Absolute Stability of a Class of Trilateral Haptic Systems

Trilateral haptic systems can be modeled as three-port networks. We present a criterion for absolute stability of a general class of three-port networks. Traditionally, existing (i.e, Llewellyn´s) criteria 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 termination would need to be assumed known for it to reduce to a two-port network. This is restrictive because, for absolute stability, all three terminations of the three-port network must be allowed to be arbitrary (while passive). Extending Llewellyn´s criterion, we present closed-form necessary and sufficient conditions for absolute stability of a general class of three-port networks. We first find a symmetrization condition under which a general asymmetric impedance (or admittance) matrix Z_3×3 has a symmetric equivalent Z_eq from a network stability perspective. Then, via the equivalence of passivity and absolute stability for reciprocal networks, an absolute stability condition for the original nonreciprocal network is derived. To demonstrate the convenience and utility of using this criterion for both analysis and design, it is applied to the problem of designing stabilizing controllers for dual-user haptic teleoperation systems, with simulations and experiments validating the criterion.