Zero Moment Point (ZMP)

A walking robot is not standing still, so the simple rule of keeping the center of mass over the feet is not enough: accelerating the body also pushes on the ground. The zero moment point is where the ground's push effectively acts. As long as it stays inside the foot, the foot stays flat. This idea drove humanoid walking for decades, from Honda's ASIMO onward.

Center of mass forward offset Body acceleration forward Center of mass height Foot length

The cart-table model

Treat the body as a point mass at height $z_c$ above the ground. If its horizontal position is $x$ and acceleration $\ddot x$, the zero moment point on the ground is

$$p_{\text{ZMP}} = x - \frac{z_c}{g}\,\ddot x$$

The foot stays flat on the ground as long as $p_{\text{ZMP}}$ lies between the heel and the toe.

What to notice

Standing still, the ZMP sits right under the center of mass. Accelerate the body forward and the ZMP moves backward toward the heel, since the feet must push forward on the ground. A taller center of mass makes the same acceleration move the ZMP further, which is why tall robots must move gently. Push the ZMP off the foot and the robot rolls over its heel or toe. ZMP-based walking controllers plan body motion so the ZMP always stays well inside the feet.