Simple Kalman Filter Implementation

The Kalman filter is a recursive algorithm for state estimation, widely applied in navigation, robotics, sensor fusion, and other domains. Its core principle involves optimally estimating system states by combining system models with sensor measurements while accounting for noise influence. The algorithm operates through iterative prediction and correction cycles.

For beginners, implementing a simple Kalman filter helps understand fundamental concepts. In MATLAB, the implementation logic follows these steps:

Define System Model: Establish state transition equations and observation equations that describe how the system evolves over time and how measurements reflect system states. For example, a constant velocity model would use position and velocity as state variables with linear transition matrices.

Initialize Parameters: Set initial state estimates, process noise covariance (Q), measurement noise covariance (R), and state covariance (P). These parameters significantly impact filter convergence speed and accuracy. In code, Q represents model uncertainty while R reflects sensor accuracy.

Prediction Step: Predict the next state and covariance using the system model. This phase relies solely on system dynamics without incorporating measurement data. The MATLAB implementation typically involves matrix multiplication for state propagation and covariance prediction using the state transition matrix.

Update Step: Upon receiving new measurements, calculate the Kalman gain (K), then correct the predicted values using measurement data to obtain refined state estimates and covariance. The Kalman gain optimally weights the prediction and measurement based on their respective uncertainties.

Beginners can simplify the problem using one-dimensional states (e.g., position estimation in constant motion) to gradually understand the alternating prediction-update cycle. By adjusting Q and R parameters, users can observe the filter's noise sensitivity and response characteristics. Higher Q values make the filter more responsive to measurements, while higher R values increase reliance on predictions.

In practical applications, Kalman filters can be extended to nonlinear systems through variants like Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF), though the fundamental prediction-correction framework remains consistent.