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Zero Moment Point

Zero Moment Point

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Zero Moment Point is a concept related with dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...

 and control of legged locomotion, e.g., for humanoid robot
Humanoid robot
A humanoid robot or an anthropomorphic robot is a robot with its overall appearance, based on that of the human body, allowing interaction with made-for-human tools or environments. In general humanoid robots have a torso with a head, two arms and two legs, although some forms of humanoid robots...

s. It specifies the point with respect to which dynamic reaction force at the contact of the foot with the ground does not produce any moment
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 in the horizontal direction, i.e. the point where total of vertical inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

 and gravity forces equals 0 (zero). The concept assumes the contact area is planar and has sufficiently high friction to keep the feet from sliding.


This concept was introduced in January 1968 by Miomir Vukobratović
Miomir Vukobratovic
Miomir Vukobratović is a Serbian mechanical engineer and pioneer in humanoid robots. His major interest is in the development of efficient modeling and control of robot dynamics.-Education:He received the B.Sc. and Ph.D...

 at The Third All-Union Congress of Theoretical and Applied Mechanics in Moscow. In the following works and papers that were produced between 1970 and 1972 it would then be called zero moment point and would be spread around the world.

The zero moment point is a very important concept in the motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

 planning for biped robots. Since they have only two points of contact with the floor and they are supposed to walk
WALK may refer to:*WALK , a radio station licensed to East Patchogue, New York, United States*WALK-FM, a radio station licensed to Patchogue, New York, United States...

, “run
Running is a means of terrestrial locomotion allowing humans and other animals to move rapidly on foot. It is simply defined in athletics terms as a gait in which at regular points during the running cycle both feet are off the ground...

” or “jump
Jumping or leaping is a form of locomotion or movement in which an organism or non-living mechanical system propels itself through the air along a ballistic trajectory...

” (in the motion context), their motion has to be planned concerning the dynamical stability of their whole body. This is not an easy task, especially because the upper body of the robot (torso) has larger mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

 and inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

 than the legs which are supposed to support and move the robot. This can be compared to the problem of balancing an inverted pendulum
Inverted pendulum
An inverted pendulum is a pendulum which has its mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart and pole...


The trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

 of a walking robot is planned using the angular momentum equation
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

 to ensure that the generated joint
A joint is the location at which two or more bones make contact. They are constructed to allow movement and provide mechanical support, and are classified structurally and functionally.-Classification:...

 trajectories guarantee the dynamical postural stability of the robot, which usually is quantified by the distance of the zero moment point in the boundaries of a predefined stability region. The position of the zero moment point is affected by the referred mass and inertia of the robot’s torso, since its motion generally requires large ankle torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

s to maintain a satisfactory dynamical postural stability.

One approach to solve this problem consists in using small trunk motions to stabilize the posture of the robot. However, some new planning methods are being developed to define the trajectories of the legs’ links in such a way that the torso of the robot is naturally steered in order to reduce the ankle torque needed to compensate its motion. If the trajectory planning for the leg links is well succeeded, then the zero moment point won’t move out of the predefined stability region and the motion of the robot will become smoother, mimicking a natural trajectory.

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:


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.

See also

  • Honda's Asimo
    is a humanoid robot created by Honda. Introduced in 2000, ASIMO, which is an acronym for "Advanced Step in Innovative MObility", was created to be a helper to people. With aspirations of helping people who lack full mobility, ASIMO is used to encourage young people to study science and mathematics...

     robot, which uses ZMP control.
    TOPIO is a bipedal humanoid robot designed to play table tennis against a human being. It has been developed since 2005 by TOSY, a robotics firm in Vietnam. It was publicly demonstrated at the Tokyo International Robot Exhibition on November 28, 2007...

  • Hubo
    HUBO is a walking humanoid robot, head mounted on a life-size walking bipedal frame, developed by the Korea Advanced Institute of Science and Technology and released on January 6, 2005...