Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 184, pp. 98–105, 2011
Advanced Guidance Scheme for Lunar Descent and Landing from Orbital Speed Conditions
By Ibrahim Mustafa MEHEDI and Takashi KUBOTA
Department of Electrical Engineering and Information Systems, The University of Tokyo, Tokyo, Japan
(Received April 8th, 2010)
Precise landing technology is one of the most important technologies for future lunar or planetary exploration missions. To achieve a precise landing, an advanced guidance scheme is necessary. This paper outlines a comparison of different solution methods for motion control equations utilized in guidance schemes for lunar descent, and proposes an advanced solution that allows a full depiction of descent vehicle motion from orbital states down to the final landing event. In the conventional solution methods, there exist some poor assumptions such as during descent, constant vertical gravitational acceleration is the only other force acting on the descent vehicle. This inadequate postulation limits the validity of the system solutions within a very low altitude terminal descent area; that is, close to the lunar surface. In this paper, an advanced descent solution is proposed where the centrifugal acceleration term is retained along with the gravitational acceleration term. It allows a complete representation of the descent module motion from orbital speed conditions down to the final landing state. Mathematical derivations of the new scheme are verified in terms of a conven- tional scheme, and comparative simulation results for a fully integrated solution, conventional schemes and a proposed advanced scheme are demonstrated to test the performance.
Key Words: Guidance, Lunar Descent, Centrifugal Acceleration
Nomenclature vector can be identified and inserted as an input of attitude controller that can maintain the thrust vector parallel to u: lander velocity vector magnitude or speed the velocity vector instantaneously but in the opposite direc- gl: lunar gravitational acceleration tion, as shown in Fig. 1. The great benefit of using gravity- N: ratio of thrust F and vehicle mass m turn descent is to guarantee an upright landing and optimize y: real-time altitude of lander from lunar surface fuel consumption.8) x: horizontal span The lunar guidance scheme takes a horizontally oriented c: cross range distance spacecraft from orbital speeds at a point of hundreds of kilo- yl: lunar radius meters from the desired landing point to an almost vertical t: time orientation and very low speed. Implemented guidance : velocity vector pitch angle relative to the local vertical schemes for lunar landing date back to the Apollo era.5) : angle of thrust vector relative to reverse direction of Although the Apollo lunar descent guidance schemes lander velocity worked well to meet the criteria of the 1960s, they can not : cross range angle fulfill the goal of lunar exploration that encompasses the : thrust roll angle desire to easily and cheaply explore many locations on Subscripts the moon. 0: initial The primary task of the descent scheme is to easily and D: descent efficiently solve spacecraft motion equations that help to generate reference trajectories for lunar descent and landing. 