Flywheel Inverted pendulum torque calculation

[Baba Greb]

Hey all,
I started building my own flywheel inverted pendulum (i'm using a standard PID controller for the balancing).
I came across many articles regarding the differential equation of the pendulum.
Most article develop the motion equation using the Euler–Lagrange, few chose the Newtonian way.
What I can't figure out is the definition of the torque that all Euler–Lagrange articles made:
image 1
https://www.dropbox.com/s/2ijg9tu2aj3t12w/1.PNG?dl=0

image 2
https://www.dropbox.com/s/n09hxwwf51dbj4f/2.PNG?dl=0

Why is the torque for the wheel different from the rod?
why isn't the torque a single value given by the motor mounted on the wheel?

I would expect it to be the same torque only in the reversed direction like in a Newtonian development:
image 3
https://www.dropbox.com/s/eog1da7j4d8tecy/3.PNG?dl=0

Please help :)

 

[eljose79]

Hello,

Thank you for sharing your experience with building your own flywheel inverted pendulum and your question about the torque in the Euler-Lagrange method. As a scientist who has studied and researched the dynamics of pendulum systems, I would be happy to provide some insight into your question.

Firstly, it is important to note that the Euler-Lagrange method and the Newtonian method are two different approaches to solving the dynamics of a system. The Newtonian method uses the principles of Newton's laws of motion to derive the equations of motion, while the Euler-Lagrange method uses the principle of least action to derive the equations of motion. Both methods lead to the same equations of motion, but the approach and interpretation of the equations may differ.

In the Euler-Lagrange method, the torque is defined as the product of the generalized coordinate and the generalized force. In the case of the flywheel inverted pendulum, the generalized coordinate is the angle of the pendulum and the generalized force is the torque applied to the pendulum. Therefore, the torque in the Euler-Lagrange method is the torque applied to the pendulum to maintain its equilibrium.

On the other hand, in the Newtonian method, the torque is defined as the product of the moment of inertia and the angular acceleration. In the case of the flywheel inverted pendulum, the moment of inertia and the angular acceleration of the flywheel are different from that of the pendulum rod. This is because the flywheel has its own moment of inertia and angular acceleration due to its rotation, while the pendulum rod has a different moment of inertia and angular acceleration due to its oscillation. Therefore, the torque in the Newtonian method is the torque required to maintain the rotation and oscillation of the flywheel and pendulum, respectively.

In summary, the torque in the Euler-Lagrange method and the Newtonian method has different definitions and interpretations. The torque in the Euler-Lagrange method is the torque applied to the pendulum, while the torque in the Newtonian method is the torque required to maintain the rotation and oscillation of the flywheel and pendulum. I hope this helps clarify your understanding of the torque in the Euler-Lagrange method. Please let me know if you have any further questions.