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Error Analysis and Trajectory Correction Maneuvers of Lunar Transfer Orbit
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 37 (2013) 440–454
Error Analysis and Trajectory Correction Maneuvers of Lunar Transfer Orbit† 1
ZHAO Yu-hui1,2
HOU Xi-yun1
LIU Lin1,3
School of Astronomy and Space Science, Nanjing University, Nanjing 210093 Purple Mountain Observatory of Chinese Academy of Sciences, Nanjing 210008 3 Key Laboratory of Science and Technology on Aerospace Flight Dynamics, Beijing Aerospace Control Center, Beijing 100094 2
Abstract For a returnable lunar probe, this paper studies the characteristics of both the Earth-Moon transfer orbit and the return orbit. On the basis of the error propagation matrix, the linear equation to estimate the first midcourse trajectory correction maneuver (TCM) is figured out. Numerical simulations are performed, and the features of error propagation in lunar transfer orbit are given. The advantages, disadvantages, and applications of two TCM strategies are discussed, and the computation of the second TCM of the return orbit is also simulated under the conditions at the reentry time. Key words: planets and satellites: detection—methods: analytical—methods: numerical
1. INTRODUCTION Compared to the Earth satellites, a lunar probe travels a much larger distance and takes a much longer time to fly. Due to the errors in the mechanical model and the Telemetry, Tracking and Control (TT&C), the actual trajectory always deviates a little from the designed trajectory[1−3]. Even worse, the deviation may increase as the probe travels. To ensure the entry into the targeted circumlunar orbit and the reentry into the Earth atmosphere, it is necessary to analyze the divergence of the initial errors at the injection times † Supported by National Natural Science Foundation of China (No. 11003009, 10903002, 11203015, 11033009). Received 2012–08–21; revised version 2012–09–19 A translation of Acta Astron. Sin. Vol. 54, No. 3, pp. 261–273, 2013 [email protected]
0275-1062/13/$-see frontfront matter © 2013 B.V. All Science rights reserved. cElsevier 0275-1062/01/$-see matter 2013 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2013.10.006 PII:
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of these two segments of transfer orbits, as well as the errors accumulated in the transfer orbits, and according to these errors to make the corresponding midcourse trajectory correction maneuvers (TCMs)[4−6] . Based on the dynamical property of the transfer orbit of the lunar probe and the character of the error propagation matrix, this paper investigates the error propagations in the Earth-Moon and Moon-Earth transfer orbits. From the error transfer matrix, a linear equation to estimate the velocity increment for the first midcourse corrections in these two transfer trajectories is established. Using numerical experiments, the error propagation characters of the transfer trajectory under the real dynamical model is revealed. The properties and applications of two TCM methods, namely the target point method and the target orbit method, are discussed. Finally, taking the reentry constraint conditions into account, the second midcourse TCM for the Moon-Earth return trajectory is analyzed and computed.
2. MECHANICAL MODEL AND ERROR PROPAGATION EQUATION Consider the restricted three-body problem consisting of the Earth, the Moon and a probe. The Earth-Moon transfer orbit is the trajectory along which the probe departs from the massive primary (namely the Earth), escapes from its gravisphere by acceleration, and finally enters the gravisphere of the secondary primary (namely the Moon). And the MoonEarth transfer orbit is the opposite. In Fig.1, we show a sketch of the transfer orbit, in which M1 and M2 represent the masses of the two primaries in the restricted three-body problem. Apparently, the error propagations in these two transfer trajectories are similar to each other, even after taking into account of the various perturbations in the real system. Nevertheless, the Moon-Earth trajectory starts from a specific circumlunar orbit and some constraints must be satisfied for the probe’s reentry to the Earth atmosphere within a definite area. These initial and ending constraint conditions of the Moon-Earth trajectory are quite different from those of the Earth-Moon trajectory. Thus the error propagations and midcourse corrections of these two transfer trajectories differ from each other. There are two types of errors in the lunar probe’s transfer trajectory. One is the systematic error (mainly the error of the mechanical model) and the other one is the error arising from the orbit injection and TT&C. Assume that the mechanical model for the lunar probe is ˙ = F(X, t) , X (1) where X = (x, y, z, x, ˙ y, ˙ z) ˙ T is the probe’s state, while F(X, t) is the right-hand function. ¯ for the design of the nominal transfer orbit is denoted by The mechanical model F ˙ =F ¯ (X, t) = F (X, t) − F(X, t) . X
(2)
In which F(X, t) is the model error. If only F(X, t) is small enough, the error caused by it in the transfer orbit is negligible. The linear error propagation equation under the probe’s mechanical model of Eq.(2) reads X(t) = At0 ,t · X(t0 ) ¯ (3) ˙ t0 ,t = ∂ F · At0 ,t , At0 ,t = I6×6 , A ∂X
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Fig. 1 The sketch of the lunar probe’s transfer orbit in the restricted three-body problem of the Earth-Moon system
where I6×6 is the unit matrix, X(t0 ) and X(t) represent the deviations of the real probe state from the nominal trajectory respectively at the initial moment t0 and at the moment t afterwards, and At0 ,t is the state transition matrix (STM), i.e. the error propagation matrix mentioned above, which can be obtained through integration of the second equation in Eq.(3). The purpose of the error analysis is to find the variation of the error at the terminal time X(t) that is caused by the initial error X(t0 ). No matter the injection error or the TT&C error, we denote their values at t0 as X(t0 ). For example the measuring error, the errors in different directions (radial or transverse) may be distributed onto the different components of X(t0 ). It is the same for the injection error and control error. If the error propagation time is not too long and the magnitude of the initial error is not too large, Eq.(3) gives a good description of the variation of X(t) with X(t0 ), i.e., the different distributions of X(t0 ) in different directions result in the different variations of X(t) in different directions. Due to the strong instability of the transfer orbit, the nonlinear terms which have been ignored in our analysis may increase significantly as the integration time increases. Therefore, if the propagation time is too long or the initial error X(t0 ) is too large, Eq.(3) is no longer applicable, and an integration of the complete equation of motion is needed to make a reasonable estimation of the error X(t).
