# Safety Control for Satellite Formation Flying via High-Order Control Barrier Functions
## 1 Introduction  
Satellite formation flying allows for coordinated missions where multiple satellites work together to achieve objectives that a single satellite could not accomplish [1]. Compared with single-satellite missions, satellite formation technology can improve system mission flexibility, redundancy, and overall performance through joint observation, data sharing, and functional collaboration among multiple satellites. This technology has demonstrated wide application prospects in various fields such as earth observation, scientific exploration, communications, and navigation [2, 3, 4]. To achieve the coordinated operation of multiple satellites within a formation, precise control of the relative motion between satellites is required. Dynamic modeling of relative motion is the foundation for realizing formation control. The Clohessy-Wiltshire equations are a set of classical linear relative motion dynamic equations used to describe the relative motion between formation satellites on near-circular orbits [5]. Due to their linearization characteristics and the existence of analytical solutions, the Clohessy-Wiltshire equation has low computational complexity and is widely used in orbit control problems for formation flying. However, the Clohessy-Wiltshire equation assumes that the motion of formation satellites relative to the leader satellite is a linear system, neglecting higher-order nonlinear terms. In cases where formation satellites are far apart or operate for extended periods, the errors resulting from the linear assumptions can significantly increase. To address the limitations of the Clohessy-Wiltshire equation, researchers have proposed a series of modification schemes, such as adopting nonlinear dynamic models or using numerical integrationmethods to combine linear and nonlinear terms for correction. For nonlinear dynamic problems, decomposition methods are commonly used to separate the linear and nonlinear terms in the dynamic equations and solve them using high-precision numerical methods to ensure the accuracy and stability of the control system. This method can effectively reduce system errors and meet the requirements of high-precision formation control [6]. Numerous formation flying control design results have also been found in the literature when the nonlinear dynamics are considered [7; 8; 9; 10]. To name a few, the authors of [11] provided a neural network-based adaptive sliding mode control scheme for satellite formation flying, and the negative effects of uncertainties together with external disturbances on the relative position and attitude were handled. A second-order sliding mode controller was developed for satellite formation flying in [12], where the derivation of the switching function of the sliding mode controller was not used. Alternatively, the authors of [13] designed a nominal controller by applying [PERSON] and [PERSON]'s approach, based on which adaptive laws and robust control schemes have been designed to deal with system uncertainty and possible initial condition deviation from the constraints. Considering the collision avoidance of satellites, several safety control schemes have been reported; see, for example, [14; 15; 16; 17]. Although related safety control results for satellite formation flying exist in the literature, a more effective approach, namely, control barrier function-based safety control, has not been fully exploited. This method can handle complex safety constraints and strictly guarantee global safety. [PERSON] et al. in [18] provided an introduction of the control barrier function and discussed its application of designing safety control schemes. Using the control barrier function method, great progress has been made in safety control [19; 20; 21]. A control barrier function-based safety control algorithm was developed for a constrained robot in [22].