I’m currently learning the game engine Godot . As an introductory project, I am going to create a boids -style simulation that uses some light Lie group theory. Let’s first review the basic ingredients of a Godot project. If I got any of the details of Godot wrong, let me know! Basics Nodes are Class Instances A node refers to an instance/object of a class. Since each node is the instance of some class, there is a mapping ...

A review of the basic concepts that go into the design and training of VLA models.

I’ve been going through Russ Tedrake’s notes on robotics, which got me thinking about their so-called monogram notation. The result of this deliberation was a new notation for spatial velocities that bridges the gap between abstract Lie groups and their applications to robotics and computer graphics.

A principal bundle is a fiber bundle, that additionally has the right-action of a group that preserves fibers. Beginning with coset spaces, we look at some interesting examples of principal bundles and the things we can do with them.

Most students’ first time seeing an infinite-dimensional space is when they learn about the Fourier transform. Yet, I always hesitate to play around with them because I do not know how (and moreover, whether!) the theorems that I know from finite-dimensional linear algebra can be extended to the infinite-dimensional setting.

Some results about the differentials (i.e., pushforwards) of the exponential and logarithm maps of a Lie group. I rely extensively on the interpretation of tangent vectors as equivalence classes of curves.

Poincaré’s 1901 paper introduces (in just a humble 3 pages) the Euler-Poincaré equations, which are the specialization of the Euler-Lagrange equations to the case where a Lie group acts on the configuration manifold. I work through Poincaré’s paper without making too many identifications.

The Fourier transform maps a complex-valued function to a function on its Pontryagin dual space. Generalizations of this concept to non-Abelian compact and locally compact Lie groups are reviewed.

Despite having encountered the Lagrangian and Hamiltonian formalisms of mechanics several times in a variety of engineering and physics settings, I had never been able to retain it in my memory. In this post, I would like to collect my thoughts on their differential geometric treatment, which assumes a very simple and memorable form once we introduce the language of symplectic geometry.

I will construct the Haar measure for SO(3) and pull it back to the axis-angle parametrization of orthogonal matrices. The goal is to be able to integrate (and thereby define probability densities) on SO(3).