In general, the prediction stage in the particle filters uses the dynamic model to predict the state probability density forward from one measurement time to the next. Particularly in particle filters based mobile beacon-assisted localization, the dynamic model is needed for unknown nodes to provide enough variability in choosing new samples, i.e., which is used to limit a sample impoverishment phenomenon. MBL (or A-MBL) and Hang et al. [14] adopt uniform distribution and normal distribution as the dynamic model, respectively. To the best of our knowledge, no theoretical analysis or experimental evaluations are given to explain why this particular dynamic model should be adopted, however. In fact, on one hand, the choice of the dynamic model will affect the accuracy of localization results. On the other hand, the approach to simulating random variables of the dynamic model will affect the efficiency of localization process. To this end, an appropriate dynamic model should improve the location accuracy when locating unknown nodes and time efficiency when simulating random variables.

In general, the update operation uses the latest measurement to modify the prediction probability density. Particularly in particle filters based mobile beacon-assisted localization, the unknown node filters the impossible samples based on new observations. MBL (or A-MBL) only relies on the 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 [16]. Typically, network protocols maintain information about their neighbors in order to make informed decisions for routing, aggregation, and dissemination [17]. Some proposed approaches, such as Mobile and Static sensor network Localization (MSL) [16], use observations from the neighbors of unknown nodes, which may have greater communication costs, to increase efficiency and accuracy. It is still unknown whether or not to use observations from the neighbors in mobile beacon-assisted approaches, however. It is known that adopting neighbors’ observation should improve the location accuracy when locating unknown nodes and communication efficiency when collecting observations.

The parameters in the dynamic model will affect the accuracy and efficiency of particle filter-based mobile beacon-assisted localization. As an example concerning the number of samples [3,16,18], under the same conditions, keeping more samples improves efficiency at the beginning of localization. The time complexity of the update stage and the memory requirements to keep samples are both linear in the number of samples needed for the estimation, however. Another example about the parameter α in the dynamic model [3,18], a greater value of parameter in the dynamic model improves the efficiency at the beginning of localization and the accuracy at the other extreme. Though adopting predefined adjustment tables in A-MBL is convenient and effective, obtaining these tables is difficult. To this end, the key question in A-MBL is how to determine the number of samples and the value of parameter in the dynamic model to achieve more flexibility. It is known that the approach should be unrelated to the deployment region, the location time and the number of unknown nodes, i.e., the approach should be self-adapting.

To address above problems, we first describe three different dynamic models and some related approaches to simulating random variables of the dynamic models. The theoretical analysis and experimental evaluation will be given to explain why the particular dynamic model should be adopted. Then, we describe how to make use of the neighbors’ observations in the update stage. Next, we give the condition whether the unknown node use the observation from the neighbors to improve the accuracy. Finally, we analysis and compare some approaches to judge the localization to reach the stable phase. As a result, a self-adapting mechanism will be proposed.

Major contributions of this paper are as follows:

In order to improve the time efficiency in the prediction stage of particle filter-based mobile beacon-assisted localization, we give the theoretical analysis and experimental evaluations to suggest which probability distribution should be adopted in the dynamic model.

The rest of this paper is organized as follows: Section 2 describes the algorithm of A-MBL. Section 3 analyses and discusses the different dynamic models, the neighbors’ observation, and the self-adapting mechanism. Section 4 shows and discusses our evaluation results. Section 5 gives an overview of related works. Finally, Section 6 concludes our work.

Similar to the assumption in [3], 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., a GPS, which allows it to know its location at all times. 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 beacon is capable of moving a distance in a time step (v_b) 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. We do not assume very tightly synchronized clocks. In a realistic deployment, it would be necessary to deal with network collisions and account for missed messages [18].

A-MBL uses the particle filter approach (also called sequential Monte Carlo method) to perform Bayesian filter based localization 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 [19].

Let { < l t i ,   w t i > ,   i = 1 , … , N } denotes a random measure that characterizes the posterior density p(l_t | o_t) which denotes the current location estimate l_t conditioned on the observation o_t from the beacon at time t, where L t = { l t i ,   i = 1 , … , N } denotes a set of support samples (or called particles) with associated weights W t = { w t i ,   i = 1 , … , N } at time t and N denotes the number of samples of an unknown node.

The details of the A-MBL are as follows:

Initialization: In this stage, all unknown nodes have no information about their locations. The initial set of samples Equation 1 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 2. The weight equal to one represents the importance of corresponding sample, which infers one of the location estimates of the unknown node: (1) L 0 = { l 0 i | l 0 i ∼ p ( l 0 ) = 1 S ,   i = 1 , … , N } (2) w 0 i = 1 ,   i = 1 , … , Nwhere L₀ denotes the initial set of each sample, p(l₀) denotes initial location probability density, the symbol ∼ denotes sample generated sign, i.e., the samples on the left side are generated from the probability density on the right side, and w 0 i denotes the initial weight of each sample.

