'm/s',
lower='-5',
upper='5',
method='utm',
activator='eps1')
ocp.scale(m='x', rad=1)
guess_maker = beluga.guess_generator(
'ones',
start=[1.0], # Starting values for states in order
costate_guess=0.1,
control_guess=[0.35],
use_control_guess=True)
continuation_steps = beluga.init_continuation()
continuation_steps.add_step('bisection') \
.num_cases(10, 'log') \
.const('eps1', 2e-1)
beluga.add_logger(file_level=logging.DEBUG, display_level=logging.INFO)
bvp_solver = beluga.bvp_algorithm('spbvp')
beluga.solve(ocp=ocp,
method='indirect',
optim_options={
'control_method': 'differential',
'analytical_jacobian': True
},
def test_brachistochrone_shooting():
from math import pi
import beluga
from beluga.ivpsol import Trajectory
from beluga.bvpsol import Solution
ocp = beluga.OCP('brachisto')
# Define independent variables
ocp.independent('t', 's')
# Define equations of motion
ocp.state('x', 'v*cos(theta)', 'm') \
.state('y', 'v*sin(theta)', 'm') \
.state('v', 'g*sin(theta)', 'm/s')
# Define controls
ocp.control('theta', 'rad')
# Define constants
ocp.constant('g', -9.81, 'm/s^2')
# Define costs
ocp.path_cost('1', '1')
# Define constraints
ocp.constraints() \
.initial('x-x_0', 'm') \
.initial('y-y_0', 'm') \
.initial('v-v_0', 'm/s') \
.terminal('x-x_f', 'm') \
.terminal('y-y_f', 'm')
ocp.scale(m='y', s='y/v', kg=1, rad=1)
shooting_solver = beluga.bvp_algorithm('Shooting')
guess_maker = beluga.guess_generator('auto', start=[0, 0, 0], direction='forward', costate_guess=-0.1, control_guess = [-pi/2], use_control_guess=True)
continuation_steps = beluga.init_continuation()
continuation_steps.add_step('bisection') \
.num_cases(21) \
.terminal('x', 10) \
.terminal('y', -10)
sol = beluga.solve(ocp, method='icrm', bvp_algorithm=shooting_solver, steps=continuation_steps,
guess_generator=guess_maker)
assert isinstance(sol, Trajectory)
assert isinstance(sol, Solution)
assert sol.t.shape[0] == sol.y.shape[0]
assert sol.t.shape[0] == sol.u.shape[0]
assert sol.y.shape[1] == 7
assert sol.u.shape[1] == 1
y0 = sol.y[0]
yf = sol.y[-1]
assert abs(y0[0] - 0) < tol
assert abs(y0[1] - 0) < tol
assert abs(y0[2] - 0) < tol
assert abs(y0[3] + 0.0667) < tol
assert abs(y0[4] - 0.0255) < tol
assert abs(y0[5] + 0.1019) < tol
assert abs(sol.t[-1] - 1.8433) < tol
assert abs(yf[0] - 10) < tol
assert abs(yf[1] + 10) < tol
assert abs(yf[2] - 14.0071) < tol
assert abs(yf[3] + 0.0667) < tol
assert abs(yf[4] - 0.0255) < tol
assert abs(yf[5] - 0) < tol
assert abs(y0[3] - yf[3]) < tol
assert abs(y0[4] - yf[4]) < tol
sol = beluga.solve(ocp, method='traditional', bvp_algorithm=shooting_solver, steps=continuation_steps, guess_generator=guess_maker)
y0 = sol.y[0]
yf = sol.y[-1]
assert sol.t.shape[0] == sol.y.shape[0]
assert sol.t.shape[0] == sol.u.shape[0]
assert sol.y.shape[1] == 6
assert sol.u.shape[1] == 1
assert abs(y0[0] - 0) < tol
assert abs(y0[1] - 0) < tol
