KILO-EKF: Koopman-Inspired Learned Observations Extended Kalman Filter

TL;DR

This work addresses robust real-time state estimation on Lie groups with imperfect sensor models by introducing KILO-EKF, which learns a lifted, linear-Gaussian measurement model from data and integrates it into a standard EKF while preserving the prediction step. The method represents measurements in a lifted feature space via a Koopman-inspired approach, using SERFFs and handcrafted features to form $x_k = oldsymbol{p}_{oldsymbol{\xi}}(oldsymbol{\xi}_k)$ and $y_k = oldsymbol{p}_{oldsymbol{\gamma}}(oldsymbol{oldsymbol{\gamma}}_k)$ so that $y_k = oldsymbol{D}x_k + oldsymbol{n}_k$. Training yields closed-form solutions for $oldsymbol{D}$ and $oldsymbol{R}$, enabling linear-time learning in the amount of data, and inference remains real-time through a standard EKF update with Jacobians obtained via the learned lifting. The framework is specialized to SE$_2$(3) with SERFF-based liftings and yields improved localization accuracy and consistency on a quadrotor MILUV dataset using IMU, UWB, and laser measurements, outperforming miscalibrated geometry-based EKFs and matching or exceeding data-calibrated baselines. The results demonstrate that Koopman-inspired measurement learning provides a scalable, data-driven alternative to traditional sensor calibration, while preserving the efficiency and structure of recursive filtering in robotic estimation.

Abstract

We present the Koopman-Inspired Learned Observations Extended Kalman Filter (KILO-EKF), which combines a standard EKF prediction step with a correction step based on a Koopman-inspired measurement model learned from data. By lifting measurements into a feature space where they are linear in the state, KILO-EKF enables flexible modeling of complex or poorly calibrated sensors while retaining the structure and efficiency of recursive filtering. The resulting linear-Gaussian measurement model is learned in closed form from groundtruth training data, without iterative optimization or reliance on an explicit parametric sensor model. At inference, KILO-EKF performs a standard EKF update using Jacobians obtained via the learned lifting. We validate the approach on a real-world quadrotor localization task using an IMU, ultra-wideband (UWB) sensors, and a downward-facing laser. We compare against multiple EKF baselines with varying levels of sensor calibration. KILO-EKF achieves better accuracy and consistency compared to data-calibrated baselines, and significantly outperforms EKFs that rely on imperfect geometric models, while maintaining real-time inference and fast training. These results demonstrate the effectiveness of Koopman-inspired measurement learning as a scalable alternative to traditional model-based calibration.

Figures (9)

• Figure 1: Overview of the proposed KILO-EKF: the standard EKF structure is retained, while the measurement model is replaced with a Koopman-inspired, data-driven representation.
• Figure 2: System overview of the quadrotor platform miluv used to evaluate KILO-EKF, which learns data-driven measurement models for UWB ranging and laser height sensing.
• Figure 3: Trajectory estimates for the five cross-validation folds, illustrating the diversity of path geometries used in evaluation. Qualitatively, although not always visually pronounced, the proposed KILO-EKF generally tracks the groundtruth more closely than the two baselines.
• Figure 4: Violin plots of per-timestep position errors, orientation errors, and NEES across the five cross-validation folds. Each point corresponds to a single time step. KILO-EKF consistently achieves comparable or lower errors than the baseline methods across all folds, and exhibits NEES values closer to 1, indicating more consistent covariance estimates.
• Figure 5: Per-axis position and orientation error time series for Fold 2. The shaded region indicates the $3\sigma$ covariance envelope. KILO-EKF exhibits lower errors and more consistent covariance estimates, while the baseline methods show larger errors that frequently exceed the predicted covariance bounds.