def test_issue_13924():
if not numpy:
skip("numpy not installed.")
a = sympify(numpy.array([1]))
assert isinstance(a, ImmutableDenseNDimArray)
assert a[0] == 1
def test_issue_15265():
from sympy.core.sympify import sympify
from sympy.core.singleton import S
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
eqn = sin(x)
p = plot(eqn, xlim=(-S.Pi, S.Pi), ylim=(-1, 1))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(-S.Pi, S.Pi))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(sympify('-3.14'), sympify('3.14')))
p._backend.close()
p = plot(eqn, xlim=(sympify('-3.14'), sympify('3.14')), ylim=(-1, 1))
p._backend.close()
raises(ValueError,
lambda: plot(eqn, xlim=(-S.ImaginaryUnit, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.ImaginaryUnit)))
raises(ValueError,
lambda: plot(eqn, xlim=(-S.Infinity, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.Infinity)))
def test_scalar_numpy():
if not np:
skip("numpy not installed or Python too old.")
assert represent(Integer(1), format='numpy') == 1
assert represent(Float(1.0), format='numpy') == 1.0
assert represent(1.0+I, format='numpy') == 1.0+1.0j
def test_scipy_fns():
if not scipy:
skip("scipy not installed")
single_arg_sympy_fns = [erf, erfc, factorial, gamma, loggamma, digamma]
single_arg_scipy_fns = [scipy.special.erf, scipy.special.erfc,
scipy.special.factorial, scipy.special.gamma, scipy.special.gammaln,
scipy.special.psi]
numpy.random.seed(0)
for (sympy_fn, scipy_fn) in zip(single_arg_sympy_fns, single_arg_scipy_fns):
test_values = 20 * numpy.random.rand(20)
f = lambdify(x, sympy_fn(x), modules = "scipy")
assert numpy.all(abs(f(test_values) - scipy_fn(test_values)) < 1e-15)
double_arg_sympy_fns = [RisingFactorial, besselj, bessely, besseli,
besselk]
double_arg_scipy_fns = [scipy.special.poch, scipy.special.jn,
scipy.special.yn, scipy.special.iv, scipy.special.kn]
#suppress scipy warnings
import warnings
warnings.filterwarnings('ignore', '.*floating point number truncated*')
for (sympy_fn, scipy_fn) in zip(double_arg_sympy_fns, double_arg_scipy_fns):
for i in range(20):
test_values = 20 * numpy.random.rand(2)
f = lambdify((x,y), sympy_fn(x,y), modules = "scipy")
assert abs(f(*test_values) - scipy_fn(*test_values)) < 1e-15
def test_chronos_cg():
skip("Test takes too long")
delta_1, omega_2, pi_1, pi_2, mu_12 = map(lambda x: Tensor(x, rank=0), \
['delta_1', 'omega_2', 'pi_1', 'pi_2', 'mu_12'])
r_1, r_2, q_1, q_2, p_1, p_2, x_1, x_2 = map(lambda x: Tensor(x, rank=1), \
['r_1', 'r_2', 'q_1', 'q_2', 'p_1', 'p_2', 'x_1', 'x_2'])
A, R_0, P_0 = map(lambda x: Tensor(x, rank=2), ['A', 'R_0', 'P_0'])
# Specify which variables are known
knowns = [pi_1, p_1, r_1, q_1, x_1, A, R_0, P_0]
