def test_n_link_pendulum_on_cart_higher_order():
l0, m0 = symbols("l0 m0")
l1, m1 = symbols("l1 m1")
m2 = symbols("m2")
g = symbols("g")
q0, q1, q2 = dynamicsymbols("q0 q1 q2")
u0, u1, u2 = dynamicsymbols("u0 u1 u2")
F, T1 = dynamicsymbols("F T1")
kane1 = models.n_link_pendulum_on_cart(2)
massmatrix1 = Matrix([[m0 + m1 + m2, -l0*m1*cos(q1) - l0*m2*cos(q1),
-l1*m2*cos(q2)],
[-l0*m1*cos(q1) - l0*m2*cos(q1), l0**2*m1 + l0**2*m2,
l0*l1*m2*(sin(q1)*sin(q2) + cos(q1)*cos(q2))],
[-l1*m2*cos(q2),
l0*l1*m2*(sin(q1)*sin(q2) + cos(q1)*cos(q2)),
l1**2*m2]])
forcing1 = Matrix([[-l0*m1*u1**2*sin(q1) - l0*m2*u1**2*sin(q1) -
l1*m2*u2**2*sin(q2) + F],
[g*l0*m1*sin(q1) + g*l0*m2*sin(q1) -
l0*l1*m2*(sin(q1)*cos(q2) - sin(q2)*cos(q1))*u2**2],
[g*l1*m2*sin(q2) - l0*l1*m2*(-sin(q1)*cos(q2) +
sin(q2)*cos(q1))*u1**2]])
assert simplify(massmatrix1 - kane1.mass_matrix) == zeros(3)
assert simplify(forcing1 - kane1.forcing) == Matrix([0, 0, 0])
def test_pend():
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, l, g = symbols('m l g')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, -l * u * sin(q) * N.x + l * u * cos(q) * N.y)
kd = [qd - u]
FL = [(P, m * g * N.x)]
pa = Particle('pa', P, m)
BL = [pa]
KM = KanesMethod(N, [q], [u], kd)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
rhs.simplify()
assert expand(rhs[0]) == expand(-g / l * sin(q))
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(
2, 1)
def test_body_dcm():
A = Body('A')
B = Body('B')
A.frame.orient_axis(B.frame, B.frame.z, 10)
assert A.dcm(B) == Matrix([[cos(10), sin(10), 0], [-sin(10),
cos(10), 0], [0, 0, 1]])
assert A.dcm(B.frame) == Matrix([[cos(10), sin(10), 0],
[-sin(10), cos(10), 0], [0, 0, 1]])
def test_find_dynamicsymbols():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
# Test finding all dynamicsymbols
sol = {x, y.diff(), y, x.diff().diff(), z, z.diff()}
assert find_dynamicsymbols(expr) == sol
# Test finding all but those in sym_list
exclude = [x, y, z]
sol = {y.diff(), x.diff().diff(), z.diff()}
assert find_dynamicsymbols(expr, exclude) == sol
def test_find_dynamicsymbols():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
expr = Matrix([[a * x + b, x * y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
# Test finding all dynamicsymbols
sol = {x, y.diff(), y, x.diff().diff(), z, z.diff()}
assert find_dynamicsymbols(expr) == sol
# Test finding all but those in sym_list
exclude = [x, y, z]
sol = {y.diff(), x.diff().diff(), z.diff()}
assert find_dynamicsymbols(expr, exclude) == sol
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix(
[
[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)],
]
)
elif axis == 2:
return Matrix(
[
[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)],
]
)
elif axis == 3:
return Matrix(
[
[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1],
]
)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0], [0, cos(angle), -sin(angle)], [0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)], [0, 1, 0], [-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0], [sin(angle), cos(angle), 0], [0, 0, 1]])
def test_pend():
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, l, g = symbols('m l g')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, -l * u * sin(q) * N.x + l * u * cos(q) * N.y)
kd = [qd - u]
FL = [(P, m * g * N.x)]
pa = Particle('pa', P, m)
BL = [pa]
KM = KanesMethod(N, [q], [u], kd)
with warns_deprecated_sympy():
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
rhs.simplify()
assert expand(rhs[0]) == expand(-g / l * sin(q))
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(2, 1)
def test_pend():
q, u = dynamicsymbols("q u")
qd, ud = dynamicsymbols("q u", 1)
m, l, g = symbols("m l g")
N = ReferenceFrame("N")
P = Point("P")
P.set_vel(N, -l * u * sin(q) * N.x + l * u * cos(q) * N.y)
kd = [qd - u]
FL = [(P, m * g * N.x)]
pa = Particle("pa", P, m)
BL = [pa]
KM = KanesMethod(N, [q], [u], kd)
with warns_deprecated_sympy():
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
rhs.simplify()
assert expand(rhs[0]) == expand(-g / l * sin(q))
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(
2, 1)
def test_two_dof():
# This is for a 2 d.o.f., 2 particle spring-mass-damper.
# The first coordinate is the displacement of the first particle, and the
# second is the relative displacement between the first and second
# particles. Speeds are defined as the time derivatives of the particles.
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
N = ReferenceFrame('N')
P1 = Point('P1')
P2 = Point('P2')
P1.set_vel(N, u1 * N.x)
P2.set_vel(N, (u1 + u2) * N.x)
kd = [q1d - u1, q2d - u2]
# Now we create the list of forces, then assign properties to each
# particle, then create a list of all particles.
FL = [(P1, (-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2) * N.x), (P2, (-k2 *
q2 - c2 * u2) * N.x)]
pa1 = Particle('pa1', P1, m)
pa2 = Particle('pa2', P2, m)
BL = [pa1, pa2]
# Finally we create the KanesMethod object, specify the inertial frame,
# pass relevant information, and form Fr & Fr*. Then we calculate the mass
# matrix and forcing terms, and finally solve for the udots.
KM = KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd)
KM.kanes_equations(BL, FL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m)
assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 *
c2 * u2) / m)
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(4, 1)
# Make sure an error is raised if nonlinear kinematic differential
# equations are supplied.
kd = [q1d - u1**2, sin(q2d) - cos(u2)]
raises(ValueError, lambda: KanesMethod(N, q_ind=[q1, q2],
u_ind=[u1, u2], kd_eqs=kd))
def test_find_dynamicsymbols():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
# Test finding all dynamicsymbols
sol = {x, y.diff(), y, x.diff().diff(), z, z.diff()}
assert find_dynamicsymbols(expr) == sol
# Test finding all but those in sym_list
exclude_list = [x, y, z]
sol = {y.diff(), x.diff().diff(), z.diff()}
assert find_dynamicsymbols(expr, exclude=exclude_list) == sol
# Test finding all dynamicsymbols in a vector with a given reference frame
d, e, f = dynamicsymbols('d, e, f')
A = ReferenceFrame('A')
v = d * A.x + e * A.y + f * A.z
sol = {d, e, f}
assert find_dynamicsymbols(v, reference_frame=A) == sol
# Test if a ValueError is raised on supplying only a vector as input
raises(ValueError, lambda: find_dynamicsymbols(v))
def test_msubs():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
# Test simple substitution
expr = Matrix([[a * x + b, x * y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
sol = Matrix([[a + b, y], [x.diff().diff(), 1]])
sd = {x: 1, z: 1, z.diff(): 0, y.diff(): 0}
assert msubs(expr, sd) == sol
# Test smart substitution
expr = cos(x + y) * tan(x + y) + b * x.diff()
sd = {x: 0, y: pi / 2, x.diff(): 1}
assert msubs(expr, sd, smart=True) == b + 1
N = ReferenceFrame('N')
v = x * N.x + y * N.y
d = x * (N.x | N.x) + y * (N.y | N.y)
v_sol = 1 * N.y
d_sol = 1 * (N.y | N.y)
sd = {x: 0, y: 1}
assert msubs(v, sd) == v_sol
assert msubs(d, sd) == d_sol
def test_msubs():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
# Test simple substitution
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
sol = Matrix([[a + b, y],
[x.diff().diff(), 1]])
sd = {x: 1, z: 1, z.diff(): 0, y.diff(): 0}
assert msubs(expr, sd) == sol
# Test smart substitution
expr = cos(x + y)*tan(x + y) + b*x.diff()
sd = {x: 0, y: pi/2, x.diff(): 1}
assert msubs(expr, sd, smart=True) == b + 1
N = ReferenceFrame('N')
v = x*N.x + y*N.y
d = x*(N.x|N.x) + y*(N.y|N.y)
v_sol = 1*N.y
d_sol = 1*(N.y|N.y)
sd = {x: 0, y: 1}
assert msubs(v, sd) == v_sol
assert msubs(d, sd) == d_sol
def kinematic_equations(speeds, coords, rot_type, rot_order=''):
"""Gives equations relating the qdot's to u's for a rotation type.
Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z
Parameters
==========
speeds : list of length 3
The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
The coordinates used to define the orientation of the two frames.
rot_type : str
The type of rotation used to create the equations. Body, Space, or
Quaternion only
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
... order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']
"""
# Code below is checking and sanitizing input
approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131',
'212', '232', '313', '323', '1', '2', '3', '')
# make sure XYZ => 123 and rot_type is in lower case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.lower()
if not isinstance(speeds, (list, tuple)):
raise TypeError('Need to supply speeds in a list')
if len(speeds) != 3:
raise TypeError('Need to supply 3 body-fixed speeds')
if not isinstance(coords, (list, tuple)):
raise TypeError('Need to supply coordinates in a list')
if rot_type in ['body', 'space']:
if rot_order not in approved_orders:
raise ValueError('Not an acceptable rotation order')
if len(coords) != 3:
raise ValueError('Need 3 coordinates for body or space')
# Actual hard-coded kinematic differential equations
w1, w2, w3 = speeds
if w1 == w2 == w3 == 0:
return [S.Zero] * 3
q1, q2, q3 = coords
q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords]
s1, s2, s3 = [sin(q1), sin(q2), sin(q3)]
c1, c2, c3 = [cos(q1), cos(q2), cos(q3)]
if rot_type == 'body':
if rot_order == '123':
return [
q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 * c3,
q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3
]
if rot_order == '231':
return [
q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 * c3,
q3d - w1 - (-w2 * c3 + w3 * s3) * s2 / c2
]
if rot_order == '312':
return [
q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 * s3,
q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2
]
if rot_order == '132':
return [
q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 * c3,
q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2
]
if rot_order == '213':
return [
q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 * s3,
q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3
]
if rot_order == '321':
return [
q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 * s3,
q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2
]
if rot_order == '121':
return [
q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 * s3,
q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2
]
if rot_order == '131':
return [
q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 * c3,
q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2
]
if rot_order == '212':
return [
q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 * s3,
q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2
]
if rot_order == '232':
return [
q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 * c3,
q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2
]
if rot_order == '313':
return [
q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 * s3,
q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3
]
if rot_order == '323':
return [
q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 * c3,
q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3
]
if rot_type == 'space':
if rot_order == '123':
return [
q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2,
q2d - w2 * c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2
]
if rot_order == '231':
return [
q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2,
q2d + w1 * s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2
]
if rot_order == '312':
return [
q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3,
q2d - w1 * c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2
]
if rot_order == '132':
return [
q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2,
q2d - w2 * s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2
]
if rot_order == '213':
return [
q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2,
q2d - w1 * c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2
]
if rot_order == '321':
return [
q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3,
q2d - w1 * s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2
]
if rot_order == '121':
return [
q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2,
q2d - w2 * c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2
]
if rot_order == '131':
return [
q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2,
q2d - w2 * s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2
]
if rot_order == '212':
return [
q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2,
q2d - w1 * c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2
]
if rot_order == '232':
return [
q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2,
q2d + w1 * s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2
]
if rot_order == '313':
return [
q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3,
q2d - w1 * c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2
]
if rot_order == '323':
return [
q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3,
q2d - w1 * s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2
]
elif rot_type == 'quaternion':
if rot_order != '':
raise ValueError('Cannot have rotation order for quaternion')
if len(coords) != 4:
raise ValueError('Need 4 coordinates for quaternion')
# Actual hard-coded kinematic differential equations
e0, e1, e2, e3 = coords
w = Matrix(speeds + [0])
E = Matrix([[e0, -e3, e2, e1], [e3, e0, -e1, e2], [-e2, e1, e0, e3],
[-e1, -e2, -e3, e0]])
edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]])
return list(edots.T - 0.5 * w.T * E.T)
else:
raise ValueError('Not an approved rotation type for this function')
def kinematic_equations(speeds, coords, rot_type, rot_order=''):
"""Gives equations relating the qdot's to u's for a rotation type.
Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z
Parameters
==========
speeds : list of length 3
The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
The coordinates used to define the orientation of the two frames.
rot_type : str
The type of rotation used to create the equations. Body, Space, or
Quaternion only
rot_order : str
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
... order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']
"""
# Code below is checking and sanitizing input
approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131',
'212', '232', '313', '323', '1', '2', '3', '')
rot_order = str(rot_order).upper() # Now we need to make sure XYZ = 123
rot_type = rot_type.upper()
rot_order = [i.replace('X', '1') for i in rot_order]
rot_order = [i.replace('Y', '2') for i in rot_order]
rot_order = [i.replace('Z', '3') for i in rot_order]
rot_order = ''.join(rot_order)
if not isinstance(speeds, (list, tuple)):
raise TypeError('Need to supply speeds in a list')
if len(speeds) != 3:
raise TypeError('Need to supply 3 body-fixed speeds')
if not isinstance(coords, (list, tuple)):
raise TypeError('Need to supply coordinates in a list')
if rot_type.lower() in ['body', 'space']:
if rot_order not in approved_orders:
raise ValueError('Not an acceptable rotation order')
if len(coords) != 3:
raise ValueError('Need 3 coordinates for body or space')
# Actual hard-coded kinematic differential equations
q1, q2, q3 = coords
q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords]
w1, w2, w3 = speeds
s1, s2, s3 = [sin(q1), sin(q2), sin(q3)]
c1, c2, c3 = [cos(q1), cos(q2), cos(q3)]
if rot_type.lower() == 'body':
if rot_order == '123':
return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 *
c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3]
if rot_order == '231':
return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2]
if rot_order == '312':
return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 *
s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2]
if rot_order == '132':
return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 *
c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2]
if rot_order == '213':
return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 *
s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3]
if rot_order == '321':
return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 *
s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2]
if rot_order == '121':
return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 *
s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2]
if rot_order == '131':
return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2]
if rot_order == '212':
return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 *
s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2]
if rot_order == '232':
return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 *
c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2]
if rot_order == '313':
return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 *
s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3]
if rot_order == '323':
return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 *
c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3]
if rot_type.lower() == 'space':
if rot_order == '123':
return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2]
if rot_order == '231':
return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2]
if rot_order == '312':
return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2]
if rot_order == '132':
return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 *
s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2]
if rot_order == '213':
return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2]
if rot_order == '321':
return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2]
if rot_order == '121':
return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2]
if rot_order == '131':
return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 *
s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2]
if rot_order == '212':
return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2]
if rot_order == '232':
return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2]
if rot_order == '313':
return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2]
if rot_order == '323':
return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2]
elif rot_type.lower() == 'quaternion':
if rot_order != '':
raise ValueError('Cannot have rotation order for quaternion')
if len(coords) != 4:
raise ValueError('Need 4 coordinates for quaternion')
# Actual hard-coded kinematic differential equations
e0, e1, e2, e3 = coords
w = Matrix(speeds + [0])
E = Matrix([[e0, -e3, e2, e1], [e3, e0, -e1, e2], [-e2, e1, e0, e3],
[-e1, -e2, -e3, e0]])
edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]])
return list(edots.T - 0.5 * w.T * E.T)
else:
raise ValueError('Not an approved rotation type for this function')
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols('t r m g v')
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols('q1:7')
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols('u:6')
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols('u1:7')
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame('N') # Inertial frame
NO = Point('NO') # Inertial origin
A = N.orientnew('A', 'Axis', [q1, N.z]) # Yaw intermediate frame
B = A.orientnew('B', 'Axis', [q2, A.x]) # Lean intermediate frame
C = B.orientnew('C', 'Axis', [q3, B.y]) # Disc fixed frame
CO = NO.locatenew('CO', q4*N.x + q5*N.y + q6*N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1*B.x + u2*B.y + u3*B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4*C.x + u5*C.y + u6*C.z)
# Disc Ground Contact Point
P = CO.locatenew('P', r*B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_co_n_qd - CO.vel(N), uv) for uv in N])
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(N, C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m*g*A.z
# Create inertia dyadic of disc C about point CO
I = (m * r**2) / 4
J = (m * r**2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody('Disc', CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(N, [q1, q2, q3, q4, q5], [u1, u2, u3], kd_eqs=kindiffs,
q_dependent=[q6], configuration_constraints=f_c,
u_dependent=[u4, u5, u6], velocity_constraints=f_v)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t)
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qd:
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix([0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r*cos(q2)}
u_op = {u1: 0,
u2: sin(q2)*q1d + q3d,
u3: cos(q2)*q1d,
u4: -r*(sin(q2)*q1d + q3d)*cos(q3),
u5: 0,
u6: -r*(sin(q2)*q1d + q3d)*sin(q3)}
qd_op = {q2d: 0,
q4d: -r*(sin(q2)*q1d + q3d)*cos(q1),
q5d: -r*(sin(q2)*q1d + q3d)*sin(q1),
q6d: 0}
ud_op = {u1d: 4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5,
u2d: 0,
u3d: 0,
u4d: r*(sin(q2)*sin(q3)*q1d*q3d + sin(q3)*q3d**2),
u5d: r*(4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5),
u6d: -r*(sin(q2)*cos(q3)*q1d*q3d + cos(q3)*q3d**2)}
A, B = linearizer.linearize(op_point=[q_op, u_op, qd_op, ud_op], A_and_B=True, simplify=True)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix([[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1)*q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1)*q3d, 0, 0, 0, 0, cos(q1), -sin(q1), 0],
[0, 4/5, 0, 0, 0, 0, 0, 6*q3d/5],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -2*q3d, 0, 0]])
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert A.subs(upright_nominal).subs(q3d, 1/sqrt(3)).eigenvals() == {0: 8}
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Defines the orientation of this frame relative to a parent frame.
Parameters
==========
parent : ReferenceFrame
The frame that this ReferenceFrame will have its orientation matrix
defined in relation to.
rot_type : str
The type of orientation matrix that is being created. Supported
types are 'Body', 'Space', 'Quaternion', 'Axis', and 'DCM'.
See examples for correct usage.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
In case of rot_type='DCM', value must be a
sympy.matrices.MatrixBase object (or subclasses of it).
rot_order : str
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols, eye, ImmutableMatrix
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
Now we have a choice of how to implement the orientation. First is
Body. Body orientation takes this reference frame through three
successive simple rotations. Acceptable rotation orders are of length
3, expressed in XYZ or 123, and cannot have a rotation about about an
axis twice in a row.
>>> B.orient(N, 'Body', [q1, q2, q3], '123')
>>> B.orient(N, 'Body', [q1, q2, 0], 'ZXZ')
>>> B.orient(N, 'Body', [0, 0, 0], 'XYX')
Next is Space. Space is like Body, but the rotations are applied in the
opposite order.
>>> B.orient(N, 'Space', [q1, q2, q3], '312')
Next is Quaternion. This orients the new ReferenceFrame with
Quaternions, defined as a finite rotation about lambda, a unit vector,
by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.
>>> B.orient(N, 'Quaternion', [q0, q1, q2, q3])
Next is Axis. This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> B.orient(N, 'Axis', [q1, N.x + 2 * N.y])
Last is DCM (Direction Cosine Matrix). This is a rotation matrix
given manually.
