# Affine Formation Maneuver Control for Multi-Agent Based on Optimal Flight System
## 1 Introduction  
The use of unmanned aerial vehicle (UAV) swarms [1; 2] has significantly expanded the range of applications and possibilities in the unmanned field, surpassing the capabilities of single-vehicle systems. Formation-based control algorithms aim to guide all vehicles to form specific configurations while maintaining relative velocity and distance. However, relying on fixed formation patterns limits the adaptability of formation control in different environments. Hence, the notion of dynamically altering configurations in UAV formation control has garnered significant attention from researchers, particularly the maneuver control for the purpose of achieving continuous formation change [3; 4; 5; 6]. This approach possesses the ability to autonomously navigate aerial formations in cluttered and wild environments, effectively providing assistance during natural disasters like earthquakes and floods. Maneuver control can aid in formation search and guidance [7]. Additionally, researchers can inspect restricted areas such as documenting structural and cultural artifacts within historical buildings [8] or autonomously searching subterranean environments [9], which can gain situational awareness and assist specialized personnel in specific missions, such as assessing the structural integrity of collapsed buildings, tunnels, mines, exploration of a newly discovered branch in a cave network, and searching for lost persons. Formation configurations can be associated with different tasks and environments, such as collective animal behavior [10] in nature or reconnaissance and strike missions [11; 12]. However,continuous configuration changes increase system complexity and communication requirements among UAV agents, particularly when adjusting and preserving the position and orientation of each agent in motion [13]. Additionally, the effectiveness of configuration change may be hindered by various external and internal constraints. Therefore, various methods combining formation control with configuration changes have been extended to multi-agent systems, resulting in more versatile forms of unmanned vehicles, including maneuvering [13; 14], surveillance [15], reconnaissance [16], and traffic monitoring and management [17]. In addition to researching practical constraints and scenarios, distributed methods have proven to be more adaptable and consider mutual autonomy coordination for configuration changes, in contrast to centralized approaches [3; 17]. This suggests that the characteristics of the distributed strategy are better suited for the dynamic configuration changes of formations. Distributed formation control strategies closely related to configurations can be broadly categorized into position, displacement, and distance-based methods, based on the types of sensed and controlled variables used thus far. However, these methods have limited effectiveness in accurately tracking the target formation [18], which restricts configuration changes to variations in time, scales, and orientation of the target configuration [19; 20; 21]. From a control perspective, the dynamic change of configuration can be specified at the global formation level [19; 20; 21; 22] or the local level [23; 24]. The global approach involves presetting the path points of the formation's center of gravity for agents to follow, maintaining the rigid body structure by preserving relative positions and transitioning between different formation shapes. On the other hand, the local approach focuses on agent-level control, which is more adaptable to tasks and collision avoidance. The first approach involves maintaining the rigid structure of the configuration with constant constraints, requiring a configuration with a predefined change scale and nominal configuration. Ref. [6] proposed an affine-based generation method for target configuration using an interaction law with both designed positive and negative weights (different from consensus control law). [PERSON].