def get_gebm2uvlm_gains(self, data):
r"""
Provides:
- the gain matrices required to connect the linearised GEBM and UVLM
inputs/outputs
- the stiffening and damping factors to be added to the linearised
GEBM equations in order to account for non-zero aerodynamic loads at
the linearisation point.
The function produces the gain matrices:
- ``Kdisp``: gains from GEBM to UVLM grid displacements
- ``Kvel_disp``: influence of GEBM dofs displacements to UVLM grid
velocities.
- ``Kvel_vel``: influence of GEBM dofs displacements to UVLM grid
displacements.
- ``Kforces`` (UVLM->GEBM) dimensions are the transpose than the
Kdisp and Kvel* matrices. Hence, when allocation this term, ``ii``
and ``jj`` indices will unintuitively refer to columns and rows,
respectively.
And the stiffening/damping terms accounting for non-zero aerodynamic
forces at the linearisation point:
- ``Kss``: stiffness factor (flexible dof -> flexible dof) accounting
for non-zero forces at the linearisation point.
- ``Csr``: damping factor (rigid dof -> flexible dof)
- ``Crs``: damping factor (flexible dof -> rigid dof)
- ``Crr``: damping factor (rigid dof -> rigid dof)
Stiffening and damping related terms due to the non-zero aerodynamic forces at the linearisation point:
.. math::
\mathbf{F}_{A,n} = C^{AG}(\mathbf{\chi})\sum_j \mathbf{f}_{G,j} \rightarrow
\delta\mathbf{F}_{A,n} = C^{AG}_0 \sum_j \delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\chi}(C^{AG}\sum_j
\mathbf{f}_{G,j}^0)\delta\chi
The term multiplied by the variation in the quaternion, :math:`\delta\chi`, couples the forces with the rigid
body equations and becomes part of :math:`\mathbf{C}_{sr}`.
Similarly, the linearisation of the moments results in expression that contribute to the stiffness and
damping matrices.
.. math::
\mathbf{M}_{B,n} = \sum_j \tilde{X}_B C^{BA}(\Psi)C^{AG}(\chi)\mathbf{f}_{G,j}
.. math::
\delta\mathbf{M}_{B,n} = \sum_j \tilde{X}_B\left(C_0^{BG}\delta\mathbf{f}_{G,j}
+ \frac{\partial}{\partial\Psi}(C^{BA}\delta\mathbf{f}^0_{A,j})\delta\Psi
+ \frac{\partial}{\partial\chi}(C^{BA}_0 C^{AG} \mathbf{f}_{G,j})\delta\chi\right)
The linearised equations of motion for the geometrically exact beam model take the input term :math:`\delta
\mathbf{Q}_n = \{\delta\mathbf{F}_{A,n},\, T_0^T\delta\mathbf{M}_{B,n}\}`, which means that the moments
should be provided as :math:`T^T(\Psi)\mathbf{M}_B` instead of :math:`\mathbf{M}_A = C^{AB}\mathbf{M}_B`,
where :math:`T(\Psi)` is the tangential operator.
.. math::
\delta(T^T\mathbf{M}_B) = T^T_0\delta\mathbf{M}_B
+ \frac{\partial}{\partial\Psi}(T^T\delta\mathbf{M}_B^0)\delta\Psi
is the linearised expression for the moments, where the first term would correspond to the input terms to the
beam equations and the second arises due to the non-zero aerodynamic moment at the linearisation point and
must be subtracted (since it comes from the forces) to form part of :math:`\mathbf{K}_{ss}`. In addition, the
:math:`\delta\mathbf{M}_B` term depends on both :math:`\delta\Psi` and :math:`\delta\chi`, therefore those
terms would also contribute to :math:`\mathbf{K}_{ss}` and :math:`\mathbf{C}_{sr}`, respectively.
The contribution from the total forces and moments will be accounted for in :math:`\mathbf{C}_{rr}` and
:math:`\mathbf{C}_{rs}`.
.. math::
\delta\mathbf{F}_{tot,A} = \sum_n\left(C^{GA}_0 \sum_j \delta\mathbf{f}_{G,j}
+ \frac{\partial}{\partial\chi}(C^{AG}\sum_j
\mathbf{f}_{G,j}^0)\delta\chi\right)
Therefore, after running this method, the beam matrices will be updated as:
>>> K_beam[:flex_dof, :flex_dof] += Kss
>>> C_beam[:flex_dof, -rigid_dof:] += Csr
>>> C_beam[-rigid_dof:, :flex_dof] += Crs
>>> C_beam[-rigid_dof:, -rigid_dof:] += Crr
Track body option
The ``track_body`` setting restricts the UVLM grid to linear translation motions and therefore should be used to
ensure that the forces are computed using the reference linearisation frame.
The UVLM and beam are linearised about a reference equilibrium condition. The UVLM is defined in the inertial
reference frame while the beam employs the body attached frame and therefore a projection from one frame onto
another is required during the coupling process.
However, the inputs to the UVLM (i.e. the lattice grid coordinates) are obtained from the beam deformation which
is expressed in A frame and therefore the grid coordinates need to be projected onto the inertial frame ``G``.
