Proximity Approaches and Design Strategies for Non ...

Proximity Approaches and Design Strategies for Non-Cooperative Rendezvous*V-bar Hopping vs. Spiral Approach

Takahiro SASAKI,† Yu NAKAJIMA, and Toru YAMAMOTO

Research and Development Directorate, JAXA, Tsukuba, Ibaraki 305–8505, Japan

As the amount of debris in orbit increases, so does the risk of collisions and their seriousness. All nations involvedwith space operations acknowledge this growing threat. One solution receiving increased attention is active debrisremoval. The first step in a debris removal mission would be to approach the debris. In this phase, it is important to ensurepassive abort safety and to guarantee the robustness against collisions in the case of off-nominal thruster burns, that may becaused by spacecraft anomalies such as navigation sensor or actuator failures. This paper compares two types of passiveabort safe trajectories –the V-bar hopping and spiral approaches– considering the ¦V budget, the duration of operations,and variation in the line-of-sight vector to the target. This paper also proposes design strategies for determining the pa-rameters in the two candidate trajectories, considering passive abort safety. The robustness of the trajectories against col-lisions due to off-nominal thruster burns is also demonstrated through Monte Carlo simulations. The paper investigateswhich trajectories are suitable for an active debris removal mission to a non-cooperative target.

Key Words: Active Debris Removal, Formation Flying, Proximity Operations, Spacecraft

Nomenclature

a: semi-major axis, me: eccentricitye: eccentricity vectori: inclination, radi: inclination vector

Nhop: the number of hops on the V-barNspiral: the number of spirals along the V-bar

u: mean argument of latitude, radv: velocity vector w.r.t. hill coordinates, m/sx: position vector w.r.t. hill coordinates, m�: relative orbit elements£: off-nominal rate

�hop: hopping rate�spiral: spiral rate�v: impulsive delta-v, m/sª: relative ascending node, rad­: mean longitude, radº: relative perigee, radn: orbital mean motion, rad/s+: right ascension of ascending node, rad

Subscripts0: initial valued: destination valueN: normal directionR: radial directionT: tangential direction

1. Introduction

Orbital debris has become an increasingly severe problemand a challenge to all nations. To solve this problem, activedebris removal (ADR) missions are gaining progressivelymore attention. The first step in an ADR mission would beto approach the debris, which is usually a non-cooperativetarget. In a non-cooperative rendezvous, obtaining continu-ous sensor measurements for navigation is quite difficultdue to loss of sensor lock or the effect of solar reflectionon the optical camera. It is important to ensure passive abort(PA) safety and to guarantee robustness against collisions inthe case of off-nominal thruster burns, that may be caused byspacecraft anomalies such as navigation sensor or actuatorfailures.

Considerable knowledge of safe PA trajectories has beengained from previous missions that entailed both cooperativeand non-cooperative rendezvous. Rendezvous with the Inter-national Space Station (ISS), which is a cooperative target,uses one of the safe PA trajectories, the R-bar and V-bar ap-proaches. They have been extensively and safely imple-mented, especially when a precise relative navigation solu-tion is required. The Space Shuttle used both R-bar and V-bar approaches1,2) as Fig. 1 shows. The ETS-VII is an engi-neering test satellite developed by the National Space Devel-opment Agency of Japan (NASDA). It approached the targetusing V-bar hopping from 1100 m to 150 m based on rela-tive GPS navigation, and then approached along the V-barusing rendezvous laser radar (RVR) navigation in the finalphase.3) The Orbital Express was a flight demonstration pro-gram established by the Defense Advanced Research Pro-jects Agency (DARPA) to develop and validate key technol-ogies required for cost-effective servicing of next-generationsatellites. Some rendezvous scenarios have been successfully

© 2021 The Japan Society for Aeronautical and Space Sciences+Received 17 January 2020; final revision received 27 September 2020;accepted for publication 7 October 2020.†Corresponding author, [email protected]

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demonstrated using the V-bar approach.4,5) The ATV is acargo transfer vehicle developed by the European SpaceAgency (ESA) for the ISS.6) The rendezvous is split into sev-eral sub-phases including a stable hold on the V-bar and itsapproach. The HTV7) is a cargo vehicle developed by theJapan Aerospace Exploration Agency (JAXA) for the ISS.These missions are summarized in Table 1. The previousstudy2,8,9) showed that the V-bar approach is better for hold-ing, whereas the R-bar approach is better for a quick ap-proach and recovery. Unlike the V-bar approach, the R-barapproach has to deal with not only Coriolis force but alsocentrifugal force on the satellite. Furthermore, fuel consump-tion with the R-bar/V-bar hopping approach is more efficientthan that with the straight-line R-bar/V-bar approach. Basedon this, a V-bar hopping approach would seem to offer thebest balance of PA safety and fuel consumption for non-co-operative rendezvous.

