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.set_input_defaults('A_out', val=np.pi*13e-3**2/4*np.ones(nn))
self.set_input_defaults('p', val=6.5e5*np.ones(nn))
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'] = 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=10.0, defect_ref=10.0)
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=0.1,
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=20, 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=-1.0)
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()
<>:29: SyntaxWarning: invalid escape sequence '\g'
<>:29: SyntaxWarning: invalid escape sequence '\g'
/tmp/ipykernel_4245/850823774.py:29: SyntaxWarning: invalid escape sequence '\g'
axes.ravel()[i].set_ylabel(f"{state} ({unit})" if state != 'gam' else f'$\gamma$ ({unit})')
def plot_trajectory(p, exp_out, legend_loc='center right'):
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(8, 6))
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'] = 5
p.driver.opt_settings['max_iter'] = 1000
p.driver.opt_settings['mu_strategy'] = 'adaptive'
p.driver.opt_settings['nlp_scaling_method'] = 'gradient-based' # for faster convergence
p.driver.opt_settings['alpha_for_y'] = 'safer-min-dual-infeas'
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(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim_out = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')
'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.
Jacobian shape: (611, 613) (1.23% nonzero)
FWD solves: 20 REV solves: 0
Total colors vs. total size: 20 vs 613 (96.74% improvement)
Sparsity computed using tolerance: 1e-12.
Dense total jacobian for Problem 'problem' was computed 3 times.
Time to compute sparsity: 0.5023 sec
Time to compute coloring: 0.7473 sec
Memory to compute coloring: -2.2344 MB
Coloring created on: 2025-07-16 18:57:14
List of user-set options:
Name Value used
alpha_for_y = safer-min-dual-infeas yes
file_print_level = 5 yes
hessian_approximation = limited-memory yes
linear_solver = mumps yes
max_iter = 1000 yes
mu_strategy = adaptive yes
nlp_scaling_method = gradient-based yes
output_file = /home/runner/work/dymos/dymos/docs/dymos_book/examples/water_rocket/problem_out/IPOPT.out yes
print_level = 5 yes
print_user_options = yes yes
sb = yes yes
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 103007
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 0
Total number of variables............................: 613
variables with only lower bounds: 180
variables with lower and upper bounds: 67
variables with only upper bounds: 30
Total number of equality constraints.................: 610
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
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.0000000e+02 9.00e+01 4.22e-01 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -7.0820124e+01 2.49e+01 4.15e+03 -5.8 4.98e+01 - 1.84e-01 7.23e-01h 1
2 -4.4787630e+01 8.88e+00 6.71e+04 0.6 4.13e+01 - 1.00e+00 9.82e-01h 1
3 -4.9302111e+01 7.38e+01 2.80e+06 1.6 6.63e+01 - 1.00e+00 3.21e-01f 1
4 -3.9326926e+01 4.90e+01 3.99e+05 1.9 3.77e+01 - 3.43e-01 1.00e+00h 1
5 -3.2035130e+01 3.01e+01 9.92e+04 1.9 1.77e+01 - 3.95e-01 1.00e+00f 1
6 -2.9180468e+01 2.35e+01 1.84e+04 1.9 1.71e+01 - 9.02e-01 1.00e+00f 1
7 -2.7622916e+01 3.67e+00 2.00e+04 1.9 3.93e+01 - 3.16e-01 1.00e+00f 1
8 -2.7696393e+01 2.36e+01 1.23e+04 1.4 4.58e+01 - 9.99e-01 1.00e+00h 1
9 -2.7003630e+01 1.31e+00 1.97e+03 0.4 1.00e+01 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -2.7332373e+01 1.07e-01 3.06e+02 -0.6 1.88e+00 - 1.00e+00 1.00e+00h 1
11 -3.0084122e+01 2.12e-02 2.34e+02 -1.7 2.75e+00 - 1.00e+00 1.00e+00f 1
12 -3.3530125e+01 3.18e-02 2.73e+02 -2.7 3.45e+00 - 1.00e+00 1.00e+00f 1
13 -3.7138105e+01 5.50e-02 2.99e+02 -3.7 3.61e+00 - 1.00e+00 1.00e+00f 1
14 -3.7191722e+01 4.35e-03 4.19e+00 -4.9 1.38e-01 - 1.00e+00 1.00e+00h 1
15 -3.7653940e+01 6.04e-03 1.38e+00 -6.4 6.84e-01 - 1.00e+00 1.00e+00f 1
16 -4.0149550e+01 6.57e-02 2.12e+00 -7.5 2.50e+00 -2.0 1.00e+00 1.00e+00f 1
17 -6.4795521e+01 3.83e+00 1.46e+01 -5.5 8.58e+01 - 1.00e+00 2.87e-01f 1
18 -5.7291126e+01 3.44e+00 1.02e+01 -5.7 5.87e+01 - 1.00e+00 5.32e-01h 1
19 -5.7289070e+01 3.43e+00 1.02e+01 -5.2 4.51e+01 - 1.00e+00 5.52e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 -5.6677582e+01 3.18e+00 4.52e+00 -11.0 2.77e+01 - 6.82e-01 1.09e-01h 1
21 -5.3580006e+01 8.14e+00 1.93e+02 -11.0 2.30e+01 - 4.56e-02 1.00e+00H 1
22 -5.3579920e+01 8.14e+00 2.67e+00 -3.7 2.67e+00 - 1.00e+00 1.65e-04h 1
23 -5.2549642e+01 1.46e-01 2.21e+02 -4.2 1.93e+00 - 1.00e+00 1.00e+00h 1
24 -5.3014530e+01 9.27e-03 9.42e+01 -10.2 1.03e+00 - 8.25e-01 1.00e+00h 1
25 -5.3156474e+01 1.31e-03 2.56e+01 -2.9 2.17e-01 - 4.86e-01 1.00e+00f 1
26 -5.3184882e+01 9.43e-04 2.99e+00 -4.9 1.01e-01 - 1.00e+00 2.81e-01h 1
27 -5.3214864e+01 2.25e-04 3.00e-02 -5.1 3.00e-02 - 1.00e+00 1.00e+00h 1
28 -5.3237897e+01 1.51e-04 2.41e-02 -5.6 2.32e-02 - 1.00e+00 9.93e-01h 1
29 -7.5769111e+01 2.81e+01 9.30e+01 -4.8 1.51e+02 - 6.37e-01 3.41e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 -1.0861279e+01 8.59e+00 1.14e+02 -7.0 9.16e+01 - 4.93e-01 7.08e-01h 1
31 -1.0775111e+01 8.55e+00 1.14e+02 -5.4 2.83e+01 - 6.67e-03 4.45e-03h 1
32 -1.0794756e+01 5.87e+00 8.92e+01 -5.4 2.14e+01 - 2.96e-02 3.50e-01f 1
33 -1.0729049e+01 4.99e+00 9.63e+01 -5.4 1.42e+01 - 6.60e-07 1.50e-01h 1
34 -2.4955568e+01 2.03e+00 9.14e+01 -5.4 3.89e+01 - 2.38e-01 3.66e-01f 1
35 -2.9044168e+01 4.00e+00 1.06e+02 -5.4 5.34e+01 - 1.84e-06 1.52e-01f 1
36 -3.1153402e+01 4.04e+00 6.83e+01 -5.4 5.18e+01 - 6.38e-01 5.62e-02f 1
37 -3.1053505e+01 4.00e+00 6.72e+01 -5.4 1.73e+01 - 4.67e-02 9.42e-03h 1
38r-3.1053505e+01 4.00e+00 9.99e+02 0.6 0.00e+00 - 0.00e+00 3.62e-08R 2
39r-3.1053656e+01 1.23e+00 9.99e+02 -1.6 3.18e+03 - 8.01e-05 1.25e-03f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 -2.9886107e+01 1.10e+00 5.62e+00 -5.4 1.07e+01 - 3.30e-02 1.09e-01h 1
