Controlling unstable rolling phenomena

The paper addresses dynamic and control issues related to a dynamical model called the classical shimmying wheel. The classical shimmying wheel models the rolling dynamics of many physical rolling systems such as aircraft nose wheels, motorcycles, automotive systems, and tractor-trailer systems. Such a system can exhibit undesirable unstable rolling motion, that is, shimmying, which can often lead to disastrous results. Prior work with this particular model has focused on the stability of the system as well as an analysis of the qualitative nature of its dynamics, including numerical observation of possible chaotic behavior. Such behavior is observed when the rolling element is allowed to slip under certain conditions, which is intended to realistically model real physical rolling systems. In cases where the rolling dynamics of the system are unstable, the dynamics are characterized by the presence of an attractor wherein the system repeatedly switches back and forth between rolling and slipping. We present a slightly different, but more realistic, condition for the rolling element to switch from pure rolling to a slipping state and observe similar behavior. Additionally, we present a controller for the system designed using the method of feedback linearization. This controller stabilizes the purely rolling system but fails to always stabilize the system that is allowed to slip. We investigate the conditions under which the controller stabilizes the slipping system and propose an effective alternative control strategy for the slipping system for the case when the original controller fails to stabilize the system and where the uncontrolled rolling system is unstable. Finally, we investigate the stability of the system about operating points that are not equilibrium points, which models a physical system executing a turn.