Manifold Extended Kalman Filter Overview

• Manifold EKF is a geometry-aware recursive estimator for nonlinear systems that evolve on smooth manifolds such as SO(3) and SE(3).
• It leverages local tangent space operations using boxplus/boxminus and exponential maps to accurately linearize and update state estimates.
• By transporting covariances and accounting for manifold curvature, Manifold EKF enhances consistency and accuracy over classical EKF techniques.

The manifold extended Kalman filter is a family of EKF constructions for nonlinear systems whose state does not naturally live in a global Euclidean space, but on a smooth manifold such as $SO(3)$, $SE(3)$, $\mathsf{S}^3$, a Stiefel manifold, or a terrain surface. Its defining move is to keep the nominal estimate on the manifold while representing uncertainty, residuals, and linearizations in a local tangent space, typically through $\boxplus/\boxminus$ operators, exponential and logarithm maps, or equivalent local charts. In this way, the Manifold EKF preserves the geometric structure of attitude, pose, and other constrained states while retaining the recursive predict–update form of the classical EKF (Huai et al., 2023, Ge et al., 6 Jun 2025).

1. Geometric setting and state representation

In the manifold formulation, the state estimate is a point $\hat{\xi}\in\mathcal M$, while the estimation error is a tangent vector in $T_{\hat{\xi}}\mathcal M$ or, after choosing local coordinates, a vector in $\mathbb R^m$. This replaces the Euclidean assumption that both state and error are globally representable by ordinary vector addition. The core local operators are

$$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$

where $\vartheta_\xi$ is a normal-coordinate chart around $\xi$ induced by an affine connection or an equivalent local construction (Ge et al., 2023, Ge et al., 6 Jun 2025).

A recurrent claim in the recent literature is that an affine connection is the minimum geometric structure needed for a genuinely geometric EKF on a manifold, because it supplies geodesics, exponential maps, normal coordinates, and parallel transport. This allows the filter to define local Gaussian-like distributions, transport covariances between tangent spaces, and express update and reset steps intrinsically rather than in an arbitrary chart (Ge et al., 2023). In the concentrated-Gaussian viewpoint, uncertainty is represented by a Gaussian-like density in normal coordinates centered at the current estimate,

$SE(3)$0

with $SE(3)$1 and $SE(3)$2 understood as local parameters rather than intrinsic global moments (Ge et al., 6 Jun 2025).

The same structural idea appears in less abstract forms. On Lie groups, attitude error is often represented in the Lie algebra, for example through

$SE(3)$3

with $SE(3)$4 as the local attitude-error coordinate (Goodarzi et al., 2016). In quaternion-based formulations, the nominal quaternion remains on $SE(3)$5, while the covariance is tracked in a local $SE(3)$6-dimensional chart around that quaternion (Sadananda et al., 27 Sep 2025). In terrain-aware filtering, the nominal pose is reduced from a full $SE(3)$7 state to a chart coordinate on a $SE(3)$8-manifold together with a heading in the tangent plane, so that the state space becomes $SE(3)$9 rather than a free $\mathsf{S}^3$0-DoF pose space (Raab et al., 20 Aug 2025).

2. Local linearization, prediction, and iterated update

A standard manifold-EKF formulation starts from a nonlinear discrete-time process model

$\mathsf{S}^3$1

and an observation model

$\mathsf{S}^3$2

with the simplifying assumption, in one common treatment, that the measurement space is a vector space and the measurement noise is additive (Huai et al., 2023). The predicted state is

$\mathsf{S}^3$3

and the covariance is propagated through tangent-space Jacobians,

$\mathsf{S}^3$4

The crucial difference from the Euclidean EKF is that the process Jacobian is defined through $\mathsf{S}^3$5 and $\mathsf{S}^3$6,

$\mathsf{S}^3$7

so perturbations are applied and measured in tangent coordinates rather than by ordinary subtraction (Huai et al., 2023).

