An Intelligent Prediction System for Mobile Source Localization Using Time Delay Measurements

Substituting (6) into (4) yields the following expression:

where $\varepsilon_{i}=(\bm{v}^{T}\bm{v}-c^{2})n_{i}$, $i=1,2,\ldots,M$. To establish the optimization model, a new unknown vector $\bm{x}$ is defined by $\bm{x}=[\bm{u}^{T},\bm{v}^{T},\bm{u}^{T}\bm{v},\bm{v}^{T}\bm{v},\bm{d}^{T},1]^{T}\in\mathbb{R}^{M+2p+3}$, $\bm{d}=[d_{1},d_{2},\ldots,d_{M}]^{T}$. Then (7) is also represented by a linear matrix form:

where $\bm{\varepsilon}=[\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{M}]^{T}$. According to the expression of (7), $\bm{\varepsilon}\in\mathbb{R}^{M}$ and $\bm{G}\in\mathbb{R}^{M\times(M+2p+3)}$ are also defined by

Therefore, the weighted least square (WLS) solution to the prediction problem is formulated as

where $\bm{Q}\in\mathbb{R}^{M\times M}$ is the covariance matrix with respect to $\bm{\varepsilon}$, and $\bm{Q}$ is further obtained by

The equality condition (10a) is also rewritten as

where $\bm{D}_{0}=[\bm{0}_{p},\bm{I}_{p},\bm{0}_{p\times(M+3)}]$. The equality condition (10b) is also given by

where $\bm{D}_{i}=[\bm{I}_{p},\bm{0}_{p\times(M+p+2)},\bm{s}_{i}]$, $i=1,2,\ldots,M$. When the equality conditions (10a) and (10b) are equivalent to the conditions (12) and (13), (10) is rewritten as

To further express (IV-A) as a convex form, a new unknown matrix is defined by $\bm{X}=\bm{x}\bm{x}^{T}$. It is obviously shown that $\mathrm{rank(\bm{X})}=1$. Thus, (IV-A) is given by

where $\bm{C}=\bm{G}^{T}\bm{Q}^{-1}\bm{G}$. Due to the rank-one constraint of (15e), the expression of (15) is non-convex. However, dropping the rank-one condition of $\bm{X}$ yields a convex SDP form,

where $b_{i}=[\bm{0}_{1\times(M+2)},1]^{T}$, the sparse matrices $\bm{A}_{i}$ can be easily obtained by the expression of (15), $i=1,\ldots,M+3$. It is obviously shown that the problem depicted by (IV-A) is a typical SDP problem that can be effectively solved using a convex optimization package such as SEDUMI and SDPT3. Unfortunately, dropping the rank-one constraint relaxes the problem depicted by (15) and leads to produce a suboptimal SDP solution with a rank higher than 1, which implies that the solution of (IV-A) may not be the optimal solution of the original problem depicted by (15). To obtain the optimal performance, we propose the penalty function method to meet the rank-one constraint.

The convex problem depicted by (16) provides a relaxed SDP solution to the mobile source localization problem. However, the relaxed SDP solution of (16) is suboptimal due to the drop of rank-one constraint. To obtain the optimal performance, we attempt to find a rank-one solution by introducing the penalty terms. To design the PF-SDP method, we firstly prove the conclusion of Lemma 1.

To achieve the equality (18), a feasible method is to restrict the magnitude of $\bm{X}_{i,i}$ by introducing the penalty terms. Therefore, a new PF-based SDP form is given by

where $\eta$ is a strictly positive constant and called as penalty coefficient. When $\eta$ is gradually increased, $\bm{X}_{i,i}$ will sufficiently close to $\bm{X}_{i,N}^{2}$. Eventually, large enough $\eta$ will lead that $\mathrm{rank}(\bm{X})=1$.

