def tr2angvec(T, unit='rad', check=False):
r"""
Convert SO(3) or SE(3) to angle and rotation vector
:param R: SO(3) or SE(3) matrix
:type R: numpy.ndarray, shape=(3,3) or (4,4)
:param unit: 'rad' or 'deg'
:type unit: str
:param check: check that rotation matrix is valid
:type check: bool
:return: :math:`(\theta, {\bf v})`
:rtype: float, numpy.ndarray, shape=(3,)
``tr2angvec(R)`` is a rotation angle and a vector about which the rotation
acts that corresponds to the rotation part of ``R``.
By default the angle is in radians but can be changed setting `unit='deg'`.
Notes:
- If the input is SE(3) the translation component is ignored.
:seealso: :func:`~angvec2r`, :func:`~angvec2tr`, :func:`~tr2rpy`, :func:`~tr2eul`
"""
if argcheck.ismatrix(T, (4, 4)):
R = trn.t2r(T)
else:
R = T
assert isrot(R, check=check)
v = trn.vex(trlog(R))
if vec.iszerovec(v):
theta = 0
v = np.r_[0, 0, 0]
else:
theta = vec.norm(v)
v = vec.unitvec(v)
if unit == 'deg':
theta *= 180 / math.pi
return (theta, v)
def rodrigues(w, theta):
"""
Rodrigues' formula for rotation
:param w: rotation vector
:type w: array_like
:param theta: rotation angle
:type theta: float or None
"""
w = argcheck.getvector(w)
if vec.iszerovec(w):
# for a zero so(n) return unit matrix, theta not relevant
if len(w) == 1:
return np.eye(2)
else:
return np.eye(3)
if theta is None:
theta = vec.norm(w)
w = vec.unitvec(w)
skw = skew(w)
return np.eye(skw.shape[0]) + math.sin(theta) * skw + (
1.0 - math.cos(theta)) * skw @ skw
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 vec.iszerovec(tw):
return np.eye(3)
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")