Closed form algorithm of double-satellite TDOA+AOA localization based on WGS-84 model

Wanchun LI, Ruibin CHEN, Yuning GUO, Caixia FU

School of Information and Communication Engineering, University of Electronic Science and Technology of China,Chengdu 611731, China

KEYWORDS

Angle-of-Arrival (AOA);

Constrained Total Least Squares (CTLS);

Crame´r-Rao Lower Bound(CRLB);

Double-satellite positioning system;

Taylor-Series Iteration(TSI);Time Difference of Arrival(TDOA);

Weighted Least Squares(WLS);

Wideband Geodetic System(WGS-84)

Abstract In this paper, we consider the double-satellite localization under the earth ellipsoid model of the Wideband Geodetic System(WGS-84) using the Time Difference of Arrival (TDOA)and the Angle-of-Arrival (AOA). Several closed-form solution algorithms via the pseudolinearization of the measurement equations are presented to efficiently estimate the location.These algorithms include the Weighted Least Squares (WLS), the Constrained Total Least Squares(CTLS), and the Taylor-Series Iteration (TSI). Performance comparison of the proposed methods with the Crame´r-Rao Lower Bound (CRLB) in the simulation is shown to demonstrate that the proposed algorithms are feasible and have stable performance.

1. Introduction

With the development of space technology and benefit from increasing of satellite payload,Space electronic reconnaissance has drawn much attention for its advantages of wide coverage and little influence by climate.In electronic reconnaissance,the localization of radiation source is one of the most important part. Different from traditional ground, sea and low altitude electronic reconnaissance,principle and method of localization in space electronic reconnaissance have changed because of the specialty of satellite orbit and the altitude of reconnaissance platform. On the other hand, its positioning system is greatly different from that of satellite navigation because of noncooperative target. Therefore, it is rather meaningful and useful to research the geolocationtheory and the method for the space electronic reconnaissance system.1

So far, the algorithm to obtain the target location from echo signals is still a research hotspot. The paper derived a maximum likelihood estimator to estimate target’s location and velocity directly in a two-dimensional plane.2The direct method requires a good initial guess to avoid local convergence and its computational complexity is impractically high due to higher-dimensional search.3,4In the other way, we can divide the positioning process into two parts, parameter extraction and position estimation. In the second part, we study how to obtain the position of target according to various measurement.5The paper proposed a closed form algorithm to obtain the location and velocity of the target for multiple platform.6The paper proposed an algorithm which utilize the Time Difference of Arrival (TDOA) and the Angle-of-Arrival (AOA)measurement to obtain the analytic solution.7,8

Due to high altitude of satellite orbit and the high cost of satellite,there are some difference between traditional localization and one in space. Firstly, we should set a unified coordinate system to show the relative position of satellite and the target on earth. If we use the traditional method to locate the target by satellite only, the performance was hard to be guarantee.Aiming at the above problems,the paper proposed the Wideband Geodetic System (WGS-84) model can be used as constraint to improve the performance of TDOA localization algorithm.9At present, most papers pay attention to the research of TDOA+ the Frequency Difference of Arrival(FDOA)hybrid,10,11AOA12or TDOA13satellites localization.The positioning system of double-satellite combining TDOA and FDOA has a higher requirement for the construction of double-satellite in the same orbit or different orbit,and it can’t maintain relatively stable positioning accuracy for a long time.The method Using Newton iteration to solve the multivariate nonlinear equations couldn’t be convergent when the initial solution is unreliable. The AOA positioning system of singlesatellite performs with high flexibility due to that it can complete positioning task by itself, but it has a weak observability to the subsatellite target and its accuracy is affected by error of the angle measurement greatly. TDOA+AOA hybrid positioning system can solve the problem of weak observability of subsatellite points and has a more stable performance at the cost of moderate complexity of system implementation.Therefore, this paper proposes a target localization algorithm that through the pseudo linearization of the measurement equation and combining it with the time difference equation to simplify the analytic solution under the constraints of the WGS-84 earth ellipsoid model.14,15Meanwhile, we derive the Crame´r-Rao Lower Bound (CRLB) of the system positioning error and also carry out the simulation.

