```
from __future__ import print_function, division
__all__ = ['LagrangesMethod']
from sympy import diff, zeros, Matrix, eye, sympify
from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point)
[docs]class LagrangesMethod(object):
"""Lagrange's method object.
This object generates the equations of motion in a two step procedure. The
first step involves the initialization of LagrangesMethod by supplying the
Lagrangian and a list of the generalized coordinates, at the bare minimum.
If there are any constraint equations, they can be supplied as keyword
arguments. The Lagrangian multipliers are automatically generated and are
equal in number to the constraint equations.Similarly any non-conservative
forces can be supplied in a list (as described below and also shown in the
example) along with a ReferenceFrame. This is also discussed further in the
__init__ method.
Attributes
==========
mass_matrix : Matrix
The system's mass matrix
forcing : Matrix
The system's forcing vector
mass_matrix_full : Matrix
The "mass matrix" for the qdot's, qdoubledot's, and the
lagrange multipliers (lam)
forcing_full : Matrix
The forcing vector for the qdot's, qdoubledot's and
lagrange multipliers (lam)
Examples
========
This is a simple example for a one degree of freedom translational
spring-mass-damper.
In this example, we first need to do the kinematics.$
This involves creating generalized coordinates and its derivative.
Then we create a point and set its velocity in a frame::
>>> from sympy.physics.mechanics import LagrangesMethod, Lagrangian
>>> from sympy.physics.mechanics import ReferenceFrame, Particle, Point
>>> from sympy.physics.mechanics import dynamicsymbols, kinetic_energy
>>> from sympy import symbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> m, k, b = symbols('m k b')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, qd * N.x)
We need to then prepare the information as required by LagrangesMethod to
generate equations of motion.
First we create the Particle, which has a point attached to it.
Following this the lagrangian is created from the kinetic and potential
energies.
Then, a list of nonconservative forces/torques must be constructed, where
each entry in is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where
the Vectors represent the nonconservative force or torque.
>>> Pa = Particle('Pa', P, m)
>>> Pa.set_potential_energy(k * q**2 / 2.0)
>>> L = Lagrangian(N, Pa)
>>> fl = [(P, -b * qd * N.x)]
Finally we can generate the equations of motion.
First we create the LagrangesMethod object.To do this one must supply an
the Lagrangian, the list of generalized coordinates. Also supplied are the
constraint equations, the forcelist and the inertial frame, if relevant.
Next we generate Lagrange's equations of motion, such that:
Lagrange's equations of motion = 0.
We have the equations of motion at this point.
>>> l = LagrangesMethod(L, [q], forcelist = fl, frame = N)
>>> print(l.form_lagranges_equations())
Matrix([[b*Derivative(q(t), t) + 1.0*k*q(t) + m*Derivative(q(t), t, t)]])
We can also solve for the states using the 'rhs' method.
>>> print(l.rhs())
Matrix([[Derivative(q(t), t)], [(-b*Derivative(q(t), t) - 1.0*k*q(t))/m]])
Please refer to the docstrings on each method for more details.
"""
def __init__(self, Lagrangian, q_list, coneqs=None, forcelist=None, frame=None):
"""Supply the following for the initialization of LagrangesMethod
Lagrangian : Sympifyable
q_list : list
A list of the generalized coordinates
coneqs : list
A list of the holonomic and non-holonomic constraint equations.
VERY IMPORTANT NOTE- The holonomic constraints must be
differentiated with respect to time and then included in coneqs.
forcelist : list
Takes a list of (Point, Vector) or (ReferenceFrame, Vector) tuples
which represent the force at a point or torque on a frame. This
feature is primarily to account for the nonconservative forces
amd/or moments.
frame : ReferenceFrame
Supply the inertial frame. This is used to determine the
generalized forces due to non-sonservative forces.
"""
self._L = sympify(Lagrangian)
self.eom = None # initializing the eom Matrix
self._m_cd = Matrix([]) # Mass Matrix of differentiated coneqs
self._m_d = Matrix([]) # Mass Matrix of dynamic equations
self._f_cd = Matrix([]) # Forcing part of the diff coneqs
self._f_d = Matrix([]) # Forcing part of the dynamic equations
self.lam_coeffs = Matrix([]) # Initializing the coeffecients of lams
self.forcelist = forcelist
self.inertial = frame
self.lam_vec = Matrix([])
self._term1 = Matrix([])
self._term2 = Matrix([])
self._term3 = Matrix([])
self._term4 = Matrix([])
# Creating the qs, qdots and qdoubledots
q_list = list(q_list)
if not isinstance(q_list, list):
raise TypeError('Generalized coords. must be supplied in a list')
self._q = q_list
self._qdots = [diff(i, dynamicsymbols._t) for i in self._q]
self._qdoubledots = [diff(i, dynamicsymbols._t) for i in self._qdots]
self.coneqs = coneqs
[docs] def form_lagranges_equations(self):
"""Method to form Lagrange's equations of motion.
Returns a vector of equations of motion using Lagrange's equations of
the second kind.
