Aerospace and Electronic Systems - November 2018 - 32

Optimal Trajectory in the Presence of Wind
Each link contains also its associated bearing (see Figure 3b) θl,
which is given by the following formula:

 y
θ l ( N o , N d ) = arctan  
x

=
Δ
y
sin(
).cos(
φ
)

d
λ
 x = cos(φ ).sin(φ ) − sin(φ ).cos(φ ).cos(Δ )
o
d
o
d
λ

Δ
=
−
λ
λ
d
o
 λ

(4)

Based on the previous equation, one can now compute the tail
wind on each extremity of the link l (TWlo and TWld):


TWo = Wo .cos(θl − θWo ) TWd = Wd .cos(θ l − θWd )

(5)

The two tail winds are then averaged and associated to each
link:
TWl =

TWO + TWd
2

(6)

This last tail wind will be used for the cost associated to each
link in the shortest path computation.
To compute the wind optimal trajectory, we will consider for
each link the time needed by aircraft to connect node No to node Nd.
This time tl is given by:
tl =

Figure 3.

Information contained in links. (a) Wind information contained in
nodes. (b) Information contained in links.

origin node No and one destination node Nd. The grand circle distance of link l, dl is given by the following formula:
 
d l = R.arcsin Pd ∧ Po

{

}

(2)

where

Po = ( xo , yo , zo )T


Pd = ( xd , yd , zd )T

where λ is the latitude and ϕ is the longitude.
32

(3)

(7)

where Ta is the true airspeed of the aircraft.
Having a graph with layer structure, a Bellman-Ford algorithm
has been implemented for finding the shortest path between a node
at the extreme left (No) and all the nodes at the extreme right (Nd).
The algorithm is organized into three steps:
STEP 1: Initialization This step initializes distances (dist)
from source to all vertices as infinite and distance to source itself
as 0.
STEP 2: Propagation The source node is first considered and
its associated neighboring links.
Starting from the source node NO = src (first column) and for
each link associated to No the algorithm marks the neighboring
nodes of NO = src with the following rule;
C

are the Cartesian coordinates of the nodes No and Nd, ∧ is the vector product, and R is the radius of the Earth. For a given node P
(see Figure 4), the Cartesian coordinates are given by the following
formula:
 x = R.cos(φ ) * cos(λ )
 
P =  y = R.cos(φ ) *sin(λ )

 z = R.sin(λ )

dl
Ta + TWl

if dist[Nd] > dist[No] + dl then update dist[Nd] ⇒ dist[Nd] =
dist[No] + dl (keep in node Nd the node No which has been
use for this update)

Shift to the next column (column 2) and apply the same rule
to all nodes which have been updated in order to propagate the
distance update to the third column. This process is repeated until
the propagation reaches the last column (on the right).
STEP 3: Path building To compute the shortest path for the
source node src to any destination nodes on the right (dest), first
select a destination node among the nodes belonging to the last
column (column number K). Select the node NK - 1 in column K - 1
which has updated the dest node in column K. Then, select the
node NK - 2 in column K - 2 which has updated the node NK - 1 in

IEEE A&E SYSTEMS MAGAZINE

NOVEMBER 2018
Aerospace and Electronic Systems - November 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - November 2018
