"""Implementations of actuators for linked force and torque application."""
from abc import ABC, abstractmethod
from sympy import S, sympify, exp, sign
from sympy.physics.mechanics.joint import PinJoint
from sympy.physics.mechanics.loads import Torque
from sympy.physics.mechanics.pathway import PathwayBase
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy.physics.vector import ReferenceFrame, Vector
__all__ = [
'ActuatorBase',
'ForceActuator',
'LinearDamper',
'LinearSpring',
'TorqueActuator',
'DuffingSpring',
'CoulombKineticFriction',
]
[docs]
class ActuatorBase(ABC):
"""Abstract base class for all actuator classes to inherit from.
Notes
=====
Instances of this class cannot be directly instantiated by users. However,
it can be used to created custom actuator types through subclassing.
"""
def __init__(self):
"""Initializer for ``ActuatorBase``."""
pass
[docs]
@abstractmethod
def to_loads(self):
"""Loads required by the equations of motion method classes.
Explanation
===========
``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be
passed to the ``loads`` parameters of its ``kanes_equations`` method
when constructing the equations of motion. This method acts as a
utility to produce the correctly-structred pairs of points and vectors
required so that these can be easily concatenated with other items in
the list of loads and passed to ``KanesMethod.kanes_equations``. These
loads are also in the correct form to also be passed to the other
equations of motion method classes, e.g. ``LagrangesMethod``.
"""
pass
def __repr__(self):
"""Default representation of an actuator."""
return f'{self.__class__.__name__}()'
[docs]
class ForceActuator(ActuatorBase):
"""Force-producing actuator.
Explanation
===========
A ``ForceActuator`` is an actuator that produces a (expansile) force along
its length.
A force actuator uses a pathway instance to determine the direction and
number of forces that it applies to a system. Consider the simplest case
where a ``LinearPathway`` instance is used. This pathway is made up of two
points that can move relative to each other, and results in a pair of equal
and opposite forces acting on the endpoints. If the positive time-varying
Euclidean distance between the two points is defined, then the "extension
velocity" is the time derivative of this distance. The extension velocity
is positive when the two points are moving away from each other and
negative when moving closer to each other. The direction for the force
acting on either point is determined by constructing a unit vector directed
from the other point to this point. This establishes a sign convention such
that a positive force magnitude tends to push the points apart, this is the
meaning of "expansile" in this context. The following diagram shows the
positive force sense and the distance between the points::
P Q
o<--- F --->o
| |
|<--l(t)--->|
Examples
========
To construct an actuator, an expression (or symbol) must be supplied to
represent the force it can produce, alongside a pathway specifying its line
of action. Let's also create a global reference frame and spatially fix one
of the points in it while setting the other to be positioned such that it
can freely move in the frame's x direction specified by the coordinate
``q``.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (ForceActuator, LinearPathway,
... Point, ReferenceFrame)
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> force = symbols('F')
>>> pA, pB = Point('pA'), Point('pB')
>>> pA.set_vel(N, 0)
>>> pB.set_pos(pA, q*N.x)
>>> pB.pos_from(pA)
q(t)*N.x
>>> linear_pathway = LinearPathway(pA, pB)
>>> actuator = ForceActuator(force, linear_pathway)
>>> actuator
ForceActuator(F, LinearPathway(pA, pB))
Parameters
==========
force : Expr
The scalar expression defining the (expansile) force that the actuator
produces.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of a
concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
"""
def __init__(self, force, pathway):
"""Initializer for ``ForceActuator``.
Parameters
==========
force : Expr
The scalar expression defining the (expansile) force that the
actuator produces.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of
a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
"""
self.force = force
self.pathway = pathway
@property
def force(self):
"""The magnitude of the force produced by the actuator."""
return self._force
@force.setter
def force(self, force):
if hasattr(self, '_force'):
msg = (
f'Can\'t set attribute `force` to {repr(force)} as it is '
f'immutable.'
)
raise AttributeError(msg)
self._force = sympify(force, strict=True)
@property
def pathway(self):
"""The ``Pathway`` defining the actuator's line of action."""
return self._pathway
@pathway.setter
def pathway(self, pathway):
if hasattr(self, '_pathway'):
msg = (
f'Can\'t set attribute `pathway` to {repr(pathway)} as it is '
f'immutable.'
)
raise AttributeError(msg)
if not isinstance(pathway, PathwayBase):
msg = (
f'Value {repr(pathway)} passed to `pathway` was of type '
f'{type(pathway)}, must be {PathwayBase}.'