# Now try the chronos variant and repeat.
chronos_eqns = [r_2 - r_1 - delta_1 * q_1,
q_2 - A * p_2,
p_2 - r_2 + p_1 * mu_12,
q_2 - A * r_2 + q_1 * mu_12,
x_2 - x_1 - delta_1 * p_1,
omega_2 - T(r_2) * r_2,
pi_2 - T(p_2) * A * p_2,
T(R_0) * r_2,
T(r_1) * r_2,
T(P_0) * A * p_2,
T(p_1) * A * p_2,
T(p_2) * A * p_2 - T(r_2) * A * r_2 + T(mu_12) * pi_1 * mu_12,
]
run_cg_algorithms(chronos_eqns, knowns)
def test_benchmark_czichowski():
skip('This takes too much time (without gmpy)')
config.setup('GB_METHOD', 'f5b')
helper_test_benchmark_czichowski()
config.setup('GB_METHOD', 'buchberger')
helper_test_benchmark_czichowski()
def test_benchmark_coloring():
skip('takes too much time')
V = range(1, 12+1)
E = [(1,2),(2,3),(1,4),(1,6),(1,12),(2,5),(2,7),(3,8),(3,10),
(4,11),(4,9),(5,6),(6,7),(7,8),(8,9),(9,10),(10,11),
(11,12),(5,12),(5,9),(6,10),(7,11),(8,12),(3,4)]
V = [Symbol('x' + str(i)) for i in V]
E = [(V[i-1], V[j-1]) for i, j in E]
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = V
I3 = [x**3 - 1 for x in V]
Ig = [x**2 + x*y + y**2 for x, y in E]
I = I3 + Ig
assert groebner(I[:-1], V, order='lex') == [
x1 + x11 + x12,
x2 - x11,
x3 - x12,
x4 - x12,
x5 + x11 + x12,
x6 - x11,
x7 - x12,
x8 + x11 + x12,
x9 - x11,
x10 + x11 + x12,
x11**2 + x11*x12 + x12**2,
x12**3 - 1,
]
assert groebner(I, V, order='lex') == [1]
def test_issue_10295():
if not numpy:
skip("numpy not installed.")
A = numpy.array([[1, 3, -1],
[0, 1, 7]])
sA = S(A)
assert sA.shape == (2, 3)
for (ri, ci), val in numpy.ndenumerate(A):
assert sA[ri, ci] == val
B = numpy.array([-7, x, 3*y**2])
sB = S(B)
assert B[0] == -7
assert B[1] == x
assert B[2] == 3*y**2
C = numpy.arange(0, 24)
C.resize(2,3,4)
sC = S(C)
assert sC[0, 0, 0].is_integer
assert sC[0, 0, 0] == 0
a1 = numpy.array([1, 2, 3])
a2 = numpy.array([i for i in range(24)])
a2.resize(2, 4, 3)
assert sympify(a1) == ImmutableDenseNDimArray([1, 2, 3])
assert sympify(a2) == ImmutableDenseNDimArray([i for i in range(24)], (2, 4, 3))
def test_numpy_dagger():
if not np:
skip("numpy not installed or Python too old.")
a = np.matrix([[1.0, 2.0j], [-1.0j, 2.0]])
adag = a.copy().transpose().conjugate()
assert (Dagger(a) == adag).all()
def test_issue_15827():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 2, 3)
C = MatrixSymbol("C", 3, 4)
D = MatrixSymbol("D", 4, 5)
k=symbols("k")
f = lambdify(A, (2*k)*A)
g = lambdify(A, (2+k)*A)
h = lambdify(A, 2*A)
i = lambdify((B, C, D), 2*B*C*D)
assert numpy.array_equal(f(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2*k, 4*k, 6*k], [2*k, 4*k, 6*k], [2*k, 4*k, 6*k]], dtype=object))
assert numpy.array_equal(g(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[k + 2, 2*k + 4, 3*k + 6], [k + 2, 2*k + 4, 3*k + 6], \
[k + 2, 2*k + 4, 3*k + 6]], dtype=object))
assert numpy.array_equal(h(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2, 4, 6], [2, 4, 6], [2, 4, 6]]))
assert numpy.array_equal(i(numpy.array([[1, 2, 3], [1, 2, 3]]), numpy.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]), \
numpy.array([[1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]])), numpy.array([[ 120, 240, 360, 480, 600], \
[ 120, 240, 360, 480, 600]]))
def test_valued_non_diagonal_metric():
numpy = import_module("numpy")
if numpy is None:
skip("numpy not installed.")
mmatrix = Matrix(ndm_matrix)
assert NA(n0)*NA(-n0) == (NA(n0).get_matrix().T * mmatrix * NA(n0).get_matrix())[0, 0]
def test_valued_tensor_self_contraction():
numpy = import_module("numpy")
if numpy is None:
skip("numpy not installed.")
