The Hull Problem#

The Hull problem is a 1-DOF optimal control problem [Hul03]. It can be stated as:

Minimize the control effort required to move a frictionless sliding block from some initial position such that the final displacement from a pre-specified point is minimized.

State and control variables#

This system has one state variables, the position (\(x\)) of the sliding block.

This system has a single control variable (\(u\)), the velocity of the block.

The dynamics of the system are governed by

(48)#\[\begin{align} \dot{x} &= u \end{align}\]

Problem Definition#

We seek to minimize the control effort required and minimize the displacement from the origin.

(49)#\[\begin{align} \mathrm{Minimize} \, J &= 2.5x_f^2 \, + \, 0.5 \int_0^1 u^2 dt \end{align}\]

Subject to the initial conditions

(50)#\[\begin{align} t_0 &= 0.0 \\ x_0 &= 1.5 \end{align}\]

and the terminal constraints

(51)#\[\begin{align} t_f &= 10.0 \end{align}\]

Dealing with combined terminal and integral costs in Dymos#

In classic optimal control, the objective is often broken into the terminal component (the Mayer term) and the integral component (the Lagrange term). Dymos does not distinguish between the two. In this case, since the objective \(J\) consists of both a terminal cost and an integrated cost (Bolza form), we add a term to the ODE to account for the integrated quantity

(52)#\[\begin{align} \dot{x_L} &= L \\ L &= 0.5 u^2 \end{align}\]

where \(x_L\) is a state added to account for the Lagrange term.

Dymos supports the definition of simple mathematical expressions as the cost, so the final value of \(x_L\) can be added to the final value of \(2.5x^2\).

Defining the ODE#

The following code implements the equations of motion for the Hull problem. Since the rate of \(x\) is given by a control (\(u\)), there is no need to compute its rate in the ODE. Dymos can pull their values from those other states and controls. The ODE, therefore, only needs to compute the rate of change of \(x_L\) (\(L\)).

A few things to note:

  1. By providing the tag dymos.state_rate_source:{name}, we’re letting Dymos know what states need to be integrated, there’s no need to specify a rate source when using this ODE in our Phase.

  2. Pairing the above tag with dymos.state_units:{units} means we don’t have to specify units when setting properties for the state in our run script.

import numpy as np
import openmdao.api as om


class HullProblemODE(om.ExplicitComponent):
    def initialize(self):
        self.options.declare('num_nodes', types=int)

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

        self.add_input('u', val=np.zeros(nn), desc='control')

        self.add_output('L', val=np.zeros(nn), desc='Lagrangian', units='1/s')

        # Setup partials
        self.declare_partials(of='L', wrt='u', rows=np.arange(nn), cols=np.arange(nn))

    def compute(self, inputs, outputs, discrete_inputs=None, discrete_outputs=None):
        u = inputs['u']

        outputs['L'] = 0.5 * u ** 2

    def compute_partials(self, inputs, partials, discrete_inputs=None):
        u = inputs['u']
        partials['L', 'u'] = u

Solving the Hull problem with Dymos#

The following script solves the Hull problem with Dymos.

To begin, import the packages we require:

import dymos as dm
import matplotlib.pyplot as plt

We then instantiate an OpenMDAO problem and set the optimizer and its options.

The call to declare_coloring tells the optimizer to attempt to find a sparsity pattern that minimizes the work required to compute the derivatives across the model.

#
# Initialize the Problem and the optimization driver
#
p = om.Problem()
               
p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring()

Next, we add a Dymos Trajectory group to the problem’s model and add a phase to it.

In this case we’re using the Radau pseudospectral transcription to solve the problem.

#
# Create a trajectory and add a phase to it
#
traj = p.model.add_subsystem('traj', dm.Trajectory())
tx = transcription=dm.Radau(num_segments=24)
phase = traj.add_phase('phase0', dm.Phase(ode_class=HullProblemODE, transcription=tx))

At this point, we set the options on the main variables used in a Dymos phase.

In addition to time, we have two states (x, and x_L) and a single control (u).

Here we use bounds on the states themselves to constrain the initial values of x and1 x_L. From an optimization perspective, this means that we are removing the first and last values in the state histories of \(x\) and \(x_L\) from the vector of design variables. Their initial values will remain unchanged throughout the optimization process.

On the other hand, we could specify fix_initial=False, fix_final=False for these values, and Dymos would be free to change them. We would then need to put a boundary constraint in place to enforce their final values. Feel free to experiment with different ways of enforcing the boundary constraints on this problem and see how it affects performance.

The scaler values (ref) are all set to 1 here.

Also, we don’t need to specify targets for any of the variables here because their names are the targets in the top-level of the model. The rate source and units for the states are obtained from the tags in the ODE component we previously defined.

#
# Set the variables
#
phase.set_time_options(fix_initial=True, fix_duration=True)

phase.set_time_options(fix_initial=True, fix_duration=True)
phase.add_state('x', fix_initial=True, fix_final=False, rate_source='u')
phase.add_state('xL', fix_initial=True, fix_final=False, rate_source='L')
phase.add_control('u', opt=True, targets=['u'])
#
# Minimize time at the end of the phase
#
phase.add_objective('J = 2.5*x**2 + xL')

#
# Setup the Problem
#
p.setup()
--- Constraint Report [traj] ---
    --- phase0 ---
        None
<openmdao.core.problem.Problem at 0x7fdcf8917a90>

We then set the initial guesses for the variables in the problem and solve it.

We’re using the phase interp method to provide initial guesses for the states and controls. In this case, by giving it two values, it is linearly interpolating from the first value to the second value, and then returning the interpolated value at the input nodes for the given variable.

