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SSTO Earth Launch

This example is based on the Time-Optimal Launch of a Titan II example given in Appendix B of Longuski1. It finds the pitch profile for a single-stage-to-orbit launch vehicle that minimizes the time required to reach orbit insertion under constant thrust.

The free-body-diagram of the single-stage-to-orbit problem.

The vehicle dynamics are given by

\begin{align} \frac{dx}{dt} &= v_x \\ \frac{dy}{dt} &= v_y \\ \frac{dv_x}{dt} &= \frac{1}{m} (T \cos \theta - D \cos \gamma) \\ \frac{dv_y}{dt} &= \frac{1}{m} (T \sin \theta - D \sin \gamma) - g \\ \frac{dm}{dt} &= \frac{T}{g I_{sp}} \end{align}

The initial conditions are

\begin{align} x_0 &= 0 \\ y_0 &= 0 \\ v_{x0} &= 0 \\ v_{y0} &= 0 \\ m_0 &= 117000 \rm{\,kg} \end{align}

and the final conditions are

\begin{align} x_f &= \rm{free} \\ y_f &= 185 \rm{\,km} \\ v_{xf} &= V_{circ} \\ v_{yf} &= 0 \\ m_f &= \rm{free} \end{align}

Defining the ODE

Generally, one could define the ODE system as a composite group of multile components. The atmosphere component computes density (\rho). The eom component computes the state rates. Decomposing the ODE into smaller calculations makes it easier to derive the analytic derivatives.

The notional XDSM diagram for the ODE system in the SSTO problem.

However, for this example we will demonstrate the use of complex-step differentiation and define the ODE as a single component. This saves time up front in the deevlopment of the ODE at a minor cost in execution time.

The unconnected inputs to the EOM at the top of the diagram are provided by the Dymos phase as states, controls, or time values. The outputs, including the state rates, are shown on the right side of the diagram. The Dymos phases use state rate values to ensure that the integration technique satisfies the dynamics of the system.

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import openmdao.api as om
import numpy as np


class LaunchVehicleODE(om.ExplicitComponent):

    def initialize(self):
        self.options.declare('num_nodes', types=int,
                             desc='Number of nodes to be evaluated in the RHS')

        self.options.declare('g', types=float, default=9.80665,
                             desc='Gravitational acceleration, m/s**2')

        self.options.declare('rho_ref', types=float, default=1.225,
                             desc='Reference atmospheric density, kg/m**3')

        self.options.declare('h_scale', types=float, default=8.44E3,
                             desc='Reference altitude, m')

        self.options.declare('CD', types=float, default=0.5,
                             desc='coefficient of drag')

        self.options.declare('S', types=float, default=7.069,
                             desc='aerodynamic reference area (m**2)')

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

        self.add_input('y',
                       val=np.zeros(nn),
                       desc='altitude',
                       units='m')

        self.add_input('vx',
                       val=np.zeros(nn),
                       desc='x velocity',
                       units='m/s')

        self.add_input('vy',
                       val=np.zeros(nn),
                       desc='y velocity',
                       units='m/s')

        self.add_input('m',
                       val=np.zeros(nn),
                       desc='mass',
                       units='kg')

        self.add_input('theta',
                       val=np.zeros(nn),
                       desc='pitch angle',
                       units='rad')

        self.add_input('thrust',
                       val=2100000 * np.ones(nn),
                       desc='thrust',
                       units='N')

        self.add_input('Isp',
                       val=265.2 * np.ones(nn),
                       desc='specific impulse',
                       units='s')
        # Outputs
        self.add_output('xdot',
                        val=np.zeros(nn),
                        desc='velocity component in x',
                        units='m/s')

        self.add_output('ydot',
                        val=np.zeros(nn),
                        desc='velocity component in y',
                        units='m/s')

        self.add_output('vxdot',
                        val=np.zeros(nn),
                        desc='x acceleration magnitude',
                        units='m/s**2')

        self.add_output('vydot',
                        val=np.zeros(nn),
                        desc='y acceleration magnitude',
                        units='m/s**2')

        self.add_output('mdot',
                        val=np.zeros(nn),
                        desc='mass rate of change',
                        units='kg/s')

        self.add_output('rho',
                        val=np.zeros(nn),
                        desc='density',
                        units='kg/m**3')

        # Setup partials
        # Complex-step derivatives
        self.declare_coloring(wrt='*', method='cs', show_sparsity=True)

    def compute(self, inputs, outputs):

        theta = inputs['theta']
        cos_theta = np.cos(theta)
        sin_theta = np.sin(theta)
        vx = inputs['vx']
        vy = inputs['vy']
        m = inputs['m']
        F_T = inputs['thrust']
        Isp = inputs['Isp']
        y = inputs['y']

        g = self.options['g']
        rho_ref = self.options['rho_ref']
        h_scale = self.options['h_scale']

        CDA = self.options['CD'] * self.options['S']

        outputs['rho'] = rho_ref * np.exp(-y / h_scale)
        outputs['xdot'] = vx
        outputs['ydot'] = vy
        outputs['vxdot'] = (F_T * cos_theta - 0.5 * CDA * outputs['rho'] * vx**2) / m
        outputs['vydot'] = (F_T * sin_theta - 0.5 * CDA * outputs['rho'] * vy**2) / m - g
        outputs['mdot'] = -F_T / (g * Isp)

Solving the problem

import matplotlib.pyplot as plt
import openmdao.api as om
import dymos as dm

#
# Setup and solve the optimal control problem
#
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring(tol=1.0E-12)

from dymos.examples.ssto.launch_vehicle_ode import LaunchVehicleODE

#
# Initialize our Trajectory and Phase
#
traj = dm.Trajectory()

phase = dm.Phase(ode_class=LaunchVehicleODE,
                 transcription=dm.GaussLobatto(num_segments=12, order=3, compressed=False))

traj.add_phase('phase0', phase)
p.model.add_subsystem('traj', traj)

