Zero Moment Point

The point where the ground-reaction force produces no horizontal moment — the classic balance criterion for biped walking.

The zero moment point (ZMP) was introduced by Miomir Vukobratović and Davor Juričić in 1968: it's the point at which the ground-reaction force produces no moment about the horizontal axes, which on flat ground coincides with the center of pressure (where the resultant ground-reaction force acts). The center of pressure can only ever fall within the support polygon, so the ZMP criterion requires that the planned ZMP stay inside the footprint's support region at all times — otherwise the robot tips over the edge of its foot. When the center-of-mass height is constant, p = x − (z/g)ẍ, where p is the horizontal ZMP position, x and z are the center of mass's horizontal position and height, ẍ is its horizontal acceleration, and g is gravitational acceleration. This lets engineers fix the footstep locations and a ZMP trajectory first, then work backward to the center-of-mass trajectory — the approach earlier-generation humanoids such as Honda's ASIMO used to walk; today's reinforcement-learning-based locomotion controllers generally don't compute ZMP explicitly.

ExampleWhen the robot stands still, ẍ = 0 and the ZMP sits directly beneath the center of mass's vertical projection. When it wants to accelerate forward, the ZMP shifts toward the heel, moving further the harder it accelerates — and once it crosses the heel's edge, the toes lift off and the whole robot tips over backward.

Also called

ZMP, ZMP Criterion

Related

Center of Pressure (CoP) · Support Polygon · Linear Inverted Pendulum Model (LIPM) · ZMP Preview Control · Capture Point · Bipedal Locomotion

Sources

Zero moment point - Wikipedia
Underactuated Robotics (Tedrake), Ch. Simple Models of Walking: CoP and ZMP