"""
q = self._q
qd = self._qdots
qdd = self._qdoubledots
n = len(q)
#Putting the Lagrangian in a Matrix
L = Matrix([self._L])
#Determining the first term in Lagrange's EOM
self._term1 = L.jacobian(qd)
self._term1 = ((self._term1).diff(dynamicsymbols._t)).transpose()
#Determining the second term in Lagrange's EOM
self._term2 = (L.jacobian(q)).transpose()
#term1 and term2 will be there no matter what so leave them as they are
if self.coneqs is not None:
coneqs = self.coneqs
#If there are coneqs supplied then the following will be created
coneqs = list(coneqs)
if not isinstance(coneqs, list):
raise TypeError('Enter the constraint equations in a list')
o = len(coneqs)
#Creating the multipliers
self.lam_vec = Matrix(dynamicsymbols('lam1:' + str(o + 1)))
#Extracting the coeffecients of the multipliers
coneqs_mat = Matrix(coneqs)
qd = self._qdots
self.lam_coeffs = -coneqs_mat.jacobian(qd)
#Determining the third term in Lagrange's EOM
#term3 = ((self.lam_vec).transpose() * self.lam_coeffs).transpose()
self._term3 = self.lam_coeffs.transpose() * self.lam_vec
#Taking the time derivative of the constraint equations
diffconeqs = [diff(i, dynamicsymbols._t) for i in coneqs]
#Extracting the coeffecients of the qdds from the diff coneqs
diffconeqs_mat = Matrix(diffconeqs)
qdd = self._qdoubledots
self._m_cd = diffconeqs_mat.jacobian(qdd)
#The remaining terms i.e. the 'forcing' terms in diff coneqs
qddzero = dict(zip(qdd, [0] * n))
self._f_cd = -diffconeqs_mat.subs(qddzero)
else:
self._term3 = zeros(n, 1)
if self.forcelist is not None:
forcelist = self.forcelist
N = self.inertial
if not isinstance(N, ReferenceFrame):
raise TypeError('Enter a valid ReferenceFrame')
if not isinstance(forcelist, (list, tuple)):
raise TypeError('Forces must be supplied in a list of: lists'
' or tuples')
self._term4 = zeros(n, 1)
for i, v in enumerate(qd):
for j, w in enumerate(forcelist):
if isinstance(w[0], ReferenceFrame):
speed = w[0].ang_vel_in(N)
self._term4[i] += speed.diff(v, N) & w[1]
if isinstance(w[0], Point):
speed = w[0].vel(N)
self._term4[i] += speed.diff(v, N) & w[1]
else:
raise TypeError('First entry in force pair is a point'
' or frame')
else:
self._term4 = zeros(n, 1)
self.eom = self._term1 - self._term2 - self._term3 - self._term4
return self.eom
@property
[docs] def mass_matrix(self):
""" Returns the mass matrix, which is augmented by the Lagrange
multipliers, if necessary.
If the system is described by 'n' generalized coordinates and there are
no constraint equations then an n X n matrix is returned.
If there are 'n' generalized coordinates and 'm' constraint equations
have been supplied during initialization then an n X (n+m) matrix is
returned. The (n + m - 1)th and (n + m)th columns contain the
coefficients of the Lagrange multipliers.
"""
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
#The 'dynamic' mass matrix is generated by the following
self._m_d = (self.eom).jacobian(self._qdoubledots)
if len(self.lam_coeffs) != 0:
return (self._m_d).row_join((self.lam_coeffs).transpose())
else:
return self._m_d
@property
[docs] def mass_matrix_full(self):
""" Augments the coefficients of qdots to the mass_matrix. """
n = len(self._q)
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
#THE FIRST TWO ROWS OF THE MATRIX
row1 = eye(n).row_join(zeros(n, n))
row2 = zeros(n, n).row_join(self.mass_matrix)
if self.coneqs is not None:
m = len(self.coneqs)
I = eye(n).row_join(zeros(n, n + m))
below_eye = zeros(n + m, n)
A = (self.mass_matrix).col_join((self._m_cd).row_join(zeros(m, m)))
below_I = below_eye.row_join(A)
return I.col_join(below_I)
else:
A = row1.col_join(row2)
return A
@property
[docs] def forcing(self):
""" Returns the forcing vector from 'lagranges_equations' method. """
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
qdd = self._qdoubledots
qddzero = dict(zip(qdd, [0] * len(qdd)))
if self.coneqs is not None:
lam = self.lam_vec
lamzero = dict(zip(lam, [0] * len(lam)))
#The forcing terms from the eoms
self._f_d = -((self.eom).subs(qddzero)).subs(lamzero)
else:
#The forcing terms from the eoms
self._f_d = -(self.eom).subs(qddzero)
return self._f_d
@property
[docs] def forcing_full(self):
""" Augments qdots to the forcing vector above. """
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
if self.coneqs is not None:
return (Matrix(self._qdots)).col_join((self.forcing).col_join(self._f_cd))
else:
return (Matrix(self._qdots)).col_join(self.forcing)
[docs] def rhs(self, method="GE"):
""" Returns equations that can be solved numerically
Parameters
==========
method : string
The method by which matrix inversion of mass_matrix_full must be
performed such as Gauss Elimination or LU decomposition.
"""
# TODO- should probably use the matinvmul method from Kane
return ((self.mass_matrix_full).inv(method, try_block_diag=True) *
self.forcing_full)
```