# Two-Burn Orbit Raise#

This example demonstrates the use of a Trajectory to encapsulate a three-phase orbit raising maneuver with a burn-coast-burn phase sequence. This example is based on the problem provided in Enright [EC91].

The dynamics are given by

(58)#\begin{align} \frac{dr}{dt} &= v_r \\ \frac{d\theta}{dt} &= \frac{v_\theta}{r} \\ \frac{dv_r}{dt} &= \frac{v^2_\theta}{r} - \frac{1}{r^2} + a_{thrust} \sin u_1 \\ \frac{dv_\theta}{dt} &= - \frac{v_r v_\theta}{r} + a_{thrust} \cos u_1 \\ \frac{da_{thrust}}{dt} &= \frac{a^2_{thrust}}{c} \\ \frac{d \Delta v}{dt} &= a_{thrust} \end{align}

The initial conditions are

(59)#\begin{align} r &= 1 \rm{\,DU} \\ \theta &= 0 \rm{\,rad} \\ v_r &= 0 \rm{\,DU/TU}\\ v_\theta &= 1 \rm{\,DU/TU}\\ a_{thrust} &= 0.1 \rm{\,DU/TU^2}\\ \Delta v &= 0 \rm{\,DU/TU} \end{align}

and the final conditions are

(60)#\begin{align} r &= 3 \rm{\,DU} \\ \theta &= \rm{free} \\ v_r &= 0 \rm{\,DU/TU}\\ v_\theta &= \sqrt{\frac{1}{3}} \rm{\,DU/TU}\\ a_{thrust} &= \rm{free}\\ \Delta v &= \rm{free} \end{align}

## Building and running the problem#

The following code instantiates our problem, our trajectory, three phases, and links them accordingly. The spacecraft initial position, velocity, and acceleration magnitude are fixed. The objective is to minimize the delta-V needed to raise the spacecraft into a circular orbit at 3 Earth radii.

Note the call to link_phases which provides time, position, velocity, and delta-V continuity across all phases, and acceleration continuity between the first and second burn phases. Acceleration is 0 during the coast phase. Alternatively, we could have specified a different ODE for the coast phase, as in the example.

This example runs inconsistently with SLSQP but is solved handily by IPOPT and SNOPT.

Hide code cell outputs
import numpy as np
import openmdao.api as om
import openmdao.utils.units as units

# Add canonical units to OpenMDAO
MU_earth = 3.986592936294783e14
R_earth = 6378137.0

period = 2 * np.pi * np.sqrt(R_earth**3 / MU_earth)

# Add canonical time and distance units for these EOM
units.add_unit('TU', '{0}*s'.format(period))
units.add_unit('DU', '{0}*m'.format(R_earth))

class FiniteBurnODE(om.ExplicitComponent):

def initialize(self):
self.options.declare('num_nodes', types=int)

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

# Inputs
self.add_input('r',
val=np.ones(nn),
desc='radius from center of attraction',
units='DU')

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

self.add_input('vr',
val=np.zeros(nn),
desc='local vertical velocity component',
units='DU/TU')

self.add_input('vt',
val=np.zeros(nn),
desc='local horizontal velocity component',
units='DU/TU')

self.add_input('accel',
val=np.zeros(nn),
desc='acceleration due to thrust',
units='DU/TU**2')

self.add_input('u1',
val=np.zeros(nn),
desc='thrust angle above local horizontal',
units='rad')

self.add_input('c',
val=np.zeros(nn),
desc='exhaust velocity',
units='DU/TU')

self.add_output('r_dot',
val=np.ones(nn),
desc='rate of change of radius from center of attraction',
units='DU/TU')

self.add_output('theta_dot',
val=np.zeros(nn),
desc='rate of change of anomaly term',
units='rad/TU')

self.add_output('vr_dot',
val=np.zeros(nn),
desc='rate of change of local vertical velocity component',
units='DU/TU**2')

self.add_output('vt_dot',
val=np.zeros(nn),
desc='rate of change of local horizontal velocity component',
units='DU/TU**2')

self.add_output('at_dot',
val=np.zeros(nn),
desc='rate of change of acceleration due to thrust',
units='DU/TU**3')

self.add_output('deltav_dot',
val=np.zeros(nn),
desc='rate of change of delta-V',
units='DU/TU**2')

self.add_output('pos_x',
val=np.zeros(nn),
desc='x-component of position',
units='DU')

self.add_output('pos_y',
val=np.zeros(nn),
desc='x-component of position',
units='DU')

# Setup partials
ar = np.arange(self.options['num_nodes'])

# r dot is a linear function of vr, so provide the partial value here
self.declare_partials(of='r_dot', wrt='vr', rows=ar, cols=ar, val=1.0)

self.declare_partials(of='theta_dot', wrt='r', rows=ar, cols=ar)
self.declare_partials(of='theta_dot', wrt='vt', rows=ar, cols=ar)

self.declare_partials(of='vr_dot', wrt='r', rows=ar, cols=ar)
self.declare_partials(of='vr_dot', wrt='vt', rows=ar, cols=ar)
self.declare_partials(of='vr_dot', wrt='accel', rows=ar, cols=ar)
self.declare_partials(of='vr_dot', wrt='u1', rows=ar, cols=ar)

self.declare_partials(of='vt_dot', wrt='r', rows=ar, cols=ar)
self.declare_partials(of='vt_dot', wrt='vr', rows=ar, cols=ar)
self.declare_partials(of='vt_dot', wrt='vt', rows=ar, cols=ar)
self.declare_partials(of='vt_dot', wrt='accel', rows=ar, cols=ar)
self.declare_partials(of='vt_dot', wrt='u1', rows=ar, cols=ar)

self.declare_partials(of='at_dot', wrt='accel', rows=ar, cols=ar)
self.declare_partials(of='at_dot', wrt='c', rows=ar, cols=ar)

self.declare_partials(of='deltav_dot', wrt='accel', rows=ar, cols=ar, val=1.0)

self.declare_partials(of='pos_x', wrt='r', rows=ar, cols=ar)
self.declare_partials(of='pos_x', wrt='theta', rows=ar, cols=ar)

self.declare_partials(of='pos_y', wrt='r', rows=ar, cols=ar)
self.declare_partials(of='pos_y', wrt='theta', rows=ar, cols=ar)

def compute(self, inputs, outputs):
r = inputs['r']
theta = inputs['theta']
vr = inputs['vr']
vt = inputs['vt']
at = inputs['accel']
u1 = inputs['u1']
c = inputs['c']

outputs['r_dot'] = vr
outputs['theta_dot'] = vt / r
outputs['vr_dot'] = vt**2 / r - 1 / r**2 + at * np.sin(u1)
outputs['vt_dot'] = -vr * vt / r + at * np.cos(u1)
outputs['at_dot'] = at**2 / c
outputs['deltav_dot'] = at

outputs['pos_x'] = r * np.cos(theta)
outputs['pos_y'] = r * np.sin(theta)

def compute_partials(self, inputs, partials):
r = inputs['r']
theta = inputs['theta']
vr = inputs['vr']
vt = inputs['vt']
at = inputs['accel']
u1 = inputs['u1']
c = inputs['c']

su1 = np.sin(u1)
cu1 = np.cos(u1)

partials['theta_dot', 'r'] = -vt / r**2
partials['theta_dot', 'vt'] = 1.0 / r

partials['vr_dot', 'r'] = -vt**2 / r**2 + 2.0 / r**3
partials['vr_dot', 'vt'] = 2 * vt / r
partials['vr_dot', 'accel'] = su1
partials['vr_dot', 'u1'] = at * cu1

partials['vt_dot', 'r'] = vr * vt / r**2
partials['vt_dot', 'vr'] = -vt / r
partials['vt_dot', 'vt'] = -vr / r
partials['vt_dot', 'accel'] = cu1
partials['vt_dot', 'u1'] = -at * su1

partials['at_dot', 'accel'] = 2 * at / c
partials['at_dot', 'c'] = -at**2 / c**2

partials['pos_x', 'r'] = np.cos(theta)
partials['pos_x', 'theta'] = -r * np.sin(theta)

partials['pos_y', 'r'] = np.sin(theta)
partials['pos_y', 'theta'] = r * np.cos(theta)

import numpy as np
import openmdao.api as om
import dymos as dm
import matplotlib.pyplot as plt
from dymos.examples.finite_burn_orbit_raise.finite_burn_eom import FiniteBurnODE

p = om.Problem(model=om.Group())

