/

# Source code for sympy.physics.mechanics.essential

from __future__ import print_function, division

__all__ = ['ReferenceFrame', 'Vector', 'Dyadic', 'dynamicsymbols',
'MechanicsStrPrinter', 'MechanicsPrettyPrinter',
'MechanicsLatexPrinter', 'CoordinateSym']

from sympy import (
Symbol, sin, cos, eye, trigsimp, diff, sqrt, sympify,
expand, zeros, Derivative, Function, symbols, Add,
solve, S, ImmutableMatrix as Matrix)
from sympy.core import C
from sympy.core.compatibility import reduce, u, string_types
from sympy.core.function import UndefinedFunction
from sympy.printing.conventions import split_super_sub
from sympy.printing.latex import LatexPrinter
from sympy.printing.pretty.pretty import PrettyPrinter
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.str import StrPrinter
from sympy.utilities import group

See:
Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill

A more powerful way to represent a rigid body's inertia. While it is more
complex, by choosing Dyadic components to be in body fixed basis vectors,
the resulting matrix is equivalent to the inertia tensor.

"""

def __init__(self, inlist):
"""
Just like Vector's init, you shouldn't call this unless creating a

Stores a Dyadic as a list of lists; the inner list has the measure
number and the two unit vectors; the outerlist holds each unique
unit vector pair.

"""

self.args = []
if inlist == 0:
inlist = []
while len(inlist) != 0:
for i, v in enumerate(self.args):
if ((str(inlist[0][1]) == str(self.args[i][1])) and
(str(inlist[0][2]) == str(self.args[i][2]))):
self.args[i] = (self.args[i][0] +
inlist[0][0], inlist[0][1], inlist[0][2])
inlist.remove(inlist[0])
break
self.args.append(inlist[0])
inlist.remove(inlist[0])
i = 0
# This code is to remove empty parts from the list
while i < len(self.args):
if ((self.args[i][0] == 0) | (self.args[i][1] == 0) |
(self.args[i][2] == 0)):
self.args.remove(self.args[i])
i -= 1
i += 1

def __and__(self, other):
"""The inner product operator for a Dyadic and a Dyadic or Vector.

Parameters
==========

The other Dyadic or Vector to take the inner product with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> D1 = outer(N.x, N.y)
>>> D2 = outer(N.y, N.y)
>>> D1.dot(D2)
(N.x|N.y)
>>> D1.dot(N.y)
N.x

"""

for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
ol += v[0] * v2[0] * (v[2] & v2[1]) * (v[1] | v2[2])
else:
other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[1] * (v[2] & other)
return ol

def __div__(self, other):
"""Divides the Dyadic by a sympifyable expression. """
return self.__mul__(1 / other)

__truediv__ = __div__

def __eq__(self, other):
"""Tests for equality.

Is currently weak; needs stronger comparison testing

"""

if other == 0:
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
return set(self.args) == set(other.args)

def __mul__(self, other):
"""Multiplies the Dyadic by a sympifyable expression.

Parameters
==========

other : Sympafiable
The scalar to multiply this Dyadic with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> 5 * d
5*(N.x|N.x)

"""

newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1],
newlist[i][2])

def __ne__(self, other):
return not self.__eq__(other)

def __neg__(self):
return self * -1

def _latex(self, printer=None):
ar = self.args  # just to shorten things
if len(ar) == 0:
return str(0)
ol = []  # output list, to be concatenated to a string
mlp = MechanicsLatexPrinter()
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + ' + mlp.doprint(ar[i][1]) + r"\otimes " +
mlp.doprint(ar[i][2]))
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - ' +
mlp.doprint(ar[i][1]) +
r"\otimes " +
mlp.doprint(ar[i][2]))
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = mlp.doprint(ar[i][0])
arg_str = '(%s)' % arg_str
if arg_str.startswith('-'):
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + r" " +
mlp.doprint(ar[i][1]) +
r"\otimes " +
mlp.doprint(ar[i][2]))
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr

def _pretty(self, printer=None):
e = self

class Fake(object):
baseline = 0

def render(self, *args, **kwargs):
self = e
ar = self.args  # just to shorten things
mpp = MechanicsPrettyPrinter()
if len(ar) == 0:
return unicode(0)
ol = []  # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(u(" + ") +
mpp.doprint(ar[i][1]) +
u("\u2a02 ") +
mpp.doprint(ar[i][2]))
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(u(" - ") +
mpp.doprint(ar[i][1]) +
u("\u2a02 ") +
mpp.doprint(ar[i][2]))
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = mpp.doprint(ar[i][0])
arg_str = u("(%s)") % arg_str
if arg_str.startswith(u("-")):
arg_str = arg_str[1:]
str_start = u(" - ")
else:
str_start = u(" + ")
ol.append(str_start + arg_str + u(" ") +
mpp.doprint(ar[i][1]) +
u("\u2a02 ") +
mpp.doprint(ar[i][2]))
outstr = u("").join(ol)
if outstr.startswith(u(" + ")):
outstr = outstr[3:]
elif outstr.startswith(" "):
outstr = outstr[1:]
return outstr
return Fake()

def __rand__(self, other):
"""The inner product operator for a Vector or Dyadic, and a Dyadic

This is for: Vector dot Dyadic

Parameters
==========

other : Vector
The vector we are dotting with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, dot, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> dot(N.x, d)
N.x

