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")