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Control Lyapunov Functions and Hybrid Zero Dynamics. To appear
 Proc. 51st IEEE Conf. Decision and Control
, 2012
"... Abstract — Hybrid zero dynamics extends the ByrnesIsidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the ..."
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Cited by 16 (11 self)
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Abstract — Hybrid zero dynamics extends the ByrnesIsidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the system with impulse effects as well as its reset map, the restriction dynamics is called the hybrid zero dynamics. Prior results on the stabilization of periodic orbits of the hybrid zero dynamics have relied on inputoutput linearization of the transverse variables. The principal result of this paper shows how control Lyapunov functions can be used to exponentially stabilize periodic orbits of the hybrid zero dynamics, thereby significantly extending the class of stabilizing controllers. An illustration of this result on a model of a bipedal walking robot is provided. I.
Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics
"... Abstract—This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models— systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C 1 submanifold Z that is contained in the zero set of an output func ..."
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Cited by 16 (12 self)
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Abstract—This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models— systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C 1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the fullorder dynamics of the system with impulse effects have relied on inputoutput linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the fullorder dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions. I.
First steps toward underactuated humaninspired bipedal robotic walking
 In 2012 IEEE Conference on Robotics and Automation
, 2012
"... Abstract — This paper presents the first steps toward going from human data to formal controller design to experimental realization in the context of underactuated bipedal robots. Specifically, by studying experimental human walking data, we find that specific outputs of the human, i.e., functions o ..."
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Cited by 16 (13 self)
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Abstract — This paper presents the first steps toward going from human data to formal controller design to experimental realization in the context of underactuated bipedal robots. Specifically, by studying experimental human walking data, we find that specific outputs of the human, i.e., functions of the kinematics, appear to be canonical to walking and are all characterized by a single function of time, termed a human walking function. Using the human outputs and walking function, we design a humaninspired controller that drives the output of the robot to the output of the human as represented by the walking function. The main result of the paper is an optimization problem that determines the parameters of this controller so as to guarantee stable underactuated walking that is as “close ” as possible to human walking. This result is demonstrated through the simulation of a physical underactuated 2D bipedal robot, AMBER. Experimentally implementing this control on AMBER through “feedforward ” control, i.e., trajectory tracking, repeatedly results in 510 steps. I.
HumanData Based Cost of Bipedal Robotic Walking
 In 14th Int. Conf. on Hybrid Systems: Computation and Control
, 2011
"... This paper proposes a cost function constructed from human data, the humanbased cost, which is used to gauge the “humanlike”nature of robotic walking. This cost function is constructed by utilizing motion capture data from a 9 subject straight line walking experiment. Employing a novel technique ..."
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Cited by 11 (11 self)
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This paper proposes a cost function constructed from human data, the humanbased cost, which is used to gauge the “humanlike”nature of robotic walking. This cost function is constructed by utilizing motion capture data from a 9 subject straight line walking experiment. Employing a novel technique to process the data, we determine the times when the number of contact points change during the course of a step which automatically determines the ordering of discrete events or the domain breakdown along with the amount of time spent in each domain. The result is a weighted graph or walking cycle, associated with each of the subjects walking gaits. Finding a weighted cycle that minimizes the cut distance between this collection of graphs produces an optimal or universal domain graph for walking together with an optimal walking cycle. In essence, we find a single domain graph and the time spent in each domain that yields the most“natural ” and “humanlike ” bipedal walking. The humanbased cost is then defined as the cut distance from this optimal gait. The main findings of this paper are twofold: (1) when the humanbased cost is computed for subjects in the experiment it detects medical conditions that result in aberrations in their walking, and (2) when the humanbased cost is computed for existing robotic models the more humanlike walking gaits are correctly identified.
Towards the unification of locomotion and manipulation through control lyapunov functions and quadratic programs
 In Control of CyberPhysical Systems
, 2013
"... Abstract. This paper presents the first steps toward unifying locomotion controllers and algorithms with wholebody control and manipulation. A theoretical framework for this unification will be given based upon quadratic programs utilizing control Lyapunov functions. In particular, we will first ..."
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Cited by 9 (6 self)
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Abstract. This paper presents the first steps toward unifying locomotion controllers and algorithms with wholebody control and manipulation. A theoretical framework for this unification will be given based upon quadratic programs utilizing control Lyapunov functions. In particular, we will first consider output based feedback linearization strategies for locomotion together with wholebody control methods for manipulation. We will show that these two traditionally disjoint methods are equivalent through the correct choice of controller. We will then present a method for unifying these two methodologies through the use of control Lyapunov functions presented in the form of a quadratic program. In addition, it will be shown that these controllers can be combined with forcebased control to achieve locomotion and forcebased manipulation in a single framework. Finally, simulation results will be presented demonstrating the validity of the proposed framework.
A.D.: Speed regulation in 3D robotic walking through motion transitions between humaninspired partial hybrid zero dynamics
 the IEEE International Conference on Robotics and Automation
, 2013
"... Abstract — This paper employs the HumanInspired Control framework in the formal design, optimization and implementation of controllers for 3D bipedal robotic walking. In this framework, controllers drive the robot to a lowdimensional representation, termed the partial hybrid zero dynamics, which ..."
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Cited by 7 (7 self)
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Abstract — This paper employs the HumanInspired Control framework in the formal design, optimization and implementation of controllers for 3D bipedal robotic walking. In this framework, controllers drive the robot to a lowdimensional representation, termed the partial hybrid zero dynamics, which is shaped by the parameters of the outputs describing human locomotion data. The main result of this paper is the use of partial hybrid zero dynamics in an optimization problem to compute physical constraints on the robot, without integrating the dynamics of the system, and while simultaneously yielding provably stable walking controllers for a 3D robot model. Controllers corresponding to various walking speeds are obtained through a second speed regulation optimization, and formal methods are presented which provide smooth transitions between walking speeds. These formal results are demonstrated through simulation and utilized to obtain 3D walking experimentally with the NAO robot. I.
