The Mountain Car Problem#
The mountain car problem proposes a vehicle stuck in a “well.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.
The problem is a popular machine learning test case, though the methods in Dymos are capable of solving it. It first appeared in the PhD thesis of Andrew Moore in 1990. [Moo90]. The implementation here is based on that given by Melnikov, Makmal, and Briegel [MMB14].
State and control variables#
This system has two state variables, the position (\(x\)) and velocity (\(v\)) of the car.
This system has a single control variable (\(u\)), the effort put into moving. This control is contrained to the range \([-1 \, 1]\).
The dynamics of the system are governed by
(65)#\[\begin{align}
\dot{x} &= v \\
\dot{v} &= 0.001 * u - 0.0025 * \cos(3 x)
\end{align}\]
Problem Definition#
We seek to minimize the time required to exit the well in the positive direction.
(66)#\[\begin{align}
\mathrm{Minimize} \, J &= t_f
\end{align}\]
Subject to the initial conditions
(67)#\[\begin{align}
x_0 &= -0.5 \\
v_0 &= 0.0
\end{align}\]
the control constraints
(68)#\[\begin{align}
|u| \le 1
\end{align}\]
and the terminal constraints
(69)#\[\begin{align}
x_f &= 0.5 \\
v_f &\ge 0.0
\end{align}\]
Defining the ODE#
The following code implements the equations of motion for the mountain car problem.
A few things to note:
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.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.We only use compute_partials to override the values of \(\frac{\partial \dot{v}}{\partial x}\) because \(\frac{\partial \dot{v}}{\partial u}\) and \(\frac{\partial \dot{x}}{\partial v}\) are constant and their value is specified during
setup.
import numpy as np
import openmdao.api as om
class MountainCarODE(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
self.add_input('x', shape=(nn,), units='m')
self.add_input('v', shape=(nn,), units='m/s')
self.add_input('u', shape=(nn,), units='unitless')
self.add_output('x_dot', shape=(nn,), units='m/s',
tags=['dymos.state_rate_source:x', 'dymos.state_units:m'])
self.add_output('v_dot', shape=(nn,), units='m/s**2',
tags=['dymos.state_rate_source:v', 'dymos.state_units:m/s'])
ar = np.arange(nn, dtype=int)
self.declare_partials(of='x_dot', wrt='v', rows=ar, cols=ar, val=1.0)
self.declare_partials(of='v_dot', wrt='u', rows=ar, cols=ar, val=0.001)
self.declare_partials(of='v_dot', wrt='x', rows=ar, cols=ar)
def compute(self, inputs, outputs):
x = inputs['x']
v = inputs['v']
u = inputs['u']
outputs['x_dot'] = v
outputs['v_dot'] = 0.001 * u - 0.0025 * np.cos(3*x)
def compute_partials(self, inputs, partials):
x = inputs['x']
partials['v_dot', 'x'] = 3 * 0.0025 * np.sin(3 * x)
Solving the minimum-time mountain car problem with Dymos#
The following script solves the minimum-time mountain car problem with Dymos.
Note that this example requires the IPOPT optimizer via the pyoptsparse package.
Scipy’s SLSQP optimizer is generally not capable of solving this problem.
To begin, import the packages we require:
import dymos as dm
import matplotlib.pyplot as plt
from matplotlib import animation
Next, we set two constants.
U_MAX is the maximum allowable magnitude of the acceleration.
The references show this problem being solved with \(-1 \le u \le 1\).
Variable NUM_SEG is the number of equally spaced polynomial segments into which time is being divided.
Within each of these segments, the time-history of each state and control is being treated as a polynomial (we’re using the default order of 3).
# The maximum absolute value of the acceleration authority of the car
U_MAX = 1.0
# The number of segments into which the problem is discretized
NUM_SEG = 30
We then instantiate an OpenMDAO problem and set the optimizer and its options.
For IPOPT, setting option nlp_scaling_method to 'gradient-based' can substantially improve the convergence of the optimizer without the need for us to set all of the scaling manually.
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(optimizer='IPOPT')
p.driver.opt_settings['print_level'] = 0
p.driver.opt_settings['max_iter'] = 500
p.driver.opt_settings['mu_strategy'] = 'adaptive'
p.driver.opt_settings['bound_mult_init_method'] = 'mu-based'
p.driver.opt_settings['tol'] = 1.0E-8
p.driver.opt_settings['nlp_scaling_method'] = 'gradient-based' # for faster convergence
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=NUM_SEG)
phase = traj.add_phase('phase0', dm.Phase(ode_class=MountainCarODE, 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 v) and a single control (u).
There are no parameters and no polynomial controls. We could have tried to use a polynomial control here, but as we will see the solution contains large discontinuities in the control value, which make it ill-suited for a polynomial control. Polynomial controls are modeled as a single (typically low-order) polynomial across the entire phase.
We’re fixing the initial time and states to whatever values we provide before executing the problem. We will constrain the final values with nonlinear constraints in the next step.
The scaler values (ref) are all set to 1 here. We’re using IPOPT’s gradient-based scaling option and will let it work the scaling out for us.
Bounds on time duration are guesses, and the bounds on the states and controls come from the implementation in the references.
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, duration_bounds=(.05, 10000), duration_ref=1)
phase.add_state('x', fix_initial=True, fix_final=False, lower=-1.2, upper=0.5, ref=1, defect_ref=1)
phase.add_state('v', fix_initial=True, fix_final=False, lower=-0.07, upper=0.07, ref=1, defect_ref=1)
phase.add_control('u', lower=-U_MAX, upper=U_MAX, ref=1, continuity=True, rate_continuity=False)
Next we define the optimal control problem by specifying the objective, boundary constraints, and path constraints.
Why do we have a path constraint on the control u when we’ve already specified its bounds?
Excellent question!
In the Radau transcription, the \(n^{th}\) order control polynomial is governed by design variables provided at \(n\) points in the segment that do not contain the right-most endpoint.
Instead, this value is interpolated based on the values of the first \((n-1)\).
