# Cannonball Implicit Duration#

This example demonstrates determining the duration of a phase using a nonlinear solver to satisfy an endpoint condition. As in the multi-phase cannonball problem, we will simultaneously find the optimal design for the cannonball (its radius) and the optimal flight profile (its launch angle). However, here we will use a single phase that terminates at the condition

(96)#\begin{align} h(t_f) = 0 \end{align}

## The cannonball sizing component#

This component computes the area (needed to compute drag) and mass (needed to compute energy) of a cannonball of a given radius and density.

This component sits upstream of the trajectory model and feeds its outputs to the trajectory as parameters.

import numpy as np
from scipy.interpolate import interp1d

import openmdao.api as om

import dymos as dm
from dymos.models.atmosphere.atmos_1976 import USatm1976Data

#############################################
# Component for the design part of the model
#############################################
class CannonballSizeComp(om.ExplicitComponent):
"""
Compute the area and mass of a cannonball with a given radius and density.

Notes
-----
This component is not vectorized with 'num_nodes' as is the usual way
with Dymos, but is instead intended to compute a scalar mass and reference
area from scalar radius and density inputs. This component does not reside
in the ODE but instead its outputs are connected to the trajectory via
input design parameters.
"""
def setup(self):

self.add_output(name='S', shape=(1,), desc='aerodynamic reference area', units='m**2')

self.declare_partials(of='mass', wrt='dens')

def compute(self, inputs, outputs):
dens = inputs['dens']

outputs['mass'] = (4/3.) * dens * np.pi * radius ** 3
outputs['S'] = np.pi * radius ** 2

def compute_partials(self, inputs, partials):
dens = inputs['dens']

partials['mass', 'dens'] = (4/3.) * np.pi * radius ** 3
partials['mass', 'radius'] = 4. * dens * np.pi * radius ** 2

## The cannonball ODE component#

This component computes the state rates and the kinetic energy of the cannonball. By calling the declare_coloring method wrt all inputs and using method 'cs', we’re telling OpenMDAO to automatically determine the sparsity pattern of the outputs with respect to the inputs, and to automatically compute those outputs using complex-step approximation.

class CannonballODE(om.ExplicitComponent):
"""
Cannonball ODE assuming flat earth and accounting for air resistance
"""

def initialize(self):
self.options.declare('num_nodes', types=int)

def setup(self):
nn = self.options['num_nodes']

# static parameters
# 0.5 good assumption for a sphere

# time varying inputs

# state rates

# Ask OpenMDAO to compute the partial derivatives using complex-step
# with a partial coloring algorithm for improved performance, and use
# a graph coloring algorithm to automatically detect the sparsity pattern.
self.declare_coloring(wrt='*', method='cs')

alt_data = USatm1976Data.alt * om.unit_conversion('ft', 'm')[0]
rho_data = USatm1976Data.rho * om.unit_conversion('slug/ft**3', 'kg/m**3')[0]
self.rho_interp = interp1d(np.array(alt_data, dtype=complex),
np.array(rho_data, dtype=complex),
kind='linear')

def compute(self, inputs, outputs):

gam = inputs['gam']
v = inputs['v']
h = inputs['h']
m = inputs['m']
S = inputs['S']
CD = inputs['CD']

GRAVITY = 9.80665  # m/s**2

# handle complex-step gracefully from the interpolant
if np.iscomplexobj(h):
rho = self.rho_interp(inputs['h'])
else:
rho = self.rho_interp(inputs['h']).real

q = 0.5*rho*inputs['v']**2
qS = q * S
D = qS * CD
cgam = np.cos(gam)
sgam = np.sin(gam)
outputs['v_dot'] = - D/m-GRAVITY*sgam
outputs['gam_dot'] = -(GRAVITY/v)*cgam
outputs['h_dot'] = v*sgam
outputs['r_dot'] = v*cgam
outputs['ke'] = 0.5*m*v**2

