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 nos e wheels, motorcycles, automotive systems, and tractor-trailer systems. Suc h a system can exhibit undesirable unstable rolling motion, that is, shimmy ing, which can often lead to disastrous results. Prior work with this parti cular model has focused on the stability of the system as well as an analys is of the qualitative nature of its dynamics, including numerical observati on 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 rolli ng dynamics of the system are unstable, the dynamics are characterized by t he presence of an attractor wherein the system repeatedly switches back and forth between rolling and slipping. We present a slightly different, but m ore realistic, condition for the rolling element to switch from pure rollin g to a slipping state and observe similar behavior. Additionally, we presen t a controller for the system designed using the method of feedback lineari zation. This controller stabilizes the purely rolling system but fails to a lways stabilize the system that is allowed to slip. We investigate the cond itions under which the controller stabilizes the slipping system and propos e 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 st ability of the system about operating points that are not equilibrium point s, which models a physical system executing a turn.