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:

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.

3. 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']

tags=['dymos.state_rate_source:x', 'dymos.state_units:m'])
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['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
#


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
#

#
# Setup the Problem
#
p.setup()

--- Constraint Report [traj] ---
--- phase0 ---
[final]   5.0000e-01 <= x  [m]
[final]   0.0000e+00 <= v  [m/s]
[path]    -1.0000e+00 <= u <= 1.0000e+00  [unitless]

<openmdao.core.problem.Problem at 0x7f5678b24460>


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
#
p['traj.phase0.t_initial'] = 0.0
p['traj.phase0.t_duration'] = 500.0

p.set_val('traj.phase0.states:x', phase.interp('x', ys=[-0.5, 0.5]))
p.set_val('traj.phase0.states:v', phase.interp('v', ys=[0, 0.07]))
p.set_val('traj.phase0.controls:u', np.sin(phase.interp('u', ys=[0, 1.0])))

#
# Solve for the optimal trajectory
#
dm.run_problem(p, run_driver=True, simulate=True, refine_method='ph', refine_iteration_limit=5)

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 0.426565 seconds.
Total jacobian shape: (332, 271)

Jacobian shape: (332, 271)  ( 1.68% nonzero)
FWD solves: 9   REV solves: 0
Total colors vs. total size: 9 vs 271  (96.7% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.426565 sec.
Time to compute coloring: 0.210060 sec.
Memory to compute coloring: 0.250000 MB.

Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc

Solution:
--------------------------------------------------------------------------------
Total Time:                    2.9427
User Objective Time :       0.1861
User Sensitivity Time :     1.1535
Interface Time :            0.8409
Opt Solver Time:            0.7621
Calls to Objective Function :     123
Calls to Sens Function :          120

