Anchor Localization Using Distance Matrix Factorization
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Solution Overview
Problem
Current localization methods, such as GPS and signal processing with WiFi stations or beacons, face issues with precision and require strong constraints like precise absolute reference positions, limiting their effectiveness in accurately determining the location of mobile sources.
Innovation Solution
A device and method that utilize a memory to receive distance data between anchors and mobile sources, performing singular value decomposition on a matrix of Euclidean distances to calculate a transformation matrix, allowing for precise localization with reduced constraints by minimizing the difference between the product of the simplification matrix and a vector, thereby determining the coordinates of both anchors and mobile sources.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If triangulation methods are used for localization, then the system can be implemented with existing infrastructure, but the measurement precision and localization accuracy remain low
Solution Approach 1:
The patent transforms the localization problem by changing the mathematical parameters and approach. Instead of using traditional triangulation with angle measurements, it employs matrix factorization techniques (SVD, Cholesky decomposition) on distance matrices to directly compute coordinates. This parameter transformation from angular to distance-based matrix operations achieves higher precision while maintaining computational feasibility
Solution Approach 2:
The patent replaces the geometric-trigonometric mechanical approach of triangulation with an algebraic-matrix computational system. By substituting the mechanical triangulation process with matrix factorization operations, the system achieves superior localization accuracy without proportionally increasing system complexity
2Measurement precision
If precise absolute reference positions are required for triangulation, then localization accuracy can be maintained, but the ease of operation and deployment become difficult
Solution Approach 1:
The patent inverts the traditional approach by not requiring known absolute reference positions. Instead of using known anchor positions to locate mobile sources, the method simultaneously determines both anchor and mobile source coordinates through matrix factorization of distance measurements, eliminating the need for precise absolute reference positioning
Solution Approach 2:
The patent performs preliminary matrix decomposition operations (SVD, Cholesky) on the distance matrix before actual localization computation. This preliminary action prepares the data structure in advance, enabling direct coordinate calculation without iterative refinement or requiring pre-known reference positions, thereby simplifying deployment
3Measurement precision
If more anchors and locations are used to improve localization precision, then the measurement accuracy increases, but the device complexity and computational load increase
Solution Approach 1:
The patent segments the large-scale localization problem into manageable matrix operations. By decomposing the distance matrix into smaller factor matrices through SVD or Cholesky decomposition, the system can handle multiple anchors and locations efficiently. The segmentation of computational tasks allows precision to scale with N while controlling computational complexity through structured matrix factorization
Data Source
Figure 1~4
AI summary
A locating device comprises a memory (10) arranged to receive data concerning the distance between a plurality of anchors and one or more mobile sources at a plurality of locations different to each other. The number of anchors and the number of locations is such that one must be higher than four and is designated as the primary number, and the other higher than ten and is designated as the secondary number. The device further comprises a decomposer (4) arranged to calculate a decomposition, into singular values, of a matrix formed by the product of the Euclidean distance matrix derived from the distance data multiplied by two square matrices, one of which has a size equal to the primary number squared and the other is the transpose of a matrix with a size equal to the secondary number squared. The device further comprises a reducer (6) arranged to calculate a resolution vector minimising, according to the least squares method, the difference between the product of a simplification matrix multiplied by the resolution vector and a matrix formed by the product of the Euclidean distance matrix derived from the distance data multiplied by a vector of which all the elements are equal to one except the first element which is zero, and by a vector of which all the elements are zero except the first which is equal to one, and a solver (8) arranged to derive an inversion vector from the last three elements of the resolution vector and an inversion matrix formed from the other elements, in order to determine a transformation matrix such that the inversion matrix is equal to the product of the transformation matrix multiplied by its transpose, and to return a matrix of coordinates of the anchors and a matrix of coordinates of the locations from the transformation matrix, the inversion vector, the first matrix of the decomposition into singular values and the product of the second matrix and the third matrix of the decomposition into singular values.