SSTO Earth Launch#
This example is based on the Time-Optimal Launch of a Titan II example given in Appendix B of Longuski [LGuzmanP14]. It finds the pitch profile for a single-stage-to-orbit launch vehicle that minimizes the time required to reach orbit insertion under constant thrust.
The vehicle dynamics are given by
(79)#\[\begin{align}
\frac{dx}{dt} &= v_x \\
\frac{dy}{dt} &= v_y \\
\frac{dv_x}{dt} &= \frac{1}{m} (T \cos \theta - D \cos \gamma) \\
\frac{dv_y}{dt} &= \frac{1}{m} (T \sin \theta - D \sin \gamma) - g \\
\frac{dm}{dt} &= \frac{T}{g I_{sp}}
\end{align}\]
The initial conditions are
(80)#\[\begin{align}
x_0 &= 0 \\
y_0 &= 0 \\
v_{x0} &= 0 \\
v_{y0} &= 0 \\
m_0 &= 117000 \rm{\,kg}
\end{align}\]
and the final conditions are
(81)#\[\begin{align}
x_f &= \rm{free} \\
y_f &= 185 \rm{\,km} \\
v_{xf} &= V_{circ} \\
v_{yf} &= 0 \\
m_f &= \rm{free}
\end{align}\]
Defining the ODE#
Generally, one could define the ODE system as a composite group of multile components. The atmosphere component computes density (\(\rho\)). The eom component computes the state rates. Decomposing the ODE into smaller calculations makes it easier to derive the analytic derivatives.
However, for this example we will demonstrate the use of complex-step differentiation and define the ODE as a single component. This saves time up front in the deevlopment of the ODE at a minor cost in execution time.
The unconnected inputs to the EOM at the top of the diagram are provided by the Dymos phase as states, controls, or time values. The outputs, including the state rates, are shown on the right side of the diagram. The Dymos phases use state rate values to ensure that the integration technique satisfies the dynamics of the system.
import openmdao.api as om
import numpy as np
class LaunchVehicleODE(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int,
desc='Number of nodes to be evaluated in the RHS')
self.options.declare('g', types=float, default=9.80665,
desc='Gravitational acceleration, m/s**2')
self.options.declare('rho_ref', types=float, default=1.225,
desc='Reference atmospheric density, kg/m**3')
self.options.declare('h_scale', types=float, default=8.44E3,
desc='Reference altitude, m')
self.options.declare('CD', types=float, default=0.5,
desc='coefficient of drag')
self.options.declare('S', types=float, default=7.069,
desc='aerodynamic reference area (m**2)')
def setup(self):
nn = self.options['num_nodes']
self.add_input('y',
val=np.zeros(nn),
desc='altitude',
units='m')
self.add_input('vx',
val=np.zeros(nn),
desc='x velocity',
units='m/s')
self.add_input('vy',
val=np.zeros(nn),
desc='y velocity',
units='m/s')
self.add_input('m',
val=np.zeros(nn),
desc='mass',
units='kg')
self.add_input('theta',
val=np.zeros(nn),
desc='pitch angle',
units='rad')
self.add_input('thrust',
val=2100000 * np.ones(nn),
desc='thrust',
units='N')
self.add_input('Isp',
val=265.2 * np.ones(nn),
desc='specific impulse',
units='s')
# Outputs
self.add_output('xdot',
val=np.zeros(nn),
desc='velocity component in x',
units='m/s')
self.add_output('ydot',
val=np.zeros(nn),
desc='velocity component in y',
units='m/s')
self.add_output('vxdot',
val=np.zeros(nn),
desc='x acceleration magnitude',
units='m/s**2')
self.add_output('vydot',
val=np.zeros(nn),
desc='y acceleration magnitude',
units='m/s**2')
self.add_output('mdot',
val=np.zeros(nn),
desc='mass rate of change',
units='kg/s')
self.add_output('rho',
val=np.zeros(nn),
desc='density',
units='kg/m**3')
# Setup partials
# Complex-step derivatives
self.declare_coloring(wrt='*', method='cs', show_sparsity=True)
def compute(self, inputs, outputs):
theta = inputs['theta']
cos_theta = np.cos(theta)
sin_theta = np.sin(theta)
