Math & Kinematics

Velocity Kinematics

Velocity kinematics is the study of the relationship between a robot's joint velocities and the linear and angular velocities of its links, expressed through the manipulator Jacobian, which maps joint rates to the end-effector twist. The Jacobian's transpose maps end-effector wrenches back to joint torques, and configurations where the Jacobian loses rank are kinematic singularities at which certain motions become infeasible.

Why it matters for physical AI

Every Cartesian controller, teleoperation retargeter, and singularity-avoidance scheme runs on velocity kinematics; policies that command end-effector twists depend on it to translate their outputs into joint motion.