A novel contribution to an hybrid INS/GPS system
using a fuzzy approach*
1st Khoder Wassim 2nd Alwan Monzer
Faculty of Economics and Business Administration Faculty of Economics and Business Administration
Lebanese University Lebanese University
Tripoli, Lebanon Tripoli, Lebanon
wassim.khoder@ul.edu.lb monzer.alwan@ul.edu.lb
Abstract—In this paper, we present a technique based on as the extended Kalman filter (EKF) and filters based on
fuzzy logic to improve the performance of an integrated inertial sampling such as the unscented Kalman filter (UKF) [2], [3].
navigation system with GPS. The fuzzy technique proposed is The second is to use of the probabilistic approach techniques
mainly used to predict position and velocity measurements during
the absence of GPS signals. As long as GPS measurements are such as the particle filters [4], and finally we have the methods
available, the Q-SUKF [1] filter for INS / GPS integration works based on Artificial Intelligence (AI), such as the Artificial
efficiently and provides an accurate estimate of the states of Neural Networks (ANN) [5] or adaptive information systems
navigation. Nevertheless, during the disturbance of GPS signals, for neural networks and fuzzy methods (ANFIS) [6]. The
the fuzzy technique will be used with the Q-SUKF filter to Kalman filtering provides a powerful tool for creating synergy
correct the performance degradation of the algorithm. Finally,
an experimental part on the use of the fuzzy technique proposed between different sensors to hybridize. It can take advantage of
with the Q-SUKF has been validated. The aftereffects of our the benefits and characteristics of different sensors to provide
experiment have demonstrated the adequacy and the critical an hybrid inertial navigation system that performs better than
effect of the fuzzy method used. It decreases the error’s estimation unaided inertial navigation. It gives the optimal estimate by
of the position and velocity during GPS blackout periods. minimizing the mean squared error (MMSE). The extended
Index Terms—Inertial Navigation System, Global Positioning
System, Takagi-Sugeno Fuzzy Model, Fuzzy C-Means, Quater-
Kalman filter (EKF) is probably the most common and popular
nion Scaled Unscented Kalman Filter. approach to deal with non-linear system [7], [8], [9], [10],
[11]. However, it has some disadvantages. A new evaluation
method exists called ’Unscented Kalman Filter’ (UKF) [12],
I. I NTRODUCTION
[13], [14]. IT is nowadays considered a superior alternative to
The last decades have seen an increase in demand for iner- EKF in treating the errors of inertial navigation systems aided
tial navigation systems (INS) low-weight, low cost and low- by one or more additional external sensors. In [1], [15], we
power consumption, in many applications such as personal, have proposed a new type of UKF filter called Quaternion-
automotive and air navigation. The technological progress Scaled Unscented Kalman Filter (Q-SUKF) which combines
of micro-electro-mechanical systems (MEMS) showed a the scaled unscented Kalman filter (SUKF) and the use of
promised signs for the development of these systems. Com- quaternions as attitude representation parameters [1], [15].
pared to higher quality systems, a low-cost inertial naviga- This new hybridization digital filter makes it possible to obtain
tion system may cause a strong drift on the accuracy of a recalibrated state of the inertial navigation from information
the estimate of position, velocity and attitude on short time provided by external sensors. The external measurements used
intervals. This is mainly due to the great uncertainty of the are the position and / or velocity resulting from satellite
outputs of MEMS sensors and their sensitivity to changes navigation (GPS, Galileo). The Low-cost inertial navigation
in the environment. If the accuracy of low-cost INS can be systems provide accurate and reliable navigation parameter
improved, its cost can be reduced in existing applications solutions when integrated with the GPS aided sensor in the
and new applications may emerge. Like most uncertainties Q-SUKF filter. However, the performance of the integrated
exist in the behavior of the sensor errors, the calibration system will deteriorate considerably during periods of GPS
would significantly improve the accuracy. However, intensive failure. During the last decade, the Takagi-Sugeno (TS) fuzzy
calibration would increase considerably the cost. Another way systems have been used to model successfully the non-linear
to improve the accuracy would be to hybridize the inertial systems and have proved a good representation of dynamic
navigation systems with other complementary external sensors. systems [16], [17], [18], [19]. In these approaches, the non
Choosing the appropriate method of hybridization is the key linear behavior of a system is represented by a composition
for the development of Hybridised inertial navigation problem. of If-Then rules, concatenating a set of local linear sub models.
