Curve (Docstrings)#

Implementations of characteristic curves for musculotendon models.

class sympy.physics.biomechanics.curve.CharacteristicCurveCollection(tendon_force_length: CharacteristicCurveFunction, tendon_force_length_inverse: CharacteristicCurveFunction, fiber_force_length_passive: CharacteristicCurveFunction, fiber_force_length_passive_inverse: CharacteristicCurveFunction, fiber_force_length_active: CharacteristicCurveFunction, fiber_force_velocity: CharacteristicCurveFunction, fiber_force_velocity_inverse: CharacteristicCurveFunction)[source]#

Simple data container to group together related characteristic curves.

class sympy.physics.biomechanics.curve.CharacteristicCurveFunction[source]#

Base class for all musculotendon characteristic curve functions.

class sympy.physics.biomechanics.curve.FiberForceLengthActiveDeGroote2016(l_M_tilde, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11)[source]#

Active muscle fiber force-length curve based on De Groote et al., 2016 [R706].

Explanation

The function is defined by the equation:

\(fl_{\text{act}}^M = c_0 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_1}{c_2 + c_3 \tilde{l}^M}\right)^2\right) + c_4 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_5}{c_6 + c_7 \tilde{l}^M}\right)^2\right) + c_8 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_9}{c_{10} + c_{11} \tilde{l}^M}\right)^2\right)\)

with constant values of \(c0 = 0.814\), \(c1 = 1.06\), \(c2 = 0.162\), \(c3 = 0.0633\), \(c4 = 0.433\), \(c5 = 0.717\), \(c6 = -0.0299\), \(c7 = 0.2\), \(c8 = 0.1\), \(c9 = 1.0\), \(c10 = 0.354\), and \(c11 = 0.0\).

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces a active fiber force of 1 at a normalized fiber length of 1, and an active fiber force of 0 at normalized fiber lengths of 0 and 2.

Examples

The preferred way to instantiate FiberForceLengthActiveDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized muscle fiber length. We’ll create a Symbol called l_M_tilde to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import FiberForceLengthActiveDeGroote2016
>>> l_M_tilde = Symbol('l_M_tilde')
>>> fl_M = FiberForceLengthActiveDeGroote2016.with_defaults(l_M_tilde)
>>> fl_M
FiberForceLengthActiveDeGroote2016(l_M_tilde, 0.814, 1.06, 0.162, 0.0633,
0.433, 0.717, -0.0299, 0.2, 0.1, 1.0, 0.354, 0.0)

It’s also possible to populate the two constants with your own values too.

>>> from sympy import symbols
>>> c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11 = symbols('c0:12')
>>> fl_M = FiberForceLengthActiveDeGroote2016(l_M_tilde, c0, c1, c2, c3,
...     c4, c5, c6, c7, c8, c9, c10, c11)
>>> fl_M
FiberForceLengthActiveDeGroote2016(l_M_tilde, c0, c1, c2, c3, c4, c5, c6,
c7, c8, c9, c10, c11)

You don’t just have to use symbols as the arguments, it’s also possible to use expressions. Let’s create a new pair of symbols, l_M and l_M_opt, representing muscle fiber length and optimal muscle fiber length respectively. We can then represent l_M_tilde as an expression, the ratio of these.

>>> l_M, l_M_opt = symbols('l_M l_M_opt')
>>> l_M_tilde = l_M/l_M_opt
>>> fl_M = FiberForceLengthActiveDeGroote2016.with_defaults(l_M_tilde)
>>> fl_M
FiberForceLengthActiveDeGroote2016(l_M/l_M_opt, 0.814, 1.06, 0.162, 0.0633,
0.433, 0.717, -0.0299, 0.2, 0.1, 1.0, 0.354, 0.0)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> fl_M.doit(evaluate=False)
0.814*exp(-19.0519737844841*(l_M/l_M_opt
- 1.06)**2/(0.390740740740741*l_M/l_M_opt + 1)**2)
+ 0.433*exp(-12.5*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt - 0.1495)**2)
+ 0.1*exp(-3.98991349867535*(l_M/l_M_opt - 1.0)**2)

The function can also be differentiated. We’ll differentiate with respect to l_M using the diff method on an instance with the single positional argument l_M.