3. ERROR PROPAGATION MATRIX AND ERROR DIVERGENCE CHARACTER OF TRANSFER ORBIT
For the convenience of giving a general expression of the error propagation, dimensionless
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variables are adopted hereafter. Except where noted, the units in this paper are defined as ⎧ ⎨ M = mE + mM L=d , (4) ⎩ T = {d3 /[G(mE + mM )]}1/2 = 1/n where mE and mM are the masses of the Earth and the Moon, d is the average Earth-Moon distance, and n is the mean motion of the Moon around the Earth. Denoting the position and velocity components in the state X = (x, y, z, x, ˙ y, ˙ z) ˙ T of Eq. (1) by r = (x, y, z)T and T ˙r = (x, ˙ y, ˙ z) ˙ , the error propagation matrix given by Eq. (2) can be written as ⎞ ⎛ ∂ r(t) ∂ r (t) ˙ (t0 ) ∂X(t) ∂ r(t0 ) ∂ A1 A2 r = At0 ,t = = ⎝ ∂r˙ (t) ∂r˙ (t) ⎠ , (5) A3 A4 ∂X(t0 ) ∂ r(t0 )
∂ r˙ (t0 )
in which all the 4 matrices are 3 × 3 matrices. For high-thrust transfer trajectories with different constraints, the 9 elements for the four 3×3 matrices in A of Eq.(5) are generally of the same order of magnitude, and their variation ranges are relatively limited. No matter for the Earth-Moon transfer trajectory or the Moon-Earth trajectory, numerical experiments indicate that their orders of magnitude are as follows: ∂r(t) ∂r˙ (t) ∂r˙ (t) ∂r(t) : 100 , : 102 , : 102 , : 104 , ˙ ∂r(t0 ) ∂r(t0 ) ∂r(t0 ) ∂r˙ (t0 )
(6)
Apparently, for the deviations caused by the initial errors of the same order of magnitude, the position deviation propagates slower than the velocity deviation. However, for the initial errors of the same order of magnitude, the deviation due to the position error propagates much quicker than the one caused by the velocity error (measured in the normalized units). For an arbitrarily given Earth-Moon transfer trajectory and a Moon-Earth trajectory, both with the transferring time of 3 days, as an example, we illustrate in Fig.2 the temporal variation of the element A(2, 4) in the error propagation matrix. In Fig.2, the upper panel is for the Earth-Moon trajectory and the lower for the Moon-Earth case, and the abscissa is time in days. Apparently, the element changes slowly before the probe approaches to the target object, but when the probe arrives in the vicinity of the target object an abrupt variation happens, implying the strong instability of the probe orbit. Because the gravitation of the Earth is much stronger than the Moon’s, the element variation for the Moon-Earth trajectory at the end close to the Earth should be much stronger than the analogue variation for the Earth-Moon trajectory at the near-Moon end. A comparison between the curves in the two panels of Fig.2 proves this. Again using the element A(2, 4) in the error propagation matrix of the Earth-Moon transfer orbit as an example, we present in Fig.3 the variations of A(2, 4) with respect to time under the following conditions: (a) the variation of A(2, 4) during the whole transfer orbit; (b) the variation of A(2, 4) within the gravisphere of the Moon; (c) the variation of A(2, 4) in the two-body model (the probe and the Earth). From Fig.3 we know that the variation rate of A(2, 4) increases with time in the whole transferring. Although it grows up very quickly after the probe reaches the gravisphere of the Moon, as shown in Fig.3(b), the element A(2, 4) has attained the order of magnitude
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Fig. 2 The variation of A(2, 4) in both the Earth-Moon and Moon-Earth transfer orbits
(100 ) that A(2, 4) achieves after the probe’s arrival at the target object. Namely, the initial error is constantly amplified in the Earth-Moon transferring, the deviation of the orbit state caused by the initial error has grown too big to be ignored. Other elements in the error propagation matrix have the same behaviour as A(2, 4). Ignoring the gravitation of the Moon, the temporal variation of A(2, 4) in the twobody model is shown in Fig.3(c). Clearly, for a highly elliptical orbit under the two-body model, the variation of A(2, 4) with respect to the geocentric distance has the same profile as the segment of the curve in Fig.3(a) before the probe enters the gravisphere of the Moon. This clearly implies the fact again that the abrupt variation of the elements in the error propagation matrix of the Earth-Moon transfer orbit in the near-Moon end is caused by the gravitation of the Moon. Assuming that A is constant in a short duration, and using the formula given in Eq.(3), we obtain the state transition matrix (STM) as follows. At1 ,t2 = eF t ,
(7)
since the matrix F is the time derivative of the right-hand function in Eq.(2), it is proportional to 1/r3 , where r is the selenocentric distance when we discuss the effects from the gravitation of the Moon. In the two-body model, the Moon’s gravitation has been ignored so that we can regard r → ∞. Thus the exponential term is e0 → 1, and the divergence does not happen at all. But when the gravitation of the Moon is taken into account, r decreases and 1/r increases as the probe approaches to the Moon, so that the elements in the error propagation matrix diverge exponentially. Select arbitrarily an Earth-Moon transfer orbit with the injection epoch in the nearEarth end to be MJD57736.1449 (i.e., 03:28:39.36 on Dec.14, 2016), while the probe departs
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Fig. 3 The variation of A(2, 4) in the Earth-Moon transfer orbit
from an Earth-parking orbit of height 200 km and inclination 20.6◦. The transfer duration of this transfer orbit is 5 d and the targeted circumlunar orbit has the height 200 km and the inclination 45.3◦ . The maneuver method applied in both the near-Earth end and the nearMoon end is the tangent orbit maneuver in the orbital plane. At the injection time, an error of 10−4 is added to three components of both the position and velocity vectors. Taking the gravitation from the Earth, Moon, and the Sun, as well as the non-spherical effects of the Earth into account, we make the numerical integration on the transfer orbit. The deviations of the position and velocity along the orbit as the functions of geocentric distance are presented in Fig.4. In this figure, the abscissa is the geocentric distance expressed in dimensionless units, and the ordinate is the order of magnitude of the error (for details, see the figure caption). The state deviation, particularly the velocity deviation changes abruptly after the probe enters the gravisphere of the Moon, implying that the lunar gravitation results in the quick enlargement of the initial error in the near Moon end. Also from Fig.4, we know that the initial position error has the biggest effect on the velocity deviation, consistent with the result obtained from the analysis on the error propagation matrix. Additional numerical experiments also show that the error divergence in the Moon-Earth return orbit is very similar to that of the Earth-Moon orbit, the details we prefer to omit here. If the initial error X(t0 ) was not generated in the right beginning of the launch but at a moment t in the transfer process, e.g. it was just the residual error of the midcourse correction, the deviation of the real orbit from the nominal orbit at the moment of arrival at the Moon is shown in Fig.5. In Fig.5, the upper two panels present the variations of the final position error with respect to the time when the initial error occurs respectively in position (panel (a)) and in velocity (panel (b)). Two lower panels are for the velocity errors at the moment of arrival in the circumlunar orbit. The time is expressed in days and
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the error is expressed in dimensionless units for all four panels. The four curves in Fig.5 are declined with time, particularly in the short duration at beginning, they decrease very quickly. That is, the closer the probe is to the Moon when the error happens, the smaller the state deviation at the arrival will be. This character is determined by the stability of the orbit, and implies that the orbit correction should not be performed too early. For the error divergence of the Moon-Earth transfer orbit, similar properties can be found, which we will not discuss in detail.
Fig. 4 Errors versus the geocentric distance for the Earth-Moon transfer orbit. (a) Position error resulted from the initial position error; (b) Position error resulted from the initial velocity error; (c) Velocity error resulted from the initial position error; (d) Velocity error resulted from the initial velocity error.
4. FIRST MIDCOURSE CORRECTION FOR THE EARTH-MOON TRANSFER ORBIT UNDER LINEAR APPROXIMATION For both the Earth-Moon transfer orbit and the Moon-Earth transfer orbit, usually two midcourse corrections are executed. The orbit injection error is corrected during the first TCM just after the launch of the probe, and the second TCM is performed just before the arrival at the Moon (or the reentry of the Earth atmosphere) to remove the residual error caused by the first TCM and the other errors. Generally, the second TCM needs a relatively small amount of energy. Depending on the specific mission, more corrections may be arranged during the transfer flight. But the selection of the moments for two midcourse TCMs will influence directly the velocity increment necessary for completing the maneuver. In this paper we will study only the first TCM that consumes more energy. The first midcourse TCM should be made after a while when the orbit has been determined through adequate observations. Denote the orbit injection time by t0 , the state error at the injection time by X(t0 ) = (r(t0 ), r˙ (t0 ))T , and the epoch of the first midcourse TCM by t1 . The
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Fig. 5 The final errors as a function of the time when the initial errors occur. (a) The final position error due to position error; (b) The final position error due to velocity error; (c) The final velocity error due to position error; (d) The final velocity error due to velocity error.