41 -2.9868348e+01 1.10e+00 5.68e+00 -5.4 9.37e+01 - 4.98e-06 9.96e-04h 1
42 -2.9519573e+01 1.08e+00 7.80e+00 -5.4 9.73e+01 - 2.66e-01 2.05e-02h 1
43r-2.9519573e+01 1.08e+00 1.00e+03 0.0 0.00e+00 - 0.00e+00 2.41e-07R 2
44r-2.9508681e+01 7.39e-01 9.98e+02 -2.2 7.30e+02 - 6.12e-05 1.46e-03f 1
45r-2.9508681e+01 7.39e-01 9.99e+02 -0.1 0.00e+00 - 0.00e+00 3.10e-07R 6
46r-2.9479087e+01 5.61e-01 9.93e+02 0.7 3.11e+02 - 6.27e-03 2.73e-03f 1
47 -2.9259269e+01 4.27e-01 1.15e+01 -5.4 1.10e+01 - 1.08e-01 2.62e-01h 1
48r-2.9259269e+01 4.27e-01 9.99e+02 -0.4 0.00e+00 - 0.00e+00 2.92e-07R 4
49r-2.9259269e+01 4.27e-01 9.82e+02 -6.3 3.26e+02 - 1.71e-02 4.80e-08f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50r-2.9341383e+01 2.41e-01 9.67e+02 0.6 4.04e+02 - 2.63e-02 2.29e-03f 1
51 -2.9335993e+01 2.34e-01 3.80e+01 -5.4 4.74e+00 - 2.99e-01 3.05e-02h 1
52 -2.9341911e+01 2.30e-01 4.21e+01 -5.4 4.72e+00 - 1.06e-01 1.57e-02h 1
53 -2.9705430e+01 3.84e+00 1.43e+03 -5.4 9.86e+00 - 1.62e-01 9.04e-01h 1
54 -2.9571105e+01 3.77e+00 1.39e+03 -5.4 3.05e+01 - 1.00e-04 2.10e-02h 5
55 -3.1693336e+01 4.46e-01 7.76e+02 -5.4 2.85e+00 - 1.68e-03 1.00e+00h 1
56 -3.4232885e+01 7.41e-01 3.73e+02 -5.4 3.44e+00 - 4.83e-01 1.00e+00h 1
57 -3.4081356e+01 7.15e-02 1.58e+02 -5.4 1.70e+00 - 5.82e-01 1.00e+00h 1
58 -3.4271258e+01 4.15e-03 2.89e+01 -5.4 4.60e-01 - 3.03e-04 1.00e+00h 1
59 -3.4793530e+01 5.20e-03 2.64e+01 -5.4 2.39e+02 - 8.81e-02 2.18e-03f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 -6.0278068e+01 1.48e+00 1.05e+01 -5.4 1.74e+02 - 1.10e-01 1.47e-01f 1
61 -4.8509506e+01 1.73e+02 5.60e+03 0.1 1.76e+04 - 2.79e-04 6.70e-04f 1
62 -4.8634745e+01 1.17e+02 1.54e+03 -1.6 1.97e+00 - 3.07e-01 4.46e-01h 1
63 -4.8634497e+01 1.13e+02 1.77e+02 -1.6 3.38e+00 - 1.00e+00 4.02e-02h 1
64 -4.8582705e+01 4.57e+01 3.02e+03 -1.6 3.14e+00 - 1.00e+00 1.00e+00h 1
65 -4.8516247e+01 2.88e+01 6.89e+02 -1.6 2.80e+00 - 1.00e+00 6.13e-01h 1
66 -4.8724594e+01 1.77e+01 1.22e+02 -1.6 2.18e+00 - 1.00e+00 1.00e+00h 1
67 -4.9897534e+01 1.04e+01 4.26e+01 -1.6 3.56e+00 - 2.12e-01 1.00e+00h 1
68 -5.4897042e+01 3.36e+00 1.46e+01 -1.6 5.00e+00 -2.5 6.82e-01 1.00e+00h 1
69 -7.3716289e+01 4.10e+00 1.77e+02 -1.6 1.88e+01 - 9.38e-01 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 -6.4133539e+01 1.12e+00 3.51e+02 -1.3 9.58e+00 - 1.00e+00 1.00e+00h 1
71 -5.4695568e+01 6.11e-01 1.29e+02 -2.1 1.27e+01 - 2.95e-01 1.00e+00H 1
72 -5.5993090e+01 5.92e-02 7.31e+01 -1.8 3.39e+00 - 1.00e+00 1.00e+00h 1
73 -5.2706617e+01 6.09e-02 6.58e+01 -7.9 3.29e+00 - 2.97e-01 1.00e+00h 1
74 -5.4690403e+01 1.05e+01 7.76e+02 -2.1 1.35e+01 - 3.37e-01 1.00e+00f 1
75 -6.9517992e+01 8.90e+00 2.69e+02 -2.1 1.48e+01 - 1.00e+00 1.00e+00f 1
76 -6.4420727e+01 1.63e+00 5.97e+02 -2.1 5.17e+00 - 1.00e+00 1.00e+00h 1
77 -5.3012160e+01 2.60e-01 5.00e+01 -2.8 1.14e+01 - 1.00e+00 1.00e+00h 1
78 -5.2158084e+01 1.24e-01 9.28e+01 -2.9 3.24e+00 -2.1 1.00e+00 1.00e+00h 1
79 -5.2537480e+01 5.56e-03 2.14e+01 -2.9 3.79e-01 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 -5.2828835e+01 1.03e-02 4.71e+00 -2.9 2.91e-01 - 1.00e+00 1.00e+00h 1
81 -5.3248192e+01 1.57e-03 5.27e+00 -2.9 4.19e-01 - 1.00e+00 1.00e+00f 1
82 -5.3350997e+01 1.14e-01 4.09e-01 -4.4 2.12e+00 - 1.00e+00 7.31e-01h 1
83 -5.3428347e+01 5.07e-03 7.40e+00 -3.4 5.08e-01 - 1.00e+00 1.00e+00h 1
84 -5.3266136e+01 3.25e-03 5.57e-01 -4.3 4.64e-01 - 1.00e+00 9.51e-01h 1
85 -5.3400298e+01 2.90e-04 3.28e-01 -6.0 1.34e-01 - 1.00e+00 1.00e+00h 1
86 -5.3382573e+01 1.77e-04 3.62e-02 -7.9 1.77e-02 - 1.00e+00 1.00e+00h 1
87 -5.3391343e+01 8.08e-05 2.23e-03 -9.5 8.77e-03 - 1.00e+00 1.00e+00h 1
88 -5.3387327e+01 9.15e-06 2.83e-03 -11.0 4.02e-03 -2.5 1.00e+00 1.00e+00h 1
89 -5.3387868e+01 1.19e-05 1.27e-02 -11.0 1.41e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 -5.3388406e+01 1.03e-05 5.30e-05 -11.0 6.49e-04 - 1.00e+00 1.00e+00h 1
91 -5.3387909e+01 1.03e-06 2.17e-03 -11.0 4.97e-04 - 1.00e+00 1.00e+00h 1
92 -5.3387928e+01 2.19e-06 2.30e-03 -11.0 2.44e-04 - 1.00e+00 1.00e+00h 1
93 -5.3388030e+01 1.20e-06 1.78e-04 -11.0 2.05e-04 - 1.00e+00 1.00e+00h 1
94 -5.3387971e+01 1.04e-07 3.29e-05 -11.0 5.88e-05 - 1.00e+00 1.00e+00h 1
95 -5.3387977e+01 4.37e-08 7.21e-07 -11.0 6.51e-06 - 1.00e+00 1.00e+00h 1
96 -5.3387979e+01 1.42e-08 4.69e-07 -11.0 1.84e-06 - 1.00e+00 1.00e+00h 1
97 -5.3387978e+01 1.34e-09 7.71e-07 -11.0 7.12e-07 - 1.00e+00 1.00e+00h 1
98 -5.3387978e+01 1.07e-09 7.78e-07 -11.0 3.36e-07 -3.0 1.00e+00 1.00e+00h 1
99 -5.3387978e+01 2.38e-10 4.95e-07 -11.0 1.01e-07 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 -5.3387978e+01 3.11e-11 3.53e-07 -11.0 1.24e-08 - 1.00e+00 1.00e+00h 1
101 -5.3387978e+01 8.97e-12 4.27e-07 -11.0 1.46e-09 - 1.00e+00 1.00e+00H 1
102 -5.3387978e+01 3.24e-12 1.70e-07 -11.0 5.96e-10 - 1.00e+00 1.00e+00h 1
103 -5.3387978e+01 5.63e-13 3.02e-07 -11.0 1.37e-10 - 1.00e+00 1.00e+00h 1
104 -5.3387978e+01 1.50e-13 5.48e-07 -11.0 3.02e-11 - 1.00e+00 1.00e+00h 1
105 -5.3387978e+01 1.25e-13 6.49e-07 -11.0 7.93e-12 - 1.00e+00 1.00e+00h 1
106 -5.3387978e+01 5.90e-14 4.40e-07 -11.0 5.62e-13 - 1.00e+00 1.00e+00h 1
107 -5.3387978e+01 8.13e-14 9.21e-07 -11.0 1.74e-13 - 1.00e+00 1.00e+00h 1
108 -5.3387978e+01 9.52e-14 2.44e-07 -11.0 1.41e-13 - 1.00e+00 1.00e+00h 1
109 -5.3387978e+01 7.24e-14 3.59e-07 -11.0 9.62e-14 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 109
(scaled) (unscaled)
Objective...............: -5.3387978408025084e+01 -5.3387978408025084e+01
Dual infeasibility......: 3.5863296877325918e-07 3.5863296877325918e-07
Constraint violation....: 7.2375804125490088e-14 7.2375804125490088e-14
Variable bound violation: 9.9994695856864269e-09 9.9994695856864269e-09
Complementarity.........: 1.0000142303125445e-11 1.0000142303125445e-11
Overall NLP error.......: 3.5863296877325918e-07 3.5863296877325918e-07
Number of objective function evaluations = 133
Number of objective gradient evaluations = 109
