research article optimal lunar landing trajectory design...

Research ArticleOptimal Lunar Landing Trajectory Design for Hybrid Engine

Dong-Hyun Cho,1 Donghoon Kim,2 and Henzeh Leeghim3

1 IT Convergence Technology Team, KARI, Daejeon 305-806, Republic of Korea2Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA3Department of Aerospace Engineering, Chosun University, Gwangju 501-759, Republic of Korea

Correspondence should be addressed to Henzeh Leeghim; [email protected]

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 AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The lunar landing stage is usually divided into two parts: deorbit burn and powered descent phases. The optimal lunar landingproblem 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 theinitial time to obtain enough time to reduce the horizontal velocity. Due to the increment in the altitude, the lunar lander requiresmore fuel for lunar landingmissions. In this work, a hybrid engine for the lunar landingmission is introduced, and an optimal lunarlanding strategy for the hybrid engine is suggested. For this approach, it is assumed that the lunar lander retrofired the impulsivethruster to reduce the horizontal velocity rapidly at the initiated time on the powered descent phase.Then, the lunar lander reducedthe total velocity and altitude for the lunar landing by using the continuous thruster. In contradistinction to other formal optimallunar landing problems, the initial horizontal velocity andmass are not fixed at the start time.The initial free optimal control theoryis 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 optimalproblem because the dynamics is very simple, and the variousenergy-like values can be selected as a candidate for the costfunction of the optimization problems. For these reasons,many researches have been focusing on designing the optimallunar landing trajectory. In general, the lunar landing isdivided into two phases: deorbit burn and powered descentphases. In the first phase, the lunar lander falls down to reducethe altitude from the lunar parking orbit. At this phase, thetarget perilune altitude is usually chosen between 10 km and15 km from the lunar surface to avoid mountainous lunarterrain and potential guidance errors. For the deorbit burnmaneuver, the Hohmann transfer method is widely appliedby using impulsive thrusters. At the periapsis of the transferorbit, the second phase is initiated to land on the lunarsurface. The initial states of the powered descent phase canbe in general calculated by the Hohmann transfer, and thecontinuous thruster is applied to reduce the ground speed onthe second phase.

Most researches for the optimal lunar landing trajectoryproblem are focused on the powered descent phase byusing a continuous thruster. To find an optimal solution,two-dimensional (2D) dynamics has been studied by manyresearchers because of their simplicity. Park et al. designedthe optimal trajectory for the specific landing site by issuinga pseudospectral method in 2D space, and the solution isgenerally used for the optimal lunar landing researches [1].Before this research, Ramanan and Lal suggested the sameoptimal lunar landing strategies and analyzed the strategieson a case-by-case basis for the various perilune altitudes[2]. In general, the lower perilune altitude saves fuel forthe landing in the powered descent phase, but fuel for thedeorbit burn is increased. Therefore, Ramanan and Lal alsotried to find a suitable perilune altitude for whole lunarlanding phases. Cho et al. suggested a similar approach forthe initial free optimal problem, and the optimal perilunealtitude is also designed with the optimal powered descenttrajectory [3]. In addition to the research, there are manyother approaches to find the optimal lunar landing trajectory.Shan and Duan also studied 2D lunar landing problems with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 462072, 8 pageshttp://dx.doi.org/10.1155/2015/462072

2 Mathematical Problems in Engineering

variable thrust level [4], and Hawkins solved a vertical lunarlanding problem by using spacecraft rotational motions [5].For the vertical lunar landing, Cho et al. also suggested anapproach by using the path constraint [6]. Peng et al. appliedthe hybrid optimization algorithm for the optimal lunarlanding trajectory design problem [7, 8]. In the guidanceproblem, Liu et al. studied the landing guidance control byusing the optimal lunar landing trajectory [9, 10]. McInnessuggested another guidance strategy for a vertical landingby using the gravity-turn technique [11]. Zhou et al. studieda vertical lunar landing problem by using the attitude statevariable [12].