Cooperative rendezvous during formation flying (FF) mis-sions also adopt a PA safe trajectory, the spiral trajectory. Itis a powerful representative as a rendezvous trajectory and isa candidate trajectory for approaching a non-cooperative tar-get. The spiral approach, which usually considers relativeorbit elements (ROEs) between the chaser and the target de-bris, is related to the phasing of the relative eccentricity/inclination (e/i)-vector as shown in Fig. 1. It finds a path withe/i-vector separation10) and optimizes PA safety. The spiraltrajectory has been used by some FF missions with a cooper-ative rendezvous such as TanDEM-X/TerraSAR-X,10,11)

GRACE,12) or PRISMA.13,14) The TanDEM-X/TerraSAR-X

is an Earth observation mission that uses interferometry todeliver a digital-elevation map of the Earth. It exploits coop-erative FF implemented by e/i-vector separation. GRACEwas an Earth observation mission to map the gravity fieldof the Earth. It exploited cooperative FF in leader-followerconfiguration, and it used e/i-vector separation only forswapping the leader satellite at one point in the mission.PRISMA demonstrated the practicality of GPS-based, radio-frequency based, and vision-based technologies using fidu-cial markers for cooperative FF. PRISMA was also used tomimic some scenarios of non-cooperative rendezvous, butthe intersatellite link was never deactivated. The other mis-sion that includes a not fully-cooperative rendezvous is theRestore-L mission15) planned by the National Aeronauticsand Space Administration (NASA). It is equipped with arobot-controlled service and capture satellite. Since the targetsatellite (i.e., Landsat-7) is still operational, it can providecomplete attitude control and communicate with the ground.

There are very few missions that perform ADR using anon-cooperative rendezvous. The AVANTI experiment16–18)

in the FireBird missions of the German Aerospace Center(DLR) focused on a non-cooperative target. In this mission,the rendezvous was performed with a relative GNC designedto exploit e/i PA safety. The ESA is planning the e.Deorbitmission,19) the objective of which is to remove Envisat (a8-ton inactive satellite) with a challenging rendezvous, cap-ture, and de-orbit process. The plan is to perform e/i-vectorseparation. The ESA is also studying a hybrid trajectorymethod consisting of a V-bar hopping approach followedby e/i-vector separation prior to deorbit as a part of thee.Inspector mission,20) which is a rendezvous-only missionwith a cubesat. Additionally, Astroscale is now developingthe ADR satellite ELSA-d,21) which consists of two space-craft, a chaser (180 kg) and a target (20 kg). They will belaunched stacked together and will demonstrate technologiesfor rendezvous and proximity operations, including e/i-vector separation trajectories, but the target is not fullynon-cooperative because it has a docking plate. JAXA isjoining the Commercial Removal of Debris Demonstration(CRD2) project,22,23) the aim of which is developing spacedebris removal as a new, sustainable space business. The ob-jective of the CRD2 project is to remove large space debris(e.g., the upper stage of Japan’s H-IIA rocket) at low cost us-ing a small ADR satellite. These missions are summarized inTable 2. Tables 1 and 2 show that V-bar hopping should notbe used for a non-cooperative rendezvous. The AVANTI ex-periment is the only mission for which the spiral trajectory isdemonstrated on orbit for non-cooperative ADR. Further-more, quantitative trade-off research to investigate non-cooperative rendezvous has not been done yet. This paperpresents a fairly objective comparison of the V-bar hoppingapproach and the spiral approach for non-cooperative ren-dezvous, which would be encountered during ADR.

Some parameters are needed to design the candidate tra-jectories. The number of hops on the V-bar approach andthe hopping interval/duration are needed to design the V-bar hopping trajectory. Designing a spiral trajectory requires

Fig. 1. Illustrations of V-bar/R-bar hopping and spiral approaches.

Table 1. Missions with R/V-bar approach.

ISS rendezvous Rendezvous demonstrator

Cooperative Space Shuttle, HTV,ATV

ETS-VII, Orbital Express

Partially cooperative — —

Non-cooperative — —

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the number of spirals along the V-bar, which depends on thedifference in altitudes, and the magnitude of the e/i-vector.Furthermore, the hopping and spiral rates are introduced.As the ADR satellite approaches the target, the hopping in-terval becomes shorter at the constant hopping rate in theV-bar hopping approach and the size of the relative orbits be-comes smaller at the constant spiral rate every orbital revolu-tion in the spiral approach. Basically, the accuracy and reli-ability of the navigation system are improved as the ADRsatellite becomes closer to the target debris. This paperpresents analytical solutions for these parameters while con-sidering PA safety and duration of operation, and proposesdesign strategies for non-cooperative rendezvous using thecandidate trajectories. Using the proposed strategies, deter-mining the parameters for safe trajectory design becomeeasier, and quantitative trade-offs between the candidatescan be made objectively. As a result, the characteristics ofeach trajectory can be examined and their applicability tonon-cooperative rendezvous missions can be clarified.

