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.
Show 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
p.set_val('traj.burn1.t_initial', val=0.0)
p.set_val('traj.burn1.t_duration', val=2.25)
p.set_val('traj.burn1.states:r', val=burn1.interp('r', [1, 1.5]))
p.set_val('traj.burn1.states:theta', val=burn1.interp('theta', [0, 1.7]))
p.set_val('traj.burn1.states:vr', val=burn1.interp('vr', [0, 0]))
p.set_val('traj.burn1.states:vt', val=burn1.interp('vt', [1, 1]))
p.set_val('traj.burn1.states:accel', val=burn1.interp('accel', [0.1, 0]))
p.set_val('traj.burn1.states:deltav', val=burn1.interp('deltav', [0, 0.1]))
p.set_val('traj.burn1.controls:u1', val=burn1.interp('u1', [-3.5, 13.0]))
p.set_val('traj.coast.t_initial', val=2.25)
p.set_val('traj.coast.t_duration', val=3.0)
p.set_val('traj.coast.states:r', val=coast.interp('r', [1.3, 1.5]))
p.set_val('traj.coast.states:theta', val=coast.interp('theta', [2.1767, 1.7]))
p.set_val('traj.coast.states:vr', val=coast.interp('vr', [0.3285, 0]))
p.set_val('traj.coast.states:vt', val=coast.interp('vt', [0.97, 1]))
p.set_val('traj.coast.states:accel', val=coast.interp('accel', [0, 0]))
p.set_val('traj.burn2.t_initial', val=5.25)
p.set_val('traj.burn2.t_duration', val=1.75)
p.set_val('traj.burn2.states:r', val=burn2.interp('r', [1, 3.]))
p.set_val('traj.burn2.states:theta', val=burn2.interp('theta', [0, 4.0]))
p.set_val('traj.burn2.states:vr', val=burn2.interp('vr', [0, 0]))
p.set_val('traj.burn2.states:vt', val=burn2.interp('vt', [1, np.sqrt(1 / 3.)]))
p.set_val('traj.burn2.states:deltav', val=burn2.interp('deltav', [0.1, 0.2]))
p.set_val('traj.burn2.states:accel', val=burn2.interp('accel', [0.1, 0]))
p.set_val('traj.burn2.controls:u1', val=burn2.interp('u1', [0, 0]))
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
--- burn1 ---
None
--- coast ---
None
--- burn2 ---
None
INFO: checking out_of_order
INFO:check_config:checking out_of_order
INFO: checking system
INFO:check_config:checking system
INFO: checking solvers
INFO:check_config:checking solvers
INFO: checking dup_inputs
INFO:check_config:checking dup_inputs
INFO: checking missing_recorders
INFO:check_config:checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
WARNING:check_config:The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO:check_config:checking unserializable_options
INFO: checking comp_has_no_outputs
INFO:check_config:checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
INFO:check_config:checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO:check_config:checking out_of_order
INFO: checking system
INFO:check_config:checking system
INFO: checking solvers
INFO:check_config:checking solvers
INFO: checking dup_inputs
INFO:check_config:checking dup_inputs
INFO: checking missing_recorders
INFO:check_config:checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
WARNING:check_config:The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO:check_config:checking unserializable_options
INFO: checking comp_has_no_outputs
INFO:check_config:checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
INFO:check_config:checking auto_ivc_warnings
Full total jacobian was computed 3 times, taking 0.905676 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 (93.6% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.905676 sec.
Time to compute coloring: 0.147199 sec.
