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.

Hide 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
from dymos.examples.double_integrator.double_integrator_ode import DoubleIntegratorODE

# 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)

p['traj.phase0.t_initial'] = 0.0
p['traj.phase0.t_duration'] = 1.0

p.set_val('traj.phase0.states:x', phase.interp('x', ys=[0, 0.25]))
p.set_val('traj.phase0.states:v', phase.interp('v', ys=[0, 0]))
p.set_val('traj.phase0.controls:u', phase.interp('u', ys=[1, -1]))

#
# Solve the problem.
#
dm.run_problem(p, simulate=True)
--- Constraint Report [traj] ---
    --- phase0 ---
        None

INFO: checking out_of_order
INFO:check_config:checking out_of_order
INFO: checking system
INFO:check_config:checking system
INFO: checking solvers
INFO:check_config:checking solvers
INFO: checking dup_inputs
INFO:check_config:checking dup_inputs
INFO: checking missing_recorders
INFO:check_config:checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
WARNING:check_config:The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO:check_config:checking unserializable_options
INFO: checking comp_has_no_outputs
INFO:check_config:checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
   traj.phases.phase0.rhs_all
WARNING:check_config:The following Components do not have any outputs:
   traj.phases.phase0.rhs_all
INFO: checking auto_ivc_warnings
INFO:check_config:checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO:check_config:checking out_of_order
INFO: checking system
INFO:check_config:checking system
INFO: checking solvers
INFO:check_config:checking solvers
INFO: checking dup_inputs
INFO:check_config:checking dup_inputs
INFO: checking missing_recorders
INFO:check_config:checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
WARNING:check_config:The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO:check_config:checking unserializable_options
INFO: checking comp_has_no_outputs
INFO:check_config:checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
   traj.phases.phase0.rhs_all
WARNING:check_config:The following Components do not have any outputs:
   traj.phases.phase0.rhs_all
INFO: checking auto_ivc_warnings
INFO:check_config:checking auto_ivc_warnings
Full total jacobian was computed 3 times, taking 0.107326 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  (96.7% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.107326 sec.
Time to compute coloring: 0.128783 sec.
Memory to compute coloring: 0.000000 MB.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
    Objective Function: _objfunc

    Solution: 
--------------------------------------------------------------------------------
    Total Time:                    2.7295
       User Objective Time :       0.0387
       User Sensitivity Time :     0.1027
       Interface Time :            0.1123
       Opt Solver Time:            2.4758
    Calls to Objective Function :      19
    Calls to Sens Function :           19


   Objectives
      Index  Name                                                Value
          0  traj.phases.phase0.indep_states.states:x    -2.500000E-01

   Variables (c - continuous, i - integer, d - discrete)
      Index  Name                                                            Type      Lower Bound            Value      Upper Bound     Status
          0  traj.phases.phase0.indep_states.states:v_0                         c    -1.000000E+21     1.183503E-02     1.000000E+21           
          1  traj.phases.phase0.indep_states.states:v_1                         c    -1.000000E+21     2.816497E-02     1.000000E+21           
          2  traj.phases.phase0.indep_states.states:v_2                         c    -1.000000E+21     3.333333E-02     1.000000E+21           
          3  traj.phases.phase0.indep_states.states:v_3                         c    -1.000000E+21     3.333333E-02     1.000000E+21           
          4  traj.phases.phase0.indep_states.states:v_4                         c    -1.000000E+21     4.516837E-02     1.000000E+21           
          5  traj.phases.phase0.indep_states.states:v_5                         c    -1.000000E+21     6.149830E-02     1.000000E+21           
          6  traj.phases.phase0.indep_states.states:v_6                         c    -1.000000E+21     6.666667E-02     1.000000E+21           
          7  traj.phases.phase0.indep_states.states:v_7                         c    -1.000000E+21     6.666667E-02     1.000000E+21           
          8  traj.phases.phase0.indep_states.states:v_8                         c    -1.000000E+21     7.850170E-02     1.000000E+21           
          9  traj.phases.phase0.indep_states.states:v_9                         c    -1.000000E+21     9.483163E-02     1.000000E+21           
         10  traj.phases.phase0.indep_states.states:v_10                        c    -1.000000E+21     1.000000E-01     1.000000E+21           
         11  traj.phases.phase0.indep_states.states:v_11                        c    -1.000000E+21     1.000000E-01     1.000000E+21           
         12  traj.phases.phase0.indep_states.states:v_12                        c    -1.000000E+21     1.118350E-01     1.000000E+21           
         13  traj.phases.phase0.indep_states.states:v_13                        c    -1.000000E+21     1.281650E-01     1.000000E+21           
         14  traj.phases.phase0.indep_states.states:v_14                        c    -1.000000E+21     1.333333E-01     1.000000E+21           
         15  traj.phases.phase0.indep_states.states:v_15                        c    -1.000000E+21     1.333333E-01     1.000000E+21           
