## 2 Methodology
### Compatibility Check  
Typically, the longitudinal, lateral, and vertical accelerations (\(a_{x}\), \(a_{y}\), \(a_{z}\)) along the (\(x\), \(y\), \(z\))-body axes of aircraft, as well as the angle of attack \(\alpha\), Euler angles (\(\phi\), \(\theta\), and \(\psi\)), aileron deflection (\(\delta_{\text{a}}\)), elevator (\(\delta_{\text{e}}\)), rudder (\(\delta_{\text{r}}\)), stabilizer (\(\delta_{\text{s}}\)), etc., are available and recorded in the QAR or FDR of all transport aircraft. Since the recorded flight data may contain errors (or biases), compatibility checks are performed to remove them by satisfying the following kinematic equations [12,18]:  
\[\dot{\phi}=p+q\sin\phi\tan\theta+r\cos\phi\tan\theta \tag{1}\]  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]  
\[\dot{\psi}=(q\sin\phi+r\cos\phi)\sec\theta \tag{3}\]  
\[\dot{V}=(a_{x}-g\sin\theta)\cos\alpha\cos\beta+(a_{y}+g\sin\phi\cos\theta)\sin\beta \tag{4}\]  
\[+(a_{z}+g\cos\phi\cos\theta)\sin\alpha\cos\beta\]  
Figure 1: Flowchart for the development of a nonlinear and unsteady aerodynamic data database. \[\dot{\alpha}=\left[(a_{z}+g\cos\theta\cos\phi)\cos\alpha-(a_{x}-g\sin\theta)\sin \alpha\right]/(V\cos\beta) \tag{5}\]  
\[+q-\tan\beta(p\cos\alpha+r\sin\alpha)\]  
\[\dot{\beta}=\cos\beta(a_{y}+g\cos\theta\sin\phi)/V+p\sin\alpha-r\cos\alpha\]  
\[-\sin\beta[(a_{z}+g\cos\theta\cos\phi)\sin\alpha-(a_{x}-g\sin\theta)\cos \alpha]/V \tag{6}\]  
where \(g\) is the gravitational acceleration and \(V\) is the flight speed. Let the errors be denoted as \(b_{a_{x}},b_{a_{y}},b_{a_{z}},p_{y},b_{a},p_{y},b_{x},b_{y},b_{x},b_{\beta},b_{ \phi},b_{\phi},b_{\psi}\), for \(a_{x}\), \(a_{y}\), \(a_{z}\), etc., respectively. These errors are estimated by minimizing the squared sum of the differences between the two sides of Equations (1)-(6). These equations in vector form can be written as:  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]0  
where  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]1  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]2  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]3  
where superscript \"\(\overset{\rightharpoonup}{-}\)\" stands for the mean value and the subscript \"\(m\)\" indicates the measured or recorded values. The cost function is defined as:  
\[\dot{\theta}=q\cos\phi-r\sin\phi \tag{2}\]4  
where \(Q\) is a weighting diagonal matrix with elements being 1.0 and \(\overset{\rightharpoonup}{z}\) is calculated with a central difference scheme with \(\overset{\rightharpoonup}{z}_{m}\), which is the measured value of \(\overset{\rightharpoonup}{z}\). The steepest descent optimization method is adopted to minimize the cost function. As a result of analysis, variables unavailable in the QAR or FDR, such as \(\beta\), \(p\), \(q\), and \(r\), are able to be estimated. The above-mentioned unavailable variables in the QAR or FDR need initial values as a basis for correction, where the angular rates, such as \(p,q\), and \(r\), are obtained from derivatives of \(\phi\), \(\theta\), and \(\psi\) with time by using the method of monotonic cubic interpolation. This interpolation method is used to connect the flight data in a continuous curve to obtain the slope of the curve. The value of \(\beta\) cannot be obtained from the derivative. The initial value of \(\beta\) is assumed to be 0 at the time of estimation, and it is calculated when Equation (6) is satisfied.