Memory to compute coloring: 0.250000 MB.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 3.8605
User Objective Time : 0.1918
User Sensitivity Time : 3.2162
Interface Time : 0.2476
Opt Solver Time: 0.2049
Calls to Objective Function : 28
Calls to Sens Function : 28
Objectives
Index Name Value
0 traj.phases.burn2.indep_states.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.phases.burn1.indep_states.states:r_0 c -1.000000E+21 1.002411E+00 1.000000E+21
2 traj.phases.burn1.indep_states.states:r_1 c -1.000000E+21 1.002411E+00 1.000000E+21
3 traj.phases.burn1.indep_states.states:r_2 c -1.000000E+21 1.021674E+00 1.000000E+21
4 traj.phases.burn1.indep_states.states:r_3 c -1.000000E+21 1.021674E+00 1.000000E+21
5 traj.phases.burn1.indep_states.states:r_4 c -1.000000E+21 1.072668E+00 1.000000E+21
6 traj.phases.burn1.indep_states.states:r_5 c -1.000000E+21 1.072668E+00 1.000000E+21
7 traj.phases.burn1.indep_states.states:r_6 c -1.000000E+21 1.162426E+00 1.000000E+21
8 traj.phases.burn1.indep_states.states:r_7 c -1.000000E+21 1.162426E+00 1.000000E+21
9 traj.phases.burn1.indep_states.states:r_8 c -1.000000E+21 1.289870E+00 1.000000E+21
10 traj.phases.burn1.indep_states.states:theta_0 c -1.000000E+21 4.551154E-01 1.000000E+21
11 traj.phases.burn1.indep_states.states:theta_1 c -1.000000E+21 4.551154E-01 1.000000E+21
12 traj.phases.burn1.indep_states.states:theta_2 c -1.000000E+21 9.219251E-01 1.000000E+21
13 traj.phases.burn1.indep_states.states:theta_3 c -1.000000E+21 9.219251E-01 1.000000E+21
14 traj.phases.burn1.indep_states.states:theta_4 c -1.000000E+21 1.378635E+00 1.000000E+21
15 traj.phases.burn1.indep_states.states:theta_5 c -1.000000E+21 1.378635E+00 1.000000E+21
16 traj.phases.burn1.indep_states.states:theta_6 c -1.000000E+21 1.798766E+00 1.000000E+21
17 traj.phases.burn1.indep_states.states:theta_7 c -1.000000E+21 1.798766E+00 1.000000E+21
18 traj.phases.burn1.indep_states.states:theta_8 c -1.000000E+21 2.165052E+00 1.000000E+21
19 traj.phases.burn1.indep_states.states:vr_0 c -1.000000E+21 1.772243E-02 1.000000E+21
20 traj.phases.burn1.indep_states.states:vr_1 c -1.000000E+21 1.772243E-02 1.000000E+21
21 traj.phases.burn1.indep_states.states:vr_2 c -1.000000E+21 7.447792E-02 1.000000E+21
22 traj.phases.burn1.indep_states.states:vr_3 c -1.000000E+21 7.447792E-02 1.000000E+21
23 traj.phases.burn1.indep_states.states:vr_4 c -1.000000E+21 1.570336E-01 1.000000E+21
24 traj.phases.burn1.indep_states.states:vr_5 c -1.000000E+21 1.570336E-01 1.000000E+21
25 traj.phases.burn1.indep_states.states:vr_6 c -1.000000E+21 2.452608E-01 1.000000E+21
26 traj.phases.burn1.indep_states.states:vr_7 c -1.000000E+21 2.452608E-01 1.000000E+21
27 traj.phases.burn1.indep_states.states:vr_8 c -1.000000E+21 3.247035E-01 1.000000E+21
28 traj.phases.burn1.indep_states.states:vt_0 c -1.000000E+21 1.042686E+00 1.000000E+21
29 traj.phases.burn1.indep_states.states:vt_1 c -1.000000E+21 1.042686E+00 1.000000E+21
30 traj.phases.burn1.indep_states.states:vt_2 c -1.000000E+21 1.069132E+00 1.000000E+21