orbit deviation from the nominal orbit at t1 can be calculated by Eq.(3) as follows: r(t1 ) r(t0 ) X(t ) = A = A , X(t1 ) = t0 ,t1 0 t0 ,t1 r˙ (t1 ) r˙ (t0 )
(8)
where At0 ,t1 is the state transition matrix from t0 to t1 . The correction is performed at t1 . Denoting the correction by a velocity increment p, the error propagation at the arrival of the circumlunar orbit can be calculated by Eq.(3) again[4−6]: r(t2 ) r(t1 ) X(t2 ) = = At1 ,t2 X(t1 ) = At1 ,t2 . (9) r˙ (t2 ) r˙ (t1 ) + p After the orbital correction, if the probe is required to reach the same point as the nominal orbit at t2 , r(t2 ) = 0 must be satisfied, from which the velocity increment p at t1 for the first TCM can be obtained as follows[4−6]: p = (At1 ,t2 ,2 )−1 (−At1 ,t2 ,1 r(t1 )) − r˙ (t1 ) ,
(10)
where the subscripts “1, 2” in the matrices At1 ,t2 ,1 and At1 ,t2 ,2 have the same meanings as in Eq.(5). For the above-mentioned Earth-Moon orbit, adopting an ascending node error of Ω = 0.42◦ at the injection time, we plot in Fig.6 as an example the variation of the TCM velocity increment with respect to the TCM time. The impulse is calculated by using Eq.(9) and the differential correction method. In Fig.6, the solid curve represents the result of linear approximation computed from Eq.(10), while the dashed curve is the result from the differential correction. Fig.6 shows that for a given initial error, the earlier the first TCM is made, the smaller the maneuver increment is needed, provided the accurate orbit can be
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determined from observations. It is worth noting that the two curves in Fig.5 both decrease in the beginning, reflecting the property of the initial error. Close to the Earth the orbit can be described in a two-body model, in which the velocity and position deviations due to the ascending node error are related to the the perigee argument, so that the correction amplitude may decrease along with the orbit evolution. But after the probe leaves the Earth, the overall trend of the curves is the increase with time. Meanwhile, Fig. 6 also proves that the linear approximation differs only a little from the direct integration, provided the initial error is small and the midcourse TCM takes place early. Therefore, the linear approximation given in Eq.(9) can be applied to calculate the velocity increment needed by the midcourse TCM. The accuracy requirement can be attained if the orbit injection error is not too big and the integration time is not too long. On the contrary, if X(t0 ) is big and the integration time is long, the nonlinear terms that have been omitted are too large to be neglected, and the direct numerical integration should be used to find the velocity increment for the midcourse TCM. Generally, the linear analysis introduced above can be used to deal with the correction for the injection error of the Earth-Moon transfer orbit. Moreover, as shown by Eq.(10), the velocity increment required for the first midcourse TCM has a linear relation with the initial error.
Fig. 6 The velocity increment versus the maneuver time
5. TWO MIDCOURSE TCM METHODS Practically, in an ordinary launch mission, the aim of orbit correction is to send the probe to a target orbit that satisfies the requirements of the mission, rather than to a target point, because in the latter case a proper target point must be selected to ensure the energy consumption of orbit correction be not too large, which is not an easy task. At the same time, a second orbit maneuver is still needed at the target point to make sure that the probe’s real orbit coincides absolutely with the designed orbit. For the above-mentioned
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targeted circumlunar orbit, its ascending node must be in the range of 227.26◦ ∼ 231.94◦ to meet the requirements of the lunar surface exploration task. For the initial errors listed in Table 1, we calculate the velocity increments for two TCM strategies, namely the pointto-point correction and the point-to-orbit correction methods, and the results are listed in Table 2. Table 1
Table 2 Error a a i i Ω Ω ω ω v v t0 t0
Model 1 2 1 2 1 2 1 2 1 2 1 2
Initial error of the Earth-Moon transfer orbit ω/◦ ±0.8
a/km ±8000
i/◦ ±0.4
Ω/◦ ±0.8
v/(m/s) ±1
t0 /min ±5
Comparison I of two midcourse TCM strategies in the Earth-Moon transfer orbit vE /(m/s) 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47
v1 /(m/s) 12.50 18.59 2.13 1.11 11.10 10.78 10.84 10.43 11.74 11.67 9753.32 17.10
v2 /(m/s) 2.78 1.47 4.28 4.19 1.25 1576.28
vM /(m/s) 801.32 799.73 801.32 801.33 801.32 802.92 801.32 802.94 801.32 800.91 801.32 803.84
v/(m/s) 3951.07 3952.79 3939.39 3936.91 3951.17 3948.17 3950.82 3947.84 3948.78 3947.05 15265.39 3955.41
In Table 2, Model 1 and Model 2 represent respectively the target point method and the target orbit method. The first TCM epoch is selected to be 18 h after the orbit injection. In Model 1 the target point is the point on the nominal orbit 4.6 d after the injection. In Table 2, the velocities are all given in m/s, vE and vM are the velocity increments in the near-Earth end and near-Moon end respectively, and v is the summation of velocity increments of the orbit maneuvers and midcourse TCMs from entering the EarthMoon transfer orbit to forming the circumlunar orbit. The values listed in this table show that, except for the semi-major axis error, all the velocity increments required by the orbit maneuvers in the target orbit correction method are relatively small. Particularly, when the initial error is relatively large, the target orbit correction method is much more economical in maneuver energy. For the injection time error, the target point correction method does not work at all, because the velocity increment needed by the correction is too large. For comparisons, we take account of all the initial errors except that of the injection time. The point-to-point method and the point-to-orbit strategies are applied to perform the TCM. The required minimal velocity increments are 32.80 m/s and 23.95 m/s, respectively. Right before the near-Moon braking, the orbital elements of the probe’s trajectory in the selenocentric mean equatorial coordinates are listed in Table 3. Table 3 Model 1 2
Comparison II of two midcourse TCM strategies in the Earth-Moon transfer orbit a/km 7413207.16 7319192.74
e 1.26 1.26
i/◦ 45.3 45.3
Ω/◦ 231.8 231.6
ω/◦ 152.0 311.36
M/◦ 0 0
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For the target orbit strategy, the orbital elements of the transfer orbit with respect to the Moon have changed before the near-Moon braking. This is due to the fact that the adjustable variables are restricted in the differential correction. When the number of constraints is greater than the number of adjustable variables (generally there are 3 adjustable variables, namely the 3 components of the initial velocity) and the constraints are relatively strict, the obtained target orbit may not satisfy all the constraints. As for the Moon-Earth transfer orbit, in addition to the constraints of geocentric distance, reentry angle and orbital inclination, the trajectory of the probe is also confined by the reentry flight distance which is determined by the latitude of the reentry point and the ascending node of the reentry orbit. Hence, in the midcourse TCM, a proper orbit correction method should be selected according to the corresponding conditions. 6. MIDCOURSE TCM OF MOON-EARTH TRANSFER ORBIT The first midcourse TCM in the Moon-Earth transfer orbit is primarily to correct the TT&C error of the circumlunar orbit, the injection error and the error of engine ignition time. In order to reduce the error accumulation and save the maneuver energy, the first midcourse TCM should be performed early in 24 h after the near-Moon braking. On the other hand, a too early correction will make the residual error increase quickly as the orbit evolves. As a compromise, the first TCM is selected to be 18 h after the near-Moon braking in this paper. The error of the Moon-Earth transfer orbit arises mainly from the following four sides: the error of circumlunar orbital elements, the error of the distance and velocity measurements, the error of engine ignition time, and the error of engine working duration. The former two are all due to the TT&C error for the circumlunar orbit. Assuming a circular circumlunar orbit of the lunar probe with a height of 200 km and an inclination of 45◦ , of which the transfer time is 5.83 d, the orbital ascending node at the reentry time is 34.3◦ , the reentry angle is −6◦ , and the reentry latitude is ϕ = 6.14◦. Adopting the target orbit strategy and using the height of the reentry point, orbital inclination and reentry angle as the constraints, we calculate the correction and search for the transfer orbit through the differential correction method. The TT&C error is reflected in the semi-major axis, inclination and ascending node of the circumlunar orbit. For the nominal orbit mentioned above, we calculate the velocity increment required by the first midcourse TCM. For this example, the initial errors are listed in Table 4 and the results are presented in Table 5. In Table 4, the error of the semi-major axis, i.e. the height of the circumlunar orbit, is 1 km, the inclination error is 0.6◦ , and the error of the ascending node longitude is 0.8◦ . The along-trajectory velocity error due to the engine working duration error is 24 m/s, while the offset of the near-Moon braking epoch caused by the TT&C error is 30 s. After landing on the Moon, the probe launches from the lunar surface and orbits on the circumlunar orbit. In Table 4, the near-Moon maneuver duration is denoted by tr . The error bound is set to be 3 circles, that is, the probe may jump to the Moon-Earth transfer orbit within 3 circles earlier or later than the mission schedule, depending on the progress of the mission. Table 4 a/km ±1
Initial error of the Moon-Earth transfer orbit
i/◦ ±0.6
Ω/◦ ±0.8
v/(m/s) ±24
t0 /s ±30
tr /circle ±3
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Table 5
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Results of midcourse TCMs in the Moon-Earth transfer orbit Error a i Ω v t0 tr a i Ω v t0 tr
Value 1.0 km 0.6◦ 0.8◦ 8 m/s 30 s 3 circle −1.0 km −0.6◦ −0.8◦ −8 m/s −30 s −3 circle
v/(m/s) 0.5488 9.31 5.61 26.15 10.58 8.38 0.5495 9.30 5.41 26.57 9.91 12.51
ϕ/◦ 6.143 6.08 6.10 6.22 6.06 6.51 6.137 6.20 6.18 6.06 6.21 5.70
Ω/◦ 34.30 34.38 34.35 34.07 34.39 37.71 34.31 34.23 34.27 34.55 34.23 31.98
Judging from the results listed in Table 5, before entering in the Moon-Earth transfer orbit, the most influential error is the engine ignition error, which needs a velocity increment of 26 m/s to compensate in the first midcourse TCM. This is the most significant energy consumption in the TCM. After the correction, the variations of the reentry flight distance and the longitude of ascending node are still relatively large, whereas the reentry latitude changes less than 0.3◦ and the longitude of the ascending node of the instantaneous orbit at the moment of reentry varies within a range of less than 1◦ . On the contrary, the TT&C error and the engine ignition time offset have relatively small influences on the orbit at the reentry moment. The varying ranges of the latitude of reentry point and the longitude of the orbital ascending node at the reentry moment are not larger than 0.1◦ . For the probe that enters the return trajectory 3 circles later, the TCM velocity increment is about 13 m/s. But this error now influences very strongly the position of the reentry point and the longitude of ascending node. This is due to the fact that the Moon has changed its position around the injection time of the Moon-Earth transfer orbit. Thus a new nominal orbit should be adopted as the Moon-Earth return orbit. In a practical mission, the velocity increment of the first midcourse TCM should be carefully calculated after considering all the errors. Denote the initial errors at the injection time by εi and a state (position, velocity or any orbital element) of the orbit at the moment by ε, where i indicates the different types of errors. From Eq.(10), for the first TCM, the relation between the state deviation and εi can be expressed as follows.
