Improved Linear Direct Solution for Asynchronous Radio Network Localization (RNL)

Abstract

In the field of localization the linear least square solution is frequently used. This solution is compared to nonlinear solvers more effected by noise, but able to provide a position estimation without the knowledge of any starting condition. The linear least square solution is able to minimize Gaussian noise by solving an overdetermined equation with the Moore-Penrose pseudoinverse. Unfortunately this solution fails if it comes to non Gaussian noise. This publication presents a direct solution which is able to use pre-filtered data for the LPM (RNL) equation. The used input for the linear position estimation will not be the raw data but over the time filtered data, for this reason this solution will be called direct solution. It will be shown that the presented symmetrical direct solution is superior to non symmetrical direct solution and especially to the not pre-filtered linear least square solution.

Improved Time of Arrival measurement model for non‐convex optimization

“…34 Alternatively, the nonlinear system can be transformed into a linear system. 9,10 With the assumptions made in Section 2.1, it is possible to obtain a linear system. With regard to future extensions to determining the base station positions as well as the location of the object T, this article focuses on finding a solution with a non-convex optimization algorithm.…”

“…With known reference station positions, it is possible to transform the quadratic to a linear system. [9][10][11] This linear system can be used to provide an initial estimate. On the other hand, the linear system is more affected by noise, compared with the quadratic system.…”

“…On the other hand, the linear system is more affected by noise, compared with the quadratic system. 9,10 In general, local minima are not an issue in applications with known reference station positions such as GNSS. This changes if it is necessary to obtain the locations of the reference stations without additional measuring equipment, which is also known as self-calibration.…”

Self-Calibration for the Time Difference of Arrival Positioning

“…The ratio Ra linearly increased with M. In contrast to the UWB, it was necessary to calculate the offset analytically for the LPM, as well. The linear estimation that we used can be found in [18]. This linear solution was expanded by the ability to operate with the additional dimensionsλ i , λ j andλ.…”

“…The important disadvantage of the LPM equation was that all data were strongly affected by the offset O j , which changed from one measurement to the next. In [18], we showed that subtracting one measurement from another eliminated the offset and had some significant advantages for data filtering. In contrast to the UWB, it was also necessary to calculate the offset analytically for the LPM.…”