# Decentralized Consensus Protocols on \(SO(4)^{N}\) and \(Tso(4)^{N}\) with Reshaping
## 5 Conclusions  
Lyapunov-based consensus protocols with reshaping for multi-agent systems on \(SO(3)^{N}\) are leveraged and extended to multi-agent configuration synchronization on \(SO(4)^{N}\) and \(TSO(4)^{N}\). For synchronization with non-consensus equilibria, the reshaping strategy is used to destabilize these equilibria and produce configuration synchronization with almost global asymptotic stability. The control strategy using an approach based on a Morse-Bott-Lyapunov function is supported by numerical simulations of an undirected, ring communication graph and a ring lattice. The presented simulations confirm the proposed advantages of Case 1 and Case 2 reshaping on \(SO(4)^{N}\) and \(TSO(4)^{N}\) synchronization by destabilizing stable non-consensus equilibria to achieve AGAS on the consensus manifolds \(SO(4)^{N}\) and \(TSO(4)^{N}\).  
The illustrative examples used to demonstrate how reshaping destabilizes non-consensus equilibria are limited to ring graphs and ring lattices. These are only a subset of the set of circulant networks. The Kuramoto model [21] can be considered as an analog of the consensus protocols on \(SO(3)\) and \(SO(4)\)[20]. Specifically, the single-integrator Kuramoto model corresponds to kinematics-level control on \(SO(4)\) and the double-integrator Kuramoto model corresponds to \(TSO(4)\). The Kuramoto model is analyzed in [21] for circulant networks, and it is proven that there exists an upper bound on connectivity for which at least one non-consensus equilibrium is locally stable for circulant networks.  
Figure 9: Simulations on \(TSO(4)^{9}\) for (**a**) Case 1 reshaping protocol with \(\arg\max_{\theta\in[0,180]}f(\theta)=50^{\circ}\) on a 9-agent ring graph and (**b**) Case 2 reshaping protocol with \(\arg\max_{\theta\in[0,180]}f(\theta)=35^{\circ}\) on a 9-agent ring graph. Initial conditions are chosen in the region of attraction of \(p=1\) non-consensus equilibria. Right plot demonstrates consensus, while left plot does not.  
Meanwhile, [30] shows that, for ring lattices, this upper bound is \(\mu=0.6818\). Thus, for a nine-node ring lattice with \(m=3\), the consensus manifold is AGAS stable without the need for reshaping, since \(\delta=6\) and \(\mu=0.75\) in Equation (2). Similarly, for any complete graph with \(m=1\), the consensus manifold is globally stable. As shown in Section 3.3.4, increased connectivity reduces the order of reshaping required for the nine-agent ring lattice, where Case 1, as opposed to Case 2, in the nine-agent ring graph example is sufficient in achieving consensus. Based on this observation, it can be conjectured that, if a network has a subgraph consisting of a ring graph or ring lattice on N nodes, then a reshaping strategy with \(\arg\max_{\theta\in[0,180]}f(\theta)<2\pi/N\) can be developed such that almost global stability for the synchronization manifold is guaranteed.  
Conceptualization, E.A.B.; Methodology, E.A.B. and V.S.; Software, V.S.; Writing--original draft, V.S.; Writing--review & editing, E.A.B. All authors have read and agreed to the published version of the manuscript.  
This research was sponsored by the Army Research Office and was accomplished under Grant Number W911 NF-23-1-0155. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.  
Not applicable.  
Data Availability StatementData is contained within the article.  
The authors declare no conflicts of interest.