# Distributed Attitude Synchronization for Spacecraft Formation Flying via Event-Triggered Control
## 2 Preliminaries and Problem Statement
### Problem Formulation  
For simpleness, formation spacecraft can be regarded as rigid bodies. For \(i\)th spacecraft, the attitude dynamics described by Modified Rodrigues Parameters (MRPs) can be described by [26]:  
\[\dot{\sigma_{i}}=G(\sigma_{i})\omega_{i} \tag{1}\]  
\[G(\sigma_{i})=\frac{1}{2}\left[\frac{1-\left\|\sigma_{i}\right\|^{2}}{2}I_{3}+ \sigma_{i}^{\times}+\sigma_{i}\sigma_{i}^{T}\right] \tag{2}\]  
\[J_{i}\dot{\omega_{i}}=-\omega_{i}{}^{\times}J_{i}\omega_{i}+u_{i} \tag{3}\]  
where \(J_{i}\in\mathbb{R}^{3\times 3}\) refers to the inertia matrix, \(u_{i}\in\mathbb{R}^{3}\) represents the control input, \(\sigma_{i}=[\sigma_{i1},\sigma_{i2},\sigma_{i3}]^{T}\in\mathbb{R}^{3}\) and \(\omega_{i}=[\omega_{i1},\omega_{i2},\omega_{i3}]^{T}\in\mathbb{R}^{3}\) stand for the spacecraft attitude described by MRPs and the angular velocity, respectively.  
**Remark 1**.: _MRPs is expressed as \(\sigma_{i}=E_{i}\tan(\theta_{i}/4)\), where \(E_{i}\) is Euler axis of the attitude rotation and \(\theta_{i}\) is Euler angular with satisfying \(\theta_{i}\in(-2\pi,2\pi)\). It should be noted that the attitude denoted by MRPs will be singular when \(\theta_{i}\) is close to \(\pm 2\pi\). As shown in [27], a corresponding vector \(\sigma_{i}^{s}=-\sigma_{i}/\sigma_{i}^{T}\sigma_{i}\) of the original MRPs vector \(\sigma_{i}\) is introduced and the singularity problem is able to be solved by choosing to switch the MRPs when \(\sigma_{i}^{T}\sigma_{i}>1\)._  
To facilitate the design of the SFF system, the following assumptions are introduced:  
**Assumption 2**.: _The generalized angular velocity is assumed to satisfy that \(\left\|\omega\right\|\leq\delta_{1}\), \(\left\|\dot{\sigma}\right\|\leq\delta_{1}\), where \(\delta_{1}\) is a positive constant._  
**Assumption 3**.: _The generalized angular acceleration is assumed to satisfy that \(\left\|\dot{\omega}\right\|<\delta_{2}\) and \(\left\|\vec{\sigma}\right\|<\delta_{2}\), in which \(\delta_{2}>0\) is constant._  
**Assumption 4**.: _For the reference signal, the derivative of the reference angular acceleration is assumed to be bounded with satisfying \(\left\|\ddot{\omega}_{d}\right\|\leq\delta_{3}\), where \(\delta_{3}>0\) is a constant._  
**Remark 2**.: _Assumptions 2 and 4 are general which can be found in [7; 28] respectively. Assumption 3 is reasonable because the control torque of each spacecraft has an upper bound, which means that the angular acceleration is bounded._  
Define the reference attitude and the reference angular velocity as \(\sigma_{d}\) and \(\omega_{d}\), respectively. For \(i\)th spacecraft, the tracking errors of attitude and angular velocity are defined as:  
\[\sigma_{ei}=\sigma_{i}-\sigma_{d} \tag{4}\]  
\[\omega_{ei}=\omega_{i}-\omega_{d} \tag{5}\]  
Considering the SFF system subject to limited energy and computing resources, the control objective is to establish an event-based attitude coordination controller to realize attitude coordination and drive the system state errors \(\sigma_{ei}\) and \(\omega_{ei}\) to arrive at a neighborhood around the origin.