Centre of mass

The centre of mass (COM) of a body made of parts with masses $m_i$ at positions $x_i$ is

$$x_{\text{COM}} = \frac{\sum_i m_i\, x_i}{\sum_i m_i}$$

For the whole body’s motion, gravity acts as if all the mass sat at this point. The COM doesn’t have to be inside the body: in a deep backbend it can sit in the air under the arch.

Why it matters for balance

Standing still, a body is in balance when its COM is directly above the area enclosed by its feet, the support polygon. If the COM moves outside it, gravity creates a torque about the edge of the feet, and the body starts to topple.

A fighter’s weight split is a statement about the COM. “70 % on the rear foot” means the COM sits 30 % of the way from the rear foot to the lead foot. Taking moments about the rear foot, the lead foot carries the share

$$\frac{F_{\text{lead}}}{F_{\text{lead}} + F_{\text{rear}}} = \frac{d_{\text{COM}}}{d_{\text{feet}}}$$

where $d$ is the distance from the rear foot.

Moving bodies

Once the body is moving, the COM’s position alone no longer predicts a fall. Its velocity matters too: a COM that is inside the feet but moving fast will soon be outside. The capture point combines both.

In this project

The solved stance places the COM where the 70/30 split needs it (shown as a red line dropped to the floor in the balance post). Each body segment’s mass and its own centre of mass come from the de Leva (1996) tables (see anthropometry).