def test_vector_form_append_2(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
# vector 1-form
Q1 = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q1_ = st.col_stack(Q1, sp.zeros(2, 6))
w1 = pc.VectorDifferentialForm(1, XX, coeff=Q1_)
# 1-forms
Q2 = sp.Matrix([[x1, x2, x3], [x3, x1, x2]])
Q2_ = st.col_stack(Q2, sp.zeros(2, 6))
w2 = pc.VectorDifferentialForm(1, XX, coeff=Q2_)
w1.append(w2)
# vector form to compare with:
B = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3], [x1, x2, x3],
[x3, x1, x2]])
B_ = st.col_stack(B, sp.zeros(4, 6))
self.assertEqual(w1.coeff, B_)
def test_left_mul_by_2(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# matrix independent of s
M2 = sp.Matrix([[1, 0], [-C, 1]])
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
t = w.left_mul_by(M2, additional_symbols=[C])
# object to compare with:
t2 = -C * w1 + w2
self.assertEqual(t2.coeff, t.coeff.row(1).T)
def test_left_mul_by_1(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# s-dependent matrix
M1 = sp.Matrix([[1, 0], [-C * s, 1]])
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
t = w.left_mul_by(M1, s, [C])
t2 = -C * w1.dot() + w2
self.assertEqual(t2.coeff, t.coeff.row(1).T)
def test_mul(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
W = pc.VectorDifferentialForm(1, xx, coeff=Q)
W1 = s * W
W2 = W * C
self.assertEqual(W1.coeff, nct.nc_mul(s, W.coeff))
self.assertNotEqual(W1.coeff, nct.nc_mul(W.coeff, s))
self.assertEqual(W2.coeff, nct.nc_mul(W.coeff, C))
self.assertNotEqual(W2.coeff, nct.nc_mul(C, W.coeff))
alpha = pc.DifferentialForm(1, xx)
with self.assertRaises(TypeError) as cm:
alpha * W1
with self.assertRaises(TypeError) as cm:
W1 * alpha
M = sp.eye(2)
with self.assertRaises(sp.SympifyError) as cm:
M * W1
with self.assertRaises(TypeError) as cm:
W1 * M
def test_simplify(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', )
Q = sp.Matrix([[sin(x1)**2 + cos(x1)**2 - 1, 0, 0],
[0, x3 * (1 - sin(x2)**2) - cos(x2)**2 * x3, 0]])
Omega = pc.VectorDifferentialForm(1, xx, coeff=Q)
Omega2 = pc.simplify(Omega)
self.assertEqual(Omega2.coeff, 0 * Omega.coeff)
def test_vector_form_dot(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [1, 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# vector 1-forms
omega = pc.VectorDifferentialForm(1, XX, coeff=Q_)
omega_dot = omega.dot()
omega_1, omega_2 = omega.unpack()
omega_1dot = omega_1.dot()
omega_2dot = omega_2.dot()
Qdot_ = st.row_stack(omega_1dot.coeff.T, omega_2dot.coeff.T)
# vector form to compare with
omegadot_comp = pc.VectorDifferentialForm(1, XX, coeff=Qdot_)
self.assertEqual(omega_dot.coeff, omegadot_comp.coeff)
def test_difference_vector_forms(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
A = sp.Matrix([[x3 / sin(x1), 1, 0], [1, 0, x3]])
A_ = st.col_stack(A, sp.zeros(2, 6))
B = sp.Matrix([[x3 / cos(x1), 0, 1], [-tan(x1), 0, x3]])
B_ = st.col_stack(B, sp.zeros(2, 6))
# vector 1-forms
omega_a = pc.VectorDifferentialForm(1, XX, coeff=A_)
omega_b = pc.VectorDifferentialForm(1, XX, coeff=B_)
omega_c = omega_a - omega_b
# vector form to compare with
Q_ = A_ - B_
omega_comp = pc.VectorDifferentialForm(1, XX, coeff=Q_)
self.assertEqual(omega_c.coeff, omega_comp.coeff)
def test_vector_form_dot(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [1, 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# vector 1-forms
omega = pc.VectorDifferentialForm(1, XX, coeff=Q_)
with self.assertRaises(ValueError) as cm:
# coordinates are not part of basis
omega.dot().dot().dot()
def test_vector_form_get_coeff_from_idcs(self):
x1, x2, x3 = xx = st.symb_vector('x1:4')
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
w_0 = w.get_coeff_from_idcs(0)
Q_0 = Q_.col(0)
self.assertEqual(w_0, Q_0)
w_1 = w.get_coeff_from_idcs(1)
Q_1 = Q_.col(1)
self.assertEqual(w_1, Q_1)
def test_stack_to_vector_form(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
w_stacked = pc.stack_to_vector_form(w1, w2)
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
self.assertEqual(w.coeff, w_stacked.coeff)
def test_vector_k_form(self):
x1, x2, x3 = xx = st.symb_vector('x1:4')
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
w1_tilde = w.get_differential_form(0)
self.assertEqual(w1.coeff, w1_tilde.coeff)
w2_tilde = w.get_differential_form(1)
self.assertEqual(w2.coeff, w2_tilde.coeff)
def test_vector_form_subs(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [C, 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
B = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
B_ = st.col_stack(B, sp.zeros(2, 6))
# vector 1-forms
omega = pc.VectorDifferentialForm(1, XX, coeff=Q_).subs(C, -tan(x1))
self.assertEqual(B_, omega.coeff)
def test_left_mul_by_3(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
M3 = sp.Matrix([[1, 0], [-C * s**2, 1]])
# vector 1-forms
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
with self.assertRaises(Exception) as cm:
# raises NotImplemented but this might change
t = w.left_mul_by(M3, s, additional_symbols=[C])