#
# Set the options for the variables
#
phase.set_time_options(initial_bounds=(0, 0), duration_bounds=(10, 500))

phase.add_state('x', fix_initial=True, ref=1.0E5, defect_ref=10000.0,
                rate_source='xdot')
phase.add_state('y', fix_initial=True, ref=1.0E5, defect_ref=10000.0,
                rate_source='ydot')
phase.add_state('vx', fix_initial=True, ref=1.0E3, defect_ref=1000.0,
                rate_source='vxdot')
phase.add_state('vy', fix_initial=True, ref=1.0E3, defect_ref=1000.0,
                rate_source='vydot')
phase.add_state('m', fix_initial=True, ref=1.0E3, defect_ref=100.0,
                rate_source='mdot')

phase.add_control('theta', units='rad', lower=-1.57, upper=1.57, targets=['theta'])
phase.add_parameter('thrust', units='N', opt=False, val=2100000.0, targets=['thrust'])

#
# Set the options for our constraints and objective
#
phase.add_boundary_constraint('y', loc='final', equals=1.85E5, linear=True)
phase.add_boundary_constraint('vx', loc='final', equals=7796.6961)
phase.add_boundary_constraint('vy', loc='final', equals=0)

phase.add_objective('time', loc='final', scaler=0.01)

p.model.linear_solver = om.DirectSolver()

#
# Setup and set initial values
#
p.setup(check=True)

p.set_val('traj.phase0.t_initial', 0.0)
p.set_val('traj.phase0.t_duration', 150.0)
p.set_val('traj.phase0.states:x', phase.interpolate(ys=[0, 1.15E5], nodes='state_input'))
p.set_val('traj.phase0.states:y', phase.interpolate(ys=[0, 1.85E5], nodes='state_input'))
p.set_val('traj.phase0.states:vx', phase.interpolate(ys=[0, 7796.6961], nodes='state_input'))
p.set_val('traj.phase0.states:vy', phase.interpolate(ys=[1.0E-6, 0], nodes='state_input'))
p.set_val('traj.phase0.states:m', phase.interpolate(ys=[117000, 1163], nodes='state_input'))
p.set_val('traj.phase0.controls:theta', phase.interpolate(ys=[1.5, -0.76], nodes='control_input'))
p.set_val('traj.phase0.parameters:thrust', 2.1, units='MN')

#
# Solve the Problem
#
dm.run_problem(p)





#
# Get the explicitly simulated results
#
exp_out = traj.simulate()

#
# Plot the results
#
fig, axes = plt.subplots(nrows=2, ncols=1, figsize=(10, 8))

axes[0].plot(p.get_val('traj.phase0.timeseries.states:x'),
             p.get_val('traj.phase0.timeseries.states:y'),
             marker='o',
             ms=4,
             linestyle='None',
             label='solution')

axes[0].plot(exp_out.get_val('traj.phase0.timeseries.states:x'),
             exp_out.get_val('traj.phase0.timeseries.states:y'),
             marker=None,
             linestyle='-',
             label='simulation')

axes[0].set_xlabel('range (m)')
axes[0].set_ylabel('altitude (m)')
axes[0].set_aspect('equal')

axes[1].plot(p.get_val('traj.phase0.timeseries.time'),
             p.get_val('traj.phase0.timeseries.controls:theta'),
             marker='o',
             ms=4,
             linestyle='None')

axes[1].plot(exp_out.get_val('traj.phase0.timeseries.time'),
             exp_out.get_val('traj.phase0.timeseries.controls:theta'),
             linestyle='-',
             marker=None)

axes[1].set_xlabel('time (s)')
axes[1].set_ylabel('theta (deg)')

plt.suptitle('Single Stage to Orbit Solution Using Linear Tangent Guidance')
fig.legend(loc='lower center', ncol=2)

plt.show()
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 comp_has_no_outputs
INFO: checking auto_ivc_warnings
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 comp_has_no_outputs
INFO: checking auto_ivc_warnings

