Single-station location algorithm based on secondary scatterer model

Abstract

Based on secondary scatterer model, realize the location algorithm of the target by virtue of scatterer information. Firstly, use extended Kalman filtering method of resultant motion to effectively suppress the influence of quadratic item on error multiplicative amplification during the location process by circle fitting method, so as to make the estimated location of the second layer of scatterers and the distance between those scatterers and the target more accurate Secondly, serve the second layer of scatterers as virtual observation station, and use Chan algorithm to calculate the position of the first layer of scatterers. Finally, use the position of the first layer of scatterers and signal launching angle to locate the target. The simulation results verify the feasibility and effectiveness of this algorithm.

Keywords

Secondary scatterer model resultant motion extended Kalman filtering virtual observation station Chan algorithm

1. Introduction

There are many existing algorithms applied to the field of wireless location, such as particle swarm optimization and its improved algorithm [1], artificial bee colony algorithm [2], ultrasonic wave heads location algorithm [3]. Wireless location plays an important role in real life, for instance, the use of GPS for data analysis on traffic [4], the detection of the marine environment [5], construction site selection [6]. Signal observability is the foundation of wireless location. In the area where the signal observability has problems, multi-station location can not locate the target at times, this is to consider using single-station to locate the target. Single-station location technology requires only one observation station, and possesses high flexibility, good motility, concise system, no need of information synchronization and information exchange, and many other advantages, so that study on single-station location technology is increasingly becoming one of the hot issues in location field [7]. The development of signal processing technology makes the single-station location technology improve continuously and puts forward new location methods, for example, literature [8] puts forward a regularized CTLS (constrained total least squares) location algorithm combined with the angle of arrival and time-difference information for the passive location problem of the targets using single-station external radiation source; literature [9] puts forward a direct location determination algorithm using constant modulus signals of single moving antenna array pair; literature [10] puts forward a passive location algorithm of onboard single station based on phase difference.

In real environment, the signal propagation is often NLOS (non-line-of-sight) propagation, which is one of the main reasons affecting the location accuracy, and the signal propagation path between the target and scatterer and between the scatterer and single station can be considered as line-of-sight propagation, and can avoid the influence of signal NLOS propagation on location parameters during the location process; therefore, the target location frequently uses scatterer information, the estimation of scatterer information becomes the key to affect the location accuracy. Literature [11] puts forward hybrid location algorithm based on parameter reconstruction for the location problem in NLOS environment; literatures [12, 13] put forward NLOS location algorithm based on scatterer model for NLOS environment; literature [14] uses signal path parameters and the location geometrical relationship among base station, mobile station and scatterer to convert the location problem into linear constraint optimization problem, thus realizing the estimation of target location, however, this method does not give the method to obtain the scatterer position, and therefore is not suitable for practical application; literature [15] puts forward circle fitting location method to estimate the target location by mobile single station, which uses NLOS propagation information to estimate the scatterer location and scattering distance, and converts the single-station location problem into multi-station location problem. However, during the estimation process, the quadratic item will make the measuring error amplify multiplicatively and difficult to be eliminated [16]; literature [16] puts forward the extended Kalman filtering algorithm of resultant motion, which can eliminate the problem of multiplicative amplification of measuring error caused by circle fitting method [15], realize high accuracy of location parameters, and effectively reduce the location error.

The existing single-station location method by virtue of scatterer information is mainly based on the situations of the single reflection of the scatterer, which greatly restricts the location method based on scatterer model. The location algorithm of secondary scatterer model breaks the limitation based on single scatterer model. Literatures [17, 18] puts forward multiple-station location method based on secondary scatterer model; literature [19] puts forward single-station passive location method based on secondary scatterer model, which however uses circle fitting method to estimate the scatterer information and will cause larger error; this article uses the method mentioned in literature [16] to estimate the scatterer information, so as to enhance the location accuracy and make it more suitable for the secondary scatterer model. The simulation results show that the performance of algorithm used in this article is superior to the location method in literature [19].

