import matplotlib.pyplot as plt
from cartopy import crs as ccrs
from cartopy.feature import BORDERS
from openap import aero

import numpy as np
import opentop
import pandas as pd

Define objective function

Here I am using a three hump camel function to simulate a complex cost gird. You can also test with the six hump_camel function for more local minima.

def gaussian(x, y):
    return -np.exp(-(x**2) - y**2)


def three_hump_camel(x, y):
    return 2 * x**2 - 1.05 * x**4 + x**6 / 6 + x * y + y**2


def six_hump_camel(x, y):
    return (4 - 2.1 * x**2 + x**4 / 3) * x**2 + x * (y) + (-4 + 4 * (y) ** 2) * (y) ** 2

# search ranges for the hump functions
x = np.linspace(-2, 2, 50)
y = np.linspace(-1, 1, 50)
z = np.linspace(0, 13_000, 10)
ts = np.arange(0, 4 * 3600, 3600)

X, Y, Z = np.meshgrid(x, y, z)

V = np.zeros((len(x), len(y), len(z), len(ts)))

# shift the minima with time
for i, ts_ in enumerate(ts):
    v = gaussian(X - i * 0.5, Y)
    V[:, :, :, i] = v

V = V - V.min()

# re-scale to our lon/lat grid
x_ = np.interp(x, (min(x), max(x)), (-10, 30))
y_ = np.interp(y, (min(y), max(y)), (35, 60))
X, Y, Z, TS = np.meshgrid(x_, y_, z, ts)

fig, axes = plt.subplots(1, 4, figsize=(8, 6), subplot_kw={"projection": "3d"})
axes = axes.flatten()

for i in range(len(ts)):
    ax = axes[i]
    ax.plot_surface(
        X[:, :, 0, i],
        Y[:, :, 0, i],
        V[:, :, 0, i],
        edgecolor="tab:blue",
        lw=0.5,
        alpha=0.3,
    )
    ax.set_title(f"t = {i} h")

plt.tight_layout()
plt.show()

construct the cost data frame

Your own cost should have the same structure as follows

df_cost = pd.DataFrame(
    np.array([X, Y, Z, TS, V]).reshape(5, -1).T,
    columns=["longitude", "latitude", "h", "ts", "cost"],
).sort_values(["ts", "h", "latitude", "longitude"], ascending=True)

df_cost

	longitude	latitude	h	ts	cost
0	-10.000000	35.0	0.0	0.0	0.992742
40	-9.183673	35.0	0.0	0.0	0.990202
80	-8.367347	35.0	0.0	0.0	0.986874
120	-7.551020	35.0	0.0	0.0	0.982579
160	-6.734694	35.0	0.0	0.0	0.977120
...	...	...	...	...	...
99839	26.734694	60.0	13000.0	10800.0	0.642505
99879	27.551020	60.0	13000.0	10800.0	0.654778
99919	28.367347	60.0	13000.0	10800.0	0.671036
99959	29.183673	60.0	13000.0	10800.0	0.690670
99999	30.000000	60.0	13000.0	10800.0	0.712975

100000 rows × 5 columns

Set up the new objective function

General steps to set you own objective function / cost grid:

1. generate the interpolate model (linear) based on the grid data
2. define the objective function
3. pass the new objective function and interplant model to optimizer's trajectory() generator

(Note: this process can take a while, ~1 minute with a modern computer)

optimizer = opentop.Cruise("A320", "EHAM", "LGAV", m0=0.85)
optimizer.init_conditions(h_min=32_000 * aero.ft, h_max=36_000 * aero.ft)
optimizer.fix_cruise_altitude()
optimizer.setup(nodes=40)
optimizer.debug = False

# the interpolant required for the cost grid
interpolant = opentop.tools.construct_interpolant(
    longitude=df_cost.longitude.unique(),
    latitude=df_cost.latitude.unique(),
    height=df_cost.h.unique(),
    timestamp=df_cost.ts.unique(),
    grid_value=df_cost.cost,
    shape="bspline",
)


def obj_grid(x, u, dt, **kwargs):
    """The final cost is the compound of the 4d grid cost and fuel"""
    # ensure the n_dim is set to 4
    # grid_cost can be time dependent
    grid_cost = optimizer.obj_grid_cost(x, u, dt, **kwargs)
    fuel_cost = optimizer.obj_fuel(x, u, dt)
    return grid_cost * 2 + fuel_cost


# generate the flight trajectory
# interpolant is passed to trajectory() and then to obj_grid() internally
flight = optimizer.trajectory(
    objective=obj_grid,
    interpolant=interpolant,
    n_dim=4,
    time_dependent=True,
    h_min=32_000 * aero.ft,
    h_max=36_000 * aero.ft,
)

Visualize the trajectory

opentop.vis.trajectory(flight)
plt.show()

Visualize the projected 4D cost along the trajectory

fig, axes = plt.subplots(
    1,
    3,
    figsize=(8, 6),
    subplot_kw=dict(
        projection=ccrs.TransverseMercator(central_longitude=10, central_latitude=50)
    ),
)

for t in range(3):
    ax = axes[t]
    ax.set_extent([-10, 30, 35, 60])
    ax.add_feature(BORDERS, lw=0.5, color="gray")
    # ax.gridlines(draw_labels=True, color="gray", alpha=0.5, ls="--")
    ax.coastlines(resolution="50m", lw=0.5, color="gray")

    df_cost_pivot = df_cost.query(f"h==0 and ts=={t * 3600}").pivot(
        index="latitude", columns="longitude", values="cost"
    )

    lat, lon, val = (
        df_cost_pivot.index.values,
        df_cost_pivot.columns.values,
        df_cost_pivot.values,
    )

    ax.contourf(
        lon,
        lat,
        val,
        transform=ccrs.PlateCarree(),
        alpha=0.7,
        cmap="viridis",
        levels=50,
    )

    current_data = flight.query(f"{t * 3600}<ts<{t * 3600 + 600}").iloc[0]

    ax.scatter(
        current_data.longitude,
        current_data.latitude,
        color="r",
        lw=2,
        transform=ccrs.PlateCarree(),
    )

    ax.plot(
        flight.longitude, flight.latitude, color="k", lw=1, transform=ccrs.PlateCarree()
    )

    ax.scatter(
        flight.longitude.iloc[0],
        flight.latitude.iloc[0],
        c="k",
        transform=ccrs.PlateCarree(),
    )
    ax.scatter(
        flight.longitude.iloc[-1],
        flight.latitude.iloc[-1],
        c="k",
        transform=ccrs.PlateCarree(),
    )

plt.tight_layout()
plt.show()

/home/junzi/arc/code/1-public/opentop/.venv/lib/python3.13/site-packages/cartopy/mpl/feature_artist.py:143: UserWarning: facecolor will have no effect as it has been defined as "never".
  warnings.warn('facecolor will have no effect as it has been '

/home/junzi/arc/code/1-public/opentop/.venv/lib/python3.13/site-packages/cartopy/mpl/feature_artist.py:143: UserWarning: facecolor will have no effect as it has been defined as "never".
  warnings.warn('facecolor will have no effect as it has been '

/home/junzi/arc/code/1-public/opentop/.venv/lib/python3.13/site-packages/cartopy/mpl/feature_artist.py:143: UserWarning: facecolor will have no effect as it has been defined as "never".
  warnings.warn('facecolor will have no effect as it has been '