This paper first discusses the approach policy of two can-didate trajectories for non-cooperative targets of ADR mis-sions and shows the design of example nominal trajectories.This paper then considers the ¦V budget, duration of oper-ations, and variation in the line-of-sight (LoS) vector throughnumerical simulations. The robustness of candidate trajecto-ries against collisions is also demonstrated through MonteCarlo simulations, and the characteristics of the two candi-date trajectories for ADR missions to non-cooperative targetsare examined and clarified.

2. Problem Statements

This section summarizes three problem statements.Mission scenario: This paper focuses on the proximity

approaches (the V-bar hopping and spiral approaches) ofan ADR satellite from 1 km to 100m, while minimizingthe risk of collision when off-nominal thruster burn occurs.Since a delay in detecting mechanical failure is critical in aproximity operation, PA safety is an important factor duringthe rendezvous phase. In this paper, the goal of ADR satel-lites is defined by the nominal destination at 100m, whichis different between trajectories as shown in Fig. 2, whereADR satellites precisely estimate the motion of the target de-bris during the final approach along the circular path aroundthe debris and capture.

Definition of coordinates and safe regions: LVLH,R-bar/V-bar, and RTN coordinates are defined with their ori-gins set to the target’s center of mass, as shown in Fig. 2. AnLVLH coordinate is frequently used for the reference frameof the Earth observation satellite’s attitude. The LVLH and

RTN coordinates are often used in V-bar hopping and spiralapproaches, respectively. In defining the volume throughwhich a satellite can safely approach, the Keep-Out-Sphere(KOS) should be designated as a forbidden volume that thesatellite cannot enter on approach, as shown by the redsphere in Fig. 2. However, the proximity approach of a sat-ellite generally entails GNC errors, including a guidanceerror (i.e., due to modeling error in guidance algorithm), nav-igation error (i.e., in estimating orbital and attitude determi-nations) and control error (i.e., excessive/insufficient thrusterburn or control input delay). This means that the nominaldestinations for V-bar hopping and spiral approach shouldbe set while determining a safe distance, as shown in Fig. 2.

Variation in the LoS vector: ADR satellites need to keepdebris in the sensor field-of-view (FOV) to derive relativeposition and attitude. It is preferable to maintain LVLHattitude as long as the debris is kept in sensor FOV sinceattitude-pointing control usually imposes strict constraintson a satellite’s systems, such as navigation sensors (e.g., op-tical cameras and 3D LiDARs), power control, thermal main-tenance, and communications. Therefore, small variation inthe LoS vector leads to simplifying satellite system design.In this paper, the LoS vector is used as one of the compara-tive items for trade-offs.

3. Trajectory Design

This section discusses two types of trajectories (V-barhopping and spiral approach) considering trajectory safety.3.1. V-bar hopping approach

V-bar hopping would seem to be the most balanced ap-proach for a debris removal satellite in terms of PA safety

Fig. 2. Definition of coordinates, KOS, and nominal destination.

Table 2. Missions with e/i-vector separation.

Formation flying On-orbit service Active debris removal

Cooperative TanDEM-X/TerraSAR-X, GRACE — —

Partially cooperative PRISMA (partial mission) Restore-L ELSA-dNon-cooperative — — AVANTI, e.Inspector/e.Deorbit, CRD2

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and fuel consumption considering the V-bar hold for observ-ing the target and shooting many images. After enough datais collected, the ADR satellite will fly around to the capturesurface of the debris. In this approach, the satellite covers apredefined distance in small steps through ¦V impulses inthe radial direction or radial/tangential directions at the V-bar crossing.

Hill-Clohessy-Wiltshire (HCW) equation: If the orbit ofthe target debris is circular, a HCW equation24) is a usefulrepresentation for calculating the required ¦V. Since a V-bar hopping approach has no out-of-plane (�y) motion, thein-plane components in the HCW equations are only usedas follows:

xxzð�tÞvxzð�tÞ

" #¼

�11ð�tÞ �12ð�tÞ�21ð�tÞ �22ð�tÞ

" #xxzð0Þvxzð0Þ

" #ð1Þ

where

�11ð�tÞ ¼1 6ðn�t � sin n�tÞ0 4� 3 cos n�t

" #; ð2Þ

�12ð�tÞ ¼

1

nð4 sin n�t � 3n�tÞ 2

nð1� cos n�tÞ

�2

nð1� cos n�tÞ sin n�t

n

26664

37775;