The measurement update may be written as a weighted least-squares problem on the manifold,

$\mathsf{S}^3$8

This is the manifold analogue of the Gaussian MAP objective: the first term penalizes deviation from the prior and the second penalizes the measurement residual (Huai et al., 2023).

The iterated extended Kalman filter on manifolds refines this idea by repeatedly relinearizing the measurement model about the current iterate. With $\mathsf{S}^3$9, the $\boxplus/\boxminus$0-th iteration uses the local Jacobians

$\boxplus/\boxminus$1

and

$\boxplus/\boxminus$2

The resulting update is

$\boxplus/\boxminus$3

$\boxplus/\boxminus$4

$\boxplus/\boxminus$5

followed by

$\boxplus/\boxminus$6

The covariance is updated once, after the iterations terminate,

$\boxplus/\boxminus$7

This formulation emphasizes that manifold filtering is still an EKF, but with state differences, increments, and Jacobians defined relative to a tangent space (Huai et al., 2023).

3. Update geometry, covariance transport, and reset

A central issue in the modern literature is that covariance matrices on manifolds belong to specific tangent spaces. A naive coordinate implementation of the EKF fails to account for this in two places: the update step, because the measurement covariance is defined at the true output rather than the estimated output, and the reset step, because after the mean is re-centered the covariance still refers to the old tangent space (Ge et al., 2023).

To address this, geometric EKF formulations transport covariances between tangent spaces. In the affine-connection framework, the measurement covariance is parallel transported from the tangent space at the true output to the tangent space at the estimated output, and the posterior covariance is parallel transported from the tangent space at the pre-reset estimate to the tangent space at the post-reset estimate (Ge et al., 2023). In the more explicit concentrated-Gaussian development, the process-noise covariance and measurement-noise covariance are modified by Jacobians of the exponential and logarithm maps. For example, process noise at the noise-free propagated state $\boxplus/\boxminus$8 is transformed as

$\boxplus/\boxminus$9

and the reset step becomes

$\hat{\xi}\in\mathcal M$0

$\hat{\xi}\in\mathcal M$1

These are not merely bookkeeping devices: they express the fact that the posterior has been reparameterized around a new nominal point (Ge et al., 6 Jun 2025).

Curvature enters these transforms explicitly. For a symmetric affine connection, the tangential and positional Jacobians admit curvature-dependent approximations such as

$\hat{\xi}\in\mathcal M$2

and

$\hat{\xi}\in\mathcal M$3

which reduce to the identity in flat space (Ge et al., 6 Jun 2025). This suggests that the manifold EKF is not only a matter of enforcing constraints; it is also a matter of transporting uncertainty in a way that depends on curvature and, more generally, on the chosen affine connection.

Benchmarking on a simplified inertial navigation problem on $\hat{\xi}\in\mathcal M$4 shows that the full geometric EKF improves estimation accuracy and consistency over the classical EKF, especially in the transient phase. In that study, the update correction was the dominant geometric contribution, the reset correction alone was not sufficient, and the geometric iterated EKF still outperformed the classical iterated EKF, though with a more modest gain (Ge et al., 6 Jun 2025).

4. Lie-group, invariant, and error-state formulations

A large and influential subclass of manifold EKFs is built on matrix Lie groups. Here the state is a group element $\hat{\xi}\in\mathcal M$5, the local error lies in the Lie algebra, and the state correction is applied through the exponential map. In continuous-time Lie-group filtering, one derivation models process noise directly on the Lie algebra and adopts a Stratonovich interpretation of the stochastic differential equation so that solutions remain on the manifold (Maurer et al., 2 Jun 2025). The local error dynamics are then linearized in algebra coordinates, yielding an ordinary linear Kalman filter on the error.

This viewpoint encompasses left-invariant and right-invariant error definitions. For the left-invariant case,

$\hat{\xi}\in\mathcal M$6

whereas for the right-invariant case,

$\hat{\xi}\in\mathcal M$7

A key recent result is that, when the post-update covariance reset is done in full order, the left- and right-invariant EKFs are equivalent in terms of the resulting state estimate, with their covariances related by the adjoint map,

$\hat{\xi}\in\mathcal M$8

Monte Carlo simulations on aided inertial navigation confirm that the full-order left and right filters are identical, with total MAE between L-FO and R-FO equal to $\hat{\xi}\in\mathcal M$9, whereas reduced-order covariance resets break the invariance property (Maurer et al., 2 Jun 2025).