Objectives
Index  Name                                 Value
0  traj.phases.phase0.time.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.phases.phase0.indep_states.states:x_0                         c    -1.200000E+00    -4.993918E-01     5.000000E-01
2  traj.phases.phase0.indep_states.states:x_1                         c    -1.200000E+00    -4.965749E-01     5.000000E-01
3  traj.phases.phase0.indep_states.states:x_2                         c    -1.200000E+00    -4.952113E-01     5.000000E-01
4  traj.phases.phase0.indep_states.states:x_3                         c    -1.200000E+00    -4.912601E-01     5.000000E-01
5  traj.phases.phase0.indep_states.states:x_4                         c    -1.200000E+00    -4.839852E-01     5.000000E-01
6  traj.phases.phase0.indep_states.states:x_5                         c    -1.200000E+00    -4.812613E-01     5.000000E-01
7  traj.phases.phase0.indep_states.states:x_6                         c    -1.200000E+00    -4.743104E-01     5.000000E-01
8  traj.phases.phase0.indep_states.states:x_7                         c    -1.200000E+00    -4.632080E-01     5.000000E-01
9  traj.phases.phase0.indep_states.states:x_8                         c    -1.200000E+00    -4.593596E-01     5.000000E-01
10  traj.phases.phase0.indep_states.states:x_9                         c    -1.200000E+00    -4.499012E-01     5.000000E-01
11  traj.phases.phase0.indep_states.states:x_10                        c    -1.200000E+00    -4.363638E-01     5.000000E-01
12  traj.phases.phase0.indep_states.states:x_11                        c    -1.200000E+00    -4.322699E-01     5.000000E-01
13  traj.phases.phase0.indep_states.states:x_12                        c    -1.200000E+00    -4.247954E-01     5.000000E-01
14  traj.phases.phase0.indep_states.states:x_13                        c    -1.200000E+00    -4.186817E-01     5.000000E-01
15  traj.phases.phase0.indep_states.states:x_14                        c    -1.200000E+00    -4.177845E-01     5.000000E-01
16  traj.phases.phase0.indep_states.states:x_15                        c    -1.200000E+00    -4.176175E-01     5.000000E-01
17  traj.phases.phase0.indep_states.states:x_16                        c    -1.200000E+00    -4.216950E-01     5.000000E-01
18  traj.phases.phase0.indep_states.states:x_17                        c    -1.200000E+00    -4.240204E-01     5.000000E-01
19  traj.phases.phase0.indep_states.states:x_18                        c    -1.200000E+00    -4.311749E-01     5.000000E-01
20  traj.phases.phase0.indep_states.states:x_19                        c    -1.200000E+00    -4.451029E-01     5.000000E-01
21  traj.phases.phase0.indep_states.states:x_20                        c    -1.200000E+00    -4.504564E-01     5.000000E-01
22  traj.phases.phase0.indep_states.states:x_21                        c    -1.200000E+00    -4.643313E-01     5.000000E-01
23  traj.phases.phase0.indep_states.states:x_22                        c    -1.200000E+00    -4.869306E-01     5.000000E-01
24  traj.phases.phase0.indep_states.states:x_23                        c    -1.200000E+00    -4.948560E-01     5.000000E-01
25  traj.phases.phase0.indep_states.states:x_24                        c    -1.200000E+00    -5.142625E-01     5.000000E-01
26  traj.phases.phase0.indep_states.states:x_25                        c    -1.200000E+00    -5.435816E-01     5.000000E-01
27  traj.phases.phase0.indep_states.states:x_26                        c    -1.200000E+00    -5.533912E-01     5.000000E-01
28  traj.phases.phase0.indep_states.states:x_27                        c    -1.200000E+00    -5.766416E-01     5.000000E-01
29  traj.phases.phase0.indep_states.states:x_28                        c    -1.200000E+00    -6.101308E-01     5.000000E-01
30  traj.phases.phase0.indep_states.states:x_29                        c    -1.200000E+00    -6.209737E-01     5.000000E-01
31  traj.phases.phase0.indep_states.states:x_30                        c    -1.200000E+00    -6.460628E-01     5.000000E-01
32  traj.phases.phase0.indep_states.states:x_31                        c    -1.200000E+00    -6.808768E-01     5.000000E-01
33  traj.phases.phase0.indep_states.states:x_32                        c    -1.200000E+00    -6.918463E-01     5.000000E-01
34  traj.phases.phase0.indep_states.states:x_33                        c    -1.200000E+00    -7.167113E-01     5.000000E-01
35  traj.phases.phase0.indep_states.states:x_34                        c    -1.200000E+00    -7.500849E-01     5.000000E-01
36  traj.phases.phase0.indep_states.states:x_35                        c    -1.200000E+00    -7.603375E-01     5.000000E-01
37  traj.phases.phase0.indep_states.states:x_36                        c    -1.200000E+00    -7.831224E-01     5.000000E-01
38  traj.phases.phase0.indep_states.states:x_37                        c    -1.200000E+00    -8.126995E-01     5.000000E-01
39  traj.phases.phase0.indep_states.states:x_38                        c    -1.200000E+00    -8.215465E-01     5.000000E-01
40  traj.phases.phase0.indep_states.states:x_39                        c    -1.200000E+00    -8.407829E-01     5.000000E-01
41  traj.phases.phase0.indep_states.states:x_40                        c    -1.200000E+00    -8.647953E-01     5.000000E-01
42  traj.phases.phase0.indep_states.states:x_41                        c    -1.200000E+00    -8.717406E-01     5.000000E-01
43  traj.phases.phase0.indep_states.states:x_42                        c    -1.200000E+00    -8.865112E-01     5.000000E-01
44  traj.phases.phase0.indep_states.states:x_43                        c    -1.200000E+00    -9.033494E-01     5.000000E-01
45  traj.phases.phase0.indep_states.states:x_44                        c    -1.200000E+00    -9.075277E-01     5.000000E-01
46  traj.phases.phase0.indep_states.states:x_45                        c    -1.200000E+00    -9.135009E-01     5.000000E-01
47  traj.phases.phase0.indep_states.states:x_46                        c    -1.200000E+00    -9.137392E-01     5.000000E-01
48  traj.phases.phase0.indep_states.states:x_47                        c    -1.200000E+00    -9.118763E-01     5.000000E-01
49  traj.phases.phase0.indep_states.states:x_48                        c    -1.200000E+00    -9.041177E-01     5.000000E-01
50  traj.phases.phase0.indep_states.states:x_49                        c    -1.200000E+00    -8.854927E-01     5.000000E-01
51  traj.phases.phase0.indep_states.states:x_50                        c    -1.200000E+00    -8.777039E-01     5.000000E-01
52  traj.phases.phase0.indep_states.states:x_51                        c    -1.200000E+00    -8.565061E-01     5.000000E-01
53  traj.phases.phase0.indep_states.states:x_52                        c    -1.200000E+00    -8.197802E-01     5.000000E-01
54  traj.phases.phase0.indep_states.states:x_53                        c    -1.200000E+00    -8.064054E-01     5.000000E-01
