SSTO Lunar Ascent with Polynomial Controls#
This example demonstrates the use of polynomial controls in Dymos. Polynomial controls define the control profile as a single polynomial across the entire phase where the control values are specified at the Legendre Gauss Lobatto (LGL) nodes in phase dimensionless time. These controls can be of any arbitrary order greater than 1 (linear).
We’ve already demonstrated that the optimal single stage ascent in the absense of an atmosphere follows the linear tangent guidance law. In this example, we’ll change the control parameterization such that \(\tan \theta\) is provided by a polynomial control of order 1. The LGL nodes of a first order polynomial are the endpoints of the phase, thus the optimizer will be governing the value of \(\tan \theta\) at the initial and final times of the phase, and the Dymos will interpolate the values of \(\tan \theta\) to all other nodes in the Phase.
This example is equivalent to the previous linear tangent example in that we’ve reduced the problem from finding the appropriate control value at all nodes to that of finding the optimal value of just two quantities. But instead of optimizing the slope and intercept given by the parameters \(a\) and \(b\), we’re parameterizing the control using the endpoint values of the linear polynomial.
Now the guidance comp needs to convert the inverse tangent of the current value of the polynomial controls.
\[\theta = \arctan{p}\]
Solving the problem#
import numpy as np
import matplotlib.pyplot as plt
import openmdao.api as om
import dymos as dm
g = 1.61544 # lunar gravity, m/s**2
class LaunchVehicle2DEOM(om.ExplicitComponent):
"""
Simple 2D Cartesian Equations of Motion for a launch vehicle subject to thrust and drag.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
# Inputs
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')
# Setup partials
ar = np.arange(self.options['num_nodes'])
self.declare_partials(of='xdot', wrt='vx', rows=ar, cols=ar, val=1.0)
self.declare_partials(of='ydot', wrt='vy', rows=ar, cols=ar, val=1.0)
self.declare_partials(of='vxdot', wrt='vx', rows=ar, cols=ar)
self.declare_partials(of='vxdot', wrt='m', rows=ar, cols=ar)
self.declare_partials(of='vxdot', wrt='theta', rows=ar, cols=ar)
self.declare_partials(of='vxdot', wrt='thrust', rows=ar, cols=ar)
self.declare_partials(of='vydot', wrt='m', rows=ar, cols=ar)
self.declare_partials(of='vydot', wrt='theta', rows=ar, cols=ar)
self.declare_partials(of='vydot', wrt='vy', rows=ar, cols=ar)
self.declare_partials(of='vydot', wrt='thrust', rows=ar, cols=ar)
self.declare_partials(of='mdot', wrt='thrust', rows=ar, cols=ar)
self.declare_partials(of='mdot', wrt='Isp', rows=ar, cols=ar)
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']
outputs['xdot'] = vx
outputs['ydot'] = vy
outputs['vxdot'] = F_T * cos_theta / m
outputs['vydot'] = F_T * sin_theta / m - g
outputs['mdot'] = -F_T / (g * Isp)
def compute_partials(self, inputs, jacobian):
theta = inputs['theta']
cos_theta = np.cos(theta)
sin_theta = np.sin(theta)
m = inputs['m']
F_T = inputs['thrust']
Isp = inputs['Isp']
# jacobian['vxdot', 'vx'] = -CDA * rho * vx / m
jacobian['vxdot', 'm'] = -(F_T * cos_theta) / m ** 2
jacobian['vxdot', 'theta'] = -(F_T / m) * sin_theta
jacobian['vxdot', 'thrust'] = cos_theta / m
# jacobian['vydot', 'vy'] = -CDA * rho * vy / m
jacobian['vydot', 'm'] = -(F_T * sin_theta) / m ** 2
jacobian['vydot', 'theta'] = (F_T / m) * cos_theta
jacobian['vydot', 'thrust'] = sin_theta / m
jacobian['mdot', 'thrust'] = -1.0 / (g * Isp)
jacobian['mdot', 'Isp'] = F_T / (g * Isp ** 2)
class LaunchVehicleLinearTangentODE(om.Group):
"""
The LaunchVehicleLinearTangentODE for this case consists of a guidance component and
the EOM. Guidance is simply an OpenMDAO ExecComp which computes the arctangent of the
tan_theta variable.
"""
def initialize(self):
self.options.declare('num_nodes', types=int,
desc='Number of nodes to be evaluated in the RHS')
def setup(self):
nn = self.options['num_nodes']
self.add_subsystem('guidance', om.ExecComp('theta=arctan(tan_theta)',
theta={'val': np.ones(nn),
'units': 'rad'},
tan_theta={'val': np.ones(nn)}))
self.add_subsystem('eom', LaunchVehicle2DEOM(num_nodes=nn))
self.connect('guidance.theta', 'eom.theta')
#
# Setup and solve the optimal control problem
#
p = om.Problem(model=om.Group())
traj = p.model.add_subsystem('traj', dm.Trajectory())
phase = dm.Phase(ode_class=LaunchVehicleLinearTangentODE,
transcription=dm.Radau(num_segments=20, order=3, compressed=False))
traj.add_phase('phase0', phase)
phase.set_time_options(fix_initial=True, duration_bounds=(10, 1000), units='s')