>>> B.orient(N, 'DCM', eye(3))
>>> B.orient(N, 'DCM', ImmutableMatrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]]))
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
# Allow passing a rotation matrix manually.
if rot_type == 'DCM':
# When rot_type == 'DCM', then amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(amounts, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
else:
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1]])
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
rot_order = str(
rot_order).upper() # Now we need to make sure XYZ = 123
rot_type = rot_type.upper()
rot_order = [i.replace('X', '1') for i in rot_order]
rot_order = [i.replace('Y', '2') for i in rot_order]
rot_order = [i.replace('Z', '3') for i in rot_order]
rot_order = ''.join(rot_order)
if not rot_order in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = ((eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]], [axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) * sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 - q3 **
2, 2 * (q1 * q2 - q0 * q3), 2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3), q0 ** 2 - q1 ** 2 + q2 ** 2 - q3 ** 2,
2 * (q2 * q3 - q0 * q1)], [2 * (q1 * q3 - q0 * q2), 2 * (q0 *
q1 + q2 * q3), q0 ** 2 - q1 ** 2 - q2 ** 2 + q3 ** 2]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1])
* _rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1])
* _rot(a1, amounts[0]))
elif rot_type == 'DCM':
parent_orient = amounts
else:
raise NotImplementedError('That is not an implemented rotation')
#Reset the _dcm_cache of this frame, and remove it from the _dcm_caches
#of the frames it is linked to. Also remove it from the _dcm_dict of
#its parent
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
#Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
#Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
thetad = (amounts[0]).diff(dynamicsymbols._t)
wvec = thetad * amounts[1].express(parent).normalize()
elif rot_type == 'DCM':
wvec = self._w_diff_dcm(parent)
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def test_sub_qdot():
# This test solves exercises 8.12, 8.17 from Kane 1985 and defines
# some velocities in terms of q, qdot.
## --- Declare symbols ---
q1, q2, q3 = dynamicsymbols('q1:4')
q1d, q2d, q3d = dynamicsymbols('q1:4', level=1)
u1, u2, u3 = dynamicsymbols('u1:4')
u_prime, R, M, g, e, f, theta = symbols('u\' R, M, g, e, f, theta')
a, b, mA, mB, IA, J, K, t = symbols('a b mA mB IA J K t')
IA22, IA23, IA33 = symbols('IA22 IA23 IA33')
Q1, Q2, Q3 = symbols('Q1 Q2 Q3')
# --- Reference Frames ---
F = ReferenceFrame('F')
P = F.orientnew('P', 'axis', [-theta, F.y])
A = P.orientnew('A', 'axis', [q1, P.x])
A.set_ang_vel(F, u1*A.x + u3*A.z)
# define frames for wheels
B = A.orientnew('B', 'axis', [q2, A.z])
C = A.orientnew('C', 'axis', [q3, A.z])
## --- define points D, S*, Q on frame A and their velocities ---
pD = Point('D')
pD.set_vel(A, 0)
# u3 will not change v_D_F since wheels are still assumed to roll w/o slip
pD.set_vel(F, u2 * A.y)
pS_star = pD.locatenew('S*', e*A.y)
pQ = pD.locatenew('Q', f*A.y - R*A.x)
# masscenters of bodies A, B, C
pA_star = pD.locatenew('A*', a*A.y)
pB_star = pD.locatenew('B*', b*A.z)
pC_star = pD.locatenew('C*', -b*A.z)
for p in [pS_star, pQ, pA_star, pB_star, pC_star]:
p.v2pt_theory(pD, F, A)
# points of B, C touching the plane P
pB_hat = pB_star.locatenew('B^', -R*A.x)
pC_hat = pC_star.locatenew('C^', -R*A.x)
pB_hat.v2pt_theory(pB_star, F, B)
pC_hat.v2pt_theory(pC_star, F, C)
# --- relate qdot, u ---
# the velocities of B^, C^ are zero since B, C are assumed to roll w/o slip
kde = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]]
kde += [u1 - q1d]
kde_map = solve(kde, [q1d, q2d, q3d])
for k, v in list(kde_map.items()):
kde_map[k.diff(t)] = v.diff(t)
# inertias of bodies A, B, C
# IA22, IA23, IA33 are not specified in the problem statement, but are
# necessary to define an inertia object. Although the values of
# IA22, IA23, IA33 are not known in terms of the variables given in the
# problem statement, they do not appear in the general inertia terms.
inertia_A = inertia(A, IA, IA22, IA33, 0, IA23, 0)
inertia_B = inertia(B, K, K, J)
inertia_C = inertia(C, K, K, J)
# define the rigid bodies A, B, C
rbA = RigidBody('rbA', pA_star, A, mA, (inertia_A, pA_star))
rbB = RigidBody('rbB', pB_star, B, mB, (inertia_B, pB_star))
rbC = RigidBody('rbC', pC_star, C, mB, (inertia_C, pC_star))
## --- use kanes method ---
km = KanesMethod(F, [q1, q2, q3], [u1, u2], kd_eqs=kde, u_auxiliary=[u3])
forces = [(pS_star, -M*g*F.x), (pQ, Q1*A.x + Q2*A.y + Q3*A.z)]
bodies = [rbA, rbB, rbC]
# Q2 = -u_prime * u2 * Q1 / sqrt(u2**2 + f**2 * u1**2)
# -u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2) = R / Q1 * Q2
fr_expected = Matrix([
f*Q3 + M*g*e*sin(theta)*cos(q1),
Q2 + M*g*sin(theta)*sin(q1),
e*M*g*cos(theta) - Q1*f - Q2*R])
#Q1 * (f - u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2)))])
fr_star_expected = Matrix([
-(IA + 2*J*b**2/R**2 + 2*K +
mA*a**2 + 2*mB*b**2) * u1.diff(t) - mA*a*u1*u2,
-(mA + 2*mB +2*J/R**2) * u2.diff(t) + mA*a*u1**2,
0])
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
fr, fr_star = km.kanes_equations(forces, bodies)
assert (fr.expand() == fr_expected.expand())
assert ((fr_star_expected - trigsimp(fr_star)).expand() == zeros(3, 1))
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Sets the orientation of this reference frame relative to another
(parent) reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
rot_type : str
The method used to generate the direction cosine matrix. Supported
methods are:
- ``'Axis'``: simple rotations about a single common axis
- ``'DCM'``: for setting the direction cosine matrix directly
- ``'Body'``: three successive rotations about new intermediate
axes, also called "Euler and Tait-Bryan angles"
- ``'Space'``: three successive rotations about the parent
frames' unit vectors
- ``'Quaternion'``: rotations defined by four parameters which
result in a singularity free direction cosine matrix
amounts :
Expressions defining the rotation angles or direction cosine
matrix. These must match the ``rot_type``. See examples below for
details. The input types are:
- ``'Axis'``: 2-tuple (expr/sym/func, Vector)
- ``'DCM'``: Matrix, shape(3,3)
- ``'Body'``: 3-tuple of expressions, symbols, or functions
- ``'Space'``: 3-tuple of expressions, symbols, or functions
- ``'Quaternion'``: 4-tuple of expressions, symbols, or
functions
rot_order : str or int, optional
If applicable, the order of the successive of rotations. The string
``'123'`` and integer ``123`` are equivalent, for example. Required
for ``'Body'`` and ``'Space'``.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B')
>>> B2 = ReferenceFrame('B2')
Axis
----
``rot_type='Axis'`` creates a direction cosine matrix defined by a
simple rotation about a single axis fixed in both reference frames.
This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> B.orient(N, 'Axis', (q1, N.x))
The ``orient()`` method generates a direction cosine matrix and its
transpose which defines the orientation of B relative to N and vice
versa. Once orient is called, ``dcm()`` outputs the appropriate
direction cosine matrix.
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
The following two lines show how the sense of the rotation can be
defined. Both lines produce the same result.
>>> B.orient(N, 'Axis', (q1, -N.x))
>>> B.orient(N, 'Axis', (-q1, N.x))
The axis does not have to be defined by a unit vector, it can be any
vector in the parent frame.
>>> B.orient(N, 'Axis', (q1, N.x + 2 * N.y))
DCM
---
The direction cosine matrix can be set directly. The orientation of a
frame A can be set to be the same as the frame B above like so:
>>> B.orient(N, 'Axis', (q1, N.x))
>>> A = ReferenceFrame('A')
>>> A.orient(N, 'DCM', N.dcm(B))
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
**Note carefully that** ``N.dcm(B)`` **was passed into** ``orient()``
**for** ``A.dcm(N)`` **to match** ``B.dcm(N)``.
Body
----
``rot_type='Body'`` rotates this reference frame relative to the
provided reference frame by rotating through three successive simple
rotations. Each subsequent axis of rotation is about the "body fixed"
unit vectors of the new intermediate reference frame. This type of
rotation is also referred to rotating through the `Euler and Tait-Bryan
Angles <https://en.wikipedia.org/wiki/Euler_angles>`_.
For example, the classic Euler Angle rotation can be done by:
>>> B.orient(N, 'Body', (q1, q2, q3), 'XYX')
>>> B.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
This rotates B relative to N through ``q1`` about ``N.x``, then rotates
B again through q2 about B.y, and finally through q3 about B.x. It is
equivalent to:
>>> B1.orient(N, 'Axis', (q1, N.x))
>>> B2.orient(B1, 'Axis', (q2, B1.y))
>>> B.orient(B2, 'Axis', (q3, B2.x))
>>> B.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
Acceptable rotation orders are of length 3, expressed in as a string
``'XYZ'`` or ``'123'`` or integer ``123``. Rotations about an axis
twice in a row are prohibited.
>>> B.orient(N, 'Body', (q1, q2, 0), 'ZXZ')
>>> B.orient(N, 'Body', (q1, q2, 0), '121')
>>> B.orient(N, 'Body', (q1, q2, q3), 123)
Space
-----
``rot_type='Space'`` also rotates the reference frame in three
successive simple rotations but the axes of rotation are the
"Space-fixed" axes. For example:
>>> B.orient(N, 'Space', (q1, q2, q3), '312')
>>> B.dcm(N)
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
is equivalent to:
>>> B1.orient(N, 'Axis', (q1, N.z))
>>> B2.orient(B1, 'Axis', (q2, N.x))
>>> B.orient(B2, 'Axis', (q3, N.y))
>>> B.dcm(N).simplify() # doctest: +SKIP
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
It is worth noting that space-fixed and body-fixed rotations are
related by the order of the rotations, i.e. the reverse order of body
fixed will give space fixed and vice versa.