vx = inputs['vx']
vy = inputs['vy']
m = inputs['m']
F_T = inputs['thrust']
Isp = inputs['Isp']
y = inputs['y']
g = self.options['g']
rho_ref = self.options['rho_ref']
h_scale = self.options['h_scale']
CDA = self.options['CD'] * self.options['S']
outputs['rho'] = rho_ref * np.exp(-y / h_scale)
outputs['xdot'] = vx
outputs['ydot'] = vy
outputs['vxdot'] = (F_T * cos_theta - 0.5 * CDA * outputs['rho'] * vx**2) / m
outputs['vydot'] = (F_T * sin_theta - 0.5 * CDA * outputs['rho'] * vy**2) / m - g
outputs['mdot'] = -F_T / (g * Isp)
Solving the problem#
import matplotlib.pyplot as plt
import openmdao.api as om
import dymos as dm
#
# Setup and solve the optimal control problem
#
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring(tol=1.0E-12)
from dymos.examples.ssto.launch_vehicle_ode import LaunchVehicleODE
#
# Initialize our Trajectory and Phase
#
traj = dm.Trajectory()
phase = dm.Phase(ode_class=LaunchVehicleODE,
transcription=dm.GaussLobatto(num_segments=12, order=3, compressed=False))
traj.add_phase('phase0', phase)
p.model.add_subsystem('traj', traj)
#
# Set the options for the variables
#
phase.set_time_options(fix_initial=True, duration_bounds=(10, 500))
phase.add_state('x', fix_initial=True, ref=1.0E5, defect_ref=10000.0,
rate_source='xdot')
phase.add_state('y', fix_initial=True, ref=1.0E5, defect_ref=10000.0,
rate_source='ydot')
phase.add_state('vx', fix_initial=True, ref=1.0E3, defect_ref=1000.0,
rate_source='vxdot')
phase.add_state('vy', fix_initial=True, ref=1.0E3, defect_ref=1000.0,
rate_source='vydot')
phase.add_state('m', fix_initial=True, ref=1.0E3, defect_ref=100.0,
rate_source='mdot')
phase.add_control('theta', units='rad', lower=-1.57, upper=1.57, targets=['theta'])
phase.add_parameter('thrust', units='N', opt=False, val=2100000.0, targets=['thrust'])
#
# Set the options for our constraints and objective
#
phase.add_boundary_constraint('y', loc='final', equals=1.85E5, linear=True)
phase.add_boundary_constraint('vx', loc='final', equals=7796.6961)
phase.add_boundary_constraint('vy', loc='final', equals=0)
phase.add_objective('time', loc='final', scaler=0.01)
p.model.linear_solver = om.DirectSolver()
#
# Setup and set initial values
#
p.setup(check=True)
p.set_val('traj.phase0.t_initial', 0.0)
p.set_val('traj.phase0.t_duration', 150.0)
p.set_val('traj.phase0.states:x', phase.interp('x', [0, 1.15E5]))
p.set_val('traj.phase0.states:y', phase.interp('y', [0, 1.85E5]))
p.set_val('traj.phase0.states:vy', phase.interp('vy', [1.0E-6, 0]))
p.set_val('traj.phase0.states:m', phase.interp('m', [117000, 1163]))
p.set_val('traj.phase0.controls:theta', phase.interp('theta', [1.5, -0.76]))
p.set_val('traj.phase0.parameters:thrust', 2.1, units='MN')
#
# Solve the Problem
#
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
--- phase0 ---
[final] 1.8500e+05 == y [m]
[final] 7.7967e+03 == vx [m/s]
[final] 0.0000e+00 == vy [m/s]
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
Coloring for 'traj.phases.phase0.rhs_disc' (class LaunchVehicleODE)
Jacobian shape: (144, 168) (1.40% nonzero)
FWD solves: 5 REV solves: 0
Total colors vs. total size: 5 vs 168 (97.02% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.0296 sec
Time to compute coloring: 0.0153 sec
Memory to compute coloring: 0.1250 MB
Coloring for 'traj.phases.phase0.rhs_col' (class LaunchVehicleODE)
Jacobian shape: (72, 84) (2.98% nonzero)
FWD solves: 5 REV solves: 0
Total colors vs. total size: 5 vs 84 (94.05% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.0144 sec
Time to compute coloring: 0.0082 sec
Memory to compute coloring: 0.0000 MB
Full total jacobian for problem 'problem' was computed 3 times, taking 0.08346520700001747 seconds.
