Advanced Descent Solution and Trajectory Generation Scheme for Precise and Safe Lunar Landing Mission

Transcription

1 Advanced Descent Solution and Trajectory Generation Scheme for Precise and Safe Lunar Landing Mission Ibrahim M. Mehedi Department of Electrical Engineering and Information Systems, The University of Tokyo, Japan. Takashi Kubota Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan. Precise landing is a crucial requirement for future autonomous landing to investigate scientifically interesting areas of lunar surface. The primary task of precise landing scheme is to solve the spacecraft motion equations easily and efficiently, that will help to generate reference trajectory for lunar descent and landing. This paper proposed an advanced solution for lunar descent equations and reference trajectory generation scheme to circumvent complexity. Trajectory generation algorithm is developed for variable thrust to mass ratio. Mathematical modeling, algorithm design, simulations and results are presented in this paper. I. Introduction A several lunar/planetary missions were accomplished in last few decades; the guidance-navigation-control (GNC) technology is getting more important than ever. Because of long round-trip delay of communication with the ground station is extremely required to have the mission capability to perform precession autonomous descent to the selected landing site. With the help of an advanced landing technique it will be possible to reach landing sites even in areas containing hazardous terrain features like craters, rocks or slopes. Therefore, an advanced autonomous descent and trajectory generation scheme is discussed in this paper. Traditional descent solution, trajectory generation scheme and GNC mode using deep space network are not suitable for precise and safe lunar or planetary landing. It is also essential for a spacecraft to land vertically and softly on the lunar surface to ensure a safe landing mission.? In preparation for these constraints and in execution of future scientific objectives (such as to perform scientific region investigation or sample return missions), a trajectory generation scheme of lunar descent is developed. This research outlines a qualitative method of solutions of motion control equations for descent vehicle from orbital states down to the final landing event on a homogeneous spherical lunar surface. These followed by developing an analytical reference trajectory generation algorithm that will be used in real time to promptly and consistently generate two dimensional trajectories ahead of descent initiation or re-target to another landing site after descent initiation. Trajectory generation algorithm is developed for variable thrust to mass ratio. Mathematical modeling, algorithm design, simulations and results are presented in this paper. In fact this trajectory generation scheme can readily be used to develop real-time guidance algorithm for future precise and safe lunar landing mission reducing the computational burden and ensuring the safe and vertical landing capabilities. The arrangement of this paper is as follows. Outline of precession lunar landing including Earth-planet orbital transfer, parking orbit insertion and controlled decent are provided in the second section. Third section describes the historical event for successful lunar soft landing missions. Spacecraft namics and kinematics for lunar landing are described in fourth section and fifth section describes the scenario of advanced PhD Student, Department of Electrical Engineering and Information Systems, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan, AIAA Student Member. Professor, Department of Electrical Engineering and Information Systems, The University of Tokyo, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan.

2 descent solution. Sixth section illustrates reference trajectory generation scheme for safe planetary landing and finally seventh section concludes the proposed scheme. Figure 1. Different sequence of lunar landing mission II. lunar landing mission out line As shown in Fig. 1, journey from Earth to moon consists three main phases such as transfer, parking orbit insertion and decent. After spacecraft is launched, it enters in to the lunar circular parking orbit with the help of several mid-course trajectory corrections. Before starting decent, the spacecraft executes burning sequences anti-parallel direction to its velocity in order to perform some important orbit maneuvers and reaches at the point of terminal descent. Then the spacecraft will start final descent phase to land at the specified landing site on the lunar surface with in the allowable error range. III. Past and future of lunar landing mission Since 1959, several missions of spacecraft landing has been accomplished on other planets and bodies in the solar system such as Moon, Venus, Mars, Eros, Jupiter, Titan, Comet 9P, Itokawa and Enceladus. Some of them were successful soft landings and rests are both intended and unintended crash-landings explored by USSR, USA, Japan, ESA and India at different period of time. Exploring the surface of other celestial bo, Moon has got especial attention. Lunar Prospector, the last lunar surface exploration operation in 1999 by USA, has ended 24th spacecraft landing mission on Moon since 1962 starting with Ranger 4 to produce hard impact on the Moon intentionally but it hit on lunar far side due to failure of navigation system. In 1959, USSR sent the first spacecraft Luna 2 to reach the surface of the Moon, and it impacted the lunar surface west of Mare Serenitatis near the craters Aristides, Archimeded and Autolycus. USSR completed their 14th spacecraft landing mission on Moon by Luna 24 in 1976 with robotic sample return task. Following Luna 2, USSR carried several unsuccessful landing missions till to achieve first successful soft landing on Moon in 1966 by Luna 9 transmitting first photographic data from the surface of another planetary bo to Earth. Just four months after the landing of soviet Luna 9 mission, Surveyor 1 landed on an extraterrestrial bo, Moon, as first soft landing American spacecraft. This spacecraft was launched in 1966 directly in to a lunar impact trajectory and at a height of 3.4 m above the lunar surface the engines were turned off to let the spacecraft freely touchdown on the surface. Following Surveyor 1, USA completed 6 manned missions of 11 successful soft landings on lunar surface while human carried lunar rover during last 3 manned missions to the Moon surface. On the other hand USSR never carried out successful manned mission towards Moon instead 7 successful unmanned soft landing missions on lunar surface including 3 robotic sample return and 2 robotic lunar rover missions. In 1993 and 2006, Japan and ESA respectively launched lunar orbiter and both

