# Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis +
###### Abstract  
This paper addresses the estimation of accurate extreme ground impact footprints and probabilistic maps due to a total loss of control of fixed-wing unmanned aerial vehicles after a main engine failure. In this paper, we focus on the ground impact footprints that contains 95%, 99% and 99.9% of the drone impacts. These regions are defined here with density minimum volume sets and may be estimated by Monte Carlo methods. As Monte Carlo approaches lead to an underestimation of extreme ground impact footprints, we consider in this article multiple importance sampling to evaluate them. Then, we perform a reliability oriented sensitivity analysis, to estimate the most influential uncertain parameters on the ground impact position. We show the results of these estimations on a realistic drone flight scenario. U  
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Footnote †: _Applied_  
Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis +  
as the effect of the altitude of the vehicle at the failure instant. This type of impact footprints has also been obtained in [3], considering a 6 degrees-of-freedom dynamic model of a fixed-wing aircraft. The effect of wind on impact footprints has been investigated in [7]. Computation of impact locations and footprints has also been performed in [8], for both fixed-wing and multi-rotor UAVs considering different modes of failure. Some level sets are computed to provide some insights on the distribution of the impact points inside the footprint. The generation of probability maps has been investigated in other works to provide more information on the impact distributions on the ground that could be useful and reduce conservatism in risk analysis or decision making. In [9], a ballistic model with drag force is considered to represent the descent of a fixed-wing UAV and generate probability density functions of impact points. Uncertainties on drag, initial speed at the instant of failure and external wind are accounted for. Full flight dynamics of a Cesna 182 aircraft are used in [10] to compute ground impact probability maps by Monte Carlo simulations. Total loss of power is assumed and uncertainties on the initial conditions of the UAV at failure instant as well as on the deflection of unactuated control surfaces are considered. A 6 degrees-of-freedom flight mechanics model is also used in [11] for a fixed wing UAV to estimate ground impact probability maps, taking into account the influence of wind direction and speed. Real flight data have been used to model uncertainties on the turn rate and flight path angle of the vehicles for cruise-like mode at constant altitude and straight line. These uncertainties along with the ones on the actuators deflections at the instant of failure are used in the Monte Carlo process. Influence of initial altitude, speed and wind (speed and direction) are analyzed, and a full data basis has been obtained containing impact probability maps for a sampled set of values for these quantities. This data basis can be useful for risk evaluation along a given UAV flight trajectory, e.g., for mission preparation. Since Monte Carlo simulations can be time-consuming, more recent works have been dedicated to the development of surrogate models for the generation of ground impact probability maps. \(K\)-Nearest neighbors models have been considered in [10] to approximate impact probability distribution. Other techniques such as Krigging have been investigated [7] regarding impact footprints or neural networks for both generation of impact footprints [7] and probability maps [11]. In all these works, assumptions are made on the uncertainties on the variables used as inputs of the computations. Uncertainty representations are mainly based on statistic models (probability distributions) and/or bounds (intervals with no statistical assumptions). Accuracy of the resulting outputs (ground impact locations, footprints, probability maps) strongly relies on the representativeness of these assumptions.