Since this value is not a design variable, it is necessary to constrain its value separately.
We could forgo specifying any bounds on u since it’s completely covered by the path constraint, but specifying the bounds on the design variable values can sometimes help by telling the optimizer, “Don’t even bother trying values outside of this range.”.
Note that sometimes the opposite is true, and giving the optimizer the freedom to explore a larger design space, only to eventually be “reined-in” by the path constraint can sometimes be valuable.
The purpose of this interactive documentation is to let the user experiment. If you remove the path constraint, you might notice some outlying control values in the solution below.
#
# Minimize time at the end of the phase
#
phase.add_objective('time', loc='final', ref=1000)
phase.add_boundary_constraint('x', loc='final', lower=0.5)
phase.add_boundary_constraint('v', loc='final', lower=0.0)
phase.add_path_constraint('u', lower=-U_MAX, upper=U_MAX)
#
# Setup the Problem
#
p.setup()
<openmdao.core.problem.Problem at 0x7f624f10d6a0>
We then set the initial guesses for the variables in the problem and solve it.
Since fix_initial=True is set for time and the states, those values are not design variables and will remain at the values given below throughout the solution process.
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
#
phase.set_time_val(initial=0.0, duration=500.0)
phase.set_state_val('x', [-0.5, 0.5])
phase.set_state_val('v', [0, 0.07])
phase.set_control_val('u', [0, np.sin(1.0)])
#
# Solve for the optimal trajectory
#
dm.run_problem(p, run_driver=True, simulate=True, refine_method='ph', refine_iteration_limit=5)
Show code cell output
Hide code cell output
Jacobian shape: (332, 271) (1.62% nonzero)
FWD solves: 9 REV solves: 0
Total colors vs. total size: 9 vs 271 (96.68% improvement)
Sparsity computed using tolerance: 1e-25.
Dense total jacobian for Problem 'problem' was computed 3 times.
Time to compute sparsity: 0.3595 sec
Time to compute coloring: 0.2619 sec
Memory to compute coloring: 0.3750 MB
Coloring created on: 2025-07-16 18:53:03
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 3.9572
User Objective Time : 0.1383
User Sensitivity Time : 1.9241
Interface Time : 0.6307
Opt Solver Time: 1.2641
Calls to Objective Function : 93
Calls to Sens Function : 93
Objectives
Index Name Value
0 traj.phase0.t 1.027055E-01
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.phase0.t_duration_0 c 5.000000E-02 1.027055E+02 1.000000E+04
1 traj.phase0.states:x_0 c -1.200000E+00 -4.993918E-01 5.000000E-01
2 traj.phase0.states:x_1 c -1.200000E+00 -4.965749E-01 5.000000E-01
3 traj.phase0.states:x_2 c -1.200000E+00 -4.952113E-01 5.000000E-01
4 traj.phase0.states:x_3 c -1.200000E+00 -4.912601E-01 5.000000E-01
5 traj.phase0.states:x_4 c -1.200000E+00 -4.839852E-01 5.000000E-01
6 traj.phase0.states:x_5 c -1.200000E+00 -4.812613E-01 5.000000E-01
7 traj.phase0.states:x_6 c -1.200000E+00 -4.743104E-01 5.000000E-01
8 traj.phase0.states:x_7 c -1.200000E+00 -4.632080E-01 5.000000E-01
9 traj.phase0.states:x_8 c -1.200000E+00 -4.593596E-01 5.000000E-01
10 traj.phase0.states:x_9 c -1.200000E+00 -4.499012E-01 5.000000E-01
11 traj.phase0.states:x_10 c -1.200000E+00 -4.363637E-01 5.000000E-01
12 traj.phase0.states:x_11 c -1.200000E+00 -4.322699E-01 5.000000E-01
13 traj.phase0.states:x_12 c -1.200000E+00 -4.247954E-01 5.000000E-01
14 traj.phase0.states:x_13 c -1.200000E+00 -4.186817E-01 5.000000E-01
15 traj.phase0.states:x_14 c -1.200000E+00 -4.177845E-01 5.000000E-01
16 traj.phase0.states:x_15 c -1.200000E+00 -4.176174E-01 5.000000E-01
17 traj.phase0.states:x_16 c -1.200000E+00 -4.216949E-01 5.000000E-01
18 traj.phase0.states:x_17 c -1.200000E+00 -4.240203E-01 5.000000E-01
19 traj.phase0.states:x_18 c -1.200000E+00 -4.311748E-01 5.000000E-01
20 traj.phase0.states:x_19 c -1.200000E+00 -4.451029E-01 5.000000E-01
21 traj.phase0.states:x_20 c -1.200000E+00 -4.504563E-01 5.000000E-01
22 traj.phase0.states:x_21 c -1.200000E+00 -4.643312E-01 5.000000E-01
23 traj.phase0.states:x_22 c -1.200000E+00 -4.869305E-01 5.000000E-01
24 traj.phase0.states:x_23 c -1.200000E+00 -4.948559E-01 5.000000E-01
25 traj.phase0.states:x_24 c -1.200000E+00 -5.142624E-01 5.000000E-01
26 traj.phase0.states:x_25 c -1.200000E+00 -5.435816E-01 5.000000E-01
27 traj.phase0.states:x_26 c -1.200000E+00 -5.533911E-01 5.000000E-01
28 traj.phase0.states:x_27 c -1.200000E+00 -5.766415E-01 5.000000E-01
29 traj.phase0.states:x_28 c -1.200000E+00 -6.101308E-01 5.000000E-01
30 traj.phase0.states:x_29 c -1.200000E+00 -6.209737E-01 5.000000E-01
31 traj.phase0.states:x_30 c -1.200000E+00 -6.460627E-01 5.000000E-01
32 traj.phase0.states:x_31 c -1.200000E+00 -6.808767E-01 5.000000E-01
33 traj.phase0.states:x_32 c -1.200000E+00 -6.918463E-01 5.000000E-01
