Source code for sympy.physics.mechanics.kane

from __future__ import print_function, division

__all__ = ['KanesMethod']

from sympy import zeros, Matrix, diff, solve_linear_system_LU, eye
from sympy.utilities import default_sort_key
from sympy.physics.vector import (ReferenceFrame, dynamicsymbols,
        partial_velocity)
from sympy.physics.mechanics.particle import Particle
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy.physics.mechanics.functions import (msubs, find_dynamicsymbols,
        _f_list_parser)
from sympy.physics.mechanics.linearize import Linearizer
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import iterable

[docs]class KanesMethod(object): """Kane's method object. This object is used to do the "book-keeping" as you go through and form equations of motion in the way Kane presents in: Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill The attributes are for equations in the form [M] udot = forcing. Attributes ========== q, u : Matrix Matrices of the generalized coordinates and speeds bodylist : iterable Iterable of Point and RigidBody objects in the system. forcelist : iterable Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system. auxiliary : Matrix If applicable, the set of auxiliary Kane's equations used to solve for non-contributing forces. mass_matrix : Matrix The system's mass matrix forcing : Matrix The system's forcing vector mass_matrix_full : Matrix The "mass matrix" for the u's and q's forcing_full : Matrix The "forcing vector" for the u's and q's 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 speeds and coordinates and their derivatives. Then we create a point and set its velocity in a frame. >>> from sympy import symbols >>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame >>> from sympy.physics.mechanics import Point, Particle, KanesMethod >>> q, u = dynamicsymbols('q u') >>> qd, ud = dynamicsymbols('q u', 1) >>> m, c, k = symbols('m c k') >>> N = ReferenceFrame('N') >>> P = Point('P') >>> P.set_vel(N, u * N.x) Next we need to arrange/store information in the way that KanesMethod requires. The kinematic differential equations need to be stored in a dict. A list of forces/torques must be constructed, where each entry in the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the Vectors represent the Force or Torque. Next a particle needs to be created, and it needs to have a point and mass assigned to it. Finally, a list of all bodies and particles needs to be created. >>> kd = [qd - u] >>> FL = [(P, (-k * q - c * u) * N.x)] >>> pa = Particle('pa', P, m) >>> BL = [pa] Finally we can generate the equations of motion. First we create the KanesMethod object and supply an inertial frame, coordinates, generalized speeds, and the kinematic differential equations. Additional quantities such as configuration and motion constraints, dependent coordinates and speeds, and auxiliary speeds are also supplied here (see the online documentation). Next we form FR* and FR to complete: Fr + Fr* = 0. We have the equations of motion at this point. It makes sense to rearrnge them though, so we calculate the mass matrix and the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is the mass matrix, udot is a vector of the time derivatives of the generalized speeds, and forcing is a vector representing "forcing" terms. >>> KM = KanesMethod(N, q_ind=[q], u_ind=[u], kd_eqs=kd) >>> (fr, frstar) = KM.kanes_equations(FL, BL) >>> MM = KM.mass_matrix >>> forcing = KM.forcing >>> rhs = MM.inv() * forcing >>> rhs Matrix([[(-c*u(t) - k*q(t))/m]]) >>> KM.linearize(A_and_B=True, new_method=True)[0] Matrix([ [ 0, 1], [-k/m, -c/m]]) Please look at the documentation pages for more information on how to