Supersonic Interceptor Minimum Time Climb#

This example is based on the A/C Min Time to Climb example given in chapter 4 of Bryson [Bry99]. It finds the angle-of-attack history required to accelerate a supersonic interceptor from near ground level, Mach 0.4 to an altitude of 20 km and Mach 1.0.

The free-body-diagram of the min-time-climb problem.

The vehicle dynamics are given by

(62)#\[\begin{align} \frac{dv}{dt} &= \frac{T}{m} \cos \alpha - \frac{D}{m} - g \sin \gamma \\ \frac{d\gamma}{dt} &= \frac{T}{m v} \sin \alpha + \frac{L}{m v} - \frac{g \cos \gamma}{v} \\ \frac{dh}{dt} &= v \sin \gamma \\ \frac{dr}{dt} &= v \cos \gamma \\ \frac{dm}{dt} &= - \frac{T}{g I_{sp}} \end{align}\]

The initial conditions are

(63)#\[\begin{align} r_0 &= 0 \rm{\,m} \\ h_0 &= 100 \rm{\,m} \\ v_0 &= 135.964 \rm{\,m/s} \\ \gamma_0 &= 0 \rm{\,deg} \\ m_0 &= 19030.468 \rm{\,kg} \end{align}\]

and the final conditions are

(64)#\[\begin{align} h_f &= 20000 \rm{\,m} \\ M_f &= 1.0 \\ \gamma_0 &= 0 \rm{\,deg} \end{align}\]

The ODE System: min_time_climb_ode.py#

The top level ODE definition is a Group that connects several subsystems.

Hide code cell outputs
import openmdao.api as om
from .dynamic_pressure_comp_partial_coloring import DynamicPressureCompFD
from ..aero.dynamic_pressure_comp import DynamicPressureComp
from ..aero.lift_drag_force_comp import LiftDragForceComp
from ..aero.cd0_comp import CD0Comp
from ..aero.kappa_comp import KappaComp
from ..aero.cla_comp import CLaComp
from ..aero.cl_comp import CLComp
from ..aero.cd_comp import CDComp
from ..aero.mach_comp import MachComp


class AeroGroup(om.Group):
    """
    The purpose of the AeroGroup is to compute the aerodynamic forces on the
    aircraft in the body frame.

    Parameters
    ----------
    v : float
        air-relative velocity (m/s)
    sos : float
        local speed of sound (m/s)
    rho : float
        atmospheric density (kg/m**3)
    alpha : float
        angle of attack (rad)
    S : float
        aerodynamic reference area (m**2)

    """
    def initialize(self):
        self.options.declare('num_nodes', types=int,
                             desc='Number of nodes to be evaluated in the RHS')
        self.options.declare('fd', types=bool, default=False, desc='If True, use fd for partials for dynamic pressure')
        self.options.declare('partial_coloring', types=bool, default=False,
                             desc='If True and fd is True, color the approximated partials of dynamic pressure')

    def setup(self):
        nn = self.options['num_nodes']

        if self.options['fd']:
            q_comp = DynamicPressureCompFD(num_nodes=nn, partial_coloring=self.options['partial_coloring'])
        else:
            q_comp = DynamicPressureComp(num_nodes=nn)

        self.add_subsystem(name='mach_comp',
                           subsys=MachComp(num_nodes=nn),
                           promotes_inputs=['v', 'sos'],
                           promotes_outputs=['mach'])

        self.add_subsystem(name='cd0_comp',
                           subsys=CD0Comp(num_nodes=nn),
                           promotes_inputs=['mach'],
                           promotes_outputs=['CD0'])

        self.add_subsystem(name='kappa_comp',
                           subsys=KappaComp(num_nodes=nn),
                           promotes_inputs=['mach'],
                           promotes_outputs=['kappa'])

        self.add_subsystem(name='cla_comp',
                           subsys=CLaComp(num_nodes=nn),
                           promotes_inputs=['mach'],
                           promotes_outputs=['CLa'])

        self.add_subsystem(name='CL_comp',
                           subsys=CLComp(num_nodes=nn),
                           promotes_inputs=['alpha', 'CLa'],
                           promotes_outputs=['CL'])

        self.add_subsystem(name='CD_comp',
                           subsys=CDComp(num_nodes=nn),
                           promotes_inputs=['CD0', 'alpha', 'CLa', 'kappa'],
                           promotes_outputs=['CD'])

        self.add_subsystem(name='q_comp',
                           subsys=q_comp,
                           promotes_inputs=['rho', 'v'],
                           promotes_outputs=['q'])

        self.add_subsystem(name='lift_drag_force_comp',
                           subsys=LiftDragForceComp(num_nodes=nn),
                           promotes_inputs=['CL', 'CD', 'q', 'S'],
                           promotes_outputs=['f_lift', 'f_drag'])
import openmdao.api as om
from dymos.models.eom import FlightPathEOM2D
from dymos.examples.min_time_climb.prop import PropGroup
from dymos.models.atmosphere import USatm1976Comp
from dymos.examples.min_time_climb.doc.aero_partial_coloring import AeroGroup

class MinTimeClimbODE(om.Group):

    def initialize(self):
        self.options.declare('num_nodes', types=int)
        self.options.declare('fd', types=bool, default=False, desc='If True, use fd for partials')
        self.options.declare('partial_coloring', types=bool, default=False,
                             desc='If True and fd is True, color the approximated partials')

    def setup(self):
        nn = self.options['num_nodes']

        self.add_subsystem(name='atmos',
                           subsys=USatm1976Comp(num_nodes=nn, h_def='geodetic'),
                           promotes_inputs=['h'])

        self.add_subsystem(name='aero',
                           subsys=AeroGroup(num_nodes=nn,
                                            fd=self.options['fd'],
                                            partial_coloring=self.options['partial_coloring']),
                           promotes_inputs=['v', 'alpha', 'S'])

        self.connect('atmos.sos', 'aero.sos')
        self.connect('atmos.rho', 'aero.rho')

        self.add_subsystem(name='prop',
                           subsys=PropGroup(num_nodes=nn),
                           promotes_inputs=['h', 'Isp', 'throttle'])

        self.connect('aero.mach', 'prop.mach')

        self.add_subsystem(name='flight_dynamics',
                           subsys=FlightPathEOM2D(num_nodes=nn),
                           promotes_inputs=['m', 'v', 'gam', 'alpha'])

        self.connect('aero.f_drag', 'flight_dynamics.D')
        self.connect('aero.f_lift', 'flight_dynamics.L')
        self.connect('prop.thrust', 'flight_dynamics.T')

