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Research Papers

Introduction of the Foot Placement Estimator: A Dynamic Measure of Balance for Bipedal Robotics

[+] Author and Article Information
Derek L. Wight, Eric G. Kubica, David W. Wang

 University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

J. Comput. Nonlinear Dynam 3(1), 011009 (Nov 26, 2007) (10 pages) doi:10.1115/1.2815334 History: Received February 21, 2007; Revised June 14, 2007; Published November 26, 2007

The goal of most bipedal robotics research is to develop methods of achieving a dynamically balanced gait. Most current approaches focus on maintaining the balance of the system. This paper introduces a measure called the foot placement estimator (FPE) to restore balance to an unbalanced system. We begin by developing a theoretical proof to define when a biped is stable, as well as defining the region in which stability results are valid. This forms the basis for the derivation of the FPE. The results of the FPE are then extended to a complete gait cycle using the combination of a state machine and simple linear controllers. This control system is applied to a detailed and realistic simulation based on a physical robot currently under construction. Utilizing the FPE as a measure of balance allows us to create dynamically balanced gait cycles in the presence of external disturbances, including gait initiation and termination, without any precalculated trajectories.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

The parameters for the equations of motion for rotation about Points A and B

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Figure 2

Modeling the impact using conservation of angular momentum: (a) Preimpact and (b) postimpact

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Figure 3

The single variable θ is used to unify the equations of motion, which are functions of θA and θB

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Figure 4

The dotted line traces out the path of the COM. Stable Region 1 defines the necessary conditions to keep the COM within the white section.

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Figure 5

Simplifying an arbitrary biped configuration

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Figure 6

The free-body diagram when Point A is in contact with the surface

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Figure 7

A phase portrait of a simple biped using the parameters of the biped in Sec. 7 and β=60deg. The dotted line is the union of the three stable regions. The white region is the intersection of the regions of validity.

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Figure 8

A simplified biped stepping relative to the FPE. (a) Stepping closer than the FPE results in falling forward. (b) Stepping further than the FPE causes the biped to fall back onto the swing leg. (c) Stepping precisely at the FPE will perfectly balance the COM above the standing foot.

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Figure 9

Given the parameters immediately preceding impact (a), conservation of angular momentum is used to predict the velocities immediately after impact (b)

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Figure 10

The projection of the angle ϕ from the COM to the walking surface is the location of the FPE. This projection is used as a tracking reference for the swing foot until impact occurs.

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Figure 11

Overview of the state machine coordinating each of the individual joint controllers

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Figure 12

A mechanical drawing of the experimental robot under construction

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Figure 13

Simulation results from gait initiation through 15 steps to gait termination. The dashed portions of the feet position indicated when the foot is off the ground, and the solid portions indicated when the foot is on the ground.

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Figure 14

Simulation results of the biped walking in the presence of external disturbances. The dashed portions of the feet position indicated when the foot is off the ground, and the solid portions indicated when the foot is on the ground.

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