Table 1 Overview of the musculoskeletal model’s constraints.

From: Modifications to the net knee moments lead to the greatest improvements in accelerative sprinting performance: a predictive simulation study

Dynamics constraints

\(\begin{array}{c}\frac{d{\varvec{q}}}{dt}=v \end{array}\)

(1.1)

\(\begin{array}{c}\frac{d{\varvec{v}}}{dt}={{\varvec{u}}}_{\dot{{\varvec{v}}}} \end{array}\)

(1.2)

\(\begin{array}{c}\frac{d{\widetilde{{\varvec{F}}}}_{{\varvec{T}}}}{dt}={{\varvec{u}}}_{{\dot{\widetilde{{\varvec{F}}}}}_{{\varvec{T}}}} \end{array}\)

(1.3)

\(\begin{array}{c}\frac{d{\varvec{a}}}{dt}={{\varvec{u}}}_{\dot{{\varvec{a}}}} \end{array}\)

(1.4)

Path constraints

\(\begin{array}{c}M\left({\varvec{q}}\right)\cdot {{\varvec{u}}}_{\dot{{\varvec{v}}}}+C\left({\varvec{q}},{\varvec{v}}\right)+G\left({\varvec{q}}\right)-{{{\varvec{J}}}_{{\varvec{E}}{\varvec{x}}{\varvec{t}}}}^{T}\cdot Ext\left({{\varvec{u}}}_{{\varvec{G}}{\varvec{R}}{\varvec{F}}},{\varvec{A}}{\varvec{i}}{\varvec{r}}{\varvec{D}}{\varvec{r}}{\varvec{a}}{\varvec{g}}\right)-\left[\begin{array}{c}0\\{\varvec{\tau}}\end{array}\right]=0 \end{array}\)

(1.5)

\(\begin{array}{c}{\widetilde{{\varvec{F}}}}_{{\varvec{T}}}-\mathrm{cos}\left({\varvec{\theta}}\left({\varvec{q}},{\widetilde{{\varvec{F}}}}_{{\varvec{T}}}\right)\right)\cdot \left({{\varvec{F}}}_{{\varvec{C}}{\varvec{E}}}\left({\varvec{q}},{\varvec{v}},{\widetilde{{\varvec{F}}}}_{{\varvec{T}}},\boldsymbol{ }{\varvec{a}}\right)+{{\varvec{F}}}_{{\varvec{P}}{\varvec{E}}}\left({\varvec{q}},{\widetilde{{\varvec{F}}}}_{{\varvec{T}}}\right)\right)=0 \end{array}\)

(1.6)

\(\begin{array}{c}0\le {{\varvec{u}}}_{\dot{{\varvec{a}}}}+\frac{{\varvec{a}}}{{\tau }_{d}}; {\tau }_{d}=60\,ms\end{array}\)

(1.7)

\(\begin{array}{c}{{\varvec{u}}}_{\dot{{\varvec{a}}}}+\frac{{\varvec{a}}}{{\tau }_{a}}\le \frac{1}{{\tau }_{a}}; {\tau }_{a}=15\,ms \end{array}\)

(1.8)

\(\begin{array}{c}{{\varvec{H-C}}}_{{\varvec{G}}{\varvec{R}}{\varvec{F}}}\left({\varvec{q}},{\varvec{v}}\right)-{{\varvec{u}}}_{{\varvec{G}}{\varvec{R}}{\varvec{F}}}=0 \end{array}\)

(1.9)

Continuity constraints

\(\begin{array}{c}{{\varvec{x}}}_{i}^{END}-{{\varvec{x}}}_{i+1}^{1}=0 \end{array}\)

(1.10)

  1. Skeletal dynamics were enforced as per equation \((1.5)\), where \({\varvec{M}}\left({\varvec{q}}\right)\) is the mass matrix, \({\varvec{C}}\left({\varvec{q}},{\varvec{v}}\right)\) is the vector of centrifugal forces, \({\varvec{G}}\left({\varvec{q}}\right)\) is the vector of gravitational forces, \({{{\varvec{J}}}_{{\varvec{E}}{\varvec{x}}{\varvec{t}}}}^{T}\) is the transpose of the external forces Jacobian matrix, \({\varvec{E}}{\varvec{x}}{\varvec{t}}\left({{\varvec{u}}}_{{\varvec{G}}{\varvec{R}}{\varvec{F}}},{\varvec{A}}{\varvec{i}}{\varvec{r}}{\varvec{D}}{\varvec{r}}{\varvec{a}}{\varvec{g}}\right)\) is the vector of external ground reaction and air drag forces and \({\varvec{\tau}}\) is the vector of net joint moments (consisting of the moments generated by the MTUs, upper-limb and reserve actuators, and springs attached to the MTP DOFs). Contraction dynamics were imposed using the Hill model equilibrium condition \((1.6)\), where the normalised tendon force must equal the projected sum of the normalised contractile \({{\varvec{F}}}_{{\varvec{C}}{\varvec{E}}}\left({\varvec{q}},{\varvec{v}},{\widetilde{{\varvec{F}}}}_{{\varvec{T}}},\boldsymbol{ }{\varvec{a}}\right)\) and passive \({{\varvec{F}}}_{{\varvec{P}}{\varvec{E}}}\left({\varvec{q}},{\widetilde{{\varvec{F}}}}_{{\varvec{T}}}\right)\) muscle forces. Activation dynamics were enforced using the inequality constraint equations \((1.7-1.8)\), and they were derived from the original differential equation describing activation dynamics30. Equation \((1.9)\) was imposed to ensure consistency between the ground reaction forces calculated from the contact model \({{\varvec{H-C}}}_{{\varvec{GRF}}}\left({\varvec{q}},{\varvec{v}}\right)\) and the controls \({{\varvec{u}}}_{{\varvec{G}}{\varvec{R}}{\varvec{F}}}\).
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