In this paper, we develop and present a novel strategy for safe coordination of a large-scale multi-agent team with ``\textit{local deformation}" capabilities. Multi-agent coordination is defined by our proposed method as a multi-layer deformation problem specified as a Deep Neural Network (DNN) optimization problem. The proposed DNN consists of $p$ hidden layers, each of which contains artificial neurons representing unique agents. Furthermore, based on the desired positions of the agents of hidden layer $k$ ($k=1,\cdots,p-1$), the desired deformation of the agents of hidden layer $k + 1$ is planned. In contrast to the available neural network learning problems, our proposed neural network optimization receives time-invariant reference positions of the boundary agents as inputs and trains the weights based on the desired trajectory of the agent team configuration, where the weights are constrained by certain lower and upper bounds to ensure inter-agent collision avoidance. We simulate and provide the results of a large-scale quadcopter team coordination tracking a desired elliptical trajectory to validate the proposed approach.

This paper studies the problem of safe and optimal continuum deformation of a large-scale multi-agent system (MAS). We present a novel approach for MAS continuum deformation coordination that aims to achieve safe and efficient agent movement using a leader-follower multi-layer hierarchical optimization framework with a single input layer, multiple hidden layers, and a single output layer. The input layer receives the reference (material) positions of the primary leaders, the hidden layers compute the desired positions of the interior leader agents and followers, and the output layer computes the nominal position of the MAS configuration. By introducing a lower bound on the major principles of the strain field of the MAS deformation, we obtain linear inequality safety constraints and ensure inter-agent collision avoidance. The continuum deformation optimization is formulated as a quadratic programming problem. It consists of the following components: (i) decision variables that represent the weights in the first hidden layer; (ii) a quadratic cost function that penalizes deviation of the nominal MAS trajectory from the desired MAS trajectory; and (iii) inequality safety constraints that ensure inter-agent collision avoidance. To validate the proposed approach, we simulate and present the results of continuum deformation on a large-scale quadcopter team tracking a desired helix trajectory, demonstrating improvements in safety and efficiency.