2. Secondary scatterer model

The set of locating information related to the target, which consists of the scatterer location and the constraint conditions between the scatterer and target, is called the scatterer information. In this paper, the information of the scatterer includes the position of the scatterer, the distance between the single-station and the scatterer, and the distance between the scatterer and the target.

NLOS path signal is mainly formed after the signal encountering scatterer during the propagation process, and the signal generally will not pass one layer of scatterers during the propagation process. In secondary scatterer model, the signal passes through two layers of scatterers before arriving at the observation station; the secondary scatterer model is shown in Fig. 1.

Figure 1.

Schematic diagram of single-station location by secondary scatterer model.

Scatterer generally has short-time stationarity, and a tiny change of the scatterer during the observation process can be considered as measuring error, so that it is assumed that the scatterer location remains unchanged during the entire observation process. The signal is sent out by the actual target T, passes through the first layer of scatterers B and scatters, then passes through the second layer of scatterers $S_{i}$ , and finally arrives at the single moving station. The first layer of scatterer is the scatterer nearest to the target, while the second layer of scatterers are uniformly distributed in a circle with the target T as the center and with a radius of $R_{2}$ . In the above figure, the dashed line represents the motion track of single station, $\text{O}(n)$ is the location of single station at the observation point $n$ , $\text{O}(1)$ is the initial location of the single station, $\alpha_{i}(n)$ is the angle of arrival of the signal when arriving at the observation point $n$ , $h$ is the distance from target T to the first layer of scatterer B, $r_{i}$ is the distance from the first layer of scatterer to the second layer of scatterer, $p_{i}(n)$ is the distance from the second layer of scatterer to the single station. During the observation process, Doppler shift can be used for matching multipath signals, i.e. the Doppler shifts arriving at the single station through the same scatterer are same.

The signal arrival time $\tau_{i}(n)$ and signal arrival angle $\alpha_{i}(n)$ are measureable. Use the signal arrival time $\tau_{i}(n)$ can obtain the signal propagation distance $L_{i}(n)$ , i.e.

$\displaystyle L_{i}(n)=\text{C}\tau_{i}(n)=p_{i}(n)+r_{i}+h+n_{Li}(n)\quad n=1% ,2,3,\ldots,N$ (1)

In the above equation, C is the signal propagation velocity in the air, $n_{Li}(n)$ is the inherent measuring error of the system and is subject to normal distribution $N(0,\sigma_{L}^{2})$ , $i$ is the number of the second layer of scatterers, $N$ is the number of measured points of single station. The angle of arrival of the signal $\alpha_{i}(n)$ is

$\displaystyle\alpha_{i}(n)=\alpha_{i}^{0}(n)+n_{\alpha i}(n)$ (2)

In the above equation, $\alpha_{i}^{0}$ is the angle of arrival (error free) of the signal, $n_{\alpha i}(n)$ is the angle measurement error caused by system error and is subject to normal distribution $N(0,\sigma_{\alpha}^{2})$ .

The scatterer is still, so that $r_{i}$ and $h$ are constant, the distance from target T to the second layer of scatterers $(r_{i}+h)$ is called the scattering distance $c_{i}$ , the first layer of scatterer and the target T are combined and considered as one point, donated as S, the error can be attributed to the inherent measuring error of the system. From the Eq. (1), we can obtain that

$\displaystyle r_{i}+h=L_{i}(n)-\hat{p}_{i}(n)$ (3)

The scatterer location $(x_{si},y_{si})$ and the scattering distance $c_{i}$ . When the number of the second layer of scatterers $i\geqslant$ 3, and the number of observation points $N\geqslant$ 3, serve the estimated second layer of scatterers as the virtual observation station, use classic location algorithm (e.g. TDOA method and Chan algorithm) to location the first layer of scatterer $(x_{B},y_{B})$ . If the signal launching angle can be obtained, use the distance $h$ from the located first layer of scatterer to the target T to estimate the location of target T [19]. Therefore, the estimation of the location of the second layer of scatterer and the scattering distance will influence location accuracy of the algorithm.

