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Trans. JSASS Aerospace Tech. Japan Vol. 10, No. ists28, pp. Pd_11-Pd_20, 2012 Original Paper

Study on the Stationkeeping Strategy for the Libration Point Mission

1) 1) By Toshinori IKENAGA and Masayoshi UTASHIMA

1)Trajectory and Navigation Group, Japan Aerospace Exploration Agency, Ibaraki, Japan (Received June 16th, 2011)

The Japan Aerospace Exploration Agency (JAXA) is now planning the next-generation infrared astronomical mission—the Space Infrared Telescope for Cosmology and Astrophysics (). SPICA is the first Japanese libration point mission that utilizes a halo around L2 in the - system. This paper describes a study conducted on the stationkeeping strategy for that libration point mission. The main algorithm is structured with reference to a paper written by K. C. Howell et al. [1993]. In that paper, the attitude constraints of SPICA and frequent disturbances caused by unloading operation of the reaction wheels are considered in applying the stationkeeping algorithm to SPICA. Throughout the analysis, the amount of corrective delta-V for stationkeeping was revealed as being heavily influenced by thruster allocation when considering the strict attitude constraints of SPICA. The total delta-V for one of stationkeeping varies from 0.1 to 7.6 m/s, depending on onboard thruster allocation. Based on the analysis results, this paper suggests an optimal thruster allocation for SPICA-like missions.

Keywords: , , Stationkeeping

∗ Nomenclature μ : ratio of gravitational constants σ : standard deviation : position deviation between the nominal λ d : eigenvalue and estimated trajectories r Φ : state transition matrix e(t) : velocity deviation at t Subscripts I sp : specific impulse 0 : initial value J : cost function A : EMB-centered rotational coordinate m : mass system : deviation between the nominal and B : inertial coordinate system at certain r m estimated trajectories defined in the epoch of EMB-centered rotational stationkeeping algorithm coordinate system p : Lagrange point c : corrective maneuver (delta-V) r p(t) : position deviation at epoch t C : EMB-centered inertial coordinate Q(t ) : weighting diagonal matrix system D : Earth-centered inertial coordinate R(t ) : weighting diagonal matrix system : rotation matrix EM : Earth- Barycenter Rr r : position vector B S(t) : weighting diagonal matrix I : Earth-centered “mean-of-date” r v : velocity vector coordinate system r L2 : L2-centered rotational coordinate system ΔVc ()t : corrective delta-V at epoch t max : maximum value ΔV : delta-V min : minimum value r X : state vector p : onboard propellant r x : eigenvector t : epoch t γ : vernal total : total value ε : obliquity of the ecliptic track- : tracking interval θ (t ) : angle from the vernal equinox ing θ&(t) : angular velocity in rad/sec x : x-axis of the coordinate system μ : average y : y-axis of the coordinate system z : z-axis of the coordinate system μ s : gravitational constant of the Sun

μ e : gravitational constant of the Earth

μ m : gravitational constant of the Moon

Copyright© 2012 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved.

Pd_11 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. ists28 (2012)

1. Introduction To confirm the position and velocity errors caused by this approximation, the orbital information computed by the The Japan Aerospace Exploration Agency (JAXA) is now program we developed is compared with that computed by the planning the next-generation infrared astronomical Satellite Tool Kit (STK). Table 2-1-1 summarizes the mission—the Space Infrared Telescope for Cosmology and differences between them in the Earth-centered mean Astrophysics (SPICA) in cooperation with the European coordinate system of J2000 at the epoch of 23/Mar/2004 00: Space Agency (ESA). SPICA is the first Japanese libration 00: 00.0 UTC. point mission that utilizes the halo orbit around Lagrange As can be seen in Table 2-1-1, the position and velocity point 2 in the Sun-Earth system3). errors of the Sun and Moon are the order of 10 meters in The L2 libration point trajectories are unstable, thereby position, and 1.0 micrometer per in velocity, necessitating some form of trajectory control to keep respectively, which are precise enough for this study. spacecraft (S/C) close enough to their nominal paths.1) 2.2. Coordinate system This paper describes how we can apply this stationkeeping Since the trajectory handled in this study is a halo trajectory strategy to SPICA, as one case example of a libration point around L2 in the Sun-Earth system, the L2-centered rotational mission. In this study, although the Circular Restricted Three coordinate system is suitable for the input-output coordinate Body Problem (CRTBP)2, 8) is generally used for libration system. Conversely, since the force model is constructed point mission analysis, the precise force model utilizing the relative to the Earth-centered mean equator and equinox of Development Ephemerides (DE405)5) published by the Jet J2000, a coordinate transformation algorithm is necessary Propulsion Laboratory (JPL) is employed to construct a between both coordinate systems, the details of which are feasible stationkeeping strategy. The main stationkeeping described below. algorithm is structured with reference to a paper written by K. Transformation A C. Howell et al. [1993],4) and several functions are added to “Transformation A” is the coordinate transformation from meet SPICA mission requirements. the L2-centered rotational coordinate system to the The stationkeeping strategy described above is evaluated Earth-centered “mean-of-date” ecliptic coordinate system at through certain cases of trajectory maintenance simulations. the epoch of interest. The attitude constraints of SPICA and frequent disturbances y caused by unloading operation of the reaction wheels (RWs) are considered in applying SPICA. Throughout the analysis, x the amount of corrective delta-V for stationkeeping was revealed as being heavily influenced by thruster allocation vEMB L2 when considering the strict attitude constraints of SPICA. rEMB EMB Based on the analysis results, this paper suggests optimal γ thruster allocation for SPICA-like missions. Sun

