To our loved ones. A tous ceux que nous aimons. DRAFT --May 15, 2007 --DRAFT --May 15, 2007 --DR PrefaceThe objective of this book is to present systematic methods for achieving stable, agile and efficient locomotion in bipedal robots. The fundamental principles presented here can be used to improve the control of existing robots and provide guidelines for improving the mechanical design of future robots. The book also contributes to the emerging control theory of hybrid systems. Models of legged machines are fundamentally hybrid in nature, with phases modeled by ordinary differential equations interleaved with discrete transitions and reset maps. Stable walking and running correspond to the design of asymptotically stable periodic orbits in these hybrid systems and not equilibrium points. Past work has emphasized quasi-static stability criteria that are limited to flat-footed walking. This book represents a concerted effort to understand truly dynamic locomotion in planar bipedal robots, from both theoretical and practical points of view.The emphasis on sound theory becomes evident as early as Chapter 3 on modeling, where the class of robots under consideration is described by lists of hypotheses, and further hypotheses are enumerated to delineate how the robot interacts with the walking surface at impact, and even the characteristics of its gait. This careful style is repeated throughout the remainder of the book, where control algorithm design and analysis are treated. At times, the emphasis on rigor makes the reading challenging for those less mathematically inclined. Do not, however, give up hope! With the exception of Chapter 4 on the method of Poincaré sections for hybrid systems, the book is replete with concrete examples, some very simple, and others quite involved. Moreover, it is possible to cherry-pick one's way through the book in order to "just figure out how to design a controller while avoiding all the proofs." This is mapped out below and in Appendix A.The practical side of the book stems from the fact that it grew out of a project grounded in hardware. More details on this are given in the acknowledgements, but suffice it to say that every stage of the work presented here has involved the interaction of roboticists and control engineers. This interaction has led to a control theory that is closely tied to the physics of bipedal robot locomotion. The importance and advantage of doing this was first driven home to one of the authors when a multipage computation involving the Frobenius Theorem produced a quantity that one of the other authors identified as angular momentum, and she could reproduce the desired result in two lines! Fortunately, the power of control theory produced its share of eye-opening moments on the robotic side of the house, such as when days and DRAFT --May 15, 2007 --DRAFT --May 15, 2007 --DR days of simulations to tune a "physically-based" controller were replaced by a ten minute design of a PI-controller on the basis of a restricted Poincaré map, and the controller worked l...

Motivated by the problem of controlling walking in a biped with series compliant actuation, this paper develops two main theorems relating to the stabilization of periodic orbits in systems with impulse effects. First, when a periodic orbit of a system with impulse effects lies within a hybrid invariant manifold, the Jacobian linearization of the Poincaré return map results in a matrix that is block upper triangular. One diagonal block is the linearization of the return map of the hybrid zero dynamics, and the other is the product of two sensitivity matrices related to the transverse dynamics. When either sensitivity matrix is sufficiently close to zero, the stability of the return map is determined solely by the hybrid zero dynamics. The second main result of the paper details the construction of a hybrid invariant manifold by introducing impact-updated control parameters. Using the construction, entries of either (or both) of the transverse dynamics' sensitivity matrices can be made arbitrarily small. A simulation example is provided, where stable walking is achieved in a 5-link biped with series compliant actuation.

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In December 2013, 16 teams from around the world gathered at Homestead Speedway near Miami, FL to participate in the DARPA Robotics Challenge (DRC) Trials, an aggressive robotics competition partly inspired by the aftermath of the Fukushima Daiichi reactor incident. While the focus of the DRC Trials is to advance robotics for use in austere and inhospitable environments, the objectives of the DRC are to progress the areas of supervised autonomy and mobile manipulation for everyday robotics. NASA's Johnson Space Center led a team comprised of numerous partners to develop Valkyrie, NASA's first bipedal humanoid robot. Valkyrie is a 44 degree‐of‐freedom, series elastic actuator‐based robot that draws upon over 18 years of humanoid robotics design heritage. Valkyrie's application intent is aimed at not only responding to events like Fukushima, but also advancing human spaceflight endeavors in extraterrestrial planetary settings. This paper presents a brief system overview, detailing Valkyrie's mechatronic subsystems, followed by a summarization of the inverse kinematics‐based walking algorithm employed at the Trials. Next, the software and control architectures are highlighted along with a description of the operator interface tools. Finally, some closing remarks are given about the competition, and a vision of future work is provided.

Abstract-This paper introduces MABEL, a new platform for the study of bipedal locomotion in robots. One of the purposes of building the mechanism is to explore a novel powertrain design that incorporates compliance, with the objective of improving the power efficiency of the robot, both in steady state operation and in responding to disturbances. A second purpose is to inspire the development of new feedback control algorithms for running on level surfaces and walking on rough terrain. A third motivation for building the robot is science and technology outreach; indeed, it is already included in tours when K-through-12 students visit the College of Engineering at the University of Michigan. MABEL is currently walking at 1.1 m/s on a level surface, and a related monopod at Carnegie Mellon is hopping well, establishing that the testbed has the potential to realize its many objectives.

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Online optimization-based controllers are becoming increasingly prevalent as a means to control complex highdimensional nonlinear systems, e.g., bipedal and humanoid robots, due to their ability to balance multiple control objectives subject to input constraints. Motivated by these applications, the goal of this paper is to explore the continuity and smoothness properties of feedback controllers that are formulated as quadratic programs (QPs). We begin by drawing connections between these optimization-based controllers and a family of perturbed nonlinear programming problems commonly studied in operations research. With a view towards robotic systems, some existing results on perturbed nonlinear programming problems are extended and specialized to address conditions that arise when quadratic programs are used to enforce the convergence of control Lyapunov functions (CLFs).The main result of this paper is a novel set of conditions on the continuity of QPs that can be used when a subset of the constraints vanishes. A simulation study of position regulation in the compass gait biped demonstrates how the new conditions of this paper can be applied to more complex robotic systems.

New analysis and tools are presented that extend the hybrid zero dynamics (HZD) framework for the control of planar bipedal walkers. Results include (i) analysis of walking on a slope, (ii) analysis of dynamic (decoupling matrix) singularities, and (iii) an alternative method for choosing virtual constraints. A key application of the new tools is the design of controllers that render a passive bipedal gait robust to disturbances without the use of full actuationwhile still requiring zero control effort at steady-state. The new tools can also be used to design controllers for gaits having an arbitrary steady-state torque profile. Five examples are given that illustrate these and other results.