Capture point

Position and velocity determine the point

The capture point was introduced for humanoid push recovery as a way to connect centre-of-mass state with a recoverable support location. In the constant-height linear inverted pendulum model, its planar position can be written as centre-of-mass position plus centre-of-mass velocity divided by the model's natural frequency.

That velocity term is the important part. A robot can have its centre of mass above the current support polygon while already moving too quickly for the present contact to arrest. Conversely, stepping to a suitable future contact can recover a state that looks unrecoverable from the current foot alone.

Capture point and DCM coincide under common assumptions

Under the constant-height linear inverted pendulum assumptions used in HULK, the instantaneous capture point is equivalent to the planar divergent component of motion. The two names emphasise different interpretations: one asks where support should be placed to stop, while the other describes the unstable component of the simplified dynamics.

The equivalence is conditional. Variable height, angular momentum, non-coplanar contacts, hand support, and more complete centroidal models require extensions or a different state description.

Outside the current foot does not automatically mean a fall

If the capture point lies inside a feasible current support region, the model says an admissible centre of pressure can arrest the divergent motion without stepping. If it lies outside, the current contact alone cannot do so under the model. A planned step can move the future support region, so “outside the support polygon” is not by itself a fall prediction.

Friction limits, actuator saturation, delayed state estimates, swing-foot reachability, terrain geometry, and model error can all invalidate a purely geometric conclusion. The capture point is a control abstraction, not a complete hardware safety certificate.

For robot data, preserve the centre-of-mass estimate and velocity, assumed height, reference frame, active and predicted contacts, support-region construction, external-wrench compensation, filter settings, and whether the value was planned, estimated, or measured. A capture-point trace without those assumptions is not reproducible evidence of balance.