31 traj.phases.burn1.indep_states.states:vt_3 c -1.000000E+21 1.069132E+00 1.000000E+21
32 traj.phases.burn1.indep_states.states:vt_4 c -1.000000E+21 1.065074E+00 1.000000E+21
33 traj.phases.burn1.indep_states.states:vt_5 c -1.000000E+21 1.065074E+00 1.000000E+21
34 traj.phases.burn1.indep_states.states:vt_6 c -1.000000E+21 1.030094E+00 1.000000E+21
35 traj.phases.burn1.indep_states.states:vt_7 c -1.000000E+21 1.030094E+00 1.000000E+21
36 traj.phases.burn1.indep_states.states:vt_8 c -1.000000E+21 9.761629E-01 1.000000E+21
37 traj.phases.burn1.indep_states.states:accel_0 c -1.000000E+21 1.030617E-01 1.000000E+21
38 traj.phases.burn1.indep_states.states:accel_1 c -1.000000E+21 1.030617E-01 1.000000E+21
39 traj.phases.burn1.indep_states.states:accel_2 c -1.000000E+21 1.063167E-01 1.000000E+21
40 traj.phases.burn1.indep_states.states:accel_3 c -1.000000E+21 1.063167E-01 1.000000E+21
41 traj.phases.burn1.indep_states.states:accel_4 c -1.000000E+21 1.097841E-01 1.000000E+21
42 traj.phases.burn1.indep_states.states:accel_5 c -1.000000E+21 1.097841E-01 1.000000E+21
43 traj.phases.burn1.indep_states.states:accel_6 c -1.000000E+21 1.134853E-01 1.000000E+21
44 traj.phases.burn1.indep_states.states:accel_7 c -1.000000E+21 1.134853E-01 1.000000E+21
45 traj.phases.burn1.indep_states.states:accel_8 c -1.000000E+21 1.174447E-01 1.000000E+21
46 traj.phases.burn1.indep_states.states:deltav_0 c -1.000000E+21 4.523589E-02 1.000000E+21
47 traj.phases.burn1.indep_states.states:deltav_1 c -1.000000E+21 4.523589E-02 1.000000E+21
48 traj.phases.burn1.indep_states.states:deltav_2 c -1.000000E+21 9.187850E-02 1.000000E+21
49 traj.phases.burn1.indep_states.states:deltav_3 c -1.000000E+21 9.187850E-02 1.000000E+21
50 traj.phases.burn1.indep_states.states:deltav_4 c -1.000000E+21 1.400181E-01 1.000000E+21
51 traj.phases.burn1.indep_states.states:deltav_5 c -1.000000E+21 1.400181E-01 1.000000E+21
52 traj.phases.burn1.indep_states.states:deltav_6 c -1.000000E+21 1.897541E-01 1.000000E+21
53 traj.phases.burn1.indep_states.states:deltav_7 c -1.000000E+21 1.897541E-01 1.000000E+21
54 traj.phases.burn1.indep_states.states:deltav_8 c -1.000000E+21 2.411959E-01 1.000000E+21
55 traj.phases.burn1.control_group.indep_controls.controls:u1_0 c -3.000000E-01 -3.871082E-02 3.000000E-01
56 traj.phases.burn1.control_group.indep_controls.controls:u1_1 c -3.000000E-01 -2.977418E-02 3.000000E-01
57 traj.phases.burn1.control_group.indep_controls.controls:u1_2 c -3.000000E-01 -1.798095E-02 3.000000E-01
58 traj.phases.burn1.control_group.indep_controls.controls:u1_3 c -3.000000E-01 -1.798095E-02 3.000000E-01
59 traj.phases.burn1.control_group.indep_controls.controls:u1_4 c -3.000000E-01 -3.205161E-03 3.000000E-01
60 traj.phases.burn1.control_group.indep_controls.controls:u1_5 c -3.000000E-01 1.467911E-02 3.000000E-01
61 traj.phases.burn1.control_group.indep_controls.controls:u1_6 c -3.000000E-01 1.467911E-02 3.000000E-01