2. Force Model and Coordinate System Fig. 2-2-1. L2-centered rotational coordinate system (“x-y-z” system) and position and velocity vectors of EMB in the “mean-of-date” ecliptic 2.1. Force model coordinate system. In this paper, the Sun, Earth, and Moon are considered, with each orbit being precisely determined by DE405. The S/C The “mean-of-date” ecliptic coordinate system includes the trajectory is computed numerically, including the gravitational effect of the of the Earth’s rotation axis. force from each of these three celestial bodies. Conversely, the position and velocity vectors of the DE405 gives us orbital information about the planets in the Earth-Moon barycenter (EMB) in the “mean-of-date” ecliptic in the International Celestial Reference Frame coordinate system define the L2-centered rotational coordinate (ICRF). The origin of the ICRF is located at the solar system system. Fig. 2-2-1 shows the geometry of the L2-centered barycenter (SSB), and the xy plane of the ICRF corresponds to rotational coordinate system (called the “x-y-z system”). the mean equator of J2000 with a negligible difference at a yI y yD C 5) level of 0″.01. Hence, the planetary determined by y y DE405 are regarded in this paper as those relative to the mean B v yA equator and equinox of J2000. S/C x Table 2-1-1. Position and velocity errors due to the differences between xB EMB r the ICRF and the mean equator and equinox of J2000 (absolute values). xA R(t) a) Sun L2 X Y Z xC xD xI Position (m) 1.44 44.15 19.14 γ Velocity (m/s) 9.65e-6 0.43e-6 0.19e-6 Sun θ(t) b) Moon Earth X Y Z Fig. 2-2-2. Geometry of coordinate transformation from the L2-centered Position (m) 0.66 1.29 0.07 rotational coordinate system to the Earth-centered “mean-of-date” ecliptic Velocity (m/s) 3.67e-6 1.76e-6 0.66e-6 coordinate system.

Pd_12 T. IKENAGA and M. UTASHIMA : Study on the Stationkeeping Strategy for the Libration Point Mission

The equations for defining the L2-centered rotational coordinate system. This transformation is accomplished by coordinate system are shown in Eq. (2-2-1). moving the origin of the “xC-yC-zC” system (i.e., EMB) to the r r r center of the Earth. When we describe the position and x = rEMB rEMB r r r r r r r velocity vectors of the Earth and S/C in the “xC-yC-zC” system y = (()rEMB × vEMB × rEMB )(rEMB × vEMB )× rEMB r r r r r as rC_Earth, vC_Earth, rC and vC, respectively, this transformation z = ()rEMB × vEMB rEMB × vEMB (2-2-1) can be expressed as shown below in Eq. (2-2-7). r r r “Transformation A” consists of five transformation rD = rC − rC _ Earth subroutines (“TF. 1” to “TF. 5”) as described below. Fig. r r r (2-2-7) v = v − v 2-2-2 shows the geometry of the five coordinate D C C _ Earth transformations of “Transformation A.” [TF. 5] From the “xD-yD-zD” system to the “xI-yI-zI” system

[TF. 1] From the “x-y-z” system to the “xA-yA-zA” system The “xI-yI-zI” system is the Earth-centered “mean-of-date” The “xA-yA-zA” system is the EMB-centered rotational ecliptic coordinate system at the epoch of interest. The coordinate system. For this transformation, the location of the reference xy plane defined by Eq. (2-2-1) differs slightly from L2 point must be known in advance. The L2 point is one of the xy plane of the “mean-of-date” ecliptic coordinate system. the collinear Lagrange points regarding the location obtained To correct the difference, we rotate the L2-centered rotational 2, 8) from Eq. (2-2-2) below. coordinate system (called the “x-y-z” system) around the zI

5 ∗ 4 ∗ 3 ∗ 2 * * axis. Eq. (2-2-8) below expresses this rotation. p + (3 − μ )p + (3 − 2μ )p − μ p − 2μ p − μ = 0 ⎡ θ ( )θ () ⎤ (2-2-2) cos t sin t 0 r∗ ⎢ ⎥ r * x = − sinθ ()t cosθ ()t 0 x The μ appearing in Eq. (2-2-2) is the ratio of the total ⎢ ⎥ ⎢ 0 0 1⎥ gravitational constants of the Earth and Moon to those of the ⎣ ⎦ Sun, Earth, and Moon. This ratio is expressed below in Eq. ⎡ cosθ (t)sinθ ()t 0⎤ (2-2-3). r∗ ⎢ ⎥ r ∗ y = ⎢− sinθ ()t cosθ ()t 0⎥ y (2-2-8) μ = ()()μ + μ ()μ + μ + μ (2-2-3) e m s e m ⎣⎢ 0 0 1⎦⎥ The position and velocity vectors of the S/C in the 1) ⎡ θ θ ⎤ “xA-yA-zA” system are expressed below in Eq. (2-2-4). cos (t)sin ()t 0 r∗ ⎢ ⎥ r = − θ ()θ () z ⎢ sin t cos t 0⎥z ⎡PL2 ⎤ ⎡PL2 ⎤ ⎢ 0 0 1⎥ r r ⎢ ⎥ r r ⎢ ⎥ ⎣ ⎦ r = r + 0 R()t v = v + 0 R&()t (2-2-4) * * * * T * A ⎢ ⎥ A ⎢ ⎥ By assuming the three vectors of x = (x1 , x2 , x3 ) , y = ⎢ ⎥ ⎢ ⎥ * * * T * * * * T ⎣ 0 ⎦ ⎣ 0 ⎦ (y1 , y2 , y3 ) and z = (z1 , z2 , z3 ) , transformation “TF.5” is consequently expressed as shown below in Eq. (2-2-9). [TF. 2] From the “xA-yA-zA” system to the “xB-yB-zB” system ⎡x ∗ x ∗ x ∗ ⎤ The “xB-yB-zB” system is an inertial coordinate system at 1 2 3 r ⎢ ∗ ∗ ∗ ⎥ r epoch t of the rotational coordinate system “xA-yA-zA.” The rI = ⎢y1 y2 y3 ⎥rD ⎢ ∗ ∗ ∗ ⎥ position and velocity vectors of the S/C in the “xB-yB-zB” z z z 1) ⎣⎢ 1 2 3 ⎦⎥ system are expressed below in Eq. (2-2-5). (2-2-9)