Total jacobian shape: (74, 152)
Jacobian shape: (74, 152) (5.38% nonzero)
FWD solves: 12 REV solves: 0
Total colors vs. total size: 12 vs 152 (92.11% improvement)
Sparsity computed using tolerance: 1e-12
Time to compute sparsity: 0.0835 sec
Time to compute coloring: 0.0647 sec
Memory to compute coloring: 0.1250 MB
Coloring created on: 2024-03-29 20:27:12
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 1.3447
User Objective Time : 0.0686
User Sensitivity Time : 0.6677
Interface Time : 0.1594
Opt Solver Time: 0.4491
Calls to Objective Function : 51
Calls to Sens Function : 48
Objectives
Index Name Value
0 traj.phase0.t 1.431612E+00
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.phase0.t_duration_0 c 1.000000E+01 1.431612E+02 5.000000E+02
1 traj.phase0.states:x_0 c -1.000000E+16 1.423870E-03 1.000000E+16
2 traj.phase0.states:x_1 c -1.000000E+16 1.423870E-03 1.000000E+16
3 traj.phase0.states:x_2 c -1.000000E+16 5.539810E-03 1.000000E+16
4 traj.phase0.states:x_3 c -1.000000E+16 5.539810E-03 1.000000E+16
5 traj.phase0.states:x_4 c -1.000000E+16 1.299270E-02 1.000000E+16
6 traj.phase0.states:x_5 c -1.000000E+16 1.299270E-02 1.000000E+16
7 traj.phase0.states:x_6 c -1.000000E+16 2.477512E-02 1.000000E+16
8 traj.phase0.states:x_7 c -1.000000E+16 2.477512E-02 1.000000E+16
9 traj.phase0.states:x_8 c -1.000000E+16 4.205719E-02 1.000000E+16
10 traj.phase0.states:x_9 c -1.000000E+16 4.205719E-02 1.000000E+16
11 traj.phase0.states:x_10 c -1.000000E+16 6.704410E-02 1.000000E+16
12 traj.phase0.states:x_11 c -1.000000E+16 6.704410E-02 1.000000E+16
13 traj.phase0.states:x_12 c -1.000000E+16 1.024324E-01 1.000000E+16
14 traj.phase0.states:x_13 c -1.000000E+16 1.024324E-01 1.000000E+16
15 traj.phase0.states:x_14 c -1.000000E+16 1.531004E-01 1.000000E+16
16 traj.phase0.states:x_15 c -1.000000E+16 1.531004E-01 1.000000E+16
17 traj.phase0.states:x_16 c -1.000000E+16 2.286348E-01 1.000000E+16
18 traj.phase0.states:x_17 c -1.000000E+16 2.286348E-01 1.000000E+16
19 traj.phase0.states:x_18 c -1.000000E+16 3.530010E-01 1.000000E+16
20 traj.phase0.states:x_19 c -1.000000E+16 3.530010E-01 1.000000E+16
21 traj.phase0.states:x_20 c -1.000000E+16 5.931618E-01 1.000000E+16
22 traj.phase0.states:x_21 c -1.000000E+16 5.931618E-01 1.000000E+16
23 traj.phase0.states:x_22 c -1.000000E+16 1.124444E+00 1.000000E+16
24 traj.phase0.states:y_0 c -1.000000E+16 6.071941E-03 1.000000E+16
25 traj.phase0.states:y_1 c -1.000000E+16 6.071941E-03 1.000000E+16
26 traj.phase0.states:y_2 c -1.000000E+16 2.558451E-02 1.000000E+16
27 traj.phase0.states:y_3 c -1.000000E+16 2.558451E-02 1.000000E+16
28 traj.phase0.states:y_4 c -1.000000E+16 6.038537E-02 1.000000E+16
29 traj.phase0.states:y_5 c -1.000000E+16 6.038537E-02 1.000000E+16
30 traj.phase0.states:y_6 c -1.000000E+16 1.127399E-01 1.000000E+16
31 traj.phase0.states:y_7 c -1.000000E+16 1.127399E-01 1.000000E+16
32 traj.phase0.states:y_8 c -1.000000E+16 1.861696E-01 1.000000E+16
33 traj.phase0.states:y_9 c -1.000000E+16 1.861696E-01 1.000000E+16
34 traj.phase0.states:y_10 c -1.000000E+16 2.860066E-01 1.000000E+16
35 traj.phase0.states:y_11 c -1.000000E+16 2.860066E-01 1.000000E+16
36 traj.phase0.states:y_12 c -1.000000E+16 4.198521E-01 1.000000E+16
37 traj.phase0.states:y_13 c -1.000000E+16 4.198521E-01 1.000000E+16
38 traj.phase0.states:y_14 c -1.000000E+16 5.976777E-01 1.000000E+16
39 traj.phase0.states:y_15 c -1.000000E+16 5.976777E-01 1.000000E+16
40 traj.phase0.states:y_16 c -1.000000E+16 8.326284E-01 1.000000E+16
41 traj.phase0.states:y_17 c -1.000000E+16 8.326284E-01 1.000000E+16
42 traj.phase0.states:y_18 c -1.000000E+16 1.140649E+00 1.000000E+16
43 traj.phase0.states:y_19 c -1.000000E+16 1.140649E+00 1.000000E+16
44 traj.phase0.states:y_20 c -1.000000E+16 1.525001E+00 1.000000E+16
45 traj.phase0.states:y_21 c -1.000000E+16 1.525001E+00 1.000000E+16
46 traj.phase0.states:y_22 c -1.000000E+16 1.850000E+00 1.000000E+16
47 traj.phase0.states:vx_0 c -1.000000E+18 2.327869E-02 1.000000E+18
48 traj.phase0.states:vx_1 c -1.000000E+18 2.327869E-02 1.000000E+18
49 traj.phase0.states:vx_2 c -1.000000E+18 4.659191E-02 1.000000E+18
50 traj.phase0.states:vx_3 c -1.000000E+18 4.659191E-02 1.000000E+18