The least squared method can be used to solve the overdetermined equation in localization problem, but the method is just suitable for the condition that all variables is subject to the same distribution. As the improvement, the weighted least squared method can solve the problem well when the prior information of noise covariance is given. The Constrained Total Least Squares (CTLS) method have a better performance when we know the relationship between coefficient matrix and noise vector. However most practical localization models are not linear equations, which cannot be applied for the Weighted Least Squares (WLS) method directly.16Then there are Taylor-series iteration method17combining WLS method with Taylor expansion to transform the nonlinear equations to the linear equations.We show the detailed derivation of TDOA+AOA under the constraints of the WGS-84 earth ellipsoid model using WLS,Tylor and CTLS respectively in Appendices A-E.

In order to demonstrate that the double-satellite TDOA+AOA positioning system has a higher application value than the single-satellite positioning system, we compare the performance of the double-satellite with that of single-satellite in Appendix F, the simulation shows that the double-satellite positioning system has a better observability and accuracy.

To illustrate the proposed algorithm has considerable practical value, we discuss the satellite self-positioning error, the duration of observation time and the extension of the algorithm in Appendix G.We analyses how long the two satellites at the altitude of 500 km can keep the condition of baseline to have a good observation to the target and it’s enough for the real-time algorithm, the duration can be longer for the higher satellites so that it can also be applied for Medium-altitude Earth Orbit(MEO)or High-altitude Earth Orbit(HEO)satellites.For the problem of satellite self-positioning error,we can regard it as random variables and revise our measurement equations when the error is non-negligible and we provide the development thoughts. Although the proposed algorithm is for double-satellite,we can increase the number of measurement equations and apply it the multi-star positioning scene.

The rest of this paper is organized as follows: Section 2 illustrates the localization model.In Section 3,the localization algorithm is briefly described.The performance analysis is discussed in Section 4. And the localization algorithm is simulated and analyzed in Section 5. Section 6 summarizes the main conclusions of the proposed method.

In this paper, x is scalar, x is vector, A represents matrix,measured values of direction cosine angle are denoted by α and β respectively,ρ represents the measured value of normalized TDOA (distance difference), and true values are denoted by α0,β0and ρ0respectively, ‖·‖represents 2-norm based on the vector and diag ·（） is a diagonal matrix.

2. Localization model

Under the coordinate system of Earth Centered Earth Fixed(ECEF), the location of the satellite is xS，eand the position of the target radiation source is xT，e.Under the coordinate system of satellite, the position of the target radiation source is xT，b.It is known from the Ref.1,in the space-based AOA localization system, the transformation relationship is as

Fig. 1 Double-satellite TDOA+AOA localization model.

(1) The direction cosine angle of the target radiation source is measured by the double-satellite. According to the above we can obtain the expression.1

where i=1，2， Dx=［1，0，0］， Dy=［0，1，0］ and Dz=［0，0，1］. Mirefers in Appendix A. The above formula shows that the direction cosine angle of the target radiation source is a nonlinear function.

(2) The TDOA from the target radiation source to two satellites could be denoted by

where c is the speed of signal propagation. Defined by Ri=‖Mi（xSi，e-xT，e）‖ （i =1，2）.

The Eq. (4) can also be rewritten as

When the problem of the target radiation source localization is transformed into solving the target position using equations, we can get the unambiguous solution. According to the basic principle of geometric positioning,it is the space position line formed by the double-satellite AOA measurement and the space position rotation hyperboloid developed by the time difference of target signal arrival to make the line or line surface intersect and determine the WLS solution of the target. Furthermore, the location of the target radiation source can be more accuracy under the constriction of WGS-84.

3. Localization algorithm

3.1. Localization algorithm of WLS

The true value of the direction cosine angle α0i,β0i measured by the double-satellite could be formulated as

where i=1，2. Meanwhile

Transform the time difference of arriving to the distance difference, and the expression of the measured value can be derived as

where nαi，nβi，nρare independent and Gaussian distribution that

Algorithm 1. According to Eq. (6), we have

where i=1，2. i.e.

Algorithm 2. According to Eqs. (6) and (8), we have

where i=1，2. i.e.

In the same way with Algorithm 1, we have

Algorithm 3. According to Eqs. (2) and (3) we have

where i=1，2 and

In the same way as Algorithm 1, we can have

According to Eq. (13), Eq. (16), Eq. (19), we have

where the specific expressions of b, A, N and n can refer to Appendix C. Thus the WLS solution of the target radiation source could be rewritten as

where

3.2. Localization algorithm of TSI

According to Eqs. (7) and (9), we have

where

According to Best Linear Unbiased Estimation(BLUE),we can obtain

3.3. Consider constraints of WGS-84 earth ellipsoid model

According to Eq.(5),Eq.(20),the localization problem can be solved by the following equations:

Transform constraint equation into Taylor expansion form

Substitute into cxT，e=d, we can have the analysis solution λ

Finally we can have constraint solution xT，e-consaccording to Eq. (31).

