recursive Newton–Euler algorithm

The recursive Newton–Euler algorithm is a fast, organized recipe for computing inverse dynamics — that is, for working out exactly how much torque each joint of a robot arm needs to produce a desired motion. Its cleverness is that instead of solving one giant tangle of equations all at once, it walks along the chain of links in two passes, handling one link at a time. Think of an arm as a row of people holding hands: information flows out from the shoulder to the fingertip, then a tally is carried back from the fingertip to the shoulder.

The first pass moves outward, from the base of the arm to the tip. Starting from how the base is moving, it computes, link by link, how fast each successive link is turning and accelerating — because each link sits on top of the motion of the one before it. Once it reaches the end-effector, the second pass turns around and sweeps back inward, from tip to base. On this return trip it adds up the forces and torques each link must exert on its neighbor, accumulating the load as it goes, until every joint's required torque has been worked out.

Why bother with this two-pass structure? Because it is remarkably efficient. The straightforward way of writing out a robot's full equations of motion grows explosively as you add joints, but the recursive Newton–Euler approach scales gently and in proportion to the number of joints — fast enough to run hundreds or thousands of times per second on real hardware. That speed is exactly what lets a robot's controller compute its torques on the fly while it moves, which is why this algorithm is a workhorse buried inside most modern robot-control software.