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Extended Kalman Filter

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Extended Kalman Filter

The Extended Kalman Filter (EKF) is a nonlinear extension of the Kalman Filter that handles systems with nonlinear dynamics or measurement models. It linearizes the system around the current state estimate using first-order Taylor series expansion (Jacobian matrices) and then applies the standard Kalman Filter equations.

Core Idea

While the standard Kalman Filter only works for linear systems, the EKF approximates nonlinear systems by linearizing them at each time step. This makes it suitable for real-world problems like navigation, tracking, and robotics where dynamics are rarely perfectly linear.

AEGIS Application

In the AEGIS project, a 12-state Error-State EKF is used for state estimation during autonomous landing. It estimates:

  • Position (3 states)
  • Velocity (3 states)
  • Gyroscope bias (3 states)
  • Accelerometer bias (3 states)

Key Features

  • Error-state formulation: Tracks errors from a nominal state rather than the full state, improving numerical stability
  • Adaptive process-noise scaling: The velocity-noise block is scaled based on kinematic acceleration magnitude to handle high-thrust transients
  • Sensor fusion: Combines IMU (gyroscope/accelerometer), altimeter, and velocimeter data
  • Dynamic gravity: Computed from body.gravitational_parameter / r² rather than hardcoded

The EKF Equations (at a glance)

Prediction:

  • x̂ₖ⁻ = f(x̂ₖ₋₁, uₖ)
  • Pₖ⁻ = Fₖ Pₖ₋₁ Fₖᵀ + Qₖ

Update:

  • Kₖ = Pₖ⁻ Hₖᵀ (Hₖ Pₖ⁻ Hₖᵀ + R)⁻¹
  • x̂ₖ = x̂ₖ⁻ + Kₖ (zₖ - h(x̂ₖ⁻))
  • Pₖ = (I - Kₖ Hₖ) Pₖ⁻

Where F and H are Jacobian matrices of the process and measurement models.

Sources

  • AEGIS Project (src/estimation/ekf.py)
  • Cornman, L. & Mei, G. Extended Kalman Filtering. Stanford University.

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