ð3Þ

�21ð�tÞ ¼0 6nð1� cos n�tÞ0 3n sin n�t

" #; ð4Þ

�22ð�tÞ ¼4 cos n�t � 3 2 sin n�t

�2 sin n�t cos n�t

" #ð5Þ

together with xxz ¼ ½x; z�T and vxz ¼ ½vx; vz�T with respect tohill coordinates, and © is the state transition matrix of theHCW equations. The symbol n is the orbital mean motionof the target. Using Eq. (1), the required ¦V (¼ vxzð0Þ)which achieves the desired position xxzð�tÞ in �t s is ob-tained by

vxzð0Þ ¼ ��112 ð�tÞ xxzð�tÞ ��11ð�tÞxxzð0Þ

� �: ð6Þ

Note that V-bar hopping usually assumes zð0Þ ¼ zð�tÞ ¼ 0

(at the point on the V-bar). This equation provides a numer-ical solution set of the required ¦V. If rankð�12ð�tÞÞ ¼ 2,the inverse matrix ��1

12 ð�tÞ is always obtained. However,if the rankð�12ð�tÞÞ 6¼ 2 then this cannot be solved. It occurswhen detð�12ð�tÞÞ ¼ 0 or equivalently

8ð1� cos ntÞ � 3nt sin nt ¼ 0: ð7ÞNext, an analytical solution set of the required¦V is intro-

duced using the HCW equation in Eq. (1). Using the as-sumption of a V-bar hopping

zð0Þ ¼ zð�tÞ ¼ 0; ð8Þthe following solution can be obtained:

vxð0Þ ¼n sin n�t

8ð1� cos n�tÞ � 3n�t sin n�t��x ð9Þ

vzð0Þ ¼4n sin2 ðn�t=2Þ

8ð1� cos n�tÞ � 3n�t sin n�t��x ð10Þ

vxð�tÞ ¼ vxð0Þ ð11Þvzð�tÞ ¼ �vzð0Þ: ð12Þ

Note that �x ¼ jxð�tÞ � xð0Þj is a step on the V-bar. FromEqs. (9) and (10), the range for choosing �t is restricted asfollows:

8ð1� cos n�tÞ � 3n�t sin n�t > 0 ð13Þ

0 < �t <�

n: ð14Þ

Equation (13) satisfies the condition of singularity avoidancein Eq. (7). This is a PA constraint of the V-bar hopping ap-proach, which guarantees a satellite is going away from thedebris after maneuver failure. Figure 3 compares the numer-ical and analytical solutions of V-bar hopping trajectory inEqs. (6) and (9)–(10), which were found to match withinexpected error in the numerical computations. Note that thestep on the V-bar and the time are �x ¼ 0:1m and�t ¼ 2000 s, respectively, as given in the simulation.

Parameter design policy: The constraint of burn only inthe radial direction requires firing at specific times of theorbit (i.e., the second burn is exactly half an orbital period�=n after the first one). Since such a maneuver plan doesnot take into account the margin for PA safety fromEq. (14), a thruster misalignment or navigation error coulddisrupt PA safety. This paper allows for both radial and tan-gential maneuvers to ensure an arbitrary margin. Based onthe constraint for PA safety in Eq. (14) and the reference,9)

assumptions for designing the parameters are given as fol-lows:: Total hopping time ttotal can be designed according to themission.

: One hopping interval is set to �t, obtained from thenumber of hops Nhop and Eq. (14).

: Both radial and tangential maneuvers are allowed.: Each hopping duration is set proportional to distance fromthe target, thus the hopping interval becomes shorter as thechaser gets closer to the target at the constant rate �hop.

Note that �hop is the hopping rate, which is the proportionalconstant, and the number of hops Nhop is determined by

Fig. 3. Comparison of numerical and analytical solutions (�t ¼ 1=3 rev,ttotal ¼ 1 rev).

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Nhop ¼Nhop;min

ttotal

�t< Nhop; min

!

ttotal

�t

$ %ttotal

�t� Nhop;min

!8>>>>><>>>>>:

ð15Þ

where bAc is the floor function, which takes a real numberAas the input and gives the greatest integer less than or equal toA as the output. For an approach of one or two hops, the sat-ellite is likely to enter the KOS. Therefore, the minimumnumber of hops Nhop;min is defined by the GNC error andthruster maneuver error.

Example: From these assumptions, the parameters of a V-bar hopping trajectory are �hop ¼ 0:8, ttotal ¼ 6000 s (whichis described as 1 orbital period 2�=n),�t ¼ 1200 s (which isdescribed as 1/5 orbital period 2�=ð5nÞ), and Nhop;min ¼ 5.From Eq. (15), Nhop ¼ 5. Note that �t is described as a con-stant and must be shorter than 1/2 orbital period �=n inEq. (14). Using these design parameters, a desired positionset xd is calculated as follows:

xd ¼ ½xLVLHtd0 ; xLVLHt

d1 ; xLVLHtd2 ; xLVLHt

d3 ; xLVLHtd4 ; xLVLHt

d5 �ð16Þ

¼ ½�1000; �732; �518; �347; �210; �100� m: ð17ÞFigure 4 shows a nominal trajectory of the V-bar ap-

proach. The difference between the mean orbital altitudesof a satellite and a debris determine PA safety in the V-barhopping approach. Introducing the hopping rate �hop, thehopping interval becomes shorter as the chaser gets closerto the target at the constant rate. Basically, the accuracyand reliability of the navigation system are improved as theADR satellite gets closer to the target debris. Figure 5 showsan example of free-drift trajectories after maneuver failure(i.e., missed thruster burn) in a nominal trajectory of the V-bar hopping approach. The blue line shows the nominal tra-jectory of V-bar hopping and the green lines show the off-nominal trajectories in the case of maneuver failures. In thisfigure, PA is successfully achieved.