assert AB(i0, -i0) == 4
assert BA(i0, -i0) == 2
def test_valued_tensor_applyfunc():
numpy = import_module("numpy")
if numpy is None:
skip("numpy not installed.")
aA = A(i0).applyfunc(lambda x: x**2)
aB = B(i0).applyfunc(lambda x: x**3)
aB2 = B(-i0).applyfunc(lambda x: x**3)
for i in range(4):
assert aA[i] == A(i0)[i]**2
assert aB[i] == B(i1)[i]**3
assert aB*aB2 == -794
tA = A.applyfunc(lambda x: x + 33)
tB = B.applyfunc(lambda x: x + 33)
tAB = AB.applyfunc(lambda x: x + 33)
assert (tA(i0)*tA(-i0)).expand() == ((E + 33)**2 - (px + 33)**2 - (py + 33)**2 - (pz + 33)**2).expand()
assert tB(i0).get_matrix() == Matrix([33, 34, 35, 36])
assert tAB(i0, i1).get_matrix() == Matrix([
[34, 33, 33, 33],
[33, 32, 33, 33],
[33, 33, 32, 33],
[33, 33, 33, 32],
])
def test_mellin_transform_fail():
skip("Risch takes forever.")
from sympy import Max, Min
MT = mellin_transform
bpos = symbols('b', positive=True)
bneg = symbols('b', negative=True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
# TODO does not work with bneg, argument wrong. Needs changes to matching.
assert MT(expr.subs(b, -bpos), x, s) == \
((-1)**(a + 1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s)
*gamma(1 - a - 2*s)/gamma(1 - s),
(-re(a), -re(a)/2 + S(1)/2), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, -bpos), x, s) == \
(
2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*
s)*gamma(a + s)/gamma(-s + 1),
(-re(a), -re(a)/2), True)
# Test exponent 1:
assert MT(expr.subs({b: -bpos, a: 1}), x, s) == \
(-bpos**(2*s + 1)*gamma(s)*gamma(-s - S(1)/2)/(2*sqrt(pi)),
(-1, -S(1)/2), True)
def test_numpy_translation_abs():
if not numpy:
skip("numpy not installed.")
f = lambdify(x, Abs(x), "numpy")
assert f(-1) == 1
assert f(1) == 1
def test_numpy_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
#Lambdify array first, to ensure return to array as default
f = lambdify((x, y, z), A, ['numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
#Check that the types are arrays and matrices
assert isinstance(f(1, 2, 3), numpy.ndarray)
# gh-15071
class dot(Function):
pass
x_dot_mtx = dot(x, Matrix([[2], [1], [0]]))
f_dot1 = lambdify(x, x_dot_mtx)
inp = numpy.zeros((17, 3))
assert numpy.all(f_dot1(inp) == 0)
strict_kw = dict(allow_unknown_functions=False, inline=True, fully_qualified_modules=False)
p2 = NumPyPrinter(dict(user_functions={'dot': 'dot'}, **strict_kw))
f_dot2 = lambdify(x, x_dot_mtx, printer=p2)
assert numpy.all(f_dot2(inp) == 0)
p3 = NumPyPrinter(strict_kw)
# The line below should probably fail upon construction (before calling with "(inp)"):
raises(Exception, lambda: lambdify(x, x_dot_mtx, printer=p3)(inp))
def test_issue_4540():
if ON_TRAVIS:
skip("Too slow for travis.")
# Note, this integral is probably nonelementary
assert not integrate(
(sin(1/x) - x*exp(x)) /
((-sin(1/x) + x*exp(x))*x + x*sin(1/x)), x).has(Integral)
def test_matplotlib():
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if matplotlib:
plot_implicit_tests('test')
test_line_color()
else:
skip("Matplotlib not the default backend")
def test_issue_15827():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol("A", 3, 3)
f = lambdify(A, 2*A)
assert numpy.array_equal(f(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2, 4, 6], [2, 4, 6], [2, 4, 6]]))
def test_numpy_array_arg():
# Test for issue 14655 (numpy part)
if not numpy:
skip("numpy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
assert f(numpy.array([2.0, 1.0])) == 5
def test_minpoly_compose1():
skip("This test hangs.")
