IAC-04-IAA.4.11.P.05
Mission Scenarios for a Controlled Lunar Impact of a Small Satellite Nikolas Trawny, Michael Graesslin, Rene Laufer and Hans-Peter Roeser Email:
[email protected], {graesslin,laufer,roeser}@irs.uni-stuttgart.de University of Stuttgart, Institute of Space Systems Pfaffenwaldring 31, 70550 Stuttgart, Germany
Abstract The Institute of Space Systems (IRS) at the University of Stuttgart is currently planning a lunar small satellite mission. The satellite will be equipped with a 6 mN and a 100 mN electric propulsion system. At the end of its primary science mission, it will perform a controlled impact-experiment on the lunar surface, including the soft landing of a small surface unit. In this paper we present the results of a numerical simulation and optimization of possible impact trajectories starting from the satellite’s initial 100 km polar, circular orbit. The perturbing accelerations being in the same order of magnitude as the thrust, we used thrust vector control for efficient orbit manipulation. First results show that an impact using the electrical thrusters is principally feasible. Using the 6 mN thrusters by themselves for the deorbit maneuver is unadvisable due to the long thrust durations and the very low impact angle, making the impact inaccurate and difficult to control. The 100 mN thruster, however, together with an additional solid rocket motor for a final aposelene boost, yields much more favorable impact conditions at the price of a higher subsystem mass.
1
Introduction
The Institute of Space Systems (IRS) at the University of Stuttgart, Germany, is currently planning a small lunar satellite mission to be launched within this decade. Planned as an ”all electrical satellite” it will be
equipped with two different propulsion systems, one being a cluster of 4 pulsed plasma thrusters yielding 6 mN, the other an arcjet with 100 mN thrust. The spacecraft is supposed to circle the Moon in a polar orbit at 100 km altitude. As a final science ex-
periment, the probe will perform a controlled deorbit maneuver. Shortly before impact, a small surface unit is to be separated for a soft landing [3]. This paper will examine the trajectory for the impact-experiment. Analysis of the perturbing accelerations in a lunar orbit shows that they are in the same order of magnitude as those exerted by the thrusters. Therefore, emphasis lies on precise modelling of the perturbing sources and careful optimization of the thrust profile. Earlier lunar impact missions commonly used high thrust engines for the deorbit maneuvers. The final boost of Lunar Prospector, for example, had a magnitude of 45 m/s and resulted in an impact angle of 6.3◦ . Velocity increments of this size during the last halforbit are not feasible with electric thrusters. For this reason we also examined the effects of a short high thrust boost during the last aposelene passage, provided e.g. by means of a small solid rocket motor.
2
Model
Due to the particular shape of the satellite’s orbit at start time of the deorbit burn (i=90◦ , e=0), we used a formulation of the equations of motion in modified equinoctial elements. Contrary to the classical elements, they are free from singularities for polar or circular orbits. Together with the satellite’s mass the resulting state vector has the following shape: £ ¤T £ ¤T x = yT m = p f g h k L m (1)
The equations of motion can be stated in vector form as y˙ = A · ∆ + b T m ˙ = − ce
(2) (3)
In this formulation, T stands for the thrust and ce for the exhaust velocity of the propellant. For the definition of A and b, the reader is kindly asked to consult the work of Betts and Erb [1]. The perturbing accelerations ∆ are expressed in a rotating radial frame and consist of the thrust, the perturbations due to the nonspherical gravity field, third body perturbations and accelerations due to solar pressure. ∆ = ∆T + ∆G + ∆D + ∆S
(4)
For steering we used thrust vector control. The direction of the thrust is parameterized in a rotating radial frame by the unit vector u: £ ¤T u = ur uθ uh (5) The resulting acceleration can be expressed by T ∆T = u (6) m The perturbations due to the nonspherical gravity field ∆G were computed using the LP100J model from Konopliv [2]. This model, stemming largely from Clementine and Lunar Prospector data, is the best lunar gravity model currently available. Third body perturbations ∆D were modelled using the DE405 ephemerides available at the Jet Propulsion Laboratory [4].
