(a) The following coordinates were recorded using a hand-held GNSS at the beginning and end of Sauchiehall Street in Glasgow. Calculate the length of the road.
Buchanan Galleries – 55◦51′51′′N 4 ◦15′11′′W
Charing Cross – 55◦51′58′′N 4 ◦16′14′′W (8)
(b) A radar drone surveillance system is used to detect and track small airborne vehicles operating close to restricted sites such as airports, military bases and nuclear power plants. The radar antenna is mounted on a two-axis gimbal that allows it to rotate in both azimuth and elevation. Each drone within range will reflect a small portion of the radar energy from favourably aligned reflector points on the fuselage.
Using the navigation kinematic notation, show that the velocity of a reflector point p on the fuselage as seen from the antenna and resolved into antenna axes is given by,
r˙ a ap = v a eb − v a ea + C a bΩ b eb l b bp − Ω a ea r a eb − r a ea + C a b l p bp
where, Fb , Fa are free to rotate with respect to Fe and l b bp is the position of the fuselage reflector with respect to the body axes. (12)
The navigation equations expanded and resolved into the local geographic frame are,
˙v n eb = C n b f b ib + g n b (Lb , hb) − (Ω n en + 2Ω e ie) ˙v n eb
Using Trapeziodal integration to solve for velocity & position, integrate this equation over one time-step to calculate the position at time tk when the conditions shown in Table Q.1 hold. Use the Somigliana model to calculate g, assume a sampling interval of ∆t = 0.1s and assume the following initial DCM.
C n b (tk) =
0.7170 −0.6696 0.1938
0.6905 0.7204 −0.0657
−0.0956 0.1810 0.9788
Variable Values Units
˙v n eb(tk−1) [0, 0, 0] ms−2 v n eb(tk−1) [240, 0, 0] m/s
Lb(tk−1), λb(tk−1) [55.87858658, −4.676749498] deg
hb 4000m ω
b ib(tk) [0.1, 0.05, −0.2] rad/s
f b ib(tk) [0.0, 0.01, −0.05] ms−2
Table Q.2 – Previous INS estimates and current measurements
(a) GNSS systems make use of techniques developed for radio trilateration to determine receiver position. Consider the simple situation of trilateration in 2D using 3 beacons, each positioned at the following (x,y) locations,
B1(300, 250), B2(−50, 175), B3(20, −90)
If a receiver is placed at location uT = (30, 30), use an iterated linear approximation algorithm to calculate the estimated receiver position when ∥δu∥ < 0.1. Assume a nominal, initial position estimate of u = (0, 0). (10)
(b) A Kalman Filter is to be used to estimate the trajectories of the following three-state model,
Φ = 0 0 1
1 ∆t 0
0 0 1，
H = [0 1 0] ,
For this model, calculate the a´ posteriori estimated state vector xˆ + k assuming a sampling interval of ∆t = 0.1s and using the following matrices, (10)
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