# this test hangs because factor_list hangs in minpoly_op_algebraic_number
# on a polynomial of degree 96, which is factored by Sage very fast;
# one of the factors is the minimal polynomial.
ex = sqrt(1 + 2**Rational(1,3)) + sqrt(1 + 2**Rational(1,4)) + sqrt(2)
mp = minimal_polynomial(ex, x)
assert degree(mp) == 48 and mp.subs({x:0}) == -16630256576
def test_tensorflow_basic_math():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.constant(0, dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s) == 0.5
def test_tensorflow_relational():
if not tensorflow:
skip("tensorflow not installed.")
expr = x >= 0
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: 1})
def test_tensorflow_multi_min():
if not tensorflow:
skip("tensorflow not installed.")
expr = Min(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: -2}) == -2
def test_tensorflow_placeholders():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_numpy_transl():
if not numpy:
skip("numpy not installed.")
from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
for sym, nump in NUMPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert nump in numpy.__dict__
def test_issue_2236_float():
skip("This test hangs.")
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(solve(eqs, sym, rational=False, check=False, simplify=False)) == 2
def test_tensorflow_transl():
if not tensorflow:
skip("tensorflow not installed")
from sympy.utilities.lambdify import TENSORFLOW_TRANSLATIONS
for sym, tens in TENSORFLOW_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert tens in tensorflow.__dict__
def test_numpy_old_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
f = lambdify((x, y, z), A, [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
assert isinstance(f(1, 2, 3), numpy.matrix)
def test_python_div_zero_issue_11306():
if not numpy:
skip("numpy not installed.")
p = Piecewise((1 / x, y < -1), (x, y < 1), (1 / x, True))
f = lambdify([x, y], p, modules='numpy')
numpy.seterr(divide='ignore')
assert str(float(f(0,1))) == 'inf'
numpy.seterr(divide='warn')
def test_log10():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, ), log10(a), 'numpy')(100) - 2) < 1e-16
def test_exp2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, ), exp2(a), 'numpy')(5) - 32) < 1e-16
def test_log2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, ), log2(a), 'numpy')(256) - 8) < 1e-16
def test_issue_4941():
if ON_TRAVIS:
skip("Too slow for travis.")
assert not integrate(sqrt(1 + sinh(x / 20)**2), (x, -25, 25)).has(Integral)
def test_sqrt():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, ), sqrt(a), 'numpy')(4) - 2) < 1e-16
def test_issue_7130():
if ON_TRAVIS:
skip("Too slow for travis.")
i, L, a, b = symbols('i L a b')
integrand = (cos(pi * i * x / L)**2 / (a + b * x)).rewrite(exp)
assert x not in integrate(integrand, (x, 0, L)).free_symbols
def test_sampling_gamma_inverse():
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for sampling of gamma inverse.')
X = GammaInverse("x", 1, 1)
assert sample(X) in X.pspace.domain.set
def test_F95_gfortran():
if ("F95", 'gfortran') in invalid_lang_compilers:
skip("`gfortran' command didn't work as expected")
def test_hypot():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, b), hypot(a, b), 'numpy')(3, 4) - 5) < 1e-16
def test_F95_g95():
if ("F95", 'g95') in invalid_lang_compilers:
skip("`g95' command didn't work as expected")
def test_matplotlib():
if matplotlib:
plot_and_save('test')
else:
skip("Matplotlib not the default backend")
def test_to_numpy():
if not np:
skip("numpy not installed or Python too old.")
result = np.matrix([[1, 2], [3, 4]], dtype='complex')
assert (to_numpy(m) == result).all()
def test_matrix_tensor_product():
if not np:
skip("numpy not installed or Python too old.")