As the beam rotates, the projection onto the ``G`` frame of the lattice grid coordinates will result in a grid
that is not coincident with that at the linearisation reference and therefore the grid coordinates must be
projected onto the original frame, which will be referred to as ``U``. The transformation between the inertial
frame ``G`` and the ``U`` frame is a function of the rotation of the ``A`` frame and the original position:
.. math:: C^{UG}(\chi) = C^{GA}(\chi_0)C^{AG}(\chi)
Therefore, the grid coordinates obtained in ``A`` frame and projected onto the ``G`` frame can be transformed
to the ``U`` frame using
.. math:: \zeta_U = C^{UG}(\chi) \zeta_G
which allows the grid lattice coordinates to be projected onto the original linearisation frame.
In a similar fashion, the output lattice vertex forces of the UVLM are defined in the original linearisation
frame ``U`` and need to be transformed onto the inertial frame ``G`` prior to projecting them onto the ``A``
frame to use them as the input forces to the beam system.
.. math:: \boldsymbol{f}_G = C^{GU}(\chi)\boldsymbol{f}_U
The linearisation of the above relations lead to the following expressions that have to be added to the
coupling matrices:
* ``Kdisp_vel`` terms:
.. math::
\delta\boldsymbol{\zeta}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}}
\left(C^{AG}\boldsymbol{\zeta}_{G,0}\right)\delta\boldsymbol{\chi} + \delta\boldsymbol{\zeta}_G
* ``Kvel_vel`` terms:
.. math::
\delta\dot{\boldsymbol{\zeta}}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}}
\left(C^{AG}\dot{\boldsymbol{\zeta}}_{G,0}\right)\delta\boldsymbol{\chi}
+ \delta\dot{\boldsymbol{\zeta}}_G
The transformation of the forces and moments introduces terms that are functions of the orientation and
are included as stiffening and damping terms in the beam's matrices:
* ``Csr`` damping terms relating to translation forces:
.. math::
C_{sr}^{tra} -= \frac{\partial}{\partial\boldsymbol{\chi}}
\left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi}
* ``Csr`` damping terms related to moments:
.. math::
C_{sr}^{rot} -= T^\top\widetilde{\mathbf{X}}_B C^{BG}
\frac{\partial}{\partial\boldsymbol{\chi}}
\left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi}
The ``track_body`` setting.
When ``track_body`` is enabled, the UVLM grid is no longer coincident with the inertial reference frame
throughout the simulation but rather it is able to rotate as the ``A`` frame rotates. This is to simulate a free
flying vehicle, where, for instance, the orientation does not affect the aerodynamics. The UVLM defined in this
frame of reference, named ``U``, satisfies the following convention:
* The ``U`` frame is coincident with the ``G`` frame at the time of linearisation.
* The ``U`` frame rotates as the ``A`` frame rotates.
Transformations related to the ``U`` frame of reference:
* The angle between the ``U`` frame and the ``A`` frame is always constant and equal
to :math:`\boldsymbol{\Theta}_0`.
* The angle between the ``A`` frame and the ``G`` frame is :math:`\boldsymbol{\Theta}=\boldsymbol{\Theta}_0
+ \delta\boldsymbol{\Theta}`
* The projection of a vector expressed in the ``G`` frame onto the ``U`` frame is expressed by:
.. math:: \boldsymbol{v}^U = C^{GA}_0 C^{AG} \boldsymbol{v}^G
* The reverse, a projection of a vector expressed in the ``U`` frame onto the ``G`` frame, is expressed by
.. math:: \boldsymbol{v}^U = C^{GA} C^{AG}_0 \boldsymbol{v}^U
The effect this has on the aeroelastic coupling between the UVLM and the structural dynamics is that the
orientation and change of orientation of the vehicle has no effect on the aerodynamics. The aerodynamics are
solely affected by the contribution of the 6-rigid body velocities (as well as the flexible DOFs velocities).
"""
aero = data.aero
structure = data.structure
tsaero = self.uvlm.tsaero0
tsstr = self.beam.tsstruct0
Kzeta = self.uvlm.sys.Kzeta
num_dof_str = self.beam.sys.num_dof_str
num_dof_rig = self.beam.sys.num_dof_rig
num_dof_flex = self.beam.sys.num_dof_flex
use_euler = self.beam.sys.use_euler
# allocate output
Kdisp = np.zeros((3 * Kzeta, num_dof_str))