Finally, we use the dymos.run_problem method to execute the problem. This interface allows us to do some things that the standard OpenMDAO problem.run_driver interface does not. It will automatically record the final solution achieved by the optimizer in case named 'final' in a file called dymos_solution.db. By specifying simulate=True, it will automatically follow the solution with an explicit integration using scipy.solve_ivp. The results of the simulation are stored in a case named final in the file dymos_simulation.db. This explicit simulation demonstrates how the system evolved with the given controls, and serves as a check that we’re using a dense enough grid (enough segments and segments of sufficient order) to accurately represent the solution.

If those two solution didn’t agree reasonably well, we could rerun the problem with a more dense grid. Instead, we’re asking Dymos to automatically change the grid if necessary by specifying refine_method='ph'. This will attempt to repeatedly solve the problem and change the number of segments and segment orders until the solution is in reasonable agreement.

#
# Set the initial values
#
p.set_val('traj.phase0.states:x', phase.interp('x', [1.5, 1]))
p.set_val('traj.phase0.states:xL', phase.interp('xL', [0, 1]))
p.set_val('traj.phase0.t_initial', 0)
p.set_val('traj.phase0.t_duration', 10)
p.set_val('traj.phase0.controls:u', phase.interp('u', [-7, -0.14]))
#
# Solve for the optimal trajectory
#
dm.run_problem(p, run_driver=True, simulate=True)
Hide code cell output
Model viewer data has already been recorded for Driver.
Model viewer data has already been recorded for Driver.
Full total jacobian was computed 3 times, taking 0.275115 seconds.
Total jacobian shape: (191, 216) 


Jacobian shape: (191, 216)  ( 2.29% nonzero)
FWD solves: 7   REV solves: 0
Total colors vs. total size: 7 vs 216  (96.8% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.275115 sec.
Time to compute coloring: 0.109947 sec.
Memory to compute coloring: 0.000000 MB.
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
    Objective Function: _objfunc

    Solution: 
--------------------------------------------------------------------------------
    Total Time:                    3.2662
       User Objective Time :       0.1231
       User Sensitivity Time :     0.4939
       Interface Time :            0.3062
       Opt Solver Time:            2.3430
    Calls to Objective Function :      66
    Calls to Sens Function :           65


   Objectives
      Index  Name                                                            Value
          0  traj.phases.phase0.timeseries.timeseries_exec_comp.J     1.102952E-01