In these optimal lunar landing trajectories, the lunarlander frequently increased its altitude to earn enough timeto reduce the horizontal velocity. In this paper, an optimalsolution is obtained for the hybrid engine. The hybrid enginemeans that the lunar lander has an impulsive and continuousthruster, and some researches have been performed for theorbit transfer by using this hybrid engine. The HohmannSpiral Transfer (HST) is one of the examples [13, 14]. Theauthors assume that the lunar lander retrofired the impulsivethruster to reduce the horizontal velocity rapidly at theinitiated time for the powered descent phase. Then, thelunar lander reduced the total velocity and altitude for thelunar landing by using the continuous thruster. By usingthis approach, the total mass consumption can be reduced.The authors tried to find the optimal retrofired velocity andtrajectory to maximize the final landing mass. To solve thisproblem, the free initial problem for the impulsive thrusterusing the typical approach is considered.

2. Problem Definition

2.1. Assumptions. To derive the formula and perform thenumerical simulation, the following assumptions are utilized.

(i) The gravity field of the moon is uniform throughthe whole path of the lunar lander, and the moon isentirely spherical. Therefore, one can easily apply thetwo-body dynamics to this problem.

(ii) The moon rotates on its own axis with the constantangular velocity. Moreover, the lunar parking orbit(circular orbit), the lunar landing trajectory, and thelunar equator are placed in the same plane, and eachrotation direction is the same as well. Therefore, onecan easily apply the 2D dynamics.

(iii) The initial trajectory of the lunar parking orbit isthe circular orbit, and Hohmann transfer methodis used for the deorbit burn phase. Therefore, theinitial velocity of the powered descent phase can becalculated.

(iv) During the powered descent phase, the thrust of thelander is constant.

(v) There are no external perturbations.

2.2. Governing Equations. Using the above-mentionedassumptions, two-body and 2D dynamics are used. For these

T�

𝛽

u

𝜙

r

𝜔

Lunar parking orbit

Deorbitphase

Figure 1: The polar coordinate system for the lunar lander.

equations of motion, the polar coordinated system is selectedas shown in Figure 1, and the governing equations for thelunar lander are described as follows [3]:

𝑟 = V, (1)

𝜙 =

𝑢

𝑟

, (2)

�� = −

𝑢V𝑟

+

𝑇

𝑚

cos𝛽, (3)

V =𝑢

2

𝑟

−

𝜇

𝑟

2 +𝑇

𝑚

sin𝛽, (4)

�� = −

𝑇

𝐼sp𝑔, (5)

where 𝑟 and 𝜙 are the radial distance and the position anglefrom the origin of the moon, 𝑢 and V are the transverse andthe radial velocity,𝑚 is the mass property of the lander, and 𝜇is the standard gravitational parameter. The position angle isdefined from the initial position of a deorbit burn phase. Todescribe the mass flow rate (��), the specific impulse (𝐼sp) andthe gravitational acceleration on the Earth (𝑔) are adoptedin (5). To express the thruster, 𝑇 and 𝛽 are used as a thrustmagnitude and a thrust vector angle. The thrust vector angleis chosen to be a unique control input parameter during thepowered descent phase.

2.3. Cost Function. It is intended to find the landing trajec-tory tominimize the fuel consumption by using the retrofiredvelocity at the initial time of the powered descent phase.Thus,the minimum fuel problem is equivalent to the maximumfinal landing mass problem. For this reason, the optimallanding trajectory to maximize the final landing mass is

Mathematical Problems in Engineering 3

designed, and the cost function for this problem is establishedas follows:

𝐽 = −𝑚𝑓, (6)

where𝑚𝑓 is the maximum final landing mass.

2.4. BoundaryConditions. Hohmann transfermethod is usedduring the deorbit burn phase, and thus the initial horizontalvelocity state of the powered descent phase is described asfollows [15]:

𝑉0 = √𝜇(2𝑟0−

2𝑟0 + 𝑟𝑝

),(7)

where 𝑟𝑝 is the radial distance of the circular lunar parkingorbit and the value is usually 100 km. Also, 𝑟0 is the initialradial distance at the powered descent phase. In general,this altitude is the same as the perilune altitude of deorbitburn phase, and it is chosen between 10 km and 15 km dueto mountainous lunar terrain and potential guidance errors.Here, the perilune altitude is assumed as 15 km from the lunarsurface.