>>> fl_M.diff(l_M)
((-0.79798269973507*l_M/l_M_opt
+ 0.79798269973507)*exp(-3.98991349867535*(l_M/l_M_opt - 1.0)**2)
+ (10.825*(-l_M/l_M_opt + 0.717)/(l_M/l_M_opt - 0.1495)**2
+ 10.825*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt
- 0.1495)**3)*exp(-12.5*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt - 0.1495)**2)
+ (31.0166133211401*(-l_M/l_M_opt + 1.06)/(0.390740740740741*l_M/l_M_opt
+ 1)**2 + 13.6174190361677*(0.943396226415094*l_M/l_M_opt
- 1)**2/(0.390740740740741*l_M/l_M_opt
+ 1)**3)*exp(-21.4067977442463*(0.943396226415094*l_M/l_M_opt
- 1)**2/(0.390740740740741*l_M/l_M_opt + 1)**2))/l_M_opt

References

[R706] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of l_M_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(l_M_tilde, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11)[source]#

Evaluation of basic inputs.

Parameters:

l_M_tilde : Any (sympifiable)

Normalized muscle fiber length.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is 0.814.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is 1.06.

c2 : Any (sympifiable)

The third constant in the characteristic equation. The published value is 0.162.

c3 : Any (sympifiable)

The fourth constant in the characteristic equation. The published value is 0.0633.

c4 : Any (sympifiable)

The fifth constant in the characteristic equation. The published value is 0.433.

c5 : Any (sympifiable)

The sixth constant in the characteristic equation. The published value is 0.717.

c6 : Any (sympifiable)

The seventh constant in the characteristic equation. The published value is -0.0299.

c7 : Any (sympifiable)

The eighth constant in the characteristic equation. The published value is 0.2.

c8 : Any (sympifiable)

The ninth constant in the characteristic equation. The published value is 0.1.

c9 : Any (sympifiable)

The tenth constant in the characteristic equation. The published value is 1.0.

c10 : Any (sympifiable)

The eleventh constant in the characteristic equation. The published value is 0.354.

c11 : Any (sympifiable)

The tweflth constant in the characteristic equation. The published value is 0.0.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

classmethod with_defaults(l_M_tilde)[source]#

Recommended constructor that will use the published constants.

Parameters:

fl_M_act : Any (sympifiable)

Normalized passive muscle fiber force as a function of muscle fiber length.

Explanation

Returns a new instance of the inverse muscle fiber act force-length function using the four constant values specified in the original publication.

These have the values:

\(c0 = 0.814\) \(c1 = 1.06\) \(c2 = 0.162\) \(c3 = 0.0633\) \(c4 = 0.433\) \(c5 = 0.717\) \(c6 = -0.0299\) \(c7 = 0.2\) \(c8 = 0.1\) \(c9 = 1.0\) \(c10 = 0.354\) \(c11 = 0.0\)

class sympy.physics.biomechanics.curve.FiberForceLengthPassiveDeGroote2016(l_M_tilde, c0, c1)[source]#

Passive muscle fiber force-length curve based on De Groote et al., 2016 [R707].

Explanation

The function is defined by the equation:

\(fl^M_{pas} = \frac{\frac{\exp{c_1 \left(\tilde{l^M} - 1\right)}}{c_0} - 1}{\exp{c_1} - 1}\)

with constant values of \(c_0 = 0.6\) and \(c_1 = 4.0\).

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces a passive fiber force very close to 0 for all normalized fiber lengths between 0 and 1.

