GoFast Case Resolution

UNCLASSIFIED
11
UNCLASSIFIED
Inserting vFA18 = 190 m/s, the acceleration of gravity g = 9.81 m/s2, and θB = 14° into (6), the
radius of curvature Rc was found to be 14,759 m. These values are shown in Figure 4 with the
radius of curvature and position of the F/A-18 at t2.
Figure 6: The F/A-18 traveled in a curved path from its position at t1 to reach its position at t2.
To find yaw the angle φ at t2, note that the two φ’s in Figure 4 are equal. The total distance
travelled in the partial circle by the F/A-18 moving at 190 m/s over 13 seconds is 2,470 m. Had
the aircraft kept flying to complete a circle, it would have travelled 2·π·Rc or 92,736 m.
Therefore, only 2.66% of a 360° circle was completed thus φ = 9.6°. Since the yaw is
counterclockwise, away from the positive y-axis, it is a negative value, or φ = -9.6°.
To find the F/A-18’s position at t2, the Δx and Δy change in position from the location at t1 must
be found. Equations (7) and (8) use a cylindrical coordinate transformation [ref 8] to do this.
∆𝑦 = 𝑅𝑐 − 𝑅𝑐 ∙ cos(𝜑) = 14,759 ∙ (1 − cos(9.6°)) = 207 𝑚 (7)
∆𝑥 = 𝑟 ∙ sin(𝜑) = 14,759 ∙ sin(9.6°) = 2461 𝑚 (8)
Thus, at t2, the aircraft’s displacement was [2,461, -207, 0] m, or 2,461 meters ahead and 207
meters left3 of its t1 position. This will be used momentarily to find the UAP’s location at t2.
At t2 the UAP was at a range of 6,279 m from the F/A-18 such that the pointing vector, v2, is
given by m. As with t1, the LOS vector was defined as pointed in the direction of
+x-axis, not the heading of the aircraft.4 At t2 the sensor elevation and azimuth angles were β = -
35° and γ = -57°, respectively. The additional yaw of the F/A-18 of -9.6° required one additional
rotation.5 The rotation by -35° about the y-axis to the sensor elevation is given in (7a-7b).6
3 The negative is applied here to conform with our coordinate system.
4 This is not the direction the F/A-18 was pointed, otherwise there would be a y-component of the vector. The additional yaw
angle will be handled in the rotation matrices.
5 The sensor azimuth and F/A-18 yaw could be added and performed in a single rotation. In general, the aircraft would have roll
and pitch and adding the platform angle to the sensor angle may give an incorrect result.
6 The written-out step depicting the multiplication of the matrices is skipped here.