Journal of Sensors

Volume 2019, Article ID 3743475, 13 pages

https://doi.org/10.1155/2019/3743475

## Noniterative Three-Dimensional Location Estimation Using Azimuth and Elevation Measurements at Multiple Locations

Department of Information and Communication Engineering, Sejong University, Republic of Korea

Correspondence should be addressed to Joon-Ho Lee; rk.ca.gnojes@eelhnooj

Received 8 January 2019; Revised 9 April 2019; Accepted 18 July 2019; Published 4 August 2019

Academic Editor: Giovanni Diraco

Copyright © 2019 Ji Woong Paik and Joon-Ho Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Closed-form expression of three-dimensional emitter location estimation using azimuth and elevation measurements at multiple locations is presented in this paper. The three-dimensional location estimate is obtained from three-dimensional sensor locations and the azimuth and elevation measurements at each sensor location. Since the formulation is not iterative, it is not computationally intensive and does not need initial location estimate. Numerical results are presented to show the validity of the proposed scheme.

#### 1. Introduction

There has been a great deal of research on the determination of emitter location. Localization consists of two parts: measuring localization parameters between nodes and the use of these parameters to estimate location. The localization parameters can be either AOA (angle of arrival) or TOA (time of arrival).

In this paper, we consider AOA-based localization. The AOA-based localization algorithm can be classified as follows: linear least-squared (LS) estimation [1], nonlinear least-squared estimation [2, 3], total least-squared estimation [4, 5], the discrete probability density (DPD) method [6], localization algorithm for generalized bearing [7], close-form solution for positioning based on angle of arrival measurements [8], maximum likelihood (ML) estimation, and Stansfield algorithm [9, 10].

In [1], the authors presented a closed-form solution for the emitter location based on the measurements of azimuth, associated with two-dimensional localization. Note that the scheme is used for estimating the two-dimensional coordinate of an emitter, not the three-dimensional coordinates. In addition, the algorithm presented in [1] is not iterative in the sense that it is not based on Newton-based iteration. There is also a technique called total least-squared (TLS) estimation [4, 5], which is an extension of the LS estimation.

Nonlinear least-squared estimation method [2, 3] estimates the location of an emitter by minimizing the bearing errors of the LOBs (line of bearing). On the other hand, in linear least-squared estimation [1], the location estimate of an emitter is obtained by minimizing the distances between the LOBs and the emitter location. The scheme is based on Newton-type iteration, which implies that the solution is not in closed-form.

In [6], the authors presented the discrete probability density method which is based on dividing up the AOI (area of interest) into discrete intervals. This location estimation method modifies some of the erroneous results when the input data for the fix computation are LOBs.

In [7], an unconventional localization method considering the LOBs which are not based on the Cartesian coordinates is presented. This method can estimate the location of emitter in three-dimensional space. In order to estimate the location of the emitter in three dimensions, LOB is given as azimuth and elevation. In [7], a new LOB is defined by making a slant azimuth (generalized bearing) with a given azimuth and elevation. This method reduces the computational complexity by reducing the matrix and vector dimension in the localization. The scheme is based on Gauss-Newton iteration method, which implies that the solution is not in closed-form.

In [9], the method of localization using a maximum likelihood approach is presented. Further by using a simplifying assumption that the difference between measured bearings and the bearings are small enough, Stansfield algorithm has been presented [9, 10].

In [11], the authors presented the linear LSE (least-squared error) algorithm and nonlinear LSE algorithm in case of moving emitter. By estimating the initial position, initial velocity, and constant acceleration of an emitter using least-squared estimation, the position of the emitter at a specific time can be obtained.

In two-dimensional algorithm, the emitter is assumed to lie in the plane defined by the trajectory of the sensors. On the other hand, in three-dimensional algorithm, the emitter is not necessarily lie in the plane defined by the trajectory of the sensors.

As far as the authors know, there has been no study on closed-form expression of three- dimensional localization via AOA measurements of azimuth and elevation. In this paper, we derive an explicit closed-form expression for three-dimensional localization. Newton-based iterative approach for localizing three-dimensional coordinates of an emitter will be submitted as a separate manuscript [12].

The method proposed in this paper can estimate the location of the signal source by using results of multiple LOB measurements from moving antenna array. The method proposed in this paper is different from the conventional location estimation algorithm of estimating the location of the signal source through the Newton iteration method. The proposed method can estimate the location of signal source on a three-dimensional space without iteration. The scheme is an extension of the Brown algorithm [1] to three-dimensional space.

In this paper an additive noise associated with noisy LOB measurement is assumed to be zero-mean Gaussian distributed. We are concerned with estimating the location of a single stationary target by using the received signals at the moving sensor. We assume that the locations of the moving sensor are available. The validity of the scheme is illustrated using the numerical results. We assume that the position of the sensor is available without uncertainty. That is, there is no error in the estimation of sensor position. The computational cost of the proposed scheme will be compared with that of the Newton-based iterative approach.

#### 2. Noniterative Three-Dimensional Location Estimation

Let represent the location of the -th sensor:Let and denote the estimated azimuth and the estimated elevation estimated at . Then, let be defined asLocation estimate of the emitter is defined as :In the proposed algorithm, the location estimate is given by minimizing the sum of square of distances from to the line connecting and , for . Let denote the distance from to the line connecting and [13–16]:where, from (1) and (2), is used. The derivation of (4) is shown in Appendix A. By using the parallelogram shown in Figure 8, the minimum distance between a point and a straight line in three dimensions can be obtained.

An explicit expression in terms of the sensor location and the emitter location in Cartesian coordinates is

After some algebraic manipulations, it can be shown that is given byLet be defined as

Partial derivatives of with respect to , and are given by (10)-(12).

, , and result in a linear system of equations. Using these equations the location estimate can be given asDerivation of (13) is shown in Appendix B.

#### 3. Numerical Results

Numerical results illustrating the validity of the proposed formulation are presented in this section. Sensor trajectory and emitter location are illustrated in Figures 1 and 2. Figure 1 shows the linear trajectory of the moving sensor. The sensor generates a number of sensor locations along the -axis. Without loss of generality, the sensor trajectory lies along the -axis between to . The distance between adjacent sensor locations is uniform. Figure 2 shows the circular movement trajectory of the sensor. The , , and coordinates of the -th sensor location are . The distance between adjacent sensor locations is uniform. The coordinate of the first sensor location is . The number of sensor locations is arbitrarily chosen to be 10, 100, and 1000. The distance between the first sensor location and the emitter is 1000(m). in Figure 1(a) is chosen to be . AOA measurement errors are assumed to be zero-mean Gaussian distributed with standard deviation varying from to . The three-dimensional emitter location is estimated using Appendix B. The number of repetitions in the Monte-Carlo simulation is . The RMSE for iterations is defined as