from __future__ import print_function, division
from sympy.core.backend import (sympify, diff, sin, cos, Matrix, symbols,
Function, S, Symbol)
from sympy import integrate, trigsimp
from sympy.core.compatibility import reduce
from .vector import Vector, _check_vector
from .frame import CoordinateSym, _check_frame
from .dyadic import Dyadic
from .printing import vprint, vsprint, vpprint, vlatex, init_vprinting
from sympy.utilities.iterables import iterable
from sympy.utilities.misc import translate
__all__ = ['cross', 'dot', 'express', 'time_derivative', 'outer',
'kinematic_equations', 'get_motion_params', 'partial_velocity',
'dynamicsymbols', 'vprint', 'vsprint', 'vpprint', 'vlatex',
'init_vprinting']
[docs]def cross(vec1, vec2):
"""Cross product convenience wrapper for Vector.cross(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Cross product is between two vectors')
return vec1 ^ vec2
cross.__doc__ += Vector.cross.__doc__
[docs]def dot(vec1, vec2):
"""Dot product convenience wrapper for Vector.dot(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Dot product is between two vectors')
return vec1 & vec2
dot.__doc__ += Vector.dot.__doc__
[docs]def express(expr, frame, frame2=None, variables=False):
"""
Global function for 'express' functionality.
Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame.
Refer to the local methods of Vector and Dyadic for details.
If 'variables' is True, then the coordinate variables (CoordinateSym
instances) of other frames present in the vector/scalar field or
dyadic expression are also substituted in terms of the base scalars of
this frame.
Parameters
==========
expr : Vector/Dyadic/scalar(sympyfiable)
The expression to re-express in ReferenceFrame 'frame'
frame: ReferenceFrame
The reference frame to express expr in
frame2 : ReferenceFrame
The other frame required for re-expression(only for Dyadic expr)
variables : boolean
Specifies whether to substitute the coordinate variables present
in expr, in terms of those of frame
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> from sympy.physics.vector import express
>>> express(d, B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
>>> express(B.x, N)
cos(q)*N.x + sin(q)*N.y
>>> express(N[0], B, variables=True)
B_x*cos(q(t)) - B_y*sin(q(t))
"""
_check_frame(frame)
if expr == 0:
return expr
if isinstance(expr, Vector):
#Given expr is a Vector
if variables:
#If variables attribute is True, substitute
#the coordinate variables in the Vector
frame_list = [x[-1] for x in expr.args]
subs_dict = {}
for f in frame_list:
subs_dict.update(f.variable_map(frame))
expr = expr.subs(subs_dict)
#Re-express in this frame
outvec = Vector([])
for i, v in enumerate(expr.args):
if v[1] != frame:
temp = frame.dcm(v[1]) * v[0]
if Vector.simp:
temp = temp.applyfunc(lambda x:
trigsimp(x, method='fu'))
outvec += Vector([(temp, frame)])
else:
outvec += Vector([v])
return outvec
if isinstance(expr, Dyadic):
if frame2 is None:
frame2 = frame
_check_frame(frame2)
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += express(v[0], frame, variables=variables) * \
(express(v[1], frame, variables=variables) |
express(v[2], frame2, variables=variables))
return ol
else:
if variables:
#Given expr is a scalar field
frame_set = set([])
expr = sympify(expr)
#Subsitute all the coordinate variables
for x in expr.free_symbols:
if isinstance(x, CoordinateSym)and x.frame != frame:
frame_set.add(x.frame)
subs_dict = {}
for f in frame_set:
subs_dict.update(f.variable_map(frame))
return expr.subs(subs_dict)
return expr
[docs]def time_derivative(expr, frame, order=1):
"""
Calculate the time derivative of a vector/scalar field function
or dyadic expression in given frame.