         16  traj.phases.phase0.indep_states.states:v_16                        c    -1.000000E+21     1.451684E-01     1.000000E+21           
         17  traj.phases.phase0.indep_states.states:v_17                        c    -1.000000E+21     1.614983E-01     1.000000E+21           
         18  traj.phases.phase0.indep_states.states:v_18                        c    -1.000000E+21     1.666667E-01     1.000000E+21           
         19  traj.phases.phase0.indep_states.states:v_19                        c    -1.000000E+21     1.666667E-01     1.000000E+21           
         20  traj.phases.phase0.indep_states.states:v_20                        c    -1.000000E+21     1.785017E-01     1.000000E+21           
         21  traj.phases.phase0.indep_states.states:v_21                        c    -1.000000E+21     1.948316E-01     1.000000E+21           
         22  traj.phases.phase0.indep_states.states:v_22                        c    -1.000000E+21     2.000000E-01     1.000000E+21           
         23  traj.phases.phase0.indep_states.states:v_23                        c    -1.000000E+21     2.000000E-01     1.000000E+21           
         24  traj.phases.phase0.indep_states.states:v_24                        c    -1.000000E+21     2.118350E-01     1.000000E+21           
         25  traj.phases.phase0.indep_states.states:v_25                        c    -1.000000E+21     2.281650E-01     1.000000E+21           
         26  traj.phases.phase0.indep_states.states:v_26                        c    -1.000000E+21     2.333333E-01     1.000000E+21           
         27  traj.phases.phase0.indep_states.states:v_27                        c    -1.000000E+21     2.333333E-01     1.000000E+21           
         28  traj.phases.phase0.indep_states.states:v_28                        c    -1.000000E+21     2.451684E-01     1.000000E+21           
         29  traj.phases.phase0.indep_states.states:v_29                        c    -1.000000E+21     2.614983E-01     1.000000E+21           
         30  traj.phases.phase0.indep_states.states:v_30                        c    -1.000000E+21     2.666667E-01     1.000000E+21           
         31  traj.phases.phase0.indep_states.states:v_31                        c    -1.000000E+21     2.666667E-01     1.000000E+21           
         32  traj.phases.phase0.indep_states.states:v_32                        c    -1.000000E+21     2.785017E-01     1.000000E+21           
         33  traj.phases.phase0.indep_states.states:v_33                        c    -1.000000E+21     2.948316E-01     1.000000E+21           
         34  traj.phases.phase0.indep_states.states:v_34                        c    -1.000000E+21     3.000000E-01     1.000000E+21           
         35  traj.phases.phase0.indep_states.states:v_35                        c    -1.000000E+21     3.000000E-01     1.000000E+21           
         36  traj.phases.phase0.indep_states.states:v_36                        c    -1.000000E+21     3.118350E-01     1.000000E+21           
         37  traj.phases.phase0.indep_states.states:v_37                        c    -1.000000E+21     3.281650E-01     1.000000E+21           
         38  traj.phases.phase0.indep_states.states:v_38                        c    -1.000000E+21     3.333333E-01     1.000000E+21           
         39  traj.phases.phase0.indep_states.states:v_39                        c    -1.000000E+21     3.333333E-01     1.000000E+21           
         40  traj.phases.phase0.indep_states.states:v_40                        c    -1.000000E+21     3.451684E-01     1.000000E+21           
         41  traj.phases.phase0.indep_states.states:v_41                        c    -1.000000E+21     3.614983E-01     1.000000E+21           
         42  traj.phases.phase0.indep_states.states:v_42                        c    -1.000000E+21     3.666667E-01     1.000000E+21           
         43  traj.phases.phase0.indep_states.states:v_43                        c    -1.000000E+21     3.666667E-01     1.000000E+21           
         44  traj.phases.phase0.indep_states.states:v_44                        c    -1.000000E+21     3.785017E-01     1.000000E+21           
         45  traj.phases.phase0.indep_states.states:v_45                        c    -1.000000E+21     3.948316E-01     1.000000E+21           
         46  traj.phases.phase0.indep_states.states:v_46                        c    -1.000000E+21     4.000000E-01     1.000000E+21           
         47  traj.phases.phase0.indep_states.states:v_47                        c    -1.000000E+21     4.000000E-01     1.000000E+21           
         48  traj.phases.phase0.indep_states.states:v_48                        c    -1.000000E+21     4.118350E-01     1.000000E+21           
         49  traj.phases.phase0.indep_states.states:v_49                        c    -1.000000E+21     4.281650E-01     1.000000E+21           
         50  traj.phases.phase0.indep_states.states:v_50                        c    -1.000000E+21     4.333333E-01     1.000000E+21           
         51  traj.phases.phase0.indep_states.states:v_51                        c    -1.000000E+21     4.333333E-01     1.000000E+21           
         52  traj.phases.phase0.indep_states.states:v_52                        c    -1.000000E+21     4.451684E-01     1.000000E+21           
         53  traj.phases.phase0.indep_states.states:v_53                        c    -1.000000E+21     4.614983E-01     1.000000E+21           
         54  traj.phases.phase0.indep_states.states:v_54                        c    -1.000000E+21     4.666667E-01     1.000000E+21           
