Do you mean the tangent space of SO(2) as a state?

For simple angles you can use an element of SO(2) as your state. But if the algorithm calls for a difference of two states, the result would have to be an element of a real vector space (in this case a single number between -pi and pi). Analogue to this an addition of a state manifold and a vector also results in a state manifold. You can switch between those representations using the exp and log functions. In the papers these „new“ operators are described as boxplus and boxminus. And the elegant thing of this approach is that in most cases you just would replace plus with boxplus and minus with boxminus and it just works(tm)

Yeah implementing the "+" operator on the tangent space is pretty common not just for Kalman filters but also nonlinear optimization in general. The Ceres library supports LocalParameterization which does that.

Yeah LocalParametrization is exactly what they describe. The main contribution is a pretty slick mathematical model. When you use it everything just seems to work out.

Also ceres is pretty great. LocalParametrization in combination with jets works like a treat for EKFs. You never have to worry about calculating jacobians on manifolds ever again.

Looks like gibberish advertising for the project

It’s pretty hard to explain in short. But estimation algorithms don’t work on angles for example. For an algorithm an angle is just a number. So the difference between 359 and 1 degree looks pretty big. It is also not possible to just put a rotation matrix inside a state, because the algorithm has no concept of orthonormality. But if you define the change of a state in terms of vectors and the state itself as a rotation matrix you can just use off the shelf estimation algorithms like least squares and Kalman filters. You just need to redefine the plus and minus operators. Which is nothing you need a project for. You can just do it yourself.

Then you have some reading to do.

Sometimes technobabble is appropriate.