∂(r(t0 ), r˙ (t0 )) r(t1 ) r(t0 ) · · εi . (11) = A = A t ,t t ,t 0 1 0 1 r˙ (t1 ) r˙ (t0 ) ∂εi Considering the summation of all errors, we have
∂(r(t0 ), r˙ (t0 )) r(t0 ) r(t1 ) r(t1 ) = At0 ,t1 · = At0 ,t1 · | εi | ≥ . (12) r˙ (t1 ) r˙ (t0 ) r˙ (t1 ) ∂εi Since the velocity increment required by the midcourse TCM increases linearly with the deviation of orbital state when the deviation is small, the velocity increment in a practical Moon-Earth transfer orbit must be smaller than the summation of the velocity increments required by the respective errors. The second midcourse TCM of the Moon-Earth transfer orbit is designed to reduce the residual errors of the first correction. According to the discussion about the error
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propagation in Section 3, an earlier correction will require a smaller velocity increment, but if the second TCM is performed too early, the residual error inherited from this TCM will be magnified when the probe arrives at the Earth. In this case, a third correction may have to be made, in order to satisfy the longitudinal requirement on the orbit. Thus the second TCM should not be performed too early. As an example, we analyze the second midcourse TCM to a Moon-Earth transfer orbit with the transfer time of 5 d. The residual error of the first TCM is assumed to be 0.15 m/s, but the error direction is uncertain. A proper time is selected to perform the second TCM so that the velocity increment is not too large and the residual error will not be magnified too much at the reentry time. The first TCM is assumed to have taken place 18 h after the probe’s injection into the transfer orbit. We calculate and present in Fig.7 the velocity increment of the second TCM as a function of the maneuver time for the residual errors in different directions. In Fig.7, the curves from c1 to c4 are for the residual errors in the directions along the velocity, along the axes of geocentric equatorial coordinates, pointing to the Earth’s center, and in the normal direction of the orbital plane, respectively. The aim of the second TCM is to reduce the residual error of the first TCM. The velocity increment of the second TCM is quite small if the maneuver was carried out 24 h before the probe’s arrival at the Earth. As the probe approaches to the Earth, the error increases continuously and as a consequence the velocity increment required by the maneuver increases significantly. The direction of the initial error may have a certain influence on the maneuver, but it will not change the order of magnitude of the velocity increment. The largest correction component is determined by the residual error along the largest eigenvector of the state transition matrix (i.e. the error propagation matrix). As the probe approaches to the Earth, the gravitation of the Earth results in an abrupt change to the profile of the velocity increment curve. For example, our calculation gives a velocity increment of 1 km/s at the maneuver time of 139 h, which is far beyond the extent of Fig.7. For a velocity increment threshold of 1 m/s, the TCM should be performed around 108 h after the first TCM, while for a threshold of 10 m/s, the second TCM must be done around 134 h. For a given threshold of the velocity increment, numerical experiments show that the shorter the transfer time is, the later the TCM can be made, relatively.
7. CONCLUSIONS For a returnable lunar probe, by analyzing the error propagation matrix of the transfer orbit, we discuss in this paper the divergence of the error. Our investigation suggest: the first trajectory correction maneuver (TCM) should be performed as early as possible, since the initial error increases continuously as time goes by. However, the instability of the error propagation in the vicinities of both the Earth and the Moon denies either the first TCM that is too early or the second TCM that is too late. From the error propagation matrix, a linear approximation of the velocity increment for the first TCM is given, which works as well as the numerical integration method, provided the transfer time is not too long and the initial error is not too large. Through numerical experiments, we compare two trajectory correction strategies, namely the target point method and the target orbit method. The comparison suggests that the velocity increments obtained by these two strategies are basically equivalent
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Fig. 7 The velocity increment of the second TCM versus the maneuver time
to each other when the initial error is relatively small. Generally speaking, it is a little more economical in energy consumption if the the target orbit method is employed. Nevertheless, if the error is large or there is the error of launch time, a new target orbit should be adopted and corrected using the target orbit method. When the number of orbit constraints is more than the number of adjustable variables, or when the real trajectory is required to be strictly coincident with the target orbit, the target point correction method should be applied. Finally, adopting the target orbit method, we have analyzed the sources of the errors in the Moon-Earth transfer orbit through numerical experiments. Our calculations of two midcourse TCMs demonstrate that to save energy, the first TCM for reducing the influence of the injection error should be performed as early as possible. For the second TCM that is designed to correct the residual error of the first TCM, the required velocity increment increases with time slowly before the probe arrives in the vicinity of the Earth, where the velocity increment rises rapidly. Therefore, the second TCM should be performed when the probe approaches to the Earth, to avoid the residual error of the second TCM being magnified significantly as the orbit evolves. For the velocity increment threshold of 10 m/s, the second TCM should be done 3∼5 h before the reentry time, depending on the transfer time of the specific mission. In practical engineering, to select the proper epoches of midcourse TCMs, we also have to consider the constraints of the precisions of orbit measurement and determination, the navigation accuracy, the ability of the uplink command injection, etc. In fact, all the influences of these constraints can be transformed into the corresponding orbit deviations, then we can calculate the midcourse corrections using the methods introduced in this paper. We will not discuss these influences in detail because in this paper we focus on mainly the analysis of the error propagation from the angle of orbit dynamics, as well as the selection of proper midcourse TCM times.