l1 = zeros(4)
for i in range(16):
l1[i] = 2**i
l2 = zeros(4)
for i in range(16):
l2[i] = i
l3 = zeros(2)
for i in range(4):
l3[i] = i
vec = Matrix([1, 2, 3])
#test for Matrix known 4x4 matricies
numpyl1 = np.matrix(l1.tolist())
numpyl2 = np.matrix(l2.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, l2]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [l2, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for other known matrix of different dimensions
numpyl2 = np.matrix(l3.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, l3]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [l3, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for non square matrix
numpyl2 = np.matrix(vec.tolist())
numpy_product = np.kron(numpyl1, numpyl2)
args = [l1, vec]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
numpy_product = np.kron(numpyl2, numpyl1)
args = [vec, l1]
sympy_product = matrix_tensor_product(*args)
assert numpy_product.tolist() == sympy_product.tolist()
#test for random matrix with random values that are floats
random_matrix1 = np.random.rand(np.random.rand()*5 + 1, np.random.rand()*5 + 1)
random_matrix2 = np.random.rand(np.random.rand()*5 + 1, np.random.rand()*5 + 1)
numpy_product = np.kron(random_matrix1, random_matrix2)
args = [Matrix(random_matrix1.tolist()), Matrix(random_matrix2.tolist())]
sympy_product = matrix_tensor_product(*args)
assert not (sympy_product - Matrix(numpy_product.tolist())).tolist() > \
(ones(sympy_product.rows, sympy_product.cols)*epsilon).tolist()
#test for three matrix kronecker
sympy_product = matrix_tensor_product(l1, vec, l2)
numpy_product = np.kron(l1, np.kron(vec, l2))
assert numpy_product.tolist() == sympy_product.tolist()
def test_expm1():
if not np:
skip("NumPy not installed")
f = lambdify((a, ), expm1(a), 'numpy')
assert abs(f(1e-10) - 1e-10 - 5e-21) < 1e-22
def test_numpy_inverse():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A**-1, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, -2], [0, 1]]))
def test_legendre_fail():
skip("Infinite recursion kills Travis")
n = Symbol("n")
assert laguerre(-n, x) == exp(x) * laguerre(n - 1, -x)
assert laguerre(-3, x) == exp(x) * laguerre(2, -x)
def test_log1p():
if not np:
skip("NumPy not installed")
f = lambdify((a, ), log1p(a), 'numpy')
assert abs(f(1e-99) - 1e-99) < 1e-100
def test_subgroup_search2():
skip('takes too much time')
_subgroup_search(16, 17, 1)
def test_numpy_transpose():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A.T, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, 0], [2, 1]]))
def test_issue_13642():
if not numpy:
skip("numpy not installed")
f = lambdify(x, sinc(x))
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_F95_ifort():
if ("F95", 'ifort') in invalid_lang_compilers:
skip("`ifort' command didn't work as expected")
def test_C_cc():
if ("C", 'cc') in invalid_lang_compilers:
skip("`cc' command didn't work as expected")
def test_issue_15925b():
if ON_TRAVIS:
skip("Too slow for travis.")
assert not integrate(
sqrt((-12 * cos(x)**2 * sin(x))**2 + (12 * cos(x) * sin(x)**2)**2),
(x, 0, pi / 6)).has(Integral)
def test_bicycle():
if ON_TRAVIS:
skip("Too slow for travis.")
