is the bridge across that gap. It replaces jargon with code, theory with practice, and fear with curiosity.
Adjust the prediction by adding the measurement multiplied by the Kalman Gain.
% Implement the Kalman filter x_est = zeros(2, length(t)); P_est = zeros(2, 2, length(t)); x_est(:, 1) = x0; P_est(:, :, 1) = P0; for i = 2:length(t) % Prediction step x_pred = A * x_est(:, i-1); P_pred = A * P_est(:, :, i-1) * A' + Q;
% Update H = jacobian_h(x_pred); y = z(:,k) - h(x_pred); S = H * P_pred * H' + R; K = P_pred * H' / S; x_hat = x_pred + K * y; P = (eye(size(P)) - K*H) * P_pred; end is the bridge across that gap
fprintf('Step %d: Estimate = %.2f\n', k, x);
The Kalman filter has various applications, including:
The filter combines the previous estimate with the new measurement, weighting them according to their respective uncertainties. % Implement the Kalman filter x_est = zeros(2,
Instead of using complex calculus, it selects a minimal set of sample points (sigma points) around the mean and passes them through the non-linear equations. This often yields higher accuracy than the EKF for highly non-linear systems. Finding the PDF and Resources
Kim uses the analogy of a car driving down a road to explain the core concepts of and Correction . He simplifies the filter into three distinct mental steps that anyone can understand:
) : The hidden values you want to know (e.g., exact position and velocity). Measurement ( Finding the PDF and Resources Kim uses the
: Estimates how much uncertainty grew during the prediction. 2. The Update Phase (Measurement Update)
% Matrix Kalman Filter: Tracking Position and Velocity clear all; close all; clc; dt = 0.1; % Time step (seconds) N = 100; % Number of samples % System Matrices A = [1 dt; 0 1]; % State transition matrix (Pos = Pos + Vel*dt) H = [1 0]; % Measurement matrix (we only measure position) Q = [0.01 0; 0 0.01]; % Process noise covariance R = 2.25; % Measurement noise covariance (std dev of 1.5 meters) % Initial States x_est = [0; 0]; % Initial [Position; Velocity] estimate P = eye(2); % Initial error covariance matrix % Simulated True Trajectory true_pos = zeros(N, 1); measurements = zeros(N, 1); est_pos = zeros(N, 1); est_vel = zeros(N, 1); current_pos = 0; current_vel = 5; % True velocity is 5 m/s for k = 1:N % Update true positions and create noisy measurements current_pos = current_pos + current_vel * dt; true_pos(k) = current_pos; measurements(k) = current_pos + sqrt(R)*randn(); % --- 1. PREDICT --- x_pred = A * x_est; P_pred = A * P * A' + Q; % --- 2. KALMAN GAIN --- K = (P_pred * H') / (H * P_pred * H' + R); % --- 3. UPDATE --- x_est = x_pred + K * (measurements(k) - H * x_pred); P = (eye(2) - K * H) * P_pred; % Save data est_pos(k) = x_est(1); est_vel(k) = x_est(2); end % Plotting results figure; subplot(2,1,1); plot(true_pos, 'g', 'LineWidth', 2); hold on; plot(measurements, 'r.', 'MarkerSize', 8); plot(est_pos, 'b--', 'LineWidth', 2); title('Position Tracking'); ylabel('Position (m)'); legend('True', 'Noisy Sensor', 'Kalman Estimate'); grid on; subplot(2,1,2); plot(repmat(current_vel, N, 1), 'g', 'LineWidth', 2); hold on; plot(est_vel, 'b--', 'LineWidth', 2); title('Velocity Estimation'); ylabel('Velocity (m/s)'); xlabel('Time Steps'); legend('True', 'Kalman Estimate'); grid on; Use code with caution. Why Phil Kim’s Approach Works for Beginners
Choose Q, R, initial x̂ and P, then iterate predict+update each time step.