Double Integrator#
In the double integrator problem, we seek to maximize the distance traveled by a block (that starts and ends at rest) sliding without friction along a horizontal surface, with acceleration as the control.
We minimize the final time, \(t_f\), by varying the dynamic control, \(u\), subject to the dynamics:
(45)#\[\begin{align}
\frac{dx}{dt} &= v \\
\frac{dv}{dt} &= u
\end{align}\]
The initial conditions are
(46)#\[\begin{align}
x_0 &= 0 \\
v_0 &= 0
\end{align}\]
and the final conditions are
(47)#\[\begin{align}
x_f &= \rm{free} \\
v_f &= 0
\end{align}\]
The control \(u\) is constrained to fall between -1 and 1. Due to the fact that the control appears linearly in the equations of motion, we should expect bang-bang behavior in the control (alternation between its extreme values).
The ODE System: double_integrator_ode.py#
This problem is unique in that we do not actually have to calculate anything in the Dymos formulation of the ODE. We create an ExplicitComponent and provide it with the num_nodes option, but it has no inputs and no outputs. The rates for the states are entirely provided by the other states and controls.
class DoubleIntegratorODE(om.ExplicitComponent):
"""
The double integrator is a special case where the state rates are all set to other states
or parameters. Since we aren't computing any other outputs, the ODE doesn't actually
need to compute anything. OpenMDAO will warn us that the component has no outputs, but
Dymos will solve the problem just fine.
Note we still have to declare the num_nodes option in initialize so that Dymos can instantiate
the ODE.
Also note that neither time, states, nor parameters have targets, since there are no inputs
in the ODE system.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
Building and running the problem#
In order to facilitate the bang-bang behavior in the control, we disable continuity and rate continuity in the control value.
Show code cell outputs
import openmdao.api as om
class DoubleIntegratorODE(om.ExplicitComponent):
"""
The double integrator is a special case where the state rates are all set to other states
or parameters. Since we aren't computing any other outputs, the ODE doesn't actually
need to compute anything. OpenMDAO will warn us that the component has no outputs, but
Dymos will solve the problem just fine.
Note we still have to declare the num_nodes option in initialize so that Dymos can instantiate
the ODE.
Also note that neither time, states, nor parameters have targets, since there are no inputs
in the ODE system.
"""
def initialize(self):
self.options.declare('num_nodes', types=int)
import matplotlib.pyplot as plt
import openmdao.api as om
import dymos as dm
from dymos.examples.plotting import plot_results
# Initialize the problem and assign the driver
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()
# Setup the trajectory and its phase
traj = p.model.add_subsystem('traj', dm.Trajectory())
transcription = dm.Radau(num_segments=30, order=3, compressed=False)
phase = traj.add_phase('phase0',
dm.Phase(ode_class=DoubleIntegratorODE, transcription=transcription))
#
# Set the options for our variables.
#
phase.set_time_options(fix_initial=True, fix_duration=True, units='s')
phase.add_state('v', fix_initial=True, fix_final=True, rate_source='u', units='m/s')
phase.add_state('x', fix_initial=True, rate_source='v', units='m')
phase.add_control('u', units='m/s**2', scaler=0.01, continuity=False, rate_continuity=False,
rate2_continuity=False, shape=(1, ), lower=-1.0, upper=1.0)
#
# Maximize distance travelled.
#
phase.add_objective('x', loc='final', scaler=-1)
p.model.linear_solver = om.DirectSolver()
#
# Setup the problem and set our initial values.
#
p.setup(check=True)
phase.set_time_val(initial=0.0, duration=1.0)
phase.set_state_val('x', [0, 0.25])
phase.set_state_val('v', [0, 0])
phase.set_control_val('u', [1, -1])
#
# Solve the problem.
#
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
--- phase0 ---
None
'rhs_checking' is disabled for 'DirectSolver in <model> <class Group>' but that solver has redundant adjoint solves. If it is expensive to compute derivatives for this solver, turning on 'rhs_checking' may improve performance.
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
traj.phases.phase0.rhs_all
INFO: checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
traj.phases.phase0.rhs_all
INFO: checking auto_ivc_warnings
Full total jacobian for problem 'problem' was computed 3 times, taking 0.058919000000059896 seconds.