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

traj = dm.Trajectory()

traj.add_parameter('c', opt=False, val=1.5, units='DU/TU',
targets={'burn1': ['c'], 'coast': ['c'], 'burn2': ['c']})

# First Phase (burn)

burn1 = dm.Phase(ode_class=FiniteBurnODE,
transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))

burn1 = traj.add_phase('burn1', burn1)

burn1.set_time_options(fix_initial=True, duration_bounds=(.5, 10), units='TU')
burn1.add_state('r', fix_initial=True, fix_final=False, defect_scaler=100.0,
rate_source='r_dot', units='DU')
burn1.add_state('theta', fix_initial=True, fix_final=False, defect_scaler=100.0,
rate_source='theta_dot', units='rad')
burn1.add_state('vr', fix_initial=True, fix_final=False, defect_scaler=100.0,
rate_source='vr_dot', units='DU/TU')
burn1.add_state('vt', fix_initial=True, fix_final=False, defect_scaler=100.0,
rate_source='vt_dot', units='DU/TU')
burn1.add_state('accel', fix_initial=True, fix_final=False,
rate_source='at_dot', units='DU/TU**2')
burn1.add_state('deltav', fix_initial=True, fix_final=False,
rate_source='deltav_dot', units='DU/TU')
burn1.add_control('u1', rate_continuity=True, rate2_continuity=True, units='deg',
scaler=0.01, rate_continuity_scaler=0.001, rate2_continuity_scaler=0.001,
lower=-30, upper=30)
# Second Phase (Coast)
coast = dm.Phase(ode_class=FiniteBurnODE,
transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))

coast.set_time_options(initial_bounds=(0.5, 20), duration_bounds=(.5, 50), duration_ref=50,
units='TU')
coast.add_state('r', fix_initial=False, fix_final=False, defect_scaler=100.0,
rate_source='r_dot', targets=['r'], units='DU')
coast.add_state('theta', fix_initial=False, fix_final=False, defect_scaler=100.0,
rate_source='theta_dot', targets=['theta'], units='rad')
coast.add_state('vr', fix_initial=False, fix_final=False, defect_scaler=100.0,
rate_source='vr_dot', targets=['vr'], units='DU/TU')
coast.add_state('vt', fix_initial=False, fix_final=False, defect_scaler=100.0,
rate_source='vt_dot', targets=['vt'], units='DU/TU')
coast.add_state('accel', fix_initial=True, fix_final=True,
rate_source='at_dot', targets=['accel'], units='DU/TU**2')
coast.add_state('deltav', fix_initial=False, fix_final=False,
rate_source='deltav_dot', units='DU/TU')

coast.add_parameter('u1', opt=False, val=0.0, units='deg', targets=['u1'])

# Third Phase (burn)
burn2 = dm.Phase(ode_class=FiniteBurnODE,
transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))

traj.add_phase('coast', coast)
traj.add_phase('burn2', burn2)

burn2.set_time_options(initial_bounds=(0.5, 50), duration_bounds=(.5, 10), initial_ref=10,
units='TU')
burn2.add_state('r', fix_initial=False, fix_final=True, defect_scaler=100.0,
rate_source='r_dot', units='DU')
burn2.add_state('theta', fix_initial=False, fix_final=False, defect_scaler=100.0,
rate_source='theta_dot', units='rad')
burn2.add_state('vr', fix_initial=False, fix_final=True, defect_scaler=1000.0,
rate_source='vr_dot', units='DU/TU')
burn2.add_state('vt', fix_initial=False, fix_final=True, defect_scaler=1000.0,
rate_source='vt_dot', units='DU/TU')
burn2.add_state('accel', fix_initial=False, fix_final=False, defect_scaler=1.0,
rate_source='at_dot', units='DU/TU**2')
burn2.add_state('deltav', fix_initial=False, fix_final=False, defect_scaler=1.0,
rate_source='deltav_dot', units='DU/TU')

burn2.add_objective('deltav', loc='final', scaler=100.0)

burn2.add_control('u1', rate_continuity=True, rate2_continuity=True, units='deg',
scaler=0.01, lower=-90, upper=90)

burn1.add_timeseries_output('pos_x')
coast.add_timeseries_output('pos_x')
burn2.add_timeseries_output('pos_x')

burn1.add_timeseries_output('pos_y')
coast.add_timeseries_output('pos_y')
burn2.add_timeseries_output('pos_y')

# Link Phases
traj.link_phases(phases=['burn1', 'coast', 'burn2'],
vars=['time', 'r', 'theta', 'vr', 'vt', 'deltav'])

traj.link_phases(phases=['burn1', 'burn2'], vars=['accel'])

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

# Finish Problem Setup

p.setup(check=True)

# Set Initial Guesses
p.set_val('traj.parameters:c', val=1.5, units='DU/TU')

burn1 = p.model.traj.phases.burn1
burn2 = p.model.traj.phases.burn2
coast = p.model.traj.phases.coast

burn1.set_time_val(initial=0.0, duration=2.25)
burn1.set_state_val('r', [1, 1.5])
burn1.set_state_val('theta', [0, 1.7])
burn1.set_state_val('vr', [0, 0])
burn1.set_state_val('vt', [1, 1])
burn1.set_state_val('accel', [0.1, 0.0])
burn1.set_state_val('deltav', [0, 0.1])
burn1.set_control_val('u1', [-3.5, 13.0])

coast.set_time_val(initial=2.25, duration=3.0)
coast.set_state_val('r', [1.3, 1.5])
coast.set_state_val('theta', [2.1767, 1.7])
coast.set_state_val('vr', [0.3285, 0])
coast.set_state_val('vt', [0.97, 1])
coast.set_state_val('accel', [0, 0])

burn2.set_time_val(initial=5.25, duration=1.75)
burn2.set_state_val('r', [1, 3])
burn2.set_state_val('theta', [0, 4])
burn2.set_state_val('vr', [0, 0])
burn2.set_state_val('vt', [1, np.sqrt(1 / 3)])
burn2.set_state_val('accel', [0.1, 0.0])
burn2.set_state_val('deltav', [0.1, 0.2])
burn2.set_control_val('u1', [0, 0])

dm.run_problem(p, simulate=True)

--- Constraint Report [traj] ---
--- burn1 ---
None
--- coast ---
None
--- burn2 ---
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

Full total jacobian for problem 'problem' was computed 3 times, taking 0.423972416999959 seconds.
Total jacobian shape: (112, 204)