assert abs(y0[2] - 0) < tol
assert abs(y0[3] + 0.0667) < tol
assert abs(y0[4] - 0.0255) < tol
assert abs(y0[5] + 0.1019) < tol
assert abs(sol.t[-1] - 1.8433) < tol
assert abs(yf[0] - 10) < tol
assert abs(yf[1] + 10) < tol
assert abs(yf[2] - 14.0071) < tol
assert abs(yf[3] + 0.0667) < tol
assert abs(yf[4] - 0.0255) < tol
assert abs(yf[5] - 0) < tol
assert abs(y0[3] - yf[3]) < tol
assert abs(y0[4] - yf[4]) < tol
def test_planarhypersonic():
from math import pi
import beluga
ocp = beluga.OCP('planarHypersonic')
# Define independent variables
ocp.independent('t', 's')
# Define equations of motion
ocp.state('h', 'v*sin(gam)', 'm') \
.state('theta', 'v*cos(gam)/r', 'rad') \
.state('v', '-D/mass - mu*sin(gam)/r**2', 'm/s') \
.state('gam', 'L/(mass*v) + (v/r - mu/(v*r^2))*cos(gam)', 'rad')
# Define quantities used in the problem
ocp.quantity('rho', 'rho0*exp(-h/H)')
ocp.quantity('Cl', '(1.5658*alfa + -0.0000)')
ocp.quantity('Cd', '(1.6537*alfa^2 + 0.0612)')
ocp.quantity('D', '0.5*rho*v^2*Cd*Aref')
ocp.quantity('L', '0.5*rho*v^2*Cl*Aref')
ocp.quantity('r', 're+h')
# Define controls
ocp.control('alfa', 'rad')
# Define constants
ocp.constant('mu', 3.986e5 * 1e9,
'm^3/s^2') # Gravitational parameter, m^3/s^2
ocp.constant('rho0', 0.0001 * 1.2,
'kg/m^3') # Sea-level atmospheric density, kg/m^3
ocp.constant('H', 7500, 'm') # Scale height for atmosphere of Earth, m
ocp.constant('mass', 750 / 2.2046226, 'kg') # Mass of vehicle, kg
ocp.constant('re', 6378000, 'm') # Radius of planet, m
ocp.constant('Aref',
pi * (24 * .0254 / 2)**2,
'm^2') # Reference area of vehicle, m^2
ocp.constant('h_0', 80000, 'm')
ocp.constant('v_0', 4000, 'm/s')
ocp.constant('h_f', 80000, 'm')
ocp.constant('theta_f', 0, 'rad')
# Define costs
ocp.terminal_cost('-v^2', 'm^2/s^2')
# Define constraints
ocp.constraints() \
.initial('h-h_0', 'm') \
.initial('theta', 'rad') \
.initial('v-v_0', 'm/s') \
.terminal('h-h_f', 'm') \
.terminal('theta-theta_f', 'rad')
ocp.scale(m='h', s='h/v', kg='mass', rad=1)
bvp_solver = beluga.bvp_algorithm('Shooting',
algorithm='SLSQP',
tolerance=1e-6)
guess_maker = beluga.guess_generator(
'auto',
start=[80000, 0, 4000, -90 * pi / 180],
direction='forward',
costate_guess=-0.1)
continuation_steps = beluga.init_continuation()
continuation_steps.add_step('bisection') \
.num_cases(11) \
.const('h_f', 0) \
.const('theta_f', 0.01 * pi / 180)
continuation_steps.add_step('bisection') \
.num_cases(11) \
.const('theta_f', 5.0 * pi / 180)
continuation_steps.add_step('bisection') \
.num_cases(11) \
.const('rho0', 1.2)
sol = beluga.solve(ocp,
method='traditional',
bvp_algorithm=bvp_solver,
steps=continuation_steps,
guess_generator=guess_maker)
y0 = sol.y[0]
yf = sol.y[-1]
y0e = [
8.00000000e+04, 0.00000000e+00, 4.00000000e+03, 1.95069984e-02,
-1.68249327e+01, 1.21634197e+06, -2.83598229e+03, -6.15819100e-17
]
yfe = [
