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Limit cycle walkers are known as a class of walking robots capable of presenting periodic repetitive gaits without having local controllability at all times during motion. A well-known subclass of these robots is McGeer’s passive dynamic walkers solely activated by the gravity field. The mathematics governing this style of walking is hybrid and described by a set of nonlinear differential equations along with impulses. In this paper, by applying perturbation method to a simple model of these machines, we analytically prove that for this type of nonlinear impulsive system, there exist different switching surfaces, leading to the same equilibrium points (periodic solutions) with different... 

Inspired by the effects of a switching surface on the stability of passive dynamic walking (Safa and Naraghi in Robotica 33(01):195–207, 2015; Safa et al. in Nonlinear Dyn. 81(4):2127–2140, 2015), this paper suggests a new control strategy for stabilization of dynamic bipedal locomotion. It verifies that the stability improvement of a dynamic walking system is feasible while preserving the speed, step-length, period, natural dynamics, and the energy effectiveness of the gait. The proposed control policy goes behind the three primary principles: (i) The system’s switching surface has to be replaced by a new one if an external disturbance is induced. (ii) The new switching surface has to be... 

Inspired by the effects of a switching surface on the stability of passive dynamic walking (Safa and Naraghi in Robotica 33(01):195–207, 2015; Safa et al. in Nonlinear Dyn. 81(4):2127–2140, 2015), this paper suggests a new control strategy for stabilization of dynamic bipedal locomotion. It verifies that the stability improvement of a dynamic walking system is feasible while preserving the speed, step-length, period, natural dynamics, and the energy effectiveness of the gait. The proposed control policy goes behind the three primary principles: (i) The system’s switching surface has to be replaced by a new one if an external disturbance is induced. (ii) The new switching surface has to be... 

Recently, it has been proved that a different switching surface can preserve the walking trajectory while varying the walking stability [1], [2]. In this paper, by employing the simplest passive dynamic biped, we optimize the switching surface to maximize the robot's stability. Here, the stability measure is preferably the size of the basin of attraction, i.e. the collection of all possible initial conditions leading to the system's equilibrium point. Numerical investigations indicate that the maximum stability is obtained for neither the highest nor the lowest walking speed