# Semi-Analytical Planetary Landing Guidance with Constraint Equations Using Model Predictive Control
## 1 Introduction  
With the success of a series of planetary exploration missions, researchers' understanding of the planetary characteristics and planetary landing is more and more in depth [1; 2], and more appropriate potential landing target points will be found [3]. Due to the uncertain planetary environmental factors and possible faults of the lander [4; 5], the actual trajectory may deviate from the preset trajectory or even make the origin landing point unreachable. At this time, how to rapidly determine the reachable landing points from potential landing target points and plan the trajectory online brings new challenges for powered descent guidance algorithms. Therefore, rapid decision making and real-time trajectory planning methods will be the key to mission success. Powered decent guidance is essentially a trajectory optimization problem, which can be traditionally solved by indirect methods or direct methods. Based on the optimal control theory, the indirect method converts the optimal control problem into a two-point boundary value problem by introducing the costates [6; 7]. However, it is difficult to guarantee the convergence due to the difficulty in estimating the initial values of the costates [8]. In contrast, the direct method is more widely used in powered descent guidance [9], because it can handle complex dynamic models with multiple constraints and has a slightly wider convergence domain. [PERSON] et al. proposed a lossless convex programming approach and successfully applied it to powered descent guidance for Mars landing [10; 11; 12; 13]. However, for the problem of free final time, the lossless convex programming approach cannot achieve real-time planning due to the inaccuracy of final time prediction [14]. Subsequent studiesinclude sequential convex optimization [15; 16; 17; 18] and successive convex optimization [19; 20; 21; 22]. These convex optimization methods can deal with more complex constraints and dynamics (e.g., the problem with free final time and flight angle constraints) through linearization and the introduction of trust region, virtual control, and virtual buffer. The cost is the occupation of more computational resources and the decrease in computational efficiency caused by iteration. The pseudospectral convex optimization method [23; 24; 25] improves the accuracy by adopting the pseudospectral discretization, but the efficiency is not greatly improved. The research on real-time powered descent guidance control actually began in the 1960s [26], and the very famous application is the successful lunar landing of Apollo 11. Due to the limitation of the computing ability, Apollo 11 adopted the polynomial guidance method which derived an analytical expression for the acceleration command with the remaining flight time [27]. The subsequent studies further investigated the constrained optimal thrust acceleration vector profile by adopting different decoupling methods and derived elegant analytical expressions [28; 29; 30; 31; 32; 33]. The commonness of these analytical methods is to analyze the thrust acceleration rather than the thrust. As a result, these methods cannot rapidly and accurately determine the reachability of the landing point in the current state. With the development of artificial intelligence, some researchers apply learning-based methods to powered descent guidance [34; 35]. These methods can achieve high accuracy and efficiency in theory with a large and accurate training set, but the performance of these methods is not stable enough in the face of unexpected situations [14]. Model predictive control (MPC) is a popular advanced method to control dynamical systems subject to uncertain disturbances and possible faults [36; 37; 38]. In Ref. [36], a robust nonlinear multi-variable predictive control under constraints was proposed and solved using a linear matrix inequalities (LMIs) formation. [PERSON] et al. proposed a robust MPC algorithm for active suspension systems with time-varying delays and input constraints [37]. In Ref.