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Model Mathematics

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Component: environment

Component: F_SE

F_SE = cT F_max kT \ln (ⅇ^{\frac{LT - LT_r}{kT}} + 1)

Component: F_PE1

F_PE1 = c1 k1 \ln (ⅇ^{\frac{\frac{L}{L_max} - L_r1}{k1}} + 1) + eta V

Component: F_PE2

F_PE2 = c2 (ⅇ^{k2 (L - L_r2)} - 1)

Component: FL

FL = ⅇ^{- ({(| \frac{L^{beta} - 1}{omega} |)}^{rho})}

Component: FV

FV = \{\begin{matrix} \frac{V_max - V}{V_max + (cv0 + cv1 L) V} if V ≦ 0 \\ \frac{bv - ((av0 + av1 L + av2 L^{2}) V)}{bv + V} otherwise \end{matrix}

Component: Af

Af = 1 - (ⅇ^{- ({(\frac{Y S f_eff}{af nf})}^{nf})})

Component: F0

F0 = Af (FL + FV + F_PE2) + F_PE1

Component: F_CE

F_CE = F0 F_max

Component: F_total

F_total = F_SE - F_CE

Component: L_eff

\frac{d}{d time} (L_eff) = \frac{{(L - L_eff)}^{3}}{T_L (1 - Af)}

Component: S

\frac{d}{d time} (S) = \frac{as_- S}{T_s} as_= \{\begin{matrix} as1 if f_eff < 0.1 \\ as2 otherwise \end{matrix}

Component: Y

\frac{d}{d time} (Y) = \frac{1 - (c_Y (1 - (ⅇ^{\frac{- (| V |)}{V_Y}})) + Y)}{T_Y}

Component: rise_and_fall_time

\frac{d}{d time} (f_int) = \frac{f_env - f_int}{T_f} \frac{d}{d time} (f_eff) = df_eff_dt df_eff_dt = \frac{f_int - f_eff}{T_f} T_f = \{\begin{matrix} T_f1 L^{2} + T_f2 f_env if df_eff_dt \geq 0 \\ \frac{T_f3 + T_f4 Af}{L} otherwise \end{matrix}

Component: V

\frac{d}{d time} (V) = \frac{F_total}{1 mass}

Component: V0

V0 = \frac{V}{L_max}

Component: L0

L0 = \frac{L}{L_max}

Component: L

\frac{d}{d time} (L) = V

Component: user_defined_constants

Source: Derived from workspace Cheng, Brown, Loeb, 2000 at changeset 793a1c2df2bf.

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Virtual muscle: a computational approach to understanding the effects of muscle properties on motor control