⎡x ∗ x ∗ x ∗ ⎤ ⎡ 0 ⎤ r ⎢ 1 2 3 ⎥ r r r r r ⎢ ⎥ r = ∗ ∗ ∗ r = r v = v + 0 × r (2-2-5) vI ⎢ y1 y2 y3 ⎥vD B A B A ⎢ ⎥ A ⎢ ∗ ∗ ∗ ⎥ ⎢θ& ⎥ ⎢ z1 z2 z3 ⎥ ⎣ ()t ⎦ ⎣ ⎦

Transformation B [TF. 3] From the “x -y -z ” system to the “x -y -z ” system B B B C C C “Transformation B” is the coordinate transformation from This transformation entails rotation of the “x -y -z ” B B B the Earth-centered “mean-of-date” ecliptic coordinate system system around the z axis. The position and velocity vectors B at the epoch of interest to the mean equator and equinox of of the S/C in the “x -y -z ” system are expressed in Eq. C C C date at the epoch of interest.5) (2-2-6). Transformation C ⎡cosθ ()t − sin θ ()t 0⎤ r ⎢ ⎥ r “Transformation C” is the coordinate transformation from = θ θ rC ⎢sin ()t cos ()t 0⎥rB the Earth-centered mean equator and equinox of date at the

⎣⎢ 0 0 1⎦⎥ epoch of interest to the mean equator and equinox of date of (2-2-6) J2000.5) ⎡cosθ ()t − sin θ ()t 0⎤ r ⎢ ⎥ r = θ θ vC ⎢sin ()t cos ()t 0⎥vB 3. Stationkeeping Algorithm ⎣⎢ 0 0 1⎦⎥ The main stationkeeping algorithm is constructed based on 4) [TF. 4] From the “xC-yC-zC” system to the “xD-yD-zD” system a paper written by K. C. Howell et al. [1993]. This algorithm The “xD-yD-zD” system is the Earth-centered inertial minimizes the sum of squares of corrective delta-V and

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deviations between the nominal and estimated trajectories at under other conditions, the corrective maneuver is executed at the two target epochs (t1 and t2) in the Earth-centered mean another time interval ttracking. equator and equinox of J2000. [Cond. 2]

At first, let us divide the state transition matrix STM (6 x 6) Corrective maneuver ΔVc is not executed if position evaluated from initial epoch t0 to certain time t (t ≥ t0) in the deviation between the nominal and estimated trajectories is Earth-centered mean equator and equinox of J2000 by four smaller than dmin. submatrices (3 x 3) as shown below in Eq. (3-1). [Cond. 3] ⎡A B ⎤ Corrective maneuver ΔVc is not executed if current position tt 0 tt 0 Φ()t,t0 = ⎢ ⎥ (3-1) deviation at the planned epoch between the nominal and ⎢Ctt Dtt ⎥ ⎣ 0 0 ⎦ estimated trajectories is decreasing. The deviation between the nominal and estimated [Cond. 4] trajectories at epoch ti (ti ≥ t0) is defined as shown below in Corrective maneuver ΔVc is not executed if the magnitude of ΔV computed by Eq. (3-4) is smaller than ΔV . Eq. (3-2). r r r r c min m ≅ B e(t )+ B ΔV ()t + A p (t ) ti ti t0 0 ti t0 c ti t0 0 (3-2) The three vectors p(t0), e(t0), and ΔVc(t) represent the 4. Application to SPICA position deviation, velocity deviation at epoch t0, and the corrective maneuver executed at epoch t (ti ≥ t ≥ t0), The stationkeeping strategy introduced above is applied to respectively. Corrective maneuver ΔVc(t) is computed by the SPICA mission as a case study. This chapter describes the minimizing the cost function defined by Eq. (3-3). terms and conditions that must be considered to apply the r r r r r T algorithm to SPICA. J []p,e,ΔVc = ΔVc ()t Q ()t ΔVc ()t r r r r (3-3) 4.1. Attitude Constraints and Thruster Allocation + m T R(t)m + m T S ()t m t1 t1 t2 t2 o +z +1 The three matrices Q(t), R(t), and S(t) appearing in Eq. -24 o (3-3) are 3 x 3 weighting diagonal matrices.