Building and running the problem#

In the following code we follow the following process to solve the problem:

import matplotlib.pyplot as plt

import openmdao.api as om

import dymos as dm
from dymos.examples.plotting import plot_results

#
# Instantiate the problem and configure the optimization driver
#
p = om.Problem(model=om.Group())

p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()

#
# Instantiate the trajectory and phase
#
traj = dm.Trajectory()

phase = dm.Phase(ode_class=MinTimeClimbODE,
                 transcription=dm.GaussLobatto(num_segments=15, compressed=False))

traj.add_phase('phase0', phase)

p.model.add_subsystem('traj', traj)

#
# Set the options on the optimization variables
# Note the use of explicit state units here since much of the ODE uses imperial units
# and we prefer to solve this problem using metric units.
#
phase.set_time_options(fix_initial=True, duration_bounds=(50, 400),
                       duration_ref=100.0)

phase.add_state('r', fix_initial=True, lower=0, upper=1.0E6, units='m',
                ref=1.0E3, defect_ref=1.0E3,
                rate_source='flight_dynamics.r_dot')

phase.add_state('h', fix_initial=True, lower=0, upper=20000.0, units='m',
                ref=1.0E2, defect_ref=1.0E2,
                rate_source='flight_dynamics.h_dot')

phase.add_state('v', fix_initial=True, lower=10.0, units='m/s',
                ref=1.0E2, defect_ref=1.0E2,
                rate_source='flight_dynamics.v_dot')

phase.add_state('gam', fix_initial=True, lower=-1.5, upper=1.5, units='rad',
                ref=1.0, defect_ref=1.0,
                rate_source='flight_dynamics.gam_dot')

phase.add_state('m', fix_initial=True, lower=10.0, upper=1.0E5, units='kg',
                ref=1.0E3, defect_ref=1.0E3,
                rate_source='prop.m_dot')

phase.add_control('alpha', units='deg', lower=-8.0, upper=8.0, scaler=1.0,
                  rate_continuity=True, rate_continuity_scaler=100.0,
                  rate2_continuity=False)

phase.add_parameter('S', val=49.2386, units='m**2', opt=False, targets=['S'])
phase.add_parameter('Isp', val=1600.0, units='s', opt=False, targets=['Isp'])
phase.add_parameter('throttle', val=1.0, opt=False, targets=['throttle'])

#
# Setup the boundary and path constraints
#
phase.add_boundary_constraint('h', loc='final', equals=20000, scaler=1.0E-3)
phase.add_boundary_constraint('aero.mach', loc='final', equals=1.0)
phase.add_boundary_constraint('gam', loc='final', equals=0.0)

phase.add_path_constraint(name='h', lower=100.0, upper=20000, ref=20000)
phase.add_path_constraint(name='aero.mach', lower=0.1, upper=1.8)

# Minimize time at the end of the phase
phase.add_objective('time', loc='final', ref=1.0)

p.model.linear_solver = om.DirectSolver()

#
# Setup the problem and set the initial guess
#
p.setup(check=True)

p['traj.phase0.t_initial'] = 0.0
p['traj.phase0.t_duration'] = 500

p.set_val('traj.phase0.states:r', phase.interp('r', [0.0, 50000.0]))
p.set_val('traj.phase0.states:h', phase.interp('h', [100.0, 20000.0]))
p.set_val('traj.phase0.states:v', phase.interp('v', [135.964, 283.159]))
p.set_val('traj.phase0.states:gam', phase.interp('gam', [0.0, 0.0]))
p.set_val('traj.phase0.states:m', phase.interp('m', [19030.468, 10000.]))
p.set_val('traj.phase0.controls:alpha', phase.interp('alpha', [0.0, 0.0]))

#
# Solve for the optimal trajectory
#
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
    --- phase0 ---
        [final]   2.0000e+04 == h [m]
        [final]   1.0000e+00 == aero.mach [None]
        [final]   0.0000e+00 == gam [rad]
        [path]    1.0000e+02 <= h <= 2.0000e+04  [m]
        [path]    1.0000e-01 <= aero.mach <= 1.8000e+00  [None]

INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
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
WARNING: The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/driver.py:466: DriverWarning:The following design variable initial conditions are out of their specified bounds:
  traj.phase0.t_duration
    val: [500.]
    lower: 50.0
    upper: 400.0
Set the initial value of the design variable to a valid value or set the driver option['invalid_desvar_behavior'] to 'ignore'.
Full total jacobian for problem 'problem' was computed 3 times, taking 0.08966309599998112 seconds.
Total jacobian shape: (181, 191) 


Jacobian shape: (181, 191)  (5.28% nonzero)
FWD solves: 14   REV solves: 0
Total colors vs. total size: 14 vs 191  (92.67% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity:   0.0897 sec
Time to compute coloring:   0.1162 sec
Memory to compute coloring:   0.1250 MB
Coloring created on: 2024-05-22 13:41:13
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/total_jac.py:1635: DerivativesWarning:Constraints or objectives [('traj.phases.phase0->path_constraint->h', inds=[(0, 0)]), ('traj.phases.phase0->path_constraint->mach', inds=[(0, 0)])] cannot be impacted by the design variables of the problem.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
    Objective Function: _objfunc

    Solution: 
--------------------------------------------------------------------------------
    Total Time:                    5.4995
       User Objective Time :       1.2473
       User Sensitivity Time :     0.9178
       Interface Time :            0.5889
       Opt Solver Time:            2.7455
    Calls to Objective Function :     125
    Calls to Sens Function :          103


   Objectives
      Index  Name                     Value
          0  traj.phase0.t     3.252583E+02