)
raise TypeError(msg)
self._pathway = pathway
[docs]
def to_loads(self):
"""Loads required by the equations of motion method classes.
Explanation
===========
``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be
passed to the ``loads`` parameters of its ``kanes_equations`` method
when constructing the equations of motion. This method acts as a
utility to produce the correctly-structred pairs of points and vectors
required so that these can be easily concatenated with other items in
the list of loads and passed to ``KanesMethod.kanes_equations``. These
loads are also in the correct form to also be passed to the other
equations of motion method classes, e.g. ``LagrangesMethod``.
Examples
========
The below example shows how to generate the loads produced by a force
actuator that follows a linear pathway. In this example we'll assume
that the force actuator is being used to model a simple linear spring.
First, create a linear pathway between two points separated by the
coordinate ``q`` in the ``x`` direction of the global frame ``N``.
>>> from sympy.physics.mechanics import (LinearPathway, Point,
... ReferenceFrame)
>>> from sympy.physics.vector import dynamicsymbols
>>> q = dynamicsymbols('q')
>>> N = ReferenceFrame('N')
>>> pA, pB = Point('pA'), Point('pB')
>>> pB.set_pos(pA, q*N.x)
>>> pathway = LinearPathway(pA, pB)
Now create a symbol ``k`` to describe the spring's stiffness and
instantiate a force actuator that produces a (contractile) force
proportional to both the spring's stiffness and the pathway's length.
Note that actuator classes use the sign convention that expansile
forces are positive, so for a spring to produce a contractile force the
spring force needs to be calculated as the negative for the stiffness
multiplied by the length.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import ForceActuator
>>> stiffness = symbols('k')
>>> spring_force = -stiffness*pathway.length
>>> spring = ForceActuator(spring_force, pathway)
The forces produced by the spring can be generated in the list of loads
form that ``KanesMethod`` (and other equations of motion methods)
requires by calling the ``to_loads`` method.
>>> spring.to_loads()
[(pA, k*q(t)*N.x), (pB, - k*q(t)*N.x)]
A simple linear damper can be modeled in a similar way. Create another
symbol ``c`` to describe the dampers damping coefficient. This time
instantiate a force actuator that produces a force proportional to both
the damper's damping coefficient and the pathway's extension velocity.
Note that the damping force is negative as it acts in the opposite
direction to which the damper is changing in length.
>>> damping_coefficient = symbols('c')
>>> damping_force = -damping_coefficient*pathway.extension_velocity
>>> damper = ForceActuator(damping_force, pathway)
Again, the forces produces by the damper can be generated by calling
the ``to_loads`` method.
>>> damper.to_loads()
[(pA, c*Derivative(q(t), t)*N.x), (pB, - c*Derivative(q(t), t)*N.x)]
"""
return self.pathway.to_loads(self.force)
def __repr__(self):
"""Representation of a ``ForceActuator``."""
return f'{self.__class__.__name__}({self.force}, {self.pathway})'
[docs]
class LinearSpring(ForceActuator):
"""A spring with its spring force as a linear function of its length.
Explanation
===========
Note that the "linear" in the name ``LinearSpring`` refers to the fact that
the spring force is a linear function of the springs length. I.e. for a
linear spring with stiffness ``k``, distance between its ends of ``x``, and
an equilibrium length of ``0``, the spring force will be ``-k*x``, which is
a linear function in ``x``. To create a spring that follows a linear, or
straight, pathway between its two ends, a ``LinearPathway`` instance needs
to be passed to the ``pathway`` parameter.
A ``LinearSpring`` is a subclass of ``ForceActuator`` and so follows the
same sign conventions for length, extension velocity, and the direction of
the forces it applies to its points of attachment on bodies. The sign
convention for the direction of forces is such that, for the case where a
linear spring is instantiated with a ``LinearPathway`` instance as its
pathway, they act to push the two ends of the spring away from one another.