1. Introduction This paper proposes an advanced solution scheme for lunar descent equations to circumvent complexity. A fully inte- Returning to the moon is a demanding issue. Many scien- grated solution for spacecraft motion equations is indeed a tists and engineers have confirmed considerable interest in time-consuming issue, and such solutions can’t be imple- the past couple of decades.1–5,12–16) It is essential for a lunar mented using an onboard real-time trajectory generation lander to land vertically and softly on the lunar surface.6) algorithm. The Apollo target trajectory generation scheme Gravity-turn descent is one of the solutions for this purpose, is numerically complex and cumbersome. Therefore, it is and this type of descent technique requires that the lander necessary to find a qualitative solution instead of a numeri- thrust vector be oriented opposed to the velocity vector cal one. Apart from the Apollo solution, other researchers along the complete flight path of the vehicle.7) Using the have proposed solution schemes for lunar descent equations. inertial measurement unit, information about the velocity But the motion equations for spacecraft descent are that solved in conventional way have some limitations too. This Ó 2011 The Japan Society for Aeronautical and Space Sciences Aug. 2011 I. M. MEHEDI and T. KUBOTA: Advanced Guidance Scheme for Lunar Descent and Landing 99
Controlling thrust vector Attitude control
Thrust vector Lunar module Orbit
Velocity vector >>>>> IMU
Lunar surface
Fig. 1. Lunar descent technique. is because, in a conventional solution, the lunar surface is assumed to be a planar flat surface and the centrifugal accel- Fig. 2. Schematic diagram of lunar descent. eration term is ignored.6,7,11) Ignoring the centrifugal accel- eration term during lunar descent, it is assumed that a constant and vertical gravitational acceleration is the only vertical axis, is the angle of thrust vector relative to the other force acting on the descent vehicle.7) This confines reverse direction of spacecraft velocity, y is vehicle altitude the vehicle to be landed precisely on the lunar surface. from the lunar surface, yl is the lunar radius, and is the Moreover, since centrifugal forces are unaccounted for, cross range angle. the conventional solution method limits validity to regimes The other part is the kinematic states to describe the where the descent vehicle velocity is very small relative to fundamental equations of spacecraft motion. These are: the local orbital velocity. Therefore, it can only be used to y_ðtÞ¼ u cos ð4Þ describe terminal descent, when the vehicle has braked from yl orbital velocity and is close to the lunar surface. Conse- x_ðtÞ¼u sin cos ð5Þ y þ y quently, the authors demonstrate an advanced descent solu- l yl tion method for a spherical homogeneous lunar surface c_ðtÞ¼u sin sin ð6Þ y þ y where centrifugal forces are considered and descent can l be initiated from orbital speed conditions. In this paper, where x and c are the horizontal span and cross range some logical values are examined to determine a better distance, respectively. approximation for the centrifugal acceleration term without 2.1. Initial assumptions ignoring it, but the gravity is assumed to be constant in The right-hand side of the spacecraft-governing equations magnitude. These assumptions are reasonable with the are reduced to the function of velocity vector pitch angle, . descent starting from vehicle orbit.3) Compared to the con- For this purpose, some reasonable assumptions are made ventional descent method, the proposed advanced solution regarding thrust to mass ratio, thrust vector angle and lunar allows a full representation of descent module motion from gravitational acceleration force. To generate an ideal orbiting conditions down to the final vertical landing state. descent trajectory, it is rational to assume a constant value
