def trlog2(T, check=True, twist=False):
"""
Logarithm of SO(2) or SE(2) matrix
:param T: SO(2) or SE(2) matrix
:type T: numpy.ndarray, shape=(2,2) or (3,3)
:param check: check that matrix is valid
:type check: bool
:param twist: return a twist vector instead of matrix [default]
:type twist: bool
:return: logarithm
:rtype: numpy.ndarray, shape=(2,2) or (3,3)
:raises: ValueError
An efficient closed-form solution of the matrix logarithm for arguments that
are SO(2) or SE(2).
- ``trlog2(R)`` is the logarithm of the passed rotation matrix ``R`` which
will be 2x2 skew-symmetric matrix. The equivalent vector from ``vex()``
is parallel to rotation axis and its norm is the amount of rotation about
that axis.
- ``trlog(T)`` is the logarithm of the passed homogeneous transformation
matrix ``T`` which will be 3x3 augumented skew-symmetric matrix. The
equivalent vector from ``vexa()`` is the twist vector (6x1) comprising [v
w].
:seealso: :func:`~trexp`, :func:`~spatialmath.base.transformsNd.vex`,
:func:`~spatialmath.base.transformsNd.vexa`
"""
if ishom2(T, check=check):
# SE(2) matrix
if trn.iseye(T):
# is identity matrix
if twist:
return np.zeros((3, ))
else:
return np.zeros((3, 3))
else:
if twist:
return trn.vexa(scipy.linalg.logm(T))
else:
return scipy.linalg.logm(T)
elif isrot2(T, check=check):
# SO(2) rotation matrix
if twist:
return trn.vex(scipy.linalg.logm(T))
else:
return scipy.linalg.logm(T)
else:
raise ValueError("Expect SO(2) or SE(2) matrix")
def trexp2(S, theta=None):
"""
Exponential of so(2) or se(2) matrix
:param S: so(2), se(2) matrix or equivalent velctor
:type T: numpy.ndarray, shape=(2,2) or (3,3); array_like
:param theta: motion
:type theta: float
:return: 2x2 or 3x3 matrix exponential in SO(2) or SE(2)
:rtype: numpy.ndarray, shape=(2,2) or (3,3)
An efficient closed-form solution of the matrix exponential for arguments
that are so(2) or se(2).
For so(2) the results is an SO(2) rotation matrix:
- ``trexp2(S)`` is the matrix exponential of the so(3) element ``S`` which is a 2x2
skew-symmetric matrix.
- ``trexp2(S, THETA)`` as above but for an so(3) motion of S*THETA, where ``S`` is
unit-norm skew-symmetric matrix representing a rotation axis and a rotation magnitude
given by ``THETA``.
- ``trexp2(W)`` is the matrix exponential of the so(2) element ``W`` expressed as
a 1-vector (array_like).
- ``trexp2(W, THETA)`` as above but for an so(3) motion of W*THETA where ``W`` is a
unit-norm vector representing a rotation axis and a rotation magnitude
given by ``THETA``. ``W`` is expressed as a 1-vector (array_like).
For se(2) the results is an SE(2) homogeneous transformation matrix:
- ``trexp2(SIGMA)`` is the matrix exponential of the se(2) element ``SIGMA`` which is
a 3x3 augmented skew-symmetric matrix.
- ``trexp2(SIGMA, THETA)`` as above but for an se(3) motion of SIGMA*THETA, where ``SIGMA``
must represent a unit-twist, ie. the rotational component is a unit-norm skew-symmetric
matrix.
- ``trexp2(TW)`` is the matrix exponential of the se(3) element ``TW`` represented as
a 3-vector which can be considered a screw motion.
- ``trexp2(TW, THETA)`` as above but for an se(2) motion of TW*THETA, where ``TW``
must represent a unit-twist, ie. the rotational component is a unit-norm skew-symmetric
matrix.
:seealso: trlog, trexp2
"""
if argcheck.ismatrix(S, (3, 3)) or argcheck.isvector(S, 3):
# se(2) case
if argcheck.ismatrix(S, (3, 3)):
# augmentented skew matrix
tw = trn.vexa(S)
else:
# 3 vector
tw = argcheck.getvector(S)
if theta is None:
(tw, theta) = vec.unittwist2(tw)
else:
assert vec.isunittwist2(
tw), 'If theta is specified S must be a unit twist'
t = tw[0:2]
w = tw[2]
R = trn._rodrigues(w, theta)
skw = trn.skew(w)
V = np.eye(2) * theta + (1.0 - math.cos(theta)) * skw + (
theta - math.sin(theta)) * skw @ skw
return trn.rt2tr(R, V @ t)
elif argcheck.ismatrix(S, (2, 2)) or argcheck.isvector(S, 1):
# so(2) case
if argcheck.ismatrix(S, (2, 2)):
# skew symmetric matrix
w = trn.vex(S)
else:
# 1 vector
w = argcheck.getvector(S)
if theta is not None:
assert vec.isunitvec(
w), 'If theta is specified S must be a unit twist'
# do Rodrigues' formula for rotation
return trn._rodrigues(w, theta)
else:
raise ValueError(
" First argument must be SO(2), 1-vector, SE(2) or 3-vector")