>>> B.orient(N, 'Space', (q1, q2, q3), '231')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
>>> B.orient(N, 'Body', (q3, q2, q1), '132')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
Quaternion
----------
``rot_type='Quaternion'`` orients the reference frame using
quaternions. Quaternion rotation is defined as a finite rotation about
lambda, a unit vector, by an amount theta. This orientation is
described by four parameters:
- ``q0 = cos(theta/2)``
- ``q1 = lambda_x sin(theta/2)``
- ``q2 = lambda_y sin(theta/2)``
- ``q3 = lambda_z sin(theta/2)``
This type does not need a ``rot_order``.
>>> B.orient(N, 'Quaternion', (q0, q1, q2, q3))
>>> B.dcm(N)
Matrix([
[q0**2 + q1**2 - q2**2 - q3**2, 2*q0*q3 + 2*q1*q2, -2*q0*q2 + 2*q1*q3],
[ -2*q0*q3 + 2*q1*q2, q0**2 - q1**2 + q2**2 - q3**2, 2*q0*q1 + 2*q2*q3],
[ 2*q0*q2 + 2*q1*q3, -2*q0*q1 + 2*q2*q3, q0**2 - q1**2 - q2**2 + q3**2]])
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
# Allow passing a rotation matrix manually.
if rot_type == 'DCM':
# When rot_type == 'DCM', then amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(amounts, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
else:
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0], [0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)], [0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0], [0, 0, 1]])
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123 and rot_type is in upper case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = (
(eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]], [axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) * sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[
q0**2 + q1**2 - q2**2 - q3**2, 2 * (q1 * q2 - q0 * q3),
2 * (q0 * q2 + q1 * q3)
],
[
2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)
],
[
2 * (q1 * q3 - q0 * q2),
2 * (q0 * q1 + q2 * q3),
q0**2 - q1**2 - q2**2 + q3**2
]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1]) *
_rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1]) *
_rot(a1, amounts[0]))
elif rot_type == 'DCM':
parent_orient = amounts
else:
raise NotImplementedError('That is not an implemented rotation')
# Reset the _dcm_cache of this frame, and remove it from the
# _dcm_caches of the frames it is linked to. Also remove it from the
# _dcm_dict of its parent
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
# Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
# Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
thetad = (amounts[0]).diff(dynamicsymbols._t)
wvec = thetad * amounts[1].express(parent).normalize()
elif rot_type == 'DCM':
wvec = self._w_diff_dcm(parent)
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols('t r m g v')
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols('q1:7')
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols('u:6')
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols('u1:7')
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame('N') # Inertial frame
NO = Point('NO') # Inertial origin
A = N.orientnew('A', 'Axis', [q1, N.z]) # Yaw intermediate frame
B = A.orientnew('B', 'Axis', [q2, A.x]) # Lean intermediate frame
C = B.orientnew('C', 'Axis', [q3, B.y]) # Disc fixed frame
CO = NO.locatenew('CO', q4*N.x + q5*N.y + q6*N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1*B.x + u2*B.y + u3*B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4*C.x + u5*C.y + u6*C.z)
# Disc Ground Contact Point
P = CO.locatenew('P', r*B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_co_n_qd - CO.vel(N), uv) for uv in N])
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(N, C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m*g*A.z
# Create inertia dyadic of disc C about point CO
I = (m * r**2) / 4
J = (m * r**2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody('Disc', CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(N, [q1, q2, q3, q4, q5], [u1, u2, u3], kd_eqs=kindiffs,
q_dependent=[q6], configuration_constraints=f_c,
u_dependent=[u4, u5, u6], velocity_constraints=f_v)
with warns_deprecated_sympy():
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t)
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qd:
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix([0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r*cos(q2)}
u_op = {u1: 0,
u2: sin(q2)*q1d + q3d,
u3: cos(q2)*q1d,
u4: -r*(sin(q2)*q1d + q3d)*cos(q3),
u5: 0,
u6: -r*(sin(q2)*q1d + q3d)*sin(q3)}
qd_op = {q2d: 0,
q4d: -r*(sin(q2)*q1d + q3d)*cos(q1),
q5d: -r*(sin(q2)*q1d + q3d)*sin(q1),
q6d: 0}
ud_op = {u1d: 4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5,
u2d: 0,
u3d: 0,
u4d: r*(sin(q2)*sin(q3)*q1d*q3d + sin(q3)*q3d**2),
u5d: r*(4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5),
u6d: -r*(sin(q2)*cos(q3)*q1d*q3d + cos(q3)*q3d**2)}
A, B = linearizer.linearize(op_point=[q_op, u_op, qd_op, ud_op], A_and_B=True, simplify=True)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix([[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1)*q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1)*q3d, 0, 0, 0, 0, cos(q1), -sin(q1), 0],
[0, Rational(4, 5), 0, 0, 0, 0, 0, 6*q3d/5],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -2*q3d, 0, 0]])
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert A.subs(upright_nominal).subs(q3d, 1/sqrt(3)).eigenvals() == {0: 8}
def orient_axis(self, parent, axis, angle):
"""Sets the orientation of this reference frame with respect to a
parent reference frame by rotating through an angle about an axis fixed
in the parent reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
axis : Vector
Vector fixed in the parent frame about about which this frame is
rotated. It need not be a unit vector and the rotation follows the
right hand rule.
angle : sympifiable
Angle in radians by which it the frame is to be rotated.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.orient_axis(N, N.x, q1)
The ``orient_axis()`` method generates a direction cosine matrix and
its transpose which defines the orientation of B relative to N and vice
versa. Once orient is called, ``dcm()`` outputs the appropriate
direction cosine matrix:
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
>>> N.dcm(B)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
The following two lines show that the sense of the rotation can be
defined by negating the vector direction or the angle. Both lines
produce the same result.
>>> B.orient_axis(N, -N.x, q1)
>>> B.orient_axis(N, N.x, -q1)
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
amount = sympify(angle)
theta = amount
axis = _check_vector(axis)
parent_orient_axis = []
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying.')
unit_axis = axis.express(parent).normalize()
unit_col = unit_axis.args[0][0]
parent_orient_axis = (
(eye(3) - unit_col * unit_col.T) * cos(theta) +
Matrix([[0, -unit_col[2], unit_col[1]],
[unit_col[2], 0, -unit_col[0]],
[-unit_col[1], unit_col[0], 0]]) *
sin(theta) + unit_col * unit_col.T)
self._dcm(parent, parent_orient_axis)
thetad = (amount).diff(dynamicsymbols._t)
wvec = thetad*axis.express(parent).normalize()
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def test_sub_qdot2():
# This test solves exercises 8.3 from Kane 1985 and defines
# all velocities in terms of q, qdot. We check that the generalized active
# forces are correctly computed if u terms are only defined in the
# kinematic differential equations.
#
# This functionality was added in PR 8948. Without qdot/u substitution, the
# KanesMethod constructor will fail during the constraint initialization as
# the B matrix will be poorly formed and inversion of the dependent part
# will fail.
g, m, Px, Py, Pz, R, t = symbols('g m Px Py Pz R t')
q = dynamicsymbols('q:5')
qd = dynamicsymbols('q:5', level=1)
u = dynamicsymbols('u:5')
## Define inertial, intermediate, and rigid body reference frames
A = ReferenceFrame('A')
B_prime = A.orientnew('B_prime', 'Axis', [q[0], A.z])
B = B_prime.orientnew('B', 'Axis', [pi / 2 - q[1], B_prime.x])
C = B.orientnew('C', 'Axis', [q[2], B.z])
## Define points of interest and their velocities
pO = Point('O')
pO.set_vel(A, 0)
# R is the point in plane H that comes into contact with disk C.
pR = pO.locatenew('R', q[3] * A.x + q[4] * A.y)
pR.set_vel(A, pR.pos_from(pO).diff(t, A))
pR.set_vel(B, 0)
# C^ is the point in disk C that comes into contact with plane H.
pC_hat = pR.locatenew('C^', 0)
pC_hat.set_vel(C, 0)
# C* is the point at the center of disk C.
pCs = pC_hat.locatenew('C*', R * B.y)
pCs.set_vel(C, 0)
pCs.set_vel(B, 0)
# calculate velocites of points C* and C^ in frame A
pCs.v2pt_theory(pR, A, B) # points C* and R are fixed in frame B
pC_hat.v2pt_theory(pCs, A, C) # points C* and C^ are fixed in frame C
## Define forces on each point of the system
R_C_hat = Px * A.x + Py * A.y + Pz * A.z
R_Cs = -m * g * A.z
forces = [(pC_hat, R_C_hat), (pCs, R_Cs)]
## Define kinematic differential equations
# let ui = omega_C_A & bi (i = 1, 2, 3)
# u4 = qd4, u5 = qd5
u_expr = [C.ang_vel_in(A) & uv for uv in B]
u_expr += qd[3:]
kde = [ui - e for ui, e in zip(u, u_expr)]
km1 = KanesMethod(A, q, u, kde)
with warns_deprecated_sympy():
fr1, _ = km1.kanes_equations(forces, [])
## Calculate generalized active forces if we impose the condition that the
# disk C is rolling without slipping
u_indep = u[:3]
u_dep = list(set(u) - set(u_indep))
vc = [pC_hat.vel(A) & uv for uv in [A.x, A.y]]
km2 = KanesMethod(A,
q,
u_indep,
kde,
u_dependent=u_dep,
velocity_constraints=vc)
with warns_deprecated_sympy():
fr2, _ = km2.kanes_equations(forces, [])
fr1_expected = Matrix([
-R * g * m * sin(q[1]),
-R * (Px * cos(q[0]) + Py * sin(q[0])) * tan(q[1]),
R * (Px * cos(q[0]) + Py * sin(q[0])), Px, Py
])
fr2_expected = Matrix([-R * g * m * sin(q[1]), 0, 0])
assert (trigsimp(fr1.expand()) == trigsimp(fr1_expected.expand()))
assert (trigsimp(fr2.expand()) == trigsimp(fr2_expected.expand()))
def test_rolling_disc():
# Rolling Disc Example
# Here the rolling disc is formed from the contact point up, removing the
# need to introduce generalized speeds. Only 3 configuration and three
# speed variables are need to describe this system, along with the disc's
# mass and radius, and the local gravity (note that mass will drop out).
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
r, m, g = symbols('r m g')