Dynamically stable bipedal robotic walking with NAO via humaninspired hybrid zero dynamics
 In Hybrid Systems: Computation and Control
, 2012
"... This paper demonstrates the process of utilizing human locomotion data to formally design controllers that yield provably stable robotic walking and experimentally realizing these formal methods to achieve dynamically stable bipedal robotic walking on the NAO robot. Beginning with walking data, ou ..."
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Cited by 7 (7 self)
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This paper demonstrates the process of utilizing human locomotion data to formally design controllers that yield provably stable robotic walking and experimentally realizing these formal methods to achieve dynamically stable bipedal robotic walking on the NAO robot. Beginning with walking data, outputs—or functions of the kinematics—are determined that result in a lowdimensional representation of human locomotion. These same outputs can be considered on a robot, and humaninspired control is used to drive the outputs of the robot to the outputs of the human. An optimization problem is presented that determines the parameters of this controller that provide the best fit of the human data while simultaneously ensuring partial hybrid zero dynamics. The main formal result of this paper is a proof that these same parameters result in a stable hybrid periodic orbit with a fixed point that can be computed in closed form. Thus, starting with only human data we obtain a stable walking gait for the bipedal robot model. These formal results are validated through experimentation: implementing the stable walking found in simulation on NAO results in dynamically stable robotic walking that shows excellent agreement with the simulated behavior from which it was derived.
Eventbased Stabilization of Periodic Orbits for Underactuated 3D Bipedal Robots with LeftRight Symmetry
"... Abstract—Models of robotic bipedal walking are hybrid, with a differential equation describing the stance phase and a discrete map describing the impact event, that is, the nonstance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both ..."
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Cited by 7 (3 self)
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Abstract—Models of robotic bipedal walking are hybrid, with a differential equation describing the stance phase and a discrete map describing the impact event, that is, the nonstance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both continuous and discrete (eventbased) actions. This paper concentrates on the eventbased portion of the feedback design problem for 3D bipedal walking. The results are developed in the context of robustly stabilizing periodic orbits of ATRIAS 2.1, a highly underactuated 3D bipedal robot with seriescompliant actuators and point feet, against external disturbances as well as parametric and nonparametric uncertainty. It is shown that leftright symmetry of the model can be used to both simplify and improve the design of eventbased controllers. Here, the eventbased control is developed on the basis of the Poincaré map, linear matrix inequalities (LMIs), and robust optimal control (ROC). The results are illustrated by designing a controller that enhances the lateral stability of ATRIAS 2.1. I.
Robust Eventbased Stabilization of Periodic Orbits for Hybrid Systems: Application to an Underactuated 3D Bipedal Robot
"... Abstract — The first return map or Poincaré map can be viewed as a discretetime dynamical system evolving on a hyper surface that is transversal to a periodic orbit; the hyper surface is called a Poincaré section. The Poincaré map is a standard tool for assessing the stability of periodic orbits in ..."
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Cited by 6 (3 self)
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Abstract — The first return map or Poincaré map can be viewed as a discretetime dynamical system evolving on a hyper surface that is transversal to a periodic orbit; the hyper surface is called a Poincaré section. The Poincaré map is a standard tool for assessing the stability of periodic orbits in nonhybrid as well as hybrid systems. In addition, it can be used for stabilization of periodic orbits if the underlying dynamics of the system depends on a set of parameters that can be updated by a feedback law when trajectories cross the Poincaré section. This paper addresses an important practical obstacle that arises when designing feedback laws on the basis of the Jacobian linearization of the Poincaré map. In almost all practical cases, the Jacobians must be estimated numerically, and when the underlying dynamics presents a wide range of time scales, the numerical approximations of the first partial derivatives are sufficiently inaccurate that controller tuning is very difficult. Here, a robust control formalism is proposed whereby a convex set of approximations to the Jacobian linearization is systematically generated and a stabilizing controller is designed through two appropriate sets of linear matrix inequalities (LMIs). The result is illustrated on a walking gait of a 3D underactuated bipedal robot. I.
Feedback Control of a Bipedal Walker and Runner with Compliance
, 2011
"... ii ACKNOWLEDGEMENTS Well, I’m finally writing the acknowledgments. I would like to thank my advisor, Prof. Jessy Grizzle, for creating a wonderful opportunity for me to work on legged locomotion, for his dedication in nurturing me, for his patience and enthusiasm, for his fabulous intellectual suppo ..."
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Cited by 4 (2 self)
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ii ACKNOWLEDGEMENTS Well, I’m finally writing the acknowledgments. I would like to thank my advisor, Prof. Jessy Grizzle, for creating a wonderful opportunity for me to work on legged locomotion, for his dedication in nurturing me, for his patience and enthusiasm, for his fabulous intellectual support, and for giving me the freedom to explore ideas. I would like to thank my dissertation committee members, Prof. Anthony Bloch, Prof. Arthur Kuo, Prof. Harris McClamroch, and Prof. Semyon Meerkov, for their help and support. I would like to thank HaeWon Park for his many roles as collaborator, coauthor, travel companion, and friend throughout my years at the university. I would like to thank Ben Morris for his role as a fantastic mentor during my early years, for his many theoretical contributions that I routinely employ to make my work easier, and for his excellent advice “Go big, or go home, ” which helped get the running experiments rolling. I would like to thank Ioannis Poulakakis for providing inspiration, for creating the framework of compliant hybrid zero dynamics that I happily borrowed, for solving many of my technical difficulties, and for providing great help as I looked for a job. I would like to thank Jonathan Hurst for creating MABEL, which enabled this