   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-01     3.252583E+00     4.000000E+00           
          1  traj.phase0.states:r_0             c     0.000000E+00     4.337706E+00     1.000000E+03           
          2  traj.phase0.states:r_1             c     0.000000E+00     4.337706E+00     1.000000E+03           
          3  traj.phase0.states:r_2             c     0.000000E+00     1.031184E+01     1.000000E+03           
          4  traj.phase0.states:r_3             c     0.000000E+00     1.031184E+01     1.000000E+03           
          5  traj.phase0.states:r_4             c     0.000000E+00     1.587221E+01     1.000000E+03           
          6  traj.phase0.states:r_5             c     0.000000E+00     1.587221E+01     1.000000E+03           
          7  traj.phase0.states:r_6             c     0.000000E+00     2.161221E+01     1.000000E+03           
          8  traj.phase0.states:r_7             c     0.000000E+00     2.161221E+01     1.000000E+03           
          9  traj.phase0.states:r_8             c     0.000000E+00     2.800703E+01     1.000000E+03           
         10  traj.phase0.states:r_9             c     0.000000E+00     2.800703E+01     1.000000E+03           
         11  traj.phase0.states:r_10            c     0.000000E+00     3.569230E+01     1.000000E+03           
         12  traj.phase0.states:r_11            c     0.000000E+00     3.569230E+01     1.000000E+03           
         13  traj.phase0.states:r_12            c     0.000000E+00     4.461248E+01     1.000000E+03           
         14  traj.phase0.states:r_13            c     0.000000E+00     4.461248E+01     1.000000E+03           
         15  traj.phase0.states:r_14            c     0.000000E+00     5.396205E+01     1.000000E+03           
         16  traj.phase0.states:r_15            c     0.000000E+00     5.396205E+01     1.000000E+03           
         17  traj.phase0.states:r_16            c     0.000000E+00     6.365729E+01     1.000000E+03           
         18  traj.phase0.states:r_17            c     0.000000E+00     6.365729E+01     1.000000E+03           
         19  traj.phase0.states:r_18            c     0.000000E+00     7.372225E+01     1.000000E+03           
         20  traj.phase0.states:r_19            c     0.000000E+00     7.372225E+01     1.000000E+03           
         21  traj.phase0.states:r_20            c     0.000000E+00     8.422625E+01     1.000000E+03           
         22  traj.phase0.states:r_21            c     0.000000E+00     8.422625E+01     1.000000E+03           
         23  traj.phase0.states:r_22            c     0.000000E+00     9.527842E+01     1.000000E+03           
         24  traj.phase0.states:r_23            c     0.000000E+00     9.527842E+01     1.000000E+03           
         25  traj.phase0.states:r_24            c     0.000000E+00     1.058989E+02     1.000000E+03           
         26  traj.phase0.states:r_25            c     0.000000E+00     1.058989E+02     1.000000E+03           
         27  traj.phase0.states:r_26            c     0.000000E+00     1.136098E+02     1.000000E+03           
         28  traj.phase0.states:r_27            c     0.000000E+00     1.136098E+02     1.000000E+03           
         29  traj.phase0.states:r_28            c     0.000000E+00     1.199409E+02     1.000000E+03           
         30  traj.phase0.states:h_0             c     0.000000E+00     2.446029E+00     2.000000E+02           
         31  traj.phase0.states:h_1             c     0.000000E+00     2.446029E+00     2.000000E+02           
         32  traj.phase0.states:h_2             c     0.000000E+00     2.523966E+01     2.000000E+02           
         33  traj.phase0.states:h_3             c     0.000000E+00     2.523966E+01     2.000000E+02           
         34  traj.phase0.states:h_4             c     0.000000E+00     5.979602E+01     2.000000E+02           
         35  traj.phase0.states:h_5             c     0.000000E+00     5.979602E+01     2.000000E+02           
         36  traj.phase0.states:h_6             c     0.000000E+00     8.469376E+01     2.000000E+02           
         37  traj.phase0.states:h_7             c     0.000000E+00     8.469376E+01     2.000000E+02           
         38  traj.phase0.states:h_8             c     0.000000E+00     8.840914E+01     2.000000E+02           
         39  traj.phase0.states:h_9             c     0.000000E+00     8.840914E+01     2.000000E+02           
         40  traj.phase0.states:h_10            c     0.000000E+00     7.334853E+01     2.000000E+02           
         41  traj.phase0.states:h_11            c     0.000000E+00     7.334853E+01     2.000000E+02           
         42  traj.phase0.states:h_12            c     0.000000E+00     7.147469E+01     2.000000E+02           
         43  traj.phase0.states:h_13            c     0.000000E+00     7.147469E+01     2.000000E+02           
         44  traj.phase0.states:h_14            c     0.000000E+00     7.714455E+01     2.000000E+02           
         45  traj.phase0.states:h_15            c     0.000000E+00     7.714455E+01     2.000000E+02           
         46  traj.phase0.states:h_16            c     0.000000E+00     8.258633E+01     2.000000E+02           
         47  traj.phase0.states:h_17            c     0.000000E+00     8.258633E+01     2.000000E+02           
         48  traj.phase0.states:h_18            c     0.000000E+00     8.812291E+01     2.000000E+02           
         49  traj.phase0.states:h_19            c     0.000000E+00     8.812291E+01     2.000000E+02           
         50  traj.phase0.states:h_20            c     0.000000E+00     9.138648E+01     2.000000E+02           
         51  traj.phase0.states:h_21            c     0.000000E+00     9.138648E+01     2.000000E+02           
         52  traj.phase0.states:h_22            c     0.000000E+00     9.361784E+01     2.000000E+02           
         53  traj.phase0.states:h_23            c     0.000000E+00     9.361784E+01     2.000000E+02           
         54  traj.phase0.states:h_24            c     0.000000E+00     1.207569E+02     2.000000E+02           
         55  traj.phase0.states:h_25            c     0.000000E+00     1.207569E+02     2.000000E+02           
         56  traj.phase0.states:h_26            c     0.000000E+00     1.726189E+02     2.000000E+02           
         57  traj.phase0.states:h_27            c     0.000000E+00     1.726189E+02     2.000000E+02           
         58  traj.phase0.states:h_28            c     0.000000E+00     2.000000E+02     2.000000E+02          u
         59  traj.phase0.states:v_0             c     1.000000E-01     2.633882E+00     1.000000E+19           
         60  traj.phase0.states:v_1             c     1.000000E-01     2.633882E+00     1.000000E+19           
         61  traj.phase0.states:v_2             c     1.000000E-01     3.104671E+00     1.000000E+19           
         62  traj.phase0.states:v_3             c     1.000000E-01     3.104671E+00     1.000000E+19           
         63  traj.phase0.states:v_4             c     1.000000E-01     2.949158E+00     1.000000E+19           
