Course 086759 Navigation and Guidance –
Final Exam
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The exam is multiple-choice.
On each question, choose one and only one option, which you judge the most likely answer.
If you select more than one answer on the same item, then the answer will not count.
Each correct choice counts five (5) points. Each wrong choice counts zero (0) point.
Fill your answers on the table below.
You have two hours for finishing the test.
GOOD LUCK!
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For a navigation system which is known to be moving straight-and-level northwards at constant velocity, it is
find out that the velocity error in the north direction obeys the law:
∆𝑉𝑁 (𝑡) = 𝑎0 + 𝑎1 𝑡 + 𝑎2 𝑡 2
It is moreover known that navigation errors are dominated by initial condition errors, accelerometer biases and
gyro drifts and that the initial azimuth error is 𝛿𝜓(0). Use this information to answer items 1-3.
1. The initial velocity error in the north direction is:
a. 0
b. a0
c. Δ𝑉𝐸 (0)𝛿𝜓(0), where Δ𝑉𝐸 is the velocity error in the east direction
d. Depends on pitch and roll angles initial errors
2. The bias of the accelerometer in the X direction is given by
a. 𝑎1 + 𝑔𝑎2
b. 𝑎1
c. 𝑎1 − 𝑔𝛿𝜃(0)
d. 𝑎1 + 𝑔𝛿𝜙(0)
3. The drift of the gyro in the Y direction is given by:
a. Cannot be determined from the equation
b. 2𝑎2 /𝑔
c.
d.
2𝑎2
𝑔
2𝑎2
𝑔
+ 𝑔ΩE , where ΩE is the component of the earth rotational velocity in the East direction
+ 𝑔ΩN , where ΩN is the component of the earth rotational velocity in the North direction
4. For a non-accelerating trajectory, the following measurement errors do not result in velocity errors:
a. Azimuth error
b. Gyro z error
c. Bias y error
d. Both a and b
5. The “small-angle” approximation for the rotation matrix 𝑅𝑌 (𝛿𝜃) is:
a. 𝑅𝑌 (𝛿𝜃) ≈ 𝛿𝜃 ⋀
b. 𝑅𝑌 (𝛿𝜃) ≈ −𝛿𝜃 ⋀
c. 𝑅𝑌 (𝛿𝜃) ≈ 𝐼 + 𝛿𝜃 ⋀
d. 𝑅𝑌 (𝛿𝜃) ≈ 𝐼 − 𝛿𝜃 ⋀
6. The vertical channel of the inertial navigation solution for an aircraft is left to drift without external
compensation. Then:
a. The vertical errors will grow unboundedly due to accelerometer Z bias
b. The vertical errors will oscillate at the Schuler frequency
c. The vertical errors may remain constant depending on the initial conditions
d. The vertical errors will grow exponentially
7. Suppose the vertical velocity errors Δ𝑉𝐷 of an aircraft flying East at velocity 𝑉𝐸 are relatively large. Let Ω
be the earth rate. Then:
a. The interaction between vertical velocity error and VE results in Coriolis acceleration errors
b. VE and Ω result in the divergence of Δ𝑉𝐷
c. Δ𝑉𝐷 and Ω result in an acceleration error in the North direction
d. Δ𝑉𝐷 and Ω result in an acceleration error in the East direction
8. The plumb-bob gravity is:
a. Approximately aligned with the earth gravity
b. Exactly aligned with the earth gravity
c. Approximately 1.3% larger in magnitude than the earth gravity
d. Exactly 1.3% larger in magnitude than the earth gravity
9. The relationship between “down” velocity and altitude is:
a. ℎ̇ = 𝑉𝐷
b. ℎ̇ = −𝑉𝐷
10.
11.
12.
13.
c. Either a or b depending whether the system is in the North (a) or South (b) hemisphere
d. Either a or b depending on the sign of h
For an inertial navigation system standing still, the transport rate is:
a. Equal to the earth rate
b. Equal to minus the earth rate
c. Zero
d. Either a or b depending on the hemisphere on which the system is located
In general, the angular velocity 𝜔𝐿𝐼𝐿
a. Is position independent
b. Depends on the latitude
c. Depends on the longitude
d. Depends on both the latitude and the longitude
An accelerometer at rest with sensitivity axis aligned with the plumb-bob will approximately measure:
a. 0
b. -10 m/sec2
c. 10 m/sec2
d. Either b or c depending on the which side of the accelerometer points up
A gyro at rest with sensitivity axis pointing north will measure:
a. 0
b. cos
c. -cos
d. sin
Here ,  denote the earth rate and latitude, taken positive in the northern hemisphere and negative
the southern one.
14. The GPS satellites:
a. Are geo-stationary
b. Complete approximately two orbits per day
c. The number of orbits depend on the orbit inclination
d. The number of orbits depend on the orbit longitude ascending node
15. GLONASS Ephemeris advantages are:
a. They are more precise
b. They can be updated at a lower rate
c. Both a and b are true
d. Satellite position and velocity are easier to compute
16. Suppose a GPS receiver has a precise clock synchronized with GPS time. Then:
a. At least 4 satellites are required to compute a position solution
b. At most 4 satellites are required to compute a position solution
c. At least 3 satellites are required to compute a position solution
d. At least 4 satellites are required to compute a position solution
17. In order to compute position with an accuracy of the order of a few meters, one needs the following
time-of-flight accuracies:
a. Few micro-seconds
b. Few tens of nanon-seconds
c. Few nano-seconds
d. Position and time-of-flight accuracies are independent
18. The accuracy in the GPS position solution depends on:
a. The time of day and the weather
b. Weather alone
c. The distance between the satellites and the GPS receiver’s antenna
d. The constellation of satellites in-view
19. The DCM matrix R and the quaternion q represent the same rotation. Let q1= kq (where k is a
constant). Then:
a. The DCM corresponding to q1 is Rk.
b. The DCM corresponding to q1 is kR.
c. The DCM corresponding to q1 is R
d. In general, there is no simple way to compute the DCM associated with q1.
20. Navigation systems are:
a. Dull and boring
b. Completely unnecessary
c. Of only mild interest
d. The most fascinating subject I’ve seen during my studies