Water Rocket#
Author: Bernardo Bahia Monteiro (bbahia@umich.edu)
In this example, we will optimize a water rocket for range and height at the apogee, using design variables that are easily modifiable just before launch: the empty mass, the initial water volume and the launch angle. This example builds on multi-phase cannonball ane is adapted from Optimization of a Water Rocket in OpenMDAO/Dymos [BM20].
Nomenclature#
Symbol |
definition |
|---|---|
\(v_\text{out}\) |
water exit speed at the nozzle |
\(A_\text{out}\) |
nozzle area |
\(V_w\) |
water volume in the rocket |
\(p\) |
pressure in the rocket |
\(p_a\) |
ambient pressure |
\(\dot{\,}\) |
time derivative |
\(k\) |
polytropic constant |
\(V_b\) |
internal volume of the rocket |
\(\rho_w\) |
water density |
\(T\) |
thrust |
\(q\) |
dynamic pressure |
\(S\) |
cross sectional area |
\((\cdot)_0\) |
value of \((\cdot)\) at \(t=0\) |
\(t\) |
time |
Problem Formulation#
A natural objective function for a water rocket is the maximum height achieved by the rocket during flight, or the horizontal distance it travels, i.e. its range. The design of a water rocket is somewhat constrained by the soda bottle used as its engine. This means that the volume available for water and air is fixed, the initial launch pressure is limited by the bottle’s strength (since the pressure is directly related to the energy available for the rocket, it is easy to see that it should be as high as possible) and the nozzle throat area is also fixed. Given these manufacturing constraints, the design variables we are left with are the empty mass (it can be easily changed through adding ballast), the water volume at the launch, and the launch angle. With this considerations in mind, a natural formulation for the water rocket problem is
(89)#\[\begin{align}
\text{maximize} &\quad \text{range or height} \\
\text{w.r.t.} &\quad \text{empty mass, initial water volume, launch angle, trajectory} \\
\text{subject to} &\quad \text{flight dynamics} \\
&\quad \text{fluid dynamics inside the rocket} \\
&\quad 0 < \text{initial water volume} < \text{volume of bottle} \\
&\quad 0^\circ < \text{launch angle} < 90^\circ \\
&\quad 0 < \text{empty mass}
\end{align}\]
Model#
The water rocket model is divided into three basic components: a water engine, responsible for modelling the fluid dynamics inside the rocket and returning its thrust; the aerodynamics, responsible for calculating the atmospheric drag of the rocket; and the equations of motion, responsible for propagating the rocket’s trajectory in time, using Newton’s laws and the forces provided by the other two components.
In order to integrate these three basic components, some additional interfacing components are necessary: an atmospheric model to provide values of ambient pressure for the water engine and air density to the calculation of the dynamic pressure for the aerodynamic model, and a component that calculates the instantaneous mass of the rocket by summing the water mass with the rocket’s empty mass. The high level layout of this model is shown in below.
atmos, dynamic_pressure, aero and eom are the same models used in multi-phase cannonball.
The remaining components are discussed below.
Warning
The eom component has a singularity in the flight path angle derivative when the flight speed is zero.
This happens because the rotational dynamics are not modelled.
This can cause convergence problems if the launch velocity is set to zero or the launch angle is set to \(90^\circ\)
Note
Since the range of altitudes achieved by the water rocket is very small (100m), the air density and pressure are practically constant, thus the use of an atmospheric model is not necessary. However, using it makes it easier to reuse code from multi-phase cannonball.
Water engine#
The water engine is modelled by assuming that the air expansion in the rocket follows an adiabatic process and the water flow is incompressible and inviscid, i.e. it follows Bernoulli’s equation. We also make the following simplifying assumptions:
The thrust developed after the water is depleted is negligible
The area inside the bottle is much smaller than the nozzle area
The inertial forces do not affect the fluid dynamics inside the bottle
This simplified modelling can be found in Prusa[@Prusa2000]. A more rigorous formulation, which drops all these simplifying assumptions can be found in Wheeler[@Wheeler2002], Gommes[@Gommes2010], and Barria-Perotti[@BarrioPerotti2010].
The first assumption leads to an underestimation of the rocket performance, since the air left in the bottle after it is out of water is known to generate appreciable thrust[@Thorncroft2009]. This simplified model, however, produces physically meaningful results.
There are two states in this dynamical model, the water volume in the rocket \(V_w\) and the gauge pressure inside the rocket \(p\). The constitutive equations and the N2 diagram showing the model organization are shown below.
Constitutive equations of the water engine model#
Component |
Equation |
|---|---|
water_exhaust_speed |
\(v_\text{out} = \sqrt{2(p-p_a)/\rho_w}\) |
water_flow_rate |
\(\dot{V}_w = -v_\text{out} A_\text{out}\) |
pressure_rate |
\(\dot{p} = kp\frac{\dot{V_w}}{(V_b-V_w)}\) |
water_thrust |
\(T = (\rho_w v_\text{out})(v_\text{out}A_\text{out})\) |
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import openmdao.api as om
import dymos as dm
class WaterEngine(om.Group):
"""
Computes thrust and water flow for a water.
Simplifications:
- the pressure due to the water column in the non inertial frame (i.e.
under a+g acceleration) is insignificant compared to the air pressure
- the water does not have appreciable speed inside the bottle
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_subsystem(name='water_exhaust_speed',
subsys=_WaterExhaustSpeed(num_nodes=nn),
promotes=['p', 'p_a', 'rho_w'])
self.add_subsystem(name='water_flow_rate',
subsys=_WaterFlowRate(num_nodes=nn),
promotes=['A_out', 'Vdot'])
self.add_subsystem(name='pressure_rate',
subsys=_PressureRate(num_nodes=nn),
promotes=['p', 'k', 'V_b', 'V_w', 'Vdot', 'pdot'])
self.add_subsystem(name='water_thrust',
subsys=_WaterThrust(num_nodes=nn),
promotes=['rho_w', 'A_out', 'F'])
self.connect('water_exhaust_speed.v_out', 'water_flow_rate.v_out')
self.connect('water_exhaust_speed.v_out', 'water_thrust.v_out')
class _WaterExhaustSpeed(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input(name='rho_w', val=1e3*np.ones(nn), desc='water density', units='kg/m**3')
self.add_input(name='p', val=6.5e5*np.ones(nn), desc='air pressure', units='N/m**2') # 5.5bar = 80 psi
self.add_input(name='p_a', val=1.01e5*np.ones(nn), desc='air pressure', units='N/m**2')
self.add_output(name='v_out', shape=(nn,), desc='water exhaust speed', units='m/s')
ar = np.arange(nn)
self.declare_partials(of='*', wrt='*', rows=ar, cols=ar)
def compute(self, inputs, outputs):
p = inputs['p']
p_a = inputs['p_a']
rho_w = inputs['rho_w']
outputs['v_out'] = np.sqrt(2*(p-p_a)/rho_w)
def compute_partials(self, inputs, partials):
p = inputs['p']
p_a = inputs['p_a']
rho_w = inputs['rho_w']
v_out = np.sqrt(2*(p-p_a)/rho_w)
partials['v_out', 'p'] = 1/v_out/rho_w
partials['v_out', 'p_a'] = -1/v_out/rho_w
partials['v_out', 'rho_w'] = dv_outdrho_w = 1/v_out*(-(p-p_a)/rho_w**2)
class _PressureRate(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input(name='p', val=np.ones(nn), desc='air pressure', units='N/m**2')
self.add_input(name='k', val=1.4*np.ones(nn), desc='polytropic coefficient for expansion', units=None)
self.add_input(name='V_b', val=2e-3*np.ones(nn), desc='bottle volume', units='m**3')
self.add_input(name='V_w', val=1e-3*np.ones(nn), desc='water volume', units='m**3')
self.add_input(name='Vdot', shape=(nn,), desc='water flow', units='m**3/s')
self.add_output(name='pdot', shape=(nn,), desc='pressure derivative', units='N/m**2/s')
ar = np.arange(nn)
self.declare_partials(of='*', wrt='*', rows=ar, cols=ar)
def compute(self, inputs, outputs):
p = inputs['p']
k = inputs['k']
V_b = inputs['V_b']
V_w = inputs['V_w']
Vdot = inputs['Vdot']
pdot = p*k*Vdot/(V_b-V_w)
outputs['pdot'] = pdot
def compute_partials(self, inputs, partials):
p = inputs['p']
k = inputs['k']
V_b = inputs['V_b']
V_w = inputs['V_w']
Vdot = inputs['Vdot']
partials['pdot', 'p'] = k*Vdot/(V_b-V_w)
partials['pdot', 'k'] = p*Vdot/(V_b-V_w)
partials['pdot', 'V_b'] = -p*k*Vdot/(V_b-V_w)**2
partials['pdot', 'V_w'] = p*k*Vdot/(V_b-V_w)**2
partials['pdot', 'Vdot'] = p*k/(V_b-V_w)
class _WaterFlowRate(om.ExplicitComponent):
""" Computer water flow rate"""
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input(name='A_out', val=np.ones(nn), desc='nozzle outlet area', units='m**2')
self.add_input(name='v_out', val=np.zeros(nn), desc='water exhaust speed', units='m/s')
self.add_output(name='Vdot', shape=(nn,), desc='water flow', units='m**3/s')
ar = np.arange(nn)
self.declare_partials(of='*', wrt='*', rows=ar, cols=ar)
def compute(self, inputs, outputs):
A_out = inputs['A_out']
v_out = inputs['v_out']
outputs['Vdot'] = -v_out*A_out
def compute_partials(self, inputs, partials):
A_out = inputs['A_out']
v_out = inputs['v_out']
partials['Vdot', 'A_out'] = -v_out
partials['Vdot', 'v_out'] = -A_out
The _MassAdder component calculates the rocket’s instantaneous mass by
summing the water mass with the rockets empty mass, i.e.
