# SSTO Earth Launch#

This example is based on the Time-Optimal Launch of a Titan II example given in Appendix B of Longuski . 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

(84)#\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

(85)#\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

(86)#\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']

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

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

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

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

val=np.zeros(nn),
desc='pitch angle',

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

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

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

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

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

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

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

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

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)

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

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

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

rate_source='xdot')
rate_source='ydot')
rate_source='vxdot')
rate_source='vydot')
rate_source='mdot')

#
# Set the options for our constraints and objective
#

p.model.linear_solver = om.DirectSolver()

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

phase.set_time_val(initial=0.0, duration=150.0)
phase.set_state_val('x', [0, 1.15E5])
phase.set_state_val('y', [0, 1.85E5])
phase.set_state_val('vy', [1.0E-6, 0])
phase.set_state_val('m', [117000, 1163])
phase.set_control_val('theta', [1.5, -0.76])
phase.set_parameter_val('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:   0.0252 sec
Time to compute coloring:   0.0131 sec
Memory to compute coloring:   0.2500 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:   0.0122 sec
Time to compute coloring:   0.0065 sec
Memory to compute coloring:   0.0000 MB

Full total jacobian for problem 'problem' was computed 3 times, taking 0.08253424299994094 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:   0.0825 sec
Time to compute coloring:   0.0556 sec
Memory to compute coloring:   0.0000 MB
Coloring created on: 2024-08-29 21:50:59

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

Solution:
--------------------------------------------------------------------------------
Total Time:                    1.6644
User Objective Time :       0.0862
User Sensitivity Time :     0.9607
Interface Time :            0.1474
Opt Solver Time:            0.4701
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    -4.033708E-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   -4.033708E-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    1.371170E-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   -1.643805E-10    0.000000E+00              9.00000E+100
8  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -7.405642E-10    0.000000E+00              9.00000E+100
9  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00    3.453421E-10    0.000000E+00              9.00000E+100
10  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00    4.648820E-10    0.000000E+00              9.00000E+100
11  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00   -8.441018E-10    0.000000E+00              9.00000E+100
12  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00    5.812102E-10    0.000000E+00              9.00000E+100
13  traj.phase0.collocation_constraint.defects:x                   e   0.000000E+00    2.671539E-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    2.998141E-09    0.000000E+00              9.00000E+100
23  traj.phase0.collocation_constraint.defects:y                   e   0.000000E+00    2.476582E-09    0.000000E+00              9.00000E+100
24  traj.phase0.collocation_constraint.defects:y                   e   0.000000E+00    1.813846E-09    0.000000E+00              9.00000E+100
25  traj.phase0.collocation_constraint.defects:y                   e   0.000000E+00    2.495048E-10    0.000000E+00              9.00000E+100
26  traj.phase0.collocation_constraint.defects:y                   e   0.000000E+00    8.522371E-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    2.368123E-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    0.000000E+00    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    0.000000E+00    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    0.000000E+00    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    2.669709E-15    0.000000E+00              9.00000E+100
119  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    6.891574E-15    0.000000E+00              9.00000E+100
120  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    3.880391E-17    0.000000E+00              9.00000E+100
121  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -4.004563E-15    0.000000E+00              9.00000E+100
122  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    9.468154E-15    0.000000E+00              9.00000E+100
123  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    2.530015E-15    0.000000E+00              9.00000E+100
124  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    6.208625E-16    0.000000E+00              9.00000E+100
125  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    1.303811E-15    0.000000E+00              9.00000E+100
126  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -3.228485E-15    0.000000E+00              9.00000E+100
127  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00   -1.490070E-15    0.000000E+00              9.00000E+100
128  traj.phase0.continuity_comp.defect_control_rates:theta_rate    e   0.000000E+00    0.000000E+00    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   -2.220446E-16   -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   -4.612403E-17    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    1.818101E-16   -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/python3.11/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: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

Problem: problem
Driver:  pyOptSparseDriver
success     : True
iterations  : 53
runtime     : 1.9330E+00 s
model_evals : 53
model_time  : 6.2872E-02 s
deriv_evals : 49
deriv_time  : 7.5712E-01 s
exit_status : SUCCESS

Hide code cell source
sol = om.CaseReader(p.get_outputs_dir() / 'dymos_solution.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#

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.