A Modified Multidimensional Scaling with Embedded ... - CiteSeerX

Paper ID#1569234517

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A Modified Multidimensional Scaling with Embedded Particle Filter Algorithm for Cooperative Positioning of Vehicular Networks M. Efatmaneshnik, A. Tabatabaei Balaei, N. Alam, A. Dempster

Abstract—Vehicular communication technologies are on their way to be recognized as icons of modern societies. One important scientific challenge to the safety related applications of vehicular communication is indeed semi-precise positioning. Cooperative positioning is an idea for that purpose, and of course from research point of view is very attractive. From the practical point of view the attractiveness of cooperative positioning lies in its independence from any major additional infrastructure other than the vehicular communication systems. This paper introduces a new positioning algorithm for localization of mobile networks, in general, that nicely applies to vehicular networks. The algorithm is based on the well known multidimensional algorithm and shows remarkable performance compared to its counterparts in the vehicular positioning literature. Index Terms—Vehicular Networks Localization, Cooperative Positioning, DSRC, Multidimensional Scaling, GPS

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INTRODUCTION

SRC (Dedicated Short Range Communication) is a 75 MHz communication medium between 5.85 GHz and 5.925GHz. DSRC standards and technology form a platform for vehicle to vehicle as well as vehicle to infrastructure communications. This communication platform will indeed enable VANETs (Vehicular Adhoc NETworks) through which many ITS (Intelligent Transport Systems) applications including safety related, driver assistance, and traffic assignment systems would become hard realities. Safety applications transmit information about unforeseeable and potentially dangerous collisions or events [1]. Driver assistance systems include speed management for avoiding traffic jams, automated highway entering or coordination of the arrival times at an intersection [1]. Traffic assignment will Manuscript received July 15, 2009. This work was supported in part by the University of New South Wales, Faculty of Engineering Research Grant PS17403. Also support was received from School of Surveying and Spatial Information Systems/University of New South Wales. M. Efatmaneshnik is with the school of surveying/university of New South Wales, Sydney, Australia (phone: +612-938-541-90; fax:+612-931-374-93; email: [email protected]). A. Tabatabaei Balaei is with school of surveying/university of New South Wales, Sydney, Australia (email: [email protected]). N. Alam is with school of surveying/university of New South Wales, Sydney, Australia (email: [email protected]). A. Dempster is with the school of surveying/university of New South Wales, Sydney, Australia (email: [email protected]).

promote fair and cooperative road sharing between all users; this will help to resolve the chaotic situations where all users are subject to the delays in their travel times. Node/Vehicle localization in VANETs is an important part of all of the ITS applications of DSRC technology. In addition to the common cost effective integration of GPS (Global Positioning Systems), and INS (Inertial Navigation Systems), an information fusion approach to the collective network localization problem through using DSRC channel may be undertaken: cooperative localization. In general the network localization problem is studied under two interrelated main topics: static networks localization (referred to as Wireless Adhoc Networks) and Mobile Adhoc NETworks (MANETs) localization. In either case the localization problem with pseudorange measurements is usually tackled by trilateration, and multilateration to some fixed or mobile beacons (nodes with perfect or even uncertain priori known location). These correspond to when the nodes with unknown positions measure their distances (pseudoranges) respectively from three anchor points, and more than three anchor points. Contrary to the usual case Cooperative Network Localization utilizes the information of the pseudorange measurements between the nodes with unknown positions (or nodes which know their position with some uncertainty e.g. by using a GPS). In this case the pseudorange measurements are also known as dissimilarity measurements that form a dissimilarity matrix. The cooperative positioning is a new challenge for VANETs coming to effect through DSRC [1], both as a localization medium (i.e. used for ranging between the nodes with unknown positions), and an information exchange medium for nodes’ kinematic information transfers. There are two main advantages of cooperative localization: first, the collective network position accuracy increases by the nodes density, second the cooperative localization systems do not require extensive infrastructure [1]. This paper presents a novel approach for cooperative positioning that suits the needs of VANETs. In the following the literature review concerning the various network localization approaches are presented. Then filtering techniques and VANETs localization are briefly discussed. Section II presets the proposed algorithm of this paper and in section III the algorithm is put in its intended context (VANETs positioning) and its performance is evaluated within that context.

