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

Transcription

1 Design and Control of Solar Radiation Pressure Assisted Missions in the Sun-Earth System By Stefania Soldini 1) 1) Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan (Received 8 th Feb, 17) This article investigates the use of solar radiation pressure for the design and control of trajectories in the restricted threebody problem of the Sun-Earth system. A modern approach that has arisen in space mission design is the use of the invariant manifold theory for the design of trajectory manoeuvres that exploit the natural dynamics of the solar system. A spacecraft s natural dynamics are affected by environmental perturbations such as solar radiation pressure. Traditionally, the design of space missions requires any perturbations to be cancelled out through corrective manoeuvres requiring propellant and therefore the pre-storing of fuel. Invariant manifold techniques are here applied for harnessing solar radiation pressure in the design of fuel-free manoeuvres from the beginning until the end of the spacecraft s lifetime. The advantage of solar radiation pressure manoeuvres is that the spacecraft can have an unlimited source of propellant" (the Sun s radiation) consequently extending the spacecraft s life" and reducing the overall missions cost associated with the fuel budget. The size of the required reflective deployable area and the spacecraft pointing accuracy are the ultimate outcomes of this work. Along with the design of the reflective area, the definition of a control law for the station-keeping of spacecraft in libration point orbits, a methodology to transfer between quasi-periodic orbits, and an end-of-life strategy to safely dispose of the spacecraft into a graveyard orbit around 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 their goals cannot be achieved by the patched conic approximation alone. Indeed, the patched conic approach is a good approximation in the design of interplanetary transfer trajectories that make use of high-energy manoeuvres. A modern approach that has arisen in space mission design is to use Space Manifold Dynamics (SMDs) that exploits the natural dynamics of the solar system. SMD merges the knowledge of dynamical systems, celestial mechanics and astrodynamics [1, 1]. The simplest dynamical model used for the SMD approach is the Restricted Three-Body Problem (RBP). In the RBP, the motion of the spacecraft is under the mutual gravitational influence of two main celestial bodies. In this paper, the focus is in the design of space missions in the Sun-Earth system. Thus, the spacecraft motion is only influenced by the gravitational effect of the Sun and of the Earth+Moon barycentre. Unlike the patched conic approximation, in the RBP model, there exist five equilibrium points. These equilibrium points, known as libration points, are defined in a coordinate system rotating with the Sun- (Earth+Moon) [17, 4]. Currently, the libration points selected for space applications are the collinear points that are aligned with the Sun-(Earth+Moon) line. In particular, L 1, located between the line joining the Sun and the Earth+Moon barycentre and L, located in the anti-sunward direction along the Sun and the Earth+Moon barycentre line. A spacecraft placed in those points will keep a constant distance from the Sun and the Earth+Moon barycentre opening new opportunities in space mission design. Due to the unstable nature of L 1 and L, a spacecraft placed around the equilibrium points will naturally diverge from them. Thus, trajectories designed in the RBP require the spacecraft to be manoeuvred to maintain its nominal trajectory by counteracting the unwelcome environmental instabilities [19]. Among the environmental effects, the stronger perturbations are the motion of the Moon around the Earth, the eccentricity of the Earth+Moon barycentre s orbit around the Sun, the gravitational effect of passing planets and the Solar Radiation Pressure (SRP). After the gravitational effects, SRP is a significant factor in the Sun-(Earth+Moon) system, particularly when the spacecraft has extended high reflective areas, e.g. James Webb Space Telescope []. The aim of this paper is to exploit SRP to perform the required correcting manoeuvres to keep the spacecraft on the nominal trajectory. The advantage of using SRP as the source of propulsion is in the design of innovative propellant-free devices that reduce the pre-stored fuel onboard the spacecraft [1]. Therefore, SRP is a natural and unlimited source of propellant". Due to this unlimited propellant", space missions that use SRP could have longer mission lifetimes. SRP enhancing devices are applied to future LPO missions designed in the Sun- Earth system; the idea is to exploit the effect of SRP from the beginning to the end of the mission. In the field of the RBP, SRP stabilisation is applied to the: 1) development of orbital control methods for LPOs, ) transfer trajectories within the Sun-Earth system, and ) the end-of-life disposal. SRP trajectory stabilisation requires devices on board the spacecraft that can react to SRP acceleration [1]. The SRP stabilisation has been demonstrated by JAXA s Ikaros mission that 1

