Improving Connectivity of RIS-Assisted UAV Networks using RIS Partitioning and Deployment

This paper presents an innovative approach to maximizing connectivity of UAV networks using RIS deployment and partitioning, as shown in Fig. 1. In the illustrated uplink RIS-assisted UAV scenario, one user-equipment (UE) intends to transmit data to the UAVs, denoted by the set $\mathcal{K}=\{1,2,\ldots,K\}$, and RIS can aid in establishing reliable communications and support the direct link of the UE. Assuming a dense urban scenario, where direct links between the UE and some UAVs are blocked, and communication can only occur through the RIS, RIS can aid in establishing reliable communication with the blocked UAVs. To enable the RIS to coherently reflect the signal of the UE to multiple UAVs, we propose a virtual RIS partitioning approach, wherein a dedicated portion of RIS elements are configured to passively beamform the UE’s signal to one UAV. We assume that the UE and the UAVs are equipped with a single antenna, while the RIS has $N$ reflecting elements that are indexed row-by-row by $n=1,\ldots,N$. The UE and UAVs transmit at identical powers, denoted by $p$ and $P$, respectively.

In the depicted scenario of Fig. 1, we consider the setup of two UAVs $x$ and $y$, represented by UAV_j, $j\in\{x,y\}$, which are blocked and can not communicate directly with the UE and must communicate through the RIS, where UAV_x is more reliable than UAV_y. Low reliable UAV is the most critical one that causes severe connectivity degradation if it has failed due to low battery and hardware and software issues. In Fig. 1, UAV_y is not reliable since it has many connections to the network, thereby could be critical and fail at any time. Therefore, the data of UE should not be solely transmitted to UAV_y. Therefore, the RIS can concurrently boost the signal dedicated to the more reliable UAV_x and relatively enhance the QoS of UAV_y. This process involves designing certain RIS elements’ phases to significantly maximize $\text{UE}\rightarrow\text{UAV}_{x}$ channel while a few elements’ phases should be configured to align with the $\text{UE}\rightarrow\text{UAV}_{y}$ channel. The two-UAV setup is considered for analytical tractability and to draw important insights into the RIS partitioning and deployment design.

The number of RIS elements allocated for UAV_j is denoted by $N_{j}=\lceil\rho_{j}N\rceil$, where $\rho_{j}\in[0,1]$ is the RIS allocation factor such that $\sum_{j=1}^{2}\rho_{j}\leq 1$ and $\sum_{i=1}^{2}N_{i}=N$ ensure that the total number of the allocated RIS elements for partitioning is equal to the physically available number of the RIS elements. Furthermore, the reflection coefficient matrix of the RIS is modeled as $\mathbf{\Theta}=diag(G)$, where $G_{n}=|G_{n}|e^{j\theta_{n}}$ and $\theta_{n}\in[0,2\pi)$ is the phase shift of the $n$-th element, with $|G_{n}|=1$, $\forall n$.

The channels between the UE and the UAVs (RIS), between the UAVs and the UAVs, and between the RIS and the UAVs are considered to be quasi-static with flat-fading and presumed to be perfectly-known. To practically model the communication links between the UE and the RIS, between the RIS and the UAVs, and between the UE and the UAVs, we consider the Nakagami$-m$ fading model that characterizes different fading conditions, i.e., ($m=1$) for severe fading and $(m=5)$ for nearly line-of-sight (LoS). We represent the small-scale fading coefficient and path-loss for the $\text{UE}\rightarrow\text{RIS}$ channel as $\mathbf{g}^{\text{UR}}$ and $\beta^{\text{UR}}$, respectively. Likewise, the small-scale fading coefficient and large-scale path-loss for the $\text{RIS}\rightarrow\text{UAV}_{j\in\{x,y\}}$ channel are denoted as $\mathbf{g}^{\text{RK}}_{j}$ and $\beta^{\text{RK}}_{j}$, respectively. Here, $\mathbf{g}^{\text{UR}}=[g^{\text{UR}}_{n},\ldots,g^{\text{UR}}_{N}]$ and $\mathbf{g}^{\text{RK}}_{j}=[g^{\text{RK}}_{n,j},\ldots,g^{\text{RK}}_{N,j}]$. Therefore, we consider the channel from the UE to UAV_j over the $n$-th RIS element as $g^{\text{URK}}_{n,j}=g^{\text{UR}}_{n}g^{\text{RK}}_{n,j}$, where $g^{\text{UR}}_{n}=|g^{\text{UR}}_{n}|e^{-j\phi_{n}}$ denotes the channel coefficient between the UE and the $n$-th RIS element, while $g^{\text{RK}}_{n,j}=|g^{\text{RK}}_{n,j}|e^{-j\psi_{n,j}}$ is the channel coefficient between the $n$-th RIS element and UAV_j; $|g^{\text{UR}}_{n}|$ and $|g^{\text{RK}}_{n,j}|$ are the channel amplitudes, while $\phi_{n}$ and $\psi_{n,j}$ are the channel phases.

