I vector separation

Available online at www.sciencedirect.com

ScienceDirect Advances in Space Research 64 (2019) 964–981 www.elsevier.com/locate/asr

Safe deployment of cluster-flying nano-satellites using relative E/I vector separation Pengfei Liu a,⇑, Xiaoqian Chen a,b, Yong Zhao a a

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China b National Innovation Institute of Defense Technology, Beijing, China Received 31 December 2018; received in revised form 15 May 2019; accepted 22 May 2019 Available online 28 May 2019

Abstract Nano-satellites cluster flight has attracted an increasing interest in the domain of distributed spacecraft system in recent years. As the first phase of a cluster flight mission, on-orbit deployment process constitutes a great technical challenge, since not only safe relative trajectories, but also practical operational constraints must be considered. To deal with these issues, the concept of relative Eccentricity/Inclination (E/I) vector separation was utilized in the safety concept design and the sequent release parameters solving, as it provides direct insight into the safety characteristics of relative motion. Accordingly, a novel operational methodology for the safe deployment of cluster-flying nano-satellites is provided. It can deterministically generate deployment sequences that ensure safe relative trajectories between released satellite (RS) pairs, as well as that between the launch vehicle (LV) and RSs, for enough long time interval. Particularly, according to our methodology, no maneuver efforts are required for the LV and RSs either during the deployment process or after deployment. Moreover, the proposed methodology adheres to practical constraints from either the LV or ground station. A typical simulation scenario was setup for the deployment process of the pioneering cluster flight mission - Satellite Mission for Swarming and Geolocation (SAMSON). Results demonstrate the feasibility and efficiency of our methodology. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Cluster flight; Nano-satellites; Safe deployment; Relative E/I vector separation; Operational constraints

1. Introduction Cluster flying (Brown, 2006) (or cluster flight (BenYaacov et al., 2016; Wang et al., 2019; Luo et al., 2017; Liu et al., 2016; Wang and Nakasuka, 2012)), refers to the operation of multiple, cooperative satellites under the minimum and maximum relative distance (Zhang et al., 2017) constraint over long time intervals (Mazal and Gurfil, 2014). Commonly, these collaborative satellites are deployed together in a single launch, such as in SAM-

⇑ Corresponding author at: No. 109, Deya Road, KaiFu District, Changsha 410073, Hunan Province, China. E-mail addresses: [email protected] (P. Liu), chenxiaoqian@ nudt.edu.cn (X. Chen), [email protected] (Y. Zhao).

https://doi.org/10.1016/j.asr.2019.05.036 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.

SON (Gurfil et al., 2012) and NetSat (Nogueria et al., 2017) missions. From the viewpoint of relative motion, the lifetime of a typical, cluster-flying satellites system can be generally divided into four phases (Wen et al., 2015): (1) on-orbit deployment from launch system; (2) post-deployment free flying for commissioning; (3) cluster establish maneuvers; (4) long-term cluster keeping. Thus, how to safely deploy these cooperative satellites, raises the first issue for cluster flight mission design. The on-orbit deployment process of satellite is commonly an open loop and uses a predefined, time-tagged sequence, with deployment timing, release velocity magnitude and direction, and LV attitudes as inputs (Atchison and Rogers, 2016).

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In the case of single satellite deployment, specifying a feasible release sequence is quite trivial. However, when deploying multiple satellites, there comes much more challenges (Wen et al., 2015). Among them, RS-RS and RS-LV collision avoidance is the most critical one. Some research works have been done on this point. For example, Lambert et al. (2010) investigated the relationship between release parameters and collision risk via case-by-case simulations. Santoni et al. (2012) assessed the collision possibility for nano-satellites cluster by performing Monte Carlo (MC) simulations. In the work of Kilic et al. (2013), the best choice for direction, sequence, and frequencies of deployment was found to prevent collision risk between the deployed CubeSats and the upper stage of LV in QB50 mission. Optimal deployment angle and velocity was selected after detailed sensitivity and MC analysis to minimize the commissioning-phase collision probability in CYGNSS mission (Finley et al., 2014). Moreover, Atchison and Rogers (2016) presented an operational deployment methodology which minimized collision risk among large numbers of nano-satellites by changing deployment order, timing, and LV attitudes. For the deployment of cooperative multi-satellites systems, such as that in cluster flight missions, collision avoidance is necessary, but not sufficient. Other key factors should be also considered (Wen et al., 2015). First, the RS-RS safety requires a minimum relative distance to avoid collision, meanwhile limits a maximum relative range to prevent cluster dispersion. Second, deployment plans should accommodate with operational constraints from launch system, satellite platform, and ground station. For example, an on-orbit deployment process conducted by the LV depends on the limited, primary power of onboard batteries. This limits release sequence duration to less than 10 min (Atchison and Rogers, 2016). In addition, the postdeployment, free-flying relative motions among RSs should be passively safe for sufficient time, to allow for commissioning operations (Koenig and D’Amico, 2018). Moreover, for safety, all separation operations should be completed during a single ground station contact. This poses restrictions on the deployment location in orbit (Wermuth et al., 2015). Third, on the one hand, onboard fuel is usually limited for cluster-flying nanosatellites; on the other hand, post-deployment cluster establish maneuvers account for the largest part of fuel consumption during the whole mission lifetime (Wen et al., 2015). Thus, for deployment plan design, it is desirable to limit fueling-cost in the subsequent maneuver efforts as low as possible. Fourth, a practical deployment strategy design should be also compatible with various launch systems, not only those have maneuver ability, such as the LV, but also the unmaneuverable ones like the International Space Station (ISS) (Brown, 2015). Efforts have been made recently to address these challenges above. Generally speaking, none have provided an ideal solution for all of these critical aspects. Specifically, Boutonnet et al., using primer vector theory, formulated