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References 1
Szebehely V., Theory of Orbits. New York and London: Academic Press, 1967, 15-81
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Liu L., Hou X. Y., Orbit Dynamics of Deep Space Probe. Beijing: Publishing House of Electronics
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Liu L., Wang X., Orbit Dynamics of Lunar Probe. Beijing: National Defense Industry Press, 2006,
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Hou X. Y., Dynamical Properties and Applications of Equilibrium Points. Nanjing: Nanjing University,
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Battin R. H., An Introduction to the Mathematics and Methods of Astrodynamics. Reston: Aiaa,
Industry, 2012, 50-72 51-75 2008, 155-180 1999, 237-363 6
Gomez G., Jorba A., Simo C., et al., Dynamics and Mission Design Near Libration Points. Singapore: World Scientific Publishing, 2001, 1, 266-304
ELSEVIER
Chinese Astronomy and Astrophysics 37 (2013) 440–454
Error Analysis and Trajectory Correction Maneuvers of Lunar Transfer Orbit† 1
ZHAO Yu-hui1,2
HOU Xi-yun1
LIU Lin1,3
School of Astronomy and Space Science, Nanjing University, Nanjing 210093 Purple Mountain Observatory of Chinese Academy of Sciences, Nanjing 210008 3 Key Laboratory of Science and Technology on Aerospace Flight Dynamics, Beijing Aerospace Control Center, Beijing 100094 2
Abstract For a returnable lunar probe, this paper studies the characteristics of both the Earth-Moon transfer orbit and the return orbit. On the basis of the error propagation matrix, the linear equation to estimate the first midcourse trajectory correction maneuver (TCM) is figured out. Numerical simulations are performed, and the features of error propagation in lunar transfer orbit are given. The advantages, disadvantages, and applications of two TCM strategies are discussed, and the computation of the second TCM of the return orbit is also simulated under the conditions at the reentry time. Key words: planets and satellites: detection—methods: analytical—methods: numerical
1. INTRODUCTION Compared to the Earth satellites, a lunar probe travels a much larger distance and takes a much longer time to fly. Due to the errors in the mechanical model and the Telemetry, Tracking and Control (TT&C), the actual trajectory always deviates a little from the designed trajectory[1−3]. Even worse, the deviation may increase as the probe travels. To ensure the entry into the targeted circumlunar orbit and the reentry into the Earth atmosphere, it is necessary to analyze the divergence of the initial errors at the injection times † Supported by National Natural Science Foundation of China (No. 11003009, 10903002, 11203015, 11033009). Received 2012–08–21; revised version 2012–09–19 A translation of Acta Astron. Sin. Vol. 54, No. 3, pp. 261–273, 2013 [email protected]
0275-1062/13/$-see frontfront matter © 2013 B.V. All Science rights reserved. cElsevier 0275-1062/01/$-see matter 2013 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2013.10.006 PII:
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of these two segments of transfer orbits, as well as the errors accumulated in the transfer orbits, and according to these errors to make the corresponding midcourse trajectory correction maneuvers (TCMs)[4−6] . Based on the dynamical property of the transfer orbit of the lunar probe and the character of the error propagation matrix, this paper investigates the error propagations in the Earth-Moon and Moon-Earth transfer orbits. From the error transfer matrix, a linear equation to estimate the velocity increment for the first midcourse corrections in these two transfer trajectories is established. Using numerical experiments, the error propagation characters of the transfer trajectory under the real dynamical model is revealed. The properties and applications of two TCM methods, namely the target point method and the target orbit method, are discussed. Finally, taking the reentry constraint conditions into account, the second midcourse TCM for the Moon-Earth return trajectory is analyzed and computed.
2. MECHANICAL MODEL AND ERROR PROPAGATION EQUATION Consider the restricted three-body problem consisting of the Earth, the Moon and a probe. The Earth-Moon transfer orbit is the trajectory along which the probe departs from the massive primary (namely the Earth), escapes from its gravisphere by acceleration, and finally enters the gravisphere of the secondary primary (namely the Moon). And the MoonEarth transfer orbit is the opposite. In Fig.1, we show a sketch of the transfer orbit, in which M1 and M2 represent the masses of the two primaries in the restricted three-body problem. Apparently, the error propagations in these two transfer trajectories are similar to each other, even after taking into account of the various perturbations in the real system. Nevertheless, the Moon-Earth trajectory starts from a specific circumlunar orbit and some constraints must be satisfied for the probe’s reentry to the Earth atmosphere within a definite area. These initial and ending constraint conditions of the Moon-Earth trajectory are quite different from those of the Earth-Moon trajectory. Thus the error propagations and midcourse corrections of these two transfer trajectories differ from each other. There are two types of errors in the lunar probe’s transfer trajectory. One is the systematic error (mainly the error of the mechanical model) and the other one is the error arising from the orbit injection and TT&C. Assume that the mechanical model for the lunar probe is ˙ = F(X, t) , X (1) where X = (x, y, z, x, ˙ y, ˙ z) ˙ T is the probe’s state, while F(X, t) is the right-hand function. ¯ for the design of the nominal transfer orbit is denoted by The mechanical model F ˙ =F ¯ (X, t) = F (X, t) − F(X, t) . X
(2)
In which F(X, t) is the model error. If only F(X, t) is small enough, the error caused by it in the transfer orbit is negligible. The linear error propagation equation under the probe’s mechanical model of Eq.(2) reads X(t) = At0 ,t · X(t0 ) ¯ (3) ˙ t0 ,t = ∂ F · At0 ,t , At0 ,t = I6×6 , A ∂X
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Fig. 1 The sketch of the lunar probe’s transfer orbit in the restricted three-body problem of the Earth-Moon system
where I6×6 is the unit matrix, X(t0 ) and X(t) represent the deviations of the real probe state from the nominal trajectory respectively at the initial moment t0 and at the moment t afterwards, and At0 ,t is the state transition matrix (STM), i.e. the error propagation matrix mentioned above, which can be obtained through integration of the second equation in Eq.(3). The purpose of the error analysis is to find the variation of the error at the terminal time X(t) that is caused by the initial error X(t0 ). No matter the injection error or the TT&C error, we denote their values at t0 as X(t0 ). For example the measuring error, the errors in different directions (radial or transverse) may be distributed onto the different components of X(t0 ). It is the same for the injection error and control error. If the error propagation time is not too long and the magnitude of the initial error is not too large, Eq.(3) gives a good description of the variation of X(t) with X(t0 ), i.e., the different distributions of X(t0 ) in different directions result in the different variations of X(t) in different directions. Due to the strong instability of the transfer orbit, the nonlinear terms which have been ignored in our analysis may increase significantly as the integration time increases. Therefore, if the propagation time is too long or the initial error X(t0 ) is too large, Eq.(3) is no longer applicable, and an integration of the complete equation of motion is needed to make a reasonable estimation of the error X(t).
