Triangulate example for iBeacons

Question

I am looking into the possibility to use multiple iBeacons to do a \'rough\' indoor position location. The application is a kind of \'museum\' setting, and it would be easie

Answer 1

Here is an open source java library that will perform the trilateration/multilateration: https://github.com/lemmingapex/Trilateration

It uses a popular nonlinear least squares optimizer, the Levenberg-Marquardt algorithm, from Apache Commons Math.

double[][] positions = new double[][] { { 5.0, -6.0 }, { 13.0, -15.0 }, { 21.0, -3.0 }, { 12.42, -21.2 } };
double[] distances = new double[] { 8.06, 13.97, 23.32, 15.31 };

NonLinearLeastSquaresSolver solver = new NonLinearLeastSquaresSolver(new TrilaterationFunction(positions, distances), new LevenbergMarquardtOptimizer());
Optimum optimum = solver.solve();

// the answer
double[] calculatedPosition = optimum.getPoint().toArray();

// error and geometry information
RealVector standardDeviation = optimum.getSigma(0);
RealMatrix covarianceMatrix = optimum.getCovariances(0);

Most scholarly examples, like the one on wikipedia, deal with exactly three circles and assume perfectly accurate information. These circumstances allow for much simpler problem formulations with exact answers, and are usually not satisfactory for practical situations.

The problem in R² or R³ euclidean space with distances that contain measurement error, an area (ellipse) or volume (ellipsoid) of interest is usually obtained instead of a point. If a point estimate is desired instead of a region, the area centroid or volume centroid should be used. R² space requires at least 3 non-degenerate points and distances to obtain a unique region; and similarly R³ space requires at least 4 non-degenerate points and distances to obtain a unique region.