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an invariant formation deployment problem as a rendezvous process involving multiple impulsive maneuvers, followed by an optimal deployment phase with minimized fuel consumption (Boutonnet et al., 2005). But it requires numerical computation for minimum relative distances to assess collision risks among the LV and RSs. Conversely, Wen et al. (2015) provided an analytical constraint on the safe operation of cluster-flying satellites during the deployment process. However, to prevent cluster dispersion, a cluster establishment maneuver accompanied by considerable fuel consumptions, should be engaged shortly after the deployment. To enable RSs flying in a formation without any establishment maneuver, the concept of separation initialization was firstly presented by Jiang et al. (2015). Based on the relationship between release parameters and formation geometry in the Hill framework, a set of constraints for separation velocity have been derived to realize a stable formation under J2 perturbation. However, none of the practical constraints relating to the LV and ground station was considered. To deal with several practical constraints existed in the binary formation safe separation problem of AVANTI mission (Gaias and Ardaens, 2018, 2016), Wermuth et al. introduced the concept of relative E/I vector separation (D’Amico and Montenbruck, 2006) which features a direct insight into the safety characteristics of relative motion (Wermuth et al., 2015). With this feature in mind, Koenig and D’Amico (2018) extended the application of relative E/I vector separation to multiple satellite deployment scenarios, and proposed a set of safe deployment procedures for spacecraft swarms in perturbed near-circular orbits. However, the generated deployment sequence released each satellite at a quite low frequency of one per orbit, meanwhile leading to considerable fuel consumptions on the mothership satellite for each deployment. The goal of this paper is to provide an operational methodology for the safe deployment of cluster flight nano-satellites in near-circular orbits without any maneuver efforts, meanwhile taking practical operational constraints into account. To this end, a novel practical procedure to generate safe deployment sequence is proposed, on the basis of Koening’s previous work (Koenig and D’Amico, 2018), by following a four-step workflow illustrated in Fig. 1. At first, critical release parameters for each satellites in the Hill framework are represented by relative E/I vector with respect to the launch system in the relative orbital elements (ROE) space. Then, safety criteria for relative motion as well as practical operational requirements are formulated into operative constraints on the relative E/I vector. After that, feasible solutions for relative E/I vector under these operative constraints are obtained via a geometry method in the ROE space. Finally, a safe deployment sequence is obtained by converting these feasible relative E/I vector into release parameters. This paper contributes to the state-of-the-art in three aspects. First, the proposed methodology can procedurally and deterministically generate safe deployment sequences

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2005). On the other hand, the relative motion description in terms of orbital element differences, features several advantages such as a deeper geometric insight, a slow time-varying nature, as well as the easier handling of orbital perturbations and maneuver effects (Sullivan et al., 2017). Therefore, the relative motion model parameterized by ROEs (Bevilacqua and Lovell, 2014; Lovell and Tragesser, 2004; Sullivan et al., 2017, Wang et al., 2019), especially the quasi-nonsingular ROEs (Di Mauro et al., 2018; Willis and D’Amico, 2018) which contain the relative E/I vector (Lim et al., 2018; Shao et al., 2016), is preferred in this paper and shortly reviewed in the following subsection. 2.1. Relative dynamics based on ROEs (Wermuth et al., 2015): a review Fig. 1. Four-step workflow to get safe deployment sequence.

that ensures safe relative motions among the LV and RSs for sufficient long time, with neither active control maneuver during deployment nor post-deployment cluster establishment maneuver needed. Second, a rigorous safety concept design for relative motion is presented by using relative E/I vector. For that, not only the long-term safety but also the immediate safety are considered; not only the safety between each RS pair, but also that between the LV and each RS are guaranteed. Particularly, practical constraints during the deployment process are handled. Third, this methodology proposed above can be used in a wide range of cluster flight scenarios regardless of the adopted launch system, whether LV, ISS or mothership spacecraft. In the following, an overview of relative motion description based on ROEs is provided at first. Then, the safety concept design for relative motions among the LV and RSs is introduced. After that, a geometrical method to get a safe deployment sequence is described. Finally, simulation results for the deployment scenario in a typical cluster flight mission are presented and discussed, in the attempt to demonstrate the feasibility and efficiency of the overall methodology. 2. Relative motion description To perform close proximity analysis among the LV and RSs, an analytical model to predict relative trajectories should be established at first. Commonly, satellite clusters are supposed to be deployed in near circular orbits, as in missions Pleiades (LoBosco et al., 2008), SAMSON (Gurfil et al., 2012), and NetSat (Nogueria et al., 2017). Under this assumption, relative motion between two objects in space can be described by the well-known Clohessy-Wiltshire (CW) equations (Alfried et al., 2010). Although CW equations provide an analytical solution, the Cartesian formulations fail to give immediate insights into some key aspects of relative motion (D’Amico et al.,

Without loss of generality, two satellites orbiting in close proximity are considered. One is referred as chief, and the other deputy. Then a local Radial(R)-Tangential(T)-Nor mal(N) frame (D’Amico and Montenbruck, 2006) can be established with the origin at chief. According to D’Amico (2010), a set of dimensionless quasi-nonsingular ROEs can be defined as: 1 0 1 0 1 0 da da ða  ad Þ=ad B dk C B dk C B u  u þ ðX  X Þcosi C d d dC C B B C B C B C B C B dex C B decosu C B ecosx  ed cosxd C * B C B C C B B d a ¼B C¼B C¼B C B dey C B desinu C B esinx  ed sinxd C C B C B C B A @ dix A @ dicosh A @ i  id diy disinh ðX  Xd Þsinid ð1Þ where a,e,i,x,X, and M denote classical Keplerian elements and u ¼ M þ x is the mean argument of latitude. The subscript d marks quantities related to deputy. Vectors * * T T d e ¼ ðdecosu; desinuÞ and d i ¼ ðdicosh; disinhÞ define the dimensionless relative eccentricity and relative inclination vectors, with u and h respectively representing the perigee and ascending node of the relative orbit (Wermuth et al., 2015). Under the assumption of CW equations, the linearized relative motion between chief and deputy can be described by ROEs as 8 > < DrR =a ¼ da  de  cosðu  uÞ DrT =a ¼ dkðuÞ þ 2de  sinðu  uÞ : ð2Þ > : DrN =a ¼ di  sinðu  hÞ 8 > <

Differentiating Eq. (2) with respect to time yields DvR =v ¼ de  sinðu  uÞ

DvT =v ¼  32 da þ 2de  cosðu  uÞ : > : DvN =v ¼ di  cosðu  hÞ

ð3Þ

Here v denotes the orbital velocity in a circular orbit with radius a.