#
# Set the state options. We include rate_source, units, and targets here since the ODE
# is not decorated with their default values.
#
phase.add_state('x', fix_initial=True, lower=0, rate_source='eom.xdot', units='m')
phase.add_state('y', fix_initial=True, lower=0, rate_source='eom.ydot', units='m')
phase.add_state('vx', fix_initial=True, lower=0, rate_source='eom.vxdot',
units='m/s', targets=['eom.vx'])
phase.add_state('vy', fix_initial=True, rate_source='eom.vydot',
units='m/s', targets=['eom.vy'])
phase.add_state('m', fix_initial=True, rate_source='eom.mdot',
units='kg', targets=['eom.m'])
#
# The tangent of theta is modeled as a linear polynomial over the duration of the phase.
#
phase.add_polynomial_control('tan_theta', order=1, units=None, opt=True,
targets=['guidance.tan_theta'])
#
# Parameters values for thrust and specific impulse are design parameters. They are
# provided by an IndepVarComp in the phase, but with opt=False their values are not
# design variables in the optimization problem.
#
phase.add_parameter('thrust', units='N', opt=False, val=3.0 * 50000.0 * 1.61544,
targets=['eom.thrust'])
phase.add_parameter('Isp', units='s', opt=False, val=1.0E6, targets=['eom.Isp'])
#
# Set the boundary constraints. These are all states which could also be handled
# by setting fix_final=True and including the correct final value in the initial guess.
#
phase.add_boundary_constraint('y', loc='final', equals=1.85E5, linear=True)
phase.add_boundary_constraint('vx', loc='final', equals=1627.0)
phase.add_boundary_constraint('vy', loc='final', equals=0)
phase.add_objective('time', index=-1, scaler=0.01)
#
# Add theta as a timeseries output since it's not included by default.
#
phase.add_timeseries_output('guidance.theta', units='deg')
#
# Set the optimizer
#
p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()
#
# We don't strictly need to define a linear solver here since our problem is entirely
# feed-forward with no iterative loops. It's good practice to add one, however, since
# failing to do so can cause incorrect derivatives if iterative processes are ever
# introduced to the system.
#
p.model.linear_solver = om.DirectSolver()
p.setup(check=True)
#
# Assign initial guesses for the independent variables in the problem.
#
p['traj.phase0.t_initial'] = 0.0
p['traj.phase0.t_duration'] = 500.0
p['traj.phase0.states:x'] = phase.interp('x', [0, 350000.0])
p['traj.phase0.states:y'] = phase.interp('y', [0, 185000.0])
p['traj.phase0.states:vx'] = phase.interp('vx', [0, 1627.0])
p['traj.phase0.states:vy'] = phase.interp('vy', [1.0E-6, 0])
p['traj.phase0.states:m'] = phase.interp('m', [50000, 50000])
p['traj.phase0.polynomial_controls:tan_theta'] = [[0.5 * np.pi], [0.0]]
#
# Solve the problem.
#
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
--- phase0 ---
[final] 1.8500e+05 == y [m]
[final] 1.6270e+03 == vx [m/s]
[final] 0.0000e+00 == vy [m/s]
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
Coloring for 'traj.phases.phase0.rhs_all.guidance' (class ExecComp)
Jacobian shape: (80, 80) ( 1.25% nonzero)
FWD solves: 1 REV solves: 0
Total colors vs. total size: 1 vs 80 (98.8% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.006204 sec.
Time to compute coloring: 0.010058 sec.
Memory to compute coloring: 0.437500 MB.
Full total jacobian was computed 3 times, taking 0.255833 seconds.
Total jacobian shape: (303, 398)
Jacobian shape: (303, 398) ( 1.63% nonzero)
FWD solves: 10 REV solves: 0
Total colors vs. total size: 10 vs 398 (97.5% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.255833 sec.
Time to compute coloring: 0.371090 sec.
