How to derive ABA algorithm for rigid body system with rotors using gaussian constraint principle？

[xie-design]

[pwensing]

Hello!

Thanks for your interest. You can find a write-up in Chapter 4 of an undergrad thesis out of my lab by Sebastian Echeandia. Here's the file: 2020 - Sebastian Echeandia Thesis.pdf

The approach follows the same spirit as the Russian ABA derivation by Vereshchagin in 1974 that originally used Gauss Principle. It's hard to find these days, so here's the file: Vereshchagin74-Dynamics.pdf

There's an alternative derivation of ABA w/rotors in Chapter 12 of Jain's book Robot and Multibody Dynamics based on his spatial operator algebra factorization of the mass matrix and its inverse -- this also appears in a 1990 CDC paper by Jain. There's a closely related 1990 ICRA paper by Murphy et al. that follows a derivation more similar to what showed up in Featherstone's original paper.

I hope this helps!

[xie-design]

Thank you very much for your reply, which saved me a lot of time searching for the derivation ！！！

[xie-design]

Hello！

Thank you very much for the reference materials you provided. I mainly refer to an undergrad thesis out of your lab by Sebastian Echeandia. I want to confirm that in that thesis, the torque value tau in equation 4.2 is actually the output torque, and the torque tau obtained through RENA verification is the total torque. In reality, the tau we can obtain is always the total torque.

[pwensing]

Happy to save you some trouble! Let me address a few of the other questions.

In reality, the tau we can obtain is always the total torque.

Agreed. The torque that is produced between the stator and the rotor is a "total torque" in the sense that some of it goes to accelerating the rotor, while the rest is delivered to the links after the gearbox. With this we'd have $\tau_{\rm links} = N ( \tau_{\rm total} - \tau_{\rm rotor})$ where $\tau_{\rm links}$ is the torque at the output of the gearbox, $( \tau_{\rm total} - \tau_{\rm rotor})$ is the torque at the input of the gearbox, and $\tau_{\rm total}$ is the torque between the rotor and stator.

I want to confirm that in that thesis, the torque value tau in equation 4.2 is actually the output torque.

It's actually a little trickier, sorry. The torque value tau in equation 4.2 is $N \tau_{\rm total}$, that is, it's the total torque, but scaled by the gear ratio. An alternative approach would be to use $\tau_i/N$ in equation 4.2 (and account for the gear ratio suitably elsewhere), but I think most people are used to thinking about the torque as a post-gearbox quantity, so we tried to keep it as similar as what people are used to as possible.

Does that all make sense?

In the package, we follow this convention everywhere (i.e., the torque input to ABA follows the same convention as the torque output from RNEA). The unit tests show that it all checks out.