When $\eta$ is chosen to be large enough, the SDP problem depicted by (IV-B) provides a rank-one solution of the positive semidefinite $\bm{X}$. However, a concerned issue is that whether the rank-one solution is the original solution of (10) or not. It is directly shown that the problem depicted by (10) has only $M+2p+3$ variables. Since there are $M+3$ constraints among the variables in defined vector $\bm{x}$, so the problem has only $2p$ independent variables. It is also observed that the dimension of $\bm{X}$ in the problem depicted by (IV-B) is also $M+2p+3$ when $\mathrm{rank}(\bm{X})=1$. Due to the similar constraints, the problem depicted by (IV-B) has also only $2p$ independent variables. Therefore, the rank-one solution of the SDP problem depicted by (IV-B) is also the original solution of (10) since the convex SDP form of (IV-B) is derived from the non-convex problem depicted by (10).

An appropriate penalty coefficient can ensure the effective working of penalty terms. Although large enough $\eta$ will achieve the rank of the solved $\bm{X}$ to be one, too large $\eta$ may lead to a risk of excessive penalty, which will badly affect the predicted result. Hence, the choosing of $\eta$ is crucial to obtain the well performance of the prediction problem. To choose an appropriate penalty coefficient, we propose an adaptive penalty function-based SDP (AFP-SDP) algorithm to obtain a rank-one solution for the SDP optimization problem in the following.

Too large $\eta$ increases the proportion of penalty terms to the total cost function of (22), so it will weaken the original objective $\mathrm{Tr}(\bm{C}\bm{X})$. To avoid the chosen $\eta$ to be too large, $\eta$ starts from a small value and increases gradually, until the rank-one condition is satisfied. To clearly observe the effect of the penalty coefficient in the problem depicted by (22), we conducted a random test and the results are plotted in Fig.1, where $\bm{X}^{*}$ and $\mathrm{Tr}(\bm{C}\bm{X}^{*})$ represent the optimal solution and objective, respectively. In the test, $\mathrm{log}\eta$ is gradually increased from -6 to 3 (i.e. $\eta$ is gradually increased from $10^{-6}$ to $10^{3}$). As can be seen that the penalty terms almost do not work when $\mathrm{log}\eta$ is increased from -6 to -4. The localization error is gradually reduced for the working of penalty terms when $\mathrm{log}\eta$ is increased from -4 to -2. The optimal performance is achieved when $\mathrm{log}\eta$ is set to the range of (-2, 0). However, if $\mathrm{log}\eta$ is continuously to be increased, the performance will become worse. The localization error will sharply increased due to the occurrence of excessive penalty when $\mathrm{log}\eta$ is larger than 1. The occurrence of excessive penalty badly affects the localization result, so it is crucial to detect whether the penalty is excessive or not. It can be seen from Fig.2 that the optimal objective $\mathrm{Tr}(\bm{C}\bm{X}^{*})$ is simultaneously increased with the localization error when the excessive penalty occurs. Therefore, the optimal objective $\mathrm{Tr}(\bm{C}\bm{X}^{*})$ can be used to detect the occurrence of excessive penalty.

When $\eta$ is chosen to be too small, the rank of $\bm{X}^{*}$ may be larger than 1. The increasing of $\eta$ can ensure that the rank of $\bm{X}^{*}$ is gradually close to be one. To evaluate the rank of $\bm{X}^{*}$, we firstly propose an eigenvalue method according to the following threshold value,

where $\widetilde{\lambda}_{N-1}$ and $\widetilde{\lambda}_{N}$ are the $N-1$th and the $N$th eigenvalue of $\bm{X}^{*}$, $N=M+2p+3$. It is obviously shown that the rank of $\bm{X}^{*}$ tends to be one when $\tau$ is sufficiently close to zero.

To ensure the well performance of the PF-SDP, it is crucial to choose an optimal penalty coefficient which depends on various factors including the positions of sensors and mobile source, the number of sensors, and the noise level. Therefore, the optimal $\eta$ is not invariable under the different situations. In Algorithm 1, we propose an Adaptive Penalty Function-based SDP (APF-SDP) algorithm by adaptively choosing an optimal penalty coefficient.