## Generating an initial guess#

Since we will be using a nonlinear solver to converge the dynamics and the duration, a simple linear initial guess will not be sufficient. The initial guess we will use is generated from the kinematics equations that assume no atmospheric drag. We compute the state values for some duration and initial flight path angle

def initial_guess(t_dur, gam0=np.pi/3):
t = np.linspace(0, t_dur, num=100)
g = -9.80665
v0 = -0.5*g*t_dur/np.sin(gam0)
r = v0*np.cos(gam0)*t
h = v0*np.sin(gam0)*t + 0.5*g*t**2
v = np.sqrt(v0*np.cos(gam0)**2 + (v0*np.sin(gam0) + g*t)**2)
gam = np.arctan2(v0*np.sin(gam0) + g*t, v0*np.cos(gam0)**2)

guess = {'t': t, 'r': r, 'h': h, 'v': v, 'gam': gam}

return guess

## Building and running the problem#

The following code defines the components for the physical cannonball calculations and ODE problem, sets up trajectory using a single phase, and specifies the stopping condition for the phase. The initial flight path angle is free, since 45 degrees is not necessarily optimal once air resistance is taken into account.

p = om.Problem(name='cannonball_implicit_duration', model=om.Group())

p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'IPOPT'
p.driver.declare_coloring()

p.driver.opt_settings['derivative_test'] = 'first-order'
p.driver.opt_settings['mu_strategy'] = 'monotone'
p.driver.opt_settings['alpha_for_y'] = 'safer-min-dual-infeas'
p.driver.opt_settings['bound_mult_init_method'] = 'mu-based'
p.driver.opt_settings['mu_init'] = 0.01

p.set_solver_print(level=0, depth=99)

p.model.set_input_defaults('dens', val=7.87, units='g/cm**3')
ref0=0.01, ref=0.10, units='m')

phase = dm.Phase(ode_class=CannonballODE, transcription=transcription)

# Duration is bounded to be greater than 1 to ensure the nonlinear solve
# does not converge to the initial point.
phase.set_time_options(fix_initial=True, duration_bounds=(1, 100), units='s')

# All initial states except flight path angle are fixed
# The output of the ODE which provides the rate source for each state
# is obtained from the tags used on those outputs in the ODE.
# The units of the states are automatically inferred by multiplying the units
# of those rates by the time units.

upper=400000, lower=0, ref=100000)

# A duration balance is added setting altitude to zero.
# A nonlinear solver is used to find the duration of required to satisfy the condition.
phase.set_duration_balance('h', val=0.0)

# Prior to setup, this phase has the default NonlinearRunOnce and LinearRunOnce solvers.
# Because of the implicit behavior introduced by set_duration_balance, it will automatically
# use a NewtonSolver with an Armijo Goldstein linesearch.
# In this case that linesearch causes extrapolation of the atmospheric model into invalid regions.
# To account for this, we explicitly assign a NewtonSolver with a different linesearch.
phase.nonlinear_solver = om.NewtonSolver(iprint=0, solve_subsystems=True)
phase.nonlinear_solver.linesearch = None

p.model.connect('size_comp.mass', 'traj.phase.parameters:m')
p.model.connect('size_comp.S', 'traj.phase.parameters:S')

# Finish Problem Setup
p.setup()

#############################################
# Set constants and initial guesses
#############################################
p.set_val('dens', 7.87, units='g/cm**3')

p.set_val('traj.phase.parameters:CD', 0.5)

guess = initial_guess(t_dur=30.0)

phase.set_time_val(initial=0.0, duration=30.0)

phase.set_state_val('r', vals=guess['r'], time_vals=guess['t'])
phase.set_state_val('h', vals=guess['h'], time_vals=guess['t'])
phase.set_state_val('v', vals=guess['v'], time_vals=guess['t'])

#####################################################
# Run the optimization and final explicit simulation
#####################################################
dm.run_problem(p, simulate=True, make_plots=True)
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/dymos/transcriptions/transcription_base.py:390: OpenMDAOWarning:'traj.phases.phase' <class Phase>
Non-default solvers are required
implicit duration: True
solved segments: True
input initial: False
Setting traj.phases.phase.linear_solver = om.DirectSolver(iprint=2)
Explicitly set traj.phases.phase.linear_solver to override.
Set traj.phases.phase.options["auto_solvers"] = False to disable this behavior.
--- Constraint Report [traj] ---
--- phase ---
[initial] 0.0000e+00 <= ke <= 4.0000e+05  [J]

'rhs_checking' is disabled for 'DirectSolver in 'traj.phases.phase' <class Phase>' 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.

