Summary: A mathematical model of a unicycle and rider, with a uniquely realistic tyre force and moment representation, is set up with the aid of multibody modelling software. The rider’s upper body is joined to the lower body through a spherical joint, so that wheel, yaw, pitch and roll torques are available for control. The rider’s bandwidth is restricted by low-pass filters. The linear equations describing small perturbations from a straight-running state are shown, which equations derive from a parallel derivation yielding the same eigenvalues as obtained from the first method. A nonlinear simulation model and the linear model for small perturbations from a general trim (or dynamic equilibrium) state are constructed. The linear model is used to reveal the stability properties for the uncontrolled machine and rider near to straight running, and for the derivation of optimal controls. These controls minimize a cost function made up of tracking errors and control efforts. Optimal controls for near-straight-running conditions, with left/right symmetry, and more complex ones for cornering trims are included. Frequency responses of some closed-loop systems, from the former class, demonstrate excellent path-tracking qualities within bandwidth and amplitude limits. Controls are installed for path-following trials. Lane-change and clothoid manoeuvres are simulated, demonstrating good-quality tracking of longitudinal and lateral demands. Pitch torque control is little used by the rider, while yaw and roll torques are complementary, with the former being more useful in transients, while the latter has value also in steady states. Wheel torque is influential on lateral control in turning. Adaptive control by gain switching is used to enable clothoid tracking up to lateral accelerations greater than 1ms\(^{ - 2}\). General control of the motions of a virtual or robotic unicycle will be possible through the addition of more comprehensive adaptation to the control scheme described.