Implement Hill-Type Muscle Model in biomechanics

[eh111eh]

References to other Issues or PRs

N/A

Brief description of what is fixed or changed

I implement a HillTypeMuscle class that inherits from a base musculotendon class.

The Hill muscle model consists of three elements: A contractile element (CE), A nonlinear passive element (PE), and A nonlinear passive element in series with the former two (SE).

The force equilibrium equation is
$$F = F_{SE} = (F_{CE}+F_{PE})*cos\alpha$$
where $F$ is the musculotendon force, $F_{CE}$, $F_{PE}$, $F_{SE}$ are the forces produced by the contractile, passive parallel, and passive serial element, respectively, and $\alpha$ is the pennation angle.

These forces are detailed as
$$F_{CE}=aF_0f_l(l_m)f_v(v_m)$$
$$F_{PE}=F_0f_p(l_m)$$
$$F_{SE}=F_0f_t(l_t)$$
where $a$ is the muscle activation, $F_0$ is the maximum isometric force, $f_l$ and $f_v$ are dimensionless force-length and force-velocity relationships, respectively, $f_p$ and $f_t$ are dimensionless force-length relationship of the passive parallel and passive serial element, respectively, $l_m$ and $l_t$ are the lengths of muscle and tendon, respectively, and $v_m$ is the velocity of muscle contraction.

Other comments

Reference

• Lamas, M., Mouzo, F., Michaud, F. et al. Comparison of several muscle modeling alternatives for computationally intensive algorithms in human motion dynamics. Multibody Syst Dyn 54, 415–442 (2022). https://doi.org/10.1007/s11044-022-09819-y
• https://sanlab.psych.ucla.edu/wp-content/uploads/sites/38/2019/11/Morgenland_Venugopal_ECNS-with-Revised-Final.pdf
• Miller, R.H. (2018). Hill-Based Muscle Modeling. In: Müller, B., et al. Handbook of Human Motion. Springer, Cham. https://doi.org/10.1007/978-3-319-30808-1_203-2
• Hill's muscle model - Wikipedia

Release Notes

• physics.biomechanics
  • Implement Hill Type Muscle Model

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  • Implement Hill Type Muscle Model (#26443 by @eh111eh)

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#### Brief description of what is fixed or changed
I implement a **HillTypeMuscle** class that inherits from a base musculotendon class.

The Hill muscle model consists of three elements: A contractile element (CE), A nonlinear passive element (PE), and A nonlinear passive element in series with the former two (SE).

The force equilibrium equation is 
$$F = F_{SE} = (F_{CE}+F_{PE})*cos\alpha$$
where $F$ is the musculotendon force, $F_{CE}$, $F_{PE}$, $F_{SE}$ are the forces produced by the contractile, passive parallel, and passive serial element, respectively, and $\alpha$ is the pennation angle.

These forces are detailed as
$$F_{CE}=aF_0f_l(l_m)f_v(v_m)$$ 
$$F_{PE}=F_0f_p(l_m)$$ 
$$F_{SE}=F_0f_t(l_t)$$
where $a$ is the muscle activation, $F_0$ is the maximum isometric force, $f_l$ and $f_v$ are dimensionless force-length and force-velocity relationships, respectively, $f_p$ and $f_t$ are dimensionless force-length relationship of the passive parallel and passive serial element, respectively, $l_m$ and $l_t$ are the lengths of muscle and tendon, respectively, and $v_m$ is the velocity of muscle contraction.

#### Other comments
Reference
- Lamas, M., Mouzo, F., Michaud, F. et al. Comparison of several muscle modeling alternatives for computationally intensive algorithms in human motion dynamics. Multibody Syst Dyn 54, 415–442 (2022). https://doi.org/10.1007/s11044-022-09819-y
- https://sanlab.psych.ucla.edu/wp-content/uploads/sites/38/2019/11/Morgenland_Venugopal_ECNS-with-Revised-Final.pdf
- Miller, R.H. (2018). Hill-Based Muscle Modeling. In: Müller, B., et al. Handbook of Human Motion. Springer, Cham. https://doi.org/10.1007/978-3-319-30808-1_203-2
- [Hill's muscle model - Wikipedia](https://en.wikipedia.org/wiki/Hill%27s_muscle_model)

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[moorepants]

Hi, great idea to add this!

As I've mentioned in the other PRs, you'll need unit tests and to show that this muscle works. You can use the model here to demonstrate that it works:

https://docs.sympy.org/dev/tutorials/biomechanics/biomechanics.html#a-simple-musculotendon-model

We'd also like the documentation to be similar in quality as the existing muscles. So use those as a template.

I suspect you'll need to introduce new curves for the Hill functions. I recommend making a copy of all the DeGroote2016 classes and then replacing them with the equivalent Hill models.