34 traj.phase0.states:x_33 c -1.200000E+00 -7.167113E-01 5.000000E-01
35 traj.phase0.states:x_34 c -1.200000E+00 -7.500849E-01 5.000000E-01
36 traj.phase0.states:x_35 c -1.200000E+00 -7.603375E-01 5.000000E-01
37 traj.phase0.states:x_36 c -1.200000E+00 -7.831224E-01 5.000000E-01
38 traj.phase0.states:x_37 c -1.200000E+00 -8.126995E-01 5.000000E-01
39 traj.phase0.states:x_38 c -1.200000E+00 -8.215465E-01 5.000000E-01
40 traj.phase0.states:x_39 c -1.200000E+00 -8.407829E-01 5.000000E-01
41 traj.phase0.states:x_40 c -1.200000E+00 -8.647953E-01 5.000000E-01
42 traj.phase0.states:x_41 c -1.200000E+00 -8.717406E-01 5.000000E-01
43 traj.phase0.states:x_42 c -1.200000E+00 -8.865112E-01 5.000000E-01
44 traj.phase0.states:x_43 c -1.200000E+00 -9.033494E-01 5.000000E-01
45 traj.phase0.states:x_44 c -1.200000E+00 -9.075278E-01 5.000000E-01
46 traj.phase0.states:x_45 c -1.200000E+00 -9.135009E-01 5.000000E-01
47 traj.phase0.states:x_46 c -1.200000E+00 -9.137393E-01 5.000000E-01
48 traj.phase0.states:x_47 c -1.200000E+00 -9.118764E-01 5.000000E-01
49 traj.phase0.states:x_48 c -1.200000E+00 -9.041177E-01 5.000000E-01
50 traj.phase0.states:x_49 c -1.200000E+00 -8.854928E-01 5.000000E-01
51 traj.phase0.states:x_50 c -1.200000E+00 -8.777040E-01 5.000000E-01
52 traj.phase0.states:x_51 c -1.200000E+00 -8.565062E-01 5.000000E-01
53 traj.phase0.states:x_52 c -1.200000E+00 -8.197803E-01 5.000000E-01
54 traj.phase0.states:x_53 c -1.200000E+00 -8.064055E-01 5.000000E-01
55 traj.phase0.states:x_54 c -1.200000E+00 -7.727638E-01 5.000000E-01
56 traj.phase0.states:x_55 c -1.200000E+00 -7.198820E-01 5.000000E-01
57 traj.phase0.states:x_56 c -1.200000E+00 -7.016948E-01 5.000000E-01
58 traj.phase0.states:x_57 c -1.200000E+00 -6.576999E-01 5.000000E-01
59 traj.phase0.states:x_58 c -1.200000E+00 -5.923141E-01 5.000000E-01
60 traj.phase0.states:x_59 c -1.200000E+00 -5.706634E-01 5.000000E-01
61 traj.phase0.states:x_60 c -1.200000E+00 -5.197354E-01 5.000000E-01
62 traj.phase0.states:x_61 c -1.200000E+00 -4.472328E-01 5.000000E-01
63 traj.phase0.states:x_62 c -1.200000E+00 -4.239589E-01 5.000000E-01
64 traj.phase0.states:x_63 c -1.200000E+00 -3.704651E-01 5.000000E-01
65 traj.phase0.states:x_64 c -1.200000E+00 -2.970090E-01 5.000000E-01
66 traj.phase0.states:x_65 c -1.200000E+00 -2.740402E-01 5.000000E-01
67 traj.phase0.states:x_66 c -1.200000E+00 -2.222405E-01 5.000000E-01
68 traj.phase0.states:x_67 c -1.200000E+00 -1.531485E-01 5.000000E-01
69 traj.phase0.states:x_68 c -1.200000E+00 -1.319828E-01 5.000000E-01
70 traj.phase0.states:x_69 c -1.200000E+00 -8.491756E-02 5.000000E-01
71 traj.phase0.states:x_70 c -1.200000E+00 -2.341765E-02 5.000000E-01
72 traj.phase0.states:x_71 c -1.200000E+00 -4.829314E-03 5.000000E-01
73 traj.phase0.states:x_72 c -1.200000E+00 3.615594E-02 5.000000E-01
74 traj.phase0.states:x_73 c -1.200000E+00 8.913341E-02 5.000000E-01
75 traj.phase0.states:x_74 c -1.200000E+00 1.050566E-01 5.000000E-01
76 traj.phase0.states:x_75 c -1.200000E+00 1.400830E-01 5.000000E-01
77 traj.phase0.states:x_76 c -1.200000E+00 1.853435E-01 5.000000E-01
78 traj.phase0.states:x_77 c -1.200000E+00 1.989849E-01 5.000000E-01
79 traj.phase0.states:x_78 c -1.200000E+00 2.291199E-01 5.000000E-01
80 traj.phase0.states:x_79 c -1.200000E+00 2.684861E-01 5.000000E-01
81 traj.phase0.states:x_80 c -1.200000E+00 2.804879E-01 5.000000E-01
82 traj.phase0.states:x_81 c -1.200000E+00 3.072965E-01 5.000000E-01
83 traj.phase0.states:x_82 c -1.200000E+00 3.430993E-01 5.000000E-01
84 traj.phase0.states:x_83 c -1.200000E+00 3.542309E-01 5.000000E-01
85 traj.phase0.states:x_84 c -1.200000E+00 3.795258E-01 5.000000E-01
86 traj.phase0.states:x_85 c -1.200000E+00 4.143692E-01 5.000000E-01
87 traj.phase0.states:x_86 c -1.200000E+00 4.254779E-01 5.000000E-01
88 traj.phase0.states:x_87 c -1.200000E+00 4.512448E-01 5.000000E-01
89 traj.phase0.states:x_88 c -1.200000E+00 4.879814E-01 5.000000E-01
90 traj.phase0.states:x_89 c -1.200000E+00 5.000000E-01 5.000000E-01 u
91 traj.phase0.states:v_0 c -7.000000E-02 9.987135E-04 7.000000E-02
92 traj.phase0.states:v_1 c -7.000000E-02 2.356393E-03 7.000000E-02
93 traj.phase0.states:v_2 c -7.000000E-02 2.777103E-03 7.000000E-02
94 traj.phase0.states:v_3 c -7.000000E-02 3.717045E-03 7.000000E-02
95 traj.phase0.states:v_4 c -7.000000E-02 4.944779E-03 7.000000E-02
96 traj.phase0.states:v_5 c -7.000000E-02 5.312908E-03 7.000000E-02
97 traj.phase0.states:v_6 c -7.000000E-02 6.112789E-03 7.000000E-02
98 traj.phase0.states:v_7 c -7.000000E-02 7.105114E-03 7.000000E-02
99 traj.phase0.states:v_8 c -7.000000E-02 7.389280E-03 7.000000E-02
100 traj.phase0.states:v_9 c -7.000000E-02 8.059964E-03 7.000000E-02
101 traj.phase0.states:v_10 c -7.000000E-02 7.867032E-03 7.000000E-02
102 traj.phase0.states:v_11 c -7.000000E-02 7.181152E-03 7.000000E-02
