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 = msubs(kdeqs, u_zero, qdot_zero)
            k_ku = (msubs(kdeqs, qdot_zero) - f_k).jacobian(u)
            k_kqdot = (msubs(kdeqs, 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 = msubs(f_k, uaux_zero)
            self._k_ku = msubs(k_ku, 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 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))

    def _form_fr(self, fl):
        """Form the generalized active force."""
        if fl != None and (len(fl) == 0 or not iterable(fl)):
            raise ValueError('Force pairs must be supplied in an '
                'non-empty iterable or None.')

        N = self._inertial
        # pull out relevant velocities for constructing partial velocities
        vel_list, f_list = _f_list_parser(fl, N)
        vel_list = [msubs(i, 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 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 = [msubs(vel, self._qdot_u_map) for vel in vlist]
     return partial_velocity(v, self.u, N)

    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(msubs(self._f_d, 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)

    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 = [msubs(vel, 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: msubs(expr, uaux_zero)
        zero_udot_uaux = lambda expr: msubs(msubs(expr, udot_zero), 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) +
                    msubs(I & body.frame.ang_acc_in(N), 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 * msubs(Matrix(self._udot), 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

    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)

            # When calling kanes_equations, another class instance will be
            # created if auxiliary u's are present. In this case, the
            # computation of kinetic differential equation matrices will be
            # skipped as this was computed during the original KanesMethod
            # object, and the qd_u_map will not be available.
            if self._qdot_u_map is not None:
                vel = msubs(vel, self._qdot_u_map)

            self._f_nh = msubs(vel, 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:
                if self._qdot_u_map is not None:
                    acc = msubs(acc, self._qdot_u_map)
                self._f_dnh = msubs(acc, 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 _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(msubs(self._f_d, 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 = msubs(f1, subdict)
        f2 = msubs(f2, subdict)
        fh = msubs(self._f_h, subdict)
        fku = msubs(self._k_ku * Matrix(self.u), subdict)
        fkf = msubs(self._f_k, 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)
#.........这里部分代码省略.........

    def linearize(self, op_point=None, A_and_B=False, simplify=False):
        """Linearize the system about the operating point. Note that
        q_op, u_op, qd_op, ud_op must satisfy the equations of motion.
        These may be either symbolic or numeric.

        Parameters
        ----------
        op_point : dict or iterable of dicts, optional
            Dictionary or iterable of dictionaries containing the operating
            point conditions. These will be substituted in to the linearized
            system before the linearization is complete. Leave blank if you
            want a completely symbolic form. Note that any reduction in
            symbols (whether substituted for numbers or expressions with a
            common parameter) will result in faster runtime.

        A_and_B : bool, optional
            If A_and_B=False (default), (M, A, B) is returned for forming
            [M]*[q, u]^T = [A]*[q_ind, u_ind]^T + [B]r. If A_and_B=True,
            (A, B) is returned for forming dx = [A]x + [B]r, where
            x = [q_ind, u_ind]^T.

        simplify : bool, optional
            Determines if returned values are simplified before return.
            For large expressions this may be time consuming. Default is False.

        Potential Issues
        ----------------
            Note that the process of solving with A_and_B=True 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. More values may then be
            substituted in to these matrices later on. The state space form can
            then be found as A = P.T*M.LUsolve(A), B = P.T*M.LUsolve(B), where
            P = Linearizer.perm_mat.
        """

        # Run the setup if needed:
        if not self._setup_done:
            self._setup()

        # Compose dict of operating conditions
        if isinstance(op_point, dict):
            op_point_dict = op_point
        elif isinstance(op_point, collections.Iterable):
            op_point_dict = {}
            for op in op_point:
                op_point_dict.update(op)
        else:
            op_point_dict = {}

        # Extract dimension variables
        l, m, n, o, s, k = self._dims

        # Rename terms to shorten expressions
        M_qq = self._M_qq
        M_uqc = self._M_uqc
        M_uqd = self._M_uqd
        M_uuc = self._M_uuc
        M_uud = self._M_uud
        M_uld = self._M_uld
        A_qq = self._A_qq
        A_uqc = self._A_uqc
        A_uqd = self._A_uqd
        A_qu = self._A_qu
        A_uuc = self._A_uuc
        A_uud = self._A_uud
        B_u = self._B_u
        C_0 = self._C_0
        C_1 = self._C_1
        C_2 = self._C_2

        # Build up Mass Matrix
        #     |M_qq    0_nxo   0_nxk|
        # M = |M_uqc   M_uuc   0_mxk|
        #     |M_uqd   M_uud   M_uld|
        if o != 0:
            col2 = Matrix([zeros(n, o), M_uuc, M_uud])
        if k != 0:
            col3 = Matrix([zeros(n + m, k), M_uld])
        if n != 0:
            col1 = Matrix([M_qq, M_uqc, M_uqd])
            if o != 0 and k != 0:
                M = col1.row_join(col2).row_join(col3)
            elif o != 0:
                M = col1.row_join(col2)
            else:
                M = col1
        elif k != 0:
            M = col2.row_join(col3)
        else:
            M = col2
        M_eq = msubs(M, op_point_dict)

        # Build up state coefficient matrix A
        #     |(A_qq + A_qu*C_1)*C_0       A_qu*C_2|
        # A = |(A_uqc + A_uuc*C_1)*C_0    A_uuc*C_2|
        #     |(A_uqd + A_uud*C_1)*C_0    A_uud*C_2|
        # Col 1 is only defined if n != 0
        if n != 0:
            r1c1 = A_qq
#.........这里部分代码省略.........