Currently, three approaches have been identified in research on In this article, a Takagi-Sugeno fuzzy model is used to estimate
filtering methods for hybrid inertial navigation system. The the position and velocity measurements to the integrated
first is the use of the statistical approach techniques such INS/GPS system during the various GPS blackouts. The fuzzy
Copyright © 2019 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
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model requires an offline learning phase extracted from a large
number of input-output coupled data when GPS signals are
available. This phase aims to identify the parameters of the
fuzzy model used with the filter Q-SUKF. The Input-Output
data cover different dynamics and types of movement (straight
and rotation). During the learning stage, the inputs of the
fuzzy model are position, velocity predicted by the Q-SUKF
filter. The outputs of the fuzzy model are the positions and
velocity measured by the GPS. At the end of the learning stage,
the best estimates of the parameters of the fuzzy model are
achieved. These measurements allow to maintain the update
mode phase of Q-SUKF fiter. In this paper, we describe a new Fig. 1: Information accumulations for the extraction of fuzzy
hybridization filter, called Adaptive Fuzzy Logic Quaternion model (A) (FL) during the learning stage
Scaled Unscented Kalman Filter, denoted by (A) (FL) Q-
SUKF. It is based on the application of the fuzzy model with
the Q-SUKF filter. Next, an experimental part on the (A) (FL)
Q-SUKF algorithm has been validated.
II. VALIDITY OF THE Q-SUKF
The Q-SUKF is an iterative procedure of calculation in-
tended to rectify the errors of the INS through outer es-
timations given by the GPS. For whatever length of time
that the GPS estimations are accessible, the Q-SUKF works
proficiently and gives a close valuation of the navigation
parameters states. Nevertheless, during GPS blackouts, the Fig. 2: Operating mechanism of the fuzzy model (A) (FL) with
general execution of INS/GPS framework is altogether cor- the Q-SUKF during GPS blackouts.
rupted by the quick collection of blunders which influence
the inertial unit components estimation. To fix this trouble, a
fuzzy logic model expressed by (A)(FL) is suggested. The III. P ROJECTION OF THE FUZZY INFERENCE SYSTEM
fuzzy logic is a collection of scientific hypotheses which
The projected fuzzy model uses a fuzzy inference system of
manages the portrayal and control the defective information
Takagi-Sugeno type (FIS-TS) which has specific features since
(inaccurate, ambiguous or deficient). It doesn’t try to remove
it symbolizes the nonlinear systems as an interjection between
them; oppositely, it will look to keep maintain them utmost.
local linear models. The FIS-TS fuzzy model proposed can be
Then, its aim is to make flexible the representative structure
written in a general structure as:
and information’s treatment, inspiring thus from the human
mental procedures. The viable utilizations of fuzzy logic are Ri : If xk is Ai (xk )Then ỹi = aTi xk + bi , i = 1, . . . , r (1)
various. Models include: automatism, robotics, expert systems,
Where Ai (xk ) is a gaussian membership function of the input
decision support, etc. In this paper, the fuzzy logic is charac-
variable vector at observation k, xk , in the fuzzy set Ai . ai
terized as a reasoning which uses the general role of ”expert
and bi are
⇥ the⇤components of the consequent parameters vector
system” in handling the information. When GPS signals are T
accessible, this model is educed offline from a countless ⇥i = aTi bi of the i-th fuzzy rule which describes the local
amount of a paired input-output data during a period called linear model. Ri is the i-th fuzzy rule, r is the total number
learning stage. The inputs of the fuzzy model are, position and of rules and ỹi is the estimated output of the local linear
velocity, calculated by the Q-SUKF. The outputs of the fuzzy model. The proposition ” xk is Ai (xk ) ” can be defined for
model are the positions and velocities estimated by the GPS, as the different components in form of conjunction:
demonstrated in fig.1. At the final stage of the learning phase, Ri :If x1,k is Ai,1 (x1,k ) & . . . & xn,k is Ai,n (xn,k ) (2)
