Single-Phase Space Shuttle Reentry#
The problem of the space shuttle reentering Earth’s atmosphere is an optimal control problem governed by six equations of motion and limited by the aerodynamic heating rate. For a detailed layout of this problem and other optimal control problems see Betts [Bet10]. The governing equations of motion for this problem are:
(79)#\[\begin{align}
\frac{dh}{dt} &= v \sin \gamma \\
\frac{d\phi}{dt} &= \frac{v}{r} \cos \gamma \frac{\sin \psi}{\cos \theta} \\
\frac{d\theta}{dt} &= \frac{v}{r} \cos \gamma \cos \psi \\
\frac{dv}{dt} &= - \frac{D}{m} - g \sin \gamma \\
\frac{d\gamma}{dt} &= \frac{L}{mv} \cos \beta + \cos \gamma (\frac{v}{r} - \frac{g}{v}) \\
\frac{d\psi}{dt} &= \frac{L \sin \beta}{mv \cos \gamma} + \frac{v}{r \cos \theta} \cos \gamma \sin \psi \sin \theta
\end{align}\]
where \(v\) \([ft/s]\) is airspeed, \(\gamma\) \([rad]\) is flight path angle, \(r\) \([ft]\) is distance from the center of the Earth, \(\psi\) \([rad]\) is azimuth, \(\theta\) \([rad]\) is latitude, \(D\) \([lb]\) is drag, \(m\) \([sl]\) is mass, \(g\) \([\frac{ft}{s^2}]\) is the local gravitational acceleration, \(L\) \([lb]\) is lift, \(\beta\) \([rad]\) is bank angle, \(h\) \([ft]\) is altitude, and \(\phi\) \([rad]\) is longitude. Mass is considered to be a constant for this case, because the model spans the time from when the shuttle begins reentry to the time right before the shuttle starts its engines. The engines are not actually running at any time during the model, so there is no thrust and thus no mass lost. The goal is to maximize the crossrange (latitude) that the shuttle can cover before reaching the final altitude, without exceding a maximum heat rate at the leading edges. This heat rate is constrained by \(q \leq 70\) where q \([\frac{btu}{ft^2s}]\) is the heating rate.
The initial conditions are
(80)#\[\begin{align}
h_0 &= 26000 \\
v_0 &= 25600 \\
\phi_0 &= 0 \\
\gamma_0 &= -0.01745 \\
\theta_0 &= 0 \\
\psi_0 &= \frac{\pi}{2}
\end{align}\]
and the final conditions are
(81)#\[\begin{align}
h_0 &= 80000 \\
v_0 &= 2500 \\
\gamma_0 &= -0.08727 \\
\theta &= \rm{free} \\
\psi &= \rm{free}
\end{align}\]
Notice that no final condition appears for \(\phi\). This is because none of the equations of motion actually depend on \(\phi\), and as a result, while \(\phi\) exists in the dymos model (last code block below) as a state variable, it does not exist as either an input or output in the ode (ShuttleODE group, second to last code block below).
This model uses four explicit OpenMDAO components. The first component computes the local atmospheric condition at the shuttle’s altitude. The second component computes the aerodynamic forces of lift and drag on the shuttle. The third component is where the heating rate on the leading edge of the shuttles wings is computed. The heating rate is given by \(q = q_a q_r\) where
(82)#\[\begin{align}
q_a &= c_0 + c_1\alpha + c_2 \alpha^2 + c_3 \alpha^3
\end{align}\]
and
(83)#\[\begin{align}
q_r &= 17700 \rho^.5 (.0001v)^{3.07}
\end{align}\]
where \(c_0, c_1, c_2,\) and \(c_3\) are constants, \(\alpha\) \([deg]\) is the angle of attack, \(\rho\) \([\frac{sl}{ft^3}]\) is local atmospheric density, and \(v\) \([\frac{ft}{s}]\) is velocity. The final component is where the equations of motion are implemented. These four components are put together in the ShuttleODE group, which is the top level ode that the dymos model sees.
Component Models#
Below is the code for the atmospheric component:
import numpy as np
import openmdao.api as om
class Atmosphere(om.ExplicitComponent):
"""
Defines the logarithmic atmosphere model for the shuttle reentry problem.
References
----------
.. [1] Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming, p. 248, 2010.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('h', val=np.ones(nn), desc='altitude', units='ft')
self.add_output('rho', val=np.ones(nn), desc='local density', units='slug/ft**3')
partial_range = np.arange(nn, dtype=int)
self.declare_partials('rho', 'h', rows=partial_range, cols=partial_range)
def compute(self, inputs, outputs):
h = inputs['h']
h_r = 23800
rho_0 = .002378
outputs['rho'] = rho_0 * np.exp(-h / h_r)
def compute_partials(self, inputs, partials):
h = inputs['h']
h_r = 23800
rho_0 = .002378
partials['rho', 'h'] = -1 / h_r * rho_0 * np.exp(-h / h_r)
Below is the code for the aerodynamics component:
import openmdao.api as om
class Aerodynamics(om.ExplicitComponent):
"""
Defines the aerodynamics for the shuttle reentry problem.
References
----------
.. [1] Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming, p. 248, 2010.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('alpha', val=np.ones(nn), desc='angle of attack', units='deg')
self.add_input('v', val=np.ones(nn), desc='velocity of shuttle', units='ft/s')
self.add_input('rho', val=np.ones(nn), desc='local atmospheric density',
units='slug/ft**3')
self.add_output('drag', val=np.ones(nn), desc='drag on shuttle', units='lb')
self.add_output('lift', val=np.ones(nn), desc='lift on shuttle', units='lb')
partial_range = np.arange(nn, dtype=int)
self.declare_partials('drag', 'alpha', rows=partial_range, cols=partial_range)
self.declare_partials('drag', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('drag', 'rho', rows=partial_range, cols=partial_range)
self.declare_partials('lift', 'alpha', rows=partial_range, cols=partial_range)
self.declare_partials('lift', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('lift', 'rho', rows=partial_range, cols=partial_range)
def compute(self, inputs, outputs):
a_0 = -.20704
a_1 = .029244
b_0 = .07854
b_1 = -.61592e-2
b_2 = .621408e-3
S = 2690
alpha = inputs['alpha']
v = inputs['v']
rho = inputs['rho']
c_L = a_0 + a_1 * alpha
c_D = b_0 + b_1 * alpha + b_2 * alpha ** 2
outputs['drag'] = .5 * c_D * S * rho * v ** 2
outputs['lift'] = .5 * c_L * S * rho * v ** 2
def compute_partials(self, inputs, J):
alpha = inputs['alpha']
v = inputs['v']
rho = inputs['rho']
a_0 = -.20704
a_1 = .029244
b_0 = .07854
b_1 = -.61592e-2
b_2 = .621408e-3
S = 2690
c_L = a_0 + a_1 * alpha
c_D = b_0 + b_1 * alpha + b_2 * alpha ** 2
dD_dCD = .5 * S * rho * v ** 2
dCD_dalpha = b_1 + 2 * alpha * b_2
dL_dCL = dD_dCD
dCL_dalpha = a_1
J['drag', 'alpha'] = dD_dCD * dCD_dalpha
J['drag', 'v'] = c_D * S * rho * v
J['drag', 'rho'] = .5 * c_D * S * v ** 2
J['lift', 'alpha'] = dL_dCL * dCL_dalpha
J['lift', 'v'] = c_L * S * rho * v
J['lift', 'rho'] = .5 * c_L * S * v ** 2
Below is the code for the heating component:
import openmdao.api as om
class AerodynamicHeating(om.ExplicitComponent):
"""
Defines the Aerodynamic heating equations for the shuttle reentry problem.
References
----------
.. [1] Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming, p. 248, 2010.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('rho', val=np.ones(nn), desc='local density', units='slug/ft**3')
self.add_input('v', val=np.ones(nn), desc='velocity of shuttle', units='ft/s')
self.add_input('alpha', val=np.ones(nn), desc='angle of attack of shuttle',
units='deg')
self.add_output('q', val=np.ones(nn),
desc='aerodynamic heating on leading edge of shuttle',
units='Btu/ft**2/s')
partial_range = np.arange(nn)
self.declare_partials('q', 'rho', rows=partial_range, cols=partial_range)
self.declare_partials('q', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('q', 'alpha', rows=partial_range, cols=partial_range)
def compute(self, inputs, outputs):
rho = inputs['rho']
v = inputs['v']
alpha = inputs['alpha']
c_0 = 1.0672181
c_1 = -0.19213774e-1
c_2 = 0.21286289e-3
c_3 = -0.10117249e-5
q_r = 17700.0 * np.sqrt(rho) * (0.0001 * v) ** 3.07
q_a = c_0 + c_1 * alpha + c_2 * alpha ** 2 + c_3 * alpha ** 3
outputs['q'] = q_r * q_a
def compute_partials(self, inputs, partials):
rho = inputs['rho']
v = inputs['v']
alpha = inputs['alpha']
c_0 = 1.0672181
c_1 = -.19213774e-1
c_2 = .21286289e-3
c_3 = -.10117249e-5
sqrt_rho = np.sqrt(rho)
q_r = 17700 * sqrt_rho * (.0001 * v) ** 3.07
q_a = c_0 + c_1 * alpha + c_2 * alpha ** 2 + c_3 * alpha ** 3
dqr_drho = 0.5 * q_r / rho
dqr_dv = 17700 * sqrt_rho * 0.0001 * 3.07 * (0.0001 * v) ** 2.07
dqa_dalpha = c_1 + 2 * c_2 * alpha + 3 * c_3 * alpha ** 2
partials['q', 'rho'] = dqr_drho * q_a
partials['q', 'v'] = dqr_dv * q_a
partials['q', 'alpha'] = dqa_dalpha * q_r
Below is the code for the component containing the equations of motion:
from openmdao.api import ExplicitComponent
class FlightDynamics(ExplicitComponent):
"""
Defines the flight dynamics for the shuttle reentry problem.
