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TECHNICAL PAPERS

Dynamic Stability of Open-Loop Hopping

[+] Author and Article Information
Jorge G. Cham1

Center for Design Research, Stanford University, Stanford, CA 94305-2232jgcham@robotics.caltech.edu

Mark R. Cutkosky

Center for Design Research, Stanford University, Stanford, CA 94305-2232

1

Currently with the Department of Mechanical Engineering, California Institute of Technology, Pasadena, CA.

J. Dyn. Sys., Meas., Control 129(3), 275-284 (Sep 21, 2006) (10 pages) doi:10.1115/1.2718237 History: Received September 08, 2004; Revised September 21, 2006

Simulations and physical robots have shown that hopping and running are possible without sensory feedback. However, stable behavior is often limited to a certain range of the parameters of the open-loop system. Even the simplest of hopping systems can exhibit unstable behavior that results in unpredictable nonperiodic motion as system parameters are adjusted. This paper analyzes the stability of a simplified vertical hopping model driven by an open-loop, feedforward motor pattern. Periodic orbits of the resulting hybrid system are analyzed through a generalized formula for the system’s Poincare Map and Jacobian. The observed behavior is validated experimentally in a physical pneumatically actuated hopping machine. This approach leads to observations on the stability of this and similar systems, revealing inherent limitations of open-loop hopping and providing insights that can inform the design and control of dynamic legged robots capable of rapid and robust locomotion.

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Copyright © 2007 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Sprawl robots, which achieve fast and robust locomotion without sensory feedback

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Figure 2

The vertical hopping model used for analysis

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Figure 4

Illustration of Jacobian of map between two mode boundaries

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Figure 7

Work performed by the actuator is proportional to the height difference between the state at thrust activation and thrust deactivation

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Figure 8

Unstable steady-state solution for “Long Thrust” case

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Figure 9

Effect of perturbations in the timing of thrust activation on the work performed, and resulting takeoff velocity and airborne phase duration for the “Long Thrust” case

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Figure 10

Unstable steady-state solution for “Short Thrust” case for a period of 0.15. At this period, thrust is initiated well before maximum compression, and takeoff velocity is small.

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Figure 11

Effect of perturbations in the timing of thrust activation on the work performed, and resulting takeoff velocity and airborne phase duration for the “Short Thrust” case

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Figure 12

Stride periods for which unstable period-1 trajectories exist in the Short Thrust case tend to result in period-2 stable trajectories

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Figure 13

The “Dashpod,” a pneumatically actuated vertical hopper

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Figure 14

Experimental data of hopping trajectories for the Dashpod vertical hopper

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Figure 3

Sample time history of the vertical hopper

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Figure 5

Steady-state solutions as a function of stride period for the “Long Thrust” case

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Figure 6

Steady-state solutions as a function of stride period for the “Short Thrust” case

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