The optimal corrective maneuver ΔVc(t) at epoch t in the Sun Shield Earth-centered mean equator and equinox of J2000 is Telescope computed by setting the derivative of Eq. (3-3) by ΔVc equal to zero. The optimal corrective maneuver is expressed as +y shown below in Eq. (3-4). Bus Module CG Sun r T T −1 ΔVc ()t = −()Q ()t + Bt t R ()t Bt t + Bt t S ()t Bt t 1 1 2 2 T T r × [(B R()t B + B S()t B )e()t (3-4) Fig. 4-1-1. Diagram of SPICA attitude constraints. t1t t1t0 t2t t2t0 0 r + B T R()t A + B T S()t A p()t ( t1t t1t0 t2t t 2t0 ) 0 ] Since SPICA is an infrared astronomical mission with a Table 3-1 briefly explains each parameter input to the cryogenically cooled, high-resolution telescope, the attitude, stationkeeping algorithm. especially relative to the direction of the Sun, is severely constrained to avoid rising temperature due to radiant heat Corrective maneuver ΔVc(t), as computed by Eq. (3-4), is only executed when all four conditions (denoted “Cond.1” to coming from the Sun. Fig. 4-1-1 shows the attitude constraints “Cond.4”) are cleared. Detailed information about those four of the SPICA mission around the x-axis of the S/C body frame. conditions is described in the article titled “Execution There is no constraint around the y-axis, and that around the Conditions of Corrective Maneuver” below. z-axis is ±3 degrees. The thruster allocation of SPICA has yet to be determined; Table 3-1. Parameters input to stationkeeping algorithm. therefore, two types of onboard thruster allocation are

tmin Minimum time interval of delta-Vc considered in this paper. Fig. 4-1-2 shows both types, as well Time interval between planned delta-Vc as the direction of delta-V created in each case. Δt1 execution time and target time t1 Time interval between planned delta-Vc Δt2 execution time and target time t2 Time interval to execute the next delta-Vc, ttracking in case planned delta-Vc is not executed CG dmin Minimum position deviation dmax Maximum position deviation

ΔVmin Minimum value of delta-Vc

Execution Conditions of Corrective Maneuver [Cond. 1]

Corrective maneuver ΔVc computed from Eq. (3-4) is delta-VA executed within pre-determined time interval tmin. Should the corrective maneuver not be executed at the planned epoch a) Type A

Pd_14 T. IKENAGA and M. UTASHIMA : Study on the Stationkeeping Strategy for the Libration Point Mission

+z o 89 o + θ 114 - θ 89 o - θ 114 o + θ CG 1-2 2 1

θ o o 180 2-2 1-1 0 +y delta-VB1 delta-VB2 2 1 b) Type B 1-2 -114 o - θ θ - 89 o

Fig. 4-1-2. Diagram of onboard thruster allocation. o -89 - θ o θ - 114 In “Type A” thruster allocation, the plume directions of the b) Type B thrusters are toward the center axis of the S/C, which is desirable given the need to avoid contaminating the science Fig. 4-1-3. Sectors of delta-V combinations depending on delta-V to S/C-Sun direction. module. The delta-V is created in the direction of “delta-VA” in Fig. 4-1-2 a) by simultaneously firing the two thrusters on both sides. These two thrusters can also be used to control the Fig. 4-1-4 shows a geometric diagram of the dog-leg attitude of the S/C by firing the thruster on either side. maneuver described in the L2-centered rotational coordinate Conversely, “Type B” thruster allocation has two trajectory system. control thrusters attached to the lines connecting the +zL2 barycenter of the S/C with the mounting point of the thrusters. ESA’s HERSCHEL mission—the libration point mission around L2 in the Sun-Earth system7)—employed this type of thruster allocation. For “Type B,” the delta-V is created in the delta-V1 delta-V2 directions of “delta-VB1” or “delta-VB2” shown in Fig. 4-1-2 b) by firing the thruster on either side. Angle θ is the cant angle o 1 of the thruster. If “Type B” is applied to SPICA, the cant 1 o angle will be about 35 degrees. Meanwhile, the “red” thrusters cannot be used to control the attitude of the S/C, thus requiring the additional “yellow” thrusters for this purpose. delta-Vc Fig. 4-1-3 shows the sectors of delta-V combinations +xL2 described in degrees as measured from the +y direction of the S/C body frame (counterclockwise positive) around the x-axis. Fig. 4-1-4. Geometric diagram of dog-leg maneuver (described in L2-centered rotational coordinate system). In Fig. 4-1-3 a), the delta-V directed to the sectors denoted as

“1” is achieved by one “delta-V .” Conversely, the S/C A As shown in Fig. 4-1-4, if the direction of the corrective requires two “delta-V ” including 180o attitude rotation A delta-V computed by Eq. (3-4) is outside the attitude around the y-axis of the body frame (the so-called dog-leg constraints, the S/C requires two delta-Vs to achieve the maneuver) to create delta-V directed to the sectors denoted as computed corrective delta-V. Initially, the S/C inclines to the “1-1,” since these sectors are outside the attitude constraints.7) edge of the attitude constraints and creates the first delta-V, o +z o (“delta-V ”). Secondly, 180 rotation around the body frame’s 89 1 114 o y-axis creates the second delta-V (“delta-V ”). The sum of the 2 two delta-Vs becomes the corrective delta-V (“delta-V ”). c Similar to “Type A,” “Type B” also requires dog-leg 1 maneuvers if the computed corrective delta-V is outside the

attitude constraints. However, in the case of “Type B,” the

area of sectors requiring dog-leg maneuvers is smaller than o o 180 1-1 1-1 0 that of “Type A.” As shown in Fig. 4-1-3 b), sectors denoted