Memory to compute coloring: 0.789062 MB.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 1.3478
User Objective Time : 0.0152
User Sensitivity Time : 0.0572
Interface Time : 0.0543
Opt Solver Time: 1.2211
Calls to Objective Function : 6
Calls to Sens Function : 5
Objectives
Index Name Value
0 traj.phases.phase0.time.t 4.817168E+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 4.817168E+02 1.000000E+03
1 traj.phases.phase0.indep_states.states:x_0 c 0.000000E+00 6.106930E+01 1.000000E+21
2 traj.phases.phase0.indep_states.states:x_1 c 0.000000E+00 3.496393E+02 1.000000E+21
3 traj.phases.phase0.indep_states.states:x_2 c 0.000000E+00 4.914075E+02 1.000000E+21
4 traj.phases.phase0.indep_states.states:x_3 c 0.000000E+00 4.914075E+02 1.000000E+21
5 traj.phases.phase0.indep_states.states:x_4 c 0.000000E+00 9.095034E+02 1.000000E+21
6 traj.phases.phase0.indep_states.states:x_5 c 0.000000E+00 1.704975E+03 1.000000E+21
7 traj.phases.phase0.indep_states.states:x_6 c 0.000000E+00 2.010769E+03 1.000000E+21
8 traj.phases.phase0.indep_states.states:x_7 c 0.000000E+00 2.010769E+03 1.000000E+21
9 traj.phases.phase0.indep_states.states:x_8 c 0.000000E+00 2.811263E+03 1.000000E+21
10 traj.phases.phase0.indep_states.states:x_9 c 0.000000E+00 4.150586E+03 1.000000E+21
11 traj.phases.phase0.indep_states.states:x_10 c 0.000000E+00 4.632596E+03 1.000000E+21
12 traj.phases.phase0.indep_states.states:x_11 c 0.000000E+00 4.632596E+03 1.000000E+21
13 traj.phases.phase0.indep_states.states:x_12 c 0.000000E+00 5.844355E+03 1.000000E+21
14 traj.phases.phase0.indep_states.states:x_13 c 0.000000E+00 7.769668E+03 1.000000E+21
15 traj.phases.phase0.indep_states.states:x_14 c 0.000000E+00 8.441819E+03 1.000000E+21
16 traj.phases.phase0.indep_states.states:x_15 c 0.000000E+00 8.441819E+03 1.000000E+21
17 traj.phases.phase0.indep_states.states:x_16 c 0.000000E+00 1.009787E+04 1.000000E+21
18 traj.phases.phase0.indep_states.states:x_17 c 0.000000E+00 1.265747E+04 1.000000E+21
19 traj.phases.phase0.indep_states.states:x_18 c 0.000000E+00 1.353576E+04 1.000000E+21
20 traj.phases.phase0.indep_states.states:x_19 c 0.000000E+00 1.353576E+04 1.000000E+21
21 traj.phases.phase0.indep_states.states:x_20 c 0.000000E+00 1.567405E+04 1.000000E+21
22 traj.phases.phase0.indep_states.states:x_21 c 0.000000E+00 1.892357E+04 1.000000E+21
23 traj.phases.phase0.indep_states.states:x_22 c 0.000000E+00 2.002641E+04 1.000000E+21
24 traj.phases.phase0.indep_states.states:x_23 c 0.000000E+00 2.002641E+04 1.000000E+21
25 traj.phases.phase0.indep_states.states:x_24 c 0.000000E+00 2.269073E+04 1.000000E+21
26 traj.phases.phase0.indep_states.states:x_25 c 0.000000E+00 2.669437E+04 1.000000E+21
27 traj.phases.phase0.indep_states.states:x_26 c 0.000000E+00 2.804305E+04 1.000000E+21
28 traj.phases.phase0.indep_states.states:x_27 c 0.000000E+00 2.804305E+04 1.000000E+21
29 traj.phases.phase0.indep_states.states:x_28 c 0.000000E+00 3.128395E+04 1.000000E+21
30 traj.phases.phase0.indep_states.states:x_29 c 0.000000E+00 3.611577E+04 1.000000E+21
31 traj.phases.phase0.indep_states.states:x_30 c 0.000000E+00 3.773482E+04 1.000000E+21
32 traj.phases.phase0.indep_states.states:x_31 c 0.000000E+00 3.773482E+04 1.000000E+21
33 traj.phases.phase0.indep_states.states:x_32 c 0.000000E+00 4.161043E+04 1.000000E+21
34 traj.phases.phase0.indep_states.states:x_33 c 0.000000E+00 4.735527E+04 1.000000E+21
35 traj.phases.phase0.indep_states.states:x_34 c 0.000000E+00 4.927265E+04 1.000000E+21
36 traj.phases.phase0.indep_states.states:x_35 c 0.000000E+00 4.927265E+04 1.000000E+21
37 traj.phases.phase0.indep_states.states:x_36 c 0.000000E+00 5.384901E+04 1.000000E+21
38 traj.phases.phase0.indep_states.states:x_37 c 0.000000E+00 6.060228E+04 1.000000E+21
39 traj.phases.phase0.indep_states.states:x_38 c 0.000000E+00 6.284916E+04 1.000000E+21
40 traj.phases.phase0.indep_states.states:x_39 c 0.000000E+00 6.284916E+04 1.000000E+21
41 traj.phases.phase0.indep_states.states:x_40 c 0.000000E+00 6.819918E+04 1.000000E+21
42 traj.phases.phase0.indep_states.states:x_41 c 0.000000E+00 7.606414E+04 1.000000E+21
43 traj.phases.phase0.indep_states.states:x_42 c 0.000000E+00 7.867365E+04 1.000000E+21
44 traj.phases.phase0.indep_states.states:x_43 c 0.000000E+00 7.867365E+04 1.000000E+21
45 traj.phases.phase0.indep_states.states:x_44 c 0.000000E+00 8.487336E+04 1.000000E+21