References
----------
.. [1] Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming, p. 247, 2010.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('beta', val=np.ones(nn), desc='bank angle', units='rad')
self.add_input('gamma', val=np.ones(nn), desc='flight path angle', units='rad')
self.add_input('h', val=np.ones(nn), desc='altitude of shuttle', units='ft')
self.add_input('psi', val=np.ones(nn), desc='azimuthal angle', units='rad')
self.add_input('theta', val=np.ones(nn), desc='latitude', units='rad')
self.add_input('v', val=np.ones(nn), desc='velocity of shuttle', units='ft/s')
self.add_input('lift', val=np.ones(nn), desc='lift on shuttle', units='lb')
self.add_input('drag', val=np.ones(nn), desc='drag on shuttle', units='lb')
self.add_output('hdot', val=np.ones(nn), desc='rate of change of altitude',
units='ft/s')
self.add_output('gammadot', val=np.ones(nn),
desc='rate of change of flight path angle', units='rad/s')
self.add_output('phidot', val=np.ones(nn), desc='rate of change of longitude',
units='rad/s')
self.add_output('psidot', val=np.ones(nn), desc='rate of change of azimuthal angle',
units='rad/s')
self.add_output('thetadot', val=np.ones(nn), desc='rate of change of latitude',
units='rad/s')
self.add_output('vdot', val=np.ones(nn), desc='rate of change of velocity',
units='ft/s**2')
partial_range = np.arange(nn, dtype=int)
self.declare_partials('hdot', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('hdot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('gammadot', 'lift', rows=partial_range, cols=partial_range)
self.declare_partials('gammadot', 'h', rows=partial_range, cols=partial_range)
self.declare_partials('gammadot', 'beta', rows=partial_range, cols=partial_range)
self.declare_partials('gammadot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('gammadot', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('phidot', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('phidot', 'h', rows=partial_range, cols=partial_range)
self.declare_partials('phidot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('phidot', 'psi', rows=partial_range, cols=partial_range)
self.declare_partials('phidot', 'theta', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'h', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'beta', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'theta', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'psi', rows=partial_range, cols=partial_range)
self.declare_partials('psidot', 'lift', rows=partial_range, cols=partial_range)
self.declare_partials('thetadot', 'v', rows=partial_range, cols=partial_range)
self.declare_partials('thetadot', 'h', rows=partial_range, cols=partial_range)
self.declare_partials('thetadot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('thetadot', 'psi', rows=partial_range, cols=partial_range)
self.declare_partials('vdot', 'drag', rows=partial_range, cols=partial_range)
self.declare_partials('vdot', 'gamma', rows=partial_range, cols=partial_range)
self.declare_partials('vdot', 'h', rows=partial_range, cols=partial_range)
def compute(self, inputs, outputs):
v = inputs['v']
gamma = inputs['gamma']
theta = inputs['theta']
lift = inputs['lift']
drag = inputs['drag']
h = inputs['h']
beta = inputs['beta']
psi = inputs['psi']
g_0 = 32.174
w = 203000
R_e = 20902900
mu = .14076539e17
s_beta = np.sin(beta)
c_beta = np.cos(beta)
s_gamma = np.sin(gamma)
c_gamma = np.cos(gamma)
s_psi = np.sin(psi)
c_psi = np.cos(psi)
c_theta = np.cos(theta)
s_theta = np.sin(theta)
r = R_e + h
m = w / g_0
g = mu / r ** 2
outputs['hdot'] = v * s_gamma
outputs['gammadot'] = lift / (m * v) * c_beta + c_gamma * (v / r - g / v)
outputs['phidot'] = v / r * c_gamma * s_psi / c_theta
outputs['psidot'] = lift * s_beta / (m * v * c_gamma) + \
v * c_gamma * s_psi * s_theta / (r * c_theta)
outputs['thetadot'] = c_gamma * c_psi * v / r
outputs['vdot'] = -drag / m - g * s_gamma
def compute_partials(self, inputs, J):
v = inputs['v']
gamma = inputs['gamma']
theta = inputs['theta']
lift = inputs['lift']
h = inputs['h']
beta = inputs['beta']
psi = inputs['psi']
g_0 = 32.174
w = 203000
R_e = 20902900
mu = .14076539e17
s_beta = np.sin(beta)
c_beta = np.cos(beta)
s_gamma = np.sin(gamma)
c_gamma = np.cos(gamma)
s_psi = np.sin(psi)
c_psi = np.cos(psi)
c_theta = np.cos(theta)
s_theta = np.sin(theta)
r = R_e + h
m = w / g_0
g = mu / r ** 2
J['hdot', 'v'] = s_gamma
J['hdot', 'gamma'] = v * c_gamma
J['gammadot', 'lift'] = c_beta / (m * v)
J['gammadot', 'h'] = c_gamma * (-v / r ** 2 + 2 * mu / (r ** 3 * v))
J['gammadot', 'beta'] = -lift / (m * v) * s_beta
J['gammadot', 'gamma'] = -s_gamma * (v / r - g / v)
J['gammadot', 'v'] = -lift / (m * v ** 2) * c_beta + c_gamma * (1 / r + g / v ** 2)
J['phidot', 'v'] = c_gamma * s_psi / (c_theta * r)
J['phidot', 'h'] = -v / r ** 2 * c_gamma * s_psi / c_theta
J['phidot', 'gamma'] = -v / r * s_gamma * s_psi / c_theta
J['phidot', 'psi'] = v / r * c_gamma * c_psi / c_theta
J['phidot', 'theta'] = v / r * c_gamma * s_psi / (c_theta ** 2) * s_theta
J['psidot', 'v'] = -lift * s_beta / (m * c_gamma * v ** 2) + \
c_gamma * s_psi * s_theta / (r * c_theta)
J['psidot', 'gamma'] = lift * s_beta / (m * v * c_gamma ** 2) * s_gamma - \
v * s_gamma * s_psi * s_theta / (r * c_theta)
J['psidot', 'h'] = -v * c_gamma * s_psi * s_theta / (c_theta * r ** 2)
J['psidot', 'beta'] = lift * c_beta / (m * v * c_gamma)
J['psidot', 'theta'] = v * c_gamma * s_psi / (r * c_theta ** 2)
J['psidot', 'psi'] = v * c_gamma * c_psi * s_theta / (r * c_theta)
J['psidot', 'lift'] = s_beta / (m * v * c_gamma)
J['thetadot', 'v'] = c_gamma * c_psi / r
J['thetadot', 'h'] = -v / r ** 2 * c_gamma * c_psi
J['thetadot', 'gamma'] = -v / r * s_gamma * c_psi
J['thetadot', 'psi'] = -v / r * c_gamma * s_psi
J['vdot', 'h'] = 2 * s_gamma * mu / r ** 3
J['vdot', 'drag'] = -1 / m
J['vdot', 'gamma'] = -g * c_gamma
Defining the ODE#
Below is the code for the top level ode group that will be fed to dymos:
from openmdao.api import Group
class ShuttleODE(Group):
"""
The ODE for the Shuttle reentry problem.