Number of equality constraint evaluations = 133
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 114
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 0
Total seconds in IPOPT = 11.434
EXIT: Solved To Acceptable Level.
Simulating trajectory traj
Done simulating trajectory traj
Launch angle: 85.0000 deg
Flight angle at end of propulsion: 83.9875 deg
Empty mass: 0.1449 kg
Water volume: 0.9839 L
Maximum range: 16.2942 m
Maximum height: 53.3880 m
Maximum velocity: 46.1508 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'] = 5
p.driver.opt_settings['max_iter'] = 1000
p.driver.opt_settings['mu_strategy'] = 'adaptive'
p.driver.opt_settings['nlp_scaling_method'] = 'gradient-based' # for faster convergence
p.driver.opt_settings['alpha_for_y'] = 'safer-min-dual-infeas'
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(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim_out = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')
'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.
Jacobian shape: (611, 613) (1.23% nonzero)
FWD solves: 20 REV solves: 0
Total colors vs. total size: 20 vs 613 (96.74% improvement)
Sparsity computed using tolerance: 1e-12.
Dense total jacobian for Problem 'problem6' was computed 3 times.
Time to compute sparsity: 0.4807 sec
Time to compute coloring: 0.7478 sec
Memory to compute coloring: 0.1250 MB
Coloring created on: 2025-07-16 18:57:31
List of user-set options:
Name Value used
alpha_for_y = safer-min-dual-infeas yes
file_print_level = 5 yes
hessian_approximation = limited-memory yes
linear_solver = mumps yes
max_iter = 1000 yes
mu_strategy = adaptive yes
nlp_scaling_method = gradient-based yes
output_file = /home/runner/work/dymos/dymos/docs/dymos_book/examples/water_rocket/problem6_out/IPOPT.out yes
print_level = 5 yes
print_user_options = yes yes
sb = yes yes
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 103007
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 0
Total number of variables............................: 613
variables with only lower bounds: 180
variables with lower and upper bounds: 67
variables with only upper bounds: 30
Total number of equality constraints.................: 610
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
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -2.0000000e+01 9.00e+01 3.60e-01 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.3188387e+01 2.49e+01 6.36e+03 -5.8 4.98e+01 - 1.85e-01 7.23e-01h 1
2 -3.5089577e+01 7.07e+00 2.76e+04 0.6 4.19e+01 - 1.00e+00 7.63e-01h 1
3 -3.5976015e+01 6.87e+01 1.55e+06 1.8 6.90e+01 - 1.00e+00 3.48e-01f 1
4 -5.5522114e+01 1.36e+02 1.39e+05 2.3 6.51e+01 - 6.72e-01 1.00e+00f 1
5 -4.3438267e+01 5.27e+01 8.03e+04 2.3 2.98e+01 - 1.00e+00 1.00e+00f 1
6 -3.7806064e+01 1.10e+02 2.36e+05 2.1 4.99e+01 - 9.72e-01 1.00e+00f 1
7 -5.6571012e+01 1.16e+02 6.64e+04 2.6 4.77e+01 - 1.00e+00 4.78e-01h 1
8 -6.9028277e+01 2.97e+01 1.13e+05 2.4 3.16e+01 - 9.42e-01 1.00e+00h 1
9 -3.7879082e+01 1.64e+00 2.40e+04 3.2 3.11e+01 - 5.60e-01 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -5.3052216e+01 8.81e-01 1.21e+06 2.5 1.52e+01 - 9.82e-01 1.00e+00f 1
11 -5.2073297e+01 4.41e-03 1.48e+04 0.9 9.79e-01 - 1.00e+00 1.00e+00f 1
12 -6.1263024e+01 1.20e+00 2.15e+02 0.4 9.19e+00 - 9.88e-01 1.00e+00f 1
13 -7.0764256e+01 1.04e+00 4.12e+02 -0.8 9.50e+00 - 1.00e+00 1.00e+00f 1
14 -7.8629009e+01 7.41e-01 1.84e+02 -1.3 7.86e+00 - 1.00e+00 1.00e+00f 1
15 -8.1990265e+01 3.04e-01 2.89e+01 -2.5 4.32e+00 - 1.00e+00 1.00e+00f 1
16 -8.3480339e+01 1.42e-01 3.57e+01 -3.2 3.00e+00 - 1.00e+00 1.00e+00h 1
17 -8.4871615e+01 1.95e-01 4.33e+01 -3.5 1.92e+00 - 1.00e+00 1.00e+00f 1
18 -8.5188038e+01 4.68e-01 1.30e+02 -3.8 9.56e+00 - 1.00e+00 1.00e+00h 1
19 -8.4931459e+01 2.52e-02 2.47e+01 -4.5 4.10e+00 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 -8.5810579e+01 3.35e-02 1.70e+01 -5.3 4.78e+00 - 1.00e+00 1.00e+00f 1
21 -8.3431619e+01 3.09e-02 1.30e+01 -6.1 2.38e+00 - 1.00e+00 1.00e+00h 1
22 -8.5720066e+01 6.27e-02 4.14e+01 -7.0 3.34e+00 - 1.00e+00 1.00e+00f 1
23 -8.7119415e+01 8.38e-02 6.93e+00 -7.5 2.50e+00 - 1.00e+00 1.00e+00f 1
24 -8.4835237e+01 1.88e-02 1.34e+01 -8.4 5.37e+00 - 1.00e+00 1.00e+00H 1
25 -8.5172725e+01 9.68e-03 2.40e+00 -8.5 2.17e+00 - 1.00e+00 1.00e+00h 1
26 -8.5104627e+01 1.11e-04 1.41e-01 -10.3 1.21e-01 - 1.00e+00 1.00e+00h 1
27 -8.5098690e+01 9.97e-05 7.34e-02 -11.0 1.05e-02 - 1.00e+00 1.00e+00h 1
28 -8.5100329e+01 4.57e-05 6.76e-03 -11.0 6.64e-03 - 1.00e+00 1.00e+00H 1
29 -8.5096924e+01 6.62e-06 8.65e-03 -11.0 3.41e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 -8.5081796e+01 1.26e-06 4.95e-01 -11.0 2.59e-02 - 1.00e+00 1.00e+00H 1
31 -8.5081812e+01 2.31e-04 1.53e-02 -11.0 2.37e-02 - 1.00e+00 1.00e+00h 1
32 -8.5097550e+01 2.29e-05 6.88e-03 -11.0 1.57e-02 - 1.00e+00 1.00e+00h 1
33 -8.5097914e+01 1.95e-05 2.96e-03 -11.0 2.67e-03 - 1.00e+00 1.00e+00h 1
34 -8.5096570e+01 3.15e-06 6.16e-03 -11.0 2.13e-03 - 1.00e+00 1.00e+00h 1
35 -8.5096814e+01 1.24e-06 3.31e-02 -11.0 8.19e-04 - 1.00e+00 1.00e+00H 1
36 -8.5096716e+01 1.57e-07 4.95e-05 -11.0 9.84e-05 - 1.00e+00 1.00e+00h 1
37 -8.5096709e+01 3.37e-08 2.41e-07 -11.0 6.84e-06 - 1.00e+00 1.00e+00h 1
38 -8.5096712e+01 3.49e-08 3.41e-07 -11.0 3.14e-06 - 1.00e+00 1.00e+00h 1
39 -8.5096709e+01 3.34e-08 3.52e-07 -11.0 3.34e-06 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 -8.5096712e+01 5.04e-09 1.80e-06 -11.0 2.37e-06 - 1.00e+00 1.00e+00h 1
41 -8.5096712e+01 9.26e-09 1.08e-06 -11.0 2.39e-06 - 1.00e+00 1.00e+00h 1
42 -8.5096711e+01 5.84e-09 3.22e-07 -11.0 6.88e-07 - 1.00e+00 1.00e+00h 1
43 -8.5096711e+01 5.32e-09 4.12e-07 -11.0 3.79e-07 - 1.00e+00 1.00e+00h 1
44 -8.5096711e+01 5.88e-10 2.16e-07 -11.0 3.77e-07 - 1.00e+00 1.00e+00h 1
45 -8.5096711e+01 2.10e-10 8.20e-07 -11.0 4.55e-07 - 1.00e+00 1.00e+00h 1
46 -8.5096711e+01 5.60e-10 2.05e-07 -11.0 3.86e-07 - 1.00e+00 1.00e+00h 1
47 -8.5096711e+01 7.55e-10 8.51e-07 -11.0 3.28e-07 - 1.00e+00 1.00e+00H 1