Paper ID#1569234517 A. Network Positioning Literature Review A localization algorithm is the subject of this paper and is a computational algorithm that addresses the problem formulation, robustness, estimation accuracy, coordination and computational complexity, given the pseudoranges are known. Monte Carlo Localization [2], Convex Optimization [3], Iterative Multilateration [4], and Multidimensional Scaling (MDS) [5] are the most popular techniques. A cooperative localization algorithm for VANETs, however, must address certain set of problems: first it must be real time and fast; second it must be adaptive with respect to the traffic condition and/or the nodes density; third it must be robust to internodes’ connection failure. This paper is in fact an attempt to a new localization algorithm that is tried to be set in harmony with the particularities of the VANETs. In general network localization problem can be tackled by any standard estimation method such as Least Mean Squared Error (LMSE) estimation [6]-[7], Maximum Likelihood (ML) estimation [8], Maximum A Posteriori (MAP) estimation [5]. These cost functions can be minimized by suitable optimization algorithm. The cost function and the optimization algorithm together form the localization algorithm. There are however other novel techniques that will be reviewed in this section. Graph drawing is an interesting anchor free network localization technique that yields a local or relative map of the nodes positions based on network possibility problem [3]. Reference [9] for example devised a model based on imaginary embedded springs as the nodes links. The stationary state of such system would then require an energy function to be at the minimum. This minimization could yield one possible graph of the nodes locations. This algorithm was centralized, expensive in its computational cost according to authors. MDS and its Variants Multidimensional Scaling (MDS) is a general approach to information visualization for exploring similarities or dissimilarities in data and is often regarded as a powerful graph drawing technique. An MDS algorithm starts with a matrix of item–item similarities, and assigns a location to each item in N-dimensional space. MDS has found many applications in chemical modeling, economics, sociology, political science and, especially, mathematical psychology and behavioral sciences [10]. Classical MDS uses the eigenvectors related to the N first largest eigenvalues of the nodes dissimilarity matrix to construct the nodes position graph [11]. There are many spectral variants of classical MDS [11]. This is a fast and attractive technique in WSN positioning and as a result has found many applications in MANETs. It is well known that the Classical MDS solutions minimize the MSE of nodes positions; however the technique is very sensitive to noise in pseudorange measurements. Additionally the Classical MDS requires all the dissimilarity measurements and has no tolerance for missing information. As such, many variants of MDS are introduced with some weighting schemes that reduce this sensitivity and increase the robustness to the missing dissimilarity information (see [10] and references therein). These variants of MDS are mostly based on the iterative schemes that minimize a cost function rather than

2 spectral algorithms that rely on eigenvalue and/or singular value decomposition of the dissimilarity matrix. The common theme and what make one to call these algorithms MDS is that they all have analytical solutions that minimize their cost functions. One such scheme is Scaling by MAjorizing a COmplicated Function (SMACOF) that minimizes STRESS cost function presented in section II [12]. There are two variants of SMACOF: one-way MDS and multiple-way MDS. As the name suggests one-way SMACOF needs only one set of pseudorange measurements (one dissimilarity matrix) for the algorithm to converge; whereas multiple-way MDS can work with multiple dissimilarity readings [10]. Both algorithms have similar solutions with minor differences. In this paper we will build on SMACOF iterative solution and STRESS cost function since, as it will be shown in section II, STRESS cost function to the MAP formulation of position estimation problem. This non-classic MDS, on the other hand, is more accurate than both LMSE and ML network localization estimates [13]. The MAP formulation also best suits characteristics of VANETs in which nodes can be equipped with GPS that provides a priori information about nodes/vehicles locations. Because of their often fast computational response, Multidimensional Scaling Algorithms have been used for mobile network positioning. At every instance the algorithm needs to produce a new set of position estimates. Examples were MDS-MAP which was based on classical MDS [14] and mobile phones positioning based on a modified classical MDS [15]. B. Filtering In MANETs localization the motion information of the nodes and their likely routes can be used to refine the position estimates. Several mobility models, that facilitate the information fusion across the time, are discussed in [16]. These models are based on uncertainty in mobility (random walk model), uncertainty in velocity (velocity sensor model) and finally uncertainty in acceleration (random force model). The latter is usually for the inertial navigation systems. A computational algorithm that facilitates tracking and fusion of spatial information across time is referred to as a filter. There are two basic types of filters each with many variants and combinations: Kalman Filter and Particle Filters (see [16] for a comprehensive literature review). Extended Kalman Filters that accommodate nonlinearity have been vastly reported in literature and employed in localization systems. Especially their low computational complexity renders them very attractive. Particle filters also known as Sequential Monte Carlo algorithms are deemed to result in better performance in terms of accuracy of the position estimates but more complex in terms of computational complexity. There are hybrid filters as in [17] that mix the attributes of both filters to gain a compromise between the pros and cons of each filter. This paper presents a kind of particle filter for the estimation of the dissimilarity matrix at each instant based on the dissimilarity reading at the same instance and position estimates of the previous instant. This is the first time, in the literature, that the pseudoranges rather than the nodes locations are filtered for the purpose of better positions accuracy.