Examples

The preferred way to instantiate FiberForceLengthPassiveDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized muscle fiber length. We’ll create a Symbol called l_M_tilde to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import FiberForceLengthPassiveDeGroote2016
>>> l_M_tilde = Symbol('l_M_tilde')
>>> fl_M = FiberForceLengthPassiveDeGroote2016.with_defaults(l_M_tilde)
>>> fl_M
FiberForceLengthPassiveDeGroote2016(l_M_tilde, 0.6, 4.0)

It’s also possible to populate the two constants with your own values too.

>>> from sympy import symbols
>>> c0, c1 = symbols('c0 c1')
>>> fl_M = FiberForceLengthPassiveDeGroote2016(l_M_tilde, c0, c1)
>>> fl_M
FiberForceLengthPassiveDeGroote2016(l_M_tilde, c0, c1)

You don’t just have to use symbols as the arguments, it’s also possible to use expressions. Let’s create a new pair of symbols, l_M and l_M_opt, representing muscle fiber length and optimal muscle fiber length respectively. We can then represent l_M_tilde as an expression, the ratio of these.

>>> l_M, l_M_opt = symbols('l_M l_M_opt')
>>> l_M_tilde = l_M/l_M_opt
>>> fl_M = FiberForceLengthPassiveDeGroote2016.with_defaults(l_M_tilde)
>>> fl_M
FiberForceLengthPassiveDeGroote2016(l_M/l_M_opt, 0.6, 4.0)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> fl_M.doit(evaluate=False)
0.0186573603637741*(-1 + exp(6.66666666666667*(l_M/l_M_opt - 1)))

The function can also be differentiated. We’ll differentiate with respect to l_M using the diff method on an instance with the single positional argument l_M.

>>> fl_M.diff(l_M)
0.12438240242516*exp(6.66666666666667*(l_M/l_M_opt - 1))/l_M_opt

References

[R707] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of l_T_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(l_M_tilde, c0, c1)[source]#

Evaluation of basic inputs.

Parameters:

l_M_tilde : Any (sympifiable)

Normalized muscle fiber length.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is 0.6.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is 4.0.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(l_M_tilde)[source]#

Recommended constructor that will use the published constants.

Parameters:

l_M_tilde : Any (sympifiable)

Normalized muscle fiber length.

Explanation

Returns a new instance of the muscle fiber passive force-length function using the four constant values specified in the original publication.

These have the values:

\(c_0 = 0.6\) \(c_1 = 4.0\)

class sympy.physics.biomechanics.curve.FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, c0, c1)[source]#

Inverse passive muscle fiber force-length curve based on De Groote et al., 2016 [R708].

Explanation

Gives the normalized muscle fiber length that produces a specific normalized passive muscle fiber force.

The function is defined by the equation:

\({fl^M_{pas}}^{-1} = \frac{c_0 \log{\left(\exp{c_1} - 1\right)fl^M_pas + 1}}{c_1} + 1\)

with constant values of \(c_0 = 0.6\) and \(c_1 = 4.0\). This function is the exact analytical inverse of the related tendon force-length curve FiberForceLengthPassiveDeGroote2016.

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces a passive fiber force very close to 0 for all normalized fiber lengths between 0 and 1.

Examples

The preferred way to instantiate FiberForceLengthPassiveInverseDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to the normalized passive muscle fiber length-force component of the muscle fiber force. We’ll create a Symbol called fl_M_pas to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import FiberForceLengthPassiveInverseDeGroote2016
>>> fl_M_pas = Symbol('fl_M_pas')
>>> l_M_tilde = FiberForceLengthPassiveInverseDeGroote2016.with_defaults(fl_M_pas)
>>> l_M_tilde
FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, 0.6, 4.0)

It’s also possible to populate the two constants with your own values too.

