Absolute Stability of Multi-DOF Multilateral Haptic Systems

Multi-degree-of-freedom (DOF) multilateral 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 and 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. In addition, 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 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 brief, 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.