This paper describes a design of a mechanism that aims a wheel to climb obstacles, steps, or slopes with a suitable smooth path. The proposed four-bar linkage can be installed on each wheel of a vehicle, which therefore can be capable to climb stairs with suitable comfortable motion. An algorithm is formulated to design an optimum solution of the mechanism fulfilling those task requirements. A straight-line trajectory for the centre of a wheel is ensured through an easily controlled motion, and the compactness of the mechanism design makes it suitable for staircase climbing wheelchairs for aiding people with disability. Experimental tests have given satisfactory results and are shown for a practical application.