Musculotendon (Docstrings)#

Implementations of musculotendon models.

Musculotendon models are a critical component of biomechanical models, one that differentiates them from pure multibody systems. Musculotendon models produce a force dependent on their level of activation, their length, and their extension velocity. Length- and extension velocity-dependent force production are governed by force-length and force-velocity characteristics. These are normalized functions that are dependent on the musculotendon’s state and are specific to a given musculotendon model.

class sympy.physics.biomechanics.musculotendon.MusculotendonBase(name, pathway, activation_dynamics, *, musculotendon_dynamics=MusculotendonFormulation.RIGID_TENDON, tendon_slack_length=None, peak_isometric_force=None, optimal_fiber_length=None, maximal_fiber_velocity=None, optimal_pennation_angle=None, fiber_damping_coefficient=None, with_defaults=False)[source]#

Abstract base class for all musculotendon classes to inherit from.

Parameters:

name : str

The name identifier associated with the musculotendon. This name is used as a suffix when automatically generated symbols are instantiated. It must be a string of nonzero length.

pathway : PathwayBase

The pathway that the actuator follows. This must be an instance of a concrete subclass of PathwayBase, e.g. LinearPathway.

activation_dynamics : ActivationBase

The activation dynamics that will be modeled within the musculotendon. This must be an instance of a concrete subclass of ActivationBase, e.g. FirstOrderActivationDeGroote2016.

musculotendon_dynamics : MusculotendonFormulation | int

The formulation of musculotendon dynamics that should be used internally, i.e. rigid or elastic tendon model, the choice of musculotendon state etc. This must be a member of the integer enumeration MusculotendonFormulation or an integer that can be cast to a member. To use a rigid tendon formulation, set this to MusculotendonFormulation.RIGID_TENDON (or the integer value 0, which will be cast to the enumeration member). There are four possible formulations for an elastic tendon model. To use an explicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_EXPLICIT (or the integer value 1). To use an explicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_EXPLICIT (or the integer value 2). To use an implicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_IMPLICIT (or the integer value 3). To use an implicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_IMPLICIT (or the integer value 4). The default is MusculotendonFormulation.RIGID_TENDON, which corresponds to a rigid tendon formulation.

tendon_slack_length : Expr | None

The length of the tendon when the musculotendon is in its unloaded state. In a rigid tendon model the tendon length is the tendon slack length. In all musculotendon models, tendon slack length is used to normalize tendon length to give \(\tilde{l}^T = \frac{l^T}{l^T_{slack}}\).

peak_isometric_force : Expr | None

The maximum force that the muscle fiber can produce when it is undergoing an isometric contraction (no lengthening velocity). In all musculotendon models, peak isometric force is used to normalized tendon and muscle fiber force to give \(\tilde{F}^T = \frac{F^T}{F^M_{max}}\).

optimal_fiber_length : Expr | None

The muscle fiber length at which the muscle fibers produce no passive force and their maximum active force. In all musculotendon models, optimal fiber length is used to normalize muscle fiber length to give \(\tilde{l}^M = \frac{l^M}{l^M_{opt}}\).

maximal_fiber_velocity : Expr | None

The fiber velocity at which, during muscle fiber shortening, the muscle fibers are unable to produce any active force. In all musculotendon models, maximal fiber velocity is used to normalize muscle fiber extension velocity to give \(\tilde{v}^M = \frac{v^M}{v^M_{max}}\).

optimal_pennation_angle : Expr | None

The pennation angle when muscle fiber length equals the optimal fiber length.

fiber_damping_coefficient : Expr | None

The coefficient of damping to be used in the damping element in the muscle fiber model.

with_defaults : bool

Whether with_defaults alternate constructors should be used when automatically constructing child classes. Default is False.

