Robot control
Zero-moment point
Definition
The zero-moment point is a point on a chosen support plane where the net moment associated with the ground reaction wrench has zero components parallel to that plane. In flat-ground walking with the usual contact assumptions, it coincides with the center of pressure.
Also known as: ZMP, Zero moment point
Updated
Which moment becomes zero
The name does not mean that every torque acting on the robot vanishes. In three dimensions, the two moment components parallel to the support plane are zero; a moment about the plane's normal can remain. MIT's derivation distinguishes this from the simpler planar case.
For coplanar ground contacts that push without adhesion, with a nonzero total normal force, the center of pressure lies inside the convex hull of the active contact points. Under these assumptions, that center of pressure is the physical ZMP.
Planning center-of-mass motion
A walking planner can choose a ZMP trajectory inside the support polygon and derive a compatible center-of-mass trajectory. Constant center-of-mass height and negligible change in angular momentum lead to the familiar linear inverted pendulum model.
Limits of the balance criterion
A desired ZMP inside the feet is a contact-moment condition, not a complete guarantee of balance. Friction, joint torque, reachability, and future motion must also be feasible. The MIT notes warn that simplified planning can miss joint constraints. Uneven contacts, hand support, flight, and substantial angular-momentum changes require a model that represents those conditions explicitly.
Sources
Related terms
Support polygon
The support polygon is the convex hull of a robot’s active contact points or contact patches projected onto a common support plane. It describes the available support region in planar contact models.
Center of mass
The center of mass is the mass-weighted average position of a body or a collection of bodies. For an articulated robot, its position changes as the links move.
Linear inverted pendulum model
The linear inverted pendulum model approximates a walking robot by a mass moving at constant height above its support, with simplified angular-momentum dynamics. These assumptions make horizontal center-of-mass acceleration linear in the displacement from the support point.
Centroidal dynamics
Centroidal dynamics describe the motion of a multibody system’s center of mass and the evolution of its total linear and angular momentum. External forces and moments determine the rates of change of those momenta.