Approx coloring for 'traj.phases.phase0.rhs_disc' (class LaunchVehicleODE)
........................f............................................................................................................................................... 0  xdot
.........................f.............................................................................................................................................. 1  xdot
..........................f............................................................................................................................................. 2  xdot
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..........................................f............................................................................................................................. 18  xdot
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...............................................f........................................................................................................................ 23  xdot
................................................f....................................................................................................................... 24  ydot
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....................................................f................................................................................................................... 28  ydot
.....................................................f.................................................................................................................. 29  ydot
......................................................f................................................................................................................. 30  ydot
.......................................................f................................................................................................................ 31  ydot
........................................................f............................................................................................................... 32  ydot
.........................................................f.............................................................................................................. 33  ydot
..........................................................f............................................................................................................. 34  ydot
...........................................................f............................................................................................................ 35  ydot
............................................................f........................................................................................................... 36  ydot
.............................................................f.......................................................................................................... 37  ydot
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...............................................................f........................................................................................................ 39  ydot
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......................................................................f................................................................................................. 46  ydot
.......................................................................f................................................................................................ 47  ydot
........................f...............................................f.......................f.......................f............................................... 48  vxdot
.f.......................f...............................................f.......................f.......................f.............................................. 49  vxdot
..f.......................f...............................................f.......................f.......................f............................................. 50  vxdot
...f.......................f...............................................f.......................f.......................f............................................ 51  vxdot
....f.......................f...............................................f.......................f.......................f........................................... 52  vxdot
.....f.......................f...............................................f.......................f.......................f.......................................... 53  vxdot
......f.......................f...............................................f.......................f.......................f......................................... 54  vxdot
.......f.......................f...............................................f.......................f.......................f........................................ 55  vxdot
........f.......................f...............................................f.......................f.......................f....................................... 56  vxdot
.........f.......................f...............................................f.......................f.......................f...................................... 57  vxdot
..........f.......................f...............................................f.......................f.......................f..................................... 58  vxdot
...........f.......................f...............................................f.......................f.......................f.................................... 59  vxdot
............f.......................f...............................................f.......................f.......................f................................... 60  vxdot
.............f.......................f...............................................f.......................f.......................f.................................. 61  vxdot
..............f.......................f...............................................f.......................f.......................f................................. 62  vxdot
...............f.......................f...............................................f.......................f.......................f................................ 63  vxdot
................f.......................f...............................................f.......................f.......................f............................... 64  vxdot
.................f.......................f...............................................f.......................f.......................f.............................. 65  vxdot
..................f.......................f...............................................f.......................f.......................f............................. 66  vxdot
...................f.......................f...............................................f.......................f.......................f............................ 67  vxdot
....................f.......................f...............................................f.......................f.......................f........................... 68  vxdot
.....................f.......................f...............................................f.......................f.......................f.......................... 69  vxdot
......................f.......................f...............................................f.......................f.......................f......................... 70  vxdot
.......................f.......................f...............................................f.......................f.......................f........................ 71  vxdot
f...............................................f.......................f.......................f.......................f............................................... 72  vydot
.f...............................................f.......................f.......................f.......................f.............................................. 73  vydot
..f...............................................f.......................f.......................f.......................f............................................. 74  vydot
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..........................................................f.......................f.......................f.......................f..................................... 82  vydot
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...............................................................f.......................f.......................f.......................f................................ 87  vydot
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..................................................................f.......................f.......................f.......................f............................. 90  vydot
...................................................................f.......................f.......................f.......................f............................ 91  vydot
....................................................................f.......................f.......................f.......................f........................... 92  vydot
.....................................................................f.......................f.......................f.......................f.......................... 93  vydot
......................................................................f.......................f.......................f.......................f......................... 94  vydot
.......................................................................f.......................f.......................f.......................f........................ 95  vydot
........................................................................................................................f.......................f....................... 96  mdot
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...............................................................................................................................f.......................f................ 103  mdot
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.......................................................................................................................................f.......................f........ 111  mdot
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.........................................................................................................................................f.......................f...... 113  mdot
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............................................................................................................................................f.......................f... 116  mdot
.............................................................................................................................................f.......................f.. 117  mdot
..............................................................................................................................................f.......................f. 118  mdot
...............................................................................................................................................f.......................f 119  mdot
f....................................................................................................................................................................... 120  rho
.f...................................................................................................................................................................... 121  rho
..f..................................................................................................................................................................... 122  rho
...f.................................................................................................................................................................... 123  rho
....f................................................................................................................................................................... 124  rho
.....f.................................................................................................................................................................. 125  rho
......f................................................................................................................................................................. 126  rho
.......f................................................................................................................................................................ 127  rho
........f............................................................................................................................................................... 128  rho
.........f.............................................................................................................................................................. 129  rho
..........f............................................................................................................................................................. 130  rho
...........f............................................................................................................................................................ 131  rho
............f........................................................................................................................................................... 132  rho
.............f.......................................................................................................................................................... 133  rho
..............f......................................................................................................................................................... 134  rho
...............f........................................................................................................................................................ 135  rho
................f....................................................................................................................................................... 136  rho
.................f...................................................................................................................................................... 137  rho
..................f..................................................................................................................................................... 138  rho
...................f.................................................................................................................................................... 139  rho
....................f................................................................................................................................................... 140  rho
.....................f.................................................................................................................................................. 141  rho
......................f................................................................................................................................................. 142  rho
.......................f................................................................................................................................................ 143  rho
|y
                        |vx
                                                |vy
                                                                        |m
                                                                                                |theta
                                                                                                                        |thrust
                                                                                                                                                |Isp

Jacobian shape: (144, 168)  ( 1.40% nonzero)
FWD solves: 5   REV solves: 0
Total colors vs. total size: 5 vs 168  (97.0% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.052325 sec.
Time to compute coloring: 0.003627 sec.

Approx coloring for 'traj.phases.phase0.rhs_col' (class LaunchVehicleODE)
............f....................................................................... 0  xdot
.............f...................................................................... 1  xdot
..............f..................................................................... 2  xdot
...............f.................................................................... 3  xdot
................f................................................................... 4  xdot
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..................................f................................................. 22  ydot
...................................f................................................ 23  ydot
f...........f.......................f...........f...........f....................... 24  vxdot
.f...........f.......................f...........f...........f...................... 25  vxdot
..f...........f.......................f...........f...........f..................... 26  vxdot
...f...........f.......................f...........f...........f.................... 27  vxdot
....f...........f.......................f...........f...........f................... 28  vxdot
.....f...........f.......................f...........f...........f.................. 29  vxdot
......f...........f.......................f...........f...........f................. 30  vxdot
.......f...........f.......................f...........f...........f................ 31  vxdot
........f...........f.......................f...........f...........f............... 32  vxdot
.........f...........f.......................f...........f...........f.............. 33  vxdot
..........f...........f.......................f...........f...........f............. 34  vxdot
...........f...........f.......................f...........f...........f............ 35  vxdot
f.......................f...........f...........f...........f....................... 36  vydot
.f.......................f...........f...........f...........f...................... 37  vydot
..f.......................f...........f...........f...........f..................... 38  vydot
...f.......................f...........f...........f...........f.................... 39  vydot
....f.......................f...........f...........f...........f................... 40  vydot
.....f.......................f...........f...........f...........f.................. 41  vydot
......f.......................f...........f...........f...........f................. 42  vydot
.......f.......................f...........f...........f...........f................ 43  vydot
........f.......................f...........f...........f...........f............... 44  vydot
.........f.......................f...........f...........f...........f.............. 45  vydot
..........f.......................f...........f...........f...........f............. 46  vydot
...........f.......................f...........f...........f...........f............ 47  vydot
............................................................f...........f........... 48  mdot
.............................................................f...........f.......... 49  mdot
..............................................................f...........f......... 50  mdot
...............................................................f...........f........ 51  mdot
................................................................f...........f....... 52  mdot
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..................................................................f...........f..... 54  mdot
...................................................................f...........f.... 55  mdot
....................................................................f...........f... 56  mdot
.....................................................................f...........f.. 57  mdot
......................................................................f...........f. 58  mdot
.......................................................................f...........f 59  mdot
f................................................................................... 60  rho
.f.................................................................................. 61  rho
..f................................................................................. 62  rho
...f................................................................................ 63  rho
....f............................................................................... 64  rho
.....f.............................................................................. 65  rho
......f............................................................................. 66  rho
.......f............................................................................ 67  rho
........f........................................................................... 68  rho
.........f.......................................................................... 69  rho
..........f......................................................................... 70  rho
...........f........................................................................ 71  rho
|y
            |vx
                        |vy
                                    |m
                                                |theta
                                                            |thrust
                                                                        |Isp