Total jacobian shape: (181, 327)
Jacobian shape: (181, 327) (1.51% nonzero)
FWD solves: 0 REV solves: 6
Total colors vs. total size: 6 vs 181 (96.69% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.0589 sec
Time to compute coloring: 0.0835 sec
Memory to compute coloring: 0.2500 MB
Coloring created on: 2024-12-11 19:23:13
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 2.4170
User Objective Time : 0.0152
User Sensitivity Time : 0.6495
Interface Time : 0.0703
Opt Solver Time: 1.6820
Calls to Objective Function : 19
Calls to Sens Function : 19
Objectives
Index Name Value
0 traj.phase0.states:x -2.500000E-01
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.phase0.states:v_0 c -1.000000E+21 1.183503E-02 1.000000E+21
1 traj.phase0.states:v_1 c -1.000000E+21 2.816497E-02 1.000000E+21
2 traj.phase0.states:v_2 c -1.000000E+21 3.333333E-02 1.000000E+21
3 traj.phase0.states:v_3 c -1.000000E+21 3.333333E-02 1.000000E+21
4 traj.phase0.states:v_4 c -1.000000E+21 4.516837E-02 1.000000E+21
5 traj.phase0.states:v_5 c -1.000000E+21 6.149830E-02 1.000000E+21
6 traj.phase0.states:v_6 c -1.000000E+21 6.666667E-02 1.000000E+21
7 traj.phase0.states:v_7 c -1.000000E+21 6.666667E-02 1.000000E+21
8 traj.phase0.states:v_8 c -1.000000E+21 7.850170E-02 1.000000E+21
9 traj.phase0.states:v_9 c -1.000000E+21 9.483163E-02 1.000000E+21
10 traj.phase0.states:v_10 c -1.000000E+21 1.000000E-01 1.000000E+21
11 traj.phase0.states:v_11 c -1.000000E+21 1.000000E-01 1.000000E+21
12 traj.phase0.states:v_12 c -1.000000E+21 1.118350E-01 1.000000E+21
13 traj.phase0.states:v_13 c -1.000000E+21 1.281650E-01 1.000000E+21
14 traj.phase0.states:v_14 c -1.000000E+21 1.333333E-01 1.000000E+21
15 traj.phase0.states:v_15 c -1.000000E+21 1.333333E-01 1.000000E+21
16 traj.phase0.states:v_16 c -1.000000E+21 1.451684E-01 1.000000E+21
17 traj.phase0.states:v_17 c -1.000000E+21 1.614983E-01 1.000000E+21
18 traj.phase0.states:v_18 c -1.000000E+21 1.666667E-01 1.000000E+21
19 traj.phase0.states:v_19 c -1.000000E+21 1.666667E-01 1.000000E+21
20 traj.phase0.states:v_20 c -1.000000E+21 1.785017E-01 1.000000E+21
21 traj.phase0.states:v_21 c -1.000000E+21 1.948316E-01 1.000000E+21
22 traj.phase0.states:v_22 c -1.000000E+21 2.000000E-01 1.000000E+21
23 traj.phase0.states:v_23 c -1.000000E+21 2.000000E-01 1.000000E+21
24 traj.phase0.states:v_24 c -1.000000E+21 2.118350E-01 1.000000E+21
25 traj.phase0.states:v_25 c -1.000000E+21 2.281650E-01 1.000000E+21
26 traj.phase0.states:v_26 c -1.000000E+21 2.333333E-01 1.000000E+21
27 traj.phase0.states:v_27 c -1.000000E+21 2.333333E-01 1.000000E+21
28 traj.phase0.states:v_28 c -1.000000E+21 2.451684E-01 1.000000E+21
29 traj.phase0.states:v_29 c -1.000000E+21 2.614983E-01 1.000000E+21
30 traj.phase0.states:v_30 c -1.000000E+21 2.666667E-01 1.000000E+21
31 traj.phase0.states:v_31 c -1.000000E+21 2.666667E-01 1.000000E+21
32 traj.phase0.states:v_32 c -1.000000E+21 2.785017E-01 1.000000E+21
33 traj.phase0.states:v_33 c -1.000000E+21 2.948316E-01 1.000000E+21
34 traj.phase0.states:v_34 c -1.000000E+21 3.000000E-01 1.000000E+21
35 traj.phase0.states:v_35 c -1.000000E+21 3.000000E-01 1.000000E+21
36 traj.phase0.states:v_36 c -1.000000E+21 3.118350E-01 1.000000E+21
37 traj.phase0.states:v_37 c -1.000000E+21 3.281650E-01 1.000000E+21
38 traj.phase0.states:v_38 c -1.000000E+21 3.333333E-01 1.000000E+21
39 traj.phase0.states:v_39 c -1.000000E+21 3.333333E-01 1.000000E+21
40 traj.phase0.states:v_40 c -1.000000E+21 3.451684E-01 1.000000E+21
41 traj.phase0.states:v_41 c -1.000000E+21 3.614983E-01 1.000000E+21
42 traj.phase0.states:v_42 c -1.000000E+21 3.666667E-01 1.000000E+21
43 traj.phase0.states:v_43 c -1.000000E+21 3.666667E-01 1.000000E+21
44 traj.phase0.states:v_44 c -1.000000E+21 3.785017E-01 1.000000E+21
45 traj.phase0.states:v_45 c -1.000000E+21 3.948316E-01 1.000000E+21
46 traj.phase0.states:v_46 c -1.000000E+21 4.000000E-01 1.000000E+21
47 traj.phase0.states:v_47 c -1.000000E+21 4.000000E-01 1.000000E+21
48 traj.phase0.states:v_48 c -1.000000E+21 4.118350E-01 1.000000E+21
49 traj.phase0.states:v_49 c -1.000000E+21 4.281650E-01 1.000000E+21
50 traj.phase0.states:v_50 c -1.000000E+21 4.333333E-01 1.000000E+21
51 traj.phase0.states:v_51 c -1.000000E+21 4.333333E-01 1.000000E+21
52 traj.phase0.states:v_52 c -1.000000E+21 4.451684E-01 1.000000E+21
53 traj.phase0.states:v_53 c -1.000000E+21 4.614983E-01 1.000000E+21
54 traj.phase0.states:v_54 c -1.000000E+21 4.666667E-01 1.000000E+21
55 traj.phase0.states:v_55 c -1.000000E+21 4.666667E-01 1.000000E+21
