Interaction Stability Analysis from the Input-Output Viewpoints

Interaction with the environment is arguably one of the necessary actions for many robot applications such as haptic devices, manipulation, parts assembly, cooperation with humans, and the use of tools. Taxonomy of interaction behaviours is classified into three categories: cooperation, collaboration, and competition. In theory, interaction dynamics may be modelled by D'Alembert's principle or nonsmooth mechanics through seeking equality and/or inequality kinematic constraints. However, it is hard to gain these kinematic constraints in practice since they may be variable or be hardly described in a mathematical form. As a result, bond graph methodology is preferred in interaction dynamics modelling. In this paper, passivity and passivity indices with the differential operator are put forward by restricting its domain from the whole extended Hilbert function space to a set of all continuous function with finite derivative, and then the input-output stability condition, in this case, is derived. Next, mechanical impedance and admittance are defined, and a linear spatial impedance representation is given from the energetic point of view. Base on the bond graph theory, an ideal model is presented to model the idealized interaction, and invariance of port functions derived from the ideal interaction model is introduced; An interaction model is then proposed accounting for nonidealized factors and to describe cooperative, collaborative, and competitive interactions in a unified way. Finally, interaction stabilities are analyzed corresponding to different interaction models, and robustness of interaction stability is addressed based on the passivity indices.