We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edge rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability.

Stability control for walking bipeds has been considered a complex task. Even two-dimensional fore-aft stability in dynamic walking appears to be difficult to achieve. In this paper we prove the contrary, starting from the basic belief that in nature stability control must be the sum of a number of very simple rules. We study the global stability of the simplest walking model by determining the basin of attraction of the Poincare/spl acute/ map of this model. This shows that the walker, although stable, can only handle very small disturbances. It mostly falls, either forward or backward. We show that it is impossible for any form of swing leg control to solve backward falling. For the problem of forward falling, we devise a simple but very effective rule for swing leg action: "You will never fall forward if you put your swing leg fast enough in front of your stance leg. In order to prevent falling backward the next step, the swing leg shouldn't be too far in front." The effectiveness of this rule is demonstrated with our prototype "Mike.".

A new bicycle design points to the importance of mass distribution for stability.