>>> from sympy import symbols
>>> c0, c1 = symbols('c0 c1')
>>> l_M_tilde = FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, c0, c1)
>>> l_M_tilde
FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, c0, c1)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> l_M_tilde.doit(evaluate=False)
c0*log(1 + fl_M_pas*(exp(c1) - 1))/c1 + 1

The function can also be differentiated. We’ll differentiate with respect to fl_M_pas using the diff method on an instance with the single positional argument fl_M_pas.

>>> l_M_tilde.diff(fl_M_pas)
c0*(exp(c1) - 1)/(c1*(fl_M_pas*(exp(c1) - 1) + 1))

References

[R708] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of l_T_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(fl_M_pas, c0, c1)[source]#

Evaluation of basic inputs.

Parameters:

fl_M_pas : Any (sympifiable)

Normalized passive muscle fiber force.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is 0.6.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is 4.0.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(fl_M_pas)[source]#

Recommended constructor that will use the published constants.

Parameters:

fl_M_pas : Any (sympifiable)

Normalized passive muscle fiber force as a function of muscle fiber length.

Explanation

Returns a new instance of the inverse muscle fiber passive force-length function using the four constant values specified in the original publication.

These have the values:

\(c_0 = 0.6\) \(c_1 = 4.0\)

class sympy.physics.biomechanics.curve.FiberForceVelocityDeGroote2016(v_M_tilde, c0, c1, c2, c3)[source]#

Muscle fiber force-velocity curve based on De Groote et al., 2016 [R709].

Explanation

Gives the normalized muscle fiber force produced as a function of normalized tendon velocity.

The function is defined by the equation:

\(fv^M = c_0 \log{\left(c_1 \tilde{v}_m + c_2\right) + \sqrt{\left(c_1 \tilde{v}_m + c_2\right)^2 + 1}} + c_3\)

with constant values of \(c_0 = -0.318\), \(c_1 = -8.149\), \(c_2 = -0.374\), and \(c_3 = 0.886\).

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces a normalized muscle fiber force of 1 when the muscle fibers are contracting isometrically (they have an extension rate of 0).

Examples

The preferred way to instantiate FiberForceVelocityDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized muscle fiber extension velocity. We’ll create a Symbol called v_M_tilde to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import FiberForceVelocityDeGroote2016
>>> v_M_tilde = Symbol('v_M_tilde')
>>> fv_M = FiberForceVelocityDeGroote2016.with_defaults(v_M_tilde)
>>> fv_M
FiberForceVelocityDeGroote2016(v_M_tilde, -0.318, -8.149, -0.374, 0.886)

It’s also possible to populate the four constants with your own values too.

>>> from sympy import symbols
>>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3')
>>> fv_M = FiberForceVelocityDeGroote2016(v_M_tilde, c0, c1, c2, c3)
>>> fv_M
FiberForceVelocityDeGroote2016(v_M_tilde, c0, c1, c2, c3)

You don’t just have to use symbols as the arguments, it’s also possible to use expressions. Let’s create a new pair of symbols, v_M and v_M_max, representing muscle fiber extension velocity and maximum muscle fiber extension velocity respectively. We can then represent v_M_tilde as an expression, the ratio of these.

>>> v_M, v_M_max = symbols('v_M v_M_max')
>>> v_M_tilde = v_M/v_M_max
>>> fv_M = FiberForceVelocityDeGroote2016.with_defaults(v_M_tilde)
>>> fv_M
FiberForceVelocityDeGroote2016(v_M/v_M_max, -0.318, -8.149, -0.374, 0.886)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> fv_M.doit(evaluate=False)
0.886 - 0.318*log(-8.149*v_M/v_M_max - 0.374 + sqrt(1 + (-8.149*v_M/v_M_max
- 0.374)**2))

The function can also be differentiated. We’ll differentiate with respect to v_M using the diff method on an instance with the single positional argument v_M.

>>> fv_M.diff(v_M)
2.591382*(1 + (-8.149*v_M/v_M_max - 0.374)**2)**(-1/2)/v_M_max

References

[R709] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of v_M_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(v_M_tilde, c0, c1, c2, c3)[source]#

Evaluation of basic inputs.

