# Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis +
## 2 Ground Impact Simulation
### UAV Dynamics  
The model used to simulate the trajectory of the UAV to the ground is a six degrees of freedom (6 DOF) dynamic model, including full flight mechanics, and hence enabling one to incorporate the influence of wind from a dynamical point of view (and not kinematical compared to some approaches developed in the literature [12]). The model considered here is a fixed wing aircraft such as the one presented in [13]. The control input vector \(u= \begin{bmatrix}\delta_{a}&\delta_{e}&\delta_{r}&\delta_{T}\end{bmatrix} ^{\top}\) is composed of ailerons, elevators, rudder deflections, and thrust command. The state of the dynamical system to be simulated is defined as \(\chi= \begin{bmatrix}X^{\top}&V^{\top}&\eta^{\top}&\Omega^{\top}\end{bmatrix} ^{\top}\) where \(X= \begin{bmatrix}x&y&z\end{bmatrix} ^{\top}\) is the position vector defined in a local NED (north east down) frame, \(V\) and \(\Omega\) are the translation and angular velocity vectors in the aircraft body-frame, and \(\eta= \begin{bmatrix}\phi&\theta&\psi\end{bmatrix} ^{\top}\) is the vector of Euler angles (roll-pitch-yaw) describing the attitude of the UAV. The origin of the (inertial) local NED frame is chosen to correspond to \(z\)=0 (ground) and is arbitrarily chosen for \(x\) and \(y\)-components, as we are only interested in the description of the motion of UAV during its descent to the ground with respect to the vehicle position at the instant of failure (considered to be \(x=y=0\)). The rigid-body dynamics of the UAV is described as  
\[\left\{ \begin{array}{l}\dot{X}=R_{\eta}V\\ \dot{V}=-\Omega\times V+\frac{1}{m}F\\ \dot{\eta}=T_{\eta}\Omega\\ \dot{\Omega}=J^{-1}(-\Omega\times J\Omega+M)\end{array} \right. \tag{1}\]  
where \(R_{\eta}\) is the orientation matrix parametrized in terms of Z-Y-X Euler angles given by  
\[R_{\eta}=\left[ \begin{array}{ccc}c_{\theta}c_{\phi}&s_{\phi}s_{\theta}c_{ \psi}-s_{\phi}c_{\psi}&s_{\theta}s_{\psi}+s_{\theta}c_{\phi}c_{\psi}\\ c_{\theta}s_{\psi}&c_{\phi}c_{\psi}+s_{\theta}s_{\phi}s_{\psi}&s_{\theta}c_{ \phi}s_{\psi}-s_{\phi}c_{\psi}\\ -s_{\theta}&c_{\theta}s_{\phi}&c_{\theta}c_{\phi}\end{array} \right] \tag{2}\]  
and \(T_{\eta}\) is the transformation matrix defined by  
\[T_{\eta}=\left[ \begin{array}{ccc}1&s_{\theta}t_{\theta}&c_{\psi}t_{\theta} \\ 0&c_{\phi}&-s_{\phi}\\ 0&s_{\phi}/c_{\phi}&c_{\phi}/c_{\phi}\end{array} \right] \tag{3}\]  
whith the notations \(c_{\alpha}=\cos(\alpha)\), \(s_{\alpha}=\sin(\alpha)\) and \(t_{\alpha}=\tan(\alpha)\) for any given angle \(\alpha\). The inertia matrix of the UAV is denoted by \(J\) and its mass by \(m\). Values used for the UAV parameters are given in Appendix A. The resulting force F expressed in the aircraft body-frame  
\[F=F_{eng}(\delta_{T})+F_{g}(\eta)+F_{a}(V,V_{w},\eta,\Omega,\delta_{a},\delta_ {e},\delta_{r}) \tag{4}\]is composed of the thrust, due to the engine (\(F_{eng}(\delta_{T})\)) which is zero after engine failure, the gravity force (\(F_{g}(\eta)\)) and the resulting aerodynamic force \(F_{a}\), which depends on the true air speed of the UAV and then on the wind speed vector \(V_{w}\). To compute this force, a full aerodynamic model is used, such as the one described in [13] and is presented in Appendix A. Similarly, the resulting torque expressed in the aircraft body-frame  
\[M=M_{eng}(\delta_{T})+M_{a}(V,V_{w},\eta,\Omega,\delta_{a},\delta_{e},\delta_{ r}) \tag{5}\]  
is composed of the torque component \(M_{eng}(\delta_{T})\), due to the thrust of the main engine (equals to zero after engine failure) and the aerodynamic torque \(M_{a}\), which also depends on the wind speed vector \(V_{w}\). To compute this torque, a full aerodynamic model of the aircraft is also considered (see Appendix A).