We first show the algorithm convergence property in the simple one-dimensional (1-D) setup. We shall use the concept of spring-relaxation for the discussion, which is equivalent to our algorithm. We assume that there is a single location, which we called a stopping point, where all forces from beacons cancel out. This assumption is often valid for common configurations.

Consider n beacons where n ≥ 2. Each beacon is located at x_i, i = 1, 2, …, n. All distance variables are normalized to the length of the spring, i.e., the length of the spring is of one unit long. The net force applied on a node located at x is F ( x ) = ∑ i = 1 n F x i ( x )where (4) F x i ( x ) = { − x + x i − 1 , x < x 1 0 , x = x i − x + x i + 1 , x > x i

In the following, we shall prove that in 1-D setup, if there exist a unique stopping point where the net force is zero, at any other location, the net force is nonzero, and the force points towards the stopping point. In (4), we define that the leftward (resp. rightward) force carries a negative (resp. positive) sign.

Proposition 1 Consider n beacons where n ≥ 2. Each beacon is located at x_i, i = 1, 2, …, n. The net force applied on a node located at x is F ( x ) = Σ i = 1 n F x i ( x ). If F(x) = 0 produces a single unique solution, and let the solution be x̂, then ∀x < x̂, F(x) > 0 and ∀x > x̂ F(x) < 0.

Proof We first show that ∀x < x̂ F(x) > 0. By (4), F_xi (x) is a monotone decreasing function except at x = x_i where the curve steps up by two units. Thus, F(x) is also a piecewise monotone decreasing function with n upward steps occurring at x_i, i = 1, 2, …, n. In other words, at the discontinuity point x_i, if F ( x i − ) > 0, F(x_i) > 0 and F ( x i + ) > 0.

Given that as x → −∞, F_xi(x) → ∞ which implies lim_x→–∞ F(x) > 0. Since F(x) is a piecewise monotone decreasing function where at any of its discontinuity points x_i, if F ( x i − ) > 0, then F(x_i) > 0 and F ( x i + ) > 0, and x̂ is the only point that the F(x) crosses the x-axis, with condition lim_x→–∞ F(x) > 0, in the range where x ∈ (−∞, x̂), we must have F(x) > 0.

We now show that ∀x > x̂, F(x) < 0, and we shall prove this by using contradiction. Given F(x̂) = 0 as the unique solution, there exists a point x > x̂ where F(x) > 0.

Recall that F(x) is also a piecewise monotone decreasing function with n upward steps occurring at x_i, i = 1, 2, …, n where at any of its discontinuity points x_i, if F ( x i − ) > 0, then F(x_i) > 0 and F ( x i + ) > 0. As x → ∞, by (4) F_xi(x) → ∞ which implies lim_x→∞ F(x) < 0.

Given that there is a point x > x̂ where F(x) > 0, since F(x) is a piecewise monotone decreasing function, and at its discontinuity point x_i, if F ( x i − ) > 0, then F(x_i) > 0 and F ( x i + ) > 0, with F(x) > 0 and lim_x→∞ F(x) < 0, there must exist a point u ∈ (x, ∞) such that F(u) = 0 which contradict with our condition that F(x̂) = 0 is the only solution. Thus, if x > x̂ we must have F(x) < 0.

In the above discussion, we make no description on the step size of each movement of the estimated location. The analysis is valid for infinitesimal step size, or equivalently δ₁ is very small. In fact, δ₁ < 1 represents the upper bound condition for the setting of δ₁ as δ₁ = 1 corresponds to applying 100% of the net force to the movement of the estimated location.

Moreover, it is possible that a system setup gives more than one stopping points. In practice, such a setup should be avoided as it creates ambiguity in the localization. We here illustrate multiple stopping points using a two-beacon setup. Let d be the distance between the two beacons where the two beacons are located at x 1 = − d 2 and x 2 = d 2 respectively. The net force receives by a node at location x is F(x) = F_x1(x) + F_x2(x). By setting F(x) = 0, we can solve for locations where the net force is zero. One immediate solution to F(x) = 0 is x = 0. Besides, when d < 2, two other solutions can be found, which are x = ±1. The condition where d < 2 suggests that the two beacons are placed too close to each other. In practical design, beacons should be spread out in location to avoid generating multiple solutions which cause ambiguity in localization.

Similar approach can be adopted for the illustration of convergence property in a two-dimensional setup. Consider n beacons where n ≥ 2. Each beacon is located at coordinate s_i, i = 1, 2, …, n. All distance variables are normalized to the length of the spring, i.e. the length of the spring is of one unit long. We shall arbitrarily draw a straight line crossing the only stopping point, then rotate the line along with all beacons such that the line becomes horizontal and overlaps with an x-axis, and finally shift the line horizontally such that the stopping point stays at x̂ on the x-axis. After this transformation, let the xy-coordinate of the i-th beacon be (x_i, y_i), i = 1, 2, . . ., n. We now focus on the image of the forces projected on the straight line, which can be determined by (5) F x i ( x ) = ( x − x i ) ( 1 ( x − x i ) 2 + y i 2 − 1 )

We call the zone where a particular spring experiences compression a compression zone of the spring or the beacon. If the straight line does not cross the compression zone of a beacon, then we have |y_i| > 1 which ensures that the spring remains stretched along the line. It can be shown that with |y_i| > 1, F_xi(x) given in (5) is a monotone decreasing continuous function. We plot the result of (5) with x_i = 1 and y_i = 2 in Figure 2 to illustrate its monotonic decreasing characteristics.