# Code to get equations of motion for a bicycle modeled as in:
# J.P Meijaard, Jim M Papadopoulos, Andy Ruina and A.L Schwab. Linearized
# dynamics equations for the balance and steer of a bicycle: a benchmark
# and review. Proceedings of The Royal Society (2007) 463, 1955-1982
# doi: 10.1098/rspa.2007.1857
# Note that this code has been crudely ported from Autolev, which is the
# reason for some of the unusual naming conventions. It was purposefully as
# similar as possible in order to aide debugging.
# Declare Coordinates & Speeds
# Simple definitions for qdots - qd = u
# Speeds are: yaw frame ang. rate, roll frame ang. rate, rear wheel frame
# ang. rate (spinning motion), frame ang. rate (pitching motion), steering
# frame ang. rate, and front wheel ang. rate (spinning motion).
# Wheel positions are ignorable coordinates, so they are not introduced.
q1, q2, q4, q5 = dynamicsymbols('q1 q2 q4 q5')
q1d, q2d, q4d, q5d = dynamicsymbols('q1 q2 q4 q5', 1)
u1, u2, u3, u4, u5, u6 = dynamicsymbols('u1 u2 u3 u4 u5 u6')
u1d, u2d, u3d, u4d, u5d, u6d = dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)
# Declare System's Parameters
WFrad, WRrad, htangle, forkoffset = symbols('WFrad WRrad htangle forkoffset')
forklength, framelength, forkcg1 = symbols('forklength framelength forkcg1')
forkcg3, framecg1, framecg3, Iwr11 = symbols('forkcg3 framecg1 framecg3 Iwr11')
Iwr22, Iwf11, Iwf22, Iframe11 = symbols('Iwr22 Iwf11 Iwf22 Iframe11')
Iframe22, Iframe33, Iframe31, Ifork11 = symbols('Iframe22 Iframe33 Iframe31 Ifork11')
Ifork22, Ifork33, Ifork31, g = symbols('Ifork22 Ifork33 Ifork31 g')
mframe, mfork, mwf, mwr = symbols('mframe mfork mwf mwr')
# Set up reference frames for the system
# N - inertial
# Y - yaw
# R - roll
# WR - rear wheel, rotation angle is ignorable coordinate so not oriented
# Frame - bicycle frame
# TempFrame - statically rotated frame for easier reference inertia definition
# Fork - bicycle fork
# TempFork - statically rotated frame for easier reference inertia definition
# WF - front wheel, again posses a ignorable coordinate
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
R = Y.orientnew('R', 'Axis', [q2, Y.x])
Frame = R.orientnew('Frame', 'Axis', [q4 + htangle, R.y])
WR = ReferenceFrame('WR')
TempFrame = Frame.orientnew('TempFrame', 'Axis', [-htangle, Frame.y])
Fork = Frame.orientnew('Fork', 'Axis', [q5, Frame.x])
TempFork = Fork.orientnew('TempFork', 'Axis', [-htangle, Fork.y])
WF = ReferenceFrame('WF')
# Kinematics of the Bicycle First block of code is forming the positions of
# the relevant points
# rear wheel contact -> rear wheel mass center -> frame mass center +
# frame/fork connection -> fork mass center + front wheel mass center ->
# front wheel contact point
WR_cont = Point('WR_cont')
WR_mc = WR_cont.locatenew('WR_mc', WRrad * R.z)
Steer = WR_mc.locatenew('Steer', framelength * Frame.z)
Frame_mc = WR_mc.locatenew('Frame_mc', - framecg1 * Frame.x
+ framecg3 * Frame.z)
Fork_mc = Steer.locatenew('Fork_mc', - forkcg1 * Fork.x
+ forkcg3 * Fork.z)
WF_mc = Steer.locatenew('WF_mc', forklength * Fork.x + forkoffset * Fork.z)
WF_cont = WF_mc.locatenew('WF_cont', WFrad * (dot(Fork.y, Y.z) * Fork.y -
Y.z).normalize())
# Set the angular velocity of each frame.
# Angular accelerations end up being calculated automatically by
# differentiating the angular velocities when first needed.
# u1 is yaw rate
# u2 is roll rate
# u3 is rear wheel rate
# u4 is frame pitch rate
# u5 is fork steer rate
# u6 is front wheel rate
Y.set_ang_vel(N, u1 * Y.z)
R.set_ang_vel(Y, u2 * R.x)
WR.set_ang_vel(Frame, u3 * Frame.y)
Frame.set_ang_vel(R, u4 * Frame.y)
Fork.set_ang_vel(Frame, u5 * Fork.x)
WF.set_ang_vel(Fork, u6 * Fork.y)
# Form the velocities of the previously defined points, using the 2 - point
# theorem (written out by hand here). Accelerations again are calculated
# automatically when first needed.
WR_cont.set_vel(N, 0)
WR_mc.v2pt_theory(WR_cont, N, WR)
Steer.v2pt_theory(WR_mc, N, Frame)
Frame_mc.v2pt_theory(WR_mc, N, Frame)
Fork_mc.v2pt_theory(Steer, N, Fork)
WF_mc.v2pt_theory(Steer, N, Fork)
WF_cont.v2pt_theory(WF_mc, N, WF)