46 traj.phases.phase0.indep_states.states:x_45 c 0.000000E+00 9.395375E+04 1.000000E+21
47 traj.phases.phase0.indep_states.states:x_46 c 0.000000E+00 9.695801E+04 1.000000E+21
48 traj.phases.phase0.indep_states.states:x_47 c 0.000000E+00 9.695801E+04 1.000000E+21
49 traj.phases.phase0.indep_states.states:x_48 c 0.000000E+00 1.040786E+05 1.000000E+21
50 traj.phases.phase0.indep_states.states:x_49 c 0.000000E+00 1.144647E+05 1.000000E+21
51 traj.phases.phase0.indep_states.states:x_50 c 0.000000E+00 1.178897E+05 1.000000E+21
52 traj.phases.phase0.indep_states.states:x_51 c 0.000000E+00 1.178897E+05 1.000000E+21
53 traj.phases.phase0.indep_states.states:x_52 c 0.000000E+00 1.259848E+05 1.000000E+21
54 traj.phases.phase0.indep_states.states:x_53 c 0.000000E+00 1.377346E+05 1.000000E+21
55 traj.phases.phase0.indep_states.states:x_54 c 0.000000E+00 1.415942E+05 1.000000E+21
56 traj.phases.phase0.indep_states.states:x_55 c 0.000000E+00 1.415942E+05 1.000000E+21
57 traj.phases.phase0.indep_states.states:x_56 c 0.000000E+00 1.506865E+05 1.000000E+21
58 traj.phases.phase0.indep_states.states:x_57 c 0.000000E+00 1.638094E+05 1.000000E+21
59 traj.phases.phase0.indep_states.states:x_58 c 0.000000E+00 1.681011E+05 1.000000E+21
60 traj.phases.phase0.indep_states.states:x_59 c 0.000000E+00 1.681011E+05 1.000000E+21
61 traj.phases.phase0.indep_states.states:x_60 c 0.000000E+00 1.781752E+05 1.000000E+21
62 traj.phases.phase0.indep_states.states:x_61 c 0.000000E+00 1.926292E+05 1.000000E+21
63 traj.phases.phase0.indep_states.states:x_62 c 0.000000E+00 1.973350E+05 1.000000E+21
64 traj.phases.phase0.indep_states.states:x_63 c 0.000000E+00 1.973350E+05 1.000000E+21
65 traj.phases.phase0.indep_states.states:x_64 c 0.000000E+00 2.083426E+05 1.000000E+21
66 traj.phases.phase0.indep_states.states:x_65 c 0.000000E+00 2.240467E+05 1.000000E+21
67 traj.phases.phase0.indep_states.states:x_66 c 0.000000E+00 2.291382E+05 1.000000E+21
68 traj.phases.phase0.indep_states.states:x_67 c 0.000000E+00 2.291382E+05 1.000000E+21
69 traj.phases.phase0.indep_states.states:x_68 c 0.000000E+00 2.410104E+05 1.000000E+21
70 traj.phases.phase0.indep_states.states:x_69 c 0.000000E+00 2.578637E+05 1.000000E+21
71 traj.phases.phase0.indep_states.states:x_70 c 0.000000E+00 2.633083E+05 1.000000E+21
72 traj.phases.phase0.indep_states.states:x_71 c 0.000000E+00 2.633083E+05 1.000000E+21
73 traj.phases.phase0.indep_states.states:x_72 c 0.000000E+00 2.759693E+05 1.000000E+21
74 traj.phases.phase0.indep_states.states:x_73 c 0.000000E+00 2.938674E+05 1.000000E+21
75 traj.phases.phase0.indep_states.states:x_74 c 0.000000E+00 2.996321E+05 1.000000E+21
76 traj.phases.phase0.indep_states.states:x_75 c 0.000000E+00 2.996321E+05 1.000000E+21
77 traj.phases.phase0.indep_states.states:x_76 c 0.000000E+00 3.130082E+05 1.000000E+21
78 traj.phases.phase0.indep_states.states:x_77 c 0.000000E+00 3.318524E+05 1.000000E+21
79 traj.phases.phase0.indep_states.states:x_78 c 0.000000E+00 3.379071E+05 1.000000E+21
80 traj.phases.phase0.indep_states.states:y_0 c 0.000000E+00 1.072874E+02 1.000000E+21
81 traj.phases.phase0.indep_states.states:y_1 c 0.000000E+00 6.062100E+02 1.000000E+21
82 traj.phases.phase0.indep_states.states:y_2 c 0.000000E+00 8.484836E+02 1.000000E+21
83 traj.phases.phase0.indep_states.states:y_3 c 0.000000E+00 8.484836E+02 1.000000E+21
84 traj.phases.phase0.indep_states.states:y_4 c 0.000000E+00 1.555240E+03 1.000000E+21
85 traj.phases.phase0.indep_states.states:y_5 c 0.000000E+00 2.875780E+03 1.000000E+21
86 traj.phases.phase0.indep_states.states:y_6 c 0.000000E+00 3.376666E+03 1.000000E+21
87 traj.phases.phase0.indep_states.states:y_7 c 0.000000E+00 3.376666E+03 1.000000E+21
88 traj.phases.phase0.indep_states.states:y_8 c 0.000000E+00 4.672843E+03 1.000000E+21
89 traj.phases.phase0.indep_states.states:y_9 c 0.000000E+00 6.799841E+03 1.000000E+21
90 traj.phases.phase0.indep_states.states:y_10 c 0.000000E+00 7.554202E+03 1.000000E+21
91 traj.phases.phase0.indep_states.states:y_11 c 0.000000E+00 7.554202E+03 1.000000E+21
92 traj.phases.phase0.indep_states.states:y_12 c 0.000000E+00 9.427357E+03 1.000000E+21
93 traj.phases.phase0.indep_states.states:y_13 c 0.000000E+00 1.234194E+04 1.000000E+21
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284 traj.phases.phase0.indep_states.states:vy_46 c -1.000000E+21 6.238029E+02 1.000000E+21
