GPS + INS Position Fusion Accuracy Estimator
Estimates the fused (Kalman-filtered) horizontal position accuracy when combining GPS and an Inertial Navigation System (INS). Enter the individual 1-sigma position errors and the INS drift parameters to compute the optimal fused accuracy over a given time interval.
Formulas Used
1. INS Propagated Variance at time t:
σ²INS(t) = σ²INS,0 + ṡ² · t² + q · t
where ṡ is the INS velocity drift rate (m/s) and q is the process noise spectral density (m²/s³).
2. Kalman-Filter Fused Variance (N GPS updates):
1/σ²fused = 1/σ²INS(t) + N/σ²GPS
⟹ σ²fused = (σ²INS(t) · σ²GPS) / (σ²GPS + N · σ²INS(t))
3. Kalman Gain:
K = σ²INS(t) / (σ²INS(t) + σ²GPS)
4. Horizontal Accuracy Metrics (circular Gaussian):
CEP ≈ 1.1774 · σfused (50% probability)
2DRMS ≈ 2.4477 · σfused (95% probability)
Assumptions & References
- The INS position error grows as a combination of initial uncertainty, velocity-error random walk (σ²∝t²), and process noise (σ²∝t), consistent with a first-order Gauss–Markov model.
- GPS measurements are modelled as independent, zero-mean Gaussian noise with constant variance σ²GPS (AWGN assumption).
- The Kalman fusion formula 1/Pfused = 1/Pprior + N/R is the information-form update for N sequential, independent scalar measurements.
- CEP and 2DRMS conversion factors assume equal horizontal error variances in both axes (circular error distribution).
- No lever-arm, attitude, or correlated error effects are modelled; this is a simplified scalar (1-axis) covariance analysis.
- References: Groves, P.D. (2013). Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed. Artech House. | Farrell, J.A. (2008). Aided Navigation: GPS with High Rate Sensors. McGraw-Hill.