# Absolute Stability Of Multi-DOF Multi-lateral Haptic Systems

Multi-degree-of-freedom (DOF) multi-lateral haptic systems involve teleoperation of several robots in physical environments by several human operators or collaborative interaction of several human operators in a virtual environment. An m-DOF n-lateral haptic system can be modeled as an n-port network where each port (terminal) connects to a termination deﬁned by m inputs and m outputs. The stability analysis of such systems is not trivial due to dynamic coupling across the different DOFs of the robots, the human operators, and the physical/virtual environments, and unknown dynamics of the human operators and the environments exacerbate the problem. Llewellyn’s criterion only allows for absolute stability analysis of 1-DOF bilateral haptic systems (m = 1, n = 2), which can be modeled as two-port networks. The absolute stability of a general m-DOF bilateral haptic system where m > 1 cannot be obtained from m applications of Llewellyn’s criterion to each DOF of the bilateral system. Also, if we were to use Llewellyn’s criterion for absolute stability analysis of a general 1-DOF n-lateral haptic system where n > 2, we would need to couple n − 2 terminations of the n-port network to (an inﬁnite number of) known impedances in order to reduce it to an equivalent two-port network; this is a cumbersome process that involves an inﬁnite number of applications of Llewellyn’s criterion. In this paper, we present a straightforward and convenient criterion for absolute stability analysis of a class of m-DOF n-lateral haptic systems for any m ≥ 1 and n ≥ 2. As case studies, a 1-DOF trilateral and a 2-DOF bilateral haptic system are studied for absolute stability with simulations and experiments conﬁrming the theoretical stability conditions.