Spaces:
Running
Running
File size: 10,226 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 |
from sympy.core.function import expand
from sympy.core.symbol import symbols
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.matrices.dense import Matrix
from sympy.simplify.trigsimp import trigsimp
from sympy.physics.mechanics import (
PinJoint, JointsMethod, RigidBody, Particle, Body, KanesMethod,
PrismaticJoint, LagrangesMethod, inertia)
from sympy.physics.vector import dynamicsymbols, ReferenceFrame
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy import zeros
from sympy.utilities.lambdify import lambdify
from sympy.solvers.solvers import solve
t = dynamicsymbols._t # type: ignore
def test_jointsmethod():
with warns_deprecated_sympy():
P = Body('P')
C = Body('C')
Pin = PinJoint('P1', P, C)
C_ixx, g = symbols('C_ixx g')
q, u = dynamicsymbols('q_P1, u_P1')
P.apply_force(g*P.y)
with warns_deprecated_sympy():
method = JointsMethod(P, Pin)
assert method.frame == P.frame
assert method.bodies == [C, P]
assert method.loads == [(P.masscenter, g*P.frame.y)]
assert method.q == Matrix([q])
assert method.u == Matrix([u])
assert method.kdes == Matrix([u - q.diff()])
soln = method.form_eoms()
assert soln == Matrix([[-C_ixx*u.diff()]])
assert method.forcing_full == Matrix([[u], [0]])
assert method.mass_matrix_full == Matrix([[1, 0], [0, C_ixx]])
assert isinstance(method.method, KanesMethod)
def test_rigid_body_particle_compatibility():
l, m, g = symbols('l m g')
C = RigidBody('C')
b = Particle('b', mass=m)
b_frame = ReferenceFrame('b_frame')
q, u = dynamicsymbols('q u')
P = PinJoint('P', C, b, coordinates=q, speeds=u, child_interframe=b_frame,
child_point=-l * b_frame.x, joint_axis=C.z)
with warns_deprecated_sympy():
method = JointsMethod(C, P)
method.loads.append((b.masscenter, m * g * C.x))
method.form_eoms()
rhs = method.rhs()
assert rhs[1] == -g*sin(q)/l
def test_jointmethod_duplicate_coordinates_speeds():
with warns_deprecated_sympy():
P = Body('P')
C = Body('C')
T = Body('T')
q, u = dynamicsymbols('q u')
P1 = PinJoint('P1', P, C, q)
P2 = PrismaticJoint('P2', C, T, q)
with warns_deprecated_sympy():
raises(ValueError, lambda: JointsMethod(P, P1, P2))
P1 = PinJoint('P1', P, C, speeds=u)
P2 = PrismaticJoint('P2', C, T, speeds=u)
with warns_deprecated_sympy():
raises(ValueError, lambda: JointsMethod(P, P1, P2))
P1 = PinJoint('P1', P, C, q, u)
P2 = PrismaticJoint('P2', C, T, q, u)
with warns_deprecated_sympy():
raises(ValueError, lambda: JointsMethod(P, P1, P2))
def test_complete_simple_double_pendulum():
q1, q2 = dynamicsymbols('q1 q2')
u1, u2 = dynamicsymbols('u1 u2')
m, l, g = symbols('m l g')
with warns_deprecated_sympy():
C = Body('C') # ceiling
PartP = Body('P', mass=m)
PartR = Body('R', mass=m)
J1 = PinJoint('J1', C, PartP, speeds=u1, coordinates=q1,
child_point=-l*PartP.x, joint_axis=C.z)
J2 = PinJoint('J2', PartP, PartR, speeds=u2, coordinates=q2,
child_point=-l*PartR.x, joint_axis=PartP.z)
PartP.apply_force(m*g*C.x)
PartR.apply_force(m*g*C.x)
with warns_deprecated_sympy():
method = JointsMethod(C, J1, J2)
method.form_eoms()
assert expand(method.mass_matrix_full) == Matrix([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 2*l**2*m*cos(q2) + 3*l**2*m, l**2*m*cos(q2) + l**2*m],
[0, 0, l**2*m*cos(q2) + l**2*m, l**2*m]])
assert trigsimp(method.forcing_full) == trigsimp(Matrix([[u1], [u2], [-g*l*m*(sin(q1 + q2) + sin(q1)) -
g*l*m*sin(q1) + l**2*m*(2*u1 + u2)*u2*sin(q2)],
[-g*l*m*sin(q1 + q2) - l**2*m*u1**2*sin(q2)]]))
def test_two_dof_joints():
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
with warns_deprecated_sympy():
W = Body('W')
B1 = Body('B1', mass=m)
B2 = Body('B2', mass=m)
J1 = PrismaticJoint('J1', W, B1, coordinates=q1, speeds=u1)
J2 = PrismaticJoint('J2', B1, B2, coordinates=q2, speeds=u2)
W.apply_force(k1*q1*W.x, reaction_body=B1)
W.apply_force(c1*u1*W.x, reaction_body=B1)
B1.apply_force(k2*q2*W.x, reaction_body=B2)
B1.apply_force(c2*u2*W.x, reaction_body=B2)
with warns_deprecated_sympy():
method = JointsMethod(W, J1, J2)
method.form_eoms()
MM = method.mass_matrix
forcing = method.forcing
rhs = MM.LUsolve(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)
def test_simple_pedulum():
l, m, g = symbols('l m g')
with warns_deprecated_sympy():
C = Body('C')
b = Body('b', mass=m)
