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Trajectory correction for lunar flyby transfers to libration point orbits using continuous thrust

Trajectory correction for lunar flyby transfers to libration point orbits using continuous thrust Astrodynamics https://doi.org/10.1007/s42064-020-0097-2 Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 1 2 Yi Qi , Anton de Ruiter (B) 1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China 2. Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada ABSTRACT KEYWORDS Trajectory corrections for lunar yby transfers to Sun{Earth/Moon libration point orbits libration point orbit (LPOs) with continuous thrusts are investigated using an ephemeris model. The lunar trajectory correction yby transfer has special geometrical and dynamical structures; therefore, its trajectory Earth{Moon system correction strategy is considerably di erent from that of previous studies and should continuous thrust be speci cally designed. In this paper, we rst propose a control strategy based on backstepping technique the backstepping technique with a dead-band scheme using an ephemeris model. The initial error caused by the launch time error is considered. Since the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon, we adopt two sets of control parameters in two portions before and after the lunar yby, respectively. Subsequently, practical constraints owing to the navigation and propellant systems are introduced in the dynamical model of the trajectory correction. Using a prograde type 2 orbit as an example, numerical simulations show that our control strategy can eciently address trajectory corrections for lunar yby transfers Research Article with di erent practical constraints. In addition, we analyze the e ects of the navigation Received: 6 August 2020 intervals and dead-band scheme on trajectory corrections. Finally, trajectory corrections Accepted: 25 October 2020 for di erent lunar yby transfers are depicted and compared. © The Author(s) 2020 1 Introduction v requirement owing to a tangential perigee velocity error and have the risk of being unable to perform the Because of the special dynamical properties of Sun{ correction maneuver on day two. Therefore, Renk and Earth/Moon libration point orbits (LPOs), many scien- Landgraf presented an indirect strategy to mitigate the ti c and exploration missions to Sun{Earth/Moon LPOs criticality of the rst correction maneuver of the transfer have been implemented, such as ISEE-3 [1], WIND [2], towards Sun{Earth LPOs by including an intermediate, and SOHO [3] for Sun{Earth/Moon L LPOs, MAP [4], highly elliptical parking orbit [9]. Using the CHANG'E-2 GAIA [5], and CHANG'E-2 [6] for Sun{Earth/Moon L extension mission as an example, Peng et al. proposed LPOs. an ecient GPU parallel computing technique to nu- Many researchers focused on transfer problems to Sun{ merically search for transfers from a lunar orbit to the Earth/Moon LPOs. G omez et al. applied the invariant Sun{Earth L LPOs with di erent departing conditions manifolds associated with the LPOs to construct trans- using the patched elliptic restricted three-body problem fers from a low Earth orbit (LEO) to a Sun{Earth L model [10]. Their computationally ecient methodology halo orbit [7]. Based on the solution of the Lambert obtained results almost identical to those of the ephemeris problem in the restricted three-body problem described model and exhibited signi cant speedups. Qi et al. com- by the Hill equations, Sukhanov and Prado proposed a bined the technique of lunar yby using the dynamical design method for LEO-to-halo and halo-to-halo trans- fers [8]. Direct transfers to Sun{Earth LPOs increase the system approach, and they investigated lunar yby trans- B aderuiter@ryerson.ca 2 Y. Qi, A. de Ruiter fers from an LEO to Sun{Earth/Moon LPOs [11]. The Sun{Earth/Moon LPOs with continuous thrusts. Simi- trajectory correction maneuver (TCM) problem is a signi- lar to the analysis in Ref. [18], according to the special cant problem associated with transfers to LPOs since geometrical structure of lunar yby transfers, which are divided into two portions before and after lunar yby, we perturbations and errors are inevitable during practi- postulate that a new design method must be developed to cal transfer missions. Farquhar et al. studied TCMs address the trajectory correction for lunar yby transfers. in the early transfer phase section of the ISEE-3 mis- Since the design of the reference transfers (de ned as the sion [12]. Serban et al. investigated the TCM problem of transfer trajectory with no error) was adequately solved the Genesis Discovery Mission using optimal control to in the previous study [11], in this paper, we assume that compensate for launch vehicle errors, and they proposed the reference transfers are provided. two strategies to solve the TCM problem: the halo orbit In this paper, trajectory corrections for lunar yby insertion (HOI) and the manifold orbit insertion (MOI) transfers to Sun{Earth/Moon LPOs are investigated us- techniques [13]. G omez et al. presented a TCM strategy ing the ephemeris model. In contrast to the previous similar to the MOI technique for the TCM problem of study [18], the propellant system in this study was the the Genesis Mission, but they used a multiple shoot- continuous thrust rather than the impulsive thrust. We ing method instead of an optimal control procedure to assume that the initial error is the launch time error, address the TCM problem with a strong hyperbolic be- and we propose a control strategy based on the back- havior of the orbits [14]. To correct the control errors and stepping technique with a dead-band scheme. The back- orbit determination errors, Wu et al. investigated the tra- stepping technique has been widely applied in station- jectory maneuvers before the Lissajous orbit insertion of keeping [19, 20] and attitude tracking [21]. To the best of the CHANG'E-2 mission from the Moon-circling orbit to our knowledge, this is the rst time that the backstep- Sun{Earth L [6]. Xu and Xu applied the stochastic con- ping technique is used in trajectory corrections for lunar trol theory for discrete linear stochastic systems to design yby transfers. In contrast to the traditional applications a timing closed-loop TCM strategy during the transfer of the backstepping technique, such as station-keeping from an LEO to a Sun{Earth halo orbit in the circular and attitude tracking, since the perturbed transfers sig- restricted three-body problem (CRTBP) [15]. Salmani ni cantly diverge from the reference transfers after the and Busk  ens proposed a real-time control method for spacecraft passes by the Moon, we should use two sets the TCM of transfers to Sun{Earth L halo orbits in the of control parameters in two portions before and after Sun{Earth{Moon bicircular model [16]. They used an the lunar yby. Compared with Ref. [18], more practical optimal control problem to prevent disturbances such constraints in the trajectory correction will be considered as solar radiation and winds. Peng et al. researched in this paper, such as the dead-band scheme, navigation the TCM problem of transferring a spacecraft with low intervals, execution error, and limitation of the thrust thrust from an LEO to a Sun{Earth L halo orbit using engine. Therefore, the control strategy proposed herein is a receding horizon control method [17]. more realistic than that in the previous study. Through Qi and de Ruiter investigated the TCM problem of numerical simulations, we can analyze the e ects of navi- lunar yby transfers to Sun{Earth/Moon LPOs in the gation intervals and dead-band schemes on trajectory ephemeris model, and proposed several TCM strategies corrections for di erent lunar yby transfers. for lunar yby transfers under practical constraints [18]. The structure of this paper can be divided into ve As stated in Ref. [18], the lunar yby transfer is a high- parts. In Section 2, we introduce the background, in- yielding but high-risk design method because the per- cluding the ephemeris model and lunar yby transfers. turbed lunar yby transfers diverge signi cantly from In Section 3, we propose a control strategy for trajec- the reference transfers after the spacecraft passes by the tory correction using continuous thrusts. In Section 4, Moon. A two-impulse TCM must be executed before practical constraints are introduced into the control strat- the lunar yby for transfers from the Earth to LPOs to egy. In Section 5, numerical simulations are implemented promptly restrain the divergence. In this paper, we focus and analyzed. Finally, the conclusions are presented in on the trajectory correction for lunar yby transfers to Section 6. Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 3 Table 1 Spacecraft properties 2 Background Name Symbol Value 2.1 Ephemeris model Initial spacecraft mass m 500 kg Maximum thrust T 26 mN max Here, we use the J2000 Earth-centered inertial (ECI) Engine number n 2 frame to describe the motion of the spacecraft on lunar Speci c impulse I 1000 s sp yby transfers to LPOs.  is the position vector of the analyzed. Based on Ref. [11], eight types of lunar yby spacecraft in the ECI frame. P and i 2 = f Sun, transfers exist: prograde types 1{4 transfers and retro- Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, grade types 1{4 transfers. The eight types of lunar yby Neptuneg represent the position vectors of the perturbing transfers exhibit no apparent di erences in terms of fuel gravity bodies in the ECI frame. The position data of consumption. However, for L target LPOs, prograde and celestial bodies in the ECI frame can be obtained from the 1 Jet Propulsion Laboratory (JPL) ephemeris DE430 [22]. retrograde type 2 transfers require shorter transfer dura- tions than those of type 1 transfers; for L target LPOs, The equation of motion of the spacecraft with propellant prograde and retrograde type 4 transfers require shorter thrusts in the ECI frame can be expressed as transfer durations than those of type 3 transfers [11].  = GM Earth Hence, prograde and retrograde types 2 and 4 transfers kk are preferable in LPOs missions. Figure 1 shows four P P i i GM + GM + u (1) i i 3 3 types of lunar yby transfers to Sun{Earth/Moon LPOs: k P k kP k i i i2 (a) prograde type 2 transfer to an L LPO, (b) retrograde where G is the gravitational constant, M is the mass Earth type 2 transfer to an L LPO, (c) prograde type 4 trans- of the Earth, and M is the corresponding mass of the fer to an L LPO, and (d) retrograde type 4 transfer to celestial body in . u is the thrust acceleration provided an L LPO, where the black and red lines denote the by the propellant system. transfer orbits and target LPOs, respectively. These four For the continuous thrust implemented in the transfer target LPOs are actually Lissajous orbits, and we assume orbit, we can obtain that they remain unchanged in the trajectory correction problem. The transfer orbits in Fig. 1 are described in u = the dimensionless Sun{Earth/Moon rotating frame and (2) calculated in the ephemeris model. The length unit of m _ = g I 0 sp the dimensionless Sun{Earth/Moon rotating frame is the where T = (T ; T ; T ) represents the propellant thrust x y z instantaneous distance between the Sun and the Earth{ in the ECI frame; T is the magnitude of the propellant Moon barycenter (EMB). At the initials of lunar yby thrust, i.e., kTk; m is the spacecraft mass; g is the 0 transfers, a tangential maneuver v is implemented to in acceleration due to gravity at sea level, and is equal to escape the initial LEO at an altitude of 200 km. Sub- 9.80665 m/s ; and I is the speci c impulse of the engine. sp sequently, at the terminals of the lunar yby transfers, The data of the spacecraft and thrust engine are listed i.e., insertion points (blue points in Fig. 1), a tangential in Table 1. A BHT-200 Busek Hall e ect thruster can impulsive maneuver v is implemented to insert the end provide 13 mN of thrust at 200 W power and a speci c spacecraft into target LPOs. Table 2 lists the data of the impulse of 1375 s (http://www.busek.com/index htm four transfers in Fig. 1, where i denotes the inclination leo les/70000700A%20BHT-200.pdf ). Hence, if we deploy of the initial LEO in the ECI frame. two thrusters such as the BHT-200 thrusters in the space- Although lunar yby transfers to LPOs have advan- craft, the performance requirement for thrusters listed in tages in terms of fuel consumption and ight time [11], Table 1 can be realized. they are more delicate and unstable than traditional transfers without lunar ybys [18]. The design aim of 2.2 Lunar yby transfers to libration point the trajectory correction in this paper is to enable the orbits spacecraft to be inserted into the target LPO. Hence, In this paper, lunar yby transfers from an LEO at trajectory correction with continuous thrust is executed an altitude of 200 km to Sun{Earth/Moon LPOs are during the transfer. In this paper, we select the above 4 Y. Qi, A. de Ruiter (a) Prograde type 2 transfer to an L LPO (b) Retrograde type 2 transfer to an L LPO 1 1 (c) Prograde type 4 transfer to an L LPO (d) Retrograde type 4 transfer to an L LPO 2 2 Fig. 1 Four types of lunar yby transfers in the dimensionless Sun{Earth/Moon rotating frame; the black and red lines denote transfer orbits and target LPOs, respectively. Table 2 Data of the four lunar yby transfers in Fig. 1 Orbit Start time (UTC) Perilune time (UTC) Insertion time (UTC) i (deg) v (m/s) v (m/s) leo in end a 2020{08{11 14:48:16 2020{08{15 22:25:25 2021{01{13 19:30:07 24.1822 3128 5.1263 b 2020{07{25 10:24:30 2020{07{30 15:59:30 2021{01{02 01:01:19 24.1763 3133 3.0691 c 2020{07{29 02:36:13 2020{08{02 08:27:33 2020{12{31 08:54:52 24.0894 3129 2.7077 d 2020{08{10 10:43:24 2020{08{15 23:01:51 2021{01{21 04:30:06 24.1849 3134 5.6639 four lunar yby transfers as reference orbits of the tra- tory correction with continuous thrusts in the ephemeris jectory correction. In addition, we note that in Table 2, model. The control strategy is based on the backstepping the insertion maneuvers of the reference orbits v are technique, which has been widely applied in station- end considerably small; hence, a portion of the target LPO keeping [19, 20] and attitude tracking [21]. orbit after the insertion point is added into the reference Based on Eq. (1), the controlled equation of motion of orbit, and we implement continuous thrusts to achieve the spacecraft in the ephemeris model can be rewritten as the insertion instead of the small impulsive maneuvers. In other words, the reference orbit considered in this pa- x = f (t;x) + Bu(t) (3) per includes the lunar yby transfer orbit and a portion where x = [; v] is the phase state, including the position of the target LPO. and velocity state, of the spacecraft in the ECI frame; In practice, various types of errors occur at the ini- f = [v; a], where a is the gravity acceleration, i.e., the tial point introduced by the inaccuracies of the launch rst and second terms of the right-hand side in Eq. (1); vehicle, such as the velocity error [13, 18, 20] and posi- and B = [O I ] , where O and I are 3 3 33 33 33 33 tion error [20]. In this paper, we propose an initial error zero matrix and identity matrix, respectively. The state caused by the launch time error t in the initial LEO err deviation with respect to the reference orbit is x = at an altitude of 200 km. Thus, the launch time error, xx , where x = [ ; v ] is the state of the reference regardless of the delay or advance, in the initial LEO R R R R results in velocity and position errors. orbit; therefore, we can obtain x _ = f (t;x ). Thus, R R we can derive the linear dynamical equation of the state deviation as follows: 3 Control strategy In this section, we propose a control strategy for trajec- x _ = A(t;x )x + Bu(t) R Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 5 O I 33 33 algebraic Riccati equation (ARE) for the in nite-horizon = + Bu(t) (4) A (t;) O v 21 33 problem: where A is the Jacobi matrix and A (t;) = @a=@. 