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,