Energetics of Actively Powered Locomotion Using the Simplest Walking Model

[+] Author and Article Information
Arthur D. Kuo

Dept. of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109-2125email: artkuo@umich.edu

J Biomech Eng 124(1), 113-120 (Sep 17, 2001) (8 pages) doi:10.1115/1.1427703 History: Received September 20, 1999; Revised September 17, 2001
Copyright © 2002 by ASME
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(a) Configuration variables for the simplest walking model of Garcia et al. 4. (b) The velocities before toe-off, after toe-off, and after heel strike (vcm,vcm0, and vcm+, respectively) can be related by a simple geometric diagram, yielding Eq. (10). At the end of a step, the center of mass has a velocity vcm that is perpendicular to the trailing leg. Immediately before heel strike, a toe-off impulse P redirects the center of mass velocity vcm0 along a line parallel to the trailing leg. The heel strike impulse then reduces the velocity to vcm+, which is perpendicular to the leading leg. The relatively low masses of the feet implies that each impulse can be described by geometric projection along one of the legs. (c) Energy relationships needed to sustain the same speed v as a function of toe-off impulse magnitude, P. Energy must be supplied wholly by a hip torque on the stance leg when P=0, but that quantity decreases linearly with increasing P. The work done by toe-off increases quadratically with P (Eq. 11), and the energy lost at heel strike decreases quadratically. Discounting the possibility of regeneration of negative work by the hip, the least costly gait is achieved by providing all the energy with toe-off (marked with asterisk).
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Simplest walking model gaits produced by toe-off impulses or by hip torques applied to the stance leg from the torso, with the swing leg left unactuated. (a) Step period τ (left-hand axis) changes little with input energy for both types of actuation, and especially for toe-off powering. Mechanical work required per step increases with approximately the fourth power of speed v (right-hand axis), with hip actuation about four times more expensive than toe-off. (b) Periodic gaits, in terms of α and Ω, vary uniformly with energy input. (c) The result is that, in the absence of a hip spring to tune the swing leg, the step length s increases nearly linearly with speed v. (d) Eigenvalues of step-to-step transition describe passive stability of gait. For each method of actuation, there are two eigenvalues, of which one is always stable (unmarked curves). Passive stability is lost as energy levels (and speed) increase, although the addition of a hip spring easily brings the eigenvalues to magnitude less than 1.
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Scaling relationships for gaits produced by powered passive walking models. The Simplest Model (a) approximately obeys predicted relationships (i.e., the lines shown are nearly straight) between swing leg natural frequency ω, toe-off impulse magnitude P, and gait variables such as boundary conditions α and Ω and step period τ. The lines show relationships for the entire range of step length and speed combinations simulated (see Fig. 4). Gaits produced (b) have two-fold symmetry—the model looks the same walking backward or forward in space and time. Stance and swing angles are reported relative to vertical. When applied to the Anthropomorphic Model with more realistic mass parameters, the scaling relationships (c) are followed less well at low ω, and the gaits (d) are slightly less symmetrical.
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Mechanical work performed by toe-off, Wtoe, as a function of speed v and step length, s, using (a) the Simplest Model and (b) the Anthropomorphic Model. Contour lines of constant (dimensionless) mechanical work per distance (solid lines, with energy levels as labeled) increase with both speed and step length. Gaits achieved with no hip spring acting on the swing leg (ω=1, denoted by filled circles) show that increasing toe-off impulses tend to increase speed and step length in linear proportion. The effect of increasing the natural frequency ω of the swing leg is to increase speed at slightly shorter step lengths. Walking becomes less costly because the hip spring decreases collision losses. Step frequency (1/τ, denoted by dotted lines) is proportional to ω. The Simplest Model has regions that are passively unstable (shaded areas). The Anthropomorphic Model (b), using more realistic mass properties, produces results that are very similar, except that it is more difficult to achieve very short steps, and the unstable regions are less prominent. Another feature of AM is that the curved feet lower the collision losses for all gaits.



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