   Variables (c - continuous, i - integer, d - discrete)
      Index  Name                                                            Type      Lower Bound            Value      Upper Bound     Status
          0  traj.phases.phase0.indep_states.states:x_0                         c    -1.000000E+21     1.478271E+00     1.000000E+21           
          1  traj.phases.phase0.indep_states.states:x_1                         c    -1.000000E+21     1.448524E+00     1.000000E+21           
          2  traj.phases.phase0.indep_states.states:x_2                         c    -1.000000E+21     1.439054E+00     1.000000E+21           
          3  traj.phases.phase0.indep_states.states:x_3                         c    -1.000000E+21     1.417155E+00     1.000000E+21           
          4  traj.phases.phase0.indep_states.states:x_4                         c    -1.000000E+21     1.386794E+00     1.000000E+21           
          5  traj.phases.phase0.indep_states.states:x_5                         c    -1.000000E+21     1.377242E+00     1.000000E+21           
          6  traj.phases.phase0.indep_states.states:x_6                         c    -1.000000E+21     1.355564E+00     1.000000E+21           
          7  traj.phases.phase0.indep_states.states:x_7                         c    -1.000000E+21     1.325823E+00     1.000000E+21           
          8  traj.phases.phase0.indep_states.states:x_8                         c    -1.000000E+21     1.316376E+00     1.000000E+21           
          9  traj.phases.phase0.indep_states.states:x_9                         c    -1.000000E+21     1.294605E+00     1.000000E+21           
         10  traj.phases.phase0.indep_states.states:x_10                        c    -1.000000E+21     1.264427E+00     1.000000E+21           
         11  traj.phases.phase0.indep_states.states:x_11                        c    -1.000000E+21     1.254893E+00     1.000000E+21           
         12  traj.phases.phase0.indep_states.states:x_12                        c    -1.000000E+21     1.233142E+00     1.000000E+21           
         13  traj.phases.phase0.indep_states.states:x_13                        c    -1.000000E+21     1.203187E+00     1.000000E+21           
         14  traj.phases.phase0.indep_states.states:x_14                        c    -1.000000E+21     1.193684E+00     1.000000E+21           
         15  traj.phases.phase0.indep_states.states:x_15                        c    -1.000000E+21     1.171848E+00     1.000000E+21           
         16  traj.phases.phase0.indep_states.states:x_16                        c    -1.000000E+21     1.141677E+00     1.000000E+21           
         17  traj.phases.phase0.indep_states.states:x_17                        c    -1.000000E+21     1.132152E+00     1.000000E+21           
         18  traj.phases.phase0.indep_states.states:x_18                        c    -1.000000E+21     1.110416E+00     1.000000E+21           
         19  traj.phases.phase0.indep_states.states:x_19                        c    -1.000000E+21     1.080503E+00     1.000000E+21           
         20  traj.phases.phase0.indep_states.states:x_20                        c    -1.000000E+21     1.071028E+00     1.000000E+21           
         21  traj.phases.phase0.indep_states.states:x_21                        c    -1.000000E+21     1.049286E+00     1.000000E+21           
         22  traj.phases.phase0.indep_states.states:x_22                        c    -1.000000E+21     1.019250E+00     1.000000E+21           
         23  traj.phases.phase0.indep_states.states:x_23                        c    -1.000000E+21     1.009752E+00     1.000000E+21           
         24  traj.phases.phase0.indep_states.states:x_24                        c    -1.000000E+21     9.880351E-01     1.000000E+21           
         25  traj.phases.phase0.indep_states.states:x_25                        c    -1.000000E+21     9.581003E-01     1.000000E+21           
         26  traj.phases.phase0.indep_states.states:x_26                        c    -1.000000E+21     9.486212E-01     1.000000E+21           
         27  traj.phases.phase0.indep_states.states:x_27                        c    -1.000000E+21     9.268933E-01     1.000000E+21           
         28  traj.phases.phase0.indep_states.states:x_28                        c    -1.000000E+21     8.968904E-01     1.000000E+21           
         29  traj.phases.phase0.indep_states.states:x_29                        c    -1.000000E+21     8.873965E-01     1.000000E+21           
         30  traj.phases.phase0.indep_states.states:x_30                        c    -1.000000E+21     8.656686E-01     1.000000E+21           
         31  traj.phases.phase0.indep_states.states:x_31                        c    -1.000000E+21     8.357020E-01     1.000000E+21           
         32  traj.phases.phase0.indep_states.states:x_32                        c    -1.000000E+21     8.262171E-01     1.000000E+21           
         33  traj.phases.phase0.indep_states.states:x_33                        c    -1.000000E+21     8.044917E-01     1.000000E+21           
         34  traj.phases.phase0.indep_states.states:x_34                        c    -1.000000E+21     7.745071E-01     1.000000E+21           
         35  traj.phases.phase0.indep_states.states:x_35                        c    -1.000000E+21     7.650165E-01     1.000000E+21           
         36  traj.phases.phase0.indep_states.states:x_36                        c    -1.000000E+21     7.432809E-01     1.000000E+21           
         37  traj.phases.phase0.indep_states.states:x_37                        c    -1.000000E+21     7.132456E-01     1.000000E+21           
         38  traj.phases.phase0.indep_states.states:x_38                        c    -1.000000E+21     7.037192E-01     1.000000E+21           
         39  traj.phases.phase0.indep_states.states:x_39                        c    -1.000000E+21     6.818614E-01     1.000000E+21           
         40  traj.phases.phase0.indep_states.states:x_40                        c    -1.000000E+21     6.516597E-01     1.000000E+21           
         41  traj.phases.phase0.indep_states.states:x_41                        c    -1.000000E+21     6.421056E-01     1.000000E+21           
         42  traj.phases.phase0.indep_states.states:x_42                        c    -1.000000E+21     6.202547E-01     1.000000E+21           
         43  traj.phases.phase0.indep_states.states:x_43                        c    -1.000000E+21     5.901571E-01     1.000000E+21           
         44  traj.phases.phase0.indep_states.states:x_44                        c    -1.000000E+21     5.806407E-01     1.000000E+21           
         45  traj.phases.phase0.indep_states.states:x_45                        c    -1.000000E+21     5.588615E-01     1.000000E+21           
         46  traj.phases.phase0.indep_states.states:x_46                        c    -1.000000E+21     5.288345E-01     1.000000E+21           
         47  traj.phases.phase0.indep_states.states:x_47                        c    -1.000000E+21     5.193360E-01     1.000000E+21           
         48  traj.phases.phase0.indep_states.states:x_48                        c    -1.000000E+21     4.975909E-01     1.000000E+21           
         49  traj.phases.phase0.indep_states.states:x_49                        c    -1.000000E+21     4.675790E-01     1.000000E+21           
         50  traj.phases.phase0.indep_states.states:x_50                        c    -1.000000E+21     4.580733E-01     1.000000E+21           
         51  traj.phases.phase0.indep_states.states:x_51                        c    -1.000000E+21     4.362877E-01     1.000000E+21           
         52  traj.phases.phase0.indep_states.states:x_52                        c    -1.000000E+21     4.061967E-01     1.000000E+21           
