# An Efficient Trajectory Planning Method for High-Speed Interception of Invasive Drones
## 1 Introduction  
In recent years, drone technology has made significant progress, and its application areas involve monitoring, rescue, intelligent logistics, environmental monitoring, agricultural building measurement, and other aspects. However, the rapid growth of the drone market has also brought serious safety issues. Improper use or malicious behavior may pose a threat to social security and cause significant economic losses. For example, in 2018, London Gatwick Airport cancelled hundreds of flights due to drone interference [1]. In the UK, 62% of close contact incidents between small drones and aircraft in 2019 were considered high-risk [2]. To address these challenges, it has become particularly important to develop new technologies that can detect, identify, and respond to malicious drones. Therefore, developing a countermeasure system that can effectively identify and combat single or multiple threatening drones has become an urgent task [1]. Multi-rotor drones, represented by quadcopters, are known for their excellent maneuverability. They have become an ideal platform for testing advanced algorithms due to their excellent linear and angular acceleration performance brought by their high thrust to weight ratio. In recent years, quadcopter drones have achieved significant results in 3D autonomous flight [3], unknown environment map construction [4], cluster flight obstacle avoidance [5], and other areas. Some successful application cases, such as Agile Catch [6] and Rapid Aircraft Capture [7], demonstrate the enormous potential of quadcopter drones in the field of anti-drones. When executing the target arrival task, two main methods are used: (1) Focusing on guidance rate design and feedback control based on observation information. For example, methods such as the proportional guidance method [8], model predictive control [9], optimal control [10], and reinforcement learning [11; 12; 13] all focus on feedback control based on observation information to achieve guidance objectives. These methods have achieved good results in Markov-like scenarios, but it is difficult to analyze the entire task process in advance, such as energy consumption, task time, and other indicators, which poses difficulties for performance evaluation in complex task scenarios. (2) Emphasis is placed on task trajectory planning, which involves planning task trajectories [14; 15; 16] in advance and enabling the aircraft to accurately track them to achieve mission objectives. Its main task is to generate executable flight trajectories that conform to the dynamic characteristics of unmanned aerial vehicles in a short period of time. This type of method is usually able to constrain the starting and ending states or flight time based on specific task objectives, which has stronger adaptability for more complex anti-drone swarm task scenarios. There are usually two representative methods to solve trajectories: (1) Decoupling geometric and temporal information, connecting two endpoints through polynomials or spline curves, and then parameterizing the geometric trajectory in time to generate a state trajectory that respects the dynamic characteristics of the drone. This type of method cannot directly consider the dynamic characteristics of the drone to determine the flight time when planning the path. However, the later use of methods such as PID and model predictive control [17] to track the trajectory further introduces errors, leading to increased uncertainty in the final task result. (2) By leveraging the differential flatness inherent in quadrotor dynamics, both the system state and input are expressed as flat outputs along with their derivatives. Subsequently, constraints on the trajectory are extrapolated from system dynamics constraints via differential equations. Through the application of convex optimization techniques, it becomes feasible to address trajectory optimization challenges such as time optimization [18], energy efficiency [19], and minimum input jerk [6].