Parallel Axis Theorem

Gives the moment of inertia about any parallel axis from the moment of inertia about the center-of-mass axis: I = I_c + md².

The parallel axis theorem, also called the Huygens–Steiner theorem, states that a rigid body's moment of inertia I about some axis equals its moment of inertia I_c about a parallel axis through the center of mass, plus its mass m times the square of the distance d between the two axes: I = I_c + m·d². This shows the moment of inertia is smallest about an axis through the center of mass, and grows the farther the axis sits from it. In 3D there's a matrix version (Modern Robotics calls it the Steiner theorem): I_q = I_b + m(qᵀq·E − q·qᵀ), where I_b is the inertia tensor at the center of mass, q is the new reference point's position relative to the center of mass, and E is the 3×3 identity matrix. It's used constantly in robotics: URDF requires the inertia reference frame's origin to be at the center of mass, and dynamics libraries have to translate it when computing in joint frames; attaching a gripper or a payload to a flange means translating that payload's inertia before merging it in; and dynamic parameter identification also uses it to convert between different reference points.

ExampleA uniform thin rod of mass m and length L has a moment of inertia of mL²/12 about its midpoint; about one end (equivalent to a link rotating about a joint), d = L/2, giving I = mL²/12 + m(L/2)² = mL²/3 — four times as large.

Also called

Huygens–Steiner Theorem, Steiner's Theorem

Related

Moment of Inertia · Inertia Tensor · Inertial Parameters · Center of Mass (CoM) · Dynamic Parameter Identification · Unified Robot Description Format

Sources

Wikipedia: Parallel axis theorem
Modern Robotics (Lynch & Park), Theorem 8.2 Steiner's theorem