Because springs produces a contractile force and acts to pull the two ends
together towards the equilibrium length when stretched, the scalar portion
of the forces on the endpoint are negative in order to flip the sign of the
forces on the endpoints when converted into vector quantities. The
following diagram shows the positive force sense and the distance between
the points::
P Q
o<--- F --->o
| |
|<--l(t)--->|
Examples
========
To construct a linear spring, an expression (or symbol) must be supplied to
represent the stiffness (spring constant) of the spring, alongside a
pathway specifying its line of action. Let's also create a global reference
frame and spatially fix one of the points in it while setting the other to
be positioned such that it can freely move in the frame's x direction
specified by the coordinate ``q``.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (LinearPathway, LinearSpring,
... Point, ReferenceFrame)
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> stiffness = symbols('k')
>>> pA, pB = Point('pA'), Point('pB')
>>> pA.set_vel(N, 0)
>>> pB.set_pos(pA, q*N.x)
>>> pB.pos_from(pA)
q(t)*N.x
>>> linear_pathway = LinearPathway(pA, pB)
>>> spring = LinearSpring(stiffness, linear_pathway)
>>> spring
LinearSpring(k, LinearPathway(pA, pB))
This spring will produce a force that is proportional to both its stiffness
and the pathway's length. Note that this force is negative as SymPy's sign
convention for actuators is that negative forces are contractile.
>>> spring.force
-k*sqrt(q(t)**2)
To create a linear spring with a non-zero equilibrium length, an expression
(or symbol) can be passed to the ``equilibrium_length`` parameter on
construction on a ``LinearSpring`` instance. Let's create a symbol ``l``
to denote a non-zero equilibrium length and create another linear spring.
>>> l = symbols('l')
>>> spring = LinearSpring(stiffness, linear_pathway, equilibrium_length=l)
>>> spring
LinearSpring(k, LinearPathway(pA, pB), equilibrium_length=l)
The spring force of this new spring is again proportional to both its
stiffness and the pathway's length. However, the spring will not produce
any force when ``q(t)`` equals ``l``. Note that the force will become
expansile when ``q(t)`` is less than ``l``, as expected.
>>> spring.force
-k*(-l + sqrt(q(t)**2))
Parameters
==========
stiffness : Expr
The spring constant.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of a
concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
equilibrium_length : Expr, optional
The length at which the spring is in equilibrium, i.e. it produces no
force. The default value is 0, i.e. the spring force is a linear
function of the pathway's length with no constant offset.
See Also
========
ForceActuator: force-producing actuator (superclass of ``LinearSpring``).
LinearPathway: straight-line pathway between a pair of points.
"""
def __init__(self, stiffness, pathway, equilibrium_length=S.Zero):
"""Initializer for ``LinearSpring``.
Parameters
==========
stiffness : Expr
The spring constant.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of
a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
equilibrium_length : Expr, optional
The length at which the spring is in equilibrium, i.e. it produces
no force. The default value is 0, i.e. the spring force is a linear
function of the pathway's length with no constant offset.
"""
self.stiffness = stiffness
self.pathway = pathway
self.equilibrium_length = equilibrium_length
@property
def force(self):
"""The spring force produced by the linear spring."""
return -self.stiffness*(self.pathway.length - self.equilibrium_length)
@force.setter
def force(self, force):
raise AttributeError('Can\'t set computed attribute `force`.')
@property
def stiffness(self):
"""The spring constant for the linear spring."""
return self._stiffness
@stiffness.setter
def stiffness(self, stiffness):
if hasattr(self, '_stiffness'):
msg = (
f'Can\'t set attribute `stiffness` to {repr(stiffness)} as it '
f'is immutable.'
)
raise AttributeError(msg)
self._stiffness = sympify(stiffness, strict=True)
@property
def equilibrium_length(self):
"""The length of the spring at which it produces no force."""
return self._equilibrium_length
@equilibrium_length.setter
def equilibrium_length(self, equilibrium_length):
if hasattr(self, '_equilibrium_length'):
msg = (
f'Can\'t set attribute `equilibrium_length` to '
f'{repr(equilibrium_length)} as it is immutable.'
)
raise AttributeError(msg)
self._equilibrium_length = sympify(equilibrium_length, strict=True)
def __repr__(self):
"""Representation of a ``LinearSpring``."""
string = f'{self.__class__.__name__}({self.stiffness}, {self.pathway}'
if self.equilibrium_length == S.Zero:
string += ')'
else:
string += f', equilibrium_length={self.equilibrium_length})'
return string
[docs]
class LinearDamper(ForceActuator):
"""A damper whose force is a linear function of its extension velocity.
Explanation
===========
Note that the "linear" in the name ``LinearDamper`` refers to the fact that
the damping force is a linear function of the damper's rate of change in
its length. I.e. for a linear damper with damping ``c`` and extension
velocity ``v``, the damping force will be ``-c*v``, which is a linear
function in ``v``. To create a damper that follows a linear, or straight,
pathway between its two ends, a ``LinearPathway`` instance needs to be
passed to the ``pathway`` parameter.