3.2. Spiral approachThe spiral approach, which usually considers relative orbit

elements (ROEs) between the chaser and the target debris, isrelated to the phasing of the relative eccentricity/inclination(e/i)-vector and the trajectory runs spirally along the V-bar.The spiral trajectory can be used to design proximity operationgeometries characterized by passive safety and stability.10)

Relative orbit elements: ROEs �� are a useful represen-tation to describe the relative orbit and are defined as fol-lows11):

�� ¼

�a

��

�ex

�ey

�ix

�iy

26666666664

37777777775

¼

�a=a

�uþ�� cos i

�ex

�ey

�i

�� sin i

26666666664

37777777775

ð18Þ

with relative eccentricity vector �e and relative inclinationvector �i being defined as follows:

�e ¼�ex

�ey

" #¼ �e

cos �

sin�

" #; �i ¼

�ix

�iy

" #¼ �i

cos

sin

" #

ð19Þwhere a is the semi-major axis, and e and i are eccentricityand inclination, respectively. Parameters+, u, and ­ are rightascension of the ascending node, mean argument of latitude,and mean longitude, respectively. Note that the phases of therelative e/i-vectors in Eq. (19) are termed argument of peri-gee º and ascending node ª of the relative orbit. The meanargument of latitude u is sometimes used as a parameter inROEs instead of ­,11) but using ­ allows decoupling the in-plane from the out-of-plane components when introducingimpulsive maneuvers. ROEs can be geometrically character-ized as shown in Fig. 6 and are valid only for bounded rela-tive orbits (i.e., �a ¼ 0). In the R-T plane, the ratio of the ma-jor and minor axes in the ellipse is maintained at 2:1. TheR-N plane is a key factor when considering PA safety. For�e ¼ �i, the trajectory in the R-N plane is described as acircle.

Gauss equation: The Gauss equation provides the conse-

Fig. 4. Nominal V-bar hopping trajectory (ttotal ¼ 1 rev).

Fig. 5. Failed maneuver analysis of V-bar hopping.

Fig. 6. Concept of e/i-vector separation (�a ¼ 0).

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quent change of ROEs from an impulse maneuver as fol-lows11):

�ða��Þ ¼ �

a�a

a��

a�ex

a�ey

a�ix

a�iy

266666666664

377777777775¼ 1

n

0 2 0

�2 0 0

sin u 2 cos u 0

� cos u 2 sin u 0

0 0 cos u

0 0 sin u

266666666664

377777777775

�vR

�vT

�vN

264

375:

ð20Þ

This equation decouples the in-plane and out-of-plane rela-tive motions. Maneuvers in the radial or tangential directionaffect the eccentricity vector. Although propellant consump-tion with tangential maneuvers is twice as efficient as thatwith radial maneuvers, a radial maneuver has no effect onthe relative semi-major axis. Maneuvers in a normal direc-tion only affect the inclination vector, which controls theout-of-plane motion. For bounded relative motion (�a ¼ 0),the minimum collision risk is provided by parallel or anti-parallel relative e/i-vectors. In this case, the relative perigeeº and relative ascending node ª in Eqs. (19) satisfy � ¼ or� ¼ þ �.10)

Relative orbit control maneuvers: Orbital control ma-neuvers in the normal direction for out-of-plane reconfigura-tion are given by

�vN ¼ v k �i k ð21Þwhere

u ¼ arctanð��iy=��ixÞ: ð22ÞOrbital control maneuvers in the tangential direction for

in-plane reconfiguration with spiral motion are given by

�v1T ¼ þv

4ðk �e k þ k �a=a kÞ ð23Þ

�v2T ¼ �v

4ðk �e k � k �a=a kÞ ð24Þ

where

u1 ¼ arctanð��ey=��exÞ ð25Þu2 ¼ u1 þ � ð26Þ

for the maneuver locations, respectively. Orbital control ma-neuvers in the radial direction for in-plane reconfigurationwithout spiral motion are given by

�v1R ¼ þv

4ð2 k �e k þ k ��� kÞ ð27Þ

�v2R ¼ �v

4ð2 k �e k � k ��� kÞ ð28Þ

where

u1 ¼ arctanð��ey=��exÞ þ �=2 ð29Þu2 ¼ u1 þ � ð30Þ

for the maneuver locations, respectively. Note that a three-tangential maneuver scheme provides an analytical closed-

form solution for the ¦V optimal reconfiguration changingwhatever of the four in-plane ROEs25) instead of radial andtangential (RT) maneuvers in Eqs. (23)–(24) and (27)–(28). Such triple maneuvers are effective when the minimumimpulse bit of the thrusters on board the ADR satellite issmall enough. Otherwise, the control error become largedue to thrust error. Although the thruster capability is deter-mined by the satellite system design, in this paper, the RTmaneuver set is adopted considering the worst case of fuelconsumption.

