Invariant Extended Kalman Filter (InEKF)

• InEKF is a Lie-group formulation of the extended Kalman filter that redefines estimation errors via group multiplication to enable robust state estimation on manifolds.
• It linearizes error dynamics in the Lie algebra, resulting in state-estimate-independent Jacobians that enhance consistency across navigation and robotic applications.
• The reset step with covariance transport ensures equivalence between left- and right-invariant formulations, improving asymptotic performance and overall robustness.

Searching arXiv for the specified InEKF papers and related foundational work. Invariant Extended Kalman Filtering (InEKF) is a Lie-group formulation of the extended Kalman filter for state estimation problems whose dynamics and observations admit invariant structure. In its canonical form, the state evolves on a matrix Lie group, the estimation error is defined invariantly by group multiplication rather than Euclidean subtraction, and the linearization is performed in the Lie algebra. For group-affine systems, this yields log-linear error dynamics and, in the ideal setting, state-estimate-independent Jacobians, which distinguishes InEKF from standard EKF and underlies its use in inertial navigation, contact-aided legged locomotion, extended-pose estimation on $SE_2(3)$, and related robotic estimation problems (Ge et al., 6 Jul 2025). A central recent clarification is that the left- and right-invariant variants of the filter are not intrinsically different algorithms when the reset step is implemented correctly: with the appropriate covariance transport, they are stochastically equivalent and have the same performance (Ge et al., 6 Jul 2025).

1. Definition and mathematical setting

InEKF is formulated on an $n$-dimensional Lie group $G$ with identity $I$ and Lie algebra $\mathfrak{g}\cong\mathbb{R}^n$. For matrix Lie groups, $\exp_G$ is the matrix exponential and $\log_G$ its local inverse. The formalism uses wedge and vee maps $(\cdot)^\wedge:\mathbb{R}^n\to\mathfrak{g}$ and $(\cdot)^\vee:\mathfrak{g}\to\mathbb{R}^n$, left and right translations $L_X(Y)=XY$, $n$0, and the group adjoint $n$1, with matrix representation $n$2 in a chosen basis (Ge et al., 6 Jul 2025).

The class of systems most directly associated with InEKF is the class of group-affine systems. In the continuous-time left-trivialized form considered in recent analyses, the state satisfies

$n$3

with input $n$4, or, under left-invariant Gaussian process noise on the algebra,

$n$5

where $n$6 is Brownian motion, $n$7 is a constant linear map into the Lie algebra, and $n$8 is the process covariance (Ge et al., 6 Jul 2025). In discrete time, one representative group-affine model is

$n$9

The conceptual distinction from a standard EKF is that InEKF linearizes the invariant error dynamics on the Lie algebra rather than linearizing the state dynamics in ambient Euclidean coordinates. For group-affine systems with invariant inputs and equivariant outputs, the invariant error dynamics exhibit a log-linear property; this makes the prediction-step linearization global in the appropriate invariant coordinates and, in important cases, independent of the particular trajectory (Ge et al., 6 Jul 2025). This property is the main reason InEKF is often associated with improved consistency and robustness in navigation problems.

A probabilistic formulation frequently used with InEKF is the concentrated Gaussian distribution (CGD) on the Lie group. The left-concentrated Gaussian can be written as $G$0, $G$1, and the right-concentrated Gaussian as $G$2, with analogous density expressions in terms of $G$3 (Ge et al., 6 Jul 2025). These two parameterizations become important in comparing left- and right-invariant filters.

2. Invariant error definitions and filter structure

The two standard invariant errors are the left-invariant and right-invariant errors. With state estimate $G$4 and true state $G$5, they are

$G$6

These correspond to two coordinate choices for describing the estimation error on the group (Ge et al., 6 Jul 2025).

For the continuous-time noisy dynamics

$G$7

the left-invariant error satisfies

$G$8

which linearizes around $G$9 to the covariance ODE

$I$0

with

$I$1

The right-invariant error satisfies

$I$2

with corresponding covariance ODE

$I$3

where

$I$4

(Ge et al., 6 Jul 2025).

Measurements are handled either as generic outputs

$I$5

or, in an important special case, as invariant outputs of the form $I$6. For generic outputs, the linearization matrices are

$I$7

For left-invariant outputs expressed through the pseudo-measurement $I$8, the left-invariant filter has state-independent

$I$9

whereas the right-invariant filter has

$\mathfrak{g}\cong\mathbb{R}^n$0

(Ge et al., 6 Jul 2025).

The update in algebra coordinates follows the Kalman form

$\mathfrak{g}\cong\mathbb{R}^n$1

or, for pseudo-measurements, with $\mathfrak{g}\cong\mathbb{R}^n$2 replacing $\mathfrak{g}\cong\mathbb{R}^n$3 (Ge et al., 6 Jul 2025).