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C. VANETs Localization Although localization of MANETs has been researched for some time, their results cannot be applied to VANETs because of inherent differences between the two mainly arising from high nodes speed. According to [18] VANETs show frequent fragmentation, rapid topological evolution over time and short link life (e.g. less than a second for the vehicles travelling in opposite directions). As such any localization algorithm must take these factors into account since communication overhead can overwhelm the network and exhaust its channel capacity. The common interest in VANET localization is the employment of distributed localization algorithms (due to their ad hoc nature); however, a centralized, hierarchical (combination of centralized and distributed) that advocate vehicle to infrastructure communication or even a hybrid/modal algorithm have their own appeal for the higher accuracy and greater availability (i.e. reliability). Parker and Valaee [19]-[20] presented cooperative position estimation for VANETs as a distributed positioning algorithm. In [19] they introduced an iterative algorithm based on LMSE. The algorithm basically had two steps including an initiation and a refinement. The initiation of some estimate of all vehicles position estimate was done through exchanging GPS information, trilateration by using three vehicles that were GPS-equipped (this is known as Adhoc trilateration). Then each of the vehicles uses all the other nodes’ information to make more accurate position estimation: this was the refinement stage. In [20] they introduced an Extended Kalman Filter to incorporate kinematic information of the vehicles into the position estimation. The algorithm presented here follows the same steps namely initialization and refinement but, as is shown in the simulation results, can outperforms its predecessor in terms of accuracy by a considerable margin. II. NON-CLASSIC MULTIDIMENSIONAL SCALING ALGORITHM

Here is the priori information about the nodes locations. , is the accuracy weight of the range observations at time t. Note that the dissimilarity readings and their weights can be averaged over time and be used in the calculations, which might lead to more accuracy/lower iterations. di,j(Xt) is the dissimilarity/internodes distance between nodes i,j given their estimated locations at iteration t in the Xt matrix, i.e. given Xit = [xit yit ], di,j(Xt) is  

,

(2)

.

Equation (1) corresponds to the multiple-way MDS cost function that accommodates different , as the observation at the time t. The STRESS cost function is in fact the log posterior distribution of nodes positions given the dissimilarity observations log(X|{δ}) [5]. Thus the optimization of STRESS leads to the MAP estimation. As mentioned before an interesting property of S is that an analytical solution for xi can be derived via a Majorizing function. T(X,Y) majorizes F(X) when T(X,Y) ≥ F(X) for all Y and T(X,X)=F(X). The main idea in (SMACOF) as the name suggests is that a complicated function is minimized by the minimization of its majorizing function. The details of this majorization can be found in [10] and the analytical solution that minimizes Sk is (3)

. Where: 2

,

,…,

and entries of ,

1

,

(4)

,

are

/

,

,

,

A. Formal Problem Statement ,

Consider a set of N = n + m nodes in a two dimensional space with X = {X1,...,XN}, Xi Є R2, Xi =[xi yi] coordinates in two dimensional plane. Assume that there exists perfect knowledge about the last m nodes and imperfect or no knowledge about the first n nodes positions. Let ri be the accuracy weight of Xi, a priori information about the nodes locations when i < n. As a simple case, ri can be the inverse of the standard deviation of Xi. Obviously ri = 0 means that there are no a priori information about Xi. This formalization makes it simple to add/remove anchor nodes, as well as prior information (mainly taken from GPS) into the localization system. The localization problem is the estimation of coordinates X given the observed internodes distances δi,j, i,j Є {1,..,n}. The STRESS cost function for the iteration k is [10] 2

,

,

,

(1)

,

,

2 2

,

1

, ,

/

, ,

(5) /

,

,

.

We should note that this algorithm is for static sensor networks but can be modified for mobile networks. The solution is updated k times until the increment in the increasing cost function is less than a threshold: (6) Note that if only one set of dissimilarity measurements are available the algorithm still converges, by simply inserting the same dissimilarity matrix at each iteration (one-way MDS). Thus to compare the performance of this nonclassic MDS with other estimation techniques, one needs to make a new position estimation at each iteration with the dissimilarity matrix as the

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average of all dissimilarity observations up to then. Fig. 1 compares the performance of the non-classic multiple-way MDS (effectively MAP estimation), ML estimation, and LMSE estimation (see [21] for the detailed description of their cost functions). The Cramer Rao Lower Bound (CRLB) is also included in Fig. 1 which is the minimum variance of any unbiased estimation (see [22] for CRLB formulation of cooperative positioning) and bbenchmarks any estimation algorithm. A Hybrid CRLB (HCRB) can be derived for multiple-way positioning that uses several dissimilarity readings ΣP

I

(7)

.

