# Decentralized Consensus Protocols on \(SO(4)^{N}\) and \(Tso(4)^{N}\) with Reshaping
## 2 Background
### Properties of \(So(4)\)  
Rotations in higher-dimensional spaces have been a subject of considerable interest in fields such as mathematics, physics, and computer sciences. Methods of generating a rotation matrix in higher dimensions have been explored, including the use of [PERSON]'s formula and [PERSON]'s transforms, as discussed in [22; 23; 24; 25]. This paper utilizes the [PERSON] formula to map the principal rotation angles (PRAs) and planes of rotation to an element on \(SO(4)\). The decomposition of a 4D rotation matrix in terms of an exponential skew-symmetric matrix is presented in [23; 25; 26] and is also leveraged here. The intuitive three-dimensional rotation by a principal angle about a principal axis in \(SO(3)\) is replaced by planes of rotation in four dimensions on \(SO(4)\). Four-dimensional rotations can be decomposed into two PRAs and a set of two principal planes, which replace the principal angle and axis in \(SO(3)\). Methods to compute the PRAs and planes from a provided rotation matrix are discussed in [24], which is useful in classifying the rotation being applied. Meanwhile, ref. [25] presents an extensive discussion of the differences between simple, double, and isoclinic rotations. Algorithms to determine the PRAs, planes, and type of rotation are presented. This paper focuses on the most general case of double rotations to show how principal angles \(\theta_{1,2}\), associated with the absolute or relative configurations, evolve under the consensus control problem. Consider the \(4\times 4\) proper orthogonal matrix \(R\in SO(4)=\{R^{4\times 4}|R^{T}R=I_{4},|R|=1\}\). Similarly to \(SO(3)\), this matrix represents a rotational transformation in four-dimensional Euclidean space \(\mathrm{E}^{4}\). The key difference between the dimensions is that, in three dimensions, the rotation is about an axis, whereas, in four dimensions, the rotation is about a plane. All vectors in the plane are mapped to other vectors in the same plane by the rotation and are related by corresponding eigenvalues and eigenvectors of the rotation matrix. Rotations in  
Figure 1: Three communication topologies: (**a**) 5-agent ring graph, (**b**) 9-agent ring graph, and (**c**) 9-agent ring lattice. \(\mathrm{E}^{4}\) have at most two planes of rotations. The plane of rotation determines the classification of the rotation: (1) simple rotation--a single plane of rotation; (2) double rotation--two angles of rotation, each corresponding to one of two orthogonal planes of rotation. Isoclinic rotation is a specific type of double rotation where the non-zero angles of rotation about both planes are equal. Let \(R\in SO(4)\) be a double rotation in the four-dimensional Euclidean space \(xyzw\) about the \(xy\) plane by angle \(\theta_{2}\) and \(zw\) plane by angle \(\theta_{1}\). Thus, the planes about which the rotations are performed are \(E_{1}=[0_{2}\quad I_{2}]^{T}\) and \(E_{2}=[I_{2}\quad 0_{2}]^{T}\). This results in \(R\) in a particular basis with the normal form given by  
\[R= \begin{bmatrix}\cos\theta_{1}&-\sin\theta_{1}&0&0\\ \sin\theta_{1}&\cos\theta_{1}&0&0\\ 0&0&\cos\theta_{2}&-\sin\theta_{2}\\ 0&0&\sin\theta_{2}&\cos\theta_{2}\end{bmatrix}  \tag{3}\]  
The convenience of the basis above is that the eigenvalues of the matrix are readily given by \(\sigma(R)=\exp(\pm i\theta_{1}),\exp(\pm i\theta_{2})\), where \(\sigma()\) is the spectrum operator and \(\theta_{1}\) and \(\theta_{2}\) are the PRAs. Equation (3) can be decomposed into two rotations \(R_{1},R_{2}\in SO(4)\) as \(R=R_{1}R_{2}=R_{2}R_{1}=R_{1}+R_{2}-I_{4}\), where \(\sigma(R_{j})=\{\exp(\pm i\theta_{j}),1,1\}\), \(j=1,2\).