To prove the significant improvement in the new solution, for N (i.e., F=m) and gl, and control input is set to zero. three steps are performed in this study: fully integrated But in the situation of constant thrust acceleration, m is solution, conventional solution and advanced solution. not constant, and so F=m becomes variable. However, this error is removed by the real-time guidance algorithm. 2. Fundamentals of Lunar Descent Therefore, using initial values for mass and gravity is a straightforward assumption for this solution. To facilitate Fundamental three-dimensional equations of motion to the simplification of mathematical operation, roll control describe the spacecraft proposition concerning a uniform states are held to zero ( ðtÞ¼0 and _ðtÞ¼0). To activate sphere-shaped lunar body9) are divided into two parts. One the system as a planar motion and constrain motion to two is the equations of spacecraft motions for dynamic states, dimensions, the initial states are initialized to zero as follows: (cð0Þ¼0, ð0Þ¼0, and _ð0Þ¼0). Consequently, the gov- erning equations are reduced to their two-dimensional u_ðtÞ¼gl cos N cos ð1Þ forms, where Eqs. (1) and (4) remain the same. The only 1 u2 changes observed are as follows: _ðtÞ¼ gl sin N sin cos ð2Þ u y þ yl 1 u2 1 _ðtÞ¼ g sin N sin ð7Þ _ðtÞ¼ ½N sin cos ð3Þ u y þ y l u sin l y where u is the spacecraft velocity vector magnitude or x_ðtÞ¼u sin l ð8Þ y þ yl spacecraft speed, gl is the lunar gravitational acceleration, N is the ratio of thrust F and vehicle mass m, is the pitch It is reasonable to assume that y yl in order that yl= angle of the vehicle velocity vector relative to the local y þ yl 1. Then, the equation for horizontal span becomes 100 Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 184
x_ðtÞ¼u sin ð9Þ y_ð Þ¼ u cosð Þt_Dð Þð18Þ where u can be replaced from Eq. (14). For the solution of 3. Fully Integrated Solution horizontal span as a function of velocity vector pitch angle , the same procedure can be followed. To find the fully integrated numerical solutions for speed dx dx dtD u, time t, horizontal distance x and vertical range y as a func- x_ð Þ¼ ¼ d dt d tion of velocity vector pitch angle during powered D descending phase, the authors performed the following ¼ x_ðtÞt_Dð Þð19Þ mathematical derivations for simplification. From Eq. (9): du d uðg cos NÞ x_ð Þ¼u sinð Þt_ ð Þð20Þ u_ð Þ¼ ¼ l ð10Þ D 2 dt dt ðu =yl glÞ sin then 4. Conventional Descent Solution 1 u2 g cos N g du ¼ l d ð11Þ The traditional solution for descent is obtained by assum- u y l sin l ing that the lunar surface is similar to a plane, so that the This can be integrated as: lunar radius is yl !1. Therefore, the centrifugal accelera- Z Z tion term is ignored in order to obtain u2=y þ y ¼ 0.7,11) u 1 u2 g cos N l g du ¼ l d ð12Þ With this perimeter, Eq. (7) reduces to: u y l sin u0 l 0 g _ðtÞ¼ l sin ð21Þ Then following equation can be obtained: u 2 2 u =2yl u0 =2yl gl lnðu=u0Þ This reduced equation can be used to obtain a single, distin- þ 1 cos guishable differential equation with as the self-regulating gl lnðsin Þ N ln ¼ 0 ð13Þ sin variable, such that: Therefore, the equation for speed: uðg cos NÞ u_ð Þ¼u_ðtÞ= _ðtÞ¼ l ð22Þ 1 g sin uð Þ¼½ glylW 2 ð14Þ l where the Lambert W function is expressed as: then
2N 2 2 1 du cos N 1 u0 sin 1 þ cos gl þ ¼ 0 ð23Þ W ¼ ð15Þ 2 u d sin gl sin glyl u0 sin glyl e At this time, Eq. (23) can be integrated to find the descent 10,11) Now the full numerical solution for descent time tD as speed u as a function of velocity vector pitch angle : a function of velocity vector pitch angle can be N obtained by integrating the equations with the help of sin 0 tanð =2Þ gl uð Þ¼u0 ð24Þ Eqs. (1) and (10): sin tanð 0=2Þ Differentiating Eq. (24): dtD du du t_Dð Þ¼ ¼ N d d dtD du sin tanð =2Þ gl N u_ð Þ¼ ¼ u 0 cot u_ð Þ d 0 sin tanð =2Þ g sinð Þ ¼ ð16Þ 0 l u_ðtÞ ð25Þ Again, for vertical range and horizontal flight path distance, it can be written that Now the solution for time, vertical and horizontal ranges can be acquired utilizing the value of speed u in previously dy dy dtD y_ð Þ¼ ¼ derived mathematical equations to obtain the descent trajec- d dt d D tory in terms of the traditional descent solution. Therefore, ¼ y_ðtÞt_Dð Þð17Þ descent time acquired equations are: substituting the values from Eq. (4):