3.4. Localization algorithm summary

According to the derivation before, the localization algorithm process can be summarized as three steps which shows in Table 1.

4. Performance analysis and geometric precision boundary

Based on the analysis above,the system positioning accuracy is mainly influenced by the measurement error of AOA nαi, nβiand the measurement error of TDOA nρwhen we locate the target on the earth surface. Next, we calculate and analyze the CRLB of the location error under the condition of different measurement error of TDOA and AOA, and we will get the geometric precision of the method of double-satellite TDOA+AOA closed position solution under the WGS-84 model.

4.1. CRLB of location error

The measurement equation of the system is as follows.

The geometric precision boundary of target localization without the constraints of the WGS-84 earth ellipsoid model is obtained as

Table 1 Steps of proposed methods.

Considering the constraint of the WGS-84 model in Eq.(5)and according to the Ref.18when there exists additional constraint condition f（xT，e）, the CRLB could be rewritten as

where J is the gradient for estimator xT，ein constraint condition, i.e.

4.2. Performance analysis of proposed algorithm

4.2.1. WLS method

Defines residual error ΔxWLSas

Referring to first order error analysis method, we can obtain estimation and covariance of error

Theoretic analysis shows that the performance of WLS method can achieve CRLB when noise is moderate. Because the linear equations are obtained approximately by omitting the higher order terms,the performance would be worse when noise becomes significant.

4.2.2. CTLS method

Defines residual error ΔxCTLSas

Referring to first order error analysis method, we can obtain estimation and covariance of error

where

Theoretic analysis shows that the performance of CTLS method can achieve CRLB when noise is moderate. Because the linear equations are obtained approximately by omitting the higher order terms,the performance would be worse when noise becomes significant.

4.2.3. Taylor iterations

Defines residual error ΔxCTLSas

Referring to first order error analysis method, we can obtain estimation and covariance of error

Theoretic analysis shows that the performance of CTLS method can achieve CRLB when noise is moderate. Because the linear equations are obtained approximately by omitting the higher order terms,the performance would be worse when noise becomes significant.

4.2.4. Computation complexity

Considering the number of multiplications in matrix operations, WLS approaches and Taylor-series approaches both need 414 times multiplication, but Taylor series approaches need to obtain Jacobean matrix. CTLS approaches need 1243 times.Constraint solution needs more 222 times.Table 2 shows how many times matrix multiplication, matrix inversion, Jacobian solution and trig functions all proposed methods need to do.

4.3. Geometric Dilution of Precision (GDOP) of localization algorithm

GDOP is highly influenced by the geometry of the receivertransmitter Define GDOP as18

where trace means the sum of diagonal elements. GDOP represents that GDOP model for the case without the constraints of the WGS-84 earth ellipsoid, while GDOP_cons represent that GDOP model for the case with the constraints of the WGS-84 earth ellipsoid.

Table 2 Steps of proposed methods.

Assuming the orbit altitudes of the double-satellite are HS1= 550 km and HS2= 500 km,and the corresponding longitude and latitude of the sub-satellite points are（LS1，BS1）=（102°，29°） and （LS2，BS2）=（105°，33°）. The marks show in the following figure.The measurement standard deviation errors of AOA, defined as σAOA=σα=σβ, is set to 0.1°,and the measurement standard deviation errors of TDOA σTDOAis set to 1 μs. We can obtain the GDOP contour according to the CRLB on different location of double-satellite TDOA+AOA. As shown in Fig. 2(a) and (b).

Assuming the orbit altitudes of the double-satellite both are 500 km, and the corresponding longitude and latitude of the sub-satellite point are （LS1，BS1）=（102°，33°） and（LS2，BS2）=（106°，30°）. The marks show in the following Figure.σAOAis set to 0.1°,and σTDOAis set to 1 μs.We can obtain the GDOP contour according to the CRLB on different location of double-satellite TDOA+AOA. As shown in Fig. 2(c)and (d).

The unit of the numerical value marked in the figure is km,the same below.

