254 lines
8.7 KiB
Python
254 lines
8.7 KiB
Python
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import sympy
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import sys
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import unittest
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import sophus
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import functools
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class So3:
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""" 3 dimensional group of orthogonal matrices with determinant 1 """
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def __init__(self, q):
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""" internally represented by a unit quaternion q """
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self.q = q
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@staticmethod
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def exp(v):
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""" exponential map """
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theta_sq = sophus.squared_norm(v)
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theta = sympy.sqrt(theta_sq)
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return So3(
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sophus.Quaternion(
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sympy.cos(0.5 * theta),
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sympy.sin(0.5 * theta) / theta * v))
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def log(self):
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""" logarithmic map"""
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n = sympy.sqrt(sophus.squared_norm(self.q.vec))
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return 2 * sympy.atan(n / self.q.real) / n * self.q.vec
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def __repr__(self):
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return "So3:" + repr(self.q)
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def inverse(self):
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return So3(self.q.conj())
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@staticmethod
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def hat(o):
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return sympy.Matrix([[0, -o[2], o[1]],
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[o[2], 0, -o[0]],
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[-o[1], o[0], 0]])
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"""vee-operator
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It takes the 3x3-matrix representation ``Omega`` and maps it to the
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corresponding vector representation of Lie algebra.
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This is the inverse of the hat-operator, see above.
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Precondition: ``Omega`` must have the following structure:
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| 0 -c b |
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| c 0 -a |
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| -b a 0 |
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"""
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@staticmethod
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def vee(Omega):
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v = sophus.Vector3(Omega.row(2).col(1), Omega.row(0).col(2), Omega.row(1).col(0))
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return v
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def matrix(self):
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""" returns matrix representation """
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return sympy.Matrix([[
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1 - 2 * self.q.vec[1]**2 - 2 * self.q.vec[2]**2,
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2 * self.q.vec[0] * self.q.vec[1] -
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2 * self.q.vec[2] * self.q[3],
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2 * self.q.vec[0] * self.q.vec[2] +
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2 * self.q.vec[1] * self.q[3]
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], [
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2 * self.q.vec[0] * self.q.vec[1] +
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2 * self.q.vec[2] * self.q[3],
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1 - 2 * self.q.vec[0]**2 - 2 * self.q.vec[2]**2,
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2 * self.q.vec[1] * self.q.vec[2] -
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2 * self.q.vec[0] * self.q[3]
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], [
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2 * self.q.vec[0] * self.q.vec[2] -
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2 * self.q.vec[1] * self.q[3],
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2 * self.q.vec[1] * self.q.vec[2] +
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2 * self.q.vec[0] * self.q[3],
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1 - 2 * self.q.vec[0]**2 - 2 * self.q.vec[1]**2
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]])
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def __mul__(self, right):
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""" left-multiplication
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either rotation concatenation or point-transform """
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if isinstance(right, sympy.Matrix):
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assert right.shape == (3, 1), right.shape
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return (self.q * sophus.Quaternion(0, right) * self.q.conj()).vec
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elif isinstance(right, So3):
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return So3(self.q * right.q)
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assert False, "unsupported type: {0}".format(type(right))
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def __getitem__(self, key):
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return self.q[key]
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@staticmethod
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def calc_Dx_exp_x(x):
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return sympy.Matrix(4, 3, lambda r, c:
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sympy.diff(So3.exp(x)[r], x[c]))
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@staticmethod
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def Dx_exp_x_at_0():
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return sympy.Matrix([[0.5, 0.0, 0.0],
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[0.0, 0.5, 0.0],
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[0.0, 0.0, 0.5],
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[0.0, 0.0, 0.0]])
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@staticmethod
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def calc_Dx_exp_x_at_0(x):
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return So3.calc_Dx_exp_x(x).subs(x[0], 0).subs(x[1], 0).limit(x[2], 0)
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def calc_Dx_this_mul_exp_x_at_0(self, x):
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return sympy.Matrix(4, 3, lambda r, c:
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sympy.diff((self * So3.exp(x))[r], x[c]))\
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.subs(x[0], 0).subs(x[1], 0).limit(x[2], 0)
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def calc_Dx_exp_x_mul_this_at_0(self, x):
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return sympy.Matrix(3, 4, lambda r, c:
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sympy.diff((self * So3.exp(x))[c], x[r, 0]))\
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.subs(x[0], 0).subs(x[1], 0).limit(x[2], 0)
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@staticmethod
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def Dxi_x_matrix(x, i):
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if i == 0:
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return sympy.Matrix([[0, 2 * x[1], 2 * x[2]],
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[2 * x[1], -4 * x[0], -2 * x[3]],
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[2 * x[2], 2 * x[3], -4 * x[0]]])
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if i == 1:
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return sympy.Matrix([[-4 * x[1], 2 * x[0], 2 * x[3]],
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[2 * x[0], 0, 2 * x[2]],
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[-2 * x[3], 2 * x[2], -4 * x[1]]])
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if i == 2:
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return sympy.Matrix([[-4 * x[2], -2 * x[3], 2 * x[0]],
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[2 * x[3], -4 * x[2], 2 * x[1]],
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[2 * x[0], 2 * x[1], 0]])
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if i == 3:
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return sympy.Matrix([[0, -2 * x[2], 2 * x[1]],
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[2 * x[2], 0, -2 * x[0]],
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[-2 * x[1], 2 * x[0], 0]])
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@staticmethod
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def calc_Dxi_x_matrix(x, i):
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return sympy.Matrix(3, 3, lambda r, c:
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sympy.diff(x.matrix()[r, c], x[i]))
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@staticmethod
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def Dxi_exp_x_matrix(x, i):
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R = So3.exp(x)
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Dx_exp_x = So3.calc_Dx_exp_x(x)
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l = [Dx_exp_x[j, i] * So3.Dxi_x_matrix(R, j) for j in [0, 1, 2, 3]]
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return functools.reduce((lambda a, b: a + b), l)
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@staticmethod
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def calc_Dxi_exp_x_matrix(x, i):
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return sympy.Matrix(3, 3, lambda r, c:
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sympy.diff(So3.exp(x).matrix()[r, c], x[i]))
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@staticmethod
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def Dxi_exp_x_matrix_at_0(i):
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v = sophus.ZeroVector3()
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v[i] = 1
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return So3.hat(v)
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@staticmethod
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def calc_Dxi_exp_x_matrix_at_0(x, i):
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return sympy.Matrix(3, 3, lambda r, c:
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sympy.diff(So3.exp(x).matrix()[r, c], x[i])
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).subs(x[0], 0).subs(x[1], 0).limit(x[2], 0)
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class TestSo3(unittest.TestCase):
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def setUp(self):
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omega0, omega1, omega2 = sympy.symbols(
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'omega[0], omega[1], omega[2]', real=True)
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x, v0, v1, v2 = sympy.symbols('q.w() q.x() q.y() q.z()', real=True)
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p0, p1, p2 = sympy.symbols('p0 p1 p2', real=True)
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v = sophus.Vector3(v0, v1, v2)
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self.omega = sophus.Vector3(omega0, omega1, omega2)
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self.a = So3(sophus.Quaternion(x, v))
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self.p = sophus.Vector3(p0, p1, p2)
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def test_exp_log(self):
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for o in [sophus.Vector3(0., 1, 0.5),
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sophus.Vector3(0.1, 0.1, 0.1),
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sophus.Vector3(0.01, 0.2, 0.03)]:
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w = So3.exp(o).log()
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for i in range(0, 3):
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self.assertAlmostEqual(o[i], w[i])
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def test_matrix(self):
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R_foo_bar = So3.exp(self.omega)
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Rmat_foo_bar = R_foo_bar.matrix()
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point_bar = self.p
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p1_foo = R_foo_bar * point_bar
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p2_foo = Rmat_foo_bar * point_bar
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self.assertEqual(sympy.simplify(p1_foo - p2_foo),
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sophus.ZeroVector3())
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def test_derivatives(self):
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self.assertEqual(sympy.simplify(So3.calc_Dx_exp_x_at_0(self.omega) -
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So3.Dx_exp_x_at_0()),
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sympy.Matrix.zeros(4, 3))
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for i in [0, 1, 2, 3]:
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self.assertEqual(sympy.simplify(So3.calc_Dxi_x_matrix(self.a, i) -
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So3.Dxi_x_matrix(self.a, i)),
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sympy.Matrix.zeros(3, 3))
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for i in [0, 1, 2]:
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self.assertEqual(sympy.simplify(
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So3.Dxi_exp_x_matrix(self.omega, i) -
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So3.calc_Dxi_exp_x_matrix(self.omega, i)),
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sympy.Matrix.zeros(3, 3))
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self.assertEqual(sympy.simplify(
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So3.Dxi_exp_x_matrix_at_0(i) -
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So3.calc_Dxi_exp_x_matrix_at_0(self.omega, i)),
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sympy.Matrix.zeros(3, 3))
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def test_codegen(self):
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stream = sophus.cse_codegen(So3.calc_Dx_exp_x(self.omega))
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filename = "cpp_gencode/So3_Dx_exp_x.cpp"
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# set to true to generate codegen files
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if False:
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file = open(filename, "w")
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for line in stream:
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file.write(line)
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file.close()
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else:
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file = open(filename, "r")
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file_lines = file.readlines()
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for i, line in enumerate(stream):
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self.assertEqual(line, file_lines[i])
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file.close()
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stream.close
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stream = sophus.cse_codegen(
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self.a.calc_Dx_this_mul_exp_x_at_0(self.omega))
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filename = "cpp_gencode/So3_Dx_this_mul_exp_x_at_0.cpp"
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# set to true to generate codegen files
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if False:
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file = open(filename, "w")
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for line in stream:
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file.write(line)
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file.close()
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else:
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file = open(filename, "r")
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file_lines = file.readlines()
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for i, line in enumerate(stream):
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self.assertEqual(line, file_lines[i])
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file.close()
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stream.close
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if __name__ == '__main__':
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unittest.main()
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