Wireless Netw DOI 10.1007/s11276-016-1265-4
Location prediction algorithm for a nonlinear vehicular movement in VANET using extended Kalman filter Raj K. Jaiswal1 • C. D. Jaidhar1
Springer Science+Business Media New York 2016
Abstract Vehicular ad-hoc network (VANET) is an essential component of the intelligent transportation system, that facilitates the road transportation by giving a prior alert on traffic condition, collision detection warning, automatic parking and cruise control using vehicle to vehicle (V2V) and vehicle to roadside unit (V2R) communication. The accuracy of location prediction of the vehicle is a prime concern in VANET which enhances the application performance such as automatic parking, cooperative driving, routing etc. to give some examples. Generally, in a developed country, vehicle speed varies between 0 and 60 km/h in a city due to traffic rules, driving skills and traffic density. Likewise, the movement of the vehicle with steady speed is highly impractical. Subsequently, the relationship between time and speed to reach the destination is nonlinear. With reference to the previous work on location prediction in VANET, nonlinear movement of the vehicle was not considered. Thus, a location prediction algorithm should be designed by considering nonlinear movement. This paper proposes a location prediction algorithm for a nonlinear vehicular movement using extended Kalman filter (EKF). EKF is more appropriate contrasted with the Kalman filter (KF), as it is designed to work with the nonlinear system. The proposed prediction algorithm performance is measured with the real and model based mobility traces for the city and highway scenarios. Also, EKF based prediction performance is & Raj K. Jaiswal
[email protected] C. D. Jaidhar
[email protected] 1
Department of Information Technology, National Institute of Technology Karnataka, Surathkal, India
compared with KF based prediction on average Euclidean distance error (AEDE), distance error (DE), root mean square error (RMSE) and velocity error (VE). Keywords Extended Kalman filter Location Nonlinear movement Prediction VANET
1 Introduction Wireless communication is the cornerstone for many emerging networks such as M-healthcare social network [47, 48], software defined wireless networks [24, 39, 40], mobile ad-hoc network (MANET) [43, 46], wireless sensor network (WSN) [34, 41, 42], delay tolerant networks [10, 45, 49], cognitive radio network [4, 18, 44], green communications [36], mobile crowdsourcing [12] and VANET. Among these networks, VANET is designed and developed to ease the road transportation by reducing the accidents and transportation cost using ITS. The goal of ITS is to give safety and infotainment services when users drive the vehicle. Inter-vehicular communication is an important component of ITS, it initiates the formation of vehicle to vehicle (V2V) and vehicle to roadside unit (V2R) communication network [6]. In V2V communication, the vehicle communicates with another vehicle either directly or using intermediate vehicles. In the latter one, vehicle communicates directly to the fixed roadside unit (RSU). VANET render its services to ITS to accomplish the coveted goals by utilizing communication between V2V and V2R. VANET uses Dedicated Short Range Communication standard viz. IEEE 802.11p to set up the connection between nodes. IEEE 802.11a standard is modified to meet IEEE 802.11p communication standard constraints, particularly to decrease overhead operations [1].
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VANET inherits features from MANET such as multihop routing [26, 28] and self-configuring node. The following points illustrate typical characteristics of VANET [33]: •
•
•
•
Intermittent Network Connection: The frequent cut-off in the communication link increases end to end delay and packet loss. This circumstance emerges when the node speed is high. Thus, the life of a communication link in VANET hardly exists for 8–10 s. For instance, if the vehicle radio range is up to 250 m and speed is 80–100 km/h. Dynamism of Topology: Due to node movement, the topology of the network changes frequently. Thus, for a node, handling of topology change is cumbersome. For instance, a node enters into proximity of another node and leaves it quickly, incurs routing overhead, delay and convergence time. Hard Delay: VANET has a strict delay constraint to deliver safety and traffic information that alert the driver quickly so that accident is prevented. For example, heavily congested road information should spread quickly to the drivers so that they can make use of a substitute road to reach the destination. Mobility: In VANET, vehicle mobility can be anticipated as the vehicle movement is confined in one direction by the road layout and traffic signs. In any case, exact estimation of the vehicle mobility is hard, since movement of the vehicle relies on speed and driver conduct.