# The kinematics are formed by a series of simple rotations. Each simple
# rotation creates a new frame, and the next rotation is defined by the new
# frame's basis vectors. This example uses a 3-1-2 series of rotations, or
# Z, X, Y series of rotations. Angular velocity for this is defined using
# the second frame's basis (the lean frame).
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
w_R_N_qd = R.ang_vel_in(N)
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
# This is the translational kinematics. We create a point with no velocity
# in N; this is the contact point between the disc and ground. Next we form
# the position vector from the contact point to the disc's center of mass.
# Finally we form the velocity and acceleration of the disc.
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
# This is a simple way to form the inertia dyadic.
I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
# Kinematic differential equations; how the generalized coordinate time
# derivatives relate to generalized speeds.
kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L]
# Creation of the force list; it is the gravitational force at the mass
# center of the disc. Then we create the disc by assigning a Point to the
# center of mass attribute, a ReferenceFrame to the frame attribute, and mass
# and inertia. Then we form the body list.
ForceList = [(Dmc, - m * g * Y.z)]
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
BodyList = [BodyD]
# Finally we form the equations of motion, using the same steps we did
# before. Specify inertial frame, supply generalized speeds, supply
# kinematic differential equation dictionary, compute Fr from the force
# list and Fr* from the body list, compute the mass matrix and forcing
# terms, then solve for the u dots (time derivatives of the generalized
# speeds).
KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
KM.kanes_equations(ForceList, BodyList)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
kdd = KM.kindiffdict()
rhs = rhs.subs(kdd)
rhs.simplify()
assert rhs.expand() == Matrix([(6*u2*u3*r - u3**2*r*tan(q2) +
4*g*sin(q2))/(5*r), -2*u1*u3/3, u1*(-2*u2 + u3*tan(q2))]).expand()
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(6, 1)
# This code tests our output vs. benchmark values. When r=g=m=1, the
# critical speed (where all eigenvalues of the linearized equations are 0)
# is 1 / sqrt(3) for the upright case.
A = KM.linearize(A_and_B=True, new_method=True)[0]
A_upright = A.subs({r: 1, g: 1, m: 1}).subs({q1: 0, q2: 0, q3: 0, u1: 0, u3: 0})
import sympy
assert sympy.sympify(A_upright.subs({u2: 1 / sqrt(3)})).eigenvals() == {S(0): 6}
def test_replace_qdots_in_force():
# Test PR 16700 "Replaces qdots with us in force-list in kanes.py"
# The new functionality allows one to specify forces in qdots which will
# automatically be replaced with u:s which are defined by the kde supplied
# to KanesMethod. The test case is the double pendulum with interacting
# forces in the example of chapter 4.7 "CONTRIBUTING INTERACTION FORCES"
# in Ref. [1]. Reference list at end test function.
q1, q2 = dynamicsymbols("q1, q2")
qd1, qd2 = dynamicsymbols("q1, q2", level=1)
u1, u2 = dynamicsymbols("u1, u2")
l, m = symbols("l, m")
N = ReferenceFrame("N") # Inertial frame
A = N.orientnew("A", "Axis", (q1, N.z)) # Rod A frame
B = A.orientnew("B", "Axis", (q2, N.z)) # Rod B frame
O = Point("O") # Origo
O.set_vel(N, 0)
P = O.locatenew("P", (l * A.x)) # Point @ end of rod A
P.v2pt_theory(O, N, A)
Q = P.locatenew("Q", (l * B.x)) # Point @ end of rod B
Q.v2pt_theory(P, N, B)
Ap = Particle("Ap", P, m)
Bp = Particle("Bp", Q, m)
# The forces are specified below. sigma is the torsional spring stiffness
# and delta is the viscous damping coefficient acting between the two
# bodies. Here, we specify the viscous damper as function of qdots prior
# forming the kde. In more complex systems it not might be obvious which
# kde is most efficient, why it is convenient to specify viscous forces in
# qdots independently of the kde.
sig, delta = symbols("sigma, delta")
Ta = (sig * q2 + delta * qd2) * N.z
forces = [(A, Ta), (B, -Ta)]
# Try different kdes.
kde1 = [u1 - qd1, u2 - qd2]
kde2 = [u1 - qd1, u2 - (qd1 + qd2)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
# Check EOM for KM2:
# Mass and force matrix from p.6 in Ref. [2] with added forces from
# example of chapter 4.7 in [1] and without gravity.
forcing_matrix_expected = Matrix(
[
[m * l ** 2 * sin(q2) * u2 ** 2 + sig * q2 + delta * (u2 - u1)],
[m * l ** 2 * sin(q2) * -(u1 ** 2) - sig * q2 - delta * (u2 - u1)],
]
)
mass_matrix_expected = Matrix(
[[2 * m * l ** 2, m * l ** 2 * cos(q2)], [m * l ** 2 * cos(q2), m * l ** 2]]
)
assert KM2.mass_matrix.expand() == mass_matrix_expected.expand()
assert KM2.forcing.expand() == forcing_matrix_expected.expand()
# Check fr1 with reference fr_expected from [1] with u:s instead of qdots.
fr1_expected = Matrix([0, -(sig * q2 + delta * u2)])
assert fr1.expand() == fr1_expected.expand()
# Check fr2
fr2_expected = Matrix([sig * q2 + delta * (u2 - u1), -sig * q2 - delta * (u2 - u1)])
assert fr2.expand() == fr2_expected.expand()
# Specifying forces in u:s should stay the same:
Ta = (sig * q2 + delta * u2) * N.z
forces = [(A, Ta), (B, -Ta)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
assert fr1.expand() == fr1_expected.expand()
Ta = (sig * q2 + delta * (u2 - u1)) * N.z
forces = [(A, Ta), (B, -Ta)]
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
assert fr2.expand() == fr2_expected.expand()
# Test if we have a qubic qdot force:
Ta = (sig * q2 + delta * qd2 ** 3) * N.z
forces = [(A, Ta), (B, -Ta)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
fr1_cubic_expected = Matrix([0, -(sig * q2 + delta * u2 ** 3)])
assert fr1.expand() == fr1_cubic_expected.expand()
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
fr2_cubic_expected = Matrix(
[sig * q2 + delta * (u2 - u1) ** 3, -sig * q2 - delta * (u2 - u1) ** 3]
)
assert fr2.expand() == fr2_cubic_expected.expand()
def test_aux_dep():
# This test is about rolling disc dynamics, comparing the results found
# with KanesMethod to those found when deriving the equations "manually"
# with SymPy.
# The terms Fr, Fr*, and Fr*_steady are all compared between the two
# methods. Here, Fr*_steady refers to the generalized inertia forces for an
# equilibrium configuration.
# Note: comparing to the test of test_rolling_disc() in test_kane.py, this
# test also tests auxiliary speeds and configuration and motion constraints
#, seen in the generalized dependent coordinates q[3], and depend speeds
# u[3], u[4] and u[5].
# First, manual derivation of Fr, Fr_star, Fr_star_steady.
# Symbols for time and constant parameters.
# Symbols for contact forces: Fx, Fy, Fz.
t, r, m, g, I, J = symbols('t r m g I J')
Fx, Fy, Fz = symbols('Fx Fy Fz')
# Configuration variables and their time derivatives:
# q[0] -- yaw
# q[1] -- lean
# q[2] -- spin
# q[3] -- dot(-r*B.z, A.z) -- distance from ground plane to disc center in
# A.z direction
# Generalized speeds and their time derivatives:
# u[0] -- disc angular velocity component, disc fixed x direction
# u[1] -- disc angular velocity component, disc fixed y direction
# u[2] -- disc angular velocity component, disc fixed z direction
# u[3] -- disc velocity component, A.x direction
# u[4] -- disc velocity component, A.y direction
# u[5] -- disc velocity component, A.z direction
# Auxiliary generalized speeds:
# ua[0] -- contact point auxiliary generalized speed, A.x direction
# ua[1] -- contact point auxiliary generalized speed, A.y direction
# ua[2] -- contact point auxiliary generalized speed, A.z direction
q = dynamicsymbols('q:4')
qd = [qi.diff(t) for qi in q]
u = dynamicsymbols('u:6')
ud = [ui.diff(t) for ui in u]
ud_zero = dict(zip(ud, [0.] * len(ud)))
ua = dynamicsymbols('ua:3')
ua_zero = dict(zip(ua, [0.] * len(ua))) # noqa:F841
# Reference frames:
# Yaw intermediate frame: A.
# Lean intermediate frame: B.
# Disc fixed frame: C.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q[0], N.z])
B = A.orientnew('B', 'Axis', [q[1], A.x])
C = B.orientnew('C', 'Axis', [q[2], B.y])
# Angular velocity and angular acceleration of disc fixed frame
# u[0], u[1] and u[2] are generalized independent speeds.
C.set_ang_vel(N, u[0] * B.x + u[1] * B.y + u[2] * B.z)
C.set_ang_acc(
N,
C.ang_vel_in(N).diff(t, B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Velocity and acceleration of points:
# Disc-ground contact point: P.
# Center of disc: O, defined from point P with depend coordinate: q[3]
# u[3], u[4] and u[5] are generalized dependent speeds.
P = Point('P')
P.set_vel(N, ua[0] * A.x + ua[1] * A.y + ua[2] * A.z)
O = P.locatenew('O', q[3] * A.z + r * sin(q[1]) * A.y)
O.set_vel(N, u[3] * A.x + u[4] * A.y + u[5] * A.z)
O.set_acc(N, O.vel(N).diff(t, A) + cross(A.ang_vel_in(N), O.vel(N)))
# Kinematic differential equations:
# Two equalities: one is w_c_n_qd = C.ang_vel_in(N) in three coordinates
# directions of B, for qd0, qd1 and qd2.
# the other is v_o_n_qd = O.vel(N) in A.z direction for qd3.
# Then, solve for dq/dt's in terms of u's: qd_kd.
w_c_n_qd = qd[0] * A.z + qd[1] * B.x + qd[2] * B.y
v_o_n_qd = O.pos_from(P).diff(t, A) + cross(A.ang_vel_in(N), O.pos_from(P))
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv)
for uv in B] + [dot(v_o_n_qd - O.vel(N), A.z)])
qd_kd = solve(kindiffs, qd) # noqa:F841
# Values of generalized speeds during a steady turn for later substitution
# into the Fr_star_steady.
steady_conditions = solve(kindiffs.subs({qd[1]: 0, qd[3]: 0}), u)
steady_conditions.update({qd[1]: 0, qd[3]: 0})
# Partial angular velocities and velocities.
partial_w_C = [C.ang_vel_in(N).diff(ui, N) for ui in u + ua]
partial_v_O = [O.vel(N).diff(ui, N) for ui in u + ua]
partial_v_P = [P.vel(N).diff(ui, N) for ui in u + ua]