2 y The 6 th Workshop on Astrodynamics and Flight Mechanics 16, Sagamihara, Japan. utilises a -m span square solar power sail [18]. The SRP acceleration forces are enhanced by the spacecraft s reflective area, its reflectivity properties, the reflective area orientation (to the Sun), and the reduction of the spacecraft s mass. Thus, SRP devices require a light and extended highly reflective area. The acceleration needed to manoeuvre the spacecraft is controlled by mechanical variations in the former parameters, e.g. controlling the surface reflectivity [8] or by changing the area through deployable mechanisms [1]. The paper is organised as follow: Section. presents the equations of motion in the CRBP with SRP. The Hamiltonian structure preserving control law is presented in Section., while the design of transfer trajectories is shown in Section 4.. Finally, Section 5. show the end-of-life disposal strategy for future LPO satellite. with ˆN defined through the cone (α) and clock (δ) angles as: ˆN = cos α r Sun p r Sun p + sin α cos δ (r Sun p ẑ) r Sun p (r Sun p ẑ) r Sun p + sin α sin δ r Sun p ẑ r Sun p ẑ. (6).1. Displaced Equilibrium Points of the x-y plane under the SRP effect β.1. Equations of Motion x (a) Position of the equilibrium point (b) Equilibrium point for δ = 9. in the rotating system for a general Each colored line correspond to β sail attitude. from up to.1. Fig. : Computation of the equilibrium points under the SRP effect. The equation for numerically find the planar equilibrium points (x-y plane) is: Fig. 1: Synodic reference frame centered in the Sun-(Earth+Moon) barycentre. The equations of motion of the CRBP-SRP in dimensionless coordinates for the rotating frame are shown in Eq. (1) [7]: ẍ ωẏ = V x + a s x ÿ + ωẋ = V y + a s y z = V z + a s z where, µ is the mass parameter of the system. x, y, z and ẋ, ẏ, ż are the spacecraft positions and velocities in the synodic (rotating) frame and, a s x, a s y and a s z are the components along x-, y- and z-axis of the solar radiation pressure acceleration, a s. V is the total potential, V = 1 (x + y ) + (1) µ + 1 µ, () r Sun p r Earth p x + µ(1 β cos α) r Sun p y + µ(1 β cos α) r Sun p (x + µ) βµ cos α sin α y + r Sun p (1 µ) (x + µ 1) = r Earth p y + βµ cos α sin α (x + µ) + (1 µ) y = r Sun p r Earth p (7) Figure shows the position of the equilibrium point for δ = 9 for a general cone angle, α, orientation.. Hamiltonian Structure Preserving Control The Hamiltonian structure-preserving control uses the eigenstructure of the linearised equations of motion to create a control law that ensures Lyapunov stability [6]. As shown by [14], this controller aims to remove both the stable and unstable manifolds (red and green arrows in Figure ) by projecting the state position error (between the current and the target orbit) along the manifold direction. This creates an artificial centre manifold, as shown in Figure that keeps the trajectory close to the target orbit, as the eigenvalues of the linearised dynamics, are placed along the imaginary axis. where r Sun p and r Earth p are the spacecraft s distance from the Sun and the Earth respectively as shown in Figure 1 and defined as: r Sun p = (x + µ) + y + z () and r Earth p = (x + µ 1) + y + z. (4) The solar radiation pressure acceleration is defined as: a s A = P srp ˆN ˆr cr ˆN, (5) m Fig. : The effect of the Hamiltonian structure preserving control law is to replace the hyperbolic equilibrium with an artificial centre manifold.

3 The local stability impacts onto the periodic orbit stability by affecting the eigenvalues of the monodromy matrix, M. The monodromy matrix is the state transitional matrix, Φ(t +T, t ), of the system evaluated after one orbital period, T, where t is the initial time. For the Lyapunov stability, the controller should place the eigenvalues of M on the unitary circle of the complex plane [], see Figure 4. Since the HSP control aims to stabilise the system in the sense of Lyapunov, the control law is designed such as to affect the sign of b, c and of Eq. (11). Indeed, the simple Lyapunov stability can be achieved by placing the eigenvalues of the linearised dynamics on the imaginary axis, as shown in Figure 5, by adding to V rr an artificial potential, the centre manifold T. Fig. 4: Eigenvalues of the monodromy matrix with (red crosses) and without (white crosses) the effect of the HSP controller. The natural dynamics in Eq. (1) can be written in a compact form as: Ẋ = f(x). (8) In Eq. (8), f is the flow field and, X = {x, y, z, ẋ, ẏ, ż} is the state vector. The variational equations are: δ X(t) = A( X(t))δ X(t) (9) which are the linearised equations for the evaluation of the