IEEE Robotics & Automation Magazine - June 2014 - 64
Algorithm 1. Linearized uncertainty propagation.
Map1
Require: Occupancy grid map: m,
Transformation: }, x, y,
Uncertainty of the transformation: R tr .
Ensure: Transformed occupancy grid map: m'.
1: n m = T x, y, } (m)
2: m' = n m
3: for all n r = [n ri, n r j] T ! n m do
4: Calculate R r using (4)
5: h ! N (n r, R r)
6: m' ^ n ri, n r j h ! n m ^ n ri, n r j h U h ^ n ri, n r j h
7: end for
Map2
Uncertain Rotation
Rotation Alignment
Rotation Uncertainty
Uncertain Translation
Probabilistic GVD
Graph Representation
means that the center point of the kernel is h (0, 0). The mean
of the kernel is the transformed point, while its 2 # 2 covariance matrix is calculated by (4). It is important to note that for
each point of the transformed map, the kernel takes different
values. Then, in line 6, every transformed point is convolved
with the Gaussian kernel. The operator U denotes the convolution of a Gaussian kernel with the map [18] and is defined as
ml (k, l) = n m (k, l) U h (k, l)
3
=
3
/ /
j =-3 i =-3
n m (i,
j) h (k - i, l - j).
(6)
For simplicity of implementation, it is assumed that the
size of the kernel is the same as n m .
Now m' can be fused with its pair map using the entropy
filter method detailed in the "Map Fusion with the Entropy
Filter" section.
Probabilistic Map Merging with the GVD
In this section, we present the details of the map fusion process with the probabilistic GVD introduced in [1]. Consider
the case of two mobile robots, R 1 and R 2, equipped with laser
rangers exploring an environment and building occupancy
grid maps (OGMs) [3]. In an OGM representation, each cell
in map m k (i, j), k = " 1, 2 ,, is a binary random variable (RV)
where p (m k (i, j) = 1) = p (m k (i, j)) is the probability that
the cell at location ^i, jh is occupied in the map of robot k. It
is convenient to represent the OGM using the log odds representation of occupancy [3]
l k (i, j) = log
p (m k (i, j))
.
1 - p (m k (i, j))
(7)
Without loss of generality, assume that R 2 transmits its
local map, map2 to R 1 through a wireless channel. R 1 is now
responsible for incorporating the transmitted map into its
own local map, map1. There are three main challenges that
need to be overcome.
1) The relative transformation from map1 to map2 needs to
be found.
2) The uncertainty of the transformation should be accounted for.
3) The OGM probabilities from map2 need to be incorporated with the OGM probabilities of map1.
An overview of the elements of the algorithm is shown in
Figure 3(a). The subsequent sections will describe each of the
block components in detail.
64
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IEEE ROBOTICS & AUTOMATION MAGAZINE
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June 2014
(b)
Edge Matching
Translation Uncertainty
Fusion
Entropy Filter
Fused Map
(a)
Figure 3. (a) The proposed map fusion algorithm. Two input
maps, map1 and map2, are fused by finding their relative
transformation matrix. No prior information is available
regarding the relative position of the two respective robots.
(b) The experimental robots, CoroBots, each equipped with a
laser ranger and wheel encoders.
A simulated example accompanies each step of the algorithm to aid with explanation. Figure 4(a) shows the simulated
environment, where three poles are located inside a rectangular
room. The two robots map the room starting near the big pole
but moving in opposite directions. Figure 4(b) and (c) shows
the two local maps after some time has passed. Without loss of
generality, it is assumed that the second map [Figure 4(c)] is
fused into the first map [Figure 4(b)]. Free, occupied, and
unknown cells are shown by different shades of gray, using the
OGM standard. The darker the grid cell, the higher the probability of occupancy.
Uncertain Rotation Alignment
In structured environments such as urban or indoor settings,
the relative rotation between maps can be found easily using the
Radon transform, as shown by [30]. The Radon transform is the
projection of the image intensity along a radial line oriented at a
specific angle. The peak points in the Radon transform will correspond to the straight line segments in the image. As a result, it
is possible to resolve the relative rotation, }, between two
images by looking for peaks in the Radon images of both maps.
However, due to environment similarity, four rotation hypotheses are considered and only one is accepted by a similarity index
[30]. At the output of this block, there are the two aligned maps,
m 1 and m 2, with the same size of M # N, given by
m 1 = map 1,
m 2 = T0, 0, } (map 2).
(8)
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