^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper investigates the linear separation requirements for Angle-of-Arrival (AoA) and range sensors, in order to achieve the optimal performance in estimating the position of a target from multiple and typically noisy sensor measurements. We analyse the sensor-target geometry in terms of the Cramer–Rao inequality and the corresponding Fisher information matrix, in order to characterize localization performance with respect to the linear spatial distribution of sensors. Here in this paper, we consider both fixed and adjustable linear sensor arrays.

Different techniques can be used to localize an emitting or non-emitting target [

The Stansfield estimator in [

The potential performance of any particular localization algorithm is highly dependent on the relative sensor-target geometry [

Determining the optimal trajectory for a single moving platform with an AoA sensor was explored in [

Most of the existing literature concerned on the placement of AoA/range sensors around the target for optimal localization [

In this paper we provide a more rigorous characterization of the relative sensor-target geometry for linear sensor arrays based on AoA-only and range-only localization, and to the best of our knowledge, no such analysis exists in the literature.

In our approach we consider the localization problem involving a single target and multiple adjustable AoA/range sensors located as a linear array (uniform and non-uniform). In this case, the Cramer–Rao lower bound with the corresponding Fisher information determinant is used to investigate the optimality of the relative sensor-target geometry, exploring the intrinsic relation with the spatial diversity and the underlying measurement model.

The remainder of this paper is organized as follows. In Section 2, an outline of some notations and conventions together with the problem formulation for AoA-only and range-only localization is given. Section 3 provides results for optimality of the localization for linear sensor arrays that utilise the AoA sensor measurements, while Section 4 gives the results for the range sensor measurements. Section 5 provides simulation results for our theoretical derivations and the paper concludes with a discussion of the optimal configurations in Section 6. In fact, the results presented in this paper provide fundamental information about how the localization performance is affected by the sensor-target geometry for linear sensor arrays. This information is of significant value to users of multiple sensor (linear arrays) based localization systems.

Consider the ^{th}_{p}y_{p}^{T}^{th}_{i}_{si}y_{si}^{T}_{i}_{i}_{i}‖_{i}_{i}_{i}

In general, the set of measurements from _{1}_{n}^{t}_{1}_{N}^{T}_{N}_{z}

The Cramer–Rao inequality lower bounds the covariance achievable by an unbiased estimator under two mild regularity conditions. Considering the unbiased estimate

Consider the set of measurements from _{z}^{th}_{i}^{th}_{ẑ}_{ẑ}

The analysis of the optimal geometry is subjected to the following constraints:

Fixed Uniform Linear Arrays (FULA): One sensor of the linear array is fixed and the distances between consecutive sensors are equal.

Uniform Linear Arrays (ULA): The distances between consecutive sensors are equal.

Fixed Non-Uniform Linear Arrays (FNULA): One sensor of the linear array is fixed and the distances between consecutive sensors may not be equal.

Non-Uniform Linear Arrays (NULA): The distances between consecutive sensors may not be equal.

The measured value of angle (_{i}_{i}_{i}

Then using _{θ}

The Fisher information determinant for bearing-only localization can be given as,

The measured value of range (_{i}_{i}

Then using _{r}

The Fisher information determinant for range-only localization can be given as,

_{p}, y_{p}) which is b distance away from the x-axis. N number of AoA sensors (one fixed at the origin) are on the x-axis, separated by x distance from each other. The Fisher information determinant for this case is_{p}, y_{p})

_{p}, y_{p}) and the position of the fixed sensor (S_{1}) and the line on which the second sensor to be placed is known (_{1} − S_{2}‖ = ‖S_{1} − P‖)

_{1} = [x_{s1} = 0 0]^{T} ), the other on the x-axis (S_{2} = [x_{s2} 0]^{T} ). The Fisher information determinant for this case is

_{s2}, it can be shown that, det(ℐ_{x}(p)) maximizes when,

_{1}− _{2}‖ = ‖_{1}−

_{1}− _{2}‖ = ‖ _{1}− _{2}−

_{1} − _{2}‖ = ‖_{1} − _{2} −

Suppose a target (_{1}, _{2}, ,… , _{N}

Finding the optimal sensor separations becomes an (

_{k1} and S_{k2}), which are y distance apart and the remaining sensor is x distance away from the symmetric axis. Using

_{k1} and S_{k2}), which are y distance apart and the remaining sensor is x distance away from the symmetric axis. Using

_{p}, y_{p}) and b distance away from the x-axis. N number of linear range sensors (one fixed at the origin), separated by x distance from each other, are on the x-axis. The Fisher information determinant for this case is_{p}_{p}

_{p}, y_{p}_{1}P̂S_{2} = π/2)

_{1} = [_{s1}^{T}_{2} = [_{s}_{2} 0]^{T}_{p}, y_{p}

_{s}_{2}, it can be shown that, det(ℐ_{x}(p)) maximizes when

_{1}_{2} = π/2).