48 -8.5096711e+01 3.53e-10 5.26e-07 -11.0 2.37e-07 - 1.00e+00 1.00e+00H 1
49 -8.5096711e+01 1.54e-10 2.96e-07 -11.0 2.74e-08 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 -8.5096711e+01 9.48e-11 4.98e-07 -11.0 1.15e-08 - 1.00e+00 1.00e+00h 1
51 -8.5096711e+01 2.58e-11 8.63e-08 -11.0 6.89e-09 - 1.00e+00 1.00e+00h 1
52 -8.5096711e+01 7.25e-12 4.15e-07 -11.0 2.42e-09 - 1.00e+00 1.00e+00H 1
53 -8.5096711e+01 1.33e-11 3.00e-07 -11.0 2.37e-09 - 1.00e+00 1.00e+00H 1
54 -8.5096711e+01 2.14e-12 1.66e-07 -11.0 1.06e-09 - 1.00e+00 1.00e+00h 1
55 -8.5096711e+01 2.60e-13 1.55e-07 -11.0 1.83e-10 -2.0 1.00e+00 1.00e+00h 1
56 -8.5096711e+01 3.18e-13 6.07e-07 -11.0 1.33e-10 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 56
(scaled) (unscaled)
Objective...............: -8.5096711081763416e+01 -8.5096711081763416e+01
Dual infeasibility......: 6.0708479496332250e-07 6.0708479496332250e-07
Constraint violation....: 3.1817186132721855e-13 3.1817186132721855e-13
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 1.0000000000000003e-11 1.0000000000000003e-11
Overall NLP error.......: 6.0708479496332250e-07 6.0708479496332250e-07
Number of objective function evaluations = 66
Number of objective gradient evaluations = 57
Number of equality constraint evaluations = 66
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 57
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 0
Total seconds in IPOPT = 6.863
EXIT: Solved To Acceptable Level.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 6.8649
User Objective Time : 0.2812
User Sensitivity Time : 3.9673
Interface Time : 0.7824
Opt Solver Time: 1.8340
Calls to Objective Function : 67
Calls to Sens Function : 58
Objectives
Index Name Value
0 traj.descent.states:r -8.509671E+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 1.891068E+00 1.000000E+01
1 traj.ballistic_ascent.t_initial_0 c 1.000000E-03 1.127359E-01 1.000000E+00
2 traj.ballistic_ascent.t_duration_0 c 1.000000E-03 1.928639E+00 1.000000E+01
3 traj.ballistic_ascent.states:r_0 c -1.000000E+21 1.609883E+00 1.000000E+21
4 traj.ballistic_ascent.states:r_1 c -1.000000E+21 3.796505E+00 1.000000E+21
5 traj.ballistic_ascent.states:r_2 c -1.000000E+21 6.719126E+00 1.000000E+21
6 traj.ballistic_ascent.states:r_3 c -1.000000E+21 7.622675E+00 1.000000E+21
7 traj.ballistic_ascent.states:r_4 c -1.000000E+21 9.654936E+00 1.000000E+21
8 traj.ballistic_ascent.states:r_5 c -1.000000E+21 1.237962E+01 1.000000E+21
9 traj.ballistic_ascent.states:r_6 c -1.000000E+21 1.322386E+01 1.000000E+21
10 traj.ballistic_ascent.states:r_7 c -1.000000E+21 1.512589E+01 1.000000E+21
11 traj.ballistic_ascent.states:r_8 c -1.000000E+21 1.768270E+01 1.000000E+21
12 traj.ballistic_ascent.states:r_9 c -1.000000E+21 1.847644E+01 1.000000E+21
13 traj.ballistic_ascent.states:r_10 c -1.000000E+21 2.026728E+01 1.000000E+21
14 traj.ballistic_ascent.states:r_11 c -1.000000E+21 2.268007E+01 1.000000E+21
15 traj.ballistic_ascent.states:r_12 c -1.000000E+21 2.343033E+01 1.000000E+21
16 traj.ballistic_ascent.states:r_13 c -1.000000E+21 2.512518E+01 1.000000E+21
17 traj.ballistic_ascent.states:r_14 c -1.000000E+21 2.741311E+01 1.000000E+21
18 traj.ballistic_ascent.states:r_15 c -1.000000E+21 2.812555E+01 1.000000E+21
19 traj.ballistic_ascent.states:r_16 c -1.000000E+21 2.973668E+01 1.000000E+21
20 traj.ballistic_ascent.states:r_17 c -1.000000E+21 3.191524E+01 1.000000E+21
21 traj.ballistic_ascent.states:r_18 c -1.000000E+21 3.259445E+01 1.000000E+21
22 traj.ballistic_ascent.states:r_19 c -1.000000E+21 3.413183E+01 1.000000E+21
23 traj.ballistic_ascent.states:r_20 c -1.000000E+21 3.621365E+01 1.000000E+21
24 traj.ballistic_ascent.states:r_21 c -1.000000E+21 3.686338E+01 1.000000E+21
25 traj.ballistic_ascent.states:r_22 c -1.000000E+21 3.833514E+01 1.000000E+21
26 traj.ballistic_ascent.states:r_23 c -1.000000E+21 4.033049E+01 1.000000E+21
27 traj.ballistic_ascent.states:r_24 c -1.000000E+21 4.095377E+01 1.000000E+21
28 traj.ballistic_ascent.states:r_25 c -1.000000E+21 4.236652E+01 1.000000E+21
29 traj.ballistic_ascent.states:r_26 c -1.000000E+21 4.428375E+01 1.000000E+21
30 traj.ballistic_ascent.states:r_27 c -1.000000E+21 4.488304E+01 1.000000E+21
31 traj.ballistic_ascent.states:r_28 c -1.000000E+21 4.624213E+01 1.000000E+21
32 traj.ballistic_ascent.states:r_29 c -1.000000E+21 4.808798E+01 1.000000E+21
33 traj.ballistic_ascent.states:r_30 c -1.000000E+21 4.866527E+01 1.000000E+21
34 traj.ballistic_ascent.states:h_0 c -1.000000E+21 1.305634E+00 1.000000E+21
35 traj.ballistic_ascent.states:h_1 c -1.000000E+21 3.015243E+00 1.000000E+21
36 traj.ballistic_ascent.states:h_2 c -1.000000E+21 5.226005E+00 1.000000E+21
37 traj.ballistic_ascent.states:h_3 c -1.000000E+21 5.891426E+00 1.000000E+21
38 traj.ballistic_ascent.states:h_4 c -1.000000E+21 7.355373E+00 1.000000E+21
39 traj.ballistic_ascent.states:h_5 c -1.000000E+21 9.243708E+00 1.000000E+21
40 traj.ballistic_ascent.states:h_6 c -1.000000E+21 9.810733E+00 1.000000E+21
41 traj.ballistic_ascent.states:h_7 c -1.000000E+21 1.105548E+01 1.000000E+21
42 traj.ballistic_ascent.states:h_8 c -1.000000E+21 1.265423E+01 1.000000E+21
43 traj.ballistic_ascent.states:h_9 c -1.000000E+21 1.313246E+01 1.000000E+21
44 traj.ballistic_ascent.states:h_10 c -1.000000E+21 1.417867E+01 1.000000E+21
45 traj.ballistic_ascent.states:h_11 c -1.000000E+21 1.551366E+01 1.000000E+21
46 traj.ballistic_ascent.states:h_12 c -1.000000E+21 1.591067E+01 1.000000E+21
47 traj.ballistic_ascent.states:h_13 c -1.000000E+21 1.677474E+01 1.000000E+21
48 traj.ballistic_ascent.states:h_14 c -1.000000E+21 1.786652E+01 1.000000E+21
49 traj.ballistic_ascent.states:h_15 c -1.000000E+21 1.818839E+01 1.000000E+21
50 traj.ballistic_ascent.states:h_16 c -1.000000E+21 1.888344E+01 1.000000E+21
51 traj.ballistic_ascent.states:h_17 c -1.000000E+21 1.974857E+01 1.000000E+21
52 traj.ballistic_ascent.states:h_18 c -1.000000E+21 2.000017E+01 1.000000E+21
