This paper presents a new non-linear position-force control scheme for control of interaction of objects, grasped by multifingered mechanical hands, with the environment. Using the theory of input-output linearization, the hand-object dynamics in joint space is first projected along the position and force control directions, input-output linearized and decoupled. Along the position control direction(s), a state feedback controller (SFC) is used for trajectory tracking. A new dynamic SFC, which programs a variable compliance, is used along the force control direction(s). The variable compliance is appropriately modulated on the basis of the force feedback information to maintain a desired interaction force level. An optimal grasp force decomposition scheme is also developed for the control of grasping forces. The proposed scheme ensures grasp stability under dynamic conditions. The efficacy of the overall control scheme is demonstrated through numerical simulation.