# Aircraft Balanced Field Length Calculation¶

Things you'll learn through this example

• How to perform branching trajectories
• How to constraint the difference between values at the end of two different phases
• Using complex-step differentiation on a monolithic ODE component

The United States Federal Aviation Regulations Part 25 defines a balanced field length for the aircraft as the shortest field which can accommodate a "balanced takeoff". In a balanced takeoff the aircraft accelerates down the runway to some critical speed "V1".

Before achieving V1, the aircraft must be capable of rejecting the takeoff and coming to a stop before the end of the runway.

After V1, the aircraft must be capable of achieving an altitude of 35 ft above the end of the runway with a speed of V2 (the minimum safe takeoff speed or 1.2 x the stall speed) while on a single engine (for two engine aircraft).

At V1, both options must be available. The nominal phase sequence for this trajectory is:

1. Break Release to V1 (br_to_v1)

Accelerate down the runway under the power of two engines. Where V1 is some as-yet-undetermined speed.

2. V1 to Vr (v1_to_vr)

Accelerate down the runway under the power of a single engine. End at "Vr" or the rotation speed. The rotation speed here is defined as 1.2 times the stall speed.

3. Rotate (rotate)

Operating under the power of a single engine, begin pitching the nose up (increasing alpha) while rolling down the runway. In this example, the rate of change of alpha is linear over some time.

4. Climb (climb)

Still operating with one engine, begin climbing to a flight path angle of no more than 5 degrees. This phase ends when the aircraft is 35 feet above the ground with an airspeed of 1.25 x the stall speed.

5. Rejected Takeoff (rto)

Shut down all engines (zero thrust) and apply brakes (increase of runway $\mu$ coefficient to 0.3) until the aircraft airspeed is zero.

This phase is continuous in time and state with the first phase, and so forms a branch off of the nominal trajectory.

Since the RTO phase and the climb phase both must end such that they do not exceed the end of the runway, the final value of range in each of these two phases must be the same. We don't know what this value is until we've solved the problem, so we cannot simply constrain both to the same fixed value.

Instead, we'll use a trajectory linkage constraint to ensure that range at the end of the RTO phase is equal to range and the end of the climb phase.

More information on calculating the balanced field length is available in section 17.8 of Aircraft Design: A Conceptual Approach by Daniel Raymer1.

## The ODE System¶

In this problem two sets of ordinary differential equations are used: one for the aircraft motion on the runway, and one for the aircraft motion once airborne.

For simplification, we're going to assume a constant atmospheric model since the aircraft will never exceed 35 feet of altitude. Also, since the duration of the trajectory is small, we're going to assume that the vehicle fuel burn is negligible

### The Aerodynamic Model¶

Both sets of equations of motion have common aerodynamic models.

First, the lift coefficient is computed using a model which assumes linearity in lift wrt the angle of attack.

\begin{align} C_L &= C_{L0} + \frac{\alpha}{\alpha_{max}} \left(C_{L-max} - C_{L0}\right) \end{align}

Next, the drag-due-to-lift factor is computed (Equations 12.48 and 12.61 in Raymer1).

\begin{align} K_{nom} &= \frac{1}{ \pi AR e} \\ b &= \frac{span}{2} \\ K &= 33 K_{nom} \frac{ \left(\frac{h + h_w}{b} \right) ^{\frac{3}{2}}}{1.0 + 33 \left( \frac{h + h_w}{b}\right) ^{\frac{3}{2}}} \end{align}

Note the singularity in the equation for $K$ when $h + h_w$ is negative. This causes this problem to be difficult to solve using a shooting method. If the optimizer proposes a combination of initial states and a control history that results in altitude dropping significantly below zero, the propagation will fail.