Jacobian shape: (72, 84)  ( 2.98% nonzero)
FWD solves: 5   REV solves: 0
Total colors vs. total size: 5 vs 84  (94.0% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.027773 sec.
Time to compute coloring: 0.001939 sec.
Full total jacobian was computed 3 times, taking 0.502738 seconds.
Total jacobian shape: (72, 153) 


Jacobian shape: (72, 153)  ( 5.48% nonzero)
FWD solves: 12   REV solves: 0
Total colors vs. total size: 12 vs 153  (92.2% improvement)

Sparsity computed using tolerance: 1e-12
Time to compute sparsity: 0.502738 sec.
Time to compute coloring: 0.004431 sec.


Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
    Objective Function: _objfunc

    Solution: 
--------------------------------------------------------------------------------
    Total Time:                    7.6139
       User Objective Time :       0.5193
       User Sensitivity Time :     5.1795
       Interface Time :            0.9972
       Opt Solver Time:            0.9179
    Calls to Objective Function :     157
    Calls to Sens Function :           75


   Objectives
      Index  Name                                    Value          Optimum
          0  traj.phases.phase0.time.time     1.431613E+00     0.000000E+00

   Variables (c - continuous, i - integer, d - discrete)
      Index  Name                                                                Type      Lower Bound            Value      Upper Bound     Status
          0  traj.phases.phase0.time_extents.t_initial_0                            c     0.000000E+00     1.876878E-18     0.000000E+00         lu
          1  traj.phases.phase0.time_extents.t_duration_0                           c     1.000000E+01     1.431613E+02     5.000000E+02           
          2  traj.phases.phase0.indep_states.states:x_0                             c    -1.000000E+16     1.298305E-03     1.000000E+16           
          3  traj.phases.phase0.indep_states.states:x_1                             c    -1.000000E+16     1.298305E-03     1.000000E+16           
          4  traj.phases.phase0.indep_states.states:x_2                             c    -1.000000E+16     5.559628E-03     1.000000E+16           
          5  traj.phases.phase0.indep_states.states:x_3                             c    -1.000000E+16     5.559628E-03     1.000000E+16           
          6  traj.phases.phase0.indep_states.states:x_4                             c    -1.000000E+16     1.340944E-02     1.000000E+16           
          7  traj.phases.phase0.indep_states.states:x_5                             c    -1.000000E+16     1.340944E-02     1.000000E+16           
          8  traj.phases.phase0.indep_states.states:x_6                             c    -1.000000E+16     2.576500E-02     1.000000E+16           
          9  traj.phases.phase0.indep_states.states:x_7                             c    -1.000000E+16     2.576500E-02     1.000000E+16           
         10  traj.phases.phase0.indep_states.states:x_8                             c    -1.000000E+16     4.374088E-02     1.000000E+16           
         11  traj.phases.phase0.indep_states.states:x_9                             c    -1.000000E+16     4.374088E-02     1.000000E+16           
         12  traj.phases.phase0.indep_states.states:x_10                            c    -1.000000E+16     6.913213E-02     1.000000E+16           
         13  traj.phases.phase0.indep_states.states:x_11                            c    -1.000000E+16     6.913213E-02     1.000000E+16           
         14  traj.phases.phase0.indep_states.states:x_12                            c    -1.000000E+16     1.050295E-01     1.000000E+16           
         15  traj.phases.phase0.indep_states.states:x_13                            c    -1.000000E+16     1.050295E-01     1.000000E+16           
         16  traj.phases.phase0.indep_states.states:x_14                            c    -1.000000E+16     1.564984E-01     1.000000E+16           
         17  traj.phases.phase0.indep_states.states:x_15                            c    -1.000000E+16     1.564984E-01     1.000000E+16           
         18  traj.phases.phase0.indep_states.states:x_16                            c    -1.000000E+16     2.334881E-01     1.000000E+16           
         19  traj.phases.phase0.indep_states.states:x_17                            c    -1.000000E+16     2.334881E-01     1.000000E+16           
         20  traj.phases.phase0.indep_states.states:x_18                            c    -1.000000E+16     3.589406E-01     1.000000E+16           
         21  traj.phases.phase0.indep_states.states:x_19                            c    -1.000000E+16     3.589406E-01     1.000000E+16           
         22  traj.phases.phase0.indep_states.states:x_20                            c    -1.000000E+16     5.992223E-01     1.000000E+16           
         23  traj.phases.phase0.indep_states.states:x_21                            c    -1.000000E+16     5.992223E-01     1.000000E+16           
         24  traj.phases.phase0.indep_states.states:x_22                            c    -1.000000E+16     1.130587E+00     1.000000E+16           
         25  traj.phases.phase0.indep_states.states:y_0                             c    -1.000000E+16     6.072633E-03     1.000000E+16           
         26  traj.phases.phase0.indep_states.states:y_1                             c    -1.000000E+16     6.072633E-03     1.000000E+16           
         27  traj.phases.phase0.indep_states.states:y_2                             c    -1.000000E+16     2.555867E-02     1.000000E+16           
         28  traj.phases.phase0.indep_states.states:y_3                             c    -1.000000E+16     2.555867E-02     1.000000E+16           
         29  traj.phases.phase0.indep_states.states:y_4                             c    -1.000000E+16     6.030918E-02     1.000000E+16           
         30  traj.phases.phase0.indep_states.states:y_5                             c    -1.000000E+16     6.030918E-02     1.000000E+16           
