The process of positioning, using only distances from control stations, is called trilateration (or multilateration if the problem is over-determined). The observation equation is Pythagoras’s ...formula, in terms of the summed squares of coordinate differences and, thus, is nonlinear. There is one observation equation for each control station, at a minimum, which produces a system of simultaneous equations to solve. Over-determined nonlinear systems of simultaneous equations are typically solved using iterative least squares after forming the system as a truncated Taylor’s series, omitting the nonlinear terms. This paper provides a linearization of the observation equation that is not a truncated infinite series—it is exact—and, thus, is solved exactly, with full rigor, without iteration and, thus, without the need of first providing approximate coordinates to seed the iteration. However, there is a cost of requiring an additional observation beyond that required by the non-linear approach. The examples and terminology come from terrestrial land surveying, but the method is fully general: it works for, say, radio beacon positioning, as well. The approach can use slope distances directly, which avoids the possible errors introduced by atmospheric refraction into the zenith-angle observations needed to provide horizontal distances. The formulas are derived for two- and three-dimensional cases and illustrated with an example using total-station and global navigation satellite system (GNSS) data.
Abstract We have validated the performance of a prototype coordinate measurement system based on multilateration by comparing it to a laser tracker, i.e. a well-proven instrument widely used in the ...industry. After establishing the uncertainty budget of the different systems, we performed position measurements with both instruments on common targets. Using the estimated uncertainties associated with the measurements, we found that the multilateration system provided lower position uncertainties than the laser tracker: on average 18 µm versus 33 µm for distances up to 12 m. The uncertainties represented by confidence ellipsoids are compatible between the two systems: for confidence regions of 95% probability, they overlap as expected, i.e. in 94% of the cases. We also measured the length of a 0.8 m long reference scale bar with the multilateration system at an error of only 2 µm. This cross-comparison is a new and key step in the characterization of this SI-traceable multilateration system.
•We localize occupants using footstep-induced structural vibrations.•We mitigate dispersion-induced signal distortions through signal decomposition.•We locate footsteps through adaptive ...multilateration in heterogeneous floor structures.•The results show an 0.34 m localization error (6X improvements from baseline).
In this paper, we present an occupant localization approach through sensing footstep-induced floor vibrations. Occupant location information is an important part of many smart building applications, such as energy and space management in a personal home or patient tracking in a hospital room. Adoption of current occupant location sensing approaches in smart buildings (e.g., camera, radio frequency (RF), mobile devices, etc.) is often limited due to the maintenance, installment, and calibration requirements of these sensing systems. To overcome these limitations, we introduce a new approach to use footstep-induced structural vibration for step-level occupant localization. The intuition behind this approach is that footsteps induce floor vibrations which are received in different vibration sensor locations at different times. This paper focuses on localizing a single occupant within each sensing range. To localize the footsteps, we utilize the time differences of arrival (TDoA) of the footstep-induced vibrations. However, this approach involves two main challenges: (1) the vibration wave propagation in the floor is of dispersive nature (i.e., different frequency components travel at different velocities) and (2) due to floor heterogeneity, these wave propagation velocities vary in different structures as well as in different locations in a structure. These issues lead to large localization inaccuracies or calibration requirements. To address dispersion challenge, we present a decomposition-based dispersion mitigation technique which extracts specific components (which have similar propagation characteristics) for localization. To address velocity variations, we introduce an adaptive multilateration approach that employs heuristics to constrain the search space and robustly locate the footsteps when the propagation velocity is unknown. Constraining the search space overcomes the additional complexity which is resulted from adding an unknown variable (propagation velocity). We evaluated our approach using the field experiments in 3 different types of buildings (both commercial and residential) with human participants. The results show an average localization error of 0.34 m, which corresponds to a 6X reduction in error compared to a baseline method. Furthermore, our approach resulted in standard deviation of as low as 0.18 m, which corresponds to a 8.7X improvement in precision compared to the baseline approach.
This paper presents the optimal estimation for the extensively concerned linear multilateral positioning issues and further improves the accuracy of range-only tracking significantly. The traditional ...linear multilateration method has been used to achieve target positioning for range-only measurements, but the completeness of the theory and implementation effect are limited due to the imperfect model and the lack of proper statistical error analysis. Moreover, the performance of the representative nonlinear filters is not satisfactory because of the strong nonlinearity between the range observations and kinematic states of the moving target. This paper reveals the essence of the linear fusion of multiple distance sensors from a geometric perspective. The multi-sensor fusion model is reconstructed in the linear feature space by transforming the likelihood function into the probability function and separating the expectation of the compound random variable and residual. After that, a new linear multilateration method is developed based on the precise derivation of statistical characteristics corresponding to the linear fusion model, and also minimizing the squared Mahalanobis distance is introduced to replace minimizing the mean square error. The first-order moment, as a pseudo-measurement used to improve the positioning accuracy, and the second-order moment, as the covariance of the pseudo-measurement noise, are provided by this method, which has been proved to be best linear unbiased according to the Gauss-Markov theorem. Combined with the pseudo-measurement, a standard linear Kalman filter is capable of tracking the maneuvering target. Several simulations and experiments verify the superior performance of the proposed method.
