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 190 2 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.KEY WORDS-dynamics1 mechanics, design and control of legged robots1 mechanics, design and control of underactuated robots.

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A method for constructing equations of motion governing constrained systems is presented. The method, which is particularly useful when equations of motion have already been formulated, and new equations of motion, reflecting the presence of additional constraints are needed, allow the new equations to be written as a recombination of terms comprising the original equations. An explicit form in which the new dynamical equations may be cast for the purpose of numerical integration is developed, along with special cases that demonstrate how the procedure may be simplified in two commonly occurring situations. An illustrative example from the field of robotics is presented, and several areas of application are identified.

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Abstract-This work introduces a framework for the creation and analysis of efficient gaits for legged systems based on the exploitation of natural dynamics. It summarizes the theory behind hybrid dynamic modeling, the identification of optimal periodic motions with single shooting and direct collocation, and the analysis of first order stability. Three examples introduce various aspects of gait creation and analysis: a stability study of a passive dynamic walker determines the ideal position of the leg's center of mass, the cost of transportation is minimized for a prismatic monopod hopper based on series elastic actuators, and a basic controller is created for the model of a bounding robot. The presented tools and examples are freely available at www.asl.ethz.ch/people/cremy/personal/GaitCreation.