L2 stability of haptic systems with projection-based force reflection

The problem of stability of haptic interaction with virtual objects is addressed, where the force reflection is implemented using the projection-based principle. A stability condition is derived that generalizes some previously known results to the case of projection-based force reflection. It demonstrates that, in this case, an additional design parameter is brought in that allows to increase the admissible stiffness of the virtual wall and decrease the update rate without changing the damping of the haptic device. A passivity based interpretation of the result is given in terms of interconnection of generalized passive systems where the excess of passivity of haptic device compensates the shortage of passivity of the virtual wall. In particular, it is shown that the projection-based force reflection allows to arbitrarily increase the excess of passivity of the haptic device without changing its physical damping.