Let us consider a sensor network with M static sensor nodes in a 2D plane which do not have a priori known locations (called unknown nodes) and a single mobile node (called beacon), equipped with localization hardware, e.g., GPS, which allows it to know its location at all times. After random deployment of the unknown nodes in a fixed-size area, the beacon traverses the sensor network while broadcasting packets which contain the coordinates of itself and other information. Any unknown node receiving these packets from the beacon (called observation) can recognize that it is in the area around the beacon’s current location with a certain probability. With each observation in a series of different time, the unknown node’s location is bounded in the beacon’s transmission area. The accuracy can be improved when the unknown node obtains more observations from the beacon. Location estimates and observations are assumed to be available at discrete times. For dynamic state estimation, the discrete-time approach is widespread and convenient [18].

If we solve the above-mentioned localization problem with a probabilistic approach, we are interested in estimating the unknown node’s real location R at the current time-step t, given knowledge about the initial location estimate and all observations O_1:t = {O₁,…,O_t} up to the current time t. This localization problem is an instance of the Bayesian filtering problem which requires the estimation of the state of a system that changes over time using a sequence of noisy measurements made on the system [18], where we are interested in constructing the posterior density p(l_t| O_1:t) of the current location estimate l_t conditioned on all observations o_{1: t} from the beacon. To define this localization problem from a Bayesian filter perspective, we assume that we have an initial distribution, a dynamic model and an observation model: (1) p ( l 0 | o 0 ) , (2) l t = f ( l t − 1 )     for               t ≥ 1 , (3) o t = g ( l t )                 for               t ≥ 1 .

By Equation (2) and Equation (3) we mean that l_t and o_t are assumed to be generated by functions f(·) and g(·), respectively. The precise form of the functions implies via a change of variables the transition probability density p(l_t|l_0:t, o_1:t−1) and the observation probability density p(o_t|l_0:t, o_1:t−1). We denote by l_0:t = {l₀,…,l_t}, the location estimate up to time t. Note that we could assume Markov transitions and conditional independence to simplify the model due to the constraints in computing and memory power of the sensor node, i.e. current location estimate l_t is only dependent on the previous location estimate l_t−1 and the current observation o_t is only dependent on the current location estimate l_t, then the dynamic model p(l_t|l_0:t−1, o_1:t−1) = p(l_t|l_t−1) and the observation model p(o_t|l_0:t, o_1:t−1) = p(o_t|l_t). The posterior density p(l_t|o_1:t) = p(l_t|o_t) may be obtained after initialization, recursively, in two stages: a prediction stage and an update stage.

Initialization: It is assumed that the initial p(l₀|o₀) ≡ p(l_o) of the location estimate, which is also known as the prior density, is available (o₀ being the set of no observations).

Prediction: Suppose that the posterior density p(l_t−1|o_t−1) at time t−1 is available. The prediction stage involves using a dynamic model Equation (2) to obtain the prior density p(l_t|o_t−1) of the location estimate at time t via the Chapman-Kolmogorov equation: (4) p ( l t | o t − 1 ) = ∫ p ( l t | l t − 1 ) p ( l t − 1 | o t − 1 )   dl t − 1 .

Update: At time step t, an observation o_t becomes available, and this may be used to update the prior density via Bayes’ rule: (5) p ( l t | o t ) = p ( o t | l t ) p ( l t | o t − 1 ) p ( o t | o t − 1 )where the normalizing constant (6) p ( o t | o t − 1 ) = ∫ p ( o t | l t ) p ( l t | o t − 1 )   dl tdepends on the likelihood function p(o_t|l_t) defined by the observation model Equation (3). In the update stage Equation (5), the observation o_t is used to modify the prior density to obtain the required posterior density of the current location estimate.

To address the complexity of the integration step in Bayesian filter, many optimal or suboptimal approaches are proposed. The recurrence relations Equation (4) and Equation (5) form the basic for the optimal or suboptimal Bayesian solution. Solutions do exist in a restrictive set of cases [18], including the Kalman filter (optimal Bayesian filter) and particle filter (suboptimal Bayesian filter) which approximates the optimal Bayesian solution, etc.

The Kalman filter assumes that the posterior density at every time step is Gaussian and, hence, parameterized by a mean and covariance, provided that certain assumptions hold: f(·) and g(·) in the dynamic model and the observation model are known and also are a linear function. In many situation of interest, such as mobile beacon-assisted localization problem in our scenarios, the assumptions made above do not hold. The Kalman filter cannot, therefore, be used as described – approximations are necessary.