dtD du du t_Dð Þ¼ ¼ d d dtD N sin 0 tanð =2Þ gl N 1 ¼ u0 cot ð26Þ sin tanð 0=2Þ gl sinð Þ gl cos N Aug. 2011 I. M. MEHEDI and T. KUBOTA: Advanced Guidance Scheme for Lunar Descent and Landing 101 vertical range, dy dy d y_ð Þ¼ ¼ d dtD dtD 2N 2 2 g u 2 sin 0 tanð =2Þ l cot ¼ cot ¼ u0 ð27Þ gl sin tanð 0=2Þ gl and horizontal span, orbital speed to the final landing event, can be evaluated dx d qualitatively. x_ð Þ¼ dt dt D D 5. Advanced Lunar Descent Solution u2 ¼ gl 5.1. Assumptions 2N To solve the same governing equations for the proposed 2 g 2 sin 0 tanð =2Þ l 1 ¼ u0 ð28Þ advanced scheme, it is again necessary that the right-hand sin tanð 0=2Þ gl side of the equations be kept as a function of velocity vector 4.1. Descent specifications pitch angle . The assumption of a centrifugal acceleration Along with the assumptions described in Section 2.1, term is newly considered for a homogeneous spherical lunar other required descent specifications are assumed to inte- surface. The assumptions for mass and lunar gravity are grate the developed equations and to compare the lunar identical to those in Section 2.1. But for the centrifugal descent scheme simulation results obtained by the fully acceleration term, a constant value , defined as the ratio integrated and conventional solutions. The Moon has no between centrifugal acceleration and lunar gravitational atmosphere and no approximation regarding the centrifugal acceleration, can be logically chosen. Though this is notice- acceleration term is taken into consideration for conven- ably a varying value, during the reference trajectory gener- tional descent illumination. Specific descent speculations ation phase, it is reasonable to consider it as an assumption are shown in Table 1. at the initial stage because real-time guidance will compen- 4.2. Simulation results sate for the errors between the model and the environment. Figure 3 shows a comparison of different trajectory Therefore: responses for spacecraft descent to the lunar surface while u2 the governing equations are solved using the complete inte- ¼ gl ð29Þ y þ yl gration method and conventional descent illumination. The 2 computer simulation results for the fully integrated solution u gl ¼ ð1 Þgl ð30Þ is assumed to be the ideal measurement for the lunar descent y þ yl trajectory, and this ideal situation can not be expected in 5.2. Mathematical derivations realistic cases. But the authors would be pleased if responses With these assumptions, and ensuring consistency with with minimal divergence are obtained. In these simulation traditional lunar descent works,7,10,11) speed can be recog- results, it can be seen that the responses of the traditional nized by following differential equations formulated as a descent solutions and fully integrated solutions are different function of the velocity vector pitch angle : for the equations of lunar module motion. It is noticeable gl cos N that traditional descent solutions have the largest impact u_ð Þ¼u_ðtÞ= _ðtÞ¼u ð31Þ ð1 Þgl sin on the final altitude variation with respect to the fully inte- du g cos N grated solution due to the fact that the centrifugal accelera- ¼ l d ð32Þ tion term is ignored. This fact proves that the conventional u ð1 Þgl sin solution method limits validity to regimes where the descent This Eq. (32) can now be directly integrated to obtain the vehicle velocity is very small relative to the local orbital descent velocity u as a function of the velocity vector pitch velocity, and therefore, it can only be used to describe the angle : R hi gl cos N terminal descent portion. Consequently, further analysis d 0 ð1 Þgl sin using the new advanced scheme for lunar descent and land- uð Þ¼u0e ð33Þ ing needs to be performed so that the entire trajectory, from If Z g cos N l d Table 1. Lunar descent specifications. 0 ð1 Þgl sin 0 1 N ð1 Þg Lunar gravitational acceleration (g ) 1.623 m/sec2 0 l l 1 Btan C Thrust to mass ratio (N) 4 N/kg sin ð1 Þ B 2 C ¼ ln þ lnB C ð34Þ Initial lander speed (u0) 1688 m/sec sin @ A 0 tan Initial velocity vector pitch angle ( 0) 90 deg 2 102 Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 184
