This paper addresses the task space impedance control of a robot driven through the multistage nonlinear flexible transmission. The proposed controller uses limited information of the angle and the current of the motors to regulate the end point compliance at the specified set point. In particular, motor angle is employed to estimate the stationary robot link angle and joint velocity in real time. They are then used to constitute the stationary force on the attempt to cancel the robot gravity force and to form the task space interacting force according to the desired impedance characteristics. Motor current is used to infer the transmitted torque to the robot. This torque is fed back to mitigate the effect of the motor inertia from deteriorating the desired impedance. Asymptotic stability of this controller with the flexible joint robot is guaranteed with additional damping. Passivity of the system is also investigated. Simulation and experiments of the proposed control scheme on a two degrees-of-freedom (DOF) cable-pulley driven flexible joint robot model are examined.