(90)#\[\begin{align}
m = m_\text{empty}+\rho_w V_w
\end{align}\]
class _MassAdder(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('m_empty', val=np.zeros(nn), desc='empty mass', units='kg')
self.add_input('V_w', val=1e-3*np.ones(nn), desc='water volume', units='m**3')
self.add_input('rho_w', val=1e3*np.ones(nn), desc="water density", units='kg/m**3')
self.add_output('m', val=np.zeros(nn), desc='total mass', units='kg')
ar = np.arange(nn)
self.declare_partials('*', '*', cols=ar, rows=ar)
def compute(self, inputs, outputs):
outputs['m'] = inputs['m_empty'] + inputs['rho_w']*inputs['V_w']
def compute_partials(self, inputs, jacobian):
jacobian['m', 'm_empty'] = 1
jacobian['m', 'rho_w'] = inputs['V_w']
jacobian['m', 'V_w'] = inputs['rho_w']
Now these components are joined in a single group
display_source('dymos.examples.water_rocket.water_propulsion_ode.WaterPropulsionODE')
class WaterPropulsionODE(om.Group):
def initialize(self):
self.options.declare('num_nodes', types=int)
self.options.declare('ballistic', types=bool, default=False,
desc='If True, neglect propulsion system.')
def setup(self):
nn = self.options['num_nodes']
self.add_subsystem(name='atmos',
subsys=USatm1976Comp(num_nodes=nn))
if not self.options['ballistic']:
self.add_subsystem(name='water_engine',
subsys=WaterEngine(num_nodes=nn),
promotes_inputs=['rho_w'])
self.add_subsystem(name='mass_adder',
subsys=_MassAdder(num_nodes=nn),
promotes_inputs=['rho_w'])
self.add_subsystem(name='dynamic_pressure',
subsys=DynamicPressureComp(num_nodes=nn))
self.add_subsystem(name='aero',
subsys=LiftDragForceComp(num_nodes=nn))
self.add_subsystem(name='eom',
subsys=FlightPathEOM2D(num_nodes=nn))
self.connect('atmos.rho', 'dynamic_pressure.rho')
self.connect('dynamic_pressure.q', 'aero.q')
self.connect('aero.f_drag', 'eom.D')
self.connect('aero.f_lift', 'eom.L')
if not self.options['ballistic']:
self.connect('atmos.pres', 'water_engine.p_a')
self.connect('water_engine.F', 'eom.T')
self.connect('mass_adder.m', 'eom.m')
Phases#
The flight of the water rocket is split in three distinct phases: propelled ascent, ballistic ascent and ballistic descent. If the simplification of no thrust without water were lifted, there would be an extra “air propelled ascent” phase between the propelled ascent and ballistic ascent phases.
Propelled ascent: is the flight phase where the rocket still has water inside, and hence it is producing thrust. The thrust is given by the water engine model, and fed into the flight dynamic equations. It starts at launch and finishes when the water is depleted, i.e. \(V_w=0\).
Ballistic ascent: is the flight phase where the rocket is ascending (\(\gamma>0\)) but produces no thrust. This phase begins at the end of thepropelled ascent phase and ends at the apogee, defined by \(\gamma=0\).
Descent: is the phase where the rocket is descending without thrust. It begins at the end of the ballistic ascent phase and ends with ground impact, i.e. \(h=0\).
def new_propelled_ascent_phase(transcription):
propelled_ascent = dm.Phase(ode_class=WaterPropulsionODE,
transcription=transcription)
# All initial states except flight path angle and water volume are fixed
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
# Final water volume is fixed (we will set it to zero so that phase ends when bottle empties)
propelled_ascent.set_time_options(
fix_initial=True, duration_bounds=(0.001, 0.5), duration_ref=0.1, units='s')
propelled_ascent.add_state('r', units='m', rate_source='eom.r_dot',
fix_initial=True, fix_final=False, ref=100.0, defect_ref=100.0)
propelled_ascent.add_state('h', units='m', rate_source='eom.h_dot', targets=['atmos.h'],
fix_initial=True, fix_final=False, ref=100.0, defect_ref=100.0)
propelled_ascent.add_state('gam', units='deg', rate_source='eom.gam_dot', targets=['eom.gam'],
fix_initial=False, fix_final=False, lower=0, upper=85.0, ref=90)
propelled_ascent.add_state('v', units='m/s', rate_source='eom.v_dot', targets=['dynamic_pressure.v', 'eom.v'],
fix_initial=True, fix_final=False, ref=100, defect_ref=100, lower=0.01)
propelled_ascent.add_state('p', units='bar', rate_source='water_engine.pdot',
targets=['water_engine.p'], fix_initial=True, fix_final=False,
lower=1.02)
propelled_ascent.add_state('V_w', units='L', rate_source='water_engine.Vdot',
targets=['water_engine.V_w', 'mass_adder.V_w'],
fix_initial=False, fix_final=True, ref=10, defect_ref=10,
lower=0)
propelled_ascent.add_parameter(
'S', targets=['aero.S'], units='m**2')
propelled_ascent.add_parameter(
'm_empty', targets=['mass_adder.m_empty'], units='kg')
propelled_ascent.add_parameter(
'V_b', targets=['water_engine.V_b'], units='m**3')
propelled_ascent.add_timeseries_output('water_engine.F', 'T', units='N')
return propelled_ascent
def new_ballistic_ascent_phase(transcription):
ballistic_ascent = dm.Phase(ode_class=WaterPropulsionODE, transcription=transcription,
ode_init_kwargs={'ballistic': True})
# All initial states are free (they will be linked to the final stages of propelled_ascent).
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
ballistic_ascent.set_time_options(
fix_initial=False, initial_bounds=(0.001, 1), duration_bounds=(0.001, 10),
duration_ref=1, units='s')
ballistic_ascent.add_state('r', units='m', rate_source='eom.r_dot', fix_initial=False,
fix_final=False, ref=1.0, defect_ref=1.0)
ballistic_ascent.add_state('h', units='m', rate_source='eom.h_dot', targets=['atmos.h'],
fix_initial=False, fix_final=False, ref=1.0)
ballistic_ascent.add_state('gam', units='deg', rate_source='eom.gam_dot', targets=['eom.gam'],
fix_initial=False, fix_final=True, upper=89, ref=90)
ballistic_ascent.add_state('v', units='m/s', rate_source='eom.v_dot',
targets=['dynamic_pressure.v', 'eom.v'], fix_initial=False,
fix_final=False, ref=1.)
ballistic_ascent.add_parameter('S', targets=['aero.S'], units='m**2')
ballistic_ascent.add_parameter('m_empty', targets=['eom.m'], units='kg')
return ballistic_ascent
def new_descent_phase(transcription):
descent = dm.Phase(ode_class=WaterPropulsionODE, transcription=transcription,
ode_init_kwargs={'ballistic': True})
# All initial states and time are free (they will be linked to the final states of ballistic_ascent).
# Final altitude is fixed (we will set it to zero so that the phase ends at ground impact)
descent.set_time_options(initial_bounds=(.5, 100), duration_bounds=(.5, 100),
duration_ref=10, units='s')
descent.add_state('r', units='m', rate_source='eom.r_dot', fix_initial=False, fix_final=False)
descent.add_state('h', units='m', rate_source='eom.h_dot', targets=['atmos.h'],
fix_initial=False, fix_final=True)
descent.add_state('gam', units='deg', rate_source='eom.gam_dot', targets=['eom.gam'],
fix_initial=False, fix_final=False)
descent.add_state('v', units='m/s', rate_source='eom.v_dot',
targets=['dynamic_pressure.v', 'eom.v'],
fix_initial=False, fix_final=False,
ref=1)
descent.add_parameter('S', targets=['aero.S'], units='m**2')
descent.add_parameter('mass', targets=['eom.m'], units='kg')
return descent
Model parameters#
The model requires a few constant parameters. The values used are shown in the following table.
