Kalman Filter

Introduction

The Kalman Filter is an optimal recursive estimation algorithm used in robotics for predicting the internal state of a system in the presence of noise and uncertainty. It is widely used for sensor fusion, localization, and control in both mobile and aerial robotic systems.

Why Use Kalman Filter in Robotics?

  • Provides accurate estimates of position, velocity, or other states
  • Fuses noisy sensor data (e.g., IMU, GPS, encoders)
  • Tracks dynamic systems in real time
  • Handles Gaussian noise and linear system models effectively

Kalman Filter Algorithm

1. Prediction Step

Estimates the next state based on a motion model: 

x̂ₖ⁻ = A x̂ₖ₋₁ + B uₖ
Pₖ⁻ = A Pₖ₋₁ Aᵀ + Q
  • x̂ₖ⁻: Predicted state
  • Pₖ⁻: Predicted covariance
  • Q: Process noise covariance

2. Update (Correction) Step

Incorporates measurement to refine the prediction: 

Kₖ = Pₖ⁻ Hᵀ (H Pₖ⁻ Hᵀ + R)⁻¹
x̂ₖ = x̂ₖ⁻ + Kₖ (zₖ - H x̂ₖ⁻)
Pₖ = (I - Kₖ H) Pₖ⁻
  • Kₖ: Kalman gain
  • zₖ: Measurement
  • R: Measurement noise covariance

Applications in Robotics

1. Sensor Fusion

Combines data from multiple sensors like GPS, IMU, LiDAR, and wheel encoders for more reliable and accurate state estimation.

2. Localization

Kalman Filters estimate the robot's position and orientation on a map, often used in indoor and outdoor navigation systems.

3. Tracking

Tracks moving objects or people in the robot's environment using camera or radar data.

4. Control Systems

Used to estimate the internal state of a robot (like velocity or angle) required for feedback in control algorithms.

Limitations

  • Assumes linear system dynamics (Extended or Unscented Kalman Filters are needed for nonlinear systems)
  • Relies on accurate models of system and noise
  • Can diverge if the model or noise assumptions are incorrect

Kalman Filter in ROS

In ROS, Kalman Filters are commonly used through packages like:

  • robot_localization: Provides EKF and UKF implementations for fusing sensor data
  • ekf_localization_node: Estimates robot position using IMU, GPS, and odometry
  • kalman_filter: Basic implementations for teaching and testing

Conclusion

The Kalman Filter is a foundational tool in robotics, offering robust, real-time estimation of dynamic systems in uncertain environments. Whether it's used for autonomous vehicle localization or drone flight stabilization, its ability to handle noisy data makes it indispensable in modern robotics.