Moreover, for the direct links between the UE and the UAVs, let $g^{\text{UK}}_{k}$ and $\beta^{\text{UK}}_{k}$ denote the small-scale fading coefficient and path-loss for the $\text{UE}\rightarrow\text{UAV}_{k}$ channel, respectively. For the $\text{UE}\rightarrow\text{UAV}_{k}$ channel, the signal-to-noise ratio (SNR) is defined as $\gamma^{\text{(UK)}}_{k}=\frac{p|\sqrt{\beta^{\text{UK}}_{k}}g^{\text{UK}}_{k}|^{2}}{N_{0}}$, where $N_{0}$ is the additive white Gaussian noise (AWGN) variance. A typical UAV_k is assumed to be within the transmission range of the UE if $\gamma^{\text{(UK)}}_{k}\geq\gamma^{\text{UE}}_{0}$, where $\gamma^{\text{UE}}_{0}$ is the minimum SNR threshold for the UE-UAV communication links.

For simplicity, we only consider the LoS path component for each UAV-UAV link, which is well justified for UAV communications [2, 9]. Thus, the LoS path-loss between UAV_k and UAV${}_{k^{\prime}}$ can be expressed as $\Gamma_{k,k^{\prime}}=20\log\bigg{(}\frac{4\pi f_{c}d_{k,k^{\prime}}}{c}\bigg{)},$ where $d_{k,k^{\prime}}$ is the distance between UAVs $k$ and $k^{\prime}$, $f_{c}$ is the carrier frequency, $c$ is the speed of light, and $d_{k,k^{\prime}}$ is the distance between UAV_k and UAV${}_{k^{\prime}}$. The SNR in dB between UAV_k and UAV${}_{k^{\prime}}$ is $\gamma^{\text{(UAV)}}_{k,k^{\prime}}=10\log P-\Gamma_{k,k^{\prime}}-10\log N_{0}$. UAV_k and UAV${}_{k^{\prime}}$ have a successful connection provided that $\gamma^{\text{(UAV)}}_{k,k^{\prime}}\geq\gamma^{\text{UAV}}_{0}$, where $\gamma^{\text{UAV}}_{0}$ is the minimum SNR threshold for the UAV-UAV communication links.

In light of the preceding discussions, the SNR at UAV_j, $j\in\{x,y\}$, can be expressed as

where $\boldsymbol{\alpha}=[\alpha_{x},\alpha_{y},\alpha_{z}]$ is the Cartesian coordinates of the RIS. Furthermore, due to the available perfect global channel state information (CSI) and using high bit resolution of the RIS element’s phase shifter [15], we consider that a portion of RIS elements is perfectly aligned with $\text{UE}\rightarrow\text{UAV}_{j=x}$ cascaded channel, while the rest RIS elements are aligned with $\text{UE}\rightarrow\text{UAV}_{j=y}$ channel and not aligned with the $\text{UE}\rightarrow\text{UAV}_{j=x}$ channel. Hence, for the sake of analytical tractability, we ignore the impact of non-aligned links from (1)¹¹1Although we ignore the impact of non-aligned channels for tractability purposes, we show in Fig. 2 that the impact of non-aligned channels on the SNR performance is negligible..