Finally, the lift and drag are computed after computing the dynamic pressure.

\begin{align} q &= 0.5 \rho v^2 \\ L &= q S C_L \\ D &= q S \left( C_{D0} + K C_{L}^2 \right) \end{align}

### Stall Speed¶

This model relies on the ratio of the current true airspeed to stall speed ($\frac{v}{v_{stall}}$). This constraint is used to trigger the beginning of rotation and is used as a boundary constraint at the end of the initial climb. Stall speed is given by Equation 5.6 in Raymer1.

\begin{align} W &= m g \\ v_{stall} &= \sqrt{\frac{2 * W}{\rho S C_{L-max}}} \end{align}

### Runway Equations of Motion¶

The runway equations of motion are used to integrate range and speed as the vehicle rolls along the runway.

\begin{align} F_r &= mg - L \cos \alpha - T \sin \alpha \\ \dot{v} &= \frac{T \cos \alpha - D - F_r \mu_r}{m} \\ \dot{r} &= v \end{align}
State Description Units
r range $m$
v true airspeed $m s^{-1}$

### Flight Equations of Motion¶

The flight equations of motion include two additional state variables: the flight-path angle ($\gamma$) and altitude ($h$).

\begin{align} \dot{v} &= \frac{T}{m} \cos \alpha - \frac{D}{m} - g \sin \gamma \\ \dot{\gamma} &= \frac{T}{m v} \sin \alpha + \frac{L}{m v} - \frac{g \cos \gamma}{v} \\ \dot{h} &= v \sin \gamma \\ \dot{r} &= v \cos \gamma \end{align}
State Description Units
v true airspeed $m s^{-1}$
$\gamma$ flight path angle $rad$
r range $m$
h altitude $m$

### Treatment of the angle-of-attack ($\alpha$)¶

In three of the runway phases (break release to $V1$, $V1$ to $V_r$, and rejected takeoff) $\alpha$ is treated as a fixed static parameter.

In the rotation phase, $\alpha$ is treated as a polynomial control of order 1. $\alpha$ starts at an initial value of zero and increases at a linear rate until the upward forces on the aircraft due to lift and thrust cancel the downward force due to gravity.

In the climb phase, $\alpha$ is treated as a dynamic control to be chosen by the optimizer.

Phase linkage constraints are used to maintain continuity in $\alpha$ throughout the trajectory.

### Parameters¶

The following parameters define properties of the aircraft and environment for the problem.

Parameters Description Units Value
m mass $kg$ 79015.7909
g gravitational acceleration $m s^{-2}$ 9.80665
T thrust $N$ 2 x 120101.98 (nominal)
$\mu$ runway friction coefficient - 0.03 (nominal)
h altitude $m$ 0
$\rho$ atmospheric density $kg\,m^{3}$ 1.225
S aerodynamic reference area $m^2$ 124.7
CD0 zero-lift drag coefficient - 0.03
AR wing aspect ratio - 9.45
e Oswald's wing efficiency - 801
span wingspan $m$ 35.7
h_w height of wing above CoG $m$ 1.0
CL0 aerodynamic reference area - 0.5
CL_max aerodynamic reference area - 2.0

## The Optimal Control Problem¶

The following constraints and objective complete the definition of this optimal control problem.

### Objective¶

Name Phase Location Description Minimized or Maximized Ref
r rto final range Minimized 1000

### Nonlinear Boundary Constraints¶

Name Phase Description Loc Units Lower Upper Equals Ref
v_over_v_stall v1_to_vr $\frac{v}{v_{stall}}$ final - 1.2 1.2
v rto airspeed final $ms^{-1}$ 0 100
F_r rotate downforce on gear final $N$ 0 100000
h climb altitude final $ft$ 35 35
gam climb flight path angle final $rad$ 5 5
v_over_v_stall climb $\frac{v}{v_{stall}}$ final - 1.25 1.25

### Nonlinear Path Constraints¶

Name Phase Description Units Lower Upper Equals Ref
gam climb flight path angle $rad$ 0 5 5

### Phase Continuity Constraints¶

First Phase Second Phase Variables
br_to_v1[final] v1_to_vr[initial] $time$, $r$, $v$
vr_to_v1[final] rotate[initial] $time$, $r$, $v$, $\alpha$
rotate[final] climb[initial] $time$, $r$, $v$, $\alpha$
br_to_v1[final] rto[initial] $time$, $r$, $v$
climb[final] rto[final] $r$

## Source Code¶

Unlike most other Dymos examples, which use analytic derivatives, the ODE in this case is a single component. All calculations within the ODE are complex-safe and thus we can use complex-step, in conjunction with partial derivative coloring, to automatically compute the derivatives using complex-step with reasonable speed.