Parameters:

v_M_tilde : Any (sympifiable)

Normalized muscle fiber extension velocity.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is -0.318.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is -8.149.

c2 : Any (sympifiable)

The third constant in the characteristic equation. The published value is -0.374.

c3 : Any (sympifiable)

The fourth constant in the characteristic equation. The published value is 0.886.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(v_M_tilde)[source]#

Recommended constructor that will use the published constants.

Parameters:

v_M_tilde : Any (sympifiable)

Normalized muscle fiber extension velocity.

Explanation

Returns a new instance of the muscle fiber force-velocity function using the four constant values specified in the original publication.

These have the values:

\(c_0 = -0.318\) \(c_1 = -8.149\) \(c_2 = -0.374\) \(c_3 = 0.886\)

class sympy.physics.biomechanics.curve.FiberForceVelocityInverseDeGroote2016(fv_M, c0, c1, c2, c3)[source]#

Inverse muscle fiber force-velocity curve based on De Groote et al., 2016 [R710].

Explanation

Gives the normalized muscle fiber velocity that produces a specific normalized muscle fiber force.

The function is defined by the equation:

\({fv^M}^{-1} = \frac{\sinh{\frac{fv^M - c_3}{c_0}} - c_2}{c_1}\)

with constant values of \(c_0 = -0.318\), \(c_1 = -8.149\), \(c_2 = -0.374\), and \(c_3 = 0.886\). This function is the exact analytical inverse of the related muscle fiber force-velocity curve FiberForceVelocityDeGroote2016.

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces a normalized muscle fiber force of 1 when the muscle fibers are contracting isometrically (they have an extension rate of 0).

Examples

The preferred way to instantiate FiberForceVelocityInverseDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized muscle fiber force-velocity component of the muscle fiber force. We’ll create a Symbol called fv_M to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import FiberForceVelocityInverseDeGroote2016
>>> fv_M = Symbol('fv_M')
>>> v_M_tilde = FiberForceVelocityInverseDeGroote2016.with_defaults(fv_M)
>>> v_M_tilde
FiberForceVelocityInverseDeGroote2016(fv_M, -0.318, -8.149, -0.374, 0.886)

It’s also possible to populate the four constants with your own values too.

>>> from sympy import symbols
>>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3')
>>> v_M_tilde = FiberForceVelocityInverseDeGroote2016(fv_M, c0, c1, c2, c3)
>>> v_M_tilde
FiberForceVelocityInverseDeGroote2016(fv_M, c0, c1, c2, c3)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> v_M_tilde.doit(evaluate=False)
(-c2 + sinh((-c3 + fv_M)/c0))/c1

The function can also be differentiated. We’ll differentiate with respect to fv_M using the diff method on an instance with the single positional argument fv_M.

>>> v_M_tilde.diff(fv_M)
cosh((-c3 + fv_M)/c0)/(c0*c1)

References

[R710] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of fv_M that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(fv_M, c0, c1, c2, c3)[source]#

Evaluation of basic inputs.

Parameters:

fv_M : Any (sympifiable)

Normalized muscle fiber force as a function of muscle fiber extension velocity.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is -0.318.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is -8.149.

c2 : Any (sympifiable)

The third constant in the characteristic equation. The published value is -0.374.

c3 : Any (sympifiable)

The fourth constant in the characteristic equation. The published value is 0.886.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(fv_M)[source]#

Recommended constructor that will use the published constants.

Parameters:

fv_M : Any (sympifiable)

Normalized muscle fiber extension velocity.

Explanation

Returns a new instance of the inverse muscle fiber force-velocity function using the four constant values specified in the original publication.

These have the values:

\(c_0 = -0.318\) \(c_1 = -8.149\) \(c_2 = -0.374\) \(c_3 = 0.886\)

class sympy.physics.biomechanics.curve.TendonForceLengthDeGroote2016(l_T_tilde, c0, c1, c2, c3)[source]#

Tendon force-length curve based on De Groote et al., 2016 [R711].

