In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.
We propose a state feedback control design via linearization for flexible walking on flat ground. First, we generate nearly passive limit cycles, being stable or not, using impulsive toe‐off actuations. The term ‘nearly passive’ means that the dynamics is completely passive almost everywhere except at the toe‐off moment. A feature of our gait generation method is that walking gaits are characterized only by amounts of supplied energy, and we observe that other variables, including input torques, are auto‐balanced via our method. After gait generation, we design a feedback controller considering robustness and input saturation. As a result, each limit cycle can be matched with its respective controller classified only by energy levels. We have verified that walking speeds monotonically increase by adding more energy, and the ankle joint plays a significant role in compass‐gait walking. Finally, instead of applying impulsive torques, we discuss a practical issue regarding realistic control inputs that ensure stable gait transitions as energy levels are elevated.
SUMMARYIn this paper, we investigate dynamic walking as a convergence to the system's own limit cycles, not to artificially generated trajectories, which is one of the lessons in the concept of passive dynamic walking. For flexible walking, gait transitions can be performed by moving from one limit cycle to another one, and thus, the flexibility depends on the range in which limit cycles exist. To design a bipedal walker based on this approach, we explore period-1 passive limit cycles formed by natural dynamics and analyze them. We use a biped model with knees and point feet to perform numerical simulations by changing the center of mass locations of the legs. As a result, we obtain mass distributions for the maximum flexibility, which can be attained from very limited location sets. We discuss the effect of parameter variations on passive dynamic walking and how to improve robot design by analyzing walking performance. Finally, we present a practical application to a real bipedal walker, designed to exhibit more flexible walking based on this study.
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