Show how to use the An matrix in the Nonholonomic EoM chapter

[moorepants]

• Figure out how to show both methods: 1) substitute ur from the get go and 2) apply An.
• Adapt the code to support the two methods.
• Settle on new notation.

[github-actions]

Preview this pull request here https://moorepants.github.io/htmlpreview/?https://github.com/moorepants/learn-multibody-dynamics/blob/preview-pr-232/index.html

[moorepants]

@Peter230655 if you are interested, I made significant modifications to this chapter and any read through with review and feedback is very welcome. I will teach it Monday May 11 in class and have to finalize by then. It is already merged so you can see it on the website.

[Peter230655]

@Peter230655 if you are interested, I made significant modifications to this chapter and any read through with review and feedback is very welcome. I will teach it Monday May 11 in class and have to finalize by then. It is already merged so you can see it on the website.

I will start right away.

[Peter230655]

If I have any comments / questions should I put them here or in an email to you?

[moorepants]

Here is fine.

[Peter230655]

Above eq (201):
....as long as $A_r$ is invertible.
If $A_r$ is singular, likely there was a modeling error, too many dependent variables, I assume.
(I had such singularity issues when a played around with nonlinear velocity constraints)
Maybe a footnote adressing this?

Below eq (205)
...differentiation index 1
Do your students know what this is? I only learned it since I joined a lecture on DAEs
Maybe a footnote adressing this?

Above eq205)
Then using (180) and (187) the p nonholonomic.....
Maybe you could add a few steps to explain how to get from (180), (187) to (205), after all, (205) is important.

[moorepants]

Ar can be singular from unfortunate choice of coordinates and generalized speeds for a specific configuration (as well as what you mention).

I removed the mention of the differentiation index.

I think 205 arises directly from 180 and 187, there aren't really any steps between.

[Peter230655]

Ar can be singular from unfortunate choice of coordinates and generalized speeds for a specific configuration (as well as what you mention).

In my cases mentioned, I could always (as I recall) fix it by a better choice of coordinates.
I still feel that in a footnote you could mention reasons why $A_r$ may be singular, after all you indicate it may be singular.

I removed the mention of the differentiation index.

I think 205 arises directly from 180 and 187, there aren't really any steps between.

It was not obvious to me today. I will check again tomorrow and see if I will see it then.

[Peter230655]

About a page below Approach 2 (there is no equation anywhere close to point to the location) you write

Calculate $A_n$ from the nonholonomic constraints.

You give no explanation, how you do it. I assume you get it from eq(121), wq(122) (?)
Maybe a line of explanation would be useful.

[moorepants]

I can refer back to the equations in the motion.rst chapter.

[moorepants]

If A r is singular, likely there was a modeling error, too many dependent variables, I assume.

I will leave this out, as I don't have it formulated yet how to write exactly all reasons this can be invertible.