56 traj.phase0.states:v_56 c -1.000000E+21 4.785017E-01 1.000000E+21
57 traj.phase0.states:v_57 c -1.000000E+21 4.948316E-01 1.000000E+21
58 traj.phase0.states:v_58 c -1.000000E+21 5.000000E-01 1.000000E+21
59 traj.phase0.states:v_59 c -1.000000E+21 5.000000E-01 1.000000E+21
60 traj.phase0.states:v_60 c -1.000000E+21 4.881650E-01 1.000000E+21
61 traj.phase0.states:v_61 c -1.000000E+21 4.718350E-01 1.000000E+21
62 traj.phase0.states:v_62 c -1.000000E+21 4.666667E-01 1.000000E+21
63 traj.phase0.states:v_63 c -1.000000E+21 4.666667E-01 1.000000E+21
64 traj.phase0.states:v_64 c -1.000000E+21 4.548316E-01 1.000000E+21
65 traj.phase0.states:v_65 c -1.000000E+21 4.385017E-01 1.000000E+21
66 traj.phase0.states:v_66 c -1.000000E+21 4.333333E-01 1.000000E+21
67 traj.phase0.states:v_67 c -1.000000E+21 4.333333E-01 1.000000E+21
68 traj.phase0.states:v_68 c -1.000000E+21 4.214983E-01 1.000000E+21
69 traj.phase0.states:v_69 c -1.000000E+21 4.051684E-01 1.000000E+21
70 traj.phase0.states:v_70 c -1.000000E+21 4.000000E-01 1.000000E+21
71 traj.phase0.states:v_71 c -1.000000E+21 4.000000E-01 1.000000E+21
72 traj.phase0.states:v_72 c -1.000000E+21 3.881650E-01 1.000000E+21
73 traj.phase0.states:v_73 c -1.000000E+21 3.718350E-01 1.000000E+21
74 traj.phase0.states:v_74 c -1.000000E+21 3.666667E-01 1.000000E+21
75 traj.phase0.states:v_75 c -1.000000E+21 3.666667E-01 1.000000E+21
76 traj.phase0.states:v_76 c -1.000000E+21 3.548316E-01 1.000000E+21
77 traj.phase0.states:v_77 c -1.000000E+21 3.385017E-01 1.000000E+21
78 traj.phase0.states:v_78 c -1.000000E+21 3.333333E-01 1.000000E+21
79 traj.phase0.states:v_79 c -1.000000E+21 3.333333E-01 1.000000E+21
80 traj.phase0.states:v_80 c -1.000000E+21 3.214983E-01 1.000000E+21
81 traj.phase0.states:v_81 c -1.000000E+21 3.051684E-01 1.000000E+21
82 traj.phase0.states:v_82 c -1.000000E+21 3.000000E-01 1.000000E+21
83 traj.phase0.states:v_83 c -1.000000E+21 3.000000E-01 1.000000E+21
84 traj.phase0.states:v_84 c -1.000000E+21 2.881650E-01 1.000000E+21
85 traj.phase0.states:v_85 c -1.000000E+21 2.718350E-01 1.000000E+21
86 traj.phase0.states:v_86 c -1.000000E+21 2.666667E-01 1.000000E+21
87 traj.phase0.states:v_87 c -1.000000E+21 2.666667E-01 1.000000E+21
88 traj.phase0.states:v_88 c -1.000000E+21 2.548316E-01 1.000000E+21
89 traj.phase0.states:v_89 c -1.000000E+21 2.385017E-01 1.000000E+21
90 traj.phase0.states:v_90 c -1.000000E+21 2.333333E-01 1.000000E+21
91 traj.phase0.states:v_91 c -1.000000E+21 2.333333E-01 1.000000E+21
92 traj.phase0.states:v_92 c -1.000000E+21 2.214983E-01 1.000000E+21
93 traj.phase0.states:v_93 c -1.000000E+21 2.051684E-01 1.000000E+21
94 traj.phase0.states:v_94 c -1.000000E+21 2.000000E-01 1.000000E+21
95 traj.phase0.states:v_95 c -1.000000E+21 2.000000E-01 1.000000E+21
96 traj.phase0.states:v_96 c -1.000000E+21 1.881650E-01 1.000000E+21
97 traj.phase0.states:v_97 c -1.000000E+21 1.718350E-01 1.000000E+21
98 traj.phase0.states:v_98 c -1.000000E+21 1.666667E-01 1.000000E+21
99 traj.phase0.states:v_99 c -1.000000E+21 1.666667E-01 1.000000E+21
100 traj.phase0.states:v_100 c -1.000000E+21 1.548316E-01 1.000000E+21
101 traj.phase0.states:v_101 c -1.000000E+21 1.385017E-01 1.000000E+21
102 traj.phase0.states:v_102 c -1.000000E+21 1.333333E-01 1.000000E+21
103 traj.phase0.states:v_103 c -1.000000E+21 1.333333E-01 1.000000E+21
104 traj.phase0.states:v_104 c -1.000000E+21 1.214983E-01 1.000000E+21
105 traj.phase0.states:v_105 c -1.000000E+21 1.051684E-01 1.000000E+21
106 traj.phase0.states:v_106 c -1.000000E+21 1.000000E-01 1.000000E+21
107 traj.phase0.states:v_107 c -1.000000E+21 1.000000E-01 1.000000E+21
108 traj.phase0.states:v_108 c -1.000000E+21 8.816497E-02 1.000000E+21
109 traj.phase0.states:v_109 c -1.000000E+21 7.183503E-02 1.000000E+21
110 traj.phase0.states:v_110 c -1.000000E+21 6.666667E-02 1.000000E+21
111 traj.phase0.states:v_111 c -1.000000E+21 6.666667E-02 1.000000E+21
112 traj.phase0.states:v_112 c -1.000000E+21 5.483163E-02 1.000000E+21
113 traj.phase0.states:v_113 c -1.000000E+21 3.850170E-02 1.000000E+21
114 traj.phase0.states:v_114 c -1.000000E+21 3.333333E-02 1.000000E+21
115 traj.phase0.states:v_115 c -1.000000E+21 3.333333E-02 1.000000E+21
116 traj.phase0.states:v_116 c -1.000000E+21 2.149830E-02 1.000000E+21
117 traj.phase0.states:v_117 c -1.000000E+21 5.168368E-03 1.000000E+21
118 traj.phase0.states:x_0 c -1.000000E+21 7.003402E-05 1.000000E+21
119 traj.phase0.states:x_1 c -1.000000E+21 3.966326E-04 1.000000E+21
120 traj.phase0.states:x_2 c -1.000000E+21 5.555556E-04 1.000000E+21
