# Semi-Analytical Planetary Landing Guidance with Constraint Equations Using Model Predictive Control
## 3 Motion Mode Analysis and Reachability Determination  
### Motion Mode Analysis  
We first analyze the motion mode in the Y direction. As mentioned in Section 2.2, \(\mu_{y}T_{\rm max}\) is used to adjust the position and velocity in the Y direction, and the thrust direction switches at most once. Because the target displacement in the Y direction is 0, the motion modes in the Y direction are divided into only two kinds: (a) when \(v_{y0}>0\), the thrust in the Y direction should be negative first and then positive; (b) when \(v_{y0}<0\), the thrust in the Y direction should be positive first and then negative. By denoting \(t_{y}\) as the moment when the thrust direction changes, the two kinds of motion modes in the Y direction are schematically shown in Figure 2.  
Then, the motion mode in the X direction is analyzed. Based on the thrust direction change and the initial velocity \(v_{x0}\), the motion modes in the X direction are divided into three categories: (a) when \(v_{x0}<0\) and the absolute value of \(v_{x0}\) is relatively large, to meet terminal constraints, the thrust in the X direction should be positive first and then negative; (b) when \(v_{x0}<0\) and the absolute value of \(v_{x0}\) is relatively small, the thrust should be negative first and then positive; c) when \(v_{x0}\geq 0\), the thrust should be negative first and then positive. By denoting \(t_{x}\) as the moment when the thrust direction changes, the three motion modes in the X direction are schematically shown in Figure 3. Whether the value of \(v_{x0}\) is large or small will be discussed in detail in Section 3.2. We introduce two variables, \(\gamma_{y}\) and \(\gamma_{x}\), to describe the motion modes in the Y and X directions, respectively. If the thrust in the Y direction is positive first and then negative, \(\gamma_{y}=1\); if the thrust is negative first and then positive, \(\gamma_{y}=-1\); if \(v_{y0}=0\), \(\gamma_{y}=0\). The value of \(\gamma_{x}\) is assigned in the same way. Taking the derivation in  
Figure 3: Three motion modes in the X direction. Figure 2: Two motion modes in the Y direction. example, we will show the derivation process of explicit constraint equations in detail. The dynamic equation in the X direction is as follows:  
\[\dot{v}_{\mathrm{x}}(t)= \begin{cases}0,t\in[0,t_{1}]\\ \gamma_{\mathrm{x}}\dfrac{\mu_{\mathrm{x}}T_{\mathrm{max}}}{m_{0}-\eta(t-t_{1} )},t\in(t_{1},t_{\mathrm{x}}]\\ -\gamma_{\mathrm{x}}\dfrac{\mu_{\mathrm{x}}T_{\mathrm{max}}}{m_{0}-\eta(t-t_{1 })},t\in(t_{\mathrm{x}},t_{\mathrm{f}}]\end{cases}  \tag{24}\]  
By integration, the explicit expression of \(v_{\mathrm{x}}(t)\) is obtained as follows:  
\[v_{\mathrm{x}}(t)= \begin{cases}v_{\mathrm{x}0,t}\in[0,t_{1}]\\ \gamma_{\mathrm{x}}\dfrac{\mu_{\mathrm{x}}T_{\mathrm{max}}}{\eta}\left[\ln \biggl{(}\dfrac{m_{0}}{\eta}\biggr{)}-\ln\biggl{(}\dfrac{m_{0}}{\eta}-(t-t_{1 })\biggr{)}\right]+v_{\mathrm{x}0},t\in(t_{1},t_{\mathrm{x}}]\\ \gamma_{\mathrm{x}}\dfrac{\mu_{\mathrm{x}}T_{\mathrm{max}}}{\eta}\left[\ln \biggl{(}\dfrac{m_{0}}{\eta}-(t-t_{1})\biggr{)}-\ln\biggl{(}\dfrac{m_{0}}{\eta }-(t_{\mathrm{x}}-t_{1})\biggr{)}\right]+v_{\mathrm{x}}(t_{\mathrm{x}}),t\in( t_{\mathrm{x}},t_{\mathrm{f}}]\end{cases}