# Stability Analysis Of Trilateral Haptic Collaboration

This paper presents a criterion for absolute stability of a general class of three-port networks. Trilateral haptic systems, which have recently found many interesting applications, can be modeled as three-port networks. Traditionally, existing criteria (Llewellyn’s criterion) have facilitated the stability analysis of bilateral haptic systems modeled as two-port networks. If the same criteria were to be used for stability analysis of a three-port network, its third port would need to be assumed known for it to reduce to a two-port network. However, this is restrictive because, according to the definition of absolute stability, all three terminations of the three-port network must be allowed to be arbitrary (while passive).

In this paper, extending Llewellyn’s criterion, we present closedform necessary and sufficient conditions for absolute stability of a general class of three-port networks – the three terminations need to be passive but are otherwise arbitrary. To this end, we first find a symmetrization condition under which a general asymmetric impedance (or admittance) matrix Z_{33} has an equivalent symmetric counterpart Z_{eq}; this Z_{eq} models a reciprocal three-port network with the same stability characterization as the general nonreciprocal three-port network modeled by Z. Then, based on the equivalence of passivity and absolute stability for the equivalent reciprocal network, an absolute stability condition for the original nonreciprocal network is derived. To show how the resulting absolute stability criterion can be utilized at the system design stage, we have applied it to the problem of designing controllers for triple-user collaborative haptic virtual environment systems. The validity of the resulting absolute stability conditions have been verified via simulations.