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Proportional-derivative (PD) control is often used to coordinate the two copies of the virtual environment in distributed two-users networked haptic cooperation. However, a distributed PD controller designed for force interactions between two users may destabilize the haptic cooperation among multiple users because the effective coordination gain for each local copy of the virtual environment increases with the participant count. This paper proposes the average position (AP) strategy to upper bound the effective stiffness for the shared virtual object (SVO) coordination and, thus, to increase the stability of distributed multi-user haptic cooperation. The paper first motivates the AP strategy via continuous-time analysis of the autonomous dynamics of an SVO distributed among N users connected across a network with infinite bandwidth and no communication delay. We then investigate the effect of AP coordination on distributed multi-user haptic interactions over a network with limited bandwidth and constant and small communication delay via multi-rate stability and performance analyses of cooperative manipulations of an SVO by up to five operators. The paper shows that AP coordination: (1) has bounded effective coordination gain; (2) increases the stability region of distributed multi-user haptic cooperation compared to conventional PD coordination; and (3) renders less viscous SVO dynamics to operators than PD coordination. Three-users experimental manipulations of a shared virtual cube validate the analysis.

Average-Position Coordination for Distributed Multi-User Networked Haptic Cooperation