103 traj.phase0.states:v_12 c -7.000000E-02 5.108843E-03 7.000000E-02
104 traj.phase0.states:v_13 c -7.000000E-02 2.165345E-03 7.000000E-02
105 traj.phase0.states:v_14 c -7.000000E-02 1.221823E-03 7.000000E-02
106 traj.phase0.states:v_15 c -7.000000E-02 -9.448477E-04 7.000000E-02
107 traj.phase0.states:v_16 c -7.000000E-02 -3.913370E-03 7.000000E-02
108 traj.phase0.states:v_17 c -7.000000E-02 -4.839336E-03 7.000000E-02
109 traj.phase0.states:v_18 c -7.000000E-02 -6.919803E-03 7.000000E-02
110 traj.phase0.states:v_19 c -7.000000E-02 -9.664862E-03 7.000000E-02
111 traj.phase0.states:v_20 c -7.000000E-02 -1.049505E-02 7.000000E-02
112 traj.phase0.states:v_21 c -7.000000E-02 -1.231170E-02 7.000000E-02
113 traj.phase0.states:v_22 c -7.000000E-02 -1.459419E-02 7.000000E-02
114 traj.phase0.states:v_23 c -7.000000E-02 -1.525502E-02 7.000000E-02
115 traj.phase0.states:v_24 c -7.000000E-02 -1.664514E-02 7.000000E-02
116 traj.phase0.states:v_25 c -7.000000E-02 -1.825856E-02 7.000000E-02
117 traj.phase0.states:v_26 c -7.000000E-02 -1.869027E-02 7.000000E-02
118 traj.phase0.states:v_27 c -7.000000E-02 -1.952984E-02 7.000000E-02
119 traj.phase0.states:v_28 c -7.000000E-02 -2.033778E-02 7.000000E-02
120 traj.phase0.states:v_29 c -7.000000E-02 -2.050693E-02 7.000000E-02
121 traj.phase0.states:v_30 c -7.000000E-02 -2.073869E-02 7.000000E-02
122 traj.phase0.states:v_31 c -7.000000E-02 -2.070879E-02 7.000000E-02
123 traj.phase0.states:v_32 c -7.000000E-02 -2.061684E-02 7.000000E-02
124 traj.phase0.states:v_33 c -7.000000E-02 -2.026466E-02 7.000000E-02
125 traj.phase0.states:v_34 c -7.000000E-02 -1.947379E-02 7.000000E-02
126 traj.phase0.states:v_35 c -7.000000E-02 -1.915480E-02 7.000000E-02
127 traj.phase0.states:v_36 c -7.000000E-02 -1.831126E-02 7.000000E-02
128 traj.phase0.states:v_37 c -7.000000E-02 -1.691377E-02 7.000000E-02
129 traj.phase0.states:v_38 c -7.000000E-02 -1.642126E-02 7.000000E-02
130 traj.phase0.states:v_39 c -7.000000E-02 -1.521386E-02 7.000000E-02
131 traj.phase0.states:v_40 c -7.000000E-02 -1.339010E-02 7.000000E-02
132 traj.phase0.states:v_41 c -7.000000E-02 -1.278055E-02 7.000000E-02
133 traj.phase0.states:v_42 c -7.000000E-02 -1.141416E-02 7.000000E-02
134 traj.phase0.states:v_43 c -7.000000E-02 -8.458173E-03 7.000000E-02
135 traj.phase0.states:v_44 c -7.000000E-02 -6.915090E-03 7.000000E-02
136 traj.phase0.states:v_45 c -7.000000E-02 -2.911314E-03 7.000000E-02
137 traj.phase0.states:v_46 c -7.000000E-02 2.630496E-03 7.000000E-02
138 traj.phase0.states:v_47 c -7.000000E-02 4.382061E-03 7.000000E-02
139 traj.phase0.states:v_48 c -7.000000E-02 8.376856E-03 7.000000E-02
140 traj.phase0.states:v_49 c -7.000000E-02 1.381961E-02 7.000000E-02
141 traj.phase0.states:v_50 c -7.000000E-02 1.551747E-02 7.000000E-02
142 traj.phase0.states:v_51 c -7.000000E-02 1.934066E-02 7.000000E-02
143 traj.phase0.states:v_52 c -7.000000E-02 2.441559E-02 7.000000E-02
144 traj.phase0.states:v_53 c -7.000000E-02 2.595996E-02 7.000000E-02
145 traj.phase0.states:v_54 c -7.000000E-02 2.935249E-02 7.000000E-02
146 traj.phase0.states:v_55 c -7.000000E-02 3.363027E-02 7.000000E-02
147 traj.phase0.states:v_56 c -7.000000E-02 3.486800E-02 7.000000E-02
148 traj.phase0.states:v_57 c -7.000000E-02 3.745593E-02 7.000000E-02
149 traj.phase0.states:v_58 c -7.000000E-02 4.039299E-02 7.000000E-02
150 traj.phase0.states:v_59 c -7.000000E-02 4.115260E-02 7.000000E-02
151 traj.phase0.states:v_60 c -7.000000E-02 4.256515E-02 7.000000E-02
152 traj.phase0.states:v_61 c -7.000000E-02 4.374409E-02 7.000000E-02
153 traj.phase0.states:v_62 c -7.000000E-02 4.392749E-02 7.000000E-02
154 traj.phase0.states:v_63 c -7.000000E-02 4.401872E-02 7.000000E-02
155 traj.phase0.states:v_64 c -7.000000E-02 4.344021E-02 7.000000E-02
156 traj.phase0.states:v_65 c -7.000000E-02 4.310093E-02 7.000000E-02
157 traj.phase0.states:v_66 c -7.000000E-02 4.208286E-02 7.000000E-02
158 traj.phase0.states:v_67 c -7.000000E-02 4.021940E-02 7.000000E-02
159 traj.phase0.states:v_68 c -7.000000E-02 3.954271E-02 7.000000E-02
160 traj.phase0.states:v_69 c -7.000000E-02 3.788101E-02 7.000000E-02
161 traj.phase0.states:v_70 c -7.000000E-02 3.542495E-02 7.000000E-02
162 traj.phase0.states:v_71 c -7.000000E-02 3.463052E-02 7.000000E-02
163 traj.phase0.states:v_72 c -7.000000E-02 3.281297E-02 7.000000E-02
164 traj.phase0.states:v_73 c -7.000000E-02 3.037524E-02 7.000000E-02
165 traj.phase0.states:v_74 c -7.000000E-02 2.963561E-02 7.000000E-02
166 traj.phase0.states:v_75 c -7.000000E-02 2.801759E-02 7.000000E-02
167 traj.phase0.states:v_76 c -7.000000E-02 2.599539E-02 7.000000E-02
168 traj.phase0.states:v_77 c -7.000000E-02 2.541413E-02 7.000000E-02
169 traj.phase0.states:v_78 c -7.000000E-02 2.419781E-02 7.000000E-02
170 traj.phase0.states:v_79 c -7.000000E-02 2.279996E-02 7.000000E-02