Kdisp_vel = np.zeros(
(3 * Kzeta, num_dof_str)) # Orientation is in velocity DOFs
Kvel_disp = np.zeros((3 * Kzeta, num_dof_str))
Kvel_vel = np.zeros((3 * Kzeta, num_dof_str))
Kforces = np.zeros((num_dof_str, 3 * Kzeta))
Kss = np.zeros((num_dof_flex, num_dof_flex))
Csr = np.zeros((num_dof_flex, num_dof_rig))
Crs = np.zeros((num_dof_rig, num_dof_flex))
Crr = np.zeros((num_dof_rig, num_dof_rig))
Krs = np.zeros((num_dof_rig, num_dof_flex))
# get projection matrix A->G
# (and other quantities indep. from nodal position)
Cga = algebra.quat2rotation(tsstr.quat) # NG 6-8-19 removing .T
Cag = Cga.T
# for_pos=tsstr.for_pos
for_vel = tsstr.for_vel[:3]
for_rot = tsstr.for_vel[3:]
skew_for_rot = algebra.skew(for_rot)
Der_vel_Ra = np.dot(Cga, skew_for_rot)
Faero = np.zeros(3)
FaeroA = np.zeros(3)
# GEBM degrees of freedom
jj_for_tra = range(num_dof_str - num_dof_rig,
num_dof_str - num_dof_rig + 3)
jj_for_rot = range(num_dof_str - num_dof_rig + 3,
num_dof_str - num_dof_rig + 6)
if use_euler:
jj_euler = range(num_dof_str - 3, num_dof_str)
euler = algebra.quat2euler(tsstr.quat)
tsstr.euler = euler
else:
jj_quat = range(num_dof_str - 4, num_dof_str)
jj = 0 # nodal dof index
for node_glob in range(structure.num_node):
### detect bc at node (and no. of dofs)
bc_here = structure.boundary_conditions[node_glob]
if bc_here == 1: # clamp (only rigid-body)
dofs_here = 0
jj_tra, jj_rot = [], []
# continue
elif bc_here == -1 or bc_here == 0: # (rigid+flex body)
dofs_here = 6
jj_tra = 6 * structure.vdof[node_glob] + np.array([0, 1, 2],
dtype=int)
jj_rot = 6 * structure.vdof[node_glob] + np.array([3, 4, 5],
dtype=int)
else:
raise NameError('Invalid boundary condition (%d) at node %d!' \
% (bc_here, node_glob))
jj += dofs_here
# retrieve element and local index
ee, node_loc = structure.node_master_elem[node_glob, :]
# get position, crv and rotation matrix
Ra = tsstr.pos[node_glob, :] # in A FoR, w.r.t. origin A-G
Rg = np.dot(Cag.T, Ra) # in G FoR, w.r.t. origin A-G
psi = tsstr.psi[ee, node_loc, :]
psi_dot = tsstr.psi_dot[ee, node_loc, :]
Cab = algebra.crv2rotation(psi)
Cba = Cab.T
Cbg = np.dot(Cab.T, Cag)
Tan = algebra.crv2tan(psi)
track_body = self.settings['track_body']
### str -> aero mapping
# some nodes may be linked to multiple surfaces...
for str2aero_here in aero.struct2aero_mapping[node_glob]:
# detect surface/span-wise coordinate (ss,nn)
nn, ss = str2aero_here['i_n'], str2aero_here['i_surf']
# print('%.2d,%.2d'%(nn,ss))
# surface panelling
M = aero.aero_dimensions[ss][0]
N = aero.aero_dimensions[ss][1]
Kzeta_start = 3 * sum(self.uvlm.sys.MS.KKzeta[:ss])
shape_zeta = (3, M + 1, N + 1)
for mm in range(M + 1):
# get bound vertex index
ii_vert = [
Kzeta_start + np.ravel_multi_index(
(cc, mm, nn), shape_zeta) for cc in range(3)
]
# get position vectors
zetag = tsaero.zeta[ss][:, mm,
nn] # in G FoR, w.r.t. origin A-G
zetaa = np.dot(Cag, zetag) # in A FoR, w.r.t. origin A-G
Xg = zetag - Rg # in G FoR, w.r.t. origin B
Xb = np.dot(Cbg, Xg) # in B FoR, w.r.t. origin B
# get rotation terms
Xbskew = algebra.skew(Xb)
XbskewTan = np.dot(Xbskew, Tan)
# get velocity terms
zetag_dot = tsaero.zeta_dot[ss][:, mm, nn] - Cga.dot(
for_vel) # in G FoR, w.r.t. origin A-G
zetaa_dot = np.dot(
Cag, zetag_dot) # in A FoR, w.r.t. origin A-G
# get aero force
faero = tsaero.forces[ss][:3, mm, nn]
Faero += faero
faero_a = np.dot(Cag, faero)
FaeroA += faero_a
maero_g = np.cross(Xg, faero)
maero_b = np.dot(Cbg, maero_g)
### ---------------------------------------- allocate Kdisp
if bc_here != 1:
# wrt pos - Eq 25 second term
Kdisp[np.ix_(ii_vert, jj_tra)] += Cga
# wrt psi - Eq 26
Kdisp[np.ix_(ii_vert,
jj_rot)] -= np.dot(Cbg.T, XbskewTan)
# w.r.t. position of FoR A (w.r.t. origin G)
# null as A and G have always same origin in SHARPy
# # ### w.r.t. quaternion (attitude changes)
if use_euler:
Kdisp_vel[np.ix_(ii_vert, jj_euler)] += \
algebra.der_Ceuler_by_v(tsstr.euler, zetaa)
# Track body - project inputs as for A not moving
if track_body:
Kdisp_vel[np.ix_(ii_vert, jj_euler)] += \
Cga.dot(algebra.der_Peuler_by_v(tsstr.euler, zetag))
else:
# Equation 25
# Kdisp[np.ix_(ii_vert, jj_quat)] += \
# algebra.der_Cquat_by_v(tsstr.quat, zetaa)
Kdisp_vel[np.ix_(ii_vert, jj_quat)] += \
algebra.der_Cquat_by_v(tsstr.quat, zetaa)
# Track body - project inputs as for A not moving
if track_body:
Kdisp_vel[np.ix_(ii_vert, jj_quat)] += \
Cga.dot(algebra.der_CquatT_by_v(tsstr.quat, zetag))
### ------------------------------------ allocate Kvel_disp
if bc_here != 1:
# # wrt pos
Kvel_disp[np.ix_(ii_vert, jj_tra)] += Der_vel_Ra
# wrt psi (at zero psi_dot)
Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \
np.dot(Cga,
np.dot(skew_for_rot,
np.dot(Cab, XbskewTan)))
# # wrt psi (psi_dot contributions - verified)
Kvel_disp[np.ix_(ii_vert, jj_rot)] += np.dot(
Cbg.T,
np.dot(algebra.skew(np.dot(XbskewTan, psi_dot)),
Tan))
if np.linalg.norm(psi) >= 1e-6:
Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \
np.dot(Cbg.T,
np.dot(Xbskew,
algebra.der_Tan_by_xv(psi, psi_dot)))
# # w.r.t. position of FoR A (w.r.t. origin G)
# # null as A and G have always same origin in SHARPy
# # ### w.r.t. quaternion (attitude changes) - Eq 30
if use_euler:
Kvel_vel[np.ix_(ii_vert, jj_euler)] += \
algebra.der_Ceuler_by_v(tsstr.euler, zetaa_dot)
# Track body if ForA is rotating
if track_body:
Kvel_vel[np.ix_(ii_vert, jj_euler)] += \
Cga.dot(algebra.der_Peuler_by_v(tsstr.euler, zetag_dot))
else:
Kvel_vel[np.ix_(ii_vert, jj_quat)] += \
algebra.der_Cquat_by_v(tsstr.quat, zetaa_dot)
# Track body if ForA is rotating
if track_body:
Kvel_vel[np.ix_(ii_vert, jj_quat)] += \
Cga.dot(algebra.der_CquatT_by_v(tsstr.quat, zetag_dot))
### ------------------------------------- allocate Kvel_vel
if bc_here != 1:
# wrt pos_dot
Kvel_vel[np.ix_(ii_vert, jj_tra)] += Cga
# # wrt crv_dot
Kvel_vel[np.ix_(ii_vert,
jj_rot)] -= np.dot(Cbg.T, XbskewTan)
# # wrt velocity of FoR A
Kvel_vel[np.ix_(ii_vert, jj_for_tra)] += Cga
Kvel_vel[np.ix_(ii_vert, jj_for_rot)] -= \
np.dot(Cga, algebra.skew(zetaa))
# wrt rate of change of quaternion: not implemented!
### -------------------------------------- allocate Kforces
if bc_here != 1:
# nodal forces
Kforces[np.ix_(jj_tra, ii_vert)] += Cag
# nodal moments
Kforces[np.ix_(jj_rot, ii_vert)] += \
np.dot(Tan.T, np.dot(Cbg, algebra.skew(Xg)))
# or, equivalently, np.dot( algebra.skew(Xb),Cbg)
# total forces
Kforces[np.ix_(jj_for_tra, ii_vert)] += Cag
# total moments
Kforces[np.ix_(jj_for_rot, ii_vert)] += \
np.dot(Cag, algebra.skew(zetag))
# quaternion equation
# null, as not dep. on external forces
### --------------------------------------- allocate Kstiff
### flexible dof equations (Kss and Csr)
if bc_here != 1:
# nodal forces
if use_euler:
if not track_body:
Csr[jj_tra, -3:] -= algebra.der_Peuler_by_v(
tsstr.euler, faero)
# Csr[jj_tra, -3:] -= algebra.der_Ceuler_by_v(tsstr.euler, Cga.T.dot(faero))
else:
if not track_body:
Csr[jj_tra, -4:] -= algebra.der_CquatT_by_v(
tsstr.quat, faero)
# Track body
# if track_body:
# Csr[jj_tra, -4:] -= algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero))
### moments
TanTXbskew = np.dot(Tan.T, Xbskew)
# contrib. of TanT (dpsi) - Eq 37 - Integration of UVLM and GEBM
Kss[np.ix_(jj_rot, jj_rot)] -= algebra.der_TanT_by_xv(
psi, maero_b)
# contrib of delta aero moment (dpsi) - Eq 36
Kss[np.ix_(jj_rot, jj_rot)] -= \
np.dot(TanTXbskew, algebra.der_CcrvT_by_v(psi, np.dot(Cag, faero)))
# contribution of delta aero moment (dquat)
if use_euler:
if not track_body:
Csr[jj_rot, -3:] -= \
np.dot(TanTXbskew,
np.dot(Cba,
algebra.der_Peuler_by_v(tsstr.euler, faero)))
# if track_body:
# Csr[jj_rot, -3:] -= \
# np.dot(TanTXbskew,
# np.dot(Cbg,
# algebra.der_Peuler_by_v(tsstr.euler, Cga.T.dot(faero))))
else:
if not track_body:
Csr[jj_rot, -4:] -= \
np.dot(TanTXbskew,
np.dot(Cba,
algebra.der_CquatT_by_v(tsstr.quat, faero)))
# Track body
# if track_body:
# Csr[jj_rot, -4:] -= \
# np.dot(TanTXbskew,
# np.dot(Cbg,
# algebra.der_CquatT_by_v(tsstr.quat, Cga.T.dot(faero))))
### rigid body eqs (Crs and Crr)
if bc_here != 1:
# Changed Crs to Krs - NG 14/5/19
# moments contribution due to delta_Ra (+ sign intentional)
Krs[3:6, jj_tra] += algebra.skew(faero_a)
# moment contribution due to delta_psi (+ sign intentional)
Krs[3:6,