         64  traj.phase0.states:v_5             c     1.000000E-01     2.949158E+00     1.000000E+19           
         65  traj.phase0.states:v_6             c     1.000000E-01     2.842674E+00     1.000000E+19           
         66  traj.phase0.states:v_7             c     1.000000E-01     2.842674E+00     1.000000E+19           
         67  traj.phase0.states:v_8             c     1.000000E-01     3.248801E+00     1.000000E+19           
         68  traj.phase0.states:v_9             c     1.000000E-01     3.248801E+00     1.000000E+19           
         69  traj.phase0.states:v_10            c     1.000000E-01     3.941025E+00     1.000000E+19           
         70  traj.phase0.states:v_11            c     1.000000E-01     3.941025E+00     1.000000E+19           
         71  traj.phase0.states:v_12            c     1.000000E-01     4.243043E+00     1.000000E+19           
         72  traj.phase0.states:v_13            c     1.000000E-01     4.243043E+00     1.000000E+19           
         73  traj.phase0.states:v_14            c     1.000000E-01     4.395001E+00     1.000000E+19           
         74  traj.phase0.states:v_15            c     1.000000E-01     4.395001E+00     1.000000E+19           
         75  traj.phase0.states:v_16            c     1.000000E-01     4.563111E+00     1.000000E+19           
         76  traj.phase0.states:v_17            c     1.000000E-01     4.563111E+00     1.000000E+19           
         77  traj.phase0.states:v_18            c     1.000000E-01     4.737651E+00     1.000000E+19           
         78  traj.phase0.states:v_19            c     1.000000E-01     4.737651E+00     1.000000E+19           
         79  traj.phase0.states:v_20            c     1.000000E-01     4.969663E+00     1.000000E+19           
         80  traj.phase0.states:v_21            c     1.000000E-01     4.969663E+00     1.000000E+19           
         81  traj.phase0.states:v_22            c     1.000000E-01     5.194006E+00     1.000000E+19           
         82  traj.phase0.states:v_23            c     1.000000E-01     5.194006E+00     1.000000E+19           
         83  traj.phase0.states:v_24            c     1.000000E-01     4.826572E+00     1.000000E+19           
         84  traj.phase0.states:v_25            c     1.000000E-01     4.826572E+00     1.000000E+19           
         85  traj.phase0.states:v_26            c     1.000000E-01     3.708917E+00     1.000000E+19           
         86  traj.phase0.states:v_27            c     1.000000E-01     3.708917E+00     1.000000E+19           
         87  traj.phase0.states:v_28            c     1.000000E-01     2.950719E+00     1.000000E+19           
         88  traj.phase0.states:gam_0           c    -1.500000E+00     1.014493E-01     1.500000E+00           
         89  traj.phase0.states:gam_1           c    -1.500000E+00     1.014493E-01     1.500000E+00           
         90  traj.phase0.states:gam_2           c    -1.500000E+00     5.979443E-01     1.500000E+00           
         91  traj.phase0.states:gam_3           c    -1.500000E+00     5.979443E-01     1.500000E+00           
         92  traj.phase0.states:gam_4           c    -1.500000E+00     4.557325E-01     1.500000E+00           
         93  traj.phase0.states:gam_5           c    -1.500000E+00     4.557325E-01     1.500000E+00           
         94  traj.phase0.states:gam_6           c    -1.500000E+00     3.191609E-01     1.500000E+00           
         95  traj.phase0.states:gam_7           c    -1.500000E+00     3.191609E-01     1.500000E+00           
         96  traj.phase0.states:gam_8           c    -1.500000E+00    -1.775656E-01     1.500000E+00           
         97  traj.phase0.states:gam_9           c    -1.500000E+00    -1.775656E-01     1.500000E+00           
         98  traj.phase0.states:gam_10          c    -1.500000E+00    -1.196090E-01     1.500000E+00           
         99  traj.phase0.states:gam_11          c    -1.500000E+00    -1.196090E-01     1.500000E+00           
        100  traj.phase0.states:gam_12          c    -1.500000E+00     4.688152E-02     1.500000E+00           
        101  traj.phase0.states:gam_13          c    -1.500000E+00     4.688152E-02     1.500000E+00           
        102  traj.phase0.states:gam_14          c    -1.500000E+00     5.957514E-02     1.500000E+00           
        103  traj.phase0.states:gam_15          c    -1.500000E+00     5.957514E-02     1.500000E+00           
        104  traj.phase0.states:gam_16          c    -1.500000E+00     5.533961E-02     1.500000E+00           
        105  traj.phase0.states:gam_17          c    -1.500000E+00     5.533961E-02     1.500000E+00           
        106  traj.phase0.states:gam_18          c    -1.500000E+00     4.987172E-02     1.500000E+00           
        107  traj.phase0.states:gam_19          c    -1.500000E+00     4.987172E-02     1.500000E+00           
        108  traj.phase0.states:gam_20          c    -1.500000E+00     1.063274E-02     1.500000E+00           
        109  traj.phase0.states:gam_21          c    -1.500000E+00     1.063274E-02     1.500000E+00           
        110  traj.phase0.states:gam_22          c    -1.500000E+00     7.519137E-02     1.500000E+00           
        111  traj.phase0.states:gam_23          c    -1.500000E+00     7.519137E-02     1.500000E+00           
        112  traj.phase0.states:gam_24          c    -1.500000E+00     4.620378E-01     1.500000E+00           
        113  traj.phase0.states:gam_25          c    -1.500000E+00     4.620378E-01     1.500000E+00           
        114  traj.phase0.states:gam_26          c    -1.500000E+00     6.283286E-01     1.500000E+00           
        115  traj.phase0.states:gam_27          c    -1.500000E+00     6.283286E-01     1.500000E+00           
        116  traj.phase0.states:gam_28          c    -1.500000E+00     0.000000E+00     1.500000E+00           
        117  traj.phase0.states:m_0             c     1.000000E-02     1.883945E+01     1.000000E+02           
        118  traj.phase0.states:m_1             c     1.000000E-02     1.883945E+01     1.000000E+02           
        119  traj.phase0.states:m_2             c     1.000000E-02     1.863480E+01     1.000000E+02           
        120  traj.phase0.states:m_3             c     1.000000E-02     1.863480E+01     1.000000E+02           
        121  traj.phase0.states:m_4             c     1.000000E-02     1.847308E+01     1.000000E+02           
        122  traj.phase0.states:m_5             c     1.000000E-02     1.847308E+01     1.000000E+02           
        123  traj.phase0.states:m_6             c     1.000000E-02     1.835400E+01     1.000000E+02           
        124  traj.phase0.states:m_7             c     1.000000E-02     1.835400E+01     1.000000E+02           
        125  traj.phase0.states:m_8             c     1.000000E-02     1.825101E+01     1.000000E+02           
        126  traj.phase0.states:m_9             c     1.000000E-02     1.825101E+01     1.000000E+02           