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According to Eqs. (2) and (3), the immediate relative motion at u ¼ u0 is illustrated as a tilted ellipse in Fig. 2. As indicated in this figure, the relative motion of deputy with respect to chief is naturally decoupled into a harmonic oscillation perpendicular to the orbital plane and a 2*1 ellipse in-plane motion. While ade and adi determine the dimension of relative motion, relative semi-major axis  ada and relative mean longitude adk ¼ a du þ diy coti locate the mean offsets in radial and along-track direction, respectively. Moreover, the instantaneous phases of inplane and out-of-plane motion are respectively represented by angle u  u and u  h, whereas relative phase angle g ¼ h  u governs the orientation and shape of relative motion in the R-N plane. Particularly, the R-N projection is an ellipse centered in ðada; 0Þ for g 2 f0; pg, while gets dwindled and tilted to a line as g ! p2 (Larbi and Stoll, 2016). Furthermore, if a non-zero relative semi-major axis exists, the in-plane ellipse is instantaneous and drifts in the along-track direction with the dimensionless relative mean longitude,dk, varying with time according to dkðuÞ ¼ 1:5ðu  u0 Þda0 þ dk0

ð4Þ

if no orbital perturbations are considered (Wermuth et al., 2015). The inversion of linear relative motion model above provides a direct relationship between an impulsive maneuver, or an instantaneous velocity change, located at the mean argument of latitude uM and the consequent change of ROEs as follows: aDda ¼ þ2dvT =n aDdk ¼ 2dvR =n aDdex ¼ þðdvR =nÞsinuM þ 2ðdvT =nÞcosuM aDdey ¼ ðdvR =nÞcosuM þ 2ðdvT =nÞsinuM aDdix ¼ þðdvN =nÞcosuM

ð5Þ

aDdiy ¼ þðdvN =nÞsinuM where n denotes the mean motion of chief, and delta-v is expressed in the R-T-N frame as (Wermuth et al., 2015): * T D v ¼ ðdvR ; dvT ; dvN Þ ð6Þ

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The natural motion model in Eq. (2) can be further extended by considering J2 perturbations on orbital elements. Specifically, the Earth’s equatorial bulge induces a variety of short periodic, long-periodic and secular perturbations of orbital elements for an LEO satellite. For satellites flying in close proximity, periodic perturbations are essentially cancelled. The resulting secular variations of ROEs can be derived from Brower’s theory (Alfried et al., 2010) as follows (D’Amico, 2010): 9 8 da > > > > > > > dk  21 csinð2iÞdix ðu  u0 Þ > > > > > 2 > > > > = < 0 decos u þ u ð ð u  u Þ Þ * 0 ð7Þ d a J 2 ð uÞ ¼ > desinðu þ u0 ðu  u0 ÞÞ > > > > > > > > > > > dix > > > > ; : 2 diy þ 3csin ðiÞdix ðu  u0 Þ Here  2 J 2 RE c¼ 2 a u0 ¼

 du 3  ¼ c 5cos2 i  1 du 2

ð8Þ ð9Þ

and RE marks Earth’s radius. Due to its direct reflection on the geometric nature of relative motion, this model above provides a powerful tool to handle collision avoidance problem in proximity operation missions. Generally, collision avoidance can be guaranteed if the relative separation between co-orbiting satellites is no smaller than a specified safe threshold all the times (Koenig and D’Amico, 2018). Moreover, it is well known that uncertainty in predicting the in-track separation of two spacecraft is much higher than that for the radial and cross-track components (D’Amico, 2010). To minimum the collision risk in the presence of in-track position uncertainties, the two involved spacecrafts should be properly separated in radial and cross-track direction. According to D’Amico (2010), this requirement can be achieved by relative E/I vector separation, which ensures a passive minimum separation of d perpendicular to the flight direction. Particularly, when da ¼ 0 is assumed for

Fig. 2. Sketch of ROEs and their relation to instantaneous (no drift) relative motion in the R-T-N frame at u ¼ u0 .

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a stable formation, the necessary and sufficient condition to satisfy a minimum separation of d in the R-N plane can be derived as (Zhou et al., 2017) pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d 2 2 2 4 4 2 2 ðDdeÞ þ ðDdiÞ  ðDdeÞ þ ðDdiÞ  2ðDdeÞ ðDdiÞ cosð2DgÞ P 2 aC

ð10Þ

Here Dde and Ddi represent the size of relative E/I vector, while Dg denotes the relative phase angle between these two involved satellites. Besides, aC marks their common semi-major axis. 2.2. Relative motion between the LV and each RS To investigate the relative motion between the LV and each RS in the deployment process, the LV is defined as chief which indicates the origin of R-T-N frame while each RS is referred as deputy. Before a satellite is deployed, the * relative state, ad a , between this satellite and the LV is null. Then, the separation maneuver with an instantaneous * T velocity change of D v ¼ ðdvR ; dvT ; dvN Þ is executed at the mean argument of latitude u0 , resulting in the following variations of ROEs (Wermuth et al., 2015): 8 naDda ¼ 2dvT > > > > > naDdk ¼ 2dvR > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > < naDde ¼ dv2R þ 4dv2T ð11Þ dvR tanðu0  uÞ ¼ 2dv > > T > > > naDdi ¼ jdvN j > > > > : tanh ¼ tanu ¼ Ddiy 0

Ddix

Similar to Fig. 2, the immediate relative motion patterns at separation time are illustrated in Fig. 3, according to different delta-v profiles. In this figure, relative trajectories in the R-T plane are depicted in dashed lines to represent that in-plane ellipses are instantaneous, whereas that in the R-N plane are displayed as bold line to indicate that out-ofplane relative trajectories are stationary. From these patterns of relative motion, the following features can be exploited to support safety concept design described in the next section: (1) When non-zero relative semi-major axis exists, the R-T plane relative trajectory exhibits an instantaneous ellipse drift at a rate nearly three times the size of dvT (Atchison and Rogers, 2016), but with an opposite direction. For example, a satellite deployed by an separation * velocity D v with component in +T direction, will move forward the LV at first, but soon cross over and eventually behind it due to the in-track drift (Atchison and Rogers, 2016). This constitutes an unsafe relative motion pattern in early-orbit operations (Wen et al., 2015), or the socalled ‘‘first-period Close Approach” (Atchison and Rogers, 2016), which should be carefully handled in the safety concept design for cluster flight missions, as suggested by Wen et al. (2015).

