Pathway (Docstrings)¶
Implementations of pathways for use by actuators.
- class sympy.physics.mechanics.pathway.LinearPathway(*attachments)[source]¶
Linear pathway between a pair of attachment points.
- Parameters:
attachments : tuple[Point, Point]
Pair of
Point
objects between which the linear pathway spans. Constructor expects two points to be passed, e.g.LinearPathway(Point('pA'), Point('pB'))
. More or fewer points will cause an error to be thrown.
Explanation
A linear pathway forms a straight-line segment between two points and is the simplest pathway that can be formed. It will not interact with any other objects in the system, i.e. a
LinearPathway
will intersect other objects to ensure that the path between its two ends (its attachments) is the shortest possible.A linear pathway is made up of two points that can move relative to each other, and a pair of equal and opposite forces acting on the points. 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. The following diagram shows the positive force sense and the distance between the points:
P Q o<--- F --->o | | |<--l(t)--->|
Examples
>>> from sympy.physics.mechanics import LinearPathway
To construct a pathway, two points are required to be passed to the
attachments
parameter as atuple
.>>> from sympy.physics.mechanics import Point >>> pA, pB = Point('pA'), Point('pB') >>> linear_pathway = LinearPathway(pA, pB) >>> linear_pathway LinearPathway(pA, pB)
The pathway created above isn’t very interesting without the positions and velocities of its attachment points being described. Without this its not possible to describe how the pathway moves, i.e. its length or its extension velocity.
>>> from sympy.physics.mechanics import ReferenceFrame >>> from sympy.physics.vector import dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> pB.set_pos(pA, q*N.x) >>> pB.pos_from(pA) q(t)*N.x
A pathway’s length can be accessed via its
length
attribute.>>> linear_pathway.length sqrt(q(t)**2)
Note how what appears to be an overly-complex expression is returned. This is actually required as it ensures that a pathway’s length is always positive.
A pathway’s extension velocity can be accessed similarly via its
extension_velocity
attribute.>>> linear_pathway.extension_velocity sqrt(q(t)**2)*Derivative(q(t), t)/q(t)
- property extension_velocity¶
Exact analytical expression for the pathway’s extension velocity.
- property length¶
Exact analytical expression for the pathway’s length.
- to_loads(force)[source]¶
Loads required by the equations of motion method classes.
- Parameters:
force : Expr
Magnitude of the force acting along the length of the pathway. As per the sign conventions for the pathway length, pathway extension velocity, and pair of point forces, if this
Expr
is positive then the force will act to push the pair of points away from one another (it is expansile).
Explanation
KanesMethod
requires a list ofPoint
-Vector
tuples to be passed to theloads
parameters of itskanes_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 toKanesMethod.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 in a linear actuator that produces an expansile force
F
. First, create a linear actuator between two points separated by the coordinateq
in thex
direction of the global frameN
.>>> 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) >>> linear_pathway = LinearPathway(pA, pB)
Now create a symbol
F
to describe the magnitude of the (expansile) force that will be produced along the pathway. The list of loads thatKanesMethod
requires can be produced by calling the pathway’sto_loads
method withF
passed as the only argument.>>> from sympy import symbols >>> F = symbols('F') >>> linear_pathway.to_loads(F) [(pA, - F*q(t)/sqrt(q(t)**2)*N.x), (pB, F*q(t)/sqrt(q(t)**2)*N.x)]
- class sympy.physics.mechanics.pathway.ObstacleSetPathway(*attachments)[source]¶
Obstacle-set pathway between a set of attachment points.
- Parameters:
attachments : tuple[Point, Point]
The set of
Point
objects that define the segmented obstacle-set pathway.
Explanation
An obstacle-set pathway forms a series of straight-line segment between pairs of consecutive points in a set of points. It is similiar to multiple linear pathways joined end-to-end. It will not interact with any other objects in the system, i.e. an
ObstacleSetPathway
will intersect other objects to ensure that the path between its pairs of points (its attachments) is the shortest possible.Examples
To construct an obstacle-set pathway, three or more points are required to be passed to the
attachments
parameter as atuple
.>>> from sympy.physics.mechanics import ObstacleSetPathway, Point >>> pA, pB, pC, pD = Point('pA'), Point('pB'), Point('pC'), Point('pD') >>> obstacle_set_pathway = ObstacleSetPathway(pA, pB, pC, pD) >>> obstacle_set_pathway ObstacleSetPathway(pA, pB, pC, pD)
The pathway created above isn’t very interesting without the positions and velocities of its attachment points being described. Without this its not possible to describe how the pathway moves, i.e. its length or its extension velocity.
