Aerospace and Electronic Systems - November 2018 - 61

Kosari and Kassaei

Figure 6.

Transformation of the exterior of a triangle.

lected arbitrarily. The parameters C1 and C2 are complex constant
numbers that must be computed based on the given points. Except
for certain cases and simple geometric shapes, the integral term in
(28) cannot be solved analytically and must be calculated numerically. Figure 6 also shows a typical result of applying this mapping
function on a rectangular area in which k1 = 500, k2 = 1,000, and
h = 1,500.
The mapped equation of motion, according to (21), could be
expressed as follows.
 ( w − a )α1 ( w − a )α3
1
3
w = 
α

( w) 2


 (V cos γ + iV sin γ )


(29)

Equation (29) together with (3),
(4), (8), (9) along with the performance
index functions of (11), (15), and (24),
form the optimal TOAF trajectoryplanning problem formulation.
Table 4 presents the results of optimal paths regarding a triangular barricade for a similar flight scenario as
for the circular obstacle case study.
Comparing the results in Table 3 with
the one in Table 4, note the reduced
value of ToF and vertical acceleration
cost for the triangular obstacle against
a circular obstacle case. Figures 7 and
8 show the optimal paths and the corresponding state and control variables
histories. All graphs are plotted in
terms of aircraft down-range.
As it is observed, for time optimal
cases, the aircraft flies with maximum
thrust available and small AoA that
means the elevator control surface
would take small values. However, for
minimum acceleration and minimum
height scenarios, the throttle setting
changes between the minimum and
maximum thrust while the aircraft AoA
is higher; that means the control surfaces take larger values. These observations are quantitatively verified in Table
4. Figure 8 shows the corresponding
flight path angle and speed histories regarding the optimal trajectories. Since the final speed is free, for the two cases of minimum
time, the final speed is close to the maximum value.

COMPLICATED SVOMS
By using the Schwartz-Christoffel theorem, we can model obstacles that are more complex. It can be shown that in SchwartzChristoffel transformation, the correspondence of three points can
be prescribed arbitrarily [24]. In other words, A1, A2, and A3 on
the boundaries of a polygon in z-plane can be associated with any
points a1, a2, and a3 on the real axis of w-plane (see Figure 3). If the
SVOM polygon has more than three vertices, determination of a4,
a5, ..., an might be difficult and usually is computed numerically.
Here, we developed a Schwartz-Christoffel Obstacle Model Gen-

Table 4.

Results of TOAF Trajectory Planning for Triangle SVOM
Objective

Time of Flight, sec.

Mean Height, m

Vertical Acc. Cost

CPU Calculation Time, sec.

Time

54.0

680.1

1,061.4

5.4

Height

65.7

581.5

2,815.6

5.6

Vertical acc.

82.9

727.8

259.4

12.5

NOVEMBER 2018

IEEE A&E SYSTEMS MAGAZINE

Aerospace and Electronic Systems - November 2018

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