55  traj.phases.phase0.indep_states.states:x_54                        c    -1.200000E+00    -7.727637E-01     5.000000E-01
56  traj.phases.phase0.indep_states.states:x_55                        c    -1.200000E+00    -7.198818E-01     5.000000E-01
57  traj.phases.phase0.indep_states.states:x_56                        c    -1.200000E+00    -7.016946E-01     5.000000E-01
58  traj.phases.phase0.indep_states.states:x_57                        c    -1.200000E+00    -6.576998E-01     5.000000E-01
59  traj.phases.phase0.indep_states.states:x_58                        c    -1.200000E+00    -5.923140E-01     5.000000E-01
60  traj.phases.phase0.indep_states.states:x_59                        c    -1.200000E+00    -5.706632E-01     5.000000E-01
61  traj.phases.phase0.indep_states.states:x_60                        c    -1.200000E+00    -5.197353E-01     5.000000E-01
62  traj.phases.phase0.indep_states.states:x_61                        c    -1.200000E+00    -4.472326E-01     5.000000E-01
63  traj.phases.phase0.indep_states.states:x_62                        c    -1.200000E+00    -4.239588E-01     5.000000E-01
64  traj.phases.phase0.indep_states.states:x_63                        c    -1.200000E+00    -3.704650E-01     5.000000E-01
65  traj.phases.phase0.indep_states.states:x_64                        c    -1.200000E+00    -2.970089E-01     5.000000E-01
66  traj.phases.phase0.indep_states.states:x_65                        c    -1.200000E+00    -2.740401E-01     5.000000E-01
67  traj.phases.phase0.indep_states.states:x_66                        c    -1.200000E+00    -2.222404E-01     5.000000E-01
68  traj.phases.phase0.indep_states.states:x_67                        c    -1.200000E+00    -1.531484E-01     5.000000E-01
69  traj.phases.phase0.indep_states.states:x_68                        c    -1.200000E+00    -1.319827E-01     5.000000E-01
70  traj.phases.phase0.indep_states.states:x_69                        c    -1.200000E+00    -8.491750E-02     5.000000E-01
71  traj.phases.phase0.indep_states.states:x_70                        c    -1.200000E+00    -2.341760E-02     5.000000E-01
72  traj.phases.phase0.indep_states.states:x_71                        c    -1.200000E+00    -4.829274E-03     5.000000E-01
73  traj.phases.phase0.indep_states.states:x_72                        c    -1.200000E+00     3.615597E-02     5.000000E-01
74  traj.phases.phase0.indep_states.states:x_73                        c    -1.200000E+00     8.913343E-02     5.000000E-01
75  traj.phases.phase0.indep_states.states:x_74                        c    -1.200000E+00     1.050566E-01     5.000000E-01
76  traj.phases.phase0.indep_states.states:x_75                        c    -1.200000E+00     1.400830E-01     5.000000E-01
77  traj.phases.phase0.indep_states.states:x_76                        c    -1.200000E+00     1.853435E-01     5.000000E-01
78  traj.phases.phase0.indep_states.states:x_77                        c    -1.200000E+00     1.989849E-01     5.000000E-01
79  traj.phases.phase0.indep_states.states:x_78                        c    -1.200000E+00     2.291198E-01     5.000000E-01
80  traj.phases.phase0.indep_states.states:x_79                        c    -1.200000E+00     2.684861E-01     5.000000E-01
81  traj.phases.phase0.indep_states.states:x_80                        c    -1.200000E+00     2.804879E-01     5.000000E-01
82  traj.phases.phase0.indep_states.states:x_81                        c    -1.200000E+00     3.072965E-01     5.000000E-01
83  traj.phases.phase0.indep_states.states:x_82                        c    -1.200000E+00     3.430993E-01     5.000000E-01
84  traj.phases.phase0.indep_states.states:x_83                        c    -1.200000E+00     3.542309E-01     5.000000E-01
85  traj.phases.phase0.indep_states.states:x_84                        c    -1.200000E+00     3.795258E-01     5.000000E-01
86  traj.phases.phase0.indep_states.states:x_85                        c    -1.200000E+00     4.143692E-01     5.000000E-01
87  traj.phases.phase0.indep_states.states:x_86                        c    -1.200000E+00     4.254779E-01     5.000000E-01
88  traj.phases.phase0.indep_states.states:x_87                        c    -1.200000E+00     4.512448E-01     5.000000E-01
89  traj.phases.phase0.indep_states.states:x_88                        c    -1.200000E+00     4.879813E-01     5.000000E-01
90  traj.phases.phase0.indep_states.states:x_89                        c    -1.200000E+00     5.000000E-01     5.000000E-01          u
91  traj.phases.phase0.indep_states.states:v_0                         c    -7.000000E-02     9.987135E-04     7.000000E-02
92  traj.phases.phase0.indep_states.states:v_1                         c    -7.000000E-02     2.356393E-03     7.000000E-02
93  traj.phases.phase0.indep_states.states:v_2                         c    -7.000000E-02     2.777103E-03     7.000000E-02
94  traj.phases.phase0.indep_states.states:v_3                         c    -7.000000E-02     3.717045E-03     7.000000E-02
95  traj.phases.phase0.indep_states.states:v_4                         c    -7.000000E-02     4.944779E-03     7.000000E-02
96  traj.phases.phase0.indep_states.states:v_5                         c    -7.000000E-02     5.312908E-03     7.000000E-02
97  traj.phases.phase0.indep_states.states:v_6                         c    -7.000000E-02     6.112790E-03     7.000000E-02
98  traj.phases.phase0.indep_states.states:v_7                         c    -7.000000E-02     7.105115E-03     7.000000E-02
99  traj.phases.phase0.indep_states.states:v_8                         c    -7.000000E-02     7.389281E-03     7.000000E-02
100  traj.phases.phase0.indep_states.states:v_9                         c    -7.000000E-02     8.059965E-03     7.000000E-02
101  traj.phases.phase0.indep_states.states:v_10                        c    -7.000000E-02     7.867031E-03     7.000000E-02
102  traj.phases.phase0.indep_states.states:v_11                        c    -7.000000E-02     7.181148E-03     7.000000E-02
103  traj.phases.phase0.indep_states.states:v_12                        c    -7.000000E-02     5.108835E-03     7.000000E-02
104  traj.phases.phase0.indep_states.states:v_13                        c    -7.000000E-02     2.165338E-03     7.000000E-02
105  traj.phases.phase0.indep_states.states:v_14                        c    -7.000000E-02     1.221816E-03     7.000000E-02
106  traj.phases.phase0.indep_states.states:v_15                        c    -7.000000E-02    -9.448555E-04     7.000000E-02
107  traj.phases.phase0.indep_states.states:v_16                        c    -7.000000E-02    -3.913376E-03     7.000000E-02
108  traj.phases.phase0.indep_states.states:v_17                        c    -7.000000E-02    -4.839342E-03     7.000000E-02
109  traj.phases.phase0.indep_states.states:v_18                        c    -7.000000E-02    -6.919809E-03     7.000000E-02
