‫מערכות נווט והנחייה‬
‫סמסטר אביב – תשע"א‬
‫טכניון – מכון טכנולוגי לישראל‬
‫הפקולטה להנדסת אוירונוטיקה וחלל‬
12–07–2011
‫בחן אמצע – מועד א‬
.‫ דקות‬120 ‫ זמן המבחן‬1.
(25 points) An inertial measurement unit at rest at latitude = 32o, longitude = 35o,
measures the acceleration:
𝑎𝑥 = 𝑎𝑥𝑇 + 𝑏𝑥
𝑎𝑦 = 𝑎𝑦𝑇 + 𝑏𝑦
𝑎𝑧 = 𝑎𝑧𝑇 + 𝑏𝑧
Explain how you can compute the leveling angles and the errors in these angles
resulting from accelerometer biases.
Solution:
The IMU is at rest, hence its accelerometers measure gravity:
 0  b x 
a  C  0   b y  
 g  bz 
B
B
L
0
 a x  1

a   0 c

 y 
 a z  0  s
0  c

s   0
c   s
0  s   0  bx 
1
0   0   b y  
0 c   g  bz 
 s  bx 


g  s c   b y 
c c  bz 


Nominal leveling angles can be computed by ignoring the accelerometer biases:
  atan2 a y , a z 

  atan2 a x , a y2  a x2

Errors can be estimated by linearization around the “true” values, e.g., by writing the
measurement 𝑎𝑥 = 𝑎𝑥𝑇 + 𝑏𝑥 . Then, for instance,
T    atan2 a yT , a zT  
 
atan2 a y , a z 
a y
by 
atan2 a y , a z 
a z
bz
a yT
a zT
by  2
bz
2
2
a  a yT
a zT  a yT
2
zT
Which can further be approximated using the measured accelerations.
2. (25 points) For a given tactical IMU the position error in the North direction can be
approximated by the function:
𝛿𝑥(𝑡) = 0.3𝑡 + 0.2𝑡 2 + 0.01𝑡 3
The leveling angles are known to be =0.3o and =-0.2o. Compute the initial velocity
error in the north direction, the bias X and the drift X.
Solution:
Over short periods of time, position errors can be approximated using:
1
1
𝛿𝑥(𝑡) = 𝑣𝑥 (0)𝑡 + (−𝑔𝛿𝜃 + 𝑏𝑥 )𝑡 2 + 𝑔𝑑𝑦 𝑡 3
2
6
The first and second term can be used, together with the information on 𝛿𝜃 to compute
the initial velocity error and the bias X. Drift X cannot be estimated using the information
provided, just drift Y.
3. (25 points) Consider the 2D version of the GPS problem, where pseudo-measurements:
𝑝𝑟1 = 9.544
𝑝𝑟2 = 11.630
𝑝𝑟3 = 11.198
are made to three satellites located at
𝑆𝑉1 = [5 10]
𝑆𝑉2 = [−5 10]
𝑆𝑉3 = [0 12]
a. Carry out a full iteration of the nonlinear least squares optimization to compute an
estimate for the location and the clock bias.
b. Compute the GDOP using the quantities computed for the iteration.
Solution:
The solution requires carrying out the computations detailed during the last lecture.
4. (25 points) An inertial system located at the Equator and has an initial orientation of 0, 0,
0 (roll, pitch and yaw). The system has no sensor errors or initialization errors except for
an initial azimuth error of o. Compute the evolution of the orientation errors as a
function of time. Compute the evolution of the velocity errors as a function of time.
Solution:
Consider the equation in the handout:
d
 LIC    LI  CBLcB
dt


Since sensors have no errors, δωB =0. Also, since the system is located at the Equator
the velocity is zero and hence:

C
LI
 
  0 
 0 
The corresponding error (also from the handout) is given by:

 0

1
LI  
 RM
 0



1
RN
0
0

0

 0


0 v L   0
 



0
 R N



0 0
0 0 p L

0 0

Therefore:
𝛿∅̇
0
1 −𝛿𝑣𝑦
Ω 0
[ 𝛿𝜃 ] = Ω [−𝛿𝜓] + [ 𝛿𝑣𝑥 ] − [ 0 ]
𝑅
𝑅 𝛿𝑝
𝛿𝜓
𝛿𝜃
𝑥
0
Differentiating the last equation:
𝛿𝜓̈ = Ω𝛿𝜃̇ −
Ω
𝛿𝑣 = −Ω2 δψ
𝑅 𝑥
Which can be solved for the initial azimuth error. Velocity errors can be computed
likewise.