         53  traj.phases.phase0.indep_states.states:x_53                        c    -1.000000E+21     3.966684E-01     1.000000E+21           
         54  traj.phases.phase0.indep_states.states:x_54                        c    -1.000000E+21     3.748501E-01     1.000000E+21           
         55  traj.phases.phase0.indep_states.states:x_55                        c    -1.000000E+21     3.447892E-01     1.000000E+21           
         56  traj.phases.phase0.indep_states.states:x_56                        c    -1.000000E+21     3.352962E-01     1.000000E+21           
         57  traj.phases.phase0.indep_states.states:x_57                        c    -1.000000E+21     3.136003E-01     1.000000E+21           
         58  traj.phases.phase0.indep_states.states:x_58                        c    -1.000000E+21     2.836652E-01     1.000000E+21           
         59  traj.phases.phase0.indep_states.states:x_59                        c    -1.000000E+21     2.741664E-01     1.000000E+21           
         60  traj.phases.phase0.indep_states.states:x_60                        c    -1.000000E+21     2.523507E-01     1.000000E+21           
         61  traj.phases.phase0.indep_states.states:x_61                        c    -1.000000E+21     2.222161E-01     1.000000E+21           
         62  traj.phases.phase0.indep_states.states:x_62                        c    -1.000000E+21     2.127007E-01     1.000000E+21           
         63  traj.phases.phase0.indep_states.states:x_63                        c    -1.000000E+21     1.909824E-01     1.000000E+21           
         64  traj.phases.phase0.indep_states.states:x_64                        c    -1.000000E+21     1.610963E-01     1.000000E+21           
         65  traj.phases.phase0.indep_states.states:x_65                        c    -1.000000E+21     1.516347E-01     1.000000E+21           
         66  traj.phases.phase0.indep_states.states:x_66                        c    -1.000000E+21     1.299375E-01     1.000000E+21           
         67  traj.phases.phase0.indep_states.states:x_67                        c    -1.000000E+21     9.996178E-02     1.000000E+21           
         68  traj.phases.phase0.indep_states.states:x_68                        c    -1.000000E+21     9.047482E-02     1.000000E+21           
         69  traj.phases.phase0.indep_states.states:x_69                        c    -1.000000E+21     6.876274E-02     1.000000E+21           
         70  traj.phases.phase0.indep_states.states:x_70                        c    -1.000000E+21     3.882258E-02     1.000000E+21           
         71  traj.phases.phase0.indep_states.states:x_71                        c    -1.000000E+21     2.934811E-02     1.000000E+21           
         72  traj.phases.phase0.indep_states.states:xL_0                        c    -1.000000E+21     1.595767E-03     1.000000E+21           
         73  traj.phases.phase0.indep_states.states:xL_1                        c    -1.000000E+21     3.763248E-03     1.000000E+21           
         74  traj.phases.phase0.indep_states.states:xL_2                        c    -1.000000E+21     4.457413E-03     1.000000E+21           
         75  traj.phases.phase0.indep_states.states:xL_3                        c    -1.000000E+21     6.078314E-03     1.000000E+21           
         76  traj.phases.phase0.indep_states.states:xL_4                        c    -1.000000E+21     8.336200E-03     1.000000E+21           
         77  traj.phases.phase0.indep_states.states:xL_5                        c    -1.000000E+21     9.042362E-03     1.000000E+21           
         78  traj.phases.phase0.indep_states.states:xL_6                        c    -1.000000E+21     1.063069E-02     1.000000E+21           
         79  traj.phases.phase0.indep_states.states:xL_7                        c    -1.000000E+21     1.279729E-02     1.000000E+21           
         80  traj.phases.phase0.indep_states.states:xL_8                        c    -1.000000E+21     1.348791E-02     1.000000E+21           
         81  traj.phases.phase0.indep_states.states:xL_9                        c    -1.000000E+21     1.508993E-02     1.000000E+21           
         82  traj.phases.phase0.indep_states.states:xL_10                       c    -1.000000E+21     1.732064E-02     1.000000E+21           
         83  traj.phases.phase0.indep_states.states:xL_11                       c    -1.000000E+21     1.802420E-02     1.000000E+21           
         84  traj.phases.phase0.indep_states.states:xL_12                       c    -1.000000E+21     1.962315E-02     1.000000E+21           
         85  traj.phases.phase0.indep_states.states:xL_13                       c    -1.000000E+21     2.182112E-02     1.000000E+21           
         86  traj.phases.phase0.indep_states.states:xL_14                       c    -1.000000E+21     2.252006E-02     1.000000E+21           
         87  traj.phases.phase0.indep_states.states:xL_15                       c    -1.000000E+21     2.413150E-02     1.000000E+21           
         88  traj.phases.phase0.indep_states.states:xL_16                       c    -1.000000E+21     2.636126E-02     1.000000E+21           
         89  traj.phases.phase0.indep_states.states:xL_17                       c    -1.000000E+21     2.706348E-02     1.000000E+21           
         90  traj.phases.phase0.indep_states.states:xL_18                       c    -1.000000E+21     2.866017E-02     1.000000E+21           
         91  traj.phases.phase0.indep_states.states:xL_19                       c    -1.000000E+21     3.085194E-02     1.000000E+21           
         92  traj.phases.phase0.indep_states.states:xL_20                       c    -1.000000E+21     3.154682E-02     1.000000E+21           
         93  traj.phases.phase0.indep_states.states:xL_21                       c    -1.000000E+21     3.314444E-02     1.000000E+21           
         94  traj.phases.phase0.indep_states.states:xL_22                       c    -1.000000E+21     3.535428E-02     1.000000E+21           
         95  traj.phases.phase0.indep_states.states:xL_23                       c    -1.000000E+21     3.605249E-02     1.000000E+21           
         96  traj.phases.phase0.indep_states.states:xL_24                       c    -1.000000E+21     3.764650E-02     1.000000E+21           
         97  traj.phases.phase0.indep_states.states:xL_25                       c    -1.000000E+21     3.984146E-02     1.000000E+21           
         98  traj.phases.phase0.indep_states.states:xL_26                       c    -1.000000E+21     4.053686E-02     1.000000E+21           
         99  traj.phases.phase0.indep_states.states:xL_27                       c    -1.000000E+21     4.213246E-02     1.000000E+21           
        100  traj.phases.phase0.indep_states.states:xL_28                       c    -1.000000E+21     4.433741E-02     1.000000E+21           
        101  traj.phases.phase0.indep_states.states:xL_29                       c    -1.000000E+21     4.503499E-02     1.000000E+21           
        102  traj.phases.phase0.indep_states.states:xL_30                       c    -1.000000E+21     4.663060E-02     1.000000E+21           
        103  traj.phases.phase0.indep_states.states:xL_31                       c    -1.000000E+21     4.883023E-02     1.000000E+21           
        104  traj.phases.phase0.indep_states.states:xL_32                       c    -1.000000E+21     4.952649E-02     1.000000E+21           
        105  traj.phases.phase0.indep_states.states:xL_33                       c    -1.000000E+21     5.112172E-02     1.000000E+21           
        106  traj.phases.phase0.indep_states.states:xL_34                       c    -1.000000E+21     5.332400E-02     1.000000E+21           