Moreover, for feasible optimization in this paper, we can represent the RIS elements’ allocation portions $\boldsymbol{\rho}=\{\rho_{1},\rho_{2}\}$ to denote the RIS section allocated to the UAVs_j, $j\in\{x,y\}$. Hence, $N_{j}$ in the first summation term in (1) can also be presented as $N_{j}=\lceil\rho_{j}N\rceil$. Additionally and without loss of generality, we rewrite the summation term of $\sum_{n=1}^{\lceil\rho_{j}N\rceil}(.)$ as $\rho_{j}\sum_{n=1}^{N}(.)$²²2Note that real-valued RIS portions may result in non-integer RIS partition allocation. In this case, the RIS allocates some elements for exact portion allocated for UAVs $j\in\{x,y\}$.. We further rewrite $\gamma_{j}(\boldsymbol{\alpha},\boldsymbol{\rho})$ as

where $Q=\sum_{n=1}^{N}Q_{n}=\sum_{n=1}^{N}|g^{\text{UR}}_{n}||g^{\text{RK}}_{n,j}|$ is the $N$-element double-Nakagami$-m$ that is independent and identically distributed (i.i.d.) random variable (RV) with parameters $m_{1}$, $m_{2}$, $\Omega_{1}$, and $\Omega_{2}$, i.e., the distribution of the product of two RVs following the Nakagami$-m$ distribution with the probability density function (PDF) is given in [12].

The total channel gain for $\text{UE}\rightarrow\text{UAV}_{j}$ cascaded channels in (II-C) can be maximized through the optimization of the reflection matrix $\mathbf{\Theta}$. Since RIS’s controller has perfect knowledge of the CSI, it has the capability to determine suitable phase shifts that can effectively nullify the phases of the respective $\text{UE}\rightarrow\text{RIS}$ and $\text{RIS}\rightarrow\text{UAV}_{j}$ channels as $\theta_{n}=\phi_{n}+\psi_{n,j}$, $\forall n,j\in\{x,y\}$ [12, 15]. Specifically, for the $n$-th RIS element, given $\phi_{n}$ and $\psi_{n,j}$, the RIS’s controller can perfectly align $\theta_{n}$ with $\phi_{n}$ and $\psi_{n,j}$ to nullify their effect, which provides the maximum channel gain value of $\text{UE}\rightarrow\text{UAV}_{j}$ link.

In the remaining of this subsection, we present some numerical results to assess the equivalence of the analytical SNR expressions in (1) and (II-C). All channels are modeled with Nakagami$-m$ fading distribution. Therefore, we set $m=1$ to emulate the Rayleigh channel for the $\text{UE}\rightarrow\text{UAV}_{k}$ links, and $m_{1}=5$, and $m_{2}=1$, $\Omega_{1}=\Omega_{2}=1$ for $\text{UE}\rightarrow\text{RIS}$ and $\text{RIS}\rightarrow\text{UAV}_{j}$ links, respectively. Unless stated otherwise, the following system parameters are considered: $p=23$ dBm, $P=30$ dBm, $N=100$, $N_{0}=-120$ dBm. For the purposes of this part, the fixed 3D Cartesian coordinates in meters for the UE are $(318,220,0)$, while UAV_x, UAV_y, and the RIS are located at $(460,340,200)$, $(370,14,200)$, and $(0,0,120)$, respectively.