A ``LinearDamper`` is a subclass of ``ForceActuator`` and so follows the
same sign conventions for length, extension velocity, and the direction of
the forces it applies to its points of attachment on bodies. The sign
convention for the direction of forces is such that, for the case where a
linear damper is instantiated with a ``LinearPathway`` instance as its
pathway, they act to push the two ends of the damper away from one another.
Because dampers produce a force that opposes the direction of change in
length, when extension velocity is positive the scalar portions of the
forces applied at the two endpoints are negative in order to flip the sign
of the forces on the endpoints wen converted into vector quantities. When
extension velocity is negative (i.e. when the damper is shortening), the
scalar portions of the fofces applied are also negative so that the signs
cancel producing forces on the endpoints that are in the same direction as
the positive sign convention for the forces at the endpoints of the pathway
(i.e. they act to push the endpoints away from one another). The following
diagram shows the positive force sense and the distance between the
points::
P Q
o<--- F --->o
| |
|<--l(t)--->|
Examples
========
To construct a linear damper, an expression (or symbol) must be supplied to
represent the damping coefficient of the damper (we'll use the symbol
``c``), alongside a pathway specifying its line of action. Let's also
create a global reference frame and spatially fix one of the points in it
while setting the other to be positioned such that it can freely move in
the frame's x direction specified by the coordinate ``q``. The velocity
that the two points move away from one another can be specified by the
coordinate ``u`` where ``u`` is the first time derivative of ``q``
(i.e., ``u = Derivative(q(t), t)``).
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (LinearDamper, LinearPathway,
... Point, ReferenceFrame)
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> damping = symbols('c')
>>> pA, pB = Point('pA'), Point('pB')
>>> pA.set_vel(N, 0)
>>> pB.set_pos(pA, q*N.x)
>>> pB.pos_from(pA)
q(t)*N.x
>>> pB.vel(N)
Derivative(q(t), t)*N.x
>>> linear_pathway = LinearPathway(pA, pB)
>>> damper = LinearDamper(damping, linear_pathway)
>>> damper
LinearDamper(c, LinearPathway(pA, pB))
This damper will produce a force that is proportional to both its damping
coefficient and the pathway's extension length. Note that this force is
negative as SymPy's sign convention for actuators is that negative forces
are contractile and the damping force of the damper will oppose the
direction of length change.
>>> damper.force
-c*sqrt(q(t)**2)*Derivative(q(t), t)/q(t)
Parameters
==========
damping : Expr
The damping constant.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of a
concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
See Also
========
ForceActuator: force-producing actuator (superclass of ``LinearDamper``).
LinearPathway: straight-line pathway between a pair of points.
"""
def __init__(self, damping, pathway):
"""Initializer for ``LinearDamper``.
Parameters
==========
damping : Expr
The damping constant.
pathway : PathwayBase
The pathway that the actuator follows. This must be an instance of
a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``.
"""
self.damping = damping
self.pathway = pathway
@property
def force(self):
"""The damping force produced by the linear damper."""
return -self.damping*self.pathway.extension_velocity
@force.setter
def force(self, force):
raise AttributeError('Can\'t set computed attribute `force`.')
@property
def damping(self):
"""The damping constant for the linear damper."""
return self._damping
@damping.setter
def damping(self, damping):
if hasattr(self, '_damping'):
msg = (
f'Can\'t set attribute `damping` to {repr(damping)} as it is '
f'immutable.'
)
raise AttributeError(msg)
self._damping = sympify(damping, strict=True)
def __repr__(self):
"""Representation of a ``LinearDamper``."""
return f'{self.__class__.__name__}({self.damping}, {self.pathway})'
[docs]
class TorqueActuator(ActuatorBase):
"""Torque-producing actuator.
Explanation
===========
A ``TorqueActuator`` is an actuator that produces a pair of equal and
opposite torques on a pair of bodies.
Examples
========
To construct a torque actuator, an expression (or symbol) must be supplied
to represent the torque it can produce, alongside a vector specifying the
axis about which the torque will act, and a pair of frames on which the
torque will act.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (ReferenceFrame, RigidBody,
... TorqueActuator)
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> torque = symbols('T')
>>> axis = N.z
>>> parent = RigidBody('parent', frame=N)
>>> child = RigidBody('child', frame=A)
>>> bodies = (child, parent)
>>> actuator = TorqueActuator(torque, axis, *bodies)
>>> actuator
TorqueActuator(T, axis=N.z, target_frame=A, reaction_frame=N)
Note that because torques actually act on frames, not bodies,
``TorqueActuator`` will extract the frame associated with a ``RigidBody``
when one is passed instead of a ``ReferenceFrame``.