285 traj.phases.phase0.indep_states.states:vy_47 c -1.000000E+21 6.238029E+02 1.000000E+21
286 traj.phases.phase0.indep_states.states:vy_48 c -1.000000E+21 6.225753E+02 1.000000E+21
287 traj.phases.phase0.indep_states.states:vy_49 c -1.000000E+21 6.165660E+02 1.000000E+21
288 traj.phases.phase0.indep_states.states:vy_50 c -1.000000E+21 6.135687E+02 1.000000E+21
289 traj.phases.phase0.indep_states.states:vy_51 c -1.000000E+21 6.135687E+02 1.000000E+21
290 traj.phases.phase0.indep_states.states:vy_52 c -1.000000E+21 6.046602E+02 1.000000E+21
291 traj.phases.phase0.indep_states.states:vy_53 c -1.000000E+21 5.876022E+02 1.000000E+21
292 traj.phases.phase0.indep_states.states:vy_54 c -1.000000E+21 5.810350E+02 1.000000E+21
293 traj.phases.phase0.indep_states.states:vy_55 c -1.000000E+21 5.810350E+02 1.000000E+21
294 traj.phases.phase0.indep_states.states:vy_56 c -1.000000E+21 5.639032E+02 1.000000E+21
295 traj.phases.phase0.indep_states.states:vy_57 c -1.000000E+21 5.355666E+02 1.000000E+21
296 traj.phases.phase0.indep_states.states:vy_58 c -1.000000E+21 5.254907E+02 1.000000E+21
297 traj.phases.phase0.indep_states.states:vy_59 c -1.000000E+21 5.254907E+02 1.000000E+21
298 traj.phases.phase0.indep_states.states:vy_60 c -1.000000E+21 5.005069E+02 1.000000E+21
299 traj.phases.phase0.indep_states.states:vy_61 c -1.000000E+21 4.618879E+02 1.000000E+21
300 traj.phases.phase0.indep_states.states:vy_62 c -1.000000E+21 4.487218E+02 1.000000E+21
301 traj.phases.phase0.indep_states.states:vy_63 c -1.000000E+21 4.487218E+02 1.000000E+21
302 traj.phases.phase0.indep_states.states:vy_64 c -1.000000E+21 4.169867E+02 1.000000E+21
303 traj.phases.phase0.indep_states.states:vy_65 c -1.000000E+21 3.698427E+02 1.000000E+21
304 traj.phases.phase0.indep_states.states:vy_66 c -1.000000E+21 3.541771E+02 1.000000E+21
305 traj.phases.phase0.indep_states.states:vy_67 c -1.000000E+21 3.541771E+02 1.000000E+21
306 traj.phases.phase0.indep_states.states:vy_68 c -1.000000E+21 3.170705E+02 1.000000E+21
307 traj.phases.phase0.indep_states.states:vy_69 c -1.000000E+21 2.632979E+02 1.000000E+21
308 traj.phases.phase0.indep_states.states:vy_70 c -1.000000E+21 2.457161E+02 1.000000E+21
309 traj.phases.phase0.indep_states.states:vy_71 c -1.000000E+21 2.457161E+02 1.000000E+21
310 traj.phases.phase0.indep_states.states:vy_72 c -1.000000E+21 2.045276E+02 1.000000E+21
311 traj.phases.phase0.indep_states.states:vy_73 c -1.000000E+21 1.457738E+02 1.000000E+21
312 traj.phases.phase0.indep_states.states:vy_74 c -1.000000E+21 1.267600E+02 1.000000E+21
313 traj.phases.phase0.indep_states.states:vy_75 c -1.000000E+21 1.267600E+02 1.000000E+21
314 traj.phases.phase0.indep_states.states:vy_76 c -1.000000E+21 8.253103E+01 1.000000E+21
315 traj.phases.phase0.indep_states.states:vy_77 c -1.000000E+21 2.007713E+01 1.000000E+21
316 traj.phases.phase0.indep_states.states:vy_78 c -1.000000E+21 -2.646978E-23 1.000000E+21
317 traj.phases.phase0.indep_states.states:m_0 c -1.000000E+21 4.999872E+04 1.000000E+21
318 traj.phases.phase0.indep_states.states:m_1 c -1.000000E+21 4.999695E+04 1.000000E+21
319 traj.phases.phase0.indep_states.states:m_2 c -1.000000E+21 4.999639E+04 1.000000E+21
320 traj.phases.phase0.indep_states.states:m_3 c -1.000000E+21 4.999639E+04 1.000000E+21
321 traj.phases.phase0.indep_states.states:m_4 c -1.000000E+21 4.999510E+04 1.000000E+21
322 traj.phases.phase0.indep_states.states:m_5 c -1.000000E+21 4.999333E+04 1.000000E+21
323 traj.phases.phase0.indep_states.states:m_6 c -1.000000E+21 4.999277E+04 1.000000E+21
324 traj.phases.phase0.indep_states.states:m_7 c -1.000000E+21 4.999277E+04 1.000000E+21
325 traj.phases.phase0.indep_states.states:m_8 c -1.000000E+21 4.999149E+04 1.000000E+21
326 traj.phases.phase0.indep_states.states:m_9 c -1.000000E+21 4.998972E+04 1.000000E+21
327 traj.phases.phase0.indep_states.states:m_10 c -1.000000E+21 4.998916E+04 1.000000E+21
328 traj.phases.phase0.indep_states.states:m_11 c -1.000000E+21 4.998916E+04 1.000000E+21
329 traj.phases.phase0.indep_states.states:m_12 c -1.000000E+21 4.998788E+04 1.000000E+21
330 traj.phases.phase0.indep_states.states:m_13 c -1.000000E+21 4.998611E+04 1.000000E+21
331 traj.phases.phase0.indep_states.states:m_14 c -1.000000E+21 4.998555E+04 1.000000E+21
332 traj.phases.phase0.indep_states.states:m_15 c -1.000000E+21 4.998555E+04 1.000000E+21