        127  traj.phase0.states:m_10            c     1.000000E-02     1.811459E+01     1.000000E+02           
        128  traj.phase0.states:m_11            c     1.000000E-02     1.811459E+01     1.000000E+02           
        129  traj.phase0.states:m_12            c     1.000000E-02     1.794316E+01     1.000000E+02           
        130  traj.phase0.states:m_13            c     1.000000E-02     1.794316E+01     1.000000E+02           
        131  traj.phase0.states:m_14            c     1.000000E-02     1.776707E+01     1.000000E+02           
        132  traj.phase0.states:m_15            c     1.000000E-02     1.776707E+01     1.000000E+02           
        133  traj.phase0.states:m_16            c     1.000000E-02     1.759239E+01     1.000000E+02           
        134  traj.phase0.states:m_17            c     1.000000E-02     1.759239E+01     1.000000E+02           
        135  traj.phase0.states:m_18            c     1.000000E-02     1.741882E+01     1.000000E+02           
        136  traj.phase0.states:m_19            c     1.000000E-02     1.741882E+01     1.000000E+02           
        137  traj.phase0.states:m_20            c     1.000000E-02     1.724476E+01     1.000000E+02           
        138  traj.phase0.states:m_21            c     1.000000E-02     1.724476E+01     1.000000E+02           
        139  traj.phase0.states:m_22            c     1.000000E-02     1.706462E+01     1.000000E+02           
        140  traj.phase0.states:m_23            c     1.000000E-02     1.706462E+01     1.000000E+02           
        141  traj.phase0.states:m_24            c     1.000000E-02     1.690433E+01     1.000000E+02           
        142  traj.phase0.states:m_25            c     1.000000E-02     1.690433E+01     1.000000E+02           
        143  traj.phase0.states:m_26            c     1.000000E-02     1.682842E+01     1.000000E+02           
        144  traj.phase0.states:m_27            c     1.000000E-02     1.682842E+01     1.000000E+02           
        145  traj.phase0.states:m_28            c     1.000000E-02     1.680487E+01     1.000000E+02           
        146  traj.phase0.controls:alpha_0       c    -8.000000E+00     5.400658E+00     8.000000E+00           
        147  traj.phase0.controls:alpha_1       c    -8.000000E+00     2.645922E+00     8.000000E+00           
        148  traj.phase0.controls:alpha_2       c    -8.000000E+00     2.027262E+00     8.000000E+00           
        149  traj.phase0.controls:alpha_3       c    -8.000000E+00     2.027262E+00     8.000000E+00           
        150  traj.phase0.controls:alpha_4       c    -8.000000E+00     2.080183E+00     8.000000E+00           
        151  traj.phase0.controls:alpha_5       c    -8.000000E+00     1.340193E+00     8.000000E+00           
        152  traj.phase0.controls:alpha_6       c    -8.000000E+00     1.340193E+00     8.000000E+00           
        153  traj.phase0.controls:alpha_7       c    -8.000000E+00     7.773049E-01     8.000000E+00           
        154  traj.phase0.controls:alpha_8       c    -8.000000E+00     1.361534E+00     8.000000E+00           
        155  traj.phase0.controls:alpha_9       c    -8.000000E+00     1.361534E+00     8.000000E+00           
        156  traj.phase0.controls:alpha_10      c    -8.000000E+00     1.892584E+00     8.000000E+00           
        157  traj.phase0.controls:alpha_11      c    -8.000000E+00     1.170161E+00     8.000000E+00           
        158  traj.phase0.controls:alpha_12      c    -8.000000E+00     1.170161E+00     8.000000E+00           
        159  traj.phase0.controls:alpha_13      c    -8.000000E+00     4.830056E-01     8.000000E+00           
        160  traj.phase0.controls:alpha_14      c    -8.000000E+00     1.119857E+00     8.000000E+00           
        161  traj.phase0.controls:alpha_15      c    -8.000000E+00     1.119857E+00     8.000000E+00           
        162  traj.phase0.controls:alpha_16      c    -8.000000E+00     1.997080E+00     8.000000E+00           
        163  traj.phase0.controls:alpha_17      c    -8.000000E+00     2.031038E+00     8.000000E+00           
        164  traj.phase0.controls:alpha_18      c    -8.000000E+00     2.031038E+00     8.000000E+00           
        165  traj.phase0.controls:alpha_19      c    -8.000000E+00     1.681271E+00     8.000000E+00           
        166  traj.phase0.controls:alpha_20      c    -8.000000E+00     1.407316E+00     8.000000E+00           
        167  traj.phase0.controls:alpha_21      c    -8.000000E+00     1.407316E+00     8.000000E+00           
        168  traj.phase0.controls:alpha_22      c    -8.000000E+00     1.239354E+00     8.000000E+00           
        169  traj.phase0.controls:alpha_23      c    -8.000000E+00     1.207563E+00     8.000000E+00           
        170  traj.phase0.controls:alpha_24      c    -8.000000E+00     1.207563E+00     8.000000E+00           
        171  traj.phase0.controls:alpha_25      c    -8.000000E+00     1.240086E+00     8.000000E+00           
        172  traj.phase0.controls:alpha_26      c    -8.000000E+00     1.265067E+00     8.000000E+00           
        173  traj.phase0.controls:alpha_27      c    -8.000000E+00     1.265067E+00     8.000000E+00           
        174  traj.phase0.controls:alpha_28      c    -8.000000E+00     1.260861E+00     8.000000E+00           
        175  traj.phase0.controls:alpha_29      c    -8.000000E+00     1.205823E+00     8.000000E+00           
        176  traj.phase0.controls:alpha_30      c    -8.000000E+00     1.205823E+00     8.000000E+00           
        177  traj.phase0.controls:alpha_31      c    -8.000000E+00     1.150393E+00     8.000000E+00           
        178  traj.phase0.controls:alpha_32      c    -8.000000E+00     1.145010E+00     8.000000E+00           
        179  traj.phase0.controls:alpha_33      c    -8.000000E+00     1.145010E+00     8.000000E+00           
        180  traj.phase0.controls:alpha_34      c    -8.000000E+00     1.350321E+00     8.000000E+00           
        181  traj.phase0.controls:alpha_35      c    -8.000000E+00     1.926970E+00     8.000000E+00           
        182  traj.phase0.controls:alpha_36      c    -8.000000E+00     1.926970E+00     8.000000E+00           
        183  traj.phase0.controls:alpha_37      c    -8.000000E+00     2.729913E+00     8.000000E+00           
        184  traj.phase0.controls:alpha_38      c    -8.000000E+00     3.614100E+00     8.000000E+00           
        185  traj.phase0.controls:alpha_39      c    -8.000000E+00     3.614100E+00     8.000000E+00           
        186  traj.phase0.controls:alpha_40      c    -8.000000E+00     3.938262E+00     8.000000E+00           
        187  traj.phase0.controls:alpha_41      c    -8.000000E+00     3.061127E+00     8.000000E+00           
        188  traj.phase0.controls:alpha_42      c    -8.000000E+00     3.061127E+00     8.000000E+00           
        189  traj.phase0.controls:alpha_43      c    -8.000000E+00    -9.747930E-02     8.000000E+00           
        190  traj.phase0.controls:alpha_44      c    -8.000000E+00    -6.617732E+00     8.000000E+00           

