Source code for sympy.physics.mechanics.lagrange

from __future__ import print_function, division

__all__ = ['LagrangesMethod']

from sympy import diff, zeros, Matrix, eye, sympify
from sympy.physics.vector import dynamicsymbols, ReferenceFrame
from sympy.physics.mechanics.functions import (find_dynamicsymbols, msubs,
from sympy.physics.mechanics.linearize import Linearizer
from sympy.utilities import default_sort_key
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import iterable

[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 the generalized coordinates, at the bare minimum. If there are any constraint equations, they can be supplied as keyword arguments. The Lagrange multipliers are automatically generated and are equal in number to the constraint equations. Similarly any non-conservative forces can be supplied in an iterable (as described below and also shown in the example) along with a ReferenceFrame. This is also discussed further in the __init__ method. Attributes ========== q, u : Matrix Matrices of the generalized coordinates and speeds forcelist : iterable Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system. 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 their derivatives. 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, an iterable of nonconservative forces/torques must be constructed, where each item is a (Point, Vector) or (ReferenceFrame, Vector) tuple, with the Vectors representing the nonconservative forces or torques. >>> Pa = Particle('Pa', P, m) >>> Pa.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 the Lagrangian, and the generalized coordinates. The constraint equations, the forcelist, and the inertial frame may also be provided, 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, qs, coneqs=None, forcelist=None, frame=None, hol_coneqs=None, nonhol_coneqs=None): """Supply the following for the initialization of LagrangesMethod Lagrangian : Sympifyable qs: array_like The generalized coordinates hol_coneqs: array_like, optional The holonomic constraint equations nonhol_coneqs: array_like, optional The nonholonomic constraint equations forcelist : iterable, optional Takes an iterable 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 and/or moments. frame : ReferenceFrame, optional Supply the inertial frame. This is used to determine the generalized forces due to non-conservative forces. """ self._L = Matrix([sympify(Lagrangian)]) self.eom = None 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() # The coeffecients of the multipliers forcelist = forcelist if forcelist else [] if not iterable(forcelist): raise TypeError('Force pairs must be supplied in an iterable.') self._forcelist = forcelist if frame and not isinstance(frame, ReferenceFrame): raise TypeError('frame must be a valid ReferenceFrame') 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 if not iterable(qs): raise TypeError('Generalized coordinates must be an iterable') self._q = Matrix(qs) self._qdots = self.q.diff(dynamicsymbols._t) self._qdoubledots = self._qdots.diff(dynamicsymbols._t) # Deal with constraint equations if coneqs: SymPyDeprecationWarning("The `coneqs` kwarg is deprecated in " "favor of `hol_coneqs` and `nonhol_coneqs`. Please " "update your code").warn() self.coneqs = coneqs else: mat_build = lambda x: Matrix(x) if x else Matrix() hol_coneqs = mat_build(hol_coneqs) nonhol_coneqs = mat_build(nonhol_coneqs) self.coneqs = Matrix([hol_coneqs.diff(dynamicsymbols._t), nonhol_coneqs]) self._hol_coneqs = hol_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. """ qds = self._qdots qdd_zero = dict((i, 0) for i in self._qdoubledots) n = len(self.q) # Internally we represent the EOM as four terms: # EOM = term1 - term2 - term3 - term4 = 0 # First term self._term1 = self._L.jacobian(qds) self._term1 = self._term1.diff(dynamicsymbols._t).T # Second term self._term2 = self._L.jacobian(self.q).T # Third term if self.coneqs: coneqs = self.coneqs m = len(coneqs) # Creating the multipliers self.lam_vec = Matrix(dynamicsymbols('lam1:' + str(m + 1))) self.lam_coeffs = -coneqs.jacobian(qds) self._term3 = self.lam_coeffs.T * self.lam_vec # Extracting the coeffecients of the qdds from the diff coneqs diffconeqs = coneqs.diff(dynamicsymbols._t) self._m_cd = diffconeqs.jacobian(self._qdoubledots) # The remaining terms i.e. the 'forcing' terms in diff coneqs self._f_cd = -diffconeqs.subs(qdd_zero) else: self._term3 = zeros(n, 1) # Fourth term if self.forcelist: N = self.inertial self._term4 = zeros(n, 1) for i, qd in enumerate(qds): flist = zip(*_f_list_parser(self.forcelist, N)) self._term4[i] = sum(v.diff(qd, N) & f for (v, f) in flist) else: self._term4 = zeros(n, 1) # Form the dynamic mass and forcing matrices without_lam = self._term1 - self._term2 - self._term4 self._m_d = without_lam.jacobian(self._qdoubledots) self._f_d = -without_lam.subs(qdd_zero) # Form the EOM self.eom = without_lam - self._term3 return self.eom
[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') if self.coneqs: return (self._m_d).row_join(self.lam_coeffs.T) else: return self._m_d
[docs] def mass_matrix_full(self): """Augments the coefficients of qdots to the mass_matrix.""" if self.eom is None: raise ValueError('Need to compute the equations of motion first') n = len(self.q) m = len(self.coneqs) row1 = eye(n).row_join(zeros(n, n + m)) row2 = zeros(n, n).row_join(self.mass_matrix) if self.coneqs: row3 = zeros(m, n).row_join(self._m_cd).row_join(zeros(m, m)) return row1.col_join(row2).col_join(row3) else: return row1.col_join(row2)
[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') return self._f_d
[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: return self._qdots.col_join(self.forcing).col_join(self._f_cd) else: return self._qdots.col_join(self.forcing)
