## 2 Problem Statement  
Firstly, the reference coordinate system \(O\)-_xyz_ shown in Figure 2 is established. The origin of the coordinate system is fixed on the landing platform, the \(x\) axis is aligned to the reverse direction of the local gravity acceleration, and the \(y\) axis and \(z\) axis are aligned to the east and north direction of the local ground plane, respectively. Considering the flight time of the powered landing is short, the inertial centrifugal force and Coriolis force caused by the rotation of the reference coordinate system are negligible. It is worth noting that the acceleration provided by the thruster is the main driving force that affects the motion of the rocket, while the aerodynamic force is small relative to the thrust. Therefore, the dynamic model only considers the effect of thrust. In order to evaluate the robustness of the proposed guidance algorithm, an aerodynamic model is added to the closed-loop simulation. The dynamic equation of the 3-DOF motion of the vehicle can be expressed as follows:  
\[ \begin{array}{l}\dot{r}=v\\ \dot{v}=g+\frac{T}{m}\\ \dot{m}=-\frac{\left|\dot{T}\right|}{l_{\theta}\delta_{0}}\end{array}  \tag{1}\]  
Figure 2: Definition of the reference coordinate system for powered landing problem. where \(r,v\) and \(m\) are the position vector, velocity vector and mass of the vehicle, respectively. \(I_{\rm sp}\) is the specific impulse of the engine, and \(g_{0}\) is the earth's gravitational constant and \(g_{0}\)= 9.8065 m/s\({}^{2}\). \(g\) is the gravitational acceleration vector. The initial altitude of the powered landing phase is usually about 5 km, and the variation of the gravitational acceleration is negligible. Therefore, \(g\) can be regarded as a constant vector:  
\[g=(-g_{0},0,0)^{T} \tag{2}\]  
\(T\) is the thrust vector and \(\|T\|\) is the thrust magnitude. Due to the performance limitation of the rocket engine, \(\|T\|\) needs to satisfy the following constraints:  
\[T_{\rm min}\leq\|T\|\leq T_{\rm max} \tag{3}\]  
where \(T_{\rm min}\) and \(T_{\rm max}\) are the lower and upper bounds of the thrust magnitude, respectively. In order to ensure that the vehicle can avoid obstacles and terrain near the landing site, it is also necessary to add the glideslope constraints, as shown in Figure 2:  
\[\sqrt{r_{y}^{2}(t)+r_{z}^{2}(t)}-r_{x}(t)\tan\gamma_{\rm max}\leq 0 \tag{4}\]  
where \(\gamma_{\rm max}\) is the maximum glideslope angle and \(\gamma_{\rm max}\ \in(0,\pi/2)\). \(r_{x}\), \(r_{y}\) and \(r_{z}\) are the three components of the position vector \(r\), respectively. Given the constraint, the range of the horizontal motion decreases with altitude, and lateral maneuvers become small as the rocket approaches the landing site. Assuming that the thrust direction is parallel to the longitudinal axis of the rocket, the attitude of the rocket can be limited by constraining the direction of the thrust. The following thrust pointing constraints need to be added, as shown in Figure 2:  
\[\sqrt{T_{y}^{2}(t)+T_{z}^{2}(t)}-T_{x}(t)\tan\theta_{\rm max}\ \leq 0 \tag{5}\]  
where \(\theta_{\rm max}\) is the maximum thrust pointing angle and \(\theta_{\rm max}\ \in(0,\pi/2)\). \(T_{x}\), \(T_{y}\) and \(T_{z}\) are the three components of the thrust vector, respectively. Given the constraint, the horizontal components of the thrust direction are constrained to a small value, thereby equivalently constraining the attitude of the vehicle. The initial and terminal constraints are listed as follows:  
\[\left\{ \begin{array}{l}r(t_{0})=r_{0}\\ v(t_{0})=v_{0}\\ m(t_{0})=m_{0}\\ r(t_{t})=0\\ v(t_{t})=0\end{array} \right. \tag{6}\]  
where \(t_{0}\) and \(t_{t}\) are the initial time and the terminal time of the powered landing, respectively. \(r_{0}\), \(v_{0}\) and \(m_{0}\) are the position, velocity and mass at the initial time, respectively, which can be obtained by the navigation system. Since the landing platform is located at the origin of the coordinate system, the position and velocity vector at the terminal time are both zero vectors.