The particle filter solutions offer a number of significant advantages compared with other techniques currently available, including the Kalman filter. These advantages arise principally from the generality of the approach, which allows inference of full posterior distributions in general state-space models, which may be both nonlinear and non-Gaussian.

Thus, in this paper, we use a particle filter (also called sequential Monte Carlo method) to perform a Bayesian filter on a sample representation. The key idea is to represent the required posterior density by a set of random samples with associated weights and to compute estimates based on these samples and weights [18]. As the number of samples becomes very large, this characterization of particle filter becomes an equivalent representation to the usual functional description of the posterior density, and the particle filter approaches the suboptimal Bayesian estimate.

In order to develop the details of the algorithm, let { < l t i , w t i > , i = 1 , ... , N } denotes a random measure that characterizes the posterior density p(l_t|o_t), where L t = { l t i , i = 1 , ... , N }, is a set of support samples (or called particles) with associated weights W t = { w t i , i = 1 , ... , N }. The weights are normalized such that ∑ i = 1 N   w t i = 1. Then, the posterior density at t can be approximated as: (7) p ( l t | o t ) ≈ ∑ i = 1 N   w t i δ ( l t − l t i )where δ(·) is Dirac delta function (8) δ ( l t − l t i ) = { 1             if       l t = l t i 0               otherwisesamples lⁱ are easily generated from a proposal density (or called importance density) q(·): (9) l i ∼ q ( l )where the symbol ∼ denotes sample generated sign, i.e., the samples on the left side are generated from the probability density on the right side. Weights are defined by: (10) w t i ∝ p ( l t i | o t ) q ( l t i | o t )where the symbol ∝ is used to denote proportionality up to a normalization constant.

The proposal density is chosen to factorize such that: (11) q ( l t | o t ) = q ( l t | l t − 1 , o t ) q ( l t − 1 | o t − 1 )

To derive the weight update equation, p(l_t|o_t) from Equation (5) is expressed via Bayes’ rule: (12) p ( l t | o t ) = p ( o t | l t ) p ( l t | o t − 1 ) p ( o t | o t − 1 ) ∝ p ( o t | l t ) p ( l t | l t − 1 ) p ( l t − 1 | o t − 1 ) p ( o t | o t − 1 ) ∝ p ( o t | l t ) p ( l t | l t − 1 ) p ( l t − 1 | o t − 1 ) .

The weight update equation can then be shown to be (the proposal density is chosen to be dynamic model Equation (2), i.e. q ( l t i | l t − 1 i , o t ) = p ( l t i | l t − 1 i )): (13) w t i ∝ p ( l t i | o t ) q ( l t i | o t ) ∝ p ( o t | l t i ) p ( l t | l t − 1 i ) p ( l t − 1 i | o t − 1 ) q ( l t i | l t − 1 i , o t ) q ( l t − 1 i | o t − 1 ) ∝ p ( o t | l t i ) p ( l t i | l t − 1 i ) q ( l t i | l t − 1 i , o t ) w t − 1 i ∝ p ( o t |       l t i ) w t − 1 i .

The generic particle filter proceeds for localization are as follows:

Initialization: N samples and weights are chosen from the initial distribution Equation (14) and the initial observation Equation (15), respectively. (14) l 0 i ∼ p ( l 0 | o 0 ) , (15) w 0 i = p ( o 0 | l 0 i ) .

Assumption: We assume that all unknown nodes are randomly deployed in an area of size S and each sensor (unknown node or beacon) has the same ideal radio range r. The algorithm does not assume very tightly synchronized clocks. The beacon is capable of moving a distance v_b in a time step in any direction where 0 ≤ v_b ≤ v_max. The beacon knows v_max, but it does not know the value of v_b or the direction of movement in any time step. At time t, every unknown node within the radio range of the beacon will hear a location announcement from that beacon. In a realistic deployment, it would be necessary to deal with network collisions and account for missed messages [19].