21 T 1 S P + PS Q + PR P = O (10) 33 Since the phase state of the reference orbit x (t) is Subsequently, the control gain of subsystem (8), k, known, A(t;x ) = A(t) only depends on time. The can be obtained using k = R P . Therefore, in the classical optimal linear quadratic regulator (LQR) control numerical computation of k, we only require to solve an technique to minimize the quadratic cost function ARE once. T T J = [x (t)Qx(t) + u (t)Ru(t)]dt For the original system (3), the control input can be expressed as requires a solution to the time-varying Riccati equation. T=m = u = (A S )z kz (t) 21 1 2 To avoid repetitively solving the Riccati equation, we can = (A S + kS)x kx = K x use the backstepping technique proposed by Nazari et 21 1 2 (11) al. to reduce the original system into a time-invariant subsystem for which it is easier to obtain the control law where A can be obtained from the reference orbit, i.e., stabilizing this subsystem [19]. Based on the result of the A (t;) = @a =@ . 21 R R subsystem, we can derive the control law that stabilizes Based on the above analysis, the performance of the the original system through backstepping. backstepping technique depends on the selection of pa- According to Refs. [19, 23], the backstepping transfor- rameters S, Q, and R. When the values of these three mation is expressed as parameters are known, the control strategy is speci ed. In the station-keeping problem [19, 20], S, Q, and R are z = x 1 1 (5) xed during the entire mission. However, lunar yby z = x Sx 2 2 1 transfers are considerably di erent from LPOs in the where x =  and x = vv are the position 1 R 2 R station-keeping problem. The most noticeable di erence and velocity deviations with respect to the reference orbit, is that the geometrical structure and orbital environment respectively, and S is a constant negative de nite matrix. of lunar yby transfers are divided into two portions According to Eq. (4), we obtain before and after the lunar yby. Based on Ref. [24], the prior-lunar yby can be approximately analyzed in x _ = x 1 2 (6) the Earth{Moon CRTBP, while the post-lunar yby can x _ = A (t)x + u 2 21 1 be approximately studied in the Sun{Earth CRTBP. In Hence, taking the derivative of Eq. (5) with respect to Ref. [18], an error analysis before and after lunar yby t, we obtain: indicated that, compared with the transfer without a z _ = Sz + z 1 1 2 lunar yby, the initial error of the lunar yby transfer (7) is ampli ed by the lunar yby. Therefore, we postulate z _ = (A (t) S )z Sz + u 2 21 1 2 that two sets of the parameters S, Q, and R should be Let u = (A S )z + u ; thus, we can obtain a 21 1 1 adopted in two portions before and after the lunar yby. time-invariant subsystem: First, we estimate the magnitude of S, i.e., kSk. As mentioned earlier, we use the launch time error (t ) z _ = Sz + u (t) (8) 2 2 1 err to create initial velocity and position errors, and when For the time-invariant subsystem (8), we assume that t is known, the perturbed orbit is speci ed and can be err u (t) = kz (t), where k should satisfy the minimization 1 2 obtained using numerical integration. Using the prograde of the quadratic cost function type 2 transfer in Fig. 1(a) as an example, we assume that T T the launch time on the initial LEO of altitude 200 km is J = [z (t)Qz (t) + u (t)Ru (t)]dt (9) 2 1 2 1 postponed for 10 s, i.e., t = 10 s. In addition, owing err where Q and R are constant weighting matrices, Q 2 to the limitation of navigation technology, the trajectory 33 33 R is a positive semi-de nite matrix, and R 2 R is a correction cannot be performed until a minimum of 12 positive de nite matrix. To calculate k, we must solve an hours after launch. Numerical computation indicates 6 Y. Qi, A. de Ruiter that when the continuous thrust begins to operate 12 estimation equation of Q and R and set Q = 10 I . 33 hours after the launch, the position and velocity errors For the portion before the lunar yby, we adopt a larger from the reference orbit here are approximately 1545 km Q compared with the estimation equation to increase and 24 m/s, respectively. Hence, kx k is approximately the convergence rate of the perturbed orbit; therefore, 0.024 km/s, and kx k is approximately 1545 km. Since we set Q = 0:6I . To distinguish these two values of 1 33 z and z in Eq. (5) are the equivalent parameters of Q before and after the lunar yby, we denote them as 1 2 x and x , respectively, based on the second equation Q and Q , respectively. For the application scope of 1 2 1 2 of Eq. (5), we estimate that kSk should be smaller than Q , we postulate that it should encompass the portion 0:024=1545  1:5 10 . The purpose of the backstep- before the lunar yby and the process of the lunar yby. ping control is to drive z to zero asymptotically. Based Based on the geometrical structure of the lunar yby on Eq. (5), if we set z = 0, x _ = x = Sx , which transfers (Fig. 1), the region in which the distance of 2 1 2 1 has the solution x (t) = exp (St)x (0). Hence, the the spacecraft to the EMB is smaller than 1.1a (1 + e ) 1 1 m m magnitude of S can also determine the rate of convergence can seemingly satisfy the above requirement, where a of the position error. denotes the semi-major axis of the lunar orbit (approxi- Second, we discuss the relationship between Q and R. mately 3:8476 10 km) and e is the eccentricity of the In the cost function (J ) in Eq. (9), the terms z Qz and 2 lunar orbit (approximately 0.0549). Outside this region, u Ru represent the performance indices of the trajec- 1 Q is appropriate. 1 2 tory correction and fuel consumption, respectively. For For instance, we use an idealistic simulation to com- a xed kRk, a larger kQk can increase the convergence pare the e ects of Q on the trajectory correction of the rate of the perturbed orbit to the reference orbit, but prograde type 2 transfer in Fig. 1(a). The idealistic simu- the thrust acceleration (u) will also increase; otherwise, lation requires that practical errors, such as navigation or the decrease in kQk can retard the convergence rate of execution errors, and the limitation of the thrust engine, the perturbed orbit and reduce the thrust acceleration. such as the maximum thruster T , are not considered. max 6 6 As previously discussed, kz k is the same as kx k, i.e., 2 2 If t = 10 s, S = 4  10 I , R = 10 I , and err 33 33 approximately 24 m/s. For the thrust engine in Ta- Q = 10 I , Fig. 2 shows performances of two types 2 33 ble 1, the maximum magnitude of the thrust acceleration of trajectory corrections for Q = 0:6I and 10 I , 1 33 33 5 2 kuk is approximately T =m = 5:2 10 m/s . We max 0 where d and v denote the distance and velocity dif- T T assume that the terms z Qz and u Ru have simi- 2 2 1 1 ference of the perturbed orbit from the reference orbit, lar magnitudes to balance the trajectory correction and respectively. In Fig. 2, the large divergence of the per- thrust acceleration. Subsequently, we can estimate that turbed and reference orbits in the beginning is caused kQk  4:5 10 kRk. However because the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon [18], the conver- gence rate of the perturbed orbit before the lunar yby, rather than the decrease in thrust acceleration, should be prioritized. As stated earlier, the magnitudes of S, Q, and R can a ect the convergence rate of the perturbed orbit. For convenience, we x S and R at constant values and use the magnitude of Q to change the rate of convergence for portions before and after the lunar yby. Based on the above discussion, S is a negative de nite matrix and kSk should be smaller than 1:5  10 ; therefore, for the entire transfer, we set S = 4 10 I . R is a 33 positive de nite matrix, and numerical computation indi- cates that R = 10 I is feasible for the entire transfer. 33 Fig. 2 Performances of two types of trajectory corrections For the portion after the lunar yby, we use the above for Q = 0:6I and 10 I . 1 33 33 Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 7 by the initial launch time error; the other large diver- In the transfer mission to a Sun{Earth/Moon LPO, gence of the perturbed and reference orbits at t = 153 the navigation system cannot provide updates in real days is caused by the insertion maneuver in the reference time and requires a navigation interval. Let t and T i ni orbit. As the gure shows, the trajectory correction with denote the navigation epoch and navigation interval time, kQk = 0:6 requires a larger T at the beginning of the respectively. Based on Eq. (4), in the navigation interval trajectory correction, but after that, its thrust T , d, and [t ; t + T ], the state error can be obtained using a i i ni v are all smaller than those for kQk = 10 . This forward numerical integration of the linear dynamical result con rms our earlier inference that a larger kQk system as follows: can increase the convergence rate of the perturbed orbit BT (t) x _ = A(t;x )x + to the reference orbit, but the thrust acceleration (u) will also increase. In addition, numerical computation m _ = (12) indicates that the propellant mass losses, i.e., m , for g I loss 0 sp trajectory corrections with kQ k = 0:6 and 10 I are 1 33 x(t ) = x i ni 4.1691 and 4.7528 kg, respectively. Therefore, for the where x is the state error with respect to the ref- ni total fuel consumption, using a larger kQ k to promptly erence orbit obtained from the navigation system at t . reduce the position and velocity errors before the lunar In the navigation interval [t ; t + T ], x obtained i i ni yby is more reasonable. from the linear dynamical system (12) is the pseudo state In addition, as stated in Ref. [20], a dead-band scheme error rather than the actual state error. In Eq. (12), can avoid thrusters operating continuously for long-term A(t;x ) = A(t) only depends on time and can be ob- maneuvers. Figure 3 shows the schematic of a dead- tained a priori; thus, the computational cost of the nu- band scheme. It assumes that thrusters shut down when merical integration of the linear system (12) is lower the distance of the spacecraft from the reference orbit than that of the actual ephemeris model and is more decreases to a lower boundary (which is denoted by d ) low computationally implementable on a ight processor. from a farther position; subsequently, thrusters start up The navigation error is denoted by " = [" " " " when the distance from the reference orbit increases to an n X Y Z _ " " ] . Thus, the actual state error obtained from the upper boundary (which is denoted by d ) from a closer _ _ up Y Z navigation system at time t can be expressed as position. The interval times between t and t in Fig. 3 i 1 2 is the idle time of the thrusters. In this paper, unless x = x (t ) + " (13) ni real i n explicitly stated, d and d are set to 5 and 20 km, low up respectively. where x (t ) is the actual state error without errors. real i Similar to the analysis in Q, we consider that using two values of the navigation interval time (T ) before and ni after the lunar yby is necessary because the lunar yby can enlarge the divergence of the perturbed transfer from the reference transfer [18]. Thus, a short T before the ni lunar yby can increase the accuracy of the pseudo state error in the portion before the lunar yby. Let T and ni1 T represent the values of T before and after the ni2 ni lunar yby, respectively, with T < T . For the ni1 ni2 application scope of T , we assume that it is identical ni1 Fig. 3 Schematic of a dead-band scheme. to that of Q , i.e., the distance of the spacecraft to the EMB should be smaller than 1.1a (1 + e ). m m Based on Eq. (11) and the pseudo state error (x), the 4 Practical constraints idealistic thrust in the navigation interval [t ; t + T ] i i ni In this section, practical constraints during trajectory can be expressed as correction are introduced. Two sources of practical con- T (t) = mK (t;x )x (14) straints exist: the navigation and propellant systems. 