# Sets the inertias of each body. Uses the inertia frame to construct the
# inertia dyadics. Wheel inertias are only defined by principle moments of
# inertia, and are in fact constant in the frame and fork reference frames;
# it is for this reason that the orientations of the wheels does not need
# to be defined. The frame and fork inertias are defined in the 'Temp'
# frames which are fixed to the appropriate body frames; this is to allow
# easier input of the reference values of the benchmark paper. Note that
# due to slightly different orientations, the products of inertia need to
# have their signs flipped; this is done later when entering the numerical
# value.
Frame_I = (inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0, Iframe31), Frame_mc)
Fork_I = (inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
WR_I = (inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
WF_I = (inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)
# Declaration of the RigidBody containers. ::
BodyFrame = RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
BodyFork = RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
BodyWR = RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
BodyWF = RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)
# The kinematic differential equations; they are defined quite simply. Each
# entry in this list is equal to zero.
kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]
# The nonholonomic constraints are the velocity of the front wheel contact
# point dotted into the X, Y, and Z directions; the yaw frame is used as it
# is "closer" to the front wheel (1 less DCM connecting them). These
# constraints force the velocity of the front wheel contact point to be 0
# in the inertial frame; the X and Y direction constraints enforce a
# "no-slip" condition, and the Z direction constraint forces the front
# wheel contact point to not move away from the ground frame, essentially
# replicating the holonomic constraint which does not allow the frame pitch
# to change in an invalid fashion.
conlist_speed = [WF_cont.vel(N) & Y.x, WF_cont.vel(N) & Y.y, WF_cont.vel(N) & Y.z]
# The holonomic constraint is that the position from the rear wheel contact
# point to the front wheel contact point when dotted into the
# normal-to-ground plane direction must be zero; effectively that the front
# and rear wheel contact points are always touching the ground plane. This
# is actually not part of the dynamic equations, but instead is necessary
# for the lineraization process.
conlist_coord = [WF_cont.pos_from(WR_cont) & Y.z]
# The force list; each body has the appropriate gravitational force applied
# at its mass center.
FL = [(Frame_mc, -mframe * g * Y.z),
(Fork_mc, -mfork * g * Y.z),
(WF_mc, -mwf * g * Y.z),
(WR_mc, -mwr * g * Y.z)]
BL = [BodyFrame, BodyFork, BodyWR, BodyWF]
# The N frame is the inertial frame, coordinates are supplied in the order
# of independent, dependent coordinates, as are the speeds. The kinematic
# differential equation are also entered here. Here the dependent speeds
# are specified, in the same order they were provided in earlier, along
# with the non-holonomic constraints. The dependent coordinate is also
# provided, with the holonomic constraint. Again, this is only provided
# for the linearization process.
KM = KanesMethod(N, q_ind=[q1, q2, q5],
q_dependent=[q4], configuration_constraints=conlist_coord,
u_ind=[u2, u3, u5],
u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
kd_eqs=kd)
(fr, frstar) = KM.kanes_equations(FL, BL)
# This is the start of entering in the numerical values from the benchmark
# paper to validate the eigen values of the linearized equations from this
# model to the reference eigen values. Look at the aforementioned paper for
# more information. Some of these are intermediate values, used to
# transform values from the paper into the coordinate systems used in this
# model.
PaperRadRear = 0.3
PaperRadFront = 0.35
HTA = evalf.N(pi / 2 - pi / 10)
TrailPaper = 0.08
rake = evalf.N(-(TrailPaper*sin(HTA)-(PaperRadFront*cos(HTA))))
PaperWb = 1.02
PaperFrameCgX = 0.3
PaperFrameCgZ = 0.9
PaperForkCgX = 0.9
PaperForkCgZ = 0.7
FrameLength = evalf.N(PaperWb*sin(HTA)-(rake-(PaperRadFront-PaperRadRear)*cos(HTA)))
FrameCGNorm = evalf.N((PaperFrameCgZ - PaperRadRear-(PaperFrameCgX/sin(HTA))*cos(HTA))*sin(HTA))
FrameCGPar = evalf.N((PaperFrameCgX / sin(HTA) + (PaperFrameCgZ - PaperRadRear - PaperFrameCgX / sin(HTA) * cos(HTA)) * cos(HTA)))
tempa = evalf.N((PaperForkCgZ - PaperRadFront))
tempb = evalf.N((PaperWb-PaperForkCgX))
tempc = evalf.N(sqrt(tempa**2+tempb**2))
PaperForkL = evalf.N((PaperWb*cos(HTA)-(PaperRadFront-PaperRadRear)*sin(HTA)))
ForkCGNorm = evalf.N(rake+(tempc * sin(pi/2-HTA-acos(tempa/tempc))))
ForkCGPar = evalf.N(tempc * cos((pi/2-HTA)-acos(tempa/tempc))-PaperForkL)
