Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single stance phase of walking

Abstract

Despite the overall complexity of legged locomotion, the motion of the center of mass (COM) itself is relatively simple, and can be qualitatively described by simple mechanical models. The spring-loaded inverted pendulum (SLIP) is one such model, and describes both the COM motion and the ground reaction forces (GRFs) during running. Similarly, walking can be modeled by two SLIPlike legs (double SLIP or DSLIP). However, DSLIP has many limitations and is unlikely to serve as a quantitative model for walking. As a first step to obtaining a quantitative model for walking, we explored the ability of SLIP to model the single stance phase of walking across the entire range of walking speeds. We show that SLIP can be employed to quantitatively model the single stance phase except for two exceptions: first, it predicts larger horizontal GRFs than empirically observed. A new model -angular and radial spring-loaded inverted pendulum (ARSLIP) can overcome this deficit. Second, even the single stance phase has active elements, and therefore a quantitative model of locomotion would require active elements. Surprisingly, the leg spring undergoes a contraction-extension-contraction-extension (CECE) during walking; this cycling is partly responsible for the M-shaped GRFs produced during walking. The CECE cycle also lengthens the stance duration allowing the COM to travel passively for a longer time, and decreases the velocity redirection between the beginning and end of a step. A combination of ARSLIP along with active mechanisms during transition from one step to the next is necessary to describe walking.

Drosophila uses a tripod gait across all walking speeds, and the geometry of the tripod is important for speed control

“…As the walking speed increases and stance duration decreases, the vertical oscillations must occur faster by making the effective leg stiffer (because the time needed for vertical oscillation increases with spring stiffness). This increase in stiffness has indeed been observed in humans (Antoniak et al, 2019). We found a similar increase in stiffness in flies (Figure 6D).…”

“…The magnitude of these oscillations is small and reaches a maximum of 10% (of its length at fixed point) about the fixed point. The non-dimensional radial spring constant (see methods for definition) is ~2 compared to >10 for humans (Antoniak et al, 2019). The median was 1.1x10 -8 Nm/radian.…”

“…There are two mechanisms by which the system can become mechanically stiffer. Either a fly could make each leg stiffer just as has been shown in humans (Antoniak et al, 2019;Kim and Park, 2011), or it could change the geometry of the tripod to make the overall system stiffer. The data in Figure 7 is consistent with the second idea that the changes in the geometry of the tripod are the dominant component by which flies control the stiffness of a and s and thereby changes its walking speed.…”

“…This difference makes sense. At the large Fr numbers employed by cockroaches, the CoM kinematics is dominated by the angular speed with which the body moves about its leg (Antoniak et al, 2019), and it is rather insensitive to the magnitude of s. On the other hand, and as demonstrated for flies in this study, at lower speeds, the mechanics is dominated by the spring constants a and s.…”