Values for parameters in the water rocket model
Parameter |
Value |
Unit |
Reference |
|---|---|---|---|
\(C_D\) |
0.3450 |
- |
|
\(S\) |
\(\pi 106^2/4\) |
\(mm^2\) |
|
\(k\) |
1.2 |
- |
|
\(A_\text{out}\) |
\(\pi22^2/4\) |
\(mm^2\) |
[air13] |
\(V_b\) |
2 |
L |
|
\(\rho_w\) |
1000 |
\(kg/m^3\) |
|
\(p_0\) |
6.5 |
bar |
|
\(v_0\) |
0.1 |
\(m/s\) |
|
\(h_0\) |
0 |
\(m\) |
|
\(r_0\) |
0 |
\(m\) |
Values for the bottle volume \(V_b\), its cross-sectional area \(S\) and the nozzle area \(A_\text{out}\) are determined by the soda bottle that makes the rocket primary structure, and thus are not easily modifiable by the designer. The polytropic coefficient \(k\) is a function of the moist air characteristics inside the rocket. The initial speed \(v_0\) must be set to a value higher than zero, otherwise the flight dynamic equations become singular. This issue arises from the angular dynamics of the rocket not being modelled. The drag coefficient \(C_D\) is sensitive to the aerodynamic design, but can be optimized by a single discipline analysis. The initial pressure \(p_0\) should be maximized in order to obtain the maximum range or height for the rocket. It is limited by the structural properties of the bottle, which are modifiable by the designer, since the bottle needs to be available commercially. Finally, the starting point of the rocket is set to the origin.
Putting it all together#
The different phases must be combined in a single trajectory, and linked in a sequence. Here we also define the design variables.
display_source('dymos.examples.water_rocket.phases.new_water_rocket_trajectory')
def new_water_rocket_trajectory(objective):
tx_prop = dm.Radau(num_segments=50, order=3, compressed=True)
tx_bal = dm.Radau(num_segments=10, order=3, compressed=True)
tx_desc = dm.Radau(num_segments=10, order=3, compressed=True)
traj = dm.Trajectory()
# Add phases to trajectory
propelled_ascent = traj.add_phase('propelled_ascent', new_propelled_ascent_phase(tx_prop))
ballistic_ascent = traj.add_phase('ballistic_ascent', new_ballistic_ascent_phase(tx_bal))
descent = traj.add_phase('descent', new_descent_phase(tx_desc))
# Link phases
traj.link_phases(phases=['propelled_ascent', 'ballistic_ascent'], vars=['*'])
traj.link_phases(phases=['ballistic_ascent', 'descent'], vars=['*'])
# Set objective function
if objective == 'height':
ballistic_ascent.add_objective('h', loc='final', ref=-1.0)
elif objective == 'range':
descent.add_objective('r', loc='final', ref=-100.0)
else:
raise ValueError(f"objective='{objective}' is not defined. Try using 'height' or 'range'")
# Add design parameters to the trajectory.
traj.add_parameter('CD',
targets={'propelled_ascent': ['aero.CD'],
'ballistic_ascent': ['aero.CD'],
'descent': ['aero.CD']},
val=0.3450, units=None, opt=False)
traj.add_parameter('CL',
targets={'propelled_ascent': ['aero.CL'],
'ballistic_ascent': ['aero.CL'],
'descent': ['aero.CL']},
val=0.0, units=None, opt=False)
traj.add_parameter('T',
targets={'ballistic_ascent': ['eom.T'],
'descent': ['eom.T']},
val=0.0, units='N', opt=False)
traj.add_parameter('alpha',
targets={'propelled_ascent': ['eom.alpha'],
'ballistic_ascent': ['eom.alpha'],
'descent': ['eom.alpha']},
val=0.0, units='deg', opt=False)
traj.add_parameter('m_empty', units='kg', val=0.15,
targets={'propelled_ascent': 'm_empty',
'ballistic_ascent': 'm_empty',
'descent': 'mass'},
lower=0.001, upper=1, ref=0.1,
opt=False)
traj.add_parameter('V_b', units='m**3', val=2e-3,
targets={'propelled_ascent': 'V_b'},
opt=False)
traj.add_parameter('S', units='m**2', val=np.pi*106e-3**2/4, opt=False)
traj.add_parameter('A_out', units='m**2', val=np.pi*22e-3**2/4.,
targets={'propelled_ascent': ['water_engine.A_out']},
opt=False)
traj.add_parameter('k', units=None, val=1.2, opt=False,
targets={'propelled_ascent': ['water_engine.k']})
# Make parameter m_empty a design variable
traj.set_parameter_options('m_empty', opt=True)
return traj, {'propelled_ascent': propelled_ascent,
'ballistic_ascent': ballistic_ascent,
'descent': descent}
Helper Functions to Access the Results#
from collections import namedtuple
def summarize_results(water_rocket_problem):
p = water_rocket_problem
Entry = namedtuple('Entry', 'value unit')
summary = {
'Launch angle': Entry(p.get_val('traj.propelled_ascent.timeseries.gam', units='deg')[0, 0], 'deg'),
'Flight angle at end of propulsion': Entry(p.get_val('traj.propelled_ascent.timeseries.gam',
units='deg')[-1, 0], 'deg'),
'Empty mass': Entry(p.get_val('traj.parameters:m_empty', units='kg')[0], 'kg'),
'Water volume': Entry(p.get_val('traj.propelled_ascent.timeseries.V_w', 'L')[0, 0], 'L'),
'Maximum range': Entry(p.get_val('traj.descent.timeseries.r', units='m')[-1, 0], 'm'),
'Maximum height': Entry(p.get_val('traj.ballistic_ascent.timeseries.h', units='m')[-1, 0], 'm'),
'Maximum velocity': Entry(p.get_val('traj.propelled_ascent.timeseries.v', units='m/s')[-1, 0], 'm/s'),
}
return summary
colors={'pa': 'tab:blue', 'ba': 'tab:orange', 'd': 'tab:green'}
def plot_propelled_ascent(p, exp_out):
fig, ax = plt.subplots(2, 2, sharex=True, figsize=(12, 6))
t_imp = p.get_val('traj.propelled_ascent.timeseries.time', 's')
t_exp = exp_out.get_val('traj.propelled_ascent.timeseries.time', 's')
c = colors['pa']
ax[0,0].plot(t_imp, p.get_val('traj.propelled_ascent.timeseries.p', 'bar'), '.', color=c)
ax[0,0].plot(t_exp, exp_out.get_val('traj.propelled_ascent.timeseries.p', 'bar'), '-', color=c)
ax[0,0].set_ylabel('p (bar)')
ax[0,0].set_ylim(bottom=0)
ax[1,0].plot(t_imp, p.get_val('traj.propelled_ascent.timeseries.V_w', 'L'), '.', color=c)
ax[1,0].plot(t_exp, exp_out.get_val('traj.propelled_ascent.timeseries.V_w', 'L'), '-', color=c)
ax[1,0].set_ylabel('$V_w$ (L)')
ax[0,1].plot(t_imp, p.get_val('traj.propelled_ascent.timeseries.T', 'N'), '.', color=c)
ax[0,1].plot(t_exp, exp_out.get_val('traj.propelled_ascent.timeseries.T', 'N'), '-', color=c)
ax[0,1].set_ylabel('T (N)')
ax[0,1].set_ylim(bottom=0)
ax[1,1].plot(t_imp, p.get_val('traj.propelled_ascent.timeseries.v', 'm/s'), '.', color=c)
ax[1,1].plot(t_exp, exp_out.get_val('traj.propelled_ascent.timeseries.v', 'm/s'), '-', color=c)
ax[1,1].set_ylabel('v (m/s)')
ax[1,1].set_ylim(bottom=0)
ax[1,0].set_xlabel('t (s)')
ax[1,1].set_xlabel('t (s)')
for i in range(4):
ax.ravel()[i].grid(True, alpha=0.2)
fig.tight_layout()
def plot_states(p, exp_out, legend_loc='right', legend_ncol=3):
fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(12, 4), sharex=True)
states = ['r', 'h', 'v', 'gam']
units = ['m', 'm', 'm/s', 'deg']
phases = ['propelled_ascent', 'ballistic_ascent', 'descent']
time_imp = {'ballistic_ascent': p.get_val('traj.ballistic_ascent.timeseries.time'),
'propelled_ascent': p.get_val('traj.propelled_ascent.timeseries.time'),
'descent': p.get_val('traj.descent.timeseries.time')}
time_exp = {'ballistic_ascent': exp_out.get_val('traj.ballistic_ascent.timeseries.time'),
'propelled_ascent': exp_out.get_val('traj.propelled_ascent.timeseries.time'),