Parameters
==========
torque : Expr
The scalar expression defining the torque that the actuator produces.
axis : Vector
The axis about which the actuator applies torques.
target_frame : ReferenceFrame | RigidBody
The primary frame on which the actuator will apply the torque.
reaction_frame : ReferenceFrame | RigidBody | None
The secondary frame on which the actuator will apply the torque. Note
that the (equal and opposite) reaction torque is applied to this frame.
"""
def __init__(self, torque, axis, target_frame, reaction_frame=None):
"""Initializer for ``TorqueActuator``.
Parameters
==========
torque : Expr
The scalar expression defining the torque that the actuator
produces.
axis : Vector
The axis about which the actuator applies torques.
target_frame : ReferenceFrame | RigidBody
The primary frame on which the actuator will apply the torque.
reaction_frame : ReferenceFrame | RigidBody | None
The secondary frame on which the actuator will apply the torque.
Note that the (equal and opposite) reaction torque is applied to
this frame.
"""
self.torque = torque
self.axis = axis
self.target_frame = target_frame
self.reaction_frame = reaction_frame
[docs]
@classmethod
def at_pin_joint(cls, torque, pin_joint):
"""Alternate construtor to instantiate from a ``PinJoint`` instance.
Examples
========
To create a pin joint the ``PinJoint`` class requires a name, parent
body, and child body to be passed to its constructor. It is also
possible to control the joint axis using the ``joint_axis`` keyword
argument. In this example let's use the parent body's reference frame's
z-axis as the joint axis.
>>> from sympy.physics.mechanics import (PinJoint, ReferenceFrame,
... RigidBody, TorqueActuator)
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> parent = RigidBody('parent', frame=N)
>>> child = RigidBody('child', frame=A)
>>> pin_joint = PinJoint(
... 'pin',
... parent,
... child,
... joint_axis=N.z,
... )
Let's also create a symbol ``T`` that will represent the torque applied
by the torque actuator.
>>> from sympy import symbols
>>> torque = symbols('T')
To create the torque actuator from the ``torque`` and ``pin_joint``
variables previously instantiated, these can be passed to the alternate
constructor class method ``at_pin_joint`` of the ``TorqueActuator``
class. It should be noted that a positive torque will cause a positive
displacement of the joint coordinate or that the torque is applied on
the child body with a reaction torque on the parent.
>>> actuator = TorqueActuator.at_pin_joint(torque, pin_joint)
>>> actuator
TorqueActuator(T, axis=N.z, target_frame=A, reaction_frame=N)
Parameters
==========
torque : Expr
The scalar expression defining the torque that the actuator
produces.
pin_joint : PinJoint
The pin joint, and by association the parent and child bodies, on
which the torque actuator will act. The pair of bodies acted upon
by the torque actuator are the parent and child bodies of the pin
joint, with the child acting as the reaction body. The pin joint's
axis is used as the axis about which the torque actuator will apply
its torque.
"""
if not isinstance(pin_joint, PinJoint):
msg = (
f'Value {repr(pin_joint)} passed to `pin_joint` was of type '
f'{type(pin_joint)}, must be {PinJoint}.'
)
raise TypeError(msg)
return cls(
torque,
pin_joint.joint_axis,
pin_joint.child_interframe,
pin_joint.parent_interframe,
)
@property
def torque(self):
"""The magnitude of the torque produced by the actuator."""
return self._torque
@torque.setter
def torque(self, torque):
if hasattr(self, '_torque'):
msg = (
f'Can\'t set attribute `torque` to {repr(torque)} as it is '
f'immutable.'
)
raise AttributeError(msg)
self._torque = sympify(torque, strict=True)
@property
def axis(self):
"""The axis about which the torque acts."""
return self._axis
@axis.setter
def axis(self, axis):
if hasattr(self, '_axis'):
msg = (
f'Can\'t set attribute `axis` to {repr(axis)} as it is '
f'immutable.'
)
raise AttributeError(msg)
if not isinstance(axis, Vector):
msg = (
f'Value {repr(axis)} passed to `axis` was of type '
f'{type(axis)}, must be {Vector}.'
)
raise TypeError(msg)
self._axis = axis
@property
def target_frame(self):
"""The primary reference frames on which the torque will act."""
return self._target_frame
@target_frame.setter
def target_frame(self, target_frame):
if hasattr(self, '_target_frame'):
msg = (
f'Can\'t set attribute `target_frame` to {repr(target_frame)} '
f'as it is immutable.'