62 traj.phases.burn1.control_group.indep_controls.controls:u1_7 c -3.000000E-01 3.450564E-02 3.000000E-01
63 traj.phases.burn1.control_group.indep_controls.controls:u1_8 c -3.000000E-01 5.510823E-02 3.000000E-01
64 traj.phases.burn1.control_group.indep_controls.controls:u1_9 c -3.000000E-01 5.510823E-02 3.000000E-01
65 traj.phases.burn1.control_group.indep_controls.controls:u1_10 c -3.000000E-01 7.549970E-02 3.000000E-01
66 traj.phases.burn1.control_group.indep_controls.controls:u1_11 c -3.000000E-01 9.469292E-02 3.000000E-01
67 traj.phases.burn1.control_group.indep_controls.controls:u1_12 c -3.000000E-01 9.469292E-02 3.000000E-01
68 traj.phases.burn1.control_group.indep_controls.controls:u1_13 c -3.000000E-01 1.125798E-01 3.000000E-01
69 traj.phases.burn1.control_group.indep_controls.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.phases.burn2.indep_states.states:r_0 c -1.000000E+21 2.985662E+00 1.000000E+21
73 traj.phases.burn2.indep_states.states:r_1 c -1.000000E+21 2.992648E+00 1.000000E+21
74 traj.phases.burn2.indep_states.states:r_2 c -1.000000E+21 2.992648E+00 1.000000E+21
75 traj.phases.burn2.indep_states.states:r_3 c -1.000000E+21 2.996948E+00 1.000000E+21
76 traj.phases.burn2.indep_states.states:r_4 c -1.000000E+21 2.996948E+00 1.000000E+21
77 traj.phases.burn2.indep_states.states:r_5 c -1.000000E+21 2.999153E+00 1.000000E+21
78 traj.phases.burn2.indep_states.states:r_6 c -1.000000E+21 2.999153E+00 1.000000E+21
79 traj.phases.burn2.indep_states.states:r_7 c -1.000000E+21 2.999922E+00 1.000000E+21
80 traj.phases.burn2.indep_states.states:r_8 c -1.000000E+21 2.999922E+00 1.000000E+21
81 traj.phases.burn2.indep_states.states:theta_0 c -1.000000E+21 4.156692E+00 1.000000E+21
82 traj.phases.burn2.indep_states.states:theta_1 c -1.000000E+21 4.193917E+00 1.000000E+21
83 traj.phases.burn2.indep_states.states:theta_2 c -1.000000E+21 4.193917E+00 1.000000E+21
84 traj.phases.burn2.indep_states.states:theta_3 c -1.000000E+21 4.233604E+00 1.000000E+21
85 traj.phases.burn2.indep_states.states:theta_4 c -1.000000E+21 4.233604E+00 1.000000E+21
86 traj.phases.burn2.indep_states.states:theta_5 c -1.000000E+21 4.275859E+00 1.000000E+21
87 traj.phases.burn2.indep_states.states:theta_6 c -1.000000E+21 4.275859E+00 1.000000E+21
88 traj.phases.burn2.indep_states.states:theta_7 c -1.000000E+21 4.320784E+00 1.000000E+21
89 traj.phases.burn2.indep_states.states:theta_8 c -1.000000E+21 4.320784E+00 1.000000E+21
90 traj.phases.burn2.indep_states.states:theta_9 c -1.000000E+21 4.368465E+00 1.000000E+21
91 traj.phases.burn2.indep_states.states:vr_0 c -1.000000E+21 3.335965E-02 1.000000E+21
92 traj.phases.burn2.indep_states.states:vr_1 c -1.000000E+21 2.176356E-02 1.000000E+21
93 traj.phases.burn2.indep_states.states:vr_2 c -1.000000E+21 2.176356E-02 1.000000E+21
94 traj.phases.burn2.indep_states.states:vr_3 c -1.000000E+21 1.234448E-02 1.000000E+21
95 traj.phases.burn2.indep_states.states:vr_4 c -1.000000E+21 1.234448E-02 1.000000E+21
96 traj.phases.burn2.indep_states.states:vr_5 c -1.000000E+21 5.372873E-03 1.000000E+21