51 traj.phase0.states:vx_4 c -1.000000E+18 7.980599E-02 1.000000E+18
52 traj.phase0.states:vx_5 c -1.000000E+18 7.980599E-02 1.000000E+18
53 traj.phase0.states:vx_6 c -1.000000E+18 1.191955E-01 1.000000E+18
54 traj.phase0.states:vx_7 c -1.000000E+18 1.191955E-01 1.000000E+18
55 traj.phase0.states:vx_8 c -1.000000E+18 1.737234E-01 1.000000E+18
56 traj.phase0.states:vx_9 c -1.000000E+18 1.737234E-01 1.000000E+18
57 traj.phase0.states:vx_10 c -1.000000E+18 2.487371E-01 1.000000E+18
58 traj.phase0.states:vx_11 c -1.000000E+18 2.487371E-01 1.000000E+18
59 traj.phase0.states:vx_12 c -1.000000E+18 3.510681E-01 1.000000E+18
60 traj.phase0.states:vx_13 c -1.000000E+18 3.510681E-01 1.000000E+18
61 traj.phase0.states:vx_14 c -1.000000E+18 5.108624E-01 1.000000E+18
62 traj.phase0.states:vx_15 c -1.000000E+18 5.108624E-01 1.000000E+18
63 traj.phase0.states:vx_16 c -1.000000E+18 7.858760E-01 1.000000E+18
64 traj.phase0.states:vx_17 c -1.000000E+18 7.858760E-01 1.000000E+18
65 traj.phase0.states:vx_18 c -1.000000E+18 1.384927E+00 1.000000E+18
66 traj.phase0.states:vx_19 c -1.000000E+18 1.384927E+00 1.000000E+18
67 traj.phase0.states:vx_20 c -1.000000E+18 2.869862E+00 1.000000E+18
68 traj.phase0.states:vx_21 c -1.000000E+18 2.869862E+00 1.000000E+18
69 traj.phase0.states:vx_22 c -1.000000E+18 7.796696E+00 1.000000E+18
70 traj.phase0.states:vy_0 c -1.000000E+18 1.045598E-01 1.000000E+18
71 traj.phase0.states:vy_1 c -1.000000E+18 1.045598E-01 1.000000E+18
72 traj.phase0.states:vy_2 c -1.000000E+18 2.250738E-01 1.000000E+18
73 traj.phase0.states:vy_3 c -1.000000E+18 2.250738E-01 1.000000E+18
74 traj.phase0.states:vy_4 c -1.000000E+18 3.613861E-01 1.000000E+18
75 traj.phase0.states:vy_5 c -1.000000E+18 3.613861E-01 1.000000E+18
76 traj.phase0.states:vy_6 c -1.000000E+18 5.211650E-01 1.000000E+18
77 traj.phase0.states:vy_7 c -1.000000E+18 5.211650E-01 1.000000E+18
78 traj.phase0.states:vy_8 c -1.000000E+18 7.172279E-01 1.000000E+18
79 traj.phase0.states:vy_9 c -1.000000E+18 7.172279E-01 1.000000E+18
80 traj.phase0.states:vy_10 c -1.000000E+18 9.670879E-01 1.000000E+18
81 traj.phase0.states:vy_11 c -1.000000E+18 9.670879E-01 1.000000E+18
82 traj.phase0.states:vy_12 c -1.000000E+18 1.290602E+00 1.000000E+18
83 traj.phase0.states:vy_13 c -1.000000E+18 1.290602E+00 1.000000E+18
84 traj.phase0.states:vy_14 c -1.000000E+18 1.708746E+00 1.000000E+18
85 traj.phase0.states:vy_15 c -1.000000E+18 1.708746E+00 1.000000E+18
86 traj.phase0.states:vy_16 c -1.000000E+18 2.253313E+00 1.000000E+18
87 traj.phase0.states:vy_17 c -1.000000E+18 2.253313E+00 1.000000E+18
88 traj.phase0.states:vy_18 c -1.000000E+18 2.923999E+00 1.000000E+18
89 traj.phase0.states:vy_19 c -1.000000E+18 2.923999E+00 1.000000E+18
90 traj.phase0.states:vy_20 c -1.000000E+18 3.379547E+00 1.000000E+18
91 traj.phase0.states:vy_21 c -1.000000E+18 3.379547E+00 1.000000E+18
92 traj.phase0.states:vy_22 c -1.000000E+18 -2.421144E-23 1.000000E+18
93 traj.phase0.states:m_0 c -1.000000E+18 1.073668E+02 1.000000E+18
94 traj.phase0.states:m_1 c -1.000000E+18 1.073668E+02 1.000000E+18
95 traj.phase0.states:m_2 c -1.000000E+18 9.773367E+01 1.000000E+18
96 traj.phase0.states:m_3 c -1.000000E+18 9.773367E+01 1.000000E+18
97 traj.phase0.states:m_4 c -1.000000E+18 8.810050E+01 1.000000E+18
98 traj.phase0.states:m_5 c -1.000000E+18 8.810050E+01 1.000000E+18
99 traj.phase0.states:m_6 c -1.000000E+18 7.846734E+01 1.000000E+18
100 traj.phase0.states:m_7 c -1.000000E+18 7.846734E+01 1.000000E+18
101 traj.phase0.states:m_8 c -1.000000E+18 6.883417E+01 1.000000E+18
102 traj.phase0.states:m_9 c -1.000000E+18 6.883417E+01 1.000000E+18
103 traj.phase0.states:m_10 c -1.000000E+18 5.920100E+01 1.000000E+18
104 traj.phase0.states:m_11 c -1.000000E+18 5.920100E+01 1.000000E+18
105 traj.phase0.states:m_12 c -1.000000E+18 4.956784E+01 1.000000E+18
106 traj.phase0.states:m_13 c -1.000000E+18 4.956784E+01 1.000000E+18
107 traj.phase0.states:m_14 c -1.000000E+18 3.993467E+01 1.000000E+18
108 traj.phase0.states:m_15 c -1.000000E+18 3.993467E+01 1.000000E+18
109 traj.phase0.states:m_16 c -1.000000E+18 3.030151E+01 1.000000E+18
110 traj.phase0.states:m_17 c -1.000000E+18 3.030151E+01 1.000000E+18
111 traj.phase0.states:m_18 c -1.000000E+18 2.066834E+01 1.000000E+18
