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Two adaptive nonlinear controllers are proposed for the coupling of haptic devices with impedance-type and admittance-type virtual environments, respectively. Rigid contacts in admittance-type environments are modeled either as a stiff spring or a constraint on the haptic device motion. Both controllers employ user position and force measurements to replace the natural dynamics of the haptic interface with that of an adjustable mass-damper tool. The transparency and stability of the resulting systems are investigated using a Lyapunov analysis and by taking into account uncertain nonlinear dynamics for the haptic device, and uncertain mass-spring-damper type dynamics for the user and virtual environment. It is shown analytically that low-pass filtering of selected terms in the control signal can significantly reduce a stability related lower bound on the achievable synthesized mass of the haptic interface in a discrete-time implementation of the controllers. An optimization problem is formulated and solved to balance impedance reduction against noise amplification in choosing the filter gain and bandwidth. The proposed controllers as well as a conventional penalty-based method are compared in a set of experiments. The results indicate that the controller with an admittance-type constraint-based rigid environment has far superior performance in terms of the range of impedances that it can stably display to the user.