         31  traj.phases.phase0.indep_states.states:y_6                             c    -1.000000E+16     1.125910E-01     1.000000E+16           
         32  traj.phases.phase0.indep_states.states:y_7                             c    -1.000000E+16     1.125910E-01     1.000000E+16           
         33  traj.phases.phase0.indep_states.states:y_8                             c    -1.000000E+16     1.859332E-01     1.000000E+16           
         34  traj.phases.phase0.indep_states.states:y_9                             c    -1.000000E+16     1.859332E-01     1.000000E+16           
         35  traj.phases.phase0.indep_states.states:y_10                            c    -1.000000E+16     2.857302E-01     1.000000E+16           
         36  traj.phases.phase0.indep_states.states:y_11                            c    -1.000000E+16     2.857302E-01     1.000000E+16           
         37  traj.phases.phase0.indep_states.states:y_12                            c    -1.000000E+16     4.195051E-01     1.000000E+16           
         38  traj.phases.phase0.indep_states.states:y_13                            c    -1.000000E+16     4.195051E-01     1.000000E+16           
         39  traj.phases.phase0.indep_states.states:y_14                            c    -1.000000E+16     5.971841E-01     1.000000E+16           
         40  traj.phases.phase0.indep_states.states:y_15                            c    -1.000000E+16     5.971841E-01     1.000000E+16           
         41  traj.phases.phase0.indep_states.states:y_16                            c    -1.000000E+16     8.317417E-01     1.000000E+16           
         42  traj.phases.phase0.indep_states.states:y_17                            c    -1.000000E+16     8.317417E-01     1.000000E+16           
         43  traj.phases.phase0.indep_states.states:y_18                            c    -1.000000E+16     1.139717E+00     1.000000E+16           
         44  traj.phases.phase0.indep_states.states:y_19                            c    -1.000000E+16     1.139717E+00     1.000000E+16           
         45  traj.phases.phase0.indep_states.states:y_20                            c    -1.000000E+16     1.524809E+00     1.000000E+16           
         46  traj.phases.phase0.indep_states.states:y_21                            c    -1.000000E+16     1.524809E+00     1.000000E+16           
         47  traj.phases.phase0.indep_states.states:y_22                            c    -1.000000E+16     1.850000E+00     1.000000E+16           
         48  traj.phases.phase0.indep_states.states:vx_0                            c    -1.000000E+18     2.254143E-02     1.000000E+18           
         49  traj.phases.phase0.indep_states.states:vx_1                            c    -1.000000E+18     2.254143E-02     1.000000E+18           
         50  traj.phases.phase0.indep_states.states:vx_2                            c    -1.000000E+18     4.973761E-02     1.000000E+18           
         51  traj.phases.phase0.indep_states.states:vx_3                            c    -1.000000E+18     4.973761E-02     1.000000E+18           
         52  traj.phases.phase0.indep_states.states:vx_4                            c    -1.000000E+18     8.315862E-02     1.000000E+18           
         53  traj.phases.phase0.indep_states.states:vx_5                            c    -1.000000E+18     8.315862E-02     1.000000E+18           
         54  traj.phases.phase0.indep_states.states:vx_6                            c    -1.000000E+18     1.254797E-01     1.000000E+18           
         55  traj.phases.phase0.indep_states.states:vx_7                            c    -1.000000E+18     1.254797E-01     1.000000E+18           
         56  traj.phases.phase0.indep_states.states:vx_8                            c    -1.000000E+18     1.782948E-01     1.000000E+18           
         57  traj.phases.phase0.indep_states.states:vx_9                            c    -1.000000E+18     1.782948E-01     1.000000E+18           
         58  traj.phases.phase0.indep_states.states:vx_10                           c    -1.000000E+18     2.516474E-01     1.000000E+18           
         59  traj.phases.phase0.indep_states.states:vx_11                           c    -1.000000E+18     2.516474E-01     1.000000E+18           
         60  traj.phases.phase0.indep_states.states:vx_12                           c    -1.000000E+18     3.569134E-01     1.000000E+18           
         61  traj.phases.phase0.indep_states.states:vx_13                           c    -1.000000E+18     3.569134E-01     1.000000E+18           
         62  traj.phases.phase0.indep_states.states:vx_14                           c    -1.000000E+18     5.190824E-01     1.000000E+18           
         63  traj.phases.phase0.indep_states.states:vx_15                           c    -1.000000E+18     5.190824E-01     1.000000E+18           
         64  traj.phases.phase0.indep_states.states:vx_16                           c    -1.000000E+18     8.005666E-01     1.000000E+18           
         65  traj.phases.phase0.indep_states.states:vx_17                           c    -1.000000E+18     8.005666E-01     1.000000E+18           
         66  traj.phases.phase0.indep_states.states:vx_18                           c    -1.000000E+18     1.387267E+00     1.000000E+18           
         67  traj.phases.phase0.indep_states.states:vx_19                           c    -1.000000E+18     1.387267E+00     1.000000E+18           
         68  traj.phases.phase0.indep_states.states:vx_20                           c    -1.000000E+18     2.871159E+00     1.000000E+18           
         69  traj.phases.phase0.indep_states.states:vx_21                           c    -1.000000E+18     2.871159E+00     1.000000E+18           
         70  traj.phases.phase0.indep_states.states:vx_22                           c    -1.000000E+18     7.796696E+00     1.000000E+18           
         71  traj.phases.phase0.indep_states.states:vy_0                            c    -1.000000E+18     1.045338E-01     1.000000E+18           
