This paper presents the mechanical design, the locomotion and the associated dynamic model of a new robotic wheelchair on climbing winding stairs. The prototype stair-climbing robotic wheelchair is constructed comprising a pair of rotational multi-limbed structures pivotally mounted on opposite sides of a support base so that the robotic wheelchair can ascend and descend stairs; especially, capability of climbing winding stairs is addressed. Based on the skid-steering analysis, the dynamic models for climbing winding stairs are developed for the trajectory planning and motion analysis. These models are required to ensure a passenger's safety in such a way that the robotic wheelchair is operated in an open mode. Numerical and experimental results are reported that show the behavior of the prototype as it climbs a winding stair in a dynamic turning.