Elliptic and hyperbolic localizations using minimum measurement solutions

Highlights

• The newly derived minimum measurement solution is more computationally efficient than those published in the literature, complexity is always an important factor for practical realization.

• The solution derivation gives the measurement intersection condition that can be exploited for outlier detection, which is simple and easy to apply.

• The positioning estimator we proposed is algebraic and closed-form. It is shown by both analysis and simulation to achieve the CRLB performance.

• A new scheme for grouping measurements to improve the noise tolerance of the proposed estimator is developed.

• The proposed estimator performs better than the existing closed-form solutions and has a noise threshold comparable to the iterative Maximum Likelihood Estimator before performance deviates from the CRLB.

Abstract

Localization of an object using a number of sensors is often challenged by outlier observations and solution finding. This paper derives a new algebraic positioning solution using a minimum number of measurements, and from which to develop an outlier detector and an object location estimator. Two measurements are sufficient in 2-D and three in 3-D to yield a solution if they are consistent. The derived minimum measurement solution is exact and reduces the computation to the roots of a quadratic equation. The solution derivation leads to simple criteria to ascertain if the line of positions from two measurements intersect. The intersection condition enables us to establish an outlier detector based on graph processing. By partitioning the overdetermined set of measurements first to obtain the individual minimum measurement solutions, we propose a best linear unbiased estimator to form the final location estimate. Analysis supports the proposed estimator in reaching the CRLB accuracy under Gaussian noise. A measurement partitioning scheme is developed to improve performance when the noise level becomes large. We mainly use elliptic time delay measurements for presentation, and the derived results are applicable to the hyperbolic time difference measurements as well. Both the 2-D and 3-D scenarios are considered.

Introduction

Determining the location of an object using the measurements from a number of spatially distributed sensors has been a subject for research over the years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Commonly used measurements are time delay for active scenario [12] and time differences for passive environment [13]. The research interest is in part due to the fundamental nature of the problem that has a wide range of applications, as well as the non-linear nature of the problem that makes it interesting to tackle.

In a practical environment, often not all the positioning measurements are good and some maybe outliers. Outliers can come from non-line of sight (NLOS) propagation [14], [15], [16], the environment or the malfunctioning of some sensors [17], [18]. Outliers could cause tremendous damage to positioning accuracy. Various techniques have been proposed to mitigate the resulting detrimental effect [19], [20], [21]. The work [21] used a residual weighting scheme to de-emphasize the measurements that may come from NLOS. In the paper [22], the authors applied the residual test to identify and discard the NLOS measurements but the performance is highly sensitive to the detection threshold that was determined experimentally. The paper [23] formulated several hypotheses based on the Maximum Likelihood approach to detect the NLOS measurements and the performance is dependent on the availability of noise power and the NLOS error statistics. Another study [24] developed an outlier rejection method for device-free localization using RSS but it only functions properly under small noise. The paper [25] applies the Random Sample Consensus (RANSAC) technique for angle of arrival (AOA) or time difference of arrival (TDOA) localization in the presence of outliers. RANSAC is a competitive approach to handle the presence of outlier measurements for robust estimation. Nevertheless it is computationally demanding and requires the knowledge of the maximum level of non-outlier measurement error.

Although the measurement equations are nonlinear, exact algebraic positioning solution exists in the specific case of critically determined scenario in which the number of measurements is equal to the number of unknowns. In such a case, the pioneer work [26] illustrated the location fix is a focus of a conic formed by the measurements. Fang [27] later showed that navigation fix of an object on earth by TDOA can be reduced to the solution of a quadratic equation by using the station baseline plane as reference, but location fix in the 3-D space requires solving a quartic equation. Using an intermediate variable, the study in [28] developed a solution based on Spherical-Intersection that requires the root of a quadratic equation only. Nevertheless, the method is not robust and fails to produce a reasonable solution for some sensor arrangements, due to the need to invert a matrix formed directly by the sensor positions. More recently, the papers [30], [29] examine the location fix and provide a solution from a statistical point of view. In practice more measurements than unknowns are used to mitigate the noise effect and increase the positioning accuracy. Some exact solutions for the overdetermined case can be found in [31], [32], [33], which formulate the optimization problem as a polynomial system that is solved by numerical computer algebra methods or polynomial continuation techniques. Nevertheless, this paper provides an exact and simple algebraic solution which is more computationally efficient and does not rely on numerical technique.