         72  traj.phases.phase0.indep_states.states:vy_1                            c    -1.000000E+18     1.045338E-01     1.000000E+18           
         73  traj.phases.phase0.indep_states.states:vy_2                            c    -1.000000E+18     2.246571E-01     1.000000E+18           
         74  traj.phases.phase0.indep_states.states:vy_3                            c    -1.000000E+18     2.246571E-01     1.000000E+18           
         75  traj.phases.phase0.indep_states.states:vy_4                            c    -1.000000E+18     3.609698E-01     1.000000E+18           
         76  traj.phases.phase0.indep_states.states:vy_5                            c    -1.000000E+18     3.609698E-01     1.000000E+18           
         77  traj.phases.phase0.indep_states.states:vy_6                            c    -1.000000E+18     5.203537E-01     1.000000E+18           
         78  traj.phases.phase0.indep_states.states:vy_7                            c    -1.000000E+18     5.203537E-01     1.000000E+18           
         79  traj.phases.phase0.indep_states.states:vy_8                            c    -1.000000E+18     7.166933E-01     1.000000E+18           
         80  traj.phases.phase0.indep_states.states:vy_9                            c    -1.000000E+18     7.166933E-01     1.000000E+18           
         81  traj.phases.phase0.indep_states.states:vy_10                           c    -1.000000E+18     9.668202E-01     1.000000E+18           
         82  traj.phases.phase0.indep_states.states:vy_11                           c    -1.000000E+18     9.668202E-01     1.000000E+18           
         83  traj.phases.phase0.indep_states.states:vy_12                           c    -1.000000E+18     1.289622E+00     1.000000E+18           
         84  traj.phases.phase0.indep_states.states:vy_13                           c    -1.000000E+18     1.289622E+00     1.000000E+18           
         85  traj.phases.phase0.indep_states.states:vy_14                           c    -1.000000E+18     1.707016E+00     1.000000E+18           
         86  traj.phases.phase0.indep_states.states:vy_15                           c    -1.000000E+18     1.707016E+00     1.000000E+18           
         87  traj.phases.phase0.indep_states.states:vy_16                           c    -1.000000E+18     2.249040E+00     1.000000E+18           
         88  traj.phases.phase0.indep_states.states:vy_17                           c    -1.000000E+18     2.249040E+00     1.000000E+18           
         89  traj.phases.phase0.indep_states.states:vy_18                           c    -1.000000E+18     2.929061E+00     1.000000E+18           
         90  traj.phases.phase0.indep_states.states:vy_19                           c    -1.000000E+18     2.929061E+00     1.000000E+18           
         91  traj.phases.phase0.indep_states.states:vy_20                           c    -1.000000E+18     3.383754E+00     1.000000E+18           
         92  traj.phases.phase0.indep_states.states:vy_21                           c    -1.000000E+18     3.383754E+00     1.000000E+18           
         93  traj.phases.phase0.indep_states.states:vy_22                           c    -1.000000E+18    -1.100394E-19     1.000000E+18           
         94  traj.phases.phase0.indep_states.states:m_0                             c    -1.000000E+18     1.073668E+02     1.000000E+18           
         95  traj.phases.phase0.indep_states.states:m_1                             c    -1.000000E+18     1.073668E+02     1.000000E+18           
         96  traj.phases.phase0.indep_states.states:m_2                             c    -1.000000E+18     9.773365E+01     1.000000E+18           
         97  traj.phases.phase0.indep_states.states:m_3                             c    -1.000000E+18     9.773365E+01     1.000000E+18           
         98  traj.phases.phase0.indep_states.states:m_4                             c    -1.000000E+18     8.810048E+01     1.000000E+18           
         99  traj.phases.phase0.indep_states.states:m_5                             c    -1.000000E+18     8.810048E+01     1.000000E+18           
        100  traj.phases.phase0.indep_states.states:m_6                             c    -1.000000E+18     7.846731E+01     1.000000E+18           
        101  traj.phases.phase0.indep_states.states:m_7                             c    -1.000000E+18     7.846731E+01     1.000000E+18           
        102  traj.phases.phase0.indep_states.states:m_8                             c    -1.000000E+18     6.883413E+01     1.000000E+18           
        103  traj.phases.phase0.indep_states.states:m_9                             c    -1.000000E+18     6.883413E+01     1.000000E+18           
        104  traj.phases.phase0.indep_states.states:m_10                            c    -1.000000E+18     5.920096E+01     1.000000E+18           
        105  traj.phases.phase0.indep_states.states:m_11                            c    -1.000000E+18     5.920096E+01     1.000000E+18           
        106  traj.phases.phase0.indep_states.states:m_12                            c    -1.000000E+18     4.956779E+01     1.000000E+18           
        107  traj.phases.phase0.indep_states.states:m_13                            c    -1.000000E+18     4.956779E+01     1.000000E+18           
        108  traj.phases.phase0.indep_states.states:m_14                            c    -1.000000E+18     3.993462E+01     1.000000E+18           
        109  traj.phases.phase0.indep_states.states:m_15                            c    -1.000000E+18     3.993462E+01     1.000000E+18           
        110  traj.phases.phase0.indep_states.states:m_16                            c    -1.000000E+18     3.030144E+01     1.000000E+18           
        111  traj.phases.phase0.indep_states.states:m_17                            c    -1.000000E+18     3.030144E+01     1.000000E+18           
