Math & Kinematics

Riemannian Geometry

Riemannian geometry is the mathematics of smooth manifolds equipped with metrics that define distance, geodesics, and curvature. Robot orientation and pose spaces, such as the rotation group SO(3) and rigid-motion group SE(3), are curved manifolds, so interpolation, averaging, filtering, and optimization over them must respect geodesic structure rather than treat coordinates as Euclidean. Riemannian motion policies extend the framework to compose reactive controllers across task spaces.

Why it matters for physical AI

Orientation errors, pose distributions, and trajectory optimization all live on curved manifolds, and geometry-aware formulations avoid the singularities and distortions that plague naive Euclidean treatments of rotation.