3. Location algorithm

Literature [16] analyzes the relative motion relation between the single station and phony target, and uses the resultant motion EKF algorithm to enhance the estimated accuracy of scatterer location and scattering distance. Combine the first layer of scatterer and target T and treat as one point, this article uses the algorithm mentioned in literature [16] to estimate the location and scattering distance of the second layer of scatterers.

3.1 Estimation of the location and scattering distance of the second layer of scatterers

Suppose that the single station makes uniform linear motion along a fixed direction, the velocity components of the single station along X axis and Y axis are $v_{OX}$ and $v_{OY}$ respectively, the motion of the phony target is the composition of uniform linear motion and circular motion, and the linear motion of phony target and the motion of single station are of same velocity and opposite directions.

Take $\mathbf{x}(n)=[x_{Vi}(n),y_{Vi}(n),c_{i}(n)]^{T}$ as the state vector and $\mathbf{y}(n)=[L_{i}(n),\alpha_{i}(n)]^{T}$ as the measurement vector, the measurement equation is

$\displaystyle\begin{bmatrix}L_{i}(n)\\ \alpha_{i}(n)\\ \end{bmatrix}=\begin{bmatrix}\sqrt{x_{Vi}(n)^{2}+y_{Vi}(n)^{2}}\\ tg^{-1}(y_{Vi}(n)/x_{Vi}(n)\\ \end{bmatrix}+\begin{bmatrix}n_{Li}\\ n_{\alpha i}\\ \end{bmatrix}$ (4)

In the above equation, $(x_{Vi}(n),y_{Vi}(n))$ are the position coordinates of phony target.

The state equation is

$\displaystyle\begin{bmatrix}x_{Vi}(n)\\ y_{Vi}(n)\\ c_{i}(n)\\ \end{bmatrix}=\begin{bmatrix}x_{Vi}(n-1)\\ y_{Vi}(n-1)\\ c_{i}(n-1)\\ \end{bmatrix}+\begin{bmatrix}v_{ViX}(n-1)\\ v_{ViY}(n-1)\\ 0\\ \end{bmatrix}T_{s}+\begin{bmatrix}\omega_{X}(n-1)\\ \omega_{Y}(n-1)\\ 0\\ \end{bmatrix}$ (5)

In the above equation, $x_{ViX}$ and $y_{ViY}$ are the velocity components of the phony target along X axis and Y axis respectively, $T_{s}$ is the measurement period, $\omega_{X}(n-1)$ and $\omega_{Y}(n-1)$ correspond to the agitation errors of $x_{ViX}$ and $y_{ViY}$ and are subject to the normal distributions $N(0,\sigma_{VX}^{2})$ and $N(0,\sigma_{VY}^{2})$ respectively, the agitation error is the error caused by the small fluctuation of the scatterer. Based on the relative motion theory, we obtain that

$\displaystyle\left\{\begin{array}[]{l}v_{ViX}=-v_{OX}+v_{CiX}(n)\\ v_{ViY}=-v_{OY}+v_{CiY}(n)\\ \end{array}\right.$ (6)

In the above equation, the unknown parameter vectors, $v_{CiX}$ and $v_{CiY}$ , are the velocity components of phony target during the circular motion. Use the motion relationship between the single station and the phony target to obtain the total velocity component of the motion of the phony target,

$\displaystyle\left\{\begin{array}[]{l}v_{ViX}(n)=\dot{L}_{i}(n)\cos(\alpha_{i}% (n))-L_{i}(n)\sin(\alpha_{i}(n))\dot{\alpha}_{i}(n)\\ v_{ViY}(n)=\dot{L}_{i}(n)\sin(\alpha_{i}(n))+L_{i}(n)\cos(\alpha_{i}(n))\dot{% \alpha}_{i}(n)\\ \end{array}\right.$ (7)

