1
Article
2
3
A multi-sensor fusion algorithm based on LSTM for
aiding INS during GPS outages
4
Wei Fang 1, Jinguang Jiang 1, 2, *, Peihui Yan1, Jingnan Liu1,2 and Haiyong Luo3
5
6
7
8
9
10
1
11
Received: date; Accepted: date; Published: date
12
13
14
15
16
17
18
19
20
21
22
23
24
Abstract: Aiming to improve the accuracy of navigation performance during global positioning
system (GPS) outages, a method based on long short‐term memory (LSTM) for aiding inertial
navigation system (INS) which employs sensors of three‐axis accelerometers and three‐axis
gyroscopes is explored to generate the pseudo GPS position increment information acting as
substitute of GPS signal. Almost all existing artificial intelligence (AI) based methods, like
multilayer perceptron neural network (MLP) and support vector regression (SVR), are based on
relating INS error with INS output only with fixed number of time instants and do not consider the
dependence of output on the past vehicle dynamic information. Therefore, the LSTM algorithm,
which is a kind of dynamic neural network that can construct relationship among present and past
information, is adopted to attain a more stable and reliable navigation solution during a period of
GPS outages. A set of reality data was employed to evaluate the proposed algorithm. Test results
indicate that LSTM algorithm can promote navigation accuracy better than MLP and SVR
algorithms during a long period of GPS outages.
25
26
Keywords: LSTM; INS/GPS integrated navigation system; GPS outage
GNSS Research Center, Wuhan University, No.129 Luoyu Road, Wuhan 430079, China
National Engineering Research Center for Satellite Positioning System, 30 Jiangda Road, , Wuhan 430079,
China
3 Institute of Computing Technology, Chinese Academy of Sciences, No.6 Kexueyuan South Road
Zhongguancun, Haidian District, Beijing 100190, China
* Correspondence: jinguang@whu.edu.cn; Tel.: +086‐027‐68778971
2
27
1. Introduction
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
Inertial navigation system (INS) and Global navigation satellite system (GNSS), are two main
and important approaches for providing position and attitude information for geographical
reference[1]. The inertial navigation system is a self‐contained system with a high accuracy in short
period of time and not affected by environment. But standalone INS solution will degrade over time
because of the noise in the raw IMU measurements[2]. In this study, the signal from global
positioning system (GPS) is adopted to provide high precision navigation solutions. Under decent
visibility conditions, GPS can provide continuous and accurate navigation information over a long
period of time. However, GPS alone cannot give reliable positions all time since the satellite signal
may be blocked or corrupted due to high buildings, viaducts, tunnels, mountains, multi‐path
reflections and bad weather conditions[3,4,5]. Due to the complementary properties, INS and GPS
are integrated by Kalman filter (KF) for providing continuous and high precision navigation[3‐6]. In
an integrated INS/GPS system, GPS calibrate INS through its error estimation process in KF[7,8]. In
the other hand, INS bridges GPS signal gap, assists the GPS signal reacquisition after an outage and
improves the performance of detecting and correcting GPS cycle slips[9,10]. However, KF can’t
update information from GPS measurements when the vehicle is running in city canyons or tunnels
where GPS signal is blocked. Meanwhile, the INS/GPS integrated system change into pure INS,
whose position error will diverge over a period of time[2‐ 4,11]. Therefore, an improved fusion
Remote Sens. 2019, 11, x; doi: FOR PEER REVIEW
www.mdpi.com/journal/remotesensing
Remote Sens. 2019, 11, x FOR PEER REVIEW
2 of 22
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
algorithm need to be explored to improve the navigation performance of INS when GPS signal is
lost.
With the development of Artificial Intelligence (AI) and big data, a lot of AI methods have been
explored to improve navigation accuracy during GPS outages[12‐18]. Nowadays, artificial neural
networks (ANN) is the most widely used methods to model a complex nonlinear problem. Many
researchers have built a variety of neural networks for aiding INS when GPS signal is lost. Rashad
and his team first used radial basis function neural network to model INS position and position error
95
2. INS/GPS loosely-coupled integrate navigation system
between INS and GPS[12,13]. EI‐Sheimy used time, velocity and yaw as inputs to model the position
error and velocity, showing a more stable and accurate result[14]. In later works, an improved
autoregressive delay–dependent neural network model was applied, whose inputs are the current
and past 1‐step samples of the vehicle dynamic information (position and velocity) from IMU and
INS, while output is INS position error
[15]. Another method based on ensemble model has
been explored to promote the generalization of algorithm by utilizing a lot of weak learners to
construct a strong learner[7,15,17]. Besides, support vector machine and genetic algorithm was also
explored to overcome the over‐fitting and local‐minimum problems of neural networks[18]. Lately,
factor graph optimizations are widely used for multi‐sensors fusion in autonomous systems [19].
This method uses factor graph model to represent joint probability distribution function. An efficient
incremental inference algorithm over the factor graph is applied, which yields a near‐optimal
inertial navigation system.
However, almost all above methods are based on static neural network, like Radical Basis
Function Neural Network (RBFNN) or MLP. The main idea behind all of above methods is to model
the latest vehicle dynamic information by training AI model when GPS signal is available. Almost all
existing AI‐based models try to improve the navigation solutions during GPS outages by relating
INS position or velocity error to only previous INS output. The major drawback of which is that they
cannot store more past vehicle dynamic information when dealing with sequential process in the
input data (INS attitude and velocity). Therefore, under condition of long period of GPS outages,
any of existing AI‐based models may not have capability of providing a reliable and stable
navigation solution[20]. Furthermore, the major limitation of the MLP for a sliding time window
input sequence is the rapidly increase of computational complexity since the input layer has to have
number of neurons equal to the number of past samples[21]. For all above reasons, this study
suggests that LSTM neural network identifying a nonlinear dynamic process can solve above
drawbacks of those models, Which have capabilities to perform highly nonlinear dynamic mapping
and store past information[22]. LSTM is also a basic neuron in recurrent neural network (RNN)
which can select and store significant information[23], and has been widely used in variety dynamic
process, such as natural language process (NPL)[24], and time serial prediction[25,26].