(2) In the R-N plane, maintaining a specified minimum separation can be made possible for quite a long time interval in each orbit, by a proper separation of relative E/I vector. However, relative separation will decrease to zero periodically and determinately at u ¼ u0 þ k  2p(k ¼ 1; 2; 3    N ). In order to guarantee a safe separation between the LV and each RS, it is necessary to ensure that they are safely separated in the R-T plane at these times. 2.3. Relative motion between each pair of RSs To analyze the relative motion between two arbitrary nano-satellites which are successively released from the LV, satellites C and D are defined without loss of generality. Among them, satellite C denotes the early-released one and marks the origin of R-T-N frame. Then, the relative motion between satellite D and satellite C can be modeled as the consequence of a double impulsive maneuver process. Specifically, before satellite C is released, there exists no relative motion. When satellite C is deployed at time tC , the relative motion is established as a result of impulsive * velocity D v C from the viewpoint of the LV or satellite D, meanwhile an equivalent relative motion is induced by * D v C from the viewpoint of satellite C. When satellite D is separated from the LV at time tD ¼ tC þ Dt, another * impulsive velocity D v D further changes the relative motion pattern between these two satellites. As a summary, the rel* ative state of satellite D with respect to satellite C , ad a , can be represented by: 8 * > > ad a ¼ 0 t 6 tC > >  *  < * ad a ¼ Uðt;tC ÞM ðtC Þ D v C tC < t 6 tD >    *  > * > > : ad a ¼ Uðt;tC ÞM ðtC Þ D* v C þ Uðt;tD ÞM ðtD Þ D v D t > tD ð12Þ

with the state transition 2 1 6  3 nð t  t Þ 0 6 2 6 6 0 Uðt; t0 Þ ¼ 6 6 0 6 6 4 0 0

matrix 0 1

0 0

0 0

0

1

0

0 0

0 0

1 0

3 0 0 0 07 7 7 0 07 7 0 07 7 7 1 05

0

0

0

0 1

and the control input matrix 3 2 0 2 0 6 2 0 0 7 7 6 7 6 sinuðtÞ 2cosuðtÞ 0 7 16 7 M ðt Þ ¼ 6 n6 0 7 7 6 cosuðtÞ 2sinuðtÞ 7 6 4 0 0 cosuðtÞ 5 0 0 sinuðtÞ

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Fig. 3. Eight relative motion patterns between the LV and RSs for different release-velocity profiles.

To further illustrate this analysis above, Fig. 4 gives a conceptional sketch of the relative motion between satellite C and satellite D in the ROE space. From Fig. 4 and Eq. (12), two conclusions with respect to the relative motion between each RS pair can be drown as follows: (1) The relative motion commences at the moment when the later-released satellite is deployed. Relative separation between this pair of RSs at this time is the same as that between the LV and the early-released one. (2) The resulting relative motion pattern is governed by the final relative state in the ROE space. It is induced by the two separation maneuvers corresponding to these two satellites. 3. Safety concept design As mentioned earlier, the primary challenge of the deployment process in cluster flight mission is to maintain the immediate and long-term safety among the LV and RSs, in terms of avoiding collision and preventing cluster dispersion, while adhering to a series of practical constraints. With this in mind, the safety concept proposed here provides a method to achieve safe relative motions between the LV and each RS, as well as between each pair of RSs. More importantly, the safety concept and practical constraints are formulated into a set of constraints for relative E/I vector, making it easier to find feasible deployment sequences. The main idea behind this safety concept is absorbed from Larsson’s previous work (Larsson et al., 2008), and summarized as follows. For the safety between the LV and each RS, a ‘‘separation guidance” strategy is applied

to monotonically increase the separation distance and enable the relative trajectory exiting an avoidance region within a prescribed time. When the relative position is outside the avoidance region, it is not allowed to reenter it. On the other hand, for the safety between each RS couple, a ‘‘nominal guidance” strategy is adopted. That is, relative motion between RS pairs should commence and stay outside the avoidance region and, yet, remain inside a bounded range during the commissioning phase. 3.1. Avoidance region and keep-in-zone The avoidance region is defined as a cylinder centered at the origin of R–T-N frame - namely the LV or RS, with radius D, and length L (see Fig. 5). Here, D represents a safety threshold for the minimum separation perpendicular to the flight direction, L is an in-track separation which should be selected large enough to provide inherent safety independently from cluster geometry and relative navigation errors (D’Amico et al., 2013). The keep-in-zone (KIZ), as shown in Fig. 5, is defined as a sphere with radius R, and specifies the maximum relative range allowed for cluster-flying satellites. 3.2. Separation guidance Two criteria, to ensure immediate and long-term safety between the LV and each RS, are defined as follows: (1) Each RS should exit the avoidance region of LV in a short time during which an increasing relative separation should be guaranteed;

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Fig. 4. Sketch of relative motion between RS pairs in the ROE space.

Fig. 5. Avoidance region and keep-in-zone definitions.

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(2) Upon exiting the avoidance region of LV, each RS should never re-enter it. To meet these criteria, safe interval and monotonic interval in the R-N plane are defined respectively in the following. According to Eq. (2) and Fig. 2, relative trajectories between the LV and RSs are outside the avoidance region and thus naturally safe if: aL  jdik  sinðuk  hk Þj P D

ð13Þ

or aL  jdak  dek cosðuk  hk þ gk Þj P D

ð14Þ

where aL denotes the semi-major axis of LV, the subscript k(k 2 ½1; N ) marks quantities related to the kth satellite released from the LV, N stands for the total number of nano-satellites deployed in a single launch. When jaL dik j > D

ð15Þ

and jaL dak j > D

ð16Þ

are assumed, safety boundaries for angle ðu  hÞ in normal direction and radial direction can be obtained as 8 sin1 ðD=aL dik Þ > > > < p  sin1 ðD=a di Þ L k  ðu  hÞN ;k ¼ 1 > > > p þ sin ðD=aL dik Þ : 2p  sin1 ðD=aL dik Þ p=2  gk  ðu  hÞR;k ¼ 3p=2  gk As reflected in Fig. 6, these safety boundaries above divide the ðu  hÞ pie into different sectors, or time intervals. In some of these time intervals, either Eq. (15) or Eq. (16) is satisfied. They are called as safe intervals, for normal direction and radial direction respectively. Conversely, the rest of time intervals, in which none of Eqs. (15) and (16) is fulfilled, are unsafe intervals. As shown in Fig. 6, safe intervals are represented by white sectors, while the unsafe ones are filled with gray. Once angle

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ðu  hÞ related to any RS is located in a white sector, it is reasonable to conclude that the relative motion between the LV and this RS is safe at this moment. On the other hand, monotonic boundaries in radial and normal direction are respectively derived from DvN ¼ vdi  cosðu  hÞ ¼ 0

