Math & Kinematics

Pseudoinverse

The pseudoinverse, formally the Moore-Penrose generalized inverse, extends matrix inversion to non-square and rank-deficient matrices, yielding the least-squares solution of minimum norm. In robotics its central use is the Jacobian pseudoinverse, which maps desired end-effector velocities to joint velocities for redundant manipulators, with damped least-squares variants trading tracking accuracy for bounded joint speeds near singularities.

Why it matters for physical AI

Jacobian-pseudoinverse mappings run inside virtually every Cartesian controller and teleoperation pipeline, and their singularity behavior explains characteristic failures such as joint-velocity blowups near workspace boundaries.