The planar compass-gait biped robot is recognized by its simple passive morphological structure and by a complex dynamic walking system modeled by an impulsive hybrid nonlinear dynamics. Such complexity inhibits investigating the biped locomotive mechanism by means of the Poincaré map. Nevertheless, it is very difficult (and even impossible) to establish the exact closed form of the Poincaré mapping. This paper is concerned with the development of an improved closed-form analytical expression of the impact-to-impact Poincaré map for analyzing the complex walking behavior of the passive-dynamics biped walker and its stability. Our methodology consists in linearizing and then approximating the hybrid dynamics of the compass-gait biped robot around a predefined hybrid limit cycle. Based on the second-order Taylor series expansion, we design first an enhanced closed-form expression of the Poincaré map as well as an analytical expression for the computation of the step period of the bipedal locomotion. Furthermore, a simplification of these developed expressions is presented by decreasing the dimension. We provide also an expression of the Jacobian matrix for investigating the stability of the period-1 fixed point of the designed simplified Poincaré map. At the end of this work, we illustrate some simulation results in order to evaluate the validity of the new developed expression of the impact-to-impact Poincaré map in analyzing the stability and the complex walking behavior of passive-dynamics compass biped robot.

平面罗盘步态两足机器人以其简单的被动形态结构和以脉冲混合非线性动力学为模型的复杂动态行走系统而得到认可。这种复杂性阻碍了通过Poincaré映射来研究Biped机车的机制。但是，很难（甚至不可能）建立庞加莱映射的精确封闭形式。本文关注的是一种改进的闭合形式的解析式Poincaré映射解析表达式的开发，该映射用于分析被动动力学Biped步行者的复杂步行行为及其稳定性。我们的方法包括线性化然后在预定的混合极限周期内近似罗盘步态两足机器人的混合动力。基于二阶泰勒级数展开式，我们首先设计庞加莱图的增强的封闭形式表达式，以及用于计算双足运动步长的解析表达式。此外，通过减小尺寸来简化这些展开的表达式。我们还提供了Jacobian矩阵的表达式，以调查设计的简化庞加莱图的1期固定点的稳定性。在本文的最后，我们举例说明了一些仿真结果，以评估新近开发的“碰撞-碰撞”庞加莱映射表达式在分析被动动力学罗盘两足机器人的稳定性和复杂行走行为时的有效性。此外，通过减小尺寸来简化这些展开的表达式。我们还提供了Jacobian矩阵的表达式，以调查设计的简化庞加莱图的1期固定点的稳定性。在本文的最后，我们举例说明了一些仿真结果，以评估新近开发的“碰撞-碰撞”庞加莱映射表达式在分析被动动力学罗盘两足机器人的稳定性和复杂行走行为时的有效性。此外，通过减小尺寸来简化这些展开的表达式。我们还提供了Jacobian矩阵的表达式，以调查设计的简化庞加莱图的1期固定点的稳定性。在本文的最后，我们举例说明了一些仿真结果，以评估新近开发的“碰撞-碰撞”庞加莱映射表达式在分析被动动力学罗盘两足机器人的稳定性和复杂行走行为时的有效性。