)
raise AttributeError(msg)
if isinstance(target_frame, RigidBody):
target_frame = target_frame.frame
elif not isinstance(target_frame, ReferenceFrame):
msg = (
f'Value {repr(target_frame)} passed to `target_frame` was of '
f'type {type(target_frame)}, must be {ReferenceFrame}.'
)
raise TypeError(msg)
self._target_frame = target_frame
@property
def reaction_frame(self):
"""The primary reference frames on which the torque will act."""
return self._reaction_frame
@reaction_frame.setter
def reaction_frame(self, reaction_frame):
if hasattr(self, '_reaction_frame'):
msg = (
f'Can\'t set attribute `reaction_frame` to '
f'{repr(reaction_frame)} as it is immutable.'
)
raise AttributeError(msg)
if isinstance(reaction_frame, RigidBody):
reaction_frame = reaction_frame.frame
elif (
not isinstance(reaction_frame, ReferenceFrame)
and reaction_frame is not None
):
msg = (
f'Value {repr(reaction_frame)} passed to `reaction_frame` was '
f'of type {type(reaction_frame)}, must be {ReferenceFrame}.'
)
raise TypeError(msg)
self._reaction_frame = reaction_frame
[docs]
def to_loads(self):
"""Loads required by the equations of motion method classes.
Explanation
===========
``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be
passed to the ``loads`` parameters of its ``kanes_equations`` method
when constructing the equations of motion. This method acts as a
utility to produce the correctly-structred pairs of points and vectors
required so that these can be easily concatenated with other items in
the list of loads and passed to ``KanesMethod.kanes_equations``. These
loads are also in the correct form to also be passed to the other
equations of motion method classes, e.g. ``LagrangesMethod``.
Examples
========
The below example shows how to generate the loads produced by a torque
actuator that acts on a pair of bodies attached by a pin joint.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (PinJoint, ReferenceFrame,
... RigidBody, TorqueActuator)
>>> torque = symbols('T')
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> parent = RigidBody('parent', frame=N)
>>> child = RigidBody('child', frame=A)
>>> pin_joint = PinJoint(
... 'pin',
... parent,
... child,
... joint_axis=N.z,
... )
>>> actuator = TorqueActuator.at_pin_joint(torque, pin_joint)
The forces produces by the damper can be generated by calling the
``to_loads`` method.
>>> actuator.to_loads()
[(A, T*N.z), (N, - T*N.z)]
Alternatively, if a torque actuator is created without a reaction frame
then the loads returned by the ``to_loads`` method will contain just
the single load acting on the target frame.
>>> actuator = TorqueActuator(torque, N.z, N)
>>> actuator.to_loads()
[(N, T*N.z)]
"""
loads = [
Torque(self.target_frame, self.torque*self.axis),
]
if self.reaction_frame is not None:
loads.append(Torque(self.reaction_frame, -self.torque*self.axis))
return loads
def __repr__(self):
"""Representation of a ``TorqueActuator``."""
string = (
f'{self.__class__.__name__}({self.torque}, axis={self.axis}, '
f'target_frame={self.target_frame}'
)
if self.reaction_frame is not None:
string += f', reaction_frame={self.reaction_frame})'
else:
string += ')'
return string
[docs]
class DuffingSpring(ForceActuator):
"""A nonlinear spring based on the Duffing equation.
Explanation
===========
Here, ``DuffingSpring`` represents the force exerted by a nonlinear spring based on the Duffing equation:
F = -beta*x-alpha*x**3, where x is the displacement from the equilibrium position, beta is the linear spring constant,
and alpha is the coefficient for the nonlinear cubic term.
Parameters
==========
linear_stiffness : Expr
The linear stiffness coefficient (beta).
nonlinear_stiffness : Expr
The nonlinear stiffness coefficient (alpha).
pathway : PathwayBase
The pathway that the actuator follows.
equilibrium_length : Expr, optional
The length at which the spring is in equilibrium (x).
"""
def __init__(self, linear_stiffness, nonlinear_stiffness, pathway, equilibrium_length=S.Zero):
self.linear_stiffness = sympify(linear_stiffness, strict=True)
self.nonlinear_stiffness = sympify(nonlinear_stiffness, strict=True)
self.equilibrium_length = sympify(equilibrium_length, strict=True)
if not isinstance(pathway, PathwayBase):
raise TypeError("pathway must be an instance of PathwayBase.")
self._pathway = pathway
@property
def linear_stiffness(self):
return self._linear_stiffness
@linear_stiffness.setter
def linear_stiffness(self, linear_stiffness):
if hasattr(self, '_linear_stiffness'):
msg = (
f'Can\'t set attribute `linear_stiffness` to '
f'{repr(linear_stiffness)} as it is immutable.'