Explanation

Gives the normalized tendon force produced as a function of normalized tendon length.

The function is defined by the equation:

\(fl^T = c_0 \exp{c_3 \left( \tilde{l}^T - c_1 \right)} - c_2\)

with constant values of \(c_0 = 0.2\), \(c_1 = 0.995\), \(c_2 = 0.25\), and \(c_3 = 33.93669377311689\).

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces no force when the tendon is in an unstrained state. It also produces a force of 1 normalized unit when the tendon is under a 5% strain.

Examples

The preferred way to instantiate TendonForceLengthDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized tendon length. We’ll create a Symbol called l_T_tilde to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import TendonForceLengthDeGroote2016
>>> l_T_tilde = Symbol('l_T_tilde')
>>> fl_T = TendonForceLengthDeGroote2016.with_defaults(l_T_tilde)
>>> fl_T
TendonForceLengthDeGroote2016(l_T_tilde, 0.2, 0.995, 0.25,
33.93669377311689)

It’s also possible to populate the four constants with your own values too.

>>> from sympy import symbols
>>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3')
>>> fl_T = TendonForceLengthDeGroote2016(l_T_tilde, c0, c1, c2, c3)
>>> fl_T
TendonForceLengthDeGroote2016(l_T_tilde, c0, c1, c2, c3)

You don’t just have to use symbols as the arguments, it’s also possible to use expressions. Let’s create a new pair of symbols, l_T and l_T_slack, representing tendon length and tendon slack length respectively. We can then represent l_T_tilde as an expression, the ratio of these.

>>> l_T, l_T_slack = symbols('l_T l_T_slack')
>>> l_T_tilde = l_T/l_T_slack
>>> fl_T = TendonForceLengthDeGroote2016.with_defaults(l_T_tilde)
>>> fl_T
TendonForceLengthDeGroote2016(l_T/l_T_slack, 0.2, 0.995, 0.25,
33.93669377311689)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> fl_T.doit(evaluate=False)
-0.25 + 0.2*exp(33.93669377311689*(l_T/l_T_slack - 0.995))

The function can also be differentiated. We’ll differentiate with respect to l_T using the diff method on an instance with the single positional argument l_T.

>>> fl_T.diff(l_T)
6.787338754623378*exp(33.93669377311689*(l_T/l_T_slack - 0.995))/l_T_slack

References

[R711] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of l_T_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(l_T_tilde, c0, c1, c2, c3)[source]#

Evaluation of basic inputs.

Parameters:

l_T_tilde : Any (sympifiable)

Normalized tendon length.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is 0.2.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is 0.995.

c2 : Any (sympifiable)

The third constant in the characteristic equation. The published value is 0.25.

c3 : Any (sympifiable)

The fourth constant in the characteristic equation. The published value is 33.93669377311689.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(l_T_tilde)[source]#

Recommended constructor that will use the published constants.

Parameters:

l_T_tilde : Any (sympifiable)

Normalized tendon length.

Explanation

Returns a new instance of the tendon force-length function using the four constant values specified in the original publication.

These have the values:

\(c_0 = 0.2\) \(c_1 = 0.995\) \(c_2 = 0.25\) \(c_3 = 33.93669377311689\)

class sympy.physics.biomechanics.curve.TendonForceLengthInverseDeGroote2016(fl_T, c0, c1, c2, c3)[source]#

Inverse tendon force-length curve based on De Groote et al., 2016 [R712].

Explanation

Gives the normalized tendon length that produces a specific normalized tendon force.

The function is defined by the equation:

\({fl^T}^{-1} = frac{\log{\frac{fl^T + c_2}{c_0}}}{c_3} + c_1\)

with constant values of \(c_0 = 0.2\), \(c_1 = 0.995\), \(c_2 = 0.25\), and \(c_3 = 33.93669377311689\). This function is the exact analytical inverse of the related tendon force-length curve TendonForceLengthDeGroote2016.