Parameter design policy: To design the parameters, thefollowing assumptions apply:: The trajectory in the R-N plane is described as a circle(�e ¼ �i).

: �e and �i in the initial relative orbit is twice as large as thenominal destination.

: The spiraling relative trajectory becomes smaller by theconstant value ��e and ��i as a spiral rate �spiral.

: The number of spiral rotations Nspiral depends on theduration of operation.

Note that the spiral rate is defined as �spiral ¼ ½��e; ��i�T , andthe components (��e and ��i) are obtained by

��e ¼ �e0 � �ed

Nspiral

ð31Þ

��i ¼ �i0 � �id

Nspiral

ð32Þ

where the subscripts 0 and d are the initial and destinationvalues, respectively.

For spiral motion, a larger spiral bias �a provides a largerapproach rate to the debris. Satellites approach by 3��a perorbital period. Setting the distance of approach �x is givenas follows:

�a ¼ �x

3�Nspiralð33Þ

where Nspiral is the number of spirals within �x. Note that inthis paper �x is equal to 900m (¼ 1000� 100). Figure 7shows the relationship between initial/destination relativeorbits and spiral motion. From this figure, the initial e/i val-ues �e0=�i0 and the destination e/i values �ed=�id are deter-mined by

Fig. 7. PA safe definition of the radius in the spiral approach.

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a�e0 ¼ a�i0 > a�ed ¼ a�id � �aþ ðrKOS þ rmargÞ: ð34Þ

Note that the relationship between initial relative e/i valuesand destination one is usually satisfied by �e0 > �ed and�i0 > �id with the spiral rate �spiral, since the amount of nav-igation error depends on the distance between the chaser andtarget.

Example: Figure 8 shows the nominal trajectory of thespiral approach. Note that the figures are described as theLVLH frame instead of the RTN notation. Figure 9 showsan example of free-drift trajectories after maneuver failurein a nominal trajectory of the spiral approach. The blue lineshows the nominal trajectory of a spiral approach and thegreen lines show the off-nominal trajectories after maneuverfailures. Spiral motion passes by the target debris with a tra-jectory around the origin as a point on the V-bar. When thee/i-vector is separated (parallel or anti-parallel), the PAsafety of the spiral approach is guaranteed. Operationally,the size of the relative orbits may depend on the reliability

of the relative navigation solutions and additional constraintson the FOV of the sensors. However, this paper fixes a�a

with Eq. (33) and the respect of PA safety is obtained by set-ting the initial and final sizes of the e/i-vectors to complywith Eq. (34) since, in this paper, no specific sensor config-uration is assumed in order to implement a fair comparison.

4. Numerical Simulations

This section compares the¦V budgets and variation in theLoS vector of two trajectories through numerical simula-tions, while fairly considering the duration of operations(e.g., the number of hops or spirals). The robustness of twotrajectories against collisions due to off-nominal thrusterburn is also demonstrated through Monte Carlo simulations.4.1. Comparison of ¦V budget

This subsection compares the ¦V budget in the case of V-bar hopping and that in a spiral approach. The initial/finalpositions of the V-bar hopping approach are set to xLVLHt

0 ¼

(a) Spiral approach (3D)

(c) Spiral approach (X–Z)

(b) Spiral approach (X–Y)

(d) Spiral approach (Y–Z)

Fig. 8. Nominal trajectory of a spiral approach.

(a) Spiral approach (3D)

(c) Spiral approach (X–Z)

(b) Spiral approach (X–Y)

(d) Spiral approach (Y–Z)

Fig. 9. Failed maneuver analysis of a spiral approach.

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½�1000; 0; 0�T m and xLVLHtd ¼ ½�100; 0; 0�T m, and the

initial/final velocities are set to vLVLHt0 ¼ ½0; 0; 0�T m/s

and vLVLHtd ¼ ½0; 0; 0�T m/s, respectively. The initial/final

e/i-vectors of the spiral approach are set to �e0 ¼ �i0 ¼ 100

with � ¼ �1000 and �ed ¼ �id ¼ 50 with � ¼ �100. Theinitial/final velocities in the spiral approach are set tovLVLHt0 ¼ ½0; 0; �0:1�T m/s and vLVLHt

d ¼ ½0; 0; 0:05�T m/s,respectively. Tables 3 and 4 show the ¦V budget for eachmaneuver in the V-bar hopping (Nhop ¼ 5) and spiral(Nspiral ¼ 5) approaches, respectively. Note that the spiralapproach only considers ��� correction in the final radialmaneuver set. Table 5 shows the total ¦V budget for eachinterval and shows that the ¦V budget of the spiral approachis much less than that of the V-bar hopping approach. On theother hand, unlike the spiral approach, the V-bar hopping ap-proach has the advantage of a flexible duration of operationby choosing the number of hops properly. The number ofhops is determined using Eq. (15). The spiral trajectory withNspiral takesNspiral þ 0:5 rev since the burn-spacing ³ (¼ 0:5

rev) of tangential maneuvers is considered for keeping thefinal destination orbit.4.2. Comparison of variation in the LoS vector