        112  traj.phases.phase0.indep_states.states:m_18                            c    -1.000000E+18     2.066827E+01     1.000000E+18           
        113  traj.phases.phase0.indep_states.states:m_19                            c    -1.000000E+18     2.066827E+01     1.000000E+18           
        114  traj.phases.phase0.indep_states.states:m_20                            c    -1.000000E+18     1.103510E+01     1.000000E+18           
        115  traj.phases.phase0.indep_states.states:m_21                            c    -1.000000E+18     1.103510E+01     1.000000E+18           
        116  traj.phases.phase0.indep_states.states:m_22                            c    -1.000000E+18     1.401924E+00     1.000000E+18           
        117  traj.phases.phase0.control_group.indep_controls.controls:theta_0       c    -1.570000E+00     1.476545E+00     1.570000E+00           
        118  traj.phases.phase0.control_group.indep_controls.controls:theta_1       c    -1.570000E+00     1.469435E+00     1.570000E+00           
        119  traj.phases.phase0.control_group.indep_controls.controls:theta_2       c    -1.570000E+00     1.463782E+00     1.570000E+00           
        120  traj.phases.phase0.control_group.indep_controls.controls:theta_3       c    -1.570000E+00     1.463782E+00     1.570000E+00           
        121  traj.phases.phase0.control_group.indep_controls.controls:theta_4       c    -1.570000E+00     1.458416E+00     1.570000E+00           
        122  traj.phases.phase0.control_group.indep_controls.controls:theta_5       c    -1.570000E+00     1.452169E+00     1.570000E+00           
        123  traj.phases.phase0.control_group.indep_controls.controls:theta_6       c    -1.570000E+00     1.452169E+00     1.570000E+00           
        124  traj.phases.phase0.control_group.indep_controls.controls:theta_7       c    -1.570000E+00     1.444403E+00     1.570000E+00           
        125  traj.phases.phase0.control_group.indep_controls.controls:theta_8       c    -1.570000E+00     1.434481E+00     1.570000E+00           
        126  traj.phases.phase0.control_group.indep_controls.controls:theta_9       c    -1.570000E+00     1.434481E+00     1.570000E+00           
        127  traj.phases.phase0.control_group.indep_controls.controls:theta_10      c    -1.570000E+00     1.425291E+00     1.570000E+00           
        128  traj.phases.phase0.control_group.indep_controls.controls:theta_11      c    -1.570000E+00     1.419720E+00     1.570000E+00           
        129  traj.phases.phase0.control_group.indep_controls.controls:theta_12      c    -1.570000E+00     1.419720E+00     1.570000E+00           
        130  traj.phases.phase0.control_group.indep_controls.controls:theta_13      c    -1.570000E+00     1.412441E+00     1.570000E+00           
        131  traj.phases.phase0.control_group.indep_controls.controls:theta_14      c    -1.570000E+00     1.398129E+00     1.570000E+00           
        132  traj.phases.phase0.control_group.indep_controls.controls:theta_15      c    -1.570000E+00     1.398129E+00     1.570000E+00           
        133  traj.phases.phase0.control_group.indep_controls.controls:theta_16      c    -1.570000E+00     1.380133E+00     1.570000E+00           
        134  traj.phases.phase0.control_group.indep_controls.controls:theta_17      c    -1.570000E+00     1.361799E+00     1.570000E+00           
        135  traj.phases.phase0.control_group.indep_controls.controls:theta_18      c    -1.570000E+00     1.361799E+00     1.570000E+00           
        136  traj.phases.phase0.control_group.indep_controls.controls:theta_19      c    -1.570000E+00     1.340829E+00     1.570000E+00           
        137  traj.phases.phase0.control_group.indep_controls.controls:theta_20      c    -1.570000E+00     1.314923E+00     1.570000E+00           
        138  traj.phases.phase0.control_group.indep_controls.controls:theta_21      c    -1.570000E+00     1.314923E+00     1.570000E+00           
        139  traj.phases.phase0.control_group.indep_controls.controls:theta_22      c    -1.570000E+00     1.280833E+00     1.570000E+00           
        140  traj.phases.phase0.control_group.indep_controls.controls:theta_23      c    -1.570000E+00     1.235312E+00     1.570000E+00           
        141  traj.phases.phase0.control_group.indep_controls.controls:theta_24      c    -1.570000E+00     1.235312E+00     1.570000E+00           
        142  traj.phases.phase0.control_group.indep_controls.controls:theta_25      c    -1.570000E+00     1.174094E+00     1.570000E+00           
        143  traj.phases.phase0.control_group.indep_controls.controls:theta_26      c    -1.570000E+00     1.092916E+00     1.570000E+00           
        144  traj.phases.phase0.control_group.indep_controls.controls:theta_27      c    -1.570000E+00     1.092916E+00     1.570000E+00           
        145  traj.phases.phase0.control_group.indep_controls.controls:theta_28      c    -1.570000E+00     9.611379E-01     1.570000E+00           
        146  traj.phases.phase0.control_group.indep_controls.controls:theta_29      c    -1.570000E+00     7.481216E-01     1.570000E+00           
        147  traj.phases.phase0.control_group.indep_controls.controls:theta_30      c    -1.570000E+00     7.481216E-01     1.570000E+00           
        148  traj.phases.phase0.control_group.indep_controls.controls:theta_31      c    -1.570000E+00     4.306035E-01     1.570000E+00           
        149  traj.phases.phase0.control_group.indep_controls.controls:theta_32      c    -1.570000E+00    -1.467929E-02     1.570000E+00           
        150  traj.phases.phase0.control_group.indep_controls.controls:theta_33      c    -1.570000E+00    -1.467929E-02     1.570000E+00           
        151  traj.phases.phase0.control_group.indep_controls.controls:theta_34      c    -1.570000E+00    -4.580936E-01     1.570000E+00           
        152  traj.phases.phase0.control_group.indep_controls.controls:theta_35      c    -1.570000E+00    -7.700061E-01     1.570000E+00           

