Introduction¶
Preface¶
The bicycle is indeed a curious contraption that has greatly affected the lives of human’s since the early 19th century. There are probably more bicycles in the world than any other kind of vehicle. The bicycle has a notable history and it helped pave the way for the industrial revolution, the automobile, the airplane and even played a role in the emancipation of western women. Yet it is often an overlooked item in this day and age, especially in the United States of America, where the automobile is the dominant form of transportation and the bicycle is mostly considered a child’s toy. But in other parts of the world the bicycle can be viewed as a person’s stepping stone to progress or the most convenient way to get around. And what may be even more special about the bicycle is that in its elegant simplicity it still embodies the solutions to many of the world’s transportation problems, whether it be on the congested 12 lane freeways of Los Angeles or in a rural African village.
In my life, I have become an ever stronger proponent of the use of the bicycle as an appropriate mode of transportation. It is the most energy efficient way for a human to travel [WP04]. It has crept into all parts of my life with much of my time being spent thinking about different aspects of bicycles and bicycling. But this dissertation is concerned with how we actually balance on the blasted thing. Balancing, in general, may seem like a trivial task because we can all do it without consciously thinking about it, but the fact that the best engineers in the world are still baffled by the intricacies of balancing human-like robots give an idea of the difficulty of the subject. And to muddle it even more, humans are capable of much more advanced balancing acts like tight rope walking in which we invoke our very active control. Riding a bicycle falls somewhere between simply standing and these more extreme acts. The bicycle effectively disconnects us from the ground and forces us to use different control strategies to stay upright. The bicycle’s complex dynamics, the difficulty of the task, limited control actuation methods, and the ubiquity of the machine make the bicycle an ideal candidate platform for human control studies.
The Bicycle and Rider as a Dynamic System¶
The bicycle can be classified as a lightweight single track vehicle that is fundamentally made up of four components: two wheels, a front frame, and a rear frame. The wheels are connected to the respective frames by revolute joints and the two frames are connected to one another by a revolute joint such that the wheels are in-line with one another. This revolute joint between the two frames is necessary for balance and directional control of the vehicle. It is easy to show that locking the steering on a bicycle almost completely removes its ability to balance or be balanced, and certainly to be guided in a desired direction, no matter how the rider moves their body.
The bicycle has been studied by many scientists over the years. It is a rich dynamic system that is difficult to model accurately. [MPRS07] did an excellent job of sorting through 140 years of bicycle dynamics papers and providing a benchmarked bicycle model that finally verified the correct linearized equations of motion of the basic model, usually attributed as the Whipple Model [Whi99]. The model is able to predict both the non-minimum phase behavior and speed dependent stability and is now considered the foundation to all more detailed models.
I’ll briefly mention some of the common bicycle models and do so by dividing them into two main categories: models that do not exhibit open loop stability and models that do. All of these models can be extended by adding additional dynamics such as tire-road interactions, frame flexibility, and human biomechanics. These extensions can have effects on the stability and control of the complete system.
Simple Models¶
Typically a one degree of freedom model that produces a roll equation of motion is used to model a bicycle in its most basic form. This model has been derived and analyzed by many including, but not limited to, [TY48], [Kar04], and [ASKL05]. These models do not have great fidelity with regard to predicting the bicycle’s open-loop, speed-dependent stability but they are able to predict the non-minimum phase behavior. This situates them to be good candidates for basic control studies ([Get94], [CHA96], [Kar04], [ASKL05], [LS06]) as they predict the necessity of steering into the roll for stabilization and control. Controllers based on these models have also been successfully implemented on actual experimental control models [STW02] with some success. Beyond this paragraph, I will not be discussing these low order models any further.
Whipple Model¶
The lowest order model that has had some reasonable experimental validation [KSM08] is one which is able to predict speed dependent stability, and includes a complete physical description of the four basic rigid bodies that constitute a bicycle. The model is now typically referred to as the “Whipple Model”. This is in honor of Francis J. W. Whipple, the first author to publish a correct derivation of the linear equations of motion of this particular bicycle model [Whi99]. This model will be used as the basis for all further studies proposed in this dissertation. Many researchers over the past century have attempted to derive and analyze this model but very few have been successful. [MPRS07] give a complete historical review of uncontrolled bicycle research which made use of the historical comparisons in the thesis by [Han88]. [MPRS07] also benchmarked the Whipple Model by deriving the linearized equations of motion by using four independent methods (two independent pen and paper calculations and two different dynamic software packages). Furthermore, [BMCP07] benchmarked various torque-free circular motions in the non-linear case with two additional independent derivations of the equations of motion. There has been a series of recent validation attempts ([Koo06], [KSM08], [KS09], [Ste09], [ER10], [ER11]) for the Whipple model in particular and the evidence for it’s ability to describe the motion of the bicycle with no rider around the stable speed range is strong. This is important because it may be the lowest order model with the ability to predict the dynamics. In this dissertation, I make use of both the [MPRS07] model and my own derivation of the Whipple Model.
Complex Models¶
With modern dynamic tools it is relatively easy to add more degrees of freedom, flexible bodies, and more detailed forcing functions to the Whipple model with the intent of pushing the model’s ability to accurately predict bicycle and motorcycle motion. For example, the typical motorcycle is modeled with more realistic empirically derived tire-road interactions and a full suspension.
The most cited models typically have some reference to the model developed by Robin S. Sharp [Sha71]. This model extends the Whipple model concepts to include tire compliance and side slip. The model has been refined over the years to improve accuracy by adding frame flexibility, rider models and improving the tire models [SLG99] , [SL01], [SEL04] with Pacejka-style [Pac06] tire models being a popular choice. Sharp was also the first to give names to the eigenmodes of the Whipple Model [Sha75]. He and David Limebeer give a review of bicycle and motorcycle modeling in [LS06] covering much of their work. Other notable studies include ones developed by [Koe83] and the Italian group lead by Vittore Cossalter [CL02].
The motorcycle researchers have more experimental data validation of their models than in bicycle studies, and their more complicated models in general do a very good job of predicting the high speed motorcycle dynamics[1]. This is due to the fact that more work has been done to understand and measure the phenomena, that the high speed dynamics are easier to predict, and that the human’s biomechanical motions play a smaller role in the vehicle motion.
Conclusion¶
Albert Einstein once said “Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.” With the wide variety of models available, I’ve generally taken the approach of trying to use the simplest models possible to predict the measured motion in my experiments rather than adding great complexity. In my case, this model is often the Whipple model[2] with or without various rider biomechanical models which attempt to account for the large affect the rider’s freedom of movement can contribute to the system dynamics.
Footnotes
| [1] | For example, [BBCL03] is great example. |
| [2] | Not to say that the Whipple Model is not complex, au contraire. |
