Stability Analysis of Passive Dynamic Walking of Quadrupeds

We introduce a detailed numerical simulation and analysis framework to extend the principles of passive dynamic walking to quadrupedal locomotion. Non-linear limit cycle methods are used to identify possible gaits and to analyze the stability and efficiency of quadrupedal passive dynamic walking. In doing so, special attention is paid to issues that are inherent to quadrupedal locomotion, such as the occurrence of simultaneous contact collisions and the implications of the phase difference between front and back leg pairs. Limit cycles identified within this framework correspond to periodic gaits and can be placed into two categories: in-phase gaits in which front and back legs hit the ground at roughly the same time, and out-of-phase gaits with a +/- 90A phase shift between the back and front leg pairs. The latter are, in comparison, energetically more efficient but exhibit one unstable eigenvalue that leads to a phase divergence and results in a gait-transition to a less efficient in-phase gait. A detailed analysis examines the influence of various parameters on stability and locomotion speed, with the ultimate goal of determining a stable solution for the energy-efficient, out-of-phase gait. This was achieved through the use of a wobbling mass, i.e. an additional mass that is elastically attached to the main body of the quadruped. The methods, results, and gaits presented in this paper additionally provide a point of departure for the exploration of the considerably richer range of quadrupedal locomotion found in nature.