This subsection compares the variation in the LoS vector.Figure 10 shows the relative positions, which are the direc-tion of the target debris from the ADR satellites for in-planeand out-of-plane, as obtained by

�in-plane ¼ arctanzLVLHrel

xLVLHrel

!; ð35Þ

�out-of-plane ¼ arctanyLVLHrel

xLVLHrel

!: ð36Þ

The maximum direction angle of the V-bar hopping ap-proach is much less than that of the spiral approach. SensorFOV requirements can be given by

�FOV ¼ max �in-plane;�out-of-plane� �

: ð37ÞFigure 10(a) presents a target direction angle varying within10 deg, which implies that, by using sensors with 10 degFOV or larger, the attitude can be kept steadily pointing to-wards the V-bar direction during the rendezvous. Small var-iations in the LoS vector allow a simpler design of the atti-tude modes of the satellite and, consequently, of the wholespace system.4.3. Monte Carlo simulations

Monte Carlo (MC) simulations are executed to clarify therobustness against collisions in the case of off-nominalthrust. This paper considers both excessive and insufficientthruster burns. To model the resulting uncertainties, a ran-

Table 3. ¦V budget of V-bar hopping (Nhop ¼ 5).

¦V [m/s]X-th MNV x (T) y (¹N) z (¹R)

1 0.1156 — 0.20182 ¹0.0271 — 0.36253 ¹0.0177 — 0.29364 ¹0.0206 — 0.23525 ¹0.0134 — 0.18556 ¹0.0538 — 0.0825

Table 4. ¦V budget of spiral approach (Nspiral ¼ 5).

¦V [m/s]X-th MNV u [rad] R (¹z) T (þx) N (¹y)

1 0 — ¹0.00515 —

2 ³ — ¹0.00515 —

3 0 ¹0.00674 — —

4 ³ ¹0.00674 — —

5 ��=2 — — 0.013496 0 ¹0.00674 — —

7 ³ ¹0.00674 — —

8 ��=2 — — 0.013499 0 ¹0.00674 — —

10 ³ ¹0.00674 — —

11 ��=2 — — 0.0134912 0 ¹0.01046 — —

13 ³ 0.0303 — —

14 ��=2 — — 0.0134915 0 — 0.00515 —

16 ³ — 0.00515 —

Table 5. Comparison of ¦V budgets in two trajectories.

V-bar hopping Spiralttotal [rev] Nhop ¦V [m/s] Nspiral "V [m/s]

1.0 5 1.6112 — —

1.5 6 1.2881 1 0.210932.0 8 1.2756 — —

2.5 10 1.2662 2 0.159413.0 12 1.2584 — —

3.5 14 1.2558 3 0.142244.0 16 1.2542 — —

4.5 18 1.2532 4 0.133655.0 20 1.2519 — —

5.5 22 1.2524 5 0.12850

(b) Target direction using spiral approach

(a) Target direction using V-bar hopping approach

Fig. 10. Comparison of the target directions.

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143©2021 JSASS

dom variable £ with a standard uniform distribution U ð0; 2Þis added for each maneuver point as follows:

�Vthr ¼ � ��Vtrue ð38Þwhere

� ¼

0 : passive abort

�̂; 0 < �̂ < 1 : insufficient maneuver

1 : desired maneuver

~�; ~� > 1 : excessive maneuver:

8>>>><>>>>:

ð39Þ

Since the guidance profile is interrupted as soon as an off-nominal maneuver occurs, thruster burn is no longer exe-cuted. Thus, the PA safety of the trajectories can be estimatedin order to confirm the passive trajectory after an off-nominalmaneuver. If a trajectory of an ADR satellite invades theKOS because of an off-nominal thruster burn, the result isdeemed to be unsafe.

