# Advanced Control Strategies for Securing UAV Systems: A Cyber-Physical Approach
## 2 Advanced Sliding Mode Control Framework for UAV Security
### Rotor Dynamics and Flapping Motion  
The rotor dynamics, particularly the flapping motion, significantly influence the stability and control of the helicopter. The blade flapping motion is described by the flapping angle \(\beta\), influenced by aerodynamic forces, inertial forces, and control inputs. The flapping motion is modeled as follows:  
\[\ddot{\beta}+\omega_{\beta}^{2}\beta=0, \tag{5}\]  
where \(\omega_{\beta}\) is the natural frequency of the flap. The blade response considering aerodynamic forces can be represented as:  
\[\ddot{\beta}+2\zeta\omega_{\beta}\dot{\beta}+\omega_{\beta}^{2}\beta=M_{\beta}, \tag{6}\]  
where \(M_{\beta}\) is the aerodynamic flap moment, \(\zeta\) is the damping ratio, and \(\omega_{\beta}\) is the flapping frequency.  
Aerodynamic Damping and Control Inputs  
The control inputs, including the longitudinal and lateral cyclic pitches and the collective pitch, play crucial roles in the helicopter's rotational dynamics. These inputs modify the orientation of the thrust vector, impacting the helicopter's roll, pitch, and yaw motions. The dynamics are captured by [30]:  
\[\dot{\theta}_{a} =-\frac{1}{\tau_{m}}\theta_{a}+\frac{k_{\beta}}{2\Omega I_{\beta} }\theta_{b}+\frac{1}{\tau_{m}}\theta_{a}+d_{a}, \tag{7}\] \[\dot{\theta}_{b} =-\frac{1}{\tau_{m}}\theta_{b}+\frac{k_{\beta}}{2\Omega I_{\beta} }\theta_{a}+\frac{1}{\tau_{m}}\theta_{b}+d_{b}, \tag{8}\]  
where \(\theta_{a}\) and \(\theta_{b}\) are the longitudinal and lateral cyclic inputs, \(k_{\beta}\) is the blade root stiffness, \(I_{\beta}\) is the blade moment of inertia about the flap hinge, \(\Omega\) is the rotor angular speed, and \(\tau_{m}\) is the rotor time constant. The terms \(d_{a}\) and \(d_{b}\) represent unknown disturbances.  
Rotor Dynamics and Flapping Motion  
The rotor dynamics are crucial in determining the helicopter's behavior. The main rotor blades are not rigid and exhibit flapping motion, which can be represented by the flapping angles \(a\) and \(b\) for longitudinal and lateral directions, respectively. The equations can describe these dynamics:  
\[\dot{a} =\gamma_{m}\Omega\Bigg{(}\frac{-4k_{\beta}}{I_{\beta}\Omega^{2} }-8\Bigg{)}a-p+\gamma_{m}\Omega\bigg{(}\frac{-\gamma_{m}}{\gamma_{m}^{2}+64} \bigg{)}B_{m} \tag{9}\] \[\dot{b} =\gamma_{m}\Omega\bigg{(}\frac{\gamma_{m}}{\gamma_{m}^{2}+64} \bigg{)}b-q+\gamma_{m}\Omega\bigg{(}\frac{8}{\gamma_{m}^{2}+64}\bigg{)}A_{m},\]  
where \(\gamma_{m}\) is the Lock number, \(\Omega\) is the rotor angular velocity, \(I_{\beta}\) is the blade moment of inertia, \(k_{\beta}\) is the blade stiffness coefficient, and \(A_{m},B_{m}\) are cyclic pitch angles [31].  
The dominant forces and moments on the helicopter come from the main rotor and can be expressed as follows:  
\[F_{x} =-T\sin a, \tag{10}\] \[F_{y} =T\sin b,\] \[F_{z} =-T\cos a\cos b,\] \[M_{x} =(k_{\beta}+TH_{mr})b,\] \[M_{y} =(k_{\beta}+TH_{mr})a,\]  
where \(T\) is the thrust, \(H_{mr}\) is the main rotor height, and \((a,b)\) are the flapping angles.  
Considerations for Control Design  
The inclusion of rotor dynamics and actuator dynamics, as well as aerodynamic damping, is essential for accurate modeling and control design. These dynamics can significantly affect the helicopter's response to control inputs, making it critical to incorporate them into the control strategy to ensure stability and performance.