Foot–ground contact modeling within human gait simulations: from Kelvin–Voigt to hyper-volumetric models

Abstract

This study describes the development of a multibody foot–ground contact model consisting of spherical volumetric models for the surfaces of the foot. The developed model is two-dimensional and consists of two segments, the hind-foot, mid-foot, and fore-foot as one rigid body and the phalanges collectively as the second rigid body. The model has four degrees of freedom: ankle $x$ and $y$, foot orientation, and metatarsal-phalangeal joint angle. Three different types of contact elements are targeted: Kelvin–Voigt, linear volumetric, and hyper-volumetric. The models are kinematically driven at the ankle and the metatarsal joints, and simulated horizontal and vertical ground reaction forces as well as center of pressure location are compared against experimental quantities acquired from barefoot measurements during a human gait cycle. Parameter identification is performed for finding optimal contact parameters and locations of the contact elements. The hyper-volumetric foot–ground contact model was found to be a suitable choice for foot/ground interaction modeling within human gait simulations; this model showed 75 % and 62 % improvement on the matching quality over the point contact and linear volumetric models, respectively.

Appendix: Optimal contact parameters

Optimal parameters of the three different contact models are presented here.

Table 2 Optimal contact parameters of the spring–damper elements: $k_{S}$ is the spring stiffness, $a_{S}$ is the pseudo-damping, $L_{0}$ is the spring initial length, $n_{S}$ is the nonlinearity exponent, $\mu_{f}$ is the asymptotic friction coefficient, and $v_{s}$ is a shape parameter for approximation of the dry Coulomb friction. Parameters $dx$ and $dy$ for characteristic points H, P, and T are expressed in local frames AH, AP, and PT, respectively

Table 3 Optimal contact parameters of the linear volumetric elements: $k_{V}$ is the volumetric stiffness, $a_{V}$ is the volumetric pseudo-damping, $R_{V}$ is the radius of the sphere element, $\mu_{f}$ is the asymptotic friction coefficient, $v_{s}$ is a shape parameter for approximation of the dry Coulomb friction, and $dx$ and $dy$ are local coordinates of the optimal positions of contact elements

Table 4 Optimal contact parameters of the nonlinear volumetric elements: $k_{h}$ is the nonlinear volumetric pseudo-stiffness, $a_{h}$ is the nonlinear volumetric pseudo-damping, $R_{V}$ is the radius of the sphere element, ℋ is the nonlinearity exponent of the volume, $\mu_{f}$ is the asymptotic friction coefficient, $v_{s}$ is a shape parameter for approximation of the dry Coulomb friction, and $dx$ and $dy$ are local coordinates of the optimal positions of contact elements