ð17Þ

DvR ¼ vde  sinðu  h þ gÞ ¼ 0

ð18Þ

and defined as p=2 D ð u  hÞ N ¼ 3p=2 pg D ð u  hÞ R ¼ 2p  g In Fig. 6, monotonic boundaries are marked by black dashed line with blue circle – line AR C R and line EF . Among them, monotonic boundary AR C R divides the ðu  hÞ pie into two monotonic intervals in radial direction with arc AR EC R and arc C R FAR . Monotonic boundary EF , accompanied by line OP , divide the ðu  hÞ pie into four monotonic intervals in normal direction as arcs OE, EP , PF and FO. During the monotonic interval, whether in radial or normal direction, relative separation between RS and the LV varies with time monotonically, either increasing or decreasing. For safety concept design, only the increasing monotonic intervals, arc AR EC R , OE ,and PF , are of concern. Due to the fact that unsafe relative motion patterns typically occur in tangential direction in early-orbit operations, the short-term (namely the first orbit) safety criterion should be guaranteed in the R-N plane. To this end, the safe interval in radial direction and normal direction should complement each other, to allow an enough long time interval for safe relative motion in the first orbit. That means, endpoint BR should be located in arc AN BN (see Fig. 7). This corresponds to the following constraint: p   ðu  hÞN ¼ sin1 ðD=adiÞ 6 ðu  hÞR ¼  g 2  1 6 p  ðu  hÞN ¼ p  sin ðD=adiÞ ð19Þ

Fig. 6. Safe and monotonic intervals for radial (left) and normal direction (right).

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namely p p sin ðD=aL dik Þ  6 gk 6  sin1 ðD=aL dik Þ 2 2 1

ð20Þ

Furthermore, arc OAN should be included in the increasing monotonic intervals for both radial and normal directions, to ensure a monotonically increasing relative separation within the avoidance region. Thus, as shown in Fig. 7, arc OAN should be set inside arc AR EC R . This is equivalent to the following constraints: (

gk 6 0

p  gk P ðu  hÞN ;k ¼ sin1 ðD=aL dik Þ

) 0 6 gk 6 p  sin1 ðD=aL dik Þ

ð21Þ Taking Eqs. (20) and (21) into consideration, constraints for the short-term safety between the LV and RSs can be summarized as follows: 0 6 gk 6 p2  sin1 ðD=aL dik Þ

k 2 ½1; N 

ð22Þ

Relative separation in the R-N plane is not sufficient for long-term safety since it periodically decreases to zero, as mentioned in Section 2.2. Thus, the relative motion in tangential direction must be considered to fill the gap. In fact, the in-track relative motion after early orbits is dominated by the tangential drift caused by non-zero dak , and exhibits a secular behavior which provides inherent safety. By exploiting this feature, the long-term safety can be naturally ensured, if the in-track separation has already left the avoidance region before relative motion in the R-N plane comes to crisis at nearly one orbit after the deployment. This assumption can be formulated into a constraint as follows:

 

3

aL dak  2p  ðu  hÞ > 2aL dek þ L ð23Þ N ;k

2 This constraint can be further simplified into:



3

aL dak  3 p P 2aL dek þ L

2

2

ð24Þ

In summary, the immediate safety between the LV and each RS can be ensured by a relative separation in the RN plane with constraints in Eqs. (15), (16) and (22), while the secular drift in tangential direction provides inherent safety in long terms if Eq. (24) is satisfied. 3.3. Nominal guidance In the deployment process or during the commissioning phase of cluster flight missions, the safety between each RS couple, e.g. satellites C and D, involves two primary aspects. On the one hand, relative motion should be bounded to prevent cluster dispersion; on the other hand, collision risks should be avoided. Similar to separation guidance, safety criteria for each RS pair are defined as follows: (1) Relative motion should commence outside the avoidance region; (2) Relative trajectory should never enter the avoidance region during the deployment process or the sequent commissioning phase; (3) Post-deployment natural relative motion should be stable enough to be maintained in the KIZ before clusterkeeping maneuvers engaged. According to Sections 2.3 and 3.2, criterion (1) can be satisfied by setting 

Du ¼ uD  uC ¼ uD  hC P ðu  hÞN   D 1 ¼ sin aL  diC

ð25Þ

It means that late-released satellites should be deployed outside the avoidance region of early-released ones. Here, subscript C and D respectively denote quantities related to satellites C and D. Criterion (2) and (3) can be realized by the passive stable relative motion established via a proper separation of relative E/I vector (see Section 2.1). Specifically, to achieve a stable relative motion between each RS pair, the mean semi-major axis and inclination of involved satellites must be equal, namely, ðdaÞD;C ¼ ðdaÞD;L  ðdaÞC;L ¼ 0       dix D;C ¼ dix D;L  dix C;L ¼ 0

ð26Þ ð27Þ

Since the periodic perturbations affecting the relative orbital elements for satellites flying in close proximity can be essentially cancelled (Montenbruck et al., 2006), these constraints can be reasonably expressed by osculating orbit elements as follows: Fig. 7. Harmonious safe and monotonic intervals for radial and normal direction.

ðdaÞD;C ¼ ðdaÞD;L  ðdaÞC;L ¼ 0

ð28Þ

ðdix ÞD;C ¼ ðdix ÞD;L  ðdix ÞC;L ¼ 0

ð29Þ

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On the other hand, to avoid collision between each RS couple, the relative E/I vector between the two involved satellites must be properly configured in the ROE space to satisfy the following constraint: pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 2 2 4 4 2 2 ðDdeÞ þ ðDdiÞ  ðDdeÞ þ ðDdiÞ  2ðDdeÞ ðDdiÞ cosð2DgÞ 2

P C;D

D aC

ð30Þ

3.4. Practical constraints Aside from relative motion safety, the deployment process in cluster flight mission also confronts with operational constraints from the LV, satellites and ground station. Among them, the most common one is that, the total duration of deployment process should be constrained to approximately 10 min due to the limited LV primary power from onboard batteries (Atchison and Rogers, 2016). What is more, the deployment process must finished during once contact with ground-station. These constraints above actually restrict deployment locations along the orbit, and can be expressed by ROEs as u0 6 uk ¼ hk 6 uf ; k 2 ½1; N 

ð31Þ

hN  h1 ¼ uN  u1 6 n  tmax

ð32Þ

where u0 and uf respectively denotes the mean argument of latitude for the LV at the beginning and ending of the ground station contact. tmax represents the longest time duration allowed for the deployment process. As a summary, Table 1 lists operational constraints which should be considered for the safe deployment design in cluster flight missions. Each has been formulated into a corresponding constraint on the relative E/ I vector for each RS with respect to the LV. In this way, step 2 in Fig. 1 is completed. The task of next section is to find a feasible configuration for these relative E/I vectors under constraints above and then translate them into deployment sequences.