121 traj.phase0.states:x_3 c -1.000000E+21 5.555556E-04 1.000000E+21
122 traj.phase0.states:x_4 c -1.000000E+21 1.020091E-03 1.000000E+21
123 traj.phase0.states:x_5 c -1.000000E+21 1.891020E-03 1.000000E+21
124 traj.phase0.states:x_6 c -1.000000E+21 2.222222E-03 1.000000E+21
125 traj.phase0.states:x_7 c -1.000000E+21 2.222222E-03 1.000000E+21
126 traj.phase0.states:x_8 c -1.000000E+21 3.081259E-03 1.000000E+21
127 traj.phase0.states:x_9 c -1.000000E+21 4.496519E-03 1.000000E+21
128 traj.phase0.states:x_10 c -1.000000E+21 5.000000E-03 1.000000E+21
129 traj.phase0.states:x_11 c -1.000000E+21 5.000000E-03 1.000000E+21
130 traj.phase0.states:x_12 c -1.000000E+21 6.253537E-03 1.000000E+21
131 traj.phase0.states:x_13 c -1.000000E+21 8.213129E-03 1.000000E+21
132 traj.phase0.states:x_14 c -1.000000E+21 8.888889E-03 1.000000E+21
133 traj.phase0.states:x_15 c -1.000000E+21 8.888889E-03 1.000000E+21
134 traj.phase0.states:x_16 c -1.000000E+21 1.053693E-02 1.000000E+21
135 traj.phase0.states:x_17 c -1.000000E+21 1.304085E-02 1.000000E+21
136 traj.phase0.states:x_18 c -1.000000E+21 1.388889E-02 1.000000E+21
137 traj.phase0.states:x_19 c -1.000000E+21 1.388889E-02 1.000000E+21
138 traj.phase0.states:x_20 c -1.000000E+21 1.593143E-02 1.000000E+21
139 traj.phase0.states:x_21 c -1.000000E+21 1.897968E-02 1.000000E+21
140 traj.phase0.states:x_22 c -1.000000E+21 2.000000E-02 1.000000E+21
141 traj.phase0.states:x_23 c -1.000000E+21 2.000000E-02 1.000000E+21
142 traj.phase0.states:x_24 c -1.000000E+21 2.243704E-02 1.000000E+21
143 traj.phase0.states:x_25 c -1.000000E+21 2.602963E-02 1.000000E+21
144 traj.phase0.states:x_26 c -1.000000E+21 2.722222E-02 1.000000E+21
145 traj.phase0.states:x_27 c -1.000000E+21 2.722222E-02 1.000000E+21
146 traj.phase0.states:x_28 c -1.000000E+21 3.005376E-02 1.000000E+21
147 traj.phase0.states:x_29 c -1.000000E+21 3.419068E-02 1.000000E+21
148 traj.phase0.states:x_30 c -1.000000E+21 3.555556E-02 1.000000E+21
149 traj.phase0.states:x_31 c -1.000000E+21 3.555556E-02 1.000000E+21
150 traj.phase0.states:x_32 c -1.000000E+21 3.878160E-02 1.000000E+21
151 traj.phase0.states:x_33 c -1.000000E+21 4.346285E-02 1.000000E+21
152 traj.phase0.states:x_34 c -1.000000E+21 4.500000E-02 1.000000E+21
153 traj.phase0.states:x_35 c -1.000000E+21 4.500000E-02 1.000000E+21
154 traj.phase0.states:x_36 c -1.000000E+21 4.862054E-02 1.000000E+21
155 traj.phase0.states:x_37 c -1.000000E+21 5.384612E-02 1.000000E+21
156 traj.phase0.states:x_38 c -1.000000E+21 5.555556E-02 1.000000E+21
157 traj.phase0.states:x_39 c -1.000000E+21 5.555556E-02 1.000000E+21
158 traj.phase0.states:x_40 c -1.000000E+21 5.957060E-02 1.000000E+21
159 traj.phase0.states:x_41 c -1.000000E+21 6.534051E-02 1.000000E+21
160 traj.phase0.states:x_42 c -1.000000E+21 6.722222E-02 1.000000E+21
161 traj.phase0.states:x_43 c -1.000000E+21 6.722222E-02 1.000000E+21
162 traj.phase0.states:x_44 c -1.000000E+21 7.163177E-02 1.000000E+21
163 traj.phase0.states:x_45 c -1.000000E+21 7.794601E-02 1.000000E+21
164 traj.phase0.states:x_46 c -1.000000E+21 8.000000E-02 1.000000E+21
165 traj.phase0.states:x_47 c -1.000000E+21 8.000000E-02 1.000000E+21
166 traj.phase0.states:x_48 c -1.000000E+21 8.480405E-02 1.000000E+21
167 traj.phase0.states:x_49 c -1.000000E+21 9.166262E-02 1.000000E+21
168 traj.phase0.states:x_50 c -1.000000E+21 9.388889E-02 1.000000E+21
169 traj.phase0.states:x_51 c -1.000000E+21 9.388889E-02 1.000000E+21
170 traj.phase0.states:x_52 c -1.000000E+21 9.908744E-02 1.000000E+21
171 traj.phase0.states:x_53 c -1.000000E+21 1.064903E-01 1.000000E+21
172 traj.phase0.states:x_54 c -1.000000E+21 1.088889E-01 1.000000E+21
173 traj.phase0.states:x_55 c -1.000000E+21 1.088889E-01 1.000000E+21
174 traj.phase0.states:x_56 c -1.000000E+21 1.144819E-01 1.000000E+21
175 traj.phase0.states:x_57 c -1.000000E+21 1.224292E-01 1.000000E+21
176 traj.phase0.states:x_58 c -1.000000E+21 1.250000E-01 1.000000E+21
177 traj.phase0.states:x_59 c -1.000000E+21 1.250000E-01 1.000000E+21
178 traj.phase0.states:x_60 c -1.000000E+21 1.308475E-01 1.000000E+21
179 traj.phase0.states:x_61 c -1.000000E+21 1.386859E-01 1.000000E+21
180 traj.phase0.states:x_62 c -1.000000E+21 1.411111E-01 1.000000E+21
181 traj.phase0.states:x_63 c -1.000000E+21 1.411111E-01 1.000000E+21
182 traj.phase0.states:x_64 c -1.000000E+21 1.465641E-01 1.000000E+21
183 traj.phase0.states:x_65 c -1.000000E+21 1.538581E-01 1.000000E+21
184 traj.phase0.states:x_66 c -1.000000E+21 1.561111E-01 1.000000E+21
185 traj.phase0.states:x_67 c -1.000000E+21 1.561111E-01 1.000000E+21