53 traj.ballistic_ascent.states:h_19 c -1.000000E+21 2.053684E+01 1.000000E+21
54 traj.ballistic_ascent.states:h_20 c -1.000000E+21 2.118879E+01 1.000000E+21
55 traj.ballistic_ascent.states:h_21 c -1.000000E+21 2.137413E+01 1.000000E+21
56 traj.ballistic_ascent.states:h_22 c -1.000000E+21 2.176113E+01 1.000000E+21
57 traj.ballistic_ascent.states:h_23 c -1.000000E+21 2.221099E+01 1.000000E+21
58 traj.ballistic_ascent.states:h_24 c -1.000000E+21 2.233338E+01 1.000000E+21
59 traj.ballistic_ascent.states:h_25 c -1.000000E+21 2.257793E+01 1.000000E+21
60 traj.ballistic_ascent.states:h_26 c -1.000000E+21 2.283496E+01 1.000000E+21
61 traj.ballistic_ascent.states:h_27 c -1.000000E+21 2.289716E+01 1.000000E+21
62 traj.ballistic_ascent.states:h_28 c -1.000000E+21 2.300536E+01 1.000000E+21
63 traj.ballistic_ascent.states:h_29 c -1.000000E+21 2.307744E+01 1.000000E+21
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485 traj.propelled_ascent.states:v_52 c 1.000000E-04 3.431436E-01 1.000000E+19
486 traj.propelled_ascent.states:v_53 c 1.000000E-04 3.475604E-01 1.000000E+19
487 traj.propelled_ascent.states:v_54 c 1.000000E-04 3.579701E-01 1.000000E+19
488 traj.propelled_ascent.states:v_55 c 1.000000E-04 3.730835E-01 1.000000E+19
489 traj.propelled_ascent.states:v_56 c 1.000000E-04 3.780671E-01 1.000000E+19
490 traj.propelled_ascent.states:v_57 c 1.000000E-04 3.898870E-01 1.000000E+19
491 traj.propelled_ascent.states:v_58 c 1.000000E-04 4.072489E-01 1.000000E+19
492 traj.propelled_ascent.states:v_59 c 1.000000E-04 4.130291E-01 1.000000E+19
493 traj.propelled_ascent.states:p_0 c 1.020000E+00 6.305226E+00 1.000000E+21
494 traj.propelled_ascent.states:p_1 c 1.020000E+00 6.058536E+00 1.000000E+21
495 traj.propelled_ascent.states:p_2 c 1.020000E+00 5.985224E+00 1.000000E+21
496 traj.propelled_ascent.states:p_3 c 1.020000E+00 5.825217E+00 1.000000E+21
497 traj.propelled_ascent.states:p_4 c 1.020000E+00 5.620757E+00 1.000000E+21
498 traj.propelled_ascent.states:p_5 c 1.020000E+00 5.559612E+00 1.000000E+21
499 traj.propelled_ascent.states:p_6 c 1.020000E+00 5.425540E+00 1.000000E+21
500 traj.propelled_ascent.states:p_7 c 1.020000E+00 5.252980E+00 1.000000E+21
501 traj.propelled_ascent.states:p_8 c 1.020000E+00 5.201111E+00 1.000000E+21
502 traj.propelled_ascent.states:p_9 c 1.020000E+00 5.086938E+00 1.000000E+21
503 traj.propelled_ascent.states:p_10 c 1.020000E+00 4.939114E+00 1.000000E+21
504 traj.propelled_ascent.states:p_11 c 1.020000E+00 4.894489E+00 1.000000E+21
505 traj.propelled_ascent.states:p_12 c 1.020000E+00 4.795947E+00 1.000000E+21
506 traj.propelled_ascent.states:p_13 c 1.020000E+00 4.667720E+00 1.000000E+21
507 traj.propelled_ascent.states:p_14 c 1.020000E+00 4.628871E+00 1.000000E+21
508 traj.propelled_ascent.states:p_15 c 1.020000E+00 4.542850E+00 1.000000E+21
509 traj.propelled_ascent.states:p_16 c 1.020000E+00 4.430435E+00 1.000000E+21
510 traj.propelled_ascent.states:p_17 c 1.020000E+00 4.396270E+00 1.000000E+21
511 traj.propelled_ascent.states:p_18 c 1.020000E+00 4.320446E+00 1.000000E+21
512 traj.propelled_ascent.states:p_19 c 1.020000E+00 4.220990E+00 1.000000E+21
513 traj.propelled_ascent.states:p_20 c 1.020000E+00 4.190682E+00 1.000000E+21
514 traj.propelled_ascent.states:p_21 c 1.020000E+00 4.123281E+00 1.000000E+21
515 traj.propelled_ascent.states:p_22 c 1.020000E+00 4.034589E+00 1.000000E+21
516 traj.propelled_ascent.states:p_23 c 1.020000E+00 4.007498E+00 1.000000E+21
517 traj.propelled_ascent.states:p_24 c 1.020000E+00 3.947142E+00 1.000000E+21
518 traj.propelled_ascent.states:p_25 c 1.020000E+00 3.867497E+00 1.000000E+21
519 traj.propelled_ascent.states:p_26 c 1.020000E+00 3.843118E+00 1.000000E+21
520 traj.propelled_ascent.states:p_27 c 1.020000E+00 3.788721E+00 1.000000E+21
521 traj.propelled_ascent.states:p_28 c 1.020000E+00 3.716757E+00 1.000000E+21
522 traj.propelled_ascent.states:p_29 c 1.020000E+00 3.694690E+00 1.000000E+21
523 traj.propelled_ascent.states:p_30 c 1.020000E+00 3.645379E+00 1.000000E+21
524 traj.propelled_ascent.states:p_31 c 1.020000E+00 3.580000E+00 1.000000E+21
525 traj.propelled_ascent.states:p_32 c 1.020000E+00 3.559918E+00 1.000000E+21
526 traj.propelled_ascent.states:p_33 c 1.020000E+00 3.514988E+00 1.000000E+21
527 traj.propelled_ascent.states:p_34 c 1.020000E+00 3.455298E+00 1.000000E+21
528 traj.propelled_ascent.states:p_35 c 1.020000E+00 3.436936E+00 1.000000E+21
529 traj.propelled_ascent.states:p_36 c 1.020000E+00 3.395809E+00 1.000000E+21
530 traj.propelled_ascent.states:p_37 c 1.020000E+00 3.341071E+00 1.000000E+21
531 traj.propelled_ascent.states:p_38 c 1.020000E+00 3.324210E+00 1.000000E+21
532 traj.propelled_ascent.states:p_39 c 1.020000E+00 3.286405E+00 1.000000E+21
533 traj.propelled_ascent.states:p_40 c 1.020000E+00 3.236007E+00 1.000000E+21
534 traj.propelled_ascent.states:p_41 c 1.020000E+00 3.220464E+00 1.000000E+21
535 traj.propelled_ascent.states:p_42 c 1.020000E+00 3.185582E+00 1.000000E+21
536 traj.propelled_ascent.states:p_43 c 1.020000E+00 3.139009E+00 1.000000E+21
537 traj.propelled_ascent.states:p_44 c 1.020000E+00 3.124630E+00 1.000000E+21
538 traj.propelled_ascent.states:p_45 c 1.020000E+00 3.092332E+00 1.000000E+21
539 traj.propelled_ascent.states:p_46 c 1.020000E+00 3.049152E+00 1.000000E+21
540 traj.propelled_ascent.states:p_47 c 1.020000E+00 3.035806E+00 1.000000E+21
541 traj.propelled_ascent.states:p_48 c 1.020000E+00 3.005807E+00 1.000000E+21
542 traj.propelled_ascent.states:p_49 c 1.020000E+00 2.965648E+00 1.000000E+21
543 traj.propelled_ascent.states:p_50 c 1.020000E+00 2.953225E+00 1.000000E+21
544 traj.propelled_ascent.states:p_51 c 1.020000E+00 2.925280E+00 1.000000E+21
545 traj.propelled_ascent.states:p_52 c 1.020000E+00 2.887826E+00 1.000000E+21
546 traj.propelled_ascent.states:p_53 c 1.020000E+00 2.876230E+00 1.000000E+21
547 traj.propelled_ascent.states:p_54 c 1.020000E+00 2.850127E+00 1.000000E+21