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4. From safety criteria to safe deployment sequence: A geometrical method After analysis and design in previous sections, the safe deployment issue for cluster-flying nano-satellites can be finally formulated into a mathematical problem to find feasible solutions for relative E/I vector in a constrained ROE space (see Fig. 8), which is defined by the constraints listed in Table 1. The multi-dimensional and non-linearity of the constraints listed in Table 1 creates challenges for finding feasible solutions of the relative E/I vector directly from Fig. 8. For simplification, three assumptions were made as follows: (1) k d! e k was assumed to be equal for each RS. Thus, the endpoint of relative eccentricity vector, for each RS with respect to the LV, lies in an arc AB in Fig. 9. (2) The relative phase angle,g, was set to the same value for each RS. Since the relationship da ¼ de  cosg can be derived from Eq. (5), ðDdaÞC;D ¼ 0 in constraint #6 of Table 1 was naturally satisfied. (3) For each adjacent RS pair, Ddix was set to zero meanwhile Ddiy was assumed to be equal, as shown in Fig. 9. Under these assumptions above, a deterministic procedure to obtain the safe deployment sequence which satisfies all the operative constraints of Table 1, is proposed in Algorithm 1. As illustrated in Fig. 10, this procedure generally consists of four steps: Step 1, a feasible g can be initially selected according to Eq. (22). Meanwhile, due to the fact that the y-component of relative inclination vector, Ddiy , between each adjacent RS pair has the equal size, the phase angle, uk , for each RS’s relative inclination vector with respect to the LV, can be deterministically obtained as follows:   k1  ðtanu1  tanuN Þ uk ¼ hk ¼ tan tanu1  N 1 1

ð33Þ

Table 1 Operational constraints for the safe deployment of nano-satellites in cluster flight missions. Constraint

Identifier

Description

Consequence

LV-RS relative motion LV-RS relative motion LV-RS relative motion

#1 #2 #3

Immediate safety in radial direction. Immediate safety in normal direction. Monotonically increasing relative distance in avoid region.

jaL dak j > D jaL dik j > D   0 6 gk 6 p2  sin1 aLDdik

LV-RS relative motion

#4

Secular safety in tangential direction.

3

aL dak  3p P 2aL dek þ L 2 2

RS-RS relative motion

#5

Later-released satellites outside avoidance region of early-released ones.

DuC;D ¼ uD  uC P sin1 aL DdiC

RS-RS relative motion

#6

Stable relative motion to prevent cluster dispersion.

RS-RS relative motion

#7

Relative E/I vector separation for collision avoidance.

Time limit Location of deployment

#8 #9

Limited onboard battery power. Deployment process during a single ground-station contact.

ðDdaÞC;D ¼ 0; ðDdix ÞC;D ¼ 0 ðcosDgÞC;D P ðaC DdeÞDðaC DdiÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  ðaC DdeÞ2 þ ðaC DdiÞ2  D2 uN  u1 6 n  tmax u0 6 uk 6 uf

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Step 2, the maximum relative phase angle subtended by relative E/I vector of each RS pair can be determined. As seen in Fig. 10(b), the relative eccentricity vector between each RS couple is essentially a secant line in arc AB, whose slope is constrained by that of tangent lines AC and BD. On the other hand, the relative inclination vector for each RS couple is a vertical line (see Fig. 10(a)). Therefore, the maximum relative phase angle subtended relative eccentricity and inclination vector of each RS couple can be obtained by comparing the absolute value of Dgmax;1 and Dgmax;2 in Fig. 10(b), namely,



 

Dgmax ¼ max Dgmax;1 ; Dgmax;2

ð34Þ Step 3, as reflected in Fig. 10(c), the minimum modulus of relative inclination vector between each RS and the LV can be determined from Eqs. (15), (22) and (25). A feasible size for the relative inclination of each RS can thus be selected. Step 4, as shown in Fig. 10(d), the size of relative eccentricity vector for each RS with respect to the LV can be

selected after obtaining the minimum value according to Eqs. (16) and (24). Finally, using these obtained relative E/I vector for each RS, the safe deployment parameters can be calculated as follows: DvR;k ¼ n  ade  sing DvT ;k ¼ n  ade  cosg 2 DvN ;k ¼ n  adik qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dvk ¼ Dv2R;k þ Dv2T ;k þ Dv2N ;k   Dv ak ¼ tan1 DvTR;k;k !

ð35Þ

DvN ;k ffi bk ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 DvR;k þDvT ;k

Dtk ¼ Dt0 þ ðuk  u0 Þ=n Algorithm 1 (Procedures to generate safe deployment sequence).

1: Inputs: ½a; e; i; x; M; X,D,L,R,u0 ,uf ,N ,tmax 2: Choose h1 and hN that satisfy Eqs. (31) and (32), calculate hk (k 2 ½2; N  1) according to Eq. (33), and then determine the minimum value Dhmin for Dhk;kþ1 ¼ hkþ1  hk (k 2 ½1; N  1); 3: Guess a feasible g that satisfies the left side of Eq. (22); 4: Determine the value interval, ½umin ; umax , for phase angle, u, by ½umin ; umax  ¼ ½hmin  g; hmax  g (36) 5: Compute k min , the minimum value for cos ð u  p Þ with u in the value interval ½ u ; u ; j j, min max * 6: Compute  size of d i k as follows:    the minimum 1 1 N1 dix;k min ¼ aDk  max cosg ; sinðDh ; (37) min Þ tanðhN Þtanðh1 Þ ðdik Þmin ¼ minð

ðdix;k Þmin jcoshk j

Þ

according to the right * side of Eq. (22), meanwhile considering Eqs. (15) and (25); 7: Choose a feasible d i k size for each RS, according to ðdik Þmin and Eq. (29); * 8: Determine the minimum Dd e k sizes by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi u u ðDdiy Þ2  aD u k ðDdek Þmin ¼ aDk  t  2 2 ðkmin Ddiy Þ  aDk * which is derived from Eq. (30), and also the minimum sizes of d e k from the relationship   ðDdek Þmin D L   ; aL cosg ; a  9pcosg2 ðdek Þmin ¼ max Dh Þ L ð4 2sin min 2 according to Eqs. (16) and (24). * 9: Choose a feasible d e k size for each RS, based on the minimum value obtained in Step 8; 10: Calculate the final deployment parameters for each RS according to Eq. (35).