Prediction: In this stage, we adopt a dynamic model in which the unknown node is capable of moving a distance in a time step (v_node) in any direction where 0 ≤ v_node ≤ α. The unknown node knows α, 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: (3) 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     i = 1 , … , N }where P_t represents the approximation of prior density at time t after the prediction stage, L_t−1 represents the approximation of posterior density at time t−1, and the transition equation for each sample described as follows: (4) p ( l t | l t − 1 ) = { 1 π α 2 if     filter ( R ) = TRUE 0 if     filter ( R ) = FALSEwhere filter(R) denotes the unknown node receives the current location announcement of the beacon, but did not receive the location announcement from the beacon’s previous location, or the unknown node received the preceding location announcement from the beacon, but does not receive the location announcement from the beacon’s current location. For more details about filter(R) see [3].

Update: In this stage, the unknown node filters the impossible samples based on new observations. The unknown node updates samples as follows: (5) U t = { l t i | l t i   where   l t i ∈ P t       and       w ( l t i ) = 1 }where U_t represents the 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 ). The weight of sample is determined by the filter condition: (6) w t = p ( o t | l t ) = { 1 if     filter ( R ) = TRUE 0 if     filter ( R ) = FALSE

Resampling: A-MBL adopts 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.

Adapting: A-MBL adopts two predefined adjustment tables, one for the number of samples N, and the other for the parameter α. Once some record in the table is matched, the number of samples and the value of α in the unknown node will be adjusted according to the corresponding time.

The complete A-MBL for every unknown node is shown in [3].

In this section, we use the term “random numbers” to mean independent random variables from a probability distribution, i.e., we use random numbers to simulate random variables from some probability distribution. For example, let random variable Z is uniformly distributed over (0, 1). If U is uniformly distributed over (0, 1) random numbers, then Equation 7 denotes random variables Z are simulated by random numbers U: (7) Z = UWe denote the horizontal X and vertical Y location of the i-th sample of an unknown node at time t by l t i = ( X , Y ).

The aim of combined with the observation from the neighbors is to improve performance, and then get results in faster convergence of the algorithms. To this end, we modify the sampling procedure in MBL to use observations from the neighbors, i.e., each unknown node not only relies on observations from the beacon, but also from the neighbors that have better estimates.

As the approaches described in [16], after choosing a sample from the unknown node, its weight is determined using the beacon and neighbors’ observation. The weight of a sample lⁱ chosen for the unknown node p, is described as w_lⁱ (p), which is computed as follows: corresponding to the k-th neighbor q_k of the unknown node p, we set a partial weight for the sample lⁱ, w′_lⁱ (q_k). The weight w_lⁱ (p) of sample lⁱ is the product of the partial weights w′_lⁱ (q_k) obtained corresponding to each neighbor q of the unknown node p. That is: (15) w l i   ( p ) = ∏ k = 1 n w l i ′   ( q k )where n is the number of neighbors of the unknown node p.

The partial weights w′_lⁱ (q_k) of the sample lⁱ corresponding to observations from the beacon or the neighbors of unknown node are computed as follows:

If the unknown node receives the observation from the beacon, then the partial weight w′_lⁱ (q_k) of sample lⁱ will be computed by the Equation 6 as MBL (or A-MBL).

To summarize the inter-dependencies of the weights amongst the relevant node, we visualize the observations in Figure 2. In Figure 2, the black dot denotes the current unknown node and the red dots denote the samples of it. The green dots denote the neighbor of the unknown node and the blue dots denote the samples of neighbors. The dotted lines between the red dot and green dot denote the partial weight w′_lⁱ (q_k) of sample lⁱ. The dash dot line between the red dot and the blue dot denote the weights w ( q k j ) of the sample q k j of the neighbor q_k.

The sample lⁱ is kept if w_lⁱ (p) is greater than a threshold value β.

An important question is that the right conditions should be met in order to combine with the observation from neighbors is to improve accuracy and obtain faster convergence of the algorithms.

Nagpal et al. [22] have claimed that πr/4n_d is a lower bound for the error in any range-free localization algorithm in static sensor networks, where n_d is the node density, the average number of nodes in one hop transmission range, i.e., n_d is the average number of neighbors to an unknown node. In our scenarios, using a single mobile beacon that knows its position is broadly equivalent to using many static beacons each broadcasting once. We may consider each observation from mobile beacon to an unknown node as one constraint (bounded) from the static beacon in the radio range of the unknown node. Thus, the number of observations to the unknown node in mobile beacon-assisted approach is equal to the number of constraints from static neighbor beacons of the unknown node, i.e., the number of observations in mobile beacon-assisted approach is equal the node density, n_d. As a result, if the number of observations from the mobile beacon to the unknown node is larger than n_d, the neighbors’ observation will not improve the precision of the unknown nodes.