perform linearization and how to deal with dependent coordinates & speeds, and how do deal with bringing non-contributing forces into evidence. """ def __init__(self, frame, q_ind, u_ind, kd_eqs=None, q_dependent=None, configuration_constraints=None, u_dependent=None, velocity_constraints=None, acceleration_constraints=None, u_auxiliary=None): """Please read the online documentation. """ if not isinstance(frame, ReferenceFrame): raise TypeError('An intertial ReferenceFrame must be supplied') self._inertial = frame self._fr = None self._frstar = None self._forcelist = None self._bodylist = None self._initialize_vectors(q_ind, q_dependent, u_ind, u_dependent, u_auxiliary) self._initialize_constraint_matrices(configuration_constraints, velocity_constraints, acceleration_constraints) self._initialize_kindiffeq_matrices(kd_eqs) def _initialize_vectors(self, q_ind, q_dep, u_ind, u_dep, u_aux): """Initialize the coordinate and speed vectors.""" none_handler = lambda x: Matrix(x) if x else Matrix() # Initialize generalized coordinates q_dep = none_handler(q_dep) if not iterable(q_ind): raise TypeError('Generalized coordinates must be an iterable.') if not iterable(q_dep): raise TypeError('Dependent coordinates must be an iterable.') q_ind = Matrix(q_ind) self._qdep = q_dep self._q = Matrix([q_ind, q_dep]) self._qdot = self.q.diff(dynamicsymbols._t) # Initialize generalized speeds u_dep = none_handler(u_dep) if not iterable(u_ind): raise TypeError('Generalized speeds must be an iterable.') if not iterable(u_dep): raise TypeError('Dependent speeds must be an iterable.') u_ind = Matrix(u_ind) self._udep = u_dep self._u = Matrix([u_ind, u_dep]) self._udot = self.u.diff(dynamicsymbols._t) self._uaux = none_handler(u_aux) def _initialize_constraint_matrices(self, config, vel, acc): """Initializes constraint matrices.""" # Define vector dimensions o = len(self.u) m = len(self._udep) p = o - m none_handler = lambda x: Matrix(x) if x else Matrix() # Initialize configuration constraints config = none_handler(config) if len(self._qdep) != len(config): raise ValueError('There must be an equal number of dependent ' 'coordinates and configuration constraints.') self._f_h = none_handler(config) # Initialize velocity and acceleration constraints vel = none_handler(vel) acc = none_handler(acc) if len(vel) != m: raise ValueError('There must be an equal number of dependent ' 'speeds and velocity constraints.') if acc and (len(acc) != m): raise ValueError('There must be an equal number of dependent ' 'speeds and acceleration constraints.') if vel: u_zero = dict((i, 0) for i in self.u) udot_zero = dict((i, 0) for i in self._udot) self._f_nh = vel.subs(u_zero) self._k_nh = (vel - self._f_nh).jacobian(self.u) # If no acceleration constraints given, calculate them. if not acc: self._f_dnh = (self._k_nh.diff(dynamicsymbols._t) * self.u + self._f_nh.diff(dynamicsymbols._t)) self._k_dnh = self._k_nh else: self._f_dnh = acc.subs(udot_zero) self._k_dnh = (acc - self._f_dnh).jacobian(self._udot) # Form of non-holonomic constraints is B*u + C = 0. # We partition B into independent and dependent columns: # Ars is then -B_dep.inv() * B_ind, and it relates dependent speeds # to independent speeds as: udep = Ars*uind, neglecting the C term. B_ind = self._k_nh[:, :p] B_dep = self._k_nh[:, p:o] self._Ars = -B_dep.LUsolve(B_ind) else: self._f_nh = Matrix() self._k_nh = Matrix() self._f_dnh = Matrix() self._k_dnh = Matrix() self._Ars = Matrix() def _initialize_kindiffeq_matrices(self, kdeqs): """Initialize the kinematic differential equation matrices.""" if kdeqs: if len(self.q) != len(kdeqs): raise ValueError('There must be an equal number of kinematic ' 'differential equations and