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?

 

[alphy]

Hello Benzun,

Thank you for sharing your work on the robot with the spinning 3D laser scanner. It sounds like a fascinating project!

To address your question about understanding the distribution of points from a different location, it would be helpful to have more information about the specific setup and parameters of your robot and laser scanner. However, in general, understanding the frames and origin in the SO2 manifold can help with this.

The SO2 manifold is a mathematical representation of the space of 2D rotations. In other words, it describes all possible orientations of a 2D object. By using this manifold, you are able to parametrize the points collected by your laser scanner in a way that is independent of the robot's location and orientation. This allows for a uniform distribution of points, regardless of where the robot is located.

To understand the points from a different location, you can use the concept of transformation between frames in the SO2 manifold. This involves using mathematical operations to translate and rotate the points collected from one location to another. This can help you visualize how the points would appear from different perspectives.

As for resources to help you understand manifolds and their applications, I would recommend books such as "Introduction to Manifolds" by Loring W. Tu and "Differential Geometry of Curves and Surfaces" by Manfredo P. Do Carmo. You can also find online lectures and notes on the subject from universities such as MIT and Stanford.

I hope this helps and I wish you the best of luck with your project. If you have any further questions or would like to collaborate, please feel free to reach out.

Best,