variations δ X(t). In Eq. (9), A( X(t)) is the Jacobian matrix of the flow field, f, evaluated along the reference trajectory. For the linearised equations, solving the eigenvalues of the variational equations matrix, A( X(t)), is an approximation of solving the eigenvalue problem of the STM, Φ(t, t ). The variational equations of Eq. (9) are [7]: [ d δr dt δṙ ] [ I = ω J V rr ] [ δr δṙ ] J = [ 1 1 ], (1) V rr is the Jacobian matrix of the potential acceleration in Eq. () and ω J is the term associated to the Coriolis acceleration. In Eq. (1), δr and δṙ are the state position and velocity errors, respectively. The eigenvalues of the linearised dynamics are the solutions of the characteristic equation D(λ) = A λi =, where the characteristic polynomial is: Λ + bλ + c = where, b = 4ω V xx V yy c = V xxv yy V xy = b 4c. (11) As exploited by [14], the solutions of Eq. (11) are affected by the sign of. When, > the system produces two real and unequal roots; while, when < there are two complex and conjugate solutions. The change in the stability of the eigenvalues is evident for high amplitude orbits where it is possible to identify two cases along the trajectory where the eigenvalues are couples of real and pure imaginary numbers (saddle centre equilibrium, i.e., when b <, > and c < ), or where the eigenvalues are couples of complex numbers and conjugate pairs (stable unstable foci, i.e., when b <, < and c < ). Fig. 5: Eigenvalues of the linearised dynamics with (green crosses) and without (white crosses) the effect of the HSP controller. The artificial centre manifold, T, is constructed from the linear combination of the projection tensors u k u T k (eigenvectors of the stable and unstable manifolds) and the gains. This linear combination is selected such as b c, c c and c, which are the indexes of stability affected by the control law, are all greater than zero [14]. {b c > & c c > & c > } is the condition of simple Lyapunov stability; where, the HSP control is added to the dynamics in Eq. (1) as an additional control acceleration, a c, that will be modelled as SRP acceleration. Thus, a c is given by the actuators model, a s. a c is obtained by multiplying T by the state position error between the target orbit and the actual spacecraft trajectory δr, a c = T δr. (1) The acceleration, a c, affects the linearised dynamics and A( X(t)), in Eq. (1) which turns into A c ( X(t)): A c ( X(t)) = [ I V rr + T ω J ]. (1) For the hyperbolic center equilibrium the control law is: ] a c = σ G 1 [u 1 u T 1 + u u T δr, (14) while, it turns into: a c = ( σ + γ ) { G [ u 1 u 1 T + u u T ] + G [ u u T + u 4 u 4 T ]} δr, (15) for complex and conjugate pairs. In conclusion, the HSP control algorithm is designed such that: { a c Equation [14] if > Scheers et al. (a) = Equation [15] if < Soldini et al. (16b). (16) Figure 6 shows in blue the target Halo orbit and in green the controlled orbit. The initial offset from the target orbit is set of 15 km in position and m/s in velocity. In the case of no error in the HOI manoeuvre, the pointing requirements and the area needed to control the spacecraft are very tiny as shown in Table 1 for the case of A = 7 m. This shows that the HSP control

4 z [km] x x y [km] x [km] x 1 8 Fig. 6: Controlled halo orbit with the HSP control law. requires very little acceleration to stabilise the orbit. This is confirmed by Figure 7 that compares the same case scenario with (Figures 7(a), 7(c) and 7(e)) and without (Figures 7(b), 7(d) and 7(f)) an initial injection error. Note that, when the injection error is not considered, the controller acceleration is zero at the first orbital period because the spacecraft is precisely placed on the target orbit. x axis A A α δ offset [km] [m ] [m ] [ ] [ ] ±.5 ± ± 1.5 ± ± 9.5 ± ± 5.5 ± ± ± ±.8 ± 1. Table 1: Area and orientation angles required for different initial reflective area and injection errors. Area [m ] (a) Area required: offset of 4 km in x axis. α [deg] (c) In-plane angle, α: offset of 4 km in x axis. δ [deg] (e) Out-of-plane angle, δ: offset of 4 km in x axis. Area [m ] (b) Area required: no offset in x axis. α [deg] 1 1 x (d) In-plane angle, α: no offset in x axis. δ [deg] x (f) Out-of-plane angle, δ: no offset in x axis. Table 1 shows that for A = m and an injection error of 5 km the solution exist and it requires reasonable variations in the area required and in the orientation angles. It also shows that, in this case, the controller should be limited to changes in just the area and the in-plane angle, α, since the variations in the out-of-plane angle, δ, are very tiny; thus are not feasible. Fig. 7: Comparison of the effect of the initial injection error along the x-axis on to reflective area and orientations angle required for stabilising the orbit when an initial area of 7 m is selected..1. Actuator Design As shown in Table 1 for a near perfect reflective area 1 of A = m, the total area variation needed to meet the control