## 4 Design and Simulation Flight of Control Law for Jumping and Steep Descent of Drones
### Design of UAV Takeoff and Landing Control Law  
The longitudinal attitude control of the drone mainly consists of a pitch angle control loop, including pitch rate feedback and pitch angle feedback. The controller block diagram is shown in Figure 28:  
The pitch angle control adopts a cascade PID control method, with the inner loop being pitch angle velocity PI control and the outer loop being pitch angle P control. The control law is as follows:  
\[\left\{ \begin{array}{l}q_{c}=k_{\theta}(\theta_{c}-\theta)\\ \delta_{e}=(k_{q}+k_{qi};\frac{1}{s})(q-q_{c})\end{array} \right. \tag{18}\]  
where \(k_{\theta}\) is the proportional coefficient of the pitch angle controller, \(k_{q}\) and \(k_{qi}\) are the proportional coefficient and integral coefficient of the pitch angular velocity controller, respectively; \(k_{\theta}=1.5\), \(k_{qi}=0.01\), \(k_{q}=1.5\). The outer-loop longitudinal position controller includes height control and speed control. The height control input is the desired height and current height, and the output is the desired pitch angle. The speed control input is the desired speed and current drone airspeed, and the output is the desired throttle opening. The longitudinal position control block diagram based on PID is shown in Figure 29:  
Figure 28: Schematic diagram of longitudinal attitude control loop based on PID. Figure 29: Schematic diagram of longitudinal position control circuit based on PID. The drone altitude control adopts PI control, and the control law is as follows:  
\[\theta_{c}=(k_{h}+k_{hi}\cdot\frac{1}{s})(h-h_{c}) \tag{19}\]  
\[k_{h}=1.5,\ k_{hi}=0.02\]  
The speed control of the drone adopts PI control, and the control law is as follows:  
\[\delta_{t}=(k_{v}+k_{vi}\cdot\frac{1}{s})(v_{c}-v)\cdot k_{g_{t}} \tag{20}\]  
\[k_{v}=0.5,\ k_{vi}=1.5,\ k_{g_{t}}=0.01\]  
The lateral heading control of unmanned aerial vehicles includes roll control and yaw control. Roll control tracks the roll angle command signal through roll angle feedback and roll angle velocity feedback. The control logic is the same as the pitch angle control, using cascade PID control, with the inner loop being PI control of roll angle velocity and the outer loop being P control of roll angle. Yaw control improves heading stability through sideslip angle feedback At the same time, the deviation signal generated by the yaw rate signal and the yaw rate coordination signal during rolling is used to realize coordinated turning, and the damping characteristics are improved. The joint control of rudder and dynamic differential is adopted. The controller block diagram is shown in Figure 30:  
The control law design for the lateral heading roll control circuit is as follows:  
\[\left\{ \begin{array}{l}p_{c}=k_{\phi}(\phi_{c}-\phi)\\ \delta_{a}=(k_{p}+k_{pi}\cdot\frac{1}{s})(p-p_{c})\end{array} \right. \tag{21}\]  
where \(k_{\phi}=2.5\), \(k_{pi}=0.01\), and \(k_{p}=0.5\). The lateral yaw control circuit is divided into rudder control and dynamic differential control. The input of the overall loop is the expected yaw angle \(\psi_{c}\) output by the outer-loop lateral heading position control, as well as the expected roll angular velocity of the unmanned aerial vehicle. The expected roll angular velocity is set as 0, and the  
Figure 30: Block diagram of lateral heading attitude control circuit. difference between the current yaw angle \(\psi\) and the expected yaw angle \(\psi_{c}\) is the yaw angle error, \(\psi-\psi_{c}\), with the following values:  
\[\psi-\psi_{c}=\left\{ \begin{array}{ll}(\psi-\psi_{c})+360,\qquad\psi-\psi_{c} <-180\\ \psi-\psi_{c},\qquad-180\leq\psi-\psi_{c}\leq 180\\ (\psi-\psi_{c})-360,\qquad\psi-\psi_{c}>180\end{array} \right. \tag{22}\]  
Based on the results \(\psi-\psi_{c}\), it is necessary to allocate weights between rudder control and dynamic differential control, designed \(k_{\psi,\delta_{r}}\) as the weight coefficients for rudder control yaw and \(k_{\psi,\delta_{r}}=1-k_{\psi,\delta_{r}}\) as the dynamic differential control yaw.