Math & Kinematics

Jacobian Matrix

The Jacobian matrix in robotics is the matrix of partial derivatives mapping joint velocities to end-effector linear and angular velocity, with each column giving the end-effector twist produced by unit motion of one joint. Its transpose maps end-effector wrenches to joint torques, its rank drops at kinematic singularities, and its pseudoinverse drives differential inverse kinematics and operational-space control.

Why it matters for physical AI

Jacobian computations run inside virtually every manipulator controller, converting task-space commands from planners or learned policies into joint motion while exposing singularities and manipulability limits.