What is Invariant Extended Kalman Filter?
An Invariant Extended Kalman Filter is a symmetry-aware state estimator that defines estimation error on a Lie Group and exploits group-affine dynamics to simplify error propagation.
Quick Facts
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How It Works
Define error through group composition
For group state X and estimate X_hat, a right-invariant error can use X X_hat^-1, while a left-invariant error can use X_hat^-1 X. The logarithm map converts a local group error into Lie Algebra coordinates where covariance is represented. Prediction composes the estimate with a group increment rather than adding global coordinates.
Left and right conventions change Jacobians, adjoint mappings, noise injection, and measurement residuals. Mixing formulas across conventions can produce code that compiles yet reports covariance in the wrong tangent frame.
Exploit group-affine error dynamics
Barrau and Bonnabel's stability analysis identifies a class of systems whose invariant error is autonomous and whose logarithm follows an exact linear differential equation. This removes some dependence of error propagation on the estimated trajectory and supports stronger local stability reasoning than a generic EKF.
The result is conditional, not universal. Bias augmentation, calibration states, time-varying gravity, contact switches, or arbitrary observation functions may break strict group-affine structure or require a larger group and carefully derived invariant measurements.
Validate geometry, observability, and consistency
GTSAM's EKF variants documentation separates manifold, Lie Group, and invariant filters and exposes their different prediction contracts. That distinction is essential: a retraction-based EKF preserves the state manifold but does not necessarily have state-independent invariant error dynamics.
Test exponential and logarithm maps near chart boundaries, adjoint and Jacobian identities, left/right residual signs, noise frames, and covariance transport. Use Monte Carlo normalized estimation error squared, innovation statistics, unobservable-direction tests, and large-initial-error trials. Better consistency on one navigation model is not a guarantee for a different group or sensor model.
Key Characteristics
- Represents the state on a Lie Group and uncertainty in a tangent space
- Defines left- or right-invariant estimation errors through group composition
- Can obtain state-independent error dynamics for group-affine systems
- Uses exponential, logarithm, adjoint, and retraction operations explicitly
- Requires measurement and noise models consistent with the chosen invariant frame
- Is distinct from both a generic manifold EKF and an Iterated EKF
Common Use Cases
- Inertial navigation with rotation, velocity, and position states
- Robot pose estimation on SE(2) or SE(3)
- Contact-aided state estimation for legged robots
- SLAM formulations with symmetry-aware error coordinates
- Consistency studies under large orientation or pose initialization errors
Example
Loading code...Frequently Asked Questions
How is an Invariant EKF different from a standard EKF?
A standard EKF typically subtracts coordinates and linearizes dynamics around the estimated trajectory. An InEKF defines error through Lie Group composition; for group-affine dynamics, its invariant error can evolve independently of the estimated trajectory.
Is an InEKF the same as an Iterated EKF?
No. An InEKF exploits state-space symmetry and Lie Group error coordinates. An Iterated EKF repeatedly relinearizes one nonlinear measurement update. Both may be abbreviated IEKF, so implementations should use unambiguous names such as InEKF and Iterated EKF.
Does using quaternions make an EKF invariant?
No. A quaternion stores rotation without a singular local Euler-angle chart, but invariance depends on the chosen group error, group-affine system structure, covariance coordinates, noise injection, and measurement model. Quaternion normalization alone is insufficient.
Should left- or right-invariant error be used?
The choice depends on the dynamics, measurement symmetry, frame conventions, and which error yields simpler or autonomous dynamics. It changes residuals, adjoints, Jacobians, and covariance interpretation; the two forms cannot be mixed casually.
How should an InEKF be tested?
Verify group identities and finite-difference Jacobians, test frame transformations and branch boundaries, run Monte Carlo consistency metrics, inspect unobservable directions, and compare against a standard manifold EKF under identical models, data, initialization, and tuning.