3. ERROR PROPAGATION MATRIX AND ERROR DIVERGENCE CHARACTER OF TRANSFER ORBIT
For the convenience of giving a general expression of the error propagation, dimensionless
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variables are adopted hereafter. Except where noted, the units in this paper are defined as ⎧ ⎨ M = mE + mM L=d , (4) ⎩ T = {d3 /[G(mE + mM )]}1/2 = 1/n where mE and mM are the masses of the Earth and the Moon, d is the average Earth-Moon distance, and n is the mean motion of the Moon around the Earth. Denoting the position and velocity components in the state X = (x, y, z, x, ˙ y, ˙ z) ˙ T of Eq. (1) by r = (x, y, z)T and T ˙r = (x, ˙ y, ˙ z) ˙ , the error propagation matrix given by Eq. (2) can be written as ⎞ ⎛ ∂ r(t) ∂ r (t) ˙ (t0 ) ∂X(t) ∂ r(t0 ) ∂ A1 A2 r = At0 ,t = = ⎝ ∂r˙ (t) ∂r˙ (t) ⎠ , (5) A3 A4 ∂X(t0 ) ∂ r(t0 )
∂ r˙ (t0 )
in which all the 4 matrices are 3 × 3 matrices. For high-thrust transfer trajectories with different constraints, the 9 elements for the four 3×3 matrices in A of Eq.(5) are generally of the same order of magnitude, and their variation ranges are relatively limited. No matter for the Earth-Moon transfer trajectory or the Moon-Earth trajectory, numerical experiments indicate that their orders of magnitude are as follows: ∂r(t) ∂r˙ (t) ∂r˙ (t) ∂r(t) : 100 , : 102 , : 102 , : 104 , ˙ ∂r(t0 ) ∂r(t0 ) ∂r(t0 ) ∂r˙ (t0 )
(6)
Apparently, for the deviations caused by the initial errors of the same order of magnitude, the position deviation propagates slower than the velocity deviation. However, for the initial errors of the same order of magnitude, the deviation due to the position error propagates much quicker than the one caused by the velocity error (measured in the normalized units). For an arbitrarily given Earth-Moon transfer trajectory and a Moon-Earth trajectory, both with the transferring time of 3 days, as an example, we illustrate in Fig.2 the temporal variation of the element A(2, 4) in the error propagation matrix. In Fig.2, the upper panel is for the Earth-Moon trajectory and the lower for the Moon-Earth case, and the abscissa is time in days. Apparently, the element changes slowly before the probe approaches to the target object, but when the probe arrives in the vicinity of the target object an abrupt variation happens, implying the strong instability of the probe orbit. Because the gravitation of the Earth is much stronger than the Moon’s, the element variation for the Moon-Earth trajectory at the end close to the Earth should be much stronger than the analogue variation for the Earth-Moon trajectory at the near-Moon end. A comparison between the curves in the two panels of Fig.2 proves this. Again using the element A(2, 4) in the error propagation matrix of the Earth-Moon transfer orbit as an example, we present in Fig.3 the variations of A(2, 4) with respect to time under the following conditions: (a) the variation of A(2, 4) during the whole transfer orbit; (b) the variation of A(2, 4) within the gravisphere of the Moon; (c) the variation of A(2, 4) in the two-body model (the probe and the Earth). From Fig.3 we know that the variation rate of A(2, 4) increases with time in the whole transferring. Although it grows up very quickly after the probe reaches the gravisphere of the Moon, as shown in Fig.3(b), the element A(2, 4) has attained the order of magnitude
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Fig. 2 The variation of A(2, 4) in both the Earth-Moon and Moon-Earth transfer orbits
(100 ) that A(2, 4) achieves after the probe’s arrival at the target object. Namely, the initial error is constantly amplified in the Earth-Moon transferring, the deviation of the orbit state caused by the initial error has grown too big to be ignored. Other elements in the error propagation matrix have the same behaviour as A(2, 4). Ignoring the gravitation of the Moon, the temporal variation of A(2, 4) in the twobody model is shown in Fig.3(c). Clearly, for a highly elliptical orbit under the two-body model, the variation of A(2, 4) with respect to the geocentric distance has the same profile as the segment of the curve in Fig.3(a) before the probe enters the gravisphere of the Moon. This clearly implies the fact again that the abrupt variation of the elements in the error propagation matrix of the Earth-Moon transfer orbit in the near-Moon end is caused by the gravitation of the Moon. Assuming that A is constant in a short duration, and using the formula given in Eq.(3), we obtain the state transition matrix (STM) as follows. At1 ,t2 = eF t ,
(7)
since the matrix F is the time derivative of the right-hand function in Eq.(2), it is proportional to 1/r3 , where r is the selenocentric distance when we discuss the effects from the gravitation of the Moon. In the two-body model, the Moon’s gravitation has been ignored so that we can regard r → ∞. Thus the exponential term is e0 → 1, and the divergence does not happen at all. But when the gravitation of the Moon is taken into account, r decreases and 1/r increases as the probe approaches to the Moon, so that the elements in the error propagation matrix diverge exponentially. Select arbitrarily an Earth-Moon transfer orbit with the injection epoch in the nearEarth end to be MJD57736.1449 (i.e., 03:28:39.36 on Dec.14, 2016), while the probe departs
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Fig. 3 The variation of A(2, 4) in the Earth-Moon transfer orbit
from an Earth-parking orbit of height 200 km and inclination 20.6◦. The transfer duration of this transfer orbit is 5 d and the targeted circumlunar orbit has the height 200 km and the inclination 45.3◦ . The maneuver method applied in both the near-Earth end and the nearMoon end is the tangent orbit maneuver in the orbital plane. At the injection time, an error of 10−4 is added to three components of both the position and velocity vectors. Taking the gravitation from the Earth, Moon, and the Sun, as well as the non-spherical effects of the Earth into account, we make the numerical integration on the transfer orbit. The deviations of the position and velocity along the orbit as the functions of geocentric distance are presented in Fig.4. In this figure, the abscissa is the geocentric distance expressed in dimensionless units, and the ordinate is the order of magnitude of the error (for details, see the figure caption). The state deviation, particularly the velocity deviation changes abruptly after the probe enters the gravisphere of the Moon, implying that the lunar gravitation results in the quick enlargement of the initial error in the near Moon end. Also from Fig.4, we know that the initial position error has the biggest effect on the velocity deviation, consistent with the result obtained from the analysis on the error propagation matrix. Additional numerical experiments also show that the error divergence in the Moon-Earth return orbit is very similar to that of the Earth-Moon orbit, the details we prefer to omit here. If the initial error X(t0 ) was not generated in the right beginning of the launch but at a moment t in the transfer process, e.g. it was just the residual error of the midcourse correction, the deviation of the real orbit from the nominal orbit at the moment of arrival at the Moon is shown in Fig.5. In Fig.5, the upper two panels present the variations of the final position error with respect to the time when the initial error occurs respectively in position (panel (a)) and in velocity (panel (b)). Two lower panels are for the velocity errors at the moment of arrival in the circumlunar orbit. The time is expressed in days and
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the error is expressed in dimensionless units for all four panels. The four curves in Fig.5 are declined with time, particularly in the short duration at beginning, they decrease very quickly. That is, the closer the probe is to the Moon when the error happens, the smaller the state deviation at the arrival will be. This character is determined by the stability of the orbit, and implies that the orbit correction should not be performed too early. For the error divergence of the Moon-Earth transfer orbit, similar properties can be found, which we will not discuss in detail.
Fig. 4 Errors versus the geocentric distance for the Earth-Moon transfer orbit. (a) Position error resulted from the initial position error; (b) Position error resulted from the initial velocity error; (c) Velocity error resulted from the initial position error; (d) Velocity error resulted from the initial velocity error.
4. FIRST MIDCOURSE CORRECTION FOR THE EARTH-MOON TRANSFER ORBIT UNDER LINEAR APPROXIMATION For both the Earth-Moon transfer orbit and the Moon-Earth transfer orbit, usually two midcourse corrections are executed. The orbit injection error is corrected during the first TCM just after the launch of the probe, and the second TCM is performed just before the arrival at the Moon (or the reentry of the Earth atmosphere) to remove the residual error caused by the first TCM and the other errors. Generally, the second TCM needs a relatively small amount of energy. Depending on the specific mission, more corrections may be arranged during the transfer flight. But the selection of the moments for two midcourse TCMs will influence directly the velocity increment necessary for completing the maneuver. In this paper we will study only the first TCM that consumes more energy. The first midcourse TCM should be made after a while when the orbit has been determined through adequate observations. Denote the orbit injection time by t0 , the state error at the injection time by X(t0 ) = (r(t0 ), r˙ (t0 ))T , and the epoch of the first midcourse TCM by t1 . The
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Fig. 5 The final errors as a function of the time when the initial errors occur. (a) The final position error due to position error; (b) The final position error due to velocity error; (c) The final velocity error due to position error; (d) The final velocity error due to velocity error.