requirements is between m, thus the variation in the total reflective area A f as to be between -4.5 m and 1.5 m. For each reflective flap, the initial A f has to be reduced by.5 m (.66 m.47 m) or increased by.75 m (.1.47 m). In summary, the minimum flap area is of A f min = m, the maximum flap area is of A f max = m, while the nominal flap area is A f = 4.44 m. The best sail substrate is kapton and has a surface density of 7.1 g/m. The best choice of the surface coating is aluminium with a surface density of 1.5 g/m [1]. The maximum area of one flap is of m and the mass of the flap material is 4.5 g (.45 kg). To support the flap material, a m of mast structure, as shown in Figure 9, is required that has a linear mass of 7 g/m [11], which totals 56 g (.56 kg). The total mass of one flap is therefore 6.5 g, and the total mass of the two flaps is 1.1 kg. Allowing a % mass margin [1], the 1 Note that here A is the contribution of the solar array and the additional flaps area at the nominal condition, thus A = 4 A sa + A f. Fig. 8: Actuators configuration for a class of spacecraft like SOHO mission. total mass of the reflective actuator system is kg. Figure 9 shows the front and the back view of the reflective actuator flap with a maximum area of A f max = m. Shahid and Kumar [15] proposed a sliding-mode control for LPO spacecraft enhanced by solar radiation pressure. In Shahid and Kumar [15] case, the initial area required for the control is around 4 m. For a 1 kg spacecraft, it was proposed by Shahid and Kumar [15] the use of a solar sail with a final mass of 6 kg. In this paper, two additional flaps to the spacecraft solar array is proposed with a total area of m and an additional mass of kg. Electro-chromic devices are here proposed to design a reflective actuator flap for variable geometry control. The flap is cov- 4

5 Fig. 9: Front and back views of the reflective actuator flap for A f max = m. ered by pixel of reflective control device. The control law can thus be transformed in electric impulse to switch on (highly reflective pixel, in yellow in Figure 1) and off (absorption pixel, in gray in Figure 1) the reflective control devices. In this case, the shape of the flap is kept fixed and the effect of variable geometry is obtained by changing the surface luminosity of the flaps. Figure 1 shows how to modify the reflective area of the flap through reflective control device. The advantage of this method is in allowing the change of the reflective area without mechanical moving parts. This method could also add flexibility in the control law mission design. It would be possible to adjust the requirements of the control law by reshaping the on/off switching configuration. The main disadvantage is related to the effect of degradation of the material in the space environment, thus a margin in the area should be included to compensate effect of degradations during the all duration of the mission. The definition of departure Lissajous orbit requires the knowledge of the amplitudes A x (or A y ) and A z which are function of A and A 4. Note that the frequencies in Eq. (17) depend on the eigenvalues of the matrix of the linear system associated to the selected libration point which, at the same time, depend on the selected lightness parameter, β. We will use two ways to represent Lissajous orbits: a Cartesian reference frame centred at the center of mass (or at a libration point), and the plane of effective phases (Φ, Ψ). Once the amplitudes, the equilibrium point, the SRP parameters are selected (for instance β), and the initial phases Φ and Ψ are fixed, it is possible to compute the states of the Lissajous orbit at any epoch, t. Conversely, it can be convenient to fix the time t (for instance t = ) and vary Φ and Ψ between and π. Most of the current and future space missions that make use of Lissajous orbits have amplitudes in A y varying between 81, km to 75, km and between 9, km and 45, km in A z. In this paper the focus is on square Lissajous orbits around SL 1 (A y = A z). Three amplitudes where selected of 5, km, 5, km and 75, km for the departure Lissajous orbits. Once the departure Lissajous orbit is selected, an SRP manoeuvre given along the departure Lissajous orbit was investigated to compute the values of the amplitudes of the unstable, A 1, and stable, A, components of the target Lissajous orbit. Fig. 1: Concept of pixel Reflective Control Device (RCD). 4. Design of Transfer Trajectories The procedure used to design a transfer trajectory assisted by a SRP manoeuvre, using the geometry of the phase space around of two libration points, is described in the following: 1) Design of the departure Lissajous orbit: The departure Lissajous orbit can be computed using Eq. (17): ξ(t) = A 1 e λ1t + A e λt + A e λt + A 4 e λ4t η(t) = B 1 e λ1t + B e λt + B e λt + B 4 e λ4t ζ(t) = A 5 e λ5t + A 6 e λ6t ξ(t) = λ 1 A 1 e λ1t + λ A e λt + λ A e λt +λ 4 A 4 e λ4t. η(t) = λ 1 B 1 e λ1t + λ B e λt + λ B e λt +λ 4 B 4 e λ4t ζ(t) = λ 5 A 5 e λ5t + λ 6 A 6 e λ6t. (17) Fig. 11: Example of Lissajous orbit. ) Transformation between the two reference systems centered at the libration points due to the SRP manoeuvre. A spacecraft placed at a departure Lissajous orbit associated to the libration point for a selected initial lightness parameter β, will follow the geometry of SL i (β ). When a SRP manoeuvre is given, the lightness parameter will became β M, and the motion of the spacecraft will be driven by the geometry of the new equilibrium point, SL i (β M ). It is possible to choose the amplitude in x or y since their relationship is A y = ka x. 5