the best assessment of the parameters of the Fuzzy model
Then ỹi = aTi xk + bi , i = 1, . . . , r
are accomplished. When a GPS blackout happens, the fuzzy
model (A) (FL) reproduce instead an evaluated measures of Where n is the dimension of the vector xk . The components
position and velocity which should be the GPS estimations if of xk are the three elements of the position pnk = [ k k hk ],
they were accessible. Thus, the Q-SUKF keeps on utilizing the and the velocity vkn . The choice of these parameters as input
conditions of the estimation update phase, as demonstrated in of the fuzzy model is convenient because they are the main
fig.2. The Q-SUKF filer is refered to (A) (FL) Q-SUKF when factors to affect the prospected outputs of the fuzzy model
it is utilized with the proposed fuzzy model. (position and velocity vectors). In addition, these states are all
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determined in the navigation frame and easily obtained from function can be seen as a measure of the total variance
the prediction phase of the Q-SUKF filter. Two classes have of xk with respect to the centers Vi . The minimization of
been assigned iteratively to each component of the entry vector JF CM (X; U, V ) is a non-linear optimization problem that can
where a Gaussian function has been implemented to represent be solved by different methods; the most used is the Fuzzy
the membership degree to each class. Based on the number of C-Means (FCM) algorithm [22]. It can be achieved by finding
entries equal to 6 and the class number equal to 2, the number the cluster centroid vectors and the standard deviation of the
of rules is therefore equal to 26 = 64. The estimated outputs membership Gaussian functions iteratively [24]:
of the fuzzy model are the position and velocity vectors, pn N N
X X
and vn , expressed in the navigation frame and which can be µlik 1 xk µlik 1 (xj,k l 2
Vi,j )
calculated from the equation (1) of the FIS as follows: 2(l)
r
Vil = k=1
N
, i,j =
k=1
N
(6)
X X X
T
i (xk )(ai xk + bi ) µlik 1 µlik 1
ỹk = i=1 r (3) k=1 k=1
X 1 i c, 1 j n
i (xk )
i=1 Where the membership degree µik is calculated as follows:
where i (xk ) denotes the degree of fulfillment of the i-th rule: 1
µik = c 1 i c, 1 j n (7)
n
Y X
i (xk ) = Ai,j (xj,k ) (4) (Dik /Djk )2/(m 1)
j=1 j=1
n
! c
X
Y 1 (xj,k Vi,j )2
= exp avec µik 2< 0, 1 > et µik = 1
2 2
j=1 i,j i=1
m 2 [1, 1) is the fuzziness parameter of the partition. The
Vi,j , i,j
2
represent the center and the variance of the Gaussian
parameter m influences the form of the classes in the data
fuzzy membership functions respectively. To identify the FIS
space of the system. When m approaches 1, the shape of the
of the fuzzy model, the antecedent parameters Vi,j , i,j 2
, and
membership function of each class is close to be Boolean
the consequent parameters, ⇥i , must be determined.
function (m 2 {0, 1}). The partition can range from a hard
A. Determination of Antecedent Parameters partition (m = 1) to a completely fuzzy partition (m ! 1)
Abonyi in [20] has proposed the Fuzzy C-Means classifi- when there is no changes significantly of the fuzzy partition
cation algorithm (FCM) to identify the antecedent parameters matrix between two successive iterations. Although the choice
of Takagi-Sugeno fuzzy model. The FCM algorithm aims to of m depends on the data [25], usually this parameter is
divide the data points into homogeneous classes or groups. initialized to a value between 1.5 and 2.5. The iterative process
Thus, the points in the same class are as similar as possible stops when the partition becomes stable, i.e., when it no longer
while points in different classes are as dissimilar as possible. changes significantly, between two successive iterations. This
The FCM algorithm, which issued from the works of [21] is generally expressed by checking the expression (8) where
and improved later by [22], constitutes an important reference the left term indicate a matrix norm and the coefficient ✏
among the different methods of fuzzy coalescence [23] based defines the convergence threshold:
on the minimization of the objective function, of the form:
U (l) U (l 1) < ✏ (8)
c X
X N
2
JF CM (X; U, V ) = µik .Dik (5)
The expression U (l) represents the fuzzy partition matrix of
i=1 k=1
the l-iteration.