orbit deviation from the nominal orbit at t1 can be calculated by Eq.(3) as follows: r(t1 ) r(t0 ) X(t ) = A = A , X(t1 ) = t0 ,t1 0 t0 ,t1 r˙ (t1 ) r˙ (t0 )
(8)
where At0 ,t1 is the state transition matrix from t0 to t1 . The correction is performed at t1 . Denoting the correction by a velocity increment p, the error propagation at the arrival of the circumlunar orbit can be calculated by Eq.(3) again[4−6]: r(t2 ) r(t1 ) X(t2 ) = = At1 ,t2 X(t1 ) = At1 ,t2 . (9) r˙ (t2 ) r˙ (t1 ) + p After the orbital correction, if the probe is required to reach the same point as the nominal orbit at t2 , r(t2 ) = 0 must be satisfied, from which the velocity increment p at t1 for the first TCM can be obtained as follows[4−6]: p = (At1 ,t2 ,2 )−1 (−At1 ,t2 ,1 r(t1 )) − r˙ (t1 ) ,
(10)
where the subscripts “1, 2” in the matrices At1 ,t2 ,1 and At1 ,t2 ,2 have the same meanings as in Eq.(5). For the above-mentioned Earth-Moon orbit, adopting an ascending node error of Ω = 0.42◦ at the injection time, we plot in Fig.6 as an example the variation of the TCM velocity increment with respect to the TCM time. The impulse is calculated by using Eq.(9) and the differential correction method. In Fig.6, the solid curve represents the result of linear approximation computed from Eq.(10), while the dashed curve is the result from the differential correction. Fig.6 shows that for a given initial error, the earlier the first TCM is made, the smaller the maneuver increment is needed, provided the accurate orbit can be
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determined from observations. It is worth noting that the two curves in Fig.5 both decrease in the beginning, reflecting the property of the initial error. Close to the Earth the orbit can be described in a two-body model, in which the velocity and position deviations due to the ascending node error are related to the the perigee argument, so that the correction amplitude may decrease along with the orbit evolution. But after the probe leaves the Earth, the overall trend of the curves is the increase with time. Meanwhile, Fig. 6 also proves that the linear approximation differs only a little from the direct integration, provided the initial error is small and the midcourse TCM takes place early. Therefore, the linear approximation given in Eq.(9) can be applied to calculate the velocity increment needed by the midcourse TCM. The accuracy requirement can be attained if the orbit injection error is not too big and the integration time is not too long. On the contrary, if X(t0 ) is big and the integration time is long, the nonlinear terms that have been omitted are too large to be neglected, and the direct numerical integration should be used to find the velocity increment for the midcourse TCM. Generally, the linear analysis introduced above can be used to deal with the correction for the injection error of the Earth-Moon transfer orbit. Moreover, as shown by Eq.(10), the velocity increment required for the first midcourse TCM has a linear relation with the initial error.
Fig. 6 The velocity increment versus the maneuver time
5. TWO MIDCOURSE TCM METHODS Practically, in an ordinary launch mission, the aim of orbit correction is to send the probe to a target orbit that satisfies the requirements of the mission, rather than to a target point, because in the latter case a proper target point must be selected to ensure the energy consumption of orbit correction be not too large, which is not an easy task. At the same time, a second orbit maneuver is still needed at the target point to make sure that the probe’s real orbit coincides absolutely with the designed orbit. For the above-mentioned
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targeted circumlunar orbit, its ascending node must be in the range of 227.26◦ ∼ 231.94◦ to meet the requirements of the lunar surface exploration task. For the initial errors listed in Table 1, we calculate the velocity increments for two TCM strategies, namely the pointto-point correction and the point-to-orbit correction methods, and the results are listed in Table 2. Table 1
Table 2 Error a a i i Ω Ω ω ω v v t0 t0
Model 1 2 1 2 1 2 1 2 1 2 1 2
Initial error of the Earth-Moon transfer orbit ω/◦ ±0.8
a/km ±8000
i/◦ ±0.4
Ω/◦ ±0.8
v/(m/s) ±1
t0 /min ±5
Comparison I of two midcourse TCM strategies in the Earth-Moon transfer orbit vE /(m/s) 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47 3134.47
v1 /(m/s) 12.50 18.59 2.13 1.11 11.10 10.78 10.84 10.43 11.74 11.67 9753.32 17.10
v2 /(m/s) 2.78 1.47 4.28 4.19 1.25 1576.28
vM /(m/s) 801.32 799.73 801.32 801.33 801.32 802.92 801.32 802.94 801.32 800.91 801.32 803.84
v/(m/s) 3951.07 3952.79 3939.39 3936.91 3951.17 3948.17 3950.82 3947.84 3948.78 3947.05 15265.39 3955.41
In Table 2, Model 1 and Model 2 represent respectively the target point method and the target orbit method. The first TCM epoch is selected to be 18 h after the orbit injection. In Model 1 the target point is the point on the nominal orbit 4.6 d after the injection. In Table 2, the velocities are all given in m/s, vE and vM are the velocity increments in the near-Earth end and near-Moon end respectively, and v is the summation of velocity increments of the orbit maneuvers and midcourse TCMs from entering the EarthMoon transfer orbit to forming the circumlunar orbit. The values listed in this table show that, except for the semi-major axis error, all the velocity increments required by the orbit maneuvers in the target orbit correction method are relatively small. Particularly, when the initial error is relatively large, the target orbit correction method is much more economical in maneuver energy. For the injection time error, the target point correction method does not work at all, because the velocity increment needed by the correction is too large. For comparisons, we take account of all the initial errors except that of the injection time. The point-to-point method and the point-to-orbit strategies are applied to perform the TCM. The required minimal velocity increments are 32.80 m/s and 23.95 m/s, respectively. Right before the near-Moon braking, the orbital elements of the probe’s trajectory in the selenocentric mean equatorial coordinates are listed in Table 3. Table 3 Model 1 2
Comparison II of two midcourse TCM strategies in the Earth-Moon transfer orbit a/km 7413207.16 7319192.74
e 1.26 1.26
i/◦ 45.3 45.3
Ω/◦ 231.8 231.6
ω/◦ 152.0 311.36
M/◦ 0 0
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For the target orbit strategy, the orbital elements of the transfer orbit with respect to the Moon have changed before the near-Moon braking. This is due to the fact that the adjustable variables are restricted in the differential correction. When the number of constraints is greater than the number of adjustable variables (generally there are 3 adjustable variables, namely the 3 components of the initial velocity) and the constraints are relatively strict, the obtained target orbit may not satisfy all the constraints. As for the Moon-Earth transfer orbit, in addition to the constraints of geocentric distance, reentry angle and orbital inclination, the trajectory of the probe is also confined by the reentry flight distance which is determined by the latitude of the reentry point and the ascending node of the reentry orbit. Hence, in the midcourse TCM, a proper orbit correction method should be selected according to the corresponding conditions. 6. MIDCOURSE TCM OF MOON-EARTH TRANSFER ORBIT The first midcourse TCM in the Moon-Earth transfer orbit is primarily to correct the TT&C error of the circumlunar orbit, the injection error and the error of engine ignition time. In order to reduce the error accumulation and save the maneuver energy, the first midcourse TCM should be performed early in 24 h after the near-Moon braking. On the other hand, a too early correction will make the residual error increase quickly as the orbit evolves. As a compromise, the first TCM is selected to be 18 h after the near-Moon braking in this paper. The error of the Moon-Earth transfer orbit arises mainly from the following four sides: the error of circumlunar orbital elements, the error of the distance and velocity measurements, the error of engine ignition time, and the error of engine working duration. The former two are all due to the TT&C error for the circumlunar orbit. Assuming a circular circumlunar orbit of the lunar probe with a height of 200 km and an inclination of 45◦ , of which the transfer time is 5.83 d, the orbital ascending node at the reentry time is 34.3◦ , the reentry angle is −6◦ , and the reentry latitude is ϕ = 6.14◦. Adopting the target orbit strategy and using the height of the reentry point, orbital inclination and reentry angle as the constraints, we calculate the correction and search for the transfer orbit through the differential correction method. The TT&C error is reflected in the semi-major axis, inclination and ascending node of the circumlunar orbit. For the nominal orbit mentioned above, we calculate the velocity increment required by the first midcourse TCM. For this example, the initial errors are listed in Table 4 and the results are presented in Table 5. In Table 4, the error of the semi-major axis, i.e. the height of the circumlunar orbit, is 1 km, the inclination error is 0.6◦ , and the error of the ascending node longitude is 0.8◦ . The along-trajectory velocity error due to the engine working duration error is 24 m/s, while the offset of the near-Moon braking epoch caused by the TT&C error is 30 s. After landing on the Moon, the probe launches from the lunar surface and orbits on the circumlunar orbit. In Table 4, the near-Moon maneuver duration is denoted by tr . The error bound is set to be 3 circles, that is, the probe may jump to the Moon-Earth transfer orbit within 3 circles earlier or later than the mission schedule, depending on the progress of the mission. Table 4 a/km ±1
