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Research Papers

# A Simple Mass-Spring Model With Roller Feet Can Induce the Ground Reactions Observed in Human Walking

[+] Author and Article Information
Ben R. Whittington

Department of Mechanical Engineering, University of Wisconsin–Madison, Madison, WI 53706

Darryl G. Thelen

Department of Mechanical Engineering, Department of Biomedical Engineering, and Department of Orthopedics and Rehabilitation, University of Wisconsin–Madison, Madison, WI 53706thelen@engr.wisc.edu

J Biomech Eng 131(1), 011013 (Nov 26, 2008) (8 pages) doi:10.1115/1.3005147 History: Received July 27, 2007; Revised June 17, 2008; Published November 26, 2008

## Abstract

It has previously been shown that a bipedal model consisting of a point mass supported by spring limbs can be tuned to simulate periodic human walking. In this study, we incorporated roller feet into the spring-mass model and evaluated the effect of roller radius, impact angle, and limb stiffness on spatiotemporal gait characteristics, ground reactions, and center-of-pressure excursions. We also evaluated the potential of the improved model to predict speed-dependent changes in ground reaction forces and center-of-pressure excursions observed during normal human walking. We were able to find limit cycles that exhibited gait-like motion across a wide spectrum of model parameters. Incorporation of the roller foot $(R=0.3m)$ reduced the magnitude of peak ground reaction forces and allowed for forward center-of-pressure progression, making the model more consistent with human walking. At a fixed walking speed, increasing the limb impact angle reduced the cadence and prolonged stance duration. Increases in either limb stiffness or impact angle tended to result in more oscillatory vertical ground reactions. Simultaneous modulation of the limb impact angle and limb stiffness was needed to induce speed-related changes in ground reactions that were consistent with those measured during normal human walking, with better quantitative agreement achieved at slower speeds. We conclude that a simple mass-spring model with roller feet can well describe ground reaction forces, and hence center of mass motion, observed during normal human walking.

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Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 1

Each limb was represented by a translational spring, with stiffness K and slack length L0, that was rigidly coupled to a roller of radius R on one side and pinned to a point mass M on the other side. The point mass was assumed to be at the height H of the center of mass of the body with the limb spring in an unstretched upright configuration. Simulations were performed assuming normative values of H=1m and M=80kg. Roller radii of 0.0m, 0.1m, 0.2m, 0.3m, and 0.4m were considered.

Figure 2

Simulations were started with the trailing limb in an upright configuration and the forward limb oriented at the impact angle. The double support phase started when the leading limb contacted the ground and then continued until the trailing limb length reached its undeflected length. The second single support phase continued until the limb reached an upright configuration, signaling the end of the half gait cycle. We searched for limit cycle solutions in which the limb length, forward velocity, and vertical velocity at the end of the half gait cycle replicated the initial conditions.

Figure 3

The simple mass-spring model of walking is conservative, such that there was no change in total energy across a half gait cycle. The increase in gravitational potential energy during the first half of single support occurs as a result of both limb spring lengthening (decrease in spring potential energy) and a reduction in forward velocity (decrease in kinetic energy). A 0.3m radius roller was used to generate this 1.35m∕s walking speed simulation. Abbreviations: HS—heel strike, OTO—opposite toe-off, and OHS—opposite heel strike.

Figure 4

Inclusion of a roller in the mass-spring model decreased the magnitude of the peak ground reaction forces in both the anterior (Fx) and vertical (Fy) directions. The forward progression of the center of pressure, as measured by the COP excursion during stance, increased approximately linearly with the radius of the roller.

Figure 5

Shown are the limb stiffness and impact angles that resulted in limit cycle solutions when the 0.3m roller radius model walked with an average velocity of 1.35m∕s. Each solution exhibits specific (a) step length, (b) cadence, and (c) total energy. The highest energy states were associated with high cadence–low step length solutions, achieved via coupling high limb stiffness with small impact angles.

Figure 6

Variations in limb stiffness and impact angle altered the ground reactions that were induced when walking with a 0.3m radius roller at an average velocity of 1.35m∕s. An increase in impact angle resulted in larger peak ground reaction forces in both the anterior (Fx) and vertical (Fy) directions and also substantially prolonged the stance period. Increasing limb stiffness tended to induce slightly larger vertical ground reactions.

Figure 7

Use of the model parameters given in Table 1 induced anterior (Fx) and vertical (Fy) ground reaction forces that most closely resembled average experimental (expt) forces at the slow walker speed. At the faster walking speed, the simulated ground reactions exhibited greater variation in stance than seen experimentally. For all speeds, a roller radius of 0.3m resulted in reasonable approximations of the COP excursion.

Figure 8

The velocity of the mass M can be decomposed by components in the forward (x) direction and both along and perpendicular to the trailing limb

Figure 9

The ground reaction force vectors Fx and Fy for each limb are computed knowing the spring force FL and the angle ϕ that the ground reaction force vector forms with the vertical

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