Math & Kinematics

Lie Group

A Lie group is a mathematical group that is also a smooth manifold, so that group multiplication and inversion are differentiable operations. Robotics relies chiefly on SO(3), the group of 3D rotations, and SE(3), the group of rigid-body transformations, whose associated Lie algebras provide local linearizations used via the exponential and logarithm maps. This machinery underlies state estimation on manifolds, trajectory optimization, and modern kinematics formulations such as product of exponentials.

Why it matters for physical AI

Correctly handling rotation geometry, rather than treating poses as flat vectors, is essential for stable state estimators, well-conditioned policy action spaces, and physically sensible interpolation of end-effector motions.