8 Y. Qi, A. de Ruiter If the engine limitation is considered, T (t) can be rewritten as >0; T < T min max T (t) = T; T > T (15) max T; T 2 [T ; T ] min max where T and T are the minimum and maximum min max magnitudes of the thrust, respectively. Let " , (i = X; Y; Z ) denote the execution error. Thus, the control input executed in Eq. (12) is 0; T < T min T (t) = (16) E T ; T > T m min Fig. 4 Statistical results of trajectory corrections with dif- where E = diag[1 + " ; 1 + " ; 1 + " ]. m T T T X Y Z ferent navigation intervals and launch time errors t . err Using Eqs. (12){(16), a control strategy with practi- cal constraints in the navigation interval [t ; t + T ] is i i ni are all considered. Figure 4 shows the statistical results of established. We assume that both the navigation and exe- trajectory corrections with di erent navigation intervals, cution errors are white noise processes and have standard where T represents the total idle time. For the control idle normal distributions with zero means. The parameters strategy with a dead-band scheme, the total idle time of practical constraints are listed in Table 3, for which T includes two parts: the idle time of the dead-band idle the parameters of the navigation system were obtained scheme and the idle time of T < T , but for the control min from the navigation uncertainties of the ARTEMIS mis- strategy without a dead-band scheme, only the latter sion [25]. T is obtained from Table 1, and T is max min exists. For each simulation with particular values of t err equal to 1% of the maximum thrust of a single thruster. and (T ; T ), 100 sample points with practical ni1 ni2 In addition, we assume that the initial launch time error constraints were selected. As mentioned earlier, the navi- is known in the trajectory correction. gation and execution errors are white noise processes and Table 3 Practical constraints have standard normal distributions with zero means. As Parameter Value Fig. 4 shows, for the particular launch time error t , err trajectory corrections with di erent navigation intervals Navigation errors 1 km X ;Y ;Z had almost similar mean fuel consumptions or mass losses n n n 1 cm/s _ _ _ X ;Y ;Z n n n m ; therefore, the marks for (0; 0) and (0:5; 2) days ap- loss Execution errors peared covered by the marks for (0:5; 5) days. However, 2% T ;T ;T X Y Z the trajectory correction with a longer navigation inter- Engine limitation val had a longer T . In addition, the gure show that idle T 26 mN max trajectory corrections with jt j = 1 s had a lower fuel err T 0.13 mN min consumption or m but had a higher T than those loss idle with jt j = 5 and 10 s. However, the di erences between err the simulation results with jt j = 5 and 10 s were not 5 Numerical simulations err apparent. Based on the results in Fig. 4, we postulate 5.1 E ects of navigation intervals that (T ; T ) = (0:5; 5) days is a well-balanced ni1 ni2 option between the fuel cost and idle time. Hence, unless In this subsection, we investigate the e ects of navigation explicitly stated, T and T were set to 0.5 and intervals on the trajectory correction of the lunar yby ni1 ni2 5 days, respectively, in the subsequent simulations. transfer using Monte Carlo simulations. Speci cally, the prograde type 2 transfer shown in Fig. 1(a) is used as the Figure 5 depicts the details of a trajectory correction reference orbit in numerical simulations of this subsec- for t = 10 s and (T ; T ) = (0; 0) days. In err ni1 ni2 tion, and the dead-band scheme and practical constraints Fig. 5(a), the dashed and solid lines are the perturbed Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 9 (a) Perturbed and correction orbits (b) Performance of trajectory correction (c) A zoom of T curve (d) A zoom of d curve Fig. 5 Trajectory correction for the prograde type 2 orbit without navigation intervals. and correction orbits, respectively. In the correction orbit, one more long thrust segment with a larger T appeared because of the insertion maneuver in the reference orbit, blue and red arcs are idle and thrust arcs, respectively. but we can observe that this thrust segment was shorter The switching point indicated by a black point is the than the rst one from t = 0.5{21 days, and this d position at which the Q value changes from Q to Q . 1 2 was much smaller than the rst one. Furthermore, the Based on the application scope of Q , it is the point on the magni cations of the T and d curves in Fig. 5(b) are lunar yby transfer whose distance to the EMB is equal shown in Figs. 5(c) and 5(d), respectively. These two to 1.1a (1+e ). As the gure shows, the perturbed orbit m m gures indicate that the engine limitation and dead- diverged from the reference transfer and passed through band scheme performed their functions in the trajectory the target LPO, but the correction orbit could remain correction. around the reference orbit, and it was inserted into the As a comparison, Fig. 6 shows the details of a trajectory target LPO. In Fig. 5(b), green and red zones indicate the correction with t = 10 s and (T ; T ) = (0:5; 5) err ni1 ni2 thrust segment of the dead-band scheme and the actual days. In Fig. 6(a), the dashed and solid lines are the thrust segment for T > T , respectively. The dashed min perturbed and correction orbits, respectively. In the vertical lines at t = 5:142 days in this gure indicate the correction orbit, the blue and red arcs are the idle and switching epoch of the kQ k value. This gure shows thrust arcs, respectively. The switching point of the kQ k that from the beginning of the correction (t = 0:5 days) value is indicated by a black point. As the gure shows, to approximately t = 21 days, the low-thrust engines this correction orbit is considerably similar to that in continuously operated with a relatively large T . During Fig. 5(a). Similarly, the green and red zones in Fig. 6(b) this time, the divergence of the correction orbit from are the thrust segment of the dead-band scheme and the reference orbit d initially increased and subsequently the actual thrust segment with T > T , respectively. min decreased signi cantly. Correspondingly, the spacecraft The dashed vertical lines indicate the switching epoch of mass signi cantly decreased owing to fuel consumption the kQ k value. By comparing Figs. 5(b) and 6(b), we during this time. Subsequently, because the divergence observe that for the two long thrust segments during t = of the correction orbit from the reference orbit decreased 0.5{21 and 155{171 days, the two trajectory corrections signi cantly, several fragmentary short-thrust arcs with with di erent navigation intervals were similar. However, small T values were distributed in the correction orbit from t = 21{155 days, we observed that the actual thrust 21 days later. When t was approximately 155 days, segments of the correction orbit with navigation intervals 10 Y. Qi, A. de Ruiter (a) Perturbed and correction orbits (b) Performance of trajectory correction (c) A zoom of T curve (d) A zoom of d curve Fig. 6 Trajectory correction for the prograde type 2 orbit with (T ; T ) = (0:5; 5) days. ni1 ni2 in Fig. 6(b) were much more fragmentary and shorter 5.2 E ects of the dead-band scheme than those without navigation intervals in Fig. 5(b). This In this subsection, we investigate the e ects of the dead- conclusion can be con rmed by the magni cations of the band scheme on the trajectory correction. As discussed T and d curves in Figs. 6(c) and 6(d). We postulate that in Section 3, the dead-band scheme is determined by the position and velocity deviations could not decrease the lower and upper boundaries of the idle segment, rapidly to a small range owing to navigation intervals and d and d , respectively. The previous subsection dis- low up further resulted in longer thrust segments of the dead- cusses numerical simulations with the dead-band scheme band scheme (green zones in Fig. 6(b)). In Fig. 6(d), of (d ; d ) = (5; 20) km. In this subsection, two more low up d is the actual position deviation with respect to the real types of dead-band schemes of (d ; d ) = (0; 0) km and low up reference orbit, and d is the pseudo position deviation pse (10, 10) km are shown. Speci cally, the trajectory cor- obtained using Eq. (12): The curve of d uctuated pse rection with (d ; d ) = (0; 0) km was the one without low up around that of d in a small range. At the navigation real the dead-band scheme because, based on the de nition time (see the black points in Fig. 6(d)), the state of of the dead-band scheme in Fig. 3, the entire transfer spacecraft with navigation errors was obtained by the had no idle segments. In the numerical simulations in navigation system. Subsequently, during the navigation this subsection, the prograde type 2 transfer in Fig. 1(a) interval, the control strategy with practical constraints was used as the reference orbit, and practical constraints proposed in the last section was conducted. d , obtained pse listed in Table 3 were all considered. The navigation from the linear dynamical system, gradually left from interval was set as (T ; T ) = (0:5; 5) days, and ni1 ni2 d over time. However, at the next navigation time, the real the initial launch time error was set as t = 10 s. err de ected state of the spacecraft was partially corrected Figures 7(a) and 7(b) show the performances of two by the navigation system. In addition, the numerical trajectory correction scenarios with (d ; d ) = (0; 0) low up computation indicated that mass losses in the trajectory and (10, 10) km, respectively. In these gures, green corrections in Figs. 5 and 6 were considerably similar and red zones are the thrust segments of the dead-band (approximately 2 kg), but the idle time T of the latter scheme and the actual thrust segment with T > T , re- idle min was approximately 10 days larger than that of the former, spectively. The dashed vertical lines indicate the switch- which corresponded with the result shown in Fig. 4. ing epoch of the kQ k value. d is the real position 1 real Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 11 (i.e., green zones in Figs. 6 and 7) rather than the T , d, and m curves. We implemented Monte Carlo simulations to observe the di erences in statistics results with di erent dead- band schemes. For each simulation, 100 sample points with practical constraints were selected. As mentioned earlier, the navigation and execution errors are white noise processes and have standard normal distributions with zero means. Table 4 lists the Monte Carlo results of three di erent dead-band schemes, where Std. is the abbreviation of the standard deviation of the words. (a) (d ; d ) = (0; 0) km low up The table indicates that the mean values and standard deviations of their mass losses were considerably similar, but the mean idle time (T ) of (d ; d ) = (5; 20) km idle low up was larger than those of the other two dead-band schemes. This result can be explained by the observation indicated in Figs. 6 and 7, because the trajectory correction with (d ; d ) = (5; 20) km had the shortest total thrust low up segment of the dead-band scheme (i.e., the total length of green zones). Therefore, the trajectory correction with (d ; d ) = (5; 20) km had more idle time than those of low up the other two dead-band schemes. Table 4 Monte Carlo results of trajectory corrections with (b) (d ; d ) = (10; 10) km di erent dead-band schemes up low (d ; d ) (km) (5, 20) (0, 0) (10, 10) Fig. 7 Performances of two trajectory correction scenario low up with di erent dead-band schemes. Mean m (kg) 2.0393 2.0391 2.0353 loss Std. m (kg) 0.0073 0.0077 0.0072 loss deviation with respect to the reference orbit, and d pse Mean T (day) 147.8935 118.8867 120.0013 idle is the pseudo position deviation obtained by Eq. (12). Std. T (day) 2.4905 1.9880 2.3296 idle Comparing with Fig. 6(b), we observe that the T , d, and m curves are not noticeably di erent with di erent 5.3 Examples of di erent lunar yby trans- dead-band schemes, respectively. The green zones indi- fers cate thrust segments of the dead-band scheme. Based In this subsection, trajectory corrections for other lunar on the de nition of the dead-band scheme, dead-band yby transfers in Fig. 1 are analyzed and compared. In schemes with di erent (d ; d ) had di erent distri- low up the numerical simulations of this subsection, the initial butions of thrust segments, and this conclusion can be launch time error was set as t = 10 s, and the dead- err con rmed by the green regions in Figs. 6 and 7. We band scheme of (d ; d ) = (5; 20) km was used in the low up observed that the most signi cant variation among them trajectory corrections. The practical constraints listed in was the distribution of green zones. As stated earlier, Table 3 were all considered, and the navigation interval the trajectory correction without dead-band scheme, i.e., was set as (T ; T ) = (0:5; 5) days. In addition, ni1 ni2 (d ; d ) = (0; 0) km, had no idle segments during the low up control parameters in the backstepping technique, such entire transfer; therefore, the green zone encompassed as S, R, Q , and Q , were identical to the setting for 1 2 the entire transfer. The trajectory correction with the prograde type 2 in Section 3. We adopted the same dead-band scheme of (d ; d ) = (10; 10) km had idle low up application scope of Q as that in Section 3. segments but its idle segments seemed shorter than those First, we compare trajectory corrections for the four in Fig. 6. In summary, the dead-band scheme on the tra- lunar yby transfers in Fig. 1 by the Monte Carlo simu- jectory correction primarily a ected the thrust segment lations. In the Monte Carlo simulations, for a particular 12 Y. Qi, A. de Ruiter reference orbit, 100 sample points with the initial launch time error and practical constraints were selected. As mentioned earlier, the navigation and maneuver errors listed in Table 3 were white noise processes and had standard normal distributions with zero means. Table 5 shows the Monte Carlo results, where Std. is the abbre- viation of the standard deviation of the words. As the table shows, trajectory corrections for prograde types 2 and 4 had similar mean mass losses, which were smaller than those for the retrograde types 2 and 4. In particular, (a) Perturbed and correction orbits the perturbed retrograde type 4 transfer had signi cantly higher fuel consumption or mass loss than the other three lunar yby transfers. The table indicates that prograde type 2 had the largest mean (T ), and retrograde type idle 4 had the smallest mean. However, note that the direct comparison of T is meaningless for di erent lunar yby idle transfers because their total ight times (TOFs) di ered. The numerical computation indicated that the TOFs of prograde type 