6 For a Sun-pointing reflective area, we have: ξ = ξ + x SLi (β ) x SLi (β M ) η = η ζ = ζ ξ = ξ η = η ζ = ζ. (18) Once the state vector ξ = (ξ, η, ζ, ξ, η, ζ ) of the spacecraft, in the synodic reference frame centered at SL i (β M ), is known, it is possible to find the amplitudes associated to the target Lissajous orbit. This can be done through the inverse transformation given by Eq. (19): ξ η ξ η = M A 1 A A A 4, (19) where, M = [M 1 M M M 4 ] is the matrix with columns: e e k M 1 = 1 e λ1t k λ 1 e λ1t, M = e λt λ e λt k 1 λ 1 e λ1t k λ e λt M = M 4 = e αt cos(ωt) e αt [R(k 4 ) cos(ωt) + I(k ) sin(ωt)] e αt [α cos(ωt) ω sin(ωt)] e αt {R(k 4 ) [α cos(ωt) ω sin(ωt)] +I(k ) [α sin(ωt) + ω cos(ωt)]} e αt sin(ωt) e αt [I(k 4 ) cos(ωt) + R(k ) sin(ωt)] e αt [α sin(ωt) + ω cos(ωt)] e αt {I(k 4 ) [α cos(ωt) ω sin(ωt)] +R(k ) [α sin(ωt) + ω cos(ωt)]}. () (1) Figure 1 shows the location and the stability character of the libration point SL 1 when β =.5, the clock angle δ is equal to 9 and the cone angle is varied from ±9. When the libration point is in the positive y-axis, α is negative and the equilibrium point has a stable focus, while it becomes unstable in the negative y-axis when α has positive values. When α is or 9, the equilibrium point is on the x-axis and it is of the saddle center center type. Specifically, for α = the spacecraft is Sun-pointing; while, for α = 9, the effect of the SRP acceleration vanishes and the equilibrium point corresponds to the solution of the CRBP dynamics. Recall that Eq. (19) give the state vector of the spacecraft in the synodic system centered at the libration point as a Fig. 1: Stable and Unstable focus for β =.5, δ = 9 and α varied between ±9. function of the amplitudes, and can be written in a compact way as: ξ = M(β, α =, δ = 9, t) A. () In this equation, the matrix M is defined as in Eq. (19) and depends on time t and the lightness parameter β (for a Sun-pointing reflective area), ξ denotes the state vector of the spacecraft in a synodic reference frame centered at the libration point, and A = (A 1, A, A, A 4, A 5, A 6 ) is the vector of the amplitudes. The inverse transformation of Eq. () can be written in a compact way as: A = M 1 (β, α =, δ = 9, t) ξ. () ) Computation of the amplitudes of the target Lissajous orbit once the SRP manoeuvre is given. Using Eq. () for the spacecraft s state vector ξ (computed by means of Eq. (18)) at the time t M when the SRP manoeuvre is applied, it is possible to find the amplitudes of the target Lissajous orbit, A according to: A = M 1 (β M, α =, δ = 9, t M ) ξ. (4) In case the unstable manifold of the target Lissajous orbit is zero (A 1 = ), it is possible to follow the stable manifold of the target orbit. The amplitudes, the frequencies and the phases of the target Lissajous orbit can be thus computed. The amplitudes of the target Lissajous are given by: A x = A + A 4, A z = A 5 + A 6, (5) while, the phases are given by: ( ) ( ) Φ = arctan A 4 A, Ψ = arctan A 6 A. (6) 5 The in-plane, ω, and out-of-plane, ν, frequencies are function of β M ; thus, the target Lissajous orbit is given 6

7 by: ξ (t) = A x cos(ω t + Φ ), η (t) = k A x sin(ω t + Φ ), ζ (t) = A z cos(ν t + Ψ ), ξ (t) = ω A x sin(ω t + Φ ), η (t) = k ω A x cos(ω t + Φ ), ζ (t) = ν A z sin(ν t + Ψ ), (7) which, in the synodic reference frame centered at the departure libration point SL i (β ) can be obtained by applying the inverse transformation of Eq. (18). In the above equation, the effective phases are Φ = ω t + Φ and Ψ = ν t + Ψ, thus, the transfer from the departure to the target Lissajous orbit, as a consequence of the SRP manoeuvre, will be seen in the effective phases space as an instantaneous change in the effective phases (from Φ and Ψ to Φ and Ψ ). (a) Transfer strategy in the synodic reference frame centered at the Sun-Earth s center of mass. 4) Computation of the spacecraft s trajectory after the SRP manoeuvre: According to Eq. (), the trajectory followed by the spacecraft after the SRP manoeuvre is given by: ξ (t) = M(β M, α =, δ = 9, t) A, (8) where t t M is the time along the trajectory after the manoeuvre. (b) Transfer strategy in the phase space. Fig. 1: Transfer strategy using the invariant manifolds. 