   Constraints (i - inequality, e - equality)
      Index  Name                                                               Type          Lower           Value           Upper    Status  Lagrange Multiplier (N/A)
          0  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00    5.225831E-12    0.000000E+00              9.00000E+100
          1  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -1.940483E-11    0.000000E+00              9.00000E+100
          2  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00    1.940636E-11    0.000000E+00              9.00000E+100
          3  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -3.162097E-11    0.000000E+00              9.00000E+100
          4  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00    6.232129E-12    0.000000E+00              9.00000E+100
          5  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -1.363761E-11    0.000000E+00              9.00000E+100
          6  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00    4.733666E-11    0.000000E+00              9.00000E+100
          7  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -7.353019E-11    0.000000E+00              9.00000E+100
          8  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -1.526192E-10    0.000000E+00              9.00000E+100
          9  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -3.090928E-10    0.000000E+00              9.00000E+100
         10  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00    5.336198E-10    0.000000E+00              9.00000E+100
         11  traj.phases.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -5.187669E-11    0.000000E+00              9.00000E+100
         12  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    9.714589E-11    0.000000E+00              9.00000E+100
         13  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    7.911920E-11    0.000000E+00              9.00000E+100
         14  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00   -5.665917E-09    0.000000E+00              9.00000E+100
         15  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00   -8.417522E-09    0.000000E+00              9.00000E+100
         16  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00   -1.364228E-08    0.000000E+00              9.00000E+100
         17  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00   -4.778158E-09    0.000000E+00              9.00000E+100
         18  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    1.727269E-08    0.000000E+00              9.00000E+100
         19  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    1.123585E-08    0.000000E+00              9.00000E+100
         20  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    3.508263E-09    0.000000E+00              9.00000E+100
         21  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    8.057483E-12    0.000000E+00              9.00000E+100
         22  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    4.045766E-10    0.000000E+00              9.00000E+100
         23  traj.phases.phase0.collocation_constraint.defects:y                   e   0.000000E+00    1.054428E-10    0.000000E+00              9.00000E+100
         24  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    1.265261E-10    0.000000E+00              9.00000E+100
         25  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    3.711793E-10    0.000000E+00              9.00000E+100
         26  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    1.295709E-10    0.000000E+00              9.00000E+100
         27  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    1.977848E-10    0.000000E+00              9.00000E+100
         28  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    3.827392E-10    0.000000E+00              9.00000E+100
         29  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    2.393391E-10    0.000000E+00              9.00000E+100
         30  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    7.883499E-10    0.000000E+00              9.00000E+100
         31  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    4.100651E-10    0.000000E+00              9.00000E+100
         32  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    5.475177E-09    0.000000E+00              9.00000E+100
         33  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    2.300446E-09    0.000000E+00              9.00000E+100
         34  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    2.834350E-09    0.000000E+00              9.00000E+100
         35  traj.phases.phase0.collocation_constraint.defects:vx                  e   0.000000E+00    1.475680E-09    0.000000E+00              9.00000E+100
         36  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    1.164710E-09    0.000000E+00              9.00000E+100
         37  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    2.739043E-09    0.000000E+00              9.00000E+100
         38  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    9.870230E-09    0.000000E+00              9.00000E+100
         39  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    3.412989E-08    0.000000E+00              9.00000E+100
         40  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    7.160891E-08    0.000000E+00              9.00000E+100
         41  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    1.008939E-07    0.000000E+00              9.00000E+100
         42  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    8.249984E-08    0.000000E+00              9.00000E+100
         43  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    3.286087E-08    0.000000E+00              9.00000E+100
         44  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    1.912811E-08    0.000000E+00              9.00000E+100
         45  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    2.822363E-09    0.000000E+00              9.00000E+100
         46  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00    1.553544E-09    0.000000E+00              9.00000E+100
         47  traj.phases.phase0.collocation_constraint.defects:vy                  e   0.000000E+00   -8.211546E-10    0.000000E+00              9.00000E+100
         48  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -6.781479E-15    0.000000E+00              9.00000E+100
         49  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -6.781479E-15    0.000000E+00              9.00000E+100
         50  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    4.747036E-14    0.000000E+00              9.00000E+100
         51  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -1.559740E-13    0.000000E+00              9.00000E+100
         52  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -6.781479E-15    0.000000E+00              9.00000E+100
         53  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    4.747036E-14    0.000000E+00              9.00000E+100
         54  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    2.034444E-14    0.000000E+00              9.00000E+100
         55  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    4.747036E-14    0.000000E+00              9.00000E+100
         56  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -2.034444E-14    0.000000E+00              9.00000E+100
         57  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00   -6.781479E-15    0.000000E+00              9.00000E+100
         58  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    6.781479E-15    0.000000E+00              9.00000E+100
         59  traj.phases.phase0.collocation_constraint.defects:m                   e   0.000000E+00    6.781479E-15    0.000000E+00              9.00000E+100
         60  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    2.842171E-14    0.000000E+00              9.00000E+100
         61  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         62  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         63  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         64  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         65  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         66  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         67  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         68  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         69  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         70  traj.phases.phase0.continuity_comp.defect_states:x                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         71  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         72  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         73  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         74  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         75  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         76  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         77  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         78  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         79  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         80  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         81  traj.phases.phase0.continuity_comp.defect_states:y                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         82  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         83  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         84  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         85  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         86  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         87  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         88  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         89  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         90  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         91  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         92  traj.phases.phase0.continuity_comp.defect_states:vx                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         93  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         94  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         95  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         96  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         97  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         98  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         99  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        100  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        101  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        102  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        103  traj.phases.phase0.continuity_comp.defect_states:vy                   e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        104  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        105  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        106  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        107  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        108  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        109  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        110  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        111  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        112  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        113  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        114  traj.phases.phase0.continuity_comp.defect_states:m                    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        115  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00   -1.416523E-16    0.000000E+00              9.00000E+100
        116  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        117  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        118  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        119  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        120  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        121  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        122  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        123  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        124  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00   -1.299364E-17    0.000000E+00              9.00000E+100
        125  traj.phases.phase0.continuity_comp.defect_controls:theta              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        126  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    2.661950E-15    0.000000E+00              9.00000E+100
        127  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -6.612191E-15    0.000000E+00              9.00000E+100
        128  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -1.427985E-15    0.000000E+00              9.00000E+100
        129  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        130  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -9.157729E-15    0.000000E+00              9.00000E+100
        131  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -1.645287E-15    0.000000E+00              9.00000E+100
        132  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -3.538919E-15    0.000000E+00              9.00000E+100
        133  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -8.071219E-15    0.000000E+00              9.00000E+100
        134  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -1.241726E-16    0.000000E+00              9.00000E+100
        135  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -2.483452E-15    0.000000E+00              9.00000E+100
        136  traj.phases.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    9.933808E-16    0.000000E+00              9.00000E+100
        137  traj.phases.phase0.final_boundary_constraints.final_value:y           e   1.850000E+05    1.850000E+05    1.850000E+05              9.00000E+100
        138  traj.phases.phase0.final_boundary_constraints.final_value:vx          e   7.796696E+03    7.796696E+03    7.796696E+03              9.00000E+100
        139  traj.phases.phase0.final_boundary_constraints.final_value:vy          e   0.000000E+00   -1.100394E-16    0.000000E+00              9.00000E+100

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Simulating trajectory traj
Done simulating trajectory traj

References


  1. James M Longuski, José J Guzmán, and John E Prussing. Optimal control with aerospace applications. Springer, 2014.