2, prograde type 4, retrograde type 2, and retrograde type 4 were 202.3167, 193.9958, 200.7637, and 202.0079 days, respectively. We postulate that T =TOF idle can re ect the proportion of idle time during the whole transfer, which is also listed in Table 5. We can observe (b) Performance of trajectory correction that prograde type 2 had the largest proportion of idle Fig. 8 Trajectory correction for the prograde type 4 orbit. time, and retrograde type 4 had the smallest one. The thrust segment of the trajectory correction for the retro- dashed and solid lines are the perturbed and correction grade type 4 occupied approximately half of the transfer orbits, respectively. In the correction orbit, the blue time; therefore, it had the largest mass loss. Although and red arcs are the idle and thrust arcs, respectively. prograde type 2 had the largest proportion of idle time, The switching point of the kQ k value is indicated by its TOF was larger than the other three cases. There- a black point. As the gure shows, the perturbed orbit fore, its mass loss was not the lowest. In addition, based passed through the target LPO, but the correction or- on Eq. (2), the mass loss was determined by the thrust bit could remain around the reference orbit, and it was magnitude and thrust time. Hence, even if the prograde inserted into the target LPO successively. In Fig. 8(b), type 2 had the shortest thrust time, its total mass loss the green and red zones are the thrust segment of the may not have been the lowest. dead-band scheme and the actual thrust segment with Subsequently, we discuss some trajectory correction T > T , respectively. The dashed vertical lines indicate examples for di erent lunar yby transfers in detail. Fig- min ure 8 depicts the details of the trajectory correction for the switching epoch of the kQ k value. d is the actual 1 real the prograde type 4 orbit in Fig. 1(c). In Fig. 8(a), the position deviation with respect to the reference orbit, Table 5 Monte Carlo results of trajectory corrections for di erent lunar yby transfers Transfer type Prograde 2 Prograde 4 Retrograde 2 Retrograde 4 Mean m (kg) 2.0393 2.0080 3.7705 11.2331 loss Std. m (kg) 0.0073 0.0102 0.0306 0.0853 loss Mean T (day) 147.8935 138.4160 137.9849 107.7510 idle Std. T (day) 2.4905 3.4646 2.7536 2.5604 idle Mean T /TOF 0.7310 0.7135 0.6873 0.5334 idle Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 13 and d is the pseudo position deviation obtained by prograde lunar yby orbits in Figs. 6 and 8, we observe pse that both T and d curves of the retrograde type 2 orbit Eq. (12): We observed that the T curve in Fig. 8(b) was are signi cantly di erent. The trajectory correction for similar to that in Fig. 6(b), but the former had longer the retrograde type 2 orbit had a longer thrust segment green zones than the latter. In addition, we observed that at most times, the curve of d uctuated around with T = T than the above two prograde lunar yby pse max that of d in a small range. However, when t was orbits because its d curve could not rapidly decrease to a real approximately 160 days, d was clearly larger than small value after the lunar yby similar to the prograde real d . Numerical computation indicated that the insertion lunar yby orbits. As Fig. 9(b) shows, the d curve has two pse maneuver occurred at t = 155:23 days, and the latest peaks before the switching point. Numerical computation indicated that the rst peak occurred before the lunar navigation epoch occurred at t = 155 days. Because d pse yby, and the second one appears before the spacecraft was calculated over almost 5 days according to the state passed through the lunar orbit. We consider that this before the insertion maneuver, its curve diverged from phenomenon can be explained by the special shape of d until the next navigation epoch t = 160 days. real retrograde lunar yby transfers. As Fig. 9(a) shows, the Figure 9 shows the detail of a trajectory correction for retrograde lunar yby orbit experienced a lunar yby and the retrograde type 2 orbit in Fig. 1(b). In Fig. 9(a), an Earth yby, successively, before the switching point. the dashed and solid lines denote the perturbed and The Earth yby, similar to the lunar yby, could also correction orbits, respectively. In the correction orbit, result in a signi cant divergence of the perturbed orbit the blue and red arcs are idle and thrust arcs, respectively. from the reference transfers. Therefore, the d curve has The switching point of the kQ k value is represented by two peaks before the switching point. For the retrograde a black point. The correction orbit could remain around type 2 orbit, a longer ight time in the lunar orbit resulted the reference orbit and was inserted into the target LPO in a second increase in d before the switching point. In successively. Compared with trajectory corrections for addition, we observed that when the insertion maneuver occurred at t = 160:7 days, the divergence of d from pse d was smaller than that in Fig. 8, because the next real navigation epoch at t = 163 days was close to the insertion maneuver point. Figure 10 depicts the details of a trajectory correction for the retrograde type 4 orbit in Fig. 1(d). In Fig. 10(a), the dashed and solid lines are the perturbed and correc- tion orbits, respectively. In the correction orbit, the blue and red arcs are the idle and thrust arcs, respectively. The switching point of the kQ k value is represented by (a) Perturbed and correction orbits a black point. The correction orbit could remain around the reference orbit and was inserted into the target LPO successively. Compared with trajectory corrections for the other three lunar yby orbits, we observed that the trajectory correction for the retrograde type 4 orbit had a much longer thrust segment with T = T (approx- max imately 40 days) because its d curve could not rapidly decrease to a small value after the lunar yby, and it even increased to approximately 3:2 10 km after the switching point. Moreover, the numerical computation indicated that a long ight time in the lunar orbit re- sulted in an increase in d before and after the switching (b) Performance of trajectory correction point. In addition, numerical results indicated that, in the navigation interval t 2 [163; 168] days, the insertion Fig. 9 Trajectory correction for the retrograde type 2 orbit. 14 Y. Qi, A. de Ruiter method is feasible and ecient for problems with dif- ferent practical constraints. In contrast to the previous reference on this problem, the propellant system in this study was the continuous thrust rather than the impul- sive thrust. In addition, compared with a previous study, more practical constraints in the trajectory correction were considered in this study, such as the dead-band scheme, navigation intervals, execution error, and the limitation of the thrust engine. Therefore, the control strategy proposed herein is more realistic than that in the previous study. A control strategy based on the backstep- ping technique with a dead-band scheme was proposed (a) Perturbed and correction orbits using the ephemeris model. Since the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon, we postulated that using two sets of the parameters S, Q, and R is necessary in two portions before and after the lunar yby. Based on numerical analysis and computation, we selected a larger Q in the portion of the prior-lunar yby to increase the convergence rate of the perturbed orbit to the reference orbit, although it could result in a larger thrust accel- eration. For the post-lunar yby, we adopted a smaller Q to reduce the thrust acceleration. According to the distance of the spacecraft to the Earth{Moon barycenter, (b) Performance of trajectory correction the application scope of the larger Q is de ned. Fig. 10 Trajectory correction for the retrograde type 4 Using a prograde type 2 orbit as the reference orbit, the orbit. e ects of navigation intervals on trajectory corrections are maneuver occurred at t = 163:6 days and the divergence discussed using the results of numerical simulations. Nu- of d from d exceeded 2000 km, which was much pse real merical computation showed that for a particular launch larger than the other three lunar yby orbits. time error, trajectory corrections with di erent navi- gation intervals have almost similar mass losses, but the 6 Conclusions trajectory correction with longer navigation intervals has a longer idle time. Numerical results indicated that In this paper, trajectory corrections for lunar yby trans- (T ; T ) = (0:5; 5) days is a well-balanced option ni1 ni2 fers to Sun{Earth/Moon LPOs with continuous thrusts between fuel consumption and idle time. The detailed were investigated using the ephemeris model. Although analysis showed that the actual thrust segments of the many studies on trajectory corrections for transfers to correction orbit with navigation intervals are much more LPOs have been conducted, trajectory corrections for fragmentary and shorter than those without navigation lunar yby transfers are insucient and signi cantly dif- intervals because the former has longer thrust segments ferent from those without lunar yby because of their spe- of the dead-band scheme. Subsequently, we implemented cial geometrical and dynamical structures. Lunar yby numerical simulations to observe the di erences in tra- transfer is a high-yielding but high-risk design method; jectory corrections with di erent dead-band schemes. A therefore, its TCM strategy should be speci cally de- dead-band scheme can avoid thrusters operating contin- signed. In this paper, we rst introduce the backstepping uously for long-term maneuvers. Numerical computation technique into the trajectory correction for lunar yby transfers to LPOs with practical constraints. Numeri- showed that the e ects of the dead-band scheme on the cal simulations indicated that our trajectory correction trajectory correction primarily appeared in the thrust Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 15 segments of the dead-band scheme. Monte Carlo simula- [8] Sukhanov, A., Prado, A. F. B. A. Lambert problem solution in the hill model of motion. Celestial Mechanics tions indicated that mass losses with di erent dead-band and Dynamical Astronomy, 2004, 90(3{4): 331{354. schemes were signi cantly similar. Finally, trajectory [9] Renk, F., Landgraf, M. Sun{Earth Libration point trans- corrections for di erent lunar yby transfers are shown fer options with intermediate HEO. Acta Astronautica, and compared. Monte Carlo simulations showed that 2012, 74: 1{19. trajectory corrections for prograde types 2 and 4 had [10] Peng, H., Bai, X. L., Masdemont, J. J., G omez, G., Xu, S. similar mean mass losses, which were smaller than those J. Libration transfer design using patched elliptic three- for retrograde types 2 and 4. In particular, the perturbed body models and graphics processing units. Journal of retrograde type 4 transfer had signi cantly higher fuel Guidance, Control, and Dynamics, 2017, 40(12): 3155{ consumption or mass loss than the other three lunar 3166. [11] Qi, Y., Xu, S. J., Qi, R. Transfer from Earth to libration yby transfers. We observed that prograde type 2 had point orbit using lunar gravity assist. Acta Astronautica, the largest proportion of idle time, and retrograde type 2017, 133: 145{157. 4 had the smallest one. Compared with the prograde [12] Farquhar, R. W., Muhonen, D. P., Newman, C. R., lunar yby orbits, the retrograde lunar yby orbits had Heubergerg, H. S. Trajectories and orbital maneuvers a longer ight time in the lunar orbit, which resulted in for the rst libration-point satellite. Journal of Guidance larger position deviations with respect to the reference and Control, 1980, 3(6): 549{554. orbits and higher fuel consumption. [13] Serban, R., Koon, W. S., Lo, M. W., Marsden, J. E., Pet- zold, L. R., Ross, S. D., Wilson, R. S. Halo orbit mission correction maneuvers using optimal control. Automatica, Acknowledgements 2002, 38(4): 571{583. This work was supported by the Canada Research Chair [14] G omez, G., Marcote, M., Masdemont, J. J. Trajectory Program under Grant No. 950-230883. correction manoeuvres in the transfer to libration point orbits. Acta Astronautica, 2005, 56(7): 652{669. [15] Xu, M., Xu, S. J. Trajectory and correction maneuver References during the transfer from earth to halo orbit. Chinese [1] Farquhar, R. W. The ight of ISEE-3/ICE: Origins, Journal of Aeronautics, 2008, 21(3): 200{206. mission history, and a legacy. The Journal of the Astro- [16] Salmani, M., Busk  ens, C. Real-time control of optimal nautical Sciences, 2001, 49(1): 23{73. low-thrust transfer to the Sun{Earth L1 halo orbit in the [2] Ogilvie, K. W., Desch, M. D. The WIND spacecraft and bicircular four-body problem. Acta Astronautica, 2011, its early scienti c results. Advances in Space Research, 69(9{10): 882{891. 1997, 20(4{5): 559{568. [17] Peng, H. J., Gao, Q., Wu, Z. G., Zhong, W. X. Optimal [3] Domingo, V., Fleck, B., Poland, A. I. The SOHO mission: guidance based on receding horizon control for low-thrust An overview. Solar Physics, 1995, 162(1{2): 1{37. transfer to libration point orbits. Advances in Space [4] Bennett, C. L., Bay, M., Halpern, M., Hinshaw, G., Jack- Research, 2013, 51(11): 2093{2111. son, C., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., [18] Qi, Y., de Ruiter, A. Study of correction maneuver for Page, L. et al. The microwave anisotropy probe mission. lunar yby transfers in the real ephemeris. 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[7] G omez, G., Jorba, A., Masdemont, J., Sim o, C. A dy- [21] Jiang, Y., Hu, Q. L., Ma, G. F. Adaptive backstep- namical systems approach for the analysis of the SOHO ping fault-tolerant control for exible spacecraft with mission. In: Proceedings of the 3rd International Sym- unknown bounded disturbances and actuator failures. posium on Spacecraft Flight Dynamics, 1991: 449{454. ISA Transactions, 2010, 49(1): 57{69. 