"""

other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[2] * (v[1] & other)
return ol

def __rsub__(self, other):
return (-1 * self) + other

def __rxor__(self, other):
"""For a cross product in the form: Vector x Dyadic

Parameters
==========

other : Vector
The Vector that we are crossing this Dyadic with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(N.y, d)
- (N.z|N.x)

"""

other = _check_vector(other)
for i, v in enumerate(self.args):
ol += v[0] * ((other ^ v[1]) | v[2])
return ol

def __str__(self, printer=None):
"""Printing method. """
ar = self.args  # just to shorten things
if len(ar) == 0:
return str(0)
ol = []  # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + (' + str(ar[i][1]) + '|' + str(ar[i][2]) + ')')
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - (' + str(ar[i][1]) + '|' + str(ar[i][2]) + ')')
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = MechanicsStrPrinter().doprint(ar[i][0])
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*(' + str(ar[i][1]) +
'|' + str(ar[i][2]) + ')')
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr

def __sub__(self, other):
"""The subtraction operator. """

def __xor__(self, other):
"""For a cross product in the form: Dyadic x Vector.

Parameters
==========

other : Vector
The Vector that we are crossing this Dyadic with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(d, N.y)
(N.x|N.z)

"""

other = _check_vector(other)
for i, v in enumerate(self.args):
ol += v[0] * (v[1] | (v[2] ^ other))
return ol

_sympystr = __str__
_sympyrepr = _sympystr
__repr__ = __str__
__rmul__ = __mul__

[docs]    def express(self, frame1, frame2=None):
"""Expresses this Dyadic in alternate frame(s)

The first frame is the list side expression, the second frame is the
right side; if Dyadic is in form A.x|B.y, you can express it in two
different frames. If no second frame is given, the Dyadic is
expressed in only one frame.

Calls the global express function

Parameters
==========

frame1 : ReferenceFrame
The frame to express the left side of the Dyadic in
frame2 : ReferenceFrame
If provided, the frame to express the right side of the Dyadic in

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.express(B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)

"""
from sympy.physics.mechanics import express
return express(self, frame1, frame2)

[docs]    def doit(self, **hints):
"""Calls .doit() on each term in the Dyadic"""

[docs]    def dt(self, frame):
"""Take the time derivative of this Dyadic in a frame.

This function calls the global time_derivative method

Parameters
==========

frame : ReferenceFrame
The frame to take the time derivative in

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.dt(B)
- q'*(N.y|N.x) - q'*(N.x|N.y)

"""
from sympy.physics.mechanics import time_derivative
return time_derivative(self, frame)

[docs]    def simplify(self):
for v in self.args:
return out

[docs]    def subs(self, *args, **kwargs):

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = s * (N.x|N.x)
>>> a.subs({s: 2})
2*(N.x|N.x)

"""

dot = __and__
cross = __xor__

[docs]class CoordinateSym(Symbol):
"""
A coordinate symbol/base scalar associated wrt a Reference Frame.

Ideally, users should not instantiate this class. Instances of
this class must only be accessed through the corresponding frame
as 'frame[index]'.

CoordinateSyms having the same frame and index parameters are equal
(even though they may be instantiated separately).

Parameters
==========

name : string
The display name of the CoordinateSym

frame : ReferenceFrame
The reference frame this base scalar belongs to

index : 0, 1 or 2
The index of the dimension denoted by this coordinate variable

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, CoordinateSym
>>> A = ReferenceFrame('A')
>>> A[1]
A_y
>>> type(A[0])
<class 'sympy.physics.mechanics.essential.CoordinateSym'>
>>> a_y = CoordinateSym('a_y', A, 1)
>>> a_y == A[1]
True

"""

def __new__(cls, name, frame, index):
obj = super(CoordinateSym, cls).__new__(cls, name)
_check_frame(frame)
if index not in range(0, 3):
raise ValueError("Invalid index specified")
obj._id = (frame, index)
return obj

@property
def frame(self):
return self._id[0]

def __eq__(self, other):
#Check if the other object is a CoordinateSym of the same frame
#and same index
if isinstance(other, CoordinateSym):
if other._id == self._id:
return True
return False

def __ne__(self, other):
return not self.__eq__(other)

def __hash__(self):
return tuple((self._id[0].__hash__(), self._id[1])).__hash__()

[docs]class ReferenceFrame(object):
"""A reference frame in classical mechanics.

ReferenceFrame is a class used to represent a reference frame in classical
mechanics. It has a standard basis of three unit vectors in the frame's
x, y, and z directions.

It also can have a rotation relative to a parent frame; this rotation is
defined by a direction cosine matrix relating this frame's basis vectors to
the parent frame's basis vectors.  It can also have an angular velocity
vector, defined in another frame.

"""

def __init__(self, name, indices=None, latexs=None, variables=None):
"""ReferenceFrame initialization method.

A ReferenceFrame has a set of orthonormal basis vectors, along with
orientations relative to other ReferenceFrames and angular velocities
relative to other ReferenceFrames.

