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Multibody Optimisations: From Kinematic Constraints to Knee Contact Forces and Ligament Forces

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Biomechanics of Anthropomorphic Systems

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 124))

Abstract

Musculoskeletal models are widely used in biomechanics today to better understand muscle and joint function. Musculo-tendon forces as well as joint contact forces and ligament forces can be estimated within an inverse dynamics computational framework. Using a musculoskeletal model of the lower limb, this chapter presents the different optimisations required to drive the model with experimental data and to compute these forces and their interactions. In these optimisations, the development of anatomical constraints representing, for example, the medial and lateral tibiofemoral contacts or the cruciate ligaments is crucial both to inverse kinematics and to inverse dynamics. Some emblematic results are presented for knee contact forces and ligament forces during gait, illustrating the couplings between joint degrees of freedom and the interactions between forces acting both in muscles and in joints.

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Abbreviations

T :

matrix transpose

\( \dag \) :

matrix pseudo-inverse

\( {\mathbf{E}}_{3 \times 3} \) :

identity matrix

i :

index for segment

j :

index for skin or virtual marker (in inverse kinematics) or muscle (in inverse dynamics)

ui, vi, wi:

anterior, superior and lateral axes of segment

\( {\mathbf{r}}_{{P_{i} }} , \, {\mathbf{r}}_{{D_{i} }} \) :

positions of the proximal (Pi) and distal (Di) endpoints

\( \left( {P_{i} ,{\mathbf{u}}_{i} ,\underbrace {{\left( {{\mathbf{r}}_{{P_{i} }} - {\mathbf{r}}_{{D_{i} }} } \right)}}_{{{\mathbf{v}}_{i} }},{\mathbf{w}}_{i} } \right) \) :

non-orthonormal segment coordinate system

\( \left( {P_{i} ,{\mathbf{X}}_{i} ,{\mathbf{Y}}_{i} ,{\mathbf{Z}}_{i} } \right) \) :

orthonormal segment coordinate system

B i :

transformation matrix from the non-orthonormal to the orthonormal segment coordinate system

αi, βi, γi:

constant angles between the axes of the non-orthonormal segment coordinate system

L i :

segment length (between proximal and distal endpoints)

Q i :

natural coordinates (2 position and 2 direction vectors)

\( {\varvec{\Phi}}_{i}^{r} \) :

rigid body constraints

\( {\mathbf{r}}_{{M_{i}^{j} }} , \, {\mathbf{r}}_{{V_{i}^{j} }} \) :

position of skin or virtual marker (\( M_{i}^{j} \) or \( V_{i}^{j} \))

\( \left( {n_{i} } \right)_{u} , \, \left( {n_{i} } \right)_{v} , \, \left( {n_{i} } \right)_{w} \) :

coordinates in the non-orthonormal segment coordinate system

\( {\mathbf{N}}_{i}^{{}} \) :

interpolation matrix

\( {\varvec{\Phi}}^{k} \) :

kinematic constraints

d, θ:

model parameter (i.e. ligament length, angle between hinge axes)

\( {\varvec{\Phi}}^{m} \) :

driving constraints

G :

matrix of generalised masses

\( {\mathbf{Q}}, \, {\dot{\mathbf{Q}}}, \, {\ddot{\mathbf{Q}}}: \) :

vectors of generalised positions, velocities and accelerations for all segments

\( {\mathbf{K}} \) :

Jacobian matrix of the constraints

\( {\varvec{\uplambda}} \) :

vector of Lagrange multipliers

R :

vector of generalised ground reaction

\( {\mathbf{Z}}_{{{\mathbf{K}}_{2}^{T} }} \) :

orthogonal basis of the null space of \( {\mathbf{K}}_{2}^{T} \)(corresponding to a subset of the constraints)

P :

vector of generalised weights

L :

matrix of generalised muscular lever arms

f :

vector of musculo-tendon forces

f, J:

objective functions

W :

optimisation weights

\( {\mathbf{F}}_{0}^{{\mathbf{R}}} , \, {\mathbf{M}}_{0}^{{\mathbf{R}}} \) :

ground reaction force and moment vectors at the centre of pressure (P0)

\( f_{u}^{{{\mathbf{M}}_{0}^{{\mathbf{R}}} }} , \, f_{v}^{{{\mathbf{M}}_{0}^{{\mathbf{R}}} }} , \, f_{w}^{{{\mathbf{M}}_{0}^{{\mathbf{R}}} }} \) :

forces applied about the axes of the non-orthonormal foot coordinate system representing the ground reaction moment.

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Authors and Affiliations

  1. LBMC UMR_T9406, Univ Lyon, Université Claude Bernard Lyon 1, IFSTTAR, F69622, Lyon, France

    Raphael Dumas & Laurence Cheze

  2. Centre National de Rééducation Fonctionnelle et de Réadaptation—Rehazenter, Laboratoire d’Analyse du Mouvement et de la Posture (LAMP), Luxembourg, Luxembourg

    Florent Moissenet

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  1. Raphael Dumas
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Correspondence to Raphael Dumas .

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Editors and Affiliations

  1. Tokyo, Japan

    Gentiane Venture

  2. LAAS-CNRS , Toulouse, France

    Jean-Paul Laumond

  3. Toulouse, France

    Bruno Watier

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Dumas, R., Cheze, L., Moissenet, F. (2019). Multibody Optimisations: From Kinematic Constraints to Knee Contact Forces and Ligament Forces. In: Venture, G., Laumond, JP., Watier, B. (eds) Biomechanics of Anthropomorphic Systems. Springer Tracts in Advanced Robotics, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-93870-7_4

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