# Advanced Control Strategies for Securing UAV Systems: A Cyber-Physical Approach
## 2 Advanced Sliding Mode Control Framework for UAV Security
### Helicopter Dynamics  
A set of non-linear differential equations typically represents the dynamic model of a helicopter, which is defined with respect to two primary coordinate systems: the inertial frame and the body frame, as depicted in Figure 1.  
* Inertial Frame (\(\mathbf{X}_{e}\), \(\mathbf{Y}_{e}\), \(\mathbf{Z}_{b}\)): The inertial frame is a fixed coordinate system that serves as a reference for the helicopter's position and orientation in space. The axes \(\mathbf{X}_{e}\), \(\mathbf{Y}_{e}\), and \(\mathbf{Z}_{e}\) are typically aligned with the north-east-down (NED) or Earth-centered Earth-fixed (ECEF) directions, providing a global reference for the helicopter's movement. Forces such as the gravitational force \(\mathbf{W}\) act in this frame, influencing the helicopter's position and orientation. Also, the velocity vector \(\mathbf{V}\) represents the helicopter's speed and direction of movement relative to the inertial frame. * Body Frame (\(\mathbf{X}_{b}\), \(\mathbf{Y}_{b}\), \(\mathbf{Z}_{b}\)): The body frame is a moving coordinate system that is fixed to the helicopter. The axes \(\mathbf{X}_{b}\), \(\mathbf{Y}_{b}\), and \(\mathbf{Z}_{b}\) are aligned with the helicopter's forward, right, and downward directions, respectively. This frame is used to describe the helicopter's dynamics relative to its own structure. The forces and moments, such as thrust (\(\mathbf{T}\)), angular velocities (\(\mathbf{\Omega}\)), and moments (\(\mathbf{M}\)), are represented in this frame, influencing the helicopter's movement and stability. The thrust vector \(\mathbf{T}\) generated by the helicopter's propeller and the resulting moments \(\mathbf{M}\) around the helicopter's center of mass are critical for controlling the helicopter's movement and stability in three-dimensional space. The figure also introduces key aerodynamic elements such as the angle of attack (\(\alpha\)), which affects the lift and drag forces on the helicopter, and the resultant aerodynamic force components along the \(\mathbf{X}_{b}\), \(\mathbf{Y}_{b}\), and \(\mathbf{Z}_{b}\) axes. Additionally, the rotational angles (roll \(\phi\), pitch \(\theta\), and yaw \(\psi\)) define the orientation of the helicopter relative to the Body Frame. In this dynamic model, the state vector \(\mathbf{x}\) includes positions, velocities, attitudes, and angular rates, all of which are expressed in the body frame. Control inputs \(\mathbf{u}\), such as the collective pitch, longitudinal cyclic pitch, lateral cyclic pitch, and tail rotor pitch, are also defined relative to the body frame. The helicopter's dynamics are influenced by forces and moments that are calculated in the body frame but can be transformed into the inertial frame for a global perspective on the helicopter's motion. The state vector \(\mathbf{x}\) for a  
Figure 1: The inertial frame vs. body frame of the helicopter. Different colors are used for a clearer distinction between the components of each frame. helicopter includes positions, velocities, attitudes, and angular rates. The control inputs \(\mathbf{u}\) include the collective pitch, longitudinal cyclic pitch, lateral cyclic pitch, and tail rotor pitch. The helicopter's dynamics can be described as:  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{B}(\mathbf{x})\mathbf{u}+ \mathbf{d}(t), \tag{1}\]  
where \(\mathbf{x}=[\phi,\theta,\psi,p,q,r,u,v,w,x,y,z]^{T}\) represents the state vector; \(\phi,\theta,\psi\) are the roll, pitch, and yaw angles; \(p,q,r\) are the angular rates; \(u,v,w\) are the velocities in the body frame; \(x,y,z\) are the positions in the inertial frame; \(\mathbf{B}(\mathbf{x})\) is the control effectiveness matrix, reflecting the non-linear dynamics of the system and mapping the control inputs \(\mathbf{u}\) to the changes in the state vector \(\mathbf{x}\); \(\mathbf{u}=[u_{1},u_{2},u_{3},u_{4}]^{T}\) are the control inputs; and \(\mathbf{d}(t)\) represents external disturbances and uncertainties.