Initialization: In this stage, all unknown nodes have no information about their locations. The initial set of samples Equation (17) for each unknown node is chosen randomly from the whole deployed area and represented by a set of uniformly distributed samples with equal weights Equation (18). The weight equal to one represents the importance of corresponding sample, which infers one of the location estimates of the unknown node. The beacon’s current initial position Equation (19) is also chosen randomly from the whole deployment area. The beacon’s previous initial position (for further use) is chosen randomly from out of the deployment area and out of the radio range of any unknown node Equation (20): (17) L 0 = { l 0 i | l 0 i ∼ p ( l 0 ) = 1 S ,     1 ≤ i ≤ N } , (18) w 0 i = 1 ,         1 ≤ i ≤ N , (19) beacon c ∼ 1 S , (20) beacon p = ( beacon p ∼ 1 S ¯ ) ∧ ( d ( beacon p , R ) > r ) .where L₀ denotes the initial set of each sample, w 0 i denotes the initial weight of each sample, beacon_c denotes the beacon’s current location, d(beacon_p,R) denotes the distance between locations beacon_p and R, R denotes the unknown node’s real location, S̄ denotes the area out of the deployment area, and beacon_p represents the beacon’s previous location.

Prediction: In this stage, we adopt a dynamic model that the unknown node is capable of moving a distance v_node in a time step in any direction where 0 ≤ v_node ≤ α. The unknown node knows v_node, but it does not know the value of v_node or the direction of movement in any time step. Then, the unknown node generates new samples as follows: (21) P t = { l t i | l t i   is selected from   p ( l t i | l t − 1 i ) ,   where   l t − 1 i ∈ L t − 1   for all   1 ≤ i ≤ N } ,where P_t represents approximation of prior density at time t after prediction stage, L_t−1 represents approximation of posterior density at time t−1, and the transition equation for each sample described as follows: (22) p ( l t | l t − 1 ) = { 1 π α 2                 if         filter ( R ) = TRUE       0                                         if     filter ( R ) = FALSE .

The parameter α is needed for unknown nodes to provide enough variability in choosing new samples, i.e., the parameter α is used to limit a sample impoverishment phenomenon. Every time step, the beacon randomly moves a distance v_b in any direction from the previous location. A new sample is generated from each current sample by randomly choosing a point within a circle centered at the current location of the sample and the radius α when the filter(R) equal to TRUE, where filter(R) represents the filter condition of real location R of the unknown node. The details of condition filter(R) will be described in next stage (update stage). In general, the smaller the parameter a, the higher the accuracy of localization is obtained, but the longer time the stable phase of localization process is achieved. The appropriate value α is a tradeoff between the high precision of localization and the short period of localization time. Here, the appropriate value α could be determined empirically, such as α=0.1r.

Update: In this stage, the unknown node filters the impossible samples based on new observations. The unknown node updates samples as follows: (23) U t = { l t i | l t i   where   l t i ∈ P t   and   w ( l t i ) = 1 }where U_t represents approximation of posterior density at time t after the update stage, and the weight w ( l t i ) will be obtained by p ( o t | l t i ).

In order to state the description of observation p ( o t | l t i ), we define four states for every unknown node during the localization process.

Outsider: The unknown node is out of the radio range of the beacon, i.e., (24) d ( beacon c , R ) > r .

Insider: The unknown node is within the radio range of the beacon, i.e., (25) d ( beacon c , R ) ≤ r .

Arriver: The unknown node receives the current location announcement of the beacon, but did not receive the location announcement from the beacon’s previous location, i.e., (26) d ( beacon c , R ) ≤ r ∧ d ( beacon p , R ) > r .

Leaver: The unknown node received the preceding location announcement from the beacon, but does not receive the location announcement from the beacon’s current location, i.e., (27) d ( beacon c , R ) > r ∧ d ( beacon p , R ) ≤ r .

In this paper, we only rely on observations from the beacon. This has two advantages. First, the number of unknown nodes will not affect the accuracy of localization. Second, the computation and communication costs drop drastically, since nodes are no longer involved in the localization of other nodes [9]. There are two possible ways to gather observations from the beacon:

Once the unknown node is in Insider state, it gathers this observation, i.e. the filter condition of the real location R for any unknown node is: (28) filter ( R ) = { TRUE                     if         d ( beacon c , R ) ≤ r   FALSE                                                                                                           otherwise .

From an implementation perspective, in the first way, the beacon just transmits information about its own current location. Once the unknown node hears a location announcement from that beacon, an observation (also called constraint) is built to update.

In the second way, the beacon transmits both its current location and its location at the previous time step in each announcement. The unknown node needs to save state (Insider or Outsider) in previous time step with a tag: (31) tag = { TRUE                     if       state = Insider FALSE               if     state = Outsider .

