On the Stability and Control of the Bicycle

Abstract

After some brief history, a mathematical model of a bicycle that has become a benchmark is described. The symbolic equations of motion of the bicycle are given in two forms and the equations are interpreted, with special reference to stability. The mechanics of autostabilization are discussed in detail. The relationship between design and behavior is shown to be heavily speed-dependent and complex. Using optimal linear preview control theory, rider control of the bicycle is studied. It is shown that steering control by an optimal rider, especially at low speeds, is powerful in comparison with a bicycle’s self-steering. This observation leads to the expectation that riders will be insensitive to variations in design, as has been observed in practice. Optimal preview speed control is also demonstrated. Extensions to the basic treatment of bicycle dynamics in the benchmark case are considered so that the modeling includes more realistic representations of tires, frames, and riders. The implications for stability predictions are discussed and it is shown that the moderate-speed behavior is altered little by the elaborations. Rider control theory is applied to the most realistic of the models considered and the results indicate a strong similarity between the benchmark case and the complex one, where they are directly comparable. In the complex case, steering control by rider-lean-torque is feasible and the results indicate that, when this is combined with steer-torque control, it is completely secondary. When only rider-lean-torque control is possible, extended preview is necessary, high-gain control is required, and the controls are relatively complex. Much that is known about the stability and control of bicycles is collected and explained, together with new material relating to modeling accuracy, bicycle design, and rider control.