97 traj.phases.burn2.indep_states.states:vr_6 c -1.000000E+21 5.372873E-03 1.000000E+21
98 traj.phases.burn2.indep_states.states:vr_7 c -1.000000E+21 1.148444E-03 1.000000E+21
99 traj.phases.burn2.indep_states.states:vr_8 c -1.000000E+21 1.148444E-03 1.000000E+21
100 traj.phases.burn2.indep_states.states:vt_0 c -1.000000E+21 4.217795E-01 1.000000E+21
101 traj.phases.burn2.indep_states.states:vt_1 c -1.000000E+21 4.510118E-01 1.000000E+21
102 traj.phases.burn2.indep_states.states:vt_2 c -1.000000E+21 4.510118E-01 1.000000E+21
103 traj.phases.burn2.indep_states.states:vt_3 c -1.000000E+21 4.812208E-01 1.000000E+21
104 traj.phases.burn2.indep_states.states:vt_4 c -1.000000E+21 4.812208E-01 1.000000E+21
105 traj.phases.burn2.indep_states.states:vt_5 c -1.000000E+21 5.123842E-01 1.000000E+21
106 traj.phases.burn2.indep_states.states:vt_6 c -1.000000E+21 5.123842E-01 1.000000E+21
107 traj.phases.burn2.indep_states.states:vt_7 c -1.000000E+21 5.444541E-01 1.000000E+21
108 traj.phases.burn2.indep_states.states:vt_8 c -1.000000E+21 5.444541E-01 1.000000E+21
109 traj.phases.burn2.indep_states.states:accel_0 c -1.000000E+21 1.174447E-01 1.000000E+21
110 traj.phases.burn2.indep_states.states:accel_1 c -1.000000E+21 1.198378E-01 1.000000E+21
111 traj.phases.burn2.indep_states.states:accel_2 c -1.000000E+21 1.198378E-01 1.000000E+21
112 traj.phases.burn2.indep_states.states:accel_3 c -1.000000E+21 1.223305E-01 1.000000E+21
113 traj.phases.burn2.indep_states.states:accel_4 c -1.000000E+21 1.223305E-01 1.000000E+21
114 traj.phases.burn2.indep_states.states:accel_5 c -1.000000E+21 1.249291E-01 1.000000E+21
115 traj.phases.burn2.indep_states.states:accel_6 c -1.000000E+21 1.249291E-01 1.000000E+21
116 traj.phases.burn2.indep_states.states:accel_7 c -1.000000E+21 1.276404E-01 1.000000E+21
117 traj.phases.burn2.indep_states.states:accel_8 c -1.000000E+21 1.276404E-01 1.000000E+21
118 traj.phases.burn2.indep_states.states:accel_9 c -1.000000E+21 1.304721E-01 1.000000E+21
119 traj.phases.burn2.indep_states.states:deltav_0 c -1.000000E+21 2.411959E-01 1.000000E+21
120 traj.phases.burn2.indep_states.states:deltav_1 c -1.000000E+21 2.714535E-01 1.000000E+21
121 traj.phases.burn2.indep_states.states:deltav_2 c -1.000000E+21 2.714535E-01 1.000000E+21
122 traj.phases.burn2.indep_states.states:deltav_3 c -1.000000E+21 3.023341E-01 1.000000E+21
123 traj.phases.burn2.indep_states.states:deltav_4 c -1.000000E+21 3.023341E-01 1.000000E+21
124 traj.phases.burn2.indep_states.states:deltav_5 c -1.000000E+21 3.338638E-01 1.000000E+21
125 traj.phases.burn2.indep_states.states:deltav_6 c -1.000000E+21 3.338638E-01 1.000000E+21
126 traj.phases.burn2.indep_states.states:deltav_7 c -1.000000E+21 3.660705E-01 1.000000E+21
127 traj.phases.burn2.indep_states.states:deltav_8 c -1.000000E+21 3.660705E-01 1.000000E+21
128 traj.phases.burn2.indep_states.states:deltav_9 c -1.000000E+21 3.989839E-01 1.000000E+21
129 traj.phases.burn2.control_group.indep_controls.controls:u1_0 c -9.000000E-01 1.545347E-02 9.000000E-01