)
raise AttributeError(msg)
self._linear_stiffness = sympify(linear_stiffness, strict=True)
@property
def nonlinear_stiffness(self):
return self._nonlinear_stiffness
@nonlinear_stiffness.setter
def nonlinear_stiffness(self, nonlinear_stiffness):
if hasattr(self, '_nonlinear_stiffness'):
msg = (
f'Can\'t set attribute `nonlinear_stiffness` to '
f'{repr(nonlinear_stiffness)} as it is immutable.'
)
raise AttributeError(msg)
self._nonlinear_stiffness = sympify(nonlinear_stiffness, strict=True)
@property
def pathway(self):
return self._pathway
@pathway.setter
def pathway(self, pathway):
if hasattr(self, '_pathway'):
msg = (
f'Can\'t set attribute `pathway` to {repr(pathway)} as it is '
f'immutable.'
)
raise AttributeError(msg)
if not isinstance(pathway, PathwayBase):
msg = (
f'Value {repr(pathway)} passed to `pathway` was of type '
f'{type(pathway)}, must be {PathwayBase}.'
)
raise TypeError(msg)
self._pathway = pathway
@property
def equilibrium_length(self):
return self._equilibrium_length
@equilibrium_length.setter
def equilibrium_length(self, equilibrium_length):
if hasattr(self, '_equilibrium_length'):
msg = (
f'Can\'t set attribute `equilibrium_length` to '
f'{repr(equilibrium_length)} as it is immutable.'
)
raise AttributeError(msg)
self._equilibrium_length = sympify(equilibrium_length, strict=True)
@property
def force(self):
"""The force produced by the Duffing spring."""
displacement = self.pathway.length - self.equilibrium_length
return -self.linear_stiffness * displacement - self.nonlinear_stiffness * displacement**3
@force.setter
def force(self, force):
if hasattr(self, '_force'):
msg = (
f'Can\'t set attribute `force` to {repr(force)} as it is '
f'immutable.'
)
raise AttributeError(msg)
self._force = sympify(force, strict=True)
def __repr__(self):
return (f"{self.__class__.__name__}("
f"{self.linear_stiffness}, {self.nonlinear_stiffness}, {self.pathway}, "
f"equilibrium_length={self.equilibrium_length})")
[docs]
class CoulombKineticFriction(ForceActuator):
r"""Coulomb kinetic friction with Stribeck and viscous effects.
Explanation
===========
This represents a Coulomb kinetic friction with the Stribeck and viscous effect,
described by the function:
.. math::
F = (\mu_k f_n + (\mu_s - \mu_k) f_n e^{-(\frac{v}{v_s})^2}) \text{sign}(v) + \sigma v
where :math:`\mu_k` is the coefficient of kinetic friction, :math:`\mu_s` is the
coefficient of static friction, :math:`f_n` is the normal force, :math:`v` is the
relative velocity, :math:`v_s` is the Stribeck friction coefficient, and
:math:`\sigma` is the viscous friction constant.
The default friction force is :math:`F = \mu_k f_n`.
When specified, the actuator includes:
- Stribeck effect: :math:`(\mu_s - \mu_k) f_n e^{-(\frac{v}{v_s})^2}`
- Viscous effect: :math:`\sigma v`
Notes
=====
The actuator makes the following assumptions:
- The actuator assumes relative motion is non-zero.
- The normal force is assumed to be a non-negative scalar.
- The resultant friction force is opposite to the velocity direction.
- Each point in the pathway is fixed within separate objects that are sliding relative to each other. In other words, these two points are fixed in the mutually sliding objects.
This actuator has been tested for straightforward motions, like a block sliding
on a surface.
The friction force is defined to always oppose the direction of relative velocity :math:`v`.
Specifically:
- The default Coulomb friction force :math:`\mu_k f_n \text{sign}(v)` is opposite to :math:`v`.
- The Stribeck effect :math:`(\mu_s - \mu_k) f_n e^{-(\frac{v}{v_s})^2} \text{sign}(v)` is also opposite to :math:`v`.
- The viscous friction term :math:`\sigma v` is opposite to :math:`v`.
Examples
========
The below example shows how to generate the loads produced by a Coulomb kinetic
friction actuator in a mass-spring system with friction.