186 traj.phase0.states:x_68 c -1.000000E+21 1.611696E-01 1.000000E+21
187 traj.phase0.states:x_69 c -1.000000E+21 1.679193E-01 1.000000E+21
188 traj.phase0.states:x_70 c -1.000000E+21 1.700000E-01 1.000000E+21
189 traj.phase0.states:x_71 c -1.000000E+21 1.700000E-01 1.000000E+21
190 traj.phase0.states:x_72 c -1.000000E+21 1.746640E-01 1.000000E+21
191 traj.phase0.states:x_73 c -1.000000E+21 1.808694E-01 1.000000E+21
192 traj.phase0.states:x_74 c -1.000000E+21 1.827778E-01 1.000000E+21
193 traj.phase0.states:x_75 c -1.000000E+21 1.827778E-01 1.000000E+21
194 traj.phase0.states:x_76 c -1.000000E+21 1.870473E-01 1.000000E+21
195 traj.phase0.states:x_77 c -1.000000E+21 1.927083E-01 1.000000E+21
196 traj.phase0.states:x_78 c -1.000000E+21 1.944444E-01 1.000000E+21
197 traj.phase0.states:x_79 c -1.000000E+21 1.944444E-01 1.000000E+21
198 traj.phase0.states:x_80 c -1.000000E+21 1.983194E-01 1.000000E+21
199 traj.phase0.states:x_81 c -1.000000E+21 2.034361E-01 1.000000E+21
200 traj.phase0.states:x_82 c -1.000000E+21 2.050000E-01 1.000000E+21
201 traj.phase0.states:x_83 c -1.000000E+21 2.050000E-01 1.000000E+21
202 traj.phase0.states:x_84 c -1.000000E+21 2.084805E-01 1.000000E+21
203 traj.phase0.states:x_85 c -1.000000E+21 2.130529E-01 1.000000E+21
204 traj.phase0.states:x_86 c -1.000000E+21 2.144444E-01 1.000000E+21
205 traj.phase0.states:x_87 c -1.000000E+21 2.144444E-01 1.000000E+21
206 traj.phase0.states:x_88 c -1.000000E+21 2.175304E-01 1.000000E+21
207 traj.phase0.states:x_89 c -1.000000E+21 2.215585E-01 1.000000E+21
208 traj.phase0.states:x_90 c -1.000000E+21 2.227778E-01 1.000000E+21
209 traj.phase0.states:x_91 c -1.000000E+21 2.227778E-01 1.000000E+21
210 traj.phase0.states:x_92 c -1.000000E+21 2.254693E-01 1.000000E+21
211 traj.phase0.states:x_93 c -1.000000E+21 2.289530E-01 1.000000E+21
212 traj.phase0.states:x_94 c -1.000000E+21 2.300000E-01 1.000000E+21
213 traj.phase0.states:x_95 c -1.000000E+21 2.300000E-01 1.000000E+21
214 traj.phase0.states:x_96 c -1.000000E+21 2.322970E-01 1.000000E+21
215 traj.phase0.states:x_97 c -1.000000E+21 2.352364E-01 1.000000E+21
216 traj.phase0.states:x_98 c -1.000000E+21 2.361111E-01 1.000000E+21
217 traj.phase0.states:x_99 c -1.000000E+21 2.361111E-01 1.000000E+21
218 traj.phase0.states:x_100 c -1.000000E+21 2.380136E-01 1.000000E+21
219 traj.phase0.states:x_101 c -1.000000E+21 2.404086E-01 1.000000E+21
220 traj.phase0.states:x_102 c -1.000000E+21 2.411111E-01 1.000000E+21
221 traj.phase0.states:x_103 c -1.000000E+21 2.411111E-01 1.000000E+21
222 traj.phase0.states:x_104 c -1.000000E+21 2.426191E-01 1.000000E+21
223 traj.phase0.states:x_105 c -1.000000E+21 2.444698E-01 1.000000E+21
224 traj.phase0.states:x_106 c -1.000000E+21 2.450000E-01 1.000000E+21
225 traj.phase0.states:x_107 c -1.000000E+21 2.450000E-01 1.000000E+21
226 traj.phase0.states:x_108 c -1.000000E+21 2.461135E-01 1.000000E+21
227 traj.phase0.states:x_109 c -1.000000E+21 2.474199E-01 1.000000E+21
228 traj.phase0.states:x_110 c -1.000000E+21 2.477778E-01 1.000000E+21
229 traj.phase0.states:x_111 c -1.000000E+21 2.477778E-01 1.000000E+21
230 traj.phase0.states:x_112 c -1.000000E+21 2.484967E-01 1.000000E+21
231 traj.phase0.states:x_113 c -1.000000E+21 2.492588E-01 1.000000E+21
232 traj.phase0.states:x_114 c -1.000000E+21 2.494444E-01 1.000000E+21
233 traj.phase0.states:x_115 c -1.000000E+21 2.494444E-01 1.000000E+21
234 traj.phase0.states:x_116 c -1.000000E+21 2.497689E-01 1.000000E+21
235 traj.phase0.states:x_117 c -1.000000E+21 2.499866E-01 1.000000E+21
236 traj.phase0.states:x_118 c -1.000000E+21 2.500000E-01 1.000000E+21
237 traj.phase0.controls:u_0 c -1.000000E-02 1.000000E-02 1.000000E-02 u
238 traj.phase0.controls:u_1 c -1.000000E-02 1.000000E-02 1.000000E-02 u
239 traj.phase0.controls:u_2 c -1.000000E-02 1.000000E-02 1.000000E-02 u
240 traj.phase0.controls:u_3 c -1.000000E-02 1.000000E-02 1.000000E-02 u
241 traj.phase0.controls:u_4 c -1.000000E-02 1.000000E-02 1.000000E-02 u
242 traj.phase0.controls:u_5 c -1.000000E-02 1.000000E-02 1.000000E-02 u
243 traj.phase0.controls:u_6 c -1.000000E-02 1.000000E-02 1.000000E-02 u
244 traj.phase0.controls:u_7 c -1.000000E-02 1.000000E-02 1.000000E-02 u
245 traj.phase0.controls:u_8 c -1.000000E-02 1.000000E-02 1.000000E-02 u
246 traj.phase0.controls:u_9 c -1.000000E-02 1.000000E-02 1.000000E-02 u
247 traj.phase0.controls:u_10 c -1.000000E-02 1.000000E-02 1.000000E-02 u
248 traj.phase0.controls:u_11 c -1.000000E-02 1.000000E-02 1.000000E-02 u
249 traj.phase0.controls:u_12 c -1.000000E-02 1.000000E-02 1.000000E-02 u
