## 2 UAV Dynamics Model
### Climbing Performance Analysis and Leap Control Strategy Design
(2) Analysis of the climbing envelope characteristics of unmanned aerial vehicles  
Based on the above results of balancing the stable climb of the drone, we find the boundary points of the balancing state and draw the velocity climb angle envelope, velocity angle of attack envelope, and pitch angle throttle envelope of the drone climbing process, as shown in Figure 18.  
According to the minimum climb angle boundary, when the drone's flight speed is less than 7 m/s, it does not have the ability to fly at a small angle. As the flight speed increases, the minimum climb angle boundary gradually decreases from around 20\({}^{\circ}\) to zero. According to the maximum climb angle boundary, when the flight speed of the drone is less than 11 m/s, the drone has the ability to climb vertically at a low speed. As the flight speed increases, its maximum climb angle boundary gradually decreases to 0. The trend of pitch angle changing with flight speed is basically consistent with the climb angle, but its reachable boundary range is smaller than the climb angle. According to the flight speed angle-of-attack flight envelope, when the drone climbs at a low speed and low climb angle,  
Figure 18: Stable climbing balance envelope. a high-angle-of-attack state will occur. Therefore, it is recommended to require drones to perform flight missions within this angle of attack range, and the minimum climbing speed for drones is 7 m/s. According to the pitch angle throttle envelope obtained from the drone climb balancing, when the drone pitch angle is around 9\({}^{\circ}\), the available throttle range that supports drone level flight or climb is the largest. As the pitch angle of the drone decreases, its minimum throttle boundary rapidly increases. As the pitch angle of the drone increases, its minimum throttle boundary gradually increases, and the maximum throttle boundary rapidly decreases from 1 when the pitch angle approaches 90\({}^{\circ}\). (3) Design of Takeoff and Climbing Control Strategy for Drones  
Based on the analysis of the balancing results of the drone climbing state, the trend of each balancing state variable with speed, and the achievable boundary of the balancing state variable in the previous text, the climbing angle range of 30\({}^{\circ}\)-90\({}^{\circ}\) is selected to further analyze the drone power and climbing rate, in order to find the optimal drone climbing strategy. The motor power, climb rate, and total energy output of the motor during the stable climb process of the drone vary with the flight speed as shown in the Figure 19.  
Analyzing the curve of motor power variation with speed, it can be seen that its trend is basically consistent with the throttle opening. When the climb angle approaches 90\({}^{\circ}\), the motor power increases monotonically with the increase in drone flight speed. When the climb angle is slightly small, the motor power decreases first and then increases with the increase in the drone's flight speed. Each climb angle corresponds to a minimum required power and minimum power speed. Analyzing the curve of climb rate with speed variation, it can be seen that for the same climb angle, the climb rate increases correspondingly with  
Figure 19: Power, climb rate, and energy variation curves with speed. the increase in flight speed. At the same flight speed, the larger the climb angle, the higher the climb rate. Based on the above analysis, this article will design three takeoff climb strategies for MY-UAV, namely, minimum-power climb, maximum-climb-rate climb, and minimum-energy climb. To obtain the minimum-power climb strategy, search for the minimum power point in the upper-left figure, which corresponds to the unmanned aerial vehicle climb state point with a flight speed of 11 m/s and a climb angle of 30\({}^{\circ}\). When the drone climbs steadily in this state, the required power is minimized.