110  traj.phases.phase0.indep_states.states:v_19                        c    -7.000000E-02    -9.664867E-03     7.000000E-02
111  traj.phases.phase0.indep_states.states:v_20                        c    -7.000000E-02    -1.049505E-02     7.000000E-02
112  traj.phases.phase0.indep_states.states:v_21                        c    -7.000000E-02    -1.231170E-02     7.000000E-02
113  traj.phases.phase0.indep_states.states:v_22                        c    -7.000000E-02    -1.459420E-02     7.000000E-02
114  traj.phases.phase0.indep_states.states:v_23                        c    -7.000000E-02    -1.525502E-02     7.000000E-02
115  traj.phases.phase0.indep_states.states:v_24                        c    -7.000000E-02    -1.664514E-02     7.000000E-02
116  traj.phases.phase0.indep_states.states:v_25                        c    -7.000000E-02    -1.825856E-02     7.000000E-02
117  traj.phases.phase0.indep_states.states:v_26                        c    -7.000000E-02    -1.869027E-02     7.000000E-02
118  traj.phases.phase0.indep_states.states:v_27                        c    -7.000000E-02    -1.952984E-02     7.000000E-02
119  traj.phases.phase0.indep_states.states:v_28                        c    -7.000000E-02    -2.033778E-02     7.000000E-02
120  traj.phases.phase0.indep_states.states:v_29                        c    -7.000000E-02    -2.050692E-02     7.000000E-02
121  traj.phases.phase0.indep_states.states:v_30                        c    -7.000000E-02    -2.073869E-02     7.000000E-02
122  traj.phases.phase0.indep_states.states:v_31                        c    -7.000000E-02    -2.070879E-02     7.000000E-02
123  traj.phases.phase0.indep_states.states:v_32                        c    -7.000000E-02    -2.061684E-02     7.000000E-02
124  traj.phases.phase0.indep_states.states:v_33                        c    -7.000000E-02    -2.026466E-02     7.000000E-02
125  traj.phases.phase0.indep_states.states:v_34                        c    -7.000000E-02    -1.947378E-02     7.000000E-02
126  traj.phases.phase0.indep_states.states:v_35                        c    -7.000000E-02    -1.915480E-02     7.000000E-02
127  traj.phases.phase0.indep_states.states:v_36                        c    -7.000000E-02    -1.831126E-02     7.000000E-02
128  traj.phases.phase0.indep_states.states:v_37                        c    -7.000000E-02    -1.691377E-02     7.000000E-02
129  traj.phases.phase0.indep_states.states:v_38                        c    -7.000000E-02    -1.642126E-02     7.000000E-02
130  traj.phases.phase0.indep_states.states:v_39                        c    -7.000000E-02    -1.521386E-02     7.000000E-02
131  traj.phases.phase0.indep_states.states:v_40                        c    -7.000000E-02    -1.339010E-02     7.000000E-02
132  traj.phases.phase0.indep_states.states:v_41                        c    -7.000000E-02    -1.278055E-02     7.000000E-02
133  traj.phases.phase0.indep_states.states:v_42                        c    -7.000000E-02    -1.141416E-02     7.000000E-02
134  traj.phases.phase0.indep_states.states:v_43                        c    -7.000000E-02    -8.458169E-03     7.000000E-02
135  traj.phases.phase0.indep_states.states:v_44                        c    -7.000000E-02    -6.915081E-03     7.000000E-02
136  traj.phases.phase0.indep_states.states:v_45                        c    -7.000000E-02    -2.911298E-03     7.000000E-02
137  traj.phases.phase0.indep_states.states:v_46                        c    -7.000000E-02     2.630510E-03     7.000000E-02
138  traj.phases.phase0.indep_states.states:v_47                        c    -7.000000E-02     4.382076E-03     7.000000E-02
139  traj.phases.phase0.indep_states.states:v_48                        c    -7.000000E-02     8.376872E-03     7.000000E-02
140  traj.phases.phase0.indep_states.states:v_49                        c    -7.000000E-02     1.381962E-02     7.000000E-02
141  traj.phases.phase0.indep_states.states:v_50                        c    -7.000000E-02     1.551749E-02     7.000000E-02
142  traj.phases.phase0.indep_states.states:v_51                        c    -7.000000E-02     1.934068E-02     7.000000E-02
143  traj.phases.phase0.indep_states.states:v_52                        c    -7.000000E-02     2.441560E-02     7.000000E-02
144  traj.phases.phase0.indep_states.states:v_53                        c    -7.000000E-02     2.595997E-02     7.000000E-02
145  traj.phases.phase0.indep_states.states:v_54                        c    -7.000000E-02     2.935250E-02     7.000000E-02
146  traj.phases.phase0.indep_states.states:v_55                        c    -7.000000E-02     3.363028E-02     7.000000E-02
147  traj.phases.phase0.indep_states.states:v_56                        c    -7.000000E-02     3.486801E-02     7.000000E-02
148  traj.phases.phase0.indep_states.states:v_57                        c    -7.000000E-02     3.745594E-02     7.000000E-02
149  traj.phases.phase0.indep_states.states:v_58                        c    -7.000000E-02     4.039299E-02     7.000000E-02
150  traj.phases.phase0.indep_states.states:v_59                        c    -7.000000E-02     4.115260E-02     7.000000E-02
151  traj.phases.phase0.indep_states.states:v_60                        c    -7.000000E-02     4.256515E-02     7.000000E-02
152  traj.phases.phase0.indep_states.states:v_61                        c    -7.000000E-02     4.374409E-02     7.000000E-02
153  traj.phases.phase0.indep_states.states:v_62                        c    -7.000000E-02     4.392749E-02     7.000000E-02
154  traj.phases.phase0.indep_states.states:v_63                        c    -7.000000E-02     4.401872E-02     7.000000E-02
155  traj.phases.phase0.indep_states.states:v_64                        c    -7.000000E-02     4.344021E-02     7.000000E-02
156  traj.phases.phase0.indep_states.states:v_65                        c    -7.000000E-02     4.310092E-02     7.000000E-02
157  traj.phases.phase0.indep_states.states:v_66                        c    -7.000000E-02     4.208285E-02     7.000000E-02
158  traj.phases.phase0.indep_states.states:v_67                        c    -7.000000E-02     4.021939E-02     7.000000E-02
159  traj.phases.phase0.indep_states.states:v_68                        c    -7.000000E-02     3.954270E-02     7.000000E-02
160  traj.phases.phase0.indep_states.states:v_69                        c    -7.000000E-02     3.788101E-02     7.000000E-02
161  traj.phases.phase0.indep_states.states:v_70                        c    -7.000000E-02     3.542495E-02     7.000000E-02
162  traj.phases.phase0.indep_states.states:v_71                        c    -7.000000E-02     3.463051E-02     7.000000E-02
163  traj.phases.phase0.indep_states.states:v_72                        c    -7.000000E-02     3.281297E-02     7.000000E-02