250 traj.phase0.controls:u_13 c -1.000000E-02 1.000000E-02 1.000000E-02 u
251 traj.phase0.controls:u_14 c -1.000000E-02 1.000000E-02 1.000000E-02 u
252 traj.phase0.controls:u_15 c -1.000000E-02 1.000000E-02 1.000000E-02 u
253 traj.phase0.controls:u_16 c -1.000000E-02 1.000000E-02 1.000000E-02 u
254 traj.phase0.controls:u_17 c -1.000000E-02 1.000000E-02 1.000000E-02 u
255 traj.phase0.controls:u_18 c -1.000000E-02 1.000000E-02 1.000000E-02 u
256 traj.phase0.controls:u_19 c -1.000000E-02 1.000000E-02 1.000000E-02 u
257 traj.phase0.controls:u_20 c -1.000000E-02 1.000000E-02 1.000000E-02 u
258 traj.phase0.controls:u_21 c -1.000000E-02 1.000000E-02 1.000000E-02 u
259 traj.phase0.controls:u_22 c -1.000000E-02 1.000000E-02 1.000000E-02 u
260 traj.phase0.controls:u_23 c -1.000000E-02 1.000000E-02 1.000000E-02 u
261 traj.phase0.controls:u_24 c -1.000000E-02 1.000000E-02 1.000000E-02 u
262 traj.phase0.controls:u_25 c -1.000000E-02 1.000000E-02 1.000000E-02 u
263 traj.phase0.controls:u_26 c -1.000000E-02 1.000000E-02 1.000000E-02 u
264 traj.phase0.controls:u_27 c -1.000000E-02 1.000000E-02 1.000000E-02 u
265 traj.phase0.controls:u_28 c -1.000000E-02 1.000000E-02 1.000000E-02 u
266 traj.phase0.controls:u_29 c -1.000000E-02 1.000000E-02 1.000000E-02 u
267 traj.phase0.controls:u_30 c -1.000000E-02 1.000000E-02 1.000000E-02 u
268 traj.phase0.controls:u_31 c -1.000000E-02 1.000000E-02 1.000000E-02 u
269 traj.phase0.controls:u_32 c -1.000000E-02 1.000000E-02 1.000000E-02 u
270 traj.phase0.controls:u_33 c -1.000000E-02 1.000000E-02 1.000000E-02 u
271 traj.phase0.controls:u_34 c -1.000000E-02 1.000000E-02 1.000000E-02 u
272 traj.phase0.controls:u_35 c -1.000000E-02 1.000000E-02 1.000000E-02 u
273 traj.phase0.controls:u_36 c -1.000000E-02 1.000000E-02 1.000000E-02 u
274 traj.phase0.controls:u_37 c -1.000000E-02 1.000000E-02 1.000000E-02 u
275 traj.phase0.controls:u_38 c -1.000000E-02 1.000000E-02 1.000000E-02 u
276 traj.phase0.controls:u_39 c -1.000000E-02 1.000000E-02 1.000000E-02 u
277 traj.phase0.controls:u_40 c -1.000000E-02 1.000000E-02 1.000000E-02 u
278 traj.phase0.controls:u_41 c -1.000000E-02 1.000000E-02 1.000000E-02 u
279 traj.phase0.controls:u_42 c -1.000000E-02 1.000000E-02 1.000000E-02 u
280 traj.phase0.controls:u_43 c -1.000000E-02 1.000000E-02 1.000000E-02 u
281 traj.phase0.controls:u_44 c -1.000000E-02 1.000000E-02 1.000000E-02 u
282 traj.phase0.controls:u_45 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
283 traj.phase0.controls:u_46 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
284 traj.phase0.controls:u_47 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
285 traj.phase0.controls:u_48 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
286 traj.phase0.controls:u_49 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
287 traj.phase0.controls:u_50 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
288 traj.phase0.controls:u_51 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
289 traj.phase0.controls:u_52 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
290 traj.phase0.controls:u_53 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
291 traj.phase0.controls:u_54 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
292 traj.phase0.controls:u_55 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
293 traj.phase0.controls:u_56 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
294 traj.phase0.controls:u_57 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
295 traj.phase0.controls:u_58 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
296 traj.phase0.controls:u_59 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
297 traj.phase0.controls:u_60 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
298 traj.phase0.controls:u_61 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
299 traj.phase0.controls:u_62 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
300 traj.phase0.controls:u_63 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
301 traj.phase0.controls:u_64 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
302 traj.phase0.controls:u_65 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
303 traj.phase0.controls:u_66 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
304 traj.phase0.controls:u_67 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
305 traj.phase0.controls:u_68 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
306 traj.phase0.controls:u_69 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
