Professor Richard Klein and his students built a bicycle with a rather interesting feature: no one was able to ride it. A prize was offered. Hundreds of students and cycling enthusiasts attempted it. Years passed, and the prize money grew. Klein’s rear-steered bicycle became a canonical example of how non-minimum phase systems can be difficult and sometimes nearly impossible to control. It has been lauded as a particularly effective educational example in which students can experience the loss of controllability in a seemingly simple, albeit unorthodox bicycle. The primary result of the work reported here is a demonstration that it is possible for a human of modest athletic ability to ride Klein’s unridable bicycle, to keep it balanced, and to control its direction of travel. There is a secret to riding Klein’s rear-steer bicycle. The secret is revealed through an exploration of the dynamics and control of the bike that contains three elements: (1) modeling the physics of the actively steered bicycle as an inverted pendulum riding atop a carriage; (2) recognizing that the steer kinematics leads to competing physical mechanisms which an aspiring rider might exploit; and (3) examining limitations of controllability and stabilizability of the system from a state space perspective. From this vantage point, one can devise a novel strategy for riding the so-called “unridable” bike and solving Klein’s puzzle. The work adds a new chapter on the dynamics and control of the rear-steered bicycle that students can study. Introduction & Background In 1986, Richard Klein and his students made an unusual modification to a rather normal bicycle [1,2]. A photograph of the bicycle is shown in Fig 1. They placed the handlebar where the seat normally goes and moved the seat to where the handlebar normally belongs. They also moved the chain ring and wheel drive sprocket to the other side of the bike. Therefore, when a rider sits on the seat, places their hands on the handlebar in front of them, and begins pedaling, the bicycle begins moving in the direction that the rider is facing, as expected. What makes the bicycle different is that the wheel which steers the bike is the rear wheel. The unsteered wheel, which is now at the front of the bicycle, is mechanically connected to the pedals via drive chain. Klein gave a name to this bike: Rear-Steered Bike I (RSB1). For this bicycle, the handlebar and fork are separated from each other and placed at opposite ends of the bike as shown in the photo of Fig 1. The two parts are connected to each other mechanically via the steer chain wrapped around two steer sprockets of October 10, 2024 1/34 handlebar steered wheelsaddle steer chain drive chainfork non-steered wheel Direction of travel.steer axisFig 1. Klein’s “unridable bicycle.” A photograph of the rear-steered bicycle that Richard Klein and others deemed unridable. equal diameter. If the steer chain is wrapped in a simple loop configuration as shown in Fig 2A, then a left turn of the handlebar by angle δcauses the fork and the steered wheel to rotate the same angle δand in the same direction. However, since the steered wheel is at the rear of the bicycle, such a leftward turn of the handlebar causes the entire bike to turn to the right. Another way to configure the steer chain is shown in Fig 2B. Here, the chain makes a figure eight shaped loop. As a result, the fork turns in opposite direction as the handlebar. In this case, the bicycle turns in the same direction that the handlebar is rotated. Either configuration is possible on Rear-Steered Bike I (RSB1). /g71/g71 /g71 /g16/g71 steer chain handlebar (turned left)path of front wheel path of front wheelpath of rear wheel path of rear wheel(bike turns right) (bike turns left)rear wheel (steered)(A) (B) Fig 2. Steer chain configuration. A: When the steer chair wraps around the sprockets in a simple loop, a leftward turn of the handlebar causes the rear-steered bicycle to execute a rightward turn. B: When the steer chain is in a figure 8 configuration, the bicycle turns in the same direction that the handlebar is turned. From a control theory perspective, it does not matter which way the steer chain is configured. If it is possible to keep the bicycle balanced with the chain in one configuration, then, in theory, it should be possible to stabilize the bike with the exact same control strategy, except for turning the handlebar in the opposite direction. The October 10, 2024 2/34 only problem is that the humans do not operate as symmetrically as bicycles do. Therefore, the purpose of being able to reconfigure the chain, as illustrated in Fig 2, is for the convenience of riders to choose what is most comfortable for themselves. Regardless of which way the steer chain is configured, the simple act of making the rear wheel the one that steers, while keeping other geometric features of the