With the objective to maintain good performance of INS during GPS outages, a novel AI
method based on LSTM, is proposed to overcome the drawbacks of methods discussed above. When
the GPS signal is available, the velocity, yaw, angular rate and specific force information of INS are
as input features for training the LSTM model, while the output of model is position increment of
GPS. Once the GPS signal is lost for a short time, the information of INS which are discussed above
will be fed into LSTM model to generate the pseudo GPS increment, and then obtaining the pseudo
GPS position for KF to correct the INS navigation results. Actual test data is used to evaluate the
novel algorithm of LSTM, which is also compared with traditional MLP and SVR algorithms. Test
results indicate that LSTM algorithm can promote navigation accuracy better than MLP and SVR
algorithms during a period of GPS outages.
This paper is organized as follows: Section 2 introduces the traditional INS/GPS
loosely‐coupled integrated system, LSTM architecture and training method for time serials
prediction are illustrated in section 3. Section 4 demonstrates an improved model of LSTM fusion
algorithm aiding for INS, the road experimental testing that validates the proposed algorithm is
presented and discussed in section 5. Finally, conclusions are given in section 6.
Remote Sens. 2019, 11, x FOR PEER REVIEW
96
3 of 22
A loosely‐coupled integrated navigation system is illustrated in Figure 1
97
98
99
100
101
102
103
104
105
106
Figure 1 shows the common architecture of loosely‐coupled GPS/INS system. V, P and A
represent the velocity, position and attitude, while subscript indicates that whether they are from
INS or GPS. Position P contains three directions, including L, λ and h which represent latitude,
longitude and height, while attitude A contains three values, including , and which represent
pitch, roll and yaw. δV, δP and δA represent estimated errors from the output of KF, respectively.
Loosely‐coupled model is a widely used method of navigation which utilizes position error
measured from GPS and INS to correct the navigation result of INS. It is easy for understanding and
operating compared with tightly‐coupled or ultra‐tightly coupled model[27].
107
2.1 INS mechanization procedure
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
The INS is the central equipment either in GPS/INS or INS/AI. The vehicle’s velocity, position
and attitude, which are used as input features of AI algorithm, are attained from the measurements
of the IMU[28]. The position and attitude information is integrating from the original acceleration
and rate measurements. Above integration procedure which is also called mechanization of raw
measurements is usually performed in a certain reference coordinate system[29]. The most common
mechanization frames are the local level frame (n‐frame) and Earth Centered Earth Fixed reference
frame (e‐frame).
The local level frame (n‐frame) is defined along the East (E), the North (N), and the vertical (UP)
to the earth’s surface while their origin is located inside the moving vehicle. The origin of Earth
Centered Earth Fixed reference frame (e‐frame) is located at the center of mass of the earth with one
axis pointing toward the mean meridian of Greenwich (X), the second is parallel to the spin axis of
the earth (Z), and the third axis is completing a right‐handed orthogonal frame (Y). On the other
hand, all inertial sensors produce measurements relative to an inertial frame (i‐frame) resolved
along the instrument’s sensitive axis. Furthermore, the frame of choice for near‐Earth environments
is the earth‐centered inertial (ECI) frame. this is defined by:
1.the origin is at the center of mass of the earth.
2.The z‐axis is along axis of the earth’s rotation through the conventional terrestrial pole (CTP).
3.The x‐axis is in the equatorial plane pointing towards the vernal equinox.
4.The y‐axis completes a right‐handed system.
The measurements contain the specific force
from accelerometer unit and angular
increment
from gyroscope. The superscript b represents the vehicle’s body frame (b‐frame),
the b‐frame is defined along the forward (X), the transverse (Y) and the vertical (Z) direction of
body platform. Meanwhile the subscript ib indicates the measurements from IMU are in b‐frame
related to i‐frame. Actually, IMU measurements are given in sensor frame (s‐frame), but the
installation imperfection is ignored, which assumes that s‐frame is perfectly aligned with b‐frame.
In pure INS mode, INS uses current measurements and previous value to update next time
value. The procedure of update in velocity and position of pure INS mode is below:
=
+Δ , +Δ ⁄ ,
(1)
Figure 1. loosely‐coupled GPS/INS integrate navigation system
Remote Sens. 2019, 11, x FOR PEER REVIEW
4 of 22
136
137
L( ) = Lφ(
)+
λ( ) = λ(
)+
( )+
1
2
(
)
) +ℎ
Lφ(
(
)
−
(2)
138
139
140
=
141
ℎ=
142
143
144
( )+ (
( )+ℎ
1
1
2
)
2 L( ) + L(
1
)
2 ℎ( ) + ℎ(
)
(
)
−
(3)
(4)
(5)
In the attitude update, the DCM is improved by the estimated attitude errors obtained by
the kalman filtering algorithm. In Equation (6),
is the estimated attitude errors by kalman
filter,
is estimated attitude of the next time by INS mechanization.
145
= [Ι − ( ×)]
(6)
146
2.2 INS/GPS integration procedure
147
148
149
150
151
152
153
154
155
INS only is seldom useful since the inertial sensors biases and fixed‐step integration errors
will cause the navigation solutions to diverge quickly. In addition, these bias errors are random and
highly systematic in nature and are need to be modeled by stochastic processes. This means that
INS requires external information to update for overcoming the divergence of navigation solution.
A typical method to improve the performance of INS is optimally combining INS and GPS
position, while the kalman filter is utilized to incorporate the INS and GPS. In this case, a common
state vector is defined to model errors between INS and GPS. The state vector includes INS errors
and sensors bias.
Ignoring some small influences, the linear error equation of INS are[30]:
156
157
̇ =−
̇ =
+
×
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
+
(7)
−
− (2
+
)×
+
δ ̇=
158
159
×
δ ̇=(
)
+(
δℎ̇ =
(8)
(9)
)
−
(
)
(10)
(11)
where the superscript n indicates the navigation frame (n‐frame) and subscript e donates the earth
frame (e‐frame).
and
are the earth self‐rotation angular rate and its error which are
transformed from e‐frame to i‐frame and projected into n‐frame.
and
are angular rate
and its error transformed from n‐frame to e‐frame and projected into n‐frame. In equation (7),
=
+
,
means attitude vector and
is called the direction cosine matrix (DCM)
which is used to transform coordinates from b‐frame to n‐frame. Equation (8) indicates the velocity
error equation where
is accelerometer zero offset error and
is gravity error. From
equation (9) to equation (11) represent the position error where δL and δλ represent position error
in latitude and longitude, and δh represents the position error in height, the
and
denote
radius of curvature in longitude and latitude, respectively.