112 traj.phase0.states:m_19 c -1.000000E+18 2.066834E+01 1.000000E+18
113 traj.phase0.states:m_20 c -1.000000E+18 1.103517E+01 1.000000E+18
114 traj.phase0.states:m_21 c -1.000000E+18 1.103517E+01 1.000000E+18
115 traj.phase0.states:m_22 c -1.000000E+18 1.402009E+00 1.000000E+18
116 traj.phase0.controls:theta_0 c -1.570000E+00 1.477517E+00 1.570000E+00
117 traj.phase0.controls:theta_1 c -1.570000E+00 1.468233E+00 1.570000E+00
118 traj.phase0.controls:theta_2 c -1.570000E+00 1.474148E+00 1.570000E+00
119 traj.phase0.controls:theta_3 c -1.570000E+00 1.474148E+00 1.570000E+00
120 traj.phase0.controls:theta_4 c -1.570000E+00 1.477669E+00 1.570000E+00
121 traj.phase0.controls:theta_5 c -1.570000E+00 1.461203E+00 1.570000E+00
122 traj.phase0.controls:theta_6 c -1.570000E+00 1.461203E+00 1.570000E+00
123 traj.phase0.controls:theta_7 c -1.570000E+00 1.442523E+00 1.570000E+00
124 traj.phase0.controls:theta_8 c -1.570000E+00 1.439401E+00 1.570000E+00
125 traj.phase0.controls:theta_9 c -1.570000E+00 1.439401E+00 1.570000E+00
126 traj.phase0.controls:theta_10 c -1.570000E+00 1.438089E+00 1.570000E+00
127 traj.phase0.controls:theta_11 c -1.570000E+00 1.424836E+00 1.570000E+00
128 traj.phase0.controls:theta_12 c -1.570000E+00 1.424836E+00 1.570000E+00
129 traj.phase0.controls:theta_13 c -1.570000E+00 1.406383E+00 1.570000E+00
130 traj.phase0.controls:theta_14 c -1.570000E+00 1.389470E+00 1.570000E+00
131 traj.phase0.controls:theta_15 c -1.570000E+00 1.389470E+00 1.570000E+00
132 traj.phase0.controls:theta_16 c -1.570000E+00 1.375128E+00 1.570000E+00
133 traj.phase0.controls:theta_17 c -1.570000E+00 1.364390E+00 1.570000E+00
134 traj.phase0.controls:theta_18 c -1.570000E+00 1.364390E+00 1.570000E+00
135 traj.phase0.controls:theta_19 c -1.570000E+00 1.348778E+00 1.570000E+00
136 traj.phase0.controls:theta_20 c -1.570000E+00 1.319811E+00 1.570000E+00
137 traj.phase0.controls:theta_21 c -1.570000E+00 1.319811E+00 1.570000E+00
138 traj.phase0.controls:theta_22 c -1.570000E+00 1.283485E+00 1.570000E+00
139 traj.phase0.controls:theta_23 c -1.570000E+00 1.245798E+00 1.570000E+00
140 traj.phase0.controls:theta_24 c -1.570000E+00 1.245798E+00 1.570000E+00
141 traj.phase0.controls:theta_25 c -1.570000E+00 1.187473E+00 1.570000E+00
142 traj.phase0.controls:theta_26 c -1.570000E+00 1.089238E+00 1.570000E+00
143 traj.phase0.controls:theta_27 c -1.570000E+00 1.089238E+00 1.570000E+00
144 traj.phase0.controls:theta_28 c -1.570000E+00 9.418841E-01 1.570000E+00
145 traj.phase0.controls:theta_29 c -1.570000E+00 7.362030E-01 1.570000E+00
146 traj.phase0.controls:theta_30 c -1.570000E+00 7.362030E-01 1.570000E+00
147 traj.phase0.controls:theta_31 c -1.570000E+00 4.319487E-01 1.570000E+00
148 traj.phase0.controls:theta_32 c -1.570000E+00 -1.112473E-02 1.570000E+00
149 traj.phase0.controls:theta_33 c -1.570000E+00 -1.112473E-02 1.570000E+00
150 traj.phase0.controls:theta_34 c -1.570000E+00 -4.570600E-01 1.570000E+00
151 traj.phase0.controls:theta_35 c -1.570000E+00 -7.698997E-01 1.570000E+00
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier (N/A)
0 traj.phases.phase0->final_boundary_constraint->y e 1.850000E+05 1.850000E+05 1.850000E+05 9.00000E+100
1 traj.phases.phase0->final_boundary_constraint->vx e 7.796696E+03 7.796696E+03 7.796696E+03 9.00000E+100
2 traj.phases.phase0->final_boundary_constraint->vy e 0.000000E+00 -2.421144E-20 0.000000E+00 9.00000E+100
3 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.188094E-10 0.000000E+00 9.00000E+100
4 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.371169E-10 0.000000E+00 9.00000E+100
5 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.557595E-10 0.000000E+00 9.00000E+100
6 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.888822E-10 0.000000E+00 9.00000E+100
7 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.643806E-10 0.000000E+00 9.00000E+100
8 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -7.405641E-10 0.000000E+00 9.00000E+100
9 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 3.453420E-10 0.000000E+00 9.00000E+100
10 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.648821E-10 0.000000E+00 9.00000E+100
11 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.441020E-10 0.000000E+00 9.00000E+100
