# Semi-Analytical Planetary Landing Guidance with Constraint Equations Using Model Predictive Control
## 3 Motion Mode Analysis and Reachability Determination
### Reachability Analysis and Motion Mode Judgment
If \(\mu_{x,\min}\leq 1\), the second step of reachability analysis is carried out, which is to obtain the minimum \(\mu_{y}\) required to complete adjustments in the Y direction. For convenience, we introduce a new variable \(t_{1}^{*}\) equal to 0 or \(t_{\rm f,max}-m_{\rm fuel}/\eta\) when the fuel is not enough. By setting \(t_{\rm f}\) to \(t_{\rm f,max}\) and \(t_{1}\) to \(t_{1}^{*}\), the number of variables in the Y-direction constraint equations reduces to 2. In this situation, the time interval for the Y direction adjustments is the longest, and the start-time for the adjustments is the earliest. As a result, the value of \(\mu_{y}\) obtained by solving Equations (30) and (31) under this setting is the smallest, denoted as \(\mu_{y,\min}\). If \(\sqrt{\mu_{x,\min}^{2}+\mu_{y,\min}^{2}}>1\), the lander cannot complete the adjustments in the Y direction, causing the target landing point to be unable to be safely reached. If \(\sqrt{\mu_{x,\min}^{2}+\mu_{y,\min}^{2}}\leq 1\), the final step of reachability analysis will be performed. In this step, the maximum available \(\mu_{x}\), denoted as \(\mu_{x,\max}\), is first calculated, equal to \(\sqrt{1-\mu_{x,\min}^{2}-\mu_{y,\min}^{2}}\). Then, whether the \(v_{x}\) can decelerate to 0 within \(t_{\rm f,max}\) is verified. We introduce a new variable \(t_{x,\rm d}\) to represent the time when the velocity in the X direction decelerates to 0, as shown in Figure 5. The value of \(t_{x,\rm d}\) can be obtained by solving  
Figure 4: Motion schematic diagram in the Z direction. Equation (39). If \(t_{1}^{*}+t_{\text{x,d}}>t_{\text{t,max}}\), it indicates that the velocity in the X direction cannot decelerate to 0 within \(t_{\text{f,max}}\), even with the maximum available \(\mu_{\text{x}}\). \[|v_{\text{x0}}|-\frac{\mu_{\text{x,max}}T_{\text{max}}}{\eta}\biggl{[}\ln\biggl{(} \frac{m_{0}}{\eta}\biggr{)}-\ln\biggl{(}\frac{m_{0}}{\eta}-(t_{\text{x,d}}-t_{ 1})\biggr{)}\biggr{]}=0 \tag{39}\]  
If \(t_{1}^{*}+t_{\text{x,d}}\leq t_{\text{f,max}}\), the reachable boundaries in the X direction, \(x_{1}\) and \(x_{\text{r}}\), will be calculated under the condition of \(v_{\text{x,f}}=0\). As shown in Figure 6, the red curve \(C_{\text{I}}\) and purple curve \(C_{\text{r}}\) represent two critical cases. The area surrounded by \(C_{\text{I}}\) and the X-axis plus \(x_{0}+v_{\text{x0}}t_{1}^{*}\) equals \(x_{1}\). Similarly, the area surrounded by \(C_{\text{r}}\) and the X-axis plus \(x_{0}+v_{\text{x0}}t_{1}^{*}\) equals \(x_{\text{r}}\). If 0 is between \(x_{1}\) and \(x_{\text{r}}\), the target landing point can be safely reached; if not, it cannot be. In summary, the overall flow chart of reachability analysis is shown in \[\frac{\partial G}{\partial t_{1}} =-\frac{\mu_{x}T_{\max}}{\eta}\left[\ln\!\left(\frac{m_{0}}{\eta} \right)-\ln\!\left(\frac{m_{0}}{\eta}-(t_{\rm f}-t_{1})\right)\right]<0\]0  
\[\frac{\partial G}{\partial t_{1}} =-\frac{\mu_{x}T_{\max}}{\eta}\left[\ln\!\left(\frac{m_{0}}{\eta} \right)-\ln\!\left(\frac{m_{0}}{\eta}-(t_{\rm f}-t_{1})\right)\right]<0\]1 Schematic diagram of \(t_{\text{x,d}}\). \[\frac{\partial G}{\partial t_{1}} =-\frac{\mu_{x}T_{\max}}{\eta}\left[\ln\!\left(\frac{m_{0}}{\eta} \right)-\ln\!\left(\frac{m_{0}}{\eta}-(t_{\rm f}-t_{1})\right)\right]<0\]2 Two critical cases in the X direction. \[\frac{\partial G}{\partial t_{1}} =-\frac{\mu_{x}T_{\max}}{\eta}\left[\ln\!\left(\frac{m_{0}}{\eta} \right)-\ln\!\left(\frac{m_{0}}{\eta}-(t_{\rm f}-t_{1})\right)\right]<0\]3 Flow chart of reachability analysis. When the target landing point can be safely reached, a motion mode judgment is needed to determine the values of \(\gamma_{\mathrm{x}}\) and \(\gamma_{\mathrm{y}}\) in the constraint equations. The motion modes in the Y and Z direction are easy to judge, as shown in Section 3.1. When \(v_{\mathrm{x0}}>0\), the motion mode in the X direction is also easy to judge. The difficulty is to judge the motion mode in the X direction when \(v_{\mathrm{x0}}<0\).