Jacobian shape: (112, 204)  (3.64% nonzero)
FWD solves: 13   REV solves: 0
Total colors vs. total size: 13 vs 204  (93.63% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity:   0.4240 sec
Time to compute coloring:   0.0637 sec
Memory to compute coloring:   0.2500 MB
Coloring created on: 2024-08-29 21:47:52

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

Solution:
--------------------------------------------------------------------------------
Total Time:                    2.1860
User Objective Time :       0.1033
User Sensitivity Time :     1.8228
Interface Time :            0.1413
Opt Solver Time:            0.1186
Calls to Objective Function :      28
Calls to Sens Function :           28

Objectives
Index  Name                                Value
0  traj.burn2.states:deltav     3.989839E+01

Variables (c - continuous, i - integer, d - discrete)
Index  Name                         Type      Lower Bound            Value      Upper Bound     Status
0  traj.burn1.t_duration_0         c     5.000000E-01     2.228030E+00     1.000000E+01
1  traj.burn1.states:r_0           c    -1.000000E+21     1.002411E+00     1.000000E+21
2  traj.burn1.states:r_1           c    -1.000000E+21     1.002411E+00     1.000000E+21
3  traj.burn1.states:r_2           c    -1.000000E+21     1.021674E+00     1.000000E+21
4  traj.burn1.states:r_3           c    -1.000000E+21     1.021674E+00     1.000000E+21
5  traj.burn1.states:r_4           c    -1.000000E+21     1.072668E+00     1.000000E+21
6  traj.burn1.states:r_5           c    -1.000000E+21     1.072668E+00     1.000000E+21
7  traj.burn1.states:r_6           c    -1.000000E+21     1.162426E+00     1.000000E+21
8  traj.burn1.states:r_7           c    -1.000000E+21     1.162426E+00     1.000000E+21
9  traj.burn1.states:r_8           c    -1.000000E+21     1.289870E+00     1.000000E+21
10  traj.burn1.states:theta_0       c    -1.000000E+21     4.551154E-01     1.000000E+21
11  traj.burn1.states:theta_1       c    -1.000000E+21     4.551154E-01     1.000000E+21
12  traj.burn1.states:theta_2       c    -1.000000E+21     9.219251E-01     1.000000E+21
13  traj.burn1.states:theta_3       c    -1.000000E+21     9.219251E-01     1.000000E+21
14  traj.burn1.states:theta_4       c    -1.000000E+21     1.378635E+00     1.000000E+21
15  traj.burn1.states:theta_5       c    -1.000000E+21     1.378635E+00     1.000000E+21
16  traj.burn1.states:theta_6       c    -1.000000E+21     1.798766E+00     1.000000E+21
17  traj.burn1.states:theta_7       c    -1.000000E+21     1.798766E+00     1.000000E+21
18  traj.burn1.states:theta_8       c    -1.000000E+21     2.165052E+00     1.000000E+21
19  traj.burn1.states:vr_0          c    -1.000000E+21     1.772243E-02     1.000000E+21
20  traj.burn1.states:vr_1          c    -1.000000E+21     1.772243E-02     1.000000E+21
21  traj.burn1.states:vr_2          c    -1.000000E+21     7.447792E-02     1.000000E+21
22  traj.burn1.states:vr_3          c    -1.000000E+21     7.447792E-02     1.000000E+21
23  traj.burn1.states:vr_4          c    -1.000000E+21     1.570336E-01     1.000000E+21
24  traj.burn1.states:vr_5          c    -1.000000E+21     1.570336E-01     1.000000E+21
25  traj.burn1.states:vr_6          c    -1.000000E+21     2.452608E-01     1.000000E+21
26  traj.burn1.states:vr_7          c    -1.000000E+21     2.452608E-01     1.000000E+21
27  traj.burn1.states:vr_8          c    -1.000000E+21     3.247035E-01     1.000000E+21
28  traj.burn1.states:vt_0          c    -1.000000E+21     1.042686E+00     1.000000E+21
29  traj.burn1.states:vt_1          c    -1.000000E+21     1.042686E+00     1.000000E+21
30  traj.burn1.states:vt_2          c    -1.000000E+21     1.069132E+00     1.000000E+21
31  traj.burn1.states:vt_3          c    -1.000000E+21     1.069132E+00     1.000000E+21
32  traj.burn1.states:vt_4          c    -1.000000E+21     1.065074E+00     1.000000E+21
33  traj.burn1.states:vt_5          c    -1.000000E+21     1.065074E+00     1.000000E+21
34  traj.burn1.states:vt_6          c    -1.000000E+21     1.030094E+00     1.000000E+21
35  traj.burn1.states:vt_7          c    -1.000000E+21     1.030094E+00     1.000000E+21
36  traj.burn1.states:vt_8          c    -1.000000E+21     9.761629E-01     1.000000E+21
37  traj.burn1.states:accel_0       c    -1.000000E+21     1.030617E-01     1.000000E+21
38  traj.burn1.states:accel_1       c    -1.000000E+21     1.030617E-01     1.000000E+21
39  traj.burn1.states:accel_2       c    -1.000000E+21     1.063167E-01     1.000000E+21
40  traj.burn1.states:accel_3       c    -1.000000E+21     1.063167E-01     1.000000E+21
41  traj.burn1.states:accel_4       c    -1.000000E+21     1.097841E-01     1.000000E+21
42  traj.burn1.states:accel_5       c    -1.000000E+21     1.097841E-01     1.000000E+21
43  traj.burn1.states:accel_6       c    -1.000000E+21     1.134853E-01     1.000000E+21
44  traj.burn1.states:accel_7       c    -1.000000E+21     1.134853E-01     1.000000E+21
45  traj.burn1.states:accel_8       c    -1.000000E+21     1.174447E-01     1.000000E+21
46  traj.burn1.states:deltav_0      c    -1.000000E+21     4.523589E-02     1.000000E+21
47  traj.burn1.states:deltav_1      c    -1.000000E+21     4.523589E-02     1.000000E+21
48  traj.burn1.states:deltav_2      c    -1.000000E+21     9.187850E-02     1.000000E+21
49  traj.burn1.states:deltav_3      c    -1.000000E+21     9.187850E-02     1.000000E+21
50  traj.burn1.states:deltav_4      c    -1.000000E+21     1.400181E-01     1.000000E+21
51  traj.burn1.states:deltav_5      c    -1.000000E+21     1.400181E-01     1.000000E+21
52  traj.burn1.states:deltav_6      c    -1.000000E+21     1.897541E-01     1.000000E+21
53  traj.burn1.states:deltav_7      c    -1.000000E+21     1.897541E-01     1.000000E+21
54  traj.burn1.states:deltav_8      c    -1.000000E+21     2.411959E-01     1.000000E+21
55  traj.burn1.controls:u1_0        c    -3.000000E-01    -3.871082E-02     3.000000E-01
56  traj.burn1.controls:u1_1        c    -3.000000E-01    -2.977418E-02     3.000000E-01
57  traj.burn1.controls:u1_2        c    -3.000000E-01    -1.798095E-02     3.000000E-01
58  traj.burn1.controls:u1_3        c    -3.000000E-01    -1.798095E-02     3.000000E-01
59  traj.burn1.controls:u1_4        c    -3.000000E-01    -3.205161E-03     3.000000E-01
60  traj.burn1.controls:u1_5        c    -3.000000E-01     1.467911E-02     3.000000E-01