3. Reset, covariance transport, and equivalence of left and right InEKF

A defining result of current InEKF theory is that the correction should not be understood as complete at the algebra-level update alone. Once the reference state $\mathfrak{g}\cong\mathbb{R}^n$4 is changed by injecting the correction through the exponential map, the covariance must be transported to the new tangent coordinates. This is the reset step. Recent work argues that omitting this step breaks equivalence between left- and right-invariant formulations and degrades asymptotic performance (Ge et al., 6 Jul 2025).

In the formulation of Ge et al., the observer reference is unchanged during the algebraic “update” and is modified only during reset. The left-invariant reset uses right multiplication,

$\mathfrak{g}\cong\mathbb{R}^n$5

whereas the right-invariant reset uses left multiplication,

$\mathfrak{g}\cong\mathbb{R}^n$6

The corresponding covariance transport is

$\mathfrak{g}\cong\mathbb{R}^n$7

and

$\mathfrak{g}\cong\mathbb{R}^n$8

(Ge et al., 6 Jul 2025).

The recent equivalence theorem states that if initial left and right concentrated Gaussians are equivalent, specifically

$\mathfrak{g}\cong\mathbb{R}^n$9

then predict, update, and reset preserve that equivalence at all later times, provided the reset is applied correctly (Ge et al., 6 Jul 2025). In this precise sense, the left- and right-IEKF with reset are identical in effect.

This result directly addresses a common robotics belief that the handedness of the filter should be matched to the handedness of the measurement model. The recent analysis concludes that such matching is unnecessary for equal performance if the reset step is implemented properly (Ge et al., 6 Jul 2025). Without reset, however, the two implementations can differ, and empirical transient differences may appear.

A plausible implication is that part of the historical disagreement over left versus right InEKF arose less from intrinsic estimator differences than from inconsistent post-update covariance handling. That interpretation is explicitly supported by the reset-based equivalence proof and associated simulations (Ge et al., 6 Jul 2025).

4. Relation to EKF, MEKF, and iterated invariant filtering

InEKF is often contrasted with both the standard EKF and the multiplicative EKF (MEKF). Standard EKF linearizes in Euclidean coordinates and may suffer from trajectory-dependent linearization and inconsistency on manifolds. MEKF uses multiplicative perturbations for attitude, especially on $\exp_G$0, but may omit a manifold-consistent covariance reset, which can induce bias or inconsistency (Ge et al., 6 Jul 2025).

By exploiting group invariances and the log-linear property, InEKF aims to preserve correct unobservable subspaces and to reduce estimate-dependent linearization error. This point is emphasized in contact-aided legged state estimation, where the local observability matrix derived from the invariant linearization is consistent with the underlying nonlinear system, and yaw and absolute position emerge as the unobservable directions without ad hoc nonlinear observability analysis (Hartley et al., 2018, Hartley et al., 2019).

The relation between InEKF and other error-state filters has itself become a topic of study. The covariance-transformation-based ESKF framework argues that while InEKF exhibits trajectory-independent error dynamics in the ideal group-affine setting, its covariance propagation is equivalent to that of an EKF under an appropriate change of error coordinates. That framework further claims that mixed global-frame and body-frame observations can compromise the practical benefits of a single fixed invariant choice, motivating covariance transformations tailored to each measurement type (Han et al., 1 Nov 2025). This suggests that the advantages of InEKF may depend as much on coordinate alignment in the update as on the propagation model alone. Because this interpretation comes from a later comparative framework rather than from the original invariant-filter literature, it is best treated as a refinement of the standard viewpoint rather than a contradiction.

An important extension is the Iterated Invariant Extended Kalman Filter. IterIEKF performs a Gauss–Newton measurement refinement in Lie-algebra coordinates while retaining the invariant Jacobian structure. For invariant measurements, it improves accuracy in the low-noise regime and exhibits “hard-encoding” behavior analogous to the linear Kalman filter: in the noise-free limit, the updated estimate lies on the observation manifold and the posterior covariance removes uncertainty in the observed directions (Goffin et al., 2024). This iteration is particularly beneficial when measurements are highly accurate.

Another extension concerns deterministic equality constraints treated as noise-free pseudo-measurements. Invariant filtering with such pseudo-measurements admits a limiting Kalman gain based on the Moore–Penrose pseudoinverse,

$\exp_G$1

and the posterior covariance satisfies

$\exp_G$2

meaning that uncertainty collapses exactly in the constrained directions (Goffin et al., 2024). This fits naturally with the geometric viewpoint of InEKF, where the posterior distribution remains on the constraint manifold.

