The article investigates the differential flatness of a swarm of unmanned aerial vehicles (UAVs), provides the differential flatness mapping for relative motion, and uses this as a foundation to design a distributed formation controller. In terms of motion planning, it generates real-time desired formation trajectories and configurations by solving constrained optimization problems. In terms of motion control, it utilizes the differential flatness mapping to map motion commands into desired states and control inputs for each UAV. Subsequently, it designs a distributed formation controller based on local error feedback to track the desired motion trajectories. To address the stability of collective motion, this article employs Lyapunov stability theory to prove the stability of the closed-loop system and provides conditions for selecting control parameters. Finally, simulations validate the effectiveness of the formation control method in an unknown obstacle environment.