Hill¶

Module: solid

Category: discretematerial

Type string: "Hill"

Parameters¶

Name	Description	Default	Range
`Vmax`	Vmax	1	\(\in \mathbb{R}\)
`ac`	ac	0	\(\in \mathbb{R}\)
`Fmax`	Fmax	1	\(\in \mathbb{R}\)
`Ksh`	Ksh	1	\(\in \mathbb{R}\)
`Lmax`	Lmax	1	\(\in \mathbb{R}\)
`L0`	L0	0	\(\in \mathbb{R}\)
`Sv`	Sv		N/A
`Ftl`	Ftl		N/A
`Fvl`	Fvl		N/A

Description¶

The force in the Hill discrete element is the sum of the passive element and the active element.

\[ F=F_{p}+F_{a} \]

The passive force is given by,

\[ F_{p}=\begin{cases} F_{max}\left(\exp\left(K_{sh}\left(l-1\right)/L_{max}-1\right)/\exp\left(K_{sh}-1\right)\right) & ,l>1\\ 0 & ,l\leqslant1 \end{cases} \]

where \(l\) is the relative stretch defined by, \(l=l_m/l_0\) and \(l_m\) is the discrete element length and \(l_0\) is either the L0 parameter, which sets the reference length of the muscle, or the initial discrete element length (if L0 is set to zero).

The Max Force parameter (Fmax) is the maximum force the muscle can produce. Physiologically, this happens under 1) tetanized neural activation 2) at the optimum muscle length and 3) isometric contraction (contraction without movement). From a modelling perspective, this is the tensile force developed in the discrete element at the optimum length (Max Length) and 100% activation.

The Max Length parameter (Lmax) is the optimum muscle length. This corresponds to the peak of the force-length curve.

The shape parameter (\(K_{sh}\) = Ksh) is a dimensionless parameter that controls the rate of the rise of the exponential function that defines the passive force-length relationship for the parallel elastic component of the Hill model.

The active force is given by,

\[ F_a=ac\,F_{max}\,F_{tl}\left(l\right)\,F_{tv}\left(v\right) \]

Here, \(v\), a measure of the relative discrete element's growth speed, is defined by,

\[ v=v_m/\left(V_{max}\,S_v\left(ac\right)\right) \]

where \(v_m\) is the actual discrete element's growth speed and the Maximum Shortening Velocity parameter (Vmax) defines the maximum shortening velocity for the muscle.

The Activation Level parameter (ac) defines the normalized activation level of the muscle, which can vary between 0 (no activation, passive properties) to 1 (maximum activation). This parameter can be set as a constant or varied as a function of time via a load curve.

The shortening velocity-force curve (Sv) defines a scale factor that optionally scales the Maximum Shortening Velocity property (Vmax) as a function of the current activation level. This curve can be defined using the Curve Editor. The x-axis should be the activation level (varying between 0 and 1, no units), and the y-axis should be the scale factor (no units). If this curve is not defined, the value entered for Vmax is assumed constant as a function of activation level.

The normalized tension-length curve (Ftl) (Figure 1) defines the relationship between normalized length (x-axis) and normalized tension (y-axis). This curve can be defined using the Curve Editor. A normalized length of 1 means that the current length is the same as the initial length (L0). A normalized force of 1 means that the force is equal to the Max Force property (Fmax).

Figure 1. Example data for the normalized tension-length curve (`Ftl`).

The normalized force-velocity curve (Fvl) (Figure 2) defines the relationship between normalized velocity (x-axis) and normalized velocity (y-axis). This curve can be defined using the Curve Editor. A normalized velocity of 0 means that the muscle is neither shortening nor lengthening, while a normalized velocity of -1 and +1 indicate maximal velocities of shortening and lengthening, respectively. A normalized force of 1 means that the force at the current velocity of shortening/lengthening is equal to the Max Force property (Fmax).

The properties Sv, Ftl, and Ftv, are optional and will evaluate to 1 if omitted. They can be defined as load curves. See the example below.

Example:

<discrete_material id="1" name="test" type="Hill">
  <Vmax>1</Vmax>
  <ac>0.1</ac>
  <Fmax>50</Fmax>
  <Ksh>5</Ksh>
  <Lmax>1.5</Lmax>
  <L0>10</L0>
  <Ftl type="point">
    <interpolate>smooth</interpolate>               
    <points>
      <pt>0.0, 0</pt>
      <pt>0.1, 0.000258139</pt>
      <pt>0.2, 0.00161616</pt>
      <pt>0.3, 0.00740118</pt>
      <pt>0.4, 0.0272783</pt>
      <pt>0.5, 0.0820396</pt>
      <pt>0.6, 0.201851</pt>
      <pt>0.7, 0.406524</pt>
      <pt>0.8, 0.670275</pt>
      <pt>0.9, 0.904792</pt>
      <pt>1.0, 0.999955</pt>
      <pt>1.1, 0.904792</pt>
      <pt>1.2, 0.670275</pt>
      <pt>1.3, 0.406524</pt>
      <pt>1.4, 0.201851</pt>
      <pt>1.5, 0.0820396</pt>
      <pt>1.6, 0.0272783</pt>
      <pt>1.7, 0.00740118</pt>
      <pt>1.8, 0.00161616</pt>
      <pt>1.9, 0.000258139</pt>
      <pt>2.0, 0</pt>
    </points>
  </Ftl> 
</discrete_material>