171 traj.phase0.states:v_80 c -7.000000E-02 2.242906E-02 7.000000E-02
172 traj.phase0.states:v_81 c -7.000000E-02 2.171339E-02 7.000000E-02
173 traj.phase0.states:v_82 c -7.000000E-02 2.104175E-02 7.000000E-02
174 traj.phase0.states:v_83 c -7.000000E-02 2.090773E-02 7.000000E-02
175 traj.phase0.states:v_84 c -7.000000E-02 2.074700E-02 7.000000E-02
176 traj.phase0.states:v_85 c -7.000000E-02 2.086973E-02 7.000000E-02
177 traj.phase0.states:v_86 c -7.000000E-02 2.099435E-02 7.000000E-02
178 traj.phase0.states:v_87 c -7.000000E-02 2.144097E-02 7.000000E-02
179 traj.phase0.states:v_88 c -7.000000E-02 2.244081E-02 7.000000E-02
180 traj.phase0.states:v_89 c -7.000000E-02 2.285374E-02 7.000000E-02
181 traj.phase0.controls:u_0 c -1.000000E+00 9.999996E-01 1.000000E+00 u
182 traj.phase0.controls:u_1 c -1.000000E+00 9.999999E-01 1.000000E+00 u
183 traj.phase0.controls:u_2 c -1.000000E+00 9.999998E-01 1.000000E+00 u
184 traj.phase0.controls:u_3 c -1.000000E+00 9.999997E-01 1.000000E+00 u
185 traj.phase0.controls:u_4 c -1.000000E+00 9.999999E-01 1.000000E+00 u
186 traj.phase0.controls:u_5 c -1.000000E+00 9.999998E-01 1.000000E+00 u
187 traj.phase0.controls:u_6 c -1.000000E+00 9.999996E-01 1.000000E+00 u
188 traj.phase0.controls:u_7 c -1.000000E+00 9.999999E-01 1.000000E+00 u
189 traj.phase0.controls:u_8 c -1.000000E+00 9.999995E-01 1.000000E+00 u
190 traj.phase0.controls:u_9 c -1.000000E+00 9.999992E-01 1.000000E+00 u
191 traj.phase0.controls:u_10 c -1.000000E+00 9.999997E-01 1.000000E+00 u
192 traj.phase0.controls:u_11 c -1.000000E+00 -2.836272E-01 1.000000E+00
193 traj.phase0.controls:u_12 c -1.000000E+00 -9.999915E-01 1.000000E+00
194 traj.phase0.controls:u_13 c -1.000000E+00 -9.999994E-01 1.000000E+00 l
195 traj.phase0.controls:u_14 c -1.000000E+00 -9.999998E-01 1.000000E+00 l
196 traj.phase0.controls:u_15 c -1.000000E+00 -9.999974E-01 1.000000E+00
197 traj.phase0.controls:u_16 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
198 traj.phase0.controls:u_17 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
199 traj.phase0.controls:u_18 c -1.000000E+00 -9.999990E-01 1.000000E+00 l
200 traj.phase0.controls:u_19 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
201 traj.phase0.controls:u_20 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
202 traj.phase0.controls:u_21 c -1.000000E+00 -9.999995E-01 1.000000E+00 l
203 traj.phase0.controls:u_22 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
204 traj.phase0.controls:u_23 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
205 traj.phase0.controls:u_24 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
206 traj.phase0.controls:u_25 c -1.000000E+00 -1.000000E+00 1.000000E+00 l
207 traj.phase0.controls:u_26 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
208 traj.phase0.controls:u_27 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
209 traj.phase0.controls:u_28 c -1.000000E+00 -1.000000E+00 1.000000E+00 l
210 traj.phase0.controls:u_29 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
211 traj.phase0.controls:u_30 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
212 traj.phase0.controls:u_31 c -1.000000E+00 -1.000000E+00 1.000000E+00 l
213 traj.phase0.controls:u_32 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
214 traj.phase0.controls:u_33 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
215 traj.phase0.controls:u_34 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
216 traj.phase0.controls:u_35 c -1.000000E+00 -9.999998E-01 1.000000E+00 l
217 traj.phase0.controls:u_36 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
218 traj.phase0.controls:u_37 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
219 traj.phase0.controls:u_38 c -1.000000E+00 -9.999997E-01 1.000000E+00 l
220 traj.phase0.controls:u_39 c -1.000000E+00 -9.999995E-01 1.000000E+00 l
221 traj.phase0.controls:u_40 c -1.000000E+00 -9.999999E-01 1.000000E+00 l
222 traj.phase0.controls:u_41 c -1.000000E+00 -9.999995E-01 1.000000E+00 l
223 traj.phase0.controls:u_42 c -1.000000E+00 -9.999992E-01 1.000000E+00 l
224 traj.phase0.controls:u_43 c -1.000000E+00 -9.999996E-01 1.000000E+00 l
225 traj.phase0.controls:u_44 c -1.000000E+00 2.836235E-01 1.000000E+00
226 traj.phase0.controls:u_45 c -1.000000E+00 9.999857E-01 1.000000E+00
227 traj.phase0.controls:u_46 c -1.000000E+00 9.999990E-01 1.000000E+00
228 traj.phase0.controls:u_47 c -1.000000E+00 9.999998E-01 1.000000E+00 u
229 traj.phase0.controls:u_48 c -1.000000E+00 9.999958E-01 1.000000E+00
230 traj.phase0.controls:u_49 c -1.000000E+00 9.999998E-01 1.000000E+00 u
231 traj.phase0.controls:u_50 c -1.000000E+00 9.999999E-01 1.000000E+00 u
232 traj.phase0.controls:u_51 c -1.000000E+00 9.999985E-01 1.000000E+00
233 traj.phase0.controls:u_52 c -1.000000E+00 9.999999E-01 1.000000E+00 u
234 traj.phase0.controls:u_53 c -1.000000E+00 9.999999E-01 1.000000E+00 u