How to determine the number of samples and the value of parameter α in the dynamic model to achieve more flexibility is the key question to A-MBL. On the one hand, likelihood-based adaptation [23], KLD-Sampling adaptation [24], and coefficient of variation [3] do not judge the accuracy of MBL to reach the stable phase. On the other hand, the values in such predefined adjustment tables in A-MBL are related to the localization time, i.e., different scale deployment of the sensor nodes requires different predefined adjustment tables. Obtaining these tables is hard work. The intuitive answer to those questions is that MBL needs a self-adapting mechanism to achieve more flexibility, and the corresponding approach is called Self-Adapting MBL (SA-MBL).

The following major factors should be considered to satisfy the self-adapting mechanism:

The approach should judge the localization to reach the stable phase as the effect of pre-defined table in A-MBL.

In other words, in order to better judge the stable phase of localization, we need some stable attribute obtained from the unknown nodes themselves.

We know that the locations of samples are in continuous convergence as new observations are incorporated in the initialization phase. When the accuracy of MBL reaches the stable phase, the distribution of samples will also reach a stable phase. To this end, we hope to find the statistical distribution of samples which can be closer to above-mentioned objectives.

We suppose that the samples of an unknown node are sampling from a SUD when we adopt the SUD as the dynamic model. Then, we want to obtain the parameters in the SUD of the samples when the localization reaches the stable phase. We adopt the Method of Moments Estimators (MME) to estimate the parameter of the SUD.

To a single unknown node, the horizontal X and vertical Y location of the i-th sample have two unknown parameters, respectively. Let the horizontal location X of the i-th sample have two unknown parameters [a, b]. X¹, X²,…,X^N are horizontal locations of samples l¹, l²,…,l^N of size N, respectively. Then, the MME for a and b is as follows: (17) a = X ¯ − 3 N   ∑ i = 1 N ( X i − X ¯ ) 2 b = X ¯ + 3 N   ∑ i = 1 N ( X i − X ¯ ) 2where: (18) X ¯ = ∑ i = 1 N X i

We adopt Equation 19 to judge the accuracy of MBL to reach the stable phase, where AME denotes Average of Moments Estimators. When AME reaches the stable phase, the number of samples and the parameters in the dynamic model will be adjusted. Once the parameters in the dynamic model have been adjusted, the precision will be improved and AME will reach a new stable phase. The above stability, adjustment and re-stability process does not stop, until the localization obtains the desired positioning accuracy: (19) AME = 1 M   ∑ i = 1 M ( b i − a i )where M is the number of unknown nodes, a_i and b_i denote the parameters of the uniform distribution of the i-th unknown node.

From an implementation perspective, Equation 17 will be computed in the unknown node at a certain time interval. All unknown nodes do not require send the result of (b – a) to the beacon at one time, but each unknown node sends the value when it contacts the beacon. Equation 19 will be computed in the beacon and the beacon only maintains (b – a) of the recently M contacted unknown nodes.

In this section, we will evaluate the accuracy of different probability densities adopted as the dynamic model in the prediction stage for MBL. The graph in Figure 3 shows the comparison of accuracy for SUD_MBL, CUD_MBL, and ND_MBL, where SUD_MBL, CUD_MBL, and ND_MBL denote different dynamic models SUD, CUD, and ND as shown in Equation 8, 10, and 12 adopted by MBL respectively in the prediction stage.

As shown in Figure 3, on the one hand, for SUD_MBL and CUD_MBL, the curvatures of the two curves are always very close to each other, i.e., different dynamic models SUD and CUD will not affect the accuracy of MBL. On the other hand, in the initialization phase, the convergence of SUD_MBL and CUD_MBL is much faster than ND_MBL, but, in the stable phase, the accuracy of ND_MBL is a little bit higher than in SUD_MBL and CUD_MBL. In order to compare the accuracy of those different dynamic models for MBL from the quantitative point of view, we give Table 2 to show the accuracy of MBL under three different dynamic models. These experimental results are the average of 20 executions with different pseudorandom number generator seeds. In Table 2, ND_MBL shows nearly 20%–25% better accuracy when compared to the SUD_MBL and CUD_MBL. Shown in Section 3.1.3, we have known that SUD_MBL is more efficient than ND_MBL. Taking into account both the efficiency and accuracy of localization, which the dynamic model should we choose?