The expressions of velocity component in uniform linear motion are

$\displaystyle\left\{\begin{array}[]{l}v_{OX}=-[\dot{L}_{i}(n)\cos(\alpha_{i}(n% ))-(L_{i}(n)-c_{i})\sin(\alpha_{i}(n))\dot{\alpha}_{i}(n)]\\ v_{OY}=-[\dot{L}_{i}(n)\sin(\alpha_{i}(n))+(L_{i}(n)-c_{i})\cos(\alpha_{i}(n))% \dot{\alpha}_{i}(n)]\\ \end{array}\right.$ (8)

For the Eq. (6), the expressions of velocity component in circular motion are

$\displaystyle\left\{\begin{array}[]{l}v_{CiX}(n)=-c_{i}\sin(\alpha_{i}(n))\dot% {\alpha}_{i}(n)\\ v_{CiY}(n)=c_{i}\cos(\alpha_{i}(n))\dot{\alpha}_{i}(n)\\ \end{array}\right.$ (9)

In the Eqs (7)–(9), $\dot{L}_{i}(n)$ and $\dot{\alpha}_{i}(n)$ are the derivation of the variable to the time, $\dot{L}_{i}(n)$ and $\dot{\alpha}_{i}(n)$ can be obtained

$\displaystyle\left\{\begin{array}[]{l}\displaystyle\dot{L}_{i}(n)=\frac{dL_{i}% }{dn}\mathop{\lim}\limits_{\begin{array}[]{l}\text{\@setsize{\scriptsize}{9.5% pt}{\viiipt}{\@viiipt}Density of information density enough}\\ \text{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}to measure density in % time zone }\scriptstyle m[n-1]=1\\ \end{array}}\\ \quad\frac{L_{i}(n)-L_{i}(n-1)}{1}=L_{i}(n)-L_{i}(n-1)\\ \displaystyle\dot{\alpha}_{i}(n)=\frac{v_{OX}\sin(\alpha_{i}(n))-v_{OY}\cos(% \alpha_{i}(n))}{L_{i}(n)-c_{i}}\\ \end{array}\right.$ (10)

From Eqs (9) and (10), we can obtain the velocity expression of the circular motion of phony target. The equation of EKF can be written as

$\displaystyle\left\{\begin{array}[]{l}\mathbf{x}(n)=\mathbf{A}(n,n-1)\mathbf{x% }(n-1)+\mathbf{u}(n-1)+\mathbf{w}(n-1)\\ \mathbf{y}(n)=\mathbf{H}(n)\mathbf{x}(n)+\mathbf{v}(n)\\ \end{array}\right.$ (11)

In the above equation,

$\displaystyle\mathbf{A}(n,n\!-\!1)=\begin{bmatrix}1&0&-\sin(\alpha_{i}(n\!-\!1% ))\dot{\alpha}_{i}(n\!-\!1)\\ 0&1&\cos(\alpha_{i}(n\!-\!1))\dot{\alpha}_{i}(n\!-\!1)\\ 0&0&1\\ \end{bmatrix},\mathbf{u}(n\!-\!1)=-\begin{bmatrix}v_{OX}\\ v_{OY}\\ 0\\ \end{bmatrix}T_{s},$ $\displaystyle\mathbf{w}(n\!-\!1)=\begin{bmatrix}\omega_{X}(n\!-\!1)\\ \omega_{Y}(n\!-\!1)\\ 0\\ \end{bmatrix},\mathbf{v}=\begin{bmatrix}n_{Li}\\ n_{\alpha i}\\ \end{bmatrix},\mathbf{H}(n)=\begin{bmatrix}\cos(\alpha_{i}(n))&\sin(\alpha_{i}% ((n))&0\\ -\sin(\alpha_{i}(n))/L_{i}(n)&\cos(\alpha_{i}(n))/L_{i}(n)&0\\ \end{bmatrix}.$

Take the solution obtained by circle fitting method [15] as the initial value of the state vector of resultant motion EKF algorithm, we can obtain that