12 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.812105E-10 0.000000E+00 9.00000E+100
13 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.671527E-10 0.000000E+00 9.00000E+100
14 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -6.157080E-09 0.000000E+00 9.00000E+100
15 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 4.185466E-10 0.000000E+00 9.00000E+100
16 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 9.614961E-09 0.000000E+00 9.00000E+100
17 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 2.809379E-08 0.000000E+00 9.00000E+100
18 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 3.174746E-08 0.000000E+00 9.00000E+100
19 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 1.376446E-08 0.000000E+00 9.00000E+100
20 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 -3.754024E-08 0.000000E+00 9.00000E+100
21 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 -5.322523E-08 0.000000E+00 9.00000E+100
22 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 2.998142E-09 0.000000E+00 9.00000E+100
23 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 2.476581E-09 0.000000E+00 9.00000E+100
24 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 1.813847E-09 0.000000E+00 9.00000E+100
25 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 2.495046E-10 0.000000E+00 9.00000E+100
26 traj.phase0.collocation_constraint.defects:y e 0.000000E+00 8.522368E-09 0.000000E+00 9.00000E+100
27 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 3.499577E-09 0.000000E+00 9.00000E+100
28 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 3.213875E-09 0.000000E+00 9.00000E+100
29 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 8.760738E-09 0.000000E+00 9.00000E+100
30 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 1.050294E-08 0.000000E+00 9.00000E+100
31 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 1.502428E-08 0.000000E+00 9.00000E+100
32 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 3.872966E-08 0.000000E+00 9.00000E+100
33 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 1.357671E-08 0.000000E+00 9.00000E+100
34 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 3.378725E-09 0.000000E+00 9.00000E+100
35 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 2.245357E-08 0.000000E+00 9.00000E+100
36 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 6.435004E-09 0.000000E+00 9.00000E+100
37 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 2.368121E-09 0.000000E+00 9.00000E+100
38 traj.phase0.collocation_constraint.defects:vx e 0.000000E+00 2.374244E-08 0.000000E+00 9.00000E+100
39 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 2.920377E-08 0.000000E+00 9.00000E+100
40 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 1.258180E-08 0.000000E+00 9.00000E+100
41 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -2.350687E-08 0.000000E+00 9.00000E+100
42 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -1.084222E-07 0.000000E+00 9.00000E+100
43 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -1.702733E-07 0.000000E+00 9.00000E+100
44 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -2.750436E-08 0.000000E+00 9.00000E+100
45 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -1.483104E-08 0.000000E+00 9.00000E+100
46 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 2.808630E-08 0.000000E+00 9.00000E+100
47 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 6.222977E-08 0.000000E+00 9.00000E+100
48 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 9.789008E-09 0.000000E+00 9.00000E+100
49 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 1.080768E-09 0.000000E+00 9.00000E+100
50 traj.phase0.collocation_constraint.defects:vy e 0.000000E+00 -2.001000E-08 0.000000E+00 9.00000E+100
51 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 6.781475E-15 0.000000E+00 9.00000E+100
52 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 6.781475E-15 0.000000E+00 9.00000E+100
53 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -8.815917E-14 0.000000E+00 9.00000E+100
54 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 7.459622E-14 0.000000E+00 9.00000E+100