jj_rot] += np.dot(algebra.skew(faero_a),
algebra.der_Ccrv_by_v(psi, Xb))
if use_euler:
if not track_body:
# total force
Crr[:3, -3:] -= algebra.der_Peuler_by_v(
tsstr.euler, faero)
# total moment contribution due to change in euler angles
Crr[3:6, -3:] -= algebra.der_Peuler_by_v(
tsstr.euler, np.cross(zetag, faero))
Crr[3:6, -3:] += np.dot(
np.dot(Cag, algebra.skew(faero)),
algebra.der_Peuler_by_v(
tsstr.euler, np.dot(Cab, Xb)))
else:
if not track_body:
# total force
Crr[:3, -4:] -= algebra.der_CquatT_by_v(
tsstr.quat, faero)
# total moment contribution due to quaternion
Crr[3:6, -4:] -= algebra.der_CquatT_by_v(
tsstr.quat, np.cross(zetag, faero))
Crr[3:6, -4:] += np.dot(
np.dot(Cag, algebra.skew(faero)),
algebra.der_CquatT_by_v(
tsstr.quat, np.dot(Cab, Xb)))
# # Track body
# if track_body:
# # NG 20/8/19 - is the Cag needed here? Verify
# Crr[:3, -4:] -= Cag.dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero)))
#
# Crr[3:6, -4:] -= Cag.dot(algebra.skew(zetag).dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero))))
# Crr[3:6, -4:] += Cag.dot(algebra.skew(faero)).dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(zetag)))
# transfer
self.Kdisp = Kdisp
self.Kvel_disp = Kvel_disp
self.Kdisp_vel = Kdisp_vel
self.Kvel_vel = Kvel_vel
self.Kforces = Kforces
# stiffening factors
self.Kss = Kss
self.Krs = Krs
self.Csr = Csr
self.Crs = Crs
self.Crr = Crr
def test_quat_wrt_rot(self):
"""
We define:
- G: initial frame
- A: frame obtained upon rotation, Cga, defined by the quaternion q0
- B: frame obtained upon further rotation, Cab, of A defined by
the "infinitesimal" Cartesian rotation vector dcrv
The test verifies that:
1. the total rotation matrix Cgb(q0+dq) is equal to
Cgb = Cga(q0) Cab(dcrv)
where
dq = algebra.der_quat_wrt_crv(q0)
2. the difference between analytically computed delta quaternion, dq,
and the numerical delta
dq_num = algebra.crv2quat(algebra.rotation2crv(Cgb_ref))-q0
is comparable to the step used to compute the delta dcrv
3. The equality:
d(Cga(q0)*v)*dq = Cga(q0) * d(Cab*dv)*dcrv
where d(Cga(q0)*v) and d(Cab*dv) are the derivatives computed through
algebra.der_Cquat_by_v and algebra.der_Ccrv_by_v
for a random vector v.
Warning:
- The relation dcrv->dquat is not uniquely defined. However, the
resulting rotation matrix is, namely:
Cga(q0+dq)=Cga(q0)*Cab(dcrv)
"""
### case 1: simple rotation about the same axis
# linearisation point
a0 = 30. * np.pi / 180
n0 = np.array([0, 0, 1])
n0 = n0 / np.linalg.norm(n0)
q0 = algebra.crv2quat(a0 * n0)
Cga = algebra.quat2rotation(q0)
# direction of perturbation
n2 = n0
A = np.array([1e-2, 1e-3, 1e-4, 1e-5, 1e-6])
for a in A:
drot = a * n2
# build rotation manually
atot = a0 + a
Cgb_exp = algebra.crv2rotation(atot * n0) # ok
# build combined rotation
Cab = algebra.crv2rotation(drot)
Cgb_ref = np.dot(Cga, Cab)
# verify expected vs combined rotation matrices
assert np.linalg.norm(Cgb_exp - Cgb_ref) / a < 1e-8, \
'Verify test case - these matrices need to be identical'
# verify analytical rotation matrix
dq_an = np.dot(algebra.der_quat_wrt_crv(q0), drot)
Cgb_an = algebra.quat2rotation(q0 + dq_an)
erel_rot = np.linalg.norm(Cgb_an - Cgb_ref) / a
assert erel_rot < 3e-3, \
'Relative error of rotation matrix (%.2e) too large!' % erel_rot
# verify delta quaternion
erel_dq = np.linalg.norm(Cgb_an - Cgb_ref)
dq_num = algebra.crv2quat(algebra.rotation2crv(Cgb_ref)) - q0
erel_dq = np.linalg.norm(dq_num -
dq_an) / np.linalg.norm(dq_an) / a
assert erel_dq < .3, \
'Relative error delta quaternion (%.2e) too large!' % erel_dq
# verify algebraic relation
v = np.ones((3, ))
D1 = algebra.der_Cquat_by_v(q0, v)
D2 = algebra.der_Ccrv_by_v(np.zeros((3, )), v)
res = np.dot(D1, dq_num) - np.dot(np.dot(Cga, D2), drot)
erel_res = np.linalg.norm(res) / a
assert erel_res < 5e-1 * a, \
'Relative error of residual (%.2e) too large!' % erel_res
### case 2: random rotation
# linearisation point
a0 = 30. * np.pi / 180
n0 = np.array([-2, -1, 1])
n0 = n0 / np.linalg.norm(n0)
q0 = algebra.crv2quat(a0 * n0)
Cga = algebra.quat2rotation(q0)
# direction of perturbation
n2 = np.array([0.5, 1., -2.])
