# An Adaptive Launch Control for Balloon-Borne UAVs with Large Wingspans
## 5 Pull Up Control Design for Balloon Launch Large-Scale Aircraft
### Adaptive Control Strategy  
Based on the characteristics of the launch and pull-up process, we divide the control process into four stages. The corresponding control flow chart is shown in Figure 8. Concretely, _Phase 1 involves pitch angle tracking controls_. The aircraft rotates around the celestial axis before launching, which leads to uncertainties in heading and horizontal motions. Therefore, in this phase, the longitudinal channel executes pitch angle tracking controls, leaving the roll and heading uncontrolled. _Phase 2 involves heading stability-augmentation control_. In this phase, the aircraft is pulled up according to the pitch-angle command, and the aircraft's heading is definite. The heading stability-augmentation control is utilized to control the roll angle of the aircraft at 0 degrees. _Phase 3 involves the fixed pitch  
Figure 6: Natural frequency of the aircraft’s fuselage at each order.  
Figure 7: Flutter velocity and flutter frequency at different heights.  
angle directional glide control._ As the engine increases its output continuously, the required airspeed is provided by gliding at a fixed angle in order to ensure that the flight speed of the aircraft is within the safety envelope. At this moment, the aircraft flies with a fixed heading and a fixed glide angle and controls the roll angle at 0 degree. _Phase 4 involves cruise flight controls._ The UAV shifts to the cruise flight mode; i.e., the first three channels track and control the UAV according to control commands. The pull-up process is completed.  
The PID control is adopted for longitudinal control and lateral directional control [34]. The structure diagram of thepitch controller is shown in Figure 9. The pitch angle's error, \(\Delta\theta\), is calculated by subtracting the pitch angle \(\theta\) from the pitch command \(\theta_{c}\). Control input \(\delta_{e}\) is finally obtained via the proportional integral of the pitch error \(\Delta\theta\) and the damping feedback by the pitch rate, \(q\). For roll control, it is consistent with pitch control. The structure diagram of yaw control is shown in Figure 10. The difference in yaw control is based on the proportional integral of the yaw angle's error, \(\Delta\psi\), and the yaw rate's feedback, \(r\). The feedback of roll rate \(p\) and roll angle \(\phi\) increased to reduce the sideslip angle. Since the dynamic pressure changes rapidly and greatly during the pull-up process, the proportional coefficient, integral coefficient, and differential coefficient automatically adapt to the dynamic pressure. These coefficients are mathematically described as follows:  
\[\delta_{e}=Kp_{\phi}\Delta\theta+K\!i_{\phi}\Delta\theta+K\!\omega_{\phi}q \tag{4}\]  
\[\delta_{a}=Kp_{\phi}\Delta\phi+K\!i_{\phi}\Delta\phi+K\!\omega_{\phi}p \tag{5}\]  
\[\delta_{r}=Kp_{\psi}\Delta\psi+K\!i_{\psi}\Delta\psi+K\!\omega_{\psi}r+Kp_{ \phi 2}\Delta\phi+K\!\omega_{\phi 2}p \tag{6}\]  
where \(\delta_{e}\), \(\delta_{a}\), and \(\delta_{r}\) are control inputs of the elevator control surface, aileron control surface, and rudder control surface, respectively. \(Kp_{\phi}\), \(Kp_{\theta}\), and \(Kp_{\psi}\) are proportional coefficients of the roll error, pitch error, and yaw error, respectively. \(K\!i_{\phi}\), \(K\!i_{\theta}\), and \(K\!i_{\psi}\) are integral coefficients of the roll error, pitch error, and yaw error, respectively. \(K\!\omega_{\theta}\), \(K\!\omega_{\phi}\), and \(K\!\omega_{\psi}\) are differential coefficients of the roll rate, pitch rate, and yaw rate, respectively. \(Kp_{\phi 2}\) and \(K\!\omega_{\phi 2}\) are coordination control coefficients.  
Figure 8: Four stages’ control strategy on the pull-up process.