Optimal Measurement of Drone Swarm in RSS-based Passive Localization with Region Constraints

Consider a cooperate localization on a stationary target via UAVs, as illustrated in Fig. 1. The target is located at an unknown position on the ground, denoted as $\mathbf{s}=(x,y,0)$. $N$ UAVs search the radio-frequency (RF) signal emitted from the target and measure the RSS.

Each UAV takes multiple measurements while flying or hovering. The measurement amount of the $i$-th UAV is denoted as $M_{i}$. The position of the $i$-th UAV at the $j$-th measurement is denoted as $\mathbf{u}_{i,j}=(x_{i,j},y_{i,j},z_{i,j}),~{}1\leq i\leq N,1\leq j\leq M_{i}$. Note that each UAV is aware of its own geographical position using the global positioning system (GPS). Therefore, the distance from this position to the target can be expressed by

According to the well-known radio propagation path loss model (in decibels) [15], the $j$-th measured RSS (in dB) of UAV $i$ can be expressed by

where $p_{0}$ denotes the signal power of the source and $\gamma$ denotes the path loss exponent (PLE). It is assumed that the parameters $p_{0}$ and $\gamma$ are known (determined using calibration or prior knowledge) [29, 30, 3]. The measurement noise is denoted as $\eta_{i,j}$, and $\eta_{i,j}\in\mathcal{N}(0,\sigma^{2}_{i})$.

Based on the measurements, $\mathbf{s}$ can be inferred using maximum likelihood (ML) estimation. Stacking all measurements of UAV $i$ can form a $M_{i}$-dimensional column vector, shown as follows.

Accordingly, the probability distribution function (PDF) of the target position is given by

where $\mathbf{N}_{i}=\sigma_{i}^{2}\mathbf{I}_{M_{i}\times M_{i}}$ is the covariance matrix of $\bm{\eta}_{i}$. As usual, measurement noises among different UAVs are independent. Therefore, the joint PDF based on the whole measurements is given by

In this paper, we aim to find optimal geometrical configurations in such localization task. Fisher information matrix (FIM) is well-known to represent the amount of information contained in noisy measurements. Therefore, maximizing the determinant of the FIM, i.e., D-optimality criterion [23, 27, 28], is employed as the optimality metric. Based on the the joint PDF, the FIM can be derived by

Meanwhile, as shown in Fig. 1, a practical condition is considered where UAVs are only able to hover or fly on the limited area while measuring. Due to the safety factor (not detected by the enemy) or geographical limitations, the minimum flying height and the minimum distance to the target must be guaranteed. Corresponding mathematical expressions are given by

Moreover, assume all UAVs have a max flying speed, denoted as $c_{\mathrm{max}}$. The time interval of two effective measurements is denoted as $t_{0}$. This results in the following constraint.

Let $\mathbf{r}$ and $\mathbf{h}$ denote the collection of horizontal distances to the target and flying heights, respectively. $\beta$ denotes the collection of horizontal UAV-target angles where $\tan(\beta_{i,j})=\frac{x_{i,j}-x}{y_{i,j}-y}$. It can be verified that the $u_{i,j}$ can be derived from specific $r_{i,j}$, $h_{i,j}$ and $\beta_{i,j}$. To sum up, the mathematical form of this problem is as follows

In this setting, the FIM in (A), derived in Appendix A, can be simplified to

where $\mathbf{g}_{i}=\begin{bmatrix}\cos{\beta_{i}}&\sin{\beta_{i}}\end{bmatrix}$.

For the symmetric matrix $\mathbf{G}\in\mathbf{R}^{2\times 2}$, we have

Therefore, maximizing the $\mathrm{det}(\mathbf{F})$ is equal to minimizing the $\|\mathbf{G}\|^{2}$. Using the tight frame theory [31, 32, 33] as introduced in [23], there exists optimal placements minimizing $\|\mathbf{G}\|^{2}$ in both regular and irregular cases [23]. Accordingly, optimal geometries are summarized in Theorem 13 and Theorem 2, combining with the focused settings. For convenience, let $\max\{M_{i}\sigma_{i}^{-2}\frac{r_{i}^{2}}{d_{i}^{4}}|i=1,\cdots N\}=M_{k}\sigma_{k}^{-2}\frac{r_{k}^{2}}{d_{k}^{4}}$.

After trivial algebraic operations, the max information amounts in both cases are obtained, given by

Due to the split property of the max determinant of FIM, shown in formula (16) and (17), it is straightforward that the optimal horizontal distance and height of UAVs are

In this configuration, the distance between UAVs to target are all equal to $\sqrt{(r^{\ast})^{2}+(h^{\ast})^{2}}$, denoted as $d^{\ast}$. However, this result is only applicable to the special regular or irregular case. To solve problem $\mathrm{P}$, further analysis is presented in the following.

For convenience, let $\max\{M_{i}\sigma_{i}^{-2}|i=1,\cdots N\}=M_{t}\sigma_{t}^{-2}$. Besides, we define $\varpi_{a}=M_{t}\sigma_{t}^{-2}\frac{r_{t}^{2}}{d_{t}^{4}}$, $\varpi_{b}=\sum_{i=1,i\neq t}^{N}M_{i}\sigma_{i}^{-2}\frac{r_{i}^{2}}{d_{i}^{4}}$, $\varpi^{*}_{a}=M_{t}\sigma_{t}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}}$, $\varpi_{b}^{*}=\sum_{i=1,i\neq t}^{N}M_{i}\sigma_{i}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}}$ and $c=\left(\frac{10\gamma}{\ln{10}}\right)^{4}$. Moreover, (16) is simplified as $\Psi_{1}=\frac{1}{4}c(\varpi_{a}+\varpi_{b})^{2}$ and (17) is simplified as $\Psi_{2}=c\varpi_{a}\varpi_{b}$.

However, there is a candidate in regular cases. Since $\Psi_{1}(\varpi_{a},\varpi_{b})$ is an increasing function with respect to $\varpi_{a}$ and $\varpi_{b})$, $\Psi_{1}$ reaches the max value at an extreme point where $\varpi_{a}$ and $\varpi_{b}$ are max in the domain of definition. It is easy to verify that the max $\varpi_{a}$ and $\varpi_{b}$ satisfy $\varpi_{a}=\varpi_{b}=\sum_{i=1,i\neq t}^{N}M_{i}\sigma_{i}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}}$. It means that except the $t$-th UAV, the other UAV satisfies the boundary conditions of distance and height, while the $t$-th UAV compromises with them to satisfy the regular condition. Therefore, the max determinant of $\mathbf{F}$ in regular cases is equal to $c(\sum_{i=1,i\neq t}^{N}M_{i}\sigma_{i}^{-2})^{2}\frac{(r^{\ast})^{4}}{(d^{\ast})^{8}}$.

At last, comparing the max determinant of $\mathbf{F}$ in two cases yields

From the above, the configuration satisfies (18a), (18b) and (15) is the optimal solution of problem $\mathrm{P}$. $\square$

Assume each UAV cannot complete a full circle flying with respect to the target during the localization task, i.e., $t_{0}M_{i}c_{max}<2\pi r^{\ast},~{}~{}i=1\cdots N$. We consider a special case where $M_{i}=M,i=1\cdots N$ and $\max\{\sigma_{i}^{-2}|i=1,\cdots N\}\leq\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}^{-2}$ (a regular case).

The FIM in (A) is rewritten as

Note that $\mathbf{G}_{j}$ is just a special form of $\mathbf{G}$ defined in (10). Therefore, according to Theorem 3, the optimal configuration satisfies (18a), (18b) and (13) with $M_{i}=1$. In this configuration, $\mathbf{G}_{j}=\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}}\mathbf{I}$. It shows that the determinant of FIM is equal to $\frac{1}{4}\left(\frac{10\gamma}{\ln{10}}\right)^{4}(\sum_{i=1}^{N}M\sigma_{i}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}})^{2}$, which is the max value of the objective function in problem $\mathrm{P}$.

To satisfy (13) with $M_{i}=1$, the optimal UAV-target horizontal angle configuration at one measurement can be obtained using algorithm 1 proposed in [23]. Let $[\beta_{1,0}~{}~{}\beta_{2,0}~{}~{}\cdots~{}~{}\beta_{N,0}]$ be one of them. According to the principle of equivalent placements [23], rotating the overall angles around the target maintains the optimal property. Mathematically, $[\beta_{1,0}+\beta~{}~{}\beta_{2,0}+\beta~{}~{}\cdots~{}~{}\beta_{N,0}+\beta]$ is also the optimal angle configuration. Considering the flying ability of UAVs, the optimal UAV-target horizontal angle configuration is given by

From the above, the following theorem is proposed.

Assume each UAV can complete a full horizontal circle flying with respect to the target during the localization task, i.e., $t_{0}M_{i}c_{\mathrm{max}}\geq 2\pi r^{\ast},~{}i=1\cdots N$. Let $\beta_{0,i}$ denote the $i$-th UAV-target horizontal angle at the starting position.

The FIM in (A) is rewritten as

Note that $\mathbf{G}_{i}$ is just a special form of $\mathbf{G}$ defined in (10). Therefore, according to Theorem 3, the optimal configuration satisfies (18a), (18b) for $i$-th UAV. In this way, $\mathbf{G}_{j}=\sigma_{i}^{-2}\frac{(r^{\ast})^{2}}{(d^{\ast})^{4}}\sum_{j=1}^{M_{i}}\mathbf{g}_{i,j}\mathbf{g}_{i,j}^{T}$. The problem belongs to the equally-weighted optimal placements [34, 23]. Accordingly, when $M_{i}=2$, the right angle structure is optimal, given by

When $M_{i}>2$, the uniform angular array (UAA) is optimal, given by

Under this configuration, the determinant of FIM is expressed by

It is easy to verify that $\left|\mathbf{F}^{*}\right|$ in (25) reaches the max value of the objective function in problem (II-B).