   Constraints (i - inequality, e - equality)
      Index  Name                                                        Type          Lower           Value           Upper    Status  Lagrange Multiplier (N/A)
          0  traj.phases.phase0->final_boundary_constraint->h               e   2.000000E+01    2.000000E+01    2.000000E+01              9.00000E+100
          1  traj.phases.phase0->final_boundary_constraint->mach            e   1.000000E+00    1.000000E+00    1.000000E+00              9.00000E+100
          2  traj.phases.phase0->final_boundary_constraint->gam             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
          3  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    9.626006E-11    0.000000E+00              9.00000E+100
          4  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    1.368485E-10    0.000000E+00              9.00000E+100
          5  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    2.190367E-11    0.000000E+00              9.00000E+100
          6  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    7.029563E-10    0.000000E+00              9.00000E+100
          7  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    2.832644E-09    0.000000E+00              9.00000E+100
          8  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    4.682115E-10    0.000000E+00              9.00000E+100
          9  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    4.226232E-09    0.000000E+00              9.00000E+100
         10  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    7.052647E-09    0.000000E+00              9.00000E+100
         11  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    5.649101E-09    0.000000E+00              9.00000E+100
         12  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    1.387491E-08    0.000000E+00              9.00000E+100
         13  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    2.312780E-09    0.000000E+00              9.00000E+100
         14  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    2.635287E-09    0.000000E+00              9.00000E+100
         15  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    1.094939E-08    0.000000E+00              9.00000E+100
         16  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    4.446381E-09    0.000000E+00              9.00000E+100
         17  traj.phase0.collocation_constraint.defects:r                   e   0.000000E+00    4.249218E-09    0.000000E+00              9.00000E+100
         18  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    5.610073E-10    0.000000E+00              9.00000E+100
         19  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    7.633561E-11    0.000000E+00              9.00000E+100
         20  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    1.025378E-08    0.000000E+00              9.00000E+100
         21  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    4.863386E-09    0.000000E+00              9.00000E+100
         22  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    8.137654E-09    0.000000E+00              9.00000E+100
         23  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    7.240722E-09    0.000000E+00              9.00000E+100
         24  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00   -7.944418E-09    0.000000E+00              9.00000E+100
         25  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    1.765343E-08    0.000000E+00              9.00000E+100
         26  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    6.359085E-08    0.000000E+00              9.00000E+100
         27  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00   -3.293200E-08    0.000000E+00              9.00000E+100
         28  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00   -7.797214E-09    0.000000E+00              9.00000E+100
         29  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    1.576673E-08    0.000000E+00              9.00000E+100
         30  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    9.957876E-08    0.000000E+00              9.00000E+100
         31  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00    6.840287E-09    0.000000E+00              9.00000E+100
         32  traj.phase0.collocation_constraint.defects:h                   e   0.000000E+00   -2.149199E-08    0.000000E+00              9.00000E+100
         33  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    2.938043E-11    0.000000E+00              9.00000E+100
         34  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    4.970220E-10    0.000000E+00              9.00000E+100
         35  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    3.110575E-10    0.000000E+00              9.00000E+100
         36  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    2.049737E-09    0.000000E+00              9.00000E+100
         37  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    5.414231E-13    0.000000E+00              9.00000E+100
         38  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    3.931003E-10    0.000000E+00              9.00000E+100
         39  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    5.179050E-10    0.000000E+00              9.00000E+100
         40  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    1.776728E-09    0.000000E+00              9.00000E+100
         41  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    4.975366E-09    0.000000E+00              9.00000E+100
         42  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    1.467696E-09    0.000000E+00              9.00000E+100
         43  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    1.765678E-09    0.000000E+00              9.00000E+100
         44  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    8.495687E-10    0.000000E+00              9.00000E+100
         45  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    2.938751E-10    0.000000E+00              9.00000E+100
         46  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    3.655916E-09    0.000000E+00              9.00000E+100
         47  traj.phase0.collocation_constraint.defects:v                   e   0.000000E+00    2.912654E-09    0.000000E+00              9.00000E+100
         48  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00    6.511938E-12    0.000000E+00              9.00000E+100
         49  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -4.033627E-10    0.000000E+00              9.00000E+100
         50  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00    1.268022E-10    0.000000E+00              9.00000E+100
         51  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -2.257090E-09    0.000000E+00              9.00000E+100
         52  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -5.075324E-10    0.000000E+00              9.00000E+100
         53  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -4.854688E-10    0.000000E+00              9.00000E+100
         54  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -6.164998E-10    0.000000E+00              9.00000E+100
         55  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -7.899697E-10    0.000000E+00              9.00000E+100
         56  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -4.984145E-09    0.000000E+00              9.00000E+100
         57  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -1.044732E-09    0.000000E+00              9.00000E+100
         58  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -1.340575E-09    0.000000E+00              9.00000E+100
         59  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -8.149447E-10    0.000000E+00              9.00000E+100
         60  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -6.668502E-10    0.000000E+00              9.00000E+100
         61  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -3.941808E-09    0.000000E+00              9.00000E+100
         62  traj.phase0.collocation_constraint.defects:gam                 e   0.000000E+00   -4.597333E-10    0.000000E+00              9.00000E+100
         63  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -7.270526E-13    0.000000E+00              9.00000E+100
         64  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -2.701155E-12    0.000000E+00              9.00000E+100
         65  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    2.392460E-12    0.000000E+00              9.00000E+100
         66  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    1.576112E-11    0.000000E+00              9.00000E+100
         67  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    3.565121E-12    0.000000E+00              9.00000E+100
         68  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    4.465064E-11    0.000000E+00              9.00000E+100
         69  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -1.964800E-12    0.000000E+00              9.00000E+100
         70  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -1.277913E-11    0.000000E+00              9.00000E+100
         71  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -1.031078E-12    0.000000E+00              9.00000E+100
         72  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -2.965965E-11    0.000000E+00              9.00000E+100
         73  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -1.112948E-11    0.000000E+00              9.00000E+100
         74  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    1.662836E-13    0.000000E+00              9.00000E+100
         75  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    5.191721E-12    0.000000E+00              9.00000E+100