61  traj.burn1.controls:u1_6        c    -3.000000E-01     1.467911E-02     3.000000E-01
62  traj.burn1.controls:u1_7        c    -3.000000E-01     3.450564E-02     3.000000E-01
63  traj.burn1.controls:u1_8        c    -3.000000E-01     5.510823E-02     3.000000E-01
64  traj.burn1.controls:u1_9        c    -3.000000E-01     5.510823E-02     3.000000E-01
65  traj.burn1.controls:u1_10       c    -3.000000E-01     7.549970E-02     3.000000E-01
66  traj.burn1.controls:u1_11       c    -3.000000E-01     9.469292E-02     3.000000E-01
67  traj.burn1.controls:u1_12       c    -3.000000E-01     9.469292E-02     3.000000E-01
68  traj.burn1.controls:u1_13       c    -3.000000E-01     1.125798E-01     3.000000E-01
69  traj.burn1.controls:u1_14       c    -3.000000E-01     1.290522E-01     3.000000E-01
70  traj.burn2.t_initial_0          c     5.000000E-02     9.614472E-01     5.000000E+00
71  traj.burn2.t_duration_0         c     5.000000E-01     1.275260E+00     1.000000E+01
72  traj.burn2.states:r_0           c    -1.000000E+21     2.985662E+00     1.000000E+21
73  traj.burn2.states:r_1           c    -1.000000E+21     2.992648E+00     1.000000E+21
74  traj.burn2.states:r_2           c    -1.000000E+21     2.992648E+00     1.000000E+21
75  traj.burn2.states:r_3           c    -1.000000E+21     2.996948E+00     1.000000E+21
76  traj.burn2.states:r_4           c    -1.000000E+21     2.996948E+00     1.000000E+21
77  traj.burn2.states:r_5           c    -1.000000E+21     2.999153E+00     1.000000E+21
78  traj.burn2.states:r_6           c    -1.000000E+21     2.999153E+00     1.000000E+21
79  traj.burn2.states:r_7           c    -1.000000E+21     2.999922E+00     1.000000E+21
80  traj.burn2.states:r_8           c    -1.000000E+21     2.999922E+00     1.000000E+21
81  traj.burn2.states:theta_0       c    -1.000000E+21     4.156692E+00     1.000000E+21
82  traj.burn2.states:theta_1       c    -1.000000E+21     4.193917E+00     1.000000E+21
83  traj.burn2.states:theta_2       c    -1.000000E+21     4.193917E+00     1.000000E+21
84  traj.burn2.states:theta_3       c    -1.000000E+21     4.233604E+00     1.000000E+21
85  traj.burn2.states:theta_4       c    -1.000000E+21     4.233604E+00     1.000000E+21
86  traj.burn2.states:theta_5       c    -1.000000E+21     4.275859E+00     1.000000E+21
87  traj.burn2.states:theta_6       c    -1.000000E+21     4.275859E+00     1.000000E+21
88  traj.burn2.states:theta_7       c    -1.000000E+21     4.320784E+00     1.000000E+21
89  traj.burn2.states:theta_8       c    -1.000000E+21     4.320784E+00     1.000000E+21
90  traj.burn2.states:theta_9       c    -1.000000E+21     4.368465E+00     1.000000E+21
91  traj.burn2.states:vr_0          c    -1.000000E+21     3.335965E-02     1.000000E+21
92  traj.burn2.states:vr_1          c    -1.000000E+21     2.176356E-02     1.000000E+21
93  traj.burn2.states:vr_2          c    -1.000000E+21     2.176356E-02     1.000000E+21
94  traj.burn2.states:vr_3          c    -1.000000E+21     1.234448E-02     1.000000E+21
95  traj.burn2.states:vr_4          c    -1.000000E+21     1.234448E-02     1.000000E+21
96  traj.burn2.states:vr_5          c    -1.000000E+21     5.372873E-03     1.000000E+21
97  traj.burn2.states:vr_6          c    -1.000000E+21     5.372873E-03     1.000000E+21
98  traj.burn2.states:vr_7          c    -1.000000E+21     1.148444E-03     1.000000E+21
99  traj.burn2.states:vr_8          c    -1.000000E+21     1.148444E-03     1.000000E+21
100  traj.burn2.states:vt_0          c    -1.000000E+21     4.217795E-01     1.000000E+21
101  traj.burn2.states:vt_1          c    -1.000000E+21     4.510118E-01     1.000000E+21
102  traj.burn2.states:vt_2          c    -1.000000E+21     4.510118E-01     1.000000E+21
103  traj.burn2.states:vt_3          c    -1.000000E+21     4.812208E-01     1.000000E+21
104  traj.burn2.states:vt_4          c    -1.000000E+21     4.812208E-01     1.000000E+21
105  traj.burn2.states:vt_5          c    -1.000000E+21     5.123842E-01     1.000000E+21
106  traj.burn2.states:vt_6          c    -1.000000E+21     5.123842E-01     1.000000E+21
107  traj.burn2.states:vt_7          c    -1.000000E+21     5.444541E-01     1.000000E+21
108  traj.burn2.states:vt_8          c    -1.000000E+21     5.444541E-01     1.000000E+21
109  traj.burn2.states:accel_0       c    -1.000000E+21     1.174447E-01     1.000000E+21
110  traj.burn2.states:accel_1       c    -1.000000E+21     1.198378E-01     1.000000E+21
111  traj.burn2.states:accel_2       c    -1.000000E+21     1.198378E-01     1.000000E+21
112  traj.burn2.states:accel_3       c    -1.000000E+21     1.223305E-01     1.000000E+21
113  traj.burn2.states:accel_4       c    -1.000000E+21     1.223305E-01     1.000000E+21
114  traj.burn2.states:accel_5       c    -1.000000E+21     1.249291E-01     1.000000E+21
115  traj.burn2.states:accel_6       c    -1.000000E+21     1.249291E-01     1.000000E+21
116  traj.burn2.states:accel_7       c    -1.000000E+21     1.276404E-01     1.000000E+21
117  traj.burn2.states:accel_8       c    -1.000000E+21     1.276404E-01     1.000000E+21
118  traj.burn2.states:accel_9       c    -1.000000E+21     1.304721E-01     1.000000E+21
119  traj.burn2.states:deltav_0      c    -1.000000E+21     2.411959E-01     1.000000E+21
120  traj.burn2.states:deltav_1      c    -1.000000E+21     2.714535E-01     1.000000E+21
121  traj.burn2.states:deltav_2      c    -1.000000E+21     2.714535E-01     1.000000E+21
122  traj.burn2.states:deltav_3      c    -1.000000E+21     3.023341E-01     1.000000E+21
123  traj.burn2.states:deltav_4      c    -1.000000E+21     3.023341E-01     1.000000E+21
124  traj.burn2.states:deltav_5      c    -1.000000E+21     3.338638E-01     1.000000E+21
125  traj.burn2.states:deltav_6      c    -1.000000E+21     3.338638E-01     1.000000E+21
126  traj.burn2.states:deltav_7      c    -1.000000E+21     3.660705E-01     1.000000E+21
127  traj.burn2.states:deltav_8      c    -1.000000E+21     3.660705E-01     1.000000E+21
128  traj.burn2.states:deltav_9      c    -1.000000E+21     3.989839E-01     1.000000E+21
129  traj.burn2.controls:u1_0        c    -9.000000E-01     1.545347E-02     9.000000E-01
130  traj.burn2.controls:u1_1        c    -9.000000E-01     1.315257E-02     9.000000E-01
131  traj.burn2.controls:u1_2        c    -9.000000E-01     1.113035E-02     9.000000E-01
132  traj.burn2.controls:u1_3        c    -9.000000E-01     1.113035E-02     9.000000E-01