235 traj.phase0.controls:u_54 c -1.000000E+00 9.999994E-01 1.000000E+00 u
236 traj.phase0.controls:u_55 c -1.000000E+00 9.999999E-01 1.000000E+00 u
237 traj.phase0.controls:u_56 c -1.000000E+00 9.999999E-01 1.000000E+00 u
238 traj.phase0.controls:u_57 c -1.000000E+00 9.999997E-01 1.000000E+00 u
239 traj.phase0.controls:u_58 c -1.000000E+00 1.000000E+00 1.000000E+00 u
240 traj.phase0.controls:u_59 c -1.000000E+00 9.999999E-01 1.000000E+00 u
241 traj.phase0.controls:u_60 c -1.000000E+00 9.999997E-01 1.000000E+00 u
242 traj.phase0.controls:u_61 c -1.000000E+00 1.000000E+00 1.000000E+00 u
243 traj.phase0.controls:u_62 c -1.000000E+00 9.999999E-01 1.000000E+00 u
244 traj.phase0.controls:u_63 c -1.000000E+00 9.999998E-01 1.000000E+00 u
245 traj.phase0.controls:u_64 c -1.000000E+00 1.000000E+00 1.000000E+00 u
246 traj.phase0.controls:u_65 c -1.000000E+00 9.999999E-01 1.000000E+00 u
247 traj.phase0.controls:u_66 c -1.000000E+00 9.999998E-01 1.000000E+00 u
248 traj.phase0.controls:u_67 c -1.000000E+00 1.000000E+00 1.000000E+00 u
249 traj.phase0.controls:u_68 c -1.000000E+00 9.999998E-01 1.000000E+00 u
250 traj.phase0.controls:u_69 c -1.000000E+00 9.999997E-01 1.000000E+00 u
251 traj.phase0.controls:u_70 c -1.000000E+00 9.999999E-01 1.000000E+00 u
252 traj.phase0.controls:u_71 c -1.000000E+00 9.999998E-01 1.000000E+00 u
253 traj.phase0.controls:u_72 c -1.000000E+00 9.999997E-01 1.000000E+00 u
254 traj.phase0.controls:u_73 c -1.000000E+00 9.999999E-01 1.000000E+00 u
255 traj.phase0.controls:u_74 c -1.000000E+00 9.999998E-01 1.000000E+00 u
256 traj.phase0.controls:u_75 c -1.000000E+00 9.999996E-01 1.000000E+00 u
257 traj.phase0.controls:u_76 c -1.000000E+00 9.999999E-01 1.000000E+00 u
258 traj.phase0.controls:u_77 c -1.000000E+00 9.999997E-01 1.000000E+00 u
259 traj.phase0.controls:u_78 c -1.000000E+00 9.999994E-01 1.000000E+00 u
260 traj.phase0.controls:u_79 c -1.000000E+00 9.999999E-01 1.000000E+00 u
261 traj.phase0.controls:u_80 c -1.000000E+00 9.999995E-01 1.000000E+00 u
262 traj.phase0.controls:u_81 c -1.000000E+00 9.999992E-01 1.000000E+00 u
263 traj.phase0.controls:u_82 c -1.000000E+00 9.999998E-01 1.000000E+00 u
264 traj.phase0.controls:u_83 c -1.000000E+00 9.999992E-01 1.000000E+00 u
265 traj.phase0.controls:u_84 c -1.000000E+00 9.999987E-01 1.000000E+00
266 traj.phase0.controls:u_85 c -1.000000E+00 9.999996E-01 1.000000E+00 u
267 traj.phase0.controls:u_86 c -1.000000E+00 9.999985E-01 1.000000E+00
268 traj.phase0.controls:u_87 c -1.000000E+00 9.999976E-01 1.000000E+00
269 traj.phase0.controls:u_88 c -1.000000E+00 9.999989E-01 1.000000E+00
270 traj.phase0.controls:u_89 c -1.000000E+00 9.999905E-01 1.000000E+00
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier
0 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.110223E-16 0.000000E+00 1.23904E-02
1 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 3.819428E-16 0.000000E+00 7.05561E-02
2 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.454143E-17 0.000000E+00 6.43473E-02
3 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 8.834051E-17 0.000000E+00 2.00810E-02
4 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.105235E-16 0.000000E+00 1.03452E-01
5 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.963310E-16 0.000000E+00 8.54789E-02
6 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.370391E-15 0.000000E+00 2.60098E-02
7 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.132050E-15 0.000000E+00 1.27261E-01
8 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.898162E-15 0.000000E+00 9.91060E-02
9 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.525409E-15 0.000000E+00 2.96571E-02
10 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.823747E-15 0.000000E+00 1.39931E-01
11 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 7.007852E-15 0.000000E+00 1.04120E-01
12 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.781478E-15 0.000000E+00 3.07332E-02
13 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.624885E-15 0.000000E+00 1.40595E-01
14 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.518683E-15 0.000000E+00 1.00337E-01
15 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.611849E-15 0.000000E+00 2.92247E-02
16 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.138411E-15 0.000000E+00 1.29559E-01
17 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.166019E-15 0.000000E+00 8.82394E-02
18 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.115539E-15 0.000000E+00 2.52929E-02
19 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.397634E-15 0.000000E+00 1.07675E-01
20 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.897983E-15 0.000000E+00 6.85802E-02
21 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.618857E-15 0.000000E+00 1.91749E-02
22 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.452569E-15 0.000000E+00 7.62222E-02