         76  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    2.246183E-10    0.000000E+00              9.00000E+100
         77  traj.phase0.collocation_constraint.defects:m                   e   0.000000E+00    2.561286E-11    0.000000E+00              9.00000E+100
         78  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         79  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         80  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         81  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         82  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         83  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         84  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         85  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         86  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         87  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         88  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         89  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         90  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         91  traj.phase0.continuity_comp.defect_states:r                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         92  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         93  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         94  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         95  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         96  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         97  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         98  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         99  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        100  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        101  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        102  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        103  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        104  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        105  traj.phase0.continuity_comp.defect_states:h                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        106  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        107  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        108  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        109  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        110  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        111  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        112  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        113  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        114  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        115  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        116  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        117  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        118  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        119  traj.phase0.continuity_comp.defect_states:v                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        120  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        121  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        122  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        123  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        124  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        125  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        126  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        127  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        128  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        129  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        130  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        131  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00   -1.387779E-17    0.000000E+00              9.00000E+100
        132  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        133  traj.phase0.continuity_comp.defect_states:gam                  e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        134  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        135  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        136  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        137  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        138  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        139  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        140  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        141  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        142  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        143  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        144  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        145  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        146  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        147  traj.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        148  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00   -6.770799E-13    0.000000E+00              9.00000E+100
        149  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00   -9.027731E-13    0.000000E+00              9.00000E+100
        150  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    1.128466E-12    0.000000E+00              9.00000E+100
        151  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        152  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    9.027731E-13    0.000000E+00              9.00000E+100
        153  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    1.015620E-12    0.000000E+00              9.00000E+100
        154  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00   -2.821166E-13    0.000000E+00              9.00000E+100
        155  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    5.007570E-13    0.000000E+00              9.00000E+100
        156  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    3.314870E-13    0.000000E+00              9.00000E+100
        157  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    5.078099E-13    0.000000E+00              9.00000E+100
        158  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    1.163731E-13    0.000000E+00              9.00000E+100
        159  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00   -4.513866E-13    0.000000E+00              9.00000E+100
        160  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    6.770799E-13    0.000000E+00              9.00000E+100
        161  traj.phase0.continuity_comp.defect_control_rates:alpha_rate    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        162  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        163  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        164  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        165  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        166  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        167  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        168  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        169  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        170  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        171  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        172  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        173  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        174  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        175  traj.phase0.continuity_comp.defect_controls:alpha              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        176  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    5.000000E-03    1.000000E+00         l    9.00000E+100
        177  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    5.000000E-03    1.000000E+00         l    9.00000E+100
        178  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    1.223015E-02    1.000000E+00              9.00000E+100
        179  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    1.223015E-02    1.000000E+00              9.00000E+100
        180  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.914293E-02    1.000000E+00              9.00000E+100
        181  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    1.261983E-01    1.000000E+00              9.00000E+100
        182  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    1.261983E-01    1.000000E+00              9.00000E+100
        183  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    2.186847E-01    1.000000E+00              9.00000E+100
        184  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    2.989801E-01    1.000000E+00              9.00000E+100
        185  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    2.989801E-01    1.000000E+00              9.00000E+100
        186  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.667273E-01    1.000000E+00              9.00000E+100
        187  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.234688E-01    1.000000E+00              9.00000E+100
        188  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.234688E-01    1.000000E+00              9.00000E+100
        189  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.526223E-01    1.000000E+00              9.00000E+100
        190  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.420457E-01    1.000000E+00              9.00000E+100
        191  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.420457E-01    1.000000E+00              9.00000E+100
        192  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.029903E-01    1.000000E+00              9.00000E+100
