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A Compliant Hybrid Zero Dynamics Controller for Stable, Efficient and Fast Bipedal Walking on MABEL
"... The planar bipedal testbed MABEL contains springs in its drivetrain for the purpose of enhancing both energy efficiency and agility of dynamic locomotion. While the potential energetic benefits of springs are well documented in the literature, feedback control designs that effectively realize this ..."
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Cited by 44 (22 self)
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The planar bipedal testbed MABEL contains springs in its drivetrain for the purpose of enhancing both energy efficiency and agility of dynamic locomotion. While the potential energetic benefits of springs are well documented in the literature, feedback control designs that effectively realize this potential are lacking. In this paper, we extend and apply the methods of virtual constraints and hybrid zero dynamics, originally developed for rigid robots with a single degree of underactuation, to MABEL, a bipedal walker with a novel compliant transmission and multiple degrees of underactuation. A timeinvariant feedback controller is designed such that the closedloop system respects the natural compliance of the openloop system and realizes exponentially stable walking gaits. Five experiments are presented that highlight different aspects of MABEL and the feedback design method, ranging from basic elements such as stable walking and robustness under perturbations, to energy efficiency and a bipedal robot walking speed record of1.5 m/s (3.4 mph). The experiments also compare two feedback implementations of the virtual constraints, one based on PD control as in [1], and a second that implements the full hybrid zero dynamics controller. On MABEL, the full hybrid zero dynamics controller yields a much more faithful realization of the desired virtual constraints and was instrumental in achieving more rapid walking.
3D bipedal robotic walking: Models, feedback control, and open problems
 PROCEEDINGS OF THE IFAC SYMPOSIUM ON NONLINEAR CONTROL SYSTEMS
, 2010
"... The fields of control and robotics are contributing to the development of bipedal robots that can realize walking motions with the stability and agility of a human being. Dynamic models for bipeds are hybrid in nature. They contain both continuous and discrete elements, with switching events that ..."
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Cited by 25 (16 self)
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The fields of control and robotics are contributing to the development of bipedal robots that can realize walking motions with the stability and agility of a human being. Dynamic models for bipeds are hybrid in nature. They contain both continuous and discrete elements, with switching events that are spatially driven by unilateral constraints at ground contact and impulselike forces that occur at foot touchdown. Control laws for these machines must be hybrid as well. The goals of this paper are threefold: highlight certain properties of the models which greatly influence the control law design; present two control design approaches; and indicate some of the many open problems.
Reductionbased control of threedimensional bipedal walking robots,”
 Int. J. of Robotics Research,
, 2010
"... Abstract This paper develops the concept of reductionbased control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We prove a geometric property of general serialchain robots termed recursive cyclicity, identifying the inherent robot symmetrie ..."
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Cited by 16 (4 self)
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Abstract This paper develops the concept of reductionbased control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We prove a geometric property of general serialchain robots termed recursive cyclicity, identifying the inherent robot symmetries that we exploit with the Subrobot Theorem. This shows that any serialchain robot can be decomposed for arbitrarily lowerdimensional analysis and control. We apply this method to construct stable directional 3D walking gaits for a 4DOF hipped bipedal robot. The controlled reduction decouples the biped's sagittalplane motion from the yaw and lean modes, and on the sagittal subsystem we use passivitybased control to produce known planar limit cycles on flat ground. The unstable yaw and lean modes are separately controlled to 2periodic orbits through their shaped momenta. We numerically verify the existence of stable 2periodic straightwalking limit cycles and demonstrate turning capabilities for the controlled biped.
Virtual holonomic constraints for EulerLagrange systems
 IEEE Trans. Automat. Contr
"... Abstract: This paper investigates virtual holonomic constraints for EulerLagrange systems with n degreesoffreedom and n − 1 controls. The constraints have the form q1 = φ1(qn),..., qn−1 = φn−1(qn), where qn is a cyclic configuration variable, so their enforcement corresponds to the stabilization ..."
