Kalman Filter Noise Covariance Calculator
ANA›Life Services Authority›National Calculator Authority›Kalman Filter Noise Covariance Calculator
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Kalman Filter Noise Covariance Calculator
Calculate the process noise covariance matrix (Q) and measurement noise covariance matrix (R) for a Kalman filter, along with the steady-state Kalman gain (K).
Process Noise Std Dev σ_a (acceleration noise, m/s²)
Measurement Noise Std Dev σ_z (position noise, m)
Time Step Δt (seconds)
Motion Model
Constant Velocity (2-state: position, velocity) Constant Acceleration (3-state: position, velocity, acceleration)
Calculate ...
function kalCalc() { const sigmaA = parseFloat(document.getElementById('kal-sigma-a').value); const sigmaZ = parseFloat(document.getElementById('kal-sigma-z').value); const dt = parseFloat(document.getElementById('kal-dt').value); const model = document.getElementById('kal-model').value; const res = document.getElementById('kal-result');
if (isNaN(sigmaA) || sigmaA Process noise σ_a must be a positive number.'; return; } if (isNaN(sigmaZ) || sigmaZ Measurement noise σ_z must be a positive number.'; return; } if (isNaN(dt) || dt Time step Δt must be a positive number.'; return; }
const sa2 = sigmaA * sigmaA; const sz2 = sigmaZ * sigmaZ; const dt2 = dt * dt; const dt3 = dt2 * dt; const dt4 = dt3 * dt;
let Q, R, H, label, stateLabels;
// Measurement noise covariance (scalar for 1D position measurement) R = sz2;
if (model === 'cv') { // Constant Velocity model // State: [position, velocity] // Q = sigma_a^2 * G * G^T where G = [dt^2/2, dt]^T // Q = sigma_a^2 * [[dt^4/4, dt^3/2], [dt^3/2, dt^2]] Q = [ [sa2 * dt4 / 4, sa2 * dt3 / 2], [sa2 * dt3 / 2, sa2 * dt2 ] ]; H = [[1, 0]]; // observation matrix (measure position only) stateLabels = ['pos', 'vel']; label = '2×2 (Constant Velocity)'; } else { // Constant Acceleration model // State: [position, velocity, acceleration] // G = [dt^2/2, dt, 1]^T // Q = sigma_a^2 * G * G^T const dt5 = dt4 * dt; const dt6 = dt5 * dt; Q = [ [sa2 * dt4 / 4, sa2 * dt3 / 2, sa2 * dt2 / 2], [sa2 * dt3 / 2, sa2 * dt2, sa2 * dt ], [sa2 * dt2 / 2, sa2 * dt, sa2 ] ]; H = [[1, 0, 0]]; stateLabels = ['pos', 'vel', 'acc']; label = '3×3 (Constant Acceleration)'; }
// Steady-state Kalman gain via discrete algebraic Riccati equation (DARE) // Solved iteratively: P_{k+1} = FPF^T + Q, K = PH^T(HPH^T + R)^-1, P = (I-KH)P // F matrix let n = stateLabels.length; let F = []; for (let i = 0; i Array(c).fill(0)); for(let i=0;iA.map(r=>r[j])); } function matAdd(A,B) { return A.map((r,i)=>r.map((v,j)=>v+B[i][j])); } function matScale(A,s) { return A.map(r=>r.map(v=>v*s)); } function eye(n) { return Array.from({length:n},(,i)=>Array.from({length:n},(,j)=>i===j?1:0)); }
// Iterate DARE let P = matScale(Q, 1); for (let iter = 0; iter s+vH[0][j],0); // scalar const S = HPHt + R; // scalar innovation covariance const PHt = matMul(P, matT(H)); // n×1 const K = PHt.map(r=>[r[0]/S]); // n×1 // P = (I - K H) P const KH = matMul(K, H); // n×n const IKH = matAdd(eye(n), matScale(KH,-1)); const Pupd = matMul(IKH, P); // P = F Pupd F^T + Q const FP = matMul(F, Pupd); const FPFt= matMul(FP, matT(F)); const Pnew= matAdd(FPFt, Q); // Check convergence let diff=0; for(let i=0;is+vH[0][j],0); const S = HPHt + R; const PHt = matMul(P, matT(H)); const Kss = PHt.map(r=>r[0]/S);
// Format matrix as HTML table function fmtMatrix(M, rowLabels, colLabels, title) { let html = `${title}
;
html += '';
colLabels.forEach(c=>{ html+=${c}; });
html += '';
M.forEach((row,i)=>{
html +=${rowLabels[i]};
row.forEach(v=>{ html+=${v.toExponential(4)}`; });
html += '';
});
html += '';
return html;
}
let html = **Results — ${label} Model**;
html += `Inputs: σ_a = ${sigmaA} m/s², σ_z = ${sigmaZ} m, Δt = ${dt} s
;
html += fmtMatrix(Q, stateLabels, stateLabels, 'Process Noise Covariance Q');
html +=Measurement Noise Covariance R (scalar): ${R.toExponential(4)} m²
;
html += fmtMatrix(P, stateLabels, stateLabels, 'Steady-State Error Covariance P (DARE solution)');
html +=Steady-State Kalman Gain K:
;
Kss.forEach((k,i)=>{ html+=``; });
html += ``;
html +=Innovation Covariance S = H·P·Hᵀ + R = ${S.toExponential(4)} m²
;
html +=Signal-to-Noise Ratio (σ_a / σ_z) = ${(sigmaA/sigmaZ).toFixed(4)}
`;
res.innerHTML = html; }
#### Formulas
Process Noise Covariance Q (discrete-time, piecewise constant white acceleration):
Q = σ_a² · Γ · Γᵀ
Constant Velocity: Γ = [Δt²/2, Δt]ᵀ → Q = σ_a² · [[Δt⁴/4, Δt³/2], [Δt³/2, Δt²]]
Constant Acceleration: Γ = [Δt²/2, Δt, 1]ᵀ → Q = σ_a² · [[Δt⁴/4, Δt³/2, Δt²/2], [Δt³/2, Δt², Δt], [Δt²/2, Δt, 1]]
Measurement Noise Covariance R: R = σ_z²
Steady-State Kalman Gain (via Discrete Algebraic Riccati Equation — DARE):
P = F·P·Fᵀ + Q (predict)
K = P·Hᵀ·(H·P·Hᵀ + R)⁻¹ (gain)
P = (I − K·H)·P (update) — iterated until convergence
H = [1, 0, ...] (position-only measurement)
#### Assumptions & References
- Reference: Grewal & Andrews, Kalman Filtering: Theory and Practice, 4th ed., Wiley, 2015.
- Reference: Bar-Shalom, Li & Kirubarajan, Estimation with Applications to Tracking and Navigation, Wiley, 2001.
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