Since there is significant commonality between the ODEs for the runway roll and the climb, this implementation uses a single ODE class with an option mode that can be set to either 'runway' or 'climb'. Based on the value of mode, the component conditionally changes its inputs and outputs.

### BalancedFieldODEComp¶

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 class BalancedFieldODEComp(om.ExplicitComponent): """ The ODE System for an aircraft takeoff climb. Computes the rates for states v (true airspeed) gam (flight path angle) r (range) and h (altitude). References ---------- .. [1] Raymer, Daniel. Aircraft design: a conceptual approach. American Institute of Aeronautics and Astronautics, Inc., 2012. """ def initialize(self): self.options.declare('num_nodes', types=int) self.options.declare('g', types=(float, int), default=9.80665, desc='gravitational acceleration (m/s**2)') self.options.declare('mode', values=('runway', 'climb'), desc='mode of operation (ground roll or flight)') def setup(self): nn = self.options['num_nodes'] # Scalar (constant) inputs self.add_input('rho', val=1.225, desc='atmospheric density at runway', units='kg/m**3') self.add_input('S', val=124.7, desc='aerodynamic reference area', units='m**2') self.add_input('CD0', val=0.03, desc='zero-lift drag coefficient', units=None) self.add_input('CL0', val=0.5, desc='zero-alpha lift coefficient', units=None) self.add_input('CL_max', val=2.0, desc='maximum lift coefficient for linear fit', units=None) self.add_input('alpha_max', val=np.radians(10), desc='angle of attack at CL_max', units='rad') self.add_input('h_w', val=1.0, desc='height of the wing above the CG', units='m') self.add_input('AR', val=9.45, desc='wing aspect ratio', units=None) self.add_input('e', val=0.801, desc='Oswald span efficiency factor', units=None) self.add_input('span', val=35.7, desc='Wingspan', units='m') self.add_input('T', val=1.0, desc='thrust', units='N') # Dynamic inputs (can assume a different value at every node) self.add_input('m', shape=(nn,), desc='aircraft mass', units='kg') self.add_input('v', shape=(nn,), desc='aircraft true airspeed', units='m/s') self.add_input('h', shape=(nn,), desc='altitude', units='m') self.add_input('alpha', shape=(nn,), desc='angle of attack', units='rad') # Outputs self.add_output('CL', shape=(nn,), desc='lift coefficient', units=None) self.add_output('q', shape=(nn,), desc='dynamic pressure', units='Pa') self.add_output('L', shape=(nn,), desc='lift force', units='N') self.add_output('D', shape=(nn,), desc='drag force', units='N') self.add_output('K', val=np.ones(nn), desc='drag-due-to-lift factor', units=None) self.add_output('F_r', shape=(nn,), desc='runway normal force', units='N') self.add_output('v_dot', shape=(nn,), desc='rate of change of speed', units='m/s**2', tags=['state_rate_source:v']) self.add_output('r_dot', shape=(nn,), desc='rate of change of range', units='m/s', tags=['state_rate_source:r']) self.add_output('W', shape=(nn,), desc='aircraft weight', units='N') self.add_output('v_stall', shape=(nn,), desc='stall speed', units='m/s') self.add_output('v_over_v_stall', shape=(nn,), desc='stall speed ratio', units=None) # Mode-dependent IO if self.options['mode'] == 'runway': self.add_input('mu_r', val=0.05, desc='runway friction coefficient', units=None) else: self.add_input('gam', shape=(nn,), desc='flight path angle', units='rad') self.add_output('gam_dot', shape=(nn,), desc='rate of change of flight path angle', units='rad/s', tags=['state_rate_source:gam']) self.add_output('h_dot', shape=(nn,), desc='rate of change of altitude', units='m/s', tags=['state_rate_source:h']) self.declare_coloring(wrt='*', method='cs') def compute(self, inputs, outputs, discrete_inputs=None, discrete_outputs=None): g = self.options['g'] # Compute factor k to include ground effect on lift rho = inputs['rho'] v = inputs['v'] S = inputs['S'] CD0 = inputs['CD0'] m = inputs['m'] T = inputs['T'] h = inputs['h'] h_w = inputs['h_w'] span = inputs['span'] AR = inputs['AR'] CL0 = inputs['CL0'] alpha = inputs['alpha'] alpha_max = inputs['alpha_max'] CL_max = inputs['CL_max'] e = inputs['e'] outputs['W'] = W = m * g outputs['v_stall'] = v_stall = np.sqrt(2 * W / rho / S / CL_max) outputs['v_over_v_stall'] = v / v_stall outputs['CL'] = CL = CL0 + (alpha / alpha_max) * (CL_max - CL0) K_nom = 1.0 / (np.pi * AR * e) b = span / 2.0 fact = ((h + h_w) / b) ** 1.5 outputs['K'] = K = K_nom * 33 * fact / (1.0 + 33 * fact) outputs['q'] = q = 0.5 * rho * v ** 2 outputs['L'] = L = q * S * CL outputs['D'] = D = q * S * (CD0 + K * CL ** 2) # Compute the downward force on the landing gear calpha = np.cos(alpha) salpha = np.sin(alpha) # Runway normal force outputs['F_r'] = F_r = m * g - L * calpha - T * salpha # Compute the dynamics if self.options['mode'] == 'climb': gam = inputs['gam'] cgam = np.cos(gam) sgam = np.sin(gam) outputs['v_dot'] = (T * calpha - D) / m - g * sgam outputs['gam_dot'] = (T * salpha + L) / (m * v) - (g / v) * cgam outputs['h_dot'] = v * sgam outputs['r_dot'] = v * cgam else: outputs['v_dot'] = (T * calpha - D - F_r * inputs['mu_r']) / m outputs['r_dot'] = v 

