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.