# Affine Formation Maneuver Control for Multi-Agent Based on Optimal Flight System
## 1 Introduction
It is important to note that the virtual leader is not a physical vehicle, but an imaginary point used as a reference for the movement of the formation. The construction of the virtual leader is similar to Leonard [3]. The virtual leader abstraction offers a significant advantage by eliminating the need for individual node trajectories, focusing solely on planning the path of the virtual leader through areas of interest for the entire team. This approach greatly reduces communication requirements. Additionally, the motion of the virtual leader is limited to a distance that prevents any of the vehicles from experiencing excessive total forces in the direction of the virtual leader's motion. Furthermore, the communication requirements among agents may vary depending on the level of precision needed for configuration changes. Each agent updates its trajectory within the formation based on the positions of other agents, obstacles, and the virtual leader each time step. The position of the virtual only needs to be transmitted to each agent once per update at most, regardless of whether there is a need to recalculate the position to adjust the local trajectory. This approach ensures that the bandwidth requirements for all agents interacting with their positions remain manageable in small formations (less than 20 agents), as it considers only horizontal motion consensus. However, for larger groups, the position information would be limited to agents within a nearby radius. This paper investigates the generality of formation-desired configurations and proposes an adaptable algorithm for local trajectory planning. The algorithm ensures the continuous change of geometric patterns and improves the consistency of local trajectory planning with realistic constraints. Consequently, the formation control relies less on prior information and enhances the ability to avoid obstacles and other agents in real-time, particularly in situations involving sudden obstacles. Additionally, the combination of formation control methods and optimal techniques expands the utilization of formation configuration variations and definitions to adapt to more complex environments. The contributions of this article could be summarized as follows:  
1. A novel non-linear and dynamical system for formation configurations changing is created by the combination of the global and local control law, which could relieve the dependence on the rigid body of formation. 2. An APF-based adjustment metric is developed for local trajectory planning through various forces. It could solve the problem of local minimum value of APF and update the local navigation to filter the discrete value due to a sudden change of velocity. 3. A novel optimal framework has been integrated, combining a virtual leader to consider dynamic feasibility, obstacle avoidance, and flexibility. This framework can improve the elasticity of configuration variations in potential collision. The paper is organized as follows. Section 2 describes the preliminaries. In Section 3, the description of the interaction forces among agents, virtual leader, and obstacles is presented. The description and proof of the main results for local control laws are given in Section 4. The results of the simulation and discussions are drawn in Section 5. Section 6 concludes the entire paper and provides a glimpse into future prospects.