3.1 Step length on a compliant surface

The figure below from McMahon compares step length () on a rigid surface (left panel) to step length () on a compliant surface (right panel). The illustration shows the limb in multiple positions during stance phase: (i) at initial ground contact with the knee fully extended; (ii) in mid-stance with the knee flexed; and (iii) at toe-off with the knee again fully extended.

In mid-stance on the rigid surface (left panel), the height of the hip above the ground in mid-stance is , where  is the leg length and  is modeled as a constant, independent of running speed, set by the central nervous system to control the body’s trajectory.

The additional deflection of a compliant surface is characterized by , which represents the peak deflection of the surface in mid-stance. However,  can be approximated as the static deflection that would be observed if the runner were simply standing on the surface.

Figure 3.1 Schematic of a step on a rigid surface (left) and compliant surface (right). Solid line shows the stance leg, broken line shows the swing leg moving forward. Because the foot descends a distance into the compliant track, the step length on the compliant track is necessarily greater. Figure adapted from Figure 5 (McMahon, T. A., & Greene, P. R. (1979). The influence of track compliance on running. Journal of Biomechanics 12(12), 893–904.)

(a) Using the figures and description of  above, derive the expression below for the step length of the runner on a compliant surface, , in terms of the runner’s mass, , runner’s leg length, , the rigid surface step length, , surface stiffness, , and local gravitational acceleration, .

(b) Choose some realistic values for , , , , and . What is the predicted value for ? How does this equation predict how  will vary with decreasing ? Discuss how this prediction compares with your intuition.