# Decentralized Consensus Protocols on \(SO(4)^{N}\) and \(Tso(4)^{N}\) with Reshaping
## 1 Introduction  
In the past few decades, multi-agent systems have gained attention due to a wide range of applications, such as robotics, unmanned aerial vehicle swarms, and satellite formations. A key challenge is achieving consensus, where all agents in the network achieve a common (or desired relative) state, despite differences in initial conditions and communication constraints. Various approaches to the consensus problem have been pursued, such as consensus via optimization [1], where algorithms for global synchronization and balancing on connected compact manifolds become symmetric with respect to the absolute position on a manifold, allowing focus on the relative configuration. Geometric approaches for non-linear consensus problems are presented in [2], and global convergence issues not present in linear models are examined. Additional factors such as time delays in communication between agents may also be considered [3]. In all previous research on consensus, the importance of the communication topology is emphasized [4]. Lie groups, including \(SO(n)\), are used in machine learning methods by modeling transformations in a smooth and structured manner. Applications that require rotationalinvariance, such as robotics, computer vision, and physics simulations, can benefit from consensus-based distributed learning on \(SO(3)\), as presented in [5]. Decentralized learning frameworks can be employed to train models that respect the symmetries of rotational invariance and equivariance, such as in [6; 7]. Consensus problems with multiple agents can learn representations that are equivariant to rotations, ensuring consistent performance regardless of orientation. For example, a reinforcement learning neural network on a robotic arm control system should be equivariant such that, if the arm's position is rotated, the control outputs should rotate accordingly. Further discussion of this topic is presented in [8]. Some applications on the Lie group \(SO(4)\) in machine learning include representations of higher-dimensional rotations in theoretical physics and geometric deep learning, where data are represented on a manifold with higher-dimensional symmetries. The literature on this topic is limited, providing the opportunity for potential research in developing neural network architectures that are invariant to rotations in four dimensions. However, Ref. [9] explores the continuity of rotation representations in neural networks and discusses the general rotation on \(SO(n)\). The mentioned paper provides insights into the methodologies of representing higher-dimensional neural networks, which can be useful when incorporating \(SO(4)\) symmetries in machine learning models. Rotation matrices, which are elements of the Lie group \(SO(3)\), have gained interest in the development of attitude control techniques due to their non-singular and unique representation of rigid body attitude that results in globally continuous control laws [10]. Rigid body attitude consensus (or attitude synchronization) on \(SO(3)^{N}\) and the tangent bundle \(TSO(3)^{N}\) focuses on the cooperative control of the orientations of agents that share information within a network, ensuring that the multi-agent system can function in unison. The concept of consensus control on \(SO(n)^{N}\) or its tangent bundle \(TSO(n)^{N}\) can be interpreted with an understanding of attitude consensus on \(SO(3)^{N}\) and \(TSO(3)^{N}\). A thorough explanation of attitude using rotation matrices on \(SO(3)\) is presented in [10], including the use of [PERSON]'s formula, which maps the principal angle and axis to the rotation matrix, and Lyapunov-based stability analysis [10]. Various approaches have been taken to achieve consensus on special orthogonal groups. This includes a method similar to ref. [1] that reduces a non-linear consensus problem to the optimization of linear functionals over the convex hull of the group of rotation matrices on \(SO(n)\)[11].