This paper first derives an algebraic positioning fix for the critically determined situation using elliptic measurements [34]. Comparing to the solutions in [27], [30], we reduce complexity by applying the roots of a quadratic polynomial only rather than quartic, do not introduce extraneous solutions, and provide a direct and simpler derivation with geometric interpretation. In addition, the solution is exact, more general and without the use of station baseline plane as reference. Unlike the previous work [28], the proposed solution works for arbitrary sensor arrangements without having robustness issue (except linear arrangement in the 3-D scenario in which position fix is impossible). Most important, the new proposed solution enables us to deduce the conditions for the intersection of line of positions (LOPs) defined by the measurements in 2-D and 3-D for both elliptic and hyperbolic localizations, which have not appeared before in the literature. The paper [29] did a thorough analysis and provided the compatibility conditions of two TDOAs in 2-D and their results are consistent with ours.

An immediate use of the intersection condition is that it enables us to detect which of the measurements are outliers by taking a clustering approach from the graph theory. The clustering technique was used in the papers [35], [36], but for time of arrival (TOA) measurements with circle loci only. Here the outlier detection is applied to 2-D and 3-D for elliptic and hyperbolic localization. The resulting outlier detector has lower complexity, is less restrictive to apply and can perform better than comparative methods [21], [22], [23], [24], [25].

We next propose a new estimator for the common scenario of having more measurements than unknowns. The proposed estimator partitions the measurements, generates the individual critically determined solutions and combines them together using the Best Linear Unbiased Estimator (BLUE) to form the final. The estimator is algebraic and in closed-form, does not approximate the measurement equation and performs better than the existing closed-form solutions. Theoretical analysis supports the proposed method in achieving the CRLB performance under Gaussian noise, before the thresholding effect [37] sets in. A partitioning scheme is also developed to increase the noise tolerance of the final solution, based on the volume of the η-confidence ellipsoid.

The proposed estimator shares some similarity with the divide and conquer framework by Abel [38]. The theory from [38] requires all measurements be independent and the total of them be an integer multiple of the minimum number of measurements in order to achieve the optimum performance. The proposed estimator does not have these limitations and reaches the CRLB performance for Gaussian noise as supported by analysis and simulations. Furthermore, we develop a measurement partitioning scheme to improve performance in the high noise region, which is new and has not been considered before in the literature.

The paper uses elliptic time delay measurements [34] for presentation. All the developments and results apply to hyperbolic TDOA measurements as well, with the changes indicated wherever necessary. Both 2-D and 3-D localizations are included.

The contributions of the paper include

• The minimum measurement solutions for elliptic and hyperbolic localizations that are algebraic, closed-form, robust and require the roots of a quadratic equation only;

• The intersection conditions for two LOPs in 2-D and in 3-D, and their application for outlier detection using the spectral graph technique;

• An estimator based on BLUE to combine individual minimum measurement solutions for the more common scenario of having more measurements than unknowns (over-determined situation);

• Performance analysis to show that the estimator by combining individual minimum measurement solutions is able to reach the CRLB performance under Gaussian noise;

• A scheme using the η-confidence ellipsoid to select the individual measurement solutions to combine that provides higher noise tolerance.

We first introduce the localization scenario in Section 2, and derive the new critically determined algebraic solutions and deduce the conditions for LOP intersection in Section 3. Section 4 deduces the outlier measurement detection method and evaluates the performance. Section 5 devises the positioning estimator in the overdetermined situation from the minimum measurement solutions, conducts analysis and proposes a measurement partitioning scheme to improve performance at high noise level. Section 6 presents simulations and Section 7 gives the conclusion.

We shall use the common notations that bold lower and upper case letters represent column vectors and matrices. a(i) and a(i, j) are the i-th and the (i, j)-th elements of a and A. a(i: j) is a subvector containing the i-th to the j-th element of a, and A(i: j, k: l) is a submatrix constructed from the elements of A in rows i to j and columns k to l. I_N is an identity matrix of size N, the subscript may be omitted if the size is clear from the context. 0 represents a matrix of zero that may not necessarily be square. diag{ (•), ⋅⋅⋅, (*)} is a block diagonal matrix with diagonal blocks (•), ⋅⋅⋅, (*). det(•) and trace(•) are the determinant and trace operations and ‖•‖ is the Euclidean norm. (•)† is the pseudo-inverse of the matrix (•). (•)^o denotes the true value of (•) and $\Delta (•)=(•)-{(•)}^o$. ⌈a⌉ is the ceiling operation for the scalar a and ⌊a⌋ the floor operation. a! is the factorial of the positive integer a.