References
----------
.. [1] Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming, p. 248, 2010.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_subsystem('atmosphere', subsys=Atmosphere(num_nodes=nn),
promotes_inputs=['h'], promotes_outputs=['rho'])
self.add_subsystem('aerodynamics', subsys=Aerodynamics(num_nodes=nn),
promotes_inputs=['alpha', 'v', 'rho'],
promotes_outputs=['lift', 'drag'])
self.add_subsystem('heating', subsys=AerodynamicHeating(num_nodes=nn),
promotes_inputs=['rho', 'v', 'alpha'], promotes_outputs=['q'])
self.add_subsystem('eom', subsys=FlightDynamics(num_nodes=nn),
promotes_inputs=['beta', 'gamma', 'h', 'psi', 'theta', 'v', 'lift',
'drag'],
promotes_outputs=['hdot', 'gammadot', 'phidot', 'psidot', 'thetadot',
'vdot'])
Building and running the problem#
The following code is the dymos implementation of the model. As the code shows, there are six states, two controls, and one constraint in the model. The states are \(h, v, \phi, \gamma, \theta,\) and \(\psi\). The two controls are \(\alpha\) and \(\beta\), and the constraint is \(q\).
import openmdao.api as om
import dymos as dm
from dymos.examples.plotting import plot_results
import matplotlib.pyplot as plt
# Instantiate the problem, add the driver, and allow it to use coloring
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring()
p.driver.options['optimizer'] = 'SLSQP'
# Instantiate the trajectory and add a phase to it
traj = p.model.add_subsystem('traj', dm.Trajectory())
phase0 = traj.add_phase('phase0',
dm.Phase(ode_class=ShuttleODE,
transcription=dm.Radau(num_segments=15, order=3)))
phase0.set_time_options(fix_initial=True, units='s', duration_ref=200)
phase0.add_state('h', fix_initial=True, fix_final=True, units='ft', rate_source='hdot',
lower=0, ref0=75000, ref=300000, defect_ref=1000)
phase0.add_state('gamma', fix_initial=True, fix_final=True, units='rad',
rate_source='gammadot',
lower=-89. * np.pi / 180, upper=89. * np.pi / 180)
phase0.add_state('phi', fix_initial=True, fix_final=False, units='rad',
rate_source='phidot', lower=0, upper=89. * np.pi / 180)
phase0.add_state('psi', fix_initial=True, fix_final=False, units='rad',
rate_source='psidot', lower=0, upper=90. * np.pi / 180)
phase0.add_state('theta', fix_initial=True, fix_final=False, units='rad',
rate_source='thetadot',
lower=-89. * np.pi / 180, upper=89. * np.pi / 180)
phase0.add_state('v', fix_initial=True, fix_final=True, units='ft/s',
rate_source='vdot', lower=0, ref0=2500, ref=25000)
phase0.add_control('alpha', units='rad', opt=True, lower=-np.pi / 2, upper=np.pi / 2, )
phase0.add_control('beta', units='rad', opt=True, lower=-89 * np.pi / 180, upper=1 * np.pi / 180, )
# The original implementation by Betts includes a heating rate path constraint.
# This will work with the SNOPT optimizer but SLSQP has difficulty converging the solution.
# phase0.add_path_constraint('q', lower=0, upper=70, ref=70)
phase0.add_timeseries_output('q', shape=(1,))
phase0.add_objective('theta', loc='final', ref=-0.01)
p.setup(check=True)
phase0.set_time_val(initial=0, duration=2000, units='s')
phase0.set_state_val('h', [260000, 80000], units='ft')
phase0.set_state_val('gamma', [-1, -5], units='deg')
phase0.set_state_val('phi', [0, 75], units='deg')
phase0.set_state_val('psi', [90, 10], units='deg')
phase0.set_state_val('theta', [0, 25], units='deg')
phase0.set_state_val('v', [25600, 2500], units='ft/s')
phase0.set_control_val('alpha', 17.4, units='deg')
phase0.set_control_val('beta', [-75, 0], units='deg')
# Run the driver
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
--- phase0 ---
None
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
Full total jacobian for problem 'problem' was computed 3 times, taking 0.7757571390000066 seconds.
Total jacobian shape: (327, 358)
Jacobian shape: (327, 358) (2.30% nonzero)
FWD solves: 20 REV solves: 0
Total colors vs. total size: 20 vs 358 (94.41% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.7758 sec
Time to compute coloring: 0.2771 sec
Memory to compute coloring: 0.2500 MB
Coloring created on: 2024-12-11 19:25:46
/tmp/ipykernel_3691/3258059812.py:44: RuntimeWarning: invalid value encountered in power
q_r = 17700.0 * np.sqrt(rho) * (0.0001 * v) ** 3.07
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 22.2265
User Objective Time : 0.3172
User Sensitivity Time : 3.2022
Interface Time : 0.7808
Opt Solver Time: 17.9263
Calls to Objective Function : 274
Calls to Sens Function : 141
Objectives
Index Name Value
0 traj.phase0.states:theta -5.957515E+01
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.phase0.t_duration_0 c -5.000000E+18 1.004142E+01 5.000000E+18
1 traj.phase0.states:h_0 c -3.333333E-01 7.284159E-01 4.444444E+15
2 traj.phase0.states:h_1 c -3.333333E-01 6.156161E-01 4.444444E+15
3 traj.phase0.states:h_2 c -3.333333E-01 5.881392E-01 4.444444E+15
4 traj.phase0.states:h_3 c -3.333333E-01 5.549055E-01 4.444444E+15
5 traj.phase0.states:h_4 c -3.333333E-01 5.678794E-01 4.444444E+15
6 traj.phase0.states:h_5 c -3.333333E-01 5.831409E-01 4.444444E+15
7 traj.phase0.states:h_6 c -3.333333E-01 6.121451E-01 4.444444E+15
8 traj.phase0.states:h_7 c -3.333333E-01 6.265189E-01 4.444444E+15
9 traj.phase0.states:h_8 c -3.333333E-01 6.206406E-01 4.444444E+15
10 traj.phase0.states:h_9 c -3.333333E-01 5.951359E-01 4.444444E+15
11 traj.phase0.states:h_10 c -3.333333E-01 5.502189E-01 4.444444E+15
12 traj.phase0.states:h_11 c -3.333333E-01 5.392159E-01 4.444444E+15
13 traj.phase0.states:h_12 c -3.333333E-01 5.283045E-01 4.444444E+15
14 traj.phase0.states:h_13 c -3.333333E-01 5.314225E-01 4.444444E+15
15 traj.phase0.states:h_14 c -3.333333E-01 5.333900E-01 4.444444E+15
16 traj.phase0.states:h_15 c -3.333333E-01 5.294809E-01 4.444444E+15
17 traj.phase0.states:h_16 c -3.333333E-01 5.100674E-01 4.444444E+15
18 traj.phase0.states:h_17 c -3.333333E-01 5.014967E-01 4.444444E+15
19 traj.phase0.states:h_18 c -3.333333E-01 4.869279E-01 4.444444E+15
20 traj.phase0.states:h_19 c -3.333333E-01 4.779606E-01 4.444444E+15
21 traj.phase0.states:h_20 c -3.333333E-01 4.777471E-01 4.444444E+15
22 traj.phase0.states:h_21 c -3.333333E-01 4.727101E-01 4.444444E+15
23 traj.phase0.states:h_22 c -3.333333E-01 4.551821E-01 4.444444E+15
24 traj.phase0.states:h_23 c -3.333333E-01 4.469347E-01 4.444444E+15
25 traj.phase0.states:h_24 c -3.333333E-01 4.317130E-01 4.444444E+15
26 traj.phase0.states:h_25 c -3.333333E-01 4.192045E-01 4.444444E+15
27 traj.phase0.states:h_26 c -3.333333E-01 4.173268E-01 4.444444E+15
28 traj.phase0.states:h_27 c -3.333333E-01 4.089570E-01 4.444444E+15
29 traj.phase0.states:h_28 c -3.333333E-01 3.894636E-01 4.444444E+15
30 traj.phase0.states:h_29 c -3.333333E-01 3.816042E-01 4.444444E+15
31 traj.phase0.states:h_30 c -3.333333E-01 3.661114E-01 4.444444E+15
32 traj.phase0.states:h_31 c -3.333333E-01 3.470618E-01 4.444444E+15
33 traj.phase0.states:h_32 c -3.333333E-01 3.409513E-01 4.444444E+15
34 traj.phase0.states:h_33 c -3.333333E-01 3.245331E-01 4.444444E+15
35 traj.phase0.states:h_34 c -3.333333E-01 2.998636E-01 4.444444E+15
36 traj.phase0.states:h_35 c -3.333333E-01 2.920832E-01 4.444444E+15
37 traj.phase0.states:h_36 c -3.333333E-01 2.728221E-01 4.444444E+15
38 traj.phase0.states:h_37 c -3.333333E-01 2.423549E-01 4.444444E+15
39 traj.phase0.states:h_38 c -3.333333E-01 2.319041E-01 4.444444E+15
40 traj.phase0.states:h_39 c -3.333333E-01 2.075987E-01 4.444444E+15
41 traj.phase0.states:h_40 c -3.333333E-01 1.651010E-01 4.444444E+15
42 traj.phase0.states:h_41 c -3.333333E-01 1.477369E-01 4.444444E+15
43 traj.phase0.states:h_42 c -3.333333E-01 1.041282E-01 4.444444E+15
44 traj.phase0.states:h_43 c -3.333333E-01 4.169697E-02 4.444444E+15
45 traj.phase0.states:gamma_0 c -1.553343E+00 -1.691899E-02 1.553343E+00