coordinates.') kdeqs = Matrix(kdeqs) u = self.u qdot = self._qdot # Dictionaries setting things to zero u_zero = dict((i, 0) for i in u) uaux_zero = dict((i, 0) for i in self._uaux) qdot_zero = dict((i, 0) for i in qdot) f_k = kdeqs.subs(u_zero).subs(qdot_zero) k_ku = (kdeqs.subs(qdot_zero) - f_k).jacobian(u) k_kqdot = (kdeqs.subs(u_zero) - f_k).jacobian(qdot) f_k = k_kqdot.LUsolve(f_k) k_ku = k_kqdot.LUsolve(k_ku) k_kqdot = eye(len(qdot)) self._qdot_u_map = solve_linear_system_LU( Matrix([k_kqdot.T, -(k_ku * u + f_k).T]).T, qdot) self._f_k = f_k.subs(uaux_zero) self._k_ku = k_ku.subs(uaux_zero) self._k_kqdot = k_kqdot else: self._qdot_u_map = None self._f_k = Matrix() self._k_ku = Matrix() self._k_kqdot = Matrix() def _form_fr(self, fl): """Form the generalized active force.""" if not iterable(fl): raise TypeError('Force pairs must be supplied in an iterable.') N = self._inertial # pull out relevant velocities for constructing partial velocities vel_list, f_list = _f_list_parser(fl, N) vel_list = [i.subs(self._qdot_u_map) for i in vel_list] # Fill Fr with dot product of partial velocities and forces o = len(self.u) b = len(f_list) FR = zeros(o, 1) partials = partial_velocity(vel_list, self.u, N) for i in range(o): FR[i] = sum(partials[j][i] & f_list[j] for j in range(b)) # In case there are dependent speeds if self._udep: p = o - len(self._udep) FRtilde = FR[:p, 0] FRold = FR[p:o, 0] FRtilde += self._Ars.T * FRold FR = FRtilde self._forcelist = fl self._fr = FR return FR def _form_frstar(self, bl): """Form the generalized inertia force.""" if not iterable(bl): raise TypeError('Bodies must be supplied in an iterable.') t = dynamicsymbols._t N = self._inertial # Dicts setting things to zero udot_zero = dict((i, 0) for i in self._udot) uaux_zero = dict((i, 0) for i in self._uaux) uauxdot = [diff(i, t) for i in self._uaux] uauxdot_zero = dict((i, 0) for i in uauxdot) # Dictionary of q' and q'' to u and u' q_ddot_u_map = dict((k.diff(t), v.diff(t)) for (k, v) in self._qdot_u_map.items()) q_ddot_u_map.update(self._qdot_u_map) # Fill up the list of partials: format is a list with num elements # equal to number of entries in body list. Each of these elements is a # list - either of length 1 for the translational components of # particles or of length 2 for the translational and rotational # components of rigid bodies. The inner most list is the list of # partial velocities. def get_partial_velocity(body): if isinstance(body, RigidBody): vlist = [body.masscenter.vel(N), body.frame.ang_vel_in(N)] elif isinstance(body, Particle): vlist = [body.point.vel(N),] else: raise TypeError('The body list may only contain either ' 'RigidBody or Particle as list elements.') v = [vel.subs(self._qdot_u_map) for vel in vlist] return partial_velocity(v, self.u, N) partials = [get_partial_velocity(body) for body in bl] # Compute fr_star in two components: # fr_star = -(MM*u' + nonMM) o = len(self.u) MM = zeros(o, o) nonMM = zeros(o, 1) zero_uaux = lambda expr: expr.subs(uaux_zero) zero_udot_uaux = lambda expr: expr.subs(udot_zero).subs(uaux_zero) for i, body in enumerate(bl): if isinstance(body, RigidBody): M = zero_uaux(body.mass) I = zero_uaux(body.central_inertia) vel = zero_uaux(body.masscenter.vel(N)) omega = zero_uaux(body.frame.ang_vel_in(N)) acc = zero_udot_uaux(body.masscenter.acc(N)) inertial_force = (M.diff(t) * vel + M * acc) inertial_torque = zero_uaux((I.dt(body.frame) & omega) + (I & body.frame.ang_acc_in(N)).subs(udot_zero) + (omega ^ (I & omega))) for j in range(o): tmp_vel = zero_uaux(partials[i][0][j]) tmp_ang = zero_uaux(I & partials[i][1][j]) for k in range(o): # translational MM[j, k] += M * (tmp_vel & partials[i][0][k]) # rotational MM[j, k] += (tmp_ang & partials[i][1][k]) nonMM[j] += inertial_force & partials[i][0][j] nonMM[j] += inertial_torque & partials[i][1][j] else: M = zero_uaux(body.mass) vel = zero_uaux(body.point.vel(N)) acc = zero_udot_uaux(body.point.acc(N)) inertial_force = (M.diff(t) * vel + M * acc) for j in range(o): temp = zero_uaux(partials[i][0][j]) for k in range(o): MM[j, k] += M * (temp & partials[i][0][k]) nonMM[j] += inertial_force & partials[i][0][j] # Compose fr_star out of MM and nonMM MM = zero_uaux(msubs(MM, q_ddot_u_map)) nonMM = msubs(msubs(nonMM, q_ddot_u_map), udot_zero, uauxdot_zero, uaux_zero) fr_star = -(MM * Matrix(self._udot).subs(uauxdot_zero) + nonMM) # If there are dependent speeds, we need to find fr_star_tilde if self._udep: p = o - len(self._udep) fr_star_ind = fr_star[:p, 0] fr_star_dep = fr_star[p:o, 0] fr_star = fr_star_ind + (self._Ars.T * fr_star_dep) # Apply the same to MM MMi = MM[:p, :] MMd = MM[p:o, :] MM = MMi + (self._Ars.T * MMd) self._bodylist = bl self._frstar = fr_star self._k_d = MM self._f_d = -msubs(self._fr + self._frstar, udot_zero) return fr_star
[docs] def to_linearizer(self): """Returns an instance of the Linearizer class, initiated from the data in the KanesMethod 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).""" if (self._fr is None) or (self._frstar is None): raise ValueError('Need to compute Fr, Fr* first.') # Get required equation components. The Kane's method class breaks # these into pieces. Need to reassemble f_c = self._f_h if self._f_nh and self._k_nh: f_v = self._f_nh + self._k_nh*Matrix(self.u) else: f_v = Matrix() if self._f_dnh and self._k_dnh: f_a = self._f_dnh + self._k_dnh*Matrix(self._udot) else: f_a = Matrix() # Dicts to sub to zero, for splitting up expressions u_zero = dict((i, 0) for i in self.u) ud_zero = dict((i, 0) for i in self._udot) qd_zero = dict((i, 0) for i in self._qdot) qd_u_zero = dict((i, 0) for i in Matrix([self._qdot, self.u])) # Break the kinematic differential eqs apart into f_0 and f_1 f_0 = msubs(self._f_k, u_zero) + self._k_kqdot*Matrix(self._qdot) f_1 = msubs(self._f_k, qd_zero) + self._k_ku*Matrix(self.u) # Break the dynamic differential eqs into f_2 and f_3 f_2 = msubs(self._frstar, qd_u_zero) f_3 = msubs(self._frstar, ud_zero) + self._fr f_4 = zeros(len(f_2), 1) # Get the required vector components q = self.q u = self.u if self._qdep: q_i = q[:-len(self._qdep)] else: q_i = q q_d = self._qdep if self._udep: u_i = u[:-len(self._udep)] else: u_i = u u_d = self._udep # Form dictionary to set auxiliary speeds & their derivatives to 0. uaux = self._uaux uauxdot = uaux.diff(dynamicsymbols._t) uaux_zero = dict((i, 0) for i in Matrix([uaux, uauxdot])) # Checking for dynamic symbols outside the dynamic differential # equations; throws error if there is. sym_list = set(Matrix([q, self._qdot, u, self._udot, uaux, uauxdot])) if any(find_dynamicsymbols(i, sym_list) for i in [self._k_kqdot, self._k_ku, self._f_k, self._k_dnh, self._f_dnh, self._k_d]): raise ValueError('Cannot have dynamicsymbols outside dynamic \ forcing vector.') # Find all other dynamic symbols, forming the forcing vector r. # Sort r to make it canonical. r = list(find_dynamicsymbols(self._f_d.subs(uaux_zero), sym_list)) 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)