Global navigation satellite system, such as global positioning system (GPS) is being used in VANET to retrieve the location information of the vehicle which consists of latitude, longitude and velocity of the vehicle running on Earth’s surface. In VANET, this information is used in emergency and safety applications to improve the efficiency such as collision warning, emergency braking, automatic parking and so on. In addition, location information can also be used in service based applications. The accuracy of the GPS receiver is a noteworthy issue in a navigation system. As the accuracy of the receiver gets affected by trees, lofty buildings, bridge and tunnels in the city. In addition, other environmental factors such as line of sight, signal obstruction, fading and interference also affects the accuracy [20]. In general, GPS provides the location of an object with an accuracy of 5–30 m distance. In the literature, different prediction algorithms are proposed to compensate the location error such as dead reckoning, cellular and video/audio imaging [29]. In VANET, location accuracy is the most critical information for some applications. The emergency and safety applications need a high level of precision in location prediction, unlike service based applications. The
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precision of location prediction required for a few applications is outlined in Table 1. This study presents a location prediction algorithm for a nonlinear vehicular movement using EKF. The significant contributions of this work are: • • •
• •
So far, no one has proposed location prediction algorithm in VANET using EKF as per our knowledge. It analyzes the nonlinear vehicular movement, particularly in the city. The proposed algorithm performance is computed for the real and model based traces for both city and highway scenarios. The performance of the location prediction using EKF is compared with the KF based prediction. The outcomes show that the EKF based prediction gives high accuracy in location estimation as compared to the KF based prediction.
In this paper, vehicle and node, prediction and estimation, location and position are used interchangeably and resemble the same meaning. The remaining part of the paper is organized as follows: Sect. 2 discusses the previous research with reference to the location prediction, Sect. 3 explains about system model used in experiments. Section 4 briefs about KF and EKF. The location prediction algorithm is explained in Sect. 5. Sections 6 and 7 discusses the implementation and evaluation settings and results of the prediction algorithm respectively. The comparison of proposed algorithm is shown in Sect. 8 and followed by conclusion in Sect. 9.
2 Related literature In the literature reviews, various location prediction algorithms are proposed which focuses more on a WSN or MANET. Only limited work has been done towards the Table 1 Required precision of localization in VANET applications Application
Low
Medium
High
Emergency and safety Adaptive cruise control
U
Cooperative intersection safety
U
Platooning
U
Collision warning
U
Blind crossing
U
Vision enhancement
U
Services Routing
U
Data dissemination
U
Accident detection
U
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location prediction in VANET. Following are the gist of the research conducted in location prediction irrespective of the area. Hu et al. [16] proposed a prediction algorithm based on adaptive KF using fading memory and variance estimation. They found that their algorithm performs better compared to KF based prediction. Xiao et al. [38] proposed a location prediction technique using the grey theory for a nonlinear system. Their prediction results are better compared to a linear system. Li et al. [22] used auto-regression method to predict the mobility of a MANET node. Their model predicts the location of the node by considering a linear movement and mobility is predicted only when the topology is changed compared to the trajectory of the node. Liu et al. [25] proposed a distributed location estimate algorithm which improves the accuracy of location prediction using cooperative inter-vehicle distance. They designed the algorithm on the basis of GPS pseudo-range information which measures the distance between satellite and receiver. This algorithm performs better only when the signal strength is good. However, location prediction gets affected as GPS receiver could not receive the signal in the city due to trees, tunnel, buildings, fading and environmental loss [21]. Drawil et al. [9] proposed inter-vehicle communication assisted localization (IVCAL) algorithm which uses KF to predict the vehicle location in VANET. IVCAL minimizes the impact of multi-path signal effect. Khan et al. [19] evaluated the positioning and tracking performance of EKF in WSN. In their results, they found that the EKF is more suitable for localization of a node where distance measurement is affected by noise. Chen et al. [8] proposed a modified EKF to predict the location of a node in WSN. Their algorithm predicts the location when communication is affected by packet drop and insufficient coverage. Rad et al. [30] developed a cooperative localization algorithm for mobile WSN using KF. Their algorithm exploits location of the anchor node to linearize the nonlinear distance measured for the location of an unknown node. Mo et al. [27] developed a mobility assisted location management protocol (MALM) for VANET which provides the location services to the vehicles. MALM practices the historical data and KF to compute the vehicle location. Theodoros et al. [3] proposed a short-memory adaptive location predictor for mobile applications which predicts the location in the absence of historical mobility information. They used linear regression model for prediction and fuzzy controller to achieve adaptation capability. Alam et al. [2] proposed cooperative positioning which fuses data from different sensor sources to improve the performance of relative positioning in VANET. Sun et al. [35] proposed location prediction model for VANET where the vehicle does not have GPS equipment. Prediction of the location