We can draw from the pictures above that all the GDOP drawings of localization error of double-satellite TDOA+AOA hybrid localization method is roughly symmetrical about sub-satellite point, only exist certain deformation and deflection in some regions. In other words, the positioning error is stable in a wide range of latitude and longitude.Through the comparison in Fig.2,we can get a conclusion that the GDOP drawing of double-satellite TDOA+AOA hybrid localization error with the constraints of the WGS-84 earth ellipsoid model is obviously better than the GDOP without the constraints.

5. Simulation analysis of proposed localization algorithm

5.1. Simulation of algorithm performance

Assuming the orbit altitudes of the double-satellite are HS1=800 km and HS2=600 km respectively, and the corresponding longitude and latitude of the sub-satellite points are（LS1，BS1）=（101°，30°）and（LS2，BS2）=（108°，36°）respectively. Locate the target on the earth surface with TDOA+AOA hybrid algorithm and the latitude and longitude of target is set as（104°，31°）.Then we use the method of localization in Section 3 and the CRLB calculation method in Section 4 and compare the variation of the measurement error of AOA and TDOA through simulation.In the simulation of Fig.3(a),σTDOAis set to 20 μs, and σAOAvaries from 0.1°to 1°. In the simulation of Fig. 3(b), σAOAis set to 0.4°, and σTDOAvaries 10 μs from to100 μs.

Fig. 2 GDOP contour of double-satellite TDOA+AOA localization.

We can see from Fig. 3, the localization method proposed in this paper can achieve CRLB when the measurement error is moderate. The performance of the TSI method with constraints is better than that of the WLS method with constraints. When the measurements error becomes significant,the effect of pseudo linearization becomes larger because the omitted quadratic term can’t be omitted. The least squares approaches are unreliable when noise is significant, and the Taylor iteration method performs best. The localization solutions with WGS-84 constraint are better than others without constraint because of addition of prior information.Generally,our proposed idea can be realized through Taylor-series iteration with WGS-84 constraint.

Fig. 3 Root Mean Square Error (RMSE) for different proposed method.

Fig. 4 RMSE for different proposed method with iterations.

5.2. Performance improvement by using iterations

The location of satellites and target is set as the same as Section 5.1. In the simulation of following Fig. 4(a), σTDOAis set to 20 μs,and σAOAvaries from 0.1°to 1°.In the simulation of following Fig. 4(b), σAOAis set to 0.4°, and σTDOAvaries from 10 μs to 100 μs.

We can see from Fig.4,when the error of system measurement is comparatively large, the precision of the 2WLS solution is better than that without iteration. It has been proved that iterations can improve performance when the error of system measurement is significant in Fig. 4. But actually,WLS estimator can meet the requirements in most of the time.

6. Conclusions

A closed form algorithm of double-satellite TDOA+AOA localization is proposed in this paper. The simulation shows that WLS and TSI method with constraints in our methods both can approximate CRLB of the target localization consistently in the same orbit and different orbits of the doublesatellite. Though the performance of the TSI method is better than that of the WLS method, its computation complexity is higher.Thus we can choose one of them referring to the actual condition. When the error of system measurement is significant, we can still use several iterations to improve the performance of the WLS localization algorithm. In the meanwhile of high precision, double-satellite localization has a lower requirement for resource and position of stations.

Acknowledgements

This work was supported by Meteorological information and Signal Processing Key Laboratory of Sichuan Higher Education Institutes of Chengdu University of Information Technology, China (No. QXXCSYS201702).

Appendix A. The expression of the target position under the platform star coordinate system of b could be denoted by

Appendix B. The pseudo linearization of the measurement equation and the detailed derivation process in localization Algorithms 1, 2 and 3 are shown as follows:

(1) Corresponding to Eq. (11) in Algorithm 1

where i=1，2. We have

And due to the first order Taylor expansion at measurement value

We have

It could be rewritten as

(2) Corresponding to Eq. (14) in Algorithm 2

We have

Using first-order Taylor expansion with the measurement value αiand βi, we can obtain

We have

It could be rewritten as

(3) Corresponding to Eq. (17) in Algorithm 3

where i=1，2 and

i.e.

It could be rewritten as

where i=1，2, and

Appendix C. The concrete expressions of each matrix in Eq. (20) are shown as

b-AxT，e=Nn

where

where

Appendix D. The specific expression of matrix G in Eqs. (26)and (34) is shown as

Appendix E. According to Eq. (20), We have

Rewrite Eq. (E1) into the formation with n

where

The localization problem can be modeled as a constraint equation as following

where

According to the constraint equation in Eq. (E10) we can have

Define

Define the gradient vector of L about x as

where

Substitute Eq. (E16) into Eq. (E15), yields

The Hessian matrix of L about x is shown as following

Thus we can have

Appendix F.We use the CRLB to compare the performance of a single-satellite AOA positioning system with that of a double-satellite TDOA+AOA positioning system.