307 traj.phase0.controls:u_70 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
308 traj.phase0.controls:u_71 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
309 traj.phase0.controls:u_72 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
310 traj.phase0.controls:u_73 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
311 traj.phase0.controls:u_74 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
312 traj.phase0.controls:u_75 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
313 traj.phase0.controls:u_76 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
314 traj.phase0.controls:u_77 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
315 traj.phase0.controls:u_78 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
316 traj.phase0.controls:u_79 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
317 traj.phase0.controls:u_80 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
318 traj.phase0.controls:u_81 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
319 traj.phase0.controls:u_82 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
320 traj.phase0.controls:u_83 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
321 traj.phase0.controls:u_84 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
322 traj.phase0.controls:u_85 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
323 traj.phase0.controls:u_86 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
324 traj.phase0.controls:u_87 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
325 traj.phase0.controls:u_88 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
326 traj.phase0.controls:u_89 c -1.000000E-02 -1.000000E-02 1.000000E-02 l
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier (N/A)
0 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.551115E-18 0.000000E+00 9.00000E+100
1 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
2 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.850372E-18 0.000000E+00 9.00000E+100
3 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.035409E-17 0.000000E+00 9.00000E+100
4 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.110223E-17 0.000000E+00 9.00000E+100
5 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.590520E-17 0.000000E+00 9.00000E+100
6 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -4.440892E-17 0.000000E+00 9.00000E+100
7 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.295260E-17 0.000000E+00 9.00000E+100
8 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.145632E-17 0.000000E+00 9.00000E+100
9 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.590520E-17 0.000000E+00 9.00000E+100
10 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -3.700743E-18 0.000000E+00 9.00000E+100
11 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.700743E-17 0.000000E+00 9.00000E+100
12 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.850372E-17 0.000000E+00 9.00000E+100
13 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.850372E-18 0.000000E+00 9.00000E+100
14 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.960595E-17 0.000000E+00 9.00000E+100
15 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.036208E-16 0.000000E+00 9.00000E+100
16 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.181041E-17 0.000000E+00 9.00000E+100
17 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 4.440892E-17 0.000000E+00 9.00000E+100
18 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.145632E-17 0.000000E+00 9.00000E+100
19 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.551115E-17 0.000000E+00 9.00000E+100
20 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.035409E-17 0.000000E+00 9.00000E+100
21 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.220446E-17 0.000000E+00 9.00000E+100
22 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -5.551115E-18 0.000000E+00 9.00000E+100
23 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 8.511710E-17 0.000000E+00 9.00000E+100
24 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -4.070818E-17 0.000000E+00 9.00000E+100
25 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.221245E-16 0.000000E+00 9.00000E+100
26 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 9.992007E-17 0.000000E+00 9.00000E+100
27 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.349972E-16 0.000000E+00 9.00000E+100
28 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.220446E-17 0.000000E+00 9.00000E+100
29 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.220446E-17 0.000000E+00 9.00000E+100
