Lissajous Motion Near Lagrangian Point in Radial Solar Sail

J. Astrophys. Astr. (2018) 39:72 © Indian Academy of Sciences https://doi.org/10.1007/s12036-018-9563-0

Lissajous motion near Lagrangian point L2 in radial solar sail

ARUN KUMAR YADAV1,∗ , BADAM SINGH KUSHVAH1 and UDAY DOLAS2

1 Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand 826004, India. 2C. S. A. Govt. P. G. College, Sehore, M.P., India. ∗Corresponding author. E-mail: [email protected]

MS received 5 June 2018; accepted 9 September 2018; published online 28 November 2018

Abstract. An attempt was made to study the dynamics close to the collinear libration point L2 of the radial solar sail circular-restricted three-body problem (RSCRTBP) in the Sun–Jupiter System, where the third massless body is a solar sail. We analyse the qausi-periodic (Lissajous solutions) orbits about the libration point L2.The Lindstedt–Poincaré approximation for the qausi-periodic orbits was used for numerical simulations. We utilized linear quadratic regulator (LQR) to stabilize the full nonlinear model, and linear state-feedback controller was designed to stabilize the trajectory.

Keywords. Solar sail—Lissajous orbit—feedback control—linear quadratic regulator (LQR).

1. Introduction around the Sun, and the Interstellar Heliopause Probe is a mission to measure the interstellar medium. The The idea of sailing in space has been around for years: lightness number for these missions (McInnes 1993) in the 1970s NASA considered using Sun-powered sails are β = 0.02 for Geostorm, β = 0.1 for the solar polar as a potential impetus framework for playing out a orbiter, β = 0.3−0.6 for the interstellar Heliopause meet with comet Halley during its 1986 flyby. Solar Probe and β = 0.001 for IKAROS (Tsuda et al. 2013). sail spacecrafts use radiation pressure on a reflective Lissajous orbits around the libration point L2 in the sail that can be oriented relative to the Sun-line for Sun–planet system have attracted increasing interest propulsion. With the IKAROS interplanatery solar sail because of the advantage that the spacecrafts near L2 mission (Tsuda et al. 2013) and the upcoming launch have an almost unobstructed view of celestial objects of near-asteroid scout in 2018, such crafts have become and are not influenced by geomagnetic and atmospheric a reality. In solar sail technology, Macdonald and forces (Shahid & Kumar 2010). L2 libration point is a McInnes (2011) discussed the example of near-term, good location for a space telescope, and this keeps the mid-term and far-term solar sail missions. Near-term temperature of instrument exceptionally stable (Li et al. solar sail advancement is through coupling of currently 2016). Yuan et al. (2018) investigated periodic orbits of mature solar sails. Mid-term solar sail can be used solar sail equipped with reflectance control device in the to deliver a spacecraft to a true solar polar orbit in Earth–Moon system. Grøtte and Holzinger (2017) stud- approximately 5 years (Goldstein et al. 1998; Macdon- ied solar sail equilibria with albedo radiation pressure ald et al. 2006). Moreover, far-term missions such as an in the circular-restricted three-body problem. Moreover, asteroid rendezvous/sample return require sails with an Hu et al. (2014) discussed the attitude stability criteria areal density of 5–6 g/m2 and films with a thickness of of axisymmetric solar sail. Sood (2012) discussed, in his approximately 1–2 μm(Garner et al. 1999). The Solar thesis, solar sail applications for mission design in the Polar Orbiter (SPO) mission is a good example of the Sun–planet systems from the perspective of the circu- type of high-energy inner-solar system mission enabled lar restricted three-body problem. Verrier et al. (2014) by solar sail propulsion. Geosail is a mission to put a evaluated a halo family in the radial solar sail circular- sunlight based sail in Earth’s Magnetosail; solar polar restricted three-body problem. Furthermore, Farquhar orbiter is a mission to put a solar sail in a polar orbit and Kamel (1973) studied the quasi-periodic orbits 72 Page 2 of 9 J. Astrophys. Astr. (2018) 39:72