>>> import sympy as sm
>>> from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point,
... LinearPathway, CoulombKineticFriction, LinearSpring, KanesMethod, Particle)
>>> x, v = dynamicsymbols('x, v', real=True)
>>> m, g, k, mu_k, mu_s, v_s, sigma = sm.symbols('m, g, k, mu_k, mu_s, v_s, sigma')
>>> N = ReferenceFrame('N')
>>> O, P = Point('O'), Point('P')
>>> O.set_vel(N, 0)
>>> P.set_pos(O, x*N.x)
>>> pathway = LinearPathway(O, P)
>>> friction = CoulombKineticFriction(mu_k, m*g, pathway, v_s=v_s, sigma=sigma, mu_s=mu_k)
>>> spring = LinearSpring(k, pathway)
>>> block = Particle('block', point=P, mass=m)
>>> kane = KanesMethod(N, (x,), (v,), kd_eqs=(x.diff() - v,))
>>> friction.to_loads()
[(O, (g*m*mu_k*sign(sign(x(t))*Derivative(x(t), t)) + sigma*sign(x(t))*Derivative(x(t), t))*x(t)/Abs(x(t))*N.x), (P, (-g*m*mu_k*sign(sign(x(t))*Derivative(x(t), t)) - sigma*sign(x(t))*Derivative(x(t), t))*x(t)/Abs(x(t))*N.x)]
>>> loads = friction.to_loads() + spring.to_loads()
>>> fr, frstar = kane.kanes_equations([block], loads)
>>> eom = fr + frstar
>>> eom
Matrix([[-k*x(t) - m*Derivative(v(t), t) + (-g*m*mu_k*sign(v(t)*sign(x(t))) - sigma*v(t)*sign(x(t)))*x(t)/Abs(x(t))]])
Parameters
==========
f_n : sympifiable
The normal force between the surfaces. It should always be a non-negative scalar.
mu_k : sympifiable
The coefficient of kinetic friction.
pathway : PathwayBase
The pathway that the actuator follows.
v_s : sympifiable, optional
The Stribeck friction coefficient.
sigma : sympifiable, optional
The viscous friction coefficient.
mu_s : sympifiable, optional
The coefficient of static friction. Defaults to mu_k, meaning the Stribeck effect evaluates to 0 by default.
References
==========
.. [Moore2022] https://moorepants.github.io/learn-multibody-dynamics/loads.html#friction.
.. [Flores2023] Paulo Flores, Jorge Ambrosio, Hamid M. Lankarani,
"Contact-impact events with friction in multibody dynamics: Back to basics",
Mechanism and Machine Theory, vol. 184, 2023. https://doi.org/10.1016/j.mechmachtheory.2023.105305.
.. [Rogner2017] I. Rogner, "Friction modelling for robotic applications with planar motion",
Chalmers University of Technology, Department of Electrical Engineering, 2017.
"""
def __init__(self, mu_k, f_n, pathway, *, v_s=None, sigma=None, mu_s=None):
self._mu_k = sympify(mu_k, strict=True) if mu_k is not None else 1
self._mu_s = sympify(mu_s, strict=True) if mu_s is not None else self._mu_k
self._f_n = sympify(f_n, strict=True)
self._sigma = sympify(sigma, strict=True) if sigma is not None else 0
self._v_s = sympify(v_s, strict=True) if v_s is not None or v_s == 0 else 0.01
self.pathway = pathway
@property
def mu_k(self):
"""The coefficient of kinetic friction."""
return self._mu_k
@property
def mu_s(self):
"""The coefficient of static friction."""
return self._mu_s
@property
def f_n(self):
"""The normal force between the surfaces."""
return self._f_n
@property
def sigma(self):
"""The viscous friction coefficient."""
return self._sigma
@property
def v_s(self):
"""The Stribeck friction coefficient."""
return self._v_s
@property
def force(self):
v = self.pathway.extension_velocity
f_c = self.mu_k * self.f_n
f_max = self.mu_s * self.f_n
stribeck_term = (f_max - f_c) * exp(-(v / self.v_s)**2) if self.v_s is not None else 0
viscous_term = self.sigma * v if self.sigma is not None else 0
return (f_c + stribeck_term) * -sign(v) - viscous_term
@force.setter
def force(self, force):
raise AttributeError('Can\'t set computed attribute `force`.')
def __repr__(self):
return (f'{self.__class__.__name__}({self._mu_k}, {self._mu_s} '
f'{self._f_n}, {self.pathway}, {self._v_s}, '
f'{self._sigma})')