Figure 1 indicates that KF uses position error measured from GPS and INS to estimate the INS’s
error. The KF with 15‐states vector of X in equation (13) is adopted for integrating GPS and INS. The
state equation is:
̇=
+
(12)
where F and G are state transition matrix and system noise coefficient matrix separately, while W is
process noise vector.
The system states vector X is:
Remote Sens. 2019, 11, x FOR PEER REVIEW
178
179
180
181
182
183
184
=
,
,
5 of 22
,
,
,
,
=
=
189
,
,
,
,
(13)
+
(14)
+
,
=
×
×
0
×
].
(15)
+
(16)
The process of prediction and update of KF are:
Prediction stage:
192
=
193
194
,
where V is observation noise vector and is the observation matrix denoted as [0
Let 15 states KF rewrite in discrete time:
188
190
191
, ℎ,
Shown above,
, , are misalignment angles calculated in equation (7) in n‐frame.
, , are
velocity errors of three axes calculated from equation (8) in n‐frame, while
, , ℎ represent the
position errors of latitude, longitude and height calculated from equation (9‐11) in n‐frame.
, , , , ,
represent accelerometer biases and gyroscope biases in three axes in b‐frame.
The position differences between INS and GPS are defined as measured value, denoted by Z.
the measurement equation of which is shown as follows:
185
186
187
,
,
=
,
,
,
(17)
,
+
,
,
(18)
Update stage:
195
=
,
196
=
,
197
,
+
=[ −
−
]
+
(19)
,
(20)
,
(21)
198
199
200
Where
is the state transition matrix from state k‐1 to state k,
is Kalman gain matrix,
,
and
are variance and co‐variance matrices of state and observation. More knowledge about
kalman filter can be found in [31].
201
3. Structure of memory unit of LSTM
202
203
204
Recurrent neural network is now widely used in prediction sequence‐based tasks such as
pedestrians’ trajectories predicting[32] and vehicles trajectories prediction[33]. Unlike traditional
multilayer perceptron neural networks, a basic RNN has a loop to allow information to persist.
205
206
207
208
209
210
Figure 2. an unrolled recurrent neural network
In Figure 2, the chain‐like architecture indicates that RNN is related to sequences and lists.
However, traditional RNN is lack of the capability to handle the long‐term dependency[34].
Whereas, Long short‐term memory networks(LSTMs) is a special RNN with the ability of learning
long‐term dependency, the structure of which is shown in Figure 3
Remote Sens. 2019, 11, x FOR PEER REVIEW
211
212
213
214
215
216
217
218
6 of 22
Figure 3. LSTM block architecture
In above structure, the input is the known data, and the output is the forecast result
. There
are three gates in the memory unit, including input gate, forget gate and output gate, which are all
indicated by green circles. Moreover, the state of the cell is represented by , the input of every
gate is the preprocessed data
and previous state value of the memory ℎ . The blue circle
indicates for confluences, which stand for multiplications and additions. Based on information flow
in the structure of memory unit, the mathematical model of LSTM can be summarized as bellows:
219
=
[ℎ
,
]+
220
= (
[ℎ
,
]+
221
^ =
222
=
223
= (
224
[ℎ
ℎ(
ℎ =
,
(22)
)
(23)
]+
°
+ °^
[ℎ
,
°
ℎ( )
]+
)
(24)
(25)
)
(26)
(27)
225
226
227
While, the ‘°’ donates the Hadamard product and
means sigmoid non‐linearity function.
Multiple LSTMs can be stacked and temporally concatenated to form more complex structures. Such
models have been applied to solve many real‐life sequence modeling problems[35].
228
4. LSTM algorithm aiding INS/GPS navigation system during GPS outages
229
230
231
232
233
234
235
236
237
238
239
240
241
The main idea of AI‐aided INS/GPS integrated system is to mimic the mathematical
relationship between particular navigation information in INS/GPS integrated system and vehicle
dynamical data from IMU and INS for maintaining high navigation accuracy during GPS outages.
When GPS signal is available, the AI module is trained to learn the relationship. Once the GPS signal
is lost, the well‐trained AI module will predict the required information to aid INS.
More recently, several AI modules have been explored to find the relationship. Almost these
models can be divided into three classes by different outputs. The first is
−
, which relates
the information of INS with the position error between GPS and INS to estimate the error[36]. The
second is
− , which relates the information of INS with state vector
of KF[37]. Both above
models have a problem that the estimated information includes not only the data of INS, but also the
data of GPS. Since the navigation error of INS will propagate along the time and INS/GPS system
integrated by KF can’t be absolutely accurate, mixing data of GPS and INS which is acting as outputs
of AI will bring in additional errors, which results in reducing the predicting accuracy. In order to
Remote Sens. 2019, 11, x FOR PEER REVIEW
242
243
244
245
246
247
7 of 22
avoid mixed operation of the information from INS and GPS, a model based on estimating and
predicting the increment of GPS position, which is denoted as
−
is selected, and then
generates the information of pseudo GPS position.
In this paper,
−
is adopted as model framework because of the ability of avoiding
mixed operation of INS and GPS. The output of this model is the increment of position
, which
can be obtained by differential equation of INS[38‐39]. The formula is as follow:
248
∝
249
=
( )− 2
( )+
( ) ×
( )+
(28)
( )− 2
( )+
( ) ×
( )+
(29)
250
251
252
253
While,
is specific force of body frame to inertial frame,
is angular rate of earth frame to
inertial frame,
is angular rate from navigation frame to earth frame,
is velocity and
is
gravity. The superscript indicates that these vectors are projected into navigation frame n. These
values can be calculated by follows:
254
=
−
255
=
(30)
+
+
+
+
−
(31)
−
256
257
258
While,
is the output of accelerometer unit,
is direction cosine matrix to transform
coordinates from b‐frame to n‐frame. , and represent pitch, roll and yaw. The derivation of
with time is:
259
260
261
262
=
=
−
(
+
(32)
)
(33)
While,
is the output of gyro,
× means the anti‐symmetric matrix of
. The
and
can be calculated by latitude ( ), longitude ( ) and velocity ( ). The formulas are as follows:
263
=[
264
=
265
266
267
268
269
270
271
272
273
×
, 0, −
,
,
]
(34)
(35)
According to equations (28‐35), position increment
is mainly influenced by
,
,
,
, ,
. In condition of land vehicles, the body frame is almost in local level. 、
which means that the value of pitch ( ) and roll ( ) are very small,
will be mainly influenced by
value of yaw ( ). From equations(34‐35),
is mainly related with latitude
, longitude
and velocity
, while
is mainly influenced by earth rotation angular rate
. During a
period of time of GPS outages, the value of
,
and
will not change a lot. Therefore,
,
and
have an imperceptible impact on
,
. In conclusion, the main factors which influence
the value of
mostly will be
,
,
and yaw ( ), which is also the reason of choosing them
as the input features.
