## 3 Proposed Method
### State Estimation Based on the UKF  
As shown in Section 2.1, the air data system exhibits strong nonlinearity, and the standard Kalman filter is unable to accurately describe it. The unscented Kalman filter combines the Kalman filter with the unscented Transform. By designing weighted points, it approximates the sampling points of the target, and then the propagation of these weighted points through nonlinear functions is calculated without the need for the linearization of nonlinear parts, but through the probability distribution instead [39]. Compared to the extended Kalman filter (EKF), which uses the Taylor expansion for linearization [40], the UKF approximates the posterior probability density of the state using a set of deterministic samples, thereby reducing linearization errors. First and foremost, the state equation and measurement equation of the ADS need to be established. Let the state vector be represented as \(\mathbf{x}=[V,\alpha,\beta,\phi,\theta,\psi]^{\mathrm{T}}\). Considering the measurement noise of the INS, the measurement noises of three-dimensional acceleration and angular velocity are denoted as \(\mathbf{w}\), \(\mathbf{w}=\left[w_{A_{x}},w_{A_{y}},w_{A_{x}},w_{p},w_{q},w_{r}\right]^{\mathrm{T}}\). Therefore, the input vector is represented as \(\mathbf{u}=\left[A_{xm},A_{ym},A_{zm},p_{m},q_{m},r_{m}\right]^{\mathrm{T}},\mathbf{u} =\left[A_{x},A_{y},A_{z},p,q,r\right]^{\mathrm{T}}+\mathbf{w}\). Based on Equations (7) and (8), the state equation can be obtained as follows:  
\[\overset{\bullet}{\mathbf{x}}=f(\mathbf{x},\mathbf{u})+\mathbf{G}(\mathbf{x})\mathbf{w}+E(\mathbf{x})\mathbf{d} \tag{9}\]  
Figure 3: Overall scheme of the proposed method. where the disturbance vector \(\mathbf{d}=[d_{u},d_{v},d_{w}]^{\mathrm{T}}\), \(f(\cdot)\) is the nonlinear function, and noise matrix \(G(x)\) and disturbance matrix \(E(x)\) can be, respectively, represented as follows:  
\[G(x)= \begin{bmatrix}-\cos\alpha\cos\beta&-\sin\beta&-\sin\alpha\cos\beta&0&0&0\\ \frac{\sin\alpha}{V\cos\beta}&0&-\frac{\cos\alpha}{V\cos\beta}&\cos\alpha\tan \beta&-1&\sin\alpha\tan\beta\\ \frac{\cos\alpha\sin\beta}{V}&-\frac{\cos\beta}{V}&\frac{\sin\alpha\sin\beta} {V}&-\sin\alpha&0&\cos\alpha\\ 0&0&0&-1&-\sin\phi\tan\theta&-\cos\phi\tan\theta\\ 0&0&0&0&-\cos\phi&\sin\phi\\ 0&0&0&0&-\frac{\sin\phi}{\cos\theta}&-\frac{\cos\phi}{\cos\theta}\end{bmatrix}  \tag{10}\]  
\[E(x)= \begin{bmatrix}\cos\alpha\cos\beta&\sin\beta&\sin\alpha\cos\beta\\ -\frac{\sin\alpha}{V\cos\beta}&0&\frac{\cos\alpha}{V\cos\beta}\\ -\frac{\cos\alpha\sin\beta}{V}&\frac{\cos\beta}{V}&-\frac{\sin\alpha\sin\beta} {V}\\ 0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}  \tag{11}\]  
The measurement equation of the system is:  
\[\mathbf{z}=h\mathbf{x}+\mathbf{v} \tag{12}\]  
where measurement vector \(\mathbf{z}=[V_{\mathrm{m}},\alpha_{\mathrm{m}},\beta_{\mathrm{m}},\phi_{\mathrm{ m}},\theta_{\mathrm{m}},\psi_{\mathrm{m}}]^{\mathrm{T}}\), \(\mathbf{v}\) represents the corresponding measurement noise, \(\mathbf{h}=\mathbf{I}_{6\times 6}\). To conclude, the state equation and measurement equation of the ADS model are as follows:  
\[\left\{ \begin{array}{l}\overset{\bullet}{\mathbf{x}}=f(\mathbf{x},\mathbf{u})+G(\mathbf{x} )\mathbf{w}+E(\mathbf{x})\mathbf{d}\\ \mathbf{z}=h\mathbf{x}+\mathbf{v}\end{array} \right. \tag{13}\]  
Then, the computation process of the UKF is as follows:  
1. Compute the sigma point set: \[\left\{ \begin{array}{l}\mathbf{x}_{0,k-1}=\overset{\wedge}{\mathbf{x}}_{k-1}\\ \mathbf{x}_{i,k-1}=\overset{\wedge}{\mathbf{x}}_{k-1}+\left(\sqrt{(n+\lambda)P_{i,k- 1}}\right)_{i}\\ \mathbf{x}_{i+n,k-1}=\overset{\wedge}{\mathbf{x}}_{k-1}-\left(\sqrt{(n+\lambda)P_{i,k -1}}\right)_{i}\end{array} \right.\] (14)  
where \(n\) is the data dimension, \(\lambda\[\left\{