This result agrees with the geometrical relationships obtained in [

Suppose a target (_{1}, _{2},…, _{N}

Finding the optimal sensor separation becomes an (

Suppose a target (_{1}, _{2},…, _{N}

Finding the optimal sensor separation becomes an

Consider a sensor-target geometry where one sensor (_{1}) is fixed at the origin and the other sensors (_{2},_{3}, … _{N}^{T}

It can be seen from the figure that when the number of sensors is increased, the Fisher information determinant value increases and the inter-sensor distance decreases for optimal localization, which is unique for a given number of sensors.

Consider a sensor-target geometry as depicted in the _{1}) and (_{2}) are located anywhere on the ^{T}_{s}_{1} and _{s}_{2} attain these values, the geometry of the sensor-target configuration is an equilateral triangle. (_{1} − _{2}‖ = ‖_{1} − _{2} −

Consider a sensor target geometry as depicted in the

It can be seen from the figure that when the number of sensors is increased, the Fisher Information determinant value increases and the distance between the sensors decreases for optimal localization, which is unique for a given number of sensors.

Consider a sensor-target geometry where one sensor (_{1}) is fixed at the origin and the other sensors (_{2},_{3}, … _{N}^{T}

It can be seen from the figure that when the number of sensors is increased, the Fisher information determinant value increases and the distance between the sensors decreases for optimal localization, which is unique for a given number of sensors.

Consider a sensor-target geometry as depicted in the _{1}) and (_{2}) are located anywhere on the ^{T}_{s}_{1} and _{s}_{2} satisfy (14)(

Consider a sensor-target geometry where all the sensors are equally separated by

It can be seen from the simulation that when the number of sensors increases, the Fisher information determinant value increases and the distance between the sensors decreases for optimal localization, which is unique for a given number of sensors.

In this paper, we have provided a characterization of optimal sensor-target geometry for linear arrays of AoA and range sensors in passive localization problems in ℝ^{2}. We have mainly discussed two generic problems of fully adjustable linear sensor arrays and the case of an array, where the sensors are free to be moved with respect to a fixed sensor. Cramer–Rao lower bound and the corresponding Fisher information matrices are used to analyze the sensor target geometry for optimal localization.

The perfect knowledge of the emitter position should be available in the theoretical development for determining optimal sensor placement. Even though in practical applications this information is not available, a rough estimate of the likely region of the emitter is sufficient in determining the sensor positions to obtain improved localization results. Hence the results of this paper can be utilized to establish guidelines for linear sensor placement leading to improved performance.

The analysis given in this paper is also related to optimal path planning and trajectory control of mobile sensors for localization, e.g., see [

This work was supported by the Commonwealth of Australia, through the Cooperative Research Centre for Advanced Automotive Technology (AutoCRC).

The authors declare no conflict of interest.

Measurement from a sensor.

Localization with two sensors (AoA/Range) on x-axis.

Localization with

Variation of Fisher information determinant value with the distance between two adjacent sensors of ULA for different number of AoA sensors (One sensor fixed).

Variation of Fisher information determinant value with the AoA sensors positions.

Variation of Fisher information determinant value with the AoA sensors positions (Contour plot).

Variation of Fisher information determinant value with the distance between two adjacent sensors of ULA for different number of AoA sensors (All adjustable).

Variation of Fisher information determinant value with the distance between two adjacent sensors of ULA for different number of range sensors (One sensor fixed).

Variation of Fisher information determinant value with the range sensors positions.

Variation of Fisher information determinant value with the range sensors positions (Contour plot).

Variation of Fisher information determinant value with the distance between two adjacent sensors of ULA for different number of range sensors (All adjustable).