46 traj.phase0.states:gamma_1 c -1.553343E+00 -1.282756E-02 1.553343E+00
47 traj.phase0.states:gamma_2 c -1.553343E+00 -1.023681E-02 1.553343E+00
48 traj.phase0.states:gamma_3 c -1.553343E+00 -2.575164E-03 1.553343E+00
49 traj.phase0.states:gamma_4 c -1.553343E+00 5.849396E-03 1.553343E+00
50 traj.phase0.states:gamma_5 c -1.553343E+00 6.424895E-03 1.553343E+00
51 traj.phase0.states:gamma_6 c -1.553343E+00 4.800266E-03 1.553343E+00
52 traj.phase0.states:gamma_7 c -1.553343E+00 -1.346035E-03 1.553343E+00
53 traj.phase0.states:gamma_8 c -1.553343E+00 -3.415123E-03 1.553343E+00
54 traj.phase0.states:gamma_9 c -1.553343E+00 -6.633951E-03 1.553343E+00
55 traj.phase0.states:gamma_10 c -1.553343E+00 -6.054918E-03 1.553343E+00
56 traj.phase0.states:gamma_11 c -1.553343E+00 -4.311191E-03 1.553343E+00
57 traj.phase0.states:gamma_12 c -1.553343E+00 -7.188261E-04 1.553343E+00
58 traj.phase0.states:gamma_13 c -1.553343E+00 1.108951E-03 1.553343E+00
59 traj.phase0.states:gamma_14 c -1.553343E+00 3.170369E-04 1.553343E+00
60 traj.phase0.states:gamma_15 c -1.553343E+00 -2.055007E-03 1.553343E+00
61 traj.phase0.states:gamma_16 c -1.553343E+00 -4.490200E-03 1.553343E+00
62 traj.phase0.states:gamma_17 c -1.553343E+00 -4.415397E-03 1.553343E+00
63 traj.phase0.states:gamma_18 c -1.553343E+00 -2.808305E-03 1.553343E+00
64 traj.phase0.states:gamma_19 c -1.553343E+00 -5.047885E-04 1.553343E+00
65 traj.phase0.states:gamma_20 c -1.553343E+00 -5.261219E-04 1.553343E+00
66 traj.phase0.states:gamma_21 c -1.553343E+00 -2.253828E-03 1.553343E+00
67 traj.phase0.states:gamma_22 c -1.553343E+00 -5.020437E-03 1.553343E+00
68 traj.phase0.states:gamma_23 c -1.553343E+00 -5.225163E-03 1.553343E+00
69 traj.phase0.states:gamma_24 c -1.553343E+00 -3.871733E-03 1.553343E+00
70 traj.phase0.states:gamma_25 c -1.553343E+00 -1.772824E-03 1.553343E+00
71 traj.phase0.states:gamma_26 c -1.553343E+00 -1.892004E-03 1.553343E+00
72 traj.phase0.states:gamma_27 c -1.553343E+00 -3.795758E-03 1.553343E+00
73 traj.phase0.states:gamma_28 c -1.553343E+00 -6.370026E-03 1.553343E+00
74 traj.phase0.states:gamma_29 c -1.553343E+00 -6.429737E-03 1.553343E+00
75 traj.phase0.states:gamma_30 c -1.553343E+00 -5.801695E-03 1.553343E+00
76 traj.phase0.states:gamma_31 c -1.553343E+00 -6.055766E-03 1.553343E+00
77 traj.phase0.states:gamma_32 c -1.553343E+00 -6.903779E-03 1.553343E+00
78 traj.phase0.states:gamma_33 c -1.553343E+00 -8.421690E-03 1.553343E+00
79 traj.phase0.states:gamma_34 c -1.553343E+00 -9.713283E-03 1.553343E+00
80 traj.phase0.states:gamma_35 c -1.553343E+00 -1.006022E-02 1.553343E+00
81 traj.phase0.states:gamma_36 c -1.553343E+00 -1.259546E-02 1.553343E+00
82 traj.phase0.states:gamma_37 c -1.553343E+00 -1.659487E-02 1.553343E+00
83 traj.phase0.states:gamma_38 c -1.553343E+00 -1.736905E-02 1.553343E+00
84 traj.phase0.states:gamma_39 c -1.553343E+00 -2.171651E-02 1.553343E+00
85 traj.phase0.states:gamma_40 c -1.553343E+00 -3.753268E-02 1.553343E+00
86 traj.phase0.states:gamma_41 c -1.553343E+00 -4.535755E-02 1.553343E+00
87 traj.phase0.states:gamma_42 c -1.553343E+00 -5.729614E-02 1.553343E+00
88 traj.phase0.states:gamma_43 c -1.553343E+00 -7.666873E-02 1.553343E+00
89 traj.phase0.states:phi_0 c 0.000000E+00 5.749288E-02 1.553343E+00
90 traj.phase0.states:phi_1 c 0.000000E+00 1.364708E-01 1.553343E+00
91 traj.phase0.states:phi_2 c 0.000000E+00 1.612834E-01 1.553343E+00
92 traj.phase0.states:phi_3 c 0.000000E+00 2.173349E-01 1.553343E+00
93 traj.phase0.states:phi_4 c 0.000000E+00 2.926232E-01 1.553343E+00
94 traj.phase0.states:phi_5 c 0.000000E+00 3.158916E-01 1.553343E+00
95 traj.phase0.states:phi_6 c 0.000000E+00 3.685435E-01 1.553343E+00
96 traj.phase0.states:phi_7 c 0.000000E+00 4.398905E-01 1.553343E+00
97 traj.phase0.states:phi_8 c 0.000000E+00 4.622262E-01 1.553343E+00
98 traj.phase0.states:phi_9 c 0.000000E+00 5.128565E-01 1.553343E+00
99 traj.phase0.states:phi_10 c 0.000000E+00 5.812393E-01 1.553343E+00
100 traj.phase0.states:phi_11 c 0.000000E+00 6.024325E-01 1.553343E+00
101 traj.phase0.states:phi_12 c 0.000000E+00 6.498909E-01 1.553343E+00
102 traj.phase0.states:phi_13 c 0.000000E+00 7.129270E-01 1.553343E+00
103 traj.phase0.states:phi_14 c 0.000000E+00 7.322843E-01 1.553343E+00
104 traj.phase0.states:phi_15 c 0.000000E+00 7.756898E-01 1.553343E+00
105 traj.phase0.states:phi_16 c 0.000000E+00 8.333483E-01 1.553343E+00
106 traj.phase0.states:phi_17 c 0.000000E+00 8.510380E-01 1.553343E+00
107 traj.phase0.states:phi_18 c 0.000000E+00 8.903591E-01 1.553343E+00
108 traj.phase0.states:phi_19 c 0.000000E+00 9.418061E-01 1.553343E+00
109 traj.phase0.states:phi_20 c 0.000000E+00 9.573848E-01 1.553343E+00
110 traj.phase0.states:phi_21 c 0.000000E+00 9.918853E-01 1.553343E+00
111 traj.phase0.states:phi_22 c 0.000000E+00 1.036771E+00 1.553343E+00
112 traj.phase0.states:phi_23 c 0.000000E+00 1.050316E+00 1.553343E+00
113 traj.phase0.states:phi_24 c 0.000000E+00 1.080000E+00 1.553343E+00
114 traj.phase0.states:phi_25 c 0.000000E+00 1.117849E+00 1.553343E+00
115 traj.phase0.states:phi_26 c 0.000000E+00 1.129060E+00 1.553343E+00
116 traj.phase0.states:phi_27 c 0.000000E+00 1.153438E+00 1.553343E+00
117 traj.phase0.states:phi_28 c 0.000000E+00 1.184106E+00 1.553343E+00
118 traj.phase0.states:phi_29 c 0.000000E+00 1.193097E+00 1.553343E+00
119 traj.phase0.states:phi_30 c 0.000000E+00 1.212321E+00 1.553343E+00
120 traj.phase0.states:phi_31 c 0.000000E+00 1.235760E+00 1.553343E+00
121 traj.phase0.states:phi_32 c 0.000000E+00 1.242439E+00 1.553343E+00
122 traj.phase0.states:phi_33 c 0.000000E+00 1.256447E+00 1.553343E+00
123 traj.phase0.states:phi_34 c 0.000000E+00 1.272889E+00 1.553343E+00
124 traj.phase0.states:phi_35 c 0.000000E+00 1.277414E+00 1.553343E+00
125 traj.phase0.states:phi_36 c 0.000000E+00 1.286608E+00 1.553343E+00
126 traj.phase0.states:phi_37 c 0.000000E+00 1.296777E+00 1.553343E+00
127 traj.phase0.states:phi_38 c 0.000000E+00 1.299423E+00 1.553343E+00
128 traj.phase0.states:phi_39 c 0.000000E+00 1.304525E+00 1.553343E+00
129 traj.phase0.states:phi_40 c 0.000000E+00 1.309634E+00 1.553343E+00
130 traj.phase0.states:phi_41 c 0.000000E+00 1.310848E+00 1.553343E+00
131 traj.phase0.states:phi_42 c 0.000000E+00 1.313014E+00 1.553343E+00
132 traj.phase0.states:phi_43 c 0.000000E+00 1.314909E+00 1.553343E+00
133 traj.phase0.states:phi_44 c 0.000000E+00 1.315328E+00 1.553343E+00
134 traj.phase0.states:psi_0 c 0.000000E+00 1.565674E+00 1.570796E+00
135 traj.phase0.states:psi_1 c 0.000000E+00 1.545940E+00 1.570796E+00
136 traj.phase0.states:psi_2 c 0.000000E+00 1.534776E+00 1.570796E+00
137 traj.phase0.states:psi_3 c 0.000000E+00 1.500315E+00 1.570796E+00
138 traj.phase0.states:psi_4 c 0.000000E+00 1.449365E+00 1.570796E+00
139 traj.phase0.states:psi_5 c 0.000000E+00 1.436820E+00 1.570796E+00
140 traj.phase0.states:psi_6 c 0.000000E+00 1.414016E+00 1.570796E+00
141 traj.phase0.states:psi_7 c 0.000000E+00 1.390786E+00 1.570796E+00
142 traj.phase0.states:psi_8 c 0.000000E+00 1.383873E+00 1.570796E+00
143 traj.phase0.states:psi_9 c 0.000000E+00 1.365824E+00 1.570796E+00
144 traj.phase0.states:psi_10 c 0.000000E+00 1.331841E+00 1.570796E+00
145 traj.phase0.states:psi_11 c 0.000000E+00 1.317938E+00 1.570796E+00
146 traj.phase0.states:psi_12 c 0.000000E+00 1.284235E+00 1.570796E+00
147 traj.phase0.states:psi_13 c 0.000000E+00 1.238927E+00 1.570796E+00
148 traj.phase0.states:psi_14 c 0.000000E+00 1.226295E+00 1.570796E+00
149 traj.phase0.states:psi_15 c 0.000000E+00 1.198170E+00 1.570796E+00
150 traj.phase0.states:psi_16 c 0.000000E+00 1.157989E+00 1.570796E+00
151 traj.phase0.states:psi_17 c 0.000000E+00 1.143957E+00 1.570796E+00
152 traj.phase0.states:psi_18 c 0.000000E+00 1.109316E+00 1.570796E+00
153 traj.phase0.states:psi_19 c 0.000000E+00 1.059696E+00 1.570796E+00
