Zero Moment Point - ZMP Computation

ZMP Computation

The resultant force of the inertia and gravity forces acting on a biped robot is expressed by the formula:

where is the total mass of the robot, is the acceleration of the gravity, is the center of mass and is the acceleration of the center of mass.

The moment in any point can be defined as:

where is the rate of angular momentum at the center of mass.

The Newton–Euler equations of the global motion of the biped robot can be written as:

where is the resultant of the contact forces at X and is the moment related with contact forces about any point X.

The Newton–Euler equations can be rewritten as:

so it’s easier to see that we have:

These equations show that the biped robot is dynamically balanced if the contact forces and the inertia and gravity forces are strictly opposite.

If an axis is defined, where the moment is parallel to the normal vector from the surface about every point of the axis, then the Zero Moment Point (ZMP) necessarily belongs to this axis, since it is by definition directed along the vector . The ZMP will then be the intersection between the axis and the ground surface such that:

with

where represents the ZMP.

Because of the opposition between the gravity and inertia forces and the contact forces mentioned before, the point (ZMP) can be defined by:

where is a point of the sole where is the normal projection of the ankle.