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Cited by 13 (6 self)
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Abstract: This paper investigates virtual holonomic constraints for EulerLagrange systems with n degreesoffreedom and n − 1 controls. The constraints have the form q1 = φ1(qn),..., qn−1 = φn−1(qn), where qn is a cyclic configuration variable, so their enforcement corresponds to the stabilization of a desired oscillatory motion. We give conditions under which such a set of constraints is feasible, meaning that it can be made invariant by feedback. We show that it is possible to systematically determine feasible virtual constraints as periodic solutions of a scalar differential equation, the virtual constraint generator. Moreover, under a symmetry assumption we show that the motion on the constraint manifold is a EulerLagrange system with one degreeoffreedom, and use this fact to complete characterize its dynamical properties. Finally, we show that if the constraint is feasible then the virtual constraint manifold can always be stabilized using inputoutput feedback linearization. 1.
Performance analysis and feedback control of ATRIAS, a 3D bipedal robot,”
 ASME J. Dyn. Sys. Meas. Control,
, 2013
"... This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is A Sphere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, spee ..."
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Cited by 11 (7 self)
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This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is A Sphere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having 6 actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits of a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (CMT), a dimensionless measure of energetic efficiency, for walking speeds ranging from 0.5 m=s ð Þ to 1.4 m=s ð Þ. A feedback controller is designed that stabilizes the 3D walking gaits, despite the high degree of underactuation of the robot. The 3D results are illustrated in simulation. In experiments on a planarized (2D) version of the robot, the controller yielded stable walking.
Subrobots: Reductionbased control with application to threedimensional bipedal walking robots,” Master’s thesis,
, 2007
"... AbstractThis paper develops the concept of reductionbased control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We introduce a geometric property of general serialchain robots termed recursive cyclicity, leading to our presentation of the Su ..."
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Cited by 11 (1 self)
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AbstractThis paper develops the concept of reductionbased control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We introduce a geometric property of general serialchain robots termed recursive cyclicity, leading to our presentation of the Subrobot Theorem. This shows that reductionbased control can arbitrarily reduce the dimensionality of any serialchain robot, so that it may be controlled as a simpler "subrobot" while separately controlling the divided coordinates through their conserved momenta. This method is applied to construct stable directional 3D walking gaits for a 4d.o.f. hipped bipedal robot. The walker's sagittalplane subsystem can be decoupled from its yaw and lean modes, and on this planar subsystem we use passivitybased control to construct limit cycles on flat ground. Due to the controlled reduction, the unstable yaw and lean modes are separately controlled to 2periodic orbits. We numerically verify the existence of stable 2periodic limit cycles and demonstrate turning capabilities for the controlled biped.
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.
3D Bipedal Walking with Knees and Feet: A Hybrid Geometric Approach
 In 48th IEEE Conf. on Decision and Control and 28th Chinese Control Conf
, 2009
"... Abstract—Motivated by the goal of obtaining moreanthropomorphic walking in bipedal robots, this paper considers a hybrid model of a 3D hipped biped with feet and locking knees. The main observation of this paper is that functional Routhian Reduction can be used to extend twodimensional walking to ..."
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Cited by 10 (8 self)
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Abstract—Motivated by the goal of obtaining moreanthropomorphic walking in bipedal robots, this paper considers a hybrid model of a 3D hipped biped with feet and locking knees. The main observation of this paper is that functional Routhian Reduction can be used to extend twodimensional walking to three dimensions—even in the presence of periods of underactuation—by decoupling the sagittal and coronal dynamics of the 3D biped. Specifically, we assume the existence of a control law that yields stable walking for the 2D sagittal component of the 3D biped. The main result of the paper is that utilizing this controller together with “reduction control laws ” yields walking in three dimensions. This result is supported through simulation. I.
Embedding Active Force Control within the Compliant Hybrid Zero Dynamics to Achieve Stable, Fast Running on MABEL
"... A mathematical formalism for designing provably stable, running gaits in bipedal robots with compliance is presented and the theoretical work is validated experimentally on MABEL, a planar bipedal testbed that contains springs in its drivetrain. The methods of virtual constraints and hybrid zero dyn ..."
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Cited by 8 (3 self)
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A mathematical formalism for designing provably stable, running gaits in bipedal robots with compliance is presented and the theoretical work is validated experimentally on MABEL, a planar bipedal testbed that contains springs in its drivetrain. The methods of virtual constraints and hybrid zero dynamics are used to design a timeinvariant feedback controller that not only respects the natural compliance of the openloop system, but also enables active force control within the compliant hybrid zero dynamics. The controller dynamically varies the effective leg stiffness throughout the gait. When implemented on MABEL, a kneedbiped running record of 3.06 m/s (10.9 kph or 6.8 mph) is achieved.
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.