5. Applications across robotics, navigation, and human motion estimation

A major application class is contact-aided inertial navigation for legged robots. In the 3D bipedal observer of Hartley et al., the state is represented on the extended pose group $\exp_G$3, which includes base orientation, velocity, position, and world-frame contact-point positions. For the perfect contact-inertial model, the error follows a log-linear autonomous differential equation, and the observable components converge with a domain of attraction independent of trajectory (Hartley et al., 2018). The later contact-aided robot-state paper further develops bias augmentation, contact addition and removal, and world-centric and robo-centric variants, while showing improved convergence relative to a quaternion EKF in simulation and experiment on Cassie (Hartley et al., 2019).

Legged locomotion in slippery terrain motivated further variants. A right-invariant velocity-update InEKF fuses inertial data, kinematic constraints, and tracking-camera velocity, while also estimating camera–robot extrinsics. Its observability analysis preserves the expected unobservable directions—global yaw and absolute position—and the method remains robust when leg kinematic velocity updates are corrupted by slip, aided by online noise adaptation for the camera measurements (Teng et al., 2021). More recently, neural augmentation has been proposed, in which an attention-based compensator applies a post-update correction to an InEKF estimate in slip-prone quadruped locomotion while preserving the base filter recursion (Lee et al., 26 Jan 2026). This suggests a broader pattern in recent work: the invariant filter serves as the geometric estimation backbone, while learned modules compensate for systematic modeling errors.

Another central application is extended-pose estimation on $\exp_G$4. A two-position-receiver IEKF fuses IMU inputs with the absolute position of one receiver and the baseline between two receivers. Because the measurement model is left invariant, the resulting Jacobians are state-estimate independent, and the baseline directly improves attitude observability, including yaw, relative to single-receiver configurations (Pavlasek et al., 2021).

In pedestrian dead reckoning, a foot-mounted IMU can be modeled on $\exp_G$5 and corrected by stance-based zero-velocity pseudo-measurements. A recent study reports that InEKF is markedly less sensitive than EKF to process-noise mistuning and is more robust in motion-capture, multistory walking, and bipedal robot experiments (Zhang et al., 15 Aug 2025). This suggests that invariant linearization can be especially valuable in repetitive contact-rich motions with intermittent pseudo-measurements.

In scan-matching SLAM, an IEKF on $\exp_G$6 has been used for odometry-aided ICP-based localization. The invariant residual is formed in the Lie algebra, while the process model uses body-frame velocities. The experimental implementation on a wheeled robot showed centimeter-scale translational RMS errors and degree-scale heading errors, illustrating that the invariant framework extends beyond pure inertial navigation into geometric registration-based SLAM (Barczyk et al., 2014).

In human motion estimation, InEKF has been adapted to problems with imperfect sensor placement. One design augments trunk state estimation by including misalignment rotation and translation as states on $\exp_G$7, preserving the group-affine process model. Despite a non-invariant forward-kinematics observation model, the approach improves trunk velocity and orientation estimation across squatting, walking, and ladder climbing motions (Zhu et al., 2022). A related design for human body-motion estimation with imperfect IMU placement similarly augments the state with placement offsets modeled as Brownian motions and reports convergence within $\exp_G$8 seconds during squatting under significant placement inaccuracy and large initial errors (Zhu et al., 2022).

In UAV and ASV state estimation, InEKF has been extended to more specialized measurement geometries. A right-invariant error-state InEKF for fixed-wing UAV full-state estimation fuses IMU, GNSS, magnetometer, barometer, pitot, and learned aerodynamic-angle estimates, yielding improved convergence under disturbances and GNSS-denied operation (Ye et al., 2023). For autonomous surface vessels, a left-invariant InEKF on $\exp_G$9 has been developed to integrate partial orientation measurements—roll and pitch from the visual horizon, yaw from dual-antenna GPS—using an “infinite covariance” treatment for the unobservable yaw direction in horizon measurements (Benham et al., 12 Jun 2025).

Nocturnal visual localization has also adopted a right-invariant InEKF for fusing IMU, odometer, and streetlight-map observations, again using Lie-group error-state machinery to preserve consistent fusion under sparse nighttime cues (Gao et al., 2024).

6. Practical implementation, numerical issues, and common misconceptions

Several implementation themes recur across the literature. First, accurate computation of $\log_G$0, $\log_G$1, $\log_G$2, and the relevant left and right Jacobians is indispensable. For $\log_G$3, Rodrigues’ formula and closed-form Jacobians are standard; for $\log_G$4 and $\log_G$5, block adjoint formulas and kinematic Jacobian constructions are typically used (Ge et al., 6 Jul 2025, Pavlasek et al., 2021). A plausible implication is that many reported differences between filter variants in practice can be attributed to numerical or discretization details rather than to structural differences in the underlying theory.