333 traj.phases.phase0.indep_states.states:m_16 c -1.000000E+21 4.998427E+04 1.000000E+21
334 traj.phases.phase0.indep_states.states:m_17 c -1.000000E+21 4.998250E+04 1.000000E+21
335 traj.phases.phase0.indep_states.states:m_18 c -1.000000E+21 4.998194E+04 1.000000E+21
336 traj.phases.phase0.indep_states.states:m_19 c -1.000000E+21 4.998194E+04 1.000000E+21
337 traj.phases.phase0.indep_states.states:m_20 c -1.000000E+21 4.998065E+04 1.000000E+21
338 traj.phases.phase0.indep_states.states:m_21 c -1.000000E+21 4.997888E+04 1.000000E+21
339 traj.phases.phase0.indep_states.states:m_22 c -1.000000E+21 4.997832E+04 1.000000E+21
340 traj.phases.phase0.indep_states.states:m_23 c -1.000000E+21 4.997832E+04 1.000000E+21
341 traj.phases.phase0.indep_states.states:m_24 c -1.000000E+21 4.997704E+04 1.000000E+21
342 traj.phases.phase0.indep_states.states:m_25 c -1.000000E+21 4.997527E+04 1.000000E+21
343 traj.phases.phase0.indep_states.states:m_26 c -1.000000E+21 4.997471E+04 1.000000E+21
344 traj.phases.phase0.indep_states.states:m_27 c -1.000000E+21 4.997471E+04 1.000000E+21
345 traj.phases.phase0.indep_states.states:m_28 c -1.000000E+21 4.997343E+04 1.000000E+21
346 traj.phases.phase0.indep_states.states:m_29 c -1.000000E+21 4.997166E+04 1.000000E+21
347 traj.phases.phase0.indep_states.states:m_30 c -1.000000E+21 4.997110E+04 1.000000E+21
348 traj.phases.phase0.indep_states.states:m_31 c -1.000000E+21 4.997110E+04 1.000000E+21
349 traj.phases.phase0.indep_states.states:m_32 c -1.000000E+21 4.996981E+04 1.000000E+21
350 traj.phases.phase0.indep_states.states:m_33 c -1.000000E+21 4.996804E+04 1.000000E+21
351 traj.phases.phase0.indep_states.states:m_34 c -1.000000E+21 4.996748E+04 1.000000E+21
352 traj.phases.phase0.indep_states.states:m_35 c -1.000000E+21 4.996748E+04 1.000000E+21
353 traj.phases.phase0.indep_states.states:m_36 c -1.000000E+21 4.996620E+04 1.000000E+21
354 traj.phases.phase0.indep_states.states:m_37 c -1.000000E+21 4.996443E+04 1.000000E+21
355 traj.phases.phase0.indep_states.states:m_38 c -1.000000E+21 4.996387E+04 1.000000E+21
356 traj.phases.phase0.indep_states.states:m_39 c -1.000000E+21 4.996387E+04 1.000000E+21
357 traj.phases.phase0.indep_states.states:m_40 c -1.000000E+21 4.996259E+04 1.000000E+21
358 traj.phases.phase0.indep_states.states:m_41 c -1.000000E+21 4.996082E+04 1.000000E+21
359 traj.phases.phase0.indep_states.states:m_42 c -1.000000E+21 4.996026E+04 1.000000E+21
360 traj.phases.phase0.indep_states.states:m_43 c -1.000000E+21 4.996026E+04 1.000000E+21
361 traj.phases.phase0.indep_states.states:m_44 c -1.000000E+21 4.995898E+04 1.000000E+21
362 traj.phases.phase0.indep_states.states:m_45 c -1.000000E+21 4.995721E+04 1.000000E+21
363 traj.phases.phase0.indep_states.states:m_46 c -1.000000E+21 4.995665E+04 1.000000E+21
364 traj.phases.phase0.indep_states.states:m_47 c -1.000000E+21 4.995665E+04 1.000000E+21
365 traj.phases.phase0.indep_states.states:m_48 c -1.000000E+21 4.995536E+04 1.000000E+21
366 traj.phases.phase0.indep_states.states:m_49 c -1.000000E+21 4.995359E+04 1.000000E+21
367 traj.phases.phase0.indep_states.states:m_50 c -1.000000E+21 4.995303E+04 1.000000E+21
368 traj.phases.phase0.indep_states.states:m_51 c -1.000000E+21 4.995303E+04 1.000000E+21
369 traj.phases.phase0.indep_states.states:m_52 c -1.000000E+21 4.995175E+04 1.000000E+21
370 traj.phases.phase0.indep_states.states:m_53 c -1.000000E+21 4.994998E+04 1.000000E+21
371 traj.phases.phase0.indep_states.states:m_54 c -1.000000E+21 4.994942E+04 1.000000E+21
372 traj.phases.phase0.indep_states.states:m_55 c -1.000000E+21 4.994942E+04 1.000000E+21
373 traj.phases.phase0.indep_states.states:m_56 c -1.000000E+21 4.994814E+04 1.000000E+21
374 traj.phases.phase0.indep_states.states:m_57 c -1.000000E+21 4.994637E+04 1.000000E+21
375 traj.phases.phase0.indep_states.states:m_58 c -1.000000E+21 4.994581E+04 1.000000E+21
376 traj.phases.phase0.indep_states.states:m_59 c -1.000000E+21 4.994581E+04 1.000000E+21
377 traj.phases.phase0.indep_states.states:m_60 c -1.000000E+21 4.994452E+04 1.000000E+21
378 traj.phases.phase0.indep_states.states:m_61 c -1.000000E+21 4.994275E+04 1.000000E+21
379 traj.phases.phase0.indep_states.states:m_62 c -1.000000E+21 4.994219E+04 1.000000E+21
380 traj.phases.phase0.indep_states.states:m_63 c -1.000000E+21 4.994219E+04 1.000000E+21
