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A new decentralized Bayesian approach for cooperative vehicle localization based on fusion of GPS and inter-vehicle distance measurements

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A new decentralized Bayesian approach for cooperative vehicle localization based on fusion of GPS and inter-vehicle distance measurements
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  IEEE INTELLIGENT TRANSPORTATION SYSTEMS MAGAZINE   • 85   • SUMMER 2015 1939-1390/15©2015IEEE Mohsen Rohani and Denis Gingras Electrical and Computer Engineering Department, Université de Sherbrooke, Canada E-mails: mohsen.rohani@usherbrooke.ca, denis.gingras@usherbrooke.ca  Vincent Vigneron Université d’Evry, France E-mail: vincent.vigneron@ufrst.univ-evry.fr  Dominique Gruyer IFSTTAR - IM - LIVIC, Satory, France E-mail: dominique.gruyer@ifsttar.fr  A New Decentralized Bayesian Approach for Cooperative Vehicle Localization Based on Fusion of GPS and VANET Based Inter-Vehicle Distance Measurement Introduction ccurate and reliable vehicle local-ization is a key component of numerous automotive and Intelligent Trans-portation System (ITS) appli-cations, including active  vehicle safety systems, real time estimation of traf-fic conditions, and high occupancy tolling [1]. Var-ious safety critical vehicle applications in particular, such as collision avoidance or mitigation, lane change management or emergency braking assistance systems, rely principally on the accurate and reliable knowledge of vehicles’ positioning  within given vicinity. Distributed algorithms as pro-posed in [2] and [3] have under- lined a recent and important interest for the collabora-tive localization. Since the number and type of sen-sors used in vehicular applications increases, it is essential to find ways to better analyze and extract useful data from these sensors and share them between vehicles  when it is relevant. Sensors which are used in localization can be divided in two categories: proprioceptive and exteroceptive sensors [4]. Propriocep-tive sensors are those which can provide  Digital Object Identifier 10.1109/MITS.2015.2408171 Date of publication: 20 April 2015   IMAGE LICENSEDBY INGRAM PUBLISHING A  IEEE INTELLIGENT TRANSPORTATION SYSTEMS MAGAZINE   • 86   • SUMMER 2015 information about the vehicle dynamic states like position,  velocity and acceleration (GPS, accelerometer, gyroscope etc.). Exteroceptive sensors provide information about the states of the environment (video camera, lidar, etc.). GPS is one of the most common positioning devices being used in  vehicle localization as it provides absolute position of the  vehicles, whereas dead reckoning sensors such as odom-eters or INS (Inertial Navigation Systems) provide relative information only. GPS signals are however subject to dif-ferent sources of noise, and degradation as well as being subject to temporary signal loss in cluttered environment. Many of the intelligent vehicles systems like safety systems can benefit from more accurate and reliable positioning [5]. Data fusion technique is one of the common ways to improve the position estimate by exploiting the information coming from multiple sensors [6–8]. In the recent years, vehicular ad hoc network (VANET) applications has become of great interest [9–11]. With the recent emergence of multi-vehicular wireless communica-tion capabilities, cooperative architectures have become an attractive alternative for solving the localization problem [1, 9, 10, 12–20]. The main goal of cooperative localization is to exploit different sources of information coming from differ-ent vehicles within a short range area, in order to enhance positioning system efficiency while keeping the computing cost at a reasonable level. In other words, vehicles share their location and environment information to others in order to increase their own global perception. Some of the most prominent approaches for cooperating localization are based on Kalman filtering [21–23], Bayesian methods [24], Markovian modeling [25], maximum likelihood methods [26] and Split covariance intersection filter [27, 28]. In addition to the information exchanging, VANET has brought the capability for communicating vehicles to use their wireless communication devices to measure the distance between each other. There are basically two methods of measuring distance