The impulsive thruster retrofired at the initial time ofthe powered descent phase. Therefore, the initial horizontalvelocity (𝑢0) is not fixed and it is one of the optimalparameters. However, there is a relationship between theinitial horizontal velocity and the initial mass from the rocketequation as follows [15]:

𝑚0 = 𝑀 exp(𝑢0 − 𝑉0𝐼sp𝑔

) , (8)

where𝑀 represents the initial total mass of the lunar lander.In addition, the starting point of the deorbit burn phase

is used as a reference point of the position angle (𝜙), andthe perilune position is the opposite site from this referenceposition. Thus, the initial position angle is set as 180 degrees.Now, the initial states of the lunar lander at the initiated timeof the powered descent phase can be described as follows:

𝑟0 = 15+ 𝑟moon,

𝜙0 = 180∘,

𝑢0 = unkown,

V0 = 0,

𝑚0 = 𝑚 exp(𝑢0 − 𝑉0𝐼sp𝑔

) ,

(9)

where 𝑟moon is the radius of the moon.For the lunar landing mission, the final radial distance

must be the same radius of the moon, and horizontal andvertical velocities must be equal to the ground speed ofthe lunar surface. However, in this paper, the final positionangle is not fixed because of the following assumptions: themoon is entirely spherical and the gravity field is uniform.

Therefore, one can change the initial position of the deorbitburn phase to adjust the specific final position angle.Thus, thefinal position angle is not important. In addition, we want tofind the optimal trajectory to maximize the final lunar mass,which is not fixed. For these reasons, the final state constraintsare described as follows:

𝑟𝑓 = 𝑟moon,

𝜙𝑓 = f ree,

𝑢𝑓 = 𝑟moon𝜔,

V𝑓 = 0,

𝑚𝑓 = free,

(10)

where the subscript 𝑓 means the value at the final landingtime and 𝜔 represents the rotation velocity of the moon.

3. Optimal Trajectory Design

3.1. Optimal Theory. To solve the defined optimal problem,the indirect method based on the calculus variation isused because of simple system dynamics. The Hamiltonianfunction is established from the system dynamics and costfunction in the previous section as follows:

𝐻 = 𝜆𝑟 𝑟 + 𝜆𝜙

𝜙 + 𝜆𝑢�� + 𝜆VV+𝜆𝑚��, (11)

where 𝜆𝑟, 𝜆𝜙, 𝜆𝑢, 𝜆V, and 𝜆𝑚 represent the costate variablesfor each state variable and the subscripts indicate the statevariables. From the optimal control theory, the dynamics ofcostates can be obtained as follows [16]:

𝜆 = −𝐻

𝑇𝑥 .

(12)

Therefore, the similar equations of each costate to theother researches can be formulated as follows:

𝜆𝑟 = 𝜆𝜙

𝑢

𝑟

2 −𝜆𝑢𝑢V𝑟

2 +𝜆V (𝑢

2

𝑟

2 − 2𝜇

𝑟

3) ,

𝜆𝜙 = 0,

𝜆𝑢 = −

𝜆𝜙

𝑟

+ 𝜆𝑢

V𝑟

− 𝜆V2𝑢𝑟

,

𝜆V = −𝜆𝑟 +𝜆𝑢𝑢

𝑟

,

𝜆𝑚 =𝑇

𝑚

2 (𝜆𝑢 cos𝛽+𝜆V sin𝛽) .

(13)

Then, the control command can also be obtained from theoptimal control theory. The control variable must satisfy thefollowing two equations to minimize the cost function:

𝜕𝐻

𝜕𝛽

= −

𝑇

𝑚

(𝜆𝑢 sin𝛽−𝜆V cos𝛽) = 0,

𝜕

2𝐻

𝜕𝛽

2 = −

𝑇

𝑚

(𝜆𝑢 cos𝛽+𝜆V sin𝛽) > 0.(14)


The control command is generated as a function of thecostates from (14) as follows:

𝛽 = tan−1 (−𝜆V

−𝜆𝑢

) . (15)

Let us find the initial values of the costates satisfyingthe boundary conditions to design the optimal lunar landingtrajectory. In the previous section, the boundary conditionsof states were introduced. Also, the boundary conditions ofthe costates can be obtained from the optimal control theory.