[docs] def linearize(self, **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. As part of the deprecation cycle, the new method will not be used unless the kwarg ``new_method`` is set to True. If the kwarg is missing, or set to false, the old linearization method will be used. After next release the need for this kwarg will be removed. For more documentation, please see the ``Linearizer`` class.""" if 'new_method' not in kwargs or not kwargs['new_method']: # User is still using old code. SymPyDeprecationWarning('The linearize class method has changed ' 'to a new interface, the old method is deprecated. To ' 'use the new method, set the kwarg `new_method=True`. ' 'For more information, read the docstring ' 'of `linearize`.').warn() return self._old_linearize() # Remove the new method flag, before passing kwargs to linearize kwargs.pop('new_method') linearizer = self.to_linearizer() result = linearizer.linearize(**kwargs) return result + (linearizer.r,)
def _old_linearize(self): """Old method to linearize the equations of motion. Returns a tuple of (f_lin_A, f_lin_B, y) for forming [M]qudot = [f_lin_A]qu + [f_lin_B]y. Deprecated in favor of new method using Linearizer class. Please change your code to use the new `linearize` method.""" if (self._fr is None) or (self._frstar is None): raise ValueError('Need to compute Fr, Fr* first.') # Note that this is now unneccessary, and it should never be # encountered; I still think it should be in here in case the user # manually sets these matrices incorrectly. for i in self.q: if self._k_kqdot.diff(i) != 0 * self._k_kqdot: raise ValueError('Matrix K_kqdot must not depend on any q.') t = dynamicsymbols._t uaux = self._uaux uauxdot = [diff(i, t) for i in uaux] # dictionary of auxiliary speeds & derivatives which are equal to zero subdict = dict(zip(uaux[:] + uauxdot[:], [0] * (len(uaux) + len(uauxdot)))) # Checking for dynamic symbols outside the dynamic differential # equations; throws error if there is. insyms = set(self.q[:] + self._qdot[:] + self.u[:] + self._udot[:] + uaux[:] + uauxdot) if any(find_dynamicsymbols(i, insyms) for i in [self._k_kqdot, self._k_ku, self._f_k, self._k_dnh, self._f_dnh, self._k_d]): raise ValueError('Cannot have dynamicsymbols outside dynamic \ forcing vector.') other_dyns = list(find_dynamicsymbols(self._f_d.subs(subdict), insyms)) # make it canonically ordered so the jacobian is canonical other_dyns.sort(key=default_sort_key) for i in other_dyns: if diff(i, dynamicsymbols._t) in other_dyns: raise ValueError('Cannot have derivatives of specified ' 'quantities when linearizing forcing terms.') o = len(self.u) # number of speeds n = len(self.q) # number of coordinates l = len(self._qdep) # number of configuration constraints m = len(self._udep) # number of motion constraints qi = Matrix(self.q[: n - l]) # independent coords qd = Matrix(self.q[n - l: n]) # dependent coords; could be empty ui = Matrix(self.u[: o - m]) # independent speeds ud = Matrix(self.u[o - m: o]) # dependent speeds; could be empty qdot = Matrix(self._qdot) # time derivatives of coordinates # with equations in the form MM udot = forcing, expand that to: # MM_full [q,u].T = forcing_full. This combines coordinates and # speeds together for the linearization, which is necessary for the # linearization process, due to dependent coordinates. f1 is the rows # from the kinematic differential equations, f2 is the rows from the # dynamic differential equations (and differentiated non-holonomic # constraints). f1 = self._k_ku * Matrix(self.u) + self._f_k f2 = self._f_d # Only want to do this if these matrices have been filled in, which # occurs when there are dependent speeds if m != 0: f2 = self._f_d.col_join(self._f_dnh) fnh = self._f_nh + self._k_nh * Matrix(self.u) f1 = f1.subs(subdict) f2 = f2.subs(subdict) fh = self._f_h.subs(subdict) fku = (self._k_ku * Matrix(self.u)).subs(subdict) fkf = self._f_k.subs(subdict) # In the code below, we are applying the chain rule by hand on these # things. All the matrices have been changed into vectors (by # multiplying the dynamic symbols which it is paired with), so we can # take the