Sensor Fusion Tutorial

Introduction to Sensor Fusion

Sensor fusion is the process of combining data from multiple sensors to achieve more accurate and reliable information than what could be obtained from any individual sensor. In robotics and autonomous systems, sensor fusion is crucial for:

  • Improving accuracy and reliability of measurements
  • Compensating for individual sensor limitations
  • Handling sensor noise and uncertainties
  • Providing redundancy in case of sensor failures

Kalman Filter Demonstration

This interactive demo shows how a Kalman filter combines noisy GPS position measurements with IMU velocity readings to estimate a robot's true position. The robot follows a circular path, and you can adjust the noise levels to see how the filter performs under different conditions.

● True Position   ● GPS Reading   ● IMU Path   ● Kalman Estimate

Kalman Filter Explanation

The Kalman filter is an optimal estimation algorithm that combines predictions based on system dynamics with noisy measurements. It maintains both a state estimate and its uncertainty, working in two steps:

  1. Prediction Step: Uses the system model to predict the next state based on previous estimates and optional control inputs:
    • State prediction: x̂ₖ₋ = Fₖx̂ₖ₋₁ + Bₖuₖ
    • Uncertainty prediction: Pₖ₋ = FₖPₖ₋₁Fₖᵀ + Qₖ
  2. Update Step: Incorporates new measurements to correct the prediction, weighing each source based on its uncertainty:
    • Innovation: yₖ = zₖ - Hₖx̂ₖ₋
    • Innovation covariance: Sₖ = HₖPₖ₋Hₖᵀ + Rₖ
    • Kalman gain: Kₖ = Pₖ₋Hₖᵀ(Sₖ)⁻¹
    • State update: x̂ₖ = x̂ₖ₋ + Kₖyₖ
    • Uncertainty update: Pₖ = (I - KₖHₖ)Pₖ₋

Where:

  • x̂ₖ₋ is the predicted state estimate
  • Pₖ₋ is the predicted error covariance
  • Fₖ is the state transition matrix
  • Bₖ is the control input matrix
  • uₖ is the control input
  • Qₖ is the process noise covariance
  • zₖ is the measurement
  • Hₖ is the measurement matrix
  • Rₖ is the measurement noise covariance
  • Kₖ is the Kalman gain
  • yₖ is the innovation (measurement residual)
  • Sₖ is the innovation covariance

Implementation Details

In this demo:

  • State vector: [x, y, vx, vy] (position and velocity)
  • GPS provides position measurements [x, y]
  • IMU provides velocity measurements [vx, vy]
  • Innovation gating is used to reject outliers
  • Covariance matrices are kept symmetric and positive definite
  • The blue ellipse shows the 95% confidence region (2σ)

Sensor Settings

Adjust the noise levels to see how they affect the filter's performance:

Current: 30

Higher values simulate less accurate GPS readings.

Current: 10

Higher values simulate more drift in IMU measurements.

Performance Metrics

Error (RMSE):

GPS: 0.00

Kalman: 0.00

The Root Mean Square Error (RMSE) shows how far the estimates deviate from the true position. Lower values indicate better accuracy.

Learning Objectives

  • Understand the principles of sensor fusion
  • Learn how Kalman filters combine multiple sensor inputs
  • Observe the effects of sensor noise on estimation accuracy
  • Compare raw sensor data with filtered estimates
  • Visualize estimation uncertainty

Additional Resources

Applications in Robotics

  • Robot localization and navigation
  • State estimation in autonomous vehicles
  • Drone flight control systems
  • Multi-sensor perception systems

Related Topics

  • Extended Kalman Filter (EKF) for nonlinear systems
  • Unscented Kalman Filter (UKF) for strong nonlinearities
  • Particle Filters for non-Gaussian noise
  • Multi-sensor calibration and synchronization