rangeangposest

Estimates the positions of multiple targets from multiple anchors at known positions combined with one-way range and angle of arrival measurements at the anchors

Since R2025a

Syntax

tgtposest = rangeangposest(rngest,angest,rngvar,angvar,anchorpos)

[tgtposest,tgtposcov] = rangeangposest(___)

Description

Estimates the position of targets using multiple anchors at known positions based on one-way range and angle of arrival (AOA) measurements at the anchors. Estimator utilizes the nonlinear weighted least-square (WLSS) algorithm Algorithms.

tgtposest = rangeangposest(rngest,angest,rngvar,angvar,anchorpos) estimates the position of targets, tgtposest, from one-way range estimates rngest and angle-of-arrival estimates, angest. The function requires known anchor positions anchorpos. rngvar is the range-measurement variance and angvar is the angle measurement variance.

[tgtposest,tgtposcov] = rangeangposest(___) also returns the target position covariance matrix tgtposcov.

Examples

Estimate Position From Range and Angle

A monostatic radar measures the range and AOA of a target in a 2-dimensional Cartesian space. Then, the radar applies the WLLS algorithm using the measured range, measured AOA, and anchor position.

rng("default")

Create the radar anchor at coordinate (20,-10).

anchorpos = [20;-10];

Create a target at coordinate (8000,400).

tgtpos = [8000;400];

Choose a range variance of 10 meters and an angle variance of 0.01 degrees.

rngvar = 10;
aoavar = 0.01;

Create a noisy range measurement.

rngest = norm(anchorpos-tgtpos) + sqrt(rngvar)*randn;

Create a noisy angle measurement.

dir = tgtpos - anchorpos;
az = cart2sph(dir(1),dir(2),0);
aoaest = rad2deg(az) + sqrt(aoavar)*randn;

Estimate the target position.

tgtposest = rangeangposest(rngest,aoaest,rngvar,aoavar,anchorpos)

tgtposest = 2×1
103 ×

    8.0003
    0.4256

Compute the RMSE error of the target position.

rmsepos = rmse(tgtposest,tgtpos);
disp(['RMS positioning error = ',num2str(rmsepos),' meters.'])

RMS positioning error = 18.1265 meters.

Estimate Target Position Covariances

Estimate the ranges and AOAs of two targets in 3-dimensional Cartesian space using monostatic radar data. Apply the WLLS algorithm using to estimate the targets positions.

rng("default")

Create the radar anchor at coordinate (200,-10,0)

radaranchorpos = [200;-10;0];

Create the two targets.

tgtpos = [80 2000; 4000 10; 10 -6];

Choose range variances of 10 meters and angle variances of 0.01 degrees.

rngvar = 10*ones(1,2);
aoavar = 0.01*ones(2,2);

Create noisy range measurements.

rngest = vecnorm(radaranchorpos-tgtpos)+sqrt(rngvar).*randn(1,2);

Create noisy AOA measurements.

dir = tgtpos - radaranchorpos;
[az,el] = cart2sph(dir(1,:),dir(2,:),dir(3,:));
aoaest = rad2deg([az;el]) + sqrt(aoavar).*randn(2,2);

Estimate target positions and covariances.

[tgtposest,tgtposcov] = rangeangposest(rngest,aoaest,rngvar,aoavar,radaranchorpos)

tgtposest = 3×2
103 ×

    0.0958    2.0058
    4.0021    0.0111
    0.0160   -0.0101

tgtposcov = 
tgtposcov(:,:,1) =

   49.0414    1.0143    0.0041
    1.0143   10.0270   -0.1561
    0.0041   -0.1561   49.0679


tgtposcov(:,:,2) =

   10.0000    0.0008   -0.0004
    0.0008    9.9343   -0.0000
   -0.0004   -0.0000    9.9346

Compute the RMSE errors of the target positions.

rmsepos = rmse(tgtposest,tgtpos);
disp(['RMS positioning error = ', num2str(rmsepos), ' meters.'])

RMS positioning error = 9.8253      4.1462 meters.

Input Arguments

rngest — range measurements (meters)

real-valued length-L row vector | real-valued length-K row vector

Range measurements, specified as a real-valued length-L row vector or real-valued length-K row vector.

• When rngest is a real-valued length-L row vector, the measurements represents range measurements to one target measured at L anchors.

• When rngest is a real-valued length-K row vector, the measurements represent range from K targets to one anchor.

Units are in meters.

Example: [1000 2000]

Data Types: single | double

angest — angle-of-arrival estimate (degrees)

real-valued length-L row vector | real-valued P-by-L matrix | real-valued length-K row vector | real-valued P-by-K matrix

• When angest is a real-valued P-by-L matrix, the measurement applies to L anchors and one target. P represents the dimension of the AOA measurement. When P is 1, the row represents the azimuth angle (between -180° to 180°) measured at L anchors. When P is 2, the first row represents the azimuth angle and the second row represents the elevation angle (between -90° to 90° ) measured at L receive anchors.

• When angest is a real-valued P-by-K matrix, the measurement applies to K targets and one anchor. P represents the dimension of the AOA measurement. When P is 1, the row represents the azimuth angle (between -180° to 180°) measured at L anchors. When P is 2, the first row represents the azimuth angle and the second row represents the elevation angle (between -90° to 90° ) measured at K targets.

Units are in degrees.

