Motivation for the ESKF in the paper example

[knightshrub]

Hello Jeremy and Joan, first of all, thank you for the great overview paper and library! It really allowed me to get a quick overview of the most important ideas behind Lie groups even though I had never heard of them before!

As someone who is quite new to robotics and state/pose estimation, I'm wondering what the motivation for using an error-state KF instead of a regular EKF is for the localization example in section V-A? I can see that tracking the pose (non-linear globally, because its part of a Lie group) using a separate "object" and then just operating on tangent vectors like in a regular KF is quite elegant, because vector addition and covariance definition now actually make sense. Is this the natural solution to building a KF state estimator using Lie group theory as presented in the paper? Is the ESKF solution a superior choice in IMU driven systems, where IMU measurements are available and need to be integrated into the state at high rate, so that interpreting IMU data as a control input circumvents the expensive EKF update step?

I'm wondering if it would be possible to define something more akin to a "normal" EKF that also allows me to track the velocity in the tangent space correctly (x velocity, y velocity and yaw rate)? So the state vector would be in R^6, with the first 3 entries being the tangent vector of X \in SE(2) at the identity (origin of coordinate system) and the last 3 entries are the x velocity, y velocity and yaw rate.

[joansola]

Hello Elia

In the Lie context, EKF and ESKF can be considered to be the same: you compute an error in the tangent space, which is the minimal space, which is the error space:

dx = K * (y - h(x) )

and then update the state with it

x = x * Exp (dx) = x (+) dx

If you overload the (+) operator with a regular + sign, then you can write

x = x + K * ( y - h(x) )

which "looks like" standard EKF.

[joansola]

As per adding velocities, no problem. You have many options, but three are the main lines:

1. SE(3) x R6
2. R3 x SO(3) x R3 x R3
3. Use Rn and Bundles

---

1. You can proceed as so

SE3 X;
Vector6d V; // Or SE3Tangent V;

2. You can proceed as so

   Vector3d p;
   SO3 R;
   Vector3d v;
   Vector3d w; // or SO3Tangent w;

3. Also, you can make use of the Rn groups in Manif, and the Bundle to build a composite state. Again, you have the two flavors:

X \in <SE3, R6>

X \in < R3, SO3, R3, R3>

[knightshrub]

Hello Joan, thanks for the clarification!

I'll think some more about a possible implementation for my use case. Alas, bundles don't seem to be supported by the Python bindings, but I think I can manage without them.

[artivis]

@knightshrub you may be interested in giving a look at kalmanif. It is a project that turns the manif filtering examples into more reusable code.

It doesn't quite support Bundle yet, I started the integration on the feature/diff_drive branch plus it has no Python bindings....

[sadsimulation]

Hi from a different github account! I came up with a formulation of my problem using the notation of your paper using the suggested solution of SE(2)xR3 as described in this document. I modeled the system noise as random accelerations in the local tangent space. Does the derivation of the Jacobians for the prediction step and the update using a twist look correct? As far as I understand it, when using a compound state, the Kalman correction vector will be in the tangent space of SE(2) and R3, so I have to use (+) for the pose update but can use regular + for the velocity update as R3's tangent space is also R3 so in this case (+) == +.

[joansola]

Looks good to me!

Only beware of the covariance of the perturbation when passing from continuous time to discrete time. Please read Appendix E in here: https://arxiv.org/pdf/1711.02508.pdf