−3
acceleration in r /(m/s²)
2
Parameter
0
−2
−4
acc. perp. to r in orbit plane /(m/s²)
x 10
0
1
2
3
4 5 real time /s
6
7
8
x 10
0
1
2
3
4 5 real time /s
6
7
8
acc. perp. to orbit plane /(m/s²)
9 4
x 10
−3
Angle between Thrust−vector and Perturbation vector /(deg)
Semi major axis
a
1837, 1
km
Eccentricity
e
0, 0
-
Inclination
im
90
◦
of
ωm
0
◦
Argument of ascending node
Ωm
0
◦
True Anomaly
ν
free
◦
Mass
m
150
kg
Argument Periselene
−2
2
Unit
4
0
−4
Value
x 10
−3
2
9
Symbol
x 10
1
Table 1: Initial Conditions in moon-relative Kepler elements
0
−1
0
1
2
3
4 5 real time /s
6
7
8
9 4
x 10
Figure 1: Comparison of the accelerations due to the arcjet and those due to the sum of perturbations resulting from the nonspherical gravity field, the third body perturbations and the solar pressure. The perturbation due to the nonspherical gravity field is dominant. Peaks in perturbations are correlated to the periselene passage as well as to the overflight of the strong lunar mascon at Mare Imbrium. 135
90 45
0
0
1
2
3
4 5 real time /s
6
7
8
9
4
x 10
For the computation of the solar radiation pressure ∆S we implemented a conical shadow model to incorporate occultation by the moon. Figure 1 shows a comparison between the effect of thrust and of perturbing accelerations on the satellite, underscoring the importance of exact modelling of the environment as well as the need for optimal use of propulsion capabilities.
3
Trajectory tion
Optimiza-
For trajectory simulation and optimization we used the software package GESOP/SOCS [5, 6]. SOCS is a direct transcription software specifically designed to handle large trajectory optimization problems. The problem is transcribed into a nonlinear programming problem by discretization. A typical impact maneuver required about 5000 grid points. Initial conditions were given by the nominal mission orbit around the moon, a polar, circular orbit in 100 km altitude. Thus, the values for the semi major axis, eccentricity and inclination were fixed. Argument of periselene, argument of ascending node were arbitrarily set to zero whereas the true anomaly remained optimizable (cf. table 1). Maneuver start time was chosen to be dur-
ing full moon, 21 November 2010. Lunar eclipse, such as on 21 December 2010 should be avoided to minimize the risk for the orbiter. Final conditions are primarily defined by the impact on the lunar surface. h1 (tf ) ≤ 0
(7)
Moreover, the impact is supposed to occur on the near side of the Moon. A more precise impact location has not yet been defined in the mission plan. For a landing on the near side we constrained the argument of the periselene ωm to lie within a certain interval defined for example by (ωm )max = +45◦ and (ωm )min = −45◦ . 1.0 −
ωm (tf ) ≥0 (ωm )max
ωm (tf ) − 1.0 ≥ 0 (ωm )min
(8)
functions. Depending on the mission priority, we identified three distinct criteria, which may later be combined by means of a weighting function. Mass being the prime limiting factor for small satellite missions, the first objective was to minimize the propellant consumption (for constant thrust equivalent to minimizing the maneuver time). With the soft landing of the surface unit in mind, the second criterion was to minimize the impact velocity in order to reduce the kinetic energy that has to be absorbed by the landing device. Finally, we attempted to maximize the impact angle, thus reducing the error ellipse around the target area and minimizing the risk of premature collision with mountain tops or crater walls.
4
Results
(9) In a first analysis we examined the use of the very low thrust PPT cluster for deorbiting. The results show that this would lead to maAs stated above, the thrust direction is neuver times of between 12 and 18 days, with given by the three components of the vecan impact quasi tangential to the lunar surtor u. To ensure that u is a unit vector, we face. These two factors imply large error elenforced lipses, due to accumulation of possible modkuk − 1 = 0 (10) elling errors over a long period of time on as path constraint. Moreover we had to intro- the one hand, and due to the strong effect duce a minimum altitude profile for some sce- of maneuver errors on the impact location on narios in order to minimize the risk of prema- the other. Under these circumstances, a conture impact. Minimum altitude profiles were trolled lunar impact cannot be achieved, and defined by fourth order polynomials, or in therefore the use of PPTs was not further exmulti-phase scenarios by limiting the height amined. of the periselene. The arcjet, however, performed reasonGiven these boundary conditions, we opti- ably well for all three optimization objecmized the descent trajectory for different cost tives, yielding mission times of about one
altitude selenocentric /km
longitude /degrees