$\displaystyle\begin{bmatrix}\hat{x}_{si}(n)\\ \hat{y}_{si}(n)\\ \end{bmatrix}=f(\mathbf{x}(n))$ (12)

3.2 Target location

As discussed in Section 2.1, we can estimate the position coordinates $(\hat{x}_{si},\hat{y}_{si})$ and scattering distance $\hat{c}_{i}$ of the second layer of scatterers, and can obtain the distance $p_{i}(n)$ between the single station and the second layer of scatterers. Serve the estimated second layer of scatterers as the virtual observation station, use classic location algorithm (this article uses Chan algorithm) to obtain the position coordinates $(x_{B},y_{B})$ of the first layer of scatterer.

Take an example when the number of the second layer of scatterers $i=$ 3, and the difference of the distance between the first scatterer in the second layer and the target T and the distance between the scatterer $i$ and the target T, $\Delta c_{1i}=\hat{c}_{i}-\hat{c}_{1}(i=2,3)$ , so that the location of the first layer of scatterer is

$\displaystyle\begin{bmatrix}x_{B}\\ y_{B}\\ \end{bmatrix}=-\begin{bmatrix}x_{21}&y_{21}\\ x_{31}&y_{31}\\ \end{bmatrix}^{-1}\left\{\begin{bmatrix}\Delta c_{12}\\ \Delta c_{13}\\ \end{bmatrix}r_{1}+\frac{1}{2}\begin{bmatrix}\Delta c_{12}^{2}-k_{2}+k_{1}\\ \Delta c_{13}^{2}-k_{3}+k_{1}\\ \end{bmatrix}\right\}$ (13)

In the above equation,

$\displaystyle r_{1}^{2}=(x_{s1}-x_{B})^{2}+(y_{s1}-y_{B})^{2}=k_{1}-2x_{B}x_{s% 1}-2y_{B}y_{s1}+x_{B}^{2}+y_{B}^{2}$ (14) $\displaystyle x_{i1}=x_{si}-x_{s1},\;y_{i1}=y_{si}-y_{s1},i=2,3,\;k_{1}=x_{s1}% ^{2}+y_{s1}^{2},\;k_{2}=x_{s2}^{2}+y_{s2}^{2},\;k_{3}=x_{s3}^{2}+y_{s3}^{2}.$

Suppose that

$\displaystyle\mathbf{A}=-\begin{bmatrix}x_{21}&y_{21}\\ x_{31}&y_{31}\\ \end{bmatrix}^{-1}\begin{bmatrix}\Delta c_{12}\\ \Delta c_{13}\\ \end{bmatrix}$ (15) $\displaystyle\mathbf{B}=-\begin{bmatrix}x_{21}&y_{21}\\ x_{31}&y_{31}\\ \end{bmatrix}^{-1}\times\frac{1}{2}\begin{bmatrix}\Delta c_{12}^{2}-k_{2}+k_{1% }\\ \Delta c_{13}^{2}-k_{3}+k_{1}\\ \end{bmatrix}$ (16)

The matrix form of Eq. (13) is

$\displaystyle\begin{bmatrix}x_{B}\\ y_{B}\\ \end{bmatrix}=\mathbf{A}r_{1}+\mathbf{B}$ (17)

i.e.

$\displaystyle\left\{\begin{array}[]{l}x_{B}=\mathbf{A}(1,1)r_{1}+\mathbf{B}(1,% 1)\\ y_{B}=\mathbf{A}(2,1)r_{1}+\mathbf{B}(2,1)\\ \end{array}\right.$ (18)

Bring Eq. (18) into Eq. (14), we obtain that

$\displaystyle r_{1}^{2}(\mathbf{A}(1,1)^{2}+\mathbf{A}(2,1)^{2}-1)+(2\mathbf{A% }(1,1)\mathbf{B}(1,1)+2\mathbf{A}(2,1)\mathbf{B}(2,1)r_{1}+$ (19) $\displaystyle\qquad\mathbf{B}(1,1)^{2}+\mathbf{B}(2,1)^{2}=0$

Suppose that: $d=(\mathbf{A}(1,1)^{2}+\mathbf{A}(2,1)^{2}-1)$ , $e=(2\mathbf{A}(1,1)\mathbf{B}(1,1)+2\mathbf{A}(2,1)\mathbf{B}(2,1))$ , $f=\mathbf{B}(1,1)^{2}+\mathbf{B}(2,1)^{2}$ .