Figures 11 and 12 show the MC simulation results andFig. 13 shows the relation between off-nominal rate andthe number of incursions into the KOS. From these figures,in the V-bar hopping approach, excessive maneuvers in thevicinity of the target may lead to invasion of the KOS. Thisresult implies that V-bar hopping is more likely to enter KOSwhen the navigation error is not sufficiently small. On theother hand, there is no case in which a satellite in a spiral ap-proach invades the KOS. The number of satellite invasionsinto the KOS are shown in Table 6. Therefore, due to off-nominal thruster burn, the spiral approach has a lower riskof collisions than the V-bar hopping approach.4.4. Discussions

Summary of the V-bar hopping approach: The V-barhopping approach would seem to offer the best balance be-tween PA safety and fuel consumption for non-cooperativerendezvous missions for the R-bar and V-bar approaches.In this paper, the hopping approach moves the satellite a pre-defined distance in small steps through ¦V impulses in theradial and tangential directions at the V-bar crossing. Thiswould be performed when the close-range navigation systemhas verified a reliable and sufficiently precise solution.

Summary of the spiral approach: The spiral approachdefines a trajectory with eccentricity/inclination (e/i)-vectorseparation and evaluates the PA safety of the ADR satellitegeometrically. This approach has the advantages of requiringsmall ¦V corrections and robustness against collisions dueto the off-nominal thruster burns to which V-bar hopping

Fig. 11. Monte Carlo simulation using V-bar hopping. (b) Spiral approach (X–Z plane)

(a) Spiral approach (X–Y plane)

(c) Spiral approach (Y–Z plane)

Fig. 12. Monte Carlo simulation using the spiral approach.

Fig. 13. Invasions of KOS.

Table 6. The number of times the satellite invades the KOS.

The number of invasions xx/100X-th off-nominal MNV V-bar hopping Spiral approach

1 0/100 0/1002 0/100 0/1003 2/100 0/1004 5/100 0/1005 9/100 0/1006 — 0/1007 — 0/1008 — 0/1009 — 0/100

10 — 0/10011 — 0/10012 — 0/10013 — 0/10014 — 0/10015 — 0/100

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144©2021 JSASS

is subject. However, the variation in the LoS vector is large,which implies strict constraints on satellite design.

General discussion: Trajectory safety, partially guaran-teed by robustness, has an impact on non-cooperative ren-dezvous such as many ADR missions. However, a large var-iation in the LoS vector leads to strict constraints on asatellite’s systems, such as GNC sensors, power control,thermal management, and communications. Table 7 com-pares the two candidate trajectories. From the criteria, it isapparent which trajectory should be adopted for a particularmission. The V-bar hopping approach is suitable for mis-sions under the following conditions:: The amount of ¦V is not strictly constrained;: The onboard relative navigation system has already con-verged to a reliable and robust solution; and

: The requirements of satellite design are strictly determinedby mission specifications per the configuration of missioninstruments.

On the other hand, spiral approaches are suited to missionswhere:: The amount of¦V is strictly constrained due to the size ofthe chaser (e.g., ADR satellite);

: The navigation system has not been verified to provide areliable, sufficiently precise solution in advance; and

: An amount of operational time is available to collectenough data from the sensors and verify the image proc-essing and navigation algorithms in the various in-orbitconditions.

Therefore, the type of approach trajectory should be selectedaccording to the mission and related conditions.

5. Conclusion

This paper shows the quantitative trade-offs between theV-bar hopping and spiral approach trajectories while consid-ering PA safety in non-cooperative rendezvous missions.The orbital dynamics, control maneuvers and abort charac-teristics of two candidate trajectories are introduced and de-veloped. Furthermore, design strategies for approach trajec-tories for ARD missions are proposed. It then comparedthe ¦V budget, duration of operation, and variation in theLoS vector through numerical simulations. The robustnessagainst collisions due to off-nominal thruster burn was alsodemonstrated through Monte Carlo simulations. The pro-posed policy for designing trajectories considering PA safetyfacilitated the design of safe approach trajectories in non-cooperative rendezvous missions and allowed for quantita-tive trade-offs for candidate trajectories.

As a result, it is shown that the V-bar hopping approachhas the advantage of clearly observing the target and ena-

bling simpler satellite system design thanks to the small var-iation in the LoS vector when the onboard relative navigationsystem has already converged to a reliable and robust navi-gation solution. On the other hand, the spiral approach hasthe advantages of a small ¦V (i.e., total ¦V budget withthe spiral approach is smaller than that with V-bar hoppingby one order of magnitude) to achieve a proximity approachand robustness against collisions that could occur because ofoff-nominal thruster burn. This paper clarifies the character-istics of two PA-safe trajectories and makes it possible to se-lect the better trajectory considering safety and the mission.

This paper focused on evaluation by thrust error in theMonte Carlo simulations and assumed that navigation is re-liable since the quality of navigation depends on the applica-tion (i.e., what sensors are used and which navigation filtersare implemented). Therefore, determining the proper dura-tion of operation in non-cooperative rendezvous missionsremains an open question that entails correcting navigationerrors in orbit and attitude determination which, in turn, de-pends on the noise inherent in navigation sensors, the navi-gation filter algorithm, and tuning.

Acknowledgments

This work was supported by JSPS Grant-in-Aid for Scientific Re-search Grant Number 19K15020.

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Matteo CeriottiAssociate Editor

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146©2021 JSASS