548 traj.propelled_ascent.states:p_55 c 1.020000E+00 2.815106E+00 1.000000E+21
549 traj.propelled_ascent.states:p_56 c 1.020000E+00 2.804255E+00 1.000000E+21
550 traj.propelled_ascent.states:p_57 c 1.020000E+00 2.779812E+00 1.000000E+21
551 traj.propelled_ascent.states:p_58 c 1.020000E+00 2.746987E+00 1.000000E+21
552 traj.propelled_ascent.states:p_59 c 1.020000E+00 2.736808E+00 1.000000E+21
553 traj.propelled_ascent.states:V_w_0 c 0.000000E+00 1.027314E-01 1.000000E+20
554 traj.propelled_ascent.states:V_w_1 c 0.000000E+00 1.002340E-01 1.000000E+20
555 traj.propelled_ascent.states:V_w_2 c 0.000000E+00 9.685978E-02 1.000000E+20
556 traj.propelled_ascent.states:V_w_3 c 0.000000E+00 9.580827E-02 1.000000E+20
557 traj.propelled_ascent.states:V_w_4 c 0.000000E+00 9.342884E-02 1.000000E+20
558 traj.propelled_ascent.states:V_w_5 c 0.000000E+00 9.020771E-02 1.000000E+20
559 traj.propelled_ascent.states:V_w_6 c 0.000000E+00 8.920254E-02 1.000000E+20
560 traj.propelled_ascent.states:V_w_7 c 0.000000E+00 8.692563E-02 1.000000E+20
561 traj.propelled_ascent.states:V_w_8 c 0.000000E+00 8.383843E-02 1.000000E+20
562 traj.propelled_ascent.states:V_w_9 c 0.000000E+00 8.287395E-02 1.000000E+20
563 traj.propelled_ascent.states:V_w_10 c 0.000000E+00 8.068740E-02 1.000000E+20
564 traj.propelled_ascent.states:V_w_11 c 0.000000E+00 7.771881E-02 1.000000E+20
565 traj.propelled_ascent.states:V_w_12 c 0.000000E+00 7.679051E-02 1.000000E+20
566 traj.propelled_ascent.states:V_w_13 c 0.000000E+00 7.468451E-02 1.000000E+20
567 traj.propelled_ascent.states:V_w_14 c 0.000000E+00 7.182212E-02 1.000000E+20
568 traj.propelled_ascent.states:V_w_15 c 0.000000E+00 7.092632E-02 1.000000E+20
569 traj.propelled_ascent.states:V_w_16 c 0.000000E+00 6.889281E-02 1.000000E+20
570 traj.propelled_ascent.states:V_w_17 c 0.000000E+00 6.612636E-02 1.000000E+20
571 traj.propelled_ascent.states:V_w_18 c 0.000000E+00 6.525999E-02 1.000000E+20
572 traj.propelled_ascent.states:V_w_19 c 0.000000E+00 6.329228E-02 1.000000E+20
573 traj.propelled_ascent.states:V_w_20 c 0.000000E+00 6.061316E-02 1.000000E+20
574 traj.propelled_ascent.states:V_w_21 c 0.000000E+00 5.977365E-02 1.000000E+20
575 traj.propelled_ascent.states:V_w_22 c 0.000000E+00 5.786608E-02 1.000000E+20
576 traj.propelled_ascent.states:V_w_23 c 0.000000E+00 5.526699E-02 1.000000E+20
577 traj.propelled_ascent.states:V_w_24 c 0.000000E+00 5.445213E-02 1.000000E+20
578 traj.propelled_ascent.states:V_w_25 c 0.000000E+00 5.259986E-02 1.000000E+20
579 traj.propelled_ascent.states:V_w_26 c 0.000000E+00 5.007455E-02 1.000000E+20
580 traj.propelled_ascent.states:V_w_27 c 0.000000E+00 4.928246E-02 1.000000E+20
581 traj.propelled_ascent.states:V_w_28 c 0.000000E+00 4.748132E-02 1.000000E+20
582 traj.propelled_ascent.states:V_w_29 c 0.000000E+00 4.502435E-02 1.000000E+20
583 traj.propelled_ascent.states:V_w_30 c 0.000000E+00 4.425338E-02 1.000000E+20
584 traj.propelled_ascent.states:V_w_31 c 0.000000E+00 4.249974E-02 1.000000E+20
585 traj.propelled_ascent.states:V_w_32 c 0.000000E+00 4.010638E-02 1.000000E+20
586 traj.propelled_ascent.states:V_w_33 c 0.000000E+00 3.935510E-02 1.000000E+20
587 traj.propelled_ascent.states:V_w_34 c 0.000000E+00 3.764576E-02 1.000000E+20
588 traj.propelled_ascent.states:V_w_35 c 0.000000E+00 3.531184E-02 1.000000E+20
589 traj.propelled_ascent.states:V_w_36 c 0.000000E+00 3.457898E-02 1.000000E+20
590 traj.propelled_ascent.states:V_w_37 c 0.000000E+00 3.291114E-02 1.000000E+20
591 traj.propelled_ascent.states:V_w_38 c 0.000000E+00 3.063296E-02 1.000000E+20
592 traj.propelled_ascent.states:V_w_39 c 0.000000E+00 2.991740E-02 1.000000E+20
593 traj.propelled_ascent.states:V_w_40 c 0.000000E+00 2.828854E-02 1.000000E+20
594 traj.propelled_ascent.states:V_w_41 c 0.000000E+00 2.606282E-02 1.000000E+20
595 traj.propelled_ascent.states:V_w_42 c 0.000000E+00 2.536354E-02 1.000000E+20
596 traj.propelled_ascent.states:V_w_43 c 0.000000E+00 2.377144E-02 1.000000E+20
597 traj.propelled_ascent.states:V_w_44 c 0.000000E+00 2.159522E-02 1.000000E+20
598 traj.propelled_ascent.states:V_w_45 c 0.000000E+00 2.091133E-02 1.000000E+20
599 traj.propelled_ascent.states:V_w_46 c 0.000000E+00 1.935397E-02 1.000000E+20
600 traj.propelled_ascent.states:V_w_47 c 0.000000E+00 1.722459E-02 1.000000E+20
601 traj.propelled_ascent.states:V_w_48 c 0.000000E+00 1.655528E-02 1.000000E+20
602 traj.propelled_ascent.states:V_w_49 c 0.000000E+00 1.503084E-02 1.000000E+20
603 traj.propelled_ascent.states:V_w_50 c 0.000000E+00 1.294591E-02 1.000000E+20
604 traj.propelled_ascent.states:V_w_51 c 0.000000E+00 1.229042E-02 1.000000E+20
605 traj.propelled_ascent.states:V_w_52 c 0.000000E+00 1.079726E-02 1.000000E+20
606 traj.propelled_ascent.states:V_w_53 c 0.000000E+00 8.754589E-03 1.000000E+20
607 traj.propelled_ascent.states:V_w_54 c 0.000000E+00 8.112271E-03 1.000000E+20
608 traj.propelled_ascent.states:V_w_55 c 0.000000E+00 6.648883E-03 1.000000E+20
609 traj.propelled_ascent.states:V_w_56 c 0.000000E+00 4.646480E-03 1.000000E+20
610 traj.propelled_ascent.states:V_w_57 c 0.000000E+00 4.016715E-03 1.000000E+20
611 traj.propelled_ascent.states:V_w_58 c 0.000000E+00 2.581733E-03 1.000000E+20
612 traj.propelled_ascent.states:V_w_59 c 0.000000E+00 6.177737E-04 1.000000E+20
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier
0 traj.linkages.propelled_ascent:time_final|ballistic_ascent:time_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -6.07447E-11
1 traj.linkages.propelled_ascent:r_final|ballistic_ascent:r_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -1.00000E+00
2 traj.linkages.propelled_ascent:h_final|ballistic_ascent:h_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -6.78563E-01
3 traj.linkages.propelled_ascent:gam_final|ballistic_ascent:gam_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -3.55552E-02