Parameters
==========

indices : list (of strings)
If custom indices are desired for console, pretty, and LaTeX
printing, supply three as a list. The basis vectors can then be
accessed with the get_item method.
latexs : list (of strings)
If custom names are desired for LaTeX printing of each basis
vector, supply the names here in a list.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, mlatex
>>> N = ReferenceFrame('N')
>>> N.x
N.x
>>> O = ReferenceFrame('O', indices=('1', '2', '3'))
>>> O.x
O['1']
>>> O['1']
O['1']
>>> P = ReferenceFrame('P', latexs=('A1', 'A2', 'A3'))
>>> mlatex(P.x)
'A1'

"""

if not isinstance(name, string_types):
raise TypeError('Need to supply a valid name')
# The if statements below are for custom printing of basis-vectors for
# each frame.
# First case, when custom indices are supplied
if indices is not None:
if not isinstance(indices, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(indices) != 3:
raise ValueError('Supply 3 indices')
for i in indices:
if not isinstance(i, string_types):
raise TypeError('Indices must be strings')
self.str_vecs = [(name + '[\'' + indices[0] + '\']'),
(name + '[\'' + indices[1] + '\']'),
(name + '[\'' + indices[2] + '\']')]
self.pretty_vecs = [(u("\033[94m\033[1m") + name.lower() + u("_") +
indices[0] + u("\033[0;0m\x1b[0;0m")),
(u("\033[94m\033[1m") + name.lower() + u("_") +
indices[1] + u("\033[0;0m\x1b[0;0m")),
(u("\033[94m\033[1m") + name.lower() + u("_") +
indices[2] + u("\033[0;0m\x1b[0;0m"))]
self.latex_vecs = [(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[0])), (r"\mathbf{\hat{%s}_{%s}}" %
(name.lower(), indices[1])),
(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[2]))]
self.indices = indices
# Second case, when no custom indices are supplied
else:
self.str_vecs = [(name + '.x'), (name + '.y'), (name + '.z')]
self.pretty_vecs = [(u("\033[94m\033[1m") + name.lower() +
u("_x\033[0;0m\x1b[0;0m")),
(u("\033[94m\033[1m") + name.lower() +
u("_y\033[0;0m\x1b[0;0m")),
(u("\033[94m\033[1m") + name.lower() +
u("_z\033[0;0m\x1b[0;0m"))]
self.latex_vecs = [(r"\mathbf{\hat{%s}_x}" % name.lower()),
(r"\mathbf{\hat{%s}_y}" % name.lower()),
(r"\mathbf{\hat{%s}_z}" % name.lower())]
self.indices = ['x', 'y', 'z']
# Different step, for custom latex basis vectors
if latexs is not None:
if not isinstance(latexs, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(latexs) != 3:
raise ValueError('Supply 3 indices')
for i in latexs:
if not isinstance(i, string_types):
raise TypeError('Latex entries must be strings')
self.latex_vecs = latexs
self.name = name
self._var_dict = {}
#The _dcm_dict dictionary will only store the dcms of parent-child
#relationships. The _dcm_cache dictionary will work as the dcm
#cache.
self._dcm_dict = {}
self._dcm_cache = {}
self._ang_vel_dict = {}
self._ang_acc_dict = {}
self._dlist = [self._dcm_dict, self._ang_vel_dict, self._ang_acc_dict]
self._cur = 0
self._x = Vector([(Matrix([1, 0, 0]), self)])
self._y = Vector([(Matrix([0, 1, 0]), self)])
self._z = Vector([(Matrix([0, 0, 1]), self)])
#Associate coordinate symbols wrt this frame
if variables is not None:
if not isinstance(variables, (tuple, list)):
raise TypeError('Supply the variable names as a list/tuple')
if len(variables) != 3:
raise ValueError('Supply 3 variable names')
for i in variables:
if not isinstance(i, string_types):
raise TypeError('Variable names must be strings')
else:
variables = [name + '_x', name + '_y', name + '_z']
self.varlist = (CoordinateSym(variables[0], self, 0), \
CoordinateSym(variables[1], self, 1), \
CoordinateSym(variables[2], self, 2))

def __getitem__(self, ind):
"""
Returns basis vector for the provided index, if the index is a string.

If the index is a number, returns the coordinate variable correspon-
-ding to that index.
"""
if not isinstance(ind, str):
if ind < 3:
return self.varlist[ind]
else:
raise ValueError("Invalid index provided")
if self.indices[0] == ind:
return self.x
if self.indices[1] == ind:
return self.y
if self.indices[2] == ind:
return self.z
else:
raise ValueError('Not a defined index')

def __iter__(self):
return iter([self.x, self.y, self.z])

def __str__(self):
"""Returns the name of the frame. """
return self.name

__repr__ = __str__

def _dict_list(self, other, num):
"""Creates a list from self to other using _dcm_dict. """
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._dlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + self.name +
' and ' + other.name)

def _w_diff_dcm(self, otherframe):
"""Angular velocity from time differentiating the DCM. """
dcm2diff = self.dcm(otherframe)
diffed = dcm2diff.diff(dynamicsymbols._t)
angvelmat = diffed * dcm2diff.T
w1 = trigsimp(expand(angvelmat[7]), recursive=True)
w2 = trigsimp(expand(angvelmat[2]), recursive=True)
w3 = trigsimp(expand(angvelmat[3]), recursive=True)
return -Vector([(Matrix([w1, w2, w3]), self)])

[docs]    def variable_map(self, otherframe):
"""
Returns a dictionary which expresses the coordinate variables
of this frame in terms of the variables of otherframe.

