Analysis of period doubling bifurcation and chaos mirror of biped passive dynamic robot gait

With a reasonable parameter configuration,the passive dynamic walking model has a stable,efficient,natural periodic gait,which depends only on gravity and inertia when walking down a slight slope.In fact,there is a delicate balance in the energy conversion in the stable periodic gait,making the gait adjustable by changing the model parameters.Poincaré mapping is combined with Newton-Raphson iteration to obtain the numerical solution of the final state of the passive dynamic walking model.In addition,a simulation on the walking gait of the model is performed by increasing the slope step by step,thereby fixing the model's parameters synchronously.Then,the gait features obtained in the different slope stages are analyzed and discussed,the intrinsic laws are revealed in depth.The results indicate that the gait can present features of a single period,doubling period,the entrance of chaos,merging of sub-bands,and so on,because of the high sensitivity of the passive dynamic walking to the slope.From a global viewpoint,the gait becomes chaotic by way of period doubling bifurcation,with a self-similar Feigenbaum fractal structure in the process.At the entrance of chaos,the gait sequence comprises a Cantor set,and during the chaotic stage,sub-bands in the final-state diagram of the robot system present as a mirror of the period doubling bifurcation.

With a reasonable parameter configuration, the passive dynamic walking model has a stable, efficient, natural periodic gait, which depends only on gravity and inertia when walking down a slight slope. In fact, there is a delicate balance in the energy conversion in the stable periodic gait, making the gait adjustable by changing the model parameters. Poincar6 mapping is combined with Newton-Raphson iteration to obtain the numerical solution of the final state of the passive dynamic walking model. In addition, a simulation on the walking gait of the model is performed by increasing the slope step by step, thereby fixing the model＇s parame- ters synchronously. Then, the gait features obtained in the different slope stages are analyzed and discussed, the intrinsic laws are revealed in depth. The results indicate that the gait can present features of a single period, doubling period, the entrance of chaos, merging of sub-bands, and so on, because of the high sensitivity of the passive dynamic walking to the slope. From a global viewpoint, the gait becomes chaotic by way of period doubling bifurcation, with a self-similar Feigenbaum fractal structure in the process. At the entrance of chaos, the gait sequence comprises a Cantor set, and during the chaotic stage, sub-bands in the fi- nal-state diagram of the robot system present as a mirror of the period doubling bifurcation.