According to Eqs. (4)-(6), the AOA positioning system of single-satellite with WGS-84 constraints can be expressed by

Denote the parameter vector by xT，Land G1can be expressed as

where

The geometric precision boundary of target for the singlesatellite AOA positioning system is obtained

According to Eqs.(34)-(36),the Fisher Information matrix of a double-satellite TDOA+AOA positioning system can be expressed as

where

Then we have

The geometric precision boundary of target for the doublesatellite TDOA+AOA positioning system is obtained as

The performance improvement between the single-satellite and the double-satellite systems can be obtained by

Suppose that both the orbit altitudes of the double-satellite are 500 km, and the corresponding longitude and latitude of the sub-satellite points are （LS1，BS1）=（102°，33°） and（LS2，BS2）=（106°，30°）. σAOAis set to 0.1°, and σTDOAis set to 1 μs. We can obtain the GDOP contour of double-satellite TDOA+AOA as shown in Fig. 2(d).

The orbit altitudes of the single-satellite are 500 km, and the corresponding longitude and latitude of the sub-satellite point are （LS1，BS1）=（102°，33°）, σAOAis set to 0.1°. We can obtain the GDOP contour of single-satellite AOA as shown in Fig. F1(a) and GDOP improvement contour comparing to double-satellite TDOA+AOA as shown in Fig. F1(b).

Appendix G.The double-satellite localization is an engineering application based on two cooperative satellite. To realize the proposed algorithm, The AOA measurements and the TDOA measurement (For non-pulse signal waveform transmission should be considering) should be transmitted from satellites to data processing center. The data transmission is a practical problem when the proposed algorithm is in application.

Fig. F1 GDOP contour of satellite AOA positioning system.

The algorithm can be applied in LEO, MEO, HEO and their hybrid localization and it is easy to be extended to multi-satellite TDOA+AOA localization. However, there are many problems in practical application, for example, the strength of received signal is enough or not to ensure accurate measurements, the target is in reconnaissance area or not.Even though the angular velocity of the LEO satellite is so high that its effective time is limited to a specific region, we can choose the appropriate satellites arbitrary to localize the target because our algorithm is real-time.

The orbit of satellite has a great effect on localization that mainly includes:

(1) Sustainability of reconnaissance

The LEO satellite takes about 2-4 h to go around the earth once. Assuming that the period around the earth is two hours and the mean radius of the earth is 6371 km.Thus,the satellite needs 565 s for running 500 km, if wanted to track the same target, we can choose appropriate satellites in different time to work out it.

(2) The deterioration of localization error caused by the orbit error

(A) As usual, the LEO satellite has a high selfpositioning accuracy which can achieve meter level, however the RMSE of target in simulation is in kilometer level. So, the effect brought from the self-positioning error can be omitted.

(B) If the self-positioning error of satellite is too large,we should consider the self-positioning error as a Gaussian observation process independent of the AOA and TDOA measurements. Because the problem can’t be described simply we just introduce its idea.

(C) Replace the real satellite position with the satellite position with error in the pseudo-linear equations of the AOA measurement and then merge (linear term of the self-positioning error)into the error of the AOA measurement (the right side of the Eq.(16)).

The follow-up process is similar to the proposed algorithm in this paper, the difference is that the station positions are regarded as random variable, the specific algorithms can be referred to in the Refs. [19,20,21].

Fig. G1 Improvement of positioning accuracy between singlesatellite and double-satellite with varying baseline.

To illustrate the improvement of target positioning accuracy by baseline length,the target is located at 108°east longitude and 35° north latitude. Satellite 1 is fixed at 102° east longitude, 33° north latitude and 500 km altitude. Satellite 2 is located at 500 km altitude. Longitude increases with 0.1°east longitude, latitude decreases with 0.1° north latitude.Finally, it reaches 105° east longitude and 32° north latitude,totaling 30 points and 30 baseline. The remaining simulation conditions are as described above.

From Fig.G1,we can see that with the increase of baseline,the positioning accuracy of double-satellite is higher than that of single-satellite,which also shows the importance of selecting appropriate satellite for positioning.