## 1 Introduction
However, the CVX-based algorithms have the disadvantage of lacking real-time performance. ML-based algorithms have the shortcoming of insufficient landing accuracy and reliability. It is worth noting that the CVX algorithm is divided into the fixed final-time algorithm and the free final-time algorithm. Since the selection of terminal time significantly impacts the feasibility and optimality of the solution, it is more practical to study the free final-time algorithm. There are two methods in the literature. One is to fix the final time first, adopting LCvx to solve the SOCP, and then regard the final time as a parameter to conduct the one-dimensional golden search [12; 23]. Another method treats the final time as a variable and deals with the dynamic constraints by SCvx [24; 25]. The relationship between algorithms is shown in Figure 1. The common disadvantage of the two methods is that the SOCP needs to be solved many times, which dramatically reduces the computational efficiency. Many scholars have optimized the above algorithms. The authors of [26] used quadratic function to approximate the relationship between landing time and landing mass, and reduced the calculation times of the SOCP problem to 4. In reference [27], the grid search method was adopted to calculate the optimal landing time, which reduced the calculation times of the SOCP problem to 6 and was successfully applied in a flight test. The authors of [28] used linear interpolation to directly calculate the optimal landing time through the current state of the rocket, but this method needs to store a large amount of data onboard. In this paper, the CVX and ML are combined by establishing the DNN predictor to assist convex optimization, thus significantly improving the efficiency of computation (see  
Figure 1: Classification of the CVX-based algorithms. the algorithm highlighted in red in Figure 1). The algorithm proposed in this work avoids iteration by introducing the optimal terminal time predictor. Given that the accuracy of the traditional analytical estimation formulas [29,30] is insufficient, inspired by the [19], the DNN is adopted to learn the relationship between state variables and optimal terminal time. The optimal terminal time can be mapped directly from the state variables when running onboard. Therefore, the solution can be obtained by solving the SOCP only once. The structure of the paper is as follows: Section 1 briefly introduces the background of the powered landing and summarizes the development of guidance algorithms based on CVX and ML. The shortcomings of the current algorithms are illustrated. Section 2 gives the mathematical description of the powered landing. In Section 3, variable substitution and lossless relaxation techniques are used to convexify the primal problem. The effect of terminal time on the feasibility and optimality of the solution is discussed. In Section 4, the DNN optimal predictor is built to predict the optimal terminal time, and the corresponding sample generation algorithm is proposed. Section 5 integrates the algorithms of Sections 3 and 4 and analyzes the optimality and real-time performance by the open-loop simulation. Section 6 proposes a closed-loop guidance algorithm framework based on MPC and analyzes the algorithm's robustness and landing accuracy. Section 7 summarizes the work.