This procedure at an unknown node is then as described by Algorithm 1. As can be seen, the tag which represented by variable StateTag is initialized to FALSE when the localization started (step 1–3). Then, we get the filter condition of location R for an unknown node (step 4–10). Finally, we get StateTag with step12–15.

The first way was adopted by Monte Carlo Localization (MCL) [19] and MSL [9]. As the approach ADO, we adopt second way in the update stage, however. We will evaluate the accuracy of them in Section 4.

Then, the weight of sample is determined by the filter condition: (32) w t = p ( o t | l t ) = { 1                 if           filter ( R ) = TRUE       0           if           filter ( R ) = FALSE .

Resampling: A common problem with particle filter is the degeneracy phenomenon. The degeneracy implies that a large computational effort is devoted to updating particles whose contribution to the approximation to p(l_t|o_l:t) is almost zero. A suitable measure of degeneracy of the algorithm is the effective sample size N_eff defined as: (33) N eff = 1 / ∑ i = 1 N   ( w t i ) 2 ,notice that N_eff ≤ N, and small N_eff indicates severe degeneracy. The method by which the effects of degeneracy can be reduced is to use resampling whenever a significant degeneracy is observed (i.e., when N_eff falls below some threshold N_T). The basic idea of resampling is to eliminate particles that have small weights and to concentrate on particle with large weights. We adopt a Systematic resampling algorithm [20] in this paper since it is simple to implement, takes O(N_s) time, and minimizes the Monte Carlo variation.

Finally, in order to give more clear description of MBL, we describe the main stages as a state machine diagram with labeled transitions, see Figure 1.

The key metric [9] for evaluating a localization algorithm is the accuracy of the location estimates or localization error. This is computed as follows: (35) Error = 1 M ∑ i = 1 M   ∥ e i − R i ∥ ,where M is the number of unknown nodes, R_i denotes the real location of the i-th unknown node, e_i denotes the location estimate of the i-th unknown sensor where i=1,…,M and □e_i − R_i□ denotes the distance between locations e_i and R_i. The errors shown in the simulation results are in terms of the radio range, i.e., the errors shown are computed by dividing the error in Equation (35) by the radio range of sensors. Most parameter settings for our simulations are those used in [19, 9]. Our results were obtained using sensors randomly distributed in a 500 units × 500 units square field, i.e. S=500×500. In our experiments, we set ideal radio range r=100, the number of unknown nodes M=100, the number of samples for an unknown node N=50, the parameter α=0.1r in the prediction stage, the maximum speed of beacon v_max=1.0r and two predefined adjustment tables (shown in Table 1 and 2) for adapting unless otherwise specified. Other simulation parameters of the unknown node are based on the MicaZ sensor node. The beacon’s movement is implemented using random waypoint mobility model.

In this section, we first evaluate MBL algorithm under various parameters configuration, such as the maximum speed of the beacon, the number of samples for an unknown node, and the impact of parameters α in the prediction stage. Then, under the same conditions (e.g. only use a single mobile beacon for localization), we compare the efficiency and accuracy of MBL, A-MBL, MSL, and ADO in different parameters configuration. In addition, we will consider a noisy environment with random noise added to measurements.

Maximum speed of the beacon: Figure 4 shows the convergence of MBL algorithm under four different v_max scenarios. In general, the faster the speed of the beacon, the quicker the stable phase is reached. Because the faster the speed of the beacon, the more the number of unknown nodes which the beacon could contact with. When v_max is greater than or equal to 0.6r, the convergence of MBL is particularly fast, and the localization process can be divided into the initialization phase and the stable phase. In the initialization phase, the estimate error decreases dramatically as new observations (the localization is improved by both the current observation and previous observations) are incorporated. In the stable phase, the impact of filter and the beacon’s mobility reach some balance, and the estimate error fluctuates around a minimum value. When v_max is greater than or equal to 0.8r, the curves of convergence (e.g. v_max=1.0r and v_max=1.2r shown in Figure 4) are very close to each other. Based on above comparison, we set v_max=1.0r as the default parameters configuration. When the time is greater than 1000 under this default parameters configuration, the error fluctuates slightly about a constant value (nearly 0.1).

Number of samples for MBL: Maintaining more samples for the MCL algorithm can improve accuracy, but requires additional memory [19]. Based on this comment, we should select an appropriate number of samples which does not affect the accuracy of localization and also does not waste memory either. Figure 5 shows the impact of sample size on location accuracy. The estimate error drops rapidly at the beginning, since a small number of samples cannot adequately reflect the probability distribution. The estimate error is fairly stable after sample size 50 and the accuracy improves only minimally by increasing the number of samples to 100. Hence, MBL is efficient in both memory and computation when the number of sample is close to 50. So, we choose 50 as default sample quantity to save memory and achieve good accuracy.

