## 4 Reconfigurability Judgment Analysis and Autonomous Control Method
### Alignment Controllability Judgment Model
#### 4.2.1 Alignment Controllability Judgment Model under Hard Constraints  
For hard constraints, all constraint boundaries are not allowed to adjust. It means that \(\varepsilon_{p_{min}}\), \(\varepsilon_{p_{max}}\), \(\varepsilon_{x_{c}}\), \(\varepsilon_{\beta}\) and \(\varepsilon_{z}\) are all zero in Equation (24), which is equivalent to \(\varepsilon_{pmin}^{\max}\), \(\varepsilon_{pmax}^{\max}\), \(\varepsilon_{x_{c}}^{\max}\), \(\varepsilon_{\beta}^{\max}\) and \(\varepsilon_{z}^{\max}\) being zero. Therefore, the controllable under hard constraints can be judged by solving a linear programming problem (LP)  
\[\begin{split}& V=\min\left(c^{T}\left[\varepsilon_{p_{min}}, \varepsilon_{p_{max}},\varepsilon_{x_{c}},\varepsilon_{\beta},\varepsilon_{z} \right]^{T},\,c^{T}=\left[1,1,1,1,1\right]\right)\\ & s.t.\Bigg{\{}\begin{array}{l}\mathbf{P_{min}}-\left[ \varepsilon_{p_{min}},\varepsilon_{p_{min}},\varepsilon_{p_{min}},\ldots, \varepsilon_{p_{min  
Similar to Equation (25), the alignment controllability under soft constraints can be judged by solving a linear programming problem (LP). \[ \begin{split} V^{\prime}=\min\left(c^{T}[\varepsilon_{p_{\min}}, \varepsilon_{p_{\max}},\varepsilon_{x_{\
u}},\varepsilon_{\beta_{\varepsilon}} ]^{T},\ c^{T}=[1,1,1,1,1]\right)\\ s.t.\left\{\begin{array}{l}\mathbf{P_{\min}}-\left[\varepsilon_{p_{ \min}},\varepsilon_{p_{\min}},\ldots,\varepsilon_{p_{\min}}\right]^{T}-\mathbf{ P_{0}}\leq\delta\mathbf{p}\leq\mathbf{P_{\max}}+\left[\varepsilon_{p_{\max}}, \varepsilon_{p_{\max}},\ldots,\varepsilon_{p_{\max}}\right]^{T}-\mathbf{P_{0} }\\ \mathbf{y_{\min}}-\left[\varepsilon_{x_{\
u}},0,0,\varepsilon_{\beta_{\varepsilon }}\right]^{T}\leq\mathbf{y}\mathbf{c}\mathbf{c}_{\mathbf{x}}+\mathbf{C} \mathbf{G}\mathbf{c}^{\text{-}}\mathbf{N}\delta\mathbf{p}(\mathbf{x})\leq \mathbf{y_{\max}}+\left[\varepsilon_{x_{\
u}},0,0,\varepsilon_{\beta_{\varepsilon }}\right]^{T}\\ \mathbf{z_{\min}}-\left[\varepsilon_{x_{\
u}},\varepsilon_{x_{\
u}}, \varepsilon_{x_{\
u}}\right]^{T}\leq\mathbf{z}\mathbf{D_{\mathbf{x}}}+\mathbf{ D_{\mathbf{c}}\mathbf{c}^{\text{-}}\mathbf{N}\delta\mathbf{p}(\mathbf{x})\leq \mathbf{z_{\min}}+\left[\varepsilon_{x_{\
u}},\varepsilon_{x_{\
u}}, \varepsilon_{x_{\
u}}\right]^{T}\\ \mathbf{0}\leq\left[\varepsilon_{p_{\min}},\varepsilon_{p_{\max}}, \varepsilon_{x_{\
u}},\varepsilon_{\beta_{\varepsilon}}\right]^{T}\end{array}\right. \end{split}  \tag{27}\]  
Combining Equations (26) and (27), the judgment condition of alignment controllability at the boundary of each soft constraint is still \(\left[\varepsilon_{p_{\min}},\varepsilon_{p_{\max}},\varepsilon_{x_{\
u}}, \varepsilon_{\beta_{\varepsilon}},\varepsilon_{z}\right]=\mathbf{0}\), corresponding \(V^{\prime}=0\); otherwise \(V^{\prime}>0\), the alignment at the boundary of each soft constraint is uncontrollable. Compared with Equation (25), some constraints of Equation (27) have wider boundaries. In practice, alignment control precision will decrease with the expansion of some soft constraints boundaries. When the system has alignment controllable under soft constraints, the smallest adjustment of each soft constraint boundary can be calculated as  
\[\begin{split}\left[\varepsilon_{p_{\min}}^{*}, \varepsilon_{p_{\max}}^{*},\varepsilon_{x_{\
u}},\varepsilon_{y_{\
u}}, \varepsilon_{z}^{*}\right]=\underset{\varepsilon_{p_{\min}},\varepsilon_{p_{ \max}},\varepsilon_{x_{\
u}},\varepsilon_{y_{\
u}},\varepsilon_{z}}{\arg_{ \varepsilon_{\min}}}\left(\min,h^{T},\left[\varepsilon_{p_{\min}},  
When the alignment system under hard and soft constraints is judged as uncontrollable, the state is classified as an **II** alignment control fault state (**II** state). At this time, the minimum safety goal cannot be achieved by changing the control algorithm. The emergency protection device should be enabled. The emergency protection device is rigidly connected with the HPDME-ASVIS through the hydraulic cylinder, whose mechanism has been described in the literature [2].