takes place using V2R communication and dead reckoning. The accuracy of the model does not fit in many critical VANET applications as it achieves minimum 8.79-m location error in prediction. Fulop et al. [13] proposed a mobility model for the cellular network using Markovian chain. It uses the movement history of the node to maximize the accuracy as compared to other prediction models. Arafat et al. [32] used Dirichlet-multinomial model under the Bayesian estimation framework to predict the future location of a non-cooperative vehicle for VANET. Feng et al. [11] proposed a location prediction algorithm in VANET using KF for a linear movement. They compared it with artificial neural network model and found that the results are better with KF. Based on the afore-discussed literature with reference to location prediction, it is found that the greater part of the work is completed with reference to WSN or MANET. As such, the research conducted for the location prediction with reference to VANET is limited. An efficient location prediction algorithm can enhance the performance of VANET applications where precision of the location prediction is a prime variable. For instance, automatic parking and collision warning system require precise location prediction. Focusing beyond these applications, the performance of the position based routing such as packet delivery ratio and data dissemination can also be improved using future location prediction. As position based routing protocol performance also get influenced by the continuous change in location. For instance, Table 2 represents the GPS locations retrieved from OpenStreetMap [14] for Fig. 1 which covers a region of around 500 500 m2 . It is observed that the location changes as often as possible for each 200–300 m circumference. These distances could be covered rapidly if the vehicle keeps running with high-speed [31]. It is also observed that the research conducted so far towards the location prediction in VANET is done only with the linear system. The movement of the vehicle either in the city or on the highway is nonlinear as speed disruption is frequent in the city limit due to traffic, speed limit and traffic signal. For instance, a vehicle finishes 10 Table 2 Traced GPS coordinates Place
Latitude
Longitude
Post Office
13.0088615
74.7933738
Canara Bank
13.0085505
74.7941301
State Bank of India
13.0089373
74.7940711
8th Block Hostel
13.0074293
74.7941167
Mega Hostel-Tower 1
13.0077351
74.7948382
Mega Hostel-Tower 2
13.0068832
74.7952084
Mega Hostel-Tower 3
13.0063213
74.7944279
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by considering an appropriate vehicular model for VANET.
3 System model
Fig. 1 National Institute of Technology hostel area
km distance in 26 min with the distinctive velocity for a particular measure of time as follows: 1. 2. 3. 4.
80 60 40 30
km/h km/h km/h km/h
for for for for
10 min. 2 min. 4 min. 11 min.
Hence, the relationship between speed and time of the vehicle in the city is nonlinear as showed in Fig. 2. In the review of literature, previous prediction algorithms did not consider the nonlinear movement of the vehicle using EKF. EKF is designed on the basis of KF to work with the nonlinear system as explained in Sect. 4. Henceforth, it is proposed to design a location prediction algorithm using EKF for a nonlinear vehicular movement
All the vehicles are assumed to be equipped with an onboard unit and omnidirectional antenna. A vehicle communicates to another vehicle or RSU using IEEE 802.11p standard. It is also assumed that GPS and inertial navigation system are in place to measure the latitude, longitude, velocity and acceleration, orientation, steering angle respectively. For the EKF the vehicle kinematics are defined considering Ackerman steering as shown in Fig. 3, wherein U1 and U2 are the steering angle of the outer tire and inner tire respectively. The orientation of the vehicle is determined by angle h [23]. In this work, the steering angle is considered to be zero as the vehicle runs either in the city or on a highway. Thereon steering angle U1 and U2 get changed only at the turning point or during the lane change. The state X of the vehicle at time t is defined by six parameters as [x, y, vx , vy , ax , ay ], where x and y represent the latitude and longitude of the vehicle. Whereas, vx , vy , ax and ay represent the x and y component of the velocity and acceleration respectively. The equation of motion of physics as in Eq. (1), measures the changes in the state of the vehicle. In Eq. (1), initial state of the vehicle is denoted by x0 while v and a represent the velocity and acceleration of the vehicle respectively. The time interval for the change in velocity is denoted by Dt.
Speed (Kmph)
100
80
60
40
20
0
0
5
10
15
20
25
30
Time (Min.)
Fig. 2 Relationship between speed and time
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Fig. 3 Vehicle kinematics
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x ¼ x0 þ v:Dt þ ðaDt2 Þ=2
Hence, the state model X of the vehicle is defined as: 3 2 x0 þ vx Dt 2 3 x 6 y0 þ vy Dt 7 7 6y7 6 7 6 7 6 7 6 v þ a Dt x 6 7 6 x0 7 6 vx 7 6 7 ¼ 6 vy0 þ ay Dt 7 ð2Þ X¼6 7 6v 7 6 7 6 y7 6 Dv 7 x 6 7 6 ax þ 7 4 ax 5 6 0 Dt 7 5 4 Dvy ay ay 0 þ Dt Dvx Dvy Here, and in (2) are the change in x and y comDt Dt ponents of the acceleration respectively.