bike unchanged, is what makes the bicycle exhibit odd behavior, and makes it exceedingly difficult to ride. The rear-steered wheel is what made Klein’s bicycle became known as the “unridable bike.” In the Supporting Information section at the end of this article, there are links to several videos. The first, S1 Video shows mechanical engineering students attempting to ride Klein’s Rear-Steered Bike I. To be fair the video shows students’ first interactions with the bike. However, it typically does not get much better after spending hours trying to wrap their minds around it. Gordon et al [3] provide a general discussion on the dynamics of rear-steered bicycles. De Jong [4] has documented an extensive history of of rear-steered bicycles, both ridable and not ridable, that goes back more than a century. Klein’s Challenge Richard Klein openly challenged the public to try to ride his Rear-Steered Bike I (RSB1). In 2005, at the time of his co-authored article in IEEE Control Systems Magazine [2], the prize money for the first person to successfully ride it was $1,000 US. By 2006, the prize had risen to $5,000 US [5]. Recognizing the potential difficulty posed by a bicycle that steers in the opposite direction that the handlebars are pointed, Klein allowed those seeking conquer the “unridable” bike to choose which configuration from Fig 2 that they wanted for the steer chain. To successfully complete Klein’s challenge, the rider had to remain seated in the saddle, with both feet on the pedals. Furthermore, both wheels had to remain contact with the ground at all times, and the bike needed to make continuous forward progress along the specified path. The path started with a straight section of 100 ft (30.5 meters) in length, followed by a 90-degree turn, followed by a second 100 ft long straight section. At all times, the bicycle had to remain a distance of 3 ft. (0.9 meters) from the center of the path. When the rider reached the end of the path, they were to allowed put their feet on the ground to manually turn the bike around. Then the rider had to ride again, this time traversing the path in the opposite direction. In an archived web postings [5,6], Klein stated that over the years, many hundreds of overly optimistic riders attempted to win the prize reward. For some of the most skilled bicyclists and unicyclists, Klein would let them borrow the bike for a month or more to develop their skills. Before the autumn of 2009, there was just one person who was able to remain upright. However, this rider ended up riding haphazardly in an open and flat parking lot as opposed to being able to follow a prescribed path, a necessary condition for winning the prize. A “Backward Brain Bicycle” It is well known that the simple act of switching the sign of the input/output relationship, so that a bike turns in the opposite direction that the handlebar is steered, is sufficient to make the task of balancing a bicycle upright extraordinarily difficult. The popular YouTube channel, Smarter Every Day , has an episode devoted to this specific topic [7] in which Destin Sandlin added a geared connection between the handlebar and fork of a normal front-steered bicycle. When the rider would turn the handlebar one way, the fork would turn the front wheel by the same angle about its steer axis, but in the opposite direction. October 10, 2024 3/34 To be clear, Sandlin and co-workers did nothing else to modify the dynamics of their “Backwards Brain Bicycle.” From a control theory perspective in which the input to the bicycle system is the rider’s steer action, and the output is the lean angle, the difference between a normal bicycle and Sandlin’s Backward Brain Bicycle is a trivial sign change. Theoretically, both bicycles are equally controllable and equally stabilizable. The rider would just need to provide the opposite steer input in response to a perceived amount of bicycle lean. In practice, however, the Backwards Brain Bicycle is difficult to ride because of the human implementing the control strategy. In Sandlin’s case, it took eight months of daily dedicated training to de-program the subconscious part of the brain that holds automatic responses to perceived lean and replace those responses with ones that produce the opposite handlebar motion. The story of the “Backward Brain Bicycle” is relevant here in that it illustrates the minor similarities, and the major differences between it and Klein’s “Rear-Steered Bike I” (RSB1). Switching direction of the steer action (with a ”Backward Brain” gear or with a reconfigurable steer chain loop in Fig 2) does nottheoretically make a bicycle more controllable or less controllable. It just changes the rider’s perspective, making balancing the bike more difficult or less difficult depending hon how the rider’s mind is trained to respond to a leaning bicycle. To accommodate different riders’ p