References
==========
https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames
Parameters
==========
expr : Vector/Dyadic/sympifyable
The expression whose time derivative is to be calculated
frame : ReferenceFrame
The reference frame to calculate the time derivative in
order : integer
The order of the derivative to be calculated
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy import Symbol
>>> q1 = Symbol('q1')
>>> u1 = dynamicsymbols('u1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> v = u1 * N.x
>>> A.set_ang_vel(N, 10*A.x)
>>> from sympy.physics.vector import time_derivative
>>> time_derivative(v, N)
u1'*N.x
>>> time_derivative(u1*A[0], N)
N_x*Derivative(u1(t), t)
>>> B = N.orientnew('B', 'Axis', [u1, N.z])
>>> from sympy.physics.vector import outer
>>> d = outer(N.x, N.x)
>>> time_derivative(d, B)
- u1'*(N.y|N.x) - u1'*(N.x|N.y)
"""
t = dynamicsymbols._t
_check_frame(frame)
if order == 0:
return expr
if order % 1 != 0 or order < 0:
raise ValueError("Unsupported value of order entered")
if isinstance(expr, Vector):
outlist = []
for i, v in enumerate(expr.args):
if v[1] == frame:
outlist += [(express(v[0], frame,
variables=True).diff(t), frame)]
else:
outlist += (time_derivative(Vector([v]), v[1]) + \
(v[1].ang_vel_in(frame) ^ Vector([v]))).args
outvec = Vector(outlist)
return time_derivative(outvec, frame, order - 1)
if isinstance(expr, Dyadic):
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += (v[0].diff(t) * (v[1] | v[2]))
ol += (v[0] * (time_derivative(v[1], frame) | v[2]))
ol += (v[0] * (v[1] | time_derivative(v[2], frame)))
return time_derivative(ol, frame, order - 1)
else:
return diff(express(expr, frame, variables=True), t, order)
[docs]def outer(vec1, vec2):
"""Outer product convenience wrapper for Vector.outer():\n"""
if not isinstance(vec1, Vector):
raise TypeError('Outer product is between two Vectors')
return vec1 | vec2
outer.__doc__ += Vector.outer.__doc__
[docs]def kinematic_equations(speeds, coords, rot_type, rot_order=''):
"""Gives equations relating the qdot's to u's for a rotation type.
Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z
Parameters
==========
speeds : list of length 3
The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
The coordinates used to define the orientation of the two frames.
rot_type : str
The type of rotation used to create the equations. Body, Space, or
Quaternion only
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
... order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']
"""
# Code below is checking and sanitizing input
approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131',
'212', '232', '313', '323', '1', '2', '3', '')
# make sure XYZ => 123 and rot_type is in lower case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.lower()
if not isinstance(speeds, (list, tuple)):
raise TypeError('Need to supply speeds in a list')
if len(speeds) != 3:
raise TypeError('Need to supply 3 body-fixed speeds')
if not isinstance(coords, (list, tuple)):
raise TypeError('Need to supply coordinates in a list')
if rot_type in ['body', 'space']:
if rot_order not in approved_orders:
raise ValueError('Not an acceptable rotation order')
if len(coords) != 3:
raise ValueError('Need 3 coordinates for body or space')
# Actual hard-coded kinematic differential equations
w1, w2, w3 = speeds
if w1 == w2 == w3 == 0:
return [S.Zero]*3
q1, q2, q3 = coords
q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords]
s1, s2, s3 = [sin(q1), sin(q2), sin(q3)]
c1, c2, c3 = [cos(q1), cos(q2), cos(q3)]
if rot_type == 'body':
if rot_order == '123':
return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 *
c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3]
if rot_order == '231':
return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2]
if rot_order == '312':
return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 *
s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2]
if rot_order == '132':
return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 *
c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2]
if rot_order == '213':
return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 *
s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3]
if rot_order == '321':
return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 *
s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2]
if rot_order == '121':
return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 *
s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2]
if rot_order == '131':
return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2]
if rot_order == '212':
return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 *
s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2]
if rot_order == '232':
return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 *
c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2]
if rot_order == '313':
return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 *
s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3]
if rot_order == '323':
return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 *
c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3]
if rot_type == 'space':
if rot_order == '123':
return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2]
if rot_order == '231':
return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2]
if rot_order == '312':
return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2]
if rot_order == '132':
return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 *
s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2]
if rot_order == '213':
return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2]
if rot_order == '321':
return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2]
if rot_order == '121':
return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2]
if rot_order == '131':
return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 *
s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2]
if rot_order == '212':
return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2]
if rot_order == '232':
return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2]
if rot_order == '313':
return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2]
if rot_order == '323':
return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2]
elif rot_type == 'quaternion':
if rot_order != '':
raise ValueError('Cannot have rotation order for quaternion')
if len(coords) != 4:
raise ValueError('Need 4 coordinates for quaternion')
# Actual hard-coded kinematic differential equations
e0, e1, e2, e3 = coords
w = Matrix(speeds + [0])
E = Matrix([[e0, -e3, e2, e1], [e3, e0, -e1, e2], [-e2, e1, e0, e3],
[-e1, -e2, -e3, e0]])
edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]])
return list(edots.T - 0.5 * w.T * E.T)
else:
raise ValueError('Not an approved rotation type for this function')
[docs]def get_motion_params(frame, **kwargs):
"""
Returns the three motion parameters - (acceleration, velocity, and
position) as vectorial functions of time in the given frame.
If a higher order differential function is provided, the lower order
functions are used as boundary conditions. For example, given the
acceleration, the velocity and position parameters are taken as
boundary conditions.
