Newtonmas meets Festivus

A series of Easter eggs for your holiday season! Since December 23rd is Festivus and December 25th is Isaac Newton's birthday (Newtonmas), it makes sense to combine the two into a discussion about the physics of pole balancing.

Inverted pendulum in dynamic equilibrium? Festivus pole. Next up, the airing of grievances!

Newtonmas poster (with fractals). The physics of planetary precession is why we celebrate today! COURTESY: Felix Andrews.

The pole balancing problem, also called the inverted pendulum, is a classic model system in the application of reinforcement learning (a form of supervised learning). The problem requires the supervisor to keep a pole banaced while the base moves back and forth along a one-dimensional plane. This should keep eveyone at the Festivus party busy until the airing of grievances! 

An example of balancing an inverted pendulum (e.g. pole) on a cart. COURTESY: MIT Signals and Systems course, Lecture 26.

Many applications of the inverted pendulum involve balancing the inverted pendulum on a cart [1]. The application of reinforcement learning is often (but not necessarily) used to drive the controller, or where to move the cart in response to inertial forces generated by the free-swinging pole. Using the cart as the base of support, motion of the pole is translated along a single degree of freedom. You may recall the last Synthetic Daisies post in which we discussed holonomic motion. The dynamic equilibrium exhibited by the inverted pendulum is a linear version of those physics.

A demonstration of the first-order Lagrangian used in pendulum mechanics can be found in this video, COURTESY: PhysicsHelps YouTube channel.

The description of this sometimes chaotic [2] motion can be described using Lagrangian mechanics, which is a more refined form of Newton's equation of motion [3]. Yet the policy required to maintain balance of the pole can be rather simple, largely involving first-order, closed-loop feedback and and an iterative function. Hence, Newtonmas is really a celebration of the physical processes that govern our holiday adventures. Happy Newtonmas to all, and to all a good (well-controlled) system!

Inverted pendular mechanics, simulated and visualized in Python. COURTESY: "An Introduction to Numerical Modeling II", Adam Dempsey.

NOTES:

[1] there are many demonstrations of this (class projects, hobbyists) on YouTube. This is also a classic benchmark for control systems design.

[2] the identification of chaos in an inverted pendulum (particularly when we move to the double pendulum case) stems from the Lagrangian representation. For more, please see the following references: 

a) Kim, S-Y. and Hu, B.   Bifurcations and transitions to chaos in an inverted pendulum. Physical Review E, 58(3), 3028-3035 (1998).

b) Duchesne, B., Fischer, C.W., Gray C.G., Jeffrey, K.R.   Chaos in the motion of an inverted pendulum: an undergraduate laboratory experiment. American Journal of Physics, 59(11), 987-992 (1991).

[3] some practical pointers to the difference between Newtonian and Lagrangian physics can be found on the Physics StackExchange here and here,