'descent': exp_out.get_val('traj.descent.timeseries.time')}
x_imp = {phase: {state: p.get_val(f"traj.{phase}.timeseries.{state}", unit)
for state, unit in zip(states, units)
}
for phase in phases
}
x_exp = {phase: {state: exp_out.get_val(f"traj.{phase}.timeseries.{state}", unit)
for state, unit in zip(states, units)
}
for phase in phases
}
for i, (state, unit) in enumerate(zip(states, units)):
axes.ravel()[i].set_ylabel(f"{state} ({unit})" if state != 'gam' else f'$\gamma$ ({unit})')
axes.ravel()[i].plot(time_imp['propelled_ascent'], x_imp['propelled_ascent'][state], '.', color=colors['pa'])
axes.ravel()[i].plot(time_imp['ballistic_ascent'], x_imp['ballistic_ascent'][state], '.', color=colors['ba'])
axes.ravel()[i].plot(time_imp['descent'], x_imp['descent'][state], '.', color=colors['d'])
h1, = axes.ravel()[i].plot(time_exp['propelled_ascent'], x_exp['propelled_ascent'][state], '-', color=colors['pa'], label='Propelled Ascent')
h2, = axes.ravel()[i].plot(time_exp['ballistic_ascent'], x_exp['ballistic_ascent'][state], '-', color=colors['ba'], label='Ballistic Ascent')
h3, = axes.ravel()[i].plot(time_exp['descent'], x_exp['descent'][state], '-', color=colors['d'], label='Descent')
if state == 'gam':
axes.ravel()[i].yaxis.set_major_locator(mpl.ticker.MaxNLocator(nbins='auto', steps=[1, 1.5, 3, 4.5, 6, 9, 10]))
axes.ravel()[i].set_yticks(np.arange(-90, 91, 45))
axes.ravel()[i].grid(True, alpha=0.2)
axes[1, 0].set_xlabel('t (s)')
axes[1, 1].set_xlabel('t (s)')
plt.figlegend(handles=[h1, h2, h3], loc=legend_loc, ncol=legend_ncol)
fig.tight_layout()
def plot_trajectory(p, exp_out, legend_loc='center right'):
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(8, 6))
time_imp = {'ballistic_ascent': p.get_val('traj.ballistic_ascent.timeseries.time'),
'propelled_ascent': p.get_val('traj.propelled_ascent.timeseries.time'),
'descent': p.get_val('traj.descent.timeseries.time')}
time_exp = {'ballistic_ascent': exp_out.get_val('traj.ballistic_ascent.timeseries.time'),
'propelled_ascent': exp_out.get_val('traj.propelled_ascent.timeseries.time'),
'descent': exp_out.get_val('traj.descent.timeseries.time')}
r_imp = {'ballistic_ascent': p.get_val('traj.ballistic_ascent.timeseries.r'),
'propelled_ascent': p.get_val('traj.propelled_ascent.timeseries.r'),
'descent': p.get_val('traj.descent.timeseries.r')}
r_exp = {'ballistic_ascent': exp_out.get_val('traj.ballistic_ascent.timeseries.r'),
'propelled_ascent': exp_out.get_val('traj.propelled_ascent.timeseries.r'),
'descent': exp_out.get_val('traj.descent.timeseries.r')}
h_imp = {'ballistic_ascent': p.get_val('traj.ballistic_ascent.timeseries.h'),
'propelled_ascent': p.get_val('traj.propelled_ascent.timeseries.h'),
'descent': p.get_val('traj.descent.timeseries.h')}
h_exp = {'ballistic_ascent': exp_out.get_val('traj.ballistic_ascent.timeseries.h'),
'propelled_ascent': exp_out.get_val('traj.propelled_ascent.timeseries.h'),
'descent': exp_out.get_val('traj.descent.timeseries.h')}
axes.plot(r_imp['propelled_ascent'], h_imp['propelled_ascent'], 'o', color=colors['pa'])
axes.plot(r_imp['ballistic_ascent'], h_imp['ballistic_ascent'], 'o', color=colors['ba'])
axes.plot(r_imp['descent'], h_imp['descent'], 'o', color=colors['d'])
h1, = axes.plot(r_exp['propelled_ascent'], h_exp['propelled_ascent'], '-', color=colors['pa'], label='Propelled Ascent')
h2, = axes.plot(r_exp['ballistic_ascent'], h_exp['ballistic_ascent'], '-', color=colors['ba'], label='Ballistic Ascent')
h3, = axes.plot(r_exp['descent'], h_exp['descent'], '-', color=colors['d'], label='Descent')
axes.set_xlabel('r (m)')
axes.set_ylabel('h (m)')
axes.set_aspect('equal', 'box')
plt.figlegend(handles=[h1, h2, h3], loc=legend_loc)
axes.grid(alpha=0.2)
fig.tight_layout()
Optimizing for Height#
from dymos.examples.water_rocket.phases import new_water_rocket_trajectory, set_sane_initial_guesses
p = om.Problem(model=om.Group())
traj, phases = new_water_rocket_trajectory(objective='height')
traj = p.model.add_subsystem('traj', traj)
p.driver = om.pyOptSparseDriver(optimizer='IPOPT', print_results=False)
p.driver.opt_settings['print_level'] = 4
p.driver.opt_settings['max_iter'] = 1000
p.driver.opt_settings['mu_strategy'] = 'monotone'
p.driver.declare_coloring(tol=1.0E-12)
# Finish Problem Setup
p.model.linear_solver = om.DirectSolver()
p.setup()
set_sane_initial_guesses(phases)
dm.run_problem(p, run_driver=True, simulate=True)
summary = summarize_results(p)
for key, entry in summary.items():
print(f'{key}: {entry.value:6.4f} {entry.unit}')
sol_out = om.CaseReader('dymos_solution.db').get_case('final')
sim_out = om.CaseReader('dymos_simulation.db').get_case('final')
--- Constraint Report [traj] ---
--- propelled_ascent ---
None
--- ballistic_ascent ---
None
--- descent ---
None
'rhs_checking' is disabled for 'DirectSolver in <model> <class Group>' but that solver has redundant adjoint solves. If it is expensive to compute derivatives for this solver, turning on 'rhs_checking' may improve performance.
Model viewer data has already been recorded for Driver.
Model viewer data has already been recorded for Driver.
Full total jacobian for problem 'problem' was computed 3 times, taking 1.7155052250000153 seconds.
Total jacobian shape: (1151, 1153)
Jacobian shape: (1151, 1153) (0.67% nonzero)
FWD solves: 20 REV solves: 0
Total colors vs. total size: 20 vs 1153 (98.27% improvement)
Sparsity computed using tolerance: 1e-12
Time to compute sparsity: 1.7155 sec
Time to compute coloring: 1.8487 sec
Memory to compute coloring: -9.6289 MB
Coloring created on: 2024-08-15 12:54:09
List of user-set options:
Name Value used
file_print_level = 5 yes
hessian_approximation = limited-memory yes
linear_solver = mumps yes
max_iter = 1000 yes
mu_strategy = monotone yes
nlp_scaling_method = user-scaling yes
output_file = reports/problem/IPOPT.out yes
print_level = 4 yes
print_user_options = yes yes
sb = yes yes
Total number of variables............................: 1153
variables with only lower bounds: 450
variables with lower and upper bounds: 157
variables with only upper bounds: 30
Total number of equality constraints.................: 1150
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
Number of Iterations....: 61
(scaled) (unscaled)
Objective...............: -5.3408199947747761e+01 -5.3408199947747761e+01
Dual infeasibility......: 5.0666083810518151e-09 5.0666083810518151e-09
Constraint violation....: 2.4749639006222610e-09 2.4749639006222610e-09
Variable bound violation: 9.8625653199846397e-09 9.8625653199846397e-09
Complementarity.........: 2.5059035596800816e-09 2.5059035596800816e-09
Overall NLP error.......: 5.0666083810518151e-09 5.0666083810518151e-09
Number of objective function evaluations = 94
Number of objective gradient evaluations = 62
Number of equality constraint evaluations = 94
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 62
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 0
Total seconds in IPOPT = 10.729
EXIT: Optimal Solution Found.
