Improved Time of Arrival measurement model for non‐convex optimization

Sidorenko, Juri; Schatz, Volker; Doktorski, Leo R. Ya.; Scherer‐Negenborn, Norbert; Arens, Michael; Hugentobler, Urs

doi:10.1002/navi.277

Cited by 3 publications

(10 citation statements)

References 32 publications

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“…In [20] we have proven that the improved objective function two F 2 has a saddle point at the local minimum of objective function two F 2 . In test scenraios with no or small noise the improved objective function onw F L1 never converges into a local minimum.…”

Section: Discussionmentioning

confidence: 86%

“…The final proof of the hypothesis was provided with the help of the Cauchy-Bunyakovsky-Schwarz inequality [7]. Alternatively, the equation 29 in [20] can also be obtained from the variance V ar(…”

Section: Reason For the Approachmentioning

confidence: 99%

“…We assume that the proven hypothesis [20] for the improved objective function two apply as well for the improved objective function one eq. (5).…”

Section: Local Optimizationmentioning

confidence: 99%

“…In our method, the non-convex problem remains non-convex. In the publication [20] it was shown that this approach, reduces the risk of convergence to a local minimum for measurements without noise. This elaboration is more focused on the effect of noise on our approach.…”

Section: Introductionmentioning

confidence: 99%

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Improved Time of Arrival Measurement Model for Non-Convex Optimization with Noisy Data

Sidorenko

Schatz

Scherer‐Negenborn

et al. 2018

2018 International Conference on Indoor Positioning and Indoor Navigation (IPIN)

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The quadratic system provided by the Time of Arrival technique can be solved analytical or by optimization algorithms. In real environments the measurements are always corrupted by noise. This measurement noise effects the analytical solution more than non-linear optimization algorithms. On the other hand it is also true that local optimization tends to find the local minimum, instead of the global minimum. This article presents an approach how this risk can be significantly reduced in noisy environments. The main idea of our approach is to transform the local minimum to a saddle point, by increasing the number of dimensions.

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Section: Discussionmentioning

confidence: 86%

Section: Reason For the Approachmentioning

confidence: 99%

“…We assume that the proven hypothesis [20] for the improved objective function two apply as well for the improved objective function one eq. (5).…”

Section: Local Optimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Time of Arrival Measurement Model for Non-Convex Optimization with Noisy Data

Sidorenko

Schatz

Scherer‐Negenborn

et al. 2018

2018 International Conference on Indoor Positioning and Indoor Navigation (IPIN)

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“…In [17], we demonstrated that if the base station positions are known and only the tag positions have to be estimated, an additional dimension in the l 2 norm of the TOA objective function transforms the local minimum to a saddle point. This fact has been proven analytically for the squared objective function and empirically for the general TOA objective function through more than 10,000 test scenarios with different constellations and initial estimates.…”

Section: Tdoa Localizationmentioning

confidence: 99%

Self-Calibration for the Time Difference of Arrival Positioning

Sidorenko

Schatz

Bulatov

et al. 2020

Sensors

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The time-difference-of-arrival (TDOA) self-calibration is an important topic for many applications, such as indoor navigation. One of the most common methods is to perform nonlinear optimization. Unfortunately, optimization often gets stuck in a local minimum. Here, we propose a method of dimension lifting by adding an additional variable into the l 2 norm of the objective function. Next to the usual numerical optimization, a partially-analytical method is suggested, which overdetermines the system of equations proportionally to the number of measurements. The effect of dimension lifting on the TDOA self-calibration is verified by experiments with synthetic and real measurements. In both cases, self-calibration is performed for two very common and often combined localization systems, the DecaWave Ultra-Wideband (UWB) and the Abatec Local Position Measurement (LPM) system. The results show that our approach significantly reduces the risk of becoming trapped in a local minimum.2 of 15 common and often combined systems are the DecaWave Ultra-Wideband (UWB) and the Abatec Local Position Measurement (LPM) system. The DecaWave UWB system is less affected by reflections, but is, due to regulations, limited in its transmitting power. The LPM system faces the opposite problem. However, here, the problem of self-localization becomes problematic, where neither the positions of the transmitters nor the receivers are known. This is analogous to the microphone-speaker problem, where systems of (sometimes redundant or self-contradicting) quadratic equations must be solved, sometimes resulting in zero and sometimes resulting in dozens of solutions for the minimum cases (see Chapter 10 in [2]). In the few related works outlined below, it is therefore commonplace to find overdetermined systems of equations. From a reliable solution and a suitable optimizer, these systems are developed to converge to the global minimum and thus successfully self-localize the sensors.The notations used in the text and in the equations are shown in Tables 1 and 2.

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Self-Calibration for the Time-of-Arrival Positioning

et al. 2020

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Self-calibration of time-of-arrival positioning systems is made difficult by the non-linearity of the relevant set of equations. This work applies dimension lifting to this problem. The objective function is extended by an additional dimension to allow the dynamics of the optimization to avoid local minima. Next to the usual numerical optimization, a partially analytical method is suggested, which makes the system of equations overdetermined proportionally to the number of measurements. Results with the lifted objective function are compared to those with the unmodified objective function. For evaluation purposes, the fractions of convergence to local minima are determined, for both synthetic data with random geometrical constellations and real measurements with a reasonable constellation of base stations. It is shown that the lifted objective function provides improved convergence in all cases, often significantly so. INDEX TERMS Dimension lifting, self-calibration, time-of-arrival (TOA).

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Improved Time of Arrival measurement model for non‐convex optimization

Cited by 3 publications

References 32 publications

Improved Time of Arrival Measurement Model for Non-Convex Optimization with Noisy Data

Improved Time of Arrival Measurement Model for Non-Convex Optimization with Noisy Data

Self-Calibration for the Time Difference of Arrival Positioning

Self-Calibration for the Time-of-Arrival Positioning

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