Here, is the Fisher information matrix [22] and ΣP is the covariance matrix of the a priori information about the nodes locations and K is the number of distinct dissimilarity measurements and/or iterations. Note that as a result of this formulation the HCRB decreases a slight amount and that is due to the fact that at each iteration, the fisher information I is used again. A gradient based constrained optimization tool was deployed for ML and LMSE optimizations. The search space was constrained to ±σ of the a priori position information, so we have referred to these algorithms as Constrained ML (CML) and Constrained LMSE (CLS). It is common to measure the performance of localization algorithms by the average error of the network nodes’ estimated positions, which is

 

.

(8)

The Error of the a priori information was set at 10 m which is consistent with the GPS error. The ranging standard deviation was set at 3 m. The result shown here is consistent with the report in [13].

The computational complexity of the nonclassic MDS is O(n×L), where L is the total number of iterations required until the stopping rule is satisfied. For controlling the number of iterations, one can either select it directly in the algorithm or change the stopping criteria. A smaller stopping criterion leads to higher number of iterations. With more iteration, MDS algorithm can have performance very close to the Cramer Rao Bound. B. Modified MDS Algorithm for Mobile Networks The one-way non-classic MDS can be directly employed for mobile networks. However, for achieving higher accuracies, multiple-way non-classic MDS is more desirable. In VANETs the network has a unique topological identity at each moment in time and at each instance only one set of ranging can be performed. As such the ranging between the nodes repeated in different moments of time may have different outcomes in mobile networks; thus the iterations envisaged in the multipleway MDS algorithm cannot be performed. Thus multiple-way non-classic MDS algorithm cannot be directly applied to VANETs. We propose, for the first time, to use a Monte Carlo Sample Generator to generate the pseudorange/dissimilarity matrices, as many as needed, for the MDS algorithm to converge. Monte Carlo methods are a class of computational algorithms based on repeated random sampling. The Monte Carlo sample generator can employ the previously established ranging model which, in here, is assumed to be a Gaussian model with known standard deviation. The STRESS function is computed based on Monte Carlo samples rather than real measurement readings (that are impossible to have). This way the multiple-way MDS algorithm is executed based on only one set of dissimilarity readings. There is a pitfall to this approach, and that’s the number of iterations for the convergence may be very high. In order to make a compromise between the number of iterations (which is directly proportional to the computational time), one can employ the importance sampling techniques. Importance Sampling Importance sampling is a general technique for estimating the properties of a particular distribution, while only having samples generated from a different distribution alternative distribution) rather than the distribution of interest. The alternative distribution must have a narrower band (lower standard deviation) in order for the importance sampling to lead to faster convergence in the Monte Carlo Estimation process. We suggest sampling from a distribution with the mean as the sole pseudoranges/dissimilarity matrix at each time and a standard deviation lower than that of the dissimilarity matrix by a factor α: .

Fig. 1. The performance comparison of three network estimation techniques for a network in 400x400 (m) grid with 10 nodes.

(9)

Where σIM is the deviation of the alternative distribution and σDSRC is the deviation of pseudorange measurements. When α is set at infinity the non-classic MDS algorithm corresponds to one-way case.

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In our simulation experiments, as expected, we observed that as α gets closer to one from the infinity (α →1+) the number of iterations for convergence at a given ε increases but the overall accuracy declines. The parameter α can be regarded as a design parameter of the localization system. C. Modified MDS with Embedded Filter Filters are per se estimation engines. For example an Extended Kalman Filter alone can be used or it can be coupled to the output of the modified MDS. However, the modified MDS opens a possibility to filter the pseudoranges between the mobile nodes rather than the positions of the mobile nodes. We propose to use the information of the last states of the nodes and their velocity in the dissimilarity matrix sample generation process. This technique can be in effect regarded equivalent to a Particle Filter. A simple and rather informal description of this filter follows. Consider the velocity sensor model as the candidate mobility model .