Coloring for 'traj.phases.phase.rhs_all' (class CannonballODE)

Jacobian shape: (500, 303)  (0.92% nonzero)
FWD solves: 6   REV solves: 0
Total colors vs. total size: 6 vs 303  (98.02% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity:   0.0895 sec
Time to compute coloring:   0.0303 sec
Memory to compute coloring:   0.5000 MB
Full total jacobian for problem 'cannonball_implicit_duration' was computed 3 times, taking 0.26112154399993415 seconds.
Total jacobian shape: (98, 100)

Jacobian shape: (98, 100)  (24.16% nonzero)
FWD solves: 27   REV solves: 0
Total colors vs. total size: 27 vs 100  (73.00% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity:   0.2611 sec
Time to compute coloring:   0.0794 sec
Memory to compute coloring:   2.1445 MB
Coloring created on: 2024-08-29 21:46:37
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc

Solution:
--------------------------------------------------------------------------------
Total Time:                   16.8610
User Objective Time :      14.2530
User Sensitivity Time :     2.2917
Interface Time :            0.1690
Opt Solver Time:            0.1472
Calls to Objective Function :     239
Calls to Sens Function :           55

Objectives
Index  Name                           Value
0  traj.phase.states:r    -3.182962E+03

Variables (c - continuous, i - integer, d - discrete)
Index  Name                       Type      Lower Bound            Value      Upper Bound     Status
0  radius_0                      c     0.000000E+00     3.540939E-01     1.000000E+00
1  traj.phase.t_duration_0       c     1.000000E+00     1.000000E+02     1.000000E+02          u
2  traj.phase.states:r_0         c    -1.000000E+21     4.358669E+02     1.000000E+21
3  traj.phase.states:r_1         c    -1.000000E+21     7.590741E+02     1.000000E+21
4  traj.phase.states:r_2         c    -1.000000E+21     1.017857E+03     1.000000E+21
5  traj.phase.states:r_3         c    -1.000000E+21     1.234831E+03     1.000000E+21
6  traj.phase.states:r_4         c    -1.000000E+21     1.422453E+03     1.000000E+21
7  traj.phase.states:r_5         c    -1.000000E+21     1.588316E+03     1.000000E+21
8  traj.phase.states:r_6         c    -1.000000E+21     1.737381E+03     1.000000E+21
9  traj.phase.states:r_7         c    -1.000000E+21     1.873054E+03     1.000000E+21
10  traj.phase.states:r_8         c    -1.000000E+21     1.997754E+03     1.000000E+21
11  traj.phase.states:r_9         c    -1.000000E+21     2.113240E+03     1.000000E+21
12  traj.phase.states:r_10        c    -1.000000E+21     2.220813E+03     1.000000E+21
13  traj.phase.states:r_11        c    -1.000000E+21     2.321446E+03     1.000000E+21
14  traj.phase.states:r_12        c    -1.000000E+21     2.415862E+03     1.000000E+21
15  traj.phase.states:r_13        c    -1.000000E+21     2.504609E+03     1.000000E+21
16  traj.phase.states:r_14        c    -1.000000E+21     2.588097E+03     1.000000E+21
17  traj.phase.states:r_15        c    -1.000000E+21     2.666639E+03     1.000000E+21
18  traj.phase.states:r_16        c    -1.000000E+21     2.740483E+03     1.000000E+21
19  traj.phase.states:r_17        c    -1.000000E+21     2.809828E+03     1.000000E+21
20  traj.phase.states:r_18        c    -1.000000E+21     2.874850E+03     1.000000E+21
21  traj.phase.states:r_19        c    -1.000000E+21     2.935708E+03     1.000000E+21
