The Linear Inverted Pendulum Model for Walking

Walking is controlled falling. The linear inverted pendulum model treats the robot as a point mass at constant height, balanced over its foot like an upside-down pendulum. It falls away from the foot faster and faster, and the only way to stop it is to put the next foot down in the right place. The capture point says exactly where.

Step to the capture point Center of mass height Push strength

The model

With the center of mass at constant height $z_c$ and the foot at $p$, the horizontal motion is

$$\ddot x = \omega^2\,(x - p), \qquad \omega = \sqrt{g / z_c}$$

The robot moves away from its foot exponentially. The capture point is the one place where, if the foot were put there, the center of mass would come to rest exactly above it:

$$\xi = x + \frac{\dot x}{\omega}$$

What to notice

Push the robot without stepping and it falls away faster and faster. With stepping on, it places its next foot at the capture point and comes to rest over it, after one step for a small push. A harder push moves the capture point further, needing a bigger step. A taller robot falls more slowly, since $\omega$ is smaller, which gives it more time to react. Many walking controllers plan their footsteps from the capture point or its relatives, such as the divergent component of motion.