5.23214346e-04, 8.72664626e-02, 2.69147623e+03, -9.38246813e-01,
5.46455659e+02, 1.21634197e+06, -5.38295257e+03, 1.67911185e-01
]
tfe = 144.5678
assert sol.t.shape[0] == sol.y.shape[0]
assert sol.t.shape[0] == sol.u.shape[0]
assert sol.y.shape[1] == 8
assert sol.u.shape[1] == 1
assert abs((y0[0] - y0e[0]) / y0e[0]) < tol
assert abs((y0[1] - y0e[1])) < tol
assert abs((y0[2] - y0e[2]) / y0e[2]) < tol
assert abs((y0[3] - y0e[3]) / y0e[3]) < tol
assert abs((y0[4] - y0e[4]) / y0e[4]) < tol
assert abs((y0[5] - y0e[5]) / y0e[5]) < tol
assert abs((y0[6] - y0e[6]) / y0e[6]) < tol
assert abs((y0[7] - y0e[7])) < tol
assert abs((sol.t[-1] - tfe) / tfe) < tol
assert abs((yf[0] - yfe[0])) < tol
assert abs((yf[1] - yfe[1]) / yfe[1]) < tol
assert abs((yf[2] - yfe[2]) / yfe[2]) < tol
assert abs((yf[3] - yfe[3]) / yfe[3]) < tol
assert abs((yf[4] - yfe[4]) / yfe[4]) < tol
assert abs((yf[5] - yfe[5]) / yfe[5]) < tol
assert abs((yf[6] - yfe[6]) / yfe[6]) < tol
assert abs((yf[7] - yfe[7]) / yfe[7]) < tol
def test_planarhypersonic():
from math import pi
import beluga
ocp = beluga.OCP('planarHypersonic')
# Define independent variables
ocp.independent('t', 's')
# Define equations of motion
ocp.state('h', 'v*sin(gam)', 'm') \
.state('theta', 'v*cos(gam)/r', 'rad') \
.state('v', '-D/mass - mu*sin(gam)/r**2', 'm/s') \
.state('gam', 'L/(mass*v) + (v/r - mu/(v*r^2))*cos(gam)', 'rad')
# Define quantities used in the problem
ocp.quantity('rho', 'rho0*exp(-h/H)')
ocp.quantity('Cl', '(1.5658*alfa + -0.0000)')
ocp.quantity('Cd', '(1.6537*alfa^2 + 0.0612)')
ocp.quantity('D', '0.5*rho*v^2*Cd*Aref')
ocp.quantity('L', '0.5*rho*v^2*Cl*Aref')
ocp.quantity('r', 're+h')
# Define controls
ocp.control('alfa', 'rad')
# Define constants
ocp.constant('mu', 3.986e5 * 1e9, 'm^3/s^2') # Gravitational parameter, m^3/s^2
ocp.constant('rho0', 0.0001 * 1.2, 'kg/m^3') # Sea-level atmospheric density, kg/m^3
ocp.constant('H', 7500, 'm') # Scale height for atmosphere of Earth, m
ocp.constant('mass', 750 / 2.2046226, 'kg') # Mass of vehicle, kg
ocp.constant('re', 6378000, 'm') # Radius of planet, m
ocp.constant('Aref', pi * (24 * .0254 / 2) ** 2, 'm^2') # Reference area of vehicle, m^2
# Define costs
ocp.terminal_cost('-v^2', 'm^2/s^2')
# Define constraints
ocp.constraints() \
.initial('h-h_0', 'm') \
.initial('theta-theta_0', 'rad') \
.initial('v-v_0', 'm/s') \
.terminal('h-h_f', 'm') \
.terminal('theta-theta_f', 'rad')
ocp.scale(m='h', s='h/v', kg='mass', rad=1)
bvp_solver = beluga.bvp_algorithm('Shooting')
guess_maker = beluga.guess_generator('auto', start=[80000, 0, 4000, -90 * pi / 180], direction='forward', costate_guess=-0.1)
continuation_steps = beluga.init_continuation()
continuation_steps.add_step('bisection') \
.num_cases(11) \
.terminal('h', 0) \
.terminal('theta', 0.01 * pi / 180)
continuation_steps.add_step('bisection') \
.num_cases(11) \