22  traj.phase.states:r_20        c    -1.000000E+21     2.992555E+03     1.000000E+21
23  traj.phase.states:r_21        c    -1.000000E+21     3.045544E+03     1.000000E+21
24  traj.phase.states:r_22        c    -1.000000E+21     3.094833E+03     1.000000E+21
25  traj.phase.states:r_23        c    -1.000000E+21     3.140584E+03     1.000000E+21
26  traj.phase.states:h_0         c    -1.000000E+21     2.675246E+02     1.000000E+21
27  traj.phase.states:h_1         c    -1.000000E+21     4.559862E+02     1.000000E+21
28  traj.phase.states:h_2         c    -1.000000E+21     5.966683E+02     1.000000E+21
29  traj.phase.states:h_3         c    -1.000000E+21     7.042126E+02     1.000000E+21
30  traj.phase.states:h_4         c    -1.000000E+21     7.866604E+02     1.000000E+21
31  traj.phase.states:h_5         c    -1.000000E+21     8.488963E+02     1.000000E+21
32  traj.phase.states:h_6         c    -1.000000E+21     8.941007E+02     1.000000E+21
33  traj.phase.states:h_7         c    -1.000000E+21     9.244542E+02     1.000000E+21
34  traj.phase.states:h_8         c    -1.000000E+21     9.415152E+02     1.000000E+21
35  traj.phase.states:h_9         c    -1.000000E+21     9.464412E+02     1.000000E+21
36  traj.phase.states:h_10        c    -1.000000E+21     9.401270E+02     1.000000E+21
37  traj.phase.states:h_11        c    -1.000000E+21     9.232973E+02     1.000000E+21
38  traj.phase.states:h_12        c    -1.000000E+21     8.965701E+02     1.000000E+21
39  traj.phase.states:h_13        c    -1.000000E+21     8.605033E+02     1.000000E+21
40  traj.phase.states:h_14        c    -1.000000E+21     8.156262E+02     1.000000E+21
41  traj.phase.states:h_15        c    -1.000000E+21     7.624607E+02     1.000000E+21
42  traj.phase.states:h_16        c    -1.000000E+21     7.015341E+02     1.000000E+21
43  traj.phase.states:h_17        c    -1.000000E+21     6.333833E+02     1.000000E+21
44  traj.phase.states:h_18        c    -1.000000E+21     5.585557E+02     1.000000E+21
45  traj.phase.states:h_19        c    -1.000000E+21     4.776046E+02     1.000000E+21
46  traj.phase.states:h_20        c    -1.000000E+21     3.910839E+02     1.000000E+21
47  traj.phase.states:h_21        c    -1.000000E+21     2.995404E+02     1.000000E+21
48  traj.phase.states:h_22        c    -1.000000E+21     2.035072E+02     1.000000E+21
49  traj.phase.states:h_23        c    -1.000000E+21     1.034977E+02     1.000000E+21
50  traj.phase.states:gam_0       c    -1.000000E+21     5.588424E-01     1.000000E+21
51  traj.phase.states:gam_1       c    -1.000000E+21     5.398186E-01     1.000000E+21
52  traj.phase.states:gam_2       c    -1.000000E+21     5.135922E-01     1.000000E+21
53  traj.phase.states:gam_3       c    -1.000000E+21     4.797863E-01     1.000000E+21
54  traj.phase.states:gam_4       c    -1.000000E+21     4.379029E-01     1.000000E+21
55  traj.phase.states:gam_5       c    -1.000000E+21     3.873723E-01     1.000000E+21
56  traj.phase.states:gam_6       c    -1.000000E+21     3.276275E-01     1.000000E+21
57  traj.phase.states:gam_7       c    -1.000000E+21     2.582142E-01     1.000000E+21
58  traj.phase.states:gam_8       c    -1.000000E+21     1.789451E-01     1.000000E+21
59  traj.phase.states:gam_9       c    -1.000000E+21     9.009145E-02     1.000000E+21
60  traj.phase.states:gam_10      c    -1.000000E+21    -7.423020E-03     1.000000E+21