Remote Sens. 2019, 11, x FOR PEER REVIEW
8 of 22
274
275
Figure 4. configuration of training process
276
277
Figure 5. configuration of predicting process
278
279
Figure 6. Architecture of LSTM neural network
280
281
Figure 4 and Figure 5 illustrate the process and architecture of the LSTM network aiding for
INS/GPS integrated navigation system. , and represent the velocity, position and attitude. The
Remote Sens. 2019, 11, x FOR PEER REVIEW
9 of 22
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
subscript indicates the data from GPS or INS. indicates the estimated errors from the output of KF.
represents the value of yaw.
When GPS signal is available, AI module works on training mode, shown in Figure 4. On one
hand, the measurements of position from GPS and INS are fed into KF as inputs to estimate the error
of INS, which is just the loosely‐coupled mode. In process of training,
,
from IMUs and
, from INS as input features are fed into AI module, while position increment from GPS
measurement as target value is adopted to compute the loss function. In training mode, AI module
try to find relationship between
with
,
,
and yaw. When GPS signal is lost, the
system changes to predicting mode, indicated in Figure 5. Under these conditions, the value of
,
,
and yaw can be still obtained from INS and IMUs, which will be fed into well‐trained AI
module to get
. After accumulating all
from beginning of predicting, pseudo GPS position
can be attained at each time. Then, the pseudo GPS position will be fed into KF as substitute of the
original position from GPS. The hybrid system will maintain navigation continuously during GPS
outages.
Figure 6 illustrates the inner details of AI module. In each time, a serial vectors of { ( −
), . . . , ( )} are as inputs fed into AI module, while ( ) represents a feature vector of
[ ( ), ( ), ( ), ( )] and n represents the time steps. After concatenating these vectors from [t‐n,
t], a matrix with the shape of (n+1, 4) is attained. Then the value of last state in last layer is sent to a
fully‐connected layer with a softmax activation function[40] as output of the AI module, which is
denoted as
( ). From Figure 4 to Figure 6, time step t is the same with GPS cycle, which is 1Hz.
( ) at
These input features [ , , , ] at t(n‐4), t(n‐3), t(n‐2), t(n‐1) and t(n) are used to predict
t(n). So this will need 4 seconds past history while GPS rate is 1 hz.
The LSTM algorithm is proposed to deal with sequential process, Meanwhile, Vehicle dynamic
process is actually a kind of sequential process. In fact, the next arriving position has a strong
relationship with current and previous information, which has been illustrated previously.
Traditional static neural network can do non‐linear map well, but can’t solve sequential process
problem as good as recurrent neural network. therefore, LSTM architecture is adopted to replace
traditional neural network to find relationship between
ℎ
,
,
and yaw, which
also appears better performance than that of static neural network. Test results will be illustrated in
next section.
312
5. Road experimental testing results
313
314
315
316
317
318
Test data was collected by a land vehicle platform which is shown in Figure 7 to evaluate the
proposed algorithm. The prototypes of INS and GPS receiver are showed in Table 1. ICM‐20602 is a
low‐cost 6‐axis MEMS motion‐tracking device. The aiming of this study is to improve the navigation
performance of low‐cost IMU during GPS outages by utilizing AI model based on LSTM. While
Ublox‐M8P is a high precise GPS module which is used as reference system. The features of devices
are illustrated in Table.1.
319
Table 1.
Device
ICM-20602
U-Blox-M8P
Devices features
Sensor
Gyroscope sensitivity error
Gyroscope noise
Accelerometer noise
velocity precise
Dynamic heading precise
Performance
1%
4% mdps/√
100 ug /√
0.05 m/s
0.03 °
Remote Sens. 2019, 11, x FOR PEER REVIEW
10 of 22
320
321
Figure 7. Test vehicle with navigation system equipment.
322
323
324
325
326
The proposed LSTM‐based method aiding for INS/GPS integrated system was tested and
analyzed both in training and prediction mode, whose procedure and diagram have been illustrated
in Figure 4 and Figure 5. The parameter turning, the influence of the number of time steps and
hidden units in LSTM model are also explored. The proposed model of AI is trained and tested in
different time slices.
327
5.1 data description
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
Figure 8 shows the trajectory of vehicle, which was conducted in China, Hubei province, at Wu
Chang. The red icon and blue icon represent the start point and end point separately, Red line
indicates where GPS signal is assumed unavailable, meanwhile the green line and blue line
represent trajectory of validation set and training set respectively. During the path of blue line, the
whole system works under loosely‐coupled mode during trajectory. At that time, the GPS data and
INS data are integrated to maintain a continuous and high precision navigation results, meanwhile
LSTM‐based model is trained to mimic the relationship between INS dynamics and GPS position
increments. The INS dynamic information include velocity, yaw, angular rate and specific force data
at the current moment and past seconds are all concatenated as input matrix for AI module. The
position increments calculated from GPS data are treated as output vector for AI module.
When vehicle was running along red line where the GPS signal was assumed outages,
navigation results only depended on pure INS reckoning. Figure 9 shows the divergence of position
error while applied pure INS reckoning when GPS signal was assumed outages. In Figure 9, the
black points represent the presence of GPS signal. When GPS signal is assumed unavailable, the
black points are broken, the position error is divergent. When GPS signal is available, INS/GPS
integrated system have a well performance of navigation. The aim of this research is to improve the
low‐cost INS performance during GPS outages by using the algorithm proposed above.
In the experiment, trajectory data from INS and GPS in Figure 8 was split into three parts: first
part from 0s to 7400s marked by blue oval is used as training set; second part from 7400s to 7600s
marked by green oval is used as validation set; third part from 7600s to 8050s marked by red oval is
used as test set. Actually, when GPS signal is available, all the data from INS will be used to train the
AI model. Once the GPS signal is unavailable, the well‐trained model will substitute GPS to supply
the position information for Kalman filter to decrease the error of INS. All of the data has been scaled
between 0 and 1 for accelerating the process of training.
