IEEE Robotics & Automation Magazine - September 2013 - 38

Measurement Model
The received signal strength varies with the relative angle of
the plane of the loop antenna with the tag. The signal is
strongest when the tag is directly aligned with this plane.
Since the antenna is mounted on a pan-tilt unit, we can
rotate it and sample the signal strength as a function of the
relative angle from the boat. We fit a smooth function to
B

A

C

R1
D
F
R2

P

O

E

R4
H N

G

M
J

I
R3
L

K

(a)
Coverage Path
Input Regions
Coverage Path
Coverage
Waypoints

0

Y(m)

-100
-200
-300
-400
-500
-100 0 100 200 300 400 500 600
X(m)
(b)
Robot Path with Nonzero Radio Signals by Frequency
Path
48351
49691

48111
48471

48271
48751

0

Y(m)

-200
-300

-500
-200 -100 0 100 200 300 400 500
X(m)
(c)
Figure 5. The coverage experiment conducted at Lake Phalen,
Minnesota: (a) the four input regions, (b) the path found by the
algorithm described in the "Localization" section, and (c) the
actual path followed by the robot during coverage. The robot
traveled a total distance of 5.6 km in about 87 min. Locations
where signals from radio tags were detected are also marked
along with their frequencies.
IEEE ROBOTICS & AUTOMATION MAGAZINE

FIM
The Cramer-Rao lower bound (CRLB) for an unbiased estimator is a lower bound on the estimation error covariance.
This lower bound is equal to the inverse of the FIM (denoted
by I ) for the k measurements. The determinant of I is
inversely proportional to the square of the area of the 1- v
uncertainty ellipse, and is commonly used as the objective
function to be maximized. For k bearing measurements with
zero-mean Gaussian noise, the determinant of I (denoted by
I ) is given as
k

-400

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Optimization of Robot Motion
Since measuring the bearing takes time (about 1 min), the
estimation must be performed using a small number, say k,
of measurements. Furthermore, these locations must be chosen in an online fashion as the measurements become available. We present three strategies to compute k sensing
locations and compare their performance in simulations and
real-world experiments.
All three active localization strategies require an initial
estimate of the tag. In [4], we presented a scheme to initialize the target based on two bearing measurements taken
from different sensing locations. Using this initial estimate,
we propose the following three strategies to determine the
next k sensing locations of the robot.

2

sin (i i - i j)
E,
I = 14 / / ;
di d j
v i =1 j =1

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38

the samples obtained and use the point of maximum value
of this function as the bearing measurement. Based on a
number of trials, we concluded that least squares fitting of a
cubic polynomial works best for computing the bearing in
our system.
Note that the bearing obtained is an infinite line (as
opposed to a directed ray), and hence there is ambiguity
in the obtained bearing. For example, if a is the direction
with maximum signal strength, then a and a + r are
both valid bearing measurements. We disambiguate by
moving along either a or a + r and checking whether the
signal strength increases or decreases. A detailed description of other methods for disambiguating the measurements is given in [17].

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september 2013

k

(2)

where i i and d i are the angle and distance from the ith sensing location to the true target location.
We impose a grid of size n # n centered at the current
k sensing locations, we
position of the robot. To compute the
n2
exhaustively consider each of the (k ) combinations as a candidate trajectory, and compute I . An optimal trajectory can
then be chosen as one with the minimum value of I .
Greedy
Instead of computing a fixed path for the k measurements, we
can use an online greedy strategy, which picks the next sensing
location based on the current estimate and uncertainty of the
position of the tag. Given the current robot and tag estimates,
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2013