Therefore, the augmented constraint function is writtenas follows:

𝐺 = −𝑚𝑓⏟⏟⏟⏟⏟⏟⏟

Perfomance

+ ]𝑇𝜓 (𝑡𝑓, 𝑥𝑓)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Final StateConstraints

+ 𝜉

𝑇𝜎 (𝑡0, 𝑥0)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Initial StateConstraints

,

(16)

where the final and initial state constraints matrices aredescribed as follows:

𝜓 = [𝑟𝑓 − 𝑟moon 𝑢𝑓 − 𝑟moon𝜔 V𝑓]

𝑇,

𝜎 =

[

[

[

[

[

[

[

[

[

[

[

𝑟0 − (15 + 𝑟moon)

𝜙0 − 180∘

V0

𝑚0 −𝑀 exp(𝑢0 − 𝑉0𝐼sp𝑔

)

]

]

]

]

]

]

]

]

]

]

]

.

(17)

Contrary to the formal optimal lunar landing problem,this augmented constraint function has the term for the initialstate constraints in (16). Due to this additional term, theboundary conditions of the costates and the Hamiltonian canbe obtained from optimal theory [16]:

𝐻𝑓 = −𝐺𝑡𝑓,

𝜆𝑓 = 𝐺𝑇𝑥𝑓,

𝜆0 = −𝐺𝑇𝑥0.

(18)

Using these equations, the boundary conditions of thecostates and Hamiltonian are obtained as follows:

𝐻𝑓 = 0,

𝜆𝜙 (𝑡𝑓) = 0,

𝜆𝑚 (𝑡𝑓) = − 1,

𝜆𝑢 (0) + 𝜆𝑚 (0)𝑀

𝐼sp𝑔exp(

𝑢0 − 𝑉0𝐼sp𝑔

) = 0.

(19)

Now, the initial costate values and initial horizontalvelocity to satisfy these boundary conditions need to be foundfrom (10), (18), and (19).

3.2. Solution of Two-Point Boundary Value Problem: ShootingMethod. In the previous section, the optimal conditionsbased on the indirectmethod are evaluated.This optimal con-trol problem can be reformulated to the two-point boundaryvalue problem. Also, it is usually solved by using parame-ter optimization methods such as the sequential quadraticprogramming, evolutionary algorithm, genetic algorithm,coevolutionary augmented Lagrangian method, and particleswarm optimization. In general, these optimal solvers aredivided into two categories: line search process and stochasticprocess. It is usually difficult to select a suitable searchboundary of the solution in stochastic process.Therefore, theshooting method is applied to find the optimal solution.

For the shooting method, the constraints matrix (ℎ)must be described at first. From the previous section, theconstraints matrix is written as follows:

ℎ =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

𝜆𝑢 (0) + 𝜆𝑚 (0)𝑀

𝐼sp𝑔exp(

𝑢0 − 𝑉0𝐼sp𝑔

)

𝑟𝑓 − 𝑟moon

𝑢𝑓 − 𝑟moon𝜔

V𝑓

𝜆𝜙 (𝑡𝑓)

𝜆𝑚 (𝑡𝑓) + 1𝐻𝑓

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

. (20)

Note that the constraint matrix is not a full matrix. Forsimplicity, the constraint matrix is described as the reducedform. The derivative of the constraint matrix is written asfollows:

𝑑ℎ =

ℎ𝑓𝑑𝑡𝑓 + ℎ𝑧0𝛿𝑧0 + ℎ𝑧𝑓𝛿𝑧𝑓 +H.O.T, (21)

where 𝑧 = [ 𝑥𝑇 𝜆𝑇 ]𝑇 represents the augmented state matrix

and 𝑡𝑓 represents the final time. Also, the subscripts 0 and 𝑓mean the values at the initial and final times. Here, the highorder term is neglected, and the equation is rewritten for onlythe initial augmented state matrix and final time by using thestate transition matrix (Φ):

𝑑ℎ = [

ℎ𝑓 ℎ𝑧0+ ℎ𝑧𝑓

Φ𝑓] [

𝑑𝑡𝑓

𝛿𝑧0] . (22)

To reduce the constraints, one can choose the derivativeof this constraints matrix as follows:

𝑑ℎ = −𝛼ℎ, 0 < 𝛼 ≤ 1. (23)

From (22) and (23), the update law of the augmented statevariables at the initial time is obtained as follows:

[

𝑑𝑡𝑓

𝛿𝑧0] = −𝛼 [

ℎ𝑓 ℎ𝑧0+ ℎ𝑧𝑓

Φ𝑓]−1ℎ. (24)


Using the update law, the optimal solution can be foundby using an iterative process:

(i) Initially guess augmented state vector (𝑧) and totalfight time (𝑡𝑓).