If Vector.simp is True, returns a simplified version of the mapped
values. Else, returns them without simplification.

Simplification of the expressions may take time.

Parameters
==========

otherframe : ReferenceFrame
The other frame to map the variables to

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, dynamicsymbols
>>> A = ReferenceFrame('A')
>>> q = dynamicsymbols('q')
>>> B = A.orientnew('B', 'Axis', [q, A.z])
>>> A.variable_map(B)
{A_x: B_x*cos(q(t)) - B_y*sin(q(t)), A_y: B_x*sin(q(t)) + B_y*cos(q(t)), A_z: B_z}

"""

_check_frame(otherframe)
if (otherframe, Vector.simp) in self._var_dict:
return self._var_dict[(otherframe, Vector.simp)]
else:
vars_matrix = self.dcm(otherframe) * Matrix(otherframe.varlist)
mapping = {}
for i, x in enumerate(self):
if Vector.simp:
mapping[self.varlist[i]] = trigsimp(vars_matrix[i], method='fu')
else:
mapping[self.varlist[i]] = vars_matrix[i]
self._var_dict[(otherframe, Vector.simp)] = mapping
return mapping

[docs]    def ang_acc_in(self, otherframe):
"""Returns the angular acceleration Vector of the ReferenceFrame.

Effectively returns the Vector:
^N alpha ^B
which represent the angular acceleration of B in N, where B is self, and
N is otherframe.

Parameters
==========

otherframe : ReferenceFrame
The ReferenceFrame which the angular acceleration is returned in.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x

"""

_check_frame(otherframe)
if otherframe in self._ang_acc_dict:
return self._ang_acc_dict[otherframe]
else:
return self.ang_vel_in(otherframe).dt(otherframe)

[docs]    def ang_vel_in(self, otherframe):
"""Returns the angular velocity Vector of the ReferenceFrame.

Effectively returns the Vector:
^N omega ^B
which represent the angular velocity of B in N, where B is self, and
N is otherframe.

Parameters
==========

otherframe : ReferenceFrame
The ReferenceFrame which the angular velocity is returned in.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x

"""

_check_frame(otherframe)
flist = self._dict_list(otherframe, 1)
outvec = Vector(0)
for i in range(len(flist) - 1):
outvec += flist[i]._ang_vel_dict[flist[i + 1]]
return outvec

[docs]    def dcm(self, otherframe):
"""The direction cosine matrix between frames.

This gives the DCM between this frame and the otherframe.
The format is N.xyz = N.dcm(B) * B.xyz
A SymPy Matrix is returned.

Parameters
==========

otherframe : ReferenceFrame
The otherframe which the DCM is generated to.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> N.dcm(A)
Matrix([
[1,       0,        0],
[0, cos(q1), -sin(q1)],
[0, sin(q1),  cos(q1)]])

"""

_check_frame(otherframe)
#Check if the dcm wrt that frame has already been calculated
if otherframe in self._dcm_cache:
return self._dcm_cache[otherframe]
flist = self._dict_list(otherframe, 0)
outdcm = eye(3)
for i in range(len(flist) - 1):
outdcm = outdcm * flist[i]._dcm_dict[flist[i + 1]]
#After calculation, store the dcm in dcm cache for faster
#future retrieval
self._dcm_cache[otherframe] = outdcm
otherframe._dcm_cache[self] = outdcm.T
return outdcm

[docs]    def orient(self, parent, rot_type, amounts, rot_order=''):
"""Defines the orientation of this frame relative to a parent frame.

Parameters
==========

parent : ReferenceFrame
The frame that this ReferenceFrame will have its orientation matrix
defined in relation to.
rot_type : str
The type of orientation matrix that is being created. Supported
types are 'Body', 'Space', 'Quaternion', and 'Axis'. See examples
for correct usage.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
rot_order : str
If applicable, the order of a series of rotations.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q0, q1, q2, q3, q4 = symbols('q0 q1 q2 q3 q4')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')

Now we have a choice of how to implement the orientation. First is
Body. Body orientation takes this reference frame through three
successive simple rotations. Acceptable rotation orders are of length
3, expressed in XYZ or 123, and cannot have a rotation about about an
axis twice in a row.

>>> B.orient(N, 'Body', [q1, q2, q3], '123')
>>> B.orient(N, 'Body', [q1, q2, 0], 'ZXZ')
>>> B.orient(N, 'Body', [0, 0, 0], 'XYX')

Next is Space. Space is like Body, but the rotations are applied in the
opposite order.

>>> B.orient(N, 'Space', [q1, q2, q3], '312')

Next is Quaternion. This orients the new ReferenceFrame with
Quaternions, defined as a finite rotation about lambda, a unit vector,
by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.

>>> B.orient(N, 'Quaternion', [q0, q1, q2, q3])

Last is Axis. This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.