## Building and running the problem¶

In the following code we define and solve the optimal control problem. Note the use of add_linkage_constraint to handle the less common phase linkage condition, where the range must be equal at the end of the rto and climb phases.

import matplotlib.pyplot as plt
import openmdao.api as om
from openmdao.utils.general_utils import set_pyoptsparse_opt
from openmdao.utils.assert_utils import assert_near_equal
import dymos as dm
from dymos.examples.balanced_field.balanced_field_ode import BalancedFieldODEComp

p = om.Problem()

_, optimizer = set_pyoptsparse_opt('IPOPT', fallback=True)

p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring()

# Use IPOPT if available, with fallback to SLSQP
p.driver.options['optimizer'] = optimizer
p.driver.options['print_results'] = False
if optimizer == 'IPOPT':
p.driver.opt_settings['print_level'] = 5
p.driver.opt_settings['derivative_test'] = 'first-order'

# First Phase: Brake release to V1 - both engines operable
ode_init_kwargs={'mode': 'runway'})
br_to_v1.set_time_options(fix_initial=True, duration_bounds=(1, 1000), duration_ref=10.0)

# Second Phase: Rejected takeoff at V1 - no engines operable
ode_init_kwargs={'mode': 'runway'})
rto.set_time_options(fix_initial=False, duration_bounds=(1, 1000), duration_ref=1.0)

# Third Phase: V1 to Vr - single engine operable
ode_init_kwargs={'mode': 'runway'})
v1_to_vr.set_time_options(fix_initial=False, duration_bounds=(1, 1000), duration_ref=1.0)

# Fourth Phase: Rotate - single engine operable
ode_init_kwargs={'mode': 'runway'})
rotate.set_time_options(fix_initial=False, duration_bounds=(1.0, 5), duration_ref=1.0)
rotate.add_polynomial_control('alpha', order=1, opt=True, units='deg', lower=0, upper=10, ref=10, val=[0, 10])

# Fifth Phase: Climb to target speed and altitude at end of runway.
ode_init_kwargs={'mode': 'climb'})
climb.set_time_options(fix_initial=False, duration_bounds=(1, 100), duration_ref=1.0)
climb.add_control('alpha', opt=True, units='deg', lower=-10, upper=15, ref=10)