Figure 1. System model for radial solar sail in Sun–Jupiter system. about the translunar libration point. Farrés and Jorba 1, the Sun–Jupiter distance is 1 and period of its orbit (2010b) computed periodic and quasi-periodic motions is 2π. We made use of a synodic reference system with of a solar sail close to SL1 in the Earth–Sun system the origin at the centre of mass of the Sun–Jupiter sys- and Waters and McInnes (2007) calculated the periodic tem such that the Sun and Jupiter are fixed on the x-axis orbits above the ecliptic in the solar-sail-restricted three- at (−μ, 0, 0) and (1 − μ, 0, 0) .Thez-axis is perpen- body problem. Simo and McInnes (2009) calculated the dicular to the ecliptic plane and the y-axis finishes an solar sail orbits at the Earth–Moon libration points. Far- orthogonal positive orientated reference framework as rés and Jorba (2010a) studied the dynamics of a solar shown in Fig. 1. To study the dynamics of solar sail, we sail near a halo orbit. Tsirogiannis et al. (2006) studied defined the lightness number β as the ratio of solar radia- the computation of the Liapunov orbits in the photo- tion pressure to the Sun’s gravitational acceleration. The gravitational RTBP with oblateness. lightness number β is a dimensionless constant which In this study, we investigated the Lissajous orbit was used to parametrize the solar sail. The efficiency of around the collinear libration point L2 for the radial a solar sail can be parameterized through the lightness solar sail circular-restricted three-body problem (RSCR number β and defines as McInnes (1993) TBP) in the Sun–Jupiter system and stablized the trajectory of system using linear quadratic regulator (LQR) Lm β = (1) scheme. This paper is organized as follows: Section 2 2πGM Ac describes the formulation of system model. In Section 3, we analyse the Lissajous orbit about the collinear libra- where L is the luminosity of the Sun, M is the mass tion points. In Section 4, we compute the gain matrix of Sun, m is mass of the spacecraft solar sail, A is area of using linear quadratic regulator on a closed-loop sys- solar sail, G is the gravitational constant and c is speed of tem. In Section 5, numerical simulations and results are light. Equation (1) is valid for perfectly reflecting solar discussed. Section 6 presents our conclusions. sail. β = 1 means that the force of gravity and the solar- based radiation pressure on the sail are equivalent, while β>1 means that the solar radiation force dominates 2. System model and β<1 means that the force of gravity dominates. The acceleration because of the solar radiation pres- To depict the dynamics of a solar sail in the Sun–Jupiter sure is in a direction normal to the surface of the sail system, we considered the CRTBP including the solar and given as radiation pressure due to solar sail. We assumed that ( − μ) the Sun and Jupiter are point masses moving in circu- = β 1 ( ˆ . )2 asail 2 rS n n (2) lar orbits about their common centre of mass, and that rS the solar sail is a massless body that is influenced by the gravitational attraction of the two bodies and solar where rS is the vector between the solar sail and the radiation pressure. Let mS and m J be the masses of the Sun, rS = | rS |. Equations of motion of the radial solar Sun and Jupiter respectively; then in normalized units, circular restricted three body problem in compact form μ = mS/(mS + m J ) is the mass of the Sun and (1 − μ) are given as is the mass of the Jupiter. Units of mass separation and time were such that the aggregate mass of the system is r¨ + 2ω × r˙ =∇U +asail (3) J. Astrophys. Astr. (2018) 39:72 Page 3 of 9 72

Table 1. Position of collinear libration points for radial solar sail CRTBP in the Sun–Jupiter system.

Collinear libration points Dimensionless Unit (x) Dimensionless Unit (y)