[docs] def to_linearizer(self, q_ind=None, qd_ind=None, q_dep=None, qd_dep=None): """Returns an instance of the Linearizer class, initiated from the data in the LagrangesMethod class. This may be more desirable than using the linearize class method, as the Linearizer object will allow more efficient recalculation (i.e. about varying operating points). Parameters ========== q_ind, qd_ind : array_like, optional The independent generalized coordinates and speeds. q_dep, qd_dep : array_like, optional The dependent generalized coordinates and speeds. """ # Compose vectors t = dynamicsymbols._t q = self.q u = self._qdots ud = u.diff(t) # Get vector of lagrange multipliers lams = self.lam_vec mat_build = lambda x: Matrix(x) if x else Matrix() q_i = mat_build(q_ind) q_d = mat_build(q_dep) u_i = mat_build(qd_ind) u_d = mat_build(qd_dep) # Compose general form equations f_c = self._hol_coneqs f_v = self.coneqs f_a = f_v.diff(t) f_0 = u f_1 = -u f_2 = self._term1 f_3 = -(self._term2 + self._term4) f_4 = -self._term3 # Check that there are an appropriate number of independent and # dependent coordinates if len(q_d) != len(f_c) or len(u_d) != len(f_v): raise ValueError(("Must supply {:} dependent coordinates, and " + "{:} dependent speeds").format(len(f_c), len(f_v))) if set(Matrix([q_i, q_d])) != set(q): raise ValueError("Must partition q into q_ind and q_dep, with " + "no extra or missing symbols.") if set(Matrix([u_i, u_d])) != set(u): raise ValueError("Must partition qd into qd_ind and qd_dep, " + "with no extra or missing symbols.") # Find all other dynamic symbols, forming the forcing vector r. # Sort r to make it canonical. insyms = set(Matrix([q, u, ud, lams])) r = list(find_dynamicsymbols(f_3, insyms)) r.sort(key=default_sort_key) # Check for any derivatives of variables in r that are also found in r. for i in r: if diff(i, dynamicsymbols._t) in r: raise ValueError('Cannot have derivatives of specified \ quantities when linearizing forcing terms.') return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i, q_d, u_i, u_d, r, lams)
[docs] def linearize(self, q_ind=None, qd_ind=None, q_dep=None, qd_dep=None, **kwargs): """Linearize the equations of motion about a symbolic operating point. If kwarg A_and_B is False (default), returns M, A, B, r for the linearized form, M*[q', u']^T = A*[q_ind, u_ind]^T + B*r. If kwarg A_and_B is True, returns A, B, r for the linearized form dx = A*x + B*r, where x = [q_ind, u_ind]^T. Note that this is computationally intensive if there are many symbolic parameters. For this reason, it may be more desirable to use the default A_and_B=False, returning M, A, and B. Values may then be substituted in to these matrices, and the state space form found as A = P.T*M.inv()*A, B = P.T*M.inv()*B, where P = Linearizer.perm_mat. In both cases, r is found as all dynamicsymbols in the equations of motion that are not part of q, u, q', or u'. They are sorted in canonical form. The operating points may be also entered using the ``op_point`` kwarg. This takes a dictionary of {symbol: value}, or a an iterable of such dictionaries. The values may be numberic or symbolic. The more values you can specify beforehand, the faster this computation will run. For more documentation, please see the ``Linearizer`` class.""" linearizer = self.to_linearizer(q_ind, qd_ind, q_dep, qd_dep) result = linearizer.linearize(**kwargs) return result + (linearizer.r,)
[docs] def solve_multipliers(self, op_point=None, sol_type='dict'): """Solves for the values of the lagrange multipliers symbolically at the specified operating point Parameters ========== op_point : dict or iterable of dicts, optional Point at which to solve at. The operating point is specified as a dictionary or iterable of dictionaries of {symbol: value}. The value may be numeric or symbolic itself. sol_type : str, optional Solution return type. Valid options are: - 'dict': A dict of {symbol : value} (default) - 'Matrix': An ordered column matrix of the solution """ # Determine number of multipliers k = len(self.lam_vec) if k == 0: raise ValueError("System has no lagrange multipliers to solve for.") # Compose dict of operating conditions if isinstance(op_point, dict): op_point_dict = op_point elif iterable(op_point): op_point_dict = {} for op in op_point: op_point_dict.update(op) elif op_point is None: op_point_dict = {} else: raise TypeError("op_point must be either a dictionary or an " "iterable of dictionaries.") # Compose the system to be solved mass_matrix = self.mass_matrix.col_join((-self.lam_coeffs.row_join( zeros(k, k)))) force_matrix = self.forcing.col_join(self._f_cd) # Sub in the operating point mass_matrix = msubs(mass_matrix, op_point_dict) force_matrix = msubs(force_matrix, op_point_dict) # Solve for the multipliers sol_list = mass_matrix.LUsolve(-force_matrix)[-k:] if sol_type == 'dict': return dict(zip(self.lam_vec, sol_list)) elif sol_type == 'Matrix': return Matrix(sol_list) else: raise ValueError("Unknown sol_type {:}.".format(sol_type))
[docs] def rhs(self, inv_method=None, **kwargs): """Returns equations that can be solved numerically Parameters ========== inv_method : str The specific sympy inverse matrix calculation method to use. For a list of valid methods, see :meth:`~sympy.matrices.matrices.MatrixBase.inv` """ if 'method' in kwargs: # The method kwarg is deprecated in favor of inv_method. SymPyDeprecationWarning(feature="method kwarg", useinstead="inv_method kwarg", deprecated_since_version="0.7.6").warn() # For now accept both inv_method = kwargs['method'] if inv_method is None: self._rhs = self.mass_matrix_full.LUsolve(self.forcing_full) else: self._rhs = (self.mass_matrix_full.inv(inv_method, try_block_diag=True) * self.forcing_full) return self._rhs
@property def q(self): return self._q @property def u(self): return self._qdots @property def forcelist(self): return self._forcelist