# Instantiate the trajectory and add phases
traj = dm.Trajectory()

# Add parameters common to multiple phases to the trajectory
desc='aircraft mass',
targets={'br_to_v1': ['m'], 'v1_to_vr': ['m'], 'rto': ['m'],
'rotate': ['m'], 'climb': ['m']})

traj.add_parameter('T_nominal', val=27000 * 2, opt=False, units='lbf', dynamic=False,
desc='nominal aircraft thrust',
targets={'br_to_v1': ['T']})

desc='thrust under a single engine',
targets={'v1_to_vr': ['T'], 'rotate': ['T'], 'climb': ['T']})

desc='thrust when engines are shut down for rejected takeoff',
targets={'rto': ['T']})

desc='nominal runway friction coefficient',
targets={'br_to_v1': ['mu_r'], 'v1_to_vr': ['mu_r'],  'rotate': ['mu_r']})

desc='runway friction coefficient under braking',
targets={'rto': ['mu_r']})

desc='runway altitude',
targets={'br_to_v1': ['h'], 'v1_to_vr': ['h'], 'rto': ['h'],
'rotate': ['h']})

desc='atmospheric density',
targets={'br_to_v1': ['rho'], 'v1_to_vr': ['rho'], 'rto': ['rho'],
'rotate': ['rho']})

desc='aerodynamic reference area',
targets={'br_to_v1': ['S'], 'v1_to_vr': ['S'], 'rto': ['S'],
'rotate': ['S'], 'climb': ['S']})

desc='zero-lift drag coefficient',
targets={f'{phase}': ['CD0'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='wing aspect ratio',
targets={f'{phase}': ['AR'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='Oswald span efficiency factor',
targets={f'{phase}': ['e'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='wingspan',
targets={f'{phase}': ['span'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='height of wing above CG',
targets={f'{phase}': ['h_w'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='zero-alpha lift coefficient',
targets={f'{phase}': ['CL0'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='maximum lift coefficient for linear fit',
targets={f'{phase}': ['CL_max'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

desc='angle of attack at maximum lift',
targets={f'{phase}': ['alpha_max'] for phase in ['br_to_v1', 'v1_to_vr',
'rto', 'rotate' 'climb']})

# Standard "end of first phase to beginning of second phase" linkages
# Alpha changes from being a parameter in v1_to_vr to a polynomial control
# in rotate, to a dynamic control in climb.
traj.link_phases(['v1_to_vr', 'rotate'], vars=['time', 'r', 'v', 'alpha'])
traj.link_phases(['rotate', 'climb'], vars=['time', 'r', 'v', 'alpha'])

# Less common "final value of r must be the match at ends of two phases".
phase_b='climb', var_b='r', loc_b='final',
ref=1000)

# Define the constraints and objective for the optimal control problem

climb.add_boundary_constraint('h', loc='final', equals=35, ref=35, units='ft', linear=True)
climb.add_boundary_constraint('gam', loc='final', equals=5, ref=5, units='deg', linear=True)

#
# Setup the problem and set the initial guess
#
p.setup(check=True)

p.set_val('traj.br_to_v1.t_initial', 0)
p.set_val('traj.br_to_v1.t_duration', 35)
p.set_val('traj.br_to_v1.states:r', br_to_v1.interpolate(ys=[0, 2500.0], nodes='state_input'))
p.set_val('traj.br_to_v1.states:v', br_to_v1.interpolate(ys=[0, 100.0], nodes='state_input'))
p.set_val('traj.br_to_v1.parameters:alpha', 0, units='deg')

p.set_val('traj.v1_to_vr.t_initial', 35)
p.set_val('traj.v1_to_vr.t_duration', 35)
p.set_val('traj.v1_to_vr.states:r', v1_to_vr.interpolate(ys=[2500, 300.0], nodes='state_input'))
p.set_val('traj.v1_to_vr.states:v', v1_to_vr.interpolate(ys=[100, 110.0], nodes='state_input'))
p.set_val('traj.v1_to_vr.parameters:alpha', 0.0, units='deg')