between communicat-ing devices. The first method is the Time-of-Flight (TOF)  which is based on the time that it takes for a signal to travel from one node to another node, where the nodes are vehi-cles in a VANET. The second method is the Radio Signal Strength (RSS) which is based on the attenuation of the sig-nal strength while traveling from the transmitter node to the receiver node [29]. Using the communication devices to measure distance between vehicles has several advantages and disadvantages over the other range measurement devices like radar and lidar. One of the advantages is that the vehicle identification problem and the data association between the received information and range measure-ments is easier to solve. Another advantage of these method is that its detection performance in the crowded areas is better since it can still be used while an object or another  vehicle blocked the line of sight between two vehicles but it does not block the line of sight between antennas. However the disadvantages of these methods over radar and lidar is that in their general form they cannot provide the relative bearing between the vehicles and they can only provide the distance. In addition to this, lidars usually can provide more accurate measurements. Although not all of the RF range measurement methods can provide the acceptable accuracy needed for our method, there are several more accurate RF based methods such as [30] and [31, 32] which can provide the needed accuracy to be used in our method. In [30] a low cost accurate radio ranging technic is pro-posed and field trials has been conducted in different envi-ronments to characterize the ranging error.In this article, we aim to improve the GPS vehicle posi-tion estimates by using available VANET based inter-vehicle distance measurements in a cluster of vehicles. The reason that we decided to use VANET based inter-vehicle distance measurements is that in addition to the mentioned advan-tages, using this method of distance measurements reduces the cost of the system as it does not need a new range sensor to be used and vehicles can use their existing communica-tion device to measure the distance.Our proposed cooperative vehicle localization method is a decentralized Bayesian approach which allows a vehicle to incorporate its GPS position estimate with other vehicles GPS data and inter-vehicle distance measurements. Unlike [24, 25], which basically have been developed for indoor robotic applications, our method is developed for outdoor usage and automotive applications. Also this method is tak-ing the advantage of using GPS which is available in out-door usages. Our method should be seen as a pre-filtering of GPS positioning measurement using inter-vehicle distances and other vehicles GPS measurements, prior the tracking algorithms such as the Extended Kalman Filter (EKF) [33]. Therefore this method has the advantage to be incorporated  with any existing ego localization algorithm which uses GPS (see Fig. 1). Furthermore, the data dependency problem, which is a common issue in probabilistic approaches [24–26] and  which leads to over convergence, has been circumvented in our approach. Another advantage of this method is the ability to use the true probability distribution model of dis-tance measurement sensors instead of using their Gauss-ian approximation which is usually being done in Kalman based methods. In addition to the performance analysis of the method, the Sensitivity of the proposed method to the  vehicle to vehicle distance measurement accuracy, com-munication latency and communication failure is also studied. The results presented in this article are based on Monte Carlo simulations and the data used in these simula-tions were provided by Pro-SiVIC software from Civitec and GPSoft Satellite Navigation toolbox for MATLAB.The focus of this article is to describe the derivation of the Bayesian cooperative vehicle localization method and demonstrate its performance by integrating this algorithm  IEEE INTELLIGENT TRANSPORTATION SYSTEMS MAGAZINE   • 87   • SUMMER 2015  with a Kalman filter to incorporate the dynamic properties of the vehicle. Proposed Method The proposed method aims to improve GPS vehicle position-ing using additional inter-vehicle distances and vehicle-to-vehicle (V2V) communication capabilities in a cluster of  vehicles. We assume that each vehicle is able