(ii) Propagate the system and costates dynamics (Equa-tions ((1)–(5)) and (13)) with the optimal control law(Equation (15)) for the total flight time (𝑡𝑓).

(iii) Calculate the constraint matrix and check it.

(A) If ‖ℎ‖ is less than tolerance value, then thisshooting method terminates (the solution isobtained).

(B) If ‖ℎ‖ is not less than tolerance value, thencalculate the updated value from (24) and addthis value to the initial guessing value. Then, goto (ii) phase with these updated values.

4. Simulation

In the previous optimal lunar landing trajectories, the lunarlander frequently increased its altitude to earn enough timeto reduce the horizontal velocity as shown in Figure 2 [1,2]. This phenomenon depends on its thrust-mass ratio.For enough thrust, this increase in the altitude is reduceddue to enough thrust. This phenomenon is plotted in theFigure 3. However, this huge thrust-mass ratio is sometimesnot physically possible because of the cost, mass budget,thruster technology, and so forth. Thus, if the horizontalvelocity at the initial time is reduced rapidly, it is possible tosave fuel.

Using the derived conditions in the previous section,the proposed optimal approach is demonstrated based onthe identical simulation conditions in [2]. For numericalsimulations, it is assumed that lunar parking orbit is thecircular orbit in the 100 km altitude. From this parkingorbit, the lunar lander falls down to 15 km altitude by usingHohmann transfer method during the deorbit burn phase.Thus, the horizontal velocity before the retrofiring maneuveris calculated using (7) as follows:

𝑉0 = 1691.96926m/s. (25)

The initial mass of lunar lander is set to 300 kg, and theconstant thrust level and specific impulse of the thruster arealso set to 440N and 310 s to calculate the mass flow rate.

Under the simulation conditions, the optimal delta veloc-ity of initial retrofiring maneuver and the optimal lunarlanding trajectory are found by using the shooting method.The final landing mass of the previous and suggested optimalsolution is listed in Table 1. Note that the suggested optimalsolution saves about 6 kg fuel during the lunar landing phasein this case, and the saved fuel is about 4% of the totalconsumed fuel. From the numerical simulation, the requiredretrofiring velocity is obtained as follows:

Δ𝑉 = 1200.9011m/s. (26)

25

20

15

10

5

00 200 400 600 800 1000 1200

Time (s)

Alti

tude

(km

)

Altitude

Figure 2: The typical optimal lunar landing trajectory [1].

30

25

20

15

10

5

0

0 200 400 600 800 140012001000

Horizontal distance (km)

Alti

tude

(km

)Increasing thrust

Figure 3:The optimal lunar landing trajectory for the ratio of thrustmass.

Table 1: Final landing mass.

Final mass(kg)

Total fuel consumption(kg)

Previous optimal solution 154.5413 145.4587Proposed optimal solution 160.6186 139.3814

Note that this value depends on the thrust-mass ratiofor the other case and is also easily obtained by using thesuggested approach. Due to this retrofiring maneuver, thelunar lander does not increase its altitude to earn enoughtime to reduce the horizontal velocity anymore, and this resultis verified by plotting the lunar landing trajectory shown inFigure 4. In this figure, there is no more increase of altitudein comparisonwith the previous approach shown in Figure 2.Also, the horizontal velocity can be rapidly reduced, and thetotal flight time is also reduced to a third of the previousflight time. These horizontal and vertical velocity results are


15

10

5

0

0 50 100 150 200 250 300

Time (s)

Alti

tude

(km

)Altitude

Figure 4: The altitude time history for the optimal solution.