p.set_val('traj.rto.t_initial', 35)
p.set_val('traj.rto.t_duration', 35)
p.set_val('traj.rto.states:r', rto.interpolate(ys=[2500, 5000.0], nodes='state_input'))
p.set_val('traj.rto.states:v', rto.interpolate(ys=[110, 0], nodes='state_input'))
p.set_val('traj.rto.parameters:alpha', 0.0, units='deg')

p.set_val('traj.rotate.t_initial', 70)
p.set_val('traj.rotate.t_duration', 5)
p.set_val('traj.rotate.states:r', rotate.interpolate(ys=[1750, 1800.0], nodes='state_input'))
p.set_val('traj.rotate.states:v', rotate.interpolate(ys=[80, 85.0], nodes='state_input'))
p.set_val('traj.rotate.polynomial_controls:alpha', 0.0, units='deg')

p.set_val('traj.climb.t_initial', 75)
p.set_val('traj.climb.t_duration', 15)
p.set_val('traj.climb.states:r', climb.interpolate(ys=[5000, 5500.0], nodes='state_input'), units='ft')
p.set_val('traj.climb.states:v', climb.interpolate(ys=[160, 170.0], nodes='state_input'), units='kn')
p.set_val('traj.climb.states:h', climb.interpolate(ys=[0, 35.0], nodes='state_input'), units='ft')
p.set_val('traj.climb.states:gam', climb.interpolate(ys=[0, 5.0], nodes='state_input'), units='deg')
p.set_val('traj.climb.controls:alpha', 5.0, units='deg')

dm.run_problem(p, run_driver=True, simulate=True)

# Test this example in Dymos' continuous integration

fig, axes = plt.subplots(2, 1, sharex=True, gridspec_kw={'top': 0.92})
for phase in ['br_to_v1', 'rto', 'v1_to_vr', 'rotate', 'climb']:
r = sim_case.get_val(f'traj.{phase}.timeseries.states:r', units='ft')
v = sim_case.get_val(f'traj.{phase}.timeseries.states:v', units='kn')
t = sim_case.get_val(f'traj.{phase}.timeseries.time', units='s')
axes[0].plot(t, r, '-', label=phase)
axes[1].plot(t, v, '-', label=phase)
fig.suptitle('Balanced Field Length')
axes[1].set_xlabel('time (s)')
axes[0].set_ylabel('range (ft)')
axes[1].set_ylabel('airspeed (kts)')
axes[0].grid(True)
axes[1].grid(True)

tv1 = sim_case.get_val('traj.br_to_v1.timeseries.time', units='s')[-1, 0]
v1 = sim_case.get_val('traj.br_to_v1.timeseries.states:v', units='kn')[-1, 0]

tf_rto = sim_case.get_val('traj.rto.timeseries.time', units='s')[-1, 0]
rf_rto = sim_case.get_val('traj.rto.timeseries.states:r', units='ft')[-1, 0]

axes[0].annotate(f'field length = {r[-1, 0]:5.1f} ft', xy=(t[-1, 0], r[-1, 0]),
xycoords='data', xytext=(0.7, 0.5),
textcoords='axes fraction', arrowprops=dict(arrowstyle='->'),
horizontalalignment='center', verticalalignment='top')

axes[0].annotate(f'', xy=(tf_rto, rf_rto),
xycoords='data', xytext=(0.7, 0.5),
textcoords='axes fraction', arrowprops=dict(arrowstyle='->'),
horizontalalignment='center', verticalalignment='top')

axes[1].annotate(f'$v1$ = {v1:5.1f} kts', xy=(tv1, v1), xycoords='data', xytext=(0.5, 0.5),
textcoords='axes fraction', arrowprops=dict(arrowstyle='->'),
horizontalalignment='center', verticalalignment='top')

plt.legend()
plt.show()