164  traj.phases.phase0.indep_states.states:v_73                        c    -7.000000E-02     3.037524E-02     7.000000E-02
165  traj.phases.phase0.indep_states.states:v_74                        c    -7.000000E-02     2.963560E-02     7.000000E-02
166  traj.phases.phase0.indep_states.states:v_75                        c    -7.000000E-02     2.801759E-02     7.000000E-02
167  traj.phases.phase0.indep_states.states:v_76                        c    -7.000000E-02     2.599539E-02     7.000000E-02
168  traj.phases.phase0.indep_states.states:v_77                        c    -7.000000E-02     2.541413E-02     7.000000E-02
169  traj.phases.phase0.indep_states.states:v_78                        c    -7.000000E-02     2.419781E-02     7.000000E-02
170  traj.phases.phase0.indep_states.states:v_79                        c    -7.000000E-02     2.279996E-02     7.000000E-02
171  traj.phases.phase0.indep_states.states:v_80                        c    -7.000000E-02     2.242907E-02     7.000000E-02
172  traj.phases.phase0.indep_states.states:v_81                        c    -7.000000E-02     2.171340E-02     7.000000E-02
173  traj.phases.phase0.indep_states.states:v_82                        c    -7.000000E-02     2.104175E-02     7.000000E-02
174  traj.phases.phase0.indep_states.states:v_83                        c    -7.000000E-02     2.090773E-02     7.000000E-02
175  traj.phases.phase0.indep_states.states:v_84                        c    -7.000000E-02     2.074701E-02     7.000000E-02
176  traj.phases.phase0.indep_states.states:v_85                        c    -7.000000E-02     2.086973E-02     7.000000E-02
177  traj.phases.phase0.indep_states.states:v_86                        c    -7.000000E-02     2.099436E-02     7.000000E-02
178  traj.phases.phase0.indep_states.states:v_87                        c    -7.000000E-02     2.144098E-02     7.000000E-02
179  traj.phases.phase0.indep_states.states:v_88                        c    -7.000000E-02     2.244082E-02     7.000000E-02
180  traj.phases.phase0.indep_states.states:v_89                        c    -7.000000E-02     2.285376E-02     7.000000E-02
181  traj.phases.phase0.control_group.indep_controls.controls:u_0       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
182  traj.phases.phase0.control_group.indep_controls.controls:u_1       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
183  traj.phases.phase0.control_group.indep_controls.controls:u_2       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
184  traj.phases.phase0.control_group.indep_controls.controls:u_3       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
185  traj.phases.phase0.control_group.indep_controls.controls:u_4       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
186  traj.phases.phase0.control_group.indep_controls.controls:u_5       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
187  traj.phases.phase0.control_group.indep_controls.controls:u_6       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
188  traj.phases.phase0.control_group.indep_controls.controls:u_7       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
189  traj.phases.phase0.control_group.indep_controls.controls:u_8       c    -1.000000E+00     1.000000E+00     1.000000E+00          u
190  traj.phases.phase0.control_group.indep_controls.controls:u_9       c    -1.000000E+00     9.999999E-01     1.000000E+00          u
191  traj.phases.phase0.control_group.indep_controls.controls:u_10      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
192  traj.phases.phase0.control_group.indep_controls.controls:u_11      c    -1.000000E+00    -2.836320E-01     1.000000E+00
193  traj.phases.phase0.control_group.indep_controls.controls:u_12      c    -1.000000E+00    -9.999990E-01     1.000000E+00          l
194  traj.phases.phase0.control_group.indep_controls.controls:u_13      c    -1.000000E+00    -9.999999E-01     1.000000E+00          l
195  traj.phases.phase0.control_group.indep_controls.controls:u_14      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
196  traj.phases.phase0.control_group.indep_controls.controls:u_15      c    -1.000000E+00    -9.999997E-01     1.000000E+00          l
197  traj.phases.phase0.control_group.indep_controls.controls:u_16      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
198  traj.phases.phase0.control_group.indep_controls.controls:u_17      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
199  traj.phases.phase0.control_group.indep_controls.controls:u_18      c    -1.000000E+00    -9.999999E-01     1.000000E+00          l
200  traj.phases.phase0.control_group.indep_controls.controls:u_19      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
201  traj.phases.phase0.control_group.indep_controls.controls:u_20      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
202  traj.phases.phase0.control_group.indep_controls.controls:u_21      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
203  traj.phases.phase0.control_group.indep_controls.controls:u_22      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
204  traj.phases.phase0.control_group.indep_controls.controls:u_23      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
205  traj.phases.phase0.control_group.indep_controls.controls:u_24      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
206  traj.phases.phase0.control_group.indep_controls.controls:u_25      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
207  traj.phases.phase0.control_group.indep_controls.controls:u_26      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
208  traj.phases.phase0.control_group.indep_controls.controls:u_27      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
209  traj.phases.phase0.control_group.indep_controls.controls:u_28      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
210  traj.phases.phase0.control_group.indep_controls.controls:u_29      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
211  traj.phases.phase0.control_group.indep_controls.controls:u_30      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
212  traj.phases.phase0.control_group.indep_controls.controls:u_31      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
213  traj.phases.phase0.control_group.indep_controls.controls:u_32      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
214  traj.phases.phase0.control_group.indep_controls.controls:u_33      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
215  traj.phases.phase0.control_group.indep_controls.controls:u_34      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
216  traj.phases.phase0.control_group.indep_controls.controls:u_35      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
217  traj.phases.phase0.control_group.indep_controls.controls:u_36      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
