def test_particle():
m, m2, v1, v2, v3, r, g, h = symbols('m m2 v1 v2 v3 r g h')
P = Point('P')
P2 = Point('P2')
p = Particle('pa', P, m)
assert p.mass == m
assert p.point == P
# Test the mass setter
p.mass = m2
assert p.mass == m2
# Test the point setter
p.point = P2
assert p.point == P2
# Test the linear momentum function
N = ReferenceFrame('N')
O = Point('O')
P2.set_pos(O, r * N.y)
P2.set_vel(N, v1 * N.x)
assert p.linear_momentum(N) == m2 * v1 * N.x
assert p.angular_momentum(O, N) == -m2 * r *v1 * N.z
P2.set_vel(N, v2 * N.y)
assert p.linear_momentum(N) == m2 * v2 * N.y
assert p.angular_momentum(O, N) == 0
P2.set_vel(N, v3 * N.z)
assert p.linear_momentum(N) == m2 * v3 * N.z
assert p.angular_momentum(O, N) == m2 * r * v3 * N.x
P2.set_vel(N, v1 * N.x + v2 * N.y + v3 * N.z)
assert p.linear_momentum(N) == m2 * (v1 * N.x + v2 * N.y + v3 * N.z)
assert p.angular_momentum(O, N) == m2 * r * (v3 * N.x - v1 * N.z)
p.set_potential_energy(m * g * h)
assert p.potential_energy == m * g * h
# TODO make the result not be system-dependent
assert p.kinetic_energy(
N) in [m2*(v1**2 + v2**2 + v3**2)/2,
m2 * v1**2 / 2 + m2 * v2**2 / 2 + m2 * v3**2 / 2]
def test_dub_pen():
# The system considered is the double pendulum. Like in the
# test of the simple pendulum above, we begin by creating the generalized
# coordinates and the simple generalized speeds and accelerations which
# will be used later. Following this we create frames and points necessary
# for the kinematics. The procedure isn't explicitly explained as this is
# similar to the simple pendulum. Also this is documented on the pydy.org
# website.
q1, q2 = dynamicsymbols('q1 q2')
q1d, q2d = dynamicsymbols('q1 q2', 1)
q1dd, q2dd = dynamicsymbols('q1 q2', 2)
u1, u2 = dynamicsymbols('u1 u2')
u1d, u2d = dynamicsymbols('u1 u2', 1)
l, m, g = symbols('l m g')
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = N.orientnew('B', 'Axis', [q2, N.z])
A.set_ang_vel(N, q1d * A.z)
B.set_ang_vel(N, q2d * A.z)
O = Point('O')
P = O.locatenew('P', l * A.x)
R = P.locatenew('R', l * B.x)
O.set_vel(N, 0)
P.v2pt_theory(O, N, A)
R.v2pt_theory(P, N, B)
ParP = Particle('ParP', P, m)
ParR = Particle('ParR', R, m)
ParP.set_potential_energy(-m * g * l * cos(q1))
ParR.set_potential_energy(-m * g * l * cos(q1) - m * g * l * cos(q2))
L = Lagrangian(N, ParP, ParR)
lm = LagrangesMethod(L, [q1, q2])
lm.form_lagranges_equations()
assert expand(l * m *
(2 * g * sin(q1) + l * sin(q1) * sin(q2) * q2dd +
l * sin(q1) * cos(q2) * q2d**2 - l * sin(q2) * cos(q1) *
q2d**2 + l * cos(q1) * cos(q2) * q2dd + 2 * l * q1dd) -
(simplify(lm.eom[0]))) == 0
assert expand((l * m *
(g * sin(q2) + l * sin(q1) * sin(q2) * q1dd - l * sin(q1) *
cos(q2) * q1d**2 + l * sin(q2) * cos(q1) * q1d**2 +
l * cos(q1) * cos(q2) * q1dd + l * q2dd)) -
(simplify(lm.eom[1]))) == 0
def test_dub_pen():