1600 Fully integrated solution Conventional solution 1400
1200
1000
800
600
400 Horizontal speed (m/sec) 200
0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) (a) Horizontal speed
Fully integrated solution 450 250 Conventional solution 400
200 350
300
150 250
200
100 (sec) Time 150
Vertical speed (m/sec) Vertical 100 50 50 Fully integrated solution Conventional solution 0 0 90 80 70 60 50 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) Alpha (deg) (b) Vertical speed (c) Time
10 400 Fully integrated solution 0 Conventional solution 350 -10 300 -20 250 -30
-40 200
-50 150 -60 100 Vertical range (km) Vertical
-70 Horizontal span (km) 50 -80 Fully ntegrated solution Conventional solution -90 0 90 80 70 60 50 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) Alpha (deg) (d) Vertical range (e) Horizontal span
Fig. 3. Comparison of fully integrated solution to conventional solution: speed, time, vertical range and horizontal span. then where >0, so that 0 1 0 1 N 1 cos 0 ð1 Þgl 0 B C 1 Btan C sin B sin 0 C sin ð1 Þ B 2 C uð Þ¼u B C ð36Þ ð Þ¼ B C ð Þ 0 @ A u u0 @ A 35 sin 0 1 cos sin 0 tan sin 2 so where ¼ 1=ð1 Þ is a measure of the centrifugal accel- 1 cos eration term. Then, the solution for current speed obtains the tanð =2Þ¼ sin shape: Assume ð1þ Þ sin 1 cos 0 1 uð Þ¼u0 ð37Þ ¼ and ¼ N=g sin 0 1 cos ð1 Þ l Aug. 2011 I. M. MEHEDI and T. KUBOTA: Advanced Guidance Scheme for Lunar Descent and Landing 103
Next, the time to go tDð Þ, horizontal span xð Þ and vertical descent solution, different values (1; 2; 3; 4; ...) for are uti- range yð Þ are resolved in a manner identical to the conven- lized in Eqs. (37), (38), (40) and (42), and these equations tional lunar descent solution, as follows: are numerically integrated with constant approximate values for g and N, whereas g ¼ 1:623 m/sec2 and N ¼ 4 N/kg. t_Dð Þ¼1= _ðtÞ l l Initial and final values for the velocity vector pitch angle uð Þ ¼ are 90 [deg] and 0 [deg], respectively, while the initial g l sin speed u0 is assumed to be the approximate orbital speed, ð1þ Þ 1 1688 m/sec. The influence of differing the constant is u0 ð1 cos 0Þ ðsin Þ ¼ demonstrated in Figs. 4(a), 4(b), 4(c), 4(d) and 4(e) for g ðsin Þ ð1þ Þ ð1 cos Þ l 0 speed, time, vertical range and horizontal span, respectively. ðsin Þ ð1þ Þ 1 In contrast to the advanced solution, the fully integrated ¼ G ; ð38Þ tD ð1 cos Þ solution to Eqs. (1), (2), (4) and (5), and the traditional lunar descent solutions to Eqs. (25), (26), (27) and (28) are per- where formed for comparison using the same approximations for , g , N, and u0 as used for Eqs. (37), (38), (40) and u0 ð1 cos 0Þ l Gt ¼ ð39Þ (42); while no estimations are made regarding the centrifu- D g ðsin Þ ð1þ Þ l 0 gal acceleration term. For horizontal span: A comprehensive evaluation of this advanced solution x_ð Þ¼x_ðtÞ= _ðtÞ using a traditional descent scheme, and a fully integrated solution for the motion equations of a lunar landing mission u2ð Þ sinð Þ ¼ is conducted in this investigation. It can be noted that vary- gl sinð Þ ing has a reasonable impact on different responses for 2 2 2 ð1þ Þ u0 ð1 cos 0Þ ðsin Þ speed, time, vertical range and horizontal span. In particular, ¼ 2 ð1þ Þ 2 it directly influences the vertical range of the descent trajec- gl ðsin 0Þ ð1 cos Þ tory. It has already been mentioned that the trajectory 2 ð1þ Þ ðsin Þ responses for the fully integrated solution are models for ¼ Gx ; ð40Þ ð1 cos Þ 2 any other solution method. The issue is to minimize the where divergence gap between other methods and the fully inte- 2 2 grated solution. From the assessment of the various values u0 ð1 cos 0Þ Gx ¼ ð41Þ for , the authors feel that the value of ¼ 2 emerges to g ðsin Þ 2 ð1þ Þ l 0 be a realistic number and improves on the different re- For vertical range: sponses of the advanced solution for speed, time, vertical y_ð Þ¼y_ðtÞ= _ðtÞ range and horizontal span when compared to conventional solutions. Additionally, accurate verification of the mathe- u2ð Þ cos ¼ matical calculations conducted for the proposed advanced gl sin scheme was obtained by investigating the responses, as 2 2 2 ð1þ Þ u0 ð1 cos 0Þ ðsin Þ cos shown in Fig. 4. In the figures, it produces the exact same ¼ 2 ð1þ Þ 2 simulation results are obtained for the conventional descent gl ðsin 0Þ ð1 cos Þ sin solution and the solution choosing a value of ¼ 1. Further- 2 ð1þ Þ 1 ðsin Þ cos more, the same assumption for conventional descent solu- ¼ Gy ; ð42Þ ð1 cos Þ 2 tions is reproduced with a value of ¼ 1 in Eq. (30), in where which the centrifugal acceleration term is ignored. 2 2 u0 ð1 cos 0Þ Gy ¼ 2 ð1þ Þ ð43Þ 7. Conclusion gl ðsin 0Þ The conventional lunar descent and landing problem 6. Performance Test for Advanced Scheme has been advanced to allow the accurate representation of lunar descent from orbital conditions. Finding a reasonable To integrate the above equations in a qualitative manner, assumption for the lunar surface and centrifugal accelera- the value for must be an integer. This entails ¼ tion, it significantly advances the sphere of validity of the 1; 2; 3; 4; .... Instead of this solution, the ratio , mentioned traditional gravity-turn solution from low-velocity terminal in Section 5.1 can be chosen as some fractional value to descent to a complete descent from orbital conditions. The make an integer. But the authors found better results by accessibility of the descent speed, time, vertical range and directly taking logical integer values to obtain qualitative in- horizontal span as a function of the velocity vector pitch tegration of the equations. Choosing a logical value directly angle can be utilized to lessen the computational burden for proves to be more precise in approximation as well. To on real-time lunar descent guidance schemes for future land- visualize the simulated results for the advanced lunar ing missions. 104 Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 184
1600 Fully integrated solution Conventional solution 1400 τ = 1 τ 1200 = 2 τ = 3 1000 τ = 4 800
600
400 Horizontal speed (m/sec) 200
0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) (a) Horizontal speed
Fully integrated solution 450 250 Conventional solution τ = 1 400 200 τ = 2 350 τ = 3 300 τ 150 = 4 250
200 Fully integrated solution 100 (sec) Time Conventional solution 150 τ = 1
Vertical speed (m/sec) Vertical 100 τ 50 = 2 50 τ = 3 τ = 4 0 0 90 80 70 60 50 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) Alpha (deg) (b) Vertical speed (c) Time
0 400
-10 350 -20 300 -30 250 -40
-50 200 Fully integrated solution -60 150 Conventional solution Fully integrated solution -70 τ Conventional solution Vertical range (km) Vertical = 1 Horizontal span (km) 100 τ = 1 -80 τ = 2 τ = 2 τ 50 -90 = 3 τ = 3 τ = 4 τ = 4 -100 0 90 80 70 60 50 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 Alpha (deg) Alpha (deg) (d) Vertical range (e) Horizontal span
Fig. 4. Comparison of advanced solution to fully integrated solution and conventional solution: speed, time, vertical range and horizon- tal span.
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