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Research Papers

Low-Dimensional Sagittal Plane Model of Normal Human Walking

[+] Author and Article Information
S. Srinivasan

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210srinivasan.54@osu.eduDepartment of Electrical Engineering, University of South Florida, Tampa, FL 33620srinivasan.54@osu.eduDepartment of Mechanical Engineering, The Ohio State University, Columbus, OH 43210srinivasan.54@osu.edu

I. A. Raptis

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210iraptis@mail.usf.eduDepartment of Electrical Engineering, University of South Florida, Tampa, FL 33620iraptis@mail.usf.eduDepartment of Mechanical Engineering, The Ohio State University, Columbus, OH 43210iraptis@mail.usf.edu

E. R. Westervelt

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210westervelt.4@osu.eduDepartment of Electrical Engineering, University of South Florida, Tampa, FL 33620westervelt.4@osu.eduDepartment of Mechanical Engineering, The Ohio State University, Columbus, OH 43210westervelt.4@osu.edu

In DS, the legs form a closed chain. For this constrained system, fewer coordinates are needed to uniquely specify the shape.

Perry (49) described the action over a step of the foot-ankle complex using three rockers based on the center of rotation of the shank with respect to the foot. The ROS is a model that uses the center of pressure (COP) to generate a single rocker to describe the action of the foot-ankle complex.

This is valid since the impact occurs over an infinitesimal duration and the joint torques are not impulsive.

J Biomech Eng 130(5), 051017 (Sep 19, 2008) (11 pages) doi:10.1115/1.2970058 History: Received March 19, 2007; Revised November 30, 2007; Published September 19, 2008

This paper applies a robotics-inspired approach to derive a low-dimensional forward-dynamic hybrid model of human walking in the sagittal plane. The low-dimensional model is derived as a subdynamic of a higher-dimensional anthropomorphic hybrid model. The hybrid model is composed of models for single support (SS) and double support (DS), with the transition from SS to DS modeled by a rigid impact to account for the impact at heel-contact. The transition from DS to SS occurs in a continuous manner. Existing gait data are used to specify, via parametrization, the low-dimensional model that is developed. The primary result is a one-degree-of-freedom model that is an exact subdynamic of the higher-dimensional anthropomorphic model and describes the dynamics of walking. The stability properties of the model are evaluated using the method of Poincaré. The low-dimensional model is validated using the measured human gait data. The validation demonstrates the observed stability of the measured gait.

FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 4

Stick figure animation over one step of the example model. Note the rolling motion of the stance foot. The swing leg is shown with dashed lines.

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Figure 5

Joint motions from the gait data in Ref. 23 (dashed) versus models 19,19 (solid) on the limit cycle (i.e., at steady state) for one step. Note that the HAT angle, qa, is unactuated, and, therefore, its evolution is not directly controlled.

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Figure 6

Comparison of the angle between the line connecting the c.m. and the ground for the gait data in Ref. 23 (dashed) versus models 19,19 (solid) on the limit cycle (i.e., at steady state) for one step. The plot indicates a close match.

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Figure 7

The joint velocities from the simulation of the dynamics

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Figure 8

Simulation results for zdi,1 in DS and zs,1 in SS over one gait cycle using the full model and the low-dimensional model. The error between the traces is on the order of the integrator tolerance, which was set to 1×10−5.

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Figure 9

Simulation results for zdi,2 in DS and zs,2 in SS over one gait cycle using the full model and the low-dimensional model. The error between the traces is on the order of the integrator tolerance, which was set to 1×10−5.

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Figure 10

The Poincaré map of the full hybrid model 45 versus its linear approximation 50

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Figure 11

Simulation of the full hybrid model, Eqs. 19,19, for 35 steps illustrating convergence to a limit cycle as predicted by the fixed point in the Poincaré map. The triangles indicate HC, the beginning of DS, and the transition to SS.

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Figure 1

Illustration of the parsimony of human gait over a normal gait cycle. Average joint angles as a percentage of gait cycle for five adult subjects with normal gait using five trials each (bold). The dashed lines are the pointwise 1 standard deviation minimum and maximum joint angles. A step is the period from heel-contact (HC) to opposite heel-contact (OHC). The gait cycle is made up of two symmetric steps with each step comprised of single support (SS) and double support (DS) phases. Toe-off (TO) signals the transition from DS to SS while HC marks the transition from SS to DS. The superscripts + and − indicate the beginning and end of each phase, respectively. (Data courtesy of Ref. 1.)

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Figure 2

Coordinates for the example anthropomorphic model in SS and DS of Sec. 6. Note the choice of coordinates as one absolute (qa) and the others relative. The relative coordinates specify the shape (posture) of the model, and the circles indicate the internal DOF. Note also the additional DOF at the trailing ankle in the DS model. The stance foot is modeled using the roll-over shape hypothesis of Hansen (31). The roll-over shape does not apply to the swing foot, and, hence, it is depicted differently. The scalar quantity chosen to represent forward progression in single support, θs, and double support, θd, corresponds to the angle shown. This angle is between the vertical axis and the line joining the hip to the point of rolling contact. The Cartesian coordinates of the rolling point, the swing heel, and the trailing toe are (xR,yR), (xh2,yh2), and (xt2,yt2), respectively.

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Figure 3

Discrete-event system corresponding to the dynamics of one step (half a normal gait cycle)

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