--- Linkage Report [traj] ---
--- br_to_v1 - v1_to_vr ---
time [final]  ==  time [initial]
r    [final]  ==  r    [initial]
v    [final]  ==  v    [initial]
----------------------------
--- v1_to_vr - rotate ---
time  [final]  ==  time  [initial]
r     [final]  ==  r     [initial]
v     [final]  ==  v     [initial]
alpha [final]  ==  alpha [initial]
----------------------------
--- rotate - climb ---
time  [final]  ==  time  [initial]
r     [final]  ==  r     [initial]
v     [final]  ==  v     [initial]
alpha [final]  ==  alpha [initial]
----------------------------
--- br_to_v1 - rto ---
time [final]  ==  time [initial]
r    [final]  ==  r    [initial]
v    [final]  ==  v    [initial]
----------------------------
--- rto - climb ---
r [final]  ==  r [final]
----------------------------
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
INFO: checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
traj.phases.br_to_v1.continuity_comp
traj.phases.rto.continuity_comp
traj.phases.v1_to_vr.continuity_comp
traj.phases.rotate.continuity_comp

INFO: checking auto_ivc_warnings
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
INFO: checking comp_has_no_outputs
WARNING: The following Components do not have any outputs:
traj.phases.br_to_v1.continuity_comp
traj.phases.rto.continuity_comp
traj.phases.v1_to_vr.continuity_comp
traj.phases.rotate.continuity_comp

INFO: checking auto_ivc_warnings

Approx coloring for 'traj.phases.br_to_v1.rhs_all' (class BalancedFieldODEComp)

Jacobian shape: (132, 60)  ( 9.79% nonzero)
FWD solves: 16   REV solves: 0
Total colors vs. total size: 16 vs 60  (73.3% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.035930 sec.
Time to compute coloring: 0.021386 sec.

Approx coloring for 'traj.phases.rto.rhs_all' (class BalancedFieldODEComp)

Jacobian shape: (132, 60)  ( 9.77% nonzero)
FWD solves: 16   REV solves: 0
Total colors vs. total size: 16 vs 60  (73.3% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.035866 sec.
Time to compute coloring: 0.021282 sec.

Approx coloring for 'traj.phases.v1_to_vr.rhs_all' (class BalancedFieldODEComp)

Jacobian shape: (132, 60)  (10.15% nonzero)
FWD solves: 16   REV solves: 0
Total colors vs. total size: 16 vs 60  (73.3% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.036514 sec.
Time to compute coloring: 0.021756 sec.

Approx coloring for 'traj.phases.rotate.rhs_all' (class BalancedFieldODEComp)

Jacobian shape: (132, 60)  (10.15% nonzero)
FWD solves: 16   REV solves: 0
Total colors vs. total size: 16 vs 60  (73.3% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.038735 sec.
Time to compute coloring: 0.022435 sec.

Approx coloring for 'traj.phases.climb.rhs_all' (class BalancedFieldODEComp)

Jacobian shape: (260, 111)  ( 5.54% nonzero)
FWD solves: 16   REV solves: 0
Total colors vs. total size: 16 vs 111  (85.6% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.073277 sec.
Time to compute coloring: 0.042579 sec.
Full total jacobian was computed 3 times, taking 1.432253 seconds.
Total jacobian shape: (179, 166)

Jacobian shape: (179, 166)  ( 3.56% nonzero)
FWD solves: 15   REV solves: 0
Total colors vs. total size: 15 vs 166  (91.0% improvement)

Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 1.432253 sec.
Time to compute coloring: 0.152265 sec.
/home/travis/build/OpenMDAO/dymos/OpenMDAO/openmdao/core/total_jac.py:1718: UserWarning:Constraints or objectives ['traj.phases.climb.path_constraints.path:gam'] cannot be impacted by the design variables of the problem.
/home/travis/build/OpenMDAO/dymos/OpenMDAO/openmdao/core/total_jac.py:1718: UserWarning:Constraints or objectives ['traj.phases.climb.path_constraints.path:gam'] cannot be impacted by the design variables of the problem.

Simulating trajectory traj
Done simulating trajectory traj


## References¶

1. Daniel Raymer. Aircraft design: a conceptual approach. American Institute of Aeronautics and Astronautics, Inc., 2012.