133  traj.burn2.controls:u1_4        c    -9.000000E-01     9.453512E-03     9.000000E-01
134  traj.burn2.controls:u1_5        c    -9.000000E-01     8.188743E-03     9.000000E-01
135  traj.burn2.controls:u1_6        c    -9.000000E-01     8.188743E-03     9.000000E-01
136  traj.burn2.controls:u1_7        c    -9.000000E-01     7.310470E-03     9.000000E-01
137  traj.burn2.controls:u1_8        c    -9.000000E-01     6.793120E-03     9.000000E-01
138  traj.burn2.controls:u1_9        c    -9.000000E-01     6.793120E-03     9.000000E-01
139  traj.burn2.controls:u1_10       c    -9.000000E-01     6.670239E-03     9.000000E-01
140  traj.burn2.controls:u1_11       c    -9.000000E-01     6.975374E-03     9.000000E-01
141  traj.burn2.controls:u1_12       c    -9.000000E-01     6.975374E-03     9.000000E-01
142  traj.burn2.controls:u1_13       c    -9.000000E-01     7.640932E-03     9.000000E-01
143  traj.burn2.controls:u1_14       c    -9.000000E-01     8.599319E-03     9.000000E-01
144  traj.coast.t_initial_0          c     5.000000E-01     2.228030E+00     2.000000E+01
145  traj.coast.t_duration_0         c     1.000000E-02     1.477288E-01     1.000000E+00
146  traj.coast.states:r_0           c    -1.000000E+21     1.289870E+00     1.000000E+21
147  traj.coast.states:r_1           c    -1.000000E+21     1.826968E+00     1.000000E+21
148  traj.coast.states:r_2           c    -1.000000E+21     1.826968E+00     1.000000E+21
149  traj.coast.states:r_3           c    -1.000000E+21     2.302606E+00     1.000000E+21
150  traj.coast.states:r_4           c    -1.000000E+21     2.302606E+00     1.000000E+21
151  traj.coast.states:r_5           c    -1.000000E+21     2.652890E+00     1.000000E+21
152  traj.coast.states:r_6           c    -1.000000E+21     2.652890E+00     1.000000E+21
153  traj.coast.states:r_7           c    -1.000000E+21     2.878553E+00     1.000000E+21
154  traj.coast.states:r_8           c    -1.000000E+21     2.878553E+00     1.000000E+21
155  traj.coast.states:r_9           c    -1.000000E+21     2.985662E+00     1.000000E+21
156  traj.coast.states:theta_0       c    -1.000000E+21     2.165052E+00     1.000000E+21
157  traj.coast.states:theta_1       c    -1.000000E+21     2.959738E+00     1.000000E+21
158  traj.coast.states:theta_2       c    -1.000000E+21     2.959738E+00     1.000000E+21
159  traj.coast.states:theta_3       c    -1.000000E+21     3.397868E+00     1.000000E+21
160  traj.coast.states:theta_4       c    -1.000000E+21     3.397868E+00     1.000000E+21
161  traj.coast.states:theta_5       c    -1.000000E+21     3.699817E+00     1.000000E+21
162  traj.coast.states:theta_6       c    -1.000000E+21     3.699817E+00     1.000000E+21
163  traj.coast.states:theta_7       c    -1.000000E+21     3.941656E+00     1.000000E+21
164  traj.coast.states:theta_8       c    -1.000000E+21     3.941656E+00     1.000000E+21
165  traj.coast.states:theta_9       c    -1.000000E+21     4.156692E+00     1.000000E+21
166  traj.coast.states:vr_0          c    -1.000000E+21     3.247035E-01     1.000000E+21
167  traj.coast.states:vr_1          c    -1.000000E+21     3.587987E-01     1.000000E+21
168  traj.coast.states:vr_2          c    -1.000000E+21     3.587987E-01     1.000000E+21
169  traj.coast.states:vr_3          c    -1.000000E+21     2.804133E-01     1.000000E+21
170  traj.coast.states:vr_4          c    -1.000000E+21     2.804133E-01     1.000000E+21
171  traj.coast.states:vr_5          c    -1.000000E+21     1.941954E-01     1.000000E+21
172  traj.coast.states:vr_6          c    -1.000000E+21     1.941954E-01     1.000000E+21
173  traj.coast.states:vr_7          c    -1.000000E+21     1.120415E-01     1.000000E+21
174  traj.coast.states:vr_8          c    -1.000000E+21     1.120415E-01     1.000000E+21
175  traj.coast.states:vr_9          c    -1.000000E+21     3.335965E-02     1.000000E+21
176  traj.coast.states:vt_0          c    -1.000000E+21     9.761629E-01     1.000000E+21
177  traj.coast.states:vt_1          c    -1.000000E+21     6.893012E-01     1.000000E+21
178  traj.coast.states:vt_2          c    -1.000000E+21     6.893012E-01     1.000000E+21
179  traj.coast.states:vt_3          c    -1.000000E+21     5.469078E-01     1.000000E+21
180  traj.coast.states:vt_4          c    -1.000000E+21     5.469078E-01     1.000000E+21
181  traj.coast.states:vt_5          c    -1.000000E+21     4.746893E-01     1.000000E+21
182  traj.coast.states:vt_6          c    -1.000000E+21     4.746893E-01     1.000000E+21
183  traj.coast.states:vt_7          c    -1.000000E+21     4.374743E-01     1.000000E+21
184  traj.coast.states:vt_8          c    -1.000000E+21     4.374743E-01     1.000000E+21
185  traj.coast.states:vt_9          c    -1.000000E+21     4.217795E-01     1.000000E+21
186  traj.coast.states:accel_0       c    -1.000000E+21     1.217790E-19     1.000000E+21
187  traj.coast.states:accel_1       c    -1.000000E+21     1.215696E-19     1.000000E+21
188  traj.coast.states:accel_2       c    -1.000000E+21     7.918339E-20     1.000000E+21
189  traj.coast.states:accel_3       c    -1.000000E+21     4.456949E-20     1.000000E+21
190  traj.coast.states:accel_4       c    -1.000000E+21    -5.252039E-22     1.000000E+21
191  traj.coast.states:accel_5       c    -1.000000E+21    -1.401990E-20     1.000000E+21
192  traj.coast.states:accel_6       c    -1.000000E+21    -1.389690E-20     1.000000E+21
193  traj.coast.states:accel_7       c    -1.000000E+21    -5.526146E-21     1.000000E+21
194  traj.coast.states:deltav_0      c    -1.000000E+21     2.411959E-01     1.000000E+21
195  traj.coast.states:deltav_1      c    -1.000000E+21     2.411959E-01     1.000000E+21
196  traj.coast.states:deltav_2      c    -1.000000E+21     2.411959E-01     1.000000E+21
197  traj.coast.states:deltav_3      c    -1.000000E+21     2.411959E-01     1.000000E+21
198  traj.coast.states:deltav_4      c    -1.000000E+21     2.411959E-01     1.000000E+21
199  traj.coast.states:deltav_5      c    -1.000000E+21     2.411959E-01     1.000000E+21
200  traj.coast.states:deltav_6      c    -1.000000E+21     2.411959E-01     1.000000E+21
201  traj.coast.states:deltav_7      c    -1.000000E+21     2.411959E-01     1.000000E+21
202  traj.coast.states:deltav_8      c    -1.000000E+21     2.411959E-01     1.000000E+21
203  traj.coast.states:deltav_9      c    -1.000000E+21     2.411959E-01     1.000000E+21