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

00 50 100 150 200 250 300

Time (s)

Hor

izon

tal v

eloc

ity (k

m/s

)

The horizontal velocity

Figure 5: The horizontal velocity profile.

plotted in Figures 5 and 6, respectively. In the vertical velocityplot, the vertical velocity is always negative, and there are nosign changes by comparing with the previous result.This alsomeans that the altitude does not increase during the wholepath. There are no sign changes in the vertical velocity, andthe control command does not exceed 180 degrees as shownin Figure 7.

To check the optimality of this simulation result, thetrend analysis is also demonstrated in Figure 8. This resultcan be obtained by applying the previous typical optimallunar landing approach in [1, 2] by changing the initial deltavelocity from 0 to 1,600m/s with 50m/s step. In this figure,the final landing mass is increasing as increasing the initialdelta velocity, and this final landing mass is maximized near1200m/s. This value is similar to the optimal result using thesuggested approach.

In this section, the simple optimal lunar landing problemby using the suggested method is demonstrated, and theoptimal initial delta velocity with optimal lunar landing

0

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07

−0.080 50 100 150 200 250 300

Time (s)

Vert

ical

vel

ocity

(km

/s)

The vertical velocity

Figure 6: The vertical velocity history.

180

170

160

150

140

130

120

1100 50 100 150 200 250 300

Time (s)

Con

trol i

nput

(deg

.)

Control input

Figure 7: The control command for the optimal lunar landing.

162

160

158

156

154

152

150

0 200 400 600 800 1000 1200 1400 1600

Initial delta velocity (m/s)

Fina

l lan

ding

mas

s (kg

)

Figure 8: The final landing mass for the various retrofired velocityat the initiated time of the powered descent phase.


1210

1200

1190

1180

1170

1160

1150

1140280 285 290 295 300 305 310

Delt

a vel

ocity

of i

mpu

lsive

thru

ster (

m/s

)

Isp for impulsive thruster (s)

Figure 9: The optimal initial retrofired velocity for the specificimpulse.

161

160

159

158

157

156

155

154280 285 290 295 300 305 310

Cos

t val

ue-fi

nal l

andi

ng m

ass (

kg)

Isp for impulsive thruster (s)

Figure 10: The optimal final landing mass for the specific impulse.

trajectory is obtained. However, this result depends on thespecific impulse of the impulsive thruster. If the impulsivethruster has a low specific impulse, the mass consumptionfor the impulsive thruster is quietly increased more thanthe reduced fuel for the continuous thruster. To check thisphenomenon, the simple trend analysis between the specificimpulse of impulsive thruster and the optimal initial deltavelocity is demonstrated in Figure 9. Actually, the specificimpulse of impulsive thruster does not exceed the continuousthruster. Thus, in this figure, the specific impulse steppeddown from 310 s which is the specific impulse of the contin-uous thruster, and the same suggested optimal approach isapplied iteratively to obtain the optimal initial delta velocity.In this figure, the initial delta velocity is also decreased asdecreasing the specific impulse of impulsive thruster becauseof the low efficiency of impulsive thruster. For this trendanalysis, the final lunar landing mass can be obtained byusing suggested optimal approach and it can be depicted inFigure 10. In this figure, the final landing mass is decreased

as decreasing the specific impulse of impulsive thruster. Ifthe specific impulse of impulsive thruster less than 280 sec issupposed to be selected for this case, it does not provide anybenefit for the final lunar landing mass.

5. Conclusions

An optimal lunar landing strategy by using the hybrid engineis suggested in this paper. The typical optimal lunar landingtrajectory sometimes requires the increase in altitude to earnthe time to reduce the horizontal velocity. This increase inaltitude is not efficient for the maximization of the finallanding mass. To reduce the horizontal velocity rapidly, theretrofiring maneuver is applied at the initial time of thepowered descent phase by using the impulsive thruster. Theinitial state free optimization theory is applied, and thesuitable and more efficient solution is obtained. As a result,it is discovered that the lunar lander has more capability tocarry out the mission. To conclude, one can makes variouslunar landing problems and also obtain various solutions byusing the initial state free optimization approach.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This researchwas supported by the project “ADevelopment ofCore Technology for Space ExplorationUsing Nano-satellite”funded by the Korea Aerospace Research Institute (KARI).The authors would like to thank KARI for their support.

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