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Feedback Control of Dynamic Bipedal Robot Locomotion
, 2007
"... The 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 rob ..."
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Cited by 130 (24 self)
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The 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 quasistatic stability criteria that are limited to flatfooted 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
Asymptotically stable running for a fivelink, fouractuator, planar bipedal robot
 International Journal of Robotics Research
, 2005
"... Abstract — Provably asymptotically stable running gaits are developed for the fivelink, fouractuator bipedal robot, RABBIT. A controller is designed so that the Poincaré return map associated with periodic running gaits can be computed on the basis of a model with impulseeffects that, perviously, ..."
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Cited by 29 (9 self)
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Abstract — Provably asymptotically stable running gaits are developed for the fivelink, fouractuator bipedal robot, RABBIT. A controller is designed so that the Poincaré return map associated with periodic running gaits can be computed on the basis of a model with impulseeffects that, perviously, had been used only for the design of walking gaits. This feedback design leads to the notion of a hybrid zero dynamics (HZD) for running, which in turn allows the existence and stability of running gaits to be determined on the basis of a scalar map. The main results are illustrated via simulations performed on models with known parameters and on models with parameter uncertainty and structural changes. Animations of the resulting running motions are available on the web. Index Terms — Bipedal robots; hybrid systems; limit cycles; underactuated; nonlinear control. I.
Hybrid Invariant Manifolds in Systems with Impulse Effects with Application to Periodic Locomotion in Bipedal Robots
"... 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 invar ..."
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Cited by 25 (14 self)
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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 impactupdated 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 5link biped with series compliant actuation.
Interconnection and Damping Assignment PassivityBased Control of Mechanical Systems With Underactuation Degree One
"... Abstract—Interconnection and damping assignment passivitybased control is a new controller design methodology developed for (asymptotic) stabilization of nonlinear systems that does not rely on, sometimes unnatural and techniquedriven, linearization or decoupling procedures but instead endows the ..."
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Cited by 21 (6 self)
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Abstract—Interconnection and damping assignment passivitybased control is a new controller design methodology developed for (asymptotic) stabilization of nonlinear systems that does not rely on, sometimes unnatural and techniquedriven, linearization or decoupling procedures but instead endows the closedloop system with a Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. We prove in this paper that for a class of mechanical systems with underactuation degree one the partial differential equations can be explicitly solved. Furthermore, we introduce a suitable parametrization of assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. Finally, we develop a speed estimator that allows the implementation of positionfeedback controllers. The new result is applied to obtain an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling, and a controller for the pendulum in a cart that can swingup the pendulum from any position in the upper half plane and stop the cart at any desired location. In both cases we obtain very simple and intuitive positionfeedback solutions. Index Terms—Energy shaping, Hamiltonian systems, nonlinear control, passivity, underactuated mechanical systems.
Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems
 Annual Reviews in Control
, 2008
"... Abstract: This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincare ́ firstreturn map and t ..."
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Cited by 16 (6 self)
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Abstract: This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincare ́ firstreturn map and the transverse linearization, and exponentially orbitally stabilizing control designs. We begin with general nonlinear systems, and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically. The paper concludes with a discussion of numerical issues related to control design via periodic Riccati equations.
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 13 (10 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.
LMI based design for the Acrobot walking ⋆
"... Abstract: This paper aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4dimensional l ..."
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Cited by 2 (0 self)
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Abstract: This paper aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4dimensional linear timevarying system having 3 timevarying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilizing a linear system with polytopic uncertainty. The current improvement is based on applying LMI methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above mentioned LMI design are demonstrated as well.
Hybrid Zero Dynamics of Planar Bipedal Walking
"... Summary. Models of bipedal robots in motion are fundamentally hybrid due to the presence of continuous phases, discrete transitions, and unilateral constraints arising from the contact forces between the robot and the ground. A major challenge in the control of bipedal robots has been to create a fe ..."
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Summary. Models of bipedal robots in motion are fundamentally hybrid due to the presence of continuous phases, discrete transitions, and unilateral constraints arising from the contact forces between the robot and the ground. A major challenge in the control of bipedal robots has been to create a feedback theory that provides systematic synthesis methods, provable correctness and computational tools for designing asymptotically stable, periodic walking motions, especially walking motions that are dynamic unlike the quasistatic, flatfooted gaits that are prevalent in today’s machines. This chapter highlights the fundamental role of zero dynamics in obtaining truly dynamic walking gaits that include underactuated phases. The theoretical analysis is verified with experimental work. 1
Noname manuscript No. (will be inserted by the editor) Ball on a Beam: Stabilization under Saturated Input Control with Large Basin of Attraction
, 2010
"... the date of receipt and acceptance should be inserted later ..."
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unknown title
, 2007
"... Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems§ ..."
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Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems§