When sensors of a network are first deployed, they may apply the Clustering Algorithm via Waiting Timer (CAWT) from [38] to partition the sensors into clusters using the waiting timer (1) WT i ( k + 1 ) = γ ⋅ WT i ( k )where WT i ( k ) is the waiting time of sensor i at time step 0 < γ < 1 is inversely proportional to the number of neighbors. If the random waiting timer expires and none of the neighboring sensors are in a cluster, then sensor i declares itself a clusterhead. It then broadcasts a message notifying its neighbors that they are assigned to join the new cluster with ID i. After applying the CAWT, there are three different kinds of sensors: (1) the clusterheads (2) sensors with an assigned cluster ID (3) sensors without an assigned cluster ID, which will join any nearby cluster later and become 2-hop sensors. Thus, the topology of the ad-hoc network is now represented by a hierarchical collection of clusters. Figure 1 (left) shows an example of the cluster formation in a random network of 100 sensors with R/ℓ = 0.175, where R is the transmission range and ℓ is the side length of the square.

When the estimation procedure starts in a cluster-based network topology, a clusterhead called sensor 1 locates itself at the origin(0, 0) and selects the left-hand or the right-hand coordinate system as the local coordinate assignment. Then sensor 1 detects its neighbors and deploys one of the neighboring sensors, sensor 2, to the x-axis at (d₁₂, 0) based on the distance information d₁₂. A third sensor is selected to be sensor 3 which has connectivity to both sensors 1 and sensor 2. Given the known positions of sensors 1 and 2 and distance information, d₁₃ and d₂₃, sensor 3 can estimate its own location to the responding coordinate system. Therefore, sensors 1, 2 and 3 considered as a group form a basis for this local coordinate system. The solvability of the network localization problem is detailed in [39], which suggests that if the three known sensors are no-three-in-line, the network localization problem is solvable and the unknown position can be determined in the two-dimensional space. Accordingly, all other sensors which are within communication range of these sensors can then estimate their positions with respect to this local coordinate system. Similarly, as the cluster of known sensors grows, the location of each of the unknown sensors can be determined from three neighboring known sensors. Thus, the sensor locations can be obtained by building a local coordinate system from the clusterhead and applying multilateration to enable ad-hoc deployed sensor nodes to accurately estimate their locations by using known sensor locations and neighboring distance measurements. Figure 1 (right) shows the estimation procedures of ad-hoc wireless sensor networks in the two-dimensional space with sufficient connectivity.

Suppose that a sensor does not know its position but is able to receive information from other sensors which are assumed to have relative local position estimates. There are many ways to “solve” this sensor location problem. This section details the Bayesian particle filter method which may be preferred because it is robust to noisy measurements, it allows for flexible information transmission, and it can be robust to lost or lossy data.

Assume the mth sensor obtains a new measurement from (at least) three sensors and estimates its own position using the particle filter. The sensor position is given by the discrete-time state equation (2) x k m = Φ x k − 1 m + Γ w kwhere x k m is the position of the sensor and w_k is an uncorrelated Gaussian diffusion term describing the uncertainty. Note that this system equation is suitable for many different systems and the only changes will be the matrixes Φ and Γ, which depend on the system model. For instance, the differences of the methodology between a moving sensor and a fixed sensor are the choices of Φ and Γ, and the rest of the methodology is the same. Hence, the same basic procedure can be used in other tasks such as target tracking. Here we assume the sensors do not move between observations, Φ is the two-dimensional identity matrix, and Γ is a zero matrix. The measurement term for the mth sensor is (3) Z k m = ∑ ℓ ∈ I m | ∥ x k m − x k ℓ ∥ − d m ℓ | + v kwhere the sum is over the nearby sensors x k ℓ, I_m is the index set of estimated known sensors, ||·|| denotes the ℓ₁-norm ranging measurement, d_mℓ represents the measured distance between the sensors and may be approximated in application by the inverse of the signal strength or by calculated from the time delay between transmission and reception [40], and the measurement noise v_k is another uncorrelated zero mean Gaussian white noise process.

Before measurements are taken at k = 1, the initial state vector is obtained by applying the distance measurements as constraints on the x and y coordinates of the unknown sensor. The idea in [7], using known sensor positions and the bounding-box algorithm to extrapolate unknown sensor positions, inspires us to choose a proper prior density for generating initial samples. Figure 2 shows how the distance information can be use to obtain the x and y coordinate bounds of the unknown sensor. Therefore, the unknown sensor combines its bounds on the coordinates to form a bounding box, which provides a good set of initial samples for the particle filtering. The particle filter method is shown in Table 1.

Due to the error caused by the location estimation algorithm (the estimation error) and the error intrinsic to the problem (noisy distance measurements), location adjustment algorithms are needed in order to improve the estimation accuracy and limit the propagation errors. After the sensor is located near to its true position, a refinement technique is applied to elevate the estimates immediately. This subsection details the operation of a distributed model for refining the location estimates based on the initial position information from Phase I.

Because the particle filter loses diversity in the samples for static models, the Metropolis-Hastings (M-H) algorithm [41] may be used to generate new samples and provide improved estimation accuracy. The basic idea of the M-H algorithm is to simulate an ergodic Markov chain whose samples are asymptotically distributed according to the target probability distribution π(·) and use a candidate proposal distribution q(x_k(i), ·) to select the candidate of the current state independently with the acceptance probability (4) α ( x k ( i ) , x k ′ ( i ) ) = min { 1 , π ( x k ′ ( i ) ) q ( x k ′ ( i ) , x k ( i ) ) π ( x k ( i ) ) q ( x k ( i ) , x k ′ ( i ) ) } .

Therefore, instead of using a centralized accumulator host to adjust sensor locations, applying the Markov chain Monte Carlo (MCMC) method on each estimated sensor right after the location estimation allows estimation error and propagation error to be reduced in a distributed way. Here we summarize the M-H algorithm with the initial value x₀(i) in Table 2. The performance evaluation and the discussion of the proposed location adjustment algorithms are depicted in Section 5.

This section shows how the geometrical and communication requirements change when merging two coordinate systems to a single one. At some point, as the position estimation proceeds, the coverage of two coordinate systems begin to overlap, at which time they may be merged together into a single coordinate system. Eventually, all sensors have been gathered into one coordinate system and the sensor location problem is solved. If there is GPS (or other absolute measures) available, then this coordinate system can be referenced to standard measurements. If there are no GPS available, then the coordinate system is relative.

Based on the refined position estimates in Phase II and the relative coordinate system in Phase III, when the measurement does not meet the required estimation accuracy (e.g., the measurement variance is larger than the accuracy threshold), the estimated sensor, say sensor m, may broadcast a fusion message to its neighbors and trigger the cooperative sensor fusion to resolve conflicts or disagreements, and to complement the observations of the environment. The cooperative estimation system can be organized by sensor m with collecting the neighboring observations or grouping nearby two sensors into a measurement system with group IDs based on the neighboring geometric information. This subsection introduces the centralized scheme, the progressive scheme, and the distributed scheme for cooperative position estimation (Figure 5).

In order to investigate the estimation performance of the position measurement, Figure 6 depicts the measurement using known positions and distance information to obtain the unknown sensor position. Notice that sensors a, b, and c are three known sensors with estimated positions (x_a, y_a), (x_b, y_b), and (x_c, y_c), respectively. Given the distance measurements, the unknown sensor position (x_e, y_e) can be computed by triangulation, which is (24) x e = d ce ′ x a + d ae ′ x c d ae ′ + d ce ′ + d ae ( y c − y a x c − x a ) sin α 1 + ( y c − y a x c − x a ) 2 (25) y e = d ce ′ y a + d ae ′ y c d ae ′ + d ce ′ ± d ae sin   α 1 + ( y c − y a x c − x a ) 2where d_ae′ = d_ae cos α, d_ce′= d_ac – d_ae′, (26) α = arccos ( d ae 2 + d ac 2 − d ce 2 2 d ae d ac )and (27) β = arccos ( d ab 2 + d ae 2 − d be 2 2 d ae d ab )

Due to the errors in distance measurement, the position and distance estimates may be described by normal random variables. Let D_ij denote the normal random variable for the distance estimate between a pair of sensors i and j with D ij ∼ N ( μ D ij , σ D ij 2 ). Let X_i and Y_i be the normal random variables for the known estimated x and y coordinates of sensor i, respectively, with X i ∼ N ( μ X i , σ X i 2 ) and Y i ∼ N ( μ Y i , σ Y i 2 ). Now we rewrite (24) as (28) X e = W 1 + W 2where the random variables (29) W 1 = D ce ′ X a + D ae ′ X c D ae ′ + D ce ′ (30) W 2 = D ae ⋅ W 3 ⋅ W 4 W 5 (31) W 3 = Y c − Y a X c − X a (32) W 4 = sin   α     = sin   ( arccos ( D ae 2 + D ac 2 − D ce 2 2 D ae D ac ) ) = 1 − ( D ae 2 + D ac 2 − D ce 2 2 D ae D ac ) 2 (33) W 5 = 1 + ( Y c − Y a X c − X a ) 2 = 1 + W 3 2 (34) W 6 = D ce ′ Y a + D ae ′ Y c D ae ′ + D ce ′

For the random variable W₁, let us first consider the distribution function of D_ae′. Based on Figure 6, the random variable D_ae′ is (35) D ae ′ = D ae cos   α = V 2 D aewhere V = D ae 2 + D ac 2 − D ce 2. Thus, the random variable V is the weighted sum of quadratic forms of independent normal random variables (i.e., a linear combination of noncentral chi-squared random variables [53,54]), which is (36) V = ∑ k = 1 3 a k U kwhere a_k represents the weight and U_k is an independent non-central chi-squared random variable with p_k degrees of freedom and a non-centrality parameter δ_k. Hence, the characteristic function of U_k is (37) φ U k ( t ) = ( 1 − 2 i t ) − pk / 2 exp ( i t δ k 1 − 2 i t )where i = (−1)^1/2, and the characteristic function of V is (38) φ V ( t ) = ∏ k = 1 3 φ U k ( a k t )with the mean μ V = Σ k = 1 3 a k ( p k + δ k ) and the variance σ V 2 = 2 Σ k = 1 3 a k 2 ( p k + 2 δ k ). In our case, p_k = 1 ∀k, a₁ = a₂ = 1, a₃ = −1, δ 1 = μ D ae 2, δ 2 = μ D ac 2, and δ 3 = μ D ce 2.

Referring to [55, 56], the normal approximation may be applied to obtain the distribution function, which yields (39) D ae ′ ∼ N ( μ D ae ′ , σ D ae ′ 2 )with (40) μ D ae ′ = μ V 2 μ D ac (41) σ D ae ′ 2 = 4 σ D ac 2 μ D ′ ae 2 + σ V 2 4 μ D ac 2

Using the analytical techniques for normal approximation, the distribution function of W₁ can be approximated by (42) W 1 ∼ N ( μ W 1 , σ W 1 2 )with (43) μ W 1 = μ D ce ′ μ X a + μ D ae ′ μ X c μ D ae ′ + μ D ce ′ (44) σ W 1 2 = ( σ D ′ ae 2 + σ D ′ ce 2 ) μ W 1 2 + μ D ′ ce 2 σ X a 2 + μ X a 2 σ D ′ ce 2 + μ D ′ ae 2 σ X c 2 + μ X c 2 σ D ′ ae 2 ( μ D ae ′ + μ D ce ′ ) 2

In order to derive the probability density function of W₂, let us first consider the random variables W₃, W₄, and W₅, respectively.

For the random variable W₃ (31), let W₃ = M₁/M₂. Therefore, the variables M₁ = Y_c – Y_a and M₂ = X_c – X_a follow a bivariate normal distribution M 1 ∼ N ( μ Y c − μ Y a , σ Y c 2 + σ Y a 2 ) and M 2 ∼ N ( μ X c − μ X a , σ X c 2 + σ X a 2 ). [55] and [57] show that as the ratios μ_M1/σ_M2 and μ_M2/σ_M2 increase and the probability that M₂ is negative tends to zero, then the probability density function of W₃ is given by (45) W 3 ∼ N ( μ W 3 , σ W 3 2 )with (46) μ W 3 = μ M 1 μ M 2 (47) σ W 3 2 = μ M 1 2 σ M 2 2 μ M 2 4 + σ M 1 2 μ M 2 2

For the random variable W₄ (32), it can be further expressed by W 4 = 1 − Q 2, where the random variable Q = V/(2D_aeD_ac) and the random variable V is defined as in (36). Hence, the density function of Q is given by (48) Q ∼ N ( μ Q , σ Q 2 )with (49) μ Q = μ v 2 μ D ae μ D ac (50) σ Q 2 = 4 μ Q 2 ( μ D ae 2 σ D ac 2 + μ D ac 2 σ D ae 2 ) + σ V 2 4 μ D ae 2 μ D ac 2With Taylor development of W₄ atμ_Q, we obtain (51) W 4 ≈ 1 − μ Q 2 − μ Q 1 − μ Q 2 ( Q − μ Q )Therefore, the density function of W₄ is approximated by W 4 ∼ N ( μ W 4 , σ W 4 2 ) with μ W 4 = 1 − μ Q 2 and σ W 4 2 = μ Q 2 σ Q 2 / ( 1 − μ Q 2 ). Similarly, for W₅ defined as in (33), with Taylor development at μ_W3 and applying the density function of W₃, the density function may be approximated by W 5 ∼ N ( μ W 5 , σ W 5 2 ) with μ W 5 = 1 + μ W 3 2 and σ W 5 2 = μ W 3 2 σ W 3 2 / ( 1 + μ W 3 2 ).

Accordingly, referring to (30) and applying the normal approximation techniques, the density function of the random variable W₂ yields (52) W 2 ∼ N ( μ W 2 , σ W 2 2 )with (53) μ W 2 = μ D ae μ W 3 μ W 4 μ W 5 (54) σ W 2 2 = μ D ae 2 μ W 3 2 ( σ W 5 2 μ W 4 2 μ W 5 4 + σ W 4 2 μ W 5 2 ) + μ W 4 2 μ W 5 2 ( μ D ae 2 σ W 3 2 + μ W 3 2 σ D ae 2 )Thus, the density function of the estimated x coordinate X_e is X e ∼ N ( μ X e , σ X e 2 ) with μ_Xe = μ_W1 + μ_W2 and σ X e 2 = σ W 1 2 + σ W 2 2, where μ_W1, σ W 1 2, μ_W2, and σ W 2 2 are described as in (43), (44), (53), and (54), respectively.

For the estimated y coordinate Y_e, it can be rewritten as (55) Y e = W 6 ± D ae ⋅ W 4 W 5Following the same analysis procedures for X_e and applying the normal approximations, the density function yields (56) Y e ∼ N ( μ Y e , σ Y e 2 )with (57) μ Y e = μ W 6 ± μ D ae μ W 4 μ W 5 (58) σ Y e 2 = σ W 6 2 + μ D ae 2 ( σ W 5 2 μ W 4 2 μ W 5 4 + σ W 4 2 μ W 5 2 ) + σ D ae 2 μ W 4 2 μ W 5 2where the distribution of W₆ yields the same results as described for W₁ by substituting Y_a and Y_c for X_a and X_c in (43) and (44), respectively.

Accordingly, the localization performance can be assessed with the uncertainty in the distance and position information. Moreover, based on the above analysis, the effect of error propagation with imperfect position and distance information may be approximately depicted. Note that this analysis applies normal approximations to describe the probability density functions of the position estimation. The numerical results will be illustrated in Section 5. in order to compare the simulation results and validate the appropriateness of normal approximations.

The relationship of mapping between the state and observations is not known precisely because of the measurement errors and the uncertainty associated with the system model. In order to evaluate estimation behavior, the distribution of the measurement term (3) for sensor m is derived to extract information about estimation accuracy.