jacobian of them. The basic operation is take the jacobian # of the f1, f2 vectors wrt all of the q's and u's. f1 is a function of # q, u, and t; f2 is a function of q, qdot, u, and t. In the code # below, we are not considering perturbations in t. So if f1 is a # function of the q's, u's but some of the q's or u's could be # dependent on other q's or u's (qd's might be dependent on qi's, ud's # might be dependent on ui's or qi's), so what we do is take the # jacobian of the f1 term wrt qi's and qd's, the jacobian wrt the qd's # gets multiplied by the jacobian of qd wrt qi, this is extended for # the ud's as well. dqd_dqi is computed by taking a taylor expansion of # the holonomic constraint equations about q*, treating q* - q as dq, # separating into dqd (depedent q's) and dqi (independent q's) and the # rearranging for dqd/dqi. This is again extended for the speeds. # First case: configuration and motion constraints if (l != 0) and (m != 0): fh_jac_qi = fh.jacobian(qi) fh_jac_qd = fh.jacobian(qd) fnh_jac_qi = fnh.jacobian(qi) fnh_jac_qd = fnh.jacobian(qd) fnh_jac_ui = fnh.jacobian(ui) fnh_jac_ud = fnh.jacobian(ud) fku_jac_qi = fku.jacobian(qi) fku_jac_qd = fku.jacobian(qd) fku_jac_ui = fku.jacobian(ui) fku_jac_ud = fku.jacobian(ud) fkf_jac_qi = fkf.jacobian(qi) fkf_jac_qd = fkf.jacobian(qd) f1_jac_qi = f1.jacobian(qi) f1_jac_qd = f1.jacobian(qd) f1_jac_ui = f1.jacobian(ui) f1_jac_ud = f1.jacobian(ud) f2_jac_qi = f2.jacobian(qi) f2_jac_qd = f2.jacobian(qd) f2_jac_ui = f2.jacobian(ui) f2_jac_ud = f2.jacobian(ud) f2_jac_qdot = f2.jacobian(qdot) dqd_dqi = - fh_jac_qd.LUsolve(fh_jac_qi) dud_dqi = fnh_jac_ud.LUsolve(fnh_jac_qd * dqd_dqi - fnh_jac_qi) dud_dui = - fnh_jac_ud.LUsolve(fnh_jac_ui) dqdot_dui = - self._k_kqdot.inv() * (fku_jac_ui + fku_jac_ud * dud_dui) dqdot_dqi = - self._k_kqdot.inv() * (fku_jac_qi + fkf_jac_qi + (fku_jac_qd + fkf_jac_qd) * dqd_dqi + fku_jac_ud * dud_dqi) f1_q = f1_jac_qi + f1_jac_qd * dqd_dqi + f1_jac_ud * dud_dqi f1_u = f1_jac_ui + f1_jac_ud * dud_dui f2_q = (f2_jac_qi + f2_jac_qd * dqd_dqi + f2_jac_qdot * dqdot_dqi + f2_jac_ud * dud_dqi) f2_u = f2_jac_ui + f2_jac_ud * dud_dui + f2_jac_qdot * dqdot_dui # Second case: configuration constraints only elif l != 0: dqd_dqi = - fh.jacobian(qd).LUsolve(fh.jacobian(qi)) dqdot_dui = - self._k_kqdot.inv() * fku.jacobian(ui) dqdot_dqi = - self._k_kqdot.inv() * (fku.jacobian(qi) + fkf.jacobian(qi) + (fku.jacobian(qd) + fkf.jacobian(qd)) * dqd_dqi) f1_q = (f1.jacobian(qi) + f1.jacobian(qd) * dqd_dqi) f1_u = f1.jacobian(ui) f2_jac_qdot = f2.jacobian(qdot) f2_q = (f2.jacobian(qi) + f2.jacobian(qd) * dqd_dqi + f2.jac_qdot * dqdot_dqi) f2_u = f2.jacobian(ui) + f2_jac_qdot * dqdot_dui # Third case: motion constraints only elif m != 0: dud_dqi = fnh.jacobian(ud).LUsolve(- fnh.jacobian(qi)) dud_dui = - fnh.jacobian(ud).LUsolve(fnh.jacobian(ui)) dqdot_dui = - self._k_kqdot.inv() * (fku.jacobian(ui) + fku.jacobian(ud) * dud_dui) dqdot_dqi = - self._k_kqdot.inv() * (fku.jacobian(qi) + fkf.jacobian(qi) + fku.jacobian(ud) * dud_dqi) f1_jac_ud = f1.jacobian(ud) f2_jac_qdot = f2.jacobian(qdot) f2_jac_ud = f2.jacobian(ud) f1_q = f1.jacobian(qi) + f1_jac_ud * dud_dqi f1_u = f1.jacobian(ui) + f1_jac_ud * dud_dui f2_q = (f2.jacobian(qi) + f2_jac_qdot * dqdot_dqi + f2_jac_ud * dud_dqi) f2_u = (f2.jacobian(ui) + f2_jac_ud * dud_dui + f2_jac_qdot * dqdot_dui) # Fourth case: No constraints else: dqdot_dui = - self._k_kqdot.inv() * fku.jacobian(ui) dqdot_dqi = - self._k_kqdot.inv() * (fku.jacobian(qi) + fkf.jacobian(qi)) f1_q = f1.jacobian(qi) f1_u = f1.jacobian(ui) f2_jac_qdot = f2.jacobian(qdot) f2_q = f2.jacobian(qi) + f2_jac_qdot * dqdot_dqi f2_u = f2.jacobian(ui) + f2_jac_qdot * dqdot_dui f_lin_A = -(f1_q.row_join(f1_u)).col_join(f2_q.row_join(f2_u)) if other_dyns: f1_oths = f1.jacobian(other_dyns) f2_oths = f2.jacobian(other_dyns) f_lin_B = -f1_oths.col_join(f2_oths) else: f_lin_B = Matrix() return (f_lin_A, f_lin_B, Matrix(other_dyns))