In a typical machine learning application, the data set is usually iterated many times using a
gradient descent algorithm. However, navigation is a real‐time application. It does not make sense to
perform multiple iterations on a same section of the road, so it seems like making no sense to
normalize the data by a section of road data in practical applications.
But in this experiment, off‐line mode is used to explore the feasibility of neural network
methods in terms of improving INS navigation solutions when GPS signal is lost. Furthermore,
Remote Sens. 2019, 11, x FOR PEER REVIEW
358
359
360
361
362
11 of 22
different features have different numerical values and sizes. Normalization and standardization are
mainly to make calculations more convenient. For example, the size of two variables are different,
while which one value is larger than the other. Then, they may cause numerical calculation error
when both of them are simultaneously variables. So scaling input data is needed for making
calculations more convenient and quickly convergence.
363
364
Figure 8. Vehicle trajectory
365
366
Figure 9. Divergence of position error with pure INS reckoning in testing path
367
5.2 Design the LSTM network
368
369
370
371
372
AI model is comprised of one LSTM layer and a fully‐connected layer activated by softmax
function. In process of designing LSTM network, there are several hyper parameters which need
tuning carefully: (1) the number of hidden units;(2) the step time length. Setting too more hidden
units and too long time steps will spend more time to make algorithm convergence and may lead an
over‐fitting problem. In the experiment, the other parameters’ value shows in Table 2.
373
Table 2.
Parameter Set
Remote Sens. 2019, 11, x FOR PEER REVIEW
374
375
376
377
378
379
Parameters
Value
Learning rate
0.01
Learning rate decay factor
0.99
L1 regulation
0.9
L2 regulation
0.99
batch size
32
Then, validation set is used to adjust the model parameters, while test set is utilized to evaluate
the actual capability of generalization. In this study, grid search method is adopted to find the best
parameter combination, while three time steps length (1, 4, 8) and three hidden units of LSTM (16,
32, 64) are selected to be evaluated. Three time steps and three hidden units will generate 9
combinations. The model performance parameter is evaluated with Root Mean Square Error (RMSE)，
given in equation (36).
380
381
382
383
384
385
386
387
388
389
390
391
392
393
12 of 22
RMSE =
∑
(Χ ,
)−
(36)
Where
and
are the predicted and the desired output and the same as N corresponding
to the GPS outage duration.
The training time and RMSE in validation were calculated and compared in each combination,
then the final parameters are selected from the best choices among these combinations. The
experimental results are displayed in Table 3 and Figure 10, every of which indicates that setting
time step and hidden units equal to 4 and 32 respectively makes the LSTM algorithm obtains less
RMSE. The low RMSE implies the high confidence of prediction method. Specifically, the different
performance of prediction with different time steps and hidden units relate to overfitting problem
which produces the large value of RMSE.
Figure 11 indicates the loss varies with epochs during training process. It can be seen that the
loss of train and validation decrease to the minimum value together after 3 epochs. Then, trends of
train and validation tend to be flat, which makes it sense that setting early stopping for saving more
time and without losing any performance.
394
Table 3. RMSE with different time steps and hidden units
Time Steps
1
4
8
1
4
8
1
4
8
Hidden Units
16
16
16
32
32
32
64
64
64
East-RMSE(m)
3.88
4.28
6.44
5.25
2.67
4.30
5.20
6.32
6.74
North-RMSE(m)
6.32
5.76
7.48
4.43
2.86
5.23
5.45
5.78
6.37
Time(s)/epoch
2.25
5.43
9.11
2.86
6.67
13.6
3.61
8.28
14.41
Remote Sens. 2019, 11, x FOR PEER REVIEW
13 of 22
395
396
Figure 10. RMSE of predicting position increment with different hidden units and time steps
397
398
Figure 11. Train and validation loss with 32 hidden units and 4 time steps
399
5.3 Experiment results
400
401
402
403
404
405
406
407
408
409
410
411
412
413
In this section, the navigation performance of the LSTM model is compared with that of MLP
and SVR, respectively. MLP uses time window length with 4 to predict future position increment,
whose input features are same with that of LSTM including
,
,
and yaw. Aiming to have
same hidden units with LSTM, MLPNN has two hidden layers, every of which has 16 hidden units
except input and output layer. SVR also uses time window with 4 to predict future position
increment with same input features of LSTM and MLPNN. The algorithm was implemented with
Python 3.6 and Tensorflow 1.9, which is a very famous deep learning framework developed by
Google[41], and trained by Adam optimizer[42].
Under the training procedure of above algorithms(LSTM, MLP and SVR), the training set
including the above features ( ,
,
and yaw) from starting time 0s to 7400s is employed to
train the models of MLP, SVR and LSTM, respectively. The validation set containing the above
features from 7400s to 7600s, is applied to estimate the effect of trained models, from which the
effectiveness of predicting position increments of three models can be distinguished. Comparison
results are shown in Figure 12 to Figure 15 and summarized in Table 4
414
Table 4.
Mean abstract error and standard deviation in different algorithms
Remote Sens. 2019, 11, x FOR PEER REVIEW
Longitude(m)
Latitude(m)
415
416
417
418
419
420
421
422
423
424
425
426
14 of 22
LSTM-MLP
MAE
Std
1.96774 1.16764
MAE
3.84456
Std
2.45674
LSTM-SVR
MAE
Std
1.99683
1.21521
SVR
MAE
Std
4.81162 3.13656
1.94119
3.45261
2.29471
2.20089
5.13401
1.21730
MLP
1.31846
2.88711
The prediction errors of the increments of the latitude and longitude based on MLP and LSTM
algorithms are shown in Figure 12 and Figure 13 respectively. The blue line represents prediction
error of LSTM algorithm, whereas red line represents that of MLP algorithm. The value of mean
absolute errors (MAE) of predicting position increments in latitude using algorithms of LSTM and
MLP are 1.94119 and 3.45261, whereas the standard deviation of every algorithm of which is 1.21730
and 2.29471 respectively. The value of mean absolute error of predicting position increments in
longitude are 1.96774 and 3.84456, meanwhile whose standard deviation is 1.16764 and 2.45674,
respectively. It can be concluded that the LSTM algorithm can make more accurate prediction of the
position increment than the traditional MLP algorithm, which indicates that the proposed
algorithm can better construct the inner relationship between the inputs and outputs.
