Capture point

The capture point (Pratt et al. 2006) answers a simple question: if I’m moving like this, where do I have to put my foot to stop?

Model the body as an inverted pendulum: all its mass at the centre of mass, at a constant height $z$, balanced on a massless leg. If the COM is at horizontal position $x$ moving at velocity $\dot{x}$, the capture point is

$$\xi = x + \frac{\dot{x}}{\omega}, \qquad \omega = \sqrt{\frac{g}{z}}$$

where $g = 9.81\ \text{m/s}^2$. It’s the COM pushed ahead by its velocity. The faster the body moves, the further out the capture point.

Why it’s the right measure

The capture point moves away from the foot it’s pivoting on, exponentially:

$$\dot{\xi} = \omega\,(\xi - p)$$

where $p$ is the point on the floor the feet are pushing from (the centre of pressure). The feet can only push from inside the support polygon. If $\xi$ is inside, the feet can place $p$ beyond it and pull it back to rest. If $\xi$ is outside, no push from the current feet can stop it: the body must step to a new spot near $\xi$, or fall.

A worked number

For a COM about 1 m high, $\omega = \sqrt{9.81 / 1} \approx 3.1\ \text{s}^{-1}$. A sideways drift of 0.2 m/s puts the capture point $0.2 / 3.1 \approx 6.5$ cm beyond the COM. If the stance is 16 cm wide and the COM starts in the middle, that drift alone uses most of the margin.

In this project

The capture point drives everything after the shove:

• the balance gate rises as the capture point moves 6 to 15 cm from where the reference puts it;
• recovery mode starts when the capture point leaves the feet, or creeps toward the edge;
• the shaping reward penalises the capture point outside the feet (see reward shaping);
• recovery ends once the capture point is 2 cm inside the feet and the body is calm.

The balance post found that sideways shoves made the body topple 0.6–2.4 s after the shove, as the capture point crept to the edge of the feet.