For the case that the straight line touches or crosses the compression zone of a beacon, we have |y_i| ≤ 1. It can be shown that F_xi(x) given in (5) decreases monotonically except in the range ( − − y i 2 + y i 4 3, − y i 2 + y i 4 3 ) where it increases monotonically for no more than two units in total. In fact, (4) in the 1-D setup is a special case of (5) where the straight line also crosses the beacon. In Figure 2, the cases of |y_i| ≤ 1 and y_i = 0 both with x_i = 1 are also plotted illustrating a similar characteristics between the case |y_i| ≤ 1 in the 2-D setup and the case y_i = 0 in the 1-D setup.

Using this setup, following the proof given in Proposition 1, it can be shown that with the existence of a unique stopping point, at any other location along the straight line, there exist an net force where its image projected on the straight line points towards the stopping point which moves the node nearer to the stopping point. As the algorithm iterates, the location of the node will eventually converged to the stopping point. Note that since we only show that the image of a force along the line connecting the stopping point and the arbitrary chosen location points towards the the stopping point, the node may not necessarily moves along the straight line back to the only stopping point. The node may move to a new location nearer to the stopping point not necessarily on the straight line, but we can redraw another line for that new location and then apply the same argument to show that the next location of the node will again be nearer to the stopping point than that of the current location.

In phase 1, the more iterations the algorithm executes, the closer the estimated location of a sensor approaches its stopping point. Due to the RSS ranging error, the eventual stopping point may not be the true location. The task of phase 1 is mainly to bring the estimated location to the stopping point, and then the location estimation continues with phase 2 to bring further from the stopping point to the true location. Therefore, for the performance of phase 1, our interest is to investigate the estimation error by measuring the distance between the estimated location of a particular sensor and its eventual stopping point rather than its true location. In other words, we measure error = ( x e , i − x s , i ) 2 + ( y e , i − y s , i ) 2where (x_e,i,y_e,i) is the estimated location and (x_s,i,y_s,i) is the stopping location of Sensor_i, i ∈ 𝕊.

Starting with some preparation experiments, we were able to narrow down the value range of τ₁ and δ₁. We then executed Algorithm 1 ten times for each pair of (τ₁, δ₁) combinations, and calculated the mean error and number of iterations by averaging over executions. In order to make the experiment more realistic, the initial guess of sensor locations are obtained randomly and independently for each piece of execution. Figure 4 shows the estimation error versus δ₁ for different τ₁ values.

As can be seen from the figure, τ₁ value directly governs the estimation error where a smaller τ₁ allows the algorithm to terminate and give a closer location to the stopping point. In other words, setting a large value for τ₁ may give undesirable estimation location produced in phase 1. In the figure, using τ₁ = 10 gives a 10 m error indicating that the produced location estimate is still 10 m away to the stopping point. Whereas using an very small τ₁ = 0.001 gives a very small error of below 1 mm indicating that the produced location estimate almost reaches the stopping point. Notice that in the setup, the average distance between stopping points and corresponding actual positions of sensors is about 16.52 m. As a result, τ₁ ≤ 0.1 looks appropriate by giving a mean error less than 0.1 m for all cases of δ₁ values.

Moreover, the figure also shows that the estimation error generally reduces as δ₁ increases for all the considered τ₁ settings. For this setup, the estimation errors reach their lowest values at around δ₁ = 0.4. The errors then jump to higher values as δ₁ increases further indicating diverging of the algorithm. In terms of the accuracy, it is the best choice to fix δ₁ = 0.4.

While the previous results suggest that τ₁ ≤ 0.1 and δ₁ = 0.4 gives a very close location to the stopping point, it is also necessary to consider the convergence speed measured by number of iterations in executions. Figure 5 plots the number of executed iterations using different combinations of τ₁ and δ₁. Since τ₁ governs the termination of iterations, it can be expected that a smaller τ₁ gives a more strict constraint on termination, thus lengthens the converging process and results in more number of iterations. Meanwhile, δ₁ directly governs the convergence speed in the way that a larger δ₁ gives a faster convergence. However, it is observed in Figure 5 that δ₁ beyond 0.4 suddenly increases the number of iterations from tens to hundreds, suggesting that δ₁ > 0.4 gives a too rapid position adjustment thus introduces a lot of oscillations in position updates. To give concurrent consideration to both the estimation error and the convergence speed, the optimal choice for phase 1 in this setup is τ₁ = 0.001 and δ₁ = 0.4.