61  traj.phase.states:gam_11      c    -1.000000E+21    -1.118970E-01     1.000000E+21
62  traj.phase.states:gam_12      c    -1.000000E+21    -2.208805E-01     1.000000E+21
63  traj.phase.states:gam_13      c    -1.000000E+21    -3.314120E-01     1.000000E+21
64  traj.phase.states:gam_14      c    -1.000000E+21    -4.404248E-01     1.000000E+21
65  traj.phase.states:gam_15      c    -1.000000E+21    -5.451909E-01     1.000000E+21
66  traj.phase.states:gam_16      c    -1.000000E+21    -6.436480E-01     1.000000E+21
67  traj.phase.states:gam_17      c    -1.000000E+21    -7.345244E-01     1.000000E+21
68  traj.phase.states:gam_18      c    -1.000000E+21    -8.172791E-01     1.000000E+21
69  traj.phase.states:gam_19      c    -1.000000E+21    -8.919361E-01     1.000000E+21
70  traj.phase.states:gam_20      c    -1.000000E+21    -9.588926E-01     1.000000E+21
71  traj.phase.states:gam_21      c    -1.000000E+21    -1.018753E+00     1.000000E+21
72  traj.phase.states:gam_22      c    -1.000000E+21    -1.072207E+00     1.000000E+21
73  traj.phase.states:gam_23      c    -1.000000E+21    -1.119950E+00     1.000000E+21
74  traj.phase.states:gam_24      c    -1.000000E+21    -1.162642E+00     1.000000E+21
75  traj.phase.states:v_0         c    -1.000000E+21     5.750200E+02     1.000000E+21
76  traj.phase.states:v_1         c    -1.000000E+21     3.997227E+02     1.000000E+21
77  traj.phase.states:v_2         c    -1.000000E+21     3.060110E+02     1.000000E+21
78  traj.phase.states:v_3         c    -1.000000E+21     2.471802E+02     1.000000E+21
79  traj.phase.states:v_4         c    -1.000000E+21     2.066205E+02     1.000000E+21
80  traj.phase.states:v_5         c    -1.000000E+21     1.769413E+02     1.000000E+21
81  traj.phase.states:v_6         c    -1.000000E+21     1.543746E+02     1.000000E+21
82  traj.phase.states:v_7         c    -1.000000E+21     1.368146E+02     1.000000E+21
83  traj.phase.states:v_8         c    -1.000000E+21     1.230084E+02     1.000000E+21
84  traj.phase.states:v_9         c    -1.000000E+21     1.121738E+02     1.000000E+21
85  traj.phase.states:v_10        c    -1.000000E+21     1.037991E+02     1.000000E+21
86  traj.phase.states:v_11        c    -1.000000E+21     9.752646E+01     1.000000E+21
87  traj.phase.states:v_12        c    -1.000000E+21     9.307779E+01     1.000000E+21
88  traj.phase.states:v_13        c    -1.000000E+21     9.020964E+01     1.000000E+21
89  traj.phase.states:v_14        c    -1.000000E+21     8.868763E+01     1.000000E+21
90  traj.phase.states:v_15        c    -1.000000E+21     8.827834E+01     1.000000E+21
91  traj.phase.states:v_16        c    -1.000000E+21     8.875205E+01     1.000000E+21
92  traj.phase.states:v_17        c    -1.000000E+21     8.989109E+01     1.000000E+21
93  traj.phase.states:v_18        c    -1.000000E+21     9.149866E+01     1.000000E+21
94  traj.phase.states:v_19        c    -1.000000E+21     9.340455E+01     1.000000E+21
95  traj.phase.states:v_20        c    -1.000000E+21     9.546736E+01     1.000000E+21
96  traj.phase.states:v_21        c    -1.000000E+21     9.757524E+01     1.000000E+21
97  traj.phase.states:v_22        c    -1.000000E+21     9.964293E+01     1.000000E+21
98  traj.phase.states:v_23        c    -1.000000E+21     1.016075E+02     1.000000E+21
99  traj.phase.states:v_24        c    -1.000000E+21     1.034255E+02     1.000000E+21