Explanation

A musculotendon generates a contractile force based on its activation, length, and shortening velocity. This abstract base class is to be inherited by all musculotendon subclasses that implement different characteristic musculotendon curves. Characteristic musculotendon curves are required for the tendon force-length, passive fiber force-length, active fiber force- length, and fiber force-velocity relationships.

property F#

Ordered column matrix of equations on the RHS of M x' = F.

Explanation

The column matrix that forms the RHS of the linear system of ordinary differential equations governing the activation dynamics:

M(x, r, t, p) x' = F(x, r, t, p).

As zeroth-order activation dynamics have no state variables, this linear system has dimension 0 and therefore F is an empty column Matrix with shape (0, 1).

property F_M_max#

Symbol or value corresponding to the peak isometric force constant.

Explanation

The maximum force that the muscle fiber can produce when it is undergoing an isometric contraction (no lengthening velocity). In all musculotendon models, peak isometric force is used to normalized tendon and muscle fiber force to give \(\tilde{F}^T = \frac{F^T}{F^M_{max}}\).

The alias peak_isometric_force can also be used to access the same attribute.

property M#

Ordered square matrix of coefficients on the LHS of M x' = F.

Explanation

The square matrix that forms part of the LHS of the linear system of ordinary differential equations governing the activation dynamics:

M(x, r, t, p) x' = F(x, r, t, p).

As zeroth-order activation dynamics have no state variables, this linear system has dimension 0 and therefore M is an empty square Matrix with shape (0, 0).

property a#

Dynamic symbol representing activation.

Explanation

The alias activation can also be used to access the same attribute.

property activation#

Dynamic symbol representing activation.

Explanation

The alias a can also be used to access the same attribute.

property activation_dynamics#

Activation dynamics model governing this musculotendon’s activation.

Explanation

Returns the instance of a subclass of ActivationBase that governs the relationship between excitation and activation that is used to represent the activation dynamics of this musculotendon.

property alpha_opt#

Symbol or value corresponding to the optimal pennation angle constant.

Explanation

The pennation angle when muscle fiber length equals the optimal fiber length.

The alias optimal_pennation_angle can also be used to access the same attribute.

property beta#

Symbol or value corresponding to the fiber damping coefficient constant.

Explanation

The coefficient of damping to be used in the damping element in the muscle fiber model.

The alias fiber_damping_coefficient can also be used to access the same attribute.

property constants#

Ordered column matrix of non-time varying symbols present in M and F.

Explanation

Only symbolic constants are returned. If a numeric type (e.g. Float) has been used instead of Symbol for a constant then that attribute will not be included in the matrix returned by this property. This is because the primary use of this property attribute is to provide an ordered sequence of the still-free symbols that require numeric values during code generation.

The alias p can also be used to access the same attribute.

abstract curves()[source]#

Return a CharacteristicCurveCollection of the curves related to the specific model.

property e#

Dynamic symbol representing excitation.

Explanation

The alias excitation can also be used to access the same attribute.

property excitation#

Dynamic symbol representing excitation.

Explanation

The alias e can also be used to access the same attribute.

property fiber_damping_coefficient#

Symbol or value corresponding to the fiber damping coefficient constant.

Explanation

The coefficient of damping to be used in the damping element in the muscle fiber model.

The alias beta can also be used to access the same attribute.

property input_vars#

Ordered column matrix of functions of time that represent the input variables.

Explanation

The alias r can also be used to access the same attribute.

property l_M_opt#

Symbol or value corresponding to the optimal fiber length constant.

Explanation

The muscle fiber length at which the muscle fibers produce no passive force and their maximum active force. In all musculotendon models, optimal fiber length is used to normalize muscle fiber length to give \(\tilde{l}^M = \frac{l^M}{l^M_{opt}}\).

The alias optimal_fiber_length can also be used to access the same attribute.

property l_T_slack#

Symbol or value corresponding to the tendon slack length constant.