Figure 12. Comparison of MLP and LSTM algorithms in terms of prediction error in latitude
427
428
429
430
431
432
Figure 13. Comparison of MLPNN and LSTM algorithms in terms of prediction error in longitude
Meanwhile, the prediction errors of position increments in latitude and longitude based on SVR
and LSTM algorithms also have been compared and the results are shown in Figure 14 and Figure
15. The blue line represents prediction error of LSTM algorithm, whereas the red line represents
prediction error of SVR algorithm. The value of mean absolute error (MAE) of predicting position
Remote Sens. 2019, 11, x FOR PEER REVIEW
433
434
435
436
437
438
439
440
15 of 22
increments in latitude using LSTM and SVR is 2.20089 and 5.13401, while standard deviation is
1.31846 and 2.88711, respectively. The value of mean absolute error of predicting position
increments in longitude are 1.99683 and 4.81162, whose standard deviation is 1.21521 and 3.13656,
respectively. It is obviously that the LSTM algorithm also has the lower prediction error than that of
SVR algorithm, which demonstrates that the proposed algorithm can make better performance of
improving the navigation solutions during GPS outages.
Figure 14. Comparison of SVR and LSTM algorithms in terms of prediction error in latitude
441
442
443
444
445
446
447
448
449
450
451
452
453
454
Figure 15. Comparison of SVR and LSTM algorithms in terms of prediction error in longitude
The performances of improving navigation results during GPS outages in different algorithms
are displayed in Figure 16 and Table 5. The red line represents the latitude and longitude position
error while GPS signal is assumed outages and the whole system only works on pure INS mode,
whereas the yellow line, blue line and green line represent the latitude and longitude error after
improved by MLP, SVR and LSTM algorithms, respectively.
Figure 16 shows position errors among latitude and longitude in different algorithms from
7200s to 8200s. Before 7685s, the GPS signal is still available and the whole system works under
loosely‐coupled mode, where the four conditions show the same navigation results. From 7685s to
8050s, the GPS signal is assumed outages and navigation performance varies in four different
methods. The red line indicates the navigation result of pure INS mode, while yellow line, blue line
and green line represents the performance of the MLP, SVR and LSTM algorithm, respectively. From
the comparison results, the proposed LSTM algorithm obviously outperforms the MLP and SVR
Remote Sens. 2019, 11, x FOR PEER REVIEW
455
456
457
458
459
460
461
462
463
464
465
466
467
algorithms, which is also much better than that in pure INS mode. The position accuracy of both
four methods gradually deteriorates with time during GPS signal outages. At 8050s, the position
error of pure INS, SVR, MLP and LSTM is 65m, 50m, 40m, 22m in latitude and 36m, 27m, 20m, 13m
in longitude, respectively. Figure 16 indicates that LSTM can improve the navigation error better
than those of other methods, and the performance of LSTM is nearly twice than that of MLP with
more stability. The MLP or SVR algorithm, which suffers from incapability of dealing with time
dependency between current and past vehicle dynamics, cannot be a proper model for aiding INS
during GPS signal outages. At the end of GPS outages, the position errors of LSTM algorithm is
successively about 30%, 40% and 50%, compared with those of the pure INS mode, SVR algorithm
and MLP algorithm respectively. At the moment of GPS signal is reacquired at 8050s, the whole
system turns to the loosely‐coupled mode right away and position error quickly converges both in
four methods.
Table 5.
Mean position error during GPS outages in different algorithms
Mean Position
error(m)
East
North
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
16 of 22
Pure INS
SVR
MLP
LSTM
22.41
10.11
16.47
8.12
13.13
6.13
7.18
3.14
Figure 16. Position error with different algorithms
When GPS signal is outages, the whole system turns to predicting mode. The measurements
from INS, which are specific force, velocity, yaw and angular rate, are send to well‐trained AI –based
model to attain predicting value of position increments. Then, accumulating these position
increments can get the pseudo‐GPS position acting as substitute of true GPS position, which will
subtract with position measurement from INS to constitute the part of input vector of Kalman filter.
Finally, kalman filter will correct the INS error to attain corrected position, attitude and velocity.
Figure 17 illustrates the cumulative distribution of position errors in three different algorithms.
It can be seen that about 80% of position error of LSTM algorithm is less than 10 m and the biggest
position error is also less than 20 m. The results in Figure 17 indicates that LSTM algorithm
outperforms than the those of other methods.
Figure 18 shows four trajectories: red line is position recorded by pure INS, blue line is true
trajectory which is the red path in Figure 8, green line, yellow line and blue line are the trajectories
improved by AI model based on LSTM, MLP and SVR, respectively. Figure 19 illustrates the
predicting value of yaw by different models. The black line indicates the true value of yaw, while red
line, blue line, yellow line and green line represent the predicting value of yaw using pure INS, SVR,
MLP and LSTM model, respectively.
Remote Sens. 2019, 11, x FOR PEER REVIEW
17 of 22
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
Starting from the lower left point, four lines represent four different navigation results, which
are illustrated above. In Figure 19, the vehicle was running on a straight road from 7600s to 7870s,
which is the red path in Figure 8, where the heading is almost not changing very much. Respectively,
in Figure 18, the position of different methods is very similar from 0 to 750m in the north direction,
which is corresponding to the straight road mentioned above. Then, the vehicle met the first bench at
about 7870s, while the prediction errors of yaw among pure INS, SVR, MLP and LSTM model came
to be divergence, which caused the divergence of position in Figure 18. This illustrates the reason
why position errors begin to be divergence after the first bench, which also explains the divergence
of position at the second bench.
Figure 18 and Figure 17 indicate that LSTM algorithm has a better performance than that of
MLP, SVR algorithms and whose navigation accuracy with AI‐based models performs better than
that of pure INS mode. Additionally, from Figure 18 and Figure 19, it can be seen that the less
predicting error of the value of yaw can attain a more accurate navigation results. In other words, the
predicting accuracy of attitude of vehicle plays an important role in aiding for INS when GPS signal
is outages.
501
502
Figure 17. Cumulative distribution of position errors in different algorithms
503
504
Figure 18. Position results with different algorithms
Remote Sens. 2019, 11, x FOR PEER REVIEW
18 of 22
505
506
507
508
509
510
511
512
513
514
515
516
517
Figure 19. Yaws results with different algorithms
In order to evaluate the generalization ability, another vehicle path including several 90 degrees’ turns is
adopted as testing path. Figure 20 indicates the field test trajectory, which includes two simulated GPS signal
outages to evaluate the proposed algorithm. The proposed LSTM model is trained during GPS signal is
available, and the trained model is then used to predict the position coordinates during GPS signal is lost. Test
results are summarized in Table 6.
