AARO GoFast Case Resolution (mirror)

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elevation and denoted by β; the angle of rotation about the z-axis is the “left and right” motion
due to aircraft yaw or sensor pointing azimuth and denoted by γ. For this FLIR sensor, the bank
(or roll) angle of the aircraft is included in the elevation and angles of the sensor and not treated
separately. The reader is encouraged to consult [ref 3] for additional explanation of these angles.
These rotations were carried out by applying 2-D rotation matrices derived from the underlying
trigonometry [ref 6]. Equations (1), (2), and (3) provide these 3x3 rotation matrices about the x-,
y-, and z-axes by the defined angles α, β, and γ respectively.
𝑅𝑥(𝛼) = [
1 0 0
0 cos(𝛼) − sin(𝛼)
0 sin(𝛼) cos(𝛼)
] (1)
𝑅𝑦(𝛽) = [
cos (𝛽) 0 sin (𝛽)
0 1 0
− sin(𝛽) 0 cos(𝛽)
] (2)
𝑅𝑧(𝛾) = [
cos (𝛾) −sin (𝛾) 0
sin (𝛾) cos(𝛾) 0
0 0 1
] (3)
These rotations are relative to the aircraft attitude. Therefore, the initial LOS was defined as a
unit vector pointing straight ahead from the F/A-18 along the +x-axis. This vector is represented
in cartesian (x, y, z) vector notation as v = . The vector’s magnitude is defined by the
range to the UAP. Properly rotating, or pointing, this vector by the given sensor angles yields
the UAP’s relative position to the aircraft.
Position of UAP at t1
The length, or magnitude, of the LOS vector at t1 was defined by multiplying the unit vector v by
the range to the target. At t1 the range was 7,408 m, thus the initial LOS vector was v1 = m.
The next step was to point this vector at the UAP. First, it was rotated by the sensor elevation
(pitch) angle β around the y-axis:
𝑅𝑦(𝛽) ∙ 𝒗𝟏 = [
cos (𝛽) 0 sin (𝛽)
0 1 0
− sin(𝛽) 0 cos(𝛽)
] [
7408
0
0
] (4𝑎)
= [
cos (−29°) 0 sin (−29°)
0 1 0
− sin(−29°) 0 cos(−29°)
] [
7408
0
0
] (4𝑏)