Inertia

For straight-line motion, inertia is just mass: a heavier object needs more force for the same acceleration. For rotation it’s the moment of inertia $I$, which depends on where the mass is:

$$I = \sum_i m_i\, r_i^2$$

where $r_i$ is each bit of mass’s distance from the rotation axis. Mass far from the axis counts much more, because of the square. A figure skater spins faster by pulling in their arms: same mass, smaller $I$.

Newton’s second law for rotation links it to torque:

$$\tau = I\,\alpha$$

Some shapes

shape (mass $m$)	axis	$I$
rod of length $L$	through its centre, across the rod	$\tfrac{1}{12} m L^2$
rod of length $L$	through one end	$\tfrac{1}{3} m L^2$
solid sphere, radius $R$	through its centre	$\tfrac{2}{5} m R^2$

A leg swinging from the hip is closer to the rod about one end: four times harder to swing than the same rod turned about its middle.

The inertia tensor

In 3D a body can turn about any axis, and its inertia is a 3 × 3 matrix, the inertia tensor. Simulators usually store it as three principal moments along the body’s natural axes, plus a rotation.

Why it matters in a fighter

Body segments’ inertia decides how fast a limb can swing and how the body reacts when struck. Getting it right is part of why a simulated motion looks human. A kick with the wrong leg inertia would look too fast or too sluggish.

In this project

Segment inertias come from the de Leva (1996) tables (see anthropometry). The simple body copies them from the muscular body, so the two have the same overall mass distribution.