To proceed with the analysis, notations and assumptions are introduced to capture the sensible measurement performance. We rewrite the measurement term (3) as (59) Z m = ∑ ℓ ∈ I m | || P m − P ℓ || − D m ℓ | + V m (60) = ∑ ℓ ∈ I m | D ^ m ℓ − D m ℓ | + V mwhere Z^m is the random measurement term, I_m is the index set of estimated known sensors, P^m is the estimated position of sensor m, P^ℓ is the estimated position of sensor ℓ, D̂_mℓ denotes the ℓ₁-norm ranging measurement obtained from estimated positions,D_mℓ is the ranging measurement between sensors m and ℓ, and V^m is the measurement noise. Suppose that Z^m, P^m, P^ℓ, D_mℓ, and V^m are assumed to be normal random variables which distributions are Z m ∼ N ( μ Z m , σ Z m 2 ), P m ∼ N ( μ P m , σ P m 2 ), P ℓ ∼ N ( μ P ℓ , σ P ℓ 2 ), D ^ m ℓ ∼ N ( μ D ^ m ℓ , σ D ^ m ℓ 2 ), D m ℓ ∼ N ( μ D m ℓ , σ D m ℓ 2 ), and V m ∼ N ( μ V m , σ V m 2 ).

Now denote F_mℓ as (61) F m ℓ = | D ^ m ℓ − D m ℓ | = | G m ℓ |where G_mℓ = D̂_mℓ – D_mℓ, which is a normal random variable with the distribution G m ℓ ∼ N ( μ G m ℓ , σ G m ℓ 2 ) with mean μ_Gmℓ = μ_D̂m – μ_Dmℓ and variance σ G m ℓ 2 = σ D ^ m ℓ 2 + σ D m ℓ 2 = σ P m 2 + σ p ℓ 2 + σ D m ℓ 2 Consider now the situation where F_mℓ = |G_mℓ|. Therefore, the distribution of the absolute measurement F_mℓ is described as the folded normal distribution [58]. The probability density function of the resulting folded normal distribution is (62) h F m ℓ ( f m ℓ ) = 1 2 π σ G m ℓ [ e − ( f m ℓ − μ G m ℓ ) 2 / 2 σ G m l 2 + e − ( f m ℓ + μ G m ℓ ) 2 / 2 σ G m ℓ 2 ] ,             f m ℓ ≥ 0  with mean (63) μ F m ℓ = 2 π σ G m ℓ e − μ G m ℓ / 2 G m ℓ 2 + μ G m ℓ [ 1 − 2 M ( − μ G m ℓ / σ G m ℓ ) ]and variance (64) σ F m ℓ 2 = μ G m ℓ 2 + σ G m ℓ 2 − μ F m ℓ 2where (65) M ( a ) = 1 2 π ∫ − ∞ a e − t 2 / 2 dt

In order to estimate the position, at least three nearby location-aware sensors are needed. Without loss of generality, let the neighboring sensor ID be ℓ = 1, 2, and 3 and assume that the folded normal random variables are independent, which is a reasonable condition in our case. Thus, the distribution of the sum of the folded normals S m = ∑ ℓ = 1 3 F m ℓ can be derived by convolution, which is (66) h S m ( s m ) = 2 π 1 ( σ F m 1 2 + σ F m 2 2 ) σ F m 3 2 ∫ 0 s m e A ( x ) [ erf( σ F m 1 B ( x ) ) + erf ( σ F m 2 B ( x ) ) ]   d x ,   s m ≥ 0with A ( x ) = − x 2 / 2 ( σ F m 1 2 + σ F m 2 2 ) + ( s m − x ) 2 / 2 σ F m 3 2 and B ( x ) = x / 2 σ F m 1 2 σ F m 2 2 ( σ F m 1 2 + σ F m 2 2 ), where erf(·) is the error function and σ F m ℓ 2 is described as in (64), which is related to the variances of position and distance estimates.

Suppose that the measurement noise is negligible. Thus the distribution of the random measurement term Z^m may be described by (66), which highly depends on the deviations of estimated anchor positions and estimated distances. Observe that the measurement term Z^m is actually a way to depict estimation accuracy of the system since the objective is to minimize the estimation error. Based on the above settings, (66) may also represent an ideal measurement performance for realistic estimation experiments. However, with the deviations of distance estimates, in order to suppress the measurement error we may obtain inaccurate localization results though using the anchor sensors with accurate position information. In other words, when applying the particle filtering methodology, the measurement term Z^m of sensor m may weight out the best possible particles due to the incorrect distance information. The measurement performance will be illustrated and further discussed in Section 5.