In Algorithm 1, two conditions is required to be judged. One is the condition $\tau<\delta$ ($\delta$ is a constant that is sufficiently close to zero), which is considered as rank-one condition and used to judge whether the rank of $\bm{X}^{*}$ is one or not. The other is the condition $\bm{C}\bm{X}<\epsilon$ ($\epsilon$ is also a threshold value determined by the chi-square distribution of $\bm{C}\bm{X}^{*}$), which is used to detect whether the penalty is excessive or not. $\eta$ starts from a small value and increases gradually with $\eta=\gamma\eta$ ($\gamma>1$), where $\gamma$ is called as step length. When $\tau<\delta$ is satisfied, it is considered to meet the rank-one condition. Then using the condition $\bm{C}\bm{X}<\epsilon$, we further detect whether the penalty is excessive or not. When meeting the condition $\bm{C}\bm{X}<\epsilon$, we accept the solution that is considered as a final solution $\bm{X}^{*}$. If not, we relax the rank-one condition with $\delta=\alpha\delta$ ($\alpha>1$) and resolve the problem depicted by (22) until two conditions hold.

The increasing of $\eta$ ensures that the rank of the SDP solution gradually tends to be one. However, the performance of the rank-one solution may be not optimal due to the occurrence of excessive penalty. Therefore, it is required to ensure that the rank-one solution has no the excessive penalty. To evaluate the improved performance of PF-SDP compared with the RSDP, the Logarithmic MSE difference (LOGMSED) is defined by

where $\mathrm{10log(MSE})_{\mathrm{PF}}$ and $\mathrm{10log(MSE})_{\mathrm{R}}$ represent the MSE in log-scale of PF-SDP and RSDP, respectively.

The first eight sensors listed in Tab. I are used to determine the initial position and velocity of the mobile source. The noise level $10\mathrm{log}\sigma^{2}$ is set to -40, -20, 0, respectively. Fig. 3(a) illustrates the LOGMSED($\bm{u}$) performance when the penalty coefficient $\eta$ is increased from $10^{-6}$ to $10^{3}$ (i.e. log$\eta$ is increased from -6 to 3). The penalty terms do not basically work when $\eta$ is too small. As can be seen that LOGMSED($\bm{u}$) is near zero when log$\eta$ is set to -6. When $\eta$ gradually increases, LOGMSED($\bm{u}$) becomes a negative value, which illustrates that the MSE of PF-SDP is smaller than that of the RSDP due to the working of penalty terms. When log$\eta$ is set to (0, 2) for $10\mathrm{log}\sigma^{2}=-40$, (-1, 1) for $10\mathrm{log}\sigma^{2}=-20$, and (-2, 0) for $10\mathrm{log}\sigma^{2}=0$, LOGMSED($\bm{u}$) reaches the least value, indicating the optimal performance of PF-SDP. Therefore, the optimal range of $\eta$ is variable due to the different noise level. When log$\eta$ is larger than the optimal range, LOGMSED($\bm{u}$) will be sharply increased, which confirms the occurrence of excessive penalty. For instance, when log$\eta$ is set to (2,3), LOGMSED($\bm{u}$) becomes a positive value for $10\mathrm{log}\sigma^{2}=0$, illustrating that the performance of PF-SDP is worse than that of the RSDP.

When the parameter setup is the same with that of Fig. 3(a), Fig. 3(b) shows the LOGMSED($\bm{v}$) with the increasing of $\eta$. It is also observed that the LOGMSED($\bm{v}$) becomes a negative value when log$\eta$ is gradually increased from -6. Then the LOGMSED($\bm{v}$) reaches a least value that manifests the optimal performance of PF-SDP. For instance, when $10\mathrm{log}\sigma^{2}$ is set to -40, the optimal log$\eta$ is set to the range of (0, 2), where the PF-SDP provides better performance and has almost 13 dB bias compared with the RSDP. When log$\eta$ is larger than the optimal range, the LOGMSED($\bm{v}$) will be sharply increased due to the occurrence of excessive penalty.