>>> from sympy import cos, sin >>> from sympy.physics.mechanics import ReferenceFrame >>> from sympy.physics.vector import dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> pO = Point('pO') >>> pA.set_pos(pO, N.y) >>> pB.set_pos(pO, -N.x) >>> pC.set_pos(pA, cos(q) * N.x - (sin(q) + 1) * N.y) >>> pD.set_pos(pA, sin(q) * N.x + (cos(q) - 1) * N.y) >>> pB.pos_from(pA) - N.x - N.y >>> pC.pos_from(pA) cos(q(t))*N.x + (-sin(q(t)) - 1)*N.y >>> pD.pos_from(pA) sin(q(t))*N.x + (cos(q(t)) - 1)*N.y
A pathway’s length can be accessed via its
length
attribute.>>> obstacle_set_pathway.length.simplify() sqrt(2)*(sqrt(cos(q(t)) + 1) + 2)
A pathway’s extension velocity can be accessed similarly via its
extension_velocity
attribute.>>> obstacle_set_pathway.extension_velocity.simplify() -sqrt(2)*sin(q(t))*Derivative(q(t), t)/(2*sqrt(cos(q(t)) + 1))
- property attachments¶
The set of points defining a pathway’s segmented path.
- property extension_velocity¶
Exact analytical expression for the pathway’s extension velocity.
- property length¶
Exact analytical expression for the pathway’s length.
- to_loads(force)[source]¶
Loads required by the equations of motion method classes.
- Parameters:
force : Expr
The force acting along the length of the pathway. It is assumed that this
Expr
represents an expansile force.
Explanation
KanesMethod
requires a list ofPoint
-Vector
tuples to be passed to theloads
parameters of itskanes_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 toKanesMethod.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 in an actuator that follows an obstacle-set pathway between four points and produces an expansile force
F
. First, create a pair of reference frames,A
andB
, in which the four pointspA
,pB
,pC
, andpD
will be located. The first two points in frameA
and the second two in frameB
. FrameB
will also be oriented such that it relates toA
via a rotation ofq
about an axisN.z
in a global frame (N.z
,A.z
, andB.z
are parallel).>>> from sympy.physics.mechanics import (ObstacleSetPathway, Point, ... ReferenceFrame) >>> from sympy.physics.vector import dynamicsymbols >>> q = dynamicsymbols('q') >>> N = ReferenceFrame('N') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'axis', (0, N.x)) >>> B = A.orientnew('B', 'axis', (q, N.z)) >>> pO = Point('pO') >>> pA, pB, pC, pD = Point('pA'), Point('pB'), Point('pC'), Point('pD') >>> pA.set_pos(pO, A.x) >>> pB.set_pos(pO, -A.y) >>> pC.set_pos(pO, B.y) >>> pD.set_pos(pO, B.x) >>> obstacle_set_pathway = ObstacleSetPathway(pA, pB, pC, pD)
Now create a symbol
F
to describe the magnitude of the (expansile) force that will be produced along the pathway. The list of loads thatKanesMethod
requires can be produced by calling the pathway’sto_loads
method withF
passed as the only argument.>>> from sympy import Symbol >>> F = Symbol('F') >>> obstacle_set_pathway.to_loads(F) [(pA, sqrt(2)*F/2*A.x + sqrt(2)*F/2*A.y), (pB, - sqrt(2)*F/2*A.x - sqrt(2)*F/2*A.y), (pB, - F/sqrt(2*cos(q(t)) + 2)*A.y - F/sqrt(2*cos(q(t)) + 2)*B.y), (pC, F/sqrt(2*cos(q(t)) + 2)*A.y + F/sqrt(2*cos(q(t)) + 2)*B.y), (pC, - sqrt(2)*F/2*B.x + sqrt(2)*F/2*B.y), (pD, sqrt(2)*F/2*B.x - sqrt(2)*F/2*B.y)]
- class sympy.physics.mechanics.pathway.PathwayBase(*attachments)[source]¶
Abstract base class for all pathway classes to inherit from.
Notes
Instances of this class cannot be directly instantiated by users. However, it can be used to created custom pathway types through subclassing.
- property attachments¶
The pair of points defining a pathway’s ends.
- abstract property extension_velocity¶
An expression representing the pathway’s extension velocity.
- abstract property length¶
An expression representing the pathway’s length.
- abstract to_loads(force)[source]¶
Loads required by the equations of motion method classes.