+y “1” and “2” only require one-time “delta-V ” or “delta-V ” B1 B2 as shown in Fig. 4-1-2 b), and in sectors denoted “1-1,” “1-2”

and “2-2,” dog-leg maneuvers are required as in “Type A.” 1 Finally in this section, Table 4-1-1 lists the (tentative)

specifications of the onboard thruster of SPICA,3) which o -114 enables us to compute the total delta-V of SPICA. -89 o

a) Type A

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Table 4-1-1. Specifications of onboard thruster of SPICA (tentative). Earth-centered mean equator and equinox of J2000 is Thrust force 23N described by a linear combination of six eigenvectors of the Specific impulse 200 sec STM computed numerically in the same coordinate system, as Mass of propellant (max) 180 kg Initial mass of S/C 3200 kg expressed in Eq. (4-3-1). T The vector (Δu0, Δv0, Δw0) is the unloading delta-V vector The total delta-V of SPICA is computed by the following at epoch t0, the six vectors x1 to x6 are the eigenvectors of the equation:9) STM, and coefficients “a” to “f” are constants. The state deviation after one period of halo orbit (about 180 ⎛ m ⎞ ⎜ 0 ⎟ days) denoted as ΔX is expressed in Eq. (4-3-2). ΔV total = gI sp ln ⎜ ⎟ (4-1-1) T m 0 − m p ⎝ ⎠ ⎛ ΔxT ⎞ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ where, m0 denotes the initial mass of S/C, mp the mass of ⎜ ΔyT ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ onboard propellant, Isp the specific impulse, and “g” the r Δz 0 2 ⎜ T ⎟ ⎜ ⎟ standard gravitational acceleration (i.e., 9.80665 m/s ). From ΔX T = = Φ()t0 + T ,t0 (4-3-2) ⎜ ΔuT ⎟ ⎜ Δu0 ⎟ Eq. (4-1-1) the total delta-V that SPICA can yield is about 113 ⎜ ⎟ ⎜ ⎟ Δ Δ m/s. ⎜ vT ⎟ ⎜ v0 ⎟ ⎜ ⎟ ⎜ ⎟ 4.2. Maneuver control error ⎝ ΔwT ⎠ ⎝ Δw0 ⎠ This study considered the maneuver (delta-V) control error Eq. (4-3-2) can be described by a linear combination of caused by attitude control error of the S/C, and the delta-V the six eigenvectors and eigenvalues (denoted as λ1 to λ6) of magnitude error caused by the onboard thrusters. the STM.r r r r r r r ΔX T = aλ1x1 + bλ2 x2 + cλ3 x3 + dλ4 x4 + eλ5 x5 + fλ6 x6 Table 4-2-1. Configuration of maneuver control error. (4-3-3) Attitude Error 0.167 o (1σ) Magnitude Error 2.5 % (1σ) From Eq. (4-3-2) and Eq. (4-3-3), coefficients “a” to “f” are obtained by the following equation:

Table 4-2-1 summarizes the configuration of the delta-V ⎛ a ⎞ ⎛ 0 ⎞ control error applied in this study, while Fig. 4-2-1 gives us a ⎜ ⎟ ⎜ ⎟ b 0 conceptual diagram of the “real” delta-V containing the ⎜ ⎟ ⎜ ⎟ c r r r r r r −1 0 delta-V control error described in Table 4-2-1. ⎜ ⎟ = ()x x x x x x ⎜ ⎟ (4-3-4) d 1 2 3 4 5 6 Δu ⎜ ⎟ ⎜ 0 ⎟ Δ magnitude error : 2.5 % (1σ) ⎜ e ⎟ ⎜ v0 ⎟ delta-V vector (planned) ⎜ ⎟ ⎜ Δ ⎟ ⎝ f ⎠ ⎝ w0 ⎠ Table 4-3-1. Equations of “a” (L2-centered rotational coordinates). o σ attitude error : 0.167 (1 ) p0 a = -0.7243Δu0 - 0.6862Δv0 + 0.0672Δw0 p1 a = -0.7847Δu0 - 0.6113Δv0 - 0.1028Δw0 p2 a = -0.9523Δu0 - 0.3010Δv0 - 0.0513Δw0 p3 a = -0.9538Δu0 - 0.2794Δv0 + 0.1101Δw0

Fig. 4-2-1. Conceptual diagram of delta-V control error. In case Az = 300,000 km, we can obtain the eigenvalues of 6) 4.3. Unloading delta-V the STM as follows: Here, λ1 is about -1500, and λ2 to λ5, are The impact of translational forces (unloading delta-V) about 1.0. Note that λ6 is the reciprocal of λ1. This tells us that caused by unloading operation of the onboard RWs was also large eigenvalue λ1 causes divergent orbital disturbances; investigated in this study. Thus, this paper also considered the therefore, if the direction of the unloading delta-V is frequency of unloading operation at a magnitude of 6.0 determined so as to set coefficient “a” equal to zero, the mm/s.6) orbital disturbances noted above will be eliminated. To reduce the impact of frequent unloading delta-Vs described above on stationkeeping of the S/C, the proper direction of unloading delta-V must be chosen. In this study, the unloading delta-V direction was determined by using the CG STM evaluated along with one period of halo orbit.6) ⎛ 0 ⎞ ⎜ ⎟ ⎜ 0 ⎟

r ⎜ ⎟ ⎜ 0 ⎟ r r r r r r ΔX 0 = = ax1 + bx2 + cx3 + dx4 + ex5 + fx6 ⎜ Δu0 ⎟ ⎜ ⎟ (4-3-1) Ψ ⎜ Δv0 ⎟ Fig. 4-3-1. Definition of thruster cant angle. ⎜ ⎟ ⎝ Δw0 ⎠ By substituting each eigenvector of the STM into Eq. First, we assume that initial state deviation ΔX0 in the (4-3-4), the equation that describes coefficient “a” by which