(38)

(39)

(40)

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Fig. 8. The constrained ROE space defined by operational constraints.

Fig. 9. Three assumptions to simplify the safe deployment problem.

So far, a novel methodology has been completely presented for the safe deployment of multiple nano-satellites in cluster flight missions. Despite its simplicity and feasibility, to make it more efficient in reality as well as more suitable for diverse scenarios, some aspects should be noticed when taking it into implementation: (1) The on-orbit deployment location should be properly selected. According to Step 1 in Algorithm 1, a difference in the ycomponent of relative inclination vector should be established for each RS couple, in order to realize the safe relative motion in the R-N plane. As reflected in Fig. 11, for the same difference in the y-component of relative inclination vector, a larger release velocity size is needed for the deployment at the mean argument of latitude near 90 or 270 than that near 0° or 180°. That is to say, safe relative motions in the R-N plane are much easier to realize if the deployment occurs near the equator. This provides a practical suggestion for the selection of launch site and ground station in cluster flight missions. (2) Multiple clusters can be iteratively deployed from the space station by adopting the proposed methodology. The limited deployment time supported by the LV determines that, only a small-size cluster of nano-satellites can

be deployed in a single launch, by using the proposed methodology. But it does not mean that this methodology will lose its value for the large-scale deployment of nanosatellites. In fact, this deployment procedure can be equally applied for multiple-satellites deployments conducted by orbital deployers mounted on a space station (e.g. ISS), such as NanoRack (Brown, 2015) or QuadPack (Rotteveel, 2014). In these scenarios, as illustrated in Fig. 12, multiple small clusters can be iteratively, and safely deployed twice an orbit by following the procedure given in Algorithm 1. These small clusters will finally form a large satellite swarm. (3) The performance of proposed methodology may be affected by release uncertainties. The safe deployment issue discussed so far is under the assumption that, all member satellites are precisely released from the LV during the deployment process without any errors. In reality, this assumption may be no longer hold, due to the inherent uncertainties commonly existed in spring release mechanisms, as well as the LV’s attitude control capability. Under the effect of release uncertainties, the relative motions among the LV and RSs may deviate from the nominal safe trajectory carried out by our methodology, and thus induce collision risks. To assess the effects

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Fig. 10. Four-step procedure to get safe deployment sequence.

of release uncertainties and evaluate the availability of our methodology in more realistic scenarios, as suggested in literature, at least three method can be adopted: (1) MC simulation; (2) the Relative Reachable Domain (RRD) method; (3) the Worst-Case (WC) analysis. Among them, both of MC simulation and the RRD method rely on numerical solving. They can only generate open-loop outputs for a given deployment sequence and determine whether it is safe or not, but provide no insight or feedback to the design side for how to get a safe deployment plan. The WC analysis, on the other hand, needs no numerical computation and can be conducted geometrically, making it feasible to be integrated into the safe deployment methodology proposed above. By WC analysis in the ROE plane, it is convenient to know whether a deployment sequence is safe or not under effects of release uncertainties, how much safety margin is now left for this deployment

sequence, and how to adjust the design plan to make an unsafe deployment sequence into a safe one. Due to space limitation, a detailed discussion on the effects of release uncertainties on the proposed methodology, is not given in this paper. It will be provided in other publications in the future. 5. Simulation and results In order to demonstrate the reliability of proposed methodology in realistic cluster flight scenarios, simulations have been conducted on a hypothetical implementation of the SAMSON mission, which is the pioneer to perform a cluster flight of three nano-satellites. Each simulation consists of two parts: (1) the open-loop deployment sequence; (2) post-deployment free-flying trajectories for the LV and RSs. For the first part, safe deployment

P. Liu et al. / Advances in Space Research 64 (2019) 964–981

Fig. 11. Comparison between deployment near the polar and that near the equator.

sequence is obtained via Algorithm 1 implemented in the MATLAB programming environment. For the second part, the sequence is validated by using AGI’s Systems Tool Kit (STK) commercial software package, with propagator and environment model configuration listed in Table 2. According to the mission requirements of SAMSON (Wen et al., 2015), orbit elements of the LV at the injection point are set as

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Fig. 13. Time history of relative distances between the LV and each RS during the first 12 h after separation.

Table 2 Propagator and environment models used for simulation. Integrator

Runge-Kutta

Step size

Short term: 3 s Long term: 30 s 21*21 World Geodetic System 1984 Earth Gravitational Model 1996 ISA-1976 Spherical, no shadow

Geopotential Atmospheric density Solar radiation pressure

Fig. 12. Two satellite clusters iteratively deployed in each orbit from the ISS.

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Table 3 Specifications for safety-related mission parameters. Mission parameter

Description

Value(unit)

u0 uf tmax D L R

Mean argument of latitude for the LV when ground station contact is established Mean argument of latitude for the LV at the end of ground station contact The longest time duration allowed for deployment process Avoidance region radius Avoidance region length Maximum relative distance allowed for cluster-flight satellites

160(deg) 200(deg) 10(minute) 200(m) 1(km) 50(km)

E0 ¼ ½ 7000km

0

63deg

0deg 0deg 160deg :

Moreover, safety related mission parameters are specified in Table 3. Taking these mission parameters defined in Table 3 as inputs, meanwhile selecting a feasible value for g, one feasible deployment sequence can be easily obtained through Algorithm 1. In our simulation, g ¼ 5 was set. The corresponding deployment sequence was provided in Table 4, and organized by the deployment timing, size and direction angle of release velocity. This deployment sequence was then numerically simulated in STK. At each time step, relative separations among the LV and RSs were computed. The time history of these separations were given in Figs. 13–16. As expected, nominal trajectories between the LV and RSs exhibit a secular drift (see Fig. 13). In fact, relative distances increase to a level of 10 km at about 1 h after deployment, and never decrease to below this level after that. In the short-term scenario, as suggested by Fig. 14, relative trajectories during the hour and a half after deployment exit avoidance region monotonically and never intersect with it again. For the safety between RS pairs, as reflected in Fig. 15, the immediate, or first orbit, relative motion between each RS couple exit the avoidance region monotonically and never re-enters. On the other hand, as shown in Fig. 16, the long-term, 30 days, relative motion for each RS pair keep inside a bounded relative distance at about 7 km meanwhile stay outside the avoidance region in the R-N plane. This ideally satisfies the mission requirement for cluster flight. Taking these two aspects above into consideration, it is reasonable to draw the conclusion that the proposed procedure provides a feasible and efficient method for the safe deployment of multi-satellites in cluster flight missions, and that the analytical models used in previous sections are reliable. Table 4 Baseline deployment sequence that ignores the effects of release uncertainties. Satellite

ti *(s)

jDvi j(m/s)

ai (deg)

bi (deg)

1 2 3

80.9 323.8 566.7

1.2030 1.1728 1.2030

9.92 9.92 9.92

59.4 58.5 59.4

*

t ¼ 0 represents the moment when the LV is at the injection point.