Figure 21 illustrates the 1st GPS outage of during 90 s. The drift in the positional error using LSTM
methodology is less when compared to MLP, SVR and pure INS methodology. The RMSE is calculated for both
the methodologies and the results are shown in below table. Thus from the value of RMSE we observe that the
percentage improvement in the positional accuracy was 21.8% of MLP. The second outage of during 60 s is
shown in Figure 22. Here the LSTM methodology obtained a reduction in RMSE from 43.3 m to 35.4 m thereby
showing a 18.6% improvement in positional error when compared to MLP.
518
519
520
521
Figure 20. New test path for evaluate turns
Remote Sens. 2019, 11, x FOR PEER REVIEW
522
Table 6.
RMSE
19 of 22
Position errors for different models
LSTM
MLP
SVR
Pure INS
Outage1(m)
43.1
55.7
94.7
103.3
Outage2(m)
35.4
43.3
61.4
66.8
523
524
Figure 21. Position results with different Algorithms in outage1
525
526
Figure 22. Position results with different Algorithms in outage2
527
6. Conclusion
528
529
530
531
532
533
534
The purpose of this paper is to reduce the accumulating positioning error of a vehicular
navigation system using an improved fusion algorithm based on LSTM when GPS signal is blocked.
Comparing with MLP algorithm and SVR algorithm, respectively, the improved fusion algorithm
of LSTM can provide a more accurate and reliable way to develop a navigation scheme during the
GPS outages, such as tunnels or under viaducts, where no GPS satellite is available.
When GPS signal is available, the whole system is under the loose‐coupled model. Meanwhile,
information of yaw, specific force, velocity and angular rate are as input features to train the
Remote Sens. 2019, 11, x FOR PEER REVIEW
20 of 22
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
recurrent network based on LSTM cell. The improved fusion algorithm based on LSTM will try to
mimic the relationship between input features and position increment from GPS during the training
procedure when GPS signal is available. Once GPS signal is unavailable, the well‐trained LSTM
model will substitute GPS signal to supply the pseudo‐GPS position information for Kalman filter to
reduce the navigation error of INS.
In this research, the input features and output of LSTM module are carefully analyzed, where
the input features including specific force, angular rate, velocity and yaw can well represent the
vehicle dynamic information. The estimated output is selected as the position increments, which
can avoid bringing in additional errors caused by mixing the information of GPS and INS. The
existing static neural networks algorithms are weaker than LSTM algorithm when dealing with
sequential process in the input data containing attitude and velocity in INS, specific force and
angular rate in IMU. The proposed improved fusion algorithm based on LSTM is widely used for its
incredible property of dealing with sequential process to solve this problem, which can obtain a
more accurate and stable prediction of navigation results.
Reality data was used to evaluate the performance of LSTM, traditional MLP and SVR. From
the test results, the proposed improved fusion algorithm based on LSTM performs better than those
of MLP, SVR algorithm and pure INS during a period of GPS outages.
552
553
554
555
556
557
558
559
560
561
562
Author Contributions: This paper is a collaborative work by all the authors, Conceptualization, J.J. and W.F;
Data curation, W.F. J.J. and P.Y.; Formal analysis, J.J. and W.F.; Funding acquisition, J.J.; Investigation, W.F. J.J.
and P.Y.; Methodology, W.F. and J.J.; Project administration, J.J.; Software, W.F., P.Y.; Supervision, J.J.;
Validation, W.F. and P.Y.; Writing—original draft, W.F. and J.J.; Writing—review &editing, W.F., J.J., J.L. and
H.L.
Funding: This research was funded by the National Key Research and Development Program of
China (2018YFB0505200 and 2018YFB0505201), the Fundamental Research Funds for the Central
Universities(2042018kf0253).
Acknowledgments: Part of this work is supported by the Collaborative Innovation Center of
Geospatial Technology, Wuhan University and provided the experimental sites and testers.
Conflicts of Interest: The authors declare that they have no conflict of interest to disclose.
563
References
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Srinivas, P.; Anil, K. Overview of architecture for GPS‐INS integration. Control, Automation & Power
Engineering (RDCAPE) 2017 Recent Developments in. IEEE 2017, 433‐438. CrossRef
Groves, P. D., Wang, L. The four key challenges of advanced multi‐sensor navigation and positioning.
IEEE in 2014 IEEE/ION Position, Location and Navigation Symposium‐PLANS 2014(pp. 773‐792).
CrossRef
Gao, S. Multi‐sensor optimal data fusion for INS/GPS/SAR integrated navigation system. Aerospace
Science and Technology, 2009, 13.4‐5, 232‐237. CrossRef
Li, X., Chen, W. Multi‐sensor fusion methodology for enhanced land vehicle positioning. Information
Fusion, 2019. 46, 51‐62. CrossRef
Havyarimana, V., Hanyurwimfura, D. A novel hybrid approach based‐SRG model for vehicle position
prediction in multi‐GPS outage conditions. Information Fusion, 2018, 41, 1‐8. CrossRef
Abdolkarimi, E. S., Mosavi, M. R. Optimization of the low‐cost INS/GPS navigation system using ANFIS
for high speed vehicle application. In 2015 Signal Processing and Intelligent Systems Conference (SPIS) IEEE,
2015 December, (pp. 93‐98). CrossRef
Adusumilli, S.; Bhatt, D. A low‐cost INS/GPS integration methodology based on random forest
regression. Expert Systems with Applications, 2013, 40(11), 4653‐4659. CrossRef
Ye, W., Liu, Z. Enhanced Kalman Filter using Noisy Input Gaussian Process Regression for Bridging GPS
Outages in a POS, 2018. The Journal of Navigation, 71(3), 565‐584. CrossRef
Yang, Y. X. Adaptive navigation and kinematic positioning, 2rd ed.; Surveying and mapping press, Beijing,
China, 2008; pp: 18‐19, 978‐7‐5030‐4005‐4.