When developing the local position estimation (Phase I), one round of local flooding is initiated in order to gather distance information. Then, the distance and position information from the nearby three nodes are applied to estimate its own position. Thus, the time complexity is 𝒪(1) round.

Suppose that the total power requirements include the power required to transmit messages E_T, the power required to receive E_R, and the power required to process E_P. Therefore, the total energy consumption for initial local positioning in the network is E L = N T ( L ) ⋅ E T + N R ( L ) ⋅ E R + N P ( L ) ⋅ E P with (67) N T ( L ) = N S ,         N R ( L ) = ∑ i = 1 N S N i ,       N P ( L ) = N Swhere N_S is the number of sensors in the network and N_i is the number of the neighboring sensors of sensor i. Thus, for a sensor node, the average energy consumption yields E_L/N_S. Given the energy consumption analysis above, the communication complexity for establishing the local coordinate system in the network and for estimating the position of a sensor node are 𝒪(N_S) and 𝒪(1), respectively.

Due to the error caused by the location estimation algorithm (the estimation error) and the error intrinsic to the problem (noisy distance measurements), location adjustment algorithms (Phase II) are needed in order to improve the estimation accuracy and limit the propagation errors. Once the position refinement is executed, one round of 1-hop flooding is performed for broadcasting the estimated position information. Hence, the time complexity and communication complexity are the same as those in Phase I.

For merging the coordinate systems of two clusters (Phase III), two border sensors (sensor p and sensor q) execute one round of local flooding for exchanging the Merge message and determining the merging cluster. Then, the border sensor, say sensor q, calculates the adjustment quantities and 4 rounds of local flooding are performed for the the process of coordinate registration and transformation in the 2-hop cluster topology. Therefore, the time complexity is 𝒪(5) rounds. Based on the operation of building the relative global localization, the energy consumption yields E G = N T ( G ) ⋅ E T + N R ( G ) ⋅ E R + N P ( G ) ⋅ E P with (68) N T ( G ) = ( N c h − 1 ) ⋅ ( 5 + | H i ( 2 ) | ) (69) N R ( G ) = ( N c h − 1 ) ⋅ ( N P + 2 ⋅ N q + N i +   ∑ k ∈ H i ( 2 ) N k + N r ) (70) N P ( G ) = N c h − 1where H i ( 2 ) denotes the index set of 1-hop cluster members of cluster i with neighboring 2-hop cluster members, sensor r is the parent node of sensorq, N_i is the number of the neighboring sensors of sensor i, and N_ch is the number of clusters in the network. Accordingly, the communication complexity due to performing coordinate registration and transformation is 𝒪((N_ch – 1)N_avg), where N_avg is the average number of neighbors of a sensor in the network.

Based on the operations described in Section 3.4, the complexity analysis of the proposed cooperative estimation approaches (Phase IV) are described, respectively. For the centralized scheme, all the neighboring sensors transmit their observations directly to the estimated sensor. Therefore, the time complexity is 𝒪(1) round. The energy consumption yields E C C = N T ( C ) ⋅ E T + N R ( C ) ⋅ E R + N P ( C ) ⋅ E P with (71) N T ( C ) = N i ,     N R ( C ) = ∑ j ∈ S b ( i ) N j ,     N P ( C ) = 1where S b ( i ) is the index set of neighboring sensors of sensor i. Thus, the communication complexity due to transmitting the observations is 𝒪(N_avg).

For the progressive scheme, the estimation groups update the estimation result sequentially based on each group’s local observation and partial decision from it’s previous groups in the sequence. Hence, the time complexity is 𝒪(2N_EG) rounds for estimation groups consisting of two sensors, where N_EG is the number of estimation groups. The energy consumption is given by E C P = N T ( P ) ⋅ E T + N R ( P ) ⋅ E R + N P ( P ) ⋅ E P with (72) N T ( P ) = 2 ⋅ N E G ,       N R ( P ) = ∑ j ∈ G b ( i ) N j ,       N P ( P ) = N E Gwhere G b ( i ) is the index set of sensors in the estimation groups of sensor i. Thus, the communication complexity due to transmitting the observations and partial decisions is 𝒪(2N_EG).