Explanation

The length of the tendon when the musculotendon is in its unloaded state. In a rigid tendon model the tendon length is the tendon slack length. In all musculotendon models, tendon slack length is used to normalize tendon length to give \(\tilde{l}^T = \frac{l^T}{l^T_{slack}}\).

The alias tendon_slack_length can also be used to access the same attribute.

property maximal_fiber_velocity#

Symbol or value corresponding to the maximal fiber velocity constant.

Explanation

The fiber velocity at which, during muscle fiber shortening, the muscle fibers are unable to produce any active force. In all musculotendon models, maximal fiber velocity is used to normalize muscle fiber extension velocity to give \(\tilde{v}^M = \frac{v^M}{v^M_{max}}\).

The alias v_M_max can also be used to access the same attribute.

property musculotendon_dynamics#

The choice of rigid or type of elastic tendon musculotendon dynamics.

Explanation

The formulation of musculotendon dynamics that should be used internally, i.e. rigid or elastic tendon model, the choice of musculotendon state etc. This must be a member of the integer enumeration MusculotendonFormulation or an integer that can be cast to a member. To use a rigid tendon formulation, set this to MusculotendonFormulation.RIGID_TENDON (or the integer value 0, which will be cast to the enumeration member). There are four possible formulations for an elastic tendon model. To use an explicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_EXPLICIT (or the integer value 1). To use an explicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_EXPLICIT (or the integer value 2). To use an implicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_IMPLICIT (or the integer value 3). To use an implicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_IMPLICIT (or the integer value 4). The default is MusculotendonFormulation.RIGID_TENDON, which corresponds to a rigid tendon formulation.

property optimal_fiber_length#

Symbol or value corresponding to the optimal fiber length constant.

Explanation

The muscle fiber length at which the muscle fibers produce no passive force and their maximum active force. In all musculotendon models, optimal fiber length is used to normalize muscle fiber length to give \(\tilde{l}^M = \frac{l^M}{l^M_{opt}}\).

The alias l_M_opt can also be used to access the same attribute.

property optimal_pennation_angle#

Symbol or value corresponding to the optimal pennation angle constant.

Explanation

The pennation angle when muscle fiber length equals the optimal fiber length.

The alias alpha_opt can also be used to access the same attribute.

property p#

Ordered column matrix of non-time varying symbols present in M and F.

Explanation

Only symbolic constants are returned. If a numeric type (e.g. Float) has been used instead of Symbol for a constant then that attribute will not be included in the matrix returned by this property. This is because the primary use of this property attribute is to provide an ordered sequence of the still-free symbols that require numeric values during code generation.

The alias constants can also be used to access the same attribute.

property peak_isometric_force#

Symbol or value corresponding to the peak isometric force constant.

Explanation

The maximum force that the muscle fiber can produce when it is undergoing an isometric contraction (no lengthening velocity). In all musculotendon models, peak isometric force is used to normalized tendon and muscle fiber force to give \(\tilde{F}^T = \frac{F^T}{F^M_{max}}\).

The alias F_M_max can also be used to access the same attribute.

property r#

Ordered column matrix of functions of time that represent the input variables.

Explanation

The alias input_vars can also be used to access the same attribute.

rhs()[source]#

Ordered column matrix of equations for the solution of M x' = F.

Explanation

The solution to the linear system of ordinary differential equations governing the activation dynamics:

M(x, r, t, p) x' = F(x, r, t, p).

As zeroth-order activation dynamics have no state variables, this linear has dimension 0 and therefore this method returns an empty column Matrix with shape (0, 1).

property state_vars#

Ordered column matrix of functions of time that represent the state variables.

Explanation

The alias x can also be used to access the same attribute.

property tendon_slack_length#

Symbol or value corresponding to the tendon slack length constant.

Explanation

The length of the tendon when the musculotendon is in its unloaded state. In a rigid tendon model the tendon length is the tendon slack length. In all musculotendon models, tendon slack length is used to normalize tendon length to give \(\tilde{l}^T = \frac{l^T}{l^T_{slack}}\).

