## 4 Design and Simulation Flight of Control Law for Jumping and Steep Descent of Drones
### Design of UAV Takeoff and Landing Control Law
The weight coefficients are multiplied by the yaw angle error to obtain the yaw angle error of the rudder circuit and the yaw angle error of the power differential circuit, respectively:  
\[\Delta\psi_{\delta_{r}}=k_{\psi,\delta_{r}}\cdot(\psi-\psi_{c}) \tag{23}\]  
\[\Delta\psi_{\delta_{r}}=k_{\psi,\delta_{r}}\cdot(\psi-\psi_{c}) \tag{24}\]  
where it is taken that \(k_{\psi,\delta_{r}}=0.4\) and \(k_{\psi,\delta_{r}}=0.6\). In addition to the yaw angle error, the rudder control circuit also introduces the current yaw rate and the expected yaw rate as inputs to obtain the yaw rate error and outputs the rudder deflection command. The control law is as follows:  
\[\left\{ \begin{array}{ll}r_{c,k_{r}}=k_{\Delta\psi,\delta_{r}}\cdot\Delta\psi _{\delta_{r}}\\ \delta_{r}=(k_{r,\delta_{r}}+k_{r,\delta_{r}}\cdot\frac{1}{\delta})(r-r_{c, \delta_{r}})+k_{\beta,\delta_{r}}\cdot\beta\end{array} \right. \tag{25}\]  
where \(k_{\Delta\psi,\delta_{r}}\) is the proportional coefficient of yaw angle error rudder, \(k_{r,\delta_{r}}\) and \(k_{r,\delta_{r}}\) are the proportional coefficient and integral coefficient of yaw angular velocity error, and \(k_{\beta,\delta_{r}}\) is the feedback coefficient of the sideslip angle; \(k_{\Delta\psi,\delta_{r}}=2.5\), \(k_{r,\delta_{r}}=0.5\), \(k_{r,\delta_{r}}=0.01\), \(k_{\beta,\delta_{r}}=2\). The input of the power differential control circuit only uses the yaw angle error control quantity, and the output is \(\Delta\delta_{t}\), the power difference between the left and right propellers of the drone. When the power differential quantity of the drone is defined as positive, the throttle opening of the left propeller is \(\delta_{t}+\Delta\delta_{t}\), and the throttle opening of the left propeller is \(\delta_{t}-\Delta\delta_{t}\); that is, a positive power difference will cause the drone to generate a negative yaw moment. The power differential only uses a proportional controller to eliminate yaw angle errors, and the control law is as follows:  
\[\Delta\delta_{t}=k_{\delta_{t}}\cdot k_{\Delta\psi,\delta_{t}}\cdot\Delta\psi _{\delta_{t}} \tag{26}\]  
where \(k_{\delta_{t}}\) is the conversion coefficient from the angle to the throttle amount and \(k_{\Delta\psi,\delta_{r}}\) is the power differential proportional coefficient of yaw angle error; \(k_{\delta_{t}}=0.01\), \(k_{\Delta\psi,\delta_{t}}=0.5\). Due to the need for precise yaw control, this article uses a traditional PI controller for the lateral heading trajectory control of the outer loop of the drone. The input is the desired position \(x_{ec}\) and the current drone position \(x_{e}\), and the output is the desired yaw angle \(\psi_{c}\) of the drone. A block diagram of the controller is shown in Figure 31:  
Figure 31: Block diagram of lateral heading position control circuit. The control law for the lateral heading trajectory of the drone's outer loop is as follows:  
\[\psi_{c}=(k_{x_{x}\psi}\cdot k_{x_{x}}+k_{x_{x}i}\cdot\frac{1}{s})(x_{e}-x_{ce})+ \psi_{c0} \tag{27}\]  
where \(\psi_{c0}\) is the initial predetermined heading angle of the drone, \(k_{x_{x}\psi}\) is the conversion coefficient from the distance to the angle, and \(k_{x_{e}}\) and \(k_{x,i}\) are the proportional coefficient and integral coefficient of the PI controller, respectively; \(k_{x_{e}}=-5\), \(k_{x,i}=-0.05\).