# Configuration constraint: f_c, the projection of radius r in A.z direction
# is q[3].
# Velocity constraints: f_v, for u3, u4 and u5.
# Acceleration constraints: f_a.
f_c = Matrix([dot(-r * B.z, A.z) - q[3]])
f_v = Matrix([
dot(O.vel(N) - (P.vel(N) + cross(C.ang_vel_in(N), O.pos_from(P))),
ai).expand() for ai in A
])
v_o_n = cross(C.ang_vel_in(N), O.pos_from(P))
a_o_n = v_o_n.diff(t, A) + cross(A.ang_vel_in(N), v_o_n)
f_a = Matrix([dot(O.acc(N) - a_o_n, ai) for ai in A]) # noqa:F841
# Solve for constraint equations in the form of
# u_dependent = A_rs * [u_i; u_aux].
# First, obtain constraint coefficient matrix: M_v * [u; ua] = 0;
# Second, taking u[0], u[1], u[2] as independent,
# taking u[3], u[4], u[5] as dependent,
# rearranging the matrix of M_v to be A_rs for u_dependent.
# Third, u_aux ==0 for u_dep, and resulting dictionary of u_dep_dict.
M_v = zeros(3, 9)
for i in range(3):
for j, ui in enumerate(u + ua):
M_v[i, j] = f_v[i].diff(ui)
M_v_i = M_v[:, :3]
M_v_d = M_v[:, 3:6]
M_v_aux = M_v[:, 6:]
M_v_i_aux = M_v_i.row_join(M_v_aux)
A_rs = -M_v_d.inv() * M_v_i_aux
u_dep = A_rs[:, :3] * Matrix(u[:3])
u_dep_dict = dict(zip(u[3:], u_dep))
# Active forces: F_O acting on point O; F_P acting on point P.
# Generalized active forces (unconstrained): Fr_u = F_point * pv_point.
F_O = m * g * A.z
F_P = Fx * A.x + Fy * A.y + Fz * A.z
Fr_u = Matrix([
dot(F_O, pv_o) + dot(F_P, pv_p)
for pv_o, pv_p in zip(partial_v_O, partial_v_P)
])
# Inertia force: R_star_O.
# Inertia of disc: I_C_O, where J is a inertia component about principal axis.
# Inertia torque: T_star_C.
# Generalized inertia forces (unconstrained): Fr_star_u.
R_star_O = -m * O.acc(N)
I_C_O = inertia(B, I, J, I)
T_star_C = -(dot(I_C_O, C.ang_acc_in(N)) \
+ cross(C.ang_vel_in(N), dot(I_C_O, C.ang_vel_in(N))))
Fr_star_u = Matrix([
dot(R_star_O, pv) + dot(T_star_C, pav)
for pv, pav in zip(partial_v_O, partial_w_C)
])
# Form nonholonomic Fr: Fr_c, and nonholonomic Fr_star: Fr_star_c.
# Also, nonholonomic Fr_star in steady turning condition: Fr_star_steady.
Fr_c = Fr_u[:3, :].col_join(Fr_u[6:, :]) + A_rs.T * Fr_u[3:6, :]
Fr_star_c = Fr_star_u[:3, :].col_join(Fr_star_u[6:, :])\
+ A_rs.T * Fr_star_u[3:6, :]
Fr_star_steady = Fr_star_c.subs(ud_zero).subs(u_dep_dict)\
.subs(steady_conditions).subs({q[3]: -r*cos(q[1])}).expand()
# Second, using KaneMethod in mechanics for fr, frstar and frstar_steady.
# Rigid Bodies: disc, with inertia I_C_O.
iner_tuple = (I_C_O, O)
disc = RigidBody('disc', O, C, m, iner_tuple)
bodyList = [disc]
# Generalized forces: Gravity: F_o; Auxiliary forces: F_p.
F_o = (O, F_O)
F_p = (P, F_P)
forceList = [F_o, F_p]
# KanesMethod.
kane = KanesMethod(N,
q_ind=q[:3],
u_ind=u[:3],
kd_eqs=kindiffs,
q_dependent=q[3:],
configuration_constraints=f_c,
u_dependent=u[3:],
velocity_constraints=f_v,
u_auxiliary=ua)
# fr, frstar, frstar_steady and kdd(kinematic differential equations).
with warns_deprecated_sympy():
(fr, frstar) = kane.kanes_equations(forceList, bodyList)
frstar_steady = frstar.subs(ud_zero).subs(u_dep_dict).subs(steady_conditions)\
.subs({q[3]: -r*cos(q[1])}).expand()
kdd = kane.kindiffdict()
assert Matrix(Fr_c).expand() == fr.expand()
assert Matrix(Fr_star_c.subs(kdd)).expand() == frstar.expand()
assert (simplify(Matrix(Fr_star_steady).expand()) == simplify(
frstar_steady.expand()))
syms_in_forcing = find_dynamicsymbols(kane.forcing)
for qdi in qd:
assert qdi not in syms_in_forcing
def test_sub_qdot2():
# This test solves exercises 8.3 from Kane 1985 and defines
# all velocities in terms of q, qdot. We check that the generalized active
# forces are correctly computed if u terms are only defined in the
# kinematic differential equations.
#
# This functionality was added in PR 8948. Without qdot/u substitution, the
# KanesMethod constructor will fail during the constraint initialization as
# the B matrix will be poorly formed and inversion of the dependent part
# will fail.
g, m, Px, Py, Pz, R, t = symbols('g m Px Py Pz R t')
q = dynamicsymbols('q:5')
qd = dynamicsymbols('q:5', level=1)
u = dynamicsymbols('u:5')
## Define inertial, intermediate, and rigid body reference frames
A = ReferenceFrame('A')
B_prime = A.orientnew('B_prime', 'Axis', [q[0], A.z])
B = B_prime.orientnew('B', 'Axis', [pi/2 - q[1], B_prime.x])
C = B.orientnew('C', 'Axis', [q[2], B.z])
## Define points of interest and their velocities
pO = Point('O')
pO.set_vel(A, 0)
# R is the point in plane H that comes into contact with disk C.
pR = pO.locatenew('R', q[3]*A.x + q[4]*A.y)
pR.set_vel(A, pR.pos_from(pO).diff(t, A))
pR.set_vel(B, 0)
# C^ is the point in disk C that comes into contact with plane H.
pC_hat = pR.locatenew('C^', 0)
pC_hat.set_vel(C, 0)
# C* is the point at the center of disk C.
pCs = pC_hat.locatenew('C*', R*B.y)
pCs.set_vel(C, 0)
pCs.set_vel(B, 0)
# calculate velocites of points C* and C^ in frame A
pCs.v2pt_theory(pR, A, B) # points C* and R are fixed in frame B
pC_hat.v2pt_theory(pCs, A, C) # points C* and C^ are fixed in frame C
## Define forces on each point of the system
R_C_hat = Px*A.x + Py*A.y + Pz*A.z
R_Cs = -m*g*A.z
forces = [(pC_hat, R_C_hat), (pCs, R_Cs)]
## Define kinematic differential equations
# let ui = omega_C_A & bi (i = 1, 2, 3)
# u4 = qd4, u5 = qd5
u_expr = [C.ang_vel_in(A) & uv for uv in B]
u_expr += qd[3:]
kde = [ui - e for ui, e in zip(u, u_expr)]
km1 = KanesMethod(A, q, u, kde)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
fr1, _ = km1.kanes_equations(forces, [])
## Calculate generalized active forces if we impose the condition that the
# disk C is rolling without slipping
u_indep = u[:3]
u_dep = list(set(u) - set(u_indep))
vc = [pC_hat.vel(A) & uv for uv in [A.x, A.y]]
km2 = KanesMethod(A, q, u_indep, kde,
u_dependent=u_dep, velocity_constraints=vc)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
fr2, _ = km2.kanes_equations(forces, [])
fr1_expected = Matrix([
-R*g*m*sin(q[1]),
-R*(Px*cos(q[0]) + Py*sin(q[0]))*tan(q[1]),
R*(Px*cos(q[0]) + Py*sin(q[0])),
Px,
Py])
fr2_expected = Matrix([
-R*g*m*sin(q[1]),
0,
0])
assert (trigsimp(fr1.expand()) == trigsimp(fr1_expected.expand()))
assert (trigsimp(fr2.expand()) == trigsimp(fr2_expected.expand()))
def test_rolling_disc():
# Rolling Disc Example
# Here the rolling disc is formed from the contact point up, removing the
# need to introduce generalized speeds. Only 3 configuration and three
# speed variables are need to describe this system, along with the disc's
# mass and radius, and the local gravity (note that mass will drop out).
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
r, m, g = symbols('r m g')
# The kinematics are formed by a series of simple rotations. Each simple
# rotation creates a new frame, and the next rotation is defined by the new
# frame's basis vectors. This example uses a 3-1-2 series of rotations, or
# Z, X, Y series of rotations. Angular velocity for this is defined using
# the second frame's basis (the lean frame).
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
w_R_N_qd = R.ang_vel_in(N)
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
# This is the translational kinematics. We create a point with no velocity
# in N; this is the contact point between the disc and ground. Next we form
# the position vector from the contact point to the disc's center of mass.
# Finally we form the velocity and acceleration of the disc.
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
# This is a simple way to form the inertia dyadic.
I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
# Kinematic differential equations; how the generalized coordinate time
# derivatives relate to generalized speeds.
kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L]
# Creation of the force list; it is the gravitational force at the mass
# center of the disc. Then we create the disc by assigning a Point to the
# center of mass attribute, a ReferenceFrame to the frame attribute, and mass
# and inertia. Then we form the body list.
ForceList = [(Dmc, -m * g * Y.z)]
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
BodyList = [BodyD]
# Finally we form the equations of motion, using the same steps we did
# before. Specify inertial frame, supply generalized speeds, supply
# kinematic differential equation dictionary, compute Fr from the force
# list and Fr* from the body list, compute the mass matrix and forcing
# terms, then solve for the u dots (time derivatives of the generalized
# speeds).
KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd)
with warns_deprecated_sympy():
KM.kanes_equations(ForceList, BodyList)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
kdd = KM.kindiffdict()
rhs = rhs.subs(kdd)
rhs.simplify()
assert rhs.expand() == Matrix([
(6 * u2 * u3 * r - u3**2 * r * tan(q2) + 4 * g * sin(q2)) / (5 * r),
-2 * u1 * u3 / 3, u1 * (-2 * u2 + u3 * tan(q2))
]).expand()
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(
6, 1)
# This code tests our output vs. benchmark values. When r=g=m=1, the
# critical speed (where all eigenvalues of the linearized equations are 0)
# is 1 / sqrt(3) for the upright case.
A = KM.linearize(A_and_B=True)[0]
A_upright = A.subs({
r: 1,
g: 1,
m: 1
}).subs({
q1: 0,
q2: 0,
q3: 0,
u1: 0,
u3: 0
})
import sympy
assert sympy.sympify(A_upright.subs({u2: 1 / sqrt(3)})).eigenvals() == {
S.Zero: 6
}
def _tan_repl_func(expr):
"""Replace tan with sin/cos."""
if isinstance(expr, tan):
return sin(*expr.args) / cos(*expr.args)
elif not expr.args or expr.is_Derivative:
return expr
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Defines the orientation of this frame relative to a parent frame.
