Zero moment point

ZMP expresses a contact-wrench condition

ISO 19649:2017 defines the zero moment point on the support polygon through the moment resulting from forces exerted by the travel surface. In the usual flat-ground formulation, the tangential components of that moment are zero at the ZMP.

Classical biped literature also constructs the corresponding point from gravity and inertia acting on the robot. Newton–Euler balance connects that formulation to the required ground-contact wrench. The concept reduces a complex whole-body motion to a point that can be compared with the support polygon, and simplified models connect a desired ZMP trajectory to centre-of-mass acceleration.

Inside the polygon is a conditional criterion

If a planned contact wrench places the ZMP inside the active support polygon, the flat contact can in principle generate the required tipping moment without rotating about an edge. A value at the boundary leaves no margin in that direction. A calculated point outside the polygon describes a wrench the assumed contact cannot realise, rather than a physical pressure point outside the foot.

The criterion relies on assumptions about the support plane and contact. It does not by itself enforce friction, prevent foot slip, respect actuator limits or prove robustness to disturbances. The historical review by Vukobratović and Borovac discusses boundary and fictitious cases that are easily misinterpreted.

ZMP and centre of pressure coincide under matched assumptions

Sardain and Bessonnet use ZMP for the gravity-plus-inertia construction and centre of pressure for the ground–foot contact construction. For dynamically balanced motion on a shared plane, the equations make the points coincide. ISO’s support-force definition instead names that coincident contact-wrench point directly as the ZMP, so a source’s convention should be stated when the distinction matters.

On uneven or non-coplanar contacts, a single conventional ZMP needs an explicitly chosen virtual support surface or a more general wrench formulation. Dataset fields should state the reference plane and frame, contact set, computation method, filtering and whether a value is planned, estimated from state or derived from measured force–torque data.