂(x)，α ₃(x)] (8)

X=[x wherein _m, y _m], $L_i\left(x\right)=\sqrt{{(x-x_b^i)}^2+{(y-y_b^i)}^2},$ ${\mathrm{\&alpha;}}_i\left(x\right)=\arctan \frac{y-y_b^i}{x-x_b^i}.$

And the actual propagation distance of each base station signal with the arrival angle is:

$\tilde{L}=L\left(x\right)+n---\left(9\right)$

Wherein n is that NLOS propagates the error cause and obeys the systematic measurement error that average is zero Gaussian Profile.

Because the existence of error, (1), (2) always can not be met, and obtain thus target function:

$\mathrm{\&epsiv;}\left(x\right)={(\tilde{L}-L\left(x\right))}^T{\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x\right))---\left(10\right)$

∑ wherein _nCovariance matrix for noise n:

∑ _n＝E{nn ^T} (11)

The coordinate that then satisfies following formula namely can be used as the location estimation value of travelling carriage:

$\hat{x}=\underset{x}{\arg \min }\mathrm{\&epsiv;}\left(x\right)---\left(12\right)$

To (9) formula at iteration initial value x ₀The place carries out linearisation:

$L\left(x\right)=L\left(x_0\right)+\mathrm{\&phi;}\left(x\right){|}_{x=x_0}(x-x_0)---\left(13\right)$

Wherein

$\mathrm{\&phi;}\left(x\right)={{\&dtri;}_x}^T\&CircleTimes;L\left(x\right)=\left[\begin{array}{c}(x-x_1)/r_1,\left({y-y}_1\right)/r_1\\ \left({x-x}_2\right)/{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000058134620000024.png"\; state="\{\{state\}\}">r</figure-callout>}}_2,\left({y-y}_2\right)/r_2\\ (x-x_3)/r_3,\left({y-y}_3\right)/r_3\\ (y-y_1)/{r_1}^2,(x-x_1)/{r_1}^2\\ \left({y-y}_2\right)/{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000058134620000024.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2,\left({x-x}_2\right)/{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000058134620000024.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2\\ \left({y-y}_3\right)/{r_3}^2,(x-x_3)/{r_3}^2\end{array}\right],---\left(14\right)$

According to (10), (13) formula, following formula is carried out iterative:

$x^{(k+1)}=x^{\left(k\right)}+{\left({\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}\mathrm{\&phi;}\left(x^{\left(k\right)}\right)\right)}^{-1}$

$..{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))---\left(15\right)$

$=x^{\left(k\right)}+A^{\left(k\right),-1}\&CenterDot;{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))$

When twice iteration result's difference during less than an arbitrarily small positive number, iteration termination obtains final travelling carriage estimated coordinates

Innovative method and existing method are carried out Computer Simulation relatively, shown in Fig. 5 (a) and 5 (b).

Fig. 5 (a) is that the average position error in different scattering object distribution maximum radius situations compares, and for serving BS 1, scattering object distribution maximum radius value is fixed as 0.15km.Can find out that this paper method mean error all is lower than existing localization method, its positioning accuracy improves.

Fig. 5 (b) has shown the cumulative distribution function curve of each algorithm.For 3 base stations, the scattering object distribution radius is respectively 0.15km, 0.25km and 0.25km around the travelling carriage.Wherein the position error of this paper method is 83.7% less than the probability of 0.1km, and improved traditional TOA/AOA algorithm, traditional TOA/AOA algorithm, LOP algorithm and improved HLOP algorithm in the existing localization method are respectively 73%, 47.1%, 31.3% and 62.7%.Hence one can see that, and the positioning performance that the geometry location that the present invention proposes is improved one's methods is better than above-mentioned existing localization method.

Claims (1)

1. the geometry location under the NLOS environment is improved one's methods, and it is characterized in that, the method comprises parameter measurement and location of mobile station two stages of estimation, and concrete grammar is as follows:

1) the parameter measurement stage:

Adopt individual reflection circle model to simulate the NLOS communication environments, according to the base station signal parameter that moving table measuring obtains, time of arrival (toa)

With the direction of arrival degree

And the geometry site of travelling carriage, scattering object and base station has:

${\widetilde{\mathrm{\&alpha;}}}_i-\arctan \left(\frac{y_i-y_b^i}{x_i-x_b^i}\right)=0---\left(1\right)$

${\tilde{L}}_i-r_b^i-r_m^i=\widetilde{L_i}-\sqrt{{(x_i-x_b^i)}^2+{(y_i-y_b^i)}^2}-\sqrt{{(x_i-x_m)}^2+{(y_i-y_m)}^2}=0---\left(2\right)$

Wherein

With

Represent respectively base station signal propagation distance and direction of arrival degree,

(x _m, y _m) represent respectively the position coordinates of base station, scattering object and travelling carriage, signal propagation distance

By time of arrival (toa)

Multiply each other with light velocity C and to obtain:

${\tilde{L}}_i={\widetilde{\mathrm{\&tau;}}}_i\&times;C---\left(3\right)$

2) location of mobile station estimating stage: according to time of arrival (toa)

And direction of arrival degree

Adopt Gauss's Newton iteration method and in conjunction with grid search method Optimized Iterative initial coordinate values, resolve the position that obtains travelling carriage, specifically may further comprise the steps:

