The 26th Workshop on Astrodynamics and Flight Mechanics 2016, Sagamihara, Japan.Copyright c2016 by S. Soldini. Published by the Secretariat of the WAFM 2016, with permission and released to the

Secretariat of the WAFM 2016 to publish in all forms.

Design and Control of Solar Radiation Pressure Assisted Missionsin the Sun-Earth System

By Stefania Soldini1)

1) Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan

(Received 28th Feb, 2017)

This article investigates the use of solar radiation pressure for the design and control of trajectories in the restricted three-body problem of the Sun-Earth system. A modern approach that has arisen in space mission design is the use of the invariantmanifold theory for the design of trajectory manoeuvres that exploit the natural dynamics of the solar system. A spacecraftsnatural dynamics are affected by environmental perturbations such as solar radiation pressure. Traditionally, the design of spacemissions requires any perturbations to be cancelled out through corrective manoeuvres requiring propellant and therefore thepre-storing of fuel. Invariant manifold techniques are here applied for harnessing solar radiation pressure in the design offuel-free manoeuvres from the beginning until the end of the spacecrafts lifetime. The advantage of solar radiation pressuremanoeuvres is that the spacecraft can have an unlimited source of propellant" (the Suns radiation) consequently extendingthe spacecrafts life" and reducing the overall missions cost associated with the fuel budget. The size of the required reflectivedeployable area and the spacecraft pointing accuracy are the ultimate outcomes of this work. Along with the design of thereflective area, the definition of a control law for the station-keeping of spacecraft in libration point orbits, a methodology totransfer between quasi-periodic orbits, and an end-of-life strategy to safely dispose of the spacecraft into a graveyard orbitaround the Sun are the major important findings.

Key Words: Orbit Control, Transfer Trajectory, End-of-Life, Libration Point Orbit and Solar radiation Pressure

1. Introduction

Space missions that require particular orbits to meet theirgoals cannot be achieved by the patched conic approximationalone. Indeed, the patched conic approach is a good approxi-mation in the design of interplanetary transfer trajectories thatmake use of high-energy manoeuvres. A modern approach thathas arisen in space mission design is to use Space ManifoldDynamics (SMDs) that exploits the natural dynamics of the so-lar system. SMD merges the knowledge of dynamical systems,celestial mechanics and astrodynamics [1, 13]. The simplestdynamical model used for the SMD approach is the RestrictedThree-Body Problem (R3BP). In the R3BP, the motion of thespacecraft is under the mutual gravitational influence of twomain celestial bodies. In this paper, the focus is in the designof space missions in the Sun-Earth system. Thus, the space-craft motion is only influenced by the gravitational effect of theSun and of the Earth+Moon barycentre. Unlike the patchedconic approximation, in the R3BP model, there exist five equi-librium points. These equilibrium points, known as librationpoints, are defined in a coordinate system rotating with the Sun-(Earth+Moon) [17, 4]. Currently, the libration points selectedfor space applications are the collinear points that are alignedwith the Sun-(Earth+Moon) line. In particular, L1, located be-tween the line joining the Sun and the Earth+Moon barycen-tre and L2, located in the anti-sunward direction along the Sunand the Earth+Moon barycentre line. A spacecraft placed inthose points will keep a constant distance from the Sun andthe Earth+Moon barycentre opening new opportunities in spacemission design. Due to the unstable nature of L1 and L2, aspacecraft placed around the equilibrium points will naturally

diverge from them. Thus, trajectories designed in the R3BP re-quire the spacecraft to be manoeuvred to maintain its nominaltrajectory by counteracting the unwelcome environmental in-stabilities [19]. Among the environmental effects, the strongerperturbations are the motion of the Moon around the Earth, theeccentricity of the Earth+Moon barycentres orbit around theSun, the gravitational effect of passing planets and the SolarRadiation Pressure (SRP). After the gravitational effects, SRPis a significant factor in the Sun-(Earth+Moon) system, partic-ularly when the spacecraft has extended high reflective areas,e.g. James Webb Space Telescope [2].

The aim of this paper is to exploit SRP to perform the re-quired correcting manoeuvres to keep the spacecraft on thenominal trajectory. The advantage of using SRP as the sourceof propulsion is in the design of innovative propellant-free de-vices that reduce the pre-stored fuel onboard the spacecraft [10].Therefore, SRP is a natural and unlimited source of propel-lant". Due to this unlimited propellant", space missions thatuse SRP could have longer mission lifetimes. SRP enhancingdevices are applied to future LPO missions designed in the Sun-Earth system; the idea is to exploit the effect of SRP from thebeginning to the end of the mission. In the field of the R3BP,SRP stabilisation is applied to the:

1) development of orbital control methods for LPOs,

2) transfer trajectories within the Sun-Earth system, and

3) the end-of-life disposal.