218  traj.phases.phase0.control_group.indep_controls.controls:u_37      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
219  traj.phases.phase0.control_group.indep_controls.controls:u_38      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
220  traj.phases.phase0.control_group.indep_controls.controls:u_39      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
221  traj.phases.phase0.control_group.indep_controls.controls:u_40      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
222  traj.phases.phase0.control_group.indep_controls.controls:u_41      c    -1.000000E+00    -9.999999E-01     1.000000E+00          l
223  traj.phases.phase0.control_group.indep_controls.controls:u_42      c    -1.000000E+00    -9.999999E-01     1.000000E+00          l
224  traj.phases.phase0.control_group.indep_controls.controls:u_43      c    -1.000000E+00    -1.000000E+00     1.000000E+00          l
225  traj.phases.phase0.control_group.indep_controls.controls:u_44      c    -1.000000E+00     2.836322E-01     1.000000E+00
226  traj.phases.phase0.control_group.indep_controls.controls:u_45      c    -1.000000E+00     9.999994E-01     1.000000E+00          u
227  traj.phases.phase0.control_group.indep_controls.controls:u_46      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
228  traj.phases.phase0.control_group.indep_controls.controls:u_47      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
229  traj.phases.phase0.control_group.indep_controls.controls:u_48      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
230  traj.phases.phase0.control_group.indep_controls.controls:u_49      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
231  traj.phases.phase0.control_group.indep_controls.controls:u_50      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
232  traj.phases.phase0.control_group.indep_controls.controls:u_51      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
233  traj.phases.phase0.control_group.indep_controls.controls:u_52      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
234  traj.phases.phase0.control_group.indep_controls.controls:u_53      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
235  traj.phases.phase0.control_group.indep_controls.controls:u_54      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
236  traj.phases.phase0.control_group.indep_controls.controls:u_55      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
237  traj.phases.phase0.control_group.indep_controls.controls:u_56      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
238  traj.phases.phase0.control_group.indep_controls.controls:u_57      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
239  traj.phases.phase0.control_group.indep_controls.controls:u_58      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
240  traj.phases.phase0.control_group.indep_controls.controls:u_59      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
241  traj.phases.phase0.control_group.indep_controls.controls:u_60      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
242  traj.phases.phase0.control_group.indep_controls.controls:u_61      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
243  traj.phases.phase0.control_group.indep_controls.controls:u_62      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
244  traj.phases.phase0.control_group.indep_controls.controls:u_63      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
245  traj.phases.phase0.control_group.indep_controls.controls:u_64      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
246  traj.phases.phase0.control_group.indep_controls.controls:u_65      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
247  traj.phases.phase0.control_group.indep_controls.controls:u_66      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
248  traj.phases.phase0.control_group.indep_controls.controls:u_67      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
249  traj.phases.phase0.control_group.indep_controls.controls:u_68      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
250  traj.phases.phase0.control_group.indep_controls.controls:u_69      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
251  traj.phases.phase0.control_group.indep_controls.controls:u_70      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
252  traj.phases.phase0.control_group.indep_controls.controls:u_71      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
253  traj.phases.phase0.control_group.indep_controls.controls:u_72      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
254  traj.phases.phase0.control_group.indep_controls.controls:u_73      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
255  traj.phases.phase0.control_group.indep_controls.controls:u_74      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
256  traj.phases.phase0.control_group.indep_controls.controls:u_75      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
257  traj.phases.phase0.control_group.indep_controls.controls:u_76      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
258  traj.phases.phase0.control_group.indep_controls.controls:u_77      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
259  traj.phases.phase0.control_group.indep_controls.controls:u_78      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
260  traj.phases.phase0.control_group.indep_controls.controls:u_79      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
261  traj.phases.phase0.control_group.indep_controls.controls:u_80      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
262  traj.phases.phase0.control_group.indep_controls.controls:u_81      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
263  traj.phases.phase0.control_group.indep_controls.controls:u_82      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
264  traj.phases.phase0.control_group.indep_controls.controls:u_83      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
265  traj.phases.phase0.control_group.indep_controls.controls:u_84      c    -1.000000E+00     9.999999E-01     1.000000E+00          u
266  traj.phases.phase0.control_group.indep_controls.controls:u_85      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
267  traj.phases.phase0.control_group.indep_controls.controls:u_86      c    -1.000000E+00     9.999998E-01     1.000000E+00          u
268  traj.phases.phase0.control_group.indep_controls.controls:u_87      c    -1.000000E+00     9.999997E-01     1.000000E+00          u
269  traj.phases.phase0.control_group.indep_controls.controls:u_88      c    -1.000000E+00     1.000000E+00     1.000000E+00          u
270  traj.phases.phase0.control_group.indep_controls.controls:u_89      c    -1.000000E+00     9.999988E-01     1.000000E+00