n2 = n2 / np.linalg.norm(n2)
A = np.array([1e-2, 1e-3, 1e-4, 1e-5, 1e-6])
for a in A:
drot = a * n2
# build combined rotation
Cab = algebra.crv2rotation(drot)
Cgb_ref = np.dot(Cga, Cab)
# verify analytical rotation matrix
dq_an = np.dot(algebra.der_quat_wrt_crv(q0), drot)
Cgb_an = algebra.quat2rotation(q0 + dq_an)
erel_rot = np.linalg.norm(Cgb_an - Cgb_ref) / a
assert erel_rot < 3e-3, \
'Relative error of rotation matrix (%.2e) too large!' % erel_rot
# verify delta quaternion
erel_dq = np.linalg.norm(Cgb_an - Cgb_ref)
dq_num = algebra.crv2quat(algebra.rotation2crv(Cgb_ref)) - q0
erel_dq = np.linalg.norm(dq_num -
dq_an) / np.linalg.norm(dq_an) / a
assert erel_dq < .3, \
'Relative error delta quaternion (%.2e) too large!' % erel_dq
# verify algebraic relation
v = np.ones((3, ))
D1 = algebra.der_Cquat_by_v(q0, v)
D2 = algebra.der_Ccrv_by_v(np.zeros((3, )), v)
res = np.dot(D1, dq_num) - np.dot(np.dot(Cga, D2), drot)
erel_res = np.linalg.norm(res) / a
assert erel_res < 5e-1 * a, \
'Relative error of residual (%.2e) too large!' % erel_res
def get_gebm2uvlm_gains(self):
"""
Gain matrix to transfer GEBM dofs to UVLM lattice vertices and stiffening
term due to non-zero forces at the linearisation point.
The function produces the matrices:
- ``Kdisp``: from GEBM to UVLM grid displacements
- ``Kvel_disp``: influence of GEBM dofs displacements to UVLM grid
velocities.
- ``Kvel_vel``: influence of GEBM dofs displacements to UVLM grid
displacements.
- ``Kforces`` (UVLM->GEBM) dimensions are the transpose than the Kdisp and
Kvel* matrices. Hence, when allocation this term, ``ii`` and ``jj`` indices
will unintuitively refer to columns and rows, respectively.
- ``Kss``: stiffness factor accounting for non-zero forces at the
linearisation point. (flexible dof -> flexible dof)
- ``Ksr``: stiffness factor accounting for non-zero forces at the
linearisation point. (rigid dof -> flexible dof)
Notes:
- The following terms have been verified against SHARPy (to ensure same sign conventions and accuracy):
- :math:`\\mathbf{C}^{AB}`
- accuracy of :math:`X^B=\\mathbf{C}^{AB}*X^A`
- accuracy of :math:`X^G` and :math:`X^A`
"""
data = self.data
aero = self.data.aero
structure = self.data.structure # data.aero.beam
tsaero = self.tsaero
tsstr = self.tsstr
# allocate output
Kdisp = np.zeros((3 * self.linuvlm.Kzeta, self.num_dof_str))
Kvel_disp = np.zeros((3 * self.linuvlm.Kzeta, self.num_dof_str))
Kvel_vel = np.zeros((3 * self.linuvlm.Kzeta, self.num_dof_str))
Kforces = np.zeros((self.num_dof_str, 3 * self.linuvlm.Kzeta))
Kss = np.zeros((self.num_dof_flex, self.num_dof_flex))
Ksr = np.zeros((self.num_dof_flex, self.num_dof_rig))
# get projection matrix A->G
# (and other quantities indep. from nodal position)
Cga = algebra.quat2rotation(tsstr.quat)
Cag = Cga.T
# for_pos=tsstr.for_pos
for_tra = tsstr.for_vel[:3]
for_rot = tsstr.for_vel[3:]
skew_for_rot = algebra.skew(for_rot)
Der_vel_Ra = np.dot(Cga, skew_for_rot)
# GEBM degrees of freedom
jj_for_tra = range(self.num_dof_str - 10, self.num_dof_str - 7)
jj_for_rot = range(self.num_dof_str - 7, self.num_dof_str - 4)
jj_quat = range(self.num_dof_str - 4, self.num_dof_str)
jj = 0 # nodal dof index
for node_glob in range(structure.num_node):
### detect bc at node (and no. of dofs)
bc_here = structure.boundary_conditions[node_glob]
if bc_here == 1: # clamp (only rigid-body)
dofs_here = 0
jj_tra, jj_rot = [], []
# continue
elif bc_here == -1 or bc_here == 0: # (rigid+flex body)
dofs_here = 6
jj_tra = 6 * structure.vdof[node_glob] + np.array([0, 1, 2],
dtype=int)
jj_rot = 6 * structure.vdof[node_glob] + np.array([3, 4, 5],
dtype=int)
# jj_tra=[jj ,jj+1,jj+2]
# jj_rot=[jj+3,jj+4,jj+5]
else:
raise NameError('Invalid boundary condition (%d) at node %d!' \
% (bc_here, node_glob))
jj += dofs_here
# retrieve element and local index
ee, node_loc = structure.node_master_elem[node_glob, :]
# get position, crv and rotation matrix
Ra = tsstr.pos[node_glob, :] # in A FoR, w.r.t. origin A-G
Rg = np.dot(Cag.T, Ra) # in G FoR, w.r.t. origin A-G
psi = tsstr.psi[ee, node_loc, :]
psi_dot = tsstr.psi_dot[ee, node_loc, :]
Cab = algebra.crv2rotation(psi)
Cba = Cab.T
Cbg = np.dot(Cab.T, Cag)