30 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.775558E-17 0.000000E+00 9.00000E+100
31 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 6.291264E-17 0.000000E+00 9.00000E+100
32 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.551115E-18 0.000000E+00 9.00000E+100
33 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.423987E-16 0.000000E+00 9.00000E+100
34 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -8.326673E-17 0.000000E+00 9.00000E+100
35 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 4.810966E-17 0.000000E+00 9.00000E+100
36 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.145632E-17 0.000000E+00 9.00000E+100
37 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.073216E-16 0.000000E+00 9.00000E+100
38 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.017704E-16 0.000000E+00 9.00000E+100
39 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.036208E-16 0.000000E+00 9.00000E+100
40 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 9.251859E-17 0.000000E+00 9.00000E+100
41 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 2.220446E-17 0.000000E+00 9.00000E+100
42 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -3.404684E-16 0.000000E+00 9.00000E+100
43 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.461794E-16 0.000000E+00 9.00000E+100
44 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -6.291264E-17 0.000000E+00 9.00000E+100
45 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -5.921189E-17 0.000000E+00 9.00000E+100
46 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -8.141636E-17 0.000000E+00 9.00000E+100
47 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.147230E-16 0.000000E+00 9.00000E+100
48 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -9.251859E-17 0.000000E+00 9.00000E+100
49 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -6.291264E-17 0.000000E+00 9.00000E+100
50 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -2.775558E-17 0.000000E+00 9.00000E+100
51 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 5.440093E-16 0.000000E+00 9.00000E+100
52 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.221245E-16 0.000000E+00 9.00000E+100
53 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -3.700743E-17 0.000000E+00 9.00000E+100
54 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 3.404684E-16 0.000000E+00 9.00000E+100
55 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -1.480297E-17 0.000000E+00 9.00000E+100
56 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.480297E-17 0.000000E+00 9.00000E+100
57 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 -8.511710E-17 0.000000E+00 9.00000E+100
58 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.258253E-16 0.000000E+00 9.00000E+100
59 traj.phase0.collocation_constraint.defects:v e 0.000000E+00 1.665335E-17 0.000000E+00 9.00000E+100
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235 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
236 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
237 traj.phase0.continuity_comp.defect_states:x e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
Exit Status
Inform Description
0 Optimization terminated successfully.
--------------------------------------------------------------------------------
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/visualization/opt_report/opt_report.py:625: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding.
ax.set_ylim([ymin_plot, ymax_plot])
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/group.py:1137: DerivativesWarning:Constraints or objectives [ode_eval.control_interp.control_rates:u_rate, ode_eval.control_interp.control_rates:u_rate2, ode_eval.control_interp.control_values:u] cannot be impacted by the design variables of the problem because no partials were defined for them in their parent component(s).
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
Problem: problem
Driver: pyOptSparseDriver
success : True
iterations : 21
runtime : 2.6058E+00 s
model_evals : 21
model_time : 1.0488E-02 s
deriv_evals : 20
deriv_time : 7.0480E-02 s
exit_status : SUCCESS
sol = om.CaseReader(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')
plot_results([('traj.phase0.timeseries.time', 'traj.phase0.timeseries.x',
'time (s)', 'x $(m)$'),
('traj.phase0.timeseries.time', 'traj.phase0.timeseries.v',
'time (s)', 'v $(m/s)$'),
('traj.phase0.timeseries.time', 'traj.phase0.timeseries.u',
'time (s)', 'u $(m/s^2)$')],
title='Double Integrator Solution\nRadau Pseudospectral Method',
p_sol=sol, p_sim=sim)
plt.show()