5) Trajectory of the spacecraft in the synodic system centered at the Sun-Earth s center of mass. Using the transformation Eq. (9): ξ η ζ = x x SLi = y = z, (9) it is now possible to express the trajectory in the synodic reference frame centred at the center of mass of the Sun- Earth system. For a Sun-pointing spacecraft y = η, z = ζ, and only the x coordinate must be modified according to: x = ξ + x SLi (β M ). Note that in case of in-plane attitude manoeuvres, β is fixed, δ = 9 and α = α M. Figure 1(a) shows the same transfer sequence shown in Figure 1(b) but in the phase space. The black circles are representative of points along the departure Lissajous orbit where the spacecraft can be injected onto the unstable manifold. One point was selected to inject the spacecraft in an unstable trajectory as shown in green. After that green point, the spacecraft will follow the unstable trajectory (black line). The red star represents the point in which the SRP manoeuvre is given that causes an instantaneous change in the phases. Finally, the spacecraft will follow the stable manifold of the target Lissajous orbit (blue line). It is quite convenient to make use of the phase space as the trajectories are simply represented as straight lines and it clearly shows the effect of the manoeuvre in the phases. Figure 14 shows a schematic representation of the transfer strategy for a Sun-pointing reflective area. The main properties of an SRP manoeuvre for a Sun-pointing area are: Fig. 14: Heteroclinic connection between the equlibrium points before (SL i) and after (SL i) the SRP manoeuvre. The effect of a change in β causes a shift along the x-axis in the position of the equilibrium point. Thus, the libration point gets closer to the Sun for high value of β; The transfer is allowed when a change in β is given from high to lower values. The spacecraft will always move in the opposite direction with the Sun due to the effect of the SRP acceleration as shown in Figure 14 for the black arrow; The transfer is possible at the heteroclinic connection when the unstable manifold of the departure Lissajous orbit (red line in Figure 14) intersect the stable manifold of the target Lissajous orbit (green line in Figure 14); A direct transfer from the departure Lissajous orbit is not possible for a geometrical reason. Indeed, the stable manifold of the target Lissajous orbit does not intersect the 7

8 departure Lissajous orbit but just its unstable manifold as shown in Figure 14; The values in the initial β and the final β M depend on the geometrical intersection at the heteroclinic point. Indeed, the heteroclinic point has to be located between the two equilibrium points as shown in Figure 14. This condition depends on the location of the equilibrium points, for instance on β and β M. for a squared solar sail is required. For a kapton substrate (7.1 g/m ) and aluminium surface coating (1.5 g/m ), the solar sail weight is kg. For the mast structure ( m) with linear density of 7 g/m an additional mass of kg has to be allocated to support the solar sail. Thus, kg are associated to the solar sail by ensuring kg allocated for the payload from the total spacecraft mass that correspond to the bus weight of JAXA Ikaros solar sail ( m-span) Deployable structure solutions The design of the transfer trajectories enhanced by solar radiation pressure acceleration requires the use of a fixed variable reflective area Sun-pointing or fixed geometry re-orientable reflective area. In the first case, the reflectivity of an initial Sun-pointing area is reduced to allow transfer between Lissajous orbits. Thus, the use of Reflective Control Devices (RCDs) can be exploited to reduce the reflectivity of a sunshade structure as demonstrated by the Ikaros mission [18]. In Earth s orbit missions, Lücking et al [9] uses the RCDs to travel in the phase space. Figure 15 shows that when the RCDs are on, the effective reflective area is white; while, when the RCDs are off (gray areas), the effective reflective area is reduced (white areas). Similar results can be achieved for solar panels through the use of flaps that aim to reduce the reflective area as shown in Figure 16. Fig. 16: Solar panels with flaps. Finally, in the case of a fixed reflective area, attitude control manoeuvres are investigated to perform the transfer between Lissajous orbits by re-orienting the pre-existing spacecraft s reflective on-board structures such as: the solar panels, the sunshade, or the spacecraft s antennas. In this case, the transfer of the spacecraft is promising for LPO class of satellite by simply re-orienting the spacecraft in opposite angular direction. This type of transfer is more efficient with respect to changes in β as just an attitude manoeuvre is required. Fig. 15: Sunshade with reflecive control devices. For example, an initial lightness number of β =.4 requires a final lightness number of β M =.. If the sunshade of the satellite is considered as the major reflective surface, a reflectivity coefficient, C