Jacobian matrices relate joint velocities to end-effector velocities, essential for inverse kinematics and control. Overview The Jacobian matrix $J$ maps joint velocities $\dot{q}$ to end-effector velocity $\dot{x}$: $$ \dot{x} = J(q) \dot{q} $$ Where: $\dot{x} = [\dot{x}, \dot{y}, \dot{z}, \omega_x, \omega_y, …