Constraints (i - inequality, e - equality)
Index  Name                                                    Type          Lower           Value           Upper    Status  Lagrange Multiplier (N/A)
0  traj.linkages.burn1:time_final|coast:time_initial          e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
1  traj.linkages.burn1:r_final|coast:r_initial                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
2  traj.linkages.burn1:theta_final|coast:theta_initial        e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
3  traj.linkages.burn1:vr_final|coast:vr_initial              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
4  traj.linkages.burn1:vt_final|coast:vt_initial              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
5  traj.linkages.burn1:deltav_final|coast:deltav_initial      e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
6  traj.linkages.coast:time_final|burn2:time_initial          e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
7  traj.linkages.coast:r_final|burn2:r_initial                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
8  traj.linkages.coast:theta_final|burn2:theta_initial        e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
9  traj.linkages.coast:vr_final|burn2:vr_initial              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
10  traj.linkages.coast:vt_final|burn2:vt_initial              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
11  traj.linkages.coast:deltav_final|burn2:deltav_initial      e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
12  traj.linkages.burn1:accel_final|burn2:accel_initial        e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
13  traj.burn1.collocation_constraint.defects:r                e   0.000000E+00    3.633115E-15    0.000000E+00              9.00000E+100
14  traj.burn1.collocation_constraint.defects:r                e   0.000000E+00    2.164409E-15    0.000000E+00              9.00000E+100
15  traj.burn1.collocation_constraint.defects:r                e   0.000000E+00   -3.092013E-16    0.000000E+00              9.00000E+100
16  traj.burn1.collocation_constraint.defects:r                e   0.000000E+00   -1.855208E-15    0.000000E+00              9.00000E+100
17  traj.burn1.collocation_constraint.defects:r                e   0.000000E+00   -1.360486E-14    0.000000E+00              9.00000E+100
18  traj.burn1.collocation_constraint.defects:theta            e   0.000000E+00   -4.947221E-15    0.000000E+00              9.00000E+100
19  traj.burn1.collocation_constraint.defects:theta            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
20  traj.burn1.collocation_constraint.defects:theta            e   0.000000E+00   -2.473610E-14    0.000000E+00              9.00000E+100
21  traj.burn1.collocation_constraint.defects:theta            e   0.000000E+00    1.484166E-14    0.000000E+00              9.00000E+100
22  traj.burn1.collocation_constraint.defects:theta            e   0.000000E+00    2.226249E-14    0.000000E+00              9.00000E+100
23  traj.burn1.collocation_constraint.defects:vr               e   0.000000E+00    4.947221E-15    0.000000E+00              9.00000E+100
24  traj.burn1.collocation_constraint.defects:vr               e   0.000000E+00   -1.422326E-14    0.000000E+00              9.00000E+100
25  traj.burn1.collocation_constraint.defects:vr               e   0.000000E+00   -1.422326E-14    0.000000E+00              9.00000E+100
26  traj.burn1.collocation_constraint.defects:vr               e   0.000000E+00    4.328818E-15    0.000000E+00              9.00000E+100
27  traj.burn1.collocation_constraint.defects:vr               e   0.000000E+00   -6.184026E-15    0.000000E+00              9.00000E+100
28  traj.burn1.collocation_constraint.defects:vt               e   0.000000E+00   -5.874825E-15    0.000000E+00              9.00000E+100
29  traj.burn1.collocation_constraint.defects:vt               e   0.000000E+00    1.345026E-14    0.000000E+00              9.00000E+100
30  traj.burn1.collocation_constraint.defects:vt               e   0.000000E+00   -4.638019E-16    0.000000E+00              9.00000E+100
31  traj.burn1.collocation_constraint.defects:vt               e   0.000000E+00   -6.802428E-15    0.000000E+00              9.00000E+100
32  traj.burn1.collocation_constraint.defects:vt               e   0.000000E+00    1.422326E-14    0.000000E+00              9.00000E+100
33  traj.burn1.collocation_constraint.defects:accel            e   0.000000E+00   -7.730032E-19    0.000000E+00              9.00000E+100
34  traj.burn1.collocation_constraint.defects:accel            e   0.000000E+00   -8.696286E-18    0.000000E+00              9.00000E+100
35  traj.burn1.collocation_constraint.defects:accel            e   0.000000E+00   -3.478515E-18    0.000000E+00              9.00000E+100
36  traj.burn1.collocation_constraint.defects:accel            e   0.000000E+00    1.043554E-17    0.000000E+00              9.00000E+100
37  traj.burn1.collocation_constraint.defects:accel            e   0.000000E+00   -1.314105E-17    0.000000E+00              9.00000E+100
38  traj.burn1.collocation_constraint.defects:deltav           e   0.000000E+00    3.092013E-18    0.000000E+00              9.00000E+100
39  traj.burn1.collocation_constraint.defects:deltav           e   0.000000E+00   -1.236805E-17    0.000000E+00              9.00000E+100
40  traj.burn1.collocation_constraint.defects:deltav           e   0.000000E+00    1.855208E-17    0.000000E+00              9.00000E+100
41  traj.burn1.collocation_constraint.defects:deltav           e   0.000000E+00    3.092013E-18    0.000000E+00              9.00000E+100
42  traj.burn1.collocation_constraint.defects:deltav           e   0.000000E+00   -1.236805E-17    0.000000E+00              9.00000E+100
43  traj.burn1.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
44  traj.burn1.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
45  traj.burn1.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
46  traj.burn1.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
47  traj.burn1.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
48  traj.burn1.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
49  traj.burn1.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
50  traj.burn1.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
51  traj.burn1.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
52  traj.burn1.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
53  traj.burn1.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
54  traj.burn1.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
55  traj.burn1.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
56  traj.burn1.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
57  traj.burn1.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
58  traj.burn1.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
59  traj.burn1.continuity_comp.defect_states:accel             e   0.000000E+00    1.387779E-17    0.000000E+00              9.00000E+100
60  traj.burn1.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
61  traj.burn1.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
62  traj.burn1.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
63  traj.burn1.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
64  traj.burn1.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
65  traj.burn1.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
66  traj.burn1.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
67  traj.burn1.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00   -3.957777E-18    0.000000E+00              9.00000E+100
68  traj.burn1.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00   -5.936665E-18    0.000000E+00              9.00000E+100
69  traj.burn1.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00   -5.936665E-18    0.000000E+00              9.00000E+100
70  traj.burn1.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00    1.187333E-17    0.000000E+00              9.00000E+100
71  traj.burn1.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
72  traj.burn1.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
73  traj.burn1.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
74  traj.burn1.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
75  traj.burn2.collocation_constraint.defects:r                e   0.000000E+00   -3.451068E-15    0.000000E+00              9.00000E+100
76  traj.burn2.collocation_constraint.defects:r                e   0.000000E+00    7.141056E-14    0.000000E+00              9.00000E+100
77  traj.burn2.collocation_constraint.defects:r                e   0.000000E+00   -3.086051E-14    0.000000E+00              9.00000E+100
78  traj.burn2.collocation_constraint.defects:r                e   0.000000E+00   -1.817341E-14    0.000000E+00              9.00000E+100
79  traj.burn2.collocation_constraint.defects:r                e   0.000000E+00    2.255085E-15    0.000000E+00              9.00000E+100
80  traj.burn2.collocation_constraint.defects:theta            e   0.000000E+00    8.848893E-15    0.000000E+00              9.00000E+100
81  traj.burn2.collocation_constraint.defects:theta            e   0.000000E+00    2.619272E-14    0.000000E+00              9.00000E+100
82  traj.burn2.collocation_constraint.defects:theta            e   0.000000E+00   -7.433070E-15    0.000000E+00              9.00000E+100
83  traj.burn2.collocation_constraint.defects:theta            e   0.000000E+00    4.459842E-14    0.000000E+00              9.00000E+100
84  traj.burn2.collocation_constraint.defects:theta            e   0.000000E+00   -2.123734E-15    0.000000E+00              9.00000E+100
85  traj.burn2.collocation_constraint.defects:vr               e   0.000000E+00    5.309336E-15    0.000000E+00              9.00000E+100
86  traj.burn2.collocation_constraint.defects:vr               e   0.000000E+00    5.309336E-15    0.000000E+00              9.00000E+100
87  traj.burn2.collocation_constraint.defects:vr               e   0.000000E+00   -4.424446E-15    0.000000E+00              9.00000E+100
88  traj.burn2.collocation_constraint.defects:vr               e   0.000000E+00    3.982002E-15    0.000000E+00              9.00000E+100
89  traj.burn2.collocation_constraint.defects:vr               e   0.000000E+00    2.654668E-15    0.000000E+00              9.00000E+100
90  traj.burn2.collocation_constraint.defects:vt               e   0.000000E+00    2.477690E-14    0.000000E+00              9.00000E+100
91  traj.burn2.collocation_constraint.defects:vt               e   0.000000E+00    4.955380E-14    0.000000E+00              9.00000E+100