Similar to that of phase 1, in Algorithm 2, the setting of τ₂ and δ₂ decides the termination condition of the loop in the algorithm, which in turns decides the location estimation error. In this phase, we investigate the evolution of location estimation error during the execution of Algorithm 2, where the location estimation error is measured by the distance between the estimated location and the true location of a particular sensor. Precisely, error = ( x e , i − x i ) 2 + ( y e , i − y i ) 2where (x_e,i,y_e,i) is the estimated location and (x_i,y_i) is the true location of Sensor_i, i ∈ 𝕊.

The evolution of location estimation error for a particular sensor is plotted in Figure 6 where each curve is obtained by averaging over ten executions. In the figure, the symbols represent the termination point of the algorithm with various τ₂ values. The figure shows three curves with different δ₂ values: (i) when δ₂ is set to a very small value, for example δ₂ = 0.001, the algorithm takes more iterations to terminate, and the error drops smoothly over the iteration loops; (ii) when δ₂ is set to a relatively large value, for example δ₂ = 0.05, the algorithm generally takes much lower computation to terminate. Over the execution, the estimated location fluctuates greatly which indicates that the algorithm operates near the instability boundary due to a high value of δ₂; (iii) when δ₂ is set to a suitable value, for example δ₂ = 0.01, the error reduces sharply in the beginning of the algorithm execution and then stays steadily with small fluctuations showing the stationary properties. This indicates that if the setting the chosen appropriately, the algorithm achieves quick convergence and stability with low error. To give concurrent consideration to both the estimation error and the convergence speed, the optimal choice for phase 2 in this setup is τ₂ = 45 and δ₂ = 0.01.

One extension of the algorithm is that we can let Algorithm 2 execute continuously by not applying τ₂ for the termination condition. This allows the algorithm to cope with a certain mobility of sensors as this modification allows the sensors to continuously update and adjust their estimated location.

Using the chosen values for parameters, we test the overall accuracy of our proposed localization solution using the measure of location estimation error given in (9). Figure 8a depicts the estimated locations from phase 1 (shown in hollow triangles) and the corresponding true locations (shown in solid triangles). The errors are obvious. In Figure 8b, we show the estimated locations from phase 2 (shown in solid triangles) and the corresponding true locations (shown in hollow triangles). The refinement due to phase 2 is clearly illustrated with many close pairs of triangles and the matched shape of sensor deployment. Numerically, the average errors in phase 1 and 2 are 15.65 m and 6.93 m respectively. Normalizing to sensor transmission range R, phase 2 estimation gives 0.35·R in average. In Figure 7 which shows the CDF of the location estimation errors, we further see that with the effort of phase 2, 90% of the location estimation are improved from more than 30 m to 10 m.

To further show the performance advantages of our proposed method, we compare the accuracy performance of our proposed solution to other solution that can be deployed to perform localization in the same environment without additional specialized localization hardware. For this performance comparison, we select the k-nearest neighbors (kNN) approach, where k neighbors are the k pre-mapped geographic points whose RF fingerprints are closet matching to that of a sensor, and each of the k neighbors is weighed inversely proportional to its Euclidean distance in signal space from the sensor whose location is being estimated [23]. We implement a script following [23], and k = 3 is used in the script for this performance comparison.

One additional requirement for kNN approach is the database construction of RSS fingerprints through a site-survey prior to the localization. For the database construction, the more sampling points are performed, the better is the performance. With the same setup of the map size, the placement of beacons and the number of sensors, as well as the same path loss model and log-normal shadowing, we conduct performance studies for the kNN approach using different number of survey points via simulation, and report the results in Table 2. For each number of survey points, we execute kNN ten times and take average over ten times for the final mean estimation error. As can be seen, only for 25,600 (i.e., 160 by 160) survey points, the kNN approach achieves just below 16 m, which is of 2.3 times the localization error of our proposed method. We also observe from the table that further increasing of the survey points to 250,000 does not have significant improvement in the accuracy. This confirms that our solution is a good RSS-based localization candidate.

Furthermore, we compare our proposed solution to maximum-likelihood estimators (MLEs) for sensor location estimation [24] in the same one-hop setup. MLE works in a similar network setup that has a number of reference devices with known coordinates, and a number of blindfolded devices whose positions are to be estimated. With the combined range information between many pairs of devices and the known locations of reference devices, a Maximum-likelihood solution for the location of all of the blindfolded devices is determined. For this performance comparison, we used the MLE script implemented by Neal Patwari which can be obtained in [25]. Using the same system setup and sensor deployment shown in Figure 8, we execute ten trials of MLE, take mean over all ten sets of estimated locations produced in different trials as the resultant location estimates, and plot these location estimates in Figure 9c. The estimated locations produced by our algorithm as well as by kNN are also plotted for comparison. As can be seen in the figure, almost none of the location estimates can reach their actual locations, nor can them form a shape similar to that of the actual sensor deployment. Numerically, the mean localization error for MLE is 35.27 m, i.e., 1.77·R, which is five times of our spring-relaxation solution.