EXAMINATION IN SENSOR FUSION
ISY’S COMPUTER ROOMS
TIME: 3 June 2009, 14–18
COURSE: TSRT14, Sensor Fusion
PROVKOD: TEN1
INSTITUTION: ISY
NUMBER OF EXERCISES: 4
NUMBER OF PAGES: 2 + cover page
RESPONSIBLE TEACHER: Fredrik Gustafsson, tel 013-282706
VISITS: 8.00, 9.00, 10.00, 11.00
COURSE ADMINISTRATOR: Ulla Salaneck, tel 013-282225, [email protected]
APPROVED TOOLS: All written material. Matlab with Signal and Systems Lab. Course
account with own code written to exercises and labs.
SOLUTIONS: Linked from the course home page after the examination.
The exam can be inspected and checked out 2009-06-17 kl 12.30-13.00 in the examiner’s room,
entrance 27, directly to the right.
PRELIMINARY GRADE LIMITS: grade 3
grade 4
grade 5
15 points
23 points
30 points
OBS! Solutions should include code and plots and clear cross references between these. Mark
all print-outs with your name.
Good luck!
1. The following questions directly relate to the two laborations. Each question should be
answered briefly with a couple of sentences, not more than 50 words.
(a) The exact time t0 when the pulse is emitted is unknown. Which are the two main
alternatives for treating this uncertain parameter in laboration 1?
(2p)
(b) What is the main difference in using WLS and EKF for localization in laboration 1
(both use exactly the same measurements)?
(2p)
(c) How can the information matrix I(x) be used to compare two configurations of speakers/microphones in laboration 2/1, and what is the interpretation of x here?
(2p)
(d) Which map in laboration 2 should give the best tracking result: the nominal map,
the map estimated from a calibration experiment or a map estimated on-line with a
SLAM algorithm? Motivate the answer.
(2p)
(e) Suppose the damping of the pendulum in laboration 2 is not negligible. How can the
damping parameter m be included in the estimation problem in tracking using EKF
and EKF-SLAM, respectively?
(2p)
2. A speaker is hanging in a rubber band attached to the ceiling, and it is pending vertically
according to the spring-damper dynamics:
Z̈t + 2ω0 η Żt + ω02 (Zt − Z0 ) = 0,
where the eigen-frequency is ω0 = 1 and the damping ratio is η = 0.1. The microphone is
centered above the origin in the horizontal plane, and the equilibrium point is Z0 = 1 m.
The initial condition
is roughly known as Z0 = 1.5m and Ż0 = 0m/s, both with a standard
√
deviation of 0.1.
There are three microphones on the ground Z = 0 located at X = −1, X = 0 and X = 1,
respectively. Each microphone detects pulses from the speaker at 5Hz with an standard
deviation of 0.5m. Microphone and speakers are assumed synchronized, so there is no
unknown range offset. Further, there are no outliers from the detector.
(a) Define a discrete time NL object for this system using ZOH sampling of the dynamics.
Hint: the matrix exponential eA is computed with expm(A) in Matlab.
(5p)
(b) Simulate N = 20 measurements, and plot Zk and yk as a function of k.
(2p)
(c) Apply a filter to estimate Zk from the simulated yk above. Plot Ẑk and the true Zk
in the same graph.
(3p)
3. Consider a sensor network with M = 3 sensors located at (0, 0), (3, 0) and (6, 0), respectively. The true target position is at x0 = (3, 4). Each sensor provides a TOA range
measurement ri = kpi − x0 k + r0 + ei , where R = Cov(ei ) = 1 and r0 is an unknown
constant. In vector form, this reads y = (r1 , r2 , r3 )T = h(x) + e.
(a) Describe the sensor model for the TDOA observations ri − rj for all i 6= j in the
generic form ȳ = h̄(x̄) + ē. What is R̄ = Cov(ē) and h̄(x0 ) numerically in this case?
(4p)
(b) What is the CRLB for this sensor model? An analytic derivation is requested for full
points, but of course matrix inverses, multiplications and similar can be computed
numerically in Matlab.
(6p)
4. Consider the example in Section 9.8.6 for the case of a constant parameter xlk :
xnk+1 = xlk xnk + vkn ,
xlk+1 = xlk ,
yk = 0.2(xnk )2 + ek ,
Here,
vkn ∼ N (0, 0.25),
ek ∼ N (0, 1),
0.1
16
0
x0 ∼ N
,
.
0.99
0 10−3
(a) What is the conditional linear model if xnk were known at all times? What is the
optimal filter for this model? Give the recursive equations for the estimator of xlk .
(4p)
(b) Simulate N = 100 measurements of the nonlinear system for the case xl = 0.85.
Apply the optimal estimator for xlk conditional on your realization of xnk (assumed
known here).