Second, discretization matters. In GNSS-aided inertial navigation, Ge et al. report that coarse discretization can create small practical differences between left- and right-invariant implementations even though the continuous/discrete-hybrid theory predicts equivalence. These differences shrink almost linearly with sub-stepping refinement, and at $\log_G$6 sub-steps per IMU sample the rotation differences are about $\log_G$7 degrees and the covariance affine-invariant Riemannian metric difference is about $\log_G$8 (Ge et al., 6 Jul 2025).

Third, the reset step is not optional in high-performance implementations. The recent equivalence analysis concludes that reset improves asymptotic calibration and consistency for all variants and should be included in all high-performance algorithms (Ge et al., 6 Jul 2025). Omitting it can produce transient apparent advantages for a handedness matched to a measurement model, but those advantages do not persist asymptotically.

Fourth, the use of invariant filtering does not imply that every observation model becomes invariant. Several successful applications explicitly note that their measurement models are not invariant—human trunk estimation with forward-kinematics misalignment is one example—yet the invariant propagation alone still improves conditioning and convergence (Zhu et al., 2022, Zhu et al., 2022). This corrects the misconception that InEKF is only useful when both process and measurement models are exactly invariant.

Fifth, bias augmentation often breaks perfect group-affinity. Contact-aided legged estimation and related navigation filters frequently treat biases in an additive Euclidean subspace appended to the Lie-group state, sometimes described as an “imperfect InEKF” (Hartley et al., 2018, Hartley et al., 2019). This weakens the strict theoretical guarantees of the bias-free invariant formulation but does not eliminate the observed empirical advantages.

Finally, InEKF should not be conflated with a single fixed handedness rule. The recent equivalence theorem demonstrates that the left/right distinction is ultimately a coordinate distinction when reset is properly implemented (Ge et al., 6 Jul 2025). At the same time, later work on covariance-transformation-based ESKF indicates that in mixed-frame sensing problems, changing the covariance representation to match each measurement can improve practical performance (Han et al., 1 Nov 2025). Taken together, these results suggest that the deeper issue is not “which handedness is correct,” but how uncertainty is represented and transported across propagation and update.

7. Historical development and current directions

The modern InEKF literature grew out of invariant observer theory and the recognition that many navigation and robotics systems possess symmetry structures best expressed on Lie groups. In contact-aided locomotion and extended-pose inertial navigation, the use of $\log_G$9 and $(\cdot)^\wedge:\mathbb{R}^n\to\mathfrak{g}$0 provided a concrete route to log-linear error dynamics and state-estimate-independent Jacobians in the prediction step (Hartley et al., 2018, Hartley et al., 2019, Pavlasek et al., 2021). These ideas then expanded to SLAM, human motion analysis, aerial and marine vehicles, pedestrian dead reckoning, and visual localization [(Barczyk et al., 2014); (Zhu et al., 2022); (Ye et al., 2023); (Benham et al., 12 Jun 2025); (Zhang et al., 15 Aug 2025)].

Recent research directions emphasize three themes. The first is theoretical clarification. The equivalence of left and right IEKF with reset, the pseudoinverse treatment of noise-free pseudo-measurements, and the iterated invariant update all refine the mathematical interpretation of invariant estimation (Ge et al., 6 Jul 2025, Goffin et al., 2024, Goffin et al., 2024).

The second is hybridization with exteroception and smoothing. Multi-sensor fusion for quadrupeds has extended InEKF and invariant smoothing to combine IMU, kinematics, LiDAR, and GPS, with observation models explicitly derived to satisfy group-affine properties. In those experiments, exteroceptive fusion significantly reduced absolute trajectory error relative to LiDAR odometry baselines, with reported improvements of up to $(\cdot)^\wedge:\mathbb{R}^n\to\mathfrak{g}$1 indoors and $(\cdot)^\wedge:\mathbb{R}^n\to\mathfrak{g}$2 outdoors (Nisticò et al., 29 Apr 2025).

The third is augmentation with learned or adaptive models. Neural slip compensation for legged robots, LSTM-based aerodynamic-angle estimation for UAVs, and adaptive noise strategies in slippery locomotion all indicate that invariant filtering is increasingly used as a geometric substrate onto which learned components are attached (Lee et al., 26 Jan 2026, Ye et al., 2023, Teng et al., 2021). This suggests a broader synthesis between symmetry-preserving estimation and data-driven residual modeling.

Overall, InEKF now occupies a distinct place in state-estimation theory: not merely as an EKF adapted to manifolds, but as a framework in which group structure, invariant error definition, and probabilistic uncertainty representation are designed jointly. Its current development is marked both by expansion into new sensing modalities and by increasingly precise analysis of when its celebrated properties—trajectory independence, consistency, and invariance to coordinate choice—do and do not hold (Ge et al., 6 Jul 2025).