16 Y. Qi, A. de Ruiter [22] Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. Anton de Ruiter received his B.E. de- S., Kuchynka, P. The planetary and lunar ephemerides gree in mechanical engineering from the DE430 and DE431. The Interplanetary Network Progress University of Canterbury in 1999, and Report, 2014, (196): 1{81. M.A.Sc. and Ph.D. degrees in aerospace [23] Deshmukh, V. S., Sinha, S. C. Control of dynamic sys- engineering from the University of tems with time-periodic coecients via the Lyapunov- Toronto in 2001 and 2005, respectively. oquet transformation and backstepping technique. Between 2006 and 2008 he was a visiting Journal of Vibration and Control, 2004, 10(10): 1517{ research fellow at the Canadian Space 1533. Agency in Montreal, and an assistant professor in the [24] Qi, Y., Xu, S. J. Study of lunar gravity assist orbits in Department of Mechanical and Aerospace Engineering at the restricted four-body problem. Celestial Mechanics Carleton University from 2009 to 2012. He is currently an and Dynamical Astronomy, 2016, 125(3): 333{361. associate professor and Canada research chair (Tier 2) in the [25] Sweetser, T. H., Broschart, S. B., Angelopoulos, V., Department of Aerospace Engineering at Ryerson University Whi en, G. J., Folta, D. C., Chung, M. K., Hatch, S. in Toronto, Canada. His research interests are in the area J., Woodard, M. A. ARTEMIS mission design. Space of guidance, navigation, and control of aerospace systems. Science Reviews, 2011, 165(1{4): 27{57. E-mail: aderuiter@ryerson.ca. Yi Qi received his Ph.D. degree in Open Access This article is licensed under a Creative Com- aeronautical and astronautical science mons Attribution 4.0 International License, which permits and technology from Beihang Univer- use, sharing, adaptation, distribution and reproduction in sity, China, in 2017. After three years any medium or format, as long as you give appropriate credit as a postdoctoral researcher in Ryerson to the original author(s) and the source, provide a link to University, he joined Beijing Institute of the Creative Commons licence, and indicate if changes were Technology as an associate professor in made. 2020. His research area includes orbital The images or other third party material in this article are dynamics and control for deep space exploration. E-mail: included in the article's Creative Commons licence, unless lushenqiyi@gmail.com. indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecoorg/ licenses/by/4.0/. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Astrodynamics Springer Journals

Trajectory correction for lunar flyby transfers to libration point orbits using continuous thrust

Astrodynamics , Volume OnlineFirst – Feb 1, 2021

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Astrodynamics https://doi.org/10.1007/s42064-020-0097-2 Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 1 2 Yi Qi , Anton de Ruiter (B) 1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China 2. Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada ABSTRACT KEYWORDS Trajectory corrections for lunar yby transfers to Sun{Earth/Moon libration point orbits libration point orbit (LPOs) with continuous thrusts are investigated using an ephemeris model. The lunar trajectory correction yby transfer has special geometrical and dynamical structures; therefore, its trajectory Earth{Moon system correction strategy is considerably di erent from that of previous studies and should continuous thrust be speci cally designed. In this paper, we rst propose a control strategy based on backstepping technique the backstepping technique with a dead-band scheme using an ephemeris model. The initial error caused by the launch time error is considered. Since the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon, we adopt two sets of control parameters in two portions before and after the lunar yby, respectively. Subsequently, practical constraints owing to the navigation and propellant systems are introduced in the dynamical model of the trajectory correction. Using a prograde type 2 orbit as an example, numerical simulations show that our control strategy can eciently address trajectory corrections for lunar yby transfers Research Article with di erent practical constraints. In addition, we analyze the e ects of the navigation Received: 6 August 2020 intervals and dead-band scheme on trajectory corrections. Finally, trajectory corrections Accepted: 25 October 2020 for di erent lunar yby transfers are depicted and compared. © The Author(s) 2020 1 Introduction v requirement owing to a tangential perigee velocity error and have the risk of being unable to perform the Because of the special dynamical properties of Sun{ correction maneuver on day two. Therefore, Renk and Earth/Moon libration point orbits (LPOs), many scien- Landgraf presented an indirect strategy to mitigate the ti c and exploration missions to Sun{Earth/Moon LPOs criticality of the rst correction maneuver of the transfer have been implemented, such as ISEE-3 [1], WIND [2], towards Sun{Earth LPOs by including an intermediate, and SOHO [3] for Sun{Earth/Moon L LPOs, MAP [4], highly elliptical parking orbit [9]. Using the CHANG'E-2 GAIA [5], and CHANG'E-2 [6] for Sun{Earth/Moon L extension mission as an example, Peng et al. proposed LPOs. an ecient GPU parallel computing technique to nu- Many researchers focused on transfer problems to Sun{ merically search for transfers from a lunar orbit to the Earth/Moon LPOs. G omez et al. applied the invariant Sun{Earth L LPOs with di erent departing conditions manifolds associated with the LPOs to construct trans- using the patched elliptic restricted three-body problem fers from a low Earth orbit (LEO) to a Sun{Earth L model [10]. Their computationally ecient methodology halo orbit [7]. Based on the solution of the Lambert obtained results almost identical to those of the ephemeris problem in the restricted three-body problem described model and exhibited signi cant speedups. Qi et al. com- by the Hill equations, Sukhanov and Prado proposed a bined the technique of lunar yby using the dynamical design method for LEO-to-halo and halo-to-halo trans- fers [8]. Direct transfers to Sun{Earth LPOs increase the system approach, and they investigated lunar yby trans- B aderuiter@ryerson.ca 2 Y. Qi, A. de Ruiter fers from an LEO to Sun{Earth/Moon LPOs [11]. The Sun{Earth/Moon LPOs with continuous thrusts. Simi- trajectory correction maneuver (TCM) problem is a signi- lar to the analysis in Ref. [18], according to the special cant problem associated with transfers to LPOs since geometrical structure of lunar yby transfers, which are divided into two portions before and after lunar yby, we perturbations and errors are inevitable during practi- postulate that a new design method must be developed to cal transfer missions. Farquhar et al. studied TCMs address the trajectory correction for lunar yby transfers. in the early transfer phase section of the ISEE-3 mis- Since the design of the reference transfers (de ned as the sion [12]. Serban et al. investigated the TCM problem of transfer trajectory with no error) was adequately solved the Genesis Discovery Mission using optimal control to in the previous study [11], in this paper, we assume that compensate for launch vehicle errors, and they proposed the reference transfers are provided. two strategies to solve the TCM problem: the halo orbit In this paper, trajectory corrections for lunar yby insertion (HOI) and the manifold orbit insertion (MOI) transfers to Sun{Earth/Moon LPOs are investigated us- techniques [13]. G omez et al. presented a TCM strategy ing the ephemeris model. In contrast to the previous similar to the MOI technique for the TCM problem of study [18], the propellant system in this study was the the Genesis Mission, but they used a multiple shoot- continuous thrust rather than the impulsive thrust. We ing method instead of an optimal control procedure to assume that the initial error is the launch time error, address the TCM problem with a strong hyperbolic be- and we propose a control strategy based on the back- havior of the orbits [14]. To correct the control errors and stepping technique with a dead-band scheme. The back- orbit determination errors, Wu et al. investigated the tra- stepping technique has been widely applied in station- jectory maneuvers before the Lissajous orbit insertion of keeping [19, 20] and attitude tracking [21]. To the best of the CHANG'E-2 mission from the Moon-circling orbit to our knowledge, this is the rst time that the backstep- Sun{Earth L [6]. Xu and Xu applied the stochastic con- ping technique is used in trajectory corrections for lunar trol theory for discrete linear stochastic systems to design yby transfers. In contrast to the traditional applications a timing closed-loop TCM strategy during the transfer of the backstepping technique, such as station-keeping from an LEO to a Sun{Earth halo orbit in the circular and attitude tracking, since the perturbed transfers sig- restricted three-body problem (CRTBP) [15]. Salmani ni cantly diverge from the reference transfers after the and Busk  ens proposed a real-time control method for spacecraft passes by the Moon, we should use two sets the TCM of transfers to Sun{Earth L halo orbits in the of control parameters in two portions before and after Sun{Earth{Moon bicircular model [16]. They used an the lunar yby. Compared with Ref. [18], more practical optimal control problem to prevent disturbances such constraints in the trajectory correction will be considered as solar radiation and winds. Peng et al. researched in this paper, such as the dead-band scheme, navigation the TCM problem of transferring a spacecraft with low intervals, execution error, and limitation of the thrust thrust from an LEO to a Sun{Earth L halo orbit using engine. Therefore, the control strategy proposed herein is a receding horizon control method [17]. more realistic than that in the previous study. Through Qi and de Ruiter investigated the TCM problem of numerical simulations, we can analyze the e ects of navi- lunar yby transfers to Sun{Earth/Moon LPOs in the gation intervals and dead-band schemes on trajectory ephemeris model, and proposed several TCM strategies corrections for di erent lunar yby transfers. for lunar yby transfers under practical constraints [18]. The structure of this paper can be divided into ve As stated in Ref. [18], the lunar yby transfer is a high- parts. In Section 2, we introduce the background, in- yielding but high-risk design method because the per- cluding the ephemeris model and lunar yby transfers. turbed lunar yby transfers diverge signi cantly from In Section 3, we propose a control strategy for trajec- the reference transfers after the spacecraft passes by the tory correction using continuous thrusts. In Section 4, Moon. A two-impulse TCM must be executed before practical constraints are introduced into the control strat- the lunar yby for transfers from the Earth to LPOs to egy. In Section 5, numerical simulations are implemented promptly restrain the divergence. In this paper, we focus and analyzed. Finally, the conclusions are presented in on the trajectory correction for lunar yby transfers to Section 6. Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 3 Table 1 Spacecraft properties 2 Background Name Symbol Value 2.1 Ephemeris model Initial spacecraft mass m 500 kg Maximum thrust T 26 mN max Here, we use the J2000 Earth-centered inertial (ECI) Engine number n 2 frame to describe the motion of the spacecraft on lunar Speci c impulse I 1000 s sp yby transfers to LPOs.  is the position vector of the analyzed. Based on Ref. [11], eight types of lunar yby spacecraft in the ECI frame. P and i 2 = f Sun, transfers exist: prograde types 1{4 transfers and retro- Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, grade types 1{4 transfers. The eight types of lunar yby Neptuneg represent the position vectors of the perturbing transfers exhibit no apparent di erences in terms of fuel gravity bodies in the ECI frame. The position data of consumption. However, for L target LPOs, prograde and celestial bodies in the ECI frame can be obtained from the 1 Jet Propulsion Laboratory (JPL) ephemeris DE430 [22]. retrograde type 2 transfers require shorter transfer dura- tions than those of type 1 transfers; for L target LPOs, The equation of motion of the spacecraft with propellant prograde and retrograde type 4 transfers require shorter thrusts in the ECI frame can be expressed as transfer durations than those of type 3 transfers [11].  = GM Earth Hence, prograde and retrograde types 2 and 4 transfers kk are preferable in LPOs missions. Figure 1 shows four P P i i GM + GM + u (1) i i 3 3 types of lunar yby transfers to Sun{Earth/Moon LPOs: k P k kP k i i i2 (a) prograde type 2 transfer to an L LPO, (b) retrograde where G is the gravitational constant, M is the mass Earth type 2 transfer to an L LPO, (c) prograde type 4 trans- of the Earth, and M is the corresponding mass of the fer to an L LPO, and (d) retrograde type 4 transfer to celestial body in . u is the thrust acceleration provided an L LPO, where the black and red lines denote the by the propellant system. transfer orbits and target LPOs, respectively. These four For the continuous thrust implemented in the transfer target LPOs are actually Lissajous orbits, and we assume orbit, we can obtain that they remain unchanged in the trajectory correction problem. The transfer orbits in Fig. 1 are described in u = the dimensionless Sun{Earth/Moon rotating frame and (2) calculated in the ephemeris model. The length unit of m _ = g I 0 sp the dimensionless Sun{Earth/Moon rotating frame is the where T = (T ; T ; T ) represents the propellant thrust x y z instantaneous distance between the Sun and the Earth{ in the ECI frame; T is the magnitude of the propellant Moon barycenter (EMB). At the initials of lunar yby thrust, i.e., kTk; m is the spacecraft mass; g is the 0 transfers, a tangential maneuver v is implemented to in acceleration due to gravity at sea level, and is equal to escape the initial LEO at an altitude of 200 km. Sub- 9.80665 m/s ; and I is the speci c impulse of the engine. sp sequently, at the terminals of the lunar yby transfers, The data of the spacecraft and thrust engine are listed i.e., insertion points (blue points in Fig. 1), a tangential in Table 1. A BHT-200 Busek Hall e ect thruster can impulsive maneuver v is implemented to insert the end provide 13 mN of thrust at 200 W power and a speci c spacecraft into target LPOs. Table 2 lists the data of the impulse of 1375 s (http://www.busek.com/index htm four transfers in Fig. 1, where i denotes the inclination leo les/70000700A%20BHT-200.pdf ). Hence, if we deploy of the initial LEO in the ECI frame. two thrusters such as the BHT-200 thrusters in the space- Although lunar yby transfers to LPOs have advan- craft, the performance requirement for thrusters listed in tages in terms of fuel consumption and ight time [11], Table 1 can be realized. they are more delicate and unstable than traditional transfers without lunar ybys [18]. The design aim of 2.2 Lunar yby transfers to libration point the trajectory correction in this paper is to enable the orbits spacecraft to be inserted into the target LPO. Hence, In this paper, lunar yby transfers from an LEO at trajectory correction with continuous thrust is executed an altitude of 200 km to Sun{Earth/Moon LPOs are during the transfer. In this paper, we select the above 4 Y. Qi, A. de Ruiter (a) Prograde type 2 transfer to an L LPO (b) Retrograde type 2 transfer to an L LPO 1 1 (c) Prograde type 4 transfer to an L LPO (d) Retrograde type 4 transfer to an L LPO 2 2 Fig. 1 Four types of lunar yby transfers in the dimensionless Sun{Earth/Moon rotating frame; the black and red lines denote transfer orbits and target LPOs, respectively. Table 2 Data of the four lunar yby transfers in Fig. 1 Orbit Start time (UTC) Perilune time (UTC) Insertion time (UTC) i (deg) v (m/s) v (m/s) leo in end a 2020{08{11 14:48:16 2020{08{15 22:25:25 2021{01{13 19:30:07 24.1822 3128 5.1263 b 2020{07{25 10:24:30 2020{07{30 15:59:30 2021{01{02 01:01:19 24.1763 3133 3.0691 c 2020{07{29 02:36:13 2020{08{02 08:27:33 2020{12{31 08:54:52 24.0894 3129 2.7077 d 2020{08{10 10:43:24 2020{08{15 23:01:51 2021{01{21 04:30:06 24.1849 3134 5.6639 four lunar yby transfers as reference orbits of the tra- tory correction with continuous thrusts in the ephemeris jectory correction. In addition, we note that in Table 2, model. The control strategy is based on the backstepping the insertion maneuvers of the reference orbits v are technique, which has been widely applied in station- end considerably small; hence, a portion of the target LPO keeping [19, 20] and attitude tracking [21]. orbit after the insertion point is added into the reference Based on Eq. (1), the controlled equation of motion of orbit, and we implement continuous thrusts to achieve the spacecraft in the ephemeris model can be rewritten as the insertion instead of the small impulsive maneuvers. In other words, the reference orbit considered in this pa- x = f (t;x) + Bu(t) (3) per includes the lunar yby transfer orbit and a portion where x = [; v] is the phase state, including the position of the target LPO. and velocity state, of the spacecraft in the ECI frame; In practice, various types of errors occur at the ini- f = [v; a], where a is the gravity acceleration, i.e., the tial point introduced by the inaccuracies of the launch rst and second terms of the right-hand side in Eq. (1); vehicle, such as the velocity error [13, 18, 20] and posi- and B = [O I ] , where O and I are 3 3 33 33 33 33 tion error [20]. In this paper, we propose an initial error zero matrix and identity matrix, respectively. The state caused by the launch time error t in the initial LEO err deviation with respect to the reference orbit is x = at an altitude of 200 km. Thus, the launch time error, xx , where x = [ ; v ] is the state of the reference regardless of the delay or advance, in the initial LEO R R R R results in velocity and position errors. orbit; therefore, we can obtain x _ = f (t;x ). Thus, R R we can derive the linear dynamical equation of the state deviation as follows: 3 Control strategy In this section, we propose a control strategy for trajec- x _ = A(t;x )x + Bu(t) R Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 5 O I 33 33 algebraic Riccati equation (ARE) for the in nite-horizon = + Bu(t) (4) A (t;) O v 21 33 problem: where A is the Jacobi matrix and A (t;) = @a=@. 21 T 1 S P + PS Q + PR P = O (10) 33 Since the phase state of the reference orbit x (t) is Subsequently, the control gain of subsystem (8), k, known, A(t;x ) = A(t) only depends on time. The can be obtained using k = R P . Therefore, in the classical optimal linear quadratic regulator (LQR) control numerical computation of k, we only require to solve an technique to minimize the quadratic cost function ARE once. T T J = [x (t)Qx(t) + u (t)Ru(t)]dt For the original system (3), the control input can be expressed as requires a solution to the time-varying Riccati equation. T=m = u = (A S )z kz (t) 21 1 2 To avoid repetitively solving the Riccati equation, we can = (A S + kS)x kx = K x use the backstepping technique proposed by Nazari et 21 1 2 (11) al. to reduce the original system into a time-invariant subsystem for which it is easier to obtain the control law where A can be obtained from the reference orbit, i.e., stabilizing this subsystem [19]. Based on the result of the A (t;) = @a =@ . 21 R R subsystem, we can derive the control law that stabilizes Based on the above analysis, the performance of the the original system through backstepping. backstepping technique depends on the selection of pa- According to Refs. [19, 23], the backstepping transfor- rameters S, Q, and R. When the values of these three mation is expressed as parameters are known, the control strategy is speci ed. In the station-keeping problem [19, 20], S, Q, and R are z = x 1 1 (5) xed during the entire mission. However, lunar yby z = x Sx 2 2 1 transfers are considerably di erent from LPOs in the where x =  and x = vv are the position 1 R 2 R station-keeping problem. The most noticeable di erence and velocity deviations with respect to the reference orbit, is that the geometrical structure and orbital environment respectively, and S is a constant negative de nite matrix. of lunar yby transfers are divided into two portions According to Eq. (4), we obtain before and after the lunar yby. Based on Ref. [24], the prior-lunar yby can be approximately analyzed in x _ = x 1 2 (6) the Earth{Moon CRTBP, while the post-lunar yby can x _ = A (t)x + u 2 21 1 be approximately studied in the Sun{Earth CRTBP. In Hence, taking the derivative of Eq. (5) with respect to Ref. [18], an error analysis before and after lunar yby t, we obtain: indicated that, compared with the transfer without a z _ = Sz + z 1 1 2 lunar yby, the initial error of the lunar yby transfer (7) is ampli ed by the lunar yby. Therefore, we postulate z _ = (A (t) S )z Sz + u 2 21 1 2 that two sets of the parameters S, Q, and R should be Let u = (A S )z + u ; thus, we can obtain a 21 1 1 adopted in two portions before and after the lunar yby. time-invariant subsystem: First, we estimate the magnitude of S, i.e., kSk. As mentioned earlier, we use the launch time error (t ) z _ = Sz + u (t) (8) 2 2 1 err to create initial velocity and position errors, and when For the time-invariant subsystem (8), we assume that t is known, the perturbed orbit is speci ed and can be err u (t) = kz (t), where k should satisfy the minimization 1 2 obtained using numerical integration. Using the prograde of the quadratic cost function type 2 transfer in Fig. 1(a) as an example, we assume that T T the launch time on the initial LEO of altitude 200 km is J = [z (t)Qz (t) + u (t)Ru (t)]dt (9) 2 1 2 1 postponed for 10 s, i.e., t = 10 s. In addition, owing err where Q and R are constant weighting matrices, Q 2 to the limitation of navigation technology, the trajectory 33 33 R is a positive semi-de nite matrix, and R 2 R is a correction cannot be performed until a minimum of 12 positive de nite matrix. To calculate k, we must solve an hours after launch. Numerical computation indicates 6 Y. Qi, A. de Ruiter that when the continuous thrust begins to operate 12 estimation equation of Q and R and set Q = 10 I . 33 hours after the launch, the position and velocity errors For the portion before the lunar yby, we adopt a larger from the reference orbit here are approximately 1545 km Q compared with the estimation equation to increase and 24 m/s, respectively. Hence, kx k is approximately the convergence rate of the perturbed orbit; therefore, 0.024 km/s, and kx k is approximately 1545 km. Since we set Q = 0:6I . To distinguish these two values of 1 33 z and z in Eq. (5) are the equivalent parameters of Q before and after the lunar yby, we denote them as 1 2 x and x , respectively, based on the second equation Q and Q , respectively. For the application scope of 1 2 1 2 of Eq. (5), we estimate that kSk should be smaller than Q , we postulate that it should encompass the portion 0:024=1545  1:5 10 . The purpose of the backstep- before the lunar yby and the process of the lunar yby. ping control is to drive z to zero asymptotically. Based Based on the geometrical structure of the lunar yby on Eq. (5), if we set z = 0, x _ = x = Sx , which transfers (Fig. 1), the region in which the distance of 2 1 2 1 has the solution x (t) = exp (St)x (0). Hence, the the spacecraft to the EMB is smaller than 1.1a (1 + e ) 1 1 m m magnitude of S can also determine the rate of convergence can seemingly satisfy the above requirement, where a of the position error. denotes the semi-major axis of the lunar orbit (approxi- Second, we discuss the relationship between Q and R. mately 3:8476 10 km) and e is the eccentricity of the In the cost function (J ) in Eq. (9), the terms z Qz and 2 lunar orbit (approximately 0.0549). Outside this region, u Ru represent the performance indices of the trajec- 1 Q is appropriate. 1 2 tory correction and fuel consumption, respectively. For For instance, we use an idealistic simulation to com- a xed kRk, a larger kQk can increase the convergence pare the e ects of Q on the trajectory correction of the rate of the perturbed orbit to the reference orbit, but prograde type 2 transfer in Fig. 1(a). The idealistic simu- the thrust acceleration (u) will also increase; otherwise, lation requires that practical errors, such as navigation or the decrease in kQk can retard the convergence rate of execution errors, and the limitation of the thrust engine, the perturbed orbit and reduce the thrust acceleration. such as the maximum thruster T , are not considered. max 6 6 As previously discussed, kz k is the same as kx k, i.e., 2 2 If t = 10 s, S = 4  10 I , R = 10 I , and err 33 33 approximately 24 m/s. For the thrust engine in Ta- Q = 10 I , Fig. 2 shows performances of two types 2 33 ble 1, the maximum magnitude of the thrust acceleration of trajectory corrections for Q = 0:6I and 10 I , 1 33 33 5 2 kuk is approximately T =m = 5:2 10 m/s . We max 0 where d and v denote the distance and velocity dif- T T assume that the terms z Qz and u Ru have simi- 2 2 1 1 ference of the perturbed orbit from the reference orbit, lar magnitudes to balance the trajectory correction and respectively. In Fig. 2, the large divergence of the per- thrust acceleration. Subsequently, we can estimate that turbed and reference orbits in the beginning is caused kQk  4:5 10 kRk. However because the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon [18], the conver- gence rate of the perturbed orbit before the lunar yby, rather than the decrease in thrust acceleration, should be prioritized. As stated earlier, the magnitudes of S, Q, and R can a ect the convergence rate of the perturbed orbit. For convenience, we x S and R at constant values and use the magnitude of Q to change the rate of convergence for portions before and after the lunar yby. Based on the above discussion, S is a negative de nite matrix and kSk should be smaller than 1:5  10 ; therefore, for the entire transfer, we set S = 4 10 I . R is a 33 positive de nite matrix, and numerical computation indi- cates that R = 10 I is feasible for the entire transfer. 33 Fig. 2 Performances of two types of trajectory corrections For the portion after the lunar yby, we use the above for Q = 0:6I and 10 I . 1 33 33 Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 7 by the initial launch time error; the other large diver- In the transfer mission to a Sun{Earth/Moon LPO, gence of the perturbed and reference orbits at t = 153 the navigation system cannot provide updates in real days is caused by the insertion maneuver in the reference time and requires a navigation interval. Let t and T i ni orbit. As the gure shows, the trajectory correction with denote the navigation epoch and navigation interval time, kQk = 0:6 requires a larger T at the beginning of the respectively. Based on Eq. (4), in the navigation interval trajectory correction, but after that, its thrust T , d, and [t ; t + T ], the state error can be obtained using a i i ni v are all smaller than those for kQk = 10 . This forward numerical integration of the linear dynamical result con rms our earlier inference that a larger kQk system as follows: can increase the convergence rate of the perturbed orbit BT (t) x _ = A(t;x )x + to the reference orbit, but the thrust acceleration (u) will also increase. In addition, numerical computation m _ = (12) indicates that the propellant mass losses, i.e., m , for g I loss 0 sp trajectory corrections with kQ k = 0:6 and 10 I are 1 33 x(t ) = x i ni 4.1691 and 4.7528 kg, respectively. Therefore, for the where x is the state error with respect to the ref- ni total fuel consumption, using a larger kQ k to promptly erence orbit obtained from the navigation system at t . reduce the position and velocity errors before the lunar In the navigation interval [t ; t + T ], x obtained i i ni yby is more reasonable. from the linear dynamical system (12) is the pseudo state In addition, as stated in Ref. [20], a dead-band scheme error rather than the actual state error. In Eq. (12), can avoid thrusters operating continuously for long-term A(t;x ) = A(t) only depends on time and can be ob- maneuvers. Figure 3 shows the schematic of a dead- tained a priori; thus, the computational cost of the nu- band scheme. It assumes that thrusters shut down when merical integration of the linear system (12) is lower the distance of the spacecraft from the reference orbit than that of the actual ephemeris model and is more decreases to a lower boundary (which is denoted by d ) low computationally implementable on a ight processor. from a farther position; subsequently, thrusters start up The navigation error is denoted by " = [" " " " when the distance from the reference orbit increases to an n X Y Z _ " " ] . Thus, the actual state error obtained from the upper boundary (which is denoted by d ) from a closer _ _ up Y Z navigation system at time t can be expressed as position. The interval times between t and t in Fig. 3 i 1 2 is the idle time of the thrusters. In this paper, unless x = x (t ) + " (13) ni real i n explicitly stated, d and d are set to 5 and 20 km, low up respectively. where x (t ) is the actual state error without errors. real i Similar to the analysis in Q, we consider that using two values of the navigation interval time (T ) before and ni after the lunar yby is necessary because the lunar yby can enlarge the divergence of the perturbed transfer from the reference transfer [18]. Thus, a short T before the ni lunar yby can increase the accuracy of the pseudo state error in the portion before the lunar yby. Let T and ni1 T represent the values of T before and after the ni2 ni lunar yby, respectively, with T < T . For the ni1 ni2 application scope of T , we assume that it is identical ni1 Fig. 3 Schematic of a dead-band scheme. to that of Q , i.e., the distance of the spacecraft to the EMB should be smaller than 1.1a (1 + e ). m m Based on Eq. (11) and the pseudo state error (x), the 4 Practical constraints idealistic thrust in the navigation interval [t ; t + T ] i i ni In this section, practical constraints during trajectory can be expressed as correction are introduced. Two sources of practical con- T (t) = mK (t;x )x (14) straints exist: the navigation and propellant systems. 