        107  traj.phases.phase0.indep_states.states:xL_35                       c    -1.000000E+21     5.402108E-02     1.000000E+21           
        108  traj.phases.phase0.indep_states.states:xL_36                       c    -1.000000E+21     5.561780E-02     1.000000E+21           
        109  traj.phases.phase0.indep_states.states:xL_37                       c    -1.000000E+21     5.782753E-02     1.000000E+21           
        110  traj.phases.phase0.indep_states.states:xL_38                       c    -1.000000E+21     5.852989E-02     1.000000E+21           
        111  traj.phases.phase0.indep_states.states:xL_39                       c    -1.000000E+21     6.014464E-02     1.000000E+21           
        112  traj.phases.phase0.indep_states.states:xL_40                       c    -1.000000E+21     6.237891E-02     1.000000E+21           
        113  traj.phases.phase0.indep_states.states:xL_41                       c    -1.000000E+21     6.308537E-02     1.000000E+21           
        114  traj.phases.phase0.indep_states.states:xL_42                       c    -1.000000E+21     6.469909E-02     1.000000E+21           
        115  traj.phases.phase0.indep_states.states:xL_43                       c    -1.000000E+21     6.691798E-02     1.000000E+21           
        116  traj.phases.phase0.indep_states.states:xL_44                       c    -1.000000E+21     6.761887E-02     1.000000E+21           
        117  traj.phases.phase0.indep_states.states:xL_45                       c    -1.000000E+21     6.922202E-02     1.000000E+21           
        118  traj.phases.phase0.indep_states.states:xL_46                       c    -1.000000E+21     7.143053E-02     1.000000E+21           
        119  traj.phases.phase0.indep_states.states:xL_47                       c    -1.000000E+21     7.212878E-02     1.000000E+21           
        120  traj.phases.phase0.indep_states.states:xL_48                       c    -1.000000E+21     7.372692E-02     1.000000E+21           
        121  traj.phases.phase0.indep_states.states:xL_49                       c    -1.000000E+21     7.593320E-02     1.000000E+21           
        122  traj.phases.phase0.indep_states.states:xL_50                       c    -1.000000E+21     7.663252E-02     1.000000E+21           
        123  traj.phases.phase0.indep_states.states:xL_51                       c    -1.000000E+21     7.823661E-02     1.000000E+21           
        124  traj.phases.phase0.indep_states.states:xL_52                       c    -1.000000E+21     8.045454E-02     1.000000E+21           
        125  traj.phases.phase0.indep_states.states:xL_53                       c    -1.000000E+21     8.115719E-02     1.000000E+21           
        126  traj.phases.phase0.indep_states.states:xL_54                       c    -1.000000E+21     8.276610E-02     1.000000E+21           
        127  traj.phases.phase0.indep_states.states:xL_55                       c    -1.000000E+21     8.497961E-02     1.000000E+21           
        128  traj.phases.phase0.indep_states.states:xL_56                       c    -1.000000E+21     8.567705E-02     1.000000E+21           
        129  traj.phases.phase0.indep_states.states:xL_57                       c    -1.000000E+21     8.726795E-02     1.000000E+21           
        130  traj.phases.phase0.indep_states.states:xL_58                       c    -1.000000E+21     8.946296E-02     1.000000E+21           
        131  traj.phases.phase0.indep_states.states:xL_59                       c    -1.000000E+21     9.016126E-02     1.000000E+21           
        132  traj.phases.phase0.indep_states.states:xL_60                       c    -1.000000E+21     9.176979E-02     1.000000E+21           
        133  traj.phases.phase0.indep_states.states:xL_61                       c    -1.000000E+21     9.399414E-02     1.000000E+21           
        134  traj.phases.phase0.indep_states.states:xL_62                       c    -1.000000E+21     9.469488E-02     1.000000E+21           
        135  traj.phases.phase0.indep_states.states:xL_63                       c    -1.000000E+21     9.628902E-02     1.000000E+21           
        136  traj.phases.phase0.indep_states.states:xL_64                       c    -1.000000E+21     9.847679E-02     1.000000E+21           
        137  traj.phases.phase0.indep_states.states:xL_65                       c    -1.000000E+21     9.916964E-02     1.000000E+21           
        138  traj.phases.phase0.indep_states.states:xL_66                       c    -1.000000E+21     1.007607E-01     1.000000E+21           
        139  traj.phases.phase0.indep_states.states:xL_67                       c    -1.000000E+21     1.029616E-01     1.000000E+21           
        140  traj.phases.phase0.indep_states.states:xL_68                       c    -1.000000E+21     1.036582E-01     1.000000E+21           
        141  traj.phases.phase0.indep_states.states:xL_69                       c    -1.000000E+21     1.052515E-01     1.000000E+21           
        142  traj.phases.phase0.indep_states.states:xL_70                       c    -1.000000E+21     1.074472E-01     1.000000E+21           
        143  traj.phases.phase0.indep_states.states:xL_71                       c    -1.000000E+21     1.081419E-01     1.000000E+21           
        144  traj.phases.phase0.control_group.indep_controls.controls:u_0       c    -1.000000E+21    -1.481643E-01     1.000000E+21           
        145  traj.phases.phase0.control_group.indep_controls.controls:u_1       c    -1.000000E+21    -1.459267E-01     1.000000E+21           
        146  traj.phases.phase0.control_group.indep_controls.controls:u_2       c    -1.000000E+21    -1.461810E-01     1.000000E+21           
        147  traj.phases.phase0.control_group.indep_controls.controls:u_3       c    -1.000000E+21    -1.470687E-01     1.000000E+21           
        148  traj.phases.phase0.control_group.indep_controls.controls:u_4       c    -1.000000E+21    -1.487109E-01     1.000000E+21           
        149  traj.phases.phase0.control_group.indep_controls.controls:u_5       c    -1.000000E+21    -1.482349E-01     1.000000E+21           
        150  traj.phases.phase0.control_group.indep_controls.controls:u_6       c    -1.000000E+21    -1.474218E-01     1.000000E+21           
        151  traj.phases.phase0.control_group.indep_controls.controls:u_7       c    -1.000000E+21    -1.458725E-01     1.000000E+21           
        152  traj.phases.phase0.control_group.indep_controls.controls:u_8       c    -1.000000E+21    -1.459583E-01     1.000000E+21           
        153  traj.phases.phase0.control_group.indep_controls.controls:u_9       c    -1.000000E+21    -1.465227E-01     1.000000E+21           
        154  traj.phases.phase0.control_group.indep_controls.controls:u_10      c    -1.000000E+21    -1.476574E-01     1.000000E+21           
        155  traj.phases.phase0.control_group.indep_controls.controls:u_11      c    -1.000000E+21    -1.477346E-01     1.000000E+21           
        156  traj.phases.phase0.control_group.indep_controls.controls:u_12      c    -1.000000E+21    -1.473995E-01     1.000000E+21           
        157  traj.phases.phase0.control_group.indep_controls.controls:u_13      c    -1.000000E+21    -1.467599E-01     1.000000E+21           
        158  traj.phases.phase0.control_group.indep_controls.controls:u_14      c    -1.000000E+21    -1.469477E-01     1.000000E+21           
        159  traj.phases.phase0.control_group.indep_controls.controls:u_15      c    -1.000000E+21    -1.472657E-01     1.000000E+21           
        160  traj.phases.phase0.control_group.indep_controls.controls:u_16      c    -1.000000E+21    -1.478279E-01     1.000000E+21           
        161  traj.phases.phase0.control_group.indep_controls.controls:u_17      c    -1.000000E+21    -1.475920E-01     1.000000E+21           