23 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.259556E-15 0.000000E+00 4.25987E-02
24 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.022002E-15 0.000000E+00 1.12705E-02
25 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.958946E-15 0.000000E+00 3.75068E-02
26 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 4.002790E-15 0.000000E+00 1.25303E-02
27 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 3.153533E-15 0.000000E+00 2.29059E-03
28 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.324365E-15 0.000000E+00 -4.57934E-03
29 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.373789E-16 0.000000E+00 -1.82447E-02
30 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.662880E-15 0.000000E+00 -6.72827E-03
31 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -4.270039E-15 0.000000E+00 -4.49200E-02
32 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -7.215712E-15 0.000000E+00 -4.58922E-02
33 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -8.308462E-15 0.000000E+00 -1.46638E-02
34 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.117693E-14 0.000000E+00 -7.86505E-02
35 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.424138E-14 0.000000E+00 -6.74047E-02
36 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.557168E-14 0.000000E+00 -2.06935E-02
37 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.869552E-14 0.000000E+00 -1.02853E-01
38 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.206286E-14 0.000000E+00 -8.15806E-02
39 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.400486E-14 0.000000E+00 -2.45504E-02
40 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.681691E-14 0.000000E+00 -1.17247E-01
41 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -2.997044E-14 0.000000E+00 -8.90302E-02
42 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -3.283594E-14 0.000000E+00 -2.64800E-02
43 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -3.866493E-14 0.000000E+00 -1.23534E-01
44 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 -1.983875E-14 0.000000E+00 -9.13645E-02
45 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.327335E-15 0.000000E+00 -2.69735E-02
46 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.683695E-14 0.000000E+00 -1.24068E-01
47 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.530072E-14 0.000000E+00 -9.01343E-02
48 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.900137E-14 0.000000E+00 -2.64578E-02
49 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 2.463438E-14 0.000000E+00 -1.20196E-01
50 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.807788E-14 0.000000E+00 -8.54446E-02
51 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.847579E-14 0.000000E+00 -2.48732E-02
52 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.813133E-14 0.000000E+00 -1.10513E-01
53 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.316645E-14 0.000000E+00 -7.53007E-02
54 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.284575E-14 0.000000E+00 -2.15620E-02
55 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.042269E-14 0.000000E+00 -9.13387E-02
56 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 6.390211E-15 0.000000E+00 -5.68799E-02
57 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 5.333094E-15 0.000000E+00 -1.57189E-02
58 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 3.432660E-15 0.000000E+00 -5.96718E-02
59 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 3.682092E-16 0.000000E+00 -2.94257E-02
60 traj.phase0.collocation_constraint.defects:x e 0.000000E+00 1.900434E-16 0.000000E+00 -7.28575E-03
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Exit Status
Inform Description
0 Solve Succeeded
--------------------------------------------------------------------------------
==================================================
Grid Refinement - Iteration 1
--------------------------------------------------
Phase: traj.phases.phase0
Refinement Options:
Allow Refinement = True
Tolerance = 0.0001
Min Order = 3
Max Order = 14
Original Grid:
Number of Segments = 30
Segment Ends = [-1.0, -0.9333, -0.8667, -0.8, -0.7333, -0.6667, -0.6, -0.5333, -0.4667, -0.4, -0.3333, -0.2667, -0.2, -0.1333, -0.0667, 0.0, 0.0667, 0.1333, 0.2, 0.2667, 0.3333, 0.4, 0.4667, 0.5333, 0.6, 0.6667, 0.7333, 0.8, 0.8667, 0.9333, 1.0]
Segment Order = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
Error = [5.33e-07, 4.891e-07, 3.975e-07, 4.766e-05, 1.152e-06, 1.168e-06, 1.12e-06, 9.752e-07, 6.726e-07, 2.295e-07, 2.314e-07, 5.567e-07, 6.702e-07, 6.065e-07, 3.587e-05, 6.767e-07, 8.229e-07, 1.404e-06, 2.037e-06, 1.943e-06, 1.052e-06, 1.925e-06, 3.68e-06, 3.871e-06, 2.654e-06, 1.3e-06, 5.762e-07, 3.263e-07, 3.517e-07, 4.995e-07]
Successfully completed grid refinement.
