## 3 Alignment Control Response Model  
It is assumed that only one air spring is inflated/deflated each time. The equivalent force on the main engine for the \(k\)-th air pressure adjustment is [6]  
\[\mathbf{F_{\Delta\delta p(k)}}=\mathbf{G_{i}^{T}}\left[-\text{sgn}\Big{(}a_{x }{}^{i}\Big{)}\Delta\delta p_{i}S_{e}\sin\theta,0,\Delta\delta p_{i}S_{e}\cos \theta,0,0,0\right]^{T} \tag{10}\]  
where \(i\) is the air spring serial number for \(k\)-th inflated/deflated, \(\Delta\delta p_{i}\) is the air pressure variation of \(i\)-th air spring, \(\theta\) is the inclination angle of the air springs, \(\text{sgn}(a_{x}{}^{i})\) is the sign function of \(a_{x}{}^{i}\)(1 if \(a_{x}{}^{i}>0\) or \(-1\) if \(a_{x}{}^{i}<0\)), and \(S_{e}\) is the effective load-bearing area of the air spring (all air springs have the same area). Ignoring the effect of air pressure variation on the global stiffness matrix, the alignment control response for the \(k\)-th air pressure adjustment can be obtained as [6]  
\[\mathbf{\Delta x_{c}}(k+1)\mathbf{=}\mathbf{G_{c}K^{-1}F_{\Delta\delta p(k)}} \tag{11}\]  
In the first \(k\) times of inflation/deflation, it is assumed that the air pressure variation for each air spring is \(\delta\mathbf{p}(k)=\left[\delta p_{1},\delta p_{2}\dots\delta p_{n}\right]^{T}\). Summing Equation (10), the equivalent force on the main engine after \(k\) times air pressure adjustment can be obtained as  
\[\mathbf{F_{\delta p(k)}}=\sum_{i=1}^{k}\mathbf{F_{\Delta\delta p(i)}}=\mathbf{ N}\delta\mathbf{p}(k) \tag{12}\]  
where each column \(\mathbf{N}\) represents the equivalent force of unit pressure variation of each air spring, which can be expressed as  
\[\mathbf{N}=\left[ \begin{matrix}\mathbf{G_{1}^{T}}\left[\begin{matrix}-\text{ sgn}\big{(}a_{x}{}^{1}\big{)}S_{e}\sin\theta\\ 0\\ S_{e}\cos\theta\\ 0\\ 0\end{matrix}\right],\dots,\mathbf{G_{n}^{T}}\left[\begin{matrix}-\text{sgn}(a _{x}{}^{n})S_{e}\sin\theta\\ 0\\ S_{e}\cos\theta\\ 0\\ 0\end{matrix}\right]\\ 0\end{matrix} \right] \tag{13}\]  
Summing Equation (11), the variation of the alignment state after \(k\) times air pressure adjustment can be obtained as  
\[\sum_{j=0}^{k}\mathbf{\Delta x_{c}}(j\mathbf{+}1)=\sum_{j=0}^{k}\mathbf{G_{c} K^{-1}F_{\Delta\delta p(j)}}=\mathbf{G_{c}K^{-1}}\sum_{j=0}^{k}\mathbf{F_{\Delta \delta p(j)}} \tag{14}\]  
After \(k\) times of air pressure adjustment, the alignment state of the main engine is  
\[\mathbf{x_{c}}(k+1)\mathbf{=}\mathbf{x_{c0}}+\sum_{j=0}^{k}\mathbf{\Delta x_ {c}}(j+1) \tag{15}\]  
where \(\mathbf{x_{c0}}\) is the initial alignment state. Substituting Equations (12) and (14) into Equation (15), the alignment state can be further expressed as  
\[\mathbf{x_{c}}(k+1)\mathbf{=}\mathbf{x_{c0}}+\mathbf{G_{c}K^{-1}N\delta p(k)} \tag{16}\]Generally, axial deformation \(y_{c}\) is not used as the alignment monitoring component and control objective [37]. From Equation (8), it can be analyzed that the larger the \(y\)-direction coordinate of the air spring, the larger the vertical deformation it is likely to produce. At the same time, the vertical deformation of the air spring is often limited by the vertical displacement at the four corners of the HPDEME-ASVIS, which can be monitored by the displacement sensor. Therefore, the alignment control vector and vertical displacement vector are  
\[\mathbf{y}=[x_{c},z_{c},\alpha,\beta,\chi]^{T}= \begin{bmatrix}1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{bmatrix}\begin{bmatrix}x_{c}\\ y_{c}\\ z_{c}\\ \alpha\\ \beta\\ \chi\end{bmatrix} =\mathbf{C}\mathbf{x_{c}} \tag{17}\]  
\[\mathbf{z}= \begin{bmatrix}z_{1}\\ z_{2}\\ z_{3}\\ z_{4}\end{bmatrix} = \begin{bmatrix}0&0&1&a_{y}^{1}-a_{y}^{c}&a_{x}^{c}-a_{x}^{ 1}&0\\ 0&0&1&a_{y}^{2}-a_{y}^{c}&a_{x}^{c}-a_{x}^{2}&0\\ 0&0&1&a_{y}^{3}-a_{y}^{c}&a_{x}^{c}-a_{x}^{3}&0\\ 0&0&1&a_{y}^{4}-a_{y}^{c}&a_{x}^{c}-a_{x}^{4}&0\\ 0&0&1&a_{y}^{4}-a_{y}^{c}&a_{x}^{c}-a_{x}^{4}&0\end{bmatrix}\begin{bmatrix}x_{c }\\ y_{c}\\ z_{c}\\ \alpha\\ \beta\\ \chi\end{bmatrix}