to estimate its position and respective covariance matrix using its embed-ded GPS receiver independently. We consider also that each  vehicle is able to estimate its distance to other vehicles, using a VANET based method and independent from their GPS signals [30–32, 34], we refer to these method as Range Sensor in the rest of this article. Finally, it is assumed that the vehicles share their information by means of a VANET. With their GPS receivers, the vehicles can calculate the pseudo ranges between them and the visible satellites. These pseudo ranges has a standard deviation which is referred as UERE v  [35]. With the following notation:  Nth Vehicle V       =  (1)  True Position of   X V  ( )      0  =  (2)  Estimated positionof   X V  ( )      = t  (3)  True distancebetween and  D V V  ( )     0  =  (4)  Estimatedd istancebetween and  D V V  ( )     = t  (5) where  X  ( )    0  and  X  ( )    t  are two dimensional vectors provid-ing the (unknown) true position and estimated position of V       respectively. The distance between two vehicles for the no-noise case is simply given by:  ,  D X X  ( ) ( ) ( ) ij i j  0 0 0 = -  (6) where  D ( )   0  is the distance between  X  ( )  0  and .  X  ( )  j  0  As said before, we assume that   ( )   t  is measured independently from  X  ( )  t  and  X  ( )  j  t  and that we have a zero mean additive noise on the GPS estimated positions:  ( , ) ( )  X N X R f x  ( ) ( ) ( )  N  N  X X  N  0  ( ) ( )  N N  +  =  (7) where  R  X  ( )  N   is the covariance matrix of the position and   f   X  ( )  N   is the probability density function of .  X  ( )     Also, we assume that the inter-vehicular distances have a zero mean additive noise:  ( , ) ( )  D N D f d  ( ) ( ) ( ) i j  ij  D Di j  02 ( ) ( ) ij  ij  +  v  =  (8) where   ( )   0  is the true distance between V     and , V    j      2 ( )   v  is the distance variance corresponding to the accuracy of the sensors. Here it is important to mention that these assump-tions (7, 8) is only made to simplify the problem; our method is not restricted to them. This method can work with any other probability distribution model which can better describe the properties of the range measurement method. Now it is desired to estimate  X  ( )   from the observations. Let us define:  , ,  X D ( ) ( ) ( )       = " ,  (9) where   ( )    is the observation of  X  ( )  j   and   ( )    by . V     There-fore  X  ( )    estimates the position    ( )   from .   ( )     ( )( )  ,cossin      D D i j  ( ) ( )( )( ) Y ij  j i j i j   ! ii = - = G  (10) where i  is the bearing of the inter-vehicle distance mea-surement. Since we assumed that our distance measure-ment device is only able to provide us with the inter-vehicle distance and not the bearings the value of i  is unknown. Therefore the solution space for    ( )     is a circle around  X  ( )  j   with the radius of .   ( )   A. The 2 Vehicles Case  Let us first consider a simple 2-vehicle scenario. From a probabilistic point of view, we have:  ( )( )( ) . ( )   f x x   f x   f x x f x  ( ) ( )( )( ) ( ) ( )  X X iY ij  X  Y ij  X X  Y ij  i X i ( ) ( )( )( ) ( ) ( ) iY ij Y ij Y ij i i e = ee  (11)Eq. (11) is an application of the Bayes theorem, where the left member is the posterior probability to observe  x  ( )   given  x  ( )     and ( )   f x x  ( ) ( )  X X   Y i  i ( ) ( ) Y i i e e  is the likelihood of the observations. Using our model (10) we find:  ( )  ( , )  F x f x d dd dx  ( ) ( ) ( ) ( ) ( )  X  Y ij  X D j ij ij  x x  j  ( ) ( ) ( )( ) ( ) Y ij  j ij  j Y ij  = 333 ---+  #  #   (12) Satellites’SignalsRange SensorGPS ReceiverVANETNewEstimatedPosition &Covariance Inside a Vehicle ProposedMethodCooperativePositionEstimatesUpdateCovarianceUpdateEstimateKalmanGainProjectto K + 1Data Signals Sent to Other Vehicles Kalman Filter  Data SignalsReceived fromOther Vehicles FIG 1  Schematic of the system, the input data (blue), the proposed method (green), the Kalman filter (orange) and the new Position Estimation.  