4 Description of KF and EKF The process model of a system is designed to estimate the state of the system with a minimal set of information. Whereas, the measurement model describes the state of the system using measurement device. The process and measurement model equations for a linear system used in KF for state estimation are defined as in Eqs. (3) and (4) respectively. ^ x^ t ¼ At x t1 þ Bt ut þ wt
ð3Þ
x^ t1
ð4Þ
zt ¼ H t
Pt ¼ ð I K t Ht Þ P t
ð1Þ
þ vt
KF estimates and corrects a linear process recursively using feedback control mechanism. It estimates the state of the system in regular intervals to obtain the feedback measurement. The KF equations are grouped into Time Update and Measurement Update equations. The steps involved in state estimation of the system are described as follows: Time update (prediction)
ð9Þ
In Eq. (5), the state transition matrix At is obtained from the previous state x^ t1 . and P Whereas, x^ t t in Eqs. (5) and (6) are the prior (predicted) estimate of the state X and process error covariance respectively. The remaining description of each symbol used in KF is given in Table 3. Kalman gain Kt in Eq. (7) minimizes the error between predicted and measured values. Hence, Kt is the most critical factor in deciding the accuracy of the filter. The measured error covariance Pt is obtained using estimated error covariance matrix P t and identity matrix I showed in Eq. (9). x^t in Eq. (8), is the estimated state of the system by KF. The estimated state x^t and error covariance Pt in cor rection steps are updated with x^ t and Pt of prediction steps respectively for the computation of the next state [37]. 4.1 EKF As explained in Sect. 4, KF is designed to work with the linear system. It applies to a vehicular system in which the movement is linear, which is not practical in real world scenarios as explained in Sect. 2.
Table 3 Description of KF symbols Symbol
Description
At
State transition matrix at t
ATt
Transpose of At at t
Bt
Model matrix, steering angleand acceleration change
Ht
Measurement matrix at t
HtT
Transpose of Ht
Location prediction
I
Identity matrix
^ x^ t ¼ At x t1 þ Bt ut þ wt
Kt P t
Kalman gain at t Error covariance at t
Pt1
Error covariance at t 1
Pt Qt
Estimated Error covariance by filter at t Process noise covariance
Rt
Measurement noise covariance
ut
Commanded input at t
vt
Measurement noise at t
wt x^ t
Process noise at t
x^ t1
Location at t 1
x^t
Location predicted by filter at t
zt
Measured location at t
ð5Þ
Error covariance T P t ¼ At Pt1 At þ Qt
ð6Þ
Measurement update (correct) Kalman gain T T K t ¼ P t Ht ð Ht Pt Ht þ Rt Þ
1
ð7Þ
Prediction on measurement zt (update) x^t ¼
x^ t
þ K t ðzt
Ht x^ t Þ
ð8Þ
Location predicted at t
Error covariance (update)
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EKF is designed to work with the nonlinear system which can be used in location prediction algorithm in VANET, it is designed on the basis of KF for a nonlinear system. EKF linearized the nonlinear system using partial differentiation which yields the Jacobian matrix to be used in the computation to estimate the current state of the system. The process and measurement models of the nonlinear system used with EKF are defined in Eqs. (10) and (11): x^ x t ¼ f ð^ t1 ; ut ; wt Þ
ð10Þ
hð^ x t1 ; vt Þ
ð11Þ
zt ¼
The computational steps involved in EKF for location prediction in VANET are explained as: Time update (prediction) Location prediction x^ x t ¼ f ð^ t1 ; ut ; wt Þ
ð12Þ
Error covariance T T P t ¼ Ft Pt1 Ft þ Wt Qt Wt
ð13Þ
Measurement update (correct) Kalman gain T T T K t ¼ P t Ht ðHt Pt Ht þ Vt Rt Vt Þ
1
Prediction on measurement zt (update) x^t ¼ x^ x t þ K t zt hð^ t1 ; vt Þ
ð14Þ
ð15Þ
Error covariance (update) Pt ¼ ð I K t Ht Þ P t
ð16Þ
As EKF is an extended form of KF most of the terms are similar to KF as explained in Sect. 4 except Ft , Wt , Ht and Vt in Eqs. (13) and (14) respectively which are the Jacobian matrix defined as: Ft ¼ Wt ¼
df ð^ x t1 ; 0; 0Þ dx df ð^ x t1 ; 0; 0Þ dw
ð17Þ ð18Þ
Ht ¼
dhð^ x t1 ; vt Þ dx
ð19Þ
Vt ¼
dhð^ x t1 ; 0; 0Þ dv
ð20Þ
Whereas, FtT , Ht T , Vt T and Wt T are the transpose of Ft , Wt , Ht and Vt . Jacobian matrix Ft is computed on partial ^ differentiation of f ð^ x t1 ; ut ; wt Þ with respect to x t1 as shown in Eq. (21).