130 traj.phases.burn2.control_group.indep_controls.controls:u1_1 c -9.000000E-01 1.315257E-02 9.000000E-01
131 traj.phases.burn2.control_group.indep_controls.controls:u1_2 c -9.000000E-01 1.113035E-02 9.000000E-01
132 traj.phases.burn2.control_group.indep_controls.controls:u1_3 c -9.000000E-01 1.113035E-02 9.000000E-01
133 traj.phases.burn2.control_group.indep_controls.controls:u1_4 c -9.000000E-01 9.453512E-03 9.000000E-01
134 traj.phases.burn2.control_group.indep_controls.controls:u1_5 c -9.000000E-01 8.188743E-03 9.000000E-01
135 traj.phases.burn2.control_group.indep_controls.controls:u1_6 c -9.000000E-01 8.188743E-03 9.000000E-01
136 traj.phases.burn2.control_group.indep_controls.controls:u1_7 c -9.000000E-01 7.310470E-03 9.000000E-01
137 traj.phases.burn2.control_group.indep_controls.controls:u1_8 c -9.000000E-01 6.793120E-03 9.000000E-01
138 traj.phases.burn2.control_group.indep_controls.controls:u1_9 c -9.000000E-01 6.793120E-03 9.000000E-01
139 traj.phases.burn2.control_group.indep_controls.controls:u1_10 c -9.000000E-01 6.670239E-03 9.000000E-01
140 traj.phases.burn2.control_group.indep_controls.controls:u1_11 c -9.000000E-01 6.975374E-03 9.000000E-01
141 traj.phases.burn2.control_group.indep_controls.controls:u1_12 c -9.000000E-01 6.975374E-03 9.000000E-01
142 traj.phases.burn2.control_group.indep_controls.controls:u1_13 c -9.000000E-01 7.640932E-03 9.000000E-01
143 traj.phases.burn2.control_group.indep_controls.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.phases.coast.indep_states.states:r_0 c -1.000000E+21 1.289870E+00 1.000000E+21
147 traj.phases.coast.indep_states.states:r_1 c -1.000000E+21 1.826968E+00 1.000000E+21
148 traj.phases.coast.indep_states.states:r_2 c -1.000000E+21 1.826968E+00 1.000000E+21
149 traj.phases.coast.indep_states.states:r_3 c -1.000000E+21 2.302606E+00 1.000000E+21
150 traj.phases.coast.indep_states.states:r_4 c -1.000000E+21 2.302606E+00 1.000000E+21
151 traj.phases.coast.indep_states.states:r_5 c -1.000000E+21 2.652890E+00 1.000000E+21
152 traj.phases.coast.indep_states.states:r_6 c -1.000000E+21 2.652890E+00 1.000000E+21
153 traj.phases.coast.indep_states.states:r_7 c -1.000000E+21 2.878553E+00 1.000000E+21
154 traj.phases.coast.indep_states.states:r_8 c -1.000000E+21 2.878553E+00 1.000000E+21
155 traj.phases.coast.indep_states.states:r_9 c -1.000000E+21 2.985662E+00 1.000000E+21
156 traj.phases.coast.indep_states.states:theta_0 c -1.000000E+21 2.165052E+00 1.000000E+21
157 traj.phases.coast.indep_states.states:theta_1 c -1.000000E+21 2.959738E+00 1.000000E+21
158 traj.phases.coast.indep_states.states:theta_2 c -1.000000E+21 2.959738E+00 1.000000E+21
159 traj.phases.coast.indep_states.states:theta_3 c -1.000000E+21 3.397868E+00 1.000000E+21
160 traj.phases.coast.indep_states.states:theta_4 c -1.000000E+21 3.397868E+00 1.000000E+21
161 traj.phases.coast.indep_states.states:theta_5 c -1.000000E+21 3.699817E+00 1.000000E+21