q = dynamicsymbols('q')
P = PinJoint('P', C, b, speeds=q.diff(t), coordinates=q,
child_point=-l * b.x, joint_axis=C.z)
b.potential_energy = - m * g * l * cos(q)
with warns_deprecated_sympy():
method = JointsMethod(C, P)
method.form_eoms(LagrangesMethod)
rhs = method.rhs()
assert rhs[1] == -g*sin(q)/l
def test_chaos_pendulum():
#https://www.pydy.org/examples/chaos_pendulum.html
mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g = symbols('mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g')
theta, phi, omega, alpha = dynamicsymbols('theta phi omega alpha')
A = ReferenceFrame('A')
B = ReferenceFrame('B')
with warns_deprecated_sympy():
rod = Body('rod', mass=mA, frame=A,
central_inertia=inertia(A, IAxx, IAxx, 0))
plate = Body('plate', mass=mB, frame=B,
central_inertia=inertia(B, IBxx, IByy, IBzz))
C = Body('C')
J1 = PinJoint('J1', C, rod, coordinates=theta, speeds=omega,
child_point=-lA * rod.z, joint_axis=C.y)
J2 = PinJoint('J2', rod, plate, coordinates=phi, speeds=alpha,
parent_point=(lB - lA) * rod.z, joint_axis=rod.z)
rod.apply_force(mA*g*C.z)
plate.apply_force(mB*g*C.z)
with warns_deprecated_sympy():
method = JointsMethod(C, J1, J2)
method.form_eoms()
MM = method.mass_matrix
forcing = method.forcing
rhs = MM.LUsolve(forcing)
xd = (-2 * IBxx * alpha * omega * sin(phi) * cos(phi) + 2 * IByy * alpha * omega * sin(phi) *
cos(phi) - g * lA * mA * sin(theta) - g * lB * mB * sin(theta)) / (IAxx + IBxx *
sin(phi)**2 + IByy * cos(phi)**2 + lA**2 * mA + lB**2 * mB)
assert (rhs[0] - xd).simplify() == 0
xd = (IBxx - IByy) * omega**2 * sin(phi) * cos(phi) / IBzz
assert (rhs[1] - xd).simplify() == 0
def test_four_bar_linkage_with_manual_constraints():
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1:4, u1:4')
l1, l2, l3, l4, rho = symbols('l1:5, rho')
N = ReferenceFrame('N')
inertias = [inertia(N, 0, 0, rho * l ** 3 / 12) for l in (l1, l2, l3, l4)]
with warns_deprecated_sympy():
link1 = Body('Link1', frame=N, mass=rho * l1,
central_inertia=inertias[0])
link2 = Body('Link2', mass=rho * l2, central_inertia=inertias[1])
link3 = Body('Link3', mass=rho * l3, central_inertia=inertias[2])
link4 = Body('Link4', mass=rho * l4, central_inertia=inertias[3])
joint1 = PinJoint(
'J1', link1, link2, coordinates=q1, speeds=u1, joint_axis=link1.z,
parent_point=l1 / 2 * link1.x, child_point=-l2 / 2 * link2.x)
joint2 = PinJoint(
'J2', link2, link3, coordinates=q2, speeds=u2, joint_axis=link2.z,
parent_point=l2 / 2 * link2.x, child_point=-l3 / 2 * link3.x)
joint3 = PinJoint(
'J3', link3, link4, coordinates=q3, speeds=u3, joint_axis=link3.z,
parent_point=l3 / 2 * link3.x, child_point=-l4 / 2 * link4.x)
loop = link4.masscenter.pos_from(link1.masscenter) \
+ l1 / 2 * link1.x + l4 / 2 * link4.x
fh = Matrix([loop.dot(link1.x), loop.dot(link1.y)])
with warns_deprecated_sympy():
method = JointsMethod(link1, joint1, joint2, joint3)
t = dynamicsymbols._t
qdots = solve(method.kdes, [q1.diff(t), q2.diff(t), q3.diff(t)])
fhd = fh.diff(t).subs(qdots)
kane = KanesMethod(method.frame, q_ind=[q1], u_ind=[u1],
q_dependent=[q2, q3], u_dependent=[u2, u3],
kd_eqs=method.kdes, configuration_constraints=fh,
velocity_constraints=fhd, forcelist=method.loads,
bodies=method.bodies)
fr, frs = kane.kanes_equations()
assert fr == zeros(1)
# Numerically check the mass- and forcing-matrix
p = Matrix([l1, l2, l3, l4, rho])
q = Matrix([q1, q2, q3])
u = Matrix([u1, u2, u3])
eval_m = lambdify((q, p), kane.mass_matrix)
eval_f = lambdify((q, u, p), kane.forcing)
eval_fhd = lambdify((q, u, p), fhd)
p_vals = [0.13, 0.24, 0.21, 0.34, 997]
q_vals = [2.1, 0.6655470375077588, 2.527408138024188] # Satisfies fh
u_vals = [0.2, -0.17963733938852067, 0.1309060540601612] # Satisfies fhd
mass_check = Matrix([[3.452709815256506e+01, 7.003948798374735e+00,
-4.939690970641498e+00],
[-2.203792703880936e-14, 2.071702479957077e-01,
2.842917573033711e-01],
[-1.300000000000123e-01, -8.836934896046506e-03,
1.864891330060847e-01]])
forcing_check = Matrix([[-0.031211821321648],
[-0.00066022608181],
[0.001813559741243]])
eps = 1e-10
assert all(abs(x) < eps for x in eval_fhd(q_vals, u_vals, p_vals))
assert all(abs(x) < eps for x in
(Matrix(eval_m(q_vals, p_vals)) - mass_check))
assert all(abs(x) < eps for x in
(Matrix(eval_f(q_vals, u_vals, p_vals)) - forcing_check))
|