Math & Kinematics

Generalized Coordinates

Generalized coordinates are a minimal set of independent variables that completely specify the configuration of a mechanical system, such as the joint angles of a serial manipulator or the base pose plus joint angles of a floating-base robot. Lagrangian mechanics expresses dynamics directly in these coordinates, yielding the standard manipulator equation involving the mass matrix, Coriolis terms, and generalized forces. Their time derivatives are generalized velocities.

Why it matters for physical AI

Every dynamics engine, whole-body controller, and trajectory optimizer operates on generalized coordinates; understanding them is prerequisite to reading the equations of motion that govern simulation and model-based control.