381 traj.phases.phase0.indep_states.states:m_64 c -1.000000E+21 4.994091E+04 1.000000E+21
382 traj.phases.phase0.indep_states.states:m_65 c -1.000000E+21 4.993914E+04 1.000000E+21
383 traj.phases.phase0.indep_states.states:m_66 c -1.000000E+21 4.993858E+04 1.000000E+21
384 traj.phases.phase0.indep_states.states:m_67 c -1.000000E+21 4.993858E+04 1.000000E+21
385 traj.phases.phase0.indep_states.states:m_68 c -1.000000E+21 4.993730E+04 1.000000E+21
386 traj.phases.phase0.indep_states.states:m_69 c -1.000000E+21 4.993553E+04 1.000000E+21
387 traj.phases.phase0.indep_states.states:m_70 c -1.000000E+21 4.993497E+04 1.000000E+21
388 traj.phases.phase0.indep_states.states:m_71 c -1.000000E+21 4.993497E+04 1.000000E+21
389 traj.phases.phase0.indep_states.states:m_72 c -1.000000E+21 4.993369E+04 1.000000E+21
390 traj.phases.phase0.indep_states.states:m_73 c -1.000000E+21 4.993192E+04 1.000000E+21
391 traj.phases.phase0.indep_states.states:m_74 c -1.000000E+21 4.993136E+04 1.000000E+21
392 traj.phases.phase0.indep_states.states:m_75 c -1.000000E+21 4.993136E+04 1.000000E+21
393 traj.phases.phase0.indep_states.states:m_76 c -1.000000E+21 4.993007E+04 1.000000E+21
394 traj.phases.phase0.indep_states.states:m_77 c -1.000000E+21 4.992830E+04 1.000000E+21
395 traj.phases.phase0.indep_states.states:m_78 c -1.000000E+21 4.992774E+04 1.000000E+21
396 traj.phases.phase0.polynomial_control_group.indep_polynomial_controls.polynomial_controls:tan_theta_0 c -1.000000E+21 2.747399E+00 1.000000E+21
397 traj.phases.phase0.polynomial_control_group.indep_polynomial_controls.polynomial_controls:tan_theta_1 c -1.000000E+21 -1.241462E+00 1.000000E+21
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 1.627000E+03 1.627000E+03 1.627000E+03 9.00000E+100
2 traj.phases.phase0->final_boundary_constraint->vy e 0.000000E+00 -2.646978E-23 0.000000E+00 9.00000E+100
3 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
4 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.557009E-14 0.000000E+00 9.00000E+100
5 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 8.557009E-14 0.000000E+00 9.00000E+100
6 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 7.701308E-13 0.000000E+00 9.00000E+100
7 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
8 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 6.845607E-13 0.000000E+00 9.00000E+100
9 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.540262E-12 0.000000E+00 9.00000E+100
10 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.026841E-12 0.000000E+00 9.00000E+100
11 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.369121E-12 0.000000E+00 9.00000E+100
12 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.449645E-12 0.000000E+00 9.00000E+100
13 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 3.422804E-13 0.000000E+00 9.00000E+100
14 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.369121E-12 0.000000E+00 9.00000E+100
15 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -3.080523E-12 0.000000E+00 9.00000E+100
16 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.369121E-12 0.000000E+00 9.00000E+100
17 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.395963E-12 0.000000E+00 9.00000E+100
18 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 7.187888E-12 0.000000E+00 9.00000E+100
19 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -3.422804E-12 0.000000E+00 9.00000E+100
20 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 4.107364E-12 0.000000E+00 9.00000E+100
21 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -6.845607E-13 0.000000E+00 9.00000E+100
22 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 7.530168E-12 0.000000E+00 9.00000E+100
23 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 7.530168E-12 0.000000E+00 9.00000E+100
24 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 4.107364E-12 0.000000E+00 9.00000E+100
25 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 8.214729E-12 0.000000E+00 9.00000E+100
26 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.053682E-12 0.000000E+00 9.00000E+100
27 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.791925E-12 0.000000E+00 9.00000E+100
28 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.163753E-11 0.000000E+00 9.00000E+100
29 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 3.422804E-12 0.000000E+00 9.00000E+100
30 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
