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16
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.
R.: An efficiently solvable quadratic program for stabilizing dynamic locomotion
 In: Proceedings of the International Conference on Robotics and Automation (ICRA). Hong Kong
, 2014
"... Abstract — We describe a wholebody dynamic walking controller implemented as a convex quadratic program. The controller solves an optimal control problem using an approximate value function derived from a simple walking model while respecting the dynamic, input, and contact constraints of the ful ..."
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Cited by 12 (5 self)
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Abstract — We describe a wholebody dynamic walking controller implemented as a convex quadratic program. The controller solves an optimal control problem using an approximate value function derived from a simple walking model while respecting the dynamic, input, and contact constraints of the full robot dynamics. By exploiting sparsity and temporal structure in the optimization with a custom activeset algorithm, we surpass the performance of the best available offtheshelf solvers and achieve 1kHz control rates for a 34DOF humanoid. We describe applications to balancing and walking tasks using the simulated Atlas robot in the DARPA Virtual Robotics Challenge. I.
From formal methods to algorithmic implementation of human inspired control on bipedal robots
 In the Tenth International WAFR
, 2012
"... Abstract This paper presents the process of translating formal theory and methods to efficient algorithms in the context of humaninspired control of bipedal robots, with the end result being experimentally realized robust and efficient robotic walking with AMBER. We begin by considering human walk ..."
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Cited by 11 (10 self)
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Abstract This paper presents the process of translating formal theory and methods to efficient algorithms in the context of humaninspired control of bipedal robots, with the end result being experimentally realized robust and efficient robotic walking with AMBER. We begin by considering human walking data and find outputs (or virtual constraints) that, when calculated from the human data, are described by simple functions of time (termed canonical walking functions). Formally, we construct a torque controller, through model inversion, that drives the outputs of the robot to the outputs of the human as represented by the canonical walking function; while these functions fit the human data well, they do not apriori guarantee robotic walking (due to do the physical differences between humans and robots). An optimization problem is presented that determines the best fit of the canonical walking function to the human data, while guaranteeing walking for a specific bipedal robot; in addition, constraints can be added that guarantee physically realizable walking. We consider a physical bipedal robot AMBER and define a simple voltage based control law—utilizing only the human outputs and canonical walking function with parameters obtained from the optimization—for which we obtain walking in simulation. Since this controller does not require model inversion, it can be implemented efficiently in software. Moreover, applying this methodology to AMBER experimentally results in robust and efficient “humanlike ” robotic walking. 1
Motion primitives for humaninspired bipedal robotic locomotion: Walking and stair climbing
 In IEEE Intl. Conf. Robotics and Automation
, 2012
"... Abstract — This paper presents an approach to the development of bipedal robotic control techniques for multiple locomotion behaviors. Insight into the fundamental behaviors of human locomotion is obtained through the examination of experimental human data for walking on flat ground, upstairs and d ..."
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Cited by 10 (9 self)
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Abstract — This paper presents an approach to the development of bipedal robotic control techniques for multiple locomotion behaviors. Insight into the fundamental behaviors of human locomotion is obtained through the examination of experimental human data for walking on flat ground, upstairs and downstairs. Specifically, it is shown that certain outputs of the human, independent of locomotion terrain, can be characterized by a single function, termed the extended canonical human function. Optimized functions of this form are tracked via feedback linearization in simulations of a planar robotic biped walking on flat ground, upstairs and downstairs — these three modes of locomotion are termed “motion primitives. ” A second optimization is presented, which yields controllers that evolve the robot from one motion primitive to another — these modes of locomotion are termed “motion transitions. ” A final simulation is given, which shows the controlled evolution of a robotic biped as it transitions through each mode of locomotion over a pyramidal staircase. I.
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.
Humaninspired underactuated bipedal robotic walking with amber on flatground, upslope and uneven terrain
 In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS
, 2012
"... Abstract — This work presents humaninspired control strategies required for achieving three motion primitives in walking— flatground, uneven terrain and upslope—in an underactuated physical bipedal robot: AMBER. Formal models and controllers which provably guarantee the stability of walking are ..."