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
Simulating trajectory traj
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
Done simulating trajectory traj
Launch angle: 85.0000 deg
Flight angle at end of propulsion: 84.0887 deg
Empty mass: 0.1449 kg
Water volume: 0.9836 L
Maximum range: 16.0259 m
Maximum height: 53.4082 m
Maximum velocity: 46.1530 m/s
Maximum Height Solution: Propulsive Phase#
plot_propelled_ascent(sol_out, sim_out)
Maximum Height Solution: Height vs. Range#
Note that the equations of motion used here are singular in vertical flight, so the launch angle (the initial flight path angle) was limited to 85 degrees.
plot_trajectory(sol_out, sim_out, legend_loc='center right')
Maximum Height Solution: State History#
plot_states(sol_out, sim_out, legend_loc='lower center', legend_ncol=3)
Optimizing for Range#
from dymos.examples.water_rocket.phases import new_water_rocket_trajectory, set_sane_initial_guesses
p = om.Problem(model=om.Group())
traj, phases = new_water_rocket_trajectory(objective='range')
traj = p.model.add_subsystem('traj', traj)
p.driver = om.pyOptSparseDriver(optimizer='IPOPT')
p.driver.opt_settings['print_level'] = 4
p.driver.opt_settings['max_iter'] = 1000
p.driver.opt_settings['mu_strategy'] = 'monotone'
p.driver.declare_coloring(tol=1.0E-12)
# Finish Problem Setup
p.model.linear_solver = om.DirectSolver()
p.setup()
set_sane_initial_guesses(phases)
dm.run_problem(p, run_driver=True, simulate=True)
summary = summarize_results(p)
for key, entry in summary.items():
print(f'{key}: {entry.value:6.4f} {entry.unit}')
sol_out = om.CaseReader('dymos_solution.db').get_case('final')
sim_out = om.CaseReader('dymos_simulation.db').get_case('final')
--- Constraint Report [traj] ---
--- propelled_ascent ---
None
--- ballistic_ascent ---
None
--- descent ---
None
'rhs_checking' is disabled for 'DirectSolver in <model> <class Group>' but that solver has redundant adjoint solves. If it is expensive to compute derivatives for this solver, turning on 'rhs_checking' may improve performance.
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/recorders/sqlite_recorder.py:226: UserWarning:The existing case recorder file, dymos_solution.db, is being overwritten.
Model viewer data has already been recorded for Driver.
Model viewer data has already been recorded for Driver.
Full total jacobian for problem 'problem6' was computed 3 times, taking 1.6616627460000473 seconds.
Total jacobian shape: (1151, 1153)
Jacobian shape: (1151, 1153) (0.67% nonzero)
FWD solves: 20 REV solves: 0
Total colors vs. total size: 20 vs 1153 (98.27% improvement)
Sparsity computed using tolerance: 1e-12
Time to compute sparsity: 1.6617 sec
Time to compute coloring: 1.8490 sec
Memory to compute coloring: 0.0000 MB
Coloring created on: 2024-08-15 12:54:28
List of user-set options:
Name Value used
file_print_level = 5 yes
hessian_approximation = limited-memory yes
linear_solver = mumps yes
max_iter = 1000 yes
mu_strategy = monotone yes
nlp_scaling_method = user-scaling yes
output_file = reports/problem6/IPOPT.out yes
print_level = 4 yes
print_user_options = yes yes
sb = yes yes
Total number of variables............................: 1153
variables with only lower bounds: 450
variables with lower and upper bounds: 157
variables with only upper bounds: 30
Total number of equality constraints.................: 1150
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
Number of Iterations....: 48
(scaled) (unscaled)
Objective...............: -8.5111443691913224e-01 -8.5111443691913224e-01
Dual infeasibility......: 5.2430748558409714e-11 5.2430748558409714e-11
Constraint violation....: 5.0821347670970957e-09 5.0821347670970957e-09
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 9.0909090909090941e-10 9.0909090909090941e-10
Overall NLP error.......: 5.0821347670970957e-09 5.0821347670970957e-09
Number of objective function evaluations = 56
Number of objective gradient evaluations = 49
Number of equality constraint evaluations = 56
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 49
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 0
Total seconds in IPOPT = 9.490
EXIT: Optimal Solution Found.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 9.4918
User Objective Time : 0.1653
User Sensitivity Time : 8.0133
Interface Time : 0.7801
Opt Solver Time: 0.5332
Calls to Objective Function : 56
Calls to Sens Function : 49
Objectives
Index Name Value
0 traj.descent.states:r -8.511144E-01
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.parameters:m_empty_0 c 1.000000E-02 1.891010E+00 1.000000E+01
1 traj.ballistic_ascent.t_initial_0 c 1.000000E-03 1.126018E-01 1.000000E+00
2 traj.ballistic_ascent.t_duration_0 c 1.000000E-03 1.928857E+00 1.000000E+01
3 traj.ballistic_ascent.states:r_0 c -1.000000E+21 1.608068E+00 1.000000E+21
4 traj.ballistic_ascent.states:r_1 c -1.000000E+21 3.795392E+00 1.000000E+21
5 traj.ballistic_ascent.states:r_2 c -1.000000E+21 6.718918E+00 1.000000E+21
6 traj.ballistic_ascent.states:r_3 c -1.000000E+21 7.622741E+00 1.000000E+21
7 traj.ballistic_ascent.states:r_4 c -1.000000E+21 9.655603E+00 1.000000E+21
8 traj.ballistic_ascent.states:r_5 c -1.000000E+21 1.238107E+01 1.000000E+21
9 traj.ballistic_ascent.states:r_6 c -1.000000E+21 1.322554E+01 1.000000E+21
10 traj.ballistic_ascent.states:r_7 c -1.000000E+21 1.512810E+01 1.000000E+21
11 traj.ballistic_ascent.states:r_8 c -1.000000E+21 1.768560E+01 1.000000E+21
12 traj.ballistic_ascent.states:r_9 c -1.000000E+21 1.847954E+01 1.000000E+21
13 traj.ballistic_ascent.states:r_10 c -1.000000E+21 2.027084E+01 1.000000E+21
14 traj.ballistic_ascent.states:r_11 c -1.000000E+21 2.268424E+01 1.000000E+21
15 traj.ballistic_ascent.states:r_12 c -1.000000E+21 2.343468E+01 1.000000E+21
16 traj.ballistic_ascent.states:r_13 c -1.000000E+21 2.512993E+01 1.000000E+21
17 traj.ballistic_ascent.states:r_14 c -1.000000E+21 2.741840E+01 1.000000E+21
18 traj.ballistic_ascent.states:r_15 c -1.000000E+21 2.813101E+01 1.000000E+21
19 traj.ballistic_ascent.states:r_16 c -1.000000E+21 2.974250E+01 1.000000E+21
20 traj.ballistic_ascent.states:r_17 c -1.000000E+21 3.192155E+01 1.000000E+21
21 traj.ballistic_ascent.states:r_18 c -1.000000E+21 3.260091E+01 1.000000E+21
22 traj.ballistic_ascent.states:r_19 c -1.000000E+21 3.413863E+01 1.000000E+21
23 traj.ballistic_ascent.states:r_20 c -1.000000E+21 3.622089E+01 1.000000E+21
24 traj.ballistic_ascent.states:r_21 c -1.000000E+21 3.687074E+01 1.000000E+21
25 traj.ballistic_ascent.states:r_22 c -1.000000E+21 3.834281E+01 1.000000E+21
26 traj.ballistic_ascent.states:r_23 c -1.000000E+21 4.033856E+01 1.000000E+21
27 traj.ballistic_ascent.states:r_24 c -1.000000E+21 4.096196E+01 1.000000E+21
28 traj.ballistic_ascent.states:r_25 c -1.000000E+21 4.237498E+01 1.000000E+21
29 traj.ballistic_ascent.states:r_26 c -1.000000E+21 4.429259E+01 1.000000E+21
30 traj.ballistic_ascent.states:r_27 c -1.000000E+21 4.489199E+01 1.000000E+21
31 traj.ballistic_ascent.states:r_28 c -1.000000E+21 4.625134E+01 1.000000E+21
32 traj.ballistic_ascent.states:r_29 c -1.000000E+21 4.809752E+01 1.000000E+21
33 traj.ballistic_ascent.states:r_30 c -1.000000E+21 4.867492E+01 1.000000E+21
34 traj.ballistic_ascent.states:h_0 c -1.000000E+21 1.304114E+00 1.000000E+21
35 traj.ballistic_ascent.states:h_1 c -1.000000E+21 3.014264E+00 1.000000E+21
36 traj.ballistic_ascent.states:h_2 c -1.000000E+21 5.225707E+00 1.000000E+21
37 traj.ballistic_ascent.states:h_3 c -1.000000E+21 5.891330E+00 1.000000E+21
38 traj.ballistic_ascent.states:h_4 c -1.000000E+21 7.355713E+00 1.000000E+21
39 traj.ballistic_ascent.states:h_5 c -1.000000E+21 9.244599E+00 1.000000E+21
40 traj.ballistic_ascent.states:h_6 c -1.000000E+21 9.811786E+00 1.000000E+21
41 traj.ballistic_ascent.states:h_7 c -1.000000E+21 1.105688E+01 1.000000E+21
42 traj.ballistic_ascent.states:h_8 c -1.000000E+21 1.265608E+01 1.000000E+21
43 traj.ballistic_ascent.states:h_9 c -1.000000E+21 1.313444E+01 1.000000E+21
44 traj.ballistic_ascent.states:h_10 c -1.000000E+21 1.418094E+01 1.000000E+21
45 traj.ballistic_ascent.states:h_11 c -1.000000E+21 1.551628E+01 1.000000E+21
46 traj.ballistic_ascent.states:h_12 c -1.000000E+21 1.591340E+01 1.000000E+21
47 traj.ballistic_ascent.states:h_13 c -1.000000E+21 1.677769E+01 1.000000E+21
48 traj.ballistic_ascent.states:h_14 c -1.000000E+21 1.786976E+01 1.000000E+21
49 traj.ballistic_ascent.states:h_15 c -1.000000E+21 1.819170E+01 1.000000E+21
50 traj.ballistic_ascent.states:h_16 c -1.000000E+21 1.888693E+01 1.000000E+21
51 traj.ballistic_ascent.states:h_17 c -1.000000E+21 1.975227E+01 1.000000E+21
52 traj.ballistic_ascent.states:h_18 c -1.000000E+21 2.000394E+01 1.000000E+21
53 traj.ballistic_ascent.states:h_19 c -1.000000E+21 2.054074E+01 1.000000E+21
54 traj.ballistic_ascent.states:h_20 c -1.000000E+21 2.119285E+01 1.000000E+21
55 traj.ballistic_ascent.states:h_21 c -1.000000E+21 2.137823E+01 1.000000E+21
56 traj.ballistic_ascent.states:h_22 c -1.000000E+21 2.176532E+01 1.000000E+21
57 traj.ballistic_ascent.states:h_23 c -1.000000E+21 2.221529E+01 1.000000E+21
58 traj.ballistic_ascent.states:h_24 c -1.000000E+21 2.233770E+01 1.000000E+21