        162  traj.phases.phase0.control_group.indep_controls.controls:u_18      c    -1.000000E+21    -1.472729E-01     1.000000E+21           
        163  traj.phases.phase0.control_group.indep_controls.controls:u_19      c    -1.000000E+21    -1.466506E-01     1.000000E+21           
        164  traj.phases.phase0.control_group.indep_controls.controls:u_20      c    -1.000000E+21    -1.465902E-01     1.000000E+21           
        165  traj.phases.phase0.control_group.indep_controls.controls:u_21      c    -1.000000E+21    -1.467639E-01     1.000000E+21           
        166  traj.phases.phase0.control_group.indep_controls.controls:u_22      c    -1.000000E+21    -1.471121E-01     1.000000E+21           
        167  traj.phases.phase0.control_group.indep_controls.controls:u_23      c    -1.000000E+21    -1.470815E-01     1.000000E+21           
        168  traj.phases.phase0.control_group.indep_controls.controls:u_24      c    -1.000000E+21    -1.469484E-01     1.000000E+21           
        169  traj.phases.phase0.control_group.indep_controls.controls:u_25      c    -1.000000E+21    -1.466855E-01     1.000000E+21           
        170  traj.phases.phase0.control_group.indep_controls.controls:u_26      c    -1.000000E+21    -1.466842E-01     1.000000E+21           
        171  traj.phases.phase0.control_group.indep_controls.controls:u_27      c    -1.000000E+21    -1.467710E-01     1.000000E+21           
        172  traj.phases.phase0.control_group.indep_controls.controls:u_28      c    -1.000000E+21    -1.469501E-01     1.000000E+21           
        173  traj.phases.phase0.control_group.indep_controls.controls:u_29      c    -1.000000E+21    -1.469742E-01     1.000000E+21           
        174  traj.phases.phase0.control_group.indep_controls.controls:u_30      c    -1.000000E+21    -1.469280E-01     1.000000E+21           
        175  traj.phases.phase0.control_group.indep_controls.controls:u_31      c    -1.000000E+21    -1.468278E-01     1.000000E+21           
        176  traj.phases.phase0.control_group.indep_controls.controls:u_32      c    -1.000000E+21    -1.468058E-01     1.000000E+21           
        177  traj.phases.phase0.control_group.indep_controls.controls:u_33      c    -1.000000E+21    -1.468269E-01     1.000000E+21           
        178  traj.phases.phase0.control_group.indep_controls.controls:u_34      c    -1.000000E+21    -1.468773E-01     1.000000E+21           
        179  traj.phases.phase0.control_group.indep_controls.controls:u_35      c    -1.000000E+21    -1.469029E-01     1.000000E+21           
        180  traj.phases.phase0.control_group.indep_controls.controls:u_36      c    -1.000000E+21    -1.469004E-01     1.000000E+21           
        181  traj.phases.phase0.control_group.indep_controls.controls:u_37      c    -1.000000E+21    -1.469755E-01     1.000000E+21           
        182  traj.phases.phase0.control_group.indep_controls.controls:u_38      c    -1.000000E+21    -1.473643E-01     1.000000E+21           
        183  traj.phases.phase0.control_group.indep_controls.controls:u_39      c    -1.000000E+21    -1.475562E-01     1.000000E+21           
        184  traj.phases.phase0.control_group.indep_controls.controls:u_40      c    -1.000000E+21    -1.478991E-01     1.000000E+21           
        185  traj.phases.phase0.control_group.indep_controls.controls:u_41      c    -1.000000E+21    -1.479304E-01     1.000000E+21           
        186  traj.phases.phase0.control_group.indep_controls.controls:u_42      c    -1.000000E+21    -1.478336E-01     1.000000E+21           
        187  traj.phases.phase0.control_group.indep_controls.controls:u_43      c    -1.000000E+21    -1.475823E-01     1.000000E+21           
        188  traj.phases.phase0.control_group.indep_controls.controls:u_44      c    -1.000000E+21    -1.473310E-01     1.000000E+21           
        189  traj.phases.phase0.control_group.indep_controls.controls:u_45      c    -1.000000E+21    -1.472745E-01     1.000000E+21           
        190  traj.phases.phase0.control_group.indep_controls.controls:u_46      c    -1.000000E+21    -1.471658E-01     1.000000E+21           
        191  traj.phases.phase0.control_group.indep_controls.controls:u_47      c    -1.000000E+21    -1.470419E-01     1.000000E+21           
        192  traj.phases.phase0.control_group.indep_controls.controls:u_48      c    -1.000000E+21    -1.470089E-01     1.000000E+21           
        193  traj.phases.phase0.control_group.indep_controls.controls:u_49      c    -1.000000E+21    -1.469835E-01     1.000000E+21           
        194  traj.phases.phase0.control_group.indep_controls.controls:u_50      c    -1.000000E+21    -1.471012E-01     1.000000E+21           
        195  traj.phases.phase0.control_group.indep_controls.controls:u_51      c    -1.000000E+21    -1.471754E-01     1.000000E+21           
        196  traj.phases.phase0.control_group.indep_controls.controls:u_52      c    -1.000000E+21    -1.473385E-01     1.000000E+21           
        197  traj.phases.phase0.control_group.indep_controls.controls:u_53      c    -1.000000E+21    -1.474746E-01     1.000000E+21           
        198  traj.phases.phase0.control_group.indep_controls.controls:u_54      c    -1.000000E+21    -1.474962E-01     1.000000E+21           
        199  traj.phases.phase0.control_group.indep_controls.controls:u_55      c    -1.000000E+21    -1.474371E-01     1.000000E+21           
        200  traj.phases.phase0.control_group.indep_controls.controls:u_56      c    -1.000000E+21    -1.470377E-01     1.000000E+21           
        201  traj.phases.phase0.control_group.indep_controls.controls:u_57      c    -1.000000E+21    -1.468346E-01     1.000000E+21           
        202  traj.phases.phase0.control_group.indep_controls.controls:u_58      c    -1.000000E+21    -1.465508E-01     1.000000E+21           
        203  traj.phases.phase0.control_group.indep_controls.controls:u_59      c    -1.000000E+21    -1.468941E-01     1.000000E+21           
        204  traj.phases.phase0.control_group.indep_controls.controls:u_60      c    -1.000000E+21    -1.471802E-01     1.000000E+21           
        205  traj.phases.phase0.control_group.indep_controls.controls:u_61      c    -1.000000E+21    -1.476597E-01     1.000000E+21           
        206  traj.phases.phase0.control_group.indep_controls.controls:u_62      c    -1.000000E+21    -1.474240E-01     1.000000E+21           
        207  traj.phases.phase0.control_group.indep_controls.controls:u_63      c    -1.000000E+21    -1.471326E-01     1.000000E+21           
        208  traj.phases.phase0.control_group.indep_controls.controls:u_64      c    -1.000000E+21    -1.465477E-01     1.000000E+21           
        209  traj.phases.phase0.control_group.indep_controls.controls:u_65      c    -1.000000E+21    -1.464033E-01     1.000000E+21           
        210  traj.phases.phase0.control_group.indep_controls.controls:u_66      c    -1.000000E+21    -1.465177E-01     1.000000E+21           
        211  traj.phases.phase0.control_group.indep_controls.controls:u_67      c    -1.000000E+21    -1.467820E-01     1.000000E+21           
        212  traj.phases.phase0.control_group.indep_controls.controls:u_68      c    -1.000000E+21    -1.468647E-01     1.000000E+21           
        213  traj.phases.phase0.control_group.indep_controls.controls:u_69      c    -1.000000E+21    -1.468228E-01     1.000000E+21           
        214  traj.phases.phase0.control_group.indep_controls.controls:u_70      c    -1.000000E+21    -1.467156E-01     1.000000E+21           
        215  traj.phases.phase0.control_group.indep_controls.controls:u_71      c    -1.000000E+21    -1.466535E-01     1.000000E+21           

