A 2nd-Order Model of the Vertical Impedance of the Locomotor System During Human Walking

Abstract

Objective: To design better shock absorbing prosthetic components, walking machines, or bionic legs, values and properties of the mechanical impedance of the locomotor system of able-bodied ambulators need to be determined. <italic>Methods:</italic> The representation of a simple 2^nd-order shock absorption component (i.e., effective body mass, <inline-formula> <tex-math notation= LaTeX >$M_{e}$ </tex-math></inline-formula>, supported by a stiffness, <inline-formula> <tex-math notation= LaTeX >${k}$ </tex-math></inline-formula>, and damping, <inline-formula> <tex-math notation= LaTeX >${B}$ </tex-math></inline-formula>, in parallel) was incorporated into a previously proposed rocker-based inverted pendulum model of walking to more accurately model the impedance associated with human walking, including the effects of shock absorption. Using system identification techniques and quantitative gait analysis data from able-bodied subjects (<inline-formula> <tex-math notation= LaTeX >${n}\\,\\,=$ </tex-math></inline-formula> 7), the properties and values of the mechanical impedance for each subject’s trial were determined. <italic>Results:</italic> It was determined that stiffness, <inline-formula> <tex-math notation= LaTeX >${k}$ </tex-math></inline-formula>, increased linearly with walking speed while damping, <inline-formula> <tex-math notation= LaTeX >${B}$ </tex-math></inline-formula>, increased with the square root of walking speed. Furthermore, the locomotor system behaves like an optimal underdamped mechanical system (i.e., damping ratio <inline-formula> <tex-math notation= LaTeX >$\\zeta $ </tex-math></inline-formula> of approximately 0.5–0.7). <italic>Conclusion:</italic> The locomotor system seems to behave like an optimal and adaptive low-pass filter in which the cutoff frequency is adjusted based on the locomotor’s cadence. It could be hypothesized that this occurs due to the principle of conservation of the passenger’s well-being: the fact that one of the locomotor’s functional responsibilities is to prevent higher frequency vibrations that are harmful for the musculoskeletal system of the passenger. <italic>Significance:</italic> This is a holistic, simplified, and accurate model of the vertical leg mechanical impedance (locomotor system) that establishes parameters during walking over a range of speeds. These results can be used for the scientific design of better shock absorption prosthetic components and lower-limb prostheses, management of shock absorption related gait pathologies, or the design of human-like walking machines and rehabilitation robots.