59 traj.ballistic_ascent.states:h_25 c -1.000000E+21 2.258232E+01 1.000000E+21
60 traj.ballistic_ascent.states:h_26 c -1.000000E+21 2.283940E+01 1.000000E+21
61 traj.ballistic_ascent.states:h_27 c -1.000000E+21 2.290162E+01 1.000000E+21
62 traj.ballistic_ascent.states:h_28 c -1.000000E+21 2.300985E+01 1.000000E+21
63 traj.ballistic_ascent.states:h_29 c -1.000000E+21 2.308194E+01 1.000000E+21
64 traj.ballistic_ascent.states:h_30 c -1.000000E+21 2.308634E+01 1.000000E+21
65 traj.ballistic_ascent.states:gam_0 c -1.111111E+19 4.265381E-01 9.888889E-01
66 traj.ballistic_ascent.states:gam_1 c -1.111111E+19 4.182287E-01 9.888889E-01
67 traj.ballistic_ascent.states:gam_2 c -1.111111E+19 4.060968E-01 9.888889E-01
68 traj.ballistic_ascent.states:gam_3 c -1.111111E+19 4.020909E-01 9.888889E-01
69 traj.ballistic_ascent.states:gam_4 c -1.111111E+19 3.926065E-01 9.888889E-01
70 traj.ballistic_ascent.states:gam_5 c -1.111111E+19 3.787859E-01 9.888889E-01
71 traj.ballistic_ascent.states:gam_6 c -1.111111E+19 3.742288E-01 9.888889E-01
72 traj.ballistic_ascent.states:gam_7 c -1.111111E+19 3.634519E-01 9.888889E-01
73 traj.ballistic_ascent.states:gam_8 c -1.111111E+19 3.477784E-01 9.888889E-01
74 traj.ballistic_ascent.states:gam_9 c -1.111111E+19 3.426181E-01 9.888889E-01
75 traj.ballistic_ascent.states:gam_10 c -1.111111E+19 3.304299E-01 9.888889E-01
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1116 traj.propelled_ascent.states:V_w_113 c 0.000000E+00 2.029473E-02 1.000000E+20
1117 traj.propelled_ascent.states:V_w_114 c 0.000000E+00 2.002278E-02 1.000000E+20
1118 traj.propelled_ascent.states:V_w_115 c 0.000000E+00 1.940144E-02 1.000000E+20
1119 traj.propelled_ascent.states:V_w_116 c 0.000000E+00 1.854722E-02 1.000000E+20
1120 traj.propelled_ascent.states:V_w_117 c 0.000000E+00 1.827761E-02 1.000000E+20
1121 traj.propelled_ascent.states:V_w_118 c 0.000000E+00 1.766158E-02 1.000000E+20
1122 traj.propelled_ascent.states:V_w_119 c 0.000000E+00 1.681463E-02 1.000000E+20
1123 traj.propelled_ascent.states:V_w_120 c 0.000000E+00 1.654731E-02 1.000000E+20
1124 traj.propelled_ascent.states:V_w_121 c 0.000000E+00 1.593648E-02 1.000000E+20
1125 traj.propelled_ascent.states:V_w_122 c 0.000000E+00 1.509666E-02 1.000000E+20
1126 traj.propelled_ascent.states:V_w_123 c 0.000000E+00 1.483157E-02 1.000000E+20
1127 traj.propelled_ascent.states:V_w_124 c 0.000000E+00 1.422585E-02 1.000000E+20
1128 traj.propelled_ascent.states:V_w_125 c 0.000000E+00 1.339300E-02 1.000000E+20
1129 traj.propelled_ascent.states:V_w_126 c 0.000000E+00 1.313011E-02 1.000000E+20
1130 traj.propelled_ascent.states:V_w_127 c 0.000000E+00 1.252938E-02 1.000000E+20
1131 traj.propelled_ascent.states:V_w_128 c 0.000000E+00 1.170337E-02 1.000000E+20
1132 traj.propelled_ascent.states:V_w_129 c 0.000000E+00 1.144263E-02 1.000000E+20
1133 traj.propelled_ascent.states:V_w_130 c 0.000000E+00 1.084680E-02 1.000000E+20
1134 traj.propelled_ascent.states:V_w_131 c 0.000000E+00 1.002749E-02 1.000000E+20
1135 traj.propelled_ascent.states:V_w_132 c 0.000000E+00 9.768857E-03 1.000000E+20
1136 traj.propelled_ascent.states:V_w_133 c 0.000000E+00 9.177831E-03 1.000000E+20
1137 traj.propelled_ascent.states:V_w_134 c 0.000000E+00 8.365097E-03 1.000000E+20
1138 traj.propelled_ascent.states:V_w_135 c 0.000000E+00 8.108533E-03 1.000000E+20
1139 traj.propelled_ascent.states:V_w_136 c 0.000000E+00 7.522220E-03 1.000000E+20
1140 traj.propelled_ascent.states:V_w_137 c 0.000000E+00 6.715937E-03 1.000000E+20
1141 traj.propelled_ascent.states:V_w_138 c 0.000000E+00 6.461401E-03 1.000000E+20
1142 traj.propelled_ascent.states:V_w_139 c 0.000000E+00 5.879712E-03 1.000000E+20
1143 traj.propelled_ascent.states:V_w_140 c 0.000000E+00 5.079759E-03 1.000000E+20
1144 traj.propelled_ascent.states:V_w_141 c 0.000000E+00 4.827216E-03 1.000000E+20
1145 traj.propelled_ascent.states:V_w_142 c 0.000000E+00 4.250065E-03 1.000000E+20
1146 traj.propelled_ascent.states:V_w_143 c 0.000000E+00 3.456327E-03 1.000000E+20
1147 traj.propelled_ascent.states:V_w_144 c 0.000000E+00 3.205738E-03 1.000000E+20
1148 traj.propelled_ascent.states:V_w_145 c 0.000000E+00 2.633045E-03 1.000000E+20
1149 traj.propelled_ascent.states:V_w_146 c 0.000000E+00 1.845409E-03 1.000000E+20
1150 traj.propelled_ascent.states:V_w_147 c 0.000000E+00 1.596741E-03 1.000000E+20
1151 traj.propelled_ascent.states:V_w_148 c 0.000000E+00 1.028424E-03 1.000000E+20
1152 traj.propelled_ascent.states:V_w_149 c 0.000000E+00 2.467821E-04 1.000000E+20
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier (N/A)
0 traj.linkages.propelled_ascent:time_final|ballistic_ascent:time_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
1 traj.linkages.propelled_ascent:r_final|ballistic_ascent:r_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
2 traj.linkages.propelled_ascent:h_final|ballistic_ascent:h_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
3 traj.linkages.propelled_ascent:gam_final|ballistic_ascent:gam_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
4 traj.linkages.propelled_ascent:v_final|ballistic_ascent:v_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
5 traj.linkages.ballistic_ascent:time_final|descent:time_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
6 traj.linkages.ballistic_ascent:r_final|descent:r_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
7 traj.linkages.ballistic_ascent:h_final|descent:h_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
8 traj.linkages.ballistic_ascent:gam_final|descent:gam_initial e 0.000000E+00 2.192714E-29 0.000000E+00 9.00000E+100
9 traj.linkages.ballistic_ascent:v_final|descent:v_initial e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
10 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -2.741071E-15 0.000000E+00 9.00000E+100
11 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -6.852676E-15 0.000000E+00 9.00000E+100
12 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -2.398437E-15 0.000000E+00 9.00000E+100
13 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -7.537944E-15 0.000000E+00 9.00000E+100
14 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.426338E-16 0.000000E+00 9.00000E+100
15 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.370535E-15 0.000000E+00 9.00000E+100
16 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -8.565845E-15 0.000000E+00 9.00000E+100
17 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.370535E-15 0.000000E+00 9.00000E+100
18 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 6.852676E-16 0.000000E+00 9.00000E+100
19 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.918749E-14 0.000000E+00 9.00000E+100
20 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -2.398437E-15 0.000000E+00 9.00000E+100
21 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
22 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 5.139507E-15 0.000000E+00 9.00000E+100
23 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -4.111606E-15 0.000000E+00 9.00000E+100
24 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.027901E-15 0.000000E+00 9.00000E+100
25 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.083704E-15 0.000000E+00 9.00000E+100
26 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.027901E-15 0.000000E+00 9.00000E+100
27 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 6.510043E-15 0.000000E+00 9.00000E+100
28 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.815959E-14 0.000000E+00 9.00000E+100
29 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -5.482141E-15 0.000000E+00 9.00000E+100
30 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -3.768972E-15 0.000000E+00 9.00000E+100
31 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -5.482141E-15 0.000000E+00 9.00000E+100
32 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.426338E-16 0.000000E+00 9.00000E+100
33 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.062165E-14 0.000000E+00 9.00000E+100
34 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -2.672544E-14 0.000000E+00 9.00000E+100
35 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.713169E-15 0.000000E+00 9.00000E+100
36 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -2.055803E-15 0.000000E+00 9.00000E+100
37 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.987276E-14 0.000000E+00 9.00000E+100
38 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.713169E-15 0.000000E+00 9.00000E+100
39 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.678906E-14 0.000000E+00 9.00000E+100
40 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -5.824775E-15 0.000000E+00 9.00000E+100
41 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -2.055803E-15 0.000000E+00 9.00000E+100
42 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.027901E-15 0.000000E+00 9.00000E+100
43 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -2.741071E-15 0.000000E+00 9.00000E+100
44 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.027901E-15 0.000000E+00 9.00000E+100
45 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.083704E-15 0.000000E+00 9.00000E+100
46 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.167409E-15 0.000000E+00 9.00000E+100