3 were intentionally impacted on lunar surface at end of missions. Chandrayaan-1 was India s first unmanned lunar probe. It was launched by the Indian Space Research Organization (ISRO) in October 2008 carrying a lunar orbiter and an impactor. The Moon impact probe impacted near Shackleton Crater at the south pole of the lunar surface at 14 November The Chinese lunar orbiter Chang e 1 executed a controlled crash onto the surface of the Moon on 1 March In future China is planning to land motorized rovers and collects samples by A similar Russian sample return mission called Phobos-Grunt ( grunt means soil in Russian) is scheduled for launch in early The Indian Space Research Organization (ISRO) hopes to land two motorized rovers - one Indian and another Russian - on the Moon in 2013, as a part of its second Chandrayaan mission IV. Spacecraft namics and kinematics Descent method is chosen like that the lander thrust vector is aligned opposite to its velocity vector at all points along the descent trajectory. This method has the useful property that a vertical landing is guaranteed. Developed fundamental three dimensional equations of motion for spacecraft descent are divided in to two parts. One is namics and other is kinematics.? Dynamics part includes the state vectors for speed, flight path angle and crossing angle. Kinematics part includes the state vectors for altitude, horizontal span and crossing distance. For a planetary or small solar system bo of radius R, the equations of motion may be summarized as R 2 dv dt = g l cos (θ) N (1) R + y dθ dt = 1 [ u 2 ] ( v y + R g l)sinθ (2) = vcosθ (3) dt Where v is spacecraft velocity vector magnitude or spacecraft speed, g l is lunar gravitational acceleration, N is ratio of thrust and vehicle mass, θ is the pitch angle of the vehicle velocity vector relative to the local vertical, y is altitude of the spacecraft from lunar surface. V. Scenario of Advanced solution scheme for lunar landing Lunar descent problem can be solved through numerically which is a complete integrated solution. This ideal but complex method takes enough time to execute and can t be implemented in real-time on-board application. The traditional solution,?,?,? different from ideal method, also bear some limitations such as: lunar surface is considered as a plane surface and centrifugal acceleration term is not included. The conventional solution can now be significantly improved by considering a homogeneous spherical lunar surface including centrifugal acceleration term in the equations of motion. To solve those governing equations qualitatively it is essential that the right hand sides of the equations are reduced as a function of velocity vector pitch angle θ.? Therefore, the equation for speed and altitude are as follow. θ g [ l cosθ N θ v(θ) = v 0 e 0 v 2 y+r dθ = dt dθ dt = τv 0 g l (1 Γ)g l sinθ ]dθ (4) (1 cosθ 0 ) 2τρ (sinθ 0 ) 2τ(1+ρ) (sinθ) 2τ(1+ρ) cosθ (1 cosθ) 2τρ sinθ Where Γ = g l and τ = 1 1 Γ is a measure of the centrifugal acceleration term. Derivations of these equations are discussed in detail and the influences of differing the constant τ is demonstrated in authors previous research.? Unlike values (1, 2, 3, 4, 5,...) for τ are employed into equations (4) and (5) and these equations are numerically integrated with constant approximate values for g l and N whereas g l = m/sec 2 and N = 5 N/kg. Initial and final values for the velocity vector pitch angle θ is taken 90 degree and 0 degree while the initial speed v 0 is considered as approximate orbital speed, 1688 m/sec. In contrast of this advanced solution, full numerical integrated resolution to equations (1), and (3) is performed without (5)

