New Coordinate System Math

[Dylan123]

Hello,

We are currently using an older faro arm and an older version of CAM to measure cylindrical parts. We are creating a new coordinate system using the 3-2-1 method (plane line circle) with the origin being centered on the part. We figured out how to achieve this using the CAM software but are trying to replicate this coordinate translation using the raw data (x y z points and ABC angles of the head)  and MATLAB.

Our end goal is to export the raw data obtained from the faro arm and be able to perform the same math the CAM software is doing when you create a new coordinate system using the 3-2-1 method. Is anyone familiar with the math behind translating/rotating the machine coordinate system to a part coordinate system?

Thanks for the help

[Zaffin_D]

I know enough to be dangerous, but I don’t understand what you are trying to do.  Can you provide an example?

[Dylan123]

I don’t really have an example. Basically the FARO arm has machine coordinates and all the points taken using it are in this coordinate system. What I want to know is what math is being done to each point that is taken after making a new coordinate system using the 3-2-1 method. 
 

we want to take the points that are in the machine coordinates of the faro arm , import them to matlab and replicate the math that is being done by the cam software. 
 

hopefully that makes it a little more clear. Thanks for the response. 

[Zaffin_D]

Ok I’m taking a shot in the dark without knowing how the data is represented, but this may get you started.

In order to move the positions between coordinate systems you need move the point to world, then move the point to the new system.  
 

This can done by multiplying the point through the inverse (same as the transpose if the matrix is orthonormal) of the original coordinate system’s matrix (that will put the point in world; X=1,0,0; Y=0,1,0; Z=0,0,1), then multiply the point in world through the new matrix to put it in that space.

It sounds like the new matrix will be a rotation matrix based on the machines angles, but again; this is a shot in the dark.

 

Hope that helps get you going.

[crazy^millman]

Reach out to Verisurf they have a solution that will work with that device inside of Mastercam and do everything like your asking and done. Cool thing is you can make the model then compare the CAD make back to the part and see if everything is where you expect it. 

[Colin Gilchrist]

On 7/16/2021 at 1:23 PM, Dylan123 said:

Hello,

We are currently using an older faro arm and an older version of CAM to measure cylindrical parts. We are creating a new coordinate system using the 3-2-1 method (plane line circle) with the origin being centered on the part. We figured out how to achieve this using the CAM software but are trying to replicate this coordinate translation using the raw data (x y z points and ABC angles of the head)  and MATLAB.

Our end goal is to export the raw data obtained from the faro arm and be able to perform the same math the CAM software is doing when you create a new coordinate system using the 3-2-1 method. Is anyone familiar with the math behind translating/rotating the machine coordinate system to a part coordinate system?

Thanks for the help

http://modernrobotics.org

 

Chapters 1-5 are great foundational material for understanding the underlying mathematics. Chapters 6, and 6.2 cover the inverse kinematic matrix math.

Luckily, a Cartesian Coordinate System uses orthogonal matrices, which belong to the SO3 Lie Group.

Mathematically, you'll need to be sure to check the New CSYS vectors are unitized, before performing any rotation or transformation on the matrix. Are you familiar with the Dot Product and Cross Product? Linear Algebra?

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/

There are a couple other archived versions of 18.06 on the MIT OCW site. 

 

 

https://en.m.wikipedia.org/wiki/Euler_angles

https://en.m.wikipedia.org/wiki/Euler's_rotation_theorem

https://en.m.wikipedia.org/wiki/3D_rotation_group

https://en.m.wikipedia.org/wiki/Charts_on_SO(3)

https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf

 

 

[Colin Gilchrist]

This has got several different equation forms for obtaining the same result > a 3D rotation. 

https://home.aero.polimi.it/trainelli/downloads/Bottasso_ThreeDimensionalRotations.pdf

And I would be remiss if I didn't mention Martin Baker, and his wonderful website Euclidean Space. 

https://www.euclideanspace.com/