>>> B.orient(N, 'Axis', [q1, N.x + 2 * N.y])

"""

_check_frame(parent)
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)

def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1]])

approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
rot_order = str(
rot_order).upper()  # Now we need to make sure XYZ = 123
rot_type = rot_type.upper()
rot_order = [i.replace('X', '1') for i in rot_order]
rot_order = [i.replace('Y', '2') for i in rot_order]
rot_order = [i.replace('Z', '3') for i in rot_order]
rot_order = ''.join(rot_order)
if not rot_order in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = ((eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]], [axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) * sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 - q3 **
2, 2 * (q1 * q2 - q0 * q3), 2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3), q0 ** 2 - q1 ** 2 + q2 ** 2 - q3 ** 2,
2 * (q2 * q3 - q0 * q1)], [2 * (q1 * q3 - q0 * q2), 2 * (q0 *
q1 + q2 * q3), q0 ** 2 - q1 ** 2 - q2 ** 2 + q3 ** 2]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1])
* _rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1])
* _rot(a1, amounts[0]))
else:
raise NotImplementedError('That is not an implemented rotation')
#Reset the _dcm_cache of this frame, and remove it from the _dcm_caches
#of the frames it is linked to. Also remove it from the _dcm_dict of
#its parent
frames = self._dcm_cache.keys()
for frame in frames:
if frame in self._dcm_dict:
del frame._dcm_dict[self]
del frame._dcm_cache[self]
#Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
#Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.mechanics.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = dynamicsymbols('u1, u2, u3')
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}

[docs]    def orientnew(self, newname, rot_type, amounts, rot_order='', variables=None,
indices=None, latexs=None):
"""Creates a new ReferenceFrame oriented with respect to this Frame.

See ReferenceFrame.orient() for acceptable rotation types, amounts,
and orders. Parent is going to be self.

Parameters
==========

newname : str
The name for the new ReferenceFrame
rot_type : str
The type of orientation matrix that is being created.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
rot_order : str
If applicable, the order of a series of rotations.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])

.orient() documentation:\n
========================

"""

newframe = ReferenceFrame(newname, variables, indices, latexs)
newframe.orient(self, rot_type, amounts, rot_order)
return newframe

orientnew.__doc__ += orient.__doc__

[docs]    def set_ang_acc(self, otherframe, value):
"""Define the angular acceleration Vector in a ReferenceFrame.

Defines the angular acceleration of this ReferenceFrame, in another.
Angular acceleration can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.

Parameters
==========

otherframe : ReferenceFrame
A ReferenceFrame to define the angular acceleration in
value : Vector
The Vector representing angular acceleration

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x

"""

if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_acc_dict.update({otherframe: value})
otherframe._ang_acc_dict.update({self: -value})

[docs]    def set_ang_vel(self, otherframe, value):
"""Define the angular velocity vector in a ReferenceFrame.

Defines the angular velocity of this ReferenceFrame, in another.
Angular velocity can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.

Parameters
==========

otherframe : ReferenceFrame
A ReferenceFrame to define the angular velocity in
value : Vector
The Vector representing angular velocity

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x

"""

if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_vel_dict.update({otherframe: value})
otherframe._ang_vel_dict.update({self: -value})

@property
[docs]    def x(self):
"""The basis Vector for the ReferenceFrame, in the x direction. """
return self._x

@property
[docs]    def y(self):
"""The basis Vector for the ReferenceFrame, in the y direction. """
return self._y

@property
[docs]    def z(self):
"""The basis Vector for the ReferenceFrame, in the z direction. """
return self._z

[docs]class Vector(object):
"""The class used to define vectors.

It along with ReferenceFrame are the building blocks of describing a
classical mechanics system in PyDy.

Attributes
==========

simp : Boolean
Let certain methods use trigsimp on their outputs

"""

simp = False

def __init__(self, inlist):
"""This is the constructor for the Vector class.  You shouldn't be
calling this, it should only be used by other functions. You should be
treating Vectors like you would with if you were doing the math by
hand, and getting the first 3 from the standard basis vectors from a
ReferenceFrame.

The only exception is to create a zero vector:
zv = Vector(0)

"""

self.args = []
if inlist == 0:
inlist = []
while len(inlist) != 0:
for i, v in enumerate(self.args):
if inlist[0][1] == self.args[i][1]:
self.args[i] = (self.args[i][0] +
inlist[0][0], inlist[0][1])
inlist.remove(inlist[0])
break
self.args.append(inlist[0])
inlist.remove(inlist[0])
i = 0
# This code is to remove empty frames from the list
while i < len(self.args):
if self.args[i][0] == Matrix([0, 0, 0]):
self.args.remove(self.args[i])
i -= 1
i += 1

def __hash__(self):
return hash(tuple(self.args))

"""The add operator for Vector. """
other = _check_vector(other)
return Vector(self.args + other.args)

def __and__(self, other):
"""Dot product of two vectors.

Returns a scalar, the dot product of the two Vectors

Parameters
==========

other : Vector
The Vector which we are dotting with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector, dot
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> dot(N.x, N.x)
1
>>> dot(N.x, N.y)
0
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> dot(N.y, A.y)
cos(q1)

"""

return NotImplemented
other = _check_vector(other)
out = S(0)
for i, v1 in enumerate(self.args):
for j, v2 in enumerate(other.args):
out += ((v2[0].T)
* (v2[1].dcm(v1[1]))
* (v1[0]))[0]
if Vector.simp:
return trigsimp(sympify(out), recursive=True)
else:
return sympify(out)

def __div__(self, other):
"""This uses mul and inputs self and 1 divided by other. """
return self.__mul__(1 / other)

__truediv__ = __div__

def __eq__(self, other):
"""Tests for equality.