123
2
df1 6 dx1 6 6 df 6 2 6 dx df 6 6 1 ¼6 : dx 6 6 : 6 6 : 6 4 df6 dx1
df1 dx2 df2 dx2 ::
:
::
: ::
:: ::
::
::
::
:: df6 dx2
::
::
:
::
3 df1 dx6 7 7 df2 7 7 ::: 7 dx6 7 :: :: 7 7 7 :: :: 7 7 :: :: 7 7 df6 5 ::: dx6 :::
ð21Þ
Where x ¼ x^ t1 , Similarly, Jacobian matrix of Wt , Ht and Vt are obtained on partial differentiation of f ð^ x x t1 ; ut ; wt Þ with respect to wt and hð^ t1 ; vt Þ with respect to x^t1 and vt respectively. In this model, control input ut and process noise wt are assumed to be zero. Hence, process model in Eq. (12) becomes f ð^ x whereas measurement model t1 ; 0; 0Þ hð^ xt1 ; vt Þ remains the same.
5 Location prediction algorithm To predict the location of a vehicle in VANET considering nonlinear movement, EKF is used in the prediction algorithm as mentioned in Fig. 4. It is implemented in the system model as described in Sect. 3. To initialize the filter, Ft , Wt , Ht , Vt , Pt , Qt and Rt parameters are initialized as follows: Jacobian matrix Ft is obtained on differentiating par^ tially to the function f ð^ x t1 ; 0; 0Þ with respect to x t1 which is state vector measured at time t 1. Dt is the sampling interval and assumed to be one. The initial value of Ft is defined as: 3 2 1 0 Dt 0 0 0 6 0 1 0 Dt 0 0 7 7 6 7 6 60 0 1 0 0 07 7 6 ð22Þ Ft ¼ 6 7 60 0 0 1 0 07 7 6 40 0 0 0 1 05 0 0
0
0
0
1
The initial value of Jacobian matrix Wt is obtained on a partial differentiation of function f ð^ x t1 ; 0; 0Þ with respect to wt since the process noise wt is assumed to be zero. Hence, the initial value of Wt is defined as: 3 2 0 0 0 0 0 0 60 0 0 0 0 07 7 6 7 6 60 0 0 0 0 07 7 6 ð23Þ Wt ¼ 6 7 60 0 0 0 0 07 7 6 40 0 0 0 0 05 0
0
0 0
0
0
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Vt , R t Kalman Gain Kt = −1 Pt− Ht T (Ht Pt− Ht T + Vt Rt Vt T ) zt
O ne step ahead prediction = f (ˆ x− xˆ− t t−1 , 0, 0) Process error covariance = Ft Pt−1 FtT P− t
xˆ− t−1 , Ft , Pt−1
Update estimate with measurement x− xˆt = xˆ− t + Kt zt − h(ˆ t−1 , vt )
Update the error covariance Pt = (I − Kt Ht )Pt−
Fig. 4 Prediction algorithm using extended Kalman filter
Similarly, the initial value of Jacobian matrix Ht and Vt are obtained on differentiating partially to hð^ x t1 ; vt Þ with respect to x^t1 and vt as mentioned in matrices (24) and (25) as: 3 2 1 0 0 0 0 0 60 1 0 0 0 07 7 6 7 6 60 0 1 0 0 07 7 6 ð24Þ Ht ¼ 6 7 60 0 0 1 0 07 7 6 40 0 0 0 1 05 0
0 0
0
0
1
and
2
eðx; xÞ 6 eðy; xÞ 6 6 6 eðvx ; xÞ Pt ¼ 6 6 eðv ; xÞ 6 y 6 4 eðax ; xÞ eðay ; xÞ
2
0 1
0 0
0 0
0 0
1 0
0 1
0
0
0
3 0 0 0 07 7 7 0 07 7 0 07 7 7 1 05
0 0
0
0
0 1
1 60 6 6 60 Vt ¼ 6 60 6 6 40
ð25Þ
The error covariance matrix Pt is obtained using matrix (26) in which non-diagonal elements represent the correlation term which is assumed to be independent and does not affect the matrix value. The diagonal elements of the matrix correspond to the covariance in the states.