We give another result as shown in Figure 4. Compared to the SUD_A-MBL which denotes A-MBL adopting SUD as the dynamic model, ND_MBL has no advantage due to the invariable parameter in the dynamic model, i.e., the accuracy of MBL with adapting mechanism are higher than ND_MBL with different dynamic model to improve accuracy. That is to say, SUD_A-MBL achieves higher accuracy and does not lose efficiency. Of course, if ND_MBL also adopts an adapting mechanism, then ND_MBL can achieve higher accuracy. We show the accuracy comparison results between SUD_A-MBL and ND_A-MBL in Figure 5. Though SUD_A-MBL and ND_A-MBL will reach almost the same accuracy at the end, ND_A-MBL needs more time to do so than SUD_A-MBL, i.e., the efficiency of ND_A-MBL is lower because it spends longer time achieving the stable phase of localization process.

The threshold value β which mentioned in Section 3.2 depends on the number of neighbors of an unknown node. In [16], MSL uses β = (0.1)^neighbor, where neighbor denotes the number of neighbors of an unknown node. In our experiments, we set neighbor = n_d, i.e., β = (0.1)^n_d. To reduce computational complexity, the unknown node will use the observation from the neighbors in some interval.

The graph in Figure 6 shows the comparison results of accuracy for SUD_MBL and NEIGHBOR_SUD_MBL when the time interval is 200, where NEIGHBOR_SUD_MBL denotes SUD_MBL relying on observation not only from the beacon, but also from its neighbors in the update stage. As shown in Figure 6, SUD_MBL and NEIGHBOR_SUD_MBL, these two curvature of the curves are always very close to each other, i.e., combined with the observation from neighbors will not improve the accuracy of MBL and have not faster convergence under these conditions.

The graph in Figure 7 shows the comparison results of accuracy for SUD_MBL and NEIGHBOR_SUD_MBL when the time interval is 20. When the time is 20, the accuracy of NEIGHBOR_SUD_MBL is suddenly improved, just because of the impact of neighbor’s observation. After the time is larger than 106, the accuracy will not be improved by a neighbor’s observation. Obviously, NEIGHBOR_SUD_MBL is more accurate than SUD_MBL from 20 to 106.

Under the same conditions, we show the result of SUD_MBL concerning the number of observations from the mobile beacon to the unknown node. Table 3 shows the average number of observations of unknown nodes from the beacon. As shown the second column in Table 3, when the average number of observations (Number = 10) is equal to n_d (n_d = 10), SUD_MBL is still in the stabilization process (the time is 106), i.e., the samples of unknown node cannot represent the location of unknown node at this time. With the increase of the number of observations from the beacon, this number will far exceed the n_d, i.e., the neighbors’ observation will not improve the accuracy of SUD_MBL.

Figure 8 shows the coverage of AME for three different α of SUD_MBL. Whether the parameter α is 0.1r, 0.05r or 0.01r, as the time goes on, the AME of the unknown node will achieve stable phase eventually. To catch the AME of the stable phase under 10 different parameters α for later use, we give the Table 4 to show the results. These experimental results are the average of 20 executions with different pseudorandom number generator seeds. Based on this table, SA-MBL will adjust the value α according to the corresponding AME. In general, when this condition | AME_c - AME_i | < ε is satisfied, where AME_c denotes the current AME maintained in the beacon and AME_i represents some value listed in the i-th row of the Table 4. Then, the current parameter α_c maintained in the unknown node will be set as α_c = α_i+1. For example, when current value of AME in the beacon is nearly 34, the parameter α will be adjusted to 0.07r. Similarly, the number of samples N will be adjusted accordingly based on Table 4.

The graph in Figure 9 shows the comparison of accuracy between A-MBL and SA-MBL. As shown in this figre, for A_MBL and SA_MBL, the curvature of the two curves are always very close to each other, i.e., MBL with Self-Adapting mechanism will get almost the same performance with A-MBL for which obtaining predefined adjustment tables is difficult. The graph in Figure 10 shows the efficiency and accuracy comparison for SA-MBL under the same conditions except for the deployment region. In order to obtain close precision, the largest the deployment region (S = 700 × 700) will spend the longest convergence time among these experiments.

In another experiment we vary the unknown node density from 2 to 20. We set the time of beacon movement as 3,000 to achieve the stable phase. As shown in Figure 11, the unknown node density will not affect the accuracy of SA-MBL. That is to say, the adapting mechanism in SA-MBL can be unrelated to the deployment region, the location time and node density, i.e., the approach is actually self-adapting. Finally, the localization error bar in Figure 11 shows that the SA-MBL is stable.