This paper deals with the problem of robust output feedback control for a class of 5DOF upper-limb exoskeleton robot. It is able to assist with shoulder, elbow and wrist joint movements. Robustness is assured for the proposed controller in the presence of model uncertainties. Ultimate boundedness is proved for the presented adaptive observer-based controller. The other advantage of the proposed method is that the implemented information for feedback is only the measurement of position. The estimation of velocity and acceleration is provided by an adaptive observer without additional sensors. By utilizing a novel adaptive logic-based switching control scheme, a trajectory tracking performance is well achieved with a tunable error bound. The simulation example also verifies the effectiveness of the proposed control scheme.