47 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -2.741071E-15 0.000000E+00 9.00000E+100
48 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
49 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 5.824775E-15 0.000000E+00 9.00000E+100
50 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.884486E-15 0.000000E+00 9.00000E+100
51 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 1.884486E-15 0.000000E+00 9.00000E+100
52 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.027901E-15 0.000000E+00 9.00000E+100
53 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -2.569754E-15 0.000000E+00 9.00000E+100
54 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.319140E-14 0.000000E+00 9.00000E+100
55 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -7.366627E-15 0.000000E+00 9.00000E+100
56 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 2.055803E-15 0.000000E+00 9.00000E+100
57 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.167409E-15 0.000000E+00 9.00000E+100
58 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 5.482141E-15 0.000000E+00 9.00000E+100
59 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.456194E-15 0.000000E+00 9.00000E+100
60 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -2.655412E-15 0.000000E+00 9.00000E+100
61 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 9.336772E-15 0.000000E+00 9.00000E+100
62 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -4.539898E-15 0.000000E+00 9.00000E+100
63 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 4.625557E-15 0.000000E+00 9.00000E+100
64 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.338726E-15 0.000000E+00 9.00000E+100
65 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 3.811801E-15 0.000000E+00 9.00000E+100
66 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -4.454240E-15 0.000000E+00 9.00000E+100
67 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.874091E-15 0.000000E+00 9.00000E+100
68 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 6.767018E-15 0.000000E+00 9.00000E+100
69 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 9.947088E-15 0.000000E+00 9.00000E+100
70 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 6.072233E-16 0.000000E+00 9.00000E+100
71 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.807042E-17 0.000000E+00 9.00000E+100
72 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -3.997395E-17 0.000000E+00 9.00000E+100
73 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 6.091268E-17 0.000000E+00 9.00000E+100
74 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 8.375493E-17 0.000000E+00 9.00000E+100
75 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 1.370535E-16 0.000000E+00 9.00000E+100
76 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -3.997395E-16 0.000000E+00 9.00000E+100
77 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 1.237289E-16 0.000000E+00 9.00000E+100
78 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -1.027901E-16 0.000000E+00 9.00000E+100
79 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.216951E-16 0.000000E+00 9.00000E+100
80 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.807042E-18 0.000000E+00 9.00000E+100
81 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -1.789310E-16 0.000000E+00 9.00000E+100
82 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 9.136902E-17 0.000000E+00 9.00000E+100
83 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 7.614085E-18 0.000000E+00 9.00000E+100
84 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -1.675099E-16 0.000000E+00 9.00000E+100
85 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -7.614085E-17 0.000000E+00 9.00000E+100
86 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 4.187747E-17 0.000000E+00 9.00000E+100
87 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -7.614085E-18 0.000000E+00 9.00000E+100
88 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -2.398437E-16 0.000000E+00 9.00000E+100
89 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.045634E-17 0.000000E+00 9.00000E+100
90 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.807042E-17 0.000000E+00 9.00000E+100
91 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -3.807042E-17 0.000000E+00 9.00000E+100
92 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 5.710564E-17 0.000000E+00 9.00000E+100
93 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -9.136902E-17 0.000000E+00 9.00000E+100
94 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 4.187747E-17 0.000000E+00 9.00000E+100
95 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 -1.903521E-17 0.000000E+00 9.00000E+100
96 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 4.187747E-17 0.000000E+00 9.00000E+100
97 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 1.142113E-17 0.000000E+00 9.00000E+100
98 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 1.142113E-17 0.000000E+00 9.00000E+100
99 traj.ballistic_ascent.collocation_constraint.defects:gam e 0.000000E+00 3.807042E-18 0.000000E+00 9.00000E+100
100 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.027901E-15 0.000000E+00 9.00000E+100
101 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -3.426338E-15 0.000000E+00 9.00000E+100
102 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 4.796873E-15 0.000000E+00 9.00000E+100
103 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.713169E-14 0.000000E+00 9.00000E+100
104 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -6.852676E-15 0.000000E+00 9.00000E+100
105 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 2.741071E-15 0.000000E+00 9.00000E+100
106 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.541852E-14 0.000000E+00 9.00000E+100
107 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -7.880578E-15 0.000000E+00 9.00000E+100
108 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -1.216350E-14 0.000000E+00 9.00000E+100
109 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.558984E-14 0.000000E+00 9.00000E+100
110 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 5.996092E-15 0.000000E+00 9.00000E+100
111 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 4.625557E-15 0.000000E+00 9.00000E+100
112 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -9.251113E-15 0.000000E+00 9.00000E+100
113 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 3.426338E-15 0.000000E+00 9.00000E+100
114 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 7.709261E-15 0.000000E+00 9.00000E+100
115 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -7.023993E-15 0.000000E+00 9.00000E+100
116 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -4.796873E-15 0.000000E+00 9.00000E+100
117 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -2.569754E-15 0.000000E+00 9.00000E+100
118 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.233482E-14 0.000000E+00 9.00000E+100
119 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 3.426338E-15 0.000000E+00 9.00000E+100
120 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 2.655412E-15 0.000000E+00 9.00000E+100
121 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.464760E-14 0.000000E+00 9.00000E+100
122 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -2.998046E-15 0.000000E+00 9.00000E+100
123 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.199218E-15 0.000000E+00 9.00000E+100
124 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 1.336272E-14 0.000000E+00 9.00000E+100
125 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -1.113560E-15 0.000000E+00 9.00000E+100
126 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 6.338726E-15 0.000000E+00 9.00000E+100
127 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -1.696037E-14 0.000000E+00 9.00000E+100
128 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 6.081750E-15 0.000000E+00 9.00000E+100
129 traj.ballistic_ascent.collocation_constraint.defects:v e 0.000000E+00 -3.426338E-16 0.000000E+00 9.00000E+100
130 traj.descent.collocation_constraint.defects:r e 0.000000E+00 -1.571315E-14 0.000000E+00 9.00000E+100
131 traj.descent.collocation_constraint.defects:r e 0.000000E+00 2.481024E-15 0.000000E+00 9.00000E+100
132 traj.descent.collocation_constraint.defects:r e 0.000000E+00 -1.736717E-14 0.000000E+00 9.00000E+100
133 traj.descent.collocation_constraint.defects:r e 0.000000E+00 2.894528E-15 0.000000E+00 9.00000E+100
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961 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.219856E-10 0.000000E+00 9.00000E+100
962 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.167822E-10 0.000000E+00 9.00000E+100
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964 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.076937E-10 0.000000E+00 9.00000E+100
965 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.028818E-10 0.000000E+00 9.00000E+100
966 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -9.645749E-11 0.000000E+00 9.00000E+100
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982 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -4.174972E-11 0.000000E+00 9.00000E+100
