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from sympy.core.backend import eye, Matrix, zeros
from sympy.physics.mechanics import dynamicsymbols
from sympy.physics.mechanics.functions import find_dynamicsymbols
__all__ = ['SymbolicSystem']
class SymbolicSystem:
"""SymbolicSystem is a class that contains all the information about a
system in a symbolic format such as the equations of motions and the bodies
and loads in the system.
There are three ways that the equations of motion can be described for
Symbolic System:
[1] Explicit form where the kinematics and dynamics are combined
x' = F_1(x, t, r, p)
[2] Implicit form where the kinematics and dynamics are combined
M_2(x, p) x' = F_2(x, t, r, p)
[3] Implicit form where the kinematics and dynamics are separate
M_3(q, p) u' = F_3(q, u, t, r, p)
q' = G(q, u, t, r, p)
where
x : states, e.g. [q, u]
t : time
r : specified (exogenous) inputs
p : constants
q : generalized coordinates
u : generalized speeds
F_1 : right hand side of the combined equations in explicit form
F_2 : right hand side of the combined equations in implicit form
F_3 : right hand side of the dynamical equations in implicit form
M_2 : mass matrix of the combined equations in implicit form
M_3 : mass matrix of the dynamical equations in implicit form
G : right hand side of the kinematical differential equations
Parameters
==========
coord_states : ordered iterable of functions of time
This input will either be a collection of the coordinates or states
of the system depending on whether or not the speeds are also
given. If speeds are specified this input will be assumed to
be the coordinates otherwise this input will be assumed to
be the states.
right_hand_side : Matrix
This variable is the right hand side of the equations of motion in
any of the forms. The specific form will be assumed depending on
whether a mass matrix or coordinate derivatives are given.
speeds : ordered iterable of functions of time, optional
This is a collection of the generalized speeds of the system. If
given it will be assumed that the first argument (coord_states)
will represent the generalized coordinates of the system.
mass_matrix : Matrix, optional
The matrix of the implicit forms of the equations of motion (forms
[2] and [3]). The distinction between the forms is determined by
whether or not the coordinate derivatives are passed in. If
they are given form [3] will be assumed otherwise form [2] is
assumed.
coordinate_derivatives : Matrix, optional
The right hand side of the kinematical equations in explicit form.
If given it will be assumed that the equations of motion are being
entered in form [3].
alg_con : Iterable, optional
The indexes of the rows in the equations of motion that contain
algebraic constraints instead of differential equations. If the
equations are input in form [3], it will be assumed the indexes are
referencing the mass_matrix/right_hand_side combination and not the
coordinate_derivatives.
output_eqns : Dictionary, optional
Any output equations that are desired to be tracked are stored in a
dictionary where the key corresponds to the name given for the
specific equation and the value is the equation itself in symbolic
form
coord_idxs : Iterable, optional
If coord_states corresponds to the states rather than the
coordinates this variable will tell SymbolicSystem which indexes of
the states correspond to generalized coordinates.
speed_idxs : Iterable, optional
If coord_states corresponds to the states rather than the
coordinates this variable will tell SymbolicSystem which indexes of
the states correspond to generalized speeds.
bodies : iterable of Body/Rigidbody objects, optional
Iterable containing the bodies of the system
loads : iterable of load instances (described below), optional
Iterable containing the loads of the system where forces are given
by (point of application, force vector) and torques are given by
(reference frame acting upon, torque vector). Ex [(point, force),
(ref_frame, torque)]
Attributes
==========
coordinates : Matrix, shape(n, 1)
This is a matrix containing the generalized coordinates of the system
speeds : Matrix, shape(m, 1)
This is a matrix containing the generalized speeds of the system
states : Matrix, shape(o, 1)
This is a matrix containing the state variables of the system
alg_con : List
This list contains the indices of the algebraic constraints in the
combined equations of motion. The presence of these constraints
requires that a DAE solver be used instead of an ODE solver.
If the system is given in form [3] the alg_con variable will be
adjusted such that it is a representation of the combined kinematics
and dynamics thus make sure it always matches the mass matrix
entered.
dyn_implicit_mat : Matrix, shape(m, m)
This is the M matrix in form [3] of the equations of motion (the mass
matrix or generalized inertia matrix of the dynamical equations of
motion in implicit form).
dyn_implicit_rhs : Matrix, shape(m, 1)
This is the F vector in form [3] of the equations of motion (the right
hand side of the dynamical equations of motion in implicit form).
comb_implicit_mat : Matrix, shape(o, o)
This is the M matrix in form [2] of the equations of motion.