8 Y. Qi, A. de Ruiter If the engine limitation is considered, T (t) can be rewritten as >0; T < T min max T (t) = T; T > T (15) max T; T 2 [T ; T ] min max where T and T are the minimum and maximum min max magnitudes of the thrust, respectively. Let " , (i = X; Y; Z ) denote the execution error. Thus, the control input executed in Eq. (12) is 0; T < T min T (t) = (16) E T ; T > T m min Fig. 4 Statistical results of trajectory corrections with dif- where E = diag[1 + " ; 1 + " ; 1 + " ]. m T T T X Y Z ferent navigation intervals and launch time errors t . err Using Eqs. (12){(16), a control strategy with practi- cal constraints in the navigation interval [t ; t + T ] is i i ni are all considered. Figure 4 shows the statistical results of established. We assume that both the navigation and exe- trajectory corrections with di erent navigation intervals, cution errors are white noise processes and have standard where T represents the total idle time. For the control idle normal distributions with zero means. The parameters strategy with a dead-band scheme, the total idle time of practical constraints are listed in Table 3, for which T includes two parts: the idle time of the dead-band idle the parameters of the navigation system were obtained scheme and the idle time of T < T , but for the control min from the navigation uncertainties of the ARTEMIS mis- strategy without a dead-band scheme, only the latter sion [25]. T is obtained from Table 1, and T is max min exists. For each simulation with particular values of t err equal to 1% of the maximum thrust of a single thruster. and (T ; T ), 100 sample points with practical ni1 ni2 In addition, we assume that the initial launch time error constraints were selected. As mentioned earlier, the navi- is known in the trajectory correction. gation and execution errors are white noise processes and Table 3 Practical constraints have standard normal distributions with zero means. As Parameter Value Fig. 4 shows, for the particular launch time error t , err trajectory corrections with di erent navigation intervals Navigation errors 1 km X ;Y ;Z had almost similar mean fuel consumptions or mass losses n n n 1 cm/s _ _ _ X ;Y ;Z n n n m ; therefore, the marks for (0; 0) and (0:5; 2) days ap- loss Execution errors peared covered by the marks for (0:5; 5) days. However, 2% T ;T ;T X Y Z the trajectory correction with a longer navigation inter- Engine limitation val had a longer T . In addition, the gure show that idle T 26 mN max trajectory corrections with jt j = 1 s had a lower fuel err T 0.13 mN min consumption or m but had a higher T than those loss idle with jt j = 5 and 10 s. However, the di erences between err the simulation results with jt j = 5 and 10 s were not 5 Numerical simulations err apparent. Based on the results in Fig. 4, we postulate 5.1 E ects of navigation intervals that (T ; T ) = (0:5; 5) days is a well-balanced ni1 ni2 option between the fuel cost and idle time. Hence, unless In this subsection, we investigate the e ects of navigation explicitly stated, T and T were set to 0.5 and intervals on the trajectory correction of the lunar yby ni1 ni2 5 days, respectively, in the subsequent simulations. transfer using Monte Carlo simulations. Speci cally, the prograde type 2 transfer shown in Fig. 1(a) is used as the Figure 5 depicts the details of a trajectory correction reference orbit in numerical simulations of this subsec- for t = 10 s and (T ; T ) = (0; 0) days. In err ni1 ni2 tion, and the dead-band scheme and practical constraints Fig. 5(a), the dashed and solid lines are the perturbed Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 9 (a) Perturbed and correction orbits (b) Performance of trajectory correction (c) A zoom of T curve (d) A zoom of d curve Fig. 5 Trajectory correction for the prograde type 2 orbit without navigation intervals. and correction orbits, respectively. In the correction orbit, one more long thrust segment with a larger T appeared because of the insertion maneuver in the reference orbit, blue and red arcs are idle and thrust arcs, respectively. but we can observe that this thrust segment was shorter The switching point indicated by a black point is the than the rst one from t = 0.5{21 days, and this d position at which the Q value changes from Q to Q . 1 2 was much smaller than the rst one. Furthermore, the Based on the application scope of Q , it is the point on the magni cations of the T and d curves in Fig. 5(b) are lunar yby transfer whose distance to the EMB is equal shown in Figs. 5(c) and 5(d), respectively. These two to 1.1a (1+e ). As the gure shows, the perturbed orbit m m gures indicate that the engine limitation and dead- diverged from the reference transfer and passed through band scheme performed their functions in the trajectory the target LPO, but the correction orbit could remain correction. around the reference orbit, and it was inserted into the As a comparison, Fig. 6 shows the details of a trajectory target LPO. In Fig. 5(b), green and red zones indicate the correction with t = 10 s and (T ; T ) = (0:5; 5) err ni1 ni2 thrust segment of the dead-band scheme and the actual days. In Fig. 6(a), the dashed and solid lines are the thrust segment for T > T , respectively. The dashed min perturbed and correction orbits, respectively. In the vertical lines at t = 5:142 days in this gure indicate the correction orbit, the blue and red arcs are the idle and switching epoch of the kQ k value. This gure shows thrust arcs, respectively. The switching point of the kQ k that from the beginning of the correction (t = 0:5 days) value is indicated by a black point. As the gure shows, to approximately t = 21 days, the low-thrust engines this correction orbit is considerably similar to that in continuously operated with a relatively large T . During Fig. 5(a). Similarly, the green and red zones in Fig. 6(b) this time, the divergence of the correction orbit from are the thrust segment of the dead-band scheme and the reference orbit d initially increased and subsequently the actual thrust segment with T > T , respectively. min decreased signi cantly. Correspondingly, the spacecraft The dashed vertical lines indicate the switching epoch of mass signi cantly decreased owing to fuel consumption the kQ k value. By comparing Figs. 5(b) and 6(b), we during this time. Subsequently, because the divergence observe that for the two long thrust segments during t = of the correction orbit from the reference orbit decreased 0.5{21 and 155{171 days, the two trajectory corrections signi cantly, several fragmentary short-thrust arcs with with di erent navigation intervals were similar. However, small T values were distributed in the correction orbit from t = 21{155 days, we observed that the actual thrust 21 days later. When t was approximately 155 days, segments of the correction orbit with navigation intervals 10 Y. Qi, A. de Ruiter (a) Perturbed and correction orbits (b) Performance of trajectory correction (c) A zoom of T curve (d) A zoom of d curve Fig. 6 Trajectory correction for the prograde type 2 orbit with (T ; T ) = (0:5; 5) days. ni1 ni2 in Fig. 6(b) were much more fragmentary and shorter 5.2 E ects of the dead-band scheme than those without navigation intervals in Fig. 5(b). This In this subsection, we investigate the e ects of the dead- conclusion can be con rmed by the magni cations of the band scheme on the trajectory correction. As discussed T and d curves in Figs. 6(c) and 6(d). We postulate that in Section 3, the dead-band scheme is determined by the position and velocity deviations could not decrease the lower and upper boundaries of the idle segment, rapidly to a small range owing to navigation intervals and d and d , respectively. The previous subsection dis- low up further resulted in longer thrust segments of the dead- cusses numerical simulations with the dead-band scheme band scheme (green zones in Fig. 6(b)). In Fig. 6(d), of (d ; d ) = (5; 20) km. In this subsection, two more low up d is the actual position deviation with respect to the real types of dead-band schemes of (d ; d ) = (0; 0) km and low up reference orbit, and d is the pseudo position deviation pse (10, 10) km are shown. Speci cally, the trajectory cor- obtained using Eq. (12): The curve of d uctuated pse rection with (d ; d ) = (0; 0) km was the one without low up around that of d in a small range. At the navigation real the dead-band scheme because, based on the de nition time (see the black points in Fig. 6(d)), the state of of the dead-band scheme in Fig. 3, the entire transfer spacecraft with navigation errors was obtained by the had no idle segments. In the numerical simulations in navigation system. Subsequently, during the navigation this subsection, the prograde type 2 transfer in Fig. 1(a) interval, the control strategy with practical constraints was used as the reference orbit, and practical constraints proposed in the last section was conducted. d , obtained pse listed in Table 3 were all considered. The navigation from the linear dynamical system, gradually left from interval was set as (T ; T ) = (0:5; 5) days, and ni1 ni2 d over time. However, at the next navigation time, the real the initial launch time error was set as t = 10 s. err de ected state of the spacecraft was partially corrected Figures 7(a) and 7(b) show the performances of two by the navigation system. In addition, the numerical trajectory correction scenarios with (d ; d ) = (0; 0) low up computation indicated that mass losses in the trajectory and (10, 10) km, respectively. In these gures, green corrections in Figs. 5 and 6 were considerably similar and red zones are the thrust segments of the dead-band (approximately 2 kg), but the idle time T of the latter scheme and the actual thrust segment with T > T , re- idle min was approximately 10 days larger than that of the former, spectively. The dashed vertical lines indicate the switch- which corresponded with the result shown in Fig. 4. ing epoch of the kQ k value. d is the real position 1 real Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 11 (i.e., green zones in Figs. 6 and 7) rather than the T , d, and m curves. We implemented Monte Carlo simulations to observe the di erences in statistics results with di erent dead- band schemes. For each simulation, 100 sample points with practical constraints were selected. As mentioned earlier, the navigation and execution errors are white noise processes and have standard normal distributions with zero means. Table 4 lists the Monte Carlo results of three di erent dead-band schemes, where Std. is the abbreviation of the standard deviation of the words. (a) (d ; d ) = (0; 0) km low up The table indicates that the mean values and standard deviations of their mass losses were considerably similar, but the mean idle time (T ) of (d ; d ) = (5; 20) km idle low up was larger than those of the other two dead-band schemes. This result can be explained by the observation indicated in Figs. 6 and 7, because the trajectory correction with (d ; d ) = (5; 20) km had the shortest total thrust low up segment of the dead-band scheme (i.e., the total length of green zones). Therefore, the trajectory correction with (d ; d ) = (5; 20) km had more idle time than those of low up the other two dead-band schemes. Table 4 Monte Carlo results of trajectory corrections with (b) (d ; d ) = (10; 10) km di erent dead-band schemes up low (d ; d ) (km) (5, 20) (0, 0) (10, 10) Fig. 7 Performances of two trajectory correction scenario low up with di erent dead-band schemes. Mean m (kg) 2.0393 2.0391 2.0353 loss Std. m (kg) 0.0073 0.0077 0.0072 loss deviation with respect to the reference orbit, and d pse Mean T (day) 147.8935 118.8867 120.0013 idle is the pseudo position deviation obtained by Eq. (12). Std. T (day) 2.4905 1.9880 2.3296 idle Comparing with Fig. 6(b), we observe that the T , d, and m curves are not noticeably di erent with di erent 5.3 Examples of di erent lunar yby trans- dead-band schemes, respectively. The green zones indi- fers cate thrust segments of the dead-band scheme. Based In this subsection, trajectory corrections for other lunar on the de nition of the dead-band scheme, dead-band yby transfers in Fig. 1 are analyzed and compared. In schemes with di erent (d ; d ) had di erent distri- low up the numerical simulations of this subsection, the initial butions of thrust segments, and this conclusion can be launch time error was set as t = 10 s, and the dead- err con rmed by the green regions in Figs. 6 and 7. We band scheme of (d ; d ) = (5; 20) km was used in the low up observed that the most signi cant variation among them trajectory corrections. The practical constraints listed in was the distribution of green zones. As stated earlier, Table 3 were all considered, and the navigation interval the trajectory correction without dead-band scheme, i.e., was set as (T ; T ) = (0:5; 5) days. In addition, ni1 ni2 (d ; d ) = (0; 0) km, had no idle segments during the low up control parameters in the backstepping technique, such entire transfer; therefore, the green zone encompassed as S, R, Q , and Q , were identical to the setting for 1 2 the entire transfer. The trajectory correction with the prograde type 2 in Section 3. We adopted the same dead-band scheme of (d ; d ) = (10; 10) km had idle low up application scope of Q as that in Section 3. segments but its idle segments seemed shorter than those First, we compare trajectory corrections for the four in Fig. 6. In summary, the dead-band scheme on the tra- lunar yby transfers in Fig. 1 by the Monte Carlo simu- jectory correction primarily a ected the thrust segment lations. In the Monte Carlo simulations, for a particular 12 Y. Qi, A. de Ruiter reference orbit, 100 sample points with the initial launch time error and practical constraints were selected. As mentioned earlier, the navigation and maneuver errors listed in Table 3 were white noise processes and had standard normal distributions with zero means. Table 5 shows the Monte Carlo results, where Std. is the abbre- viation of the standard deviation of the words. As the table shows, trajectory corrections for prograde types 2 and 4 had similar mean mass losses, which were smaller than those for the retrograde types 2 and 4. In particular, (a) Perturbed and correction orbits the perturbed retrograde type 4 transfer had signi cantly higher fuel consumption or mass loss than the other three lunar yby transfers. The table indicates that prograde type 2 had the largest mean (T ), and retrograde type idle 4 had the smallest mean. However, note that the direct comparison of T is meaningless for di erent lunar yby idle transfers because their total ight times (TOFs) di ered. The numerical computation indicated that the TOFs of prograde type 2, prograde type 4, retrograde type 2, and retrograde type 4 were 202.3167, 193.9958, 200.7637, and 202.0079 days, respectively. We postulate that T =TOF idle can re ect the proportion of idle time during the whole transfer, which is also listed in Table 5. We can observe (b) Performance of trajectory correction that prograde type 2 had the largest proportion of idle Fig. 8 Trajectory correction for the prograde type 4 orbit. time, and retrograde type 4 had the smallest one. The thrust segment of the trajectory correction for the retro- dashed and solid lines are the perturbed and correction grade type 4 occupied approximately half of the transfer orbits, respectively. In the correction orbit, the blue time; therefore, it had the largest mass loss. Although and red arcs are the idle and thrust arcs, respectively. prograde type 2 had the largest proportion of idle time, The switching point of the kQ k value is indicated by its TOF was larger than the other three cases. There- a black point. As the gure shows, the perturbed orbit fore, its mass loss was not the lowest. In addition, based passed through the target LPO, but the correction or- on Eq. (2), the mass loss was determined by the thrust bit could remain around the reference orbit, and it was magnitude and thrust time. Hence, even if the prograde inserted into the target LPO successively. In Fig. 8(b), type 2 had the shortest thrust time, its total mass loss the green and red zones are the thrust segment of the may not have been the lowest. dead-band scheme and the actual thrust segment with Subsequently, we discuss some trajectory correction T > T , respectively. The dashed vertical lines indicate examples for di erent lunar yby transfers in detail. Fig- min ure 8 depicts the details of the trajectory correction for the switching epoch of the kQ k value. d is the actual 1 real the prograde type 4 orbit in Fig. 1(c). In Fig. 8(a), the position deviation with respect to the reference orbit, Table 5 Monte Carlo results of trajectory corrections for di erent lunar yby transfers Transfer type Prograde 2 Prograde 4 Retrograde 2 Retrograde 4 Mean m (kg) 2.0393 2.0080 3.7705 11.2331 loss Std. m (kg) 0.0073 0.0102 0.0306 0.0853 loss Mean T (day) 147.8935 138.4160 137.9849 107.7510 idle Std. T (day) 2.4905 3.4646 2.7536 2.5604 idle Mean T /TOF 0.7310 0.7135 0.6873 0.5334 idle Trajectory correction for lunar yby transfers to libration point orbits using continuous thrust 13 and d is the pseudo position deviation obtained by prograde lunar yby orbits in Figs. 6 and 8, we observe pse that both T and d curves of the retrograde type 2 orbit Eq. (12): We observed that the T curve in Fig. 8(b) was are signi cantly di erent. The trajectory correction for similar to that in Fig. 6(b), but the former had longer the retrograde type 2 orbit had a longer thrust segment green zones than the latter. In addition, we observed that at most times, the curve of d uctuated around with T = T than the above two prograde lunar yby pse max that of d in a small range. However, when t was orbits because its d curve could not rapidly decrease to a real approximately 160 days, d was clearly larger than small value after the lunar yby similar to the prograde real d . Numerical computation indicated that the insertion lunar yby orbits. As Fig. 9(b) shows, the d curve has two pse maneuver occurred at t = 155:23 days, and the latest peaks before the switching point. Numerical computation indicated that the rst peak occurred before the lunar navigation epoch occurred at t = 155 days. Because d pse yby, and the second one appears before the spacecraft was calculated over almost 5 days according to the state passed through the lunar orbit. We consider that this before the insertion maneuver, its curve diverged from phenomenon can be explained by the special shape of d until the next navigation epoch t = 160 days. real retrograde lunar yby transfers. As Fig. 9(a) shows, the Figure 9 shows the detail of a trajectory correction for retrograde lunar yby orbit experienced a lunar yby and the retrograde type 2 orbit in Fig. 1(b). In Fig. 9(a), an Earth yby, successively, before the switching point. the dashed and solid lines denote the perturbed and The Earth yby, similar to the lunar yby, could also correction orbits, respectively. In the correction orbit, result in a signi cant divergence of the perturbed orbit the blue and red arcs are idle and thrust arcs, respectively. from the reference transfers. Therefore, the d curve has The switching point of the kQ k value is represented by two peaks before the switching point. For the retrograde a black point. The correction orbit could remain around type 2 orbit, a longer ight time in the lunar orbit resulted the reference orbit and was inserted into the target LPO in a second increase in d before the switching point. In successively. Compared with trajectory corrections for addition, we observed that when the insertion maneuver occurred at t = 160:7 days, the divergence of d from pse d was smaller than that in Fig. 8, because the next real navigation epoch at t = 163 days was close to the insertion maneuver point. Figure 10 depicts the details of a trajectory correction for the retrograde type 4 orbit in Fig. 1(d). In Fig. 10(a), the dashed and solid lines are the perturbed and correc- tion orbits, respectively. In the correction orbit, the blue and red arcs are the idle and thrust arcs, respectively. The switching point of the kQ k value is represented by (a) Perturbed and correction orbits a black point. The correction orbit could remain around the reference orbit and was inserted into the target LPO successively. Compared with trajectory corrections for the other three lunar yby orbits, we observed that the trajectory correction for the retrograde type 4 orbit had a much longer thrust segment with T = T (approx- max imately 40 days) because its d curve could not rapidly decrease to a small value after the lunar yby, and it even increased to approximately 3:2 10 km after the switching point. Moreover, the numerical computation indicated that a long ight time in the lunar orbit re- sulted in an increase in d before and after the switching (b) Performance of trajectory correction point. In addition, numerical results indicated that, in the navigation interval t 2 [163; 168] days, the insertion Fig. 9 Trajectory correction for the retrograde type 2 orbit. 14 Y. Qi, A. de Ruiter method is feasible and ecient for problems with dif- ferent practical constraints. In contrast to the previous reference on this problem, the propellant system in this study was the continuous thrust rather than the impul- sive thrust. In addition, compared with a previous study, more practical constraints in the trajectory correction were considered in this study, such as the dead-band scheme, navigation intervals, execution error, and the limitation of the thrust engine. Therefore, the control strategy proposed herein is more realistic than that in the previous study. A control strategy based on the backstep- ping technique with a dead-band scheme was proposed (a) Perturbed and correction orbits using the ephemeris model. Since the perturbed transfers signi cantly diverge from the reference transfers after the spacecraft passes by the Moon, we postulated that using two sets of the parameters S, Q, and R is necessary in two portions before and after the lunar yby. Based on numerical analysis and computation, we selected a larger Q in the portion of the prior-lunar yby to increase the convergence rate of the perturbed orbit to the reference orbit, although it could result in a larger thrust accel- eration. For the post-lunar yby, we adopted a smaller Q to reduce the thrust acceleration. According to the distance of the spacecraft to the Earth{Moon barycenter, (b) Performance of trajectory correction the application scope of the larger Q is de ned. Fig. 10 Trajectory correction for the retrograde type 4 Using a prograde type 2 orbit as the reference orbit, the orbit. e ects of navigation intervals on trajectory corrections are maneuver occurred at t = 163:6 days and the divergence discussed using the results of numerical simulations. Nu- of d from d exceeded 2000 km, which was much pse real merical computation showed that for a particular launch larger than the other three lunar yby orbits. time error, trajectory corrections with di erent navi- gation intervals have almost similar mass losses, but the 6 Conclusions trajectory correction with longer navigation intervals has a longer idle time. Numerical results indicated that In this paper, trajectory corrections for lunar yby trans- (T ; T ) = (0:5; 5) days is a well-balanced option ni1 ni2 fers to Sun{Earth/Moon LPOs with continuous thrusts between fuel consumption and idle time. The detailed were investigated using the ephemeris model. Although analysis showed that the actual thrust segments of the many studies on trajectory corrections for transfers to correction orbit with navigation intervals are much more LPOs have been conducted, trajectory corrections for fragmentary and shorter than those without navigation lunar yby transfers are insucient and signi cantly dif- intervals because the former has longer thrust segments ferent from those without lunar yby because of their spe- of the dead-band scheme. Subsequently, we implemented cial geometrical and dynamical structures. Lunar yby numerical simulations to observe the di erences in tra- transfer is a high-yielding but high-risk design method; jectory corrections with di erent dead-band schemes. A therefore, its TCM strategy should be speci cally de- dead-band scheme can avoid thrusters operating contin- signed. In this paper, we rst introduce the backstepping uously for long-term maneuvers. 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Control of dynamic sys- engineering from the University of tems with time-periodic coecients via the Lyapunov- Toronto in 2001 and 2005, respectively. oquet transformation and backstepping technique. Between 2006 and 2008 he was a visiting Journal of Vibration and Control, 2004, 10(10): 1517{ research fellow at the Canadian Space 1533. Agency in Montreal, and an assistant professor in the [24] Qi, Y., Xu, S. J. Study of lunar gravity assist orbits in Department of Mechanical and Aerospace Engineering at the restricted four-body problem. Celestial Mechanics Carleton University from 2009 to 2012. He is currently an and Dynamical Astronomy, 2016, 125(3): 333{361. associate professor and Canada research chair (Tier 2) in the [25] Sweetser, T. H., Broschart, S. B., Angelopoulos, V., Department of Aerospace Engineering at Ryerson University Whi en, G. J., Folta, D. C., Chung, M. K., Hatch, S. in Toronto, Canada. His research interests are in the area J., Woodard, M. A. ARTEMIS mission design. Space of guidance, navigation, and control of aerospace systems. Science Reviews, 2011, 165(1{4): 27{57. E-mail: aderuiter@ryerson.ca. Yi Qi received his Ph.D. degree in Open Access This article is licensed under a Creative Com- aeronautical and astronautical science mons Attribution 4.0 International License, which permits and technology from Beihang Univer- use, sharing, adaptation, distribution and reproduction in sity, China, in 2017. After three years any medium or format, as long as you give appropriate credit as a postdoctoral researcher in Ryerson to the original author(s) and the source, provide a link to University, he joined Beijing Institute of the Creative Commons licence, and indicate if changes were Technology as an associate professor in made. 2020. His research area includes orbital The images or other third party material in this article are dynamics and control for deep space exploration. E-mail: included in the article's Creative Commons licence, unless lushenqiyi@gmail.com. indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecoorg/ licenses/by/4.0/.

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Published: Feb 1, 2021

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