Where X is the data matrix, N is the number of observations,
µik is the Fuzzy partition of fuzzy subset i, U = [µik ] B. Determination of Consequent Parameters
is the fuzzy partition matrix of dimension c ⇥ N , V = After the learning of the antecedent parameters using equa-
[V1 , V2 , . . . , Vc ] is a matrix of cluster centroid vectors which tions (5) and (6), the equation (3) can be rewritten as follows:
must be determined, with Vi 2 Rn , 1 i c, in our case, c
X
the number of cluster c is equal to the number of rules r ỹk = ¯i (xk ).xe ⇥i
k
and Dik is the euclidean distance between the observation xk i=1
and the Cluster centroid vector Vi . In the equation (5), the
dissimilarity measure expressed by the term JF CM (X; U, V ) T
(9)
= ¯1 (xk ).xe . . . ¯c (xk ).xe ⇥1 . . . ⇥c
is the sum of the squares of the distances between each k k
observation xk and the corresponding center Vi . The effect
of this distance is weighted by the degree of activation of the
= ¯1 (xk ).xe . . . ¯c (xk ).xe ⇥
class, µik corresponding to xk . The value of the objective k k
91
⇥ ⇤
Where xek = xk 1 is the input vector to the fuzzy model inertial measurement unit, GPS and magnetometer were used.
augmented by the unit element. ⇥ is the M-dimensional The experiment was conducted using a car driving (reference
consequent parameters vector where M =c ⇥ (n + 1). ¯i is trajectory) for 30 mimutes. This reference trajectory was
defined by the following formula: generated by the function progencar of INS toolbox version 3.0
created by GPSoft. This trajectory covers different dynamic
¯i (xk ) = i (xk )
c (10) (static and kinematic) and scenarios of motion (rotation and
X
i (xk )
rectilinear). The data of the inertial navigation system (angle
i=1 and velocity increments) were simulated from the parameters
of the profile of the automobile using certain functions of the
The linear equation (9) of the consequent parameters vector
INS toolbox. These angle and velocity increments have been
can be written as follows:
corrupted with various sources of errors such as biases, scale
Z⇥ = Y (11) factors and noises in order to generate outputs close to real
data of an inertial navigation system. The characteristics of the
Given M the number of linear consequent parameters and N errors models of the inertial sensors used in the experiment
the number of learning input-output data, the dimensions of are presented in TABLE I, where the two parameters T and
the matrices Z, ⇥, Y are N ⇥M , M ⇥1 and N ⇥1 respectively. describe the first-order markov process x represented by:
As N is always greater than M , the system of linear equations
1
(9) is an underdetermined system, therefore generally there is ẋ = x+! (14)
no exact or unique solution which can be reached. To get T
there, the least squares estimation (LSE) method is exploited Where T is the correlation time of the process x and ! is a
to minimize the squared distance between the vector Y and the wiener process with variance 2 2 /T .
linear combination Z⇥ . It is a classical problem that forms the TABLE I: Characteristics of error models of the inertial
basic in many applications such as linear regression, adaptive sensors used in the experience
filtering and signal processing. The famous formula for solving
systems of underdetermined equations uses the pseudo-inverse Parameter Model Accelerometer Gyroscope
p p
matrix of ⇥ as follows [26]: Noise Random Walk 0.6 m/s/ h 3.5 deg/ h
1st Order a = 0.1 m/s2 g = 100 deg/h
⇥⇤ = (Z T Z) 1 Z T Y (12) Bias
Gauss Markov T=1hour T=1hour
Where Z T is the transpose of Z. (Z T Z) 1 Z T is the pseudo- 1st Order sa = 1000 PPM sg = 1000 PPM
Scale Factor
inverse matrix of Z if Z T Z is non-singular. Despite that the Gauss Markov T= 4 hours T= 4hours
equation (12) is expressed in few words, it is very costly in
The GPS data (position and velocity) were generated by
terms of computation time when it comes to the calculation
adding to the positions and velocities data of the reference
of the inverse of a matrix Z T Z and, in addition, it becomes
trajectory a gaussian white noise. The initial standard deviation
poorly defined if this matrix is singular. To avoid the large
of the position expressed in Cartesian coordinates in the
computation time or the problem of singularity, sequential
navigation frame is equal to 2cm in the horizontal plane and
formulas are used to calculate the least square estimation of
is equal to 4cm in the vertical plane. The initial standard
⇥. This sequential method is more efficient, especially when
deviation of the velocity expressed in the navigation frame is
M is small. If the ith row of the matrix Z in equation (11) is
equal to 0.25m/s for the horizontal components and is equal
denoted by ziT and the ith element of the vector Y is denoted
to 0.4m/s for the vertical component. Two simulations of GPS
by yiT , then ⇥ may be calculated iteratively using the following
outages with a duration of 30s have been considered along the
sequential formulas [26], [27]:
path as shown in Fig. 3.
