Analysis of a Simplified Hopping Robot

Koditschek, Daniel E.; Bühler, Martin

doi:10.1177/027836499101000601

Cited by 242 publications

(178 citation statements)

References 12 publications

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“…In further defense of our coarse abstraction we observe that these models are sufficiently complex that so far the only analytical results for work-directed controllers encompassing physical actuator models explicitly coupled to the physical body state model have been obtained for one degree of freedom bodies (e.g. such as [35]) and that we see the present analysis as a first step along the way to that more informative but far less tractable problem. We also observe that no smooth workdirected scheme has heretofore been shown to converge even on T 2 .…”

Section: B Motor Modelmentioning

confidence: 86%

A self-exciting controller for high-speed vertical running

Lynch

Clark

Koditschek

2009

2009 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Traditional legged runners and climbers have relied heavily on gait generators in the form of internal clocks or reference trajectories. In contrast, here we present physical experiments with a fast, dynamical, vertical wall climbing robot accompanying a stability proof for the controller that generates it without any need for an additional internal clock or reference signal. Specifically, we show that this "self-exciting" controller does indeed generate an "almost" globally asymptotically stable limit cycle: the attractor basin is as large as topologically possible and includes all the state space excluding a set with empty interior. We offer an empirical comparison of the resulting climbing behavior to that achieved by a more conventional clockgenerated gait trajectory tracker. The new, self-exciting gait generator exhibits a marked improvement in vertical climbing speed, in fact setting a new benchmark in dynamic climbing by achieving a vertical speed of 1.5 body lengths per second. Abstract-Traditional legged runners and climbers have relied heavily on gait generators in the form of internal clocks or reference trajectories. In contrast, here we present physical experiments with a fast, dynamical, vertical wall climbing robot accompanying a stability proof for the controller that generates it without any need for an additional internal clock or reference signal. Specifically, we show that this "self-exciting" controller does indeed generate an "almost" globally asymptotically stable limit cycle: the attractor basin is as large as topologically possible and includes all the state space excluding a set with empty interior. We offer an empirical comparison of the resulting climbing behavior to that achieved by a more conventional clock-generated gait trajectory tracker. The new, self-exciting gait generator exhibits a marked improvement in vertical climbing speed, in fact setting a new benchmark in dynamic climbing by achieving a vertical speed of 1.5 body lengths per second.

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Section: B Motor Modelmentioning

confidence: 86%

A self-exciting controller for high-speed vertical running

Lynch

Clark

Koditschek

2009

2009 IEEE/RSJ International Conference on Intelligent Robots and Systems

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“…Earlier studies of vertically constrained hopping revealed that the combination of constant energy input with viscous damping in the leg yields global, asymptotic stability as a result of the associated unimodal return map. 18 Following a similar line of inquiry, we find it most convenient to work in a new set of coordinates, the apex height, and the total mechanical energy, yielding a new return map definition, a simple coordinate change away from the map described in Sec. II C as ͫ y a ͓k + 1͔ E a ͓k + 1͔ ͬ = P ͩͫ y a ͓k͔ E a ͓k͔ ͬͪ .…”

Section: A Equilibrium Points Of the Uncontrolled Return Mapmentioning

confidence: 99%

“…[13][14][15][16][17] This led to the development of the simple yet accurate SLIP model to describe such behaviors. 18,19 Significant research effort was devoted to both the use of this model as a basis for the design of fast and efficient legged robots 1,2,20,21 and associated control strategies 22,23 as well as its analysis to reveal fundamental aspects of legged locomotory behaviors. 11 The present paper falls into the latter category and contributes by investigating the previously unaddressed question of how the presence of passive damping and actuation through a controllable hip torque affects the behavioral characteristics of running.…”

Section: Introductionmentioning

confidence: 99%

Stride-to-stride energy regulation for robust self-stability of a torque-actuated dissipative spring-mass hopper

Ankaralı

Saranlı

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Lateral stability of the spring-mass hopper suggests a two-step control strategy for running Chaos: An Interdisciplinary Journal of Nonlinear Science 19, 026106 (2009) In this paper, we analyze the self-stability properties of planar running with a dissipative springmass model driven by torque actuation at the hip. We first show that a two-dimensional, approximate analytic return map for uncontrolled locomotion with this system under a fixed touchdown leg angle policy and an open-loop ramp torque profile exhibits only marginal self-stability that does not always persist for the exact system. We then propose a per-stride feedback strategy for the hip torque that explicitly compensates for damping losses, reducing the return map to a single dimension and substantially improving the robust stability of fixed points. Subsequent presentation of simulation evidence establishes that the predictions of this approximate model are consistent with the behavior of the exact plant model. We illustrate the relevance and utility of our model both through the qualitative correspondence of its predictions to biological data as well as its use in the design of a task-level running controller. © 2010 American Institute of Physics. ͓doi:10.1063/1.3486803͔It has long been established that simple spring-mass systems, such as the well-studied spring-loaded inverted pendulum (SLIP) model, can accurately represent the dynamics of legged locomotion. However, the existing work in this domain almost exclusively focuses on lossless leg models with actuation through tunable leg stiffness, making it difficult to generalize associated results to physical systems. In this paper, we introduce a more realistic model with damping and actuation through a controllable hip torque, subsequently developing a sufficiently accurate analytic approximation to identify and characterize its limit cycles. We show that in the absence of any explicit controls, running with this model is only marginally stable, but when an "energy regulating" feedback law is introduced on the stance hip torque, an open-loop, fixed touchdown angle policy produces asymptotically stable running across a much larger range of states. We also show that the relatively understudied hip torque actuation not only provides robust stability properties, but also has interesting correspondence to data from biological runners, more accurately predicting horizontal ground reaction forces during locomotion.

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“…Though not indicated in the figure, massless springs may be attached between links and between links and the inertial reference frame; prismatic joints between links are also allowed. This class of systems clearly includes the Acrobot [2,36,52], the brachiating robots of [15,34,35,46], the gymnast robots of [32,39,57] when pivoting on a high bar, and the stance phase models of Raibert's onelegged hopper [1,6,14,26,33,42] as well as RABBIT [7][8][9][10]41]. The control objectives will be to stabilize the system about an equilibrium point or to track a set of reference trajectories with internal stability.…”

Section: Motivating Classes Of Systemsmentioning

confidence: 99%

Nonlinear control of mechanical systems with one degree of underactuation

Chevallereau¹,

Grizzle

Moog³

2004

IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004

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Abstract-Numerous robotic tasks associated with underactuation have been studied in the literature. For a large number of these in the plane, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. This paper formulates and solves two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is not conserved, conditions are found for the existence of a set of outputs that yields a system with a one-dimensional exponentially stable zero dynamics -i.e. an exponentially minimum-phase system -along with a dynamic extension that renders the system locally input-output decouplable. If the generalized momentum conjugate to the cyclic variable is conserved, a reduced system is constructed and conditions are found for the existence of a set of outputs that yields an empty zero dynamics, along with a dynamic extension that renders the system feedback linearizable. A common element in these two feedback problems is the construction of a scalar function of the configuration variables that has relative degree three with respect to one of the input components. The function arises by partially integrating the conjugate momentum. The results are illustrated on two balancing tasks and on a ballistic flip motion.

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Analysis of a Simplified Hopping Robot

Cited by 242 publications

References 12 publications

A self-exciting controller for high-speed vertical running

A self-exciting controller for high-speed vertical running

Stride-to-stride energy regulation for robust self-stability of a torque-actuated dissipative spring-mass hopper

Nonlinear control of mechanical systems with one degree of underactuation

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