Stabilization of a Riderless Bicycle A Linear-Parameter-Varying Approach

The bicycle provides transportation for leisure, recreation, and travel between home and work, throughout the world, in big cities as well as in small villages, supporting human mobility for more than a century. In addition, this widespread vehicle probably is the least expensive means of wheeled transportation. The bicycle has a debatable origin. Some history books speculate about Leonardo da Vinci’s 1490s drawings that seem to depict a mechanical device with two wheels. The first widely recognized early bicycle precursor, called celerifere, a sort of toy of the French nobility built around 1790 by Comte Mede de Sivrac of France, was a wooden scooter-like device with neither pedals nor steering. In 1817 the German Baron Karl von Drais developed the first two-wheeled vehicle. This vehicle, called the draisine and shown in Figure 1, has a steerable front wheel. To ride the draisine, a human had to push it forward using his feet against the ground. Not until in 1861 were rotary cranks and pedals added to the front wheel of the draisine by the French inventor Pierre Michaux. The name of this new kind of vehicle was velocipede, which is Latin for fast foot, although it was sometimes called boneshaker because of the uncomfortable ride due to the wooden wheels and cobblestone-covered streets of the day. Indeed, driving the velocipede, riders were able to experience the bicycle’s selfbalancing property, noticing that this two-wheeled vehicle can balance itself when moving fast enough. The bicycle was continually developed during the last quarter of the 19th century and the 20th century, leading to the high-performance modern device of today. An account of bicycle evolution can be found in [1] as well as in the Proceedings of the International Cycling History Conference, held every year since 1990 [2]. The first qualitative analysis of bicycle balancing was developed in 1869 [3], using analogies with an invertedpendulum model. A rigorous analysis of its dynamics was given by Whipple in 1899 [4] and Carvallo in 1900 [5], where the equations of motion linearized around the upright vertical equilibrium point are derived. This model was used to show that two-wheeled vehicles are statically unstable like an inverted pendulum but can balance themselves when rolling forward in a proper speed range. This dynamic behavior, called self-stabilization, is the basis of the statement by Einstein that “Life is like a bicycle. To keep your balance you must keep moving.” Modeling, analysis, and control of bicycle dynamics has been an attractive area of research in the last century as well as in recent years. Although articles on bicycle modeling regularly appeared during the first half of the 20th century [6]–[9], bicycle dynamics has received renewed attention since 1970, mainly due to the development of computers and software for numerical simulation. Bicycle dynamics has attracted the attention of the automatic control research community because of its peculiar features, such as, for example, the fact that it depends strongly on the bicycle speed and that, under certain conditions, it exhibits both open right-half plane poles and zeros [10], making the design of feedback controllers for either balancing the bicycle in the upright position or moving it along a predefined path a challenging problem. In addition, the development of the motorcycle industry in the last century has motivated researchers to develop controllers to improve handling and safety of motorcycles, exploiting the results on the analysis of the bicycle