### str -> aero mapping
# some nodes may be linked to multiple surfaces...
for str2aero_here in aero.struct2aero_mapping[node_glob]:
# detect surface/span-wise coordinate (ss,nn)
nn, ss = str2aero_here['i_n'], str2aero_here['i_surf']
# print('%.2d,%.2d'%(nn,ss))
# surface panelling
M = aero.aero_dimensions[ss][0]
N = aero.aero_dimensions[ss][1]
Kzeta_start = 3 * sum(self.linuvlm.MS.KKzeta[:ss])
shape_zeta = (3, M + 1, N + 1)
for mm in range(M + 1):
# get bound vertex index
ii_vert = [
Kzeta_start + np.ravel_multi_index(
(cc, mm, nn), shape_zeta) for cc in range(3)
]
# get aero force
faero = tsaero.forces[ss][:3, mm, nn]
# get position vectors
zetag = tsaero.zeta[ss][:, mm,
nn] # in G FoR, w.r.t. origin A-G
zetaa = np.dot(Cag, zetag) # in A FoR, w.r.t. origin A-G
Xg = zetag - Rg # in G FoR, w.r.t. origin B
Xb = np.dot(Cbg, Xg) # in B FoR, w.r.t. origin B
# get rotation terms
Xbskew = algebra.skew(Xb)
Tan = algebra.crv2tan(psi)
XbskewTan = np.dot(Xbskew, Tan)
# get velocity terms
zetag_dot = tsaero.zeta_dot[
ss][:, mm, nn] # in G FoR, w.r.t. origin A-G
zetaa_dot = np.dot(
Cag, zetag_dot) # in A FoR, w.r.t. origin A-G
### ---------------------------------------- allocate Kdisp
if bc_here != 1:
# wrt pos
Kdisp[np.ix_(ii_vert, jj_tra)] += Cga
# wrt psi
Kdisp[np.ix_(ii_vert,
jj_rot)] -= np.dot(Cbg.T, XbskewTan)
# w.r.t. position of FoR A (w.r.t. origin G)
# null as A and G have always same origin in SHARPy
# # ### w.r.t. quaternion (attitude changes)
Kdisp[np.ix_(ii_vert, jj_quat)] = \
algebra.der_Cquat_by_v(tsstr.quat, zetaa)
### ------------------------------------ allocate Kvel_disp
if bc_here != 1:
# # wrt pos
Kvel_disp[np.ix_(ii_vert, jj_tra)] += Der_vel_Ra
# wrt psi (at zero psi_dot)
Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \
np.dot(Cga,
np.dot(skew_for_rot,
np.dot(Cab, XbskewTan)))
# # wrt psi (psi_dot contributions - verified)
Kvel_disp[np.ix_(ii_vert, jj_rot)] += np.dot(
Cbg.T,
np.dot(algebra.skew(np.dot(XbskewTan, psi_dot)),
Tan))
Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \
np.dot(Cbg.T,
np.dot(Xbskew,
algebra.der_Tan_by_xv(psi, psi_dot)))
# # w.r.t. position of FoR A (w.r.t. origin G)
# # null as A and G have always same origin in SHARPy
# # ### w.r.t. quaternion (attitude changes)
Kvel_disp[np.ix_(ii_vert, jj_quat)] = \
algebra.der_Cquat_by_v(tsstr.quat, zetaa_dot)
### ------------------------------------- allocate Kvel_vel
if bc_here != 1:
# wrt pos_dot
Kvel_vel[np.ix_(ii_vert, jj_tra)] += Cga
# # wrt crv_dot
Kvel_vel[np.ix_(ii_vert,
jj_rot)] -= np.dot(Cbg.T, XbskewTan)
# # wrt velocity of FoR A
Kvel_vel[np.ix_(ii_vert, jj_for_tra)] += Cga
Kvel_vel[np.ix_(ii_vert, jj_for_rot)] -= \
np.dot(Cga, algebra.skew(zetaa))