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991 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.852385E-11 0.000000E+00 9.00000E+100
992 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -1.610450E-11 0.000000E+00 9.00000E+100
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999 traj.propelled_ascent.collocation_constraint.defects:p e 0.000000E+00 -8.578103E-13 0.000000E+00 9.00000E+100
1000 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.080219E-17 0.000000E+00 9.00000E+100
1001 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.220234E-17 0.000000E+00 9.00000E+100
1002 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.720181E-17 0.000000E+00 9.00000E+100
1003 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -8.840930E-17 0.000000E+00 9.00000E+100
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1005 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.500263E-17 0.000000E+00 9.00000E+100
1006 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.420149E-17 0.000000E+00 9.00000E+100
1007 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.380145E-17 0.000000E+00 9.00000E+100
1008 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.100221E-17 0.000000E+00 9.00000E+100
1009 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -4.000421E-19 0.000000E+00 9.00000E+100
1010 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.000632E-18 0.000000E+00 9.00000E+100
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1014 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.380251E-17 0.000000E+00 9.00000E+100
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1017 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.600168E-18 0.000000E+00 9.00000E+100
1018 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 6.280661E-17 0.000000E+00 9.00000E+100
1019 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.540373E-17 0.000000E+00 9.00000E+100
1020 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.560164E-17 0.000000E+00 9.00000E+100
1021 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 5.620592E-17 0.000000E+00 9.00000E+100
1022 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.160227E-17 0.000000E+00 9.00000E+100
1023 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.400147E-17 0.000000E+00 9.00000E+100
1024 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.180229E-17 0.000000E+00 9.00000E+100
1025 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.000211E-17 0.000000E+00 9.00000E+100
1026 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 8.800926E-18 0.000000E+00 9.00000E+100
1027 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.520686E-17 0.000000E+00 9.00000E+100
1028 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.940204E-17 0.000000E+00 9.00000E+100
1029 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -4.820507E-17 0.000000E+00 9.00000E+100
1030 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.118118E-16 0.000000E+00 9.00000E+100
1031 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.000105E-17 0.000000E+00 9.00000E+100
1032 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.000211E-18 0.000000E+00 9.00000E+100
1033 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.000632E-19 0.000000E+00 9.00000E+100
1034 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.280135E-17 0.000000E+00 9.00000E+100
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1113 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.400463E-18 0.000000E+00 9.00000E+100
1114 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -7.300768E-18 0.000000E+00 9.00000E+100
1115 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.000211E-18 0.000000E+00 9.00000E+100
1116 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.100221E-18 0.000000E+00 9.00000E+100
1117 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.900410E-18 0.000000E+00 9.00000E+100
1118 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 5.500579E-18 0.000000E+00 9.00000E+100
1119 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.000105E-17 0.000000E+00 9.00000E+100
1120 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.300453E-18 0.000000E+00 9.00000E+100
1121 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.010106E-17 0.000000E+00 9.00000E+100
1122 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.900726E-18 0.000000E+00 9.00000E+100
1123 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.000316E-18 0.000000E+00 9.00000E+100
1124 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.000421E-19 0.000000E+00 9.00000E+100
1125 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 6.700705E-18 0.000000E+00 9.00000E+100
1126 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.900200E-18 0.000000E+00 9.00000E+100
1127 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
1128 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.100116E-18 0.000000E+00 9.00000E+100
1129 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.640173E-17 0.000000E+00 9.00000E+100
1130 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.600274E-18 0.000000E+00 9.00000E+100
1131 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -4.000421E-18 0.000000E+00 9.00000E+100
1132 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.100326E-18 0.000000E+00 9.00000E+100
1133 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.400253E-18 0.000000E+00 9.00000E+100
1134 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.200232E-18 0.000000E+00 9.00000E+100
1135 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.700705E-18 0.000000E+00 9.00000E+100
1136 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -8.000842E-19 0.000000E+00 9.00000E+100
1137 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.800189E-18 0.000000E+00 9.00000E+100
1138 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.100221E-18 0.000000E+00 9.00000E+100
1139 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 8.000842E-19 0.000000E+00 9.00000E+100
1140 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.900200E-18 0.000000E+00 9.00000E+100
1141 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.600168E-18 0.000000E+00 9.00000E+100
1142 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.200232E-18 0.000000E+00 9.00000E+100
1143 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.000211E-19 0.000000E+00 9.00000E+100
1144 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.800189E-18 0.000000E+00 9.00000E+100
1145 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.000211E-19 0.000000E+00 9.00000E+100
1146 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.300242E-18 0.000000E+00 9.00000E+100
1147 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -8.000842E-19 0.000000E+00 9.00000E+100
1148 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -7.000737E-19 0.000000E+00 9.00000E+100
1149 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.000632E-19 0.000000E+00 9.00000E+100
Exit Status
Inform Description
0 Solve Succeeded
--------------------------------------------------------------------------------
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
Simulating trajectory traj
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
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/recorders/sqlite_recorder.py:226: UserWarning:The existing case recorder file, dymos_simulation.db, is being overwritten.
Done simulating trajectory traj
Launch angle: 46.0919 deg
Flight angle at end of propulsion: 38.3884 deg
Empty mass: 0.1891 kg
Water volume: 1.0265 L
Maximum range: 85.1114 m
Maximum height: 23.0863 m
Maximum velocity: 41.3116 m/s
Maximum Range Solution: Propulsive Phase#
plot_propelled_ascent(sol_out, sim_out)
Maximum Range Solution: Height vs. Range#
plot_trajectory(sol_out, sim_out, legend_loc='center')
Maximum Range Solution: State History#
plot_states(sol_out, sim_out, legend_loc='lower center')
References#
[air13]
Nozzles. 2013. URL: http://www.aircommandrockets.com/nozzles.htm.
[BM20]
Bernardo Bahia Monteiro. Optimization of a water rocket in openmdao/dymos. Aug 2020. URL: engrxiv.org/nqta8, doi:10.31224/osf.io/nqta8.
[BPBMADiazFernandezO09]
(1,2)
R Barrio-Perotti, E Blanco-Marigorta, K Argüelles-D\'ıaz, and J Fernández-Oro. Experimental evaluation of the drag coefficient of water rockets by a simple free-fall test. European Journal of Physics, 30(5):1039–1048, jul 2009. doi:10.1088/0143-0807/30/5/012.
[FGH+20]
Laura Fischer, Thomas Günther, Linda Herzig, Tobias Jarzina, Frank Klinker, Sabine Knipper, Franz-Georg Schürmann, and Marvin Wollek. On the approximation of D.I.Y. water rocket dynamics including air drag. arXiv:2001.08828, 2020. arXiv:2001.08828.
[RBM13]
Alejandro Romanelli, Italo Bove, and Federico González Madina. Air expansion in a water rocket. American Journal of Physics, 81(10):762–766, oct 2013. doi:10.1119/1.4811116.
[TRP09]
Glen E. Thorncroft, John R. Ridgely, and Christopher C. Pascual. Hydrodynamics and thrust characteristics of a water-propelled rocket. International Journal of Mechanical Engineering Education, 37(3):241–261, jul 2009. doi:10.7227/ijmee.37.3.9.