The alias l_T_slack can also be used to access the same attribute.

property v_M_max#

Symbol or value corresponding to the maximal fiber velocity constant.

Explanation

The fiber velocity at which, during muscle fiber shortening, the muscle fibers are unable to produce any active force. In all musculotendon models, maximal fiber velocity is used to normalize muscle fiber extension velocity to give \(\tilde{v}^M = \frac{v^M}{v^M_{max}}\).

The alias maximal_fiber_velocity can also be used to access the same attribute.

classmethod with_defaults(name, pathway, activation_dynamics, *, musculotendon_dynamics=MusculotendonFormulation.RIGID_TENDON, tendon_slack_length=None, peak_isometric_force=None, optimal_fiber_length=None, maximal_fiber_velocity=10.0000000000000, optimal_pennation_angle=0.0, fiber_damping_coefficient=0.100000000000000)[source]#

Recommended constructor that will use the published constants.

Parameters:

name : str

The name identifier associated with the musculotendon. This name is used as a suffix when automatically generated symbols are instantiated. It must be a string of nonzero length.

pathway : PathwayBase

The pathway that the actuator follows. This must be an instance of a concrete subclass of PathwayBase, e.g. LinearPathway.

activation_dynamics : ActivationBase

The activation dynamics that will be modeled within the musculotendon. This must be an instance of a concrete subclass of ActivationBase, e.g. FirstOrderActivationDeGroote2016.

musculotendon_dynamics : MusculotendonFormulation | int

The formulation of musculotendon dynamics that should be used internally, i.e. rigid or elastic tendon model, the choice of musculotendon state etc. This must be a member of the integer enumeration MusculotendonFormulation or an integer that can be cast to a member. To use a rigid tendon formulation, set this to MusculotendonFormulation.RIGID_TENDON (or the integer value 0, which will be cast to the enumeration member). There are four possible formulations for an elastic tendon model. To use an explicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_EXPLICIT (or the integer value 1). To use an explicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_EXPLICIT (or the integer value 2). To use an implicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_IMPLICIT (or the integer value 3). To use an implicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_IMPLICIT (or the integer value 4). The default is MusculotendonFormulation.RIGID_TENDON, which corresponds to a rigid tendon formulation.

tendon_slack_length : Expr | None

The length of the tendon when the musculotendon is in its unloaded state. In a rigid tendon model the tendon length is the tendon slack length. In all musculotendon models, tendon slack length is used to normalize tendon length to give \(\tilde{l}^T = \frac{l^T}{l^T_{slack}}\).

peak_isometric_force : Expr | None

The maximum force that the muscle fiber can produce when it is undergoing an isometric contraction (no lengthening velocity). In all musculotendon models, peak isometric force is used to normalized tendon and muscle fiber force to give \(\tilde{F}^T = \frac{F^T}{F^M_{max}}\).

optimal_fiber_length : Expr | None

The muscle fiber length at which the muscle fibers produce no passive force and their maximum active force. In all musculotendon models, optimal fiber length is used to normalize muscle fiber length to give \(\tilde{l}^M = \frac{l^M}{l^M_{opt}}\).

maximal_fiber_velocity : Expr | None

The fiber velocity at which, during muscle fiber shortening, the muscle fibers are unable to produce any active force. In all musculotendon models, maximal fiber velocity is used to normalize muscle fiber extension velocity to give \(\tilde{v}^M = \frac{v^M}{v^M_{max}}\).

optimal_pennation_angle : Expr | None

The pennation angle when muscle fiber length equals the optimal fiber length.

fiber_damping_coefficient : Expr | None

The coefficient of damping to be used in the damping element in the muscle fiber model.

Explanation

Returns a new instance of the musculotendon class using recommended values for v_M_max, alpha_opt, and beta. The values are:

\(v^M_{max} = 10\) \(\alpha_{opt} = 0\) \(\beta = \frac{1}{10}\)

The musculotendon curves are also instantiated using the constants from the original publication.

property x#

Ordered column matrix of functions of time that represent the state variables.