The rank of PF-SDP solution will gradually tend to be one as $\eta$ increases. If meeting $\tau<\delta$, it is considered as a rank-one solution. When 10log$\sigma^{2}$ is set to 0 dB, the number of rank-one solutions is shown in Tab. II, where $\delta$ is set to $10^{-5}$. As can be seen that the number of rank-one solutions is increased with log$\eta$, since larger $\eta$ provides tighter SDP cone constraints. When log$\eta$ is larger than 1, all 1000 MC runs of PF-SDP solutions meet the rank-one condition. When 10log$\sigma^{2}$ is increased from -60 dB to 20 dB, the number of rank-one solutions is also illustrated in Tab. III, where log$\eta$ is set to -1, 0, and 1, respectively. The number of rank-one solutions is also dramatically increased with 10log$\sigma^{2}$. When 10log$\sigma^{2}$ is set to -60 dB, the number of rank-one solutions is 2 for log$\eta=0$, 205 for log$\eta=1$. However, all 1000 MC runs meet the rank-one condition for log$\eta=0$ and log$\eta=1$ when 10log$\sigma^{2}$ is increased to 20 dB.

The parameters in APF-SDP are listed as follows, $\alpha=5$, $\gamma=10$, and $\delta=10^{-5}$. The first eight sensors listed in Tab. I are used to locate the mobile source which starts from the initial position (0.2, -0.4) km with a constant velocity (-1, 1) m/s. When the noise level 10log$\sigma^{2}$ is increased from -60 dB to 20 dB, Fig. 4(a) and Fig. 4(b) illustrates the logarithmic MSEs of predicted initial position and velocity, respectively. It is obviously shown that the logarithmic MSE performance becomes worse as 10log$\sigma^{2}$ increases. When log$\eta$ is set to -6 and -4, the logarithmic MSEs of PF-SDP are almost same with that of the RSDP, indicating the no working of penalty terms. However, the performance of PF-SDP becomes better when log$\eta$ is set to -2. Especially, the logarithmic MSE of PF-SDP is closer to the CRLB at large noise level, since the rank of PF-SDP solution is easier to be one. Compared with the RSDP, the AFP-SDP provides better performance in reaching the CRLB accuracy. Hence, it confirms the advantage of the APF-SDP in the position and velocity prediction.

The noise level 10log$\sigma^{2}$ is set to -40 dB and the other parameters are the same with those in Fig. 4. Fig. 5(a) and Fig. 5(b) also shows the logarithmic MSEs of predicted initial position and velocity when the propagation speed $c$ is increased from 0.1 km/s to 1 km/s. It is shown that the performance of predicted position or velocity becomes worse as the propagation speed increases. The performance of PF-SDP is always better that of the RSDP when log$\eta$ is set to -2, indicating the working of penalty terms. The APF-SDP always provides the almost CRLB performance in the prediction of position and velocity when $c$ is varied from 0.1 km/s to 1km/s. It confirms the advantages of AFP-SDP which can adaptively choose the penalty coefficient and provides optimal performance at the entire range of $c$.

Finally, we also examine the impact of sensors on the performance of these proposed algorithms when the sensors are selected from Tab. I in the order of list. The other parameter setup is also the same with that in Fig.4. Fig. 6(a) and Fig. 6(b) show the logarithmic MSEs of predicted initial position and velocity as the number of sensors varies. As can be seen that the logarithmic MSEs of PF-SDP (log$\eta$ = -2) are the same with those of RSDP at when there are only five or six sensors. However, the PF-SDP (log$\eta$ = -2) provides better performance compared with the RSDP when the number of sensors is larger than seven. The proposed APF-SDP always reach the CRLB performance. It is also shown that adding more sensors can reduce the logarithmic MSE. However, how much reduction in logarithmic MSE by using additional sensors depends on the positions of the new sensors included.