Initial error of the Moon-Earth transfer orbit
i/◦ ±0.6
Ω/◦ ±0.8
v/(m/s) ±24
t0 /s ±30
tr /circle ±3
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Table 5
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Results of midcourse TCMs in the Moon-Earth transfer orbit Error a i Ω v t0 tr a i Ω v t0 tr
Value 1.0 km 0.6◦ 0.8◦ 8 m/s 30 s 3 circle −1.0 km −0.6◦ −0.8◦ −8 m/s −30 s −3 circle
v/(m/s) 0.5488 9.31 5.61 26.15 10.58 8.38 0.5495 9.30 5.41 26.57 9.91 12.51
ϕ/◦ 6.143 6.08 6.10 6.22 6.06 6.51 6.137 6.20 6.18 6.06 6.21 5.70
Ω/◦ 34.30 34.38 34.35 34.07 34.39 37.71 34.31 34.23 34.27 34.55 34.23 31.98
Judging from the results listed in Table 5, before entering in the Moon-Earth transfer orbit, the most influential error is the engine ignition error, which needs a velocity increment of 26 m/s to compensate in the first midcourse TCM. This is the most significant energy consumption in the TCM. After the correction, the variations of the reentry flight distance and the longitude of ascending node are still relatively large, whereas the reentry latitude changes less than 0.3◦ and the longitude of the ascending node of the instantaneous orbit at the moment of reentry varies within a range of less than 1◦ . On the contrary, the TT&C error and the engine ignition time offset have relatively small influences on the orbit at the reentry moment. The varying ranges of the latitude of reentry point and the longitude of the orbital ascending node at the reentry moment are not larger than 0.1◦ . For the probe that enters the return trajectory 3 circles later, the TCM velocity increment is about 13 m/s. But this error now influences very strongly the position of the reentry point and the longitude of ascending node. This is due to the fact that the Moon has changed its position around the injection time of the Moon-Earth transfer orbit. Thus a new nominal orbit should be adopted as the Moon-Earth return orbit. In a practical mission, the velocity increment of the first midcourse TCM should be carefully calculated after considering all the errors. Denote the initial errors at the injection time by εi and a state (position, velocity or any orbital element) of the orbit at the moment by ε, where i indicates the different types of errors. From Eq.(10), for the first TCM, the relation between the state deviation and εi can be expressed as follows.
∂(r(t0 ), r˙ (t0 )) r(t1 ) r(t0 ) · · εi . (11) = A = A t ,t t ,t 0 1 0 1 r˙ (t1 ) r˙ (t0 ) ∂εi Considering the summation of all errors, we have
∂(r(t0 ), r˙ (t0 )) r(t0 ) r(t1 ) r(t1 ) = At0 ,t1 · = At0 ,t1 · | εi | ≥ . (12) r˙ (t1 ) r˙ (t0 ) r˙ (t1 ) ∂εi Since the velocity increment required by the midcourse TCM increases linearly with the deviation of orbital state when the deviation is small, the velocity increment in a practical Moon-Earth transfer orbit must be smaller than the summation of the velocity increments required by the respective errors. The second midcourse TCM of the Moon-Earth transfer orbit is designed to reduce the residual errors of the first correction. According to the discussion about the error
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propagation in Section 3, an earlier correction will require a smaller velocity increment, but if the second TCM is performed too early, the residual error inherited from this TCM will be magnified when the probe arrives at the Earth. In this case, a third correction may have to be made, in order to satisfy the longitudinal requirement on the orbit. Thus the second TCM should not be performed too early. As an example, we analyze the second midcourse TCM to a Moon-Earth transfer orbit with the transfer time of 5 d. The residual error of the first TCM is assumed to be 0.15 m/s, but the error direction is uncertain. A proper time is selected to perform the second TCM so that the velocity increment is not too large and the residual error will not be magnified too much at the reentry time. The first TCM is assumed to have taken place 18 h after the probe’s injection into the transfer orbit. We calculate and present in Fig.7 the velocity increment of the second TCM as a function of the maneuver time for the residual errors in different directions. In Fig.7, the curves from c1 to c4 are for the residual errors in the directions along the velocity, along the axes of geocentric equatorial coordinates, pointing to the Earth’s center, and in the normal direction of the orbital plane, respectively. The aim of the second TCM is to reduce the residual error of the first TCM. The velocity increment of the second TCM is quite small if the maneuver was carried out 24 h before the probe’s arrival at the Earth. As the probe approaches to the Earth, the error increases continuously and as a consequence the velocity increment required by the maneuver increases significantly. The direction of the initial error may have a certain influence on the maneuver, but it will not change the order of magnitude of the velocity increment. The largest correction component is determined by the residual error along the largest eigenvector of the state transition matrix (i.e. the error propagation matrix). As the probe approaches to the Earth, the gravitation of the Earth results in an abrupt change to the profile of the velocity increment curve. For example, our calculation gives a velocity increment of 1 km/s at the maneuver time of 139 h, which is far beyond the extent of Fig.7. For a velocity increment threshold of 1 m/s, the TCM should be performed around 108 h after the first TCM, while for a threshold of 10 m/s, the second TCM must be done around 134 h. For a given threshold of the velocity increment, numerical experiments show that the shorter the transfer time is, the later the TCM can be made, relatively.
7. CONCLUSIONS For a returnable lunar probe, by analyzing the error propagation matrix of the transfer orbit, we discuss in this paper the divergence of the error. Our investigation suggest: the first trajectory correction maneuver (TCM) should be performed as early as possible, since the initial error increases continuously as time goes by. However, the instability of the error propagation in the vicinities of both the Earth and the Moon denies either the first TCM that is too early or the second TCM that is too late. From the error propagation matrix, a linear approximation of the velocity increment for the first TCM is given, which works as well as the numerical integration method, provided the transfer time is not too long and the initial error is not too large. Through numerical experiments, we compare two trajectory correction strategies, namely the target point method and the target orbit method. The comparison suggests that the velocity increments obtained by these two strategies are basically equivalent
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Fig. 7 The velocity increment of the second TCM versus the maneuver time
to each other when the initial error is relatively small. Generally speaking, it is a little more economical in energy consumption if the the target orbit method is employed. Nevertheless, if the error is large or there is the error of launch time, a new target orbit should be adopted and corrected using the target orbit method. When the number of orbit constraints is more than the number of adjustable variables, or when the real trajectory is required to be strictly coincident with the target orbit, the target point correction method should be applied. Finally, adopting the target orbit method, we have analyzed the sources of the errors in the Moon-Earth transfer orbit through numerical experiments. Our calculations of two midcourse TCMs demonstrate that to save energy, the first TCM for reducing the influence of the injection error should be performed as early as possible. For the second TCM that is designed to correct the residual error of the first TCM, the required velocity increment increases with time slowly before the probe arrives in the vicinity of the Earth, where the velocity increment rises rapidly. Therefore, the second TCM should be performed when the probe approaches to the Earth, to avoid the residual error of the second TCM being magnified significantly as the orbit evolves. For the velocity increment threshold of 10 m/s, the second TCM should be done 3∼5 h before the reentry time, depending on the transfer time of the specific mission. In practical engineering, to select the proper epoches of midcourse TCMs, we also have to consider the constraints of the precisions of orbit measurement and determination, the navigation accuracy, the ability of the uplink command injection, etc. In fact, all the influences of these constraints can be transformed into the corresponding orbit deviations, then we can calculate the midcourse corrections using the methods introduced in this paper. We will not discuss these influences in detail because in this paper we focus on mainly the analysis of the error propagation from the angle of orbit dynamics, as well as the selection of proper midcourse TCM times.
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