eðx; yÞ eðy; yÞ
eðx; vx Þ eðy; vx Þ
eðx; vy Þ eðy; vy Þ
eðx; ax Þ eðy; ax Þ
eðvx ; yÞ
eðvx ; vx Þ
eðvx ; vy Þ
eðvx ; ax Þ
eðvy ; yÞ eðax ; yÞ
eðvy ; vx Þ eðvy ; vy Þ eðvy ; ax Þ eðax ; vx Þ eðax ; vy Þ eðax ; ax Þ
3 eðx; ay Þ eðy; ay Þ 7 7 7 eðvx ; ay Þ 7 7 eðvy ; ay Þ 7 7 7 eðax ; ay Þ 5
eðay ; yÞ
eðay ; vx Þ eðay ; vy Þ eðay ; ax Þ
eðay ; ay Þ
ð26Þ
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The initial state estimation of X is fixed at x^0 ¼ 0 and initial error covariance P0 is fixed at a large value to minimize the error gap. Thus, the initial error covariance matrix P0 is defined as: 3 2 1000 0 0 0 0 0 6 0 1000 0 0 0 0 7 7 6 7 6 6 0 0 1000 0 0 0 7 7 6 P0 ¼ 6 0 0 1000 0 0 7 7 6 0 7 6 4 0 0 0 0 1000 0 5 0
0
0
0
0
6.1 GPS traces/dataset The dataset used in our experiments is classified into the real and model based. Further, these mobility traces are categorized into the city and highway scenarios, as the city road layouts are different compared to the highway. In addition, the speed of the vehicle on the highway used to be consistent for a long time, unlike the city. •
1000 ð27Þ
In the literature reviews, various methods have been proposed to select the appropriate values of Qt and Rt . However, in this study Qt and Rt values are taken manually in an ad-hoc manner [5]. The measurement noise Rt is the device error and defined by the manufacturer. Hence, the measurement noise Rt is defined as in (28) where Rx and Ry are the change in latitude and longitude respectively. Rx 0 Rt ¼ ð28Þ 0 Ry The initial 1 Rt ¼ 0
value of measurement noise Rt is: 0 1
•
ð29Þ
The process noise Qt is the observed error in the computational process which is difficult to measure. Thus, the initial value of Qt is kept minimum to make the process error free. The initial value of Qt is defined as: 3 2 0:01 0 0 0 0 0 6 0 0:01 0 0 0 0 7 7 6 7 6 6 0 0 0:01 0 0 0 7 7 6 ð30Þ Qt ¼ 6 0 0 0:01 0 0 7 7 6 0 7 6 4 0 0 0 0 0:01 0 5
Real GPS Traces for the city and highway scenarios are retrieved from the OpenStreetMap. Flen city from Sweden is considered in the experiments. As retrieved, traces are just similar to the city like road structure. For the highway scenario, traces are retrieved haphazardly from OpenStreetMap which describes the road structure almost the same as the highway. In both the traces speed changes based on the vehicle movement. Model Based Traces are obtained for the city and highway scenarios using VANETMOBISIM simulator [15]. The Intelligent Driving Model is chosen as it supports for the acceleration/deceleration, driver behavior, traffic signal and politeness factor which make the vehicular mobility traces equal to the real-time vehicular traffic. For the highway condition, simulation is conducted for a short time to get the traces equal to the highway scenario. Due to the lack of support of the simulator for the highway. The parameters considered for trace generation for the city and highway scenarios are mentioned in Table 4. However, junction points and traffic lights are not considered in the highway scenario.
Table 4 Parameters for model based trace [17] Description
Values
Simulator
VANETMOBISIM
X dim. (m)
1000
Y dim. (m)
1000
No. of traffic lights1
05
Traffic light duration1
60 s
No. of lanes Min. speed (m/s)
2 0.5
Max. speed (m/s)
35
6 Implementation and evaluation settings
Politeness factor
0.2,0.5,0.8
Maximum acceleration
0.9 (m/s2 )
The prediction algorithm using EKF as shown in Fig. 4 and KF are implemented in C language on Intel Core i73770 CPU @ 3.40GHz on windows 7 machine. Both EKF and KF filters iterate the state estimation process for 25 times to reduce the error to get the best state estimation.
Maximum deceleration
0.6 (m/s2 )
0
0
0
0
0
0:01
The value of Qt and Rt can be changed to tune the filter to get the best estimation of the vehicle state.
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Min. congestion distance
2m
Safe headway time
2s
Length of vehicle
5m
1
Not used in highway scenario
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6.2 Parameters for performance measurement
7 Results and discussion
The efficiency of the prediction algorithm is measured on RMSE, DE, AEDE and VE. These parameters are more appropriate to measure the performance of a prediction algorithm.
The EKF based prediction results are computed for each scenario viz. real and model based traces as discussed in Sect. 7.1 and compared with KF in Sect. 8. For each subcategory of model based trace as discussed in Sect. 6.1, it is observed that the traces generated from the VANETMOBISIM do not resemble the real GPS coordinates in any form such as angle or Universal Transverse Mercator. The former one uses two-dimensional Cartesian coordinate system to depict the location of an object on Earth’s surface. The coordinates of the location on Earth’s surface measured by the GPS system varies between 180 and 180 with respect to the latitude and longitude. However, trace generated from VANETMOBISIM takes the coordinates based on defined simulation area. For instance, if the simulation area is 1000 1000 m2 then, the coordinates point varies between 0 to 1000. Based on these observations, the measurement unit for the location accuracy in meter, distance error in the meter and velocity error in km/ h are inappropriate for the traces generated from the VANETMOBISIM. Hence, the results are measured in decimal points rather than their actual measurement units.