Explanation
KanesMethod
requires a list ofPoint
-Vector
tuples to be passed to theloads
parameters of itskanes_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 toKanesMethod.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
.
- class sympy.physics.mechanics.pathway.WrappingPathway(
- attachment_1,
- attachment_2,
- geometry,
Pathway that wraps a geometry object.
- Parameters:
attachment_1 : Point
First of the pair of
Point
objects between which the wrapping pathway spans.attachment_2 : Point
Second of the pair of
Point
objects between which the wrapping pathway spans.geometry : WrappingGeometryBase
Geometry about which the pathway wraps.
Explanation
A wrapping pathway interacts with a geometry object and forms a path that wraps smoothly along its surface. The wrapping pathway along the geometry object will be the geodesic that the geometry object defines based on the two points. It will not interact with any other objects in the system, i.e. a
WrappingPathway
will intersect other objects to ensure that the path between its two ends (its attachments) is the shortest possible.To explain the sign conventions used for pathway length, extension velocity, and direction of applied forces, we can ignore the geometry with which the wrapping pathway interacts. A wrapping pathway is made up of two points that can move relative to each other, and a pair of equal and opposite forces acting on the points. 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. The following diagram shows the positive force sense and the distance between the points:
P Q o<--- F --->o | | |<--l(t)--->|
Examples
>>> from sympy.physics.mechanics import WrappingPathway
To construct a wrapping pathway, like other pathways, a pair of points must be passed, followed by an instance of a wrapping geometry class as a keyword argument. We’ll use a cylinder with radius
r
and its axis parallel toN.x
passing through a pointpO
.>>> from sympy import symbols >>> from sympy.physics.mechanics import Point, ReferenceFrame, WrappingCylinder >>> r = symbols('r') >>> N = ReferenceFrame('N') >>> pA, pB, pO = Point('pA'), Point('pB'), Point('pO') >>> cylinder = WrappingCylinder(r, pO, N.x) >>> wrapping_pathway = WrappingPathway(pA, pB, cylinder) >>> wrapping_pathway WrappingPathway(pA, pB, geometry=WrappingCylinder(radius=r, point=pO, axis=N.x))
- property extension_velocity¶
Exact analytical expression for the pathway’s extension velocity.
- property geometry¶
Geometry around which the pathway wraps.
- property length¶
Exact analytical expression for the pathway’s length.
- to_loads(force)[source]¶
Loads required by the equations of motion method classes.
- Parameters:
force : Expr
Magnitude of the force acting along the length of the pathway. It is assumed that this
Expr
represents an expansile force.
Explanation
KanesMethod
requires a list ofPoint
-Vector
tuples to be passed to theloads
parameters of itskanes_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 toKanesMethod.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 in an actuator that produces an expansile force
F
while wrapping around a cylinder. First, create a cylinder with radiusr
and an axis parallel to theN.z
direction of the global frameN
that also passes through a pointpO
.>>> from sympy import symbols >>> from sympy.physics.mechanics import (Point, ReferenceFrame, ... WrappingCylinder) >>> N = ReferenceFrame('N') >>> r = symbols('r', positive=True) >>> pO = Point('pO') >>> cylinder = WrappingCylinder(r, pO, N.z)
Create the pathway of the actuator using the
WrappingPathway
class, defined to span between two pointspA
andpB
. Both points lie on the surface of the cylinder and the location ofpB
is defined relative topA
by the dynamics symbolq
.>>> from sympy import cos, sin >>> from sympy.physics.mechanics import WrappingPathway, dynamicsymbols >>> q = dynamicsymbols('q') >>> pA = Point('pA') >>> pB = Point('pB') >>> pA.set_pos(pO, r*N.x) >>> pB.set_pos(pO, r*(cos(q)*N.x + sin(q)*N.y)) >>> pB.pos_from(pA) (r*cos(q(t)) - r)*N.x + r*sin(q(t))*N.y >>> pathway = WrappingPathway(pA, pB, cylinder)
Now create a symbol
F
to describe the magnitude of the (expansile) force that will be produced along the pathway. The list of loads thatKanesMethod
requires can be produced by calling the pathway’sto_loads
method withF
passed as the only argument.>>> F = symbols('F') >>> loads = pathway.to_loads(F) >>> [load.__class__(load.location, load.vector.simplify()) for load in loads] [(pA, F*N.y), (pB, F*sin(q(t))*N.x - F*cos(q(t))*N.y), (pO, - F*sin(q(t))*N.x + F*(cos(q(t)) - 1)*N.y)]