The Dynamics of Rolling and Balancing in Micromobility Vehicles

The importance of micromobility has become relevant in the last decades since micromobility vehicles (electric scooters, bicycles, skateboards, unicycles, etc.) provide modern solutions for the last-mile problem of urban transportation. Although the invention of these vehicles dates back to the last century, and the analysis of their dynamics also began at that time, electrification changed the game. Higher speed and more agility characterize modern micromobility vehicles, while in the meantime, non-professional riders are using them on city roads in heavy traffic. Thus, the risk of serious accidents is high. Understanding the dynamics and control of e-bikes, scooters, skateboards, and unicycles is becoming more and more important. The aim of this course is to give a perspective on the dynamics and control of micromobility vehicles. First, the classical mechanical modeling of the simplest nonholonomic systems will be presented. Starting from Newton’s second law of dynamics and introducing kinematic constraints, the mystic, speed-dependent dynamics of nonholonomic systems will be demonstrated via the analysis of the uncontrolled skateboard. The effect of human control on stability is investigated via the implementation of a linear state feedback with human reaction time. While single-track vehicles are highly unstable, with some forward speed, the system is easy to stabilize and control. The course will showcase the key fundamental results on this important topic. The theoretical background of nonholonomic systems will be given via lessons on the Lagrangian approach extended for kinematic constraints. Then, the concept of pseudo-velocities will be introduced, and the Appellian approach will be presented. The differences between the aforementioned methods will be shown via the analysis of the spatial rolling problem of a rigid wheel. A simplified model of the electric unicycle will also be introduced. The analysis of nonholonomic articulated robotic vehicles will highlight how periodic excitations can be used for driving micromobility vehicles. Namely, the motion of the Twistcar (which is a very popular kids' cart) is analyzed, and forward and backward motions are identified depending on the amplitude, frequency, and phase of the excitation. Rider modeling, path tracking, and stabilization of bicycles and e-scooters will also be presented. Modeling of the wobble mode of bicycles, including frame flexibility, transient tire-road contact forces, and the role of rider posture, will be investigated. The stability of e-scooters under braking will be identified, and different braking control strategies (ABS, optimal brake force distribution) will be highlighted. A Kalman-filter-based estimation of the tire-road friction potential and of vehicle-rider mass distribution will also be the subject of the course. The course will also discuss the development of automated single-track vehicles. Path-following control strategies will be analyzed, and experimental investigation of bicycle control will be showcased.