Pd_16 T. IKENAGA and M. UTASHIMA : Study on the Stationkeeping Strategy for the Libration Point Mission

the elements of the unloading delta-V vector are obtained. By substituting Eq. (4-3-5) into Table 4-3-1 and setting “a” Note that the eigenvectors change, however, depending on equal to zero, Fig. 4-3-3 shows the stable delta-V direction the point from which the STM is computed through one (i.e., the unloading delta-V direction). period of the halo orbit. Table 4-3-1 summarizes the equations This study considered two types of cant angle: 0o and 20o. that describe coefficient “a” with regard to the initial points of As mentioned previously, the unloading delta-V direction is computation (denoted as p0 to p3). The state vector and epoch determined as a constant in this study. Fig. 4-3-3 shows the of p0 are the same as listed in Table 5-1-1 of Section 5-1. The relation between the attitude constraint of each thruster cant state vectors and epochs of p1 to p3 are those of 45, 90, and angle and the stable delta-V direction, indicated by a red line 135 days later than those of p0, respectively. in the figure. The green and blue areas in Fig. 4-3-3 denote the attitude constraints in case of 0o and 20o cant angles, zL2 respectively. In case of the 20o cant angle, the direction in which the four Unloading delta-V “stable delta-V directions” vary the least is chosen. Conversely, in case of the 0o cant angle, two directions (i.e., yL2 ±zL2) are chosen for the unloading delta-V. In this case, we assume alternate changes in the unloading delta-V direction to θ minimize the influence of orbital disturbances caused by the unloading delta-V. φ 5. Stationkeeping Analysis

xL2 5.1. Reference Halo Trajectory Sun S/C Fig. 4-3-2. Description of unloading delta-V direction.

When we consider “real operation” of the S/C, it is impractical to compute the STM and determine the proper direction of the unloading delta-V on a daily basis. Therefore, the intermediate values of the directions for each “a = 0” (called the “the stable delta-V direction” in this paper) are employed to determine a “better” constant direction of unloading delta-V. In addition to the analysis above, the attitude constraints of SPICA and the cant angle of onboard thrusters are considered. Fig. 4-3-1 shows the definition of the thruster cant angle.

a) xy plane Unloading delta-V (cant = 20o)

Unloading delta-V (cant = 0o)

Fig. 4-3-3. Azimuth and elevation map of unloading delta-V direction.

Fig. 4-3-2 shows the unloading delta-V direction in the b) xz plane azimuth (θ) and elevation (φ) angles. From the definition shown in Fig. 4-3-2, the unloading delta-V vector is expressed Fig. 5-1-1. Reference halo trajectory with the size of Az = 300,000 km, below by Eq. (4-3-5). L2-centered rotational coordinate system.

Δ sinφ ⎛ u 0 ⎞ ⎛ ⎞ The reference halo trajectory (with a size of A = 300,000 ⎜ ⎟ ⎜ ⎟ (4-3-5) z Δv 0 = cosφ cosθ km) is used for stationkeeping analysis. Table 5-1-1 lists ⎜ ⎟ ⎜ ⎟ ⎝ Δw 0 ⎠ ⎝ cosφ sinθ ⎠

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information about the initial point of the reference trajectory. and S) in Table 5-2-2 are weighting diagonal matrices (see Chapter 3). Table 5-1-1. Initial position and velocity of the reference halo trajectory 5.3. Analysis

with size of Az = 300,000 km, L2-centered rotational coordinate system. After making the preparations described in the previous Initial Epoch: 31/Jan/2018 12:00 (UTC) section, four cases (denoted as “Analysis 1” to “Analysis 4”) X Y Z of stationkeeping simulation were conducted. In this study, we Position (m) -3.0e+5 0.0 -2.3e+5 conducted a hundred simulations in each case; changing the Velocity (m/s) 0.0 336.7 0.0 “seed” to a random number generator and calculating the average and standard deviations. Fig. 5-1-1 shows the shape of the reference halo trajectory Analysis 1: Evaluation of the algorithm’s feasibility in the xy and xz planes of the L2-centered rotational We initially evaluated the feasibility of the stationkeeping coordinate system. algorithm. In this analysis, the attitude constraints of SPICA, 5.2. Configuration delta-V control error, and unloading delta-V were not This study deals with three trajectories: a “nominal” considered. However, both injection error and O/D error were trajectory (i.e., reference halo trajectory explained in the considered. previous section,) an “actual” trajectory in which the S/C flies, and an “estimated” trajectory that the ground S/C operator can Table 5-3-1. Total corrective delta-Vs of “Analysis 1: Evaluation of the determine via orbit determination (O/D). Fig. 5-2-1 shows the algorithm’s feasibility”. relation between those three trajectories. μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 0.072 0.016 0.104 Nominal t2 t1 Delta-Vactual t Table 5-3-1 above shows the total corrective maneuver (delta-V) of a 360-day (nearly one year) simulation. Here, μ t0 denotes the average and σ the standard deviation. Injection error Estimated As shown in Table 5-3-1, the total corrective delta-V per Delta-Vplan year is negligible (i.e., 0.104 m/s/yr in μ + 2σ) considering the O/D error variance of 100 simulations. Given this result, we can say that Actual the stationkeeping algorithm employed in this study is Fig. 5-2-1. Relation between “nominal”, “actual” and “estimated” feasible. trajectories.

