This journey explains how inverted pendulums can be balanced with an open-loop vibrational controller.
All of these resources come from the excellent Twitter thread (referenced below) that @ahmedallibhoy posted on Feb. 1, 2022. The phase plane image in the thumbnail was generated by code written by Ahmed as well (also referenced below).
This Twitter thread is really all you need to click on. Everything below this is just a listing of the resources that are called out throughout the thread.
Twitter Thread: Vibrational control of mechanical systems
This Twitter thread by @ahmedallibhoy walks through an explanation of controlling an inverted pendulum with an open-loop vibrational controller.See More
This is the video that inspired the thread. Check out a vibrational controller in action!
Suspending a pendulum in its inverted position using a jig saw (Take 1, norm...
When a pendulum is attached to a jig saw, strange things can happen! :)See More
For more detailed information on the mathematics behind what's happening in the above YouTube video and Twitter thread, check out this paper ...
Averaging and Vibrational Control of Mechanical Systems
Abstract. This paper investigates averaging theory and oscillatory control for a large class of mechanical systems. A link between averaging and controllability theory is presented by...See More
... and this paper.
Vibrational control of nonlinear systems: Vibrational controllability and tr...
In the first part of this work, the criteria for the existence of stabilizing parametric oscillations have been derived. In the present paper, the problem of choosing the stabilizing...See More
If you want to generate the gifs and images in the thread yourself, Ahmed has posted all of the code on GitHub.
Jupyter Notebook: Code used to generate vibrational control of inverted pend...
Jupyter Notebook: Code used to generate vibrational control of inverted pendulum figuresSee More
These next two books were recommended by Ahmed as great sources for those who are really interested in reading more about this theory.
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields...
From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review...See More
Averaging Methods in Nonlinear Dynamical Systems
Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the...See More
This video was recommended by Mario Figueiredo in the comments of the thread. It's a well-produced video explaining some research from UCI School of Engineering that shows vibrational control in insect flight.
Vibrational Control in Insect Flight
Abstract: It is generally accepted among biology and engineering communities that insects are unstable at hover. However, existing approaches that rely on direct averaging do not fully...See More