6.2.1 RMSE It computes the error between measured and predicted location. The RMSE for a prediction algorithm with respect to the measured value is defined as the square root of the mean squared error. Equation (31) calculates the RMSE of the prediction algorithm [7]. RMSE ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 1=N ðx x^t Þ2 t¼1 t
ð31Þ
Where xt and x^t are the measured and predicted location of the vehicle respectively. N is the total number of the predictions made. 6.2.2 DE
7.1 Location prediction with extended Kalman filter
It measures the distance gap between measured and predicted location. It is computed using Euclidean distance as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð32Þ derror ¼ ðxt x^t Þ2 þ ðyt y^t Þ2
7.1.1 Location prediction based on model traces
xt , yt and x^t , y^t in Eq. (32), are the measured and predicted location of the vehicle respectively. 6.2.3 AEDE
Figure 5 shows the prediction of the vehicle location in model based city. Intuitively, it is evident that location prediction is almost equal to the measured location with a few exceptions at some points. Though the results are in favour of EKF, however, similar location precision is difficult to get in real time scenario as the difference between two locations is huge compared to real traces. In real traces,
AEDE measures the average distance error between measured and predicted location using Euclidean distance. Equation (33) used to calculate the AEDE for each scenario is explained as:
850 800
ð33Þ
i¼1
6.2.4 VE
Longitude
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It measures the difference between measured and predicted velocity of the vehicle as follows: VE ¼ Vm Vp ð34Þ In Eq. (34), Vm and Vp are the measured and predicted velocity respectively.
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Fig. 8 Vehicle mobility prediction for highway scenario based on real trace using EKF
applications which demand the highest accuracy in location prediction such as cooperative driving and collision warning system. Hence, accuracy in location prediction with real GPS traces is practical and acceptable with EKF whereas EKF based location prediction with model based traces is not feasible in real world scenarios as it coordinates system.
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8 Performance comparison with KF based prediction
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Fig. 7 Vehicle mobility prediction for city scenario based on real trace using EKF
location changes up to 10 decimal points which is more evident while comparing the latitude and longitude coordinates in Figs. 5, 6 with Figs. 7 and 8 respectively. The prediction algorithm performance is slightly less on the highway compared to the city. However, both the performances are in favour of EKF as location prediction is close to the measured location. 7.1.2 Location prediction based on real traces The accuracy of location prediction with real GPS traces for the city and highway scenarios are less compared to the model based traces as shown in Figs. 7 and 8. However, predicted location with the real traces is close to the measured location as the difference between these two locations is changed up to eight decimal points. This difference is taken as the benchmark to decide the accuracy of prediction. Hence, based on these observations EKF based prediction is an appropriate technique to be used with those
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The performance of proposed EKF based prediction is compared with the KF based prediction on DE, RMSE, AEDE and VE parameters. For both the filters, the process model, measurement model and computational steps are same as explained in Sect. 4. EKF Distance Error
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Fig. 9 Distance error (DE) in city scenario based on model trace (EKF)
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8.1 DE based comparison The following subsections compare the DE between EKF and KF based prediction.
0.27 and 0.177 are the maximum DE in the city and highway scenarios as shown in Figs. 10 and 11. Overall, EKF has less DE compared to KF in both the scenarios. 8.1.2 DE with real trace
8.1.1 DE with model trace From, Figs. 9 and 11, it is observed that EKF has 0.10 and 0.175 maximum DE in the city and highway respectively. However, 0 and 0.01 are the minimum DE with respect to highway and the city. Hence, on the highway distance error is less compared to the city using EKF based prediction. With reference to KF based prediction, it is noticed that
Based on the results, as shown in Figs. 12 and 14, it is found that the EKF has the maximum DE of 0.18 and 0.125 for the city and highway scenarios respectively. However, DE increased to 0.24 and 0.179 using KF based prediction in the city and highway as shown in Figs. 13 and 14 respectively. KF based prediction has more DE compared to the EKF with real trace.
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Fig. 10 Distance error (DE) in city scenario based on model trace (KF)
Fig. 12 Distance error (DE) in city scenario based on real trace (EKF)
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Fig. 11 Distance error (DE) in highway scenario based on model trace (EKF/KF). Note Some of the figures are drawn separately for EKF and KF due to superimposition of the images
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Fig. 13 Distance error (DE) in city scenario based on real trace (KF)
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Fig. 14 Distance error (DE) in highway scenario based on real trace (EKF/KF)
Fig. 16 Velocity error in city scenario based on model trace (KF)
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Fig. 15 Velocity error in city scenario based on model trace (EKF)
Intuitively, DE is less on the highway compared to the city with both models using EKF and KF. However, DE slightly increases with KF based prediction in all the scenarios.