Constraints (i - inequality, e - equality)
Index  Name                                               Type          Lower           Value           Upper    Status  Lagrange Multiplier (N/A)
0  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -9.663381E-12    0.000000E+00              9.00000E+100
1  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -1.284661E-11    0.000000E+00              9.00000E+100
2  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -1.136868E-11    0.000000E+00              9.00000E+100
3  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -1.000444E-11    0.000000E+00              9.00000E+100
4  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -8.412826E-12    0.000000E+00              9.00000E+100
5  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -7.275958E-12    0.000000E+00              9.00000E+100
6  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -6.366463E-12    0.000000E+00              9.00000E+100
7  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -5.002221E-12    0.000000E+00              9.00000E+100
8  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -4.320100E-12    0.000000E+00              9.00000E+100
9  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -5.002221E-12    0.000000E+00              9.00000E+100
10  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -3.637979E-12    0.000000E+00              9.00000E+100
11  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -2.728484E-12    0.000000E+00              9.00000E+100
12  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -2.728484E-12    0.000000E+00              9.00000E+100
13  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -2.273737E-12    0.000000E+00              9.00000E+100
14  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -3.183231E-12    0.000000E+00              9.00000E+100
15  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -3.183231E-12    0.000000E+00              9.00000E+100
16  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -4.547474E-13    0.000000E+00              9.00000E+100
17  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -4.547474E-13    0.000000E+00              9.00000E+100
18  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -2.273737E-12    0.000000E+00              9.00000E+100
19  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -1.364242E-12    0.000000E+00              9.00000E+100
20  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -9.094947E-13    0.000000E+00              9.00000E+100
21  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00    4.547474E-13    0.000000E+00              9.00000E+100
22  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00   -4.547474E-13    0.000000E+00              9.00000E+100
23  traj.phase.continuity_comp.defect_states:r            e   0.000000E+00    4.547474E-13    0.000000E+00              9.00000E+100
24  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -2.330580E-11    0.000000E+00              9.00000E+100
25  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -1.483613E-11    0.000000E+00              9.00000E+100
26  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -9.549694E-12    0.000000E+00              9.00000E+100
27  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -6.139089E-12    0.000000E+00              9.00000E+100
28  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -3.751666E-12    0.000000E+00              9.00000E+100
29  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -2.273737E-12    0.000000E+00              9.00000E+100
30  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -1.364242E-12    0.000000E+00              9.00000E+100
31  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00   -7.958079E-13    0.000000E+00              9.00000E+100
32  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    2.273737E-13    0.000000E+00              9.00000E+100
33  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    4.547474E-13    0.000000E+00              9.00000E+100
34  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    3.410605E-13    0.000000E+00              9.00000E+100
35  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    7.958079E-13    0.000000E+00              9.00000E+100
36  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    7.958079E-13    0.000000E+00              9.00000E+100
37  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    1.591616E-12    0.000000E+00              9.00000E+100
38  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    2.273737E-12    0.000000E+00              9.00000E+100
39  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    2.955858E-12    0.000000E+00              9.00000E+100
40  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    3.524292E-12    0.000000E+00              9.00000E+100
41  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    3.979039E-12    0.000000E+00              9.00000E+100
42  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    4.433787E-12    0.000000E+00              9.00000E+100
43  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    5.002221E-12    0.000000E+00              9.00000E+100
44  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    5.343281E-12    0.000000E+00              9.00000E+100
45  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    5.513812E-12    0.000000E+00              9.00000E+100
46  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    5.883294E-12    0.000000E+00              9.00000E+100
47  traj.phase.continuity_comp.defect_states:h            e   0.000000E+00    6.195933E-12    0.000000E+00              9.00000E+100
48  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.887379E-15    0.000000E+00              9.00000E+100
49  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.776357E-15    0.000000E+00              9.00000E+100
50  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    2.442491E-15    0.000000E+00              9.00000E+100