Equation (3.2) will be converted into

$\displaystyle r_{1}^{2}\times d+r_{1}\times e+f=0$ (20)

The solution from the above equation is

$\displaystyle r_{1}=\frac{-e\pm\sqrt{e^{2}-4df}}{2d}$ (21)

Choose correct $r_{1}$ value, and bring it into Eq. (18), we can obtain $(x_{B},y_{B})$ .

In Eq. (3), $L_{i}(n)$ , $\hat{p}_{i}(n)$ and $r_{i}(n)$ are known, so we can obtain the distance $h$ between the target T and the first layer of scatterer, and then use the location $(x_{B},y_{B})$ of the first layer of scatterer and the signal launching angle AOD to estimate the location of target T [19].

when the number of the second layer of scatterers $i>$ 3, select the corresponding Chan algorithm to location.

4. Simulation and analysis

Simulate the location algorithm discussed in this article, set the initial position of single station as the origin of coordinates, the actual position of target T is (800, 800), the unit is m, the velocity components of the single station in uniform motion are $v_{OX}=$ 1 m/s and $v_{OY}=$ 0 respectively, $T_{s}=$ 5 s, $\sigma_{VX}=$ 0.05, $\sigma_{VY}=$ 0.05, the first and second layers of scatterers are distributed in the circles with the target T as the center and $R_{1}$ and $R_{2}$ as the radiuses respectively, $R_{2}=2R_{1}$ . The error in the calculation process can be attributed to the inherent measuring error of the system. Simulate and compare the methods discussed in this article and literature [19] under different conditions, and conduct 200 independent Mote-Carlo simulation experiments respectively.

4.1 Simulation 1

Simulate the influence of different number of measured points $N$ on the location error. When $R_{1}=$ 200 m, $R_{2}=$ 400 m, the number of the second layer of scatterers $i=$ 5, $\sigma_{\alpha}=$ 0.01 rad, and $\sigma_{L}=$ 5 m, simulate the influence of the number of measured points $N$ on the location error, as shown in Fig. 2a and b. Figure 2a shows the simulation results when the number of measured points $N=$ 500, Fig. 2b shows the simulation results when the number of measured points $N=$ 100.

Figure 2.

Influence of the number of measured points of the single station on the location error.

When the number of measured points $N=$ 500, from Fig. 2a, when the number of measured points is less than (about) 400, the error of location algorithm discussed in this article is always less than the error of location algorithm discussed in literature [19]; when the number of measured points is about 100, the location error of the algorithm in this article becomes larger gradually, and after the number of measured points exceeds 400, the location error is greater than the location error of literature [19]; the error of location algorithm in literature [19] also becomes larger when the number of measured points is near 150, this is because the equation is of excessive redundancy, causing bigger matrix, and deviation occurs in the operation process. When the number of measured points $N=$ 100, from Fig. 2b, we can see that the error of location algorithm in this article is obviously less than that in literature [19], and $N=$ 100 is the optimal number of measurement points.

4.2 Simulation 2

When $R_{1}=$ 200 m, $R_{2}=$ 400 m, $\sigma_{\alpha}=$ 0.01 rad, and $\sigma_{L}=$ 5 m, simulate the influence of different number of scatterers in the second layer $i$ on the location error when the number of measured points $N$ changes within 100, the simulation results are shown in Fig. 3.

Figure 3.

The influence of the number of scatterers in the second layer on the location error.