162 traj.phases.coast.indep_states.states:theta_6 c -1.000000E+21 3.699817E+00 1.000000E+21
163 traj.phases.coast.indep_states.states:theta_7 c -1.000000E+21 3.941656E+00 1.000000E+21
164 traj.phases.coast.indep_states.states:theta_8 c -1.000000E+21 3.941656E+00 1.000000E+21
165 traj.phases.coast.indep_states.states:theta_9 c -1.000000E+21 4.156692E+00 1.000000E+21
166 traj.phases.coast.indep_states.states:vr_0 c -1.000000E+21 3.247035E-01 1.000000E+21
167 traj.phases.coast.indep_states.states:vr_1 c -1.000000E+21 3.587987E-01 1.000000E+21
168 traj.phases.coast.indep_states.states:vr_2 c -1.000000E+21 3.587987E-01 1.000000E+21
169 traj.phases.coast.indep_states.states:vr_3 c -1.000000E+21 2.804133E-01 1.000000E+21
170 traj.phases.coast.indep_states.states:vr_4 c -1.000000E+21 2.804133E-01 1.000000E+21
171 traj.phases.coast.indep_states.states:vr_5 c -1.000000E+21 1.941954E-01 1.000000E+21
172 traj.phases.coast.indep_states.states:vr_6 c -1.000000E+21 1.941954E-01 1.000000E+21
173 traj.phases.coast.indep_states.states:vr_7 c -1.000000E+21 1.120415E-01 1.000000E+21
174 traj.phases.coast.indep_states.states:vr_8 c -1.000000E+21 1.120415E-01 1.000000E+21
175 traj.phases.coast.indep_states.states:vr_9 c -1.000000E+21 3.335965E-02 1.000000E+21
176 traj.phases.coast.indep_states.states:vt_0 c -1.000000E+21 9.761629E-01 1.000000E+21
177 traj.phases.coast.indep_states.states:vt_1 c -1.000000E+21 6.893012E-01 1.000000E+21
178 traj.phases.coast.indep_states.states:vt_2 c -1.000000E+21 6.893012E-01 1.000000E+21
179 traj.phases.coast.indep_states.states:vt_3 c -1.000000E+21 5.469078E-01 1.000000E+21
180 traj.phases.coast.indep_states.states:vt_4 c -1.000000E+21 5.469078E-01 1.000000E+21
181 traj.phases.coast.indep_states.states:vt_5 c -1.000000E+21 4.746893E-01 1.000000E+21
182 traj.phases.coast.indep_states.states:vt_6 c -1.000000E+21 4.746893E-01 1.000000E+21
183 traj.phases.coast.indep_states.states:vt_7 c -1.000000E+21 4.374743E-01 1.000000E+21
184 traj.phases.coast.indep_states.states:vt_8 c -1.000000E+21 4.374743E-01 1.000000E+21
185 traj.phases.coast.indep_states.states:vt_9 c -1.000000E+21 4.217795E-01 1.000000E+21
186 traj.phases.coast.indep_states.states:accel_0 c -1.000000E+21 1.217790E-19 1.000000E+21
187 traj.phases.coast.indep_states.states:accel_1 c -1.000000E+21 1.215696E-19 1.000000E+21
188 traj.phases.coast.indep_states.states:accel_2 c -1.000000E+21 7.918339E-20 1.000000E+21
189 traj.phases.coast.indep_states.states:accel_3 c -1.000000E+21 4.456949E-20 1.000000E+21
190 traj.phases.coast.indep_states.states:accel_4 c -1.000000E+21 -5.252039E-22 1.000000E+21
191 traj.phases.coast.indep_states.states:accel_5 c -1.000000E+21 -1.401990E-20 1.000000E+21
192 traj.phases.coast.indep_states.states:accel_6 c -1.000000E+21 -1.389690E-20 1.000000E+21
193 traj.phases.coast.indep_states.states:accel_7 c -1.000000E+21 -5.526146E-21 1.000000E+21
194 traj.phases.coast.indep_states.states:deltav_0 c -1.000000E+21 2.411959E-01 1.000000E+21
