RSSI-based position estimators

Every access point radiates its signal equally in all directions, and that signal weakens with distance in a predictable way. If a device knows an access point’s position and its transmitted power, and it measures how strong that signal is when it arrives, it can work out how far away it must be — the strength lost in transit is a function of distance alone. That gives one estimated distance per access point but, on its own, only narrows the device down to a circle (2D) or sphere (3D) around it, not a single point.

The trick is to do this for several access points at once. Each one narrows the possibilities down to its own circle; the position consistent with all of them is where those circles overlap — the same multilateration principle used by GPS. In practice, RSSI readings are noisy, so the circles rarely meet at one exact point: the estimator instead looks for the position with the smallest overall inconsistency across every reading, which is exactly what the linear, non-linear, and robust estimators below solve for, with increasing tolerance for noisy or wrong readings.

Given a measured received power (dBm), a known transmitted power (dBm), a path-loss exponent , and signal frequency , the received-power model is:

where the frequency-dependent constant is:

( is the speed of light). Solving for the distance gives the conversion every RSSI estimator applies before running lateration:

Typical path-loss exponent values: free space , urban – , suburban – , indoor line-of-sight – . The non-linear and robust estimators additionally propagate the variance of , , and into a distance standard deviation via the first-order (delta method) derivative:

This conversion is implemented once, in Utils/PositionEstimatorHelper, and reused by every class below — everything past this point is generic trilateration from (known position, estimated distance) pairs.

3D robust classes mirror the 2D ones (RANSACRobustRssiPositionEstimator3D, LMedSRobustRssiPositionEstimator3D, etc.), omitted above for brevity. 2D estimators require a minimum of 3 sources; 3D estimators require 4 — one extra source acts as the reference point that lets the trilateration equations be linearized.

The linear and non-linear indoor classes are thin wrappers that convert readings to (position, distance, distance standard deviation) triples and then delegate to the generic lateration solvers in the sibling com.irurueta.navigation.lateration package (irurueta-navigation module); the robust classes delegate similarly to that package’s RobustLaterationSolver hierarchy.

LinearRssiPositionEstimator picks one of two closed-form solvers (useHomogeneousLinearSolver, default true) that both eliminate the path-loss model’s non-linearity by taking pairwise differences of the sphere/circle equations before solving.

NonLinearRssiPositionEstimator minimizes, via Levenberg-Marquardt (LevenbergMarquardtMultiDimensionFitter), the squared-distance residual between a candidate position and each known source position , fit against the squared estimated distance :

The initial position defaults to the centroid of the known source positions unless initialPosition is set explicitly. Distance readings are weighted by their standard deviation (propagated from the path-loss model above, or a fallback of ), and the fitter additionally reports a position covariance and chi-squared value that the linear solvers cannot provide.

RobustRssiPositionEstimator adds outlier tolerance on top of the linear/non-linear solvers, following the same sample-and-score pattern for every variant:

The residual is always the trilateration distance error \(\left|\lVert \mathbf{p} - \mathbf{p}_i \rVert - d_i\right|\); what differs between variants is how candidates are scored and, for two of them, how samples are prioritized:

If isResultRefined (default true), the winning candidate is always re-solved with the non-linear solver restricted to its inlier set, so every robust RSSI estimator ends up performing one final Levenberg-Marquardt pass regardless of variant. The default robust method across the library is PROMedS.

Static create(…​) factories on RobustRssiPositionEstimator2D/3D build any robust variant from a RobustEstimatorMethod enum value.

Main classes used on this page, linked to their source code and Javadoc: