Moment of Inertia

How hard an object is to spin up or slow down about a given axis, I = Σmr².

Moment of inertia describes an object's resistance to rotating about a given axis, defined as the sum, over every part of the object, of its mass times the square of its distance from that axis: I = Σmr², in kg·m²; Euler introduced this concept in 1765. It plays the same role in rotation that mass plays in straight-line motion: τ = Iα, where τ is torque and α is angular acceleration. For the same mass, the farther it's distributed from the axis, the larger the moment of inertia: a uniform rod has moment of inertia ml²/12 about its midpoint, but ml²/3 about one end. A 3D rigid body needs the full 3×3 inertia tensor to describe its moment of inertia about every possible axis. Robot designers commonly push motors toward the torso and use lightweight materials for shins and fingers precisely to reduce the moment of inertia of the far-out links, so limbs can swing faster with less effort. Note this is different from the “second moment of area” used in structural mechanics, despite the similar name.

ExampleMany bipedal and quadruped robots mount the knee motor up near the top of the thigh and drive the knee joint through a linkage or belt instead, keeping the shin light and reducing its moment of inertia during a leg swing.

Also called

Rotational Inertia, Mass Moment of Inertia

Related

Inertia Tensor · Torque · Parallel Axis Theorem · Inertial Parameters · Rigid-Body Dynamics · Reflected Inertia

Sources

Wikipedia: Moment of inertia
Modern Robotics 8.2: Dynamics of a Single Rigid Body (Part 1 of 2)