Du, S., Gao, Y. Inertial aided cycle slip detection and identification for integrated PPP GPS and
INS. Sensors, 2012, 12(11), 14344‐14362. CrossRef
Remote Sens. 2019, 11, x FOR PEER REVIEW
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
21 of 22
Xu, Z. Novel hybrid of LS‐SVM and Kalman filter for GPS/INS integration. The Journal of Navigation
2010, 63.2, 289‐299. CrossRef
Sharaf, R; Noureldin, A. Online INS/GPS integration with a radial basis function neural network. IEEE
Aerospace and Electronic Systems Magazine 2005 20.3, 8‐14. CrossRef
Sharaf, R; Noureldin, A. Sensor integration for satellite‐based vehicular navigation using neural networks.
IEEE transactions on neural networks 2007, 18.2, 589‐594. CrossRef
El‐Sheimy, N; Chiang, K‐W; Noureldin, A. The utilization of artificial neural networks for multisensor
system integration in navigation and positioning instruments. IEEE Transactions on instrumentation and
measurement 2006, 55.5, 1606‐1615. CrossRef
Jaradat, M. A. K., & Abdel‐Hafez, M. F. Non‐linear autoregressive delay‐dependent INS/GPS navigation
system using neural networks. IEEE Sensors Journal, 2017, 17(4), 1105‐1115. CrossRef
Li, J., Song, N. Improving positioning accuracy of vehicular navigation system during GPS outages
utilizing ensemble learning algorithm. Information Fusion, 2017, 35, 1‐10. CrossRef
Adusumilli, S., Bhatt, D. A novel hybrid approach utilizing principal component regression and random
forest regression to bridge the period of GPS outages. Neurocomputing, 2015 166, 185‐192. CrossRef
Tan, X. GA‐SVR and pseudo‐position‐aided GPS/INS integration during GPS outage. The Journal of
Navigation 2015, 68.4, 678‐696. CrossRef
Indelman, V., Williams, S., Kaess, M., & Dellaert, F. Information fusion in navigation systems via factor
graph based incremental smoothing. Robotics and Autonomous Systems, 2013,61(8), 721‐738. CrossRef
Malleswaran, M; V. Vaidehi; N. Sivasankari. A novel approach to the integration of GPS and INS using
recurrent neural networks with evolutionary optimization techniques. Aerospace Science and Technology
2014, 32.1, 169‐179. CrossRef
Noureldin, A; El‐Shafie, A; Bayoumi, M. GPS/INS integration utilizing dynamic neural networks for
vehicular navigation. Information Fusion 2011, 12(1), 48‐57. CrossRef
Hasan, A. M., Samsudin, K. GPS/INS integration based on dynamic ANFIS network. International Journal of
Control and Automation, 2012 5(3), 1‐21.
Hochreiter, S; Jürgen S. Long short‐term memory. Neural computation 1997, 9.8, 1735‐1780. CrossRef
Wu, Y., Schuster, M., Chen, Z. Google's neural machine translation system: Bridging the gap between
human and machine translation. arXiv preprint, 2016, arXiv:1609.08144.
Ma, X. Long short‐term memory neural network for traffic speed prediction using remote microwave
sensor data. Transportation Research Part C: Emerging Technologies 2015, 54, 187‐197. CrossRef
Duan, Y., Lv, Y., & Wang, F. Y. Travel time prediction with LSTM neural network. In 2016 IEEE 19th
International Conference on Intelligent Transportation Systems (ITSC) IEEE, 2016, November, (pp. 1053‐1058).
CrossRef
Zhang, Y. Hybrid algorithm based on MDF‐CKF and RF for GPS/INS system during GPS outages 2018,
IEEE Access 6, 35343‐35354. CrossRef
Trawny, N., Mourikis, A. I. Vision‐aided inertial navigation for pin‐point landing using observations of
mapped landmarks. Journal of Field Robotics, 2007, 24(5), 357‐378. CrossRef
Wei, M., and K. P. Schwarz. A Strapdown Inertial Algorithm Using an Earth‐Fixed Cartesian Frame.
Navigation 1990, 37.2, 153‐167. CrossRef
Groves, P. D. Principles of GNSS, inertial, and multisensor integrated navigation systems, 2013, Artech house.
Simon J; Jeffrey K Uhlmann. New extension of the Kalman filter to nonlinear systems, AeroSense
International Society for Optics and Photonics 1997. CrossRef
Fernando, T. Soft hardwired attention: An lstm framework for human trajectory prediction and abnormal
event detection. Neural networks 2018, 108, 466‐478. CrossRef
Zhao, Z., Chen, W. LSTM network: a deep learning approach for short‐term traffic forecast. IET Intelligent
Transport Systems 2017, 11(2), 68‐75. CrossRef
Saleh, K; Mohammed H; Saied, N. Intent prediction of vulnerable road users from motion trajectories
using stacked LSTM network. Intelligent Transportation Systems (ITSC), 2017 IEEE 20th International
Conference on. IEEE, 2017. CrossRef
Xingjian, S. H. I. Convolutional LSTM network: A machine learning approach for precipitation
nowcasting. Advances in neural information processing systems 2015.
Bhatt, D. A novel hybrid fusion algorithm to bridge the period of GPS outages using low‐cost INS. Expert
Systems with Applications 2014, 41.5, 2166‐2173. CrossRef
Remote Sens. 2019, 11, x FOR PEER REVIEW
640
641
642
643
644
645
646
647
648
649
650
37.
38.
39.
40.
41.
42.
Abdel‐Hamid, W., Noureldin, A. Adaptive fuzzy prediction of low‐cost inertial‐based positioning
errors. IEEE Transactions on Fuzzy Systems, 2007, 15(3), 519‐529. CrossRef
Farrell, J., & Barth, M. The global positioning system and inertial navigation. 1999 (Vol. 61). New York:
Mcgraw‐hill.
Goshen‐Meskin, D., Bar‐Itzhack, I. Y. Unified approach to inertial navigation system error
modeling. Journal of Guidance, Control, and Dynamics, (1992), 15(3), 648‐653. CrossRef
Liu, W., Wen, Y. Large‐margin softmax loss for convolutional neural networks. international conference
on machine learning, 2016, 507‐516.
Abadi, M; Paul B. Tensorflow: a system for large‐scale machine learning. In OSDI vol. 16, pp. 265‐283.
2016. CrossRef
Diederik K; Jimmy B. Adam: A method for stochastic optimization. In ICLR, 2014.
651
652
653
654
22 of 22
© 2019 by the authors. Submitted for possible open access publication under the terms
and conditions of the Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/).