Research Papers

Analyzing Bounding and Galloping Using Simple Models

[+] Author and Article Information
Kenneth J. Waldron, Paul J. Csonka

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305

J. Estremera

 Industrial Automation Institute-CSIC, 28500 La Poveda, Madrid, Spain

S. P. Singh

Australian Centre for Field Robotics, Rose St. Bldg. J04,  The University of Sydney, NSW 2006, Australia

J. Mechanisms Robotics 1(1), 011002 (Jul 30, 2008) (11 pages) doi:10.1115/1.2959095 History: Received April 04, 2008; Revised June 22, 2008; Published July 30, 2008

This paper focuses on modeling the gait characteristics of a quadrupedal gallop. There have been a number of studies of the mechanics of the stance phase in which a foot is in contact with the ground. We seek to put these studies in the context of the stride, or overall motion cycle. The model used is theoretical, and is kept simple in the interest of transparency. It is compared to empirical data from observations of animals, and to data from experiments with robots such as our KOLT machine, and results from sophisticated simulation studies. Modeling of the energy loss inherent in the interaction between the system and the environment plays a key role in the study. Results include the discovery of a hidden symmetry in the gait pattern, usually regarded as being completely asymmetrical. Another result demonstrates that the velocities with which the two front feet impact and leave the ground are different, and similarly for the rear feet. The velocities of the foot pairs mirror each other. This is consistent with empirical observation, but is at variance with the assumption used almost universally when modeling stance. A further result elicits the importance of the pitch moment of inertia and other effects that make the mammalian architecture, in which the center of mass is closer to the shoulders than to the hips, beneficial..

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Impulsive equilibrium over the duration of a stride for a bound. W is the weight of the system, and D is the drag. T is the stride duration. 2JF is the net impulse imparted by the front feet, and 2JR is that imparted by the rear feet.

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Figure 2

Free body diagram. The center of mass is assumed to be coplanar with the shoulder and hip joints. It can be argued that the location of the center of mass a small distance above or below the plane of the joints makes little difference to the dynamics. The shoulder and hip joints are assumed to be pairs of intersecting revolutes that are, respectively, parallel to the x axis of the body and orthogonal to the plane of the leg. The body reference frame is centered on the center of mass and aligned as shown. The principal axes of inertia are assumed to coincide with the x, y, and z axes.

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Figure 3

Dynamic effect of a foot, mass m, connected to the remainder of the system M by a spring. The energy lost each stance is mvz2∕2, where vz is the vertical component of system velocity.

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Figure 4

The KOLT running machine that is in operation in the Stanford Robotic Locomotion Laboratory

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Figure 5

Coordinate senses during the (a) front foot stance, and (b) rear foot stance. Point S represents the shoulder joint, and point H represents the hip joint. G is the center of mass.

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Figure 6

Variations in vertical velocity and vertical position of center of mass, and pitch angular velocity and pitch angle through a bound stride. The intervals used in the text are τFR=τR and τRF=T−τR.

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Figure 7

Fast gallop of a horse reproduced from Gambaryan (8). The right-hand frame in the top row is the spread flight phase. The corresponding frame in the bottom row is the gathered flight phase.

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Figure 8

Graphical presentation of variations of velocity and position variables through the transverse gallop stride

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Figure 9

Stances and spread and gathered flight phases of a rotary gallop as performed by a greyhound

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Figure 10

Graphical representation of variations of position and velocity through a rotary gallop stride. This figure may be compared with the corresponding figure for a transverse gallop stride (Fig. 8). The most notable differences are in the roll (x) axis angular velocities and angular positions.



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