Method for calibrating a local positioning system based on time-difference-of-arrival measurements
A method for calibrating a time difference of arrival-based local positioning system for k−D localization, k=2 or 3, includes collecting N sets of time difference of arrival measurements related to a mobile node, N≥2, each n th set of measurements being performed by Bn beacon nodes among B beacon nodes of the positioning system while the mobile node is located is a n th position within a region covered by the positioning system, Bn≥k+2, and determining optimal beacon positions that minimize an objective function depending on N residual error vectors, the calculation of each n th position of the mobile node using beacon positions and the n th set of measurements, the calculation allowing the calculation of the n th residual error vector.
1. A method for calibrating a time difference of arrival-based local positioning system for kD localization, k=2 or 3, comprising:
estimating initial beacon positions:
collecting N sets of time difference of arrival measurements related to a mobile node, N≥2, each n th set of measurements being performed by Bn beacon nodes among B beacon nodes of the positioning system while the mobile node is located is a n th position within a region covered by the positioning system, Bn≥k+2;
calculating N residual error vectors through calculation of N positions of the mobile node, by the N sets of time difference of arrival measurements and beacon positions, a calculation of the n th position of the mobile node enabling a calculation of an n th initial residual error vector, the calculation of each n th position of the mobile node using the beacon positions and the n th set of measurement;
determining optimal beacon positions that minimize an objective function depending on the N residual error vectors; and
calibrating the local positioning system based on the optical beacon positions,
wherein the objective function is defined by
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2. The method according to according to claim 1 , further comprising, between the collecting measurements and the determining optimal beacon positions, detecting outliers in the measurements and excluding said outliers from the corresponding sets of measurements.
3. The method according to claim 2 , wherein the step of detecting outliers comprises calculating N initial residual error vectors corresponding to said initial beacon positions, and using a Grubb test on said vectors.
4. The method according to according to claim 1 , wherein the step of determining optimal beacon positions comprises varying assumed beacon positions from the initial positions until the objective function meets a termination criterion.
5. The method according to claim 4 , wherein said criterion is a convergence of the objective function.
6. The method according to claim 4 , wherein if the norm of the residual error vectors corresponding to the initial beacon positions are higher than a threshold, then the method comprises a step of re-estimating initial beacon positions.
7. The method according to according to claim 1 , wherein the N mobile node positions are distributed over the whole region covered by the positioning system.
8. The method according to according to claim 1 , wherein, the determining optimal beacon positions includes calculating the mobile node positions in a coordinate system having a beacon node as point of origin, and the method further comprises a final step of rotating and/or shifting said coordinate system.
9. A method for calibrating a time difference of arrival-based local positioning system for kD localization, k=2 or 3, comprising:
estimating initial beacon positions:
collecting N sets of time difference of arrival measurements related to a mobile node, N≥2, each n th set of measurements being performed by Bn beacon nodes among B beacon nodes of the positioning system while the mobile node is located is a n th position within a region covered by the positioning system, Bn≥k+2;
calculating N residual error vectors through calculation of N positions of the mobile node, by the N sets of time difference of arrival measurements and beacon positions, a calculation of the n th position of the mobile node enabling a calculation of an n th initial residual error vector, the calculation of each n th position of the mobile node using the beacon positions and the n th set of measurement;
determining optimal beacon positions that minimize an objective function depending on the N residual error vectors; and
calibrating the local positioning system based on the optical beacon positions,
wherein the step of determining optimal beacon positions comprises varying assumed beacon positions from the initial positions until the objective function meets a termination criterion, and
wherein the method comprises a step of re-estimating initial beacon positions when the norm of the residual error vectors corresponding to the initial beacon positions are higher than a threshold.