L1 0.93120300 0 L2 1.06672345 0 L3 − 0.99376542 0 where ω is a rotational axis of the synodic reference 3. Lissajous solution about collinear libration frame and potential U is given as follows: poiints (1 − μ) μ 1 U = + + |ω ×r |2, (4) As we know the libration points are equilibrium solution rS rJ 2 for the system as shown as in Table 1. So, we linearized where rJ =|r J | andrS =|r S |, rS = the equation of motion around these points and solved 2 2 2 2 2 2 (x+μ) +y +z , rJ = (x + μ − 1) + y + z for a periodic solution around the equilibrium point. The are the Sun–sail and Jupiter–sail distances respectively periodic nature of solution can be seen by considering and nˆ = (nx , ny, nz). linearized form of equations and given as For radial direction nˆ = (1, 0, 0), Equation (4) can x¨ − 2y˙ − (1 − 2c )x = 0, be rewritten as 2 ¨ + ˙ + ( − ) = , ( − μ)( − β) μ y 2x c2 1 y 0 (11) = 1( 2 + 2) + 1 1 + . U x y (5) z¨ + c2z = 0. 2 rS rJ For radial direction nˆ =ˆr =[1, 0, 0], equations of The solution of the characterstic equation for the S − motion in the synodic reference system are x y (in-plane) motion has two real and two imagi- nary roots. The x − y solution can be expressed in the β( − μ) ¨ − ˙ = + 1 form (Richardson 1980) x 2y Ux 2 rS (6) x = A1 cos(λt) + kA2 sin(λt), (12) y¨ + 2x˙ = Uy z¨ = Uz y =−kA1 sin(λt) + kA2 cos(λt), (13) where U , U and U are the first-order partial x y z where A and A are arbitrary constants. Out-of-plane derivatives of potential U, given as 1 2 linearized motion is a simple harmonic, i.e. (1 − μ)(1 − β)(x + μ) μ(x − 1 + μ) = − − , =− (ν ) + (ν ) Ux x 3 3 z B1 sin t B2 cos t (14) rS rJ (7) where B1 and B2 are arbitrary constants. The linearized (1 − μ)(1 − β)y μy solutions for the family of periodic orbit are called Lis- U = y − − , (8) y 3 3 sajous orbits and are given by the equations rS rJ ( − μ)( − β) μ =− (λ + ), = 1 1 z − z . x Ax sin t Uz 3 3 (9) rS rJ y =−Ay cos(λt + ), (15) z = Az sin(λt + ), 2.1 Collinear libration points where Ax , Ay = kAx and Az are the three amplitudes, λ ν To find the equilibrium point the velocity and and are the frequencies for the in-plane and the out- acceleration concerning the synodic reference outline of-plane motion and , are the initial phases for the must be zero. We equate to zero the first and second in-plane and the out-of-plane motions, respectively. The derivatives of the co-ordinates relating to equilibrium linearized motion will become quasi-periodic if the in- plane and out-of-plane frequencies λ and ν respectively points. Equation (6) are evaluated for these conditions: λ are such that is generally irrational. The parameter β( − μ) ν 1 = 2λ Ux + = 0, Uy = 0, Uz = 0(10)k λ2+ − and c2 is taken from the expression, for r 2 1 c2 S L2, 72 Page 4 of 9 J. Astrophys. Astr. (2018) 39:72 ( − β)( − μ)γ n+1 e e e 1 n n 1 1 The expression for Uxx, Uyy and Uzz at the collinear cn = (−1) μ + (−1) γ 3 (1 − γ)n+1 points are given as follows: (16) (1 − μ)(1 − β) μ γ U e = 1 − − where is the distance from closed body to libration xx r 3 r 3 points. S J ( − μ)( − β)( + μ)2 +3 1 1 x 5 rS 4. Linearization near the collinear Lagrangian 3μ(x − 1 + μ)2 − , (23) points 5 rJ (1 − μ)(1 − β) μ Given the collinear libration points shown in Table 1 of U e = 1 − − yy r 3 r 3 the nonlinear system re = (xe, 0, 0), we present small S J perturbation such that 3(1 − μ)y 3μy2 + − , (24) r 5 r 5 x = xe + ξ S J (1 − μ)(1 − β) μ y = η U e =− − zz r 3 r 3 z = ζ (17) S J 3(1 − μ)(1 − β)z2 3μz2 + − . (25) Substituting Equation (17) into Equation (6), we obtain r 5 r 5 the variational equations S J ξ¨ − η˙ = e ξ + , The LQR controller is developed to stablize the 2 Uxx uξ ˙ e nonlinear system in the neighborhood of collinear η¨ − 2ξ = U η + uη, (18) yy libration points. Then, we applied a linear feedback con- ζ¨ = e ζ + . Uzz uζ trol u =−KX to Equation (18) that minimizes the quadratic cost e , e e where Uxx Uyy and Uzz denote the second-order partial derivatives of the potential function U at the collinear ∞ 1 libration points, and minJ = (X T QX + uT Ru)dt, (26) ⎡ ⎤ 2 0 ⎡ ⎤ β(1−μ) uξ r2 ⎣ ⎦ ⎢ S ⎥ where the matrices Q and R are weights of the state and u = uη = ⎣ 0 ⎦ (19) control, which are symmetric positive semi-definite and uζ 0 the weighted matrices Q and R are free to be chosen. −1 T In state-space form, we can write Equation (6)as We obtain the gain matrix K = R B P by solving algebraic Riccati equation: X˙ = AX + Bu (20) T + − −1 T + = . where X = (ξ,η,ζ,ξ,˙ η,˙ ζ)˙ A P PA PBR B P Q 0 (27) ⎡ ⎤ 000100 The closed-loop system is given as ⎢ ⎥ ⎢ 000010⎥ ⎢ ⎥ ⎢ 000001⎥ ˙ = ( − ) . A = ⎢ e ⎥ (21) X A BK X (28) ⎢Uxx 00020⎥ ⎣ 0 U e 0 −20 0⎦ yy The necessary and sufficient condition for the collinear 00U e 000, ⎡ ⎤ zz equilibrium points to be linearly stable is that the real 000 part of the eigenvalues of the matrix A− BK are all less ⎢ ⎥ ⎡ ⎤ ⎢000⎥ than or equal to zero (Biggs & McInnes 2010). We took ⎢ ⎥ uξ ⎢000⎥ ⎣ ⎦ the weighting matrices Q and R and then computed the B = ⎢ ⎥ and u = uη (22) ⎢100⎥ gain matrix K and the eigenvalue of the matrix A − BK ⎣ ⎦ uζ 010 in three different cases. Consider the case I, Q = I6×6 3 001 and R = 10 I3×3, then the gain matrix K is given as J. Astrophys. Astr. (2018) 39:72 Page 5 of 9 72 ⎡ ⎤ 259.2387 −0.8050 0 22.3186 2.9353 0 K = ⎣ 39.8745 5.2331 0 2.9353 3.6868 0 ⎦ 003.3570 0 0 2.5913 and the eigenvalues of the matrix A − BK are given as, −11.3555, −113250, −1.6755, −1.6494, −1.2856, −1.3057. In Case II, we considered Q = I6×6 and R = 10 I3×3 and obtained the gain matrix as