92  traj.burn2.collocation_constraint.defects:vt               e   0.000000E+00   -4.070491E-14    0.000000E+00              9.00000E+100
93  traj.burn2.collocation_constraint.defects:vt               e   0.000000E+00    2.831646E-14    0.000000E+00              9.00000E+100
94  traj.burn2.collocation_constraint.defects:vt               e   0.000000E+00    1.769779E-14    0.000000E+00              9.00000E+100
95  traj.burn2.collocation_constraint.defects:accel            e   0.000000E+00    6.857892E-18    0.000000E+00              9.00000E+100
96  traj.burn2.collocation_constraint.defects:accel            e   0.000000E+00    1.747656E-17    0.000000E+00              9.00000E+100
97  traj.burn2.collocation_constraint.defects:accel            e   0.000000E+00   -3.185601E-17    0.000000E+00              9.00000E+100
98  traj.burn2.collocation_constraint.defects:accel            e   0.000000E+00    1.858267E-17    0.000000E+00              9.00000E+100
99  traj.burn2.collocation_constraint.defects:accel            e   0.000000E+00   -4.424446E-18    0.000000E+00              9.00000E+100
100  traj.burn2.collocation_constraint.defects:deltav           e   0.000000E+00    3.008624E-17    0.000000E+00              9.00000E+100
101  traj.burn2.collocation_constraint.defects:deltav           e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
102  traj.burn2.collocation_constraint.defects:deltav           e   0.000000E+00    5.309336E-18    0.000000E+00              9.00000E+100
103  traj.burn2.collocation_constraint.defects:deltav           e   0.000000E+00    1.061867E-17    0.000000E+00              9.00000E+100
104  traj.burn2.collocation_constraint.defects:deltav           e   0.000000E+00   -2.123734E-17    0.000000E+00              9.00000E+100
105  traj.burn2.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
106  traj.burn2.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
107  traj.burn2.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
108  traj.burn2.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
109  traj.burn2.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
110  traj.burn2.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
111  traj.burn2.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
112  traj.burn2.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
113  traj.burn2.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
114  traj.burn2.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
115  traj.burn2.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
116  traj.burn2.continuity_comp.defect_states:vr                e   0.000000E+00   -2.168404E-19    0.000000E+00              9.00000E+100
117  traj.burn2.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
118  traj.burn2.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
119  traj.burn2.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
120  traj.burn2.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
121  traj.burn2.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
122  traj.burn2.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
123  traj.burn2.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
124  traj.burn2.continuity_comp.defect_states:accel             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
125  traj.burn2.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
126  traj.burn2.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
127  traj.burn2.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
128  traj.burn2.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
129  traj.burn2.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00    5.663291E-16    0.000000E+00              9.00000E+100
130  traj.burn2.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00   -9.202848E-16    0.000000E+00              9.00000E+100
131  traj.burn2.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00   -1.061867E-16    0.000000E+00              9.00000E+100
132  traj.burn2.continuity_comp.defect_control_rates:u1_rate    e   0.000000E+00    1.946756E-15    0.000000E+00              9.00000E+100
133  traj.burn2.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
134  traj.burn2.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
135  traj.burn2.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
136  traj.burn2.continuity_comp.defect_controls:u1              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
137  traj.coast.collocation_constraint.defects:r                e   0.000000E+00    4.100299E-15    0.000000E+00              9.00000E+100
138  traj.coast.collocation_constraint.defects:r                e   0.000000E+00   -2.460179E-14    0.000000E+00              9.00000E+100
139  traj.coast.collocation_constraint.defects:r                e   0.000000E+00    2.460179E-14    0.000000E+00              9.00000E+100
140  traj.coast.collocation_constraint.defects:r                e   0.000000E+00    1.845135E-14    0.000000E+00              9.00000E+100
141  traj.coast.collocation_constraint.defects:r                e   0.000000E+00   -3.075224E-14    0.000000E+00              9.00000E+100
142  traj.coast.collocation_constraint.defects:theta            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
143  traj.coast.collocation_constraint.defects:theta            e   0.000000E+00   -1.640120E-14    0.000000E+00              9.00000E+100
144  traj.coast.collocation_constraint.defects:theta            e   0.000000E+00   -4.305314E-14    0.000000E+00              9.00000E+100
145  traj.coast.collocation_constraint.defects:theta            e   0.000000E+00    2.050150E-14    0.000000E+00              9.00000E+100
146  traj.coast.collocation_constraint.defects:theta            e   0.000000E+00   -2.870209E-14    0.000000E+00              9.00000E+100
147  traj.coast.collocation_constraint.defects:vr               e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
148  traj.coast.collocation_constraint.defects:vr               e   0.000000E+00    1.025075E-15    0.000000E+00              9.00000E+100
149  traj.coast.collocation_constraint.defects:vr               e   0.000000E+00    2.050150E-15    0.000000E+00              9.00000E+100
150  traj.coast.collocation_constraint.defects:vr               e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
151  traj.coast.collocation_constraint.defects:vr               e   0.000000E+00   -5.125374E-16    0.000000E+00              9.00000E+100
152  traj.coast.collocation_constraint.defects:vt               e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
153  traj.coast.collocation_constraint.defects:vt               e   0.000000E+00    1.025075E-15    0.000000E+00              9.00000E+100
154  traj.coast.collocation_constraint.defects:vt               e   0.000000E+00   -8.713135E-15    0.000000E+00              9.00000E+100
155  traj.coast.collocation_constraint.defects:vt               e   0.000000E+00    1.025075E-15    0.000000E+00              9.00000E+100
156  traj.coast.collocation_constraint.defects:vt               e   0.000000E+00   -2.818956E-15    0.000000E+00              9.00000E+100
157  traj.coast.collocation_constraint.defects:accel            e   0.000000E+00    9.133423E-20    0.000000E+00              9.00000E+100
158  traj.coast.collocation_constraint.defects:accel            e   0.000000E+00   -3.178965E-20    0.000000E+00              9.00000E+100
159  traj.coast.collocation_constraint.defects:accel            e   0.000000E+00   -3.382102E-20    0.000000E+00              9.00000E+100
160  traj.coast.collocation_constraint.defects:accel            e   0.000000E+00    9.224784E-23    0.000000E+00              9.00000E+100
161  traj.coast.collocation_constraint.defects:accel            e   0.000000E+00    4.144610E-21    0.000000E+00              9.00000E+100
162  traj.coast.collocation_constraint.defects:deltav           e   0.000000E+00   -2.782304E-17    0.000000E+00              9.00000E+100
163  traj.coast.collocation_constraint.defects:deltav           e   0.000000E+00    2.764436E-17    0.000000E+00              9.00000E+100
164  traj.coast.collocation_constraint.defects:deltav           e   0.000000E+00   -2.439979E-20    0.000000E+00              9.00000E+100
165  traj.coast.collocation_constraint.defects:deltav           e   0.000000E+00    1.546544E-20    0.000000E+00              9.00000E+100
166  traj.coast.collocation_constraint.defects:deltav           e   0.000000E+00   -2.775251E-17    0.000000E+00              9.00000E+100
167  traj.coast.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
168  traj.coast.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
169  traj.coast.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
170  traj.coast.continuity_comp.defect_states:r                 e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
171  traj.coast.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
172  traj.coast.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
173  traj.coast.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
174  traj.coast.continuity_comp.defect_states:theta             e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
175  traj.coast.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
176  traj.coast.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
177  traj.coast.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
178  traj.coast.continuity_comp.defect_states:vr                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
179  traj.coast.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
180  traj.coast.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
181  traj.coast.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
182  traj.coast.continuity_comp.defect_states:vt                e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
183  traj.coast.continuity_comp.defect_states:accel             e   0.000000E+00   -2.093765E-22    0.000000E+00              9.00000E+100
184  traj.coast.continuity_comp.defect_states:accel             e   0.000000E+00   -3.461391E-20    0.000000E+00              9.00000E+100
185  traj.coast.continuity_comp.defect_states:accel             e   0.000000E+00   -1.349470E-20    0.000000E+00              9.00000E+100
186  traj.coast.continuity_comp.defect_states:accel             e   0.000000E+00    8.370756E-21    0.000000E+00              9.00000E+100
187  traj.coast.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
188  traj.coast.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
189  traj.coast.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
190  traj.coast.continuity_comp.defect_states:deltav            e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100