400
1.7 1.65 1.6 1.55 0
1
2
4
3 4 real time /s
5
6
4
x 10
2 0 −2 −4
0
1
2
200
3 4 real time /s
5
3 4 real time /s
5
6
7 4
x 10
300 200 100 0
7 selenocentric latitude /deg
1.5
flight−path angle (rel) /deg
velocity rel. to moon /(km/s)
1.75
0
1
2
100
3 4 real time /s
5
3 4 real time /s
5
6
7 4
x 10
50 0 −50 −100
0
1
2
6
7 4
x 10
150 100 50 0
0
1
2
6
7 4
x 10
Figure 2: Impact trajectory with additional boost phase
day. The shortest mission time found was 64074 seconds, with a fuel consumption of 843 grams. When optimizing impact velocity or impact angle, we found that the altitude profile had to be constrained, since otherwise the optimizer created unrealistic trajectories. The minimal impact velocity was found to be 1678 m/s. This value is slightly lower than the velocity for a circular orbit at the lunar surface level. The corresponding trajectory is very close to the lunar surface, particularly during the last orbits. For this reason, minimizing only the impact velocity is not very well suited as optimization criterion. When optimizing the impact angle, we addressed this problem by introducing a minimum altitude profile as a path constraint, guaranteeing a minimum height of 5 km above the surface up to the last orbit. We obtained a maximum impact angle of -1.1◦ , at the price of a higher impact velocity of 1723 m/s. For the previous scenarios, the impact angle was
around -0.15◦ . In an attempt to improve the impact precision by increasing the impact angle, we examined the effect of a short additional highthrust boost during the last aposelene passage, provided, for example, by a small solid rocket motor. This booster provides 21.5 m/s during its 4 second burn, thereby lowering the periselene radius by 46 km. The impact occurs at an angle of -3.6◦ at a velocity of 1704 m/s (cf. figure 2). Not only did we achieve a lower impact velocity than with the arcjet alone, but the impact angle also significantly increased. A sensitivity study showed that this additional boost phase reduces the error in impact location by a factor of ten! However, this comes at the cost of the booster’s additional mass (about 3.7 kg including propellant). The optimizer SOCS used a grid with 5824 points for the computation of the scenario with the boost phase. Of the 62256 × 62256 elements of the hessian matrix, only 0.02% differ from zero – a strong argument for the usage of SOCS which exploits the characteristics of sparse matrices.
5
Conclusions
We have presented different strategies for a deorbit maneuver leading to a controlled impact of a small lunar satellite (cf. table 2). In particular, we examined trajectories that make optimal use of the low thrust electric propulsion capacities of the satellite in the presence of the strong perturbations in the lunar orbital environment.
Strategy
Duration
∆v (m/s)
mprop (kg)
vimp (m/s)
γimp (◦ )
Tangential breaking (PPT)
12.5 days
43.2
0.261
1683
-0.01
Minimal propellant consumption (arcjet)
17.8 h
42.8
0.843
1688
-0.15
Minimal impact velocity
19.9 h
47.9
0.942
1678
-0.1
Maximal impact angle
24.9 h
60.0
1.180
1723
-1.1
Maximal impact angle with arcjet and an additional boostphase
19.4 h
68.1
2.717 + 1.9
1704
-3.6
Table 2: Summary of the mission scenarios Our results show that a controlled im- [3] H.-P. Roeser, M. Auweter-Kurtz, H. P. pact using only electric propulsion is possible. Wagner, R. Laufer, S. Podhajsky, T. WegHowever, the need for a controlled impact at mann, and F. Huber. Challenges and ina specific site with minimal risk of premanovative technologies for a low cost lunar ture collision excludes the sole use of the PPT mission. In 5th IAA International Concluster. The performance of the arcjet yields ference on Low-Cost Planetary Missions, reasonable impact conditions which can be Noordwijk, The Netherlands, Sept. 2003. further improved by an additional short boost ESTEC. IRS-03-P6. maneuver. The trade off between the different criteria for optimality as well as their ef- [4] E. M. Standish. JPL Planetary and Lunar Ephemerides DE405/LE405. Technifects on the mission and on systems’ budgets cal report, NASA Jet Propulsion Laborawill be the subject of future work. tory, Aug. 1998. Interoffice Memorandum IOM 312.F–98–048.
References
[5] The Boeing Company. Boeing socs optimization software homepage. [1] J. T. Betts and S. O. Erb. Optimal low http://www.boeing.com/phantom/socs/, thrust trajectories to the moon. SIAM last visited 26 july 2004. J. Applied Dynamical Systems, 2(2):144– 170, 2003. [6] TTI GmbH, Dept. OGC. Optimization, guidance and control homepage. [2] A. S. Konopliv, S. W. Asmar, E. Carhttp://www.gesop.de/, last visited 26 ranza, W. L. Sjogren, and D. N. Yuan. July 2004. Recent gravity models as a result of the lunar prospector mission. Icarus, 150(1):1–18, Mar. 2001.