.terminal('theta', 5.0 * pi / 180)
continuation_steps.add_step('bisection') \
.num_cases(11) \
.const('rho0', 1.2)
sol = beluga.solve(ocp, method='traditional', bvp_algorithm=bvp_solver, steps=continuation_steps, guess_generator=guess_maker)
y0 = sol.y[0]
yf = sol.y[-1]
y0e = [80000, 0, 4000, 0.0195, -16.8243, 1212433.8085, -2836.0620, 0]
yfe = [0, 0.0873, 2691.4733, -0.9383, 546.4540, 1212433.8085, -5382.9467, 0.1840]
tfe = 144.5677
assert sol.t.shape[0] == sol.y.shape[0]
assert sol.t.shape[0] == sol.u.shape[0]
assert sol.y.shape[1] == 8
assert sol.u.shape[1] == 1
assert abs((y0[0] - y0e[0]) / y0e[0]) < tol
assert abs((y0[1] - y0e[1])) < tol
assert abs((y0[2] - y0e[2]) / y0e[2]) < tol
assert abs((y0[3] - y0e[3]) / y0e[3]) < tol
assert abs((y0[4] - y0e[4]) / y0e[4]) < tol
assert abs((y0[5] - y0e[5]) / y0e[5]) < tol
assert abs((y0[6] - y0e[6]) / y0e[6]) < tol
assert abs((y0[7] - y0e[7])) < tol
assert abs((sol.t[-1] - tfe) / tfe) < tol
assert abs((yf[0] - yfe[0])) < tol
assert abs((yf[1] - yfe[1]) / yfe[1]) < tol
assert abs((yf[2] - yfe[2]) / yfe[2]) < tol
assert abs((yf[3] - yfe[3]) / yfe[3]) < tol
assert abs((yf[4] - yfe[4]) / yfe[4]) < tol
assert abs((yf[5] - yfe[5]) / yfe[5]) < tol
assert abs((yf[6] - yfe[6]) / yfe[6]) < tol
assert abs((yf[7] - yfe[7]) / yfe[7]) < tol
def test_zermelo_custom_functions():
import beluga
ocp = beluga.OCP('zermelos_problem')
def drift_x(x, y):
return 0
def drift_y(x, y):
return ((x - 5)**4 - 625) / 625
ocp.custom_function('drift_x', drift_x)
ocp.custom_function('drift_y', drift_y)
# Define independent variables
ocp.independent('t', 's')
# Define equations of motion
ocp.state('x', 'V*cos(theta) + epsilon*drift_x(x,y)', 'm') \
.state('y', 'V*sin(theta) + epsilon*drift_y(x,y)', 'm')
# Define controls
ocp.control('theta', 'rad')
# Define constants
ocp.constant('V', 10, 'm/s')
ocp.constant('epsilon', 0.001, '1')
ocp.constant('x_f', 0, 'm')
ocp.constant('y_f', 0, 'm')
# Define costs
ocp.path_cost('1', '1')
# Define constraints
ocp.constraints() \
.initial('x', 'm') \
.initial('y', 'm') \
.terminal('x-x_f', 'm') \
.terminal('y-y_f', 'm')
ocp.scale(m='x', s='x/V', rad=1)
bvp_solver = beluga.bvp_algorithm('Shooting',
derivative_method='fd',
tolerance=1e-4,
max_error=100,
max_iterations=100)
guess_maker = beluga.guess_generator('auto',
start=[0, 0],
control_guess=[0],
use_control_guess=True,
direction='forward')
continuation_steps = beluga.init_continuation()
continuation_steps.add_step('bisection') \
.num_cases(10) \
.const('x_f', 10)
continuation_steps.add_step('bisection') \
.num_cases(10) \
.const('y_f', 10)
continuation_steps.add_step('bisection') \
.num_cases(10) \
.const('epsilon', 1)
sol = beluga.solve(ocp,
method='icrm',
bvp_algorithm=bvp_solver,
steps=continuation_steps,
guess_generator=guess_maker)
from beluga.ivpsol import Trajectory
assert isinstance(sol, Trajectory)