Implementing Extended Kalman Filter for Robust Slam: a Step-by-step Tutorial

Simultaneous Localization and Mapping (SLAM) is a key technology in robotics, enabling a robot to build a map of an unknown environment while simultaneously determining its position within that map. Implementing an Extended Kalman Filter (EKF) enhances SLAM’s robustness, especially in environments with noise and uncertainties. This tutorial provides a step-by-step guide to implementing EKF for robust SLAM.

Understanding the Extended Kalman Filter

The EKF is an extension of the Kalman Filter designed to handle nonlinear systems. It linearizes the nonlinear functions around the current estimate, allowing for recursive state estimation. In SLAM, the EKF estimates both the robot’s pose and the positions of landmarks in the environment.

Step 1: Define State and Covariance

The state vector typically includes the robot’s position and orientation, along with landmark positions:

• Robot pose: x, y, θ
• Landmark positions: x_i, y_i

The initial covariance matrix represents the uncertainty in the initial estimates.

Step 2: Prediction Step

Using the robot’s motion model, predict the next state and update the covariance matrix. The nonlinear motion equations are linearized using Jacobians:

State prediction:

x_k|k-1 = f(x_k-1, u_k)

Covariance prediction:

P_k|k-1 = F_k P_k-1 F_k^T + Q_k

Step 3: Update Step

When sensor measurements are received, update the state estimate. Linearize the measurement model using Jacobians:

Measurement prediction:

z_k = h(x_k)

Innovation:

y_k = z_measured – h(x_k|k-1)

Kalman gain calculation:

K_k = P_k|k-1 H_k^T (H_k P_k|k-1 H_k^T + R_k)^-1

State update:

x_k|k = x_k|k-1 + K_k y_k

Covariance update:

P_k|k = (I – K_k H_k) P_k|k-1

Step 4: Incorporate Landmarks

Landmarks are added to the state vector as they are detected. The EKF updates the estimates based on measurements, improving localization accuracy over time.

Conclusion

Implementing EKF for SLAM involves defining the state, predicting the robot’s movement, and updating estimates with sensor data. Proper linearization and covariance management are essential for robustness in noisy environments.