s r, of 1.9 at the beginning of life was taken into account. Note that the James Webb Space Telescope have a sunshade of 64 m with a spacecraft mass of 65 kg while in this case a 7. m of sunshade can be used for SRP transfer manoeuvre of a 1 kg spacecraft. It seems quite challenging for a class of satellite as LPO spacecraft where a mass of the order of 1 kg is usually required as the transfer is enhanced by β = β β M where the transfer manoeuvre is facilitated by higher value in β. This technique seems more promising for small class of satellite like nanosatellite (1-1 kg). For the example previously shown, a total mass of the spacecraft of 85 kg (included the weight of the sail) with m-span 5. End-Of-Life Disposal Manoeuvre For current mission in LPOs (where limited amount of fuel can be actually used to perform a EOL manoeuvre), the trend is to dispose the spacecraft on a heliocentric orbit as shown in Figure 17(a). However, if the V given is not high in enough to close the gate at L [1], the spacecraft is very likely to re-enter or cross the Earth s protected regions (region of operative satellite), Figure 17(a). Alternatively, it is possible to achieve the closure in the pseudo libartion point, SL, by augmenting the area-to-mass ratio of the satellite. Note that for a Sun-pointing spacecraft, the increase in the area-to-mass ratio causes the shift of L along the x-axis closer to the Earth. An energy approach presented in [16] is used to determine the required deployable area needed to increase the energy of the spacecraft to equal the value in energy of SL. Figure 17(b) shows the effect of closure of the zero velocity curves at SL after the deployment of a reflective area. The equations of motion are given in the Circular Restricted 8

9 y y The 6 th Workshop on Astrodynamics and Flight Mechanics 16, Sagamihara, Japan SL 1 Earth SL SL 1 Earth SL Disposal (P 1 ) x x (a) Unstable manifold: the black (b) The point P 1 represent the deployment of EOL device (black line represents an Earth encounter trajectory after 9.5 years. star). The trajectory evolution is in dashed line. Fig. 17: EOL disposal concept. Image from Ref [16]. Fig. 18: Deployable area required to enhance the EOL disposal manoeuvre for Hershel, SOHO and Gaia spacecraft. Three-Body Problem (CRBP) in the synodic (rotating) reference frame with dimensionless coordinates as given in [17]. The energy of the spacecraft is an integral of motion defined as: E = 1 v 1 (x + y (1 µ) µ ) (1 β), () r Sun p r Earth p where µ is the gravitational constant for the Sun-Earth system, v the module of the spacecraft velocity, r Sun p and r Earth p are respectively the distance of the spacecraft with the Sun and the Earth with respect to the Sun-Earth center or mass. β is the lightness number that takes into account of the Solar Radiation Pressure (SRP) effect of the Sun and it is function of the area-tomass ratio of the satellite together with its reflectivity properties [1]. For a near-perfect surface (C R = ), β is given by: (a) Cone Sail. β = A m σ, (1) where A and m are the area and the mass of the satellite while σ is the Sun s luminosity (1.5 1 kg/m ) [1]. Eq. () holds true when the normal of the reflective surface is aligned with the Sun-Spacecraft line direction (Sun-pointing). The minimum area required to perform the SRP EOL manoeuvre [16] is computed by finding the minimum of: E(x SL, β min ) = E(x, β min ). () In Eq. (), the right-hand side is associated to the energy at SL while the left-hand side of the equation correspond to the energy of the spacecraft. The condition in Eq. () is visualised as in Figure 17(b). Figure 18 summarizes the required deployable area needed to enhance the EOL disposal manoeuvre. Note that spacceraft like Gaia requires the minimum deployable area Deployable EOL Devices In [16], the minimum area-to-mass ratio required along the unstable manifold associated to the LPO (gray tube in Figure 17(a)) was found to be.15 m /kg. To guarantee the closer of the zero velocity curves after the deployment, the effective area illuminated by the Sun has to be equal or above this value. As at the EOL passive control stabilisations is required, a deployable structure that is passively stabilised to maintain the (b) Truncate Cone Sail. Fig. 19: EOL passively stabilised devices: cone sail and truncate cone sail. Sun-pointing condition for several years is investigated. Following the results of [5], a cone sail is chosen as a possible EOL device candidate. In this case, two options can be followed: a cone sail is deployed through a boom and attached to the spacecraft or directly deployed for example from the spacecraft sunshade following a truncated cone shape. Gaia spacecraft shows to be ideally appropriate for an attached truncated cone shape due to a circular shape of the sunshade. Gaia spacecraft is here taken as mission scenario for the EOL device sizing. Gaia spacecraft has a dry mass (at the EOL) of 19 kg that requires a total area of A min = m. Figure 19 shows the two configurations we are going to consider. The concave side is chosen to be towards the Sun because, as shown in [5], a concave sail is always a stable configuration under the general SRP model [18]. We considered a cone sail with kapton substrate (7.1 g/m of density) with surface coating in aluminium (1.5 g/m in 9

