Advice on implementing a new group...

[jkelly-stars]

Hi all - thanks for the nice work on ManifoldsBase.jl and Manifolds.jl. I'm in the process of trying to determine how best to implement a new group - specifically, the "group of extended poses," which is in fact the semidirect product of SE(3) and $\mathbb{R}^3$. I began by defining the semidirect product of SE(3) and $\mathbb{R}^3$ as a SemidirectProductGroup, but this seems to lead to quite complex code that's not easy to parse (by me).

Notably, the group can be embedded in GL(6) and that's how I will use it, essentially exclusively. I'd like to develop a lightweight, matrix-based implementation - unsure whether to define a new group using Manifolds.jl (also unsure if there's a tutorial on implementing a new group?) or to try to use ManifoldsBase.jl only. Any advice would be appreciated; the best possible speed is important, and given the fixed size of the matrices involved, I intend to use StaticArrays.jl.

Also, a rather silly q (as a newbie) - I'm uncertain why the representation size is given as the manifold dimension + 1? What am I missing? Calling representation_size() on SO(2) gives (2, 2), so I'm somewhat confused.

J.

[kellertuer]

For now we indeed do not yet have a tutorial on how to implement a (Lie) group – but hopefully the ManifoldsBase.jl tutorial on a manifold is a good starting point – e.g. on how our functions are usually meant to be implemented. this includes the important difference between the allocating and in-place (fast!) variants we usually have in mind.
Then the code for one of the groups a good continuation.
Sure, StaticArrays should not be a problem, neither points nor tangent vectors are usually typed (more than necessary) which is exactly to be able to use other arrays formats if you would like to.

If you feel it is easier, then do not go the SemidirectProductGroup but implement your own group in the embedding to GL(6) as you write.

For the representation size – that is the array size used to represent points and tangent vectors for a certain manifold – which is unrelated to the dimension.
Example: For the sphere, your observation is correct, the dimension of the sphere is n and then its points are represented as vectors (arrays) of length n+1 (unit vectors to be precise).
SO(2) is represented as 2x2 rotation matrices, hence its representation size is (2,2)
The symmetric positive definite matrices of size $n\times n$ are a manifold of dimension $\frac{n(n+1)}{2}$, so here representation_size(M) would be (n,n) while manifold_dimension(M) returns n*(n+1)/2.

Feel free to ask further questions along the way and if you like, a PR for new manifolds and/or groups is always welcome.

[jkelly-stars]

Super thanks! As I become more familiar, I might be able to write up a tutorial. That makes good sense re: the sizing.

[kellertuer]

A tutorial on how to implement a group would be very cool indeed.

[mateuszbaran]

Hi! We don't currently have a tutorial for Lie groups but I think the best way to go is for example copy the implementation of HeisenbergGroup and modify methods to fit your group. SemidirectProductGroup is fairly advanced and not fully developed solution so I think it's fine to ignore it.

Representing that group by 6x6 matrices is one way to go but it might not be totally efficient. You can also consider using ArrayPartition of three static arrays, and derive optimized implementation using the embedding in GL. That's what I sometimes do when implementing methods for SE(n). You can also take a look how the ArrayPartition is used for special euclidean group.

Manifolds.jl interface generally can work with both SArray and MArray, though we don't always have the necessary specialized non-mutating methods for SArray -- feel free to report if you need some. We add them when there is demand.

[mateuszbaran]

By the way, I'm currently slowly moving towards robotics and IMU integration, so there will be more support for it in Manifolds.jl and the ecosystem (see RoME.jl). I think the standard Lie group approach doesn't map to the right mathematical concepts, which appear to be vector bundles instead. I can explain more if you are interested.

[jkelly-stars]

Great thanks! Good point about ArrayPartition - I'll investigate. And absolutely, would be happy to hear more about IMU preintegration (something else I'm interested in).

[mateuszbaran]

There has been a recent work and discussion here that you can read for a start: JuliaRobotics/RoME.jl#701 . IMUDeltaGroup is very similar to the group of extended poses.

[jkelly-stars]

Ah great - very helpful! (and sorry for late reply) Also I realized I mistyped above in the original post (minor detail) - it should be an inclusion in GL(5).