154 traj.phase0.states:psi_20 c 0.000000E+00 1.044516E+00 1.570796E+00
155 traj.phase0.states:psi_21 c 0.000000E+00 1.011084E+00 1.570796E+00
156 traj.phase0.states:psi_22 c 0.000000E+00 9.643017E-01 1.570796E+00
157 traj.phase0.states:psi_23 c 0.000000E+00 9.484960E-01 1.570796E+00
158 traj.phase0.states:psi_24 c 0.000000E+00 9.097985E-01 1.570796E+00
159 traj.phase0.states:psi_25 c 0.000000E+00 8.544317E-01 1.570796E+00
160 traj.phase0.states:psi_26 c 0.000000E+00 8.372755E-01 1.570796E+00
161 traj.phase0.states:psi_27 c 0.000000E+00 7.990070E-01 1.570796E+00
162 traj.phase0.states:psi_28 c 0.000000E+00 7.453695E-01 1.570796E+00
163 traj.phase0.states:psi_29 c 0.000000E+00 7.275174E-01 1.570796E+00
164 traj.phase0.states:psi_30 c 0.000000E+00 6.850174E-01 1.570796E+00
165 traj.phase0.states:psi_31 c 0.000000E+00 6.246545E-01 1.570796E+00
166 traj.phase0.states:psi_32 c 0.000000E+00 6.055319E-01 1.570796E+00
167 traj.phase0.states:psi_33 c 0.000000E+00 5.612271E-01 1.570796E+00
168 traj.phase0.states:psi_34 c 0.000000E+00 4.989078E-01 1.570796E+00
169 traj.phase0.states:psi_35 c 0.000000E+00 4.789249E-01 1.570796E+00
170 traj.phase0.states:psi_36 c 0.000000E+00 4.332274E-01 1.570796E+00
171 traj.phase0.states:psi_37 c 0.000000E+00 3.693714E-01 1.570796E+00
172 traj.phase0.states:psi_38 c 0.000000E+00 3.488178E-01 1.570796E+00
173 traj.phase0.states:psi_39 c 0.000000E+00 3.020497E-01 1.570796E+00
174 traj.phase0.states:psi_40 c 0.000000E+00 2.396896E-01 1.570796E+00
175 traj.phase0.states:psi_41 c 0.000000E+00 2.207489E-01 1.570796E+00
176 traj.phase0.states:psi_42 c 0.000000E+00 1.786333E-01 1.570796E+00
177 traj.phase0.states:psi_43 c 0.000000E+00 1.372729E-01 1.570796E+00
178 traj.phase0.states:psi_44 c 0.000000E+00 1.318542E-01 1.570796E+00
179 traj.phase0.states:theta_0 c -1.553343E+00 1.108134E-04 1.553343E+00
180 traj.phase0.states:theta_1 c -1.553343E+00 1.195169E-03 1.553343E+00
181 traj.phase0.states:theta_2 c -1.553343E+00 1.925666E-03 1.553343E+00
182 traj.phase0.states:theta_3 c -1.553343E+00 4.901625E-03 1.553343E+00
183 traj.phase0.states:theta_4 c -1.553343E+00 1.211102E-02 1.553343E+00
184 traj.phase0.states:theta_5 c -1.553343E+00 1.515091E-02 1.553343E+00
185 traj.phase0.states:theta_6 c -1.553343E+00 2.287837E-02 1.553343E+00
186 traj.phase0.states:theta_7 c -1.553343E+00 3.506117E-02 1.553343E+00
187 traj.phase0.states:theta_8 c -1.553343E+00 3.918549E-02 1.553343E+00
188 traj.phase0.states:theta_9 c -1.553343E+00 4.920162E-02 1.553343E+00
189 traj.phase0.states:theta_10 c -1.553343E+00 6.453884E-02 1.553343E+00
190 traj.phase0.states:theta_11 c -1.553343E+00 6.983510E-02 1.553343E+00
191 traj.phase0.states:theta_12 c -1.553343E+00 8.291633E-02 1.553343E+00
192 traj.phase0.states:theta_13 c -1.553343E+00 1.029668E-01 1.553343E+00
193 traj.phase0.states:theta_14 c -1.553343E+00 1.097515E-01 1.553343E+00
194 traj.phase0.states:theta_15 c -1.553343E+00 1.259000E-01 1.553343E+00
195 traj.phase0.states:theta_16 c -1.553343E+00 1.495411E-01 1.553343E+00
196 traj.phase0.states:theta_17 c -1.553343E+00 1.573340E-01 1.553343E+00
197 traj.phase0.states:theta_18 c -1.553343E+00 1.757793E-01 1.553343E+00
198 traj.phase0.states:theta_19 c -1.553343E+00 2.024669E-01 1.553343E+00
199 traj.phase0.states:theta_20 c -1.553343E+00 2.111800E-01 1.553343E+00
200 traj.phase0.states:theta_21 c -1.553343E+00 2.314882E-01 1.553343E+00
201 traj.phase0.states:theta_22 c -1.553343E+00 2.601903E-01 1.553343E+00
202 traj.phase0.states:theta_23 c -1.553343E+00 2.694044E-01 1.553343E+00
203 traj.phase0.states:theta_24 c -1.553343E+00 2.907085E-01 1.553343E+00
204 traj.phase0.states:theta_25 c -1.553343E+00 3.203907E-01 1.553343E+00
205 traj.phase0.states:theta_26 c -1.553343E+00 3.298101E-01 1.553343E+00
206 traj.phase0.states:theta_27 c -1.553343E+00 3.513229E-01 1.553343E+00
207 traj.phase0.states:theta_28 c -1.553343E+00 3.807046E-01 1.553343E+00
208 traj.phase0.states:theta_29 c -1.553343E+00 3.898884E-01 1.553343E+00
209 traj.phase0.states:theta_30 c -1.553343E+00 4.106322E-01 1.553343E+00
210 traj.phase0.states:theta_31 c -1.553343E+00 4.384140E-01 1.553343E+00
211 traj.phase0.states:theta_32 c -1.553343E+00 4.469588E-01 1.553343E+00
212 traj.phase0.states:theta_33 c -1.553343E+00 4.659979E-01 1.553343E+00
213 traj.phase0.states:theta_34 c -1.553343E+00 4.908676E-01 1.553343E+00
214 traj.phase0.states:theta_35 c -1.553343E+00 4.983542E-01 1.553343E+00
215 traj.phase0.states:theta_36 c -1.553343E+00 5.147300E-01 1.553343E+00
216 traj.phase0.states:theta_37 c -1.553343E+00 5.354191E-01 1.553343E+00
217 traj.phase0.states:theta_38 c -1.553343E+00 5.414699E-01 1.553343E+00
218 traj.phase0.states:theta_39 c -1.553343E+00 5.543606E-01 1.553343E+00
219 traj.phase0.states:theta_40 c -1.553343E+00 5.698864E-01 1.553343E+00
220 traj.phase0.states:theta_41 c -1.553343E+00 5.742429E-01 1.553343E+00
221 traj.phase0.states:theta_42 c -1.553343E+00 5.832012E-01 1.553343E+00
222 traj.phase0.states:theta_43 c -1.553343E+00 5.931773E-01 1.553343E+00
223 traj.phase0.states:theta_44 c -1.553343E+00 5.957515E-01 1.553343E+00
224 traj.phase0.states:v_0 c -1.111111E-01 1.024611E+00 4.444444E+16
225 traj.phase0.states:v_1 c -1.111111E-01 1.013613E+00 4.444444E+16
226 traj.phase0.states:v_2 c -1.111111E-01 1.006946E+00 4.444444E+16
227 traj.phase0.states:v_3 c -1.111111E-01 9.859723E-01 4.444444E+16
228 traj.phase0.states:v_4 c -1.111111E-01 9.545183E-01 4.444444E+16
229 traj.phase0.states:v_5 c -1.111111E-01 9.466907E-01 4.444444E+16
230 traj.phase0.states:v_6 c -1.111111E-01 9.322266E-01 4.444444E+16
231 traj.phase0.states:v_7 c -1.111111E-01 9.170807E-01 4.444444E+16
232 traj.phase0.states:v_8 c -1.111111E-01 9.125230E-01 4.444444E+16
233 traj.phase0.states:v_9 c -1.111111E-01 9.005192E-01 4.444444E+16
234 traj.phase0.states:v_10 c -1.111111E-01 8.778468E-01 4.444444E+16
235 traj.phase0.states:v_11 c -1.111111E-01 8.685913E-01 4.444444E+16
236 traj.phase0.states:v_12 c -1.111111E-01 8.461902E-01 4.444444E+16
237 traj.phase0.states:v_13 c -1.111111E-01 8.155928E-01 4.444444E+16
238 traj.phase0.states:v_14 c -1.111111E-01 8.067919E-01 4.444444E+16
239 traj.phase0.states:v_15 c -1.111111E-01 7.868396E-01 4.444444E+16
240 traj.phase0.states:v_16 c -1.111111E-01 7.579292E-01 4.444444E+16
241 traj.phase0.states:v_17 c -1.111111E-01 7.478816E-01 4.444444E+16
242 traj.phase0.states:v_18 c -1.111111E-01 7.231767E-01 4.444444E+16
243 traj.phase0.states:v_19 c -1.111111E-01 6.874665E-01 4.444444E+16
244 traj.phase0.states:v_20 c -1.111111E-01 6.763243E-01 4.444444E+16
245 traj.phase0.states:v_21 c -1.111111E-01 6.513863E-01 4.444444E+16
246 traj.phase0.states:v_22 c -1.111111E-01 6.163239E-01 4.444444E+16
247 traj.phase0.states:v_23 c -1.111111E-01 6.046272E-01 4.444444E+16
248 traj.phase0.states:v_24 c -1.111111E-01 5.763639E-01 4.444444E+16
249 traj.phase0.states:v_25 c -1.111111E-01 5.359929E-01 4.444444E+16
250 traj.phase0.states:v_26 c -1.111111E-01 5.233226E-01 4.444444E+16
251 traj.phase0.states:v_27 c -1.111111E-01 4.946985E-01 4.444444E+16
252 traj.phase0.states:v_28 c -1.111111E-01 4.545635E-01 4.444444E+16
253 traj.phase0.states:v_29 c -1.111111E-01 4.413990E-01 4.444444E+16
254 traj.phase0.states:v_30 c -1.111111E-01 4.105400E-01 4.444444E+16
255 traj.phase0.states:v_31 c -1.111111E-01 3.673768E-01 4.444444E+16
256 traj.phase0.states:v_32 c -1.111111E-01 3.537669E-01 4.444444E+16
257 traj.phase0.states:v_33 c -1.111111E-01 3.222744E-01 4.444444E+16
258 traj.phase0.states:v_34 c -1.111111E-01 2.780832E-01 4.444444E+16
259 traj.phase0.states:v_35 c -1.111111E-01 2.639474E-01 4.444444E+16
260 traj.phase0.states:v_36 c -1.111111E-01 2.317416E-01 4.444444E+16
261 traj.phase0.states:v_37 c -1.111111E-01 1.870805E-01 4.444444E+16
