Abstract <p>Bicycle self-stabilization is a complex phenomenon that is still not fully understood. While the Turning-Towards-a-Fall (TTF) mechanism is recognized as fundamental, little is known about how it is spontaneously driven. In particular, the role of geometric parameters, such as the fork angle, in shaping stability characteristics has not been systematically explored. A deeper understanding of these effects could provide insights into the self-stabilization mechanism and inform the design of more stable bicycles and robotic systems. To facilitate a comprehensive and quantitative analysis of the self-stabilization mechanism, we develop a dynamics-and-data-driven surrogate dynamics model based on the model structure derived from bicycle symmetries and model reduction. A novel compressive-encoding-based neural network module is proposed, to ensure high precision in evaluation and differentiation, which facilitates stability analysis via linearization. The properties exposed in the coefficients of the surrogate dynamic equations enable us to understand how the gravity-induced overturning torque can be balanced by a stabilizing torque caused by the TTF behavior, and why the TTF can occur spontaneously. Our analysis shows that the nonholonomic constraints exerted on the bicycle motion play a significant role in generating both the stabilizing and driving torques of the TTF. Meanwhile, we also reveal that the nonholonomic-constraint-related gyroscopic moment can cause stiffness and damping effects in the surrogate model. As a result, the TTF behavior performs autonomous feedback control to generate the balancing torque, and the stiffness and damping can increase the stability of the bicycle system. Finally, the influence of the fork angle on the bicycle stability performance is discussed, and limit-cycle motions are found for the bicycle in special configurations.</p> Graphic Abstract <p></p>