31 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.369121E-12 0.000000E+00 9.00000E+100
32 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 8.214729E-12 0.000000E+00 9.00000E+100
33 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.232209E-11 0.000000E+00 9.00000E+100
34 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.232209E-11 0.000000E+00 9.00000E+100
35 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 5.476486E-12 0.000000E+00 9.00000E+100
36 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -9.583850E-12 0.000000E+00 9.00000E+100
37 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.369121E-12 0.000000E+00 9.00000E+100
38 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.107364E-12 0.000000E+00 9.00000E+100
39 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.899290E-11 0.000000E+00 9.00000E+100
40 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
41 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.738243E-12 0.000000E+00 9.00000E+100
42 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.053682E-11 0.000000E+00 9.00000E+100
43 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.655013E-11 0.000000E+00 9.00000E+100
44 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.095297E-11 0.000000E+00 9.00000E+100
45 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 6.024134E-11 0.000000E+00 9.00000E+100
46 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 3.559716E-11 0.000000E+00 9.00000E+100
47 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.369121E-11 0.000000E+00 9.00000E+100
48 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.214729E-12 0.000000E+00 9.00000E+100
49 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 2.738243E-11 0.000000E+00 9.00000E+100
50 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.214729E-12 0.000000E+00 9.00000E+100
51 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.928837E-11 0.000000E+00 9.00000E+100
52 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -6.024134E-11 0.000000E+00 9.00000E+100
53 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 5.750310E-11 0.000000E+00 9.00000E+100
54 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -9.310026E-11 0.000000E+00 9.00000E+100
55 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 8.214729E-12 0.000000E+00 9.00000E+100
56 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.107364E-11 0.000000E+00 9.00000E+100
57 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.396504E-10 0.000000E+00 9.00000E+100
58 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 3.285892E-11 0.000000E+00 9.00000E+100
59 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 1.423886E-10 0.000000E+00 9.00000E+100
60 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.628713E-10 0.000000E+00 9.00000E+100
61 traj.phases.phase0.collocation_constraint.defects:x e 0.000000E+00 8.214729E-12 0.000000E+00 9.00000E+100
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391 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
392 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
393 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
394 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
395 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
396 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
397 traj.phases.phase0.continuity_comp.defect_states:m e 0.000000E+00 0.000000E+00 0.000000E+00 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: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
sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('dymos_simulation.db').get_case('final')
fig, [traj_ax, control_ax, param_ax] = plt.subplots(nrows=3, 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_ylabel(r'$\theta$ (deg)')
control_ax.grid(True)
tan_theta_sol = sol.get_val('traj.phase0.timeseries.tan_theta')
tan_theta_sim = sim.get_val('traj.phase0.timeseries.tan_theta')
param_ax.plot(sol.get_val('traj.phase0.timeseries.time'),
tan_theta_sol,
marker='o',
ms=4,
linestyle='None')
param_ax.plot(sim.get_val('traj.phase0.timeseries.time'),
tan_theta_sim,
linestyle='-',
marker=None)
param_ax.set_xlabel('time (s)')
param_ax.set_ylabel(r'$tan(\theta)$')
param_ax.grid(True)
plt.suptitle('Single Stage to Orbit Solution Using a Polynomial Control')
fig.legend(loc='lower center', ncol=2)
plt.show()
