Zero Moment Line—Universal Stability Parameter for Multi-Contact Systems in Three Dimensions

The widely used stability parameter, the zero moment point (ZMP), which is usually defined on the ground, is redefined, in this paper, in two different ways to acquire a more general form that allows its application to systems that are not supported only on the ground, and therefore, their support polygon does not extend only on the floor. This way it allows to determine the stability of humanoid and other floating-based robots that are interacting with the environment at arbitrary heights. In the first redefinition, the ZMP is represented as a line containing all possible ZMPs, called the zero moment line (ZML), while in the second redefinition, the ZMP is represented as the ZMP angle, i.e., the angle between the ZML and the vertical line, passing through the center of mass (COM) of the investigated system. The first redefinition is useful in situations when the external forces and their acting locations are known, while the second redefinition can be applied in situations when the COM of the system under study is known and can be tracked. The first redefinition of the ZMP is also applied to two different measurements performed with two force plates, two force sensors, and the Optitrack system. In the first measurement, a subject stands up from a bench and sits down while being pulled by its hands, while in the second measurement, two subjects stand still, hold on to two double handles, and lean backward. In both cases, the stability of the subjects involved in the measurements is investigated and discussed.