==================================================
Simulating trajectory traj
Done simulating trajectory traj
Problem: problem
Driver: pyOptSparseDriver
success : True
iterations : 94
runtime : 4.6878E+00 s
model_evals : 94
model_time : 8.9955E-02 s
deriv_evals : 93
deriv_time : 9.4083E-01 s
exit_status : SUCCESS
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(p.get_outputs_dir() / 'dymos_solution.db').get_case('final')
sim = om.CaseReader(traj.sim_prob.get_outputs_dir() / 'dymos_simulation.db').get_case('final')
t = sol.get_val('traj.phase0.timeseries.time')
x = sol.get_val('traj.phase0.timeseries.x')
v = sol.get_val('traj.phase0.timeseries.v')
u = sol.get_val('traj.phase0.timeseries.u')
h = np.sin(3 * x) / 3
t_sim = sim.get_val('traj.phase0.timeseries.time')
x_sim = sim.get_val('traj.phase0.timeseries.x')
v_sim = sim.get_val('traj.phase0.timeseries.v')
u_sim = sim.get_val('traj.phase0.timeseries.u')
h_sim = np.sin(3 * x_sim) / 3
Animating the Solution#
The collapsed code cell below contains the code used to produce an animation of the mountain car solution using Matplotlib.
The green area represents the hilly terrain the car is traversing. The black circle is the center of the car, and the orange arrow is the applied control.
The applied control generally has the same sign as the velocity and is ‘bang-bang’, that is, it wants to be at its maximum possible magnitude. Interestingly, the sign of the control flips shortly before the sign of the velocity changes.
Show code cell source
Hide code cell source
fig = plt.figure(constrained_layout=True, figsize=(12, 6))
gs = fig.add_gridspec(3, 2)
anim_ax = fig.add_subplot(gs[:, 0])
anim_ax.set_aspect('equal')
x_ax = fig.add_subplot(gs[0, 1:])
v_ax = fig.add_subplot(gs[1, 1:])
u_ax = fig.add_subplot(gs[2, 1:])
x_ax.set_ylabel('x')
v_ax.set_ylabel('v')
u_ax.set_ylabel('u')
u_ax.set_xlabel('t')
# set up the subplots as needed
anim_ax.set_xlim((-1.75, 0.75))
anim_ax.set_ylim((-1.25, 1.25))
anim_ax.set_xlabel('x')
anim_ax.set_ylabel('h')
x_sol_line, = x_ax.plot(t, x, 'o', ms=1, label='solution')
v_ax.plot(t, v, 'o', ms=1)
u_ax.plot(t, u, 'o', ms=1)
x_sim_line, = x_ax.plot([], [], '-', linewidth=3, label='simulation')
v_sim_line, = v_ax.plot([], [], '-', linewidth=3)
u_sim_line, = u_ax.plot([], [], '-', linewidth=3)
plt.figlegend(ncol=2, handles=[x_sol_line, x_sim_line], loc='upper center',
bbox_to_anchor=(0.78,0.98))
x_ax.grid(alpha=0.2)
txt_x = x_ax.text(0.8, 0.1, f'x = {x_sim[0, 0]:6.3f}', horizontalalignment='left',
verticalalignment='center', transform=x_ax.transAxes)
v_ax.grid(alpha=0.2)
txt_v = v_ax.text(0.8, 0.1, f'v = {v_sim[0, 0]:6.3f}', horizontalalignment='left',
verticalalignment='center', transform=v_ax.transAxes)
u_ax.grid(alpha=0.2)
txt_u = u_ax.text(0.8, 0.1, f'u = {u_sim[0, 0]:6.3f}', horizontalalignment='left',
verticalalignment='center', transform=u_ax.transAxes)
x_terrain = np.linspace(-1.75, 0.75, 100)
h_terrain = np.sin(3 * x_terrain) / 3
terrain_line, = anim_ax.plot(x_terrain, h_terrain, '-', color='tab:gray', lw=2)
terrain = anim_ax.fill_between(x_terrain, h_terrain, -1.25*np.ones_like(x_terrain), color='tab:green')
car, = anim_ax.plot([], [], 'ko', ms=12)
u_vec = anim_ax.quiver(x_sim[0] + 0.005, h_sim[0] + 0.005, u_sim[0], [0], scale=10, angles='xy', color='tab:orange')
# See https://brushingupscience.com/2019/08/01/elaborate-matplotlib-animations/ for quiver animation
ANIM_DURATION = 5
PAUSE_DURATION = 2
ANIM_FPS = 20
num_points = t_sim.size
num_frames = ANIM_DURATION * ANIM_FPS
pause_frames = PAUSE_DURATION * ANIM_FPS
idx_from_frame_num = np.linspace(0, num_points-1, num_frames, dtype=int)
def drawframe(n):
if n >= idx_from_frame_num.size:
idx = num_points - 1
else:
idx = idx_from_frame_num[n]
x = x_sim[idx]
v = v_sim[idx]
u = u_sim[idx]
h = np.sin(3 * x) / 3 + 0.025
car.set_data(x, h)
dh_dx = np.cos(3 * x)
u_vec.set_offsets(np.atleast_2d(np.asarray([x + 0.005, h + 0.005]).T))
u_vec.set_UVC(u * np.cos(dh_dx), u * np.sin(dh_dx))
x_sim_line.set_data(t_sim[:idx], x_sim[:idx])
v_sim_line.set_data(t_sim[:idx], v_sim[:idx])
u_sim_line.set_data(t_sim[:idx], u_sim[:idx])
txt_x.set_text(f'x = {x[0]:6.3f}')
txt_v.set_text(f'v = {v[0]:6.3f}')
txt_u.set_text(f'u = {u[0]:6.3f}')
return car, u_vec, x_sim_line, v_sim_line, u_sim_line
# # blit=True re-draws only the parts that have changed.
# # repeat_delay has no effect when using to_jshtml, so pad drawframe to show the final frame for PAUSE_FRAMES extra frames.
anim = animation.FuncAnimation(fig, drawframe, frames=num_frames + pause_frames, interval=1000/ANIM_FPS, blit=True)
plt.close() # Don't let jupyter display the un-animated plot
from IPython.display import HTML
js = anim.to_jshtml()
with open('mountain_car_anim.html', 'w') as f:
print(js, file=f)
HTML(filename='mountain_car_anim.html')