# Rapid Parametric Modeling and Robust Analysis for the Hypersonic Ascent Based on Gap Metrics
## 2 Model Description
### Polynomial Fitting Modeling  
The mechanical model based on the finite element method is very complicated for implementing the control design and integrated optimization for the aerospace vehicle. Hence, the surrogate models of the aerodynamic forces and moments need to be first acquired, and they are expressed by [28]  
\[\left\{ \begin{array}{l}L=qS_{\mathrm{ref}}C_{L}\\ D=qS_{\mathrm{ref}}C_{D}\\ F_{T}=qS_{\mathrm{cap}}C_{T}\\ M_{y}=qS_{\mathrm{ref}}C_{T}\end{array} \right. \tag{12}\]where \(\bar{q}\) is the dynamic pressure, and \(S_{\text{ref}}\) and \(I_{\text{ref}}\) are the reference area and length, respectively. \(z_{T}\) is the shift of the thrust axis from the center of gravity. Furthermore, the polynomial fitting method is applied to identify the specific forms concerning \(C_{L}\), \(C_{D}\), \(C_{T}\), and \(C_{m}\). The surrogate procedure consists of four steps: the design of the sample points, selection of the model structure, identification of the model parameters, and verification of the acquired model. Without loss of generality, we first select \(y=C_{L}\) as the identification parameter to expatiate this design course. The sampling points for \(C_{L}\) are constructed using the Latin hypercube sampling method with the subsequent evaluation criterion:  
\[E_{c}=\sum_{i=1}^{N}\sum_{j=i+1}^{N}\frac{1}{\left\lVert\mathbf{x}_{i}-\mathbf{x}_{j} \right\rVert^{2}} \tag{13}\]  
where \(N\) is the number of the sample points; \(x_{i}\) and \(x_{j}\) denote the positions of the \(i\)-th and \(j\)-th sample points. In principle, the selected sample points can ensure that the distance among the sample points is the maximum, matching the minimum evaluation criterion of (14). According to these obtained sample points, we consider further the surrogate model (SM) of \(C_{L}\) with the following form:  
\[y=f(\mathbf{A}_{\text{cof}},\mathbf{x}) \tag{14}\]  
where \(\mathbf{A}_{\text{cof}}\) denotes the contingent coefficients which need to be determined in terms of the trust region method. Furthermore, we define the residual errors as  
\[\Delta y=\min_{\mathbf{A}_{\text{cof}}}\ \left\lVert f(\mathbf{A}_{\text{cof}},\mathbf{x})- \mathbf{y}_{\text{obs}}\right\rVert_{2}^{2} \tag{15}\]  
where \(\mathbf{y}_{\text{obs}}\) denotes the outputs of the sample points. In particular, the SM is comprised of first-degree terms, quadratic terms, and two crossing terms. However, such polynomial items complicate the iterative computation for obtaining these contingent coefficients. Therefore, the optimum selection of these items aids in simplifying the modeling courses. In this paper, a sensitivity analysis method is adopted to choose the identified items, and this method is based on the iterative fractional factorial design (IFFD), which distinguishes the influence relation between outputs and inputs. The basic design matrix of FFD with \(k\) parameters is donated as \(\mathbf{J}_{n_{i}}\), columns of which represent the sampling level of parameters. The orientation of an iteration is defined as positive, negative, or null. A positive orientation means the sampling values of the parameter agree with the notation of the basic design matrix (high: \(+1\), low: \(-1\)), while negative means the opposite, and null represents sampling at the median \(0\). The procedure of IFFD consists of two steps of randomization [13]. The first step affects the whole set of iterations. The orientation of each iteration is randomized with a specified proportion of null orientation. The random direction matrix \(\mathbf{O}_{n_{k}}\) is identified. The parameters are randomly assigned to columns of the basic design matrix \(\mathbf{J}_{n_{i}}\) and generate the new basic design matrix \(\mathbf{J}^{\prime}_{n_{i}}\).