AbstractFundamental principles and recent methods for investigating the nonlinear dynamics of legged robot motions with respect to control, stability, and design are discussed. One of them is the still challenging problem of producing dynamically stable gaits. The generation of fast walking or running motions requires methods and algorithms adept at handling the nonlinear dynamical effects and stability issues which arise. Reduced, recursive multibody algorithms, a numerical optimal control method, and new stability and energy performance indices are presented which are well‐suited for this purpose. Difficulties and open problems are discussed along with numerical investigations into the proposed gait generation scheme. Our analysis considers both bipedal and quadrupedal gaits.