Aaron D. Ames

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

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 C1 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 full-order dynamics of the system with impulse effects have relied on input-output 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 full-order 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.

Human-Inspired Control of Bipedal Walking Robots

This paper presents a human-inspired control approach to bipedal robotic walking: utilizing human data and output functions that appear to be intrinsic to human walking in order to formally design controllers that provably result in stable robotic walking. Beginning with human walking data, outputs-or functions of the kinematics-are determined that result in a low-dimensional representation of human locomotion. These same outputs can be considered on a robot, and human-inspired control is used to drive the outputs of the robot to the outputs of the human. The main results of this paper are that, in the case of both under and full actuation, the parameters of this controller can be determined through a human-inspired optimization problem that provides the best fit of the human data while simultaneously provably guaranteeing stable robotic walking for which the initial condition can be computed in closed form. These formal results are demonstrated in simulation by considering two bipedal robots-an underactuated 2-D bipedal robot, AMBER, and fully actuated 3-D bipedal robot, NAO-for which stable robotic walking is automatically obtained using only human data. Moreover, in both cases, these simulated walking gaits are realized experimentally to obtain human-inspired bipedal walking on the actual robots.

3D Bipedal Robotic Walking: Models, Feedback Control, and Open Problems

Abstract 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 impulse-like 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.

Models, feedback control, and open problems of 3D bipedal robotic walking

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 impulse-like 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.

Is there life after Zeno? Taking executions past the breaking (Zeno) point

Understanding Zeno phenomena plays an important role in understanding hybrid systems. A natural - and intriguing - question to ask is: what happens after a Zeno point? Inspired by the construction of Filippov (1988), we propose a method for extending Zeno executions past a Zeno point for a class of hybrid systems: Lagrangian hybrid systems. We argue that after the Zeno point is reached, the hybrid system should switch to a holonomically constrained dynamical system, where the holonomic constraints are based on the unilateral constraints on the configuration space that originally defined the hybrid system. These principles are substantiated with a series of examples

Control barrier function based quadratic programs with application to adaptive cruise control

This paper develops a control methodology that unifies control barrier functions and control Lyapunov functions through quadratic programs. The result is demonstrated on adaptive cruise control, which presents both safety and performance considerations, as well as actuator bounds. We begin by presenting a novel notion of a barrier function associated with a set, formulated in the context of Lyapunov-like conditions; the existence of a barrier function satisfying these conditions implies forward invariance of the set. This formulation naturally yields a notion of control barrier function (CBF), yielding inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the simultaneous achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). These formulations are illustrated in the context of adaptive cruise control, where the control objective of achieving a desired speed is balanced by the minimum following conditions on a lead car and force-based constraints on acceleration and braking.

Dynamically stable bipedal robotic walking with NAO via human-inspired hybrid zero dynamics

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 low-dimensional representation of human locomotion. These same outputs can be considered on a robot, and human-inspired 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.

First Steps toward Automatically Generating Bipedal Robotic Walking from Human Data

This paper presents the first steps toward automatically generating robotic walking from human walking data through the use of human-inspired control. By considering experimental human walking data, we discover that certain outputs of the human, computed from the kinematics, display the same “universal” behavior; moreover, these outputs can be described by a remarkably simple class of functions, termed canonical human walking functions, with a high degree of accuracy. Utilizing these functions, we consider a 2D bipedal robot with knees, and we construct a control law that drives the outputs of the robot to the outputs of the human. Explicit conditions are derived on the parameters of the canonical human walking functions that guarantee that the zero dynamics surface is partially invariant through impact, i.e., conditions that guarantee partial hybrid zero dynamics. These conditions therefore can be used as constraints in an optimization problem that minimizes the distance between the human data and the output of the robot. In addition, we demonstrate through simulation that these conditions automatically generate a stable periodic orbit for which the fixed point can be explicitly computed. Therefore, using only human data, we are able to automatically generate a stable walking gait for a bipedal robot which is as “human-like” as possible.

Valkyrie: NASA's First Bipedal Humanoid Robot

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. NASAs Johnson Space Center led a team comprised of numerous partners to develop Valkyrie, NASAs 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. Valkyries 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 Valkyries 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.

Sufficient Conditions for the Existence of Zeno Behavior

The existence of Zeno behavior in hybrid systems is related to a certain type of equilibria, termed Zeno equilibria, that are invariant under the discrete, but not the continuous, dynamics of a hybrid system. In analogy to the standard procedure of linearizing a vector field at an equilibrium point to determine its stability, in this paper we study the local behavior of a hybrid system near a Zeno equilibrium point by considering the value of the vector field on each domain at this point, i.e., we consider constant approximations of nonlinear hybrid systems. By means of these constant approximations, we are able to derive conditions that simultaneously imply both the existence of Zeno behavior and the local exponential stability of a Zeno equilibrium point. Moreover, since these conditions are in terms of the value of the vector field on each domain at a point, they are remarkably easy to verify.