# Revisiting Llewellyns Absolute Stability Criterion For Bilateral Teleoperation Systems Under Non-passive Operator Or Environment

Stability of a haptic teleoperation system is influenced by the typically uncertain, time-varying and/or unknown dynamics of the operator and the environment. For a stability analysis that is independent of the operator and the environment dynamics, Llewellyn's absolute stability criterion proposes certain conditions on the two-port network representing the teleoperator (comprising the master, the controller and communication channel, and the slave) assuming that the terminations (i.e., the operator and the environment) are passive. These are less-than-accurate assumptions; for instance, the assumption of operator passivity has been either rebuked in the literature or at best explained in the sense that the operator does not deliberately perform actions to destabilize the system. It is desirable to extend Llewellyn's result to the cases where the operator or the environment is non-passive. This paper revisits Llewellyn's criterion and relaxes the assumption of passivity for one of the terminations. The possibly non-passive termination is realistically assumed to have a complex impedance with an upper or lower bound on its amplitude or real part, respectively. Although the proposed stability criteria are useful for any application of two-port network systems, we specifically apply them on bilateral teleoperation systems and find the stability conditions when the operator or the environment is not passive; this is a result that Llewellyn's absolute stability criterion cannot afford.