IEEE INTELLIGENT TRANSPORTATION SYSTEMS MAGAZINE   • 88   • SUMMER 2015 Differentiating with respect to  x  ( )     we obtain:  ( )  ( , )   f x f x x x   dx  ( ) ( ) ( ) ( ) ( )  X  Y ij  X D j j Y ij  j  ( ) ( ) ( ) Y ij  j ij  = - 33 -+  #   (13)Since    ( )  j   and  D ( )    are independent, we have:  ( ) ( ) ( )   f x f x f x x dx  ( ) ( ) ( ) ( ) ( )  X  Y ij  X  j  D j Y ij  j  ( ) ( ) ( ) Y ij  j ij  = - 33 -+  #   (14)Similarly we can obtain:  ( ) ( ) . ( )   f x x f x x f x x x dx  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  X X  Y ij  i X  j i D j Y ij  i j  ( ) ( ) ( ) ( ) Y ij i j ij  e e e = - e33 -+  #   (15)  X  ( )  j   is independent of    ( )   and the second term in the inte-gral (15) is the probability that V     and V    j   observe each other at the given distance ,  x x  ( ) ( )  j i -  Which becomes like a doughnut whose center is  x  ( )   (see Fig. 2). Therefore (15) becomes:  ( ) ( ). ( )   f x x f x f x x dx  ( ) ( ) ( ) ( ) ( ) ( )  X X  Y ij  i X  j  D j i j  ( ) ( ) ( ) ( ) Y ij i j ij  e  = - e33 -+  #   (16)Also we have: ( ) ( ). ( )( ). ( ). ( )   f x f x f x x dx   f x f x f x x dx dx  ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )  X  Y ij  X i X X  Y ij  i i X i X  j  D j i j i ( ) ( ) ( ) ( )( ) ( ) ( ) Y ij iY ij ii j ij  e = =- e333333 -+-+-+  #  #  #   (17)As we do not know the true position of the vehicles neither the distance between them, we use instead, the PDFs of the estimates coming from the GPS receivers and range sensors:  ( ) ( , )   f x   N X R ( ) ( ) ( ) GPSGPS GPS X i i i ( ) i =  t t  (18) where,    ( ) GPSi t  is the position estimate obtained from the GPS receiver of V     and  R ( ) GPSi t  is its estimated covariance.  ( )  ( , )   f d N D ( ) ( )  RS RS RS Dij  ij  2 ( ) ij  v =  t  (19) where,  D ( )  RSij  t  is the distance estimate between V    and V   j   given by the range sensor (  RS ) and  R 2 v  is the variance of distance measurements using range sensor.Our prior PDF for    ( )   is therefore the one of its GPS estimate:  ( ) ( )   f x f x  ( ) ( ) GPS X i X i ( ) ( ) i i =  (20)In a similar way for the inter-vehicular distances, we use the PDF of our range sensor measurement:  ( ) ( )   f d f d  ( ) ( )  RS Dij  Dij  ( ) ( ) ij  ij  =  (21)As an example, Fig. 2 shows the real positions, estimated GPS positions for V    and V      along with their PDF in red and the inter-vehicle distance PDF centered at .  x  ( )   The uncer-tainty ellipses have been drawn using a 3 sigma deviation from the center. Substituting (20) and (21) in (16) and (17)  we have:  ( ) ( ). ( )   f x x f x f x x dx  ( ) ( ) ( ) ( ) ( ) ( ) GPS RS X X  Y ij  i X  j  D j i j  ( ) ( ) ( ) ( ) Y ij i j ij  e  = - e33 -+  #   (22)And, ( ) ( ). ( )( ). ( ). ( )   f x f x f x x dx   f x f x f x x dx dx  ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) GPSGPS GPS RS X  Y ij  X i X X  Y ij  i i X i X  j  D j i j i ( )( ) ( ) ( )( ) ( ) ( ) Y ij iY ij ii j ij  e = =- e333333 -+-+-+  #  #  #   (23)Therefore by substituting (22) and (23) in (11) we obtain the following posterior PDF for : V   ( )( ). ( ). ( )( ). ( ). ( )   f x x   f x f x f x x dx dx   f x f x f x x dx  ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) GPS GPS RSGPS GPS RS X X  iY ij  X i X  j  D j i j i X i X  j  D j i j  ( ) ( )( ) ( ) ( )( ) ( ) ( ) iY ij i j ij i j ij  e  =-- e333333 -+-+-+  #  #  #   (24)The corresponding maximum a posteriori probability (MAP) estimator is given by:  argmax  X f x x  i x X iY ij  ii X Y ij  = t ^ ^ ^ ^ ^^ ^  h h h h hh h  (25) where      t  ^ h  is the new estimated position for . V    Fig. 3 shows this posterior PDF for i  1 =  and ,  j   2 =  obtained from (24). As one can foresee, such a Bayesian approach allows one vehicle to exploit the information contained in the other vehicles GPS position and in the range sensor mea-surements to improve its own position estimate and more importantly reduce its uncertainty. B. The N Vehicles Case  Now, let us consider the case of  N   1 +  vehicles present in a cluster. We assume that each vehicle has its own GPS posi-tion estimate and that all the inter-vehicle distances can be measured. We also assume that all range sensors and GPS V  i V  j X (i) FIG 2  Real positions (squares), estimated GPS positions (dots), their PDF and uncertainty ellipses (red) and inter-vehicle PDFs centered at x  i  ^ h  (blue) for . m  65 UERE  v  =  ^  h  and .. m  45 RS  v  =  ^  h  IEEE INTELLIGENT TRANSPORTATION SYSTEMS MAGAZINE   • 89   • SUMMER 2015 receivers are the same, within the vehicle cluster, leading to identical prior distributions. Again this is only to simplify the equations but in practice, for each GPS receivers and range sensors, we can use their own prior distribution. We then have: ( , ..., )   f x x x  ,...,( ) ( ) ( )  