However, given the conventional digital signal processing (DSP) algorithms, such as the Multiple Signal Classification (MUSIC) and the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), as well as sparse estimation methods of array manifold matrix [30; 31; 32; 33; 34; 35; 36; 37; 38; 39] to determine spatio-temporal data are mostly based on a priori information and may not be suitable in beamforming-based AUAS routing algorithms. Thus, techniques have to be explored (i) to understand and simplify the mathematical interpretation of models described by AI, (ii) to reduce the computational and time burden in computing spatio-temporal data, and (iii) to design accurate and low-cost ML algorithms. As of the preliminary work, we propose a mathematical model, algorithm, and error threshold of the proposed model to route a drone swarm utilizing spatio-temporal data and multi-beam beamforming communication having decentralized flying antenna array systems. Although beamforming is implemented to improve UAS networking, the conventional routing algorithms for UAS networking are based on ad hoc on-demand distance vector (AODV) and optimized link state routing (OLSR). AODV is decentralized, reactive, and has less overhead for management, which is robust to the dynamic topology. Nevertheless, the main drawbacks of AODV are the consumption of path discovery and the local optimization for routing generation [40; 41]. In contrast, OLSR is a centralized algorithm that could achieve global optimization of routing generation with the sacrifice of overhead [42]. Compatible with beamforming, a swarm-oriented routing algorithm, implemented with decentralization and efficiency, is an urgent need for AUAS networking. Thus, challenges still remain to efficiently navigate drone swarms without centralizedcommunications, obtain efficient algorithms to capture real-time spatio-temporal data in a massive and complex multiple-input multiple-output (MIMO) model, and address limitations in beamformed antenna arrays for AUAS routing. Here, we propose a mathematical model and address complexity, precision, and reliability to route an AUAS swarm while communicating with each other (without having a base station communication) through multiple spatio-temporal data based on theories of multi-beam beamforming, sensor arrays, and structured matrices. The organization along with the contribution of the paper is as follows. * In Section 2, we propose a mathematical model followed by an algorithm and flow chart to navigate a drone swarm using multi-beam beamforming decentralized antenna array systems. * In Section 3, we present
* the array received beamformed signals of drones in the drone swarm at time stamps and compare these received signals with output signals corresponding to a ground-truth action function having the same time stamps. * the analytical arithmetic complexity and numerical computational complexity of the proposed routing algorithm. * BER results based on the proposed AUAS swarm routing algorithm. * In Section 4, we discuss an ML algorithm for the completion of spatial and temporal data to obtain the AUAS swarm routing algorithm followed by a low-cost algorithm. * In Section 5, we conclude the paper.