Fig. 2 compares exact and approximated rates averaged over $10^{5}$ Matlab simulations, which are calculated using the SNRs in (1) and (II-C) with bandwidth $B=250$ KHz. For UAV_j, the RIS partitions, respectively, are $\rho_{x}$ and $\rho_{y}=1-\rho_{x}$. We consider phase shift control using phase shift quantization levels similar to [15], where parameter $b$ is the bit resolution of the RIS element’s phase shifter. In the literature [15], $b=4=\infty$ is considered for high bit resolution (perfect phase shift). From the figure, we notice that approximated rates for the UAVs tightly match the exact rates for most values of $\rho_{x}$ and $N$. Overall, these results justify our assumption to ignore the impact of non-aligned channels from the other portions of the RIS. Therefore, in the remaining of this paper, we will use the approximated SNR expression in (II-C).

We model the uplink RIS-assisted UAV model using the graph network $\mathcal{G}(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ represents the set of vertices associated with the network nodes (UAVs and UE) and $\mathcal{E}$ represents the edges (links). For a graph edge $e_{l}$, $1\leq l\leq E$, that links two nodes $(u,v)\in\mathcal{V}$, we have:

Subsequently, the weight vector $\mathbf{w}\in{[\mathbb{R}^{+}]}^{E}$ of these links is defined as $\mathbf{w}=[w_{1},w_{2},\ldots,w_{E}]$, and is given element-wise as

For $e_{l}$, let $\mathbf{a}_{l}$ be a vector, where the $u$-th and $v$-th elements in $\mathbf{a}_{l}$ are given by $a_{u,l}=1$ and $a_{v,l}=-1$, respectively, and zero otherwise. Let $\mathbf{A}$ be the incidence matrix of a graph $\mathcal{G}$ with the $l$-th column given by $\mathbf{a}_{l}$. Hence, in an undirected graph $\mathcal{G}(\mathcal{V},\mathcal{E})$, the Laplacian matrix $\mathbf{M}$ is a $V\times V$ matrix, defined as [5]:

where the entries of $\mathbf{M}$ are given element-wise by

where $deg(u)$ is the degree of node $u$, which represents the number of its neighboring nodes.

To maximize the connectivity of the uplink RIS-assisted UAV network, we choose a well-known metric, known as the algebraic connectivity [16, 5, 4, 3, 2, 10], denoted as $\lambda_{2}(\mathbf{M})$, to measure how well a network is connected. With RIS deployment and partitioning, a new graph $\mathcal{G}^{\prime}(\mathcal{V},\mathcal{E}^{\prime})$ is constructed with the same number of $V$ nodes and a larger set of edges denoted by $\mathcal{E}^{\prime}$ with $\mathcal{E}^{\prime}=\mathcal{E}\cup\mathcal{E}_{new}$, where $\mathcal{E}_{new}$ is the new edges for the $\text{UE}\rightarrow\text{UAV}_{x}$ and $\text{UE}\rightarrow\text{UAV}_{y}$ links. The gain can be realized by computing $\lambda_{2}(\mathbf{M}^{\prime})\geq\lambda_{2}(\mathbf{M})$, where $\mathbf{M}^{\prime}$ is the resulting Laplacian matrix of the new graph $\mathcal{G}^{\prime}(\mathcal{V},\mathcal{E}^{\prime})$.

In this paper, we measure the reliability of the UAVs_j∈{x,y} based on the severity of network connectivity after removing UAV_j and its connected edges to other nodes, which is defined as $\mathcal{R}_{j}=\frac{1}{\lambda_{2}(\mathcal{G}_{-j})}$, where $\mathcal{G}_{-j}$ is the sub-graph resulting from removing UAV_j and all its adjacent edges to other nodes in $\mathcal{G}$. Thus, we consider $\mathcal{R}_{y}>\mathcal{R}_{x}$.