         55  traj.phases.phase0.indep_states.states:v_55                        c    -1.000000E+21     4.666667E-01     1.000000E+21           
         56  traj.phases.phase0.indep_states.states:v_56                        c    -1.000000E+21     4.785017E-01     1.000000E+21           
         57  traj.phases.phase0.indep_states.states:v_57                        c    -1.000000E+21     4.948316E-01     1.000000E+21           
         58  traj.phases.phase0.indep_states.states:v_58                        c    -1.000000E+21     5.000000E-01     1.000000E+21           
         59  traj.phases.phase0.indep_states.states:v_59                        c    -1.000000E+21     5.000000E-01     1.000000E+21           
         60  traj.phases.phase0.indep_states.states:v_60                        c    -1.000000E+21     4.881650E-01     1.000000E+21           
         61  traj.phases.phase0.indep_states.states:v_61                        c    -1.000000E+21     4.718350E-01     1.000000E+21           
         62  traj.phases.phase0.indep_states.states:v_62                        c    -1.000000E+21     4.666667E-01     1.000000E+21           
         63  traj.phases.phase0.indep_states.states:v_63                        c    -1.000000E+21     4.666667E-01     1.000000E+21           
         64  traj.phases.phase0.indep_states.states:v_64                        c    -1.000000E+21     4.548316E-01     1.000000E+21           
         65  traj.phases.phase0.indep_states.states:v_65                        c    -1.000000E+21     4.385017E-01     1.000000E+21           
         66  traj.phases.phase0.indep_states.states:v_66                        c    -1.000000E+21     4.333333E-01     1.000000E+21           
         67  traj.phases.phase0.indep_states.states:v_67                        c    -1.000000E+21     4.333333E-01     1.000000E+21           
         68  traj.phases.phase0.indep_states.states:v_68                        c    -1.000000E+21     4.214983E-01     1.000000E+21           
         69  traj.phases.phase0.indep_states.states:v_69                        c    -1.000000E+21     4.051684E-01     1.000000E+21           
         70  traj.phases.phase0.indep_states.states:v_70                        c    -1.000000E+21     4.000000E-01     1.000000E+21           
         71  traj.phases.phase0.indep_states.states:v_71                        c    -1.000000E+21     4.000000E-01     1.000000E+21           
         72  traj.phases.phase0.indep_states.states:v_72                        c    -1.000000E+21     3.881650E-01     1.000000E+21           
         73  traj.phases.phase0.indep_states.states:v_73                        c    -1.000000E+21     3.718350E-01     1.000000E+21           
         74  traj.phases.phase0.indep_states.states:v_74                        c    -1.000000E+21     3.666667E-01     1.000000E+21           
         75  traj.phases.phase0.indep_states.states:v_75                        c    -1.000000E+21     3.666667E-01     1.000000E+21           
         76  traj.phases.phase0.indep_states.states:v_76                        c    -1.000000E+21     3.548316E-01     1.000000E+21           
         77  traj.phases.phase0.indep_states.states:v_77                        c    -1.000000E+21     3.385017E-01     1.000000E+21           
         78  traj.phases.phase0.indep_states.states:v_78                        c    -1.000000E+21     3.333333E-01     1.000000E+21           
         79  traj.phases.phase0.indep_states.states:v_79                        c    -1.000000E+21     3.333333E-01     1.000000E+21           
         80  traj.phases.phase0.indep_states.states:v_80                        c    -1.000000E+21     3.214983E-01     1.000000E+21           
         81  traj.phases.phase0.indep_states.states:v_81                        c    -1.000000E+21     3.051684E-01     1.000000E+21           
         82  traj.phases.phase0.indep_states.states:v_82                        c    -1.000000E+21     3.000000E-01     1.000000E+21           
         83  traj.phases.phase0.indep_states.states:v_83                        c    -1.000000E+21     3.000000E-01     1.000000E+21           
         84  traj.phases.phase0.indep_states.states:v_84                        c    -1.000000E+21     2.881650E-01     1.000000E+21           
         85  traj.phases.phase0.indep_states.states:v_85                        c    -1.000000E+21     2.718350E-01     1.000000E+21           
         86  traj.phases.phase0.indep_states.states:v_86                        c    -1.000000E+21     2.666667E-01     1.000000E+21           
         87  traj.phases.phase0.indep_states.states:v_87                        c    -1.000000E+21     2.666667E-01     1.000000E+21           
         88  traj.phases.phase0.indep_states.states:v_88                        c    -1.000000E+21     2.548316E-01     1.000000E+21           
         89  traj.phases.phase0.indep_states.states:v_89                        c    -1.000000E+21     2.385017E-01     1.000000E+21           
         90  traj.phases.phase0.indep_states.states:v_90                        c    -1.000000E+21     2.333333E-01     1.000000E+21           
         91  traj.phases.phase0.indep_states.states:v_91                        c    -1.000000E+21     2.333333E-01     1.000000E+21           
         92  traj.phases.phase0.indep_states.states:v_92                        c    -1.000000E+21     2.214983E-01     1.000000E+21           
         93  traj.phases.phase0.indep_states.states:v_93                        c    -1.000000E+21     2.051684E-01     1.000000E+21           