# The system considered is the double pendulum. Like in the
# test of the simple pendulum above, we begin by creating the generalized
# coordinates and the simple generalized speeds and accelerations which
# will be used later. Following this we create frames and points necessary
# for the kinematics. The procedure isn't explicitly explained as this is
# similar to the simple pendulum. Also this is documented on the pydy.org
# website.
q1, q2 = dynamicsymbols('q1 q2')
q1d, q2d = dynamicsymbols('q1 q2', 1)
q1dd, q2dd = dynamicsymbols('q1 q2', 2)
u1, u2 = dynamicsymbols('u1 u2')
u1d, u2d = dynamicsymbols('u1 u2', 1)
l, m, g = symbols('l m g')
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = N.orientnew('B', 'Axis', [q2, N.z])
A.set_ang_vel(N, q1d * A.z)
B.set_ang_vel(N, q2d * A.z)
O = Point('O')
P = O.locatenew('P', l * A.x)
R = P.locatenew('R', l * B.x)
O.set_vel(N, 0)
P.v2pt_theory(O, N, A)
R.v2pt_theory(P, N, B)
ParP = Particle('ParP', P, m)
ParR = Particle('ParR', R, m)
ParP.set_potential_energy(- m * g * l * cos(q1))
ParR.set_potential_energy(- m * g * l * cos(q1) - m * g * l * cos(q2))
L = Lagrangian(N, ParP, ParR)
lm = LagrangesMethod(L, [q1, q2])
lm.form_lagranges_equations()
assert expand(l*m*(2*g*sin(q1) + l*sin(q1)*sin(q2)*q2dd
+ l*sin(q1)*cos(q2)*q2d**2 - l*sin(q2)*cos(q1)*q2d**2
+ l*cos(q1)*cos(q2)*q2dd + 2*l*q1dd) - (simplify(lm.eom[0]))) == 0
assert expand((l*m*(g*sin(q2) + l*sin(q1)*sin(q2)*q1dd
- l*sin(q1)*cos(q2)*q1d**2 + l*sin(q2)*cos(q1)*q1d**2
+ l*cos(q1)*cos(q2)*q1dd + l*q2dd)) - (simplify(lm.eom[1]))) == 0
def test_potential_energy():
m, M, l1, g, h, H = symbols('m M l1 g h H')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
Pa.set_potential_energy(m * g * h)
A.set_potential_energy(M * g * H)
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_potential_energy():
m, M, l1, g, h, H = symbols('m M l1 g h H')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
Pa.set_potential_energy(m * g * h)
A.set_potential_energy(M * g * H)
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_potential_energy():
m, M, l1, g, h, H = symbols("m M l1 g h H")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
Pa.set_potential_energy(m * g * h)
A.set_potential_energy(M * g * H)
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_simp_pen():
# This tests that the equations generated by LagrangesMethod are identical
# to those obtained by hand calculations. The system under consideration is
# the simple pendulum.
# We begin by creating the generalized coordinates as per the requirements
# of LagrangesMethod. Also we created the associate symbols
# that characterize the system: 'm' is the mass of the bob, l is the length
# of the massless rigid rod connecting the bob to a point O fixed in the
# inertial frame.
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u ', 1)
l, m, g = symbols('l m g')
# We then create the inertial frame and a frame attached to the massless
# string following which we define the inertial angular velocity of the
# string.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q, N.z])
A.set_ang_vel(N, qd * N.z)
# Next, we create the point O and fix it in the inertial frame. We then
# locate the point P to which the bob is attached. Its corresponding
# velocity is then determined by the 'two point formula'.
O = Point('O')
O.set_vel(N, 0)
P = O.locatenew('P', l * A.x)
P.v2pt_theory(O, N, A)
# The 'Particle' which represents the bob is then created and its
# Lagrangian generated.
Pa = Particle('Pa', P, m)
Pa.set_potential_energy(- m * g * l * cos(q))
L = Lagrangian(N, Pa)
# The 'LagrangesMethod' class is invoked to obtain equations of motion.
lm = LagrangesMethod(L, [q])
lm.form_lagranges_equations()
RHS = lm.rhs()
assert RHS[1] == -g*sin(q)/l
def test_simp_pen():
