# An Efficient Trajectory Planning Method for High-Speed Interception of Invasive Drones
## 2 System Dynamics and Problem Statement
### Quadrotor Dynamics Model  
The quadcopter is modeled as a six degree of freedom rigid body and represented as two three-dimensional column vectors: a translational \(\mathbf{p}=(p_{1},p_{2},p_{3})\) along the orthogonal inertial axis, and three degrees of freedom \(\mathbf{\theta}=(\theta_{1},\theta_{2},\theta_{3})\), describing the rotation of the frame connected to the fuselage relative to the inertial frame.  
As shown in Figure 1, the control input of the system is treated as a scalar \(f_{t}\) representing the total thrust generated by the quadcopter, and a column vector \(\mathbf{\omega}=(\omega_{1},\omega_{2},\omega_{3})\) representing the body velocity in the aircraft coordinate system. Due to the relatively low moment of inertia and high torque generation ability of quadcopters, this article ignores angular dynamics and assumes that angular rate commands are perfectly tracked. As a result, the state of the quadcopter is designed as a 9-dimensional column vector consisting of position, velocity, and attitude angle.  
This article takes the differential equation [26] of a rigid body as the basis for controlling the flight of a quadcopter, and models external disturbances using the method of reference [27]:  
\[\dot{\mathbf{p}}=\mathbf{v} \tag{1}\]  
\[\dot{\mathbf{v}}=\mathbf{a}_{t}+\mathbf{a}_{e} \tag{2}\]  
\[\mathbf{a}_{t}=\mathbf{g}+\mathbf{R}\mathbf{e}_{3}f_{t} \tag{3}\]  
\[\mathbf{a}_{e}=\hat{f}_{rb}+\mathbf{R}\mathbf{e}_{3}f_{rp} \tag{4}\]  
Figure 1: Quadrotor dynamics model.  
\[\dot{\mathbf{R}}=\mathbf{R}\boldsymbol{\omega}\times \tag{5}\]  
where \(\boldsymbol{p}\) represents position, \(\boldsymbol{v}\) represents velocity, \(\boldsymbol{a}_{t}\) represents the acceleration component generated by the drone's thrust, and \(\boldsymbol{a}_{e}\) represents the acceleration component generated by external interference forces; \(\boldsymbol{g}\) is the gravitational acceleration represented in the inertial coordinate system, \(\mathbf{R}\) is the orthogonal matrix used to describe rotation, \(\boldsymbol{e}_{3}=(0,0,1)\) is a constant vector in the fixed object coordinate system, \(\hat{\boldsymbol{f}}_{rb}\) is the resistance experienced by the aircraft structure during quadcopter flight [28], \(f_{rp}\) is the resistance generated by the rotation of the propeller [29], and \(\boldsymbol{\omega}\times\) is the oblique symmetric matrix form of the vector cross product:  
\[\boldsymbol{\omega}\times=\begin{bmatrix}0&-\omega_{3}&\omega_{2}\\ \omega_{3}&0&-\omega_{1}\\ -\omega_{2}&\omega_{1}&0\end{bmatrix} \tag{6}\]  
It is important to note that this model represents an approximation of quadrotor flight and does not accurately reflect the true physical properties of quadrotors. In reality, the propeller's thrust is influenced by a variety of factors including rotational speed, wind speed, and inflow angle. The differentiation between thrust (3) and drag (4) produced by propeller rotation is challenging to effectively implement in practice. As demonstrated in Section 6, treating thrust, propeller drag, and body drag as independent variables results in minor errors; however, these are significantly less than those introduced by the previous method [6].