10 density). The correspondent weight of the cone sail for Gaia spacecraft is 1.59 kg. If we impose that the base of the cone has the same radius of the sunshade of Gaia of 4.68 m the final height of the cone is given by: h c = 1 r ( ) Amin r 4 = m. () π A toroidal boom with radios 4.68 m is selected to keep the cone in shape which have a density of -7 g/m. The boom has a perimeter of 9.5 m that correspond to a mass of kg. An additional mass of boom that connect the apex of the cone and the sunshade of the satellite has to be added accordingly for passive stabilisation as shown in Figure 19(a). The height and the radius of the cone sail and the length of the connecting boom can be optimized to reach the most stable passive stabilisation configuration. This can be found through the general SRP model [18]. The total mass is thus kg (+ the mass of a connecting boom). For the case of a truncate cone sail directly deployed from the spacecraft sunshade, an initial area A which correspond to the sunshade of the spacecraft contribute to the overall area-to-mass ratio required for the deployment. This initial area corresponds to the minor base of the truncate cone to be deployed as shown in Figure 19(b). Thus, A corresponds to 69 m (for a sunshade with radius 4.68 m). The major base of the truncated cone is selected such as R = 1.5 r that correspond to 7. m. The height of the truncated cone can be computed as: A min h tc = π (R + r) (R r) =. m. (4) The mass of the truncate cone sail lateral surface (in white in Figure 19(b)) is 1.5 kg. As for the cone sail, the major base can be kept in shape through a toroidal support ring as for reflectors and antennas. A boom of perimeter m is here required with correspondent mass of kg. The mass of the deployable device is kg. This solution is lighter than the previous option as an additional mass for a boom that connects the spacecraft and the cone sail should be taken into account. This suggests that the design and the shape of the sunshade should be carefully designed for future spacecraft in LPO to accommodate a passively stabilised EOL device. 6. Conclusions In this paper, the use of Solar Radiation Pressure (SRP) for the design and control of trajectories in the restricted three-body problem of the Sun-Earth system is explored and it has identified: a control law for the station-keeping of spacecraft in libration point orbits; a methodology to transfer between quasi-periodic orbits, and an end-of-life strategy to safely dispose of the spacecraft into a graveyard orbit around the Sun. This paper has also sought to determine where it is possible to control a spacecraft in an LPO by using solar radiation pressure manoeuvres from the beginning to the end of the mission and, consequently, to derive the structural drivers in term of: reflective area required, and spacecraft pointing accuracy. Solar radiation pressure assisted missions have been shown to be feasible for the design and control of trajectories of LPOs from the beginning to the end of the spacecraft lifetime. SRP is a natural and unlimited source of propellant. Due to this unlimited propellant, space missions that use SRP have longer mission lifetimes, potentially decreasing the number of spacecraft launched to LPOs. Reducing the number of launches will reduce the Space Agencies overall cost budget allocated for LPOs missions. A reduced number of spacecraft at LPOs will also make the space market more sustainable by limiting the potential space debris in the vicinity of the libration points. Furthermore, the single space mission cost budget is highly related to the mass of the spacecraft, where the pre-storage of onboard propellant has a significant impact on the overall single mission cost. Thus, a SRP device has the key advantage of reducing the costs relating to the propulsion system. These facts provide evidence that underlines the importance to further research into solar radiation pressure enhancing devices for LPO missions which is needed to make this technology economically accessible. Acknowledgments S. Soldini was founded by the Japan Society for the Promotion of Science grant. S. Soldini would like to acknowledge her PhD supervisors Dr C. Colombo and Dr S. Walker for their advices in the HSP control and EOL manoeuvre design as part of her PhD thesis (Southampton, UK) and Prof. J. Masdemont and Prof. G. Gómez for their supervision in the transfer trajectory design while she held an ESR position in the Astronet II training network (Barcelona, ES). 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