Fig. 14. LV-RS nominal relative trajectories in the R-N plane during the first 90 min after separation.

6. Conclusion On-orbit deployment process, as the first phase of a cluster flight mission, confronts with several challenges related to the safety of relative motion and practical operational constraints. With this in mind, this paper addresses the safe deployment issue for cluster-flying nano-satellites in nearcircular orbits. To this end, a safety concept design is firstly introduced based on the relative dynamics expressed by quasinonsingular ROEs. In the proposed safety concept, the deployment process can be regard as safe only if: (1) it enables relative trajectories among the LV and RSs exiting the 200 m * 1000 m * 200 m avoidance region monotonically and never reentering; (2) it allows a relative separation between each RS pair to be kept within a maximum intersatellite distance specified by mission requirements; (3) it accords with practical requirements from the LV, ground station, and satellite platform. To realize this safety

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Fig. 15. RS pairs’ nominal relative trajectories in the R-N plane during the first orbit after deployment.

(a) Relative distance between Satellite 1 and Satellite 3.

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concept, two safety criteria and two practical constraints are defined accordingly. The first safety criterion, separation guidance, concerns the immediate and long-term LVRS safety. The second, nominal guidance, ensures safe relative motions between each RS pair. Finally, two practical constraints restrict the total deployment process to be completed during a single ground-station contact and before the upper-stage of LV losing power. All the safety criteria and practical requirements are then formulated into constraints for relative E/I vector which can be geometrically addressed in the ROE space and subsequently translated into safe deployment parameters. That means, once a set of nano-satellite parameters and operational constraints are given, a deployment sequence can be procedurally and deterministically generated that guarantees safe relative motions among the LV and RSs for enough long time, without any maneuver efforts. The proposed methodology was finally simulated on the on-orbit deployment scenario of SAMSON, a typical cluster flight mission which involves three nano-satellites. On the one hand, the resulting time histories of relative separations between the LV and each RS reflect that, relative motion will be absolutely safe after about 1 h after deploy-

(b) Relative distance between Satellite 2 and Satellite 1.

(c) Relative distance between Satellite 3 and Satellite 2. Fig. 16. Time history of relative distances between RS pairs during 30 days after deployment.

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ment. While during the first 1 h, relative trajectories exit avoidance region monotonically and never reenter it. On the other hand, time histories of relative distances between each RS couple indicate that, all of the relative trajectories are established at the outside of avoidance region and never intersect with it any more. Moreover, the free-flying relative motion among RSs can be maintained in a bounded relative distance at about 7 km for at least 30 days. All these results above are in consistent with our concept design, thus validate the feasibility of the proposed methodology and the availability of the analytical model used. Although primarily designed for the deployment process conducted on the LV, the proposed deployment methodology can be compatibly applied with a wide range of nanosatellite clusters (Zhang and Gurfil, 2014), swarms (Edlerman and Gurfil, 2017) or even constellations (Blackwell et al., 2013; Paek et al., 2018) deployed by other launch platforms, such as the ISS or mothership spacecraft. Due to space limitation, the effect of release uncertainties on the proposed deployment methodology have not fully discussed in this paper. More detailed works on this issue will be presented in other publications in the future. References Alfried, K., Vadali, S.R., Gurfil, P., How, J., Breger, L. (2010). Spacecraft formation flying: dynamics, control and navigation, Oxford, England, U.K. Atchison, J.A., Rogers, A.Q., 2016. Operational methodology for largescale deployment of nanosatellites into Low Earth Orbit. J. Spacecraft Rockets 53, 799–810. Ben-Yaacov, O., Ivantsov, A., Gurfil, P., 2016. Covariance analysis of differential drag-based satellite cluster flight. Acta Astronautica 123, 387–396. Bevilacqua, R., Lovell, T.A., 2014. Analytical guidance for spacecraft relative motion under constant thrust using relative orbit elements. Acta Astronautica 102, 47–61. Blackwell, W.J., Allen, G., Galbraith, C., 2013. MicroMAS: a first step towards a nanosatellite constellation for global storm observation. In: 27th Annual AIAA/USU Conference on Small Satellites. Boutonnet, A., Martinot, V., Baranov, A., Escudier, B., 2005. Optimal invariant spacecraft formation deployment with collision risk management. J. Spacecraft Rockets 42, 913–920. Brown, C., 2015. Maximizing ISS utilization for small satellite deployments. 2015 CubeSat Developers’ Workshop. San Luis Obispo, CA. Brown, O., 2006. Fractionated space architectures: a vision for responsive space. In: AIAA 4th Responsive Space Conference. Los Angeles, CA. D’amico, S., 2010. Autonomous formation flying in Low Earth Orbit. Ph. D, Technical University of Delft. D’Amico, S., Ardaens, J.S., Gaias, G., Benninghoff, H., Schlepp, B., 2013. Noncooperative rendezvous using angles-only optical navigation: system design and flight results. J. Guidance Control Dyn. 36, 1576– 1595. D’Amico, S., Montenbruck, O., 2006. Proximity operations of formationflying spacecraft using an Eccentricity/Inclination vector separation. J. Guidance, Control, Dyn. 29, 554–563. D’Amico, S., Montenbruck, O., Arbinger, C., Fiedler, H., 2005. Formation flying concept for close remote sensing satellites. Adv. Astronautical Sci. 120, 831–848. di Mauro, G., Bevilacqua, R., Spiller, D., Sullivan, J., D’Amico, S., 2018. Continuous maneuvers for spacecraft formation flying reconfiguration using relative orbit elements. Acta Astronautica 153, 311–326.

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