   Constraints (i - inequality, e - equality)
      Index  Name                                                           Type          Lower           Value           Upper    Status  Lagrange Multiplier (N/A)
          0  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    6.823246E-16    0.000000E+00              9.00000E+100
          1  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.098658E-16    0.000000E+00              9.00000E+100
          2  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    8.095376E-16    0.000000E+00              9.00000E+100
          3  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    9.020562E-16    0.000000E+00              9.00000E+100
          4  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -4.163336E-16    0.000000E+00              9.00000E+100
          5  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -1.098658E-16    0.000000E+00              9.00000E+100
          6  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    6.765422E-16    0.000000E+00              9.00000E+100
          7  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -3.989864E-16    0.000000E+00              9.00000E+100
          8  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -3.295975E-16    0.000000E+00              9.00000E+100
          9  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -6.476301E-16    0.000000E+00              9.00000E+100
         10  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.792548E-16    0.000000E+00              9.00000E+100
         11  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -3.989864E-16    0.000000E+00              9.00000E+100
         12  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -5.204170E-17    0.000000E+00              9.00000E+100
         13  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    8.095376E-17    0.000000E+00              9.00000E+100
         14  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.619075E-16    0.000000E+00              9.00000E+100
         15  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -3.932040E-16    0.000000E+00              9.00000E+100
         16  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    2.486437E-16    0.000000E+00              9.00000E+100
         17  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -7.228014E-16    0.000000E+00              9.00000E+100
         18  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.370789E-16    0.000000E+00              9.00000E+100
         19  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.312965E-17    0.000000E+00              9.00000E+100
         20  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.156482E-16    0.000000E+00              9.00000E+100
         21  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    2.717733E-16    0.000000E+00              9.00000E+100
         22  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -1.040834E-16    0.000000E+00              9.00000E+100
         23  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    4.278985E-16    0.000000E+00              9.00000E+100
         24  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    3.642919E-16    0.000000E+00              9.00000E+100
         25  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -1.619075E-16    0.000000E+00              9.00000E+100
         26  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    3.238150E-16    0.000000E+00              9.00000E+100
         27  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    5.782412E-17    0.000000E+00              9.00000E+100
         28  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.370789E-16    0.000000E+00              9.00000E+100
         29  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -7.517135E-17    0.000000E+00              9.00000E+100
         30  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    4.683753E-16    0.000000E+00              9.00000E+100
         31  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -4.625929E-17    0.000000E+00              9.00000E+100
         32  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.272131E-16    0.000000E+00              9.00000E+100
         33  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.602085E-16    0.000000E+00              9.00000E+100
         34  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    1.792548E-16    0.000000E+00              9.00000E+100
         35  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.544261E-16    0.000000E+00              9.00000E+100
         36  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -1.908196E-16    0.000000E+00              9.00000E+100
         37  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00   -2.370789E-16    0.000000E+00              9.00000E+100
         38  traj.phases.phase0.collocation_constraint.defects:x               e   0.000000E+00    4.047688E-17    0.000000E+00              9.00000E+100
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        168  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    2.775558E-17    0.000000E+00              9.00000E+100
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        170  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00   -2.775558E-17    0.000000E+00              9.00000E+100
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        176  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00   -2.775558E-17    0.000000E+00              9.00000E+100
        177  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    2.775558E-17    0.000000E+00              9.00000E+100
        178  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    2.775558E-17    0.000000E+00              9.00000E+100
        179  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00   -2.775558E-17    0.000000E+00              9.00000E+100
        180  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        181  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        182  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    2.775558E-17    0.000000E+00              9.00000E+100
        183  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        184  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00   -2.775558E-17    0.000000E+00              9.00000E+100
        185  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        186  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    2.775558E-17    0.000000E+00              9.00000E+100
        187  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        188  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
        189  traj.phases.phase0.continuity_comp.defect_controls:u              e   0.000000E+00   -2.775558E-17    0.000000E+00              9.00000E+100


