Based on Equations 18 and 20, the below equation can be
derived as:
= −
∂
∂
−
= −
∂
∂
z
X
R X R R R R R R R M r
z
X
R X
c b
c b
b
c
l
b
l
l
w
l
b
c
l
b
e
l
l
c b
c b
ψ
δ
ψ
0
0
0
0
−
=
∂
∂
⋅
−
∂
∂
R R R R R R R M r
z
X
R
z
X
R M r
b
c
l
b
l
l
w
l
b
c
l
b
e
l
l
c b
c b
c e
c
l
0
0
0
0
δ
ε
δ
Ω
(22)
where
R
c
e
represents the rotation matrix from
ECEF
to the cur-
rent image frame,
X
b
is the vector
X
w
transformed in
IMU
body
frame,
Ω
b
is the skew-symmetric matrix of the vector
X
b
,
ε
is
the attitude error in the state vector. Combining Equations 17
and 22, the elements in design matrix with respect to position
and attitude error can be obtained, shown as follows
H
z
X
X
r
CR M
H
z
X
X
CR
r
c
c
e
c
c
c
b
c b
=
∂
∂
∂
∂
= −
=
∂
∂
∂
∂
=
ε
ε
Ω
(23)
So far, the inertial navigation, visual odometry and the
tightly coupled integration are illustrated. In the next section,
map matching algorithms used in this work are presented.
Map Matching
Map matching aims to project the land vehicle to the correct
road link of the digital map used (Ochieng
et al
., 2003; Taylor
et al.
, 2006; Quddus
et al
., 2007; Ren and Karimi, 2009). The
map matching algorithms can be classified into three main
categories, namely the geometric map matching, the topologi-
cal map matching, and the advanced fuzzy logic map match-
ing (Quddus, 2006; Ren and Karimi, 2012). The geometric
map matching is only based on the geometric relationship
between the vehicle localization solution and the digital map
entities. Generally, the geometric map matching algorithms
can be divided as the point-to-point, point-to-curve, and
curve-to-curve (Cossaboom
et al.
, 2012). For the topological
map matching, in addition to the geometric relationships, the
topology of the road network and the history of the vehicle
position are also taken into consideration (Velaga, Quddus,
and Bristow, 2009). In this paper, the advanced fuzzy logic
map matching algorithm (Syed and Cannon, 2004; Quddus,
Ochieng and Noland, 2007; Balazadegan Sarvrood and Amin,
2011; Ren and Karimi, 2012) is used to project the localization
solution on the corresponding road link. The implementation
procedures are illustrated in this section as follows.
The fuzzy logic map matching algorithms utilize the fuzzy
logic theory to calculate the likelihood of the road links
and the link with the maximum value of likelihood is to be
selected as the correct matched one. In this paper, Sugeno’s
Fuzzy Inference System (
FIS
) (Sugeno and Takagi, 1983) is ap-
plied. The calculation of the road link likelihood is based on
the equation given as:
Z
W Z
W
i i
i
n
i
i
n
=
=
=
∑
∑
1
1
(24)
where
i
is the
i
th
fuzzy rule,
W
i
is the membership function
output (between 0 and 1) for the
i
th
fuzzy rule,
Z
i
is the weight
assigned to the same fuzzy rule. The membership function is
a curve that defines how the inputs are mapped to a mem-
bership value between 0 and 1. The weight assigned to each
fuzzy rule depends on the output fuzzy subset. The details of
calculation procedures of Eq (24) are to be illustrated using
the Initial Map Matching Process in fuzzy logic map matching
as an example in the next paragraph.
There are two steps to implement the fuzzy logic map
matching algorithm, known as Initial Map Matching Process
(
IMP
) and Subsequent Map Matching Process (
SMP
) (Qud-
dus, 2006). Both
IMP
and
SMP
aim to project the vehicle on
the correct road link. For
IMP
, given the geodetic latitude and
longitude, the vehicle speed, the heading error (
HE
) (the dif-
ference between the vehicle heading and the link azimuth)
and the perpendicular distance (
PD
) from the vehicle position
to the road link, a total likelihood for each link can be calcu-
lated. The links around the vehicle position can be chosen as
candidates to be matched. If no link is included, the selecting
radius has to be increased to include at least one candidate.
Equation 24) is used to calculate the road link likelihood.
The correct link with the largest likelihood is determined
as the correct one. The likelihood in
IMP
is denoted as L1 in
this part. The fuzzy rules for
IMP
can be summarized in the
table below. Taking the first fuzzy rule as an example “IF
HE
is small and velocity is high THEN L1 is average”, the input
variables of this rule are
HE
and velocity, and the input fuzzy
subsets are small and high. The output variable is the correct
matching likelihood L1 and the output fuzzy subset is aver-
age. Each fuzzy linguistic object (e.g.,
HE
is small) has its own
membership function outputting value between 0 and 1 based
on the input (e.g.,
HE
). There is more than one part in one
fuzzy rule in this example while there is only one member-
ship function output for each fuzzy rule as shown in Equation
24. The fuzzy operator AND is used to combine the two parts
in one rule. In this study, the method used for AND operator
is “minimum” namely the minimum value of two member-
ship function outputs is the final output (
W
i
in Equation 24).
The weights assigned for L1 (
Z
i
in Equation 24)) high, average
and low are 100, 50, and 10, respectively(Table 1). If the mem-
bership function outputs for
HE
is small and velocity is high
are 0.7 and 0.8 respectively, the final membership function
output would be 0.7 (
W
1
) based on the AND operator adopted.
The weight assigned for L1 is average is 50 (
Z
1
).
Table 1. Fuzzy rules for
IMP
.
IF
THEN
HE
is small and velocity is high
L1 is average
HE
is large and velocity is high
L1 is low
PD
is short and
HE
is small
L1 is high
PD
is long and
HE
is large
L1 is low
With Equation 24, the road link with the largest value
is to be selected as the correct matched one. If
IMP
selects
the same link for several consecutive epochs, map match-
ing algorithm stops
IMP
and continues to
SMP
.
SMP
checks
whether the vehicle is still on the same link selected or not
(
SMP
-1) and whether the vehicle is near a junction or has just
crossed the junction (
SMP
-2). The
SMP
performance largely
depends on the performance of
IMP
. In Figure 3,
α
is the angle
between the current point, previous map matched point and
the junction;
β
is the angle between the current point, junc-
tion and the previous map matched point;
d
1
is the distance
between the previous map matched point and junction;
d
2
is
the distance traveled by car. Given the vehicle status in Figure
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
January 2018
19
