% Initial state guess x = [0; 10]; % start at 0 m, velocity 10 m/s P = eye(2); % initial uncertainty

estimated_positions(k) = x(1); end

The Kalman filter gives a smooth estimate much closer to the true position than the raw noisy measurements. 5. MATLAB Example 2: Tracking a Falling Object (Acceleration) Now let’s track an object in free fall (constant acceleration due to gravity).

% Simulate t = 0:dt:5; true_pos = 100 + 0 t + 0.5 (-9.8)*t.^2; measurements = true_pos + sqrt(R)*randn(size(t));

% Filter est_pos = zeros(size(t)); for k = 1:length(t) % Predict x = A * x; P = A * P * A' + Q;

est_pos(k) = x(1); end

% Update K = P * H' / (H * P * H' + R); x = x + K * (measurements(k) - H*x); P = (eye(3) - K*H) * P;

% Update K = P * H' / (H * P * H' + R); % Kalman gain x = x + K * (measurements(k) - H * x); P = (eye(2) - K * H) * P;

1. What is a Kalman Filter? The Kalman filter is a recursive algorithm that estimates the state of a dynamic system from a series of incomplete and noisy measurements. It was developed by Rudolf E. Kálmán in 1960.

% Run Kalman filter estimated_positions = zeros(size(measurements)); for k = 1:length(measurements) % Predict x = A * x; P = A * P * A' + Q;

% Measurement noise (GPS error) R = 10;

% Plot results plot(0:dt:50, true_position, 'g-', 'LineWidth', 2); hold on; plot(0:dt:50, measurements, 'rx'); plot(0:dt:50, estimated_positions, 'b--', 'LineWidth', 2); legend('True', 'Noisy GPS', 'Kalman Estimate'); xlabel('Time (s)'); ylabel('Position (m)'); title('Kalman Filter for Constant Velocity'); grid on;