# This tests that the equations generated by LagrangesMethod are identical
# to those obtained by hand calculations. The system under consideration is
# the simple pendulum.
# We begin by creating the generalized coordinates as per the requirements
# of LagrangesMethod. Also we created the associate symbols
# that characterize the system: 'm' is the mass of the bob, l is the length
# of the massless rigid rod connecting the bob to a point O fixed in the
# inertial frame.
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u ', 1)
l, m, g = symbols('l m g')
# We then create the inertial frame and a frame attached to the massless
# string following which we define the inertial angular velocity of the
# string.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q, N.z])
A.set_ang_vel(N, qd * N.z)
# Next, we create the point O and fix it in the inertial frame. We then
# locate the point P to which the bob is attached. Its corresponding
# velocity is then determined by the 'two point formula'.
O = Point('O')
O.set_vel(N, 0)
P = O.locatenew('P', l * A.x)
P.v2pt_theory(O, N, A)
# The 'Particle' which represents the bob is then created and its
# Lagrangian generated.
Pa = Particle('Pa', P, m)
Pa.set_potential_energy(- m * g * l * cos(q))
L = Lagrangian(N, Pa)
# The 'LagrangesMethod' class is invoked to obtain equations of motion.
lm = LagrangesMethod(L, [q])
lm.form_lagranges_equations()
RHS = lm.rhs()
assert RHS[1] == -g*sin(q)/l
def test_lagrange_2forces():
### Equations for two damped springs in serie with two forces
### generalized coordinates
qs = q1, q2 = dynamicsymbols('q1, q2')
### generalized speeds
qds = q1d, q2d = dynamicsymbols('q1, q2', 1)
### Mass, spring strength, friction coefficient
m, k, nu = symbols('m, k, nu')
N = ReferenceFrame('N')
O = Point('O')
### Two points
P1 = O.locatenew('P1', q1 * N.x)
P1.set_vel(N, q1d * N.x)
P2 = O.locatenew('P1', q2 * N.x)
P2.set_vel(N, q2d * N.x)
pP1 = Particle('pP1', P1, m)
pP1.set_potential_energy(k * q1**2 / 2)
pP2 = Particle('pP2', P2, m)
pP2.set_potential_energy(k * (q1 - q2)**2 / 2)
#### Friction forces
forcelist = [(P1, - nu * q1d * N.x),
(P2, - nu * q2d * N.x)]
lag = Lagrangian(N, pP1, pP2)
l_method = LagrangesMethod(lag, (q1, q2), forcelist=forcelist, frame=N)
l_method.form_lagranges_equations()
eq1 = l_method.eom[0]
assert eq1.diff(q1d) == nu
eq2 = l_method.eom[1]
assert eq2.diff(q2d) == nu
g]
#Specified contains the matrix for the input torques
specified = [ankle_torque, waist_torque]
#Specifies numerical constants for inertial/mass properties
#Robot Params
#numerical_constants = array([1.035, # leg_length[m]
# 36.754, # leg_mass[kg]
# 0.85, # body_length[m]
# 91.61, # body_mass[kg]
# 9.81] # acceleration due to gravity [m/s^2]
# )
numerical_constants = array([0.75,
7.0,
0.5,
8.0,
9.81])
#Set input torques to 0
numerical_specified = zeros(2)
parameter_dict = dict(zip(constants, numerical_constants))
ke_energy = simplify(kinetic_energy(inertial_frame, waistP, bodyP).subs(parameter_dict))
waistP.set_potential_energy(leg_mass*g*leg_length*cos(theta1))
bodyP.set_potential_energy(body_mass*g*(leg_length*cos(theta1)+body_length*cos(theta1+theta2)))
pe_energy = simplify(potential_energy(waistP, bodyP).subs(parameter_dict))