In this paper, our interest is to unleash the benefits of the RIS to aid the uplink connections of the UE to the blocked UAVs. Following the previous results and discussions suggest, the SNR performance improvement inherently requires the joint optimization of RIS partitioning and location to maximize the link quality between the UE and the UAVs via the RIS while ensuring their QoS constraints. Let $\gamma^{\text{RIS}}_{0}$ be the minimum SNR threshold of $\text{UE}\rightarrow\text{UAV}_{j\in\{x,y\}}$ via the RIS. The considered optimization problem is formulated as

where $\preceq$ is the pairwise inequality and $0<\zeta\leq 1$ is a design parameter that determines the QoS limit set for the least reliable UAV_y based on $\mathcal{R}_{y}$. In P0, $C_{1}^{0}$ and $C_{2}^{0}$ constitute the QoS constraints on the UAVs_j∈{x,y}, $C_{3}^{0}$ ensures that the allocated portions does not exceed unity to limit the total number of allocated RIS elements is not higher than the total number of RIS elements. Finally, $C_{4}^{0}$ specifies the domain of optimization variables. The Laplacian matrix $\mathbf{M}^{\prime}(\boldsymbol{\rho},\boldsymbol{\alpha})$ depends on the RIS location and partitions, which determine the quality of the new links. To tackle P0 over $\boldsymbol{\rho}$ and $\boldsymbol{\alpha}$, we decompose P0 into two subproblems and solve them iteratively. For given RIS location $\boldsymbol{\alpha}_{0}$, we optimize $\lambda_{2}(\mathbf{M}^{\prime}(\boldsymbol{\rho},\boldsymbol{\alpha}_{0}))$ by finding a closed-form solution of the RIS portions, denoted by $\boldsymbol{\rho}^{*}$, that pushes the UAVs_j∈{x,y} SNR to its maximum. Then, we exploit the closed-form RIS partitioning solution to find the 3D deployment of the RIS to further maximize $\lambda_{2}(\mathbf{M}^{\prime}(\boldsymbol{\rho}^{*},\boldsymbol{\alpha}))$.

The first subproblem of the RIS partitioning to maximize the sum SNR of the UAVs_j∈{x,y} can be formulated for a given RIS location $\boldsymbol{\alpha}_{0}$ by using P0 and (II-C) as follows

which can be solved by pushing the SNR of the $\text{UE}\rightarrow\text{UAV}_{x}$ link, corresponding to the most reliable UAV, to its maximum value. In what follows, we provide the closed-form solution for the RIS partitioning.

First, the sum SNR can be maximized by satisfying UAV $y$’s QoS constraint in $C_{2}^{1}$ with equality while pushing $\gamma_{x}$ to its maximum subject in $C_{1}^{1}$. Therefore, we rewrite $C_{2}^{1}$ in (8) using (II-C) as follows

The closed-form solution for $\rho_{y}$ can be derived as

Subsequently, the rest of the unused elements should be allocated to UAV_x, and therefore, all available RIS elements must be used to reach the maximum SNR of that UAV. Thus, the RIS portion allocated to UAV_x is written as

Notice that the solution $\rho^{*}_{y}(\boldsymbol{\alpha_{0}})$ and $\rho^{*}_{x}(\boldsymbol{\alpha_{0}})$ are feasible only if the constraints in (8) are satisfied. In general, the RIS partitioning closed-form solution of the two-UAV setup can be generalized to multiple-UAV setup, where $\rho^{*}_{x}$ is given as

where $C$ is any number of blocked UAVs that is greater than one. The solution first needs to find the RIS partitions that satisfy the SNR limit of the least reliable UAV and so on.

The RIS location substantially affects the network connectivity performance as it directly relates to the channel gains and path-loss for the $\text{UE}\rightarrow\text{UAV}_{j\in\{x,y\}}$ links via the RIS. Therefore, this subsection considers the RIS deployment problem formulated in P2 to maximize the SNR of those links by using the RIS portions derived in the previous subsection:

where $\boldsymbol{\rho^{*}}=[\rho^{*}_{j=x},\rho^{*}_{j=y}]$. $C_{3}^{2}$ defines the search space range for the $\alpha_{x},\alpha_{y}$ coordinates as extending from negative infinity to positive infinity, while the coordinate constraint for $\alpha_{z}$ is limited to a range between $0$ and $\infty$. We solve P2 by meta-heuristic methods that run a global search of the RIS locations and evaluate the location fitness by the proposed RIS partitioning. In this work, we use the simulated-annealing (SA) method to solve P2.