A self-powered unicycle that balances itself in three dimensions, is an interesting problem in robotics and control theory.
The theoretical work on the unicycle problem is complemented by work on the construction of actual mechanically ridden unicycles.
To first order, a self-balancing unicycle can be considered as a non-linear control system similar to that of a two-dimensional inverted pendulum with a unicycle cart at its base: however, there are many higher-order effects involved in modelling the full system. Rotation of the drive wheel itself can provide control in only one dimension (i.e., forwards and backwards): control in other dimensions generally requires other actuators, such as auxiliary pendulums, reaction wheels, or control moment gyroscopes attached to the main unicycle pendulum.
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