Invariant Kalman filtering for extended pose estimation in multi-IMU articulated rigid-body systems

Kalman filter-based algorithms are fundamental for mobile robots, as they provide a computationally efficient solution to the challenging problem of state estimation. However, they rely on two main assumptions that are difficult to satisfy in practice: (a) the system dynamics must be linear with Gaussian process noise, and (b) the measurement model must also be linear with Gaussian measurement noise. Previous works have extended assumption (a) to nonlinear spaces through the Invariant Extended Kalman Filter (IEKF), showing that it retains properties similar to those of the classical Kalman filter when the system dynamics are group-affine on a Lie group. More recently, the counterpart of assumption (b) for the same nonlinear setting was addressed in [1]. By means of the proposed Iterated Invariant Extended Kalman Filter (IterIEKF), the authors of that work demonstrated that the update step exhibits several compatibility properties of the classical linear Kalman filter. In this work, we introduce a novel open-source state estimation algorithm for legged robots based on the IterIEKF. The update step of the proposed filter relies solely on proprioceptive measurements, exploiting kinematic constraints on foot velocity during contact and base-frame velocity, making it inherently robust to environmental conditions. Through extensive numerical simulations and evaluation on real-world datasets, we demonstrate that the IterIEKF outperforms the vanilla IEKF, the SO(3)-based Kalman Filter, and its iterated variant in terms of both accuracy and consistency.

This paper presents an invariant extended Kalman filtering (InEKF) approach for real-time state estimation of humanoid robots operating on non-inertial ground using only onboard proprioceptive sensing. The proposed approach estimates the robot's base position and velocity relative to the moving ground frame without requiring direct measurements of ground motion or externally mounted sensors. By exploiting kinematic constraints at the stance foot through foot-mounted IMUs, the filter accounts for ground-induced nonlinearities in the process and measurement models while remaining fully proprioceptive. The estimator is formulated to admit a right-invariant measurement model, enabling favorable error dynamics under large initial uncertainties. Observability analysis establishes conditions under which the robot's relative base position and velocity are observable with respect to the non-inertial ground frame. Experiments with the Digit humanoid robot standing and squatting atop a swaying and pitching ground showcase a 96% speedup in convergence rate and an 80% reduction in position estimate errors over existing InEKFs. Walking experiments on a uni-axially rotating ground achieve an average estimation error of less than 9 cm for an initial error of up to 1 m.

Inertial navigation systems aided by three-dimensional landmark measurements constitute a fundamental problem in robotic perception and state estimation. Classical SO(3)-based Extended Kalman Filter (SO(3)-EKF) approaches provide practical solutions, but suffer from the false observability problem, in which the filter becomes overconfident in unobservable directions, leading to degraded estimation performance. The Invariant EKF (IEKF) addresses this limitation by reformulating the system dynamics as a group-affine system on a Lie group, although its measurement update does not fully satisfy certain state compatibility properties. More recently, the Iterated Invariant EKF (IterIEKF) was proposed to further improve the IEKF by ensuring, in the low-noise regime, that the estimated state remains on the observed state manifold while the uncertainty is confined to its tangent space. In this work, we formulate and apply the IterIEKF to landmark-based inertial 3D localization for the first time. Through numerical simulations, we show that the proposed approach outperforms the classical SO(3)-EKF, the Iterated SO(3)-EKF, and the IEKF in terms of both estimation accuracy and consistency.