Assume the target sensor m obtains the modified distance measurements (i.e., the AOA-aided TOA measurements) from neighboring seeds and estimates its own position using the particle filter. The position of the target sensor is given by the discrete-time state equation (3) x k = Φ x k − 1 + Γ λ kwhere x_k is the position of the mobile sensor and λ_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.

The measurement term for the target sensor m is (4) z k = ∑ ℓ ∈ H m | | x k m − x k ℓ | − d m ℓ | + v kwhere the sum is over the nearby seeds with location x k ℓ, d_mℓ represents the measured distance between the estimated target sensor m and Seed ℓ and may be approximated in application by the inverse of the signal strength or by calculated from the time delay between transmission and reception [36]; the measurement noise is another uncorrelated zero mean Gaussian white noise process; the set H_m is the chosen seeds for the measurement.

Particle filter is an algorithm of estimation used to estimate the unknown sensor position from state equations. The objective is to find feasible position to make the error of state vector x minimum. The state vector is represented as a set of random samples updated and propagated with the algorithm. One of the main advantages of this approach is that the mobile sensor carries along a complete distribution of estimates of its position. Therefore, the distribution is inherently a measure of the accuracy of the positioning system—hence, if a given task requires a certain accuracy, it is possible to determine if that level of accuracy is currently available. Moreover, [37] presents a case study of applying particle filters to location estimation for ubiquitous computing. Therefore, our approach may be computationally affordable by sensor nodes. The particle filter method is shown in Table 1.

In the proposed fuzzy control system, two scenarios are considered. For Scenario 1, as depicted in Figure 5, the target sensor has no AOA information. Given the initial estimate of target position in Phase II and the reference position of Seed i, the vector d ^ → i and angle θ_i (in radians) are obtained. Hence, the projections of d ^ → i onto x-axis and y-axis are (5) x ^ i proj = d ^ i ⋅  cos ( θ i ) ,     y ^ i proj = d ^ i ⋅  sin ( θ i )Instead of using the estimated target position, the modified distance measurement (i.e., d → i ′) and angle θ_i are applied to find the projections, which are (6) x i proj = d i ′ ⋅  cos ( θ i ) ,     y i proj = d i ′ ⋅  sin ( θ i )

For Scenario 2, as shown in Figure 6, the target sensor has received AOA information. Following the same operations above, we have (7) x ^ i proj = d ^ i ⋅  cos ( φ i ) ,     y ^ i proj = d ^ i ⋅  sin ( φ i ) (8) x i proj = d i ′ ⋅  cos ( Φ i ) ,     y i proj = d i ′ ⋅  sin ( Φ i )Note that angle Φ_i is captured from the AOA information and the information of mobility model, and angle φ_i is derived from the initial target position estimate and the reference position of Seed i. Accordingly, x i diff = x ^ i proj − x i proj and y i diff = y ^ i proj − y i proj are fed as inputs to the fuzzy control system for the two scenarios described above.

The proposed fuzzy control system uses fuzzy logic and gradient descent method to adjust a suitable answer for the target sensor position. The objective function to be minimized is defined by (9) E = 1 2 ( u − u d ) 2where u^d is a desired output value for an input vector, and u is a fuzzy inference value. Here, we use the projections onto the x-axis of d ^ → i and d → i ′ to explain the operation of the fuzzy system and minimize the objective function E. In this case, the desired output value for input vector x = [x₁, x₂, . . ., x_L]^T is u i d = x i proj and the fuzzy inference value is u i = x ^ i proj + δ. Therefore, E can be rewritten by (10) E = ∑ i = 1 L 1 2 ( u i − u i d ) 2 (11) = ∑ i = 1 L 1 2 [ ( x ^ i proj + δ ) − x i proj ] 2 (12) = ∑ i = 1 L 1 2 [ ( x i diff + δ ) ] 2where δ is the output of fuzzy membership (13) δ = ∑ j = 1 r q j ( x ) w j ∑ j = 1 r q j ( x )which is calculated by center of area (COA) method [38]. Note that (14) q j ( x ) = ∏ i = 1 L μ A i j ( x i ) ,     j = 1 , 2 , … , ris the defined firing strength of rule j, r is the number of fuzzy rules, μ A i j is the membership function of the precondition part, w_j is the training parameter, and the x₁, x₂, . . ., x_L are input variables.

Suppose μ A i j is a Gaussian membership function for input variable x_i of rule j. Thus, E is further given by (15) E = ∑ i = 1 L 1 2 [ ( x i diff + ∑ j = 1 r q j ( x ) w j ∑ j = 1 r q j ( x ) ) ] 2 (16) = ∑ i = 1 L 1 2 [ ( x i diff + ∑ j = 1 r ∏ i = 1 L μ A i j ( x i ) w j ∑ j = 1 r ∏ i = 1 L μ A i j ] 2with (17) μ A i j =  exp  [ − ( x i − m i j ) 2 ( σ i j ) 2 ]where i = 1, 2, . . ., L, j = 1, 2, . . ., r, and m i j and σ i j are the mean and the standard deviation of μ A i j, respectively.

In order to minimize the function E and find a better feasible estimate, the training parameters are derived based on gradient descent method [15]. For m i j, we have (18) m i j ( t + 1 ) = m i j ( t ) − η ∂ E ∂ m i j (19) = m i j ( t ) − η ∑ i = 1 L [ ( x i diff + δ ) ] ⋅ w j − δ ∑ K = 1 r q K ⋅ q j ⋅ 2 ( x i − m i j ) ( σ i j ) 2where i = 1, 2, . . ., L, j = 1, 2, . . ., r, and η is the constant step size (0 < η < 1). Similarly, the training process of σ i j is given by (20) σ i j ( t + 1 ) = σ i j ( t ) − η ∂ E ∂ σ i j (21) = σ i j ( t ) − η ∑ i = 1 L [ ( x i diff + δ ) ] ⋅ w j − δ ∑ K = 1 r q K ⋅ q j ⋅ 2 ( x i − m i j ) 2 ( σ i j ) 3and the training parameter w_j yields (22) w j ( t + 1 ) = w j ( t ) − η ∂ E ∂ w j (23) = w j ( t ) − η ∑ i = 1 L [ ( x i diff + δ ) ] ⋅ q j ∑ K = 1 r q K .

Based on the inputs as detailed in Section 3.3.1, the linguistic variables used for the input of the fuzzy logic controller system are N (negative) and P (positive). Gradient descent method is used to decide linguistic variables of N and P, which represents a measure of the difference between x ^ i proj ( y ^ i proj ) and x i proj ( y i proj ). In Figure 7, Gaussian membership functions are developed for the linguistic states. In this work, four fuzzy rules are developed and the training parameters ( m j i, σ j i, w_j) are calculated by the above gradient descent method. Table 2 expresses the fuzzy logic in terms of fuzzy IF-THEN rules, which implements mapping of input functions into output functions.

There are many methods available for doing defuzzification (e.g., Center of Area, Mean Maximum). Here, we use the Center of Area (COA) method to determine the defuzzification value for the x-coordinate. Note that the above operations are for the refinement of the x-coordinate on x-axis. Similar procedures can be performed for the y-coordinate on the y-axis.

The probability density function of g = [r̂, φ̂] is (43) f ( g ; x , y ) = 1 2 π σ r 2 ⋅  exp  [ − 1 2 σ r 2 ( r ^ − d ) 2 ] ⋅ 1 2 π σ ϕ 2 ⋅  exp  [ − 1 2 σ ϕ 2 ( ϕ ^ −  arctan  ( y − y s x − x s ) ) 2 ]Thus, the Fisher information matrix yields (44) I ( x ( t ) ) = [ cos 2 ( ϕ ) σ r 2 + sin 2 ( ϕ ) d 2 σ ϕ 2 sin ( 2 ϕ ) 2 [ 1 σ r 2 − 1 d 2 σ ϕ 2 ] sin ( 2 ϕ ) 2 [ 1 σ r 2 − 1 d 2 σ ϕ 2 ] sin 2 ( ϕ ) σ r 2 + cos 2 ( ϕ ) d 2 σ ϕ 2 ]and the Cramer-Rao bound can then be written as (45) Var ( x ( t ) ) ≥ I − 1 ( x ( t ) ) = [ I 1 , 1 ′ I 1 , 2 ′ I 2 , 1 ′ I 2 , 2 ′ ]Thus, (46) Var ( x ˜ ) ≥ I 1 , 1 ′ ,     Var ( y ˜ ) ≥ I 2 , 2 ′

Referring to the concept of measurement modification as shown in Figure 3 and normal approximation [40], the modified distance estimate d i ′ ^ in (1) may be approximated by (47) d i ′ ^ ∼ 𝒩 ∫ ( μ d i ′ , σ d i ′ 2 )with (48) μ d i ′ = d i ′ = d i 2 + Δ d i 2 − 2 d i Δ d i  cos ( θ i ) (49) σ d i ′ 2 ≈ ( 1 + 4  cos 2 ( θ i ) Δ d i 2 ) σ p i 2 + ( 1 + 4  cos 2 ( θ i ) d i 2 ) σ Δ p i 2 + 4  sin 2 ( θ i ) d i 2 Δ d i 2 σ θ i 2where (50) σ p i 2 ≈ σ d i 2 + d i 2 σ ϕ i 2 (51) σ Δ p i 2 ≈ σ Δ d i 2 + Δ d i 2 σ ϕ m 2 (52) σ θ i 2 ≈ σ ϕ i 2 + σ ϕ m 2Note that d_i, Δd_i, θ_i, φ_i, and φ_m are the estimated parameters with respect to Seed i assumed to be Gaussian distributions centered at their true values. Therefore, the distribution of r[i] is (53) f ( r [ i ] ; x , y ) = 1 ( 2 π σ i 2 ) N s / 2  exp ( − ∑ i = 1 N s ( r [ i ] − d i ′ ) 2 2 σ i 2 )where i = 1, 2, . . ., N_s and −∞ ≤ r[i] ≤ ∞. Similarly, following the definitions (28) ∼ (32) in Section 4.1 with ∂ D ∂ x = [ ∂ d 1 ′ ( x , y ) ∂ x ∂ d 2 ′ ( x , y ) ∂ x ⋯ ∂ d N s ′ ( x , y ) ∂ x ] T = [ x − x 1 ( s ) d 1 ′ x − x 2 ( s ) d 2 ′ ⋯ x − x N s ( s ) d N s ′ ] T ∂ D ∂ y = [ ∂ d 1 ′ ( x , y ) ∂ y ∂ d 2 ′ ( x , y ) ∂ y ⋯ ∂ d N s ′ ( x , y ) ∂ y ] T = [ y − y 1 ( s ) d 1 ′ y − y 2 ( s ) d 2 ′ ⋯ y − y N s ( s ) d N s ′ ] Tand σ_i = σ_d′i, we obtain (54) Var ( x ˜ ) ≥ I 1 , 1 ′ ,     Var ( y ˜ ) ≥ I 2 , 2 ′

For the first set of experiments, we consider the mobility of the target sensor ranging from 1 to 10 m/s. Assuming that the variance of the distance measurement is σ d 2 = 10 − 3, Figure 9 shows the position estimation accuracy against the target mobility, which implies that the performance of the TDOA and the proposed TOA/AOA method remain approximately stable regardless of the node moving speed. On the contrary, the estimation error of the conventional TOA method increases with the increasing target speed. Therefore, Figure 9 suggests that ATPA may be suitable for geometrical positioning in situations involving modest mobility.

Given the target speed 5 m/s and the variance of angle estimation (angle measurement in degree), the second set of experiments investigates the effect of uncertainty of angle estimation on position estimation accuracy with varying the variance of distance measurement ranging from σ d 2 = 10 − 4 ∼ 1. As expected, Figure 10 illustrates that the proposed scheme achieves better performance with a lower variance of angle estimation. Observe that with a small variance of distance measurement, the angle information dominates the accuracy of position estimation. However, with a larger variance of distance measurement, the localization accuracy is determined by the ranging error.

As shown in Figure 11, given the target speed 5 m/s and the variance of angle estimation σ ϕ 2 = 0.1, the the position estimation error increases with the increasing variance of distance measurement (ranging from σ d 2 = 10 − 4 ∼ 1). Notice that the CRLB, the performance of the TDOA method, and the performance of the proposed method merge together with a measurement noise σ_d ≥ 1. Therefore, a fundamental problem when locating mobile sensors in a network is to estimate the distance between the seed and the target sensor, since accurate location estimates highly rely on precise distance measurements.

With the target speed 5 m/s and variance σ d 2 = 10 − 3, we vary the number of seeds in the network from 3 to 10. The estimation of position is shown in Figure 12, which shows the accuracy of the position estimate. The performance improves along with the number of seeds. However, like the TDOA approach, the improvement is not significant (especially when number of seeds is greater than 5). This suggests that even a low number of seeds can also achieve good estimation accuracy.

Compared with Figure 9, given 3 seeds with random locations and the speed at 5 m/s, the estimation accuracy of Figure 12 is much lower than the one in Figure 9. This is due to the quality of the TOA measurements. Therefore, the number of TOA measurements for position estimation (i.e., the number of seeds chosen for the measurement) should be dynamically adjusted based on the estimated distance between the target mobile sensor and the seeds in order to reduce location error.

Given variance of distance estimation σ d 2 = 0.01 and the number of samples of particle filter (ranging from 25 ∼ 750 samples/area), Figure 14 shows the location adjustment of the target sensor using TOA information. Referring to the line with label - Particle Filter (TOA), it suggests that a better initial position estimate in Phase II may be obtained with a larger sample size. Based on the initial estimation, Figure 14 further shows the improvement of positioning performance when applying the refinement schemes. Observe that the performance of the adaptive fuzzy control with an increase in the number of iterative training may be superior to that of the MCMC method with a smaller sample size. This is attributed to the fact that the number of MCMC samples may have an influence on the particle set’s quality [43]. For the proposed fuzzy control method, the fitting for training data is good.

Suppose that the target sensor receives AOA and TOA information to estimate its own coordinate. The AOA measurement noise is assumed to be a Gaussian random variable and AOA is measured with degree information. Similar to the results described in Figure 14, Figure 15 show that, given the variance of distance σ d 2 = 0.01 and variance of angle σ ϕ 2 = 0, the position estimation error is suppressed with increasing the number of samples (ranging from 25 ∼ 750 samples/area). Notice that compared with the MCMC method, the adaptive fuzzy control using AOA and TOA information has less position estimation error. Moreover, compared with Figure 14, Figure 15 shows that incorporating accurate angle information may help tackle the localization problem in addition to distance measurements. Hence, one possible way to approach network localization is to include other measurements such as angle information and heading information [39] in order to suppress the computational complexity.

In Figure 16, given the variance of distance σ d 2 = 0.01 and σ d 2 = 0.5, we explore the estimation performance with varying the variance of angle information. Notice that as shown in Figure 16 (left), the proposed adaptive fuzzy control with moderate noisy angle information may dominate the localization performance under the circumstances of good distance estimation. However, as shown in Figure 16 (right), even with the moderate noisy angle information, the noisy distance information may have the predominant influence and degrade the estimation performance due to the performance loss caused by measurement uncertainties and propagation environments. Therefore, the MCMC and the proposed adaptive fuzzy control have roughly the same estimation performance in this scenario.