NewtonEuler

Newton-Euler equations describe the dynamics of a rigid body by combining Newton's law for translation with Euler's equations for rotation. They provide a compact framework to relate external forces and moments to the translational and angular accelerations of a body or link. The equations can be written in inertial or body-fixed frames, with appropriate expressions for time derivatives.

For a rigid body with mass m, center of mass G, linear velocity v_G, and angular velocity ω about G, the translational equation is F = m a_G. The rotational equation is d/dt(I ω) + ω × (I ω) = τ, where I is the inertia tensor about G, and τ is the sum of external moments about G; H = I ω is the angular momentum. In body-fixed coordinates the equations take a form that includes derivative terms due to the rotating frame.

In multibody dynamics, the Newton-Euler method is a recursive algorithm used to compute forward dynamics or inverse dynamics efficiently for systems of linked rigid bodies. It proceeds with a forward recursion to propagate velocities and accelerations along a kinematic chain, and a backward recursion to accumulate joint forces and torques. This makes it especially suitable for serial-link robotic manipulators and tree-structured mechanisms.

Historically attributed to Isaac Newton and Leonhard Euler, Newton-Euler dynamics underpin many simulations in robotics, aerospace, biomechanics, and computer graphics. They offer an alternative to Lagrangian methods and are frequently chosen when a modular, link-based formulation is advantageous.