As shown in Fig. 5-2-1, injection error is applied at epoch t0 to the “nominal” trajectory to produce the “actual” trajectory. Moreover, the O/D error is also applied to the “actual” trajectory, which produces the “estimated” trajectory. Table 5-2-1 lists the details of injection and O/D error.

Table 5-2-1. Injection and O/D error, L2-centered rotational coordinate system. X Y Z Position (km) 1.5 2.5 15.0 Velocity (mm/s) 1.0 1.0 3.0

Table 5-2-2. Configuration of stationkeeping simulation. Fig. 5-3-1. Direction of corrective maneuver, L2-centered rotational Duration 360 days coordinate system. tmin 30 days

Δ t1 40 days

Δ t2 65 days Fig. 5-3-1 shows the direction of the corrective delta-V in ttracking 2 days the L2-centered rotational coordinate system. As the figure

dmin 0 km shows, ±xL2 directions dominate as the direction of the dmax 5.0e+4 km corrective delta-V, meaning that the corrective delta-V Q Diag (5.0e + 12, 3.0e + 13, 1.0e + 13) R Diag (1.0, 0.0, 1.0) direction is almost on a straight line from the Sun to the L2 S Diag (1.0, 1.0, 1.0) point. Δ Vmin 0.01 m/s Analysis 2: Impact of Attitude Constraints This analysis evaluated the impact of the attitude The O/D error described in Table 5-2-1 is virtually the same constraints of SPICA (see Section 4-1). As mentioned in the value as obtained in actual performance achieved on the previous chapter, two types of onboard thruster allocation are previous libration point mission. considered. Injection and O/D errors were also considered, but Table 5-2-2 summarizes the configuration of stationkeeping delta-V control and unloading delta-V errors were not. Table simulation conducted in this paper. The three matrices (Q, R, 5-3-2 lists the analysis results.

Pd_18 T. IKENAGA and M. UTASHIMA : Study on the Stationkeeping Strategy for the Libration Point Mission

Table 5-3-2. Total of corrective delta-Vs of “Analysis 2: Impact of Table 5-3-4. Total of corrective delta-Vs of “Analysis 4: Impact of Attitude Constraints”. Unloading delta-V”, 20o cant Angle. a) Type A a) Attitude constraints not considered μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 1.727 0.703 3.133 1.059 0.138 1.335

b) Type B b) Attitude constraints considered (Type A) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 0.090 0.020 0.130 52.007 30.522 113.051

As is clear, the total corrective delta-V of “Type A” thruster c) Attitude constraints considered (Type B) allocation is very large. When we compare the total delta-V in μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) the value of μ + 2σ, it is about 30 times greater than that of the 1.264 0.188 1.640 result of “Analysis 1.” Conversely, the total “Type B” is small and yielded almost the same result as obtained in “Analysis 1” As shown in Table 5-3-4, the total corrective delta-V in (only 1.3 times larger). This is because the delta-V direction case b) is extremely large or about 85 times larger than that in of “Type B” is closer to the line from the Sun to the L2 point case a). The total for case b) is obviously unacceptable at on which corrective delta-V directions exist. about 113 m/s/yr, as the maximum total delta-V of SPICA is Analysis 3: Impact of Delta-V Control Error about 113 m/s. The totals for cases a) and c) also increased This analysis evaluated the impact of delta-V control error compared to the results of “Analysis 3,” although the (see Section 4-2), while not considering the unloading delta-V. amounts—about 1.3 m/s/yr in case a) and 1.6 m/s/yr in case To compare the impact of delta-V control error, three cases c)—are still small enough. were prepared. In case a), the attitude constraints of SPICA Secondly, Table 5-3-5 lists the results of this analysis in the o are not considered. Conversely, in cases b) and c), the attitude case of the 0 cant angle. As mentioned in Section 4-3, in this constraints with “Type A” and “Type B” thruster allocations case, the direction of unloading delta-V changes in ± z are considered, respectively. Table 5-3-3 lists the analysis directions in the L2-centered rotational coordinate system results. alternately.