         94  traj.phases.phase0.indep_states.states:v_94                        c    -1.000000E+21     2.000000E-01     1.000000E+21           
         95  traj.phases.phase0.indep_states.states:v_95                        c    -1.000000E+21     2.000000E-01     1.000000E+21           
         96  traj.phases.phase0.indep_states.states:v_96                        c    -1.000000E+21     1.881650E-01     1.000000E+21           
         97  traj.phases.phase0.indep_states.states:v_97                        c    -1.000000E+21     1.718350E-01     1.000000E+21           
         98  traj.phases.phase0.indep_states.states:v_98                        c    -1.000000E+21     1.666667E-01     1.000000E+21           
         99  traj.phases.phase0.indep_states.states:v_99                        c    -1.000000E+21     1.666667E-01     1.000000E+21           
        100  traj.phases.phase0.indep_states.states:v_100                       c    -1.000000E+21     1.548316E-01     1.000000E+21           
        101  traj.phases.phase0.indep_states.states:v_101                       c    -1.000000E+21     1.385017E-01     1.000000E+21           
        102  traj.phases.phase0.indep_states.states:v_102                       c    -1.000000E+21     1.333333E-01     1.000000E+21           
        103  traj.phases.phase0.indep_states.states:v_103                       c    -1.000000E+21     1.333333E-01     1.000000E+21           
        104  traj.phases.phase0.indep_states.states:v_104                       c    -1.000000E+21     1.214983E-01     1.000000E+21           
        105  traj.phases.phase0.indep_states.states:v_105                       c    -1.000000E+21     1.051684E-01     1.000000E+21           
        106  traj.phases.phase0.indep_states.states:v_106                       c    -1.000000E+21     1.000000E-01     1.000000E+21           
        107  traj.phases.phase0.indep_states.states:v_107                       c    -1.000000E+21     1.000000E-01     1.000000E+21           
        108  traj.phases.phase0.indep_states.states:v_108                       c    -1.000000E+21     8.816497E-02     1.000000E+21           
        109  traj.phases.phase0.indep_states.states:v_109                       c    -1.000000E+21     7.183503E-02     1.000000E+21           
        110  traj.phases.phase0.indep_states.states:v_110                       c    -1.000000E+21     6.666667E-02     1.000000E+21           
        111  traj.phases.phase0.indep_states.states:v_111                       c    -1.000000E+21     6.666667E-02     1.000000E+21           
        112  traj.phases.phase0.indep_states.states:v_112                       c    -1.000000E+21     5.483163E-02     1.000000E+21           
        113  traj.phases.phase0.indep_states.states:v_113                       c    -1.000000E+21     3.850170E-02     1.000000E+21           
        114  traj.phases.phase0.indep_states.states:v_114                       c    -1.000000E+21     3.333333E-02     1.000000E+21           
        115  traj.phases.phase0.indep_states.states:v_115                       c    -1.000000E+21     3.333333E-02     1.000000E+21           
        116  traj.phases.phase0.indep_states.states:v_116                       c    -1.000000E+21     2.149830E-02     1.000000E+21           
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        272  traj.phases.phase0.control_group.indep_controls.controls:u_35      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        273  traj.phases.phase0.control_group.indep_controls.controls:u_36      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        274  traj.phases.phase0.control_group.indep_controls.controls:u_37      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        275  traj.phases.phase0.control_group.indep_controls.controls:u_38      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        276  traj.phases.phase0.control_group.indep_controls.controls:u_39      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        277  traj.phases.phase0.control_group.indep_controls.controls:u_40      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        278  traj.phases.phase0.control_group.indep_controls.controls:u_41      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        279  traj.phases.phase0.control_group.indep_controls.controls:u_42      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        280  traj.phases.phase0.control_group.indep_controls.controls:u_43      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        281  traj.phases.phase0.control_group.indep_controls.controls:u_44      c    -1.000000E-02     1.000000E-02     1.000000E-02          u
        282  traj.phases.phase0.control_group.indep_controls.controls:u_45      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        283  traj.phases.phase0.control_group.indep_controls.controls:u_46      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        284  traj.phases.phase0.control_group.indep_controls.controls:u_47      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        285  traj.phases.phase0.control_group.indep_controls.controls:u_48      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        286  traj.phases.phase0.control_group.indep_controls.controls:u_49      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        287  traj.phases.phase0.control_group.indep_controls.controls:u_50      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        288  traj.phases.phase0.control_group.indep_controls.controls:u_51      