It is very import to note that this is only as good as the SymPy
equality test; False does not always mean they are not equivalent
Vectors.
If other is 0, and self is empty, returns True.
If other is 0 and self is not empty, returns False.
If none of the above, only accepts other as a Vector.

"""

if other == 0:
other = Vector(0)
other = _check_vector(other)
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False

frame = self.args[0][1]
for v in frame:
if expand((self - other) & v) != 0:
return False
return True

def __mul__(self, other):
"""Multiplies the Vector by a sympifyable expression.

Parameters
==========

other : Sympifyable
The scalar to multiply this Vector with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> b = Symbol('b')
>>> V = 10 * b * N.x
>>> print(V)
10*b*N.x

"""

newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1])
return Vector(newlist)

def __ne__(self, other):
return not self.__eq__(other)

def __neg__(self):
return self * -1

def __or__(self, other):
"""Outer product between two Vectors.

A rank increasing operation, which returns a Dyadic from two Vectors

Parameters
==========

other : Vector
The Vector to take the outer product with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)

"""

other = _check_vector(other)
for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol

def _latex(self, printer=None):
"""Latex Printing method. """
ar = self.args  # just to shorten things
if len(ar) == 0:
return str(0)
ol = []  # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].latex_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].latex_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = MechanicsStrPrinter().doprint(ar[i][0][j])
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*' +
ar[i][1].latex_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr

def _pretty(self, printer=None):
"""Pretty Printing method. """
e = self

class Fake(object):
baseline = 0

def render(self, *args, **kwargs):
self = e
ar = self.args  # just to shorten things
if len(ar) == 0:
return unicode(0)
ol = []  # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(u(" + ") + ar[i][1].pretty_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(u(" - ") + ar[i][1].pretty_vecs[j])
elif ar[i][0][j] != 0:
# If the basis vector coeff is not 1 or -1,
# we might wrap it in parentheses, for readability.
arg_str = (MechanicsPrettyPrinter().doprint(
ar[i][0][j]))
arg_str = u("(%s)") % arg_str
if arg_str[0] == u("-"):
arg_str = arg_str[1:]
str_start = u(" - ")
else:
str_start = u(" + ")
ol.append(str_start + arg_str + '*' +
ar[i][1].pretty_vecs[j])
outstr = u("").join(ol)
if outstr.startswith(u(" + ")):
outstr = outstr[3:]
elif outstr.startswith(" "):
outstr = outstr[1:]
return outstr
return Fake()

def __ror__(self, other):
"""Outer product between two Vectors.

A rank increasing operation, which returns a Dyadic from two Vectors

Parameters
==========

other : Vector
The Vector to take the outer product with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)

"""

other = _check_vector(other)
for i, v in enumerate(other.args):
for i2, v2 in enumerate(self.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol

def __rsub__(self, other):
return (-1 * self) + other

def __str__(self, printer=None):
"""Printing method. """
ar = self.args  # just to shorten things
if len(ar) == 0:
return str(0)
ol = []  # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].str_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].str_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = MechanicsStrPrinter().doprint(ar[i][0][j])
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*' + ar[i][1].str_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr

def __sub__(self, other):
"""The subraction operator. """

def __xor__(self, other):
"""The cross product operator for two Vectors.

Returns a Vector, expressed in the same ReferenceFrames as self.

Parameters
==========

other : Vector
The Vector which we are crossing with

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> N.x ^ N.y
N.z
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> A.x ^ N.y
N.z
>>> N.y ^ A.x
- sin(q1)*A.y - cos(q1)*A.z

"""

return NotImplemented
other = _check_vector(other)
if other.args == []:
return Vector(0)

def _det(mat):
"""This is needed as a little method for to find the determinant
of a list in python; needs to work for a 3x3 list.
SymPy's Matrix won't take in Vector, so need a custom function.
You shouldn't be calling this.

"""

return (mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1])
+ mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] *
mat[2][2]) + mat[0][2] * (mat[1][0] * mat[2][1] -
mat[1][1] * mat[2][0]))

outvec = Vector(0)
ar = other.args  # For brevity
for i, v in enumerate(ar):
tempx = v[1].x
tempy = v[1].y
tempz = v[1].z
tempm = ([[tempx, tempy, tempz], [self & tempx, self & tempy,
self & tempz], [Vector([ar[i]]) & tempx,
Vector([ar[i]]) & tempy, Vector([ar[i]]) & tempz]])
outvec += _det(tempm)
return outvec

_sympystr = __str__
_sympyrepr = _sympystr
__repr__ = __str__
__rand__ = __and__
__rmul__ = __mul__

[docs]    def dot(self, other):
return self & other
dot.__doc__ = __and__.__doc__

[docs]    def cross(self, other):
return self ^ other
cross.__doc__ = __xor__.__doc__

[docs]    def outer(self, other):
return self | other
outer.__doc__ = __or__.__doc__

[docs]    def diff(self, wrt, otherframe):
"""Takes the partial derivative, with respect to a value, in a frame.

Returns a Vector.