4 (a) (b) Figure 2. A comparison for the proposed advanced solutions and complete integrated solutions of lunar descent speed and altitude as a function flight path angle θ. taking any assumption regarding centrifugal acceleration term. For comparison, computer simulation results are shown in Fig. 2. It can be noted that varying τ has reasonable impact on different responses for speed and altitude for lunar descent scheme. The significant impact is observed for altitude variation. The centrifugal acceleration effectively adjusts the rate of change of the vehicle velocity vector pitch angle which impacts the direction of the velocity vector. Therefore, the term τ directly influences the vertical range of the flight path. From the assessment of the various values for τ, a value of τ = 2 emerges to be a realistic number and improves on different responses of advanced solutions for speed and altitude. VI. Trajectory space generation Using this value of τ = 2, all the equations for state variables can be re-derived to develop a targeting flight path generation algorithm as a function of horizontal span and vertical range. Let the initial horizontal span and vertical range of the complete flight path are d 0 and y 0 and terminal values of these parameters are d t and y t. Now the complete displacements in horizontal and vertical direction are characterized by and δd = d t d 0 = δy = y t y 0 = n δd i (6) i=1 n δy i (7) This would result in a preferred total horizontal span and vertical range given a set of values for n values of N i. Instead of n steps, if two steps are considered with the same objective (identify θ 0, θ t, v 0, v t, δd, δy) as stated before, the problems still remain constrained. Hence, it is easy to compute v 1 if θ 1, N 1 and N 2 are alrea known together with θ 0 and v 0 or θ 2 and v 2. From equations (6) and (7) the two steps flight path can be evaluated for cross range and altitude. A sample targeting flight path space is shown in Fig. 3. Boundary condition for flight path angles were set at 89 0 and The initial and final speeds were set at 1688 m/sec. and 8 m/sec. The gravity is considered as constant in this analysis where the values for thrust to mass ratios, N 1 and N 2 are varied from 0.1 N/kg to 10 N/kg with 0.25 N/kg increments. Fig 3 shows the impact of varying the thrust acceleration and it appears to have set of curves whereas each curve is created varying N 1 and N 2 simultaneously. i=1

5 Figure 3. Generated trajectory space for variable thrust VII. Conclusion Lunar landing mission outline and important histories are described in this investigation. Further, to achieve precise landing, traditional solution of lunar descent is advanced significantly and compared with the complete numerical solution in this paper. Using this advanced analytical solutions of lunar descent scheme, a reference trajectory generation algorithm is proposed to make more precision landing. References 1 Cheng, R., Meredith, C. and Conrad, D., Design Considerations for Surveyor Guidance, Journal of Spacecraft and Rockets, 3(11), , McInnes, C., Gravity-Turn Descent from Low Circular Orbit Conditions, Journal of Guidance Control and Dynamics, 26(1), , Klumpp, R., Apollo guidance, navigation, and control: Apollo lunar-descent guidance, Technical report, MIT Charles Stark Draper Laboratory, Mehedi, I.M. and Kubota, T., Advance descent scheme for lunar landing, 18th IFAC Symposium on Automatic Control in Aerospace, Nara, Japan, Cheng, R., Lunar terminal guidance, In Lunar Missions and Exploration, edited by C. T. Leondes and R. W. Vance, Univ. of California Engineering and Physical Sciences Extension Series, , Wiley, New York, Pashall, S.C., Bra, T., Cohanim, B.E. and Sostaric, R., A self contained method for safe and precise lunar landing, IEEE Aerospace Conference, Ueno, S. and Yamaguchi, Y., Near-Minimum Guidance Law of a Lunar Landing Module, 14th IFAC Symposium on Automatic Control in Aerospace, , Sostaric, R., Lunar descent reference trajectory, Technical report, NASA/JSC, Xing-long, L., Gaung-Ren, D. and Kok-Lay, T., Optimal Soft Landing Control for Moon Lander, Automatica, 44, , Chao, B. and Wei, Z., A Guidance and Control Solution for Small Lunar Probe Precise-Landing Mission, Acta Astronautica, 62, 44-47, Scheeres, D., Interactions between ground-based and autonomous navigation for precision landing at small solar-system bodies, Telecommunication and data acquisition progress report, , Kenji, U., Yuzo, S. and Shingo, N., Tracking control to near-optimal trajectory for a lunar lander, 23rd International Symposium on Space Technology and Science, 1, , Kenji U., Guidance law for lunar lander with input constraint, AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, Soth Carolina, McInnes, C., Direct adaptive control for gravity-turn descent, Journal of Guidance Control and Dynamics, 22(2), , Johnson, A.E. and Montgomery, J.F., Overview of Terrain Relative Navigation Approaches for Precise Lunar Landing, IEEE Aerospace Conference, Big Sky, Mont, USA, March Paschall, S.C., Bra, T., Cohanim, B.E. and Sostaric, R., A Self Contained Method for Safe and Precise Lunar Landing, IEEE Aerospace Conference, Big Sky, Mont, USA, March 2008.