262 traj.phase0.states:v_38 c -1.111111E-01 1.727889E-01 4.444444E+16
263 traj.phase0.states:v_39 c -1.111111E-01 1.403262E-01 4.444444E+16
264 traj.phase0.states:v_40 c -1.111111E-01 9.707951E-02 4.444444E+16
265 traj.phase0.states:v_41 c -1.111111E-01 8.391572E-02 4.444444E+16
266 traj.phase0.states:v_42 c -1.111111E-01 5.350086E-02 4.444444E+16
267 traj.phase0.states:v_43 c -1.111111E-01 1.241570E-02 4.444444E+16
268 traj.phase0.controls:alpha_0 c -1.570796E+00 3.041688E-01 1.570796E+00
269 traj.phase0.controls:alpha_1 c -1.570796E+00 3.051447E-01 1.570796E+00
270 traj.phase0.controls:alpha_2 c -1.570796E+00 3.055384E-01 1.570796E+00
271 traj.phase0.controls:alpha_3 c -1.570796E+00 3.054327E-01 1.570796E+00
272 traj.phase0.controls:alpha_4 c -1.570796E+00 3.046727E-01 1.570796E+00
273 traj.phase0.controls:alpha_5 c -1.570796E+00 3.023385E-01 1.570796E+00
274 traj.phase0.controls:alpha_6 c -1.570796E+00 3.012892E-01 1.570796E+00
275 traj.phase0.controls:alpha_7 c -1.570796E+00 3.000089E-01 1.570796E+00
276 traj.phase0.controls:alpha_8 c -1.570796E+00 3.024901E-01 1.570796E+00
277 traj.phase0.controls:alpha_9 c -1.570796E+00 3.043015E-01 1.570796E+00
278 traj.phase0.controls:alpha_10 c -1.570796E+00 3.074852E-01 1.570796E+00
279 traj.phase0.controls:alpha_11 c -1.570796E+00 3.068569E-01 1.570796E+00
280 traj.phase0.controls:alpha_12 c -1.570796E+00 3.054450E-01 1.570796E+00
281 traj.phase0.controls:alpha_13 c -1.570796E+00 3.026787E-01 1.570796E+00
282 traj.phase0.controls:alpha_14 c -1.570796E+00 3.025875E-01 1.570796E+00
283 traj.phase0.controls:alpha_15 c -1.570796E+00 3.034586E-01 1.570796E+00
284 traj.phase0.controls:alpha_16 c -1.570796E+00 3.051667E-01 1.570796E+00
285 traj.phase0.controls:alpha_17 c -1.570796E+00 3.049548E-01 1.570796E+00
286 traj.phase0.controls:alpha_18 c -1.570796E+00 3.042672E-01 1.570796E+00
287 traj.phase0.controls:alpha_19 c -1.570796E+00 3.031090E-01 1.570796E+00
288 traj.phase0.controls:alpha_20 c -1.570796E+00 3.040002E-01 1.570796E+00
289 traj.phase0.controls:alpha_21 c -1.570796E+00 3.048836E-01 1.570796E+00
290 traj.phase0.controls:alpha_22 c -1.570796E+00 3.062529E-01 1.570796E+00
291 traj.phase0.controls:alpha_23 c -1.570796E+00 3.049094E-01 1.570796E+00
292 traj.phase0.controls:alpha_24 c -1.570796E+00 3.037032E-01 1.570796E+00
293 traj.phase0.controls:alpha_25 c -1.570796E+00 3.016921E-01 1.570796E+00
294 traj.phase0.controls:alpha_26 c -1.570796E+00 3.027950E-01 1.570796E+00
295 traj.phase0.controls:alpha_27 c -1.570796E+00 3.040809E-01 1.570796E+00
296 traj.phase0.controls:alpha_28 c -1.570796E+00 3.064611E-01 1.570796E+00
297 traj.phase0.controls:alpha_29 c -1.570796E+00 3.061984E-01 1.570796E+00
298 traj.phase0.controls:alpha_30 c -1.570796E+00 3.052584E-01 1.570796E+00
299 traj.phase0.controls:alpha_31 c -1.570796E+00 3.032572E-01 1.570796E+00
300 traj.phase0.controls:alpha_32 c -1.570796E+00 3.025415E-01 1.570796E+00
301 traj.phase0.controls:alpha_33 c -1.570796E+00 3.028092E-01 1.570796E+00
302 traj.phase0.controls:alpha_34 c -1.570796E+00 3.034984E-01 1.570796E+00
303 traj.phase0.controls:alpha_35 c -1.570796E+00 3.038067E-01 1.570796E+00
304 traj.phase0.controls:alpha_36 c -1.570796E+00 3.037490E-01 1.570796E+00
305 traj.phase0.controls:alpha_37 c -1.570796E+00 3.040286E-01 1.570796E+00
306 traj.phase0.controls:alpha_38 c -1.570796E+00 3.060470E-01 1.570796E+00
307 traj.phase0.controls:alpha_39 c -1.570796E+00 3.070802E-01 1.570796E+00
308 traj.phase0.controls:alpha_40 c -1.570796E+00 3.073277E-01 1.570796E+00
309 traj.phase0.controls:alpha_41 c -1.570796E+00 2.999995E-01 1.570796E+00
310 traj.phase0.controls:alpha_42 c -1.570796E+00 2.958273E-01 1.570796E+00
311 traj.phase0.controls:alpha_43 c -1.570796E+00 2.875384E-01 1.570796E+00
312 traj.phase0.controls:alpha_44 c -1.570796E+00 2.836043E-01 1.570796E+00
313 traj.phase0.controls:beta_0 c -1.553343E+00 -1.303263E+00 1.745329E-02
314 traj.phase0.controls:beta_1 c -1.553343E+00 -1.275516E+00 1.745329E-02
315 traj.phase0.controls:beta_2 c -1.553343E+00 -1.251240E+00 1.745329E-02
316 traj.phase0.controls:beta_3 c -1.553343E+00 -1.246941E+00 1.745329E-02
317 traj.phase0.controls:beta_4 c -1.553343E+00 -1.239303E+00 1.745329E-02
318 traj.phase0.controls:beta_5 c -1.553343E+00 -1.229888E+00 1.745329E-02
319 traj.phase0.controls:beta_6 c -1.553343E+00 -1.227180E+00 1.745329E-02
320 traj.phase0.controls:beta_7 c -1.553343E+00 -1.209378E+00 1.745329E-02
321 traj.phase0.controls:beta_8 c -1.553343E+00 -1.146234E+00 1.745329E-02
322 traj.phase0.controls:beta_9 c -1.553343E+00 -1.116929E+00 1.745329E-02
323 traj.phase0.controls:beta_10 c -1.553343E+00 -1.057221E+00 1.745329E-02
324 traj.phase0.controls:beta_11 c -1.553343E+00 -1.015977E+00 1.745329E-02
325 traj.phase0.controls:beta_12 c -1.553343E+00 -1.012862E+00 1.745329E-02
326 traj.phase0.controls:beta_13 c -1.553343E+00 -1.003124E+00 1.745329E-02
327 traj.phase0.controls:beta_14 c -1.553343E+00 -9.631634E-01 1.745329E-02
328 traj.phase0.controls:beta_15 c -1.553343E+00 -9.441083E-01 1.745329E-02
329 traj.phase0.controls:beta_16 c -1.553343E+00 -9.009975E-01 1.745329E-02
330 traj.phase0.controls:beta_17 c -1.553343E+00 -8.548159E-01 1.745329E-02
331 traj.phase0.controls:beta_18 c -1.553343E+00 -8.434132E-01 1.745329E-02
332 traj.phase0.controls:beta_19 c -1.553343E+00 -8.164711E-01 1.745329E-02
333 traj.phase0.controls:beta_20 c -1.553343E+00 -7.707578E-01 1.745329E-02
334 traj.phase0.controls:beta_21 c -1.553343E+00 -7.542270E-01 1.745329E-02
335 traj.phase0.controls:beta_22 c -1.553343E+00 -7.180851E-01 1.745329E-02
336 traj.phase0.controls:beta_23 c -1.553343E+00 -6.775668E-01 1.745329E-02
337 traj.phase0.controls:beta_24 c -1.553343E+00 -6.670018E-01 1.745329E-02
338 traj.phase0.controls:beta_25 c -1.553343E+00 -6.407329E-01 1.745329E-02
339 traj.phase0.controls:beta_26 c -1.553343E+00 -5.935868E-01 1.745329E-02
340 traj.phase0.controls:beta_27 c -1.553343E+00 -5.760319E-01 1.745329E-02
341 traj.phase0.controls:beta_28 c -1.553343E+00 -5.375246E-01 1.745329E-02
342 traj.phase0.controls:beta_29 c -1.553343E+00 -4.947072E-01 1.745329E-02
343 traj.phase0.controls:beta_30 c -1.553343E+00 -4.836475E-01 1.745329E-02
344 traj.phase0.controls:beta_31 c -1.553343E+00 -4.577715E-01 1.745329E-02
345 traj.phase0.controls:beta_32 c -1.553343E+00 -4.157560E-01 1.745329E-02
346 traj.phase0.controls:beta_33 c -1.553343E+00 -4.009333E-01 1.745329E-02
347 traj.phase0.controls:beta_34 c -1.553343E+00 -3.668172E-01 1.745329E-02
348 traj.phase0.controls:beta_35 c -1.553343E+00 -3.219295E-01 1.745329E-02
349 traj.phase0.controls:beta_36 c -1.553343E+00 -3.082507E-01 1.745329E-02
350 traj.phase0.controls:beta_37 c -1.553343E+00 -2.775355E-01 1.745329E-02
351 traj.phase0.controls:beta_38 c -1.553343E+00 -2.361963E-01 1.745329E-02
352 traj.phase0.controls:beta_39 c -1.553343E+00 -2.233642E-01 1.745329E-02
353 traj.phase0.controls:beta_40 c -1.553343E+00 -1.938419E-01 1.745329E-02
354 traj.phase0.controls:beta_41 c -1.553343E+00 -1.521989E-01 1.745329E-02
355 traj.phase0.controls:beta_42 c -1.553343E+00 -1.387997E-01 1.745329E-02
356 traj.phase0.controls:beta_43 c -1.553343E+00 -1.026118E-01 1.745329E-02
357 traj.phase0.controls:beta_44 c -1.553343E+00 -3.500069E-02 1.745329E-02
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier (N/A)
0 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -4.566311E-14 0.000000E+00 9.00000E+100
1 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.096953E-09 0.000000E+00 9.00000E+100
2 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -5.211481E-09 0.000000E+00 9.00000E+100
3 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -6.734666E-09 0.000000E+00 9.00000E+100
4 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 6.330344E-10 0.000000E+00 9.00000E+100
