EKF v.s. ESKF or Direct v.s. Indirect KF

Last updated on May 7, 2023 pm

[TOC]

Overview

EKF and ESKF:

EKF: Extended Kalman Filter --> Direct Kalman Filter
ESKF: Error State Kalman Filter --> Indirect Kalman Filter

Estimator State Types:

True State
Nominal State
Error State

ESKF Advantages:

The orientation error-state is minimal (i.e., it has the same number of parameters as degrees of freedom), avoiding issues related to over-parametrization (or redundancy) and the consequent risk of singularity of the involved covariances matrices, resulting typically from enforcing constraints.
The error-state system is always operating close to the origin, and therefore far from possible parameter singularities, gimbal lock issues, or the like, providing a guarantee that the linearization validity holds at all times.
The error-state is always small, meaning that all second-order products are negligible. This makes the computation of Jacobians very easy and fast. Some Jacobians may even be constant or equal to available state magnitudes.
The error dynamics are slow because all the large-signal dynamics have been integrated in the nominal-state. This means that we can apply KF corrections (which are the only means to observe the errors) at a lower rate than the predictions.

EKF

Generic Problem Formulation

Consider a nonlinear, bounded, observable system with continuous process dynamics and discrete measurement as

\[ \begin{aligned} \dot{\mathbf{x}}(t) &=\mathbf{f}(\mathbf{x}(t), \mathbf{u}(t))+\Gamma \mathbf{w}(t) , \quad \mathbf{w} \sim \mathcal{N}(0, Q) \\ \mathbf{z}_{k} &=\mathbf{h}\left(\mathbf{x}_{k}\right)+\mathbf{v}_{k} , \quad \mathbf{v}_{k} \sim \mathcal{N}\left(0, R_{k}\right) \end{aligned} \]

and

\[ \mathbf{x}(0)=\mathbf{x}_{0} \]

The process and measurement noise are assumed to be zero mean, band-limited, uncorrelated, white multivariate Gaussian processes given by

\[ \begin{aligned} E\left[\mathbf{w}(t) \mathbf{w}^{T}(\tau)\right] &=Q \delta(t-\tau)=\left\{\begin{array}{lr} Q, & t=\tau \\ 0, & t \neq \tau \end{array}\right.\\ E\left[\mathbf{v}_{k} \mathbf{v}_{j}^{T}\right] &=R_{k} \delta_{k j}=\left\{\begin{array}{ll} R_{k}, & k=j \\ 0, & k \neq j \end{array}\right. \end{aligned} \]

\(Q\): continuous process noise covariance matrix
\(R\): discrete measurement noise covariance matrix

Prediction

\[ \begin{aligned} \dot{\hat{\mathbf{x}}}(t) &=\mathbf{f}(\hat{\mathbf{x}}(t), \mathbf{u}(t)) \\ \dot{P}(t) &=F(t) P(t)+P(t) F^{T}(t)+\Gamma Q \Gamma^{T} \\ F(t) &=\left.\frac{\partial \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t))}{\partial \mathbf{x}(t)}\right|_{\mathbf{x}(t)=\hat{\mathbf{x}}(t), \mathbf{u}(t)} \end{aligned} \]

\(P\): state error covariance matrix

EKF Transition to Update Stage:

\[ \begin{aligned} \hat{\mathbf{x}}_{k}^{-} &=\hat{\mathbf{x}}(k \Delta t)+\dot{\hat{\mathbf{x}}}(t) \cdot \Delta t \\ P_{k}^{-} &=P(k \Delta t)+\dot{P}(t) \cdot \Delta t \end{aligned} \]

Update

measurement process

\[ \hat{\mathbf{z}}_{k}^{-}=\mathbf{h}\left(\hat{\mathbf{x}}_{k}^{-}\right) \]

kalman gain

\[ K_{k} =P_{k}^{-} H_{k}^{T}\left(H_{k} P_{k}^{-} H_{k}^{T}+R_{k}\right)^{-1} \]

where (w.r.t true-state)

\[ \color{blue}{H_{k} =\left.\frac{\partial \mathbf{h}\left(\mathbf{x}_{k}\right)}{\partial \mathbf{x}_{k}}\right|_{\mathbf{x}_{k}=\hat{\mathbf{x}}_{k}^{-}}} \]

finally

\[ \begin{aligned} \hat{\mathbf{x}}_{k} &=\hat{\mathbf{x}}_{k}^{-}+K_{k}\left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k}^{-}\right) \\ P_{k} &=\left(I-K_{k} H_{k}\right) P_{k}^{-} \end{aligned} \]

ESKF

consider the following simplified nonlinear process model:

\[ \dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)) \]

Applying a small perturbation δx (t) around x (t) yields

\[ \dot{\mathbf{x}}(t)+\delta \dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)+\delta \mathbf{x}(t)) =\mathbf{f}(\mathbf{x}(t))+\nabla \mathbf{f}(\mathbf{x}(t)) \delta \mathbf{x}(t)+\mathcal{O}(t, \mathbf{x}(t), \delta \mathbf{x}(t)) \]

Prediction

linear time varying system with δx as the state vector as

\[ \delta \dot{\mathbf{x}}(t)=F(t) \delta \mathbf{x}(t)+\Gamma \mathbf{w}(t) \]

the error-state process model is linear !!!

and

\[ \delta \mathbf{x}_{k}=\Phi_{k-1} \delta \mathbf{x}_{k-1}+\tilde{\mathbf{w}}_{k-1} , \quad \tilde{\mathbf{w}}_{k-1}=\int_{0}^{\Delta t} \Phi(t, 0) \Gamma \mathbf{w}(t) d t \]

where \(\tilde{\mathbf{w}}\) is the discretized process noise and \(\Phi_{k-1}\) is the state transition matrix.

the error state covariance matrix is propagated using

\[ P_{k}^{-}=\Phi_{k-1} P_{k-1} \Phi_{k-1}^{T}+Q_{k-1} \]

where \(Q_{k-1}\) is the discrete time equivalent process noise covariance matrix and is given by

\[ Q_{k-1}=\int_{0}^{\Delta t} \Phi(t, 0) \Gamma Q \Gamma^{T} \Phi^{T}(t, 0) d t \]

where, \(\Phi(t, 0)=e^{F(t) t}\) is the continuous time state transition matrix and \(Q\) is the continuous time process noise covariance matrix.

Update

The measurement model is assumed to be in direct discrete time

\[ \mathbf{z}_{k}=\mathbf{h}\left(\mathbf{x}_{k}\right)+\mathbf{v}_{k} \]

kalman gain

\[ \begin{aligned} S_{k} &=R_{k}+H_{k} P_{k}^{-} H_{k}^{T} \\ K_{k} &=P_{k}^{-} H_{k}^{T} S_{k}^{-1} \end{aligned} \]

where (w.r.t error-state)

\[ \color{blue}{ \mathbf{H} \left.\triangleq \frac{\partial h}{\partial \delta \mathbf{x}}\right|_{\mathbf{x}}} \]

finally

\[ \begin{aligned} \mathcal{I}_{k} &= \mathbf{z}_{k}-\mathbf{h}\left(\hat{\mathbf{x}}_{k}^{-}\right) \\ \delta \hat{\mathbf{x}}_{k} &=K_{k} \mathcal{I}_{k} \end{aligned} \]

\[ P_{k}=\left(I-K_{k} H_{k}\right) P_{k}^{-} \]

true-state estimate

\[ \hat{\mathbf{x}}_{k} =\hat{\mathbf{x}}_{k}^{-}+\delta \hat{\mathbf{x}}_{k} \]