Example: [10 20]

Data Types: single | double

rngvar — range estimate variance (meters-squared)

[] (default) | positive real-valued length-L row vector | positive real-valued length-K row vector

Range estimate variance, specified as an empty matrix [], positive real-valued length-L row vector, or positive real-valued length-K row vector. When rngvar is empty, the variance estimates are unknown and are set to be unity for either all L anchor or all K targets. Units are in meters-squared.

Example: [10 15]

Data Types: single | double

angvar — angle estimate variance (degrees-squared)

[] | real-valued 1-by-L row vector | real-valued 2-by-L matrix

• Angle estimate variances for L anchors and one target, specified as empty [], a real-valued length-L row vector, or positive real-valued 2-by-L matrix. When angvar is a vector, it represents the azimuth angle variance estimates of one target measured at L receive anchors. When angvar is a matrix, it represents the angle variance estimates of a target measured at L receive anchors. When angvar is empty, the variance estimates are unknown and are set unity all L receive anchors. When the measurements are accurate, you can set the corresponding variance to be empty.

• Angle estimate variances for K targets and one anchor, specified as empty [], a real-valued length-K row vector, or positive real-valued 2-by-K matrix. When angvar is a vector, it represents the azimuth angle variance estimates of a target measured at K targets. When angvar is a matrix, it represents the angle variance estimates of a target measured at K targets. When angvar is empty, the variance estimates are unknown and are set unity all K targets. When the measurements are accurate, you can can set the corresponding variance to be empty.

Units are in degrees-squared.

Example: [.10 .15]

Data Types: single | double

anchorpos — Receive anchor positions (meters)

real-valued Q-by-L matrix

Receive anchor positions, specified as a real-valued Q-by-L matrix where L is the number of receive anchors. When there are multiple targets, L=1. Q represents the dimensionality of the Cartesian space. Positions may be two or three dimensional. For two dimensional positions, Q=2 and for three dimensional positions, Q=3. Each column of anchorpos represents the position of a different anchor in Q-dimensional space. The l^th column of anchorpos denotes the position of the l^th anchor in Q-dimensional Cartesian space. Units are in meters.

Example: [20 31 –16; 13 64 –44]

Data Types: single | double

Output Arguments

tgtposest — Target position estimate (meters)

real-valued length-Q column vector | real-valued length-Q-by-K matrix

Target position estimate, returned as a real-valued length-Q column vector or real-valued Q-by-K matrix where K is the number of targets. Q is the dimension of space. Units are in meters.

Data Types: single | double

tgtposcov — Estimated target position covariance (meters-squared)

real-valued Q-by-Q matrix | real-valued Q-by-Q-by-K array

Estimated target position covariance, returned as a real-valued Q-by-Q matrix or Q-by-Q-by-K array where Q is the dimensions of space.

When the input contains L anchors with one target, the tgtposcov is a Q-by-Q matrix, which is a real positive semi-definite symmetric matrix representing the estimated target position covariance.

When the input contains 1 anchor pair with K targets and one anchor, tgtposcov is a Q-by-Q-by-K array, where each page is a real-valued positive semi-definite symmetric matrix, representing the estimated target position covariance. Units are in meters-squared.

Units are in meters-squared.

Data Types: single | double

More About

Algorithms

When the measurements are noise-free, each one-way range defines a circle or a sphere where a target must lie in the 2-dimensional or 3-dimensional space, each AOA defines a straight line from an anchor towards the target, and the target position is given by the intersection of multiple spheres and straight lines.

In the presence of range and AOA estimation errors, the target position needs to be estimated from more advanced range and AOA position estimation algorithms. This function uses a weighted linear least squares (WLLS) algorithm to fuse the measurements rngest, the AOA measurements angest, and the known anchor positions anchorpos into the estimated target position tgtposest, where the weight used in WLLS algorithm is calculated from rngest, angest, the range estimation variance RNGVAR, and the angle estimation variance angvar. This function supports multiple anchors with one target, and also supports one anchor with multiple targets. The WLLS algorithm approximates maximum-likelihood estimation and can attain the position estimation Cramer-Rao lower bound (CRLB) when the range and AOA measurement errors are small.

Precision

This function supports single and double precision for input arguments. If the input argument tsoaest is single precision, the outputs are single precision. If the input argument tsoaest is double precision, the outputs are double precision. The precision of the outputs are independent of the precision of the other arguments.

References

[1] Reza Zekavat and R. Michael Buehrer, Handbook of Position Location: Theory, Practice, and Advances, 2019

[2] A. Noroozi, M. A. Sebt, S. M. Hosseini, R. Amiri and M. M. Nayebi, "Closed-Form Solution for Elliptic Localization in Distributed MIMO Radar Systems With Minimum Number of Sensors," in IEEE Transactions on Aerospace and Electronic Systems, vol. 56, no. 4, pp. 3123-3133, Aug. 2020.

[3] M. Malanowski and K. Kulpa, "Two Methods for Target Localization in Multistatic Passive Radar," in IEEE Transactions on Aerospace and Electronic Systems, vol. 48, no. 1, pp. 572-580, Jan. 2012.

[4] P. Stoica and J. Li, "Lecture Notes - Source Localization from Range-Difference Measurements" in IEEE Signal Processing Magazine, vol. 23, no. 6, pp. 63-66, Nov. 2006.

Version History

Introduced in R2025a