Solves a formulation of n-D space trilateration problem using a nonlinear least squares optimizer.

Input: positions, distances
Output: centroid with geometry and error

Uses Levenberg-Marquardt algorithm from Apache Commons Math.

double[][] positions = new double[][] { { 5.0, -6.0 }, { 13.0, -15.0 }, { 21.0, -3.0 }, { 12.4, -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[] centroid = optimum.getPoint().toArray();

// error and geometry information; may throw SingularMatrixException depending the threshold argument provided
RealVector standardDeviation = optimum.getSigma(0);
RealMatrix covarianceMatrix = optimum.getCovariances(0);

The multilateration problem can be formulated as an optimization problem and solved using Non-linear least squares methods. A well-formed solution will be an ellipse in R², or an ellipsoid in R³. If you are only interested in a maximum likelihood point estimate, the centroid is also provided. 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.

To add a dependency on Trilateration using Maven, use the following:

<dependency>
    <groupId>com.lemmingapex.trilateration</groupId>
    <artifactId>trilateration</artifactId>
    <version>1.0.2</version>
</dependency>

To add a dependency using Gradle:

dependencies {
  implementation 'com.lemmingapex.trilateration:trilateration:1.0.2'
}

./gradlew clean
./gradlew test -i

./gradlew.bat clean
./gradlew.bat test -i