Parameters
==========
parent : ReferenceFrame
The frame that this ReferenceFrame will have its orientation matrix
defined in relation to.
rot_type : str
The type of orientation matrix that is being created. Supported
types are 'Body', 'Space', 'Quaternion', 'Axis', and 'DCM'.
See examples for correct usage.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
In case of rot_type='DCM', value must be a
sympy.matrices.MatrixBase object (or subclasses of it).
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols, eye, ImmutableMatrix
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
Now we have a choice of how to implement the orientation. First is
Body. Body orientation takes this reference frame through three
successive simple rotations. Acceptable rotation orders are of length
3, expressed in XYZ or 123, and cannot have a rotation about about an
axis twice in a row.
>>> B.orient(N, 'Body', [q1, q2, q3], 123)
>>> B.orient(N, 'Body', [q1, q2, 0], 'ZXZ')
>>> B.orient(N, 'Body', [0, 0, 0], 'XYX')
Next is Space. Space is like Body, but the rotations are applied in the
opposite order.
>>> B.orient(N, 'Space', [q1, q2, q3], '312')
Next is Quaternion. This orients the new ReferenceFrame with
Quaternions, defined as a finite rotation about lambda, a unit vector,
by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.
>>> B.orient(N, 'Quaternion', [q0, q1, q2, q3])
Next is Axis. This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> B.orient(N, 'Axis', [q1, N.x + 2 * N.y])
Last is DCM (Direction Cosine Matrix). This is a rotation matrix
given manually.
>>> B.orient(N, 'DCM', eye(3))
>>> B.orient(N, 'DCM', ImmutableMatrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]]))
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
# Allow passing a rotation matrix manually.
if rot_type == 'DCM':
# When rot_type == 'DCM', then amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(amounts, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
else:
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0], [0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)], [0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0], [0, 0, 1]])
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123 and rot_type is in upper case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if not rot_order in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = (
(eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]], [axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) * sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[
q0**2 + q1**2 - q2**2 - q3**2, 2 * (q1 * q2 - q0 * q3),
2 * (q0 * q2 + q1 * q3)
],
[
2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)
],
[
2 * (q1 * q3 - q0 * q2),
2 * (q0 * q1 + q2 * q3),
q0**2 - q1**2 - q2**2 + q3**2
]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1]) *
_rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1]) *
_rot(a1, amounts[0]))
elif rot_type == 'DCM':
parent_orient = amounts
else:
raise NotImplementedError('That is not an implemented rotation')
#Reset the _dcm_cache of this frame, and remove it from the _dcm_caches
#of the frames it is linked to. Also remove it from the _dcm_dict of
#its parent
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
#Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
#Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
thetad = (amounts[0]).diff(dynamicsymbols._t)
wvec = thetad * amounts[1].express(parent).normalize()
elif rot_type == 'DCM':
wvec = self._w_diff_dcm(parent)
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def test_aux_dep():
# This test is about rolling disc dynamics, comparing the results found
# with KanesMethod to those found when deriving the equations "manually"
# with SymPy.
# The terms Fr, Fr*, and Fr*_steady are all compared between the two
# methods. Here, Fr*_steady refers to the generalized inertia forces for an
# equilibrium configuration.
# Note: comparing to the test of test_rolling_disc() in test_kane.py, this
# test also tests auxiliary speeds and configuration and motion constraints
#, seen in the generalized dependent coordinates q[3], and depend speeds
# u[3], u[4] and u[5].
# First, manual derivation of Fr, Fr_star, Fr_star_steady.
# Symbols for time and constant parameters.
# Symbols for contact forces: Fx, Fy, Fz.
t, r, m, g, I, J = symbols('t r m g I J')
Fx, Fy, Fz = symbols('Fx Fy Fz')
# Configuration variables and their time derivatives:
# q[0] -- yaw
# q[1] -- lean
# q[2] -- spin
# q[3] -- dot(-r*B.z, A.z) -- distance from ground plane to disc center in
# A.z direction
# Generalized speeds and their time derivatives:
# u[0] -- disc angular velocity component, disc fixed x direction
# u[1] -- disc angular velocity component, disc fixed y direction
# u[2] -- disc angular velocity component, disc fixed z direction
# u[3] -- disc velocity component, A.x direction
# u[4] -- disc velocity component, A.y direction
# u[5] -- disc velocity component, A.z direction
# Auxiliary generalized speeds:
# ua[0] -- contact point auxiliary generalized speed, A.x direction
# ua[1] -- contact point auxiliary generalized speed, A.y direction
# ua[2] -- contact point auxiliary generalized speed, A.z direction
q = dynamicsymbols('q:4')
qd = [qi.diff(t) for qi in q]
u = dynamicsymbols('u:6')
ud = [ui.diff(t) for ui in u]
ud_zero = dict(zip(ud, [0.]*len(ud)))
ua = dynamicsymbols('ua:3')
ua_zero = dict(zip(ua, [0.]*len(ua)))
# Reference frames:
# Yaw intermediate frame: A.
# Lean intermediate frame: B.
# Disc fixed frame: C.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q[0], N.z])
B = A.orientnew('B', 'Axis', [q[1], A.x])
C = B.orientnew('C', 'Axis', [q[2], B.y])
# Angular velocity and angular acceleration of disc fixed frame
# u[0], u[1] and u[2] are generalized independent speeds.
C.set_ang_vel(N, u[0]*B.x + u[1]*B.y + u[2]*B.z)
C.set_ang_acc(N, C.ang_vel_in(N).diff(t, B)
+ cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Velocity and acceleration of points:
# Disc-ground contact point: P.
# Center of disc: O, defined from point P with depend coordinate: q[3]
# u[3], u[4] and u[5] are generalized dependent speeds.
P = Point('P')
P.set_vel(N, ua[0]*A.x + ua[1]*A.y + ua[2]*A.z)
O = P.locatenew('O', q[3]*A.z + r*sin(q[1])*A.y)
O.set_vel(N, u[3]*A.x + u[4]*A.y + u[5]*A.z)
O.set_acc(N, O.vel(N).diff(t, A) + cross(A.ang_vel_in(N), O.vel(N)))
# Kinematic differential equations:
# Two equalities: one is w_c_n_qd = C.ang_vel_in(N) in three coordinates
# directions of B, for qd0, qd1 and qd2.
# the other is v_o_n_qd = O.vel(N) in A.z direction for qd3.
# Then, solve for dq/dt's in terms of u's: qd_kd.
w_c_n_qd = qd[0]*A.z + qd[1]*B.x + qd[2]*B.y
v_o_n_qd = O.pos_from(P).diff(t, A) + cross(A.ang_vel_in(N), O.pos_from(P))
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_o_n_qd - O.vel(N), A.z)])
qd_kd = solve(kindiffs, qd)
# Values of generalized speeds during a steady turn for later substitution
# into the Fr_star_steady.
steady_conditions = solve(kindiffs.subs({qd[1] : 0, qd[3] : 0}), u)
steady_conditions.update({qd[1] : 0, qd[3] : 0})
# Partial angular velocities and velocities.
partial_w_C = [C.ang_vel_in(N).diff(ui, N) for ui in u + ua]
partial_v_O = [O.vel(N).diff(ui, N) for ui in u + ua]
partial_v_P = [P.vel(N).diff(ui, N) for ui in u + ua]
# Configuration constraint: f_c, the projection of radius r in A.z direction
# is q[3].
# Velocity constraints: f_v, for u3, u4 and u5.
# Acceleration constraints: f_a.
f_c = Matrix([dot(-r*B.z, A.z) - q[3]])
f_v = Matrix([dot(O.vel(N) - (P.vel(N) + cross(C.ang_vel_in(N),
O.pos_from(P))), ai).expand() for ai in A])
v_o_n = cross(C.ang_vel_in(N), O.pos_from(P))
a_o_n = v_o_n.diff(t, A) + cross(A.ang_vel_in(N), v_o_n)
f_a = Matrix([dot(O.acc(N) - a_o_n, ai) for ai in A])
# Solve for constraint equations in the form of
# u_dependent = A_rs * [u_i; u_aux].
# First, obtain constraint coefficient matrix: M_v * [u; ua] = 0;
# Second, taking u[0], u[1], u[2] as independent,
# taking u[3], u[4], u[5] as dependent,
# rearranging the matrix of M_v to be A_rs for u_dependent.
# Third, u_aux ==0 for u_dep, and resulting dictionary of u_dep_dict.
M_v = zeros(3, 9)
for i in range(3):
for j, ui in enumerate(u + ua):
M_v[i, j] = f_v[i].diff(ui)
M_v_i = M_v[:, :3]
M_v_d = M_v[:, 3:6]
M_v_aux = M_v[:, 6:]
M_v_i_aux = M_v_i.row_join(M_v_aux)
A_rs = - M_v_d.inv() * M_v_i_aux
u_dep = A_rs[:, :3] * Matrix(u[:3])
u_dep_dict = dict(zip(u[3:], u_dep))