Moreover, besides this configuration, their exists other optimal geometrical configurations with beyond half circle flying for $M_{i}>2$. Let $K_{i}$ denote an arbitrarily integer number satisfying $K_{i}<M_{i}$, and $K_{i}\geq\frac{M_{i}}{2}$. Via flipping some positions in UAA that are measured after the $K_{i}$-th measurement about the target, the configuration is completed. Mathematically, in this configuration, we have

According to the principle of equivalent placements [23], this configuration is an equivalent form to the optimal full circle flying configuration, i.e, UAA. Therefore, it is optimal. Fig. 2 illustrates both of them.

Thus, the following theorem is proposed.

In this subsection, $N=3$ UAVs equipped with RSS sensors are deployed to measure the RF signal from the target while hovering. The horizontal distances to the target and heights satisfy (18a) and (18b) respectively. The $1$-th UAV is hovering with $\beta_{1}=0$. Both regular and irregular cases are considered. The setting for the regular cases are $\sigma^{2}_{1}=16~{}\mathrm{dB};\sigma^{2}_{2}=16~{}\mathrm{dB};\sigma^{2}_{3}=16~{}\mathrm{dB}$, named as case (a) and $\sigma^{2}_{1}=8~{}\mathrm{dB};\sigma^{2}_{2}=12~{}\mathrm{dB};\sigma^{2}_{3}=16~{}\mathrm{dB}$, named as case (b). Meanwhile, the setting for the irregular cases are $\sigma^{2}_{1}=2~{}\mathrm{dB};\sigma^{2}_{2}=8~{}\mathrm{dB};\sigma^{2}_{3}=16~{}\mathrm{dB}$, named as case (c) and $\sigma^{2}_{1}=2~{}\mathrm{dB};\sigma^{2}_{2}=16~{}\mathrm{dB};\sigma^{2}_{3}=16~{}\mathrm{dB}$, named as case (d).

Fig. 3 shows the the FIM determinant via the UAV-target angle configuration. In the regular case (a) (a equally-weighted placement), the optimal angle configurations are $[60^{\circ},~{}120^{\circ}]$, $[60^{\circ},~{}300^{\circ}]$, $[120^{\circ},~{}60^{\circ}]$, $[120^{\circ},~{}240^{\circ}]$, $[240^{\circ},~{}120^{\circ}]$, $[240^{\circ},~{}300^{\circ}]$, $[300^{\circ},~{}60^{\circ}]$, $[300^{\circ},~{}240^{\circ}]$. In the regular case (b), the optimal angle configurations are around $[103^{\circ},~{}72^{\circ}]$, $[103^{\circ},~{}252^{\circ}]$, $[283^{\circ},~{}102^{\circ}]$, $[283^{\circ},~{}253^{\circ}]$. It can be verified that all the optimal configurations are consistent with Theorem 13. In the irregular cases (c) and (d), the optimal angle configurations are $[90^{\circ},~{}90^{\circ}]$, $[90^{\circ},~{}270^{\circ}]$, $[270^{\circ},~{}90^{\circ}]$, $[270^{\circ},~{}270^{\circ}]$. Obviously, the results are the same as Theorem 2.

In this subsection, $N=3$ UAVs are deployed to measure the RF signal from the target while hovering. Both regular and irregular cases are considered. As for each case, the variance settings maintain the same as the last subsection (case (b) and case (c)). The optimal UAV-target angle configurations are used.

Fig. 4 shows the determinant of FIM and a low bound of root mean square error (RMSE) via hovering horizontal distance to the target. Specifically, the low bound is $\sqrt{\mathrm{Tr}(\mathrm{CRLB})}$. When UAV 2 and 3 are fixed, i.e., $r_{2}=r_{3}=h_{0}$, the optimal horizontal distance of UAV 1 is equal to $h_{0}$. When no UAV is fixed, the optimal horizontal distance of UAVs is also equal to $h_{0}$. It is concluded that the optimal UAV-target elevation angle is $45^{\circ}$ when $r_{0}\leq h_{0}$. Note that it is different to normal 2D cases where sensors should be be close to the target. The reason is that a additional dimensional exists while measuring the target on the ground via UAVs.

In this subsection, a practical scenario is considered where the proposed optimal configurations are used to refine the prior estimation. $N=15$ UAVs are deployed. Among them, the measurement noise variance of ten UAVs are set to be $12~{}\mathrm{dB}$ and the others are set to be $16~{}\mathrm{dB}$.

Fig. 5 shows the performances of the proposed configurations via variance of prior estimation error. As observed, the robustness lever is ordered as follows: full circle flying, beyond half circle flying, below half circle flying, hovering. Moreover, the uncertainty of the target has little impact on the proposed configurations, which ensures the practicality of these ideal optimal configurations.