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        289  traj.phases.phase0.control_group.indep_controls.controls:u_52      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        290  traj.phases.phase0.control_group.indep_controls.controls:u_53      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        291  traj.phases.phase0.control_group.indep_controls.controls:u_54      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        292  traj.phases.phase0.control_group.indep_controls.controls:u_55      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        293  traj.phases.phase0.control_group.indep_controls.controls:u_56      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        294  traj.phases.phase0.control_group.indep_controls.controls:u_57      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        295  traj.phases.phase0.control_group.indep_controls.controls:u_58      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        296  traj.phases.phase0.control_group.indep_controls.controls:u_59      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        297  traj.phases.phase0.control_group.indep_controls.controls:u_60      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        298  traj.phases.phase0.control_group.indep_controls.controls:u_61      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        299  traj.phases.phase0.control_group.indep_controls.controls:u_62      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        300  traj.phases.phase0.control_group.indep_controls.controls:u_63      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        301  traj.phases.phase0.control_group.indep_controls.controls:u_64      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        302  traj.phases.phase0.control_group.indep_controls.controls:u_65      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        303  traj.phases.phase0.control_group.indep_controls.controls:u_66      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        304  traj.phases.phase0.control_group.indep_controls.controls:u_67      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        305  traj.phases.phase0.control_group.indep_controls.controls:u_68      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        306  traj.phases.phase0.control_group.indep_controls.controls:u_69      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        307  traj.phases.phase0.control_group.indep_controls.controls:u_70      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        308  traj.phases.phase0.control_group.indep_controls.controls:u_71      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        309  traj.phases.phase0.control_group.indep_controls.controls:u_72      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        310  traj.phases.phase0.control_group.indep_controls.controls:u_73      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        311  traj.phases.phase0.control_group.indep_controls.controls:u_74      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        312  traj.phases.phase0.control_group.indep_controls.controls:u_75      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        313  traj.phases.phase0.control_group.indep_controls.controls:u_76      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        314  traj.phases.phase0.control_group.indep_controls.controls:u_77      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        315  traj.phases.phase0.control_group.indep_controls.controls:u_78      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        316  traj.phases.phase0.control_group.indep_controls.controls:u_79      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        317  traj.phases.phase0.control_group.indep_controls.controls:u_80      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        318  traj.phases.phase0.control_group.indep_controls.controls:u_81      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        319  traj.phases.phase0.control_group.indep_controls.controls:u_82      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        320  traj.phases.phase0.control_group.indep_controls.controls:u_83      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        321  traj.phases.phase0.control_group.indep_controls.controls:u_84      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        322  traj.phases.phase0.control_group.indep_controls.controls:u_85      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        323  traj.phases.phase0.control_group.indep_controls.controls:u_86      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        324  traj.phases.phase0.control_group.indep_controls.controls:u_87      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        325  traj.phases.phase0.control_group.indep_controls.controls:u_88      c    -1.000000E-02    -1.000000E-02     1.000000E-02          l
        326  traj.phases.phase0.control_group.indep_controls.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.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -7.401487E-18    0.000000E+00              9.00000E+100
          1  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.850372E-18    0.000000E+00              9.00000E+100
          2  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -3.700743E-18    0.000000E+00              9.00000E+100
          3  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.220446E-17    0.000000E+00              9.00000E+100