The same Lie-group machinery appears in application-oriented filters. A quadrotor EKF on $T_{\hat{\xi}}\mathcal M$0 represents attitude perturbations by

$T_{\hat{\xi}}\mathcal M$1

builds an $T_{\hat{\xi}}\mathcal M$2-dimensional Euclidean error vector

$T_{\hat{\xi}}\mathcal M$3

and reconstructs the updated manifold state through an exponential-map-based attitude correction (Goodarzi et al., 2016). This coordinate-free construction was motivated by coupled rotational and translational dynamics and was explicitly framed as avoiding Euler-angle singularities, quaternion double-cover ambiguity, unwinding phenomena, and dependence on a particular local coordinate chart (Goodarzi et al., 2016).

In attitude determination, quaternion-based manifold EKFs use $T_{\hat{\xi}}\mathcal M$4 as the state manifold and a local $T_{\hat{\xi}}\mathcal M$5-dimensional chart for the error. The state takes the form

$T_{\hat{\xi}}\mathcal M$6

the correction is performed in the local chart, and the mean quaternion is re-centered by quaternion multiplication (Sadananda et al., 27 Sep 2025). In this setting, the Manifold EKF and its magnetometer-augmented variant have been used to show how a geometry-consistent update on $T_{\hat{\xi}}\mathcal M$7 can be combined with TRIAD preprocessing to reduce the influence of disturbed magnetometer readings on pitch and roll estimation (Sadananda et al., 27 Sep 2025).

5. Application domains and reported empirical performance

The manifold-EKF label covers a range of constructions: full intrinsic filters on Lie groups, generic manifold filters using $T_{\hat{\xi}}\mathcal M$8, error-state filters on reduced manifolds, and Euclidean EKFs followed by projection onto a constraint manifold. The reported gains are therefore application-specific, but the common pattern is that geometry-aware state representation or geometry-aware correction improves feasibility, consistency, or accuracy.

System and manifold	Formulation	Reported result
Indoor position and attitude tracking on $T_{\hat{\xi}}\mathcal M$9 (Salem et al., 2 Jan 2025)	EKFRie and UKFRie incorporate the $\mathbb R^m$0 manifold structure of the rotation matrix	On a long stair path, conventional EKF and UKF achieved RMSE of $\mathbb R^m$1 m and $\mathbb R^m$2 m, while EKFRie and UKFRie achieved $\mathbb R^m$3 m and $\mathbb R^m$4 m; against EKF/UKF with the Isosceles triangle manifold, RMSE improved from $\mathbb R^m$5 cm and $\mathbb R^m$6 cm to $\mathbb R^m$7 cm and $\mathbb R^m$8 cm
IMU–acoustic indoor pose estimation with receiver-array geometry (AlSharif et al., 2024)	Standard EKF/UKF plus projection onto the isosceles triangle manifold	In the final real-world experiment, position RMSE improved from about $\mathbb R^m$9–$$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$0 cm to about $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$1 cm, and yaw RMSE from about $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$2 to about $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$3–$$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$4
Terrain-aided UGV localization on a smooth surface manifold (Raab et al., 20 Aug 2025)	Manifold Error State Extended Kalman Filter with state in $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$5	Across $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$6 Monte Carlo trials, the M-ESEKF had mean ANEES much closer to $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$7; generic settings for the constrained filter could produce ANEES values above $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$8 or divergence; positional RMSE was below $$\xi \boxplus u = \vartheta_\xi^{-1}(u), \qquad \zeta \boxminus \xi = \vartheta_\xi(\zeta),$$9 m and maximum orientation error below $\vartheta_\xi$0 rad
Attitude estimation on $\vartheta_\xi$1 with TRIAD-aided correction (Sadananda et al., 27 Sep 2025)	Quaternion-centered Manifold EKF2 with TRIAD-derived magnetometer substitute	At $\vartheta_\xi$2 roll, the error was approximately $\vartheta_\xi$3 for $\vartheta_\xi$4, $\vartheta_\xi$5 for $\vartheta_\xi$6, and $\vartheta_\xi$7 for TRIAD-aided Manifold EKF2

