Newton-Euler Equations

A pair of dynamics equations describing a rigid body's translation (F = ma) and rotation (Euler's equation) together.

The Newton-Euler equations combine Newton's second law with Euler's equation for rigid-body rotation to describe how a rigid body translates and rotates under applied forces and torques: F = ma, and τ = Iα + ω×(Iω). F is the net external force, m is mass, a is the acceleration of the center of mass; τ is the net torque about the center of mass, I is the inertia tensor, α is angular acceleration, ω is angular velocity, and ω×(Iω) is the gyroscopic term produced by the rotation itself. A robot is made of many links, and each one satisfies this pair of equations. The classic recursive Newton-Euler algorithm first sweeps from base to tip, computing each link's velocity and acceleration, then sweeps back from tip to base, computing forces and torques, ending with the torque required at every joint — its computation cost grows only linearly with the number of joints. Along with the Euler-Lagrange equations, it's one of the two main routes to deriving robot dynamics.

ExampleGiven an arm's current joint angles, joint velocities, and the desired joint accelerations, the recursive Newton-Euler algorithm computes the torque each motor should output — exactly the feedforward term used in computed-torque control.

Also called

Newton-Euler Formulation

Related

Recursive Newton-Euler Algorithm · Euler-Lagrange Equations · Inverse Dynamics · Rigid-Body Dynamics · Inertia Tensor · Computed Torque Control

Sources

Wikipedia: Newton–Euler equations
Modern Robotics 8.3: Newton-Euler Inverse Dynamics (Northwestern)