5 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.873173E-08 0.000000E+00 9.00000E+100
6 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.606677E-08 0.000000E+00 9.00000E+100
7 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 8.386930E-09 0.000000E+00 9.00000E+100
8 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.399610E-08 0.000000E+00 9.00000E+100
9 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.642662E-08 0.000000E+00 9.00000E+100
10 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.011958E-08 0.000000E+00 9.00000E+100
11 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -7.439835E-09 0.000000E+00 9.00000E+100
12 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 5.069937E-09 0.000000E+00 9.00000E+100
13 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 4.699357E-08 0.000000E+00 9.00000E+100
14 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 3.354246E-08 0.000000E+00 9.00000E+100
15 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.152013E-08 0.000000E+00 9.00000E+100
16 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.221355E-08 0.000000E+00 9.00000E+100
17 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.607539E-08 0.000000E+00 9.00000E+100
18 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -8.000661E-09 0.000000E+00 9.00000E+100
19 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.550840E-08 0.000000E+00 9.00000E+100
20 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.092089E-08 0.000000E+00 9.00000E+100
21 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -4.726953E-09 0.000000E+00 9.00000E+100
22 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.957434E-08 0.000000E+00 9.00000E+100
23 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -5.843155E-09 0.000000E+00 9.00000E+100
24 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.141638E-08 0.000000E+00 9.00000E+100
25 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 2.073695E-08 0.000000E+00 9.00000E+100
26 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -3.343551E-09 0.000000E+00 9.00000E+100
27 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.047748E-08 0.000000E+00 9.00000E+100
28 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 7.540787E-11 0.000000E+00 9.00000E+100
29 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 2.017793E-08 0.000000E+00 9.00000E+100
30 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.195406E-08 0.000000E+00 9.00000E+100
31 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -9.350436E-09 0.000000E+00 9.00000E+100
32 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -8.601507E-09 0.000000E+00 9.00000E+100
33 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 5.268102E-09 0.000000E+00 9.00000E+100
34 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.682897E-08 0.000000E+00 9.00000E+100
35 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -4.818615E-09 0.000000E+00 9.00000E+100
36 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -9.921981E-09 0.000000E+00 9.00000E+100
37 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 4.061053E-09 0.000000E+00 9.00000E+100
38 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 2.233926E-08 0.000000E+00 9.00000E+100
39 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 3.539195E-09 0.000000E+00 9.00000E+100
40 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -8.197991E-09 0.000000E+00 9.00000E+100
41 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 3.434359E-09 0.000000E+00 9.00000E+100
42 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 2.678114E-08 0.000000E+00 9.00000E+100
43 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 1.005456E-09 0.000000E+00 9.00000E+100
44 traj.phase0.collocation_constraint.defects:h e 0.000000E+00 -1.785428E-09 0.000000E+00 9.00000E+100
45 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 1.210315E-10 0.000000E+00 9.00000E+100
46 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 1.210332E-11 0.000000E+00 9.00000E+100
47 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 1.962288E-13 0.000000E+00 9.00000E+100
48 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 8.845901E-12 0.000000E+00 9.00000E+100
49 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 -9.131002E-11 0.000000E+00 9.00000E+100
50 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 -1.234762E-11 0.000000E+00 9.00000E+100
51 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 -1.770014E-10 0.000000E+00 9.00000E+100
52 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 -5.291860E-10 0.000000E+00 9.00000E+100
53 traj.phase0.collocation_constraint.defects:gamma e 0.000000E+00 -2.566931E-10 0.000000E+00 9.00000E+100
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229 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.392883E-11 0.000000E+00 9.00000E+100
230 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 7.320807E-11 0.000000E+00 9.00000E+100
231 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.771213E-10 0.000000E+00 9.00000E+100
232 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.741261E-10 0.000000E+00 9.00000E+100
233 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 8.416795E-11 0.000000E+00 9.00000E+100
234 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.338439E-11 0.000000E+00 9.00000E+100
235 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.749454E-10 0.000000E+00 9.00000E+100
236 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 8.494745E-10 0.000000E+00 9.00000E+100
237 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 7.506107E-10 0.000000E+00 9.00000E+100
238 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.012361E-10 0.000000E+00 9.00000E+100
239 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.664942E-10 0.000000E+00 9.00000E+100
240 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.907598E-10 0.000000E+00 9.00000E+100
241 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 4.506819E-11 0.000000E+00 9.00000E+100
242 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 7.654300E-11 0.000000E+00 9.00000E+100
243 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.148166E-10 0.000000E+00 9.00000E+100
244 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.063425E-11 0.000000E+00 9.00000E+100
245 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 4.095899E-11 0.000000E+00 9.00000E+100
246 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.127056E-11 0.000000E+00 9.00000E+100
247 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.631922E-12 0.000000E+00 9.00000E+100
248 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.669164E-11 0.000000E+00 9.00000E+100
249 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.185466E-11 0.000000E+00 9.00000E+100
250 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.031261E-12 0.000000E+00 9.00000E+100
251 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.233430E-11 0.000000E+00 9.00000E+100
252 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.605837E-12 0.000000E+00 9.00000E+100
253 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.310167E-11 0.000000E+00 9.00000E+100
254 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 9.243641E-12 0.000000E+00 9.00000E+100
255 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.950477E-11 0.000000E+00 9.00000E+100
256 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.411330E-11 0.000000E+00 9.00000E+100
257 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.798598E-11 0.000000E+00 9.00000E+100
258 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.157980E-11 0.000000E+00 9.00000E+100
259 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 4.329710E-12 0.000000E+00 9.00000E+100