This matrix contains a block diagonal structure where the top
left block (the first rows) represent the matrix in the
implicit form of the kinematical equations and the bottom right
block (the last rows) represent the matrix in the implicit form
of the dynamical equations.
comb_implicit_rhs : Matrix, shape(o, 1)
This is the F vector in form [2] of the equations of motion. The top
part of the vector represents the right hand side of the implicit form
of the kinemaical equations and the bottom of the vector represents the
right hand side of the implicit form of the dynamical equations of
motion.
comb_explicit_rhs : Matrix, shape(o, 1)
This vector represents the right hand side of the combined equations of
motion in explicit form (form [1] from above).
kin_explicit_rhs : Matrix, shape(m, 1)
This is the right hand side of the explicit form of the kinematical
equations of motion as can be seen in form [3] (the G matrix).
output_eqns : Dictionary
If output equations were given they are stored in a dictionary where
the key corresponds to the name given for the specific equation and
the value is the equation itself in symbolic form
bodies : Tuple
If the bodies in the system were given they are stored in a tuple for
future access
loads : Tuple
If the loads in the system were given they are stored in a tuple for
future access. This includes forces and torques where forces are given
by (point of application, force vector) and torques are given by
(reference frame acted upon, torque vector).
Example
=======
As a simple example, the dynamics of a simple pendulum will be input into a
SymbolicSystem object manually. First some imports will be needed and then
symbols will be set up for the length of the pendulum (l), mass at the end
of the pendulum (m), and a constant for gravity (g). ::
>>> from sympy import Matrix, sin, symbols
>>> from sympy.physics.mechanics import dynamicsymbols, SymbolicSystem
>>> l, m, g = symbols('l m g')
The system will be defined by an angle of theta from the vertical and a
generalized speed of omega will be used where omega = theta_dot. ::
>>> theta, omega = dynamicsymbols('theta omega')
Now the equations of motion are ready to be formed and passed to the
SymbolicSystem object. ::
>>> kin_explicit_rhs = Matrix([omega])
>>> dyn_implicit_mat = Matrix([l**2 * m])
>>> dyn_implicit_rhs = Matrix([-g * l * m * sin(theta)])
>>> symsystem = SymbolicSystem([theta], dyn_implicit_rhs, [omega],
... dyn_implicit_mat)
Notes
=====
m : number of generalized speeds
n : number of generalized coordinates
o : number of states
"""
def __init__(self, coord_states, right_hand_side, speeds=None,
mass_matrix=None, coordinate_derivatives=None, alg_con=None,
output_eqns={}, coord_idxs=None, speed_idxs=None, bodies=None,
loads=None):
"""Initializes a SymbolicSystem object"""
# Extract information on speeds, coordinates and states
if speeds is None:
self._states = Matrix(coord_states)
if coord_idxs is None:
self._coordinates = None
else:
coords = [coord_states[i] for i in coord_idxs]
self._coordinates = Matrix(coords)
if speed_idxs is None:
self._speeds = None
else:
speeds_inter = [coord_states[i] for i in speed_idxs]
self._speeds = Matrix(speeds_inter)
else:
self._coordinates = Matrix(coord_states)
self._speeds = Matrix(speeds)
self._states = self._coordinates.col_join(self._speeds)
# Extract equations of motion form
if coordinate_derivatives is not None:
self._kin_explicit_rhs = coordinate_derivatives
self._dyn_implicit_rhs = right_hand_side
self._dyn_implicit_mat = mass_matrix
self._comb_implicit_rhs = None
self._comb_implicit_mat = None
self._comb_explicit_rhs = None
elif mass_matrix is not None:
self._kin_explicit_rhs = None
self._dyn_implicit_rhs = None
self._dyn_implicit_mat = None
self._comb_implicit_rhs = right_hand_side
self._comb_implicit_mat = mass_matrix
self._comb_explicit_rhs = None
else:
self._kin_explicit_rhs = None
self._dyn_implicit_rhs = None
self._dyn_implicit_mat = None
self._comb_implicit_rhs = None
self._comb_implicit_mat = None
self._comb_explicit_rhs = right_hand_side
# Set the remainder of the inputs as instance attributes
if alg_con is not None and coordinate_derivatives is not None:
alg_con = [i + len(coordinate_derivatives) for i in alg_con]
self._alg_con = alg_con
self.output_eqns = output_eqns
# Change the body and loads iterables to tuples if they are not tuples
# already
if not isinstance(bodies, tuple) and bodies is not None:
bodies = tuple(bodies)
if not isinstance(loads, tuple) and loads is not None:
loads = tuple(loads)
self._bodies = bodies
self._loads = loads
@property
def coordinates(self):
"""Returns the column matrix of the generalized coordinates"""
if self._coordinates is None:
raise AttributeError("The coordinates were not specified.")
else:
return self._coordinates
@property
def speeds(self):
"""Returns the column matrix of generalized speeds"""
if self._speeds is None:
raise AttributeError("The speeds were not specified.")
else:
return self._speeds
@property
def states(self):
"""Returns the column matrix of the state variables"""
return self._states
@property
def alg_con(self):
"""Returns a list with the indices of the rows containing algebraic
constraints in the combined form of the equations of motion"""
return self._alg_con
@property
def dyn_implicit_mat(self):
"""Returns the matrix, M, corresponding to the dynamic equations in
implicit form, M x' = F, where the kinematical equations are not
included"""
if self._dyn_implicit_mat is None:
raise AttributeError("dyn_implicit_mat is not specified for "
"equations of motion form [1] or [2].")
