Inertia Tensor

A 3×3 symmetric matrix describing how hard a rigid body is to rotate about any given axis.

The inertia tensor (also called the inertia matrix) is the complete description of a rigid body's rotational inertia, given as a 3×3 symmetric positive-definite matrix. The diagonal entries Ixx, Iyy, Izz are the moments of inertia about the x, y, and z axes — for example Ixx = Σm(y² + z²) — while the off-diagonal entries are called products of inertia, such as Ixy = −Σm·x·y (sign convention varies by textbook), and reflect whether rotating about one axis tends to drag the body around another. It relates angular velocity ω to angular momentum through L = Iω, and rotational kinetic energy is K = ½ωᵀIω. Rotating the reference frame gives I' = RᵀIR, and changing the reference point uses the parallel axis theorem; eigendecomposition yields the principal axes, along which the products of inertia all vanish. URDF's inertial tag requires filling in each link's mass, center-of-mass location, and the six independent entries of the inertia tensor — getting these wrong will throw off both dynamics simulation and torque control.

ExampleA 2 kg point mass located at (0.1, 0, 0) m has Ixx = 0, Iyy = Izz = 2×0.1² = 0.02 kg·m², and all products of inertia equal to zero — meaning it costs nothing to spin about an x-axis passing through the point, while rotating about the y- or z-axis must overcome 0.02 kg·m² of inertia.

Also called

Inertia Matrix, Rotational Inertia Matrix, 3×3 Inertia Matrix

Related

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

Sources

Wikipedia: Moment of inertia（Inertia tensor）
Lynch & Park, Modern Robotics（预印本 PDF，8.2 节 Dynamics of a Single Rigid Body；4.2 节 URDF） (Chinese)