4 traj.linkages.propelled_ascent:v_final|ballistic_ascent:v_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -2.15605E+00
5 traj.linkages.ballistic_ascent:time_final|descent:time_initial e 0.000000E+00 0.000000E+00 0.000000E+00 2.26728E-11
6 traj.linkages.ballistic_ascent:r_final|descent:r_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -1.00000E+00
7 traj.linkages.ballistic_ascent:h_final|descent:h_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -6.75353E-01
8 traj.linkages.ballistic_ascent:gam_final|descent:gam_initial e 0.000000E+00 9.979300E-32 0.000000E+00 -4.49786E-01
9 traj.linkages.ballistic_ascent:v_final|descent:v_initial e 0.000000E+00 0.000000E+00 0.000000E+00 -1.67877E+00
10 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 0.000000E+00 0.000000E+00 2.22222E-01
11 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.027785E-15 0.000000E+00 1.02497E+00
12 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.370380E-15 0.000000E+00 7.52806E-01
13 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.062045E-14 0.000000E+00 2.22222E-01
14 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.425951E-15 0.000000E+00 1.02497E+00
15 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.027785E-15 0.000000E+00 7.52806E-01
16 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -6.851902E-16 0.000000E+00 2.22222E-01
17 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -3.425951E-16 0.000000E+00 1.02497E+00
18 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 7.879688E-15 0.000000E+00 7.52806E-01
19 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -3.083356E-15 0.000000E+00 2.22222E-01
20 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 6.166712E-15 0.000000E+00 1.02497E+00
21 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.083356E-15 0.000000E+00 7.52806E-01
22 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -6.851902E-15 0.000000E+00 2.22222E-01
23 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 8.222283E-15 0.000000E+00 1.02497E+00
24 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.781495E-14 0.000000E+00 7.52806E-01
25 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -3.425951E-16 0.000000E+00 2.22222E-01
26 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -5.481522E-15 0.000000E+00 1.02497E+00
27 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.370380E-15 0.000000E+00 7.52806E-01
28 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 2.398166E-15 0.000000E+00 2.22222E-01
29 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -9.935258E-15 0.000000E+00 1.02497E+00
30 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.233342E-14 0.000000E+00 7.52806E-01
31 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.850014E-14 0.000000E+00 2.22222E-01
32 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.096304E-14 0.000000E+00 1.02497E+00
33 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.301861E-14 0.000000E+00 7.52806E-01
34 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.528730E-14 0.000000E+00 2.22222E-01
35 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.712976E-15 0.000000E+00 1.02497E+00
36 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 -1.918533E-14 0.000000E+00 7.52806E-01
37 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 3.700027E-14 0.000000E+00 2.22222E-01
38 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 1.712976E-15 0.000000E+00 1.02497E+00
39 traj.ballistic_ascent.collocation_constraint.defects:r e 0.000000E+00 7.194497E-15 0.000000E+00 7.52806E-01
40 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 3.425951E-15 0.000000E+00 1.50792E-01
41 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.370380E-15 0.000000E+00 6.95271E-01
42 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.425951E-16 0.000000E+00 5.10428E-01
43 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 2.055571E-15 0.000000E+00 1.50654E-01
44 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.851902E-16 0.000000E+00 6.94673E-01
45 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 3.083356E-15 0.000000E+00 5.10024E-01
46 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 4.111141E-15 0.000000E+00 1.50538E-01
47 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.425951E-16 0.000000E+00 6.94168E-01
48 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 3.083356E-15 0.000000E+00 5.09682E-01
49 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.851902E-16 0.000000E+00 1.50440E-01
50 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 2.055571E-15 0.000000E+00 6.93741E-01
51 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -5.310224E-15 0.000000E+00 5.09393E-01
52 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -6.851902E-15 0.000000E+00 1.50357E-01
53 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 1.884273E-15 0.000000E+00 6.93381E-01
54 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.425951E-15 0.000000E+00 5.09151E-01
55 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 1.096304E-14 0.000000E+00 1.50287E-01
56 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.939844E-15 0.000000E+00 6.93080E-01
57 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 8.564878E-16 0.000000E+00 5.08948E-01
58 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -1.062045E-14 0.000000E+00 1.50229E-01
59 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -4.881980E-15 0.000000E+00 6.92828E-01
60 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 -3.768546E-15 0.000000E+00 5.08779E-01
61 traj.ballistic_ascent.collocation_constraint.defects:h e 0.000000E+00 4.368088E-15 0.000000E+00 1.50180E-01
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550 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 3.704796E-15 0.000000E+00 1.66440E-05
551 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.053953E-15 0.000000E+00 1.30539E-01
552 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.007777E-16 0.000000E+00 2.28995E-01