Fig. 17 Velocity error in highway scenario based on model trace (EKF/KF)
the city and highway scenario using KF as shown in Figs. 16 and 17 respectively. In general, EKF has less VE compared to KF in all the scenarios. However, KF has more VE on the highway compared to the city, unlike EKF.
8.2 VE based comparison 8.2.2 VE with real trace The following subsections compare the VE between EKF and KF based prediction. 8.2.1 VE with model trace From Figs. 15 and 17, it is learned that EKF has maximum VE of 0.125 and 0.10 in the city and highway scenarios respectively. However, VE increased to 0.138 and 0.196 in
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The maximum VE noted for EKF is 0.127 and 0.122 for the city and highway respectively as shown in Figs. 18 and 20. VE increases marginally to 0.143 and 0.131 in the city and highway using KF as shown in Figs. 19 and 20 respectively. In both the cases viz. EKF and KF highway has less VE compared to the city. However, EKF outperforms KF with a marginal difference in all the scenarios.
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Fig. 18 Velocity error in city scenario based on real trace (EKF)
Fig. 20 Velocity error in highway scenario based on real trace (EKF/ KF)
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Fig. 19 Velocity error in city scenario based on real trace (KF) Fig. 21 Root mean square error (RMSE) for latitude
In general, KF has more VE compared to EKF in all the scenarios. 8.3 RMSE based comparison In Figs. 21, 22 and 23, R and M on the x-axis are used for real and model based traces which are used in conjunction with the city and highway. From the Fig. 21, it is observed that the KF based prediction has highest 0.062 error in latitude whereas with EKF based prediction it reduced to 0.042. Figure 22 shows the error in longitude. Intuitively, it can be seen in the result that the error is most likely to be similar to an observed error for the latitude. However, it has an
exception with EKF based prediction on model based city traces wherein the error reduced to 0.022. Based on RMSE results for the latitude and longitude as shown in Figs. 21 and 22, it is evident that the EKF based prediction has less error in the predicted location compared to the KF based prediction. 8.4 AEDE based comparison The AEDE is shown in Fig. 23 for all the scenarios. It is as highest as 0.14 for KF based prediction with real traces for the highway scenario whereas with the EKF based prediction AEDE is reduced to 0.122. Overall, EKF has less AEDE compared to the KF in all the scenarios.
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Fig. 22 Root mean square error (RMSE) for longitude
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9 Conclusion Based on our experimental results, it is observed that location prediction using EKF for model based traces is almost equal to the measured location. However, coordinate system in model based traces is different from the GPS system. It is also observed in our simulation results that the performance of the prediction algorithm is better on the highway compared to the city scenario. With reference to the real traces, EKF prediction algorithm performs better on the highway compared to the city. However, it is lower than the model based traces. Nevertheless, model based traces are unrealistic to be used in real world scenarios, unlike real traces. In addition, DE is less on the highway with real and model based traces compared to the city. On comparison of DE between model and real based traces, it is found that the model based traces showed less error compared to the real traces. Velocity error is less with reference to model based traces on the highway compared to the real traces. On the basis of DE, RMSE, AEDE and VE results, it is also observed that the EKF based prediction outperforms the KF based prediction. Hence, it is concluded that the EKF based prediction should be used in VANET where location precision is more crucial, compared to KF based prediction. Though the time complexity of the proposed algorithm is discussed briefly in the Sect. 8.5, nonetheless, it is a future work to analyze the time complexity of our EKF based prediction algorithm with reference to VANET.
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Fig. 23 Average euclidean distance error (AEDE)
8.5 Analysis of time complexity The time complexity of the KF based prediction algorithm is Oðn3 Þ where n represents the number of parameters used in the state as discussed in [11]. As it involves the matrix multiplication and inverse operation. However, the time complexity is more for the EKF based prediction algorithm as it involves the computation of nonlinear function in Eqs. (12–15) and partial differentiation to compute the Jacobian matrix in addition to matrix multiplication and inversion computation. However, location precision has more priority over computational cost, as VANET computational device has enough resources to work with EKF.
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Raj K. Jaiswal has received B.E. (Computer Science and Engineering) from C.S.I.T, Durg, India in 2003 and M.E. (Computer Networking) from U.V.C.E, Bangalore, India in 2010. At present, he is pursuing Full-Time Ph.D. in Information Technology at National Institute of Technology Karnataka, Surathkal, India. His research areas of interest include Vehicular Ad-hoc Network (Routing, Location Estimation) and Network Security.
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C. D. Jaidhar is presently working as Assistant Professor at National Institute of Technology Karnataka, Surathkal, India. His primary research area includes Ad-hoc Networks, Computer Networks, Cryptography and Network Security, Security Issues in Voice over IP and Cloud Computing.