For the distributed scheme, the target sensor fuses its local estimate and the estimates received from the neighborhood. Hence, the time complexity is 𝒪(2) rounds, which are for group estimation and estimation fusion. The energy consumption is given by E C D = N T ( D ) ⋅ E T + N R ( D ) ⋅ E R + N P ( D ) ⋅ E P with (73) N T ( D ) = 2 ⋅ N E G ,       N R ( D ) = ∑ j ∈ G b ( i ) N j ,       N P ( D ) = N E G + 1Similar to the progressive scheme, the communication complexity due to transmitting the observations and group decisions is 𝒪(2N_EG).

In this section we present the result of initial local position estimation by placing three reference nodes and one unknown node over the sensing field 10 × 10 units in size and assuming that this unknown node connects with all reference nodes. The positions of the three reference nodes are (7,3), (3,8), (1,4) and the true position of this unknown node is (4,1). Figures 7 (a)–(c) show the procedures of initial position estimate by using particle filter method with the bounding box algorithm. “•” represents the reference node, “○” represents the unknown node, “·” represents the particle and “×” represents the initial estimated sensor location. The estimation error of the initial position estimate is shown in Figure 7 (d), which depicts that the estimation error converges to 0.5 quickly after a few iterations. Figure 8 shows the resulting initial position estimation for a network of 25 nodes with 3 reference nodes, which suggests that the particle filter method with bounding box algorithm provides an acceptable performance of initial position estimation.

This set of experiments compares the initial position estimation of an unknown sensor using particle filtering and the theoretical approximation derived in Section 4.2. We consider the uncertainty of position estimation and distance measurement and then use the estimated distributions from other known sensors to get the distribution of the position estimate of the unknown sensor. Let σ_D and σ_P denote the distance deviation and position deviation, respectively. Suppose that the true position of the unknown sensor m is (17,30) and the position distributions of three known sensors are N ( μ P 1 , σ P 1 2 ), N ( μ P 2 , σ P 2 2 ), and N ( μ P 3 , σ P 3 2 ) with μ_P¹ = (0, 0), μ_P² = (23, 70), μ_P³ = (50, 0), σ_Dm1 = σ_Dm2 = σ_Dm3 = σ_D, and σ_P¹ = σ_P² = σ_P³ = σ_P. In the simulation, we sample the distributions of the position estimates obtained from three known sensors and the distributions of the distance estimates 200 times in order to get the possible values for the particle filtering. Then we fuse these 200 distributions to get the possible position distribution of the unknown sensor using the Covariance Intersection (CI) method described as in Equations (22) and (23). In this set of experiments, we choose to weight each typical run equally.

Given the deviations at distance and position information, Figure 9 shows the distributions of simulation results and the theoretical approximation of initial position estimates with unbiased distance and position information (top left) and biased distance measurements (top right), respectively. Figure 9 (bottom) shows the standard deviation of the mean value of position estimates. The plot varies the variance of the distance measurement. Observe that Figures 9 demonstrates that the approximated probability density function can well describe the measurement performance, which implies that the approximation may be a sensible way to assess the estimation process of an unknown sensor under the circumstances with ranging and positioning errors of known sensors.

Due to the errors occurring in the distance estimation, the following special case is studied to explore the impact of the ranging error on the performance of random measurement term. Assume that the mean value of the distance estimation is the true distance and the neighboring reference sensors have unbiased position information, which means the distance estimation is unbiased and the ℓ₁-norm distance is μ_D̂mℓ = μ_Dmℓ = d_mℓ (i.e., μ_Gmℓ = 0). Figure 10 shows the distributions of the random measurements with position and distance deviations. Note that as substituting the true position of the estimated sensor into the measurement term (66) with unbiased distance and position estimates having small deviations, the peak value of the distribution locates near zero, which may represent an ideal measurement case since the estimated sensor owns accurate distance and position information. On the other hand, with distance and position estimates having large deviations, the true position of the estimated sensor makes the peak value of the distribution shift away from zero. Therefore, in order to suppress the estimation error and shift the peak value of the distribution back to zero, incorrect estimates may be obtained. That is, as the iterative process of the particle filter proceeds, the most possible estimates may be weighted smaller or even be weighted out due to the low value of the likelihood function at the particle and the uncertainty of distance and position information.

