Question about invariant and consistent

[Joanna-HE]

Hi, great job and thanks for your open source!

I would like to ask that what is the difference between invariant and consistent in the context of EKF?

Hope to hear from you!

[ChangwuLiu]

In the context of EKF, invariant' refers to the so-called invariant extended Kalman filter (InEKF). InEKF originates from a series of so-call symmetry-preserving observers, which means the observer mimics the symmetry of the system it observes. From my perspective, the term invariant' comes from the invariance' of dynamical systems, i.e. the transformations mapping flow into flow (this is the meaning of symmetry). Due to the symmetry-preserving properties, such observers possess certain guaranteed stability properties. Initially, the invariant filter' can only be applied to `invariant systems', a rather small class of systems. Later in Barrau and Bonnabel's paper (2015), they extend those stability results to group-affine systems, which basically cover 6-DoF pose estimation in navigation.

Inconsistency in EKF, roughly speaking, happens in partially unobservable nonlinear systems when the unobservable subspace of the EKF does not match that of the true system (system without noise).
As you could see, the invariant filter is not intended to solve the inconsistency problem from its original starting point. A rather interesting by-product is that switching the non-linear error form (re-defining the retractiong) can let the filter see the correct' unobservable directions. Benefiting from the state-estimate-independent' Jacobians in InEKF, the EKF based on the right-invariant error is actually compatible with the unobservable structure of 6-DoF SLAM problem, thus invariant filter can be adopted for consistent filter design in VIO.

[Joanna-HE]

Thanks for your patient and kind response! My problem is solved.

…

________________________________ 寄件者: Changwu Liu ***@***.***> 寄件日期: 2023年6月12日 下午 01:18 收件者: ChangwuLiu/InGVIO ***@***.***> 副本: ***@***.*** ***@***.***>; Author ***@***.***> 主旨: Re: [ChangwuLiu/InGVIO] Question about invariant and consistent (Issue #6) In the context of EKF, invariant' refers to the so-called invariant extended Kalman filter (InEKF). InEKF originates from a series of so-call symmetry-preserving observers, which means the observer mimics the symmetry of the system it observes. From my perspective, the term invariant' comes from the invariance' of dynamical systems, i.e. the transformations mapping flow into flow (this is the meaning of symmetry). Due to the symmetry-preserving properties, such observers possess certain guaranteed stability properties. Initially, the invariant filter' can only be applied to `invariant systems', a rather small class of systems. Later in Barrau and Bonnabel's paper (2015), they extend those stability results to group-affine systems, which basically cover 6-DoF pose estimation in navigation. Inconsistency in EKF, roughly speaking, happens in partially unobservable nonlinear systems when the unobservable subspace of the EKF does not match that of the true system (system without noise). As you could see, the invariant filter is not intended to solve the inconsistency problem from its original starting point. A rather interesting by-product is that switching the non-linear error form (re-defining the retractiong) can let the filter see the correct' unobservable directions. Benefiting from the state-estimate-independent' Jacobians in InEKF, the EKF based on the right-invariant error is actually compatible with the unobservable structure of 6-DoF SLAM problem, thus invariant filter can be adopted for consistent filter design in VIO. ― Reply to this email directly, view it on GitHub<#6 (comment)>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/ANJUIXQIKN247YPS2WURXHDXK2RCPANCNFSM6AAAAAAZCMMX3E>. You are receiving this because you authored the thread.Message ID: ***@***.***>