It is estimated that nearly 16 million people are affected by stroke worldwide every year. Among them, more than 60% of individuals suffer from upper limb dysfunction. Robotic rehabilitation therapy has been proven to improve these individuals' motor ability and positively affect their recovery. In this research, Desktop-Mounted Rehabilitation Robot (DMRbot) has been developed. DMRbot is an end-effector type robot with three degrees of International 337 freedom, mounted on a desktop. It can provide passive therapeutic exercises to individuals having impaired upper-limb function. DMRbot has the capability to provide rehabilitative therapy across the whole range of upper-limb workspace. Modified Denavit-Hartenberg conventions were used for kinematic, and the iterative Newton-Euler method was used to develop the dynamic model of DMRbot. The DMRbot is maneuvered to follow any trajectory within its workspace using a linear Proportional-Integral-Derivative (PID) control technique. These trajectories are generated such that the resulting robot's motion synergistically promotes rehabilitative exercises for the patient. Experimental results show that the DMRbot can provide various passive exercises to the right and left upper limbs. In addition, it has shown great promise in delivering multi-joint upper limb motions in 2D and 3D planes.