In our simulations, two proposal densities are chosen for refining the position estimates with the Metropolis-Hastings Algorithm. Proposal density I is a normal distribution centered around the current state estimate, 𝒩 ( x ¯ k , σ ɛ 1 2 ). Proposal density II is the sum of all the current samples and noise. The distribution of the noise is 𝒩 ( 0 , σ ɛ 2 2 ). Assume that σ ɛ 1 2 = σ ɛ 2 2. This subsection reports the performance of the sensor location estimate and adjustment by applying the particle filtering and a few Monte Carlo Markov Chain (MCMC) steps on each particle. Moreover, the importance of resampling after using the MCMC steps and the impact of error propagation on estimation accuracy are explored via simulation.

Figure 14 (left) shows the structure of two clusters generated from the initialized sensors A and B. There are two shared border sensors, sensor 1 and sensor 2, in the overlapping area. Figure 14 (right) shows the position estimations of local coordinate systems. Note that applying the estimation procedures, these two clusters are generated from the initialized sensors A and B, which are located at the origin in their local coordinate systems.

Once the border sensors have the information of local coordinate systems from two clusters, they transmit Merge signals. The effect will be to reorient the cluster centered on sensor B. Therefore, sensor 1 uses its own information and the merge signal received from sensor 2 to calculate the adjustment information. Then sensor 1 transmits an Adjust message to the sensors in the reoriented cluster to update their positions. Figure 15 (left) depicts the movement of the reoriented cluster using the vector d S → and Figure 15 (right) shows that the positions of the sensors in the reoriented cluster match the corresponding positions in the coordinate system of sensor A using the rotation matrix R_merge.

In this set of experiments, two cases of the measurement deviation of the known positions are considered: σ P 2 = 0.01 and σ P 2 = 1. The initial settings and group topology are illustrated in Figure 5. Without loss of generality, suppose that the prior information for each cooperative scheme is the bounding box (detailed in Section 3.1.) of the estimated sensor using three neighboring sensors (e.g., the three black nodes shown in Figure 5). For the centralized approach, the estimated sensor receives the distance and position information sent from each cooperative member and includes these information into measurement term (3). For the progressive approach, each cooperative group generates N₁ = 250 samples from the bounding box and N₂ = 250 samples from the Gaussian approximation from previous estimation group, which is described is Table 4. For the distributed scheme, each cooperative group generates N = 500 samples from the prior distribution (i.e., the bounding box) for particle filtering. Given the distance and position information of the neighboring known sensors, the best possible position measurement of the estimated sensor is obtained by combining 300 typical runs for each cooperative approach.

Observe that in the case of good distance observations and a moderate position deviation, Figure 16 depicts that the estimated sensor may still obtain acceptable position estimate without cooperation. On the other hand, with poor distance observations, the cooperative techniques may be good approaches to improve the estimation accuracy.

The analysis of energy consumption (derived in Section 4.3.) shows that the total energy consumption of these estimation schemes are close when the data of all the sensors are applied. Given an accuracy threshold, the progressive process may terminate and return results without having all the sensors being visited such that computation time and network bandwidth can be reserved. Therefore, the strength of the progressive estimation scheme is the reduction of the amount of communication and the conservation of energy. However, compared with the centralized and distributed schemes, the progressive approach may produce a larger processing delay since it may spend more time on information processing when using the data from all the sensors. Among these three cooperative estimation methods, the distributed scheme may provide an efficient way to weight out the faulty sensors and corrupted observations to suppress the estimation error. This is because if some sensors are faulty or the observations are corrupted, the fusion among all the neighboring sensors (i.e., the centralized scheme) may degrade the estimation performance as shown in Figure 16.

This subsection compares the localization algorithms from two perspectives: the algorithm perspective and the network topology perspective. Hence, the probabilistic approximation method (i.e., the particle filter) and the deterministic methods are investigated from the algorithm perspective. Moreover, the proposed hierarchical approach and other cluster-based approaches are discussed from the network topology perspective.