Mathematical Problems in Engineering

Volume 2015, Article ID 462072, 8 pages

http://dx.doi.org/10.1155/2015/462072

## Optimal Lunar Landing Trajectory Design for Hybrid Engine

^{1}IT Convergence Technology Team, KARI, Daejeon 305-806, Republic of Korea^{2}Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA^{3}Department of Aerospace Engineering, Chosun University, Gwangju 501-759, Republic of Korea

Received 15 January 2015; Revised 26 May 2015; Accepted 27 May 2015

Academic Editor: Gen Qi Xu

Copyright © 2015 Dong-Hyun Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The lunar landing stage is usually divided into two parts: deorbit burn and powered descent phases. The optimal lunar landing problem is likely to be transformed to the trajectory design problem on the powered descent phase by using continuous thrusters. The optimal lunar landing trajectories in general have variety in shape, and the lunar lander frequently increases its altitude at the initial time to obtain enough time to reduce the horizontal velocity. Due to the increment in the altitude, the lunar lander requires more fuel for lunar landing missions. In this work, a hybrid engine for the lunar landing mission is introduced, and an optimal lunar landing strategy for the hybrid engine is suggested. For this approach, it is assumed that the lunar lander retrofired the impulsive thruster to reduce the horizontal velocity rapidly at the initiated time on the powered descent phase. Then, the lunar lander reduced the total velocity and altitude for the lunar landing by using the continuous thruster. In contradistinction to other formal optimal lunar landing problems, the initial horizontal velocity and mass are not fixed at the start time. The initial free optimal control theory is applied, and the optimal initial value and lunar landing trajectory are obtained by simulation studies.

#### 1. Introduction

The lunar landing trajectory is a good example for the optimal problem because the dynamics is very simple, and the various energy-like values can be selected as a candidate for the cost function of the optimization problems. For these reasons, many researches have been focusing on designing the optimal lunar landing trajectory. In general, the lunar landing is divided into two phases: deorbit burn and powered descent phases. In the first phase, the lunar lander falls down to reduce the altitude from the lunar parking orbit. At this phase, the target perilune altitude is usually chosen between 10 km and 15 km from the lunar surface to avoid mountainous lunar terrain and potential guidance errors. For the deorbit burn maneuver, the Hohmann transfer method is widely applied by using impulsive thrusters. At the periapsis of the transfer orbit, the second phase is initiated to land on the lunar surface. The initial states of the powered descent phase can be in general calculated by the Hohmann transfer, and the continuous thruster is applied to reduce the ground speed on the second phase.

Most researches for the optimal lunar landing trajectory problem are focused on the powered descent phase by using a continuous thruster. To find an optimal solution, two-dimensional (2D) dynamics has been studied by many researchers because of their simplicity. Park et al. designed the optimal trajectory for the specific landing site by issuing a pseudospectral method in 2D space, and the solution is generally used for the optimal lunar landing researches [1]. Before this research, Ramanan and Lal suggested the same optimal lunar landing strategies and analyzed the strategies on a case-by-case basis for the various perilune altitudes [2]. In general, the lower perilune altitude saves fuel for the landing in the powered descent phase, but fuel for the deorbit burn is increased. Therefore, Ramanan and Lal also tried to find a suitable perilune altitude for whole lunar landing phases. Cho et al. suggested a similar approach for the initial free optimal problem, and the optimal perilune altitude is also designed with the optimal powered descent trajectory [3]. In addition to the research, there are many other approaches to find the optimal lunar landing trajectory. Shan and Duan also studied 2D lunar landing problems with variable thrust level [4], and Hawkins solved a vertical lunar landing problem by using spacecraft rotational motions [5]. For the vertical lunar landing, Cho et al. also suggested an approach by using the path constraint [6]. Peng et al. applied the hybrid optimization algorithm for the optimal lunar landing trajectory design problem [7, 8]. In the guidance problem, Liu et al. studied the landing guidance control by using the optimal lunar landing trajectory [9, 10]. McInnes suggested another guidance strategy for a vertical landing by using the gravity-turn technique [11]. Zhou et al. studied a vertical lunar landing problem by using the attitude state variable [12].

In these optimal lunar landing trajectories, the lunar lander frequently increased its altitude to earn enough time to reduce the horizontal velocity. In this paper, an optimal solution is obtained for the hybrid engine. The hybrid engine means that the lunar lander has an impulsive and continuous thruster, and some researches have been performed for the orbit transfer by using this hybrid engine. The Hohmann Spiral Transfer (HST) is one of the examples [13, 14]. The authors assume that the lunar lander retrofired the impulsive thruster to reduce the horizontal velocity rapidly at the initiated time for the powered descent phase. Then, the lunar lander reduced the total velocity and altitude for the lunar landing by using the continuous thruster. By using this approach, the total mass consumption can be reduced. The authors tried to find the optimal retrofired velocity and trajectory to maximize the final landing mass. To solve this problem, the free initial problem for the impulsive thruster using the typical approach is considered.

#### 2. Problem Definition

##### 2.1. Assumptions

To derive the formula and perform the numerical simulation, the following assumptions are utilized.(i)The gravity field of the moon is uniform through the whole path of the lunar lander, and the moon is entirely spherical. Therefore, one can easily apply the two-body dynamics to this problem.(ii)The moon rotates on its own axis with the constant angular velocity. Moreover, the lunar parking orbit (circular orbit), the lunar landing trajectory, and the lunar equator are placed in the same plane, and each rotation direction is the same as well. Therefore, one can easily apply the 2D dynamics.(iii)The initial trajectory of the lunar parking orbit is the circular orbit, and Hohmann transfer method is used for the deorbit burn phase. Therefore, the initial velocity of the powered descent phase can be calculated.(iv)During the powered descent phase, the thrust of the lander is constant.(v)There are no external perturbations.

##### 2.2. Governing Equations

Using the above-mentioned assumptions, two-body and 2D dynamics are used. For these equations of motion, the polar coordinated system is selected as shown in Figure 1, and the governing equations for the lunar lander are described as follows [3]:where and are the radial distance and the position angle from the origin of the moon, and are the transverse and the radial velocity, is the mass property of the lander, and is the standard gravitational parameter. The position angle is defined from the initial position of a deorbit burn phase. To describe the mass flow rate (), the specific impulse () and the gravitational acceleration on the Earth () are adopted in (5). To express the thruster, and are used as a thrust magnitude and a thrust vector angle. The thrust vector angle is chosen to be a unique control input parameter during the powered descent phase.