The invariant extended Kalman filter (IEKF) (not to be confused with the iterated extended Kalman filter) was first introduced as a version of the extended Kalman filter (EKF) for nonlinear systems possessing symmetries (or invariances),[1] then generalized and recast as an adaptation to Lie groups of the linear Kalman filtering theory.
The main benefit is that the gain and covariance equations have reduced dependence on the estimated value of the state.
Consider a system whose state is encoded at time step
Alternatively, the same theory applies to a measurement defined by a right action: The invariant extended Kalman filter is an observer
defined by the following equations if the measurement function is a left action: where
If measurement function is a right action then the update state is defined as: The discrete-time framework above was first introduced for continuous-time dynamics of the shape: where the vector field
It leads to more involved computations than the discrete-time framework, but properties are similar.
The main benefit of invariant extended Kalman filtering is the behavior of the invariant error variable, whose definition depends on the type of measurement.
For left actions we define a left-invariant error variable as: while for right actions we define a right-invariant error variable as: Indeed, replacing
by their values we obtain for left actions, after some algebra: and for right actions: We see the estimated value of the state is not involved in the equation followed by the error variable, a property of linear Kalman filtering the classical extended Kalman filter does not share, but the similarity with the linear case actually goes much further.
be a linear version of the error variable defined by the identity: Then, with
we actually have:[2] In other words, there are no higher-order terms: the dynamics is linear for the error variable
This result and error dynamics independence are at the core of theoretical properties and practical performance of IEKF.
[2] Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged.
From mathematical and engineering viewpoint, it makes sense that a filter well-designed for the considered system should preserve the same invariance properties.
The idea for the IEKF is a modification of the EKF equations to take advantage of the symmetries of the system.
The previous system with noise is said to be invariant if it is left unchanged by the action the transformations groups
, since the standard output error usually does not preserve the symmetries of the system.
Given the considered system and associated transformation group, there exists a constructive method to determine
This much simpler dependence and its consequences are the main interests of the IEKF.
Near such trajectories, we are back to the "true", i.e. linear, Kalman filter where convergence is guaranteed.
Informally, this means the IEKF converges in general at least around any slowly varying permanent trajectory, rather than just around any slowly varying equilibrium point for the EKF.
Invariant extended Kalman filters are for instance used in attitude and heading reference systems.
In such systems the orientation, velocity and/or position of a moving rigid body, e.g. an aircraft, are estimated from different embedded sensors, such as inertial sensors, magnetometers, GPS or sonars.
The use of an IEKF naturally leads[6] to consider the quaternion error
, which is often used as an ad hoc trick to preserve the constraints of the quaternion group.
The benefits of the IEKF compared to the EKF are experimentally shown for a large set of trajectories.
[7] A major application of the Invariant extended Kalman filter is inertial navigation, which fits the framework after embedding of the state (consisting of attitude matrix
[8] defined by the group operation: The problem of simultaneous localization and mapping also fits the framework of invariant extended Kalman filtering after embedding of the state (consisting of attitude matrix
for planar systems)[8] defined by the group operation: The main benefit of the Invariant extended Kalman filter in this case is solving the problem of false observability.