# Decentralized Consensus Protocols on \(SO(4)^{N}\) and \(Tso(4)^{N}\) with Reshaping
## 1 Introduction
Another optimization approach uses a gradient algorithm based on the cost function of an output space of a system and is applied to synchronization where only partial state information is shared [12]. This is useful in scenarios such as satellite communication links in a constellation where bandwidth is limited between neighboring lines. Similar to ref. [2], a geometric approach for coordinated motion in Lie groups is taken in [13], driving a swarm of simple integrator agents towards local convergence. In ref. [14], an analytical solution to closed-loop kinematics is presented and results in almost globally exponentially stabilizing control laws. An attitude tracking problem for a class of almost global asymptotic stability (AGAS) stable feedback laws is presented in [15] with a closed form solution as well, although non-consensus equilibria are not addressed. A leader-follower strategy for attitude formation tracking control is presented in [16] to achieve the desired relative attitude between agents, and the stability of the proposed tracking control law is enhanced by feedback reshaping. Similarly to the topologies explored in this paper, ref. [17] studies ring communication graphs and the almost global stabilization of consensus with input delays, destabilizing undesired non-consensus equilibria using reshaping. Rigid body pose consensus control and the impact of communication delays between agents are studied in [18] on \(TSE(3)\), where rigid body attitudes are expressed as rotation matrices, and the almost global asymptotic stability of the consensus subspace is achieved by an extended Morse-Lyapunov-Krasovski approach. An extension of the Morse-Lyapunov approach is also used in [19] to achieve the almost global asymptotic stability of the consensus subspace on Lie groups \(SO(3)\) and \(SE(3)\). In ref. [20], non-linear stability analysis and [PERSON]'s direct method are applied to design non-linear feedback control laws for rigid body control on \(TSO(3)\) and for the problem of multi-body attitude synchronization on \(TSO(3)^{N}\). The improvement in globally continuous control laws using rotation matrices and the feedback reshaping strategy is demonstrated via stability analysis and simulation. Without reshaping, the attitude control of a rigid body with initial principal rotation angles (PRAs) close to 180\({}^{\circ}\) results in poor closed-loop response performance. Additionally, multi-body attitude synchronization on five- and nine-agent ring graphs results in stable non-consensus equilibria, whereas adding feedback reshaping destabilizes these equilibria and produces attitude synchronization with almost global stability. The authors find that applying a reshaping function for the PRA designed to destabilize the non-consensus equilibria ensures the almost global stability of the consensus subspace with a fixed number of agents. The authors observe that different numbers of agents may require different reshaping functions. Consensus with AGAS is achieved as long as the relative PRA of the undesired equilibrium is greater than the angle that maximizes the reshaping function. In this paper, four-dimensional rotations are used to represent an agent's configuration, synonymous to an agent's attitude in three dimensions, which is analagous to the application of rotation-invariant neural networks in four dimensions. Consensus protocols are proposed on the Lie group \(SO(4)\) and its corresponding tangent bundle \(TSO(4)\). These protocols are proven to achieve almost global asymptotic stability when applying the feedback reshaping strategy presented in [20]. The remainder of this paper is organized as follows. Section 2 provides the background on communication topologies and the properties of \(SO(4)\). Section 3 presents a proposed consensus protocol on \(SO(4)^{N}\) with reshaping that destabilizes non-consensus equilibria and achieves almost global asymptotic stability. Section 4 extends the consensus protocol to \(TSO(4)^{N}\).