From Fig. 3, the location error of the algorithm in this article is obviously less than the error of the location algorithm in literature [19]; when the number of measured points is within 100, a larger number of scatterers in the second layer will generate smaller location error, and the location error will decrease with the increase of the number of measured points, this is because that a larger number of scatterers in the second layer will provide more effective information on the target to be estimated, and will help to improve the location accuracy; of course, the increase of the number of scatterers in the second layer will cause the increase of calculated quantity of location algorithm. From Fig. 3, when the number of scatterers in the second layer changes from 4 to 6, the location accuracy does not change obviously after the stabilization of the algorithm in this article, that is to say, when the number of scatterers in the second layer increases, the location error will decrease slightly.

4.3 Simulation 3

When the number of measured points $N=$ 100, $\sigma_{\alpha}=$ 0.01 rad, $\sigma_{L}=$ 5 m, the number of scatterers in the second layer $i=$ 5, simulate the influence on location accuracy when $R_{1}$ is 100 m, 200 m, 300 m and 400 m; the simulation results are shown in Fig. 4.

Figure 4.

Influence of scattering radius $R_{1}$ on location error.

From Fig. 4, the location error of location algorithm in this article is obviously less than that in literature [19], the location errors of both location algorithms increase with the increase of $R_{1}$ , the increase of $R_{1}$ is actually the increase of distance between the target and the first layer of scatterer, which causes the increase of the error of effective location data, in this article, the location algorithm error does not increase rapidly with the increase of $R_{1}$ , so that it is of stable performance.

4.4 Simulation 4

When $R_{1}=$ 200 m, $R_{m}=$ 400 m, the number of measured points $N=$ 100, $\sigma_{L}=$ 5 m, and the number of scatterers in the second layer $i=$ 5, simulate the influence of different angle measurement errors on location accuracy; the simulation results are shown in Fig. 5.

Figure 5.

Influence of angle measurement error on location error.

Figure 6.

Influence of distance measurement error on location error.

From Fig. 5, the location errors of both algorithms increase with the increase of angle measurement error, the location error increases with the increase of angle measurement error for the algorithm in literature [19], and the rate of change is greater than that of the algorithm in this article; the location error increases slightly with the increase of angle measurement error of the algorithm in this article, so that this algorithm is of stable performance.

4.5 Simulation 5

When $R_{1}=$ 200 m, $R_{m}=$ 400 m, the number of measured points $N=$ 100, $\sigma_{\alpha}=$ 0.01 rad, and the number of scatterers in the second layer $i=$ 5, simulate the influence of different distance measurement errors on location accuracy; the simulation results are shown in Fig. 6.

From Fig. 6, the location errors of both algorithms increase with the increase of distance measurement error, and the location error of the algorithm in this article is obviously less than that in literature [19], the error of the location algorithm in literature [19] becomes stable with the increase of the distance measurement error, while the location error of the algorithm in this article increases slightly, so that this algorithm is of stable performance.

5. Conclusion

Through comparing the algorithm in this article and the algorithm in literature [19] under different simulation conditions, the superiority and stability of the algorithm in this article is reflected, and the position accuracy is optimal when the number of observation points is $N=$ 100. The single-station location algorithm based on the secondary scatterer model breaks the restrictions of existing scatterer single station algorithm which is based on single scattering, extends the scope of application of scatterer-based location algorithm, and it lays the foundation for the single station location research under the multiple scatterer model, and is suitable for the circumstances under which the effective location conditions of the target are relatively simple and the signal measurability has serious problems. Use the relative motion theory to establish relations between the single station motion and phony target motion, and use EKF algorithm of resultant motion to make the estimation of effective location parameters more accurate, so as to finally realize higher accuracy of location target. However, the algorithm in this article is highly dependent on the measurement accuracy of scatterer position and scattering distance, and the estimation error of scatterer information will directly affect the location error of the target; therefore, the estimation of effective location parameters of the scatterer will become the key work on target location algorithm using scatterer in the future.

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