While it is possible to change the constant values, these were carefully selected in the original publication to give the characteristic curve specific and required properties. For example, the function produces no force when the tendon is in an unstrained state. It also produces a force of 1 normalized unit when the tendon is under a 5% strain.

Examples

The preferred way to instantiate TendonForceLengthInverseDeGroote2016 is using the with_defaults() constructor because this will automatically populate the constants within the characteristic curve equation with the floating point values from the original publication. This constructor takes a single argument corresponding to normalized tendon force-length, which is equal to the tendon force. We’ll create a Symbol called fl_T to represent this.

>>> from sympy import Symbol
>>> from sympy.physics.biomechanics import TendonForceLengthInverseDeGroote2016
>>> fl_T = Symbol('fl_T')
>>> l_T_tilde = TendonForceLengthInverseDeGroote2016.with_defaults(fl_T)
>>> l_T_tilde
TendonForceLengthInverseDeGroote2016(fl_T, 0.2, 0.995, 0.25,
33.93669377311689)

It’s also possible to populate the four constants with your own values too.

>>> from sympy import symbols
>>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3')
>>> l_T_tilde = TendonForceLengthInverseDeGroote2016(fl_T, c0, c1, c2, c3)
>>> l_T_tilde
TendonForceLengthInverseDeGroote2016(fl_T, c0, c1, c2, c3)

To inspect the actual symbolic expression that this function represents, we can call the doit() method on an instance. We’ll use the keyword argument evaluate=False as this will keep the expression in its canonical form and won’t simplify any constants.

>>> l_T_tilde.doit(evaluate=False)
c1 + log((c2 + fl_T)/c0)/c3

The function can also be differentiated. We’ll differentiate with respect to l_T using the diff method on an instance with the single positional argument l_T.

>>> l_T_tilde.diff(fl_T)
1/(c3*(c2 + fl_T))

References

[R712] (1,2)

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

doit(deep=True, evaluate=True, **hints)[source]#

Evaluate the expression defining the function.

Parameters:

deep : bool

Whether doit should be recursively called. Default is True.

evaluate : bool.

Whether the SymPy expression should be evaluated as it is constructed. If False, then no constant folding will be conducted which will leave the expression in a more numerically- stable for values of l_T_tilde that correspond to a sensible operating range for a musculotendon. Default is True.

**kwargs : dict[str, Any]

Additional keyword argument pairs to be recursively passed to doit.

classmethod eval(fl_T, c0, c1, c2, c3)[source]#

Evaluation of basic inputs.

Parameters:

fl_T : Any (sympifiable)

Normalized tendon force as a function of tendon length.

c0 : Any (sympifiable)

The first constant in the characteristic equation. The published value is 0.2.

c1 : Any (sympifiable)

The second constant in the characteristic equation. The published value is 0.995.

c2 : Any (sympifiable)

The third constant in the characteristic equation. The published value is 0.25.

c3 : Any (sympifiable)

The fourth constant in the characteristic equation. The published value is 33.93669377311689.

fdiff(argindex=1)[source]#

Derivative of the function with respect to a single argument.

Parameters:

argindex : int

The index of the function’s arguments with respect to which the derivative should be taken. Argument indexes start at 1. Default is 1.

inverse(argindex=1)[source]#

Inverse function.

Parameters:

argindex : int

Value to start indexing the arguments at. Default is 1.

classmethod with_defaults(fl_T)[source]#

Recommended constructor that will use the published constants.

Parameters:

fl_T : Any (sympifiable)

Normalized tendon force as a function of tendon length.

Explanation

Returns a new instance of the inverse tendon force-length function using the four constant values specified in the original publication.

These have the values:

\(c_0 = 0.2\) \(c_1 = 0.995\) \(c_2 = 0.25\) \(c_3 = 33.93669377311689\)