# wrt rate of change of quaternion: not implemented!
### -------------------------------------- allocate Kforces
if bc_here != 1:
# nodal forces
Kforces[np.ix_(jj_tra, ii_vert)] += Cbg
# nodal moments
Kforces[np.ix_(jj_rot, ii_vert)] += \
np.dot(Cbg, algebra.skew(Xg))
# or, equivalently, np.dot( algebra.skew(Xb),Cbg)
# total forces
Kforces[np.ix_(jj_for_tra, ii_vert)] += Cag
# total moments
Kforces[np.ix_(jj_for_rot, ii_vert)] += \
np.dot(Cag, algebra.skew(zetag))
# quaternion equation
# null, as not dep. on external forces
### --------------------------------------- allocate Kstiff
if bc_here != 1:
# forces
Dfdcrv = algebra.der_CcrvT_by_v(
psi, np.dot(Cag, faero))
Dfdquat = np.dot(
Cba, algebra.der_CquatT_by_v(tsstr.quat, faero))
Kss[np.ix_(jj_tra, jj_rot)] -= Dfdcrv
Ksr[jj_tra, -4:] -= Dfdquat
# moments
Kss[np.ix_(jj_rot, jj_rot)] -= np.dot(Xbskew, Dfdcrv)
Ksr[jj_rot, -4:] -= np.dot(Xbskew, Dfdquat)
# embed()
# transfer
self.Kdisp = Kdisp
self.Kvel_disp = Kvel_disp
self.Kvel_vel = Kvel_vel
self.Kforces = Kforces
# stiffening factors
self.Kss = Kss
self.Ksr = Ksr
def test_rotation_matrices_derivatives(self):
"""
Checks derivatives of rotation matrix derivatives with respect to
quaternions and Cartesian rotation vectors
Note: test only includes CRV <-> quaternions conversions
"""
### linearisation point
# fi0=np.pi/6
# nv0=np.array([1,3,1])
fi0 = 2.0 * np.pi * random.random() - np.pi
nv0 = np.array([random.random(), random.random(), random.random()])
nv0 = nv0 / np.linalg.norm(nv0)
fv0 = fi0 * nv0
qv0 = algebra.crv2quat(fv0)
ev0 = algebra.quat2euler(qv0)
# direction of perturbation
# fi1=np.pi/3
# nv1=np.array([-2,4,1])
fi1 = 2.0 * np.pi * random.random() - np.pi
nv1 = np.array([random.random(), random.random(), random.random()])
nv1 = nv1 / np.linalg.norm(nv1)
fv1 = fi1 * nv1
qv1 = algebra.crv2quat(fv1)
ev1 = algebra.quat2euler(qv1)
# linearsation point
Cga0 = algebra.quat2rotation(qv0)
Cag0 = Cga0.T
Cab0 = algebra.crv2rotation(fv0)
Cba0 = Cab0.T
Cga0_euler = algebra.euler2rot(ev0)
Cag0_euler = Cga0_euler.T
# derivatives
# xv=np.ones((3,)) # dummy vector
xv = np.array([random.random(),
random.random(),
random.random()]) # dummy vector
derCga = algebra.der_Cquat_by_v(qv0, xv)
derCag = algebra.der_CquatT_by_v(qv0, xv)
derCab = algebra.der_Ccrv_by_v(fv0, xv)
derCba = algebra.der_CcrvT_by_v(fv0, xv)
derCga_euler = algebra.der_Ceuler_by_v(ev0, xv)
derCag_euler = algebra.der_Peuler_by_v(ev0, xv)
A = np.array([1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6])
er_ag = 10.
er_ga = 10.
er_ab = 10.
er_ba = 10.
er_ag_euler = 10.
er_ga_euler = 10.
for a in A:
# perturbed
qv = a * qv1 + (1. - a) * qv0
fv = a * fv1 + (1. - a) * fv0
ev = a * ev1 + (1. - a) * ev0
dqv = qv - qv0
dfv = fv - fv0
dev = ev - ev0
Cga = algebra.quat2rotation(qv)
Cag = Cga.T
Cab = algebra.crv2rotation(fv)
Cba = Cab.T
Cga_euler = algebra.euler2rot(ev)
Cag_euler = Cga_euler.T
dCag_num = np.dot(Cag - Cag0, xv)
dCga_num = np.dot(Cga - Cga0, xv)
dCag_an = np.dot(derCag, dqv)
dCga_an = np.dot(derCga, dqv)
er_ag_new = np.max(np.abs(dCag_num - dCag_an))
er_ga_new = np.max(np.abs(dCga_num - dCga_an))
dCab_num = np.dot(Cab - Cab0, xv)
dCba_num = np.dot(Cba - Cba0, xv)
dCab_an = np.dot(derCab, dfv)
dCba_an = np.dot(derCba, dfv)
er_ab_new = np.max(np.abs(dCab_num - dCab_an))
er_ba_new = np.max(np.abs(dCba_num - dCba_an))
dCag_num_euler = np.dot(Cag_euler - Cag0_euler, xv)
dCga_num_euler = np.dot(Cga_euler - Cga0_euler, xv)
dCag_an_euler = np.dot(derCag_euler, dev)
dCga_an_euler = np.dot(derCga_euler, dev)
er_ag_euler_new = np.max(np.abs(dCag_num_euler - dCag_an_euler))
er_ga_euler_new = np.max(np.abs(dCga_num_euler - dCga_an_euler))
assert er_ga_new < er_ga, 'der_Cquat_by_v error not converging to 0'
assert er_ag_new < er_ag, 'der_CquatT_by_v error not converging to 0'
assert er_ab_new < er_ab, 'der_Ccrv_by_v error not converging to 0'
assert er_ba_new < er_ba, 'der_CcrvT_by_v error not converging to 0'
assert er_ga_euler_new < er_ga_euler, 'der_Ceuler_by_v error not converging to 0'
assert er_ag_euler_new < er_ag_euler, 'der_Peuler_by_v error not converging to 0'
er_ag = er_ag_new
er_ga = er_ga_new
er_ab = er_ab_new
er_ba = er_ba_new
er_ag_euler = er_ag_euler_new
er_ga_euler = er_ga_euler_new
assert er_ga < A[-2], 'der_Cquat_by_v error too large'
assert er_ag < A[-2], 'der_CquatT_by_v error too large'
assert er_ab < A[-2], 'der_Ccrv_by_v error too large'
assert er_ba < A[-2], 'der_CcrvT_by_v error too large'
assert er_ag_euler < A[-2], 'der_Peuler_by_v error too large'
assert er_ga_euler < A[-2], 'der_Ceuler_by_v error too large'