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Research Papers

A Discrete Control Lyapunov Function for Exponential Orbital Stabilization of the Simplest Walker

[+] Author and Article Information
Pranav A. Bhounsule

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249
e-mail: pranav.bhounsule@utsa.edu

Ali Zamani

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249

1Corresponding author.

Manuscript received January 19, 2017; final manuscript received June 15, 2017; published online August 18, 2017. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 9(5), 051011 (Aug 18, 2017) (8 pages) Paper No: JMR-17-1018; doi: 10.1115/1.4037440 History: Received January 19, 2017; Revised June 15, 2017

In this paper, we demonstrate the application of a discrete control Lyapunov function (DCLF) for exponential orbital stabilization of the simplest walking model supplemented with an actuator between the legs. The Lyapunov function is defined as the square of the difference between the actual and nominal velocity of the unactuated stance leg at the midstance position (stance leg is normal to the ramp). The foot placement is controlled to ensure an exponential decay in the Lyapunov function. In essence, DCLF does foot placement control to regulate the midstance walking velocity between successive steps. The DCLF is able to enlarge the basin of attraction by an order of magnitude and to increase the average number of steps to failure by 2 orders of magnitude over passive dynamic walking. We compare DCLF with a one-step dead-beat controller (full correction of disturbance in a single step) and find that both controllers have similar robustness. The one-step dead-beat controller provides the fastest convergence to the limit cycle while using least amount of energy per unit step. However, the one-step dead-beat controller is more sensitive to modeling errors. We also compare the DCLF with an eigenvalue-based controller for the same rate of convergence. Both controllers yield identical robustness but the DCLF is more energy-efficient and requires lower maximum torque. Our results suggest that the DCLF controller with moderate rate of convergence provides good compromise between robustness, energy-efficiency, and sensitivity to modeling errors.

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Figures

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Fig. 1

Difference between CCLF and DCLF methodologies: (a) CCLF constructs a controller that keeps the trajectory within a tube along the trajectory. (b) DCLF constructs a controller that keeps the trajectory in the basin of attraction only at the Poincaré section.

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Fig. 2

Model: The simplest slope walking model analyzed by Garcia et al. [18]

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Fig. 3

Basin of attraction: Uncontrolled/passive dynamic case (white region) and DCLF (gray region includes the white region for the uncontrolled case) for three values of c. The dashed line shows the passive limit cycle.

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Fig. 4

Rate of stabilization: The convergence is exponential for 0<c<1 and dead-beat for c = 1 for state perturbation θ˙=−0.5 for γ=0.009

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Fig. 5

Robustness to step-down disturbance for different c values: (a) average number of steps to failure, (b) average energy (absolute value of mechanical work done by the hip actuator) used per step, (c) average step length, and (d) maximum torque needed

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Fig. 6

Sensitivity of DCLF to modeling errors: (a) average number of steps to failure, (b) average energy (absolute value of mechanical work done by the hip actuator) used per step, and (c) maximum torque needed. The slope, γ, is 1.1 of its actual value.

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Fig. 7

Robustness to step-down disturbance comparing eigenvalue-based with Lyapunov-based stability: (a) average number of steps to failure, (b) average energy (absolute value of mechanical work done by the hip actuator) used per step, (c) average step length, and (d) maximum torque needed. All plots are for c = 0.9 which corresponds to an eigenvalue of 1−c=0.32.

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