The values of time at which the boundary conditions are specified
are taken from timevalue1(for position boundary condition) and
timevalue2(for velocity boundary condition).
If any of the boundary conditions are not provided, they are taken
to be zero by default (zero vectors, in case of vectorial inputs). If
the boundary conditions are also functions of time, they are converted
to constants by substituting the time values in the dynamicsymbols._t
time Symbol.
This function can also be used for calculating rotational motion
parameters. Have a look at the Parameters and Examples for more clarity.
Parameters
==========
frame : ReferenceFrame
The frame to express the motion parameters in
acceleration : Vector
Acceleration of the object/frame as a function of time
velocity : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
position : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
timevalue1 : sympyfiable
Value of time for position boundary condition
timevalue2 : sympyfiable
Value of time for velocity boundary condition
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z
"""
##Helper functions
def _process_vector_differential(vectdiff, condition, \
variable, ordinate, frame):
"""
Helper function for get_motion methods. Finds derivative of vectdiff wrt
variable, and its integral using the specified boundary condition at
value of variable = ordinate.
Returns a tuple of - (derivative, function and integral) wrt vectdiff
"""
#Make sure boundary condition is independent of 'variable'
if condition != 0:
condition = express(condition, frame, variables=True)
#Special case of vectdiff == 0
if vectdiff == Vector(0):
return (0, 0, condition)
#Express vectdiff completely in condition's frame to give vectdiff1
vectdiff1 = express(vectdiff, frame)
#Find derivative of vectdiff
vectdiff2 = time_derivative(vectdiff, frame)
#Integrate and use boundary condition
vectdiff0 = Vector(0)
lims = (variable, ordinate, variable)
for dim in frame:
function1 = vectdiff1.dot(dim)
abscissa = dim.dot(condition).subs({variable : ordinate})
# Indefinite integral of 'function1' wrt 'variable', using
# the given initial condition (ordinate, abscissa).
vectdiff0 += (integrate(function1, lims) + abscissa) * dim
#Return tuple
return (vectdiff2, vectdiff, vectdiff0)
##Function body
_check_frame(frame)
#Decide mode of operation based on user's input
if 'acceleration' in kwargs:
mode = 2
elif 'velocity' in kwargs:
mode = 1
else:
mode = 0
#All the possible parameters in kwargs
#Not all are required for every case
#If not specified, set to default values(may or may not be used in
#calculations)
conditions = ['acceleration', 'velocity', 'position',
'timevalue', 'timevalue1', 'timevalue2']
for i, x in enumerate(conditions):
if x not in kwargs:
if i < 3:
kwargs[x] = Vector(0)
else:
kwargs[x] = S(0)
elif i < 3:
_check_vector(kwargs[x])
else:
kwargs[x] = sympify(kwargs[x])
if mode == 2:
vel = _process_vector_differential(kwargs['acceleration'],
kwargs['velocity'],
dynamicsymbols._t,
kwargs['timevalue2'], frame)[2]
pos = _process_vector_differential(vel, kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)[2]
return (kwargs['acceleration'], vel, pos)
elif mode == 1:
return _process_vector_differential(kwargs['velocity'],
kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)
else:
vel = time_derivative(kwargs['position'], frame)
acc = time_derivative(vel, frame)
return (acc, vel, kwargs['position'])
[docs]def partial_velocity(vel_vecs, gen_speeds, frame):
"""Returns a list of partial velocities with respect to the provided
generalized speeds in the given reference frame for each of the supplied
velocity vectors.
The output is a list of lists. The outer list has a number of elements
equal to the number of supplied velocity vectors. The inner lists are, for
each velocity vector, the partial derivatives of that velocity vector with
respect to the generalized speeds supplied.
Parameters
==========
vel_vecs : iterable
An iterable of velocity vectors (angular or linear).
gen_speeds : iterable
An iterable of generalized speeds.
frame : ReferenceFrame
The reference frame that the partial derivatives are going to be taken
in.
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import partial_velocity
>>> u = dynamicsymbols('u')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> vel_vecs = [P.vel(N)]
>>> gen_speeds = [u]
>>> partial_velocity(vel_vecs, gen_speeds, N)
[[N.x]]
"""
if not iterable(vel_vecs):
raise TypeError('Velocity vectors must be contained in an iterable.')
if not iterable(gen_speeds):
raise TypeError('Generalized speeds must be contained in an iterable')
vec_partials = []
for vec in vel_vecs:
partials = []
for speed in gen_speeds:
partials.append(vec.diff(speed, frame, var_in_dcm=False))
vec_partials.append(partials)
return vec_partials
[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.vector 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 iterable(esses):
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 = '\''