T
Si zi+1 zi+1 Si
Si+1 = Si + ; i = 0, . . . , N 1 12000
T
1 + zi+1 Si zi+1 (13) 11000
T 10000
⇥i+1 = ⇥i + Si+1 zi+1 (yi+1 zi+1 ⇥i )
9000
Where Si is often called the covariance matrix and the esti- 8000 GPS Blackout
7000
East (m)
(2)
mated least squares ⇥⇤ is equal to ⇥N . The initial condition 6000
5000
of the equation (13) is ⇥0 = 0 and S0 =⌘I, where ⌘ is an 4000
arbitrary positive number which is large and I is the M ⇥ M 3000
GPS Blackout
2000
dimensional identity matrix. 1000 (1)
0
IV. S IMULATIONS AND RESULTS -1000
-35 000 -30 000 -25 000 -20 000 -15 000 -10 000 -5 000 0 5 000 10 000
A. Simulations North (m)
To test the effectiveness of the (A) (FL)Q-SUKF filter and
Fig. 3: Simulated trajectory with GPS outages indicated.
its impact on the accuracy of the navigation parameters calcu-
lation (specially the position and velocity), a simulated data of
92
B. Results 10
Q-SUKF
8 δ vn
To test the implementation of the proposed methodology,
Velocity Error (m/s)
6 δ ve
an initial attitude error of 60 degrees is given on each axis. 4
2
δ vd
The diagonal terms of the initial covariance matrix represent 0
-2
variances or mean squared errors. The off-diagonal terms are -4
set to be zeros. The parameters used in the Q-SUKF are given -6
-8
by scaling parameters ↵ = 0.05 and = 2, and by weight -10
230 235 240 245 250 255 260 265 270 275 280
of 0th point !0 = 0.5. The Fig. 4, Fig. 5, Fig. 6 and Fig. 7 Time (s)
demonstrate the position and velocity errors during the two (a)
periods of GPS blackouts 1, 2 respectively, before and after 6
(A)(F)Q-SUKF
the use of the proposed technique of the fuzzy model to Q- 5 δ vn
Velocity Error (m/s)
4 δ ve
SUKF. 3 δ vd
2
Q-SUKF 1
80
δx
n
0
60
-1
Position Error (m)
δ xe
40
20 δ xd -2
230 235 240 245 250 255 260 265 270 275 280
0
Time (s)
-20
-40 (b)
-60
-80
-100 Fig. 6: Velocity error estimated during GPS blackout1
230 235 240 245 250 255 260 265 270 275 280
Time (s)
(a)
(A)(FL)Q-SUKF Q-SUKF
2 10
δ Vn
1.5 8 δ Ve
1
Position Error (m)
Velocity Error (m/s)
δ Vd
6
0.5
0 4
-0.5 2
-1 δ xn
0
-1.5 δ xe
-2 -2
δ xd
-2.5 -4
-3 940 945 950 955 960 965 970 975 980 985 990
230 235 240 245 250 255 260 265 270 275 280 Time (s)
Time (s)
(a)
(b)
(A)(FL)Q-SUKF
Fig. 4: Position error estimated during GPS blackout1 δ vn
Velocity Error (m/s)
3
2.5 δ ve
2 δ vd
1.5
1
Q-SUKF 0.5
80 0
δ xn
-0.5
70 -1
Position Error (m)
60 δ xe -1.5
50 -2
δ xd 940 945 950 955 960 965 970 975 980 985 990
40
Time (s)
30
20
10
(b)
0
-10 Fig. 7: Velocity error estimated during GPS blackout2
940 945 950 955 960 965 970 975 980 985 990
Time (s)
We notice in these figures that the maximum errors of
(a) the position and velocity components have been reduced
(A)(FL)Q-SUKF
2 considerably after the application of the proposed technique of
1.5
the fuzzy model to the Q-SUKF filter during the two periods
Position Error (m)
1
0.5 of GPS blackouts. The TABLE II summarizes the percentage
0
δ xn
of the reduction of these errors.
-0.5
-1
-1.5
δ xe
δ xd
The fuzzy model applied to the Q-SUKF conduct to an
-2 important enhancement of 75.32 % and 43.90 % at least in
940 945 950 955 960 965 970 975 980 985 990
Time (s) reducing the errors estimation of position and velocity respec-
(b)
tively. Finally, in spite of the fact that these first outcomes
cannot be extrapolated, they might anticipate to give the green
Fig. 5: Position error estimated during GPS blackout2 light for future research presenting GPS blackouts in various
situations of real scenarios for the purpose of a generalization.
93
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