%% 2. KALMAN FILTER INITIALIZATION % State vector: [Position; Velocity] x_est = [0; 0]; % Initial guess: position 0, velocity 0 P_est = [100, 0; % High uncertainty in initial position 0, 10]; % Lower uncertainty in initial velocity
%% Kalman Filter x_est = [0; 0]; % [pos; vel] P_est = eye(2) * 1;
% State transition with known input (gravity) % x(k+1) = F x(k) + B u(k) F = [1, dt; 0, 1]; B = [0.5*dt^2; dt]; % Control input matrix for acceleration u = g; % Control input (gravity) If you rely solely on the GPS, your
% --- CORRECTION STEP (Using the measurement) --- z = measurements(k); % Current measurement y = z - H * x_pred; % Innovation (measurement residual) S = H * P_pred * H' + R; % Innovation covariance K = P_pred * H' / S; % Kalman Gain
Introduction: The Magic of "Noisy" Measurements Imagine you are trying to track the position of a speeding car using a GPS. Your GPS device updates every second, but the reading is never perfect—it jumps around by a few meters due to atmospheric interference or urban canyons. If you rely solely on the GPS, your tracking line will look jagged and erratic. Kálmán in 1960, the Kalman Filter is a
| Step | Equation Name | Formula (Simplified) | | :--- | :--- | :--- | | Predict | State Estimate | x_pred = F * x_prev | | Predict | Covariance Estimate | P_pred = F * P_prev * F' + Q | | Update | Kalman Gain | K = P_pred * H' / (H * P_pred * H' + R) | | Update | State Estimate (Corrected) | x_est = x_pred + K * (z - H * x_pred) | | Update | Covariance (Corrected) | P_est = (I - K * H) * P_pred |
%% 3. KALMAN FILTER LOOP for k = 1:N % --- PREDICTION STEP --- x_pred = F * x_est; % Predict state P_pred = F * P_est * F' + Q; % Predict covariance Kálmán in 1960
Invented by Rudolf E. Kálmán in 1960, the Kalman Filter is a mathematical algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, to produce estimates of unknown variables that are more accurate than those based on a single measurement alone.