(1) dwindles feasible zone

Under the NLOS environment, because travelling carriage (x _m, y _m) and each base station

Between distance

By the distance of scattering object to travelling carriage

Add that scattering object is to the distance between the base station

Consist of, namely

Then according to parameter

With

The position of rough estimate travelling carriage:

$\left\{\begin{array}{c}{\hat{x}}_{m1}^i=x_b^i+{\tilde{L}}_i\cos {\widetilde{\mathrm{\&alpha;}}}_i\\ {\hat{y}}_{m1}^i=y_b^i+{\tilde{L}}_i\sin {\widetilde{\mathrm{\&alpha;}}}_i\end{array}\right.---\left(4\right)$

$\left\{\begin{array}{c}{\hat{x}}_{m2}^i=x_b^i+({\tilde{L}}_i-R_d^i)\cos {\widetilde{\mathrm{\&alpha;}}}_i\\ {\hat{y}}_{m2}^i=y_b^i+({\tilde{L}}_i-R_d^i)\sin {\widetilde{\mathrm{\&alpha;}}}_i\end{array}\right.---\left(5\right)$

Constraints according to the proposition of GBSBCM model

Obtain the feasible zone of algorithm:

$0\&le;x_m\&le;(\sqrt{3}/2)R$

$0\&le;y_m\&le;\min \{\sqrt{3}{x}_m,-(\sqrt{3}/3)x_m+R\}$ (6)

$0\&le;r_m^i=|x_i+{\mathrm{jy}}_i-(x_m+{\mathrm{jy}}_m)|\&le;R_d^i,i=\mathrm{1,2,3}$ (7)

Wherein, R is radius of society,

Be the maximum of scattering object distribution radius corresponding to each base station, i=1,2,3;

From the coordinate of above (4), (5), choose maximum, the min coordinates value of satisfied (6), (7), obtain a feasible zone [x who dwindles _Min, x _Max] * [y _Min, y _Max];

(2) grid search is determined the iteration initial coordinate values

In the feasible zone that this dwindles, carry out grid search, choose the grid point that satisfies constraints (7) and consist of candidate's point set CPS;

Coordinate figure to all candidate points among the CPS is averaging, the iteration initial value x that is optimized ₀

(3) adopt Gaussian-Newton method to estimate the position coordinates of travelling carriage

In theory, each base station signal arrives the propagation distance of travelling carriage and arrives angle and can be expressed as:

L(x)=[L ₁(x),L ₂(x),L ₃(x),α ₁(x),α ₂(x),α ₃(x)](8)

Wherein $x=[x_m,y_m],L_i\left(x\right)=\sqrt{{(x-x_b^i)}^2+{(y-y_b^i)}^2},{\mathrm{\&alpha;}}_i\left(x\right)=\arctan \frac{y-y_b^i}{x-x_b^i}$

And the actual propagation distance of each base station signal with the arrival angle is:

$\tilde{L}=L\left(x\right)+n---\left(9\right)$

Wherein n is that NLOS propagates the error cause and obeys the systematic measurement error that average is zero Gaussian Profile;

Because the existence of error, (1), (2) always can not be met, and obtain thus target function:

$\mathrm{\&epsiv;}\left(x\right){(\tilde{L}-L\left(x\right))}^T{\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x\right))---\left(10\right)$

∑ wherein _nCovariance matrix for noise n:

∑ _n=E{nn ^T}(11)

The coordinate that then satisfies following formula namely can be used as the location estimation value of travelling carriage:

$\hat{x}=\underset{x}{\arg \min }\mathrm{\&epsiv;}\left(x\right)---\left(12\right)$

To (9) formula at iteration initial value x ₀The place carries out linearisation:

$L\left(x\right)=L\left(x_0\right)+\mathrm{\&phi;}\left(x\right){|}_{x=x_0}(x-x_0)---\left(13\right)$

Wherein

$\mathrm{\&phi;}\left(x\right)={{\&dtri;}_x}^T\&CircleTimes;L\left(x\right)=\left[\begin{array}{c}(x-x_1)/r_1,(y-y_1)/r_1\\ (x-x_2)/r_2,(y-y_2)/r_2\\ (x-x_3)/r_3,(y-y_3)/r_3\\ (y-y_1)/{r_1}^2,(x-x_1)/{r_1}^2\\ (y-y_2)/{r_2}^2,(x-x_2)/{r_2}^2\\ (y-y_3)/{r_3}^2,(x-x_3)/{r_3}^2\end{array}\right],---\left(14\right)$

According to (10), (13) formula, following formula is carried out iterative:

$x^{(k+1)}=x^{\left(k\right)}+{\left({\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}\mathrm{\&phi;}\left(x^{\left(k\right)}\right)\right)}^{-1}$

$\&CenterDot;\&CenterDot;{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))---\left(15\right)$

$=x^{\left(k\right)}+A^{\left(k\right),-1}\&CenterDot;{\mathrm{\&phi;}}^T\left(x^{\left(k\right)}\right){\mathrm{\&Sigma;}}_n^{-1}(\tilde{L}-L\left(x^{\left(k\right)}\right))$

When twice iteration result's difference during less than an arbitrarily small positive number, iteration termination obtains final travelling carriage estimated coordinates

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