To verify the performance of our proposed algorithms, we also conducted the real experiments using a mobile car equipped with Ultrasonic module (UM). Besides the UM, motion sensors are also equipped to the mobile car and used to measure the parameters including the velocity and moving direction. Nine sensors are placed at the positions listed in Tab. IV. The UM transmits the ultrasonic signals to the sensors at the initial position, then the signals are received by the sensors and reflected to the UM of mobile car. Extensive tests show that the measurement noise in the range-difference is sufficiently close to be zero-mean Gaussian distribution with an predicted noise variance $6.3\times 10^{-3}$ ms². The initial position of the mobile car is set to (1, 2) m, and the velocity of the mobile car is in the range of (1, 3) m/s.

We collect 200 sampling data that is used to predict the initial position and and velocity of the mobile car. Fig. 7(b) illustrates the cumulative distribution function (CDF) of position error with 200 runs with these different algorithms. As can be seen that 20% of the position error is larger than 0.05 m for RSDP. However, it is reduced to near 0.01 m for the APF-SDP, indicating the advantage of our APF-SDP in the performance of position prediction. Fig. 7(c) shows the CDF of velocity error using 200 sampling data. 20% of the velocity error is larger than 0.08 m/s for RSDP, but it is reduced to 0.06 m/s for our proposed APF-SDP. Therefore, the APF-SDP performs better than the RSDP in the prediction of velocity, confirming the advantage of the APF-SDP by achieving rank-one constraint.

Selecting the sensors from Tab. IV in the order of list, we also verify the performance of these proposed algorithm when the number of sensors is increased from 5 to 9. To clearly illustrate the effect of these different algorithms, we use the root mean square error (RMSE) to evaluate the performance with 200 sampling data. The position RMSE performance of these algorithms is plotted in Fig. 8(a), where “CRLB expected” represents the best achievable performance expected using the predicted noise variance. The RSDP performs not well even if the number of sensors increases. However, the proposed APF-SDP almost reaches the “CRLB expected” performance, which is consistent with the simulation results. Fig. 8(b) illustrates the velocity RMSE as the number of sensors increases. The performance of PF-SDP is better than that of the RSDP when log$\eta$ is set to -2. However, the PF-SDP can not provide the “CRLB expected” performance since it can not adaptively choose the penalty coefficient.

There are many position and velocity prediction problems in industrial Internet of things, such as the position obtaining of mobile AV and the tracking of UAV, which are illustrated in Fig. 9(a) and Fig. 9(b), respectively. Most of the current research on these problems is focused on the position obtaining of mobile source using some ranging information or the velocity prediction using motion sensors or Dropper shift measurements. The motion sensor is an extra hardware device which will increase the system cost. Since the velocity obtaining of Dropper shift measurement is subject to the error of (2, 4) m/s, it is considered to be unacceptable in some application systems. Our proposed method does not require any motion sensors or Dropper shift measurements and realize the prediction of position along with the velocity of mobile source.

Our experimental results show that the mean position RMSE of APF-SDP is smaller than 0.01 m when the UM equipped to the mobile source is used to collect the TD information. Apparently, the signal can be acoustic or electromagnetic, and the signal can travels in the underwater or underground scenarios. Due to the different propagation speed of the signal, our proposed algorithms provides distinct accuracy performance in the prediction of position and velocity, which is also demonstrated in Fig. 5. Therefore, a feasible method to improve the accuracy performance is to reduce the noise level at large propagation speed. Moreover, the precise timing is very important to ensure the performance especially when the propagation speed of electromagnetic signal reaches $3\times 10^{8}$ m/s.