Parameter α for MBL: Figure 6 shows the impact of parameter α on location accuracy. If α = 0, i.e., all samples of the unknown node always keep static in the prediction stage. As a result, it cannot provide enough variability in choosing new samples. Hence, to improve the accuracy, the algorithms should increase the number of samples for each unknown node. In Figure 6, when α=0, the number of samples is needed to reach about 5,000 in order to achieve the similar precision as α=0.1r (but this number of samples is just 50). The parameter α significantly improves the accuracy and reduces the number of samples. Based on a number of experimental results, we adopt α=0.1r and the number of samples N=50 for an unknown node as default values.

In order to compare different algorithms under the same conditions, MSL in our evaluation will only use a single mobile beacon for localization.

Efficiency: The graph in Figure 7 shows the efficiency comparison for MBL (α=0.01r), A-MBL under same conditions except the parameter α. In order to obtain close precision, MBL spends more time with a fixed α than A-MBL with adapting α obtained from predefined adjustment tables.

Accuracy: Figure 8 shows the comparison of accuracy for MBL, A-MBL, MSL and ADO under same maximum speed of the beacon (v_max). Figure 8 illustrates that the curve of ADO drops to a certain value, then to the horizon, because the Arriver and Leaver information only be used once by ADO when the beacon passes by traverse route. Even though the beacon in ADO pass by the radio ranges of the unknown node on many occasions, the accuracy of unknown node will not be improved. Thus, as the time goes on, the average accuracy of all unknown nodes does not decrease. As shown in Figure 8, MBL and MSL, these two curvature of the curves are very similar to each other. As the time goes on, these two curves both reach to the stable phase. MBL shows nearly 50% better performance and nearly equal time of localization when compared to the MSL. Before the parameter α in A-MBL adjusted, the accuracy of MBL and A-MBL are very close. Once the parameter α in A-MBL has been adjusted, the accuracy of A-MBL is further improved. The accuracy of MBL still keeps stable, however. Thus, it can be seen from the Figure 8 that A-MBL outperforms MBL, MSL and ADO.

Speed of the beacon: Figure 9 shows how the localization error varies with the changing speed of the beacon. As a result, the faster speed of the beacon, the worse accuracy of ADO is achieved. Because the faster speed of the beacon which just travels the whole deployment area of a sensor network only once, the more the numbers of unknown nodes do not be localized. With further increase in the speed of beacon, the accuracy of MBL is close to MSL, for the area of Arriver and Leaver will become smaller. Finally, when v_b ≥ 2r, the area of Arriver and Leaver is equal to zero, and then MBL will degrade to MSL. The emergence of the significant error when the beacon (v_max=20) at low speed, just because the time (t=3,000) is too short for convergence of MBL, A-MBL, and MSL. As a result of Figure 9, A-MBL always performs better than both MBL and MSL in some range of speed.

Number of the unknown nodes: In this experiment we vary the number of unknown nodes from 20 to 200. We set the time of beacon movement as 3,000 to achieve the stable phase. As shown in Figure 10, the number of unknown nodes will not affect the accuracy of A-MBL, MBL and MSL, just because each of them only use the beacon for localization in our evaluations (not from neighborhood information).

All of our previous experiments assumed an ideal scenario where location sensory data are not influenced by irregular radio range and any receiver within the radio range of sender will hear the packets from that sender. However, on the one hand, variability in actual radio transmission patterns can have a substantial impact on localization accuracy depending on the localization technique [19]. On the other hand, the packet reception depends not only on the sender, but also on the receiver.

Figure 11 shows the impact of degree of irregularity on estimate error. The MBL and A-MBL are not substantially affected. We use degree of irregularity (DOI) to denote the maximum radio range variation in the direction of radio propagation. For example, if DOI = 0.1, then the actual radio range in each direction is randomly chosen from [0.9r, 1.1r].

If the unknown node (receiver) within radio range of the beacon (sender) will not hear a location announcement (such as network collisions or missed packages) from that beacon at some time, the beacon must keep moving for longer time to send location information which the unknown node could receive them, i.e. the time to achieve final stable phase of accuracy as ideal state will be extended. The accuracy of MBL and A-MBL will not be affected in such scenarios.