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Cited by 6 (5 self)
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Abstract — This work presents humaninspired control strategies required for achieving three motion primitives in walking— flatground, uneven terrain and upslope—in an underactuated physical bipedal robot: AMBER. Formal models and controllers which provably guarantee the stability of walking are developed and verified in the simulation. Computationally tractable conditions are given that allow for the experimental implementation of these formal methods through the closed form approximation of constraints that restrict maximum torque, maximum velocity and ensure proper foot clearance. Considering the special property of the motors used in the robot, i.e., low leakage inductance and high angular speed, we approximate the motor model and translate the formal controllers satisfying these constraints into an efficient voltagebased controller that can be directly implemented on AMBER. The end result is robotic walking on AMBER for the three motion primitives that shows good agreement with the formal results from which it was derived. I.
Models, Feedback Control, and Open Problems of 3D Bipedal Robot Walking
, 2014
"... The fields of control and robotics are working toward 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 ..."
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Cited by 5 (3 self)
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The fields of control and robotics are working toward 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 governed by a combination of unilateral constraints and impulselike forces that occur at foot touchdown. Control laws for these machines must be hybrid as well. The goals of this paper are fourfold: highlight certain properties of the models which greatly influence the control law design; overview the literature; present two control design approaches in depth; and indicate some of the many open problems.
HumanInspired Control of Bipedal Robots via Control Lyapunov Functions and Quadratic Programs
"... This paper briefly presents the process of formally achieving bipedal robotic walking through controller synthesis inspired by human locomotion. Motivated by the hierarchical control present in humans, we begin by viewing the human as a “black box ” and describe outputs, or virtual constraints, tha ..."
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Cited by 5 (3 self)
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This paper briefly presents the process of formally achieving bipedal robotic walking through controller synthesis inspired by human locomotion. Motivated by the hierarchical control present in humans, we begin by viewing the human as a “black box ” and describe outputs, or virtual constraints, that appear to characterize human walking. By considering the equivalent outputs for the bipedal robot, a nonlinear controller can be constructed that drives the outputs of the robot to the outputs of the human; moreover, the parameters of this controller can be optimized so that stable robotic walking is provably achieved while simultaneously producing outputs of the robot that are as close as possible to those of a human. Finally, considering a control Lyapunov function based representation of these outputs allows for the class of controllers that provably achieve stable robotic walking can be greatly enlarged. The end result is the generation of bipedal robotic walking that is remarkably humanlike and is experimentally realizable, as evidenced by the implementation of the resulting controllers on multiple robotic platforms.
Achieving Bipedal Locomotion on Rough Terrain through HumanInspired Control
"... Abstract — This paper presents a method for achieving robotic walking on rough terrain through HumanInspired Control. This control methodology uses human data to achieve human like walking in robots by considering outputs that appear to be indicative of walking, and using nonlinear control methods ..."
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Cited by 4 (3 self)
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Abstract — This paper presents a method for achieving robotic walking on rough terrain through HumanInspired Control. This control methodology uses human data to achieve human like walking in robots by considering outputs that appear to be indicative of walking, and using nonlinear control methods to track a set of functions called Canonical Walking Functions (CWF). While this method has proven successful on a specific welldefined terrain, rough terrain walking is achieved by dynamically changing the CWF that the robot outputs should track at every step. To make the computation more tractable Extended Canonical Walking Functions (ECWF) are used to generate these desired functions instead of CWF. The state of the robot, after every nonstance foot strike, is actively sensed and a new CWF is constructed to ensure Hybrid Zero Dynamics is respected for the next step. Finally, the technique developed is implemented on different terrains in simulation. The same technique is adopted experimentally on the bipedal robot AMBER and tested on sinusoidal terrain. Experimental results show how the walking gait morphs based upon the terrain, thereby justifying the theory applied.