GPS + INS Position Fusion Accuracy Estimator
ANA›Life Services Authority›National Calculator Authority›GPS + INS Position Fusion Accuracy Estimator
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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.
GPS Horizontal Position Accuracy, σ_GPS (m, 1-sigma)
INS Initial Position Accuracy, σ_INS0 (m, 1-sigma)
INS Drift Rate (m/s, 1-sigma velocity error)
Integration Time, t (seconds)
GPS Update Rate (Hz)
INS Process Noise Spectral Density, q (m²/s³)
Calculate
Results will appear here.
function gpsCalc() { // --- Read inputs --- const sigmaGPS = parseFloat(document.getElementById('gps_gps_sigma').value); const sigmaINS0 = parseFloat(document.getElementById('gps_ins_sigma0').value); const driftRate = parseFloat(document.getElementById('gps_ins_drift').value); const t = parseFloat(document.getElementById('gps_time').value); const updateRate = parseFloat(document.getElementById('gps_gps_update_rate').value); const q = parseFloat(document.getElementById('gps_process_noise').value);
// --- Validation --- const errors = []; if (isNaN(sigmaGPS) || sigmaGPS 0."); if (isNaN(sigmaINS0) || sigmaINS0 0."); if (isNaN(q) || q 0) { document.getElementById('gps_result').innerHTML = 'Input Error:' + errors.join('') + ''; return; }
// ----------------------------------------------------------------------- // CORE FORMULAS // ----------------------------------------------------------------------- // 1. INS propagated variance at time t (horizontal, 1-axis): // σ²_INS(t) = σ²_INS0 + (drift_rate)² · t² + q · t // (random-walk position error from velocity error + process noise growth) const varINS_t = sigmaINS0 * sigmaINS0 + driftRate * driftRate * t * t + q * t; const sigmaINS_t = Math.sqrt(varINS_t);
// 2. GPS measurement variance (assumed stationary, white noise) const varGPS = sigmaGPS * sigmaGPS;
// 3. Number of GPS updates in time t const N = Math.max(1, Math.round(updateRate * t));
// 4. Kalman-filter fused variance after N GPS updates: // Each update: P_fused = (P_prior * R) / (P_prior + R) // where P_prior = varINS_t (propagated INS covariance) // R = varGPS (GPS measurement noise) // After N independent updates (sequential fusion): // 1/P_fused = 1/P_prior + N/R // => P_fused = (P_prior * R) / (R + N * P_prior) const varFused = (varINS_t * varGPS) / (varGPS + N * varINS_t); const sigmaFused = Math.sqrt(varFused);
// 5. Kalman gain at final update const K = varINS_t / (varINS_t + varGPS);
// 6. 2D (horizontal) CEP (Circular Error Probable) ≈ 1.1774 * σ_fused // (valid for equal-variance circular Gaussian) const CEP = 1.1774 * sigmaFused;
// 7. 95% horizontal accuracy (2DRMS) ≈ 2.4477 * σ_fused const twoSigma95 = 2.4477 * sigmaFused;
// 8. Improvement ratio const improvement = ((sigmaGPS - sigmaFused) / sigmaGPS * 100);
// ----------------------------------------------------------------------- // Format output // ----------------------------------------------------------------------- const fmt = (v, d=4) => isFinite(v) ? v.toFixed(d) : "N/A";
document.getElementById('gps_result').innerHTML = ` ### Fusion Results
Parameter Value
GPS 1-sigma accuracy ${fmt(sigmaGPS,3)} m
INS propagated 1-sigma at t = ${fmt(t,1)} s ${fmt(sigmaINS_t,3)} m GPS updates used (N) ${N}
Kalman Gain (K) ${fmt(K,4)}
Fused 1-sigma accuracy
${fmt(sigmaFused,3)} m CEP (50% probability circle) ${fmt(CEP,3)} m
2DRMS (95% horizontal) ${fmt(twoSigma95,3)} m Accuracy improvement over GPS-only
${fmt(improvement,1)} %
`; }
#### 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
- 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.
- 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.
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