(10)

and are the nodes position vectors at time t+1 Where and t, is the velocity vector of nodes at time t and is the uncertainty in the velocity measurements at time t. Contrary to the usual case, in which the filter directly employs the mobility model for position prediction, we built upon this model to arrive at a prediction of dissimilarity measurements at time t+1: (11)

 

,

III. COOPERATIVE POSITIONING ALGORITHM FOR VANETS Fig. 2. demonstrates the modified MDS algorithm with embedded filter. At the first moment when there are no previous dissimilarity readings the algorithm starts without the predicted dissimilarities. The estimation process for a single vehicle in VANETs is as follows: Initialization: A vehicle estimates its distance from all of its neighbours by DSRC device; then broadcasts its GPS position reading as well as all the dissimilarity measurements to all of its neighbours. A neighbour in this sense is a vehicle that is detected by the DSRC device. The set of neighbours for each of the vehicles are different. As such from the bulk of dissimilarity readings that are received, those related to own neighbouring vehicles are extracted and the rest are discarded. The vehicle estimates its position via the modified MDS without filtering. If a vehicle is not equipped with the GPS, it simply sets a priori position information weight (r) of that vehicle (weather own vehicle or any of the neighbours) as zero. Iteration: the vehicle constantly performs dissimilarity measurements and updates them by the predicted dissimilarities. If a new vehicle joins the neighbours then its pseudorange is used without filtering (updating via prediction). A new set of dissimilarity measurements

New GPS position Estimates

Mobility Model (dissimilarity prediction)

Update dissimilarity matrix

. Where subscriptions i,j reflect the nodes identity in the network, vx, and vy are the velocity of the nodes at the direction of x and y. Note that (11) can be computed by the mean values (estimated values) of all the variables (x,y, vx, and vy ) or alternatively can be computed by using samples from the distributions of the variables. The normality assumption about the underlying distributions of the variables is fair but the algorithm can work with any kind of distribution. This is indeed one of the strength of the particle filters. Upon the arrival of the dissimilarity reading at time t+1 the predicted dissimilarities can be updated via a weighted averaging: ,

,

1

,

    0

1.

(12)

The weight p can be determined based on the accuracy of the velocity readings and MDS position estimation accuracies, in a pre-processing procedure. For multiple-way modified MDS one can draw one sample from the distributions of all the variables in (11) and then update it by a sample from the distribution of δt+1. The importance sampling techniques can be employed here too. This modification can render the algorithm suitable for real time systems as VANETs.

Non-classic MDS algorithm

Final Position Estimate Fig. 2. The modified MDS algorithm with embedded particle filter.

A. Performance Evaluation and Simulation Results A simulation experiment was set up to demonstrate the performance of the presented positioning algorithm and to compare it with the Kalman Filter algorithm presented in [20]. The road map and the directions are shown in Fig. 3. was used and the simulation was performed in MatlabTM environment. The vehicles were deemed to move in the marked three streets in a circular manner, in both directions and couldn’t change lane or direction. There was only one lane at each direction. The traffic rate was set at 1200 vehicles/h, which leads to 60 vehicles circulating in the shown area. The vehicles had a maximum velocity of 60 km/h and a minimum of 40 km/h.

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6 The main purpose of this is to reduce the communication overheads for certain positioning accuracies. REFERENCES [1]

[2] [3] [4]

Fig. 3. The road map on which the simulation tests was performed. The part of W10Mile Rd that was used is roughly 900m long.

[5]

The DSRC range was set at 150 m, which means that all the vehicles within this range were regarded as neighbors. The ranging standard deviation was set at 3 m and positioning error of GPS was 10 m. Each vehicle on average had 10 neighbors. Fig. 4. Shows the result of the simulation over 100 seconds. The modified MDS, and its variant with the embedded filter, are compared to the performance of Kalman Filter. The modified MDS has a close but slightly better performance than Kalman Filter. However the modified MDS with embedded filter has achieved less error at all times.

[6]

[7]

[8] [9] [10] [11] [12] [13]

[14]

[15] [16] Fig. 4. The comparison of positioning algorithms.

IV. CONCLUSION AND FUTURE DEVELOPMENTS A modification of non-classic MDS for localization of vehicles in roads in a cooperative manner was presented. This algorithm is promising in its performance and can be further improved. This is due to the fact that the presented algorithm has many parameters including the stopping criteria, the number of iterations, and more importantly those controlling parameters of the built in importance sampling. Tuning these parameters can indeed improve the performance of the algorithm particularly in terms of its run time and computational complexity. From here on, this research can take many possible paths and further investigations. One possibility that is nicely accommodated in the presented MDS algorithm is the adaptive neighbor selection presented in [5].

[17]

[18]

[19] [20] [21] [22]

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