[docs] def kanes_equations(self, FL, BL): """ Method to form Kane's equations, Fr + Fr* = 0. Returns (Fr, Fr*). In the case where auxiliary generalized speeds are present (say, s auxiliary speeds, o generalized speeds, and m motion constraints) the length of the returned vectors will be o - m + s in length. The first o - m equations will be the constrained Kane's equations, then the s auxiliary Kane's equations. These auxiliary equations can be accessed with the auxiliary_eqs(). Parameters ========== FL : list Takes in a list of (Point, Vector) or (ReferenceFrame, Vector) tuples which represent the force at a point or torque on a frame. BL : list A list of all RigidBody's and Particle's in the system. """ if not self._k_kqdot: raise AttributeError('Create an instance of KanesMethod with ' 'kinematic differential equations to use this method.') fr = self._form_fr(FL) frstar = self._form_frstar(BL) if self._uaux: if not self._udep: km = KanesMethod(self._inertial, self.q, self._uaux, u_auxiliary=self._uaux) else: km = KanesMethod(self._inertial, self.q, self._uaux, u_auxiliary=self._uaux, u_dependent=self._udep, velocity_constraints=(self._k_nh * self.u + self._f_nh)) km._qdot_u_map = self._qdot_u_map self._km = km fraux = km._form_fr(FL) frstaraux = km._form_frstar(BL) self._aux_eq = fraux + frstaraux self._fr = fr.col_join(fraux) self._frstar = frstar.col_join(frstaraux) return (self._fr, self._frstar)
[docs] def rhs(self, inv_method=None): """ Returns the system's equations of motion in first order form. The output of this will be the right hand side of: [qdot, udot].T = f(q, u, t) Or, the equations of motion in first order form. The right hand side is what is needed by most numerical ODE integrators. 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 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
[docs] def kindiffdict(self): """Returns a dictionary mapping q' to u.""" if not self._qdot_u_map: raise AttributeError('Create an instance of KanesMethod with ' 'kinematic differential equations to use this method.') return self._qdot_u_map
@property
[docs] def auxiliary_eqs(self): """A matrix containing the auxiliary equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') if not self._uaux: raise ValueError('No auxiliary speeds have been declared.') return self._aux_eq
@property
[docs] def mass_matrix(self): """The mass matrix of the system.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') return Matrix([self._k_d, self._k_dnh])
@property
[docs] def mass_matrix_full(self): """The mass matrix of the system, augmented by the kinematic differential equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') o = len(self.u) n = len(self.q) return ((self._k_kqdot).row_join(zeros(n, o))).col_join((zeros(o, n)).row_join(self.mass_matrix))
@property
[docs] def forcing(self): """The forcing vector of the system.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') return -Matrix([self._f_d, self._f_dnh])
@property
[docs] def forcing_full(self): """The forcing vector of the system, augmented by the kinematic differential equations.""" if not self._fr or not self._frstar: raise ValueError('Need to compute Fr, Fr* first.') f1 = self._k_ku * Matrix(self.u) + self._f_k return -Matrix([f1, self._f_d, self._f_dnh])
@property def q(self): return self._q @property def u(self): return self._u @property def bodylist(self): return self._bodylist @property def forcelist(self): return self._forcelist