Inverted pendulum on a rimless wheel

The controlled rolling of a rimless wheel with a rod attached to the center of the wheel is investigated. The wheel can roll both up and down the hill. The angle of deflection of the rod from the wheel support spoke is used as a control. The optimal laws of rod control for wheel acceleration satisfying the geometric constraints of the system are found. A sequence and Poincare diagrams are constructed for the angular velocity of rotation of the wheel when changing the support spokes and the conditions for the existence of fixed points on the diagrams depending on the angles of inclination of the rod are revealed.