X X X iY iY    1 ( ) ( ) ( )         1 e  = e   ( ,..., ). ( )( ,..., ). ( )   f x x x f x dx   f x x x f x  ,...,( ) ( ) ( ) ( ) ( ),...,( ) ( ) ( ) ( )  X X X  Y iY  N    X i i X X X  Y iY  N    X i 11 ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Y iY  N    N   11 ee e33e -+  #   (26)Considering that , ...,     ( ) ( )    iN  1  are independent from each other, equation (26) becomes: ( ,..., )( ). ( )( ). ( )   f x x x   f x f x x dx   f x f x x  ,...,( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )  X X X  i Y  i Y  iN   X  i  X X   Y  i  i    i N i  X  i  X X   Y  i  i    i N  111 ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) i Y  i Y  iN i Y  i ii Y  i i 1 eee = ee33e !! =-+= %% #   (27)By substitution of (22) in (27) we have: ,...,   f x x x  ,...,( ) ( ) ( )  X X X iY iY    1 ( ) ( ) ( )         1 e  = e  ^ h   ( ) ( ) . (  ) U   f x f x f x x dx  ( ) ( ) ( ) ( ) ( ) GPS GPS RS X i X  j  D j i j  j  j i N  1 ( ) ( ) ( ) i j ij  #  - 33 ! -+= % #   (28) where U   is a normalization factor that can be computed by substituting (22) into(27):  ( ) ( ) . (  ) U f x f x f x x dx dx  ( ) ( ) ( ) ( ) ( ) ( ) GPS GPS  X i X   j  D  RS i j j  j  j i N i 1 ( ) ( ) ( ) i j ij  # = - 3333 ! -+=-+ % #  #   (29)Finally, the MAP estimator for the position of vehicle V   given the  N   other vehicles, is given by:  , , argmax  X f x x X  , , i x X X X iY iY  iN  1 ( ) iY iY iN  1( ) i e f =  e f t ^ ^ ^ ^ ^ ^ ^ h h h h h h h  (30)Since there is no analytical solution for equation (30), a par-ticle filter [33] has been applied in order to solve this equa-tion. A sample set of 1000 particles has been selected for each ,( ... )  X j     1 ( )    =  according to its PDF ( ) .   f x  ( ) GPS X   j  ( )  j   The results shown here have been obtained with 5 filter iterations.Fig.4 shows the posterior probability (28) for V  1  in a cluster of 5 vehicles. The new position estimation for V  1  is calculated using (30) and is shown as a star in the central green area. As we can see from Fig. 4 the new estimated position is much closer to the real position of V  1  and the uncertainty has been reduced as the green area illustrated in Fig. 4 is smaller in size than the red area. Since the most computationally complex part of this algorithm is solving (30) with a particle filter, therefore the order of this algo-rithm is ( ) , O N N   p 2 $  where  N   is the number of vehicles in the cluster and  N   p  is the number of particles. With proper choose of  N   p  and  N   each iteration of the algorithm can run in real-time and the result can be sent for further use in the tracking algorithms.One should note that depending on the method chosen for estimating the inter-vehicular distances, different meth-ods for identifying neighbor vehicles and associating the measured distances with the received information from  VANET can be applied. Although in the case of VANET based distance measurement methods there is no need to use any special method because vehicles can be identified and data can be associated using their communication device MAC address, However in the case of Radar and Lidar, different approaches can be utilized like the one used in [21] which is based on the Mahalanobis distance between position esti-mations. Details on association and identification methods is outside the scope of this article as we will focus only on the fusion method for cooperative localization.One important point that must be mentioned here with our Bayesian approach is the over-convergence  issue. This usually occurs when the fused information coming from different vehicles are not independent of each other. For each time step of our method, only the GPS and inter-vehicle distance measurements of the vehicles present in the clus-ter are used in the calculation of the posterior PDF. These V 2 V 1 FIG 3  Posterior probability of V  1  and its uncertainty ellipse (green), new estimated position (star on green area), real position (squares), estimated GPS position (star on red area), their PDF and uncertainty ellipses (red) for . m  65 UERE  v  =  ^  h  and .. m  35 RS  v  =  ^  h V 1 V 3 V 4 V 5 V 2 FIG 4  Posterior PDF of V  1  in a cluster of 5 vehicles (green), real position (squares), new estimated position for V  1  (star on green area) for. m  65 UERE  v  =  ^  h  and .. m  15 RS  v  =  ^  h
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