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    0.000000E+00    0.000000E+00              9.00000E+100
1  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.103053E-16    0.000000E+00              9.00000E+100
2  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    5.938857E-18    0.000000E+00              9.00000E+100
3  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -4.973792E-17    0.000000E+00              9.00000E+100
4  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.622703E-16    0.000000E+00              9.00000E+100
5  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.954581E-16    0.000000E+00              9.00000E+100
6  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    8.744966E-16    0.000000E+00              9.00000E+100
7  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.147684E-15    0.000000E+00              9.00000E+100
8  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.071176E-15    0.000000E+00              9.00000E+100
9  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.065935E-15    0.000000E+00              9.00000E+100
10  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    5.297460E-15    0.000000E+00              9.00000E+100
11  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    5.134142E-15    0.000000E+00              9.00000E+100
12  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.533620E-15    0.000000E+00              9.00000E+100
13  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.521832E-15    0.000000E+00              9.00000E+100
14  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.731874E-15    0.000000E+00              9.00000E+100
15  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.341765E-15    0.000000E+00              9.00000E+100
16  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.001951E-15    0.000000E+00              9.00000E+100
17  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.152048E-15    0.000000E+00              9.00000E+100
18  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.039210E-15    0.000000E+00              9.00000E+100
19  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.999123E-15    0.000000E+00              9.00000E+100
20  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.480170E-15    0.000000E+00              9.00000E+100
21  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.477201E-15    0.000000E+00              9.00000E+100
22  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.456415E-15    0.000000E+00              9.00000E+100
23  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.500956E-15    0.000000E+00              9.00000E+100
24  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.266371E-15    0.000000E+00              9.00000E+100
25  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.678424E-15    0.000000E+00              9.00000E+100
26  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.298338E-15    0.000000E+00              9.00000E+100
27  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.149866E-15    0.000000E+00              9.00000E+100
28  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    9.086451E-16    0.000000E+00              9.00000E+100
29  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -3.325760E-16    0.000000E+00              9.00000E+100
30  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -8.136234E-16    0.000000E+00              9.00000E+100
31  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -2.310215E-15    0.000000E+00              9.00000E+100
32  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -3.670213E-15    0.000000E+00              9.00000E+100
33  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -5.267766E-15    0.000000E+00              9.00000E+100
34  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -6.134839E-15    0.000000E+00              9.00000E+100
35  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -8.130295E-15    0.000000E+00              9.00000E+100
36  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -9.745664E-15    0.000000E+00              9.00000E+100
37  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.100470E-14    0.000000E+00              9.00000E+100
38  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.256068E-14    0.000000E+00              9.00000E+100
39  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.391474E-14    0.000000E+00              9.00000E+100
40  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.591614E-14    0.000000E+00              9.00000E+100
41  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.842530E-14    0.000000E+00              9.00000E+100
42  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -2.029901E-14    0.000000E+00              9.00000E+100
43  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -2.491944E-14    0.000000E+00              9.00000E+100
44  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -6.102175E-15    0.000000E+00              9.00000E+100
45  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.487981E-14    0.000000E+00              9.00000E+100
46  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    4.055571E-14    0.000000E+00              9.00000E+100
47  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.939586E-14    0.000000E+00              9.00000E+100
48  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.250336E-14    0.000000E+00              9.00000E+100
49  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.848676E-14    0.000000E+00              9.00000E+100
50  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.232817E-14    0.000000E+00              9.00000E+100
51  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.202231E-14    0.000000E+00              9.00000E+100
52  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.256869E-14    0.000000E+00              9.00000E+100
53  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.739595E-14    0.000000E+00              9.00000E+100
54  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.615472E-14    0.000000E+00              9.00000E+100
55  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.365447E-14    0.000000E+00              9.00000E+100
56  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.824417E-14    0.000000E+00              9.00000E+100
57  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.742461E-14    0.000000E+00              9.00000E+100
58  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.327928E-14    0.000000E+00              9.00000E+100
59  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    7.768024E-15    0.000000E+00              9.00000E+100
60  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    6.485231E-15    0.000000E+00              9.00000E+100
61  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.874407E-15    0.000000E+00              9.00000E+100
62  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.769779E-15    0.000000E+00              9.00000E+100
63  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -3.111961E-15    0.000000E+00              9.00000E+100
64  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -5.143050E-15    0.000000E+00              9.00000E+100
65  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -7.744269E-15    0.000000E+00              9.00000E+100
66  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -8.136234E-15    0.000000E+00              9.00000E+100
67  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -8.884529E-15    0.000000E+00              9.00000E+100
68  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -9.276494E-15    0.000000E+00              9.00000E+100
69  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -9.145839E-15    0.000000E+00              9.00000E+100
70  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -8.551954E-15    0.000000E+00              9.00000E+100
71  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -7.566103E-15    0.000000E+00              9.00000E+100
72  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -7.055362E-15    0.000000E+00              9.00000E+100
73  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -5.867590E-15    0.000000E+00              9.00000E+100
74  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -4.525409E-15    0.000000E+00              9.00000E+100
75  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -3.806807E-15    0.000000E+00              9.00000E+100
76  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -2.535892E-15    0.000000E+00              9.00000E+100
77  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -1.158077E-15    0.000000E+00              9.00000E+100
78  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00   -4.572920E-16    0.000000E+00              9.00000E+100
79  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    7.126628E-16    0.000000E+00              9.00000E+100
80  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.852923E-15    0.000000E+00              9.00000E+100
81  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.749691E-15    0.000000E+00              9.00000E+100
82  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    3.984973E-15    0.000000E+00              9.00000E+100
83  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    5.344971E-15    0.000000E+00              9.00000E+100
84  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    6.734663E-15    0.000000E+00              9.00000E+100
85  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    8.854835E-15    0.000000E+00              9.00000E+100
86  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.146199E-14    0.000000E+00              9.00000E+100
87  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    1.424732E-14    0.000000E+00              9.00000E+100
88  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.034652E-14    0.000000E+00              9.00000E+100
89  traj.phases.phase0.collocation_constraint.defects:x     e   0.000000E+00    2.212818E-14    0.000000E+00              9.00000E+100
90  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.080686E-15    0.000000E+00              9.00000E+100
91  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    9.984703E-17    0.000000E+00              9.00000E+100
92  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    4.378051E-16    0.000000E+00              9.00000E+100
93  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    9.567127E-16    0.000000E+00              9.00000E+100
94  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.833622E-16    0.000000E+00              9.00000E+100
95  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    6.619969E-16    0.000000E+00              9.00000E+100
96  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.189813E-15    0.000000E+00              9.00000E+100
97  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    3.418554E-16    0.000000E+00              9.00000E+100
98  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.484714E-15    0.000000E+00              9.00000E+100
99  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    2.408392E-15    0.000000E+00              9.00000E+100
100  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.618338E-15    0.000000E+00              9.00000E+100
101  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.573938E-15    0.000000E+00              9.00000E+100
102  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -4.647898E-15    0.000000E+00              9.00000E+100
103  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -4.398466E-16    0.000000E+00              9.00000E+100
104  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -5.267023E-16    0.000000E+00              9.00000E+100
105  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.981351E-15    0.000000E+00              9.00000E+100
106  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.542573E-16    0.000000E+00              9.00000E+100
107  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -4.094099E-16    0.000000E+00              9.00000E+100
108  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.076418E-15    0.000000E+00              9.00000E+100
109  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.639079E-16    0.000000E+00              9.00000E+100
110  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -5.051740E-16    0.000000E+00              9.00000E+100
111  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -9.283175E-16    0.000000E+00              9.00000E+100
112  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -4.146064E-16    0.000000E+00              9.00000E+100
113  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -7.134052E-16    0.000000E+00              9.00000E+100
114  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.037815E-15    0.000000E+00              9.00000E+100
115  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -6.306323E-16    0.000000E+00              9.00000E+100
116  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -8.949115E-16    0.000000E+00              9.00000E+100
117  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.181647E-15    0.000000E+00              9.00000E+100
118  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -7.841147E-16    0.000000E+00              9.00000E+100
119  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.016287E-15    0.000000E+00              9.00000E+100
120  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.255975E-15    0.000000E+00              9.00000E+100
121  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -8.036247E-16    0.000000E+00              9.00000E+100
122  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -9.718846E-16    0.000000E+00              9.00000E+100
123  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.214496E-15    0.000000E+00              9.00000E+100
124  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -6.732251E-16    0.000000E+00              9.00000E+100
125  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -8.650316E-16    0.000000E+00              9.00000E+100
126  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.201691E-15    0.000000E+00              9.00000E+100
127  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -5.062875E-16    0.000000E+00              9.00000E+100
128  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -9.628371E-16    0.000000E+00              9.00000E+100
129  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.442957E-15    0.000000E+00              9.00000E+100
130  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -5.062875E-16    0.000000E+00              9.00000E+100
131  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.642094E-15    0.000000E+00              9.00000E+100
132  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.597507E-15    0.000000E+00              9.00000E+100
133  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.534823E-15    0.000000E+00              9.00000E+100
134  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    3.146926E-14    0.000000E+00              9.00000E+100
135  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    4.952413E-14    0.000000E+00              9.00000E+100
136  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    2.012530E-15    0.000000E+00              9.00000E+100
137  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    3.103053E-16    0.000000E+00              9.00000E+100
138  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.513518E-14    0.000000E+00              9.00000E+100
139  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    8.351517E-16    0.000000E+00              9.00000E+100
140  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    9.398241E-16    0.000000E+00              9.00000E+100
