## 2 MIMO Beamforming-Based Mathematical Model and Routing Algorithm to Navigate a Collection of AUAS  
In this section, we propose a novel mathematical model to route an AUAS swarm based on MIMO beamforming. After the model is proposed, an algorithm is stated to route an AUAS swarm consisting of a massive number of drones. Thus, we will be able to deploy over 100 drones without using base station communication. The proposed method is different from the conventional AODV and is suitable for the 5G NR, B5G, and 6G cellular networking on the AUAS networking on a large scale. The difference between the scenarios is that the existing AODV can be deployed in a group and the members of the group can not exceed 100 [43; 44]. We assume a drone swarm consists of \(M\) number of drones, and each drone--say \(u\)--has a uniform array with \(N\) elements and \(d\) element spacing. We also assume that there are \(M\) uncorrelated signals impinging on the array from \(M\) drones (to establish communication among drones) with unique directions \(\{\theta_{i}\}_{i=1}^{M}\), amplitudes \(\{a_{i}\}_{i=1}^{M}\), and temporal variables \(\{\omega_{i}\}_{i=1}^{M}\) s.t. \(-2\pi f_{i}\leq\omega_{i}<2\pi f_{i}\), where \(f_{i}\) is the unique temporal frequency in each drone. Thus, the assumptions are made so that each drone could communicate using a unique RF beacon modulated at a low-rate digital waveform that carries a unique binary identification code. If \(\mathbf{x}^{(u)}(t)=[x(0,t),x(1,t),\cdots,x(M-1,t)]^{T}\)--say \(\mathbf{x}^{(u)}\) denotes the source signal of drone \(u\) at time \(t\), then the array received \(N\)-beamformed signal of drone \(u\) at time \(t\), i.e., \(\mathbf{y}^{(u)}(\omega_{i},a_{i},\theta_{i},t)=\sum\limits_{k=0,i=1}^{N-1M} a_{i}e^{i(\omega_{i}t-k\omega_{i}\phi_{i})}\mathbf{x}^{(u)}+\mathbf{y}^{(u)}\), where \(\psi_{i}=2\pi\frac{d}{\lambda}\sin(\theta_{i})\), \(\lambda\) denotes the wavelength of the incident signal, and \(\mathbf{j}^{(u)}(t)=[n_{0}(t),n_{1}(t),\cdots,n_{N-1}(t)]^{T}\)--say \(\mathbf{j}^{(u)}\) as the additive white Gaussian noise (AWGN), takes the vector form \(\mathbf{y}^{(u)}(\omega_{i},a_{i},\theta_{i},t)=[y(0,t),y(1,t),\cdots,y(N-1,t )]^{T}\)--referred to as \(\mathbf{y}^{(u)}\). Now, for all \(u\)'s we can rewrite \(\mathbf{y}^{(u)}\)'s as a collection of matrix-vector products describing a MIMO beamforming model, and for each drone, \(u\), described via:  
\[\mathbf{y}^{(u)}=\mathbf{V}^{(u)}_{ki}\cdot\mathbf{S}^{(u)}_{j}\cdot\mathbf{x }^{(u)}+\mathbf{j}^{(u)}, \tag{1}\]where \(\mathbf{V}_{ki}^{(u)}=\left[e^{-j\kappa\omega_{i}\psi_{i}}\right]_{k=0,i=1}^{N-1,M}\) is an \(N\times M\) matrix determined by spatial and temporal frequencies of each drone \(u\), which we call a frequency Vandermonde matrix, \(j^{2}=-1\), and \(\mathbf{S}_{i}^{(u)}=[a_{i}e^{i\omega_{i}t}]_{i=1}^{M}\) is an \(M\times M\) matrix consisting of temporal frequencies and amplitudes (Recall that \(\mathbf{V}_{ki}^{(u)}\cdot\mathbf{S}_{i}^{(u)}\) is called the array manifold matrix). Thus, we will have a set of \(M\) systems of equations, and for each drone \(u\) we have a system of \(N\) equations as given in (1). The simplest example case of the Equation (1) can be described as having three drones with each consisting of four elements in the antenna array. Thus, we could see the overall system as a collection of three systems with each consisting of four sets of equations. Thus, if each drone received signals from other drones (while communicating with itself and the other two), the collection of 12-beamformed signals (4-beamformed signals per drone) can be put into a collection of matrix-vector form as follows  
\[ \begin{bmatrix}\mathbf{y}_{4\times 1}^{(1)}\\ \mathbf{y}_{4\times 1}^{(2)}\\ \mathbf{y}_{4\times 1}^{(3)}\end{bmatrix} =