51  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    3.164136E-15    0.000000E+00              9.00000E+100
52  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    4.274359E-15    0.000000E+00              9.00000E+100
53  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    5.884182E-15    0.000000E+00              9.00000E+100
54  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    7.882583E-15    0.000000E+00              9.00000E+100
55  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    9.547918E-15    0.000000E+00              9.00000E+100
56  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.193490E-14    0.000000E+00              9.00000E+100
57  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.283956E-14    0.000000E+00              9.00000E+100
58  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.300349E-14    0.000000E+00              9.00000E+100
59  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.204592E-14    0.000000E+00              9.00000E+100
60  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    9.936496E-15    0.000000E+00              9.00000E+100
61  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    7.216450E-15    0.000000E+00              9.00000E+100
62  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    4.884981E-15    0.000000E+00              9.00000E+100
63  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    2.331468E-15    0.000000E+00              9.00000E+100
64  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    1.776357E-15    0.000000E+00              9.00000E+100
65  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    6.661338E-16    0.000000E+00              9.00000E+100
66  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    5.551115E-16    0.000000E+00              9.00000E+100
67  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    3.330669E-16    0.000000E+00              9.00000E+100
68  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    6.661338E-16    0.000000E+00              9.00000E+100
69  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00   -4.440892E-16    0.000000E+00              9.00000E+100
70  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    2.220446E-16    0.000000E+00              9.00000E+100
71  traj.phase.continuity_comp.defect_states:gam          e   0.000000E+00    4.440892E-16    0.000000E+00              9.00000E+100
72  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    2.273737E-11    0.000000E+00              9.00000E+100
73  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    8.526513E-12    0.000000E+00              9.00000E+100
74  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    4.263256E-12    0.000000E+00              9.00000E+100
75  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    2.444267E-12    0.000000E+00              9.00000E+100
76  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    1.676881E-12    0.000000E+00              9.00000E+100
77  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    1.165290E-12    0.000000E+00              9.00000E+100
78  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    7.958079E-13    0.000000E+00              9.00000E+100
79  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    5.968559E-13    0.000000E+00              9.00000E+100
80  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    3.552714E-13    0.000000E+00              9.00000E+100
81  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    1.563194E-13    0.000000E+00              9.00000E+100
82  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    2.842171E-14    0.000000E+00              9.00000E+100
83  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -5.684342E-14    0.000000E+00              9.00000E+100
84  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -1.563194E-13    0.000000E+00              9.00000E+100
85  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -1.847411E-13    0.000000E+00              9.00000E+100
86  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -2.273737E-13    0.000000E+00              9.00000E+100
87  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -1.847411E-13    0.000000E+00              9.00000E+100
88  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -9.947598E-14    0.000000E+00              9.00000E+100
89  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -7.105427E-14    0.000000E+00              9.00000E+100
90  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00   -1.421085E-14    0.000000E+00              9.00000E+100
91  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    7.105427E-14    0.000000E+00              9.00000E+100
92  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    1.136868E-13    0.000000E+00              9.00000E+100
93  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    1.705303E-13    0.000000E+00              9.00000E+100
94  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    2.557954E-13    0.000000E+00              9.00000E+100
95  traj.phase.continuity_comp.defect_states:v            e   0.000000E+00    2.415845E-13    0.000000E+00              9.00000E+100
96  traj.phases.phase->initial_boundary_constraint->ke    i   0.000000E+00    4.000000E+00    4.000000E+00         u    9.00000E+100

Exit Status
Inform  Description
0  Solve Succeeded
--------------------------------------------------------------------------------
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
Problem: cannonball_implicit_duration
Driver:  pyOptSparseDriver
success     : True
iterations  : 240
runtime     : 1.7554E+01 s
model_evals : 240
model_time  : 1.4168E+01 s
deriv_evals : 55
deriv_time  : 2.0047E+00 s
exit_status : SUCCESS

## Displaying the timeseries data#

from IPython.display import HTML

# Define the path to the HTML file
html_file_path = p.get_reports_dir() / 'traj_results_report.html'