Aerospace and Electronic Systems - November 2018 - 56

A Simplified Virtual Obstacle Avoidance Approach for Aircraft Altitude Bounded Flight Maneuvers
in accordance with the terrain profile could be done by utilizing
Joukowski or Schwarz-Christoffel mappings.

thrust, maximum and minimum AoA, and maximum and minimum flight speed. The mathematical expression of the problem
constraints is given in (8).

SOLUTION APPROACH AND METHODOLOGY

α min ≤ α ≤ α max

PROBLEM FORMULATION

0 ≤η ≤1

Assuming the flat earth model as an inertial frame of reference,
standard atmosphere, and regarding the aircraft as a point with
constant mass during maneuvers, the aircraft longitudinal equation
of motion is as follows [22]:
x = V cos γ

(1)

y = V sin γ

(2)

V = (T / m ) cos α − ( D / m ) − g sin γ

(3)

γ = (T / mV ) sin α + ( L / mV ) − ( g / V ) cos γ

(4)

The parameters x, y, V, and γ are the problem state variables,
respectively, representing the horizontal range, altitude, velocity,
and flight path angle of the aircraft, and g is gravity acceleration.
The terms L and D are aerodynamic lift and drag forces, which are
computed as follows:

1
ρV 2 Sref CD
2

(5)

(6)

In (6), CL0 is lift coefficient at zero AoA and CD0 is zero-lift drag
coefficient, CLα is lift-curve slope of aircraft and K is drag polar
constant. The aircraft thrust force T is defined by (7).
(7)

where Tmax is the maximum thrust, η is the throttle setting and 0 ≤ η
≤ 1. Function g(η) is defined based on the motor size and volume.
According to (6) and (7), the aircraft AoA α and throttle setting η
are considered as trajectory planning problem control variables.
Equations (1) through (4) state the aircraft longitudinal flight
dynamics equations of motion. These equations are bounded by the
performance characteristics of the aircraft system, like maximum
56

One of the key issues in optimal route design, especially in
TOAF scenarios, is the selection of an appropriate performance index (or the so-called cost function). In fact, the proper selection of
the cost function will affect the quality of the planned flight path.
In TOAF missions, the cost function can include the minimum
flight time, the minimum control effort, the minimum flight distance, the terrain masking, or a combination of these parameters.
The time-optimal trajectory planning cost function could be
defined as follows:
(10)

t0

CL = CL0 + CLα α

T = Tmax ( y, M ) g (η )

(9)

− RoDmax ≤ y ≤ RoCmax

J =  dt

In (5) Sref is the reference area (i.e. wing area) of the aircraft, CL
and CD are the lift and drag coefficients, respectively, and ρ is the
air density, which changes according to the standard atmospheric
model. It is common to compute the lift and drag coefficients using
the quasi-linear and the drag-polar model in terms of the angle of
attack (AoA), α and flight Mach number, M given by:

CD = CD0 + KCL 2

The other important aircraft performance parameter in TOAF
maneuvers is the maximum flight climb rate. This performance parameter is taken into account as an inequality constraint according
to (9), where RoCmax and −RoDmax are the maximum rate of climb
(RoC) and maximum rate of decent (RoD) of aircraft.

tf

1
L = ρV 2 S ref CL
2
D=

(8)

Vmin ≤ V ≤ Vmax

Sometimes, in order to avoid abrupt changes in the control surface (such as elevator) position and to improve the elevator rate
saturation problem, it is preferred that the effect of both flight time
and control effort be considered as a combined weighted cost function. Therefore, a cost function might be defined as follows, where
α is the aircraft AoA and w1 and w2 are the weighting parameters.
tf

(

)

J =  w1 + w2α 2 dt    
t0

(11)

Another cost function that is common in the TF/TA trajectory
planning is a hybrid function, including two conflicting parameters
of time and terrain masking [3], defined as follows:
tf

(

)

(

)

2
J =   1 − K p + K p y − Tr ( x )  dt ; 0 ≤ K p ≤ 1



t0

(12)

where Tr(x) is the height of terrain profile plus a ground clearance
at the down-range x position.
The other performance index that could be of interest in trajectory planning is defined based on minimizing a functional containing the square of aircraft vertical acceleration. This, in turn,
could help sustain structural loads and also ensure the comfort of
the passengers [8]. This cost function can be expressed as follows:
tf

J =  
y 2 dt
t0 ,

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NOVEMBER 2018
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