          4  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    7.401487E-18    0.000000E+00              9.00000E+100
          5  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.551115E-18    0.000000E+00              9.00000E+100
          6  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -5.181041E-17    0.000000E+00              9.00000E+100
          7  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.700743E-18    0.000000E+00              9.00000E+100
          8  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -7.401487E-18    0.000000E+00              9.00000E+100
          9  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.590520E-17    0.000000E+00              9.00000E+100
         10  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.480297E-17    0.000000E+00              9.00000E+100
         11  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.440892E-17    0.000000E+00              9.00000E+100
         12  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.625929E-17    0.000000E+00              9.00000E+100
         13  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.736152E-17    0.000000E+00              9.00000E+100
         14  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    7.401487E-18    0.000000E+00              9.00000E+100
         15  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -9.251859E-18    0.000000E+00              9.00000E+100
         16  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.700743E-17    0.000000E+00              9.00000E+100
         17  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.295260E-17    0.000000E+00              9.00000E+100
         18  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.110223E-17    0.000000E+00              9.00000E+100
         19  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.480297E-17    0.000000E+00              9.00000E+100
         20  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -3.700743E-18    0.000000E+00              9.00000E+100
         21  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.996004E-17    0.000000E+00              9.00000E+100
         22  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.921189E-17    0.000000E+00              9.00000E+100
         23  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -6.846375E-17    0.000000E+00              9.00000E+100
         24  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    4.810966E-17    0.000000E+00              9.00000E+100
         25  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    6.291264E-17    0.000000E+00              9.00000E+100
         26  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.480297E-17    0.000000E+00              9.00000E+100
         27  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.369275E-16    0.000000E+00              9.00000E+100
         28  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.220446E-17    0.000000E+00              9.00000E+100
         29  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -7.401487E-18    0.000000E+00              9.00000E+100
         30  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.165734E-16    0.000000E+00              9.00000E+100
         31  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.480297E-17    0.000000E+00              9.00000E+100
         32  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.017704E-16    0.000000E+00              9.00000E+100
         33  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -8.881784E-17    0.000000E+00              9.00000E+100
         34  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -6.846375E-17    0.000000E+00              9.00000E+100
         35  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.276756E-16    0.000000E+00              9.00000E+100
         36  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -9.621933E-17    0.000000E+00              9.00000E+100
         37  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
         38  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -3.700743E-17    0.000000E+00              9.00000E+100
         39  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.591320E-16    0.000000E+00              9.00000E+100
         40  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
         41  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.220446E-16    0.000000E+00              9.00000E+100
         42  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    2.183439E-16    0.000000E+00              9.00000E+100
         43  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    4.440892E-17    0.000000E+00              9.00000E+100
         44  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -9.251859E-17    0.000000E+00              9.00000E+100
         45  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    9.066821E-17    0.000000E+00              9.00000E+100
         46  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.700743E-18    0.000000E+00              9.00000E+100
         47  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -6.661338E-17    0.000000E+00              9.00000E+100
         48  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    7.216450E-17    0.000000E+00              9.00000E+100
         49  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.921189E-17    0.000000E+00              9.00000E+100
         50  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.480297E-16    0.000000E+00              9.00000E+100
         51  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.961394E-16    0.000000E+00              9.00000E+100