Further application classes extend beyond these examples. A Stiefel-manifold EKF uses the Riemannian logarithm for innovations and the exponential map for updates, proving that in the constant-state case the filtered mean converges to the true state in probability and the estimated scalar variance tends to zero (Figueras et al., 4 Nov 2025). A related Stiefel formulation reports significant improvement over raw manifold-valued measurements on both $\vartheta_\xi$8 and $\vartheta_\xi$9 (Figueras et al., 4 Nov 2025). A LiDAR–inertial odometry method uses an $\xi$0-style state together with $\xi$1 updates; although it is not presented as a full Lie-theoretic Manifold EKF, it is explicitly manifold-aware in its pose representation (Akai et al., 2024).

6. Conceptual boundaries, common misconceptions, and current directions

A common misconception is that “Manifold EKF” names a single canonical algorithm. The literature instead contains several distinct but overlapping designs. Some filters are fully intrinsic and are developed directly on a smooth manifold equipped with an affine connection (Ge et al., 2023, Ge et al., 6 Jun 2025). Some are Lie-group EKFs that linearize left- or right-invariant errors in the Lie algebra and apply corrections via the group exponential (Maurer et al., 2 Jun 2025). Some are error-state filters in which only the local perturbation is treated as Euclidean, while the nominal state remains on $\xi$2, $\xi$3, or a lower-dimensional manifold (Goodarzi et al., 2016, Raab et al., 20 Aug 2025). Others keep the EKF itself Euclidean and enforce geometry afterwards by projection, as in the isosceles-triangle manifold used for indoor acoustic localization (AlSharif et al., 2024). These are related approaches, but not identical ones.

A second misconception is that manifold awareness amounts only to keeping the estimate on the constraint set. The recent geometric papers argue that the deeper issue is covariance placement and transport. In this view, an EKF that merely normalizes quaternions or projects a state back to the manifold may still be geometrically inconsistent if it ignores how the measurement covariance and posterior covariance move between tangent spaces (Ge et al., 2023, Ge et al., 6 Jun 2025). Likewise, the Lie-group literature emphasizes that the covariance reset after correction is part of the correct probabilistic re-centering of the posterior, and that reduced-order resets can destroy invariance properties (Maurer et al., 2 Jun 2025).

Theoretical interpretations are also broadening. One line of work identifies the EKF with Amari’s online natural gradient in trajectory space under the pure fading-memory process-noise choice

$\xi$4

with the covariance playing the role of an inverse Fisher information matrix (Ollivier, 2019). Another line establishes convergence only under restrictive conditions, such as constant hidden state and isotropic measurement noise on Stiefel manifolds (Figueras et al., 4 Nov 2025). These results suggest that “manifold EKF” is best understood as a geometric generalization of EKF methodology rather than a single theorem-backed object with uniform guarantees across all manifolds and all noise models.

The present direction of the field is therefore twofold. One direction makes the EKF more intrinsic by incorporating affine connections, curvature, torsion, parallel transport, and full-order reset operations (Ge et al., 6 Jun 2025). The other direction specializes the filter to state-space structure and sensing geometry—Lie-group symmetry, rigid-body arrays, terrain surfaces, or TRIAD-compatible attitude sensing—to reduce linearization error or improve consistency in the problem at hand (Maurer et al., 2 Jun 2025, Raab et al., 20 Aug 2025, Sadananda et al., 27 Sep 2025). Taken together, these developments define the modern Manifold EKF as a geometry-aware recursive estimator whose essential operation is unchanged from the classical EKF, but whose state, error, and covariance are explicitly tied to the manifold on which the system actually evolves.