Parameters
==========

wrt : Symbol
What the partial derivative is taken with respect to.
otherframe : ReferenceFrame
The ReferenceFrame that the partial derivative is taken in.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector, dynamicsymbols
>>> from sympy import Symbol
>>> Vector.simp = True
>>> t = Symbol('t')
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.diff(t, N)
- q1'*A.z

"""

wrt = sympify(wrt)
_check_frame(otherframe)
outvec = Vector(0)
for i, v in enumerate(self.args):
if v[1] == otherframe:
outvec += Vector([(v[0].diff(wrt), otherframe)])
else:
if otherframe.dcm(v[1]).diff(wrt) == zeros(3, 3):
d = v[0].diff(wrt)
outvec += Vector([(d, v[1])])
else:
d = (Vector([v]).express(otherframe)).args[0][0].diff(wrt)
outvec += Vector([(d, otherframe)]).express(v[1])
return outvec

[docs]    def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.

Parameters
==========

otherframe : ReferenceFrame
The frame for this Vector to be described in

variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, Vector, dynamicsymbols
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z

"""
from sympy.physics.mechanics import express
return express(self, otherframe, variables=variables)

[docs]    def doit(self, **hints):
"""Calls .doit() on each term in the Vector"""
ov = Vector(0)
for i, v in enumerate(self.args):
ov += Vector([(v[0].applyfunc(lambda x: x.doit(**hints)), v[1])])
return ov

[docs]    def dt(self, otherframe):
"""
Returns a Vector which is the time derivative of
the self Vector, taken in frame otherframe.

Calls the global time_derivative method

Parameters
==========

otherframe : ReferenceFrame
The frame to calculate the time derivative in

"""
from sympy.physics.mechanics import time_derivative
return time_derivative(self, otherframe)

[docs]    def simplify(self):
"""Returns a simplified Vector."""
outvec = Vector(0)
for i in self.args:
outvec += Vector([(i[0].simplify(), i[1])])
return outvec

[docs]    def subs(self, *args, **kwargs):
"""Substituion on the Vector.

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = N.x * s
>>> a.subs({s: 2})
2*N.x

"""

ov = Vector(0)
for i, v in enumerate(self.args):
ov += Vector([(v[0].subs(*args, **kwargs), v[1])])
return ov

[docs]    def magnitude(self):
"""Returns the magnitude (Euclidean norm) of self."""
return sqrt(self & self)

[docs]    def normalize(self):
"""Returns a Vector of magnitude 1, codirectional with self."""
return Vector(self.args + []) / self.magnitude()

class MechanicsStrPrinter(StrPrinter):
"""String Printer for mechanics. """

def _print_Derivative(self, e):
t = dynamicsymbols._t
if (bool(sum([i == t for i in e.variables])) &
isinstance(type(e.args[0]), UndefinedFunction)):
ol = str(e.args[0].func)
for i, v in enumerate(e.variables):
ol += dynamicsymbols._str
return ol
else:
return StrPrinter().doprint(e)

def _print_Function(self, e):
t = dynamicsymbols._t
if isinstance(type(e), UndefinedFunction):
return StrPrinter().doprint(e).replace("(%s)" % t, '')
return e.func.__name__ + "(%s)" % self.stringify(e.args, ", ")

class MechanicsLatexPrinter(LatexPrinter):
"""Latex Printer for mechanics. """

def _print_Function(self, expr, exp=None):
func = expr.func.__name__
t = dynamicsymbols._t

if hasattr(self, '_print_' + func):
return getattr(self, '_print_' + func)(expr, exp)
elif isinstance(type(expr), UndefinedFunction) and (expr.args == (t,)):
name, sup, sub = split_super_sub(func)
if len(sup) != 0:
sup = r"^{%s}" % "".join(sup)
else:
sup = r""
if len(sub) != 0:
sub = r"_{%s}" % "".join(sub)
else:
sub = r""
if exp:
sup += r"^{%s}" % self._print(exp)
return r"%s" % (name + sup + sub)
else:
args = [str(self._print(arg)) for arg in expr.args]
# How inverse trig functions should be displayed, formats are:
# abbreviated: asin, full: arcsin, power: sin^-1
inv_trig_style = self._settings['inv_trig_style']
# If we are dealing with a power-style inverse trig function
inv_trig_power_case = False
# If it is applicable to fold the argument brackets
can_fold_brackets = self._settings['fold_func_brackets'] and \
len(args) == 1 and \
not self._needs_function_brackets(expr.args[0])

inv_trig_table = ["asin", "acos", "atan", "acot"]

# If the function is an inverse trig function, handle the style
if func in inv_trig_table:
if inv_trig_style == "abbreviated":
func = func
elif inv_trig_style == "full":
func = "arc" + func[1:]
elif inv_trig_style == "power":
func = func[1:]
inv_trig_power_case = True

# Can never fold brackets if we're raised to a power
if exp is not None:
can_fold_brackets = False

if inv_trig_power_case:
name = r"\operatorname{%s}^{-1}" % func
elif exp is not None:
name = r"\operatorname{%s}^{%s}" % (func, exp)
else:
name = r"\operatorname{%s}" % func

if can_fold_brackets:
name += r"%s"
else:
name += r"\left(%s\right)"

if inv_trig_power_case and exp is not None:
name += r"^{%s}" % exp

return name % ",".join(args)

def _print_Derivative(self, der_expr):
# make sure it is an the right form
der_expr = der_expr.doit()
if not isinstance(der_expr, Derivative):
return self.doprint(der_expr)