SRP trajectory stabilisation requires devices on board the space-craft that can react to SRP acceleration [10]. The SRP stabil-isation has been demonstrated by JAXAs Ikaros mission that

1

The 26th Workshop on Astrodynamics and Flight Mechanics 2016, Sagamihara, Japan.Copyright c2016 by S. Soldini. Published by the Secretariat of the WAFM 2016, with permission and released to the

Secretariat of the WAFM 2016 to publish in all forms.

utilises a 20-m span square solar power sail [18]. The SRPacceleration forces are enhanced by the spacecrafts reflectivearea, its reflectivity properties, the reflective area orientation (tothe Sun), and the reduction of the spacecrafts mass. Thus, SRPdevices require a light and extended highly reflective area. Theacceleration needed to manoeuvre the spacecraft is controlledby mechanical variations in the former parameters, e.g. control-ling the surface reflectivity [8] or by changing the area throughdeployable mechanisms [10].

The paper is organised as follow: Section 2. presents theequations of motion in the CR3BP with SRP. The Hamiltonianstructure preserving control law is presented in Section 3., whilethe design of transfer trajectories is shown in Section 4.. Finally,Section 5. show the end-of-life disposal strategy for future LPOsatellite.

2. Equations of Motion

Fig. 1: Synodic reference frame centered in the Sun-(Earth+Moon)barycentre.

The equations of motion of the CR3BP-SRP in dimensionlesscoordinates for the rotating frame are shown in Eq. (1) [7]:

x 2y = Vx + asxy + 2x = Vy + a

sy

z = Vz + asz

(1)

where, is the mass parameter of the system. x, y, z and x,y, z are the spacecraft positions and velocities in the synodic(rotating) frame and, asx, asy and asz are the components alongx-, y- and z-axis of the solar radiation pressure acceleration, as.V is the total potential,

V =1

2(x2 + y2) +

rSunp+

1 rEarthp

, (2)

where rSunp and rEarthp are the spacecrafts distance fromthe Sun and the Earth respectively as shown in Figure 1 anddefined as:

rSunp =

(x+ )2 + y2 + z2 (3)

andrEarthp =

(x+ 1)2 + y2 + z2. (4)

The solar radiation pressure acceleration is defined as:

as = PsrpA

m

N r

cRN , (5)

with N defined through the cone () and clock () angles as:

N = cosrSunp|rSunp| + sin cos

(rSunpz)rSunp|(rSunpz)rSunp|

+ sin sin rSunpz|rSunpz| .

(6)

2.1. Displaced Equilibrium Points of the x-y plane un-der the SRP effect

(a) Position of the equilibrium pointin the rotating system for a generalsail attitude.

0.985 0.99 0.995 1 1.005 1.010.015

0.01

0.005

0

0.005

0.01

0.015

x

y

0 0.02 0.04 0.06 0.08 0.1

(b) Equilibrium point for = 90.Each colored line correspond to from 0 up to 0.1.

Fig. 2: Computation of the equilibrium points under the SRP effect.

The equation for numerically find the planar equilibriumpoints (x-y plane) is:

x+(1 cos3 )r3Sunp

(x+ ) cos2 sin

r3Sunp

y + (1)r3Earthp

(x+ 1) = 0

y + (1 cos3 )

r3Sunp

y + cos2 sin

r3Sunp

(x+ ) + (1)r3Earthp

y = 0

(7)Figure 2 shows the position of the equilibrium point for = 90

for a general cone angle, , orientation.

3. Hamiltonian Structure Preserving Control

The Hamiltonian structure-preserving control uses the eigen-structure of the linearised equations of motion to create a con-trol law that ensures Lyapunov stability [6]. As shown by [14],this controller aims to remove both the stable and unstable man-ifolds (red and green arrows in Figure 3) by projecting the stateposition error (between the current and the target orbit) alongthe manifold direction. This creates an artificial centre mani-fold, as shown in Figure 3 that keeps the trajectory close to thetarget orbit, as the eigenvalues of the linearised dynamics, areplaced along the imaginary axis.

Fig. 3: The effect of the Hamiltonian structure preserving controllaw is to replace the hyperbolic equilibrium with an artificial centremanifold.

2

The 26th Workshop on Astrodynamics and Flight Mechanics 2016, Sagamihara, Japan.Copyright c2016 by S. Soldini. Published by the Secretariat of the WAFM 2016, with permission and released to the

Secretariat of the WAFM 2016 to publish in all forms.