Frames and origin in SO2 Manifold

[benzun_1999]

Hi

I am working on a robot that has a spinning 3D laser scanner. It rotates about two axis and collects data. In one axis it has full 3D rotation and in another axis it has limit rotation.

Now the read world points collect by this laser scanner is not unifomaly distributed but if parametreized in the SO2 manifold it will be uniform. Now if was in another point on the robot the points that i observe will be distributed different. Is there a way to understand how the points will be from this new location?

I am new to manifolds. I don't know if i explained the problem correctly. Can anyone point me to a book, idea, notes that can help me understand this. If you are interested I can try explaining more about it and would like to collobrate.

Thanks,
Benzun

 

[Mandelbroth]

Honestly, I don't exactly know what you mean. You are a little vague on your details.

$SO_2(\mathbb{R})$ is a very "nice" manifold (in fact, group) to work with because it has a lot of structure on it. I'm sure we can help somehow if you give us a better description of your problem.

 

[benzun_1999]

Yes. I can explain you more. I know very litte about topology and manifolds.

A rotating laser basically rotates about two axis and generates a response for every angles so in SO2(R).

If you watch that video the white line is the scan generated at every instand of time. The scan itself is composed of points obtained by rotation about another axis.

So all the points are sampled uniformly in this space. Now I need to move the origin of scan to another location in the world and find the relationship between the original scan and the new points after shifting the origin.

Using ecludean geometry i can do it but since i am sampling the world in SO3 I feel it might be faster and easier to compute the transform that occurs due to change in orgin more accurately and easily if i solve it in SO3 manifold.

 

[Mandelbroth]

Yes. I can explain you more. I know very litte about topology and manifolds.

A rotating laser basically rotates about two axis and generates a response for every angles so in SO2(R).

If you watch that video the white line is the scan generated at every instand of time. The scan itself is composed of points obtained by rotation about another axis.

So all the points are sampled uniformly in this space. Now I need to move the origin of scan to another location in the world and find the relationship between the original scan and the new points after shifting the origin.

Using ecludean geometry i can do it but since i am sampling the world in SO3 I feel it might be faster and easier to compute the transform that occurs due to change in orgin more accurately and easily if i solve it in SO3 manifold.

I apologize, but I am slightly more confused now. In the video, it looked like several white lines were being used. Also, you switch from saying "SO2" to "SO3." Your wording is slightly confusing.

I'm not an expert, but I'm going to guess that the scan from the new origin will overlap with the one from the original origin. Could you demonstrate what you are trying to say?