Newton–Euler formulation

The Newton–Euler formulation derives a robot's dynamics the direct, physical way: by tracking every force and twist acting on each individual link and insisting they balance, just as the great laws of motion demand. It uses two partner rules. Newton's law governs straight-line pushing — the net force on a link equals its mass times how fast its center is accelerating. Euler's law governs spinning — it ties the net twisting effort (torque) on a link to how stubbornly it resists turning and how fast its spin is changing. Apply both to every link, and you have captured the whole robot's motion.

The natural way to do this is link by link, like passing a message down and back along a chain. First you sweep outward from the robot's base to its tip, working out how fast each link is moving and accelerating, since each link inherits the motion of the one before it. Then you sweep back inward from the tip to the base, adding up the forces: each link must support its own inertia plus whatever it is handing back from the links beyond it. By the time you reach the base you know precisely the force and torque at every joint.

This back-and-forth recipe is wonderfully efficient, which is why a refined version of it — the recursive Newton–Euler algorithm — is what real robots run thousands of times a second to compute the torques they need on the fly. It reaches exactly the same equations of motion as the energy-based Lagrangian approach; the two are simply different routes to the identical truth. Engineers reach for the Lagrangian method when they want a clean formula to study, and for Newton–Euler when they want raw speed in a live controller.