# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 4 Applications
### Launch Vehicle Design Problem  
#### 4.2.1 Test Case Description  
This test case is derived from a launch vehicle design problem to estimate flight envelops considering modeling uncertainty. In the early design phase, in order to explore the design space at an affordable computational cost, low and medium fidelity models are used resulting in uncertainty in the estimated system performances. To evaluate the impact  
Figure 10: Boxplots for 10 repetitions of the estimation of the quantile confidence area (a) and root mean square error (b) along the 10 active-learning iterations. Figure 9: Estimation of the quantile and its confidence area before refinement (a) and after 10 active-learning iterations (b). of the modeling uncertainty on the launch vehicle, uncertainty propagation is carried out. One objective is to estimate the launch vehicle flights envelopes that describe the possible trajectories considering the modeling uncertainty. The flight envelops are defined based on quantile estimation of the trajectory state variables such as altitude, velocity, flight path angle, etc. In the current test case, the considered mission is to inject a payload of 10 tons into a 400 km circular equatorial orbit using a Two-Stage-To-Orbit (TSTO) vehicle. The launch vehicle has two stages: a first stage with nine rocket engines and a second stage with one rocket engine. All the engines use a combination of liquid oxydier and rocket propellant (RP-1). The launch pad is located at the European spaceport at Kourou in French Guiana. For one realization of the input uncertain modeling variables, the multidisciplinary analysis used to assess the vehicle performances consists in determining the optimal launch vehicle trajectory (optimal state variables) involving the solving of an optimal control problem. Therefore, uncertainty propagation is a challenge even in early design phases as it combines the computational cost of multidisciplinary analysis, uncertainty propagation and trajectory optimization. In this test case, the surrogate-based model approach presented in the previous sections is carried out. It is assumed that the architecture of the launch vehicle is defined from a baseline and the objective is to assess the launch vehicle robustness to modeling uncertainty through the determination of flight envelops (extreme quantiles). The input modeling uncertainties considered in this problem are gathered into an input uncertainty vector \(\mathbf{U}\in\mathbb{R}^{7}\). These variables and their probability distributions are summarized in Table 2. These distributions have been assessed using expert knowledge. Four uncertainties are considered on the modeling of the propulsive performance: the specific impulse (\(U_{Isp}\)) of the first and second stages and the mass flow rate (\(U_{q}\)). In addition, an uncertainty on dry masses is assumed using \(U_{m_{1}}\) and \(U_{m_{2}}\) to account for structural dry mass misknowledge in the early design phase. Eventually, for the aerodynamics model, it is assumed that the drag coefficient suffers from some uncertainty \(U_{CD}\). Figure 11: Confidence area based on the surrogate model estimation and based on Monte Carlo (MC) with 14 evaluations of \(\mathcal{Q}_{\mathbf{X}}(\cdot)\). Propulsion  
The propulsion module is in charge of computing the rocket performances for the first and second stages. The propulsion module is based on the Chemical Equilibrium with Applications (CEA) model [33]. This code determines the rocket engine performances such as specific impulse and thrust based on different input characteristics (chamber pressure, oxydizer to fuel ratio, etc.). The model involves a thermochemical analysis of complex mixtures of propellants. For a rocket engine, it involves the combustion of injected gas in the combustion chamber and the hot gas expansion in the nozzle. CEA is suited for rocket engine calculations at the conceptual and preliminary design phases.