55 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -7.459622E-14 0.000000E+00 9.00000E+100
56 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 4.747032E-14 0.000000E+00 9.00000E+100
57 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -4.747032E-14 0.000000E+00 9.00000E+100
58 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 4.747032E-14 0.000000E+00 9.00000E+100
59 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -2.034442E-14 0.000000E+00 9.00000E+100
60 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -6.781475E-15 0.000000E+00 9.00000E+100
61 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 2.034442E-14 0.000000E+00 9.00000E+100
62 traj.phase0.collocation_constraint.defects:m e 0.000000E+00 -6.781475E-15 0.000000E+00 9.00000E+100
63 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
64 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 -1.136868E-13 0.000000E+00 9.00000E+100
65 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
66 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
67 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
68 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
69 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
70 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
71 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
72 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
73 traj.phase0.continuity_comp.defect_states:x e 2.678272E-12 1.455192E-11 2.678272E-12 9.00000E+100
74 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
75 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
76 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
77 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
78 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
79 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
80 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
81 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
82 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
83 traj.phase0.continuity_comp.defect_states:y e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
84 traj.phase0.continuity_comp.defect_states:y e 8.235664E-12 2.910383E-11 8.235664E-12 9.00000E+100
85 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
86 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
87 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
88 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
89 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
90 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
91 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
92 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
93 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
94 traj.phase0.continuity_comp.defect_states:vx e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
95 traj.phase0.continuity_comp.defect_states:vx e -2.081668E-17 0.000000E+00 -2.081668E-17 9.00000E+100
96 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
97 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
98 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
99 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
100 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
101 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
102 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
103 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
104 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
105 traj.phase0.continuity_comp.defect_states:vy e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
106 traj.phase0.continuity_comp.defect_states:vy e -2.584939E-24 0.000000E+00 -2.584939E-24 9.00000E+100
107 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
108 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
109 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
110 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
111 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
112 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
113 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
114 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
115 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
116 traj.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