else:
return self._dyn_implicit_mat
@property
def dyn_implicit_rhs(self):
"""Returns the column matrix, F, corresponding to the dynamic equations
in implicit form, M x' = F, where the kinematical equations are not
included"""
if self._dyn_implicit_rhs is None:
raise AttributeError("dyn_implicit_rhs is not specified for "
"equations of motion form [1] or [2].")
else:
return self._dyn_implicit_rhs
@property
def comb_implicit_mat(self):
"""Returns the matrix, M, corresponding to the equations of motion in
implicit form (form [2]), M x' = F, where the kinematical equations are
included"""
if self._comb_implicit_mat is None:
if self._dyn_implicit_mat is not None:
num_kin_eqns = len(self._kin_explicit_rhs)
num_dyn_eqns = len(self._dyn_implicit_rhs)
zeros1 = zeros(num_kin_eqns, num_dyn_eqns)
zeros2 = zeros(num_dyn_eqns, num_kin_eqns)
inter1 = eye(num_kin_eqns).row_join(zeros1)
inter2 = zeros2.row_join(self._dyn_implicit_mat)
self._comb_implicit_mat = inter1.col_join(inter2)
return self._comb_implicit_mat
else:
raise AttributeError("comb_implicit_mat is not specified for "
"equations of motion form [1].")
else:
return self._comb_implicit_mat
@property
def comb_implicit_rhs(self):
"""Returns the column matrix, F, corresponding to the equations of
motion in implicit form (form [2]), M x' = F, where the kinematical
equations are included"""
if self._comb_implicit_rhs is None:
if self._dyn_implicit_rhs is not None:
kin_inter = self._kin_explicit_rhs
dyn_inter = self._dyn_implicit_rhs
self._comb_implicit_rhs = kin_inter.col_join(dyn_inter)
return self._comb_implicit_rhs
else:
raise AttributeError("comb_implicit_mat is not specified for "
"equations of motion in form [1].")
else:
return self._comb_implicit_rhs
def compute_explicit_form(self):
"""If the explicit right hand side of the combined equations of motion
is to provided upon initialization, this method will calculate it. This
calculation can potentially take awhile to compute."""
if self._comb_explicit_rhs is not None:
raise AttributeError("comb_explicit_rhs is already formed.")
inter1 = getattr(self, 'kin_explicit_rhs', None)
if inter1 is not None:
inter2 = self._dyn_implicit_mat.LUsolve(self._dyn_implicit_rhs)
out = inter1.col_join(inter2)
else:
out = self._comb_implicit_mat.LUsolve(self._comb_implicit_rhs)
self._comb_explicit_rhs = out
@property
def comb_explicit_rhs(self):
"""Returns the right hand side of the equations of motion in explicit
form, x' = F, where the kinematical equations are included"""
if self._comb_explicit_rhs is None:
raise AttributeError("Please run .combute_explicit_form before "
"attempting to access comb_explicit_rhs.")
else:
return self._comb_explicit_rhs
@property
def kin_explicit_rhs(self):
"""Returns the right hand side of the kinematical equations in explicit
form, q' = G"""
if self._kin_explicit_rhs is None:
raise AttributeError("kin_explicit_rhs is not specified for "
"equations of motion form [1] or [2].")
else:
return self._kin_explicit_rhs
def dynamic_symbols(self):
"""Returns a column matrix containing all of the symbols in the system
that depend on time"""
# Create a list of all of the expressions in the equations of motion
if self._comb_explicit_rhs is None:
eom_expressions = (self.comb_implicit_mat[:] +
self.comb_implicit_rhs[:])
else:
eom_expressions = (self._comb_explicit_rhs[:])
functions_of_time = set()
for expr in eom_expressions:
functions_of_time = functions_of_time.union(
find_dynamicsymbols(expr))
functions_of_time = functions_of_time.union(self._states)
return tuple(functions_of_time)
def constant_symbols(self):
"""Returns a column matrix containing all of the symbols in the system
that do not depend on time"""
# Create a list of all of the expressions in the equations of motion
if self._comb_explicit_rhs is None:
eom_expressions = (self.comb_implicit_mat[:] +
self.comb_implicit_rhs[:])
else:
eom_expressions = (self._comb_explicit_rhs[:])
constants = set()
for expr in eom_expressions:
constants = constants.union(expr.free_symbols)
constants.remove(dynamicsymbols._t)
return tuple(constants)
@property
def bodies(self):
"""Returns the bodies in the system"""
if self._bodies is None:
raise AttributeError("bodies were not specified for the system.")
else:
return self._bodies
@property
def loads(self):
"""Returns the loads in the system"""
if self._loads is None:
raise AttributeError("loads were not specified for the system.")
else:
return self._loads
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