## 2 Method  
In the conventional tracking process, it is difficult for the quadrotor to unify space and time with a closed-loop control mechanism. The UAV controller evaluates its real-time position and/or state and then calculates the PID output values corresponding to the desired values at the next moment. This pattern results in an error between the desired and actual positions. The error is then sent to the input of the closed-loop controller, and the new output values are used to force the UAVs to approach the target. Although the closed-loop control method guarantees a high spatial precision, the feedback mode makes it difficult to reach the target in a given time. To make the best of the biological and open-loop control mechanisms, position control cannot be used, and a one-to-one relationship should be established between the control commands and the desired acceleration under different situations. According to Newtonian mechanics and UAV PID control equations, a set of dynamic equations of the quadrotor trajectory can be generally expressed as  
\[\left\{ \begin{array}{l}\vec{P(t)}=\vec{P}_{0}+\int_{0}^{t}\vec{v}_{0}dt+ \int_{0}^{t}\int_{0}^{t}\vec{a}(t)dtdt\\ \vec{a}(t)=f(s,m,\vec{u}(t),\vec{v},V_{bat},\cdots\cdots)\\ \vec{u}(t)=k_{p}\vec{e}(t)+k_{i}\int_{0}^{t}\vec{e}(t)dt+k_{d}\frac{d\vec{e}( t)}{dt}\\ \vec{e}(t)=\vec{\theta}_{desired}-\vec{\theta}_{actual}\;\;\text{or}\;\;\vec{ \omega}_{desired}-\vec{\omega}_{actual}\\ \vec{\theta}_{desired}=g(\vec{Q}_{cont})\end{array} \right. \tag{1}\]  
where \(\vec{P(t)}\) represents the quadrotor position from the initial point, \(\vec{u}(t)\) is the PID control output, \(\vec{e}(t)\) is the angle error or angular velocity error, \(\vec{\theta}_{desired}\) is the desired UAV angle, and \(\vec{Q}_{cont}\) is the control commands. \(\vec{a}(t)\) represents the actual acceleration under different conditions of the UAV structure (\(s\)), UAV weight (\(m\)), PID controller output (\(\vec{u}(t)\)), velocity (\(\vec{v}\)), battery voltage, etc. The \(\vec{a}(t)\) function (\(f\)) is very complicated and changes over time; thus, it cannot be expressed as a calculus formula. From Formula (1), \(\vec{a}(t)\) comes from \(\vec{Q}_{cont}\) and the current state of the UAV. Then, \(\vec{a}(t)\) can be expressed as  
\[\vec{a}(t)=M(s,m,\vec{Q}_{cont},\vec{v},V_{bat},\ldots\ldots) \tag{2}\]  
then  
\[\vec{Q}_{cont}=M^{-1}(s,m,\vec{a}(t)_{desired},\vec{v},V_{bat},\ldots\ldots) \tag{3}\]  
The \(\vec{Q}_{cont}\) function (\(M^{-1}\)) is defined as the maneuverability model that represents the relationship between \(\vec{Q}_{cont}\) and \(\vec{a}(t)_{desired}\) under the current UAV conditions (velocity, power, mechanical structure, etc.). To implement this control method, a new open-closed-loop compound control strategy for trajectory tracking based on the proposed real-time platform is shown in Figure 1. The compound control strategy consists of three parts: a desired acceleration sequence generator, a maneuverability model, and a closed-loop controller. The maneuverability model was built from the data acquired by the real-time control and acquisition platform. The desired acceleration vector is the input of the maneuverability model, and the UAV control instructions are the output of the model. The maneuverability model considers the control instructions and the current UAV states (speed, power, mechanical structure, etc.) as the influencing factors of the acceleration, and is trained many times from the actual flight data. As a result, a one-to-one relationship is obtained between the control instructions and the desired acceleration under different situations. This method can reproduce acceleration sequences that satisfy the spatial and temporal requirements of the given trajectory. The architecture of the control strategy is divided into three parts. Firstly, the preset trajectory generator needs to generate an optimal trajectory for reaching the target point, but this trajectory is generally an ideal acceleration sequence that has no correlation with the actual UAV.