# Here is the final assembly of the numerical values. The symbol 'v' is the
# forward speed of the bicycle (a concept which only makes sense in the
# upright, static equilibrium case?). These are in a dictionary which will
# later be substituted in. Again the sign on the *product* of inertia
# values is flipped here, due to different orientations of coordinate
# systems.
v = symbols('v')
val_dict = {WFrad: PaperRadFront,
WRrad: PaperRadRear,
htangle: HTA,
forkoffset: rake,
forklength: PaperForkL,
framelength: FrameLength,
forkcg1: ForkCGPar,
forkcg3: ForkCGNorm,
framecg1: FrameCGNorm,
framecg3: FrameCGPar,
Iwr11: 0.0603,
Iwr22: 0.12,
Iwf11: 0.1405,
Iwf22: 0.28,
Ifork11: 0.05892,
Ifork22: 0.06,
Ifork33: 0.00708,
Ifork31: 0.00756,
Iframe11: 9.2,
Iframe22: 11,
Iframe33: 2.8,
Iframe31: -2.4,
mfork: 4,
mframe: 85,
mwf: 3,
mwr: 2,
g: 9.81,
q1: 0,
q2: 0,
q4: 0,
q5: 0,
u1: 0,
u2: 0,
u3: v / PaperRadRear,
u4: 0,
u5: 0,
u6: v / PaperRadFront}
# Linearizes the forcing vector; the equations are set up as MM udot =
# forcing, where MM is the mass matrix, udot is the vector representing the
# time derivatives of the generalized speeds, and forcing is a vector which
# contains both external forcing terms and internal forcing terms, such as
# centripital or coriolis forces. This actually returns a matrix with as
# many rows as *total* coordinates and speeds, but only as many columns as
# independent coordinates and speeds.
forcing_lin = KM.linearize()[0]
# As mentioned above, the size of the linearized forcing terms is expanded
# to include both q's and u's, so the mass matrix must have this done as
# well. This will likely be changed to be part of the linearized process,
# for future reference.
MM_full = KM.mass_matrix_full
MM_full_s = MM_full.subs(val_dict)
forcing_lin_s = forcing_lin.subs(KM.kindiffdict()).subs(val_dict)
MM_full_s = MM_full_s.evalf()
forcing_lin_s = forcing_lin_s.evalf()
# Finally, we construct an "A" matrix for the form xdot = A x (x being the
# state vector, although in this case, the sizes are a little off). The
# following line extracts only the minimum entries required for eigenvalue
# analysis, which correspond to rows and columns for lean, steer, lean
# rate, and steer rate.
Amat = MM_full_s.inv() * forcing_lin_s
A = Amat.extract([1, 2, 4, 6], [1, 2, 3, 5])
# Precomputed for comparison
Res = Matrix([[ 0, 0, 1.0, 0],
[ 0, 0, 0, 1.0],
[9.48977444677355, -0.891197738059089*v**2 - 0.571523173729245, -0.105522449805691*v, -0.330515398992311*v],
[11.7194768719633, -1.97171508499972*v**2 + 30.9087533932407, 3.67680523332152*v, -3.08486552743311*v]])
# Actual eigenvalue comparison
eps = 1.e-12
for i in range(6):
error = Res.subs(v, i) - A.subs(v, i)
assert all(abs(x) < eps for x in error)
def sskip():
if not runslow: skip("slow")
def test_issue_15227():
if ON_TRAVIS:
skip("Too slow for travis.")
i = integrate(log(1 - x) * log((1 + x)**2) / x, (x, 0, 1))
assert not i.has(Integral)
