By the dawn of the new millennium, robotics has undergone a major tra- formation in scope and dimensions. This expansion has been brought about bythematurityofthe?eldandtheadvancesinitsrelatedtechnologies.From a largely dominant industrial focus, robotics has been rapidly expanding into the challenges of the human world. The new generation of robots is expected to safely and dependably co-habitat with humans in homes, workplaces, and communities,providingsupportinservices,entertainment,education,heal- care, manufacturing, and assistance. Beyond its impact on physical robots, the body of knowledge robotics has produced is revealing a much wider range of applications reaching across - verse researchareas and scienti?c disciplines, such as: biomechanics, haptics, neurosciences, virtual simulation, animation, surgery, and sensor networks among others. In return, the challenges of the new emerging areas are pr- ing an abundant source of stimulation and insights for the ?eld of robotics. It is indeed at the intersection of disciplines that the most striking advances happen. The goal of the series of Springer Tracts in Advanced Robotics (STAR) is to bring, in a timely fashion, the latest advances and developments in robotics on the basis of their signi?cance and quality. It is our hope that the wider dissemination of research developments will stimulate more exchanges and collaborations among the research community and contribute to further advancement of this rapidly growing ?eld.

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Walking robots are complex machines with many degrees of freedom. Designing efficient controllers for such robots can be a daunting task, and the differential equations by themselves usually do not help much when trying to understand the dynamics. Still, research on passive dynamic walking robots has shown that it is possible to make robotic mechanisms walk very naturally and efficiently without using any control! The gap between theoretically well-understood position-controlled walking robots and experimentally-designed uncontrolled passive-dynamic walkers is nevertheless large, and extending a passive-dynamic walker to be more robust and versatile is non-trivial.

The purpose of this work is to present a set of mathematical tools that can simplify studying robotic walking motions and designing energy-efficient controllers. The authors extend classical dynamic modeling methods and view robots and controllers as energy-exchanging physical systems, which forms the basisof the so-called port-based approach. They show how such methods can be used to analyze walking mechanisms, find efficient walking trajectories, and design controllers that increase robustness and stability with minimal energy cost. Extensive examples and illustrations are used with the objective to make the mathematics intuitive and accessible to everyone with an engineering background.

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