Stiffness of a wobbling mass models analysed by a smooth orthogonal decomposition of the skin movement relative to the underlying bone
Introduction
The movement of the skin, muscles, and fat relative to the underlying bone is a well-known phenomenon. It has been described, from a kinematics point of view, as the soft tissue artefact (STA). Indeed, this relative movement has a deleterious effect on the joint kinematics estimated from skin markers and motion capture systems (Leardini et al., 2005, Peters et al., 2010). At the same time, from a dynamics point of view, the soft tissue motion, modelled as wobbling masses connected to the rigid-body model of the lower limb, is also recognised to have an effect on the joint kinetics (i.e. energy dissipation, torque reduction) (Challis and Pain, 2008, Gruber et al., 1998) during motor tasks involving impacts with the ground.
One key parameter of these wobbling mass models is the stiffness of the springs connecting them to the rigid-bodies. Most of the models of the literature include linear or non-linear springs attached to a wobbling mass that can translate (and eventually rotate) with respect to the bone (Alonso et al., 2007, Gittoes et al., 2006, Gruber et al., 1998, Günther et al., 2003, McLean et al., 2003, Pain and Challis, 2004, Wilson et al., 2006). Identification of the parameters of these wobbling mass models, based on the ground reaction forces, as well as sensitivity analyses have been widely performed (Alonso et al., 2007, Gittoes et al., 2009, Pain and Challis, 2004, Wilson et al., 2006). However, to the best of the author’s knowledge, the estimation of the stiffness parameters from the displacements of the skin relative to the underling bone measured in vivo by intra-cortical pins has not been performed yet. For this estimation, the displacements of the skin markers in the bone-embedded coordinate systems are viewed as a proxy for the wobbling mass movement.
The objective of this study was to estimate the stiffness matrix of a wobbling mass model, defined as a cluster of lumped masses undergoing translations about the three axes of the bone-embedded coordinate system, by applying a structural vibration analysis method, called smooth orthogonal decomposition (Chelidze and Zhou, 2006) to the simultaneous measurements of skin and intra-cortical pin markers (Benoit et al., 2006, Reinschmidt et al., 1997). In this method, the displacement of the skin markers relative to the underlying bone was modelled as the free undamped vibrations of a dynamical system for which the stiffness matrix can be straightforwardly identified.
Section snippets
Smooth orthogonal decomposition of the skin movement relative to the underlying bone
The STA vector, , was defined to represent the displacement that the skin marker j (j = 1:mi) associated with the segment i (i = 1 for shank and i = 2 for thigh) underwent relative to a relevant bone-embedded coordinate system and a reference position at each discrete time k (k = 1:n) during the analysed motor task (Dumas et al., 2014a). The STA of all markers on the segment i were represented using the STA field, :
A sample covariance matrix was computed from this STA field
Results
Fig. 1 represents the stiffness coefficients (i.e. median, quartiles, minimum and maximum for the five running trials of subjects R1, R2 and R3) for the translation of the maker-cluster about the X, Y and Z axes of the bone-embedded coordinate systems for the different subjects and motor tasks. The stiffness appeared generally higher for the shank (i.e. between 10.2 kN/m and 55.5 kN/m) than for the thigh (i.e. between 2.3 kN/m and 11.1 kN/m). The stiffness was higher about the X axis than about the
Discussion
This study applied a structural vibration analysis method called smooth orthogonal decomposition to estimate the stiffness matrix of a wobbling mass model from simultaneous measurements of skin and intra-cortical pin markers. The wobbling mass model consisted of a cluster of lumped masses (or, equivalently, of a concentrated mass at the cluster centroid) undergoing translations about the three axes of the bone-embedded coordinate system. Yet, Eqs. (8) and (10), allows to define stiffness matrix
Conflict of interest
The authors do not have any financial or personal relationships with other people or organizations that would have inappropriately influence this study.
Acknowledgements
The authors would like to thank A.J. van den Bogert of Cleveland State University for kindly providing the unfiltered running data. The authors are also grateful to D. Benoit for having made available the unfiltered walking, cutting and hopping data for this special issue of the Journal of Biomechanics on the soft tissue artefact.
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