# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 4 Applications
### Launch Vehicle Design Problem
#### 4.2.1 Test Case Description
It provides coupling variables (specific impulse, mass flow rate) that are transferred to the mass and sizing and the trajectory disciplines. To account for the modeling uncertainty, the nominal value of the estimated parameters by CEA is perturbed as presented in Table 2. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Name** & **Notation** & **Model (Mean, Standard Deviation)** \\ \hline Specific impulse stage 1 & \(U_{1 sp_{1}}\) & \(\mathcal{N}(0,1)\) (additive, s) \\ Specific impulse stage 2 & \(U_{1 sp_{2}}\) & \(\mathcal{N}(0,1)\) (additive, s) \\ Residual mass stage 1 & \(U_{m_{1}}\) & \(\mathcal{U}(-750,750)\) (additive, kg) \\ Residual mass stage 2 & \(U_{m_{2}}\) & \(\mathcal{U}(-250,250)\) (additive, kg) \\ Mass flow rate stage 1 & \(U_{l_{1}}\) & \(\mathcal{N}(0,5)\) (additive, kg/s) \\ Mass flow rate stage 2 & \(U_{l_{2}}\) & \(\mathcal{N}(0,5)\) (additive, kg/s) \\ Drag coefficient & \(U_{CD}\) & \(\mathcal{U}(-0.05,0.05)\) (additive, \(-\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Probability distribution of the input uncertain random vector. Figure 12: Uncertainty propagation for launch vehicle design test case using multidisciplinary simulation. Structures  
The structure discipline is in charge of estimating the dry mass of the two stages accounting for the coupling variables coming from the propulsion and trajectory disciplines. Mass Estimation Relationships (MERs) determined based on existing expandable launch vehicles are used [34] to estimate the masses of the different subsystems. All the different components of the launch vehicle (tanks, rocket engines, nozzle, avionics, electrical systems, thrust frame, interstage, etc.) are modeled. Their corresponding masses are estimated based on analytical relationships defined by the MERs. This discipline provides a fast estimation of the launch vehicle dry mass. To account for the fidelity level of the model, the dry mass of the first and second stages are assumed to be uncertain as presented in Table 2.  
Aerodynamics  
Trajectory  
The trajectory discipline is in charge of optimizing the trajectory path in order to minimize the consumption of propellant during the flight. To simulate the flight of the launch vehicle, it is necessary to numerically integrate a system of ordinary differential equations that corresponds to the equations of motion according to the time. Different approaches exist to simulate and optimize the launch vehicle flight and in the multidisciplinary process, a pseudo-spectral method [37] is used in order to find the optimal control law. The control law consists of a parameterized pitch profile as a function of time. These parameters are optimized based on a Legendre-Gauss-Lobatto collocation technique [37] and DYMOS library [38]. The equations of motion that are considered involve five state variables:  
\[f_{\text{ode}}: \begin{cases}\dot{r}&=v\sin\left(\phi\right)\\ \dot{\lambda}&=\frac{v\cos\left(\phi\right)}{r}\\ \dot{v}&=\frac{-D+T\cos\left(\theta-\phi\right)}{m}+\left(-g+\omega^{2}r \right)\sin\left(\phi\right)\\ \dot{\phi}&=\frac{L}{m\Omega}+\frac{T\sin\left(\theta-\phi\right)}{m\Omega}+ \frac{\left(\omega^{2}r-g\right)\cos\left(\phi\right)}{v}+2\omega+\frac{v\cos \left(\phi\right)}{r}\\ \dot{m}&=-q\end{cases}  \tag{29}\]  
with \(\omega\) the angular velocity of the Earth, \(\lambda\) the longitude of the launch vehicle, \(\phi\) the flight path angle, \(\mathbf{r}\) the position vector from the center of the Earth to the geometric center of the launch vehicle, \(\alpha\) the angle of attack, \(m\) the current launcher mass, \(q\) the mass flow rate, \(g\) the acceleration of gravity, \(L\) the lift force, \(D\) the drag force, \(\theta\) the pitch angle, \(T\) the thrust force and \(v\) the velocity vector. The trajectory is decomposed into five phases with appropriate guidance laws: lift-off, pitch-over maneuver, exponential decay, gravity turn and bi-linear tangent phase.