Exit Status
Inform  Description
0  Solve Succeeded
--------------------------------------------------------------------------------

/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:1104: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:u1_rate2] cannot be impacted by the design variables of the problem because no partials were defined for them in their parent component(s).
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/group.py:1104: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:u1_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

Problem: problem
Driver:  pyOptSparseDriver
success     : True
iterations  : 30
runtime     : 2.7833E+00 s
model_evals : 30
model_time  : 9.1182E-02 s
deriv_evals : 29
deriv_time  : 1.3275E+00 s
exit_status : SUCCESS


## Plotting the results#

The following code cell reads the resulting state, time, and control histories from the solution and simulation record files and plots them. It is collapsed by default but can be viewed by expanding it with the button to the right.

Hide code cell source
sol = om.CaseReader(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')

fig = plt.figure(figsize=(12, 6))
fig.suptitle('Two Burn Orbit Raise Solution')
ax_u1 = plt.subplot2grid((2, 2), (0, 0))
ax_deltav = plt.subplot2grid((2, 2), (1, 0))
ax_xy = plt.subplot2grid((2, 2), (0, 1), rowspan=2)

span = np.linspace(0, 2 * np.pi, 100)
ax_xy.plot(np.cos(span), np.sin(span), 'k--', lw=1)
ax_xy.plot(3 * np.cos(span), 3 * np.sin(span), 'k--', lw=1)
ax_xy.set_xlim(-4.5, 4.5)
ax_xy.set_ylim(-4.5, 4.5)

ax_xy.set_xlabel('x ($R_e$)')
ax_xy.set_ylabel('y ($R_e$)')
ax_xy.set_aspect('equal')
ax_xy.plot([0], [0], '*', color='gold')

ax_u1.set_ylabel('$u_1$ ($deg$)')
ax_u1.grid(True)

ax_deltav.set_xlabel('time ($TU$)')
ax_deltav.set_ylabel('${\Delta}v$ ($DU/TU$)')
ax_deltav.grid(True)

t_sol = dict((phs, sol.get_val(f'traj.{phs}.timeseries.time'.format(phs)))
for phs in ['burn1', 'coast', 'burn2'])
x_sol = dict((phs, sol.get_val(f'traj.{phs}.timeseries.pos_x'.format(phs)))
for phs in ['burn1', 'coast', 'burn2'])
y_sol = dict((phs, sol.get_val(f'traj.{phs}.timeseries.pos_y'.format(phs)))
for phs in ['burn1', 'coast', 'burn2'])
dv_sol = dict((phs, sol.get_val(f'traj.{phs}.timeseries.deltav'.format(phs)))
for phs in ['burn1', 'coast', 'burn2'])
u1_sol = dict((phs, sol.get_val(f'traj.{phs}.timeseries.u1'.format(phs), units='deg'))
for phs in ['burn1', 'burn2'])

t_exp = dict((phs, sim.get_val(f'traj.{phs}.timeseries.time'))
for phs in ['burn1', 'coast', 'burn2'])
x_exp = dict((phs, sim.get_val(f'traj.{phs}.timeseries.pos_x'))
for phs in ['burn1', 'coast', 'burn2'])
y_exp = dict((phs, sim.get_val(f'traj.{phs}.timeseries.pos_y'))
for phs in ['burn1', 'coast', 'burn2'])
dv_exp = dict((phs, sim.get_val(f'traj.{phs}.timeseries.deltav'))
for phs in ['burn1', 'coast', 'burn2'])
u1_exp = dict((phs, sim.get_val(f'traj.{phs}.timeseries.u1',
units='deg'))
for phs in ['burn1', 'burn2'])

for phs in ['burn1', 'coast', 'burn2']:
try:
ax_u1.plot(t_exp[phs], u1_exp[phs], '-', marker=None, color='C0')
ax_u1.plot(t_sol[phs], u1_sol[phs], 'o', mfc='C1', mec='C1', ms=3)
except KeyError:
pass

ax_deltav.plot(t_exp[phs], dv_exp[phs], '-', marker=None, color='C0')
ax_deltav.plot(t_sol[phs], dv_sol[phs], 'o', mfc='C1', mec='C1', ms=3)

exp_plt, = ax_xy.plot(x_exp[phs], y_exp[phs], '-', marker=None, color='C0', label='explicit')
imp_plt, = ax_xy.plot(x_sol[phs], y_sol[phs], 'o', mfc='C1', mec='C1', ms=3, label='implicit')

ax_xy.legend(handles=[imp_plt, exp_plt], labels=['solution', 'simulation'], loc='lower center', ncol=2)

plt.show()


## References#

[EC91]

Paul J Enright and Brace A Conway. Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming. Journal of Guidance, Control, and Dynamics, 14(5):981–985, 1991.