Concepts of an omnidirectional pinion wheel are described. In one embodiment, the wheel includes a hub, first and second annular rims each including inner and outer rim surfaces, spokes that extend from the hub to the first and second annular rims, a pinion ring including pinion rods that extend between the inner surfaces of the first and second annular rims, and first and second annular rings of rollers affixed on the outer surfaces of the first and second annular rims. Using an axis of freedom of the rollers, the wheel can move sideways in addition to forward and backward. Further, when used with a vertical rack gear, the wheel can provide vertical displacement by engagement between teeth of the gear and the pinion ring. Additionally, various racks and tracks with teeth for pinion ring engagement are described along with an example vehicle capable of vertical displacement using the wheels.