achieving stability of an inverted pendulum has become a common engineering challenge for researchers. there are different variations of the inverted pendulum on a cart ranging from a rod on a cart to a multiple segmented inverted pendulum on a cart. another variation places the inverted pendulum's rod or segmented rod on the end of a rotating assembly. in both, (the cart and rotating system) the inverted pendulum can only fall in a plane. the inverted pendulums in these projects can either be required to only maintain balance after an equilibrium position is achieved or be able to achieve equilibrium by itself. another platform is a two-wheeled balancing inverted pendulum. the two wheeled platform has the ability to spin on the spot offering a great deal of maneuverability. yet another variation balances on a single point. a spinning top, a unicycle, or an inverted pendulum atop a spherical ball all balance on a single point. as derived above the inverted pendulum can also be achieved by having a vertically oscillating base.