117 traj.phase0.continuity_comp.defect_states:m e -8.952838E-13 -1.818989E-12 -8.952838E-13 9.00000E+100
118 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -1.334854E-15 0.000000E+00 9.00000E+100
119 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 9.561283E-15 0.000000E+00 9.00000E+100
120 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -2.623144E-15 0.000000E+00 9.00000E+100
121 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -6.674272E-15 0.000000E+00 9.00000E+100
122 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -1.210682E-15 0.000000E+00 9.00000E+100
123 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -1.080301E-14 0.000000E+00 9.00000E+100
124 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -2.110933E-15 0.000000E+00 9.00000E+100
125 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -2.669709E-15 0.000000E+00 9.00000E+100
126 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 2.235105E-15 0.000000E+00 9.00000E+100
127 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
128 traj.phase0.continuity_comp.defect_control_rates:theta_rate e 0.000000E+00 -2.980140E-15 0.000000E+00 9.00000E+100
129 traj.phase0.continuity_comp.defect_controls:theta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
130 traj.phase0.continuity_comp.defect_controls:theta e 2.168404E-17 0.000000E+00 2.168404E-17 9.00000E+100
131 traj.phase0.continuity_comp.defect_controls:theta e -4.065758E-17 0.000000E+00 -4.065758E-17 9.00000E+100
132 traj.phase0.continuity_comp.defect_controls:theta e -1.149181E-16 -1.145955E-16 -1.149181E-16 9.00000E+100
133 traj.phase0.continuity_comp.defect_controls:theta e 3.702715E-17 1.759206E-16 3.702715E-17 9.00000E+100
134 traj.phase0.continuity_comp.defect_controls:theta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
135 traj.phase0.continuity_comp.defect_controls:theta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
136 traj.phase0.continuity_comp.defect_controls:theta e -4.177734E-18 -4.023448E-17 -4.177734E-18 9.00000E+100
137 traj.phase0.continuity_comp.defect_controls:theta e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
138 traj.phase0.continuity_comp.defect_controls:theta e 3.964971E-16 4.342420E-16 3.964971E-16 9.00000E+100
139 traj.phase0.continuity_comp.defect_controls:theta e -1.734723E-16 -1.734723E-16 -1.734723E-16 9.00000E+100
Exit Status
Inform Description
0 Optimization terminated successfully.
--------------------------------------------------------------------------------
/usr/share/miniconda/envs/test/lib/python3.10/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/python3.10/site-packages/openmdao/core/group.py:1098: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:theta_rate2] cannot be impacted by the design variables of the problem because no partials were defined for them in their parent component(s).
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
False
Show code cell source
sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('dymos_simulation.db').get_case('final')
fig, [traj_ax, control_ax] = plt.subplots(nrows=2, ncols=1, figsize=(10, 8))
traj_ax.plot(sol.get_val('traj.phase0.timeseries.x'),
sol.get_val('traj.phase0.timeseries.y'),
marker='o',
ms=4,
linestyle='None',
label='solution')
traj_ax.plot(sim.get_val('traj.phase0.timeseries.x'),
sim.get_val('traj.phase0.timeseries.y'),
marker=None,
linestyle='-',
label='simulation')
traj_ax.set_xlabel('range (m)')
traj_ax.set_ylabel('altitude (m)')
traj_ax.set_aspect('equal')
traj_ax.grid(True)
control_ax.plot(sol.get_val('traj.phase0.timeseries.time'),
sol.get_val('traj.phase0.timeseries.theta'),
marker='o',
ms=4,
linestyle='None')
control_ax.plot(sim.get_val('traj.phase0.timeseries.time'),
sim.get_val('traj.phase0.timeseries.theta'),
linestyle='-',
marker=None)
control_ax.set_xlabel('time (s)')
control_ax.set_ylabel('theta (deg)')
control_ax.grid(True)
plt.suptitle('Single Stage to Orbit Solution Using A Dynamic Control')
fig.legend(loc='lower center', ncol=2)
plt.show()
References#
[LGuzmanP14]
James M Longuski, José J Guzmán, and John E Prussing. Optimal control with aerospace applications. Springer, 1 edition, 2014. ISBN 978-1-4939-4917-5. doi:https://doi.org/10.1007/978-1-4614-8945-0.