Explanation

The alias state_vars can also be used to access the same attribute.

class sympy.physics.biomechanics.musculotendon.MusculotendonDeGroote2016(name, pathway, activation_dynamics, *, musculotendon_dynamics=MusculotendonFormulation.RIGID_TENDON, tendon_slack_length=None, peak_isometric_force=None, optimal_fiber_length=None, maximal_fiber_velocity=None, optimal_pennation_angle=None, fiber_damping_coefficient=None, with_defaults=False)[source]#

Musculotendon model using the curves of De Groote et al., 2016 [R713].

Parameters:

name : str

The name identifier associated with the musculotendon. This name is used as a suffix when automatically generated symbols are instantiated. It must be a string of nonzero length.

pathway : PathwayBase

The pathway that the actuator follows. This must be an instance of a concrete subclass of PathwayBase, e.g. LinearPathway.

activation_dynamics : ActivationBase

The activation dynamics that will be modeled within the musculotendon. This must be an instance of a concrete subclass of ActivationBase, e.g. FirstOrderActivationDeGroote2016.

musculotendon_dynamics : MusculotendonFormulation | int

The formulation of musculotendon dynamics that should be used internally, i.e. rigid or elastic tendon model, the choice of musculotendon state etc. This must be a member of the integer enumeration MusculotendonFormulation or an integer that can be cast to a member. To use a rigid tendon formulation, set this to MusculotendonFormulation.RIGID_TENDON (or the integer value 0, which will be cast to the enumeration member). There are four possible formulations for an elastic tendon model. To use an explicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_EXPLICIT (or the integer value 1). To use an explicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_EXPLICIT (or the integer value 2). To use an implicit formulation with the fiber length as the state, set this to MusculotendonFormulation.FIBER_LENGTH_IMPLICIT (or the integer value 3). To use an implicit formulation with the tendon force as the state, set this to MusculotendonFormulation.TENDON_FORCE_IMPLICIT (or the integer value 4). The default is MusculotendonFormulation.RIGID_TENDON, which corresponds to a rigid tendon formulation.

tendon_slack_length : Expr | None

The length of the tendon when the musculotendon is in its unloaded state. In a rigid tendon model the tendon length is the tendon slack length. In all musculotendon models, tendon slack length is used to normalize tendon length to give \(\tilde{l}^T = \frac{l^T}{l^T_{slack}}\).

peak_isometric_force : Expr | None

The maximum force that the muscle fiber can produce when it is undergoing an isometric contraction (no lengthening velocity). In all musculotendon models, peak isometric force is used to normalized tendon and muscle fiber force to give \(\tilde{F}^T = \frac{F^T}{F^M_{max}}\).

optimal_fiber_length : Expr | None

The muscle fiber length at which the muscle fibers produce no passive force and their maximum active force. In all musculotendon models, optimal fiber length is used to normalize muscle fiber length to give \(\tilde{l}^M = \frac{l^M}{l^M_{opt}}\).

maximal_fiber_velocity : Expr | None

The fiber velocity at which, during muscle fiber shortening, the muscle fibers are unable to produce any active force. In all musculotendon models, maximal fiber velocity is used to normalize muscle fiber extension velocity to give \(\tilde{v}^M = \frac{v^M}{v^M_{max}}\).

optimal_pennation_angle : Expr | None

The pennation angle when muscle fiber length equals the optimal fiber length.

fiber_damping_coefficient : Expr | None

The coefficient of damping to be used in the damping element in the muscle fiber model.

with_defaults : bool

Whether with_defaults alternate constructors should be used when automatically constructing child classes. Default is False.