In this article we analyze the state-of-the-art in multilateration - the family of localization methods enabled by the range difference observations. These methods are computationally efficient, ...signal-independent, and flexible with regards to the number of sensing nodes and their spatial arrangement. However, the multilateration problem does not admit a closed-form solution in the general case, and the localization performance is conditioned on the accuracy of range difference estimates. For that reason, we consider a simplified use case where multiple distributed microphones capture the signal coming from a near field sound source, and discuss their robustness to the estimation errors. In addition to surveying the relevant bibliography, we present the results of a small-scale benchmark of few ""mainstream"" multilateration algorithms, based on an in-house Room Impulse Response dataset.
Thermal effects on an uncontrolled manufacturing environment are the main barrier for accurate large machine tools. Internal and external heat sources combined with different expansion coefficients ...result in a constant thermal drift of the machine's structural loop. Thus, a characterisation method remains a challenge. This work presents a new methodology for the uncertainty assessment of a Machine Tool Integrated Inverse Multilateration approach where the ambient temperature variation is demonstrated to be a major uncertainty contributor. An “a priori” Monte-Carlo simulation-based research allows developing an appropriate measurement strategy for the use of the proposed approach minimising the influence of thermal issues.
We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, vertices in Levenshtein graphs may be strings ...(i.e., words or sequences of characters in a reference alphabet) of possibly different lengths. We study various properties of these graphs, including a necessary and sufficient condition for their shortest path distance to be identical to the edit distance, and characterize their automorphism group and determining number. We also bound the metric dimension (i.e. minimum resolving set size) of Levenshtein graphs. Regarding the latter, recall that a run is a string composed of identical characters. We construct a resolving set of two-run strings and an algorithm that computes the edit distance between a string of length k and any single-run or two-run string in O(k) operations.
Ultra-wideband (UWB) is considered the most promising radio technology for high-accuracy indoor localization because of its many desirable properties, including a sub-decimeter level ranging accuracy ...under line-of-sight (LOS) conditions, resilience to multipath fading, and low duty cycles. However, the accuracy of UWB localization deteriorates significantly in complex indoor environments due to the presence of non-light-of-sight (NLOS) propagation that may introduce a considerable positive bias in range measurements. In this paper, we present a localization method that improves the accuracy of UWB localization in mixed LOS/NLOS indoor environments by using multiple localization algorithms optimized for different localization scenarios distinguished by the number of LOS-measured distances. The method adopts a fingerprinting-based algorithm to obtain location results under NLOS-only conditions and uses the conventional multilateration algorithm when at least three LOS-measured distances are available. Additionally, the algorithm set includes two novel hybrid localization algorithms for scenarios with one or two LOS distances. These algorithms use the LOS-measured distances to limit geometrically possible locations and then employ fingerprinting to perform the final location selection. We test our approach in a realistic indoor environment over numerous experimental scenarios. The experimental results show that the proposed localization strategy reduces the mean distance error by 3 to 20 cm compared with the traditional fingerprinting-based approach.
High-precision pre-alignment of the magnet components is an important step in the construction and operation of the High Energy Photon Source (HEPS). In order to achieve 10μm pre-alignment accuracy ...of storage ring in transverse and vertical, four laser trackers were used for set up a four-station multilateration measurement system. A Rank-defect Free-network adjustment model was used to analysis the measurement accuracy of different arrangements of multilateration measurement system. By simulation, it shows when the four stations form a right-angled regular triangular pyramid the highest point measurement accuracy can be gotten. A multilateration measurement system formed by four laser tracker was built in the laboratory. Experiment results show that the absolute position measurement accuracy is within 7.1μm and the relative displacement measurement accuracy is better than 3μm in a 4 m × 1.2 m × 1.5 m volume, which can satisfy the real-time position feedback accuracy of the magnets in the process of ultra-high-precision pre-alignment. Then, a pre-alignment experiment for High Energy Photon Source based on this system was carried out, and finally the 10μm pre-alignment accuracy goal in transverse and vertical was achieved.
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