# check if expr is a dynamicsymbol
from sympy.core.function import AppliedUndef
t = dynamicsymbols._t
expr = der_expr.expr
red = expr.atoms(AppliedUndef)
syms = der_expr.variables
test1 = not all([True for i in red if i.atoms() == set([t])])
test2 = not all([(t == i) for i in syms])
if test1 or test2:
return LatexPrinter().doprint(der_expr)

# done checking
dots = len(syms)
base = self._print_Function(expr)
base_split = base.split('_', 1)
base = base_split[0]
if dots == 1:
base = r"\dot{%s}" % base
elif dots == 2:
base = r"\ddot{%s}" % base
elif dots == 3:
base = r"\dddot{%s}" % base
if len(base_split) is not 1:
base += '_' + base_split[1]
return base

class MechanicsPrettyPrinter(PrettyPrinter):
"""Pretty Printer for mechanics. """

def _print_Derivative(self, deriv):
# XXX use U('PARTIAL DIFFERENTIAL') here ?
t = dynamicsymbols._t
dots = 0
can_break = True
syms = list(reversed(deriv.variables))
x = None

while len(syms) > 0:
if syms[-1] == t:
syms.pop()
dots += 1
else:
break

f = prettyForm(binding=prettyForm.FUNC, *self._print(deriv.expr))
if not (isinstance(type(deriv.expr), UndefinedFunction)
and (deriv.expr.args == (t,))):
dots = 0
can_break = False
f = prettyForm(binding=prettyForm.FUNC,
*self._print(deriv.expr).parens())

if dots == 0:
dots = u("")
elif dots == 1:
dots = u("\u0307")
elif dots == 2:
dots = u("\u0308")
elif dots == 3:
dots = u("\u20db")
elif dots == 4:
dots = u("\u20dc")

uni_subs = [u("\u2080"), u("\u2081"), u("\u2082"), u("\u2083"), u("\u2084"),
u("\u2085"), u("\u2086"), u("\u2087"), u("\u2088"), u("\u2089"),
u("\u208a"), u("\u208b"), u("\u208c"), u("\u208d"), u("\u208e"),
u("\u208f"), u("\u2090"), u("\u2091"), u("\u2092"), u("\u2093"),
u("\u2094"), u("\u2095"), u("\u2096"), u("\u2097"), u("\u2098"),
u("\u2099"), u("\u209a"), u("\u209b"), u("\u209c"), u("\u209d"),
u("\u209e"), u("\u209f")]

fpic = f.__dict__['picture']
funi = f.__dict__['unicode']
ind = len(funi)
val = ""

for i in uni_subs:
cur_ind = funi.find(i)
if (cur_ind != -1) and (cur_ind < ind):
ind = cur_ind
val = i
if ind == len(funi):
funi += dots
else:
funi = funi.replace(val, dots + val)

if f.__dict__['picture'] == [f.__dict__['unicode']]:
fpic = [funi]
f.__dict__['picture'] = fpic
f.__dict__['unicode'] = funi

if (len(syms)) == 0 and can_break:
return f

for sym, num in group(syms, multiple=False):
s = self._print(sym)
ds = prettyForm(*s.left('d'))

if num > 1:
ds = ds**prettyForm(str(num))

if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
pform = prettyForm('d')
if len(syms) > 1:
pform = pform**prettyForm(str(len(syms)))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
return pform

def _print_Function(self, e):
t = dynamicsymbols._t
# XXX works only for applied functions
func = e.func
args = e.args
func_name = func.__name__
prettyFunc = self._print(C.Symbol(func_name))
prettyArgs = prettyForm(*self._print_seq(args).parens())
# If this function is an Undefined function of t, it is probably a
# dynamic symbol, so we'll skip the (t). The rest of the code is
# identical to the normal PrettyPrinter code
if isinstance(func, UndefinedFunction) and (args == (t,)):
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc))
else:
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform

class MechanicsTypeError(TypeError):

def __init__(self, other, type_str):
super(MechanicsTypeError, self).__init__("Expected an instance of %s, "
type_str, other, type(other)))

raise TypeError('A Dyadic must be supplied')
return other

def _check_frame(other):
if not isinstance(other, ReferenceFrame):
raise MechanicsTypeError(other, "ReferenceFrame")

def _check_vector(other):
if not isinstance(other, Vector):
raise TypeError('A Vector must be supplied')
return other

[docs]def dynamicsymbols(names, level=0):
"""Uses symbols and Function for functions of time.

Creates a SymPy UndefinedFunction, which is then initialized as a function
of a variable, the default being Symbol('t').

Parameters
==========

names : str
Names of the dynamic symbols you want to create; works the same way as
inputs to symbols
level : int
Level of differentiation of the returned function; d/dt once of t,
twice of t, etc.

Examples
========

>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy import diff, Symbol
>>> q1 = dynamicsymbols('q1')
>>> q1
q1(t)
>>> diff(q1, Symbol('t'))
Derivative(q1(t), t)

"""

esses = symbols(names, cls=Function)
t = dynamicsymbols._t
if hasattr(esses, '__iter__'):
esses = [reduce(diff, [t]*level, e(t)) for e in esses]
return esses
else:
return reduce(diff, [t]*level, esses(t))

dynamicsymbols._t = Symbol('t')
dynamicsymbols._str = '\''