Examples

This class models the musculotendon actuator parametrized by the characteristic curves described in De Groote et al., 2016 [R713]. Like all musculotendon models in SymPy’s biomechanics module, it requires a pathway to define its line of action. We’ll begin by creating a simple LinearPathway between two points that our musculotendon will follow. We’ll create a point O to represent the musculotendon’s origin and another I to represent its insertion.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import (LinearPathway, Point,
...     ReferenceFrame, dynamicsymbols)
>>> N = ReferenceFrame('N')
>>> O, I = O, P = symbols('O, I', cls=Point)
>>> q, u = dynamicsymbols('q, u', real=True)
>>> I.set_pos(O, q*N.x)
>>> O.set_vel(N, 0)
>>> I.set_vel(N, u*N.x)
>>> pathway = LinearPathway(O, I)
>>> pathway.attachments
(O, I)
>>> pathway.length
Abs(q(t))
>>> pathway.extension_velocity
sign(q(t))*Derivative(q(t), t)

A musculotendon also takes an instance of an activation dynamics model as this will be used to provide symbols for the activation in the formulation of the musculotendon dynamics. We’ll use an instance of FirstOrderActivationDeGroote2016 to represent first-order activation dynamics. Note that a single name argument needs to be provided as SymPy will use this as a suffix.

>>> from sympy.physics.biomechanics import FirstOrderActivationDeGroote2016
>>> activation = FirstOrderActivationDeGroote2016('muscle')
>>> activation.x
Matrix([[a_muscle(t)]])
>>> activation.r
Matrix([[e_muscle(t)]])
>>> activation.p
Matrix([
[tau_a_muscle],
[tau_d_muscle],
[    b_muscle]])
>>> activation.rhs()
Matrix([[((1/2 - tanh(b_muscle*(-a_muscle(t) + e_muscle(t)))/2)*(3*...]])

The musculotendon class requires symbols or values to be passed to represent the constants in the musculotendon dynamics. We’ll use SymPy’s symbols function to create symbols for the maximum isometric force F_M_max, optimal fiber length l_M_opt, tendon slack length l_T_slack, maximum fiber velocity v_M_max, optimal pennation angle alpha_opt, and fiber damping coefficient ``beta.

>>> F_M_max = symbols('F_M_max', real=True)
>>> l_M_opt = symbols('l_M_opt', real=True)
>>> l_T_slack = symbols('l_T_slack', real=True)
>>> v_M_max = symbols('v_M_max', real=True)
>>> alpha_opt = symbols('alpha_opt', real=True)
>>> beta = symbols('beta', real=True)

We can then import the class MusculotendonDeGroote2016 from the biomechanics module and create an instance by passing in the various objects we have previously instantiated. By default, a musculotendon model with rigid tendon musculotendon dynamics will be created.

>>> from sympy.physics.biomechanics import MusculotendonDeGroote2016
>>> rigid_tendon_muscle = MusculotendonDeGroote2016(
...     'muscle',
...     pathway,
...     activation,
...     tendon_slack_length=l_T_slack,
...     peak_isometric_force=F_M_max,
...     optimal_fiber_length=l_M_opt,
...     maximal_fiber_velocity=v_M_max,
...     optimal_pennation_angle=alpha_opt,
...     fiber_damping_coefficient=beta,
... )

We can inspect the various properties of the musculotendon, including getting the symbolic expression describing the force it produces using its force attribute.

>>> rigid_tendon_muscle.force
-F_M_max*(beta*(-l_T_slack + Abs(q(t)))*sign(q(t))*Derivative(q(t), t)...

When we created the musculotendon object, we passed in an instance of an activation dynamics object that governs the activation within the musculotendon. SymPy makes a design choice here that the activation dynamics instance will be treated as a child object of the musculotendon dynamics. Therefore, if we want to inspect the state and input variables associated with the musculotendon model, we will also be returned the state and input variables associated with the child object, or the activation dynamics in this case. As the musculotendon model that we created here uses rigid tendon dynamics, no additional states or inputs relating to the musculotendon are introduces. Consequently, the model has a single state associated with it, the activation, and a single input associated with it, the excitation. The states and inputs can be inspected using the x and r attributes respectively. Note that both x and r have the alias attributes of state_vars and input_vars.