        193  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.667426E-01    1.000000E+00              9.00000E+100
        194  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.667426E-01    1.000000E+00              9.00000E+100
        195  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.529900E-01    1.000000E+00              9.00000E+100
        196  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.573735E-01    1.000000E+00              9.00000E+100
        197  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.573735E-01    1.000000E+00              9.00000E+100
        198  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.706966E-01    1.000000E+00              9.00000E+100
        199  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.857227E-01    1.000000E+00              9.00000E+100
        200  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.857227E-01    1.000000E+00              9.00000E+100
        201  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    3.994530E-01    1.000000E+00              9.00000E+100
        202  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.129316E-01    1.000000E+00              9.00000E+100
        203  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.129316E-01    1.000000E+00              9.00000E+100
        204  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.269929E-01    1.000000E+00              9.00000E+100
        205  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.406146E-01    1.000000E+00              9.00000E+100
        206  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.406146E-01    1.000000E+00              9.00000E+100
        207  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.512581E-01    1.000000E+00              9.00000E+100
        208  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.569324E-01    1.000000E+00              9.00000E+100
        209  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.569324E-01    1.000000E+00              9.00000E+100
        210  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.579391E-01    1.000000E+00              9.00000E+100
        211  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.680892E-01    1.000000E+00              9.00000E+100
        212  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    4.680892E-01    1.000000E+00              9.00000E+100
        213  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    5.120660E-01    1.000000E+00              9.00000E+100
        214  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    6.037847E-01    1.000000E+00              9.00000E+100
        215  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    6.037847E-01    1.000000E+00              9.00000E+100
        216  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    7.330530E-01    1.000000E+00              9.00000E+100
        217  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    8.630944E-01    1.000000E+00              9.00000E+100
        218  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    8.630944E-01    1.000000E+00              9.00000E+100
        219  traj.phases.phase0->path_constraint->h                         i   5.000000E-03    9.610925E-01    1.000000E+00              9.00000E+100
        220  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    4.000084E-01    1.800000E+00              9.00000E+100
        221  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    5.901464E-01    1.800000E+00              9.00000E+100
        222  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    7.761622E-01    1.800000E+00              9.00000E+100
        223  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    7.761622E-01    1.800000E+00              9.00000E+100
        224  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.011492E-01    1.800000E+00              9.00000E+100
        225  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.394945E-01    1.800000E+00              9.00000E+100
        226  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.394945E-01    1.800000E+00              9.00000E+100
        227  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.347430E-01    1.800000E+00              9.00000E+100
        228  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.317176E-01    1.800000E+00              9.00000E+100
        229  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.317176E-01    1.800000E+00              9.00000E+100
        230  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.276321E-01    1.800000E+00              9.00000E+100
        231  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.286508E-01    1.800000E+00              9.00000E+100
        232  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.286508E-01    1.800000E+00              9.00000E+100
        233  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    9.731987E-01    1.800000E+00              9.00000E+100
        234  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.066851E+00    1.800000E+00              9.00000E+100
        235  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.066851E+00    1.800000E+00              9.00000E+100
        236  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.177153E+00    1.800000E+00              9.00000E+100
        237  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.267623E+00    1.800000E+00              9.00000E+100
        238  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.267623E+00    1.800000E+00              9.00000E+100
        239  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.324551E+00    1.800000E+00              9.00000E+100
        240  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.361332E+00    1.800000E+00              9.00000E+100
        241  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.361332E+00    1.800000E+00              9.00000E+100
        242  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.391048E+00    1.800000E+00              9.00000E+100
        243  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.420933E+00    1.800000E+00              9.00000E+100
        244  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.420933E+00    1.800000E+00              9.00000E+100
        245  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.453146E+00    1.800000E+00              9.00000E+100
        246  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.486339E+00    1.800000E+00              9.00000E+100
        247  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.486339E+00    1.800000E+00              9.00000E+100
        248  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.519865E+00    1.800000E+00              9.00000E+100
        249  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.555138E+00    1.800000E+00              9.00000E+100
        250  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.555138E+00    1.800000E+00              9.00000E+100
        251  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.594613E+00    1.800000E+00              9.00000E+100
        252  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.638818E+00    1.800000E+00              9.00000E+100
        253  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.638818E+00    1.800000E+00              9.00000E+100
        254  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.685059E+00    1.800000E+00              9.00000E+100
        255  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.718237E+00    1.800000E+00              9.00000E+100
        256  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.718237E+00    1.800000E+00              9.00000E+100
        257  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.716938E+00    1.800000E+00              9.00000E+100
        258  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.635775E+00    1.800000E+00              9.00000E+100
        259  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.635775E+00    1.800000E+00              9.00000E+100
        260  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.459678E+00    1.800000E+00              9.00000E+100
        261  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.256990E+00    1.800000E+00              9.00000E+100
        262  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.256990E+00    1.800000E+00              9.00000E+100
        263  traj.phases.phase0->path_constraint->mach                      i   1.000000E-01    1.081459E+00    1.800000E+00              9.00000E+100


   Exit Status
      Inform  Description
           0  Optimization terminated successfully.
--------------------------------------------------------------------------------
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/visualization/opt_report/opt_report.py:625: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding.
  ax.set_ylim([ymin_plot, ymax_plot])
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/group.py:1098: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:alpha_rate2] 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
False
sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('dymos_simulation.db').get_case('final')

plot_results([('traj.phase0.timeseries.time', 'traj.phase0.timeseries.h',
               'time (s)', 'altitude (m)'),
              ('traj.phase0.timeseries.time', 'traj.phase0.timeseries.alpha',
               'time (s)', 'alpha (deg)')],
             title='Supersonic Minimum Time-to-Climb Solution',
             p_sol=sol, p_sim=sim)

plt.show()
../../_images/00ffa1624c352012cb4fd1540aab12a5fb0876033fa4a69e92f8a565e35a824f.png

References#

[Bry99]

Arthur Earl Bryson. Dynamic optimization. Addison Wesley Longman Menlo Park, CA, 1999.