# Active forces: F_O acting on point O; F_P acting on point P.
# Generalized active forces (unconstrained): Fr_u = F_point * pv_point.
F_O = m*g*A.z
F_P = Fx * A.x + Fy * A.y + Fz * A.z
Fr_u = Matrix([dot(F_O, pv_o) + dot(F_P, pv_p) for pv_o, pv_p in
zip(partial_v_O, partial_v_P)])
# Inertia force: R_star_O.
# Inertia of disc: I_C_O, where J is a inertia component about principal axis.
# Inertia torque: T_star_C.
# Generalized inertia forces (unconstrained): Fr_star_u.
R_star_O = -m*O.acc(N)
I_C_O = inertia(B, I, J, I)
T_star_C = -(dot(I_C_O, C.ang_acc_in(N)) \
+ cross(C.ang_vel_in(N), dot(I_C_O, C.ang_vel_in(N))))
Fr_star_u = Matrix([dot(R_star_O, pv) + dot(T_star_C, pav) for pv, pav in
zip(partial_v_O, partial_w_C)])
# Form nonholonomic Fr: Fr_c, and nonholonomic Fr_star: Fr_star_c.
# Also, nonholonomic Fr_star in steady turning condition: Fr_star_steady.
Fr_c = Fr_u[:3, :].col_join(Fr_u[6:, :]) + A_rs.T * Fr_u[3:6, :]
Fr_star_c = Fr_star_u[:3, :].col_join(Fr_star_u[6:, :])\
+ A_rs.T * Fr_star_u[3:6, :]
Fr_star_steady = Fr_star_c.subs(ud_zero).subs(u_dep_dict)\
.subs(steady_conditions).subs({q[3]: -r*cos(q[1])}).expand()
# Second, using KaneMethod in mechanics for fr, frstar and frstar_steady.
# Rigid Bodies: disc, with inertia I_C_O.
iner_tuple = (I_C_O, O)
disc = RigidBody('disc', O, C, m, iner_tuple)
bodyList = [disc]
# Generalized forces: Gravity: F_o; Auxiliary forces: F_p.
F_o = (O, F_O)
F_p = (P, F_P)
forceList = [F_o, F_p]
# KanesMethod.
kane = KanesMethod(
N, q_ind= q[:3], u_ind= u[:3], kd_eqs=kindiffs,
q_dependent=q[3:], configuration_constraints = f_c,
u_dependent=u[3:], velocity_constraints= f_v,
u_auxiliary=ua
)
# fr, frstar, frstar_steady and kdd(kinematic differential equations).
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
(fr, frstar)= kane.kanes_equations(forceList, bodyList)
frstar_steady = frstar.subs(ud_zero).subs(u_dep_dict).subs(steady_conditions)\
.subs({q[3]: -r*cos(q[1])}).expand()
kdd = kane.kindiffdict()
assert Matrix(Fr_c).expand() == fr.expand()
assert Matrix(Fr_star_c.subs(kdd)).expand() == frstar.expand()
assert (simplify(Matrix(Fr_star_steady).expand()) ==
simplify(frstar_steady.expand()))
def test_sub_qdot():
# This test solves exercises 8.12, 8.17 from Kane 1985 and defines
# some velocities in terms of q, qdot.
## --- Declare symbols ---
q1, q2, q3 = dynamicsymbols('q1:4')
q1d, q2d, q3d = dynamicsymbols('q1:4', level=1)
u1, u2, u3 = dynamicsymbols('u1:4')
u_prime, R, M, g, e, f, theta = symbols('u\' R, M, g, e, f, theta')
a, b, mA, mB, IA, J, K, t = symbols('a b mA mB IA J K t')
IA22, IA23, IA33 = symbols('IA22 IA23 IA33')
Q1, Q2, Q3 = symbols('Q1 Q2 Q3')
# --- Reference Frames ---
F = ReferenceFrame('F')
P = F.orientnew('P', 'axis', [-theta, F.y])
A = P.orientnew('A', 'axis', [q1, P.x])
A.set_ang_vel(F, u1 * A.x + u3 * A.z)
# define frames for wheels
B = A.orientnew('B', 'axis', [q2, A.z])
C = A.orientnew('C', 'axis', [q3, A.z])
## --- define points D, S*, Q on frame A and their velocities ---
pD = Point('D')
pD.set_vel(A, 0)
# u3 will not change v_D_F since wheels are still assumed to roll w/o slip
pD.set_vel(F, u2 * A.y)
pS_star = pD.locatenew('S*', e * A.y)
pQ = pD.locatenew('Q', f * A.y - R * A.x)
# masscenters of bodies A, B, C
pA_star = pD.locatenew('A*', a * A.y)
pB_star = pD.locatenew('B*', b * A.z)
pC_star = pD.locatenew('C*', -b * A.z)
for p in [pS_star, pQ, pA_star, pB_star, pC_star]:
p.v2pt_theory(pD, F, A)
# points of B, C touching the plane P
pB_hat = pB_star.locatenew('B^', -R * A.x)
pC_hat = pC_star.locatenew('C^', -R * A.x)
pB_hat.v2pt_theory(pB_star, F, B)
pC_hat.v2pt_theory(pC_star, F, C)
# --- relate qdot, u ---
# the velocities of B^, C^ are zero since B, C are assumed to roll w/o slip
kde = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]]
kde += [u1 - q1d]
kde_map = solve(kde, [q1d, q2d, q3d])
for k, v in list(kde_map.items()):
kde_map[k.diff(t)] = v.diff(t)
# inertias of bodies A, B, C
# IA22, IA23, IA33 are not specified in the problem statement, but are
# necessary to define an inertia object. Although the values of
# IA22, IA23, IA33 are not known in terms of the variables given in the
# problem statement, they do not appear in the general inertia terms.
inertia_A = inertia(A, IA, IA22, IA33, 0, IA23, 0)
inertia_B = inertia(B, K, K, J)
inertia_C = inertia(C, K, K, J)
# define the rigid bodies A, B, C
rbA = RigidBody('rbA', pA_star, A, mA, (inertia_A, pA_star))
rbB = RigidBody('rbB', pB_star, B, mB, (inertia_B, pB_star))
rbC = RigidBody('rbC', pC_star, C, mB, (inertia_C, pC_star))
## --- use kanes method ---
km = KanesMethod(F, [q1, q2, q3], [u1, u2], kd_eqs=kde, u_auxiliary=[u3])
forces = [(pS_star, -M * g * F.x), (pQ, Q1 * A.x + Q2 * A.y + Q3 * A.z)]
bodies = [rbA, rbB, rbC]
# Q2 = -u_prime * u2 * Q1 / sqrt(u2**2 + f**2 * u1**2)
# -u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2) = R / Q1 * Q2
fr_expected = Matrix([
f * Q3 + M * g * e * sin(theta) * cos(q1),
Q2 + M * g * sin(theta) * sin(q1),
e * M * g * cos(theta) - Q1 * f - Q2 * R
])
#Q1 * (f - u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2)))])
fr_star_expected = Matrix([
-(IA + 2 * J * b**2 / R**2 + 2 * K + mA * a**2 + 2 * mB * b**2) *
u1.diff(t) - mA * a * u1 * u2,
-(mA + 2 * mB + 2 * J / R**2) * u2.diff(t) + mA * a * u1**2, 0
])
with warns_deprecated_sympy():
fr, fr_star = km.kanes_equations(forces, bodies)
assert (fr.expand() == fr_expected.expand())
assert ((fr_star_expected - trigsimp(fr_star)).expand() == zeros(3, 1))
def _tan_repl_func(expr):
"""Replace tan with sin/cos."""
if isinstance(expr, tan):
return sin(*expr.args) / cos(*expr.args)
elif not expr.args or expr.is_Derivative:
return expr
def test_n_link_pendulum_on_cart_inputs():
l0, m0 = symbols("l0 m0")
m1 = symbols("m1")
g = symbols("g")
q0, q1, F, T1 = dynamicsymbols("q0 q1 F T1")
u0, u1 = dynamicsymbols("u0 u1")
kane1 = models.n_link_pendulum_on_cart(1)
massmatrix1 = Matrix([[m0 + m1, -l0*m1*cos(q1)],
[-l0*m1*cos(q1), l0**2*m1]])
forcing1 = Matrix([[-l0*m1*u1**2*sin(q1) + F], [g*l0*m1*sin(q1)]])
assert simplify(massmatrix1 - kane1.mass_matrix) == zeros(2)
assert simplify(forcing1 - kane1.forcing) == Matrix([0, 0])
kane2 = models.n_link_pendulum_on_cart(1, False)
massmatrix2 = Matrix([[m0 + m1, -l0*m1*cos(q1)],
[-l0*m1*cos(q1), l0**2*m1]])
forcing2 = Matrix([[-l0*m1*u1**2*sin(q1)], [g*l0*m1*sin(q1)]])
assert simplify(massmatrix2 - kane2.mass_matrix) == zeros(2)
assert simplify(forcing2 - kane2.forcing) == Matrix([0, 0])
kane3 = models.n_link_pendulum_on_cart(1, False, True)
massmatrix3 = Matrix([[m0 + m1, -l0*m1*cos(q1)],
[-l0*m1*cos(q1), l0**2*m1]])
forcing3 = Matrix([[-l0*m1*u1**2*sin(q1)], [g*l0*m1*sin(q1) + T1]])
assert simplify(massmatrix3 - kane3.mass_matrix) == zeros(2)
assert simplify(forcing3 - kane3.forcing) == Matrix([0, 0])
kane4 = models.n_link_pendulum_on_cart(1, True, False)
massmatrix4 = Matrix([[m0 + m1, -l0*m1*cos(q1)],
[-l0*m1*cos(q1), l0**2*m1]])
forcing4 = Matrix([[-l0*m1*u1**2*sin(q1) + F], [g*l0*m1*sin(q1)]])
assert simplify(massmatrix4 - kane4.mass_matrix) == zeros(2)
assert simplify(forcing4 - kane4.forcing) == Matrix([0, 0])