>>> rigid_tendon_muscle.x
Matrix([[a_muscle(t)]])
>>> rigid_tendon_muscle.r
Matrix([[e_muscle(t)]])

To see which constants are symbolic in the musculotendon model, we can use the p or constants attribute. This returns a Matrix populated by the constants that are represented by a Symbol rather than a numeric value.

>>> rigid_tendon_muscle.p
Matrix([
[           l_T_slack],
[             F_M_max],
[             l_M_opt],
[             v_M_max],
[           alpha_opt],
[                beta],
[        tau_a_muscle],
[        tau_d_muscle],
[            b_muscle],
[     c_0_fl_T_muscle],
[     c_1_fl_T_muscle],
[     c_2_fl_T_muscle],
[     c_3_fl_T_muscle],
[ c_0_fl_M_pas_muscle],
[ c_1_fl_M_pas_muscle],
[ c_0_fl_M_act_muscle],
[ c_1_fl_M_act_muscle],
[ c_2_fl_M_act_muscle],
[ c_3_fl_M_act_muscle],
[ c_4_fl_M_act_muscle],
[ c_5_fl_M_act_muscle],
[ c_6_fl_M_act_muscle],
[ c_7_fl_M_act_muscle],
[ c_8_fl_M_act_muscle],
[ c_9_fl_M_act_muscle],
[c_10_fl_M_act_muscle],
[c_11_fl_M_act_muscle],
[     c_0_fv_M_muscle],
[     c_1_fv_M_muscle],
[     c_2_fv_M_muscle],
[     c_3_fv_M_muscle]])

Finally, we can call the rhs method to return a Matrix that contains as its elements the righthand side of the ordinary differential equations corresponding to each of the musculotendon’s states. Like the method with the same name on the Method classes in SymPy’s mechanics module, this returns a column vector where the number of rows corresponds to the number of states. For our example here, we have a single state, the dynamic symbol a_muscle(t), so the returned value is a 1-by-1 Matrix.

>>> rigid_tendon_muscle.rhs()
Matrix([[((1/2 - tanh(b_muscle*(-a_muscle(t) + e_muscle(t)))/2)*(3*...]])

The musculotendon class supports elastic tendon musculotendon models in addition to rigid tendon ones. You can choose to either use the fiber length or tendon force as an additional state. You can also specify whether an explicit or implicit formulation should be used. To select a formulation, pass a member of the MusculotendonFormulation enumeration to the musculotendon_dynamics parameter when calling the constructor. This enumeration is an IntEnum, so you can also pass an integer, however it is recommended to use the enumeration as it is clearer which formulation you are actually selecting. Below, we’ll use the FIBER_LENGTH_EXPLICIT member to create a musculotendon with an elastic tendon that will use the (normalized) muscle fiber length as an additional state and will produce the governing ordinary differential equation in explicit form.