141  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    5.703529E-15    0.000000E+00              9.00000E+100
142  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.190741E-15    0.000000E+00              9.00000E+100
143  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.019999E-15    0.000000E+00              9.00000E+100
144  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    2.397071E-15    0.000000E+00              9.00000E+100
145  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    8.136234E-16    0.000000E+00              9.00000E+100
146  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    2.806110E-16    0.000000E+00              9.00000E+100
147  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    5.300430E-16    0.000000E+00              9.00000E+100
148  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -7.891256E-16    0.000000E+00              9.00000E+100
149  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.643579E-15    0.000000E+00              9.00000E+100
150  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.635041E-15    0.000000E+00              9.00000E+100
151  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.616252E-15    0.000000E+00              9.00000E+100
152  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.920154E-15    0.000000E+00              9.00000E+100
153  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.707469E-15    0.000000E+00              9.00000E+100
154  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -3.081037E-15    0.000000E+00              9.00000E+100
155  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.481885E-15    0.000000E+00              9.00000E+100
156  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.084910E-15    0.000000E+00              9.00000E+100
157  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -2.006034E-15    0.000000E+00              9.00000E+100
158  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -1.100916E-15    0.000000E+00              9.00000E+100
159  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -6.365712E-16    0.000000E+00              9.00000E+100
160  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -6.606978E-16    0.000000E+00              9.00000E+100
161  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    7.646278E-17    0.000000E+00              9.00000E+100
162  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    4.617461E-16    0.000000E+00              9.00000E+100
163  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    7.646278E-17    0.000000E+00              9.00000E+100
164  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    5.671608E-16    0.000000E+00              9.00000E+100
165  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    9.992126E-16    0.000000E+00              9.00000E+100
166  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    2.338425E-16    0.000000E+00              9.00000E+100
167  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    7.427283E-16    0.000000E+00              9.00000E+100
168  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.235653E-15    0.000000E+00              9.00000E+100
169  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.527400E-16    0.000000E+00              9.00000E+100
170  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.011276E-15    0.000000E+00              9.00000E+100
171  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.881504E-15    0.000000E+00              9.00000E+100
172  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.252728E-16    0.000000E+00              9.00000E+100
173  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    1.875844E-15    0.000000E+00              9.00000E+100
174  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    3.445326E-15    0.000000E+00              9.00000E+100
175  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    4.392666E-16    0.000000E+00              9.00000E+100
176  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    4.069880E-15    0.000000E+00              9.00000E+100
177  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    7.093686E-15    0.000000E+00              9.00000E+100
178  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00    3.328729E-15    0.000000E+00              9.00000E+100
179  traj.phases.phase0.collocation_constraint.defects:v     e   0.000000E+00   -6.204249E-16    0.000000E+00              9.00000E+100
180  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
181  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
182  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    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   -1.110223E-16    0.000000E+00              9.00000E+100
185  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
186  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
187  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    2.220446E-16    0.000000E+00              9.00000E+100
188  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    2.220446E-16    0.000000E+00              9.00000E+100
189  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
190  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
191  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    3.330669E-16    0.000000E+00              9.00000E+100
192  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
193  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -2.220446E-16    0.000000E+00              9.00000E+100
194  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
195  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
196  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
197  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -2.220446E-16    0.000000E+00              9.00000E+100
198  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
199  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -2.220446E-16    0.000000E+00              9.00000E+100
200  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
201  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    2.220446E-16    0.000000E+00              9.00000E+100
202  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
203  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -3.330669E-16    0.000000E+00              9.00000E+100
204  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
205  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    4.440892E-16    0.000000E+00              9.00000E+100
206  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    0.000000E+00    0.000000E+00              9.00000E+100
207  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00    1.110223E-16    0.000000E+00              9.00000E+100
208  traj.phases.phase0.continuity_comp.defect_controls:u    e   0.000000E+00   -1.110223E-16    0.000000E+00              9.00000E+100
209  traj.phases.phase0->final_boundary_constraint->x        i   5.000000E-01    5.000000E-01    1.000000E+30         l    9.00000E+100
210  traj.phases.phase0->final_boundary_constraint->v        i   0.000000E+00    2.285376E-02    1.000000E+30              9.00000E+100
211  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
212  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
213  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
214  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
215  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
216  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
217  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
218  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
219  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
220  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
221  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
222  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
223  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
224  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
225  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -2.836320E-01    1.000000E+00              9.00000E+100
226  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999990E-01    1.000000E+00         l    9.00000E+100
227  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999990E-01    1.000000E+00         l    9.00000E+100
228  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
229  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
230  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999997E-01    1.000000E+00         l    9.00000E+100
231  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999997E-01    1.000000E+00         l    9.00000E+100
232  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
233  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
234  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
235  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
236  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
237  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
238  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
239  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
240  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
241  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
242  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
243  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
244  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
245  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
246  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
247  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
248  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
249  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
250  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
251  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
252  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
253  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
254  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
255  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
256  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
257  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
258  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
259  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
260  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
261  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
262  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
263  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
264  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
265  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
266  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
267  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -9.999999E-01    1.000000E+00         l    9.00000E+100
268  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00   -1.000000E+00    1.000000E+00         l    9.00000E+100
269  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    2.836322E-01    1.000000E+00              9.00000E+100
270  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999994E-01    1.000000E+00         u    9.00000E+100
271  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999994E-01    1.000000E+00         u    9.00000E+100
272  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
273  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
274  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
275  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
276  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
277  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
278  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
279  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
280  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
281  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
282  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
283  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
284  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
285  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
286  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
287  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
288  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
289  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
290  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
291  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
292  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
293  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
294  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
295  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
296  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
297  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
298  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
299  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
300  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
301  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
302  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
303  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
304  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
305  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
306  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
307  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
308  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
309  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
310  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
311  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
312  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
313  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
314  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
315  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
316  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
317  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
318  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
319  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
320  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
321  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
322  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
323  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999999E-01    1.000000E+00         u    9.00000E+100
324  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
325  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999998E-01    1.000000E+00         u    9.00000E+100
326  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999997E-01    1.000000E+00         u    9.00000E+100
327  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999997E-01    1.000000E+00         u    9.00000E+100
328  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    1.000000E+00    1.000000E+00         u    9.00000E+100
329  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999988E-01    1.000000E+00              9.00000E+100
330  traj.phases.phase0->path_constraint->u                  i  -1.000000E+00    9.999980E-01    1.000000E+00              9.00000E+100

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, 3.974e-07, 4.766e-05, 1.151e-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, 6.762e-07, 8.227e-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, 5.763e-07, 3.264e-07, 3.517e-07, 4.998e-07]
Successfully completed grid refinement.
==================================================

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')

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.

Hide code cell source
fig = plt.figure(constrained_layout=True, figsize=(12, 6))

anim_ax.set_aspect('equal')

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]
t = t_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')