195 traj.phases.coast.indep_states.states:deltav_1 c -1.000000E+21 2.411959E-01 1.000000E+21
196 traj.phases.coast.indep_states.states:deltav_2 c -1.000000E+21 2.411959E-01 1.000000E+21
197 traj.phases.coast.indep_states.states:deltav_3 c -1.000000E+21 2.411959E-01 1.000000E+21
198 traj.phases.coast.indep_states.states:deltav_4 c -1.000000E+21 2.411959E-01 1.000000E+21
199 traj.phases.coast.indep_states.states:deltav_5 c -1.000000E+21 2.411959E-01 1.000000E+21
200 traj.phases.coast.indep_states.states:deltav_6 c -1.000000E+21 2.411959E-01 1.000000E+21
201 traj.phases.coast.indep_states.states:deltav_7 c -1.000000E+21 2.411959E-01 1.000000E+21
202 traj.phases.coast.indep_states.states:deltav_8 c -1.000000E+21 2.411959E-01 1.000000E+21
203 traj.phases.coast.indep_states.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.phases.burn1.collocation_constraint.defects:r e 0.000000E+00 3.633115E-15 0.000000E+00 9.00000E+100
14 traj.phases.burn1.collocation_constraint.defects:r e 0.000000E+00 2.164409E-15 0.000000E+00 9.00000E+100
15 traj.phases.burn1.collocation_constraint.defects:r e 0.000000E+00 -3.092013E-16 0.000000E+00 9.00000E+100
16 traj.phases.burn1.collocation_constraint.defects:r e 0.000000E+00 -1.855208E-15 0.000000E+00 9.00000E+100
17 traj.phases.burn1.collocation_constraint.defects:r e 0.000000E+00 -1.360486E-14 0.000000E+00 9.00000E+100
18 traj.phases.burn1.collocation_constraint.defects:theta e 0.000000E+00 -4.947221E-15 0.000000E+00 9.00000E+100
19 traj.phases.burn1.collocation_constraint.defects:theta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
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182 traj.phases.coast.continuity_comp.defect_states:vt e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
183 traj.phases.coast.continuity_comp.defect_states:accel e 0.000000E+00 -2.093765E-22 0.000000E+00 9.00000E+100
184 traj.phases.coast.continuity_comp.defect_states:accel e 0.000000E+00 -3.461391E-20 0.000000E+00 9.00000E+100
185 traj.phases.coast.continuity_comp.defect_states:accel e 0.000000E+00 -1.349470E-20 0.000000E+00 9.00000E+100
186 traj.phases.coast.continuity_comp.defect_states:accel e 0.000000E+00 8.370756E-21 0.000000E+00 9.00000E+100
187 traj.phases.coast.continuity_comp.defect_states:deltav e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
188 traj.phases.coast.continuity_comp.defect_states:deltav e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
189 traj.phases.coast.continuity_comp.defect_states:deltav e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
190 traj.phases.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/python3.10/site-packages/openmdao/visualization/opt_report/opt_report.py:634: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding.
ax.set_ylim([ymin_plot, ymax_plot])
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
False
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.
Show code cell source
sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('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.