553 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.101426E-15 0.000000E+00 8.01137E-02
554 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.051361E-15 0.000000E+00 5.02221E-01
555 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.102722E-15 0.000000E+00 5.04892E-01
556 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.252916E-15 0.000000E+00 1.61873E-01
557 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.501944E-16 0.000000E+00 8.83035E-01
558 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.302981E-15 0.000000E+00 7.88950E-01
559 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -5.957712E-15 0.000000E+00 2.46176E-01
560 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.401815E-15 0.000000E+00 1.27699E+00
561 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.501944E-15 0.000000E+00 1.08411E+00
562 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.252916E-15 0.000000E+00 3.33895E-01
563 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.301685E-15 0.000000E+00 1.68817E+00
564 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -4.005185E-16 0.000000E+00 1.39346E+00
565 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.806221E-15 0.000000E+00 4.25951E-01
566 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.952527E-15 0.000000E+00 2.12095E+00
567 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 3.604666E-15 0.000000E+00 1.72039E+00
568 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.208036E-15 0.000000E+00 5.23362E-01
569 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.652139E-15 0.000000E+00 2.58024E+00
570 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -5.006481E-16 0.000000E+00 2.06875E+00
571 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -5.607258E-15 0.000000E+00 6.27292E-01
572 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 7.009073E-16 0.000000E+00 3.07172E+00
573 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 3.154083E-15 0.000000E+00 2.44306E+00
574 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.151491E-15 0.000000E+00 7.39115E-01
575 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.001296E-15 0.000000E+00 3.60213E+00
576 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.652139E-15 0.000000E+00 2.84878E+00
577 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 2.853694E-15 0.000000E+00 8.60490E-01
578 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -7.009073E-16 0.000000E+00 4.17970E+00
579 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.451879E-15 0.000000E+00 3.29260E+00
580 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.852398E-15 0.000000E+00 9.93464E-01
581 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.003888E-16 0.000000E+00 4.81465E+00
582 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.005185E-16 0.000000E+00 3.78295E+00
583 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.001296E-16 0.000000E+00 1.14062E+00
584 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.503240E-16 0.000000E+00 5.51996E+00
585 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.752268E-15 0.000000E+00 4.33064E+00
586 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.101426E-15 0.000000E+00 1.30528E+00
587 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -2.002592E-16 0.000000E+00 6.31250E+00
588 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.401815E-15 0.000000E+00 4.94986E+00
589 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 1.602074E-15 0.000000E+00 1.49183E+00
590 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -7.009073E-16 0.000000E+00 7.21465E+00
591 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -6.508425E-16 0.000000E+00 5.65969E+00
592 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -9.011665E-16 0.000000E+00 1.70619E+00
593 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 4.255509E-16 0.000000E+00 8.25692E+00
594 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -5.256805E-16 0.000000E+00 6.48650E+00
595 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.952527E-15 0.000000E+00 1.95658E+00
596 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.003888E-16 0.000000E+00 9.48209E+00
597 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 8.511017E-16 0.000000E+00 7.46786E+00
598 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.126458E-15 0.000000E+00 2.25476E+00
599 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 3.504536E-16 0.000000E+00 1.09523E+01
600 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.754861E-16 0.000000E+00 8.65942E+00
601 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -3.754861E-16 0.000000E+00 2.61832E+00
602 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.001296E-16 0.000000E+00 1.27614E+01
603 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 5.006481E-17 0.000000E+00 1.01474E+01
604 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 3.754861E-16 0.000000E+00 3.07474E+00
605 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 -1.501944E-16 0.000000E+00 1.50591E+01
606 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 7.509721E-17 0.000000E+00 1.20732E+01
607 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 0.000000E+00 0.000000E+00 3.66969E+00
608 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 5.006481E-17 0.000000E+00 1.80991E+01
609 traj.propelled_ascent.collocation_constraint.defects:V_w e 0.000000E+00 7.509721E-17 0.000000E+00 1.46869E+01
Exit Status
Inform Description
1 Solved To Acceptable Level
--------------------------------------------------------------------------------
Simulating trajectory traj
Done simulating trajectory traj
Launch angle: 46.8229 deg
Flight angle at end of propulsion: 38.3886 deg
Empty mass: 0.1891 kg
Water volume: 1.0273 L
Maximum range: 85.0967 m
Maximum height: 23.0818 m
Maximum velocity: 41.3029 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.