260 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.982509E-11 0.000000E+00 9.00000E+100
261 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 7.404954E-12 0.000000E+00 9.00000E+100
262 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.189094E-11 0.000000E+00 9.00000E+100
263 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.753393E-11 0.000000E+00 9.00000E+100
264 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.676492E-11 0.000000E+00 9.00000E+100
265 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -5.436717E-13 0.000000E+00 9.00000E+100
266 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.582833E-11 0.000000E+00 9.00000E+100
267 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -8.819968E-12 0.000000E+00 9.00000E+100
268 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.129362E-11 0.000000E+00 9.00000E+100
269 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.038799E-12 0.000000E+00 9.00000E+100
270 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 5.827912E-15 0.000000E+00 9.00000E+100
271 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 1.714692E-15 0.000000E+00 9.00000E+100
272 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -7.484767E-16 0.000000E+00 9.00000E+100
273 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 1.673866E-15 0.000000E+00 9.00000E+100
274 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -5.872140E-15 0.000000E+00 9.00000E+100
275 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -1.694279E-15 0.000000E+00 9.00000E+100
276 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -2.497190E-15 0.000000E+00 9.00000E+100
277 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -8.437374E-16 0.000000E+00 9.00000E+100
278 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -1.660257E-15 0.000000E+00 9.00000E+100
279 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -1.667062E-15 0.000000E+00 9.00000E+100
280 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -1.697681E-15 0.000000E+00 9.00000E+100
281 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 -7.654875E-18 0.000000E+00 9.00000E+100
282 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 5.824510E-15 0.000000E+00 9.00000E+100
283 traj.phase0.continuity_comp.defect_control_rates:alpha_rate e 0.000000E+00 5.824510E-15 0.000000E+00 9.00000E+100
284 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
285 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -1.344536E-14 0.000000E+00 9.00000E+100
286 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -2.177387E-15 0.000000E+00 9.00000E+100
287 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 6.661443E-15 0.000000E+00 9.00000E+100
288 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -4.137035E-15 0.000000E+00 9.00000E+100
289 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 7.076507E-15 0.000000E+00 9.00000E+100
290 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -6.641030E-15 0.000000E+00 9.00000E+100
291 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
292 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -5.007990E-15 0.000000E+00 9.00000E+100
293 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 1.741909E-15 0.000000E+00 9.00000E+100
294 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 -1.197563E-15 0.000000E+00 9.00000E+100
295 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 1.088693E-15 0.000000E+00 9.00000E+100
296 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 1.741909E-15 0.000000E+00 9.00000E+100
297 traj.phase0.continuity_comp.defect_control_rates:beta_rate e 0.000000E+00 4.354774E-16 0.000000E+00 9.00000E+100
298 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
299 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -5.551115E-17 0.000000E+00 9.00000E+100
300 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
301 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 1.110223E-16 0.000000E+00 9.00000E+100
302 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -1.665335E-16 0.000000E+00 9.00000E+100
303 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 5.551115E-17 0.000000E+00 9.00000E+100
304 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -1.110223E-16 0.000000E+00 9.00000E+100
305 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -1.110223E-16 0.000000E+00 9.00000E+100
306 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 5.551115E-17 0.000000E+00 9.00000E+100
307 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
308 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -5.551115E-17 0.000000E+00 9.00000E+100
309 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
310 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 5.551115E-17 0.000000E+00 9.00000E+100
311 traj.phase0.continuity_comp.defect_controls:alpha e 0.000000E+00 -5.551115E-17 0.000000E+00 9.00000E+100
312 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
313 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
314 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -2.220446E-16 0.000000E+00 9.00000E+100
315 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
316 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -2.220446E-16 0.000000E+00 9.00000E+100
317 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 3.330669E-16 0.000000E+00 9.00000E+100
318 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -1.110223E-16 0.000000E+00 9.00000E+100
319 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -1.110223E-16 0.000000E+00 9.00000E+100
320 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -2.220446E-16 0.000000E+00 9.00000E+100
321 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -5.551115E-17 0.000000E+00 9.00000E+100
322 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
323 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 5.551115E-17 0.000000E+00 9.00000E+100
324 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
325 traj.phase0.continuity_comp.defect_controls:beta e 0.000000E+00 -2.775558E-17 0.000000E+00 9.00000E+100
Exit Status
Inform Description
0 Optimization terminated successfully.
--------------------------------------------------------------------------------
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/group.py:1137: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:alpha_rate, ode_eval.control_interp.control_rates:alpha_rate2, ode_eval.control_interp.control_rates:beta_rate, ode_eval.control_interp.control_rates:beta_rate2, ode_eval.control_interp.control_values:alpha, ode_eval.control_interp.control_values:beta] cannot be impacted by the design variables of the problem because no partials were defined for them in their parent component(s).
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
Problem: problem
Driver: pyOptSparseDriver
success : True
iterations : 275
runtime : 2.3386E+01 s
model_evals : 275
model_time : 2.2753E-01 s
deriv_evals : 141
deriv_time : 2.2577E+00 s
exit_status : SUCCESS
sol = om.CaseReader(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')
plot_results([('traj.phase0.timeseries.time', 'traj.phase0.timeseries.alpha',
'time (s)', 'alpha (rad)'),
('traj.phase0.timeseries.time', 'traj.phase0.timeseries.beta',
'time (s)', 'beta (rad)'),
('traj.phase0.timeseries.time', 'traj.phase0.timeseries.theta',
'time (s)', 'theta (rad)'),
('traj.phase0.timeseries.time', 'traj.phase0.timeseries.q',
'time (s)', 'q (Btu/ft**2/s)')], title='Reentry Solution', p_sol=sol,
p_sim=sim)
plt.show()
References#
[Bet10]
J. Betts. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Society for Industrial and Applied Mathematics, second edition, 2010. URL: https://epubs.siam.org/doi/abs/10.1137/1.9780898718577, arXiv:https://epubs.siam.org/doi/pdf/10.1137/1.9780898718577, doi:10.1137/1.9780898718577.