numerical_constants = array([1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
9.81])
#Set input torques to 0
numerical_specified = zeros(3)
parameter_dict = dict(zip(constants, numerical_constants))
ke_energy = simplify(kinetic_energy(inertial_frame, bP, cP, dP))
bP.set_potential_energy(a_mass*g*a_length*cos(theta_a))
cP.set_potential_energy(b_mass*g*(a_length*cos(theta_a)+b_length*cos(theta_a+theta_b)))
dP.set_potential_energy(c_mass*g*(a_length*cos(theta_a)+b_length*cos(theta_a+theta_b)+c_length*cos(theta_a+theta_b+theta_c)))
pe_energy = simplify(potential_energy(bP, cP, dP).subs(parameter_dict))
forcing = simplify(kane.forcing)
mass_matrix = simplify(kane.mass_matrix)
zero_omega = dict(zip(speeds, zeros(3)))
torques = Matrix(specified)
# 9.81] # acceleration due to gravity [m/s^2]
# )
numerical_constants = array([0.75,
7.0,
0.5,
8.0,
9.81])
#Set input torques to 0
numerical_specified = zeros(2)
parameter_dict = dict(zip(constants, numerical_constants))
ke_energy = simplify(kinetic_energy(inertial_frame, twoP, threeP))
twoP.set_potential_energy(one_mass*g*one_length*cos(theta1))
threeP.set_potential_energy(two_mass*g*(one_length*cos(theta1)+two_length*cos(theta1+theta2)))
pe_energy = simplify(potential_energy(twoP, threeP))
forcing = simplify(kane.forcing)
mass_matrix = simplify(kane.mass_matrix)
torques = Matrix(specified)
zero_omega = dict(zip(speeds, zeros(2)))
com = simplify(forcing.subs(zero_omega) - torques)
def test_deprecated_set_potential_energy():
m, g, h = symbols('m g h')
P = Point('P')
p = Particle('pa', P, m)
with warns_deprecated_sympy():
p.set_potential_energy(m*g*h)
# part a
r1 = s*N.x
r2 = (s + l*cos(theta))*N.x + l*sin(theta)*N.y
O = Point('O')
p1 = O.locatenew('p1', r1)
p2 = O.locatenew('p2', r2)
O.set_vel(N, 0)
p1.set_vel(N, p1.pos_from(O).dt(N))
p2.set_vel(N, p2.pos_from(O).dt(N))
P1 = Particle('P1', p1, 2*m)
P2 = Particle('P2', p2, m)
P1.set_potential_energy(0)
P2.set_potential_energy(P2.mass * g * (p2.pos_from(O) & N.y))
L1 = Lagrangian(N, P1, P2)
print('{} = {}'.format('L1', msprint(L1)))
lm1 = LagrangesMethod(L1, [s, theta])
lm1.form_lagranges_equations()
rhs = lm1.rhs()
t = symbols('t')
print('{} = {}'.format(msprint(sd.diff(t)), msprint(rhs[2].simplify())))
print('{} = {}\n'.format(msprint(thetad.diff(t)), msprint(rhs[3].simplify())))
# part b
r1 = s*N.x + h*N.y
r2 = (s + l*cos(theta))*N.x + (h + l*sin(theta))*N.y
# part a
r1 = s * N.x
r2 = (s + l * cos(theta)) * N.x + l * sin(theta) * N.y
O = Point('O')
p1 = O.locatenew('p1', r1)
p2 = O.locatenew('p2', r2)
O.set_vel(N, 0)
p1.set_vel(N, p1.pos_from(O).dt(N))
p2.set_vel(N, p2.pos_from(O).dt(N))
P1 = Particle('P1', p1, 2 * m)
P2 = Particle('P2', p2, m)
P1.set_potential_energy(0)
P2.set_potential_energy(P2.mass * g * (p2.pos_from(O) & N.y))
L1 = Lagrangian(N, P1, P2)
print('{} = {}'.format('L1', msprint(L1)))
lm1 = LagrangesMethod(L1, [s, theta])
lm1.form_lagranges_equations()
rhs = lm1.rhs()
t = symbols('t')
print('{} = {}'.format(msprint(sd.diff(t)), msprint(rhs[2].simplify())))
print('{} = {}\n'.format(msprint(thetad.diff(t)), msprint(rhs[3].simplify())))
# part b
r1 = s * N.x + h * N.y
r2 = (s + l * cos(theta)) * N.x + (h + l * sin(theta)) * N.y
