Invariant State Estimation on Lie Groups

Invariant state estimation on Lie groups is a framework that exploits the symmetry properties of dynamical systems to deliver robust and consistent estimation of system states. By modelling states and error dynamics on continuous transformation groups, such as the special orthogonal group or the group of rigid‐body motions, one can derive filters whose error propagation is autonomous and independent of the system trajectory. This leads to estimators that maintain consistency even under large perturbations or poorly initialised conditions. The key innovation lies in representing both the system evolution and observation maps as group‐affine or equivariant structures, thereby ensuring that linearisation and update steps respect the underlying geometry. Practical realisations include invariant extended Kalman filters (IEKFs) and equivariant filters that have demonstrated superior convergence properties, reduced sensitivity to linearisation points and improved observability in navigation, robotics and autonomous vehicle applications. These methods have been applied successfully to inertial navigation, visual‐inertial odometry and vehicle positioning in GNSS‐denied environments, yielding significant enhancements in accuracy and stability. The generality of the approach also extends to other fields where symmetry groups govern state evolution, offering a powerful paradigm for real‐time and post‐mission estimation in complex dynamic systems.

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A novel multi state constraint equivariant filter has been introduced for visual‐inertial navigation, in which a carefully chosen symmetry group and its action include IMU biases and camera calibration states within a unified group structure. This filter achieves inherent consistency during large transients and avoids ad hoc consistency‐enforcing techniques, delivering improved estimator behaviour across several real‐world datasets. An equivariant filtering framework for inertial‐integrated navigation models the navigation dynamics on the matrix Lie group SE2(3) and proves left‐equivariance of the kinematic equations. Filters designed on both Earth‐centred and local geodetic frames show superior performance under large misalignment angles, demonstrating more stable variance estimation and higher estimation accuracy compared with traditional extended Kalman approaches. In the domain of autonomous vehicle positioning, a loosely coupled INS/GNSS integration scheme employing an invariant extended Kalman filter on a matrix Lie group has been developed. This approach eschews Jacobian dependencies inherent to linearised filters and yields substantial improvements in 2D‐position RMS and maximum errors, both in open‐sky and signal‐degraded scenarios, underscoring the practical value of invariant estimation in GNSS‐denied contexts.

Invariant State Estimation on Lie Groups publication trend

The graph below shows the total number of articles in invariant state estimation on lie groups across all publications each year (not limited to Nature Index journals).

Technical terms

Lie group: A smooth manifold endowed with a group operation, used to model continuous symmetries of a system.

Lie algebra: The tangent space at the identity of a Lie group, governing its local linear structure and used for linearisation.

Invariant extended Kalman filter (IEKF): A variant of the extended Kalman filter that leverages group symmetries to maintain autonomous error propagation and consistent updates.

Equivariant filter: A state estimator designed so that its update and propagation operations commute with the action of a symmetry group, ensuring geometric consistency.

Group‐affine system: A dynamical system whose evolution law can be expressed as an affine function on a Lie group, facilitating invariant estimation design.

References

1. MSCEqF: A Multi State Constraint Equivariant Filter for Vision-Aided Inertial Navigation. IEEE Robotics and Automation Letters (2023).
2. Equivariant filtering framework for inertial-integrated navigation. Satellite Navigation (2021).
3. Enhanced Autonomous Vehicle Positioning Using a Loosely Coupled INS/GNSS-Based Invariant-EKF Integration. Sensors (2023).