## 3 Dynamic Model and Control
### Flight Bionic Control  
According to [26], scientists discovered that geckos are able to adjust their airborne posture through their tails in the air, so that they can land on their bellies, just like a flying wingsuit athlete adjusts its center of gravity and drag to land successfully. In the same motor and propeller configuration, the faster the speed is, the higher the lift is, but the less efficient it is. Inspired by this, when robots fly in the air without adsorption, we also need to adjust the servos angle to balance the load of the four rotors, improve control performance, reduce power consumption, and extend the working time. Assume that \(W\) is the power that a single rotor requires to generate lift \(F\).  
\[\begin{array}{c}W=g(F)\\ W^{\prime}>0,W^{\prime\prime}\geq 0,F>0\end{array} \tag{28}\]  
Therefore, if the robot center of gravity and geometric center do not coincide, the front two rotors need to be increased by \(0.5\times\Delta F\) for each rotor. The lift generated by the front two rotors is \(F_{1}\), and the lift generated by the rear two rotors \(F_{2}\).  
\[\left\{\begin{array}{c}F_{1}=\frac{G}{4}-\frac{\Delta F}{2}\\ F_{2}=\frac{G}{4}+\frac{\Delta F}{2}\end{array}\right. \tag{29}\]  
The total power \(W_{tot}\) is greater than the total power in the case of uniform mass distribution.  
\[W_{tot}=2\times(g(F_{1})+g(F_{2}))\geq 4\times g(\frac{G}{4}) \tag{30}\]  
As shown in Figure 9(a), we use the flight controller to rotate the propeller at different speeds. The relationship between the lift and power of different types of propellers is calculated by the push-pull sensor and the voltage-current sensor, and the curves are fitted.  
The curves of different types of propellers show a quadratic curve relationship as shown in Figure 10b, which satisfies the condition of Equation (28).  
As shown in Figure 11, the mass of the robot (LAWCDR) can be divided into three parts, the body mass, the first joint mass, and the second joint mass (suction cup and support bars). \(L_{s}\) denotes the \(y\)-axis moment of the first joint in the body coordinate system. \(L_{v}\) denotes the \(y\)-axis moment of the second joint in the body coordinate system. During the flight phase, we optimize the servo angle by calculating the horizontal and vertical moments that the center of gravity offset causes.  
Figure 11: The robot balancing the center of gravity by adjusting the servo.  
Figure 10: Propeller lift and actual power test: (**a**) lift/power test bench (**b**) fitting curve between lift and power consumption.  
\[\left\{\begin{array}{l}argmin(\alpha_{\eta}*S_{1}+\beta_{\eta}*S_{2})\\ \par\par\par\par\par S_{1}=mgL_{1}+m_{g}L_{s}+m_{g}L_{v}=mgL_{1}+m_{g}gd_{2}sin \eta_{1}+m_{v}g(d_{2}sin\eta_{1}+d_{1}sin(\eta_{1}+\eta_{2}))\\ \par\par\par S_{2}=m_{g}gd_{2}cos\eta_{1}+m_{v}g(d_{2}cos\eta  
For the robot pitch channel, define \(\theta=x_{1}\), \(\dot{\theta}=x_{2}\), \(\dot{\theta}=\hat{x}_{1}\), \(\dot{\dot{\theta}}=\hat{x}_{2}\), where \(\hat{x}_{1}\) and \(\hat{x}_{2}\) are the observations for \(\theta\) and \(\dot{\theta}\). The pitch channel model can be rewritten as:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\frac{l_{2}}{l_{y}}u_{\theta}+v\end{array}\right. \tag{34}\]  
where \(v\) is the lump disturbance. Defining \(x_{3}=\frac{(l_{2}-l  
As shown in Figure 12, the robot adjusted the servo angle from the preparation state to the flight state at 6 s. We calculated the average value of the output, marked by the red line. The pitch channel output was reduced from 1.50 Nm to 0.56 Nm.  
The output of the extended state observer is shown in Figure 13. The initial moment observed by the ESO is \(-1.5\) Nm. After servos angle change at 6 s, it changes to 0.55 Nm, which is roughly the same amount as the change in U3. The ESO observer is able to converge within 1 s.