>>> from sympy.physics.biomechanics import MusculotendonFormulation
>>> elastic_tendon_muscle = MusculotendonDeGroote2016(
...     'muscle',
...     pathway,
...     activation,
...     musculotendon_dynamics=MusculotendonFormulation.FIBER_LENGTH_EXPLICIT,
...     tendon_slack_length=l_T_slack,
...     peak_isometric_force=F_M_max,
...     optimal_fiber_length=l_M_opt,
...     maximal_fiber_velocity=v_M_max,
...     optimal_pennation_angle=alpha_opt,
...     fiber_damping_coefficient=beta,
... )
>>> elastic_tendon_muscle.force
-F_M_max*TendonForceLengthDeGroote2016((-sqrt(l_M_opt**2*...
>>> elastic_tendon_muscle.x
Matrix([
[l_M_tilde_muscle(t)],
[        a_muscle(t)]])
>>> elastic_tendon_muscle.r
Matrix([[e_muscle(t)]])
>>> elastic_tendon_muscle.p
Matrix([
[           l_T_slack],
[             F_M_max],
[             l_M_opt],
[             v_M_max],
[           alpha_opt],
[                beta],
[        tau_a_muscle],
[        tau_d_muscle],
[            b_muscle],
[     c_0_fl_T_muscle],
[     c_1_fl_T_muscle],
[     c_2_fl_T_muscle],
[     c_3_fl_T_muscle],
[ c_0_fl_M_pas_muscle],
[ c_1_fl_M_pas_muscle],
[ c_0_fl_M_act_muscle],
[ c_1_fl_M_act_muscle],
[ c_2_fl_M_act_muscle],
[ c_3_fl_M_act_muscle],
[ c_4_fl_M_act_muscle],
[ c_5_fl_M_act_muscle],
[ c_6_fl_M_act_muscle],
[ c_7_fl_M_act_muscle],
[ c_8_fl_M_act_muscle],
[ c_9_fl_M_act_muscle],
[c_10_fl_M_act_muscle],
[c_11_fl_M_act_muscle],
[     c_0_fv_M_muscle],
[     c_1_fv_M_muscle],
[     c_2_fv_M_muscle],
[     c_3_fv_M_muscle]])
>>> elastic_tendon_muscle.rhs()
Matrix([
[v_M_max*FiberForceVelocityInverseDeGroote2016((l_M_opt*...],
[ ((1/2 - tanh(b_muscle*(-a_muscle(t) + e_muscle(t)))/2)*(3*...]])

It is strongly recommended to use the alternate with_defaults constructor when creating an instance because this will ensure that the published constants are used in the musculotendon characteristic curves.

>>> elastic_tendon_muscle = MusculotendonDeGroote2016.with_defaults(
...     'muscle',
...     pathway,
...     activation,
...     musculotendon_dynamics=MusculotendonFormulation.FIBER_LENGTH_EXPLICIT,
...     tendon_slack_length=l_T_slack,
...     peak_isometric_force=F_M_max,
...     optimal_fiber_length=l_M_opt,
... )
>>> elastic_tendon_muscle.x
Matrix([
[l_M_tilde_muscle(t)],
[        a_muscle(t)]])
>>> elastic_tendon_muscle.r
Matrix([[e_muscle(t)]])
>>> elastic_tendon_muscle.p
Matrix([
[   l_T_slack],
[     F_M_max],
[     l_M_opt],
[tau_a_muscle],
[tau_d_muscle],
[    b_muscle]])

References

[R713] (1,2,3)

De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation of direct collocation optimal control problem formulations for solving the muscle redundancy problem, Annals of biomedical engineering, 44(10), (2016) pp. 2922-2936

class sympy.physics.biomechanics.musculotendon.MusculotendonFormulation(value, names=None, *values, module=None, qualname=None, type=None, start=1, boundary=None)[source]#

Enumeration of types of musculotendon dynamics formulations.

Explanation

An (integer) enumeration is used as it allows for clearer selection of the different formulations of musculotendon dynamics.

Members

RIGID_TENDON0

A rigid tendon model.

FIBER_LENGTH_EXPLICIT1

An explicit elastic tendon model with the muscle fiber length (l_M) as the state variable.

TENDON_FORCE_EXPLICIT2

An explicit elastic tendon model with the tendon force (F_T) as the state variable.

FIBER_LENGTH_IMPLICIT3

An implicit elastic tendon model with the muscle fiber length (l_M) as the state variable and the muscle fiber velocity as an additional input variable.

TENDON_FORCE_IMPLICIT4

An implicit elastic tendon model with the tendon force (F_T) as the state variable as the muscle fiber velocity as an additional input variable.