         52  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.921189E-17    0.000000E+00              9.00000E+100
         53  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.440892E-17    0.000000E+00              9.00000E+100
         54  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.239749E-16    0.000000E+00              9.00000E+100
         55  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -3.330669E-17    0.000000E+00              9.00000E+100
         56  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.145632E-17    0.000000E+00              9.00000E+100
         57  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    4.070818E-17    0.000000E+00              9.00000E+100
         58  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.035409E-17    0.000000E+00              9.00000E+100
         59  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.517305E-16    0.000000E+00              9.00000E+100
         60  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    2.035409E-16    0.000000E+00              9.00000E+100
         61  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.480297E-17    0.000000E+00              9.00000E+100
         62  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -6.291264E-17    0.000000E+00              9.00000E+100
         63  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    8.511710E-17    0.000000E+00              9.00000E+100
         64  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.440892E-17    0.000000E+00              9.00000E+100
         65  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    6.476301E-17    0.000000E+00              9.00000E+100
         66  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    5.921189E-17    0.000000E+00              9.00000E+100
         67  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.255855E-17    0.000000E+00              9.00000E+100
         68  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.330669E-17    0.000000E+00              9.00000E+100
         69  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.070818E-17    0.000000E+00              9.00000E+100
         70  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.440892E-17    0.000000E+00              9.00000E+100
         71  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -3.700743E-18    0.000000E+00              9.00000E+100
         72  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -6.291264E-17    0.000000E+00              9.00000E+100
         73  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.850372E-17    0.000000E+00              9.00000E+100
         74  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    3.700743E-17    0.000000E+00              9.00000E+100
         75  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.480297E-17    0.000000E+00              9.00000E+100
         76  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.996004E-17    0.000000E+00              9.00000E+100
         77  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         78  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    7.401487E-18    0.000000E+00              9.00000E+100
         79  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.295260E-17    0.000000E+00              9.00000E+100
         80  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -4.070818E-17    0.000000E+00              9.00000E+100
         81  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.295260E-17    0.000000E+00              9.00000E+100
         82  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.850372E-18    0.000000E+00              9.00000E+100
         83  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -1.850372E-17    0.000000E+00              9.00000E+100
         84  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         85  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         86  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    9.251859E-18    0.000000E+00              9.00000E+100
         87  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00   -2.035409E-17    0.000000E+00              9.00000E+100
         88  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    1.850372E-18    0.000000E+00              9.00000E+100
         89  traj.phases.phase0.collocation_constraint.defects:v    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
         90  traj.phases.phase0.collocation_constraint.defects:x    e   0.000000E+00   -2.168404E-19    0.000000E+00              9.00000E+100
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        234  traj.phases.phase0.continuity_comp.defect_states:x     e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        235  traj.phases.phase0.continuity_comp.defect_states:x     e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        236  traj.phases.phase0.continuity_comp.defect_states:x     e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        237  traj.phases.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/python3.10/site-packages/openmdao/visualization/opt_report/opt_report.py:634: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding.
  ax.set_ylim([ymin_plot, ymax_plot])
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
False
sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('dymos_simulation.db').get_case('final')

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()
../../_images/4bcfe9e1e14bd17864c5b1fdbc428db9a4047e01030e81c9f4d320bcf5285084.png