Table 5-3-3. Total of corrective delta-Vs of “Analysis 3: Impact of Table 5-3-5. Total of corrective delta-Vs of “Analysis 4: Influence of o Delta-V Control Error”. Unloading delta-V”, 0 cant Angle. a) Attitude constraints is not considered a) Attitude constraints not considered μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 0.072 0.016 0.104 0.074 0.015 0.104

b) Attitude constraints is considered (Type A) b) Attitude constraints considered (Type A) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 2.868 2.426 7.720 2.801 2.412 7.625

c) Attitude constraints is considered (Type B) c) Attitude constraints considered (Type B) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) μ (m/s/yr) σ (m/s/yr) μ + 2σ (m/s/yr) 0.091 0.020 0.131 0.090 0.021 0.132

As shown in Table 5-3-3, delta-V control error has small Similar to the results obtained by the previous analysis, the impact in cases a) and c). Conversely, in case b), the total total corrective delta-V in case b) peaks among all three cases. o corrective delta-V increases to double the level of the However, in case of the 0 cant angle, the total in case b) is “Analysis 2” result due to delta-V control error. This is about 7.6 m/s/yr, which, although large, is still acceptable. because case b)—where “Type A” thruster allocation is Conversely, the amount in case c) is only 0.13 m/s/yr, which applied—is more sensitive to delta-V control error than the is negligible and only 1.3 times greater than that in case a), other two cases due to the large corrective delta-V. In fact, the where attitude constraints were not considered. standard deviation in case b) is huge compared to the average, From the results given in Tables 5-3-4 and 5-3-5, we can o o or about 85 % of said average. conclude that the 0 cant angle is obviously better than the 20 Analysis 4: Impact of Unloading delta-V cant angle to reduce the impact of unloading delta-Vs on This analysis included all of the factors that should be stationkeeping. This is because, as described in Section 4-3, considered when simulating a “real” situation. In addition to alternate changes in unloading delta-Vs occurring in ±zL2 o “Type A” and “Type B” thruster allocations, two further cases direction in case of the 0 cant angle can reduce the impact on o of the thruster cant angle (see Section 4-3) were considered to stationkeeping, while in case of the 20 cant angle, the compare the impact of the unloading delta-V. First, Table unloading delta-V directions are constant. 5-3-4 lists the analysis results in the case of a 20o cant angle. Here, the unloading delta-V direction is constant.

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6. Conclusion References

This study evaluated the feasibility of a stationkeeping 1) Masayoshi Utashima: Design of Reference Halo Trajectories algorithm structured based on a paper written by K. C. Howell around L2 Point in the Sun-Earth System (in Japanese), JAXA et al4) through certain cases of stationkeeping simulation, and Research and Development Report, JAXA-RR-05-008, 2005. 2) Masayoshi Utashima: Near Lagrange Points (in by assuming application of the algorithm to the SPICA Japanese), National Space Development Agency of Japan (NASDA) mission. Based on the analysis results given in Section 5-3, Technical Memorandum, NASDA-TMR-960033, 1997. the following was revealed: 3) SPICA Working Group: SPICA Mission Proposal ver. 2 (in Japanese), ISAS, JAXA, 2007. 1) Most corrective delta-Vs lie along a line from the Sun to 4) K. C. Howell and H. J. Pernicka: Stationkeeping Method for Libration Point Trajectories, Journal of Guidance, Control and the L2 point. Dynamics, January-February 16, No. 1 (1993). 2) Choosing proper onboard thruster allocation can reduce 5) Oliver Montenbruck and Eberhard Gill: Satellite Orbits, Models the impact of the attitude constraints of SPICA. In fact, Method and Applications, Springer, 2005. based on the results of “Analysis 2” in Section 5-3, the 6) Masaki Nakamiya and Yasuhiro Kawakatsu: Preliminary Study on Orbit Maintenance of Halo Orbits under Continuous Disturbance, total delta-V amount of “Type A” thruster allocation is 20th AAS/AIAA Space Flight Mechanics Meeting 2010, about 3.13 m/s/yr, while that of “Type B” is about 0.13 AAS-10-119. 7) Rainer Bauske: Operational Maneuver Optimization for The ESA m/s/yr. st 3) Delta-V control error increases the total corrective Mission HERSCHEL and PLANK, 21 ISSFD, Toulouse, 2009. 8) David A. Vallado: Fundamentals of Astrodynamics and delta-V in “Type A” thruster allocation due to the high Applications, Second Edition, Space Technology Library, 2004. sensitivity to delta-V control error. The results of 9) Bong Wie: Space Vehicle Dynamics and Control, 2nd Edition, “Analysis 3” in Section 5-3 reveal how the total delta-V AIAA Education Series, 2008. amount becomes more than double in “Type A” due to the impact of delta-V control error (i.e., 7.72 m/s/yr), while that of “Type B” is about 0.13 m/s/yr. 4) The frequent unloading of delta-Vs dramatically increases the total delta-V in the case of “Type A” and a 20o cant angle. The total in this case is about 113 m/s/yr,

which is excessive for SPICA in terms of the onboard

propellant. Conversely, in the case of “Type B” and the 0o

cant angle, the total is only 0.13 m/s/yr.

There are plans to develop a program in the future that can produce a long-term reference halo trajectory. In addition, the effect of the “Global Suppression Approach” developed and proposed by Dr. Masaki Nakamiya6) to suppress the impact of frequent unloading delta-Vs will also be evaluated.

Acknowledgments

I would like to express my deep gratitude to Dr. Masayoshi Utashima, co-author of this paper, for his vast and valuable advice on this study. I really appreciate his extensive knowledge and excellent skills, especially in the area of astrodynamics. I would also like to thank Dr. Masaki

Nakamiya, a researcher at the Institute of Space and Astronautical Science (ISAS), a division of JAXA, for the assistance he provided for this study. Finally, I would like to thank Dr. Nobuaki Ishii, a professor at ISAS, and the group leader of the JAXA Trajectory and Navigation Group, for giving me the opportunity to participate in this study.

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