Robotics
Invariant extended Kalman filter
Definition
An invariant extended Kalman filter defines estimation error using a symmetry of a state that evolves on a Lie group. For compatible system models, this makes the linearised error dynamics less dependent on the estimated trajectory than in a conventional extended Kalman filter.
Also known as: Invariant EKF, InEKF
Updated
Put estimation error on the system's geometry
A conventional extended Kalman filter linearises nonlinear dynamics around its current state estimate. For poses and related quantities, the state lives on a Lie group rather than an ordinary vector space. An invariant EKF defines a left-invariant or right-invariant error that respects the group's symmetry.
Barrau and Bonnabel characterise a class of systems whose invariant estimation error has autonomous, log-linear dynamics. They prove local stability around any trajectory under stated conditions analogous to those used for a linear Kalman filter. This is a conditional result for compatible systems, not a claim that every nonlinear filter becomes stable after changing coordinates.
Contact-aided filtering is a practical use
Humanoid and legged-robot estimators often combine an inertial measurement unit, joint encoders, kinematics, and detected foot contacts. Hartley and colleagues derive a contact-aided InEKF whose state evolves on a matrix Lie group. Their formulation gives log-linear error dynamics for the point-contact inertial model and exposes absolute position and yaw as unobservable under the chosen measurements.
That structure can make the filter's linearisation more consistent with the physical symmetries. The paper reports simulation and Cassie hardware comparisons against a quaternion-based EKF; those results apply to its estimator design, tuning, sensors, and experiments.
Invariance does not remove modelling errors
An InEKF still needs process and measurement models, noise assumptions, initial conditions, and covariance tuning. Contact slip, flexible links, time offsets, IMU bias behaviour, or an incorrect contact declaration can make the update inconsistent with the real robot.
The method also does not recover unobservable directions or provide a general global-convergence guarantee. A state must admit the required group structure, and the dynamics and outputs must satisfy the formulation's compatibility conditions. Invariant filtering is therefore a structured estimator design choice, not a replacement for observability analysis, calibration, or sensor validation.
Sources
Related terms
Extended Kalman filter
An extended Kalman filter is a state estimator that applies Kalman-style prediction and correction to nonlinear models by locally linearizing them. It approximates uncertainty around the current state estimate.
Contact-aided state estimation
Contact-aided state estimation estimates a legged robot's body motion by combining inertial measurements with kinematics and information about which feet or other links are in stationary contact. A trusted contact acts as a temporary reference that constrains drift.
State estimation
State estimation infers quantities describing a robot or its environment from measurements and a model. A robot state may include position, orientation, velocity, and other variables that are not all directly measured.
Rigid-body transformation
A rigid-body transformation changes a body's position and orientation without changing its shape or size. In three-dimensional robotics it consists of a proper rotation and a translation.