IEEE Robotics & Automation Magazine - June 2014 - 48
4
(b)
Figure 2. A simple 2 # 2 grid example where the shading
indicates the probability that a cell is occupied, with white being
zero and black being one. (a) A cell-refinement procedure where
a large occupied cell is divided into four smaller cells with equal
occupancy probability in each. (b) A grid-merging procedure
where four empty subcells with the same parent cell are merged
to form the parent cell.
y (m)
2
(a)
0
-2
-4
0
2
4
6
x (m)
8
10
12
4
6
8
10
Distance from Robot (m)
12
(a)
Sensing
The sensor model used for multiobject problems must consider the possibility of missing an object within the footprint
(i.e., a false negative), detecting an object that is not there
(i.e., a false positive or clutter detection), or returning noisy
estimates of true objects. To this end, we use the general
form of the FISST measurement model with Poisson clutter
detections [11] as follows:
p (Z X; q) = e - n c % l (z) mc % 1 - p d (x; q) m
z!Z
# e/
i
j
x!X
%
i (j) ! 0
p d (x j; q) g (z i (j) x j; q)
o,
l (z i (j)) (1 - p d (x j; q))
(1)
where l (z) is the clutter PHD, p d (x; q) is the detection likelihood, g (z ; x; q) is the single-object measurement model,
i: {1, f, n} " {0, 1, f, m} is a data association, and q is the
pose of the robot. Note that i (j) = 0 means that the object is
not detected, i.e., a false negative, and any element of
{1, f, m} not in the range of i ({1, f, n}) is a false positive.
Also the pose, q, is shown to emphasize the dependence of
the measurements on the robot's pose. Intuitively, this function averages over all possible data associations. This is not
prohibitively large for small numbers of objects n and measurements m where in a single data association, i, each measurement z ! Z is either said to originate from a target
x ! X or be due to a false positive. The first two terms in (1)
48
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
June 2014
pd (x)
some threshold (near one) do we divide the cell into four subcells, as shown in Figure 2. This refinement procedure continues until cells reach a minimum size, chosen to be near the
standard deviation of the measurement noise, as targets cannot be located with significantly higher precision even in a
continuous representation of the environment. If a large cell is
divided due to a series of false positive detections, we also
allow the fusion of four empty cells back into the larger parent
cell, as shown in Figure 2. These cell refinement and merging
procedures are done in such a way as to keep the probability of
an object being within the parent cell constant, where the
refinement initializes all subcells to have a uniform probability
of occupancy. See [15] for pseudocode descriptions of these
cell operations.
1
0.5
0
0
2
(b)
Figure 3. (a) The experimental results showing the true (blue
circles) and estimated (red xs) object positions as measured in
the body frame of the robot. This is superimposed on the sensor
footprint and represents approximately 600 data points. (b) The
detection likelihood as a function of the distance to the robot.
represent false positives, the next term represents false negatives, and the final term represents true detections.
In our case, the robot is equipped with a front-facing camera (label 4 from Figure 1) for object detection, which sees a
finite subset of the environment that we call the footprint.
Mathematically, the footprint is the set of cell labels that are at
least partially visible by the robot. The system runs a template-matching algorithm to detect objects of interest within
the image using shape and color and combines this with the
pitch estimate to calculate the position of the objects relative
to the robot. It then returns a list of cells occupied by objects
with a sufficiently good match.
To determine models for the detection and measurement
likelihoods, we conduct a set of experiments placing objects at
known locations in front of the robot and collecting measurements. The results of this are shown in Figure 3 overlaid on
the sensor footprint. Below this is the detection likelihood
function, p d (x c; q), where x c represents a continuous domain
position. The single-target measurement model is the position of the target corrupted by Guassian noise, g (z c ; x c; q)
= x + N (0, R), and the noise covariance R is found from
this training data.
Due to our discrete representation of the environment,
these continuous domain detection and measurement models
must be converted into a discrete form. Since the target location is uniform within a cell, we may simply average the
detection model over the cell domain to obtain the detection
likelihood for a cell
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