   Exit Status
      Inform  Description
           0  Optimization terminated successfully.
--------------------------------------------------------------------------------
Simulating trajectory traj
Model viewer data has already been recorded for Driver.
Done simulating trajectory traj
False

Plotting the solution#

The recommended practice is to obtain values from the recorded cases. While the problem object can also be queried for values, building plotting scripts that use the case recorder files as the data source means that the problem doesn’t need to be solved just to change a plot. Here we load values of various variables from the solution and simulation for use in the animation to follow.

sol = om.CaseReader('dymos_solution.db').get_case('final')
sim = om.CaseReader('dymos_simulation.db').get_case('final')

t = sol.get_val('traj.phase0.timeseries.time')
x = sol.get_val('traj.phase0.timeseries.x')
xL = sol.get_val('traj.phase0.timeseries.xL')
u = sol.get_val('traj.phase0.timeseries.u')

t_sim = sim.get_val('traj.phase0.timeseries.time')
x_sim = sim.get_val('traj.phase0.timeseries.x')
xL_sim = sim.get_val('traj.phase0.timeseries.xL')
u_sim = sim.get_val('traj.phase0.timeseries.u')

fig = plt.figure(constrained_layout=True, figsize=(12, 4))
gs = fig.add_gridspec(3, 1)

x_ax = fig.add_subplot(gs[0, 0])
xL_ax = fig.add_subplot(gs[1, 0])
u_ax = fig.add_subplot(gs[2, 0])

x_ax.set_ylabel('x ($m$)')
xL_ax.set_ylabel('xL ($m/s$)')
u_ax.set_ylabel('u ($m/s^2$)')
xL_ax.set_xlabel('t (s)')
u_ax.set_xlabel('t (s)')

x_sol_handle, = x_ax.plot(t, x, 'o', ms=1)
xL_ax.plot(t, xL, 'o', ms=1)
u_ax.plot(t, u, 'o', ms=1)

x_sim_handle, = x_ax.plot(t_sim, x_sim, '-', ms=1)
xL_ax.plot(t_sim, xL_sim, '-', ms=1)
u_ax.plot(t_sim, u_sim, '-', ms=1)

for ax in [x_ax, xL_ax, u_ax]:
    ax.grid(True, alpha=0.2)
    
plt.figlegend([x_sol_handle, x_sim_handle], ['solution', 'simulation'], ncol=2, loc='lower center');
../../_images/5ca295e99b6950bae12dab476fa175cf5aef2659a1f782085c842f3bb02a1bb6.png

References#

[Hul03]

David G Hull. Optimal Control Theory for Applications. Springer, 2003.