⎡ ⎤ 259.2508 −0.8097 0 22.3226 2.9298 0 K = ⎣ 39.8375 5.2509 0 2.9298 3.7064 0 ⎦ 003.3863 0 0 2.6216 and the eigenvalues of the matrix A − BK are given as, −11.4938, −11.1888, −1.8037, −1.5428, , −1.2096 −1.4119. In Case III, we considered Q = 10 I6×6 and R = I3×3 then compute gain matrix as

⎡ ⎤ 260.1069 −1.1597 0 22.6617 2.5453 0 K = ⎣ 37.1416 6.6569 0 2.5453 5.1977 0 ⎦ −005.2584 −004.5296 and the eigenvalues of the matrix A − BK are given as −12.9682, −9.9186, −1.10359, −3.8668, −1.0201, −3.5094. From the above, it is clear that the six eigenvalues of the matrix A− BK in all three cases are negative numbers. Hence, the collinear Libration points are linearly stable because every eigenvalue is less than or equal to zero.

(a)(b) x 104 1.5 x 104

5 1

0.5

0 L L 2 0 2 z(km) y(km)

−0.5 −5 2 1 1 −1 0 0.5 4 0 x 10 −1 4 −0.5 x 10 −1.5 −2 y(km) −1 −6000 −4000 −2000 0 2000 4000 6000 x(km) x(km) (c) (d) x 104 5 6000

4000

2000

L L 2 0 2 0 x(km) z(km)

−2000

−4000

−5 −6000 −1.5 −1 −0.5 0 0.5 1 1.5 −5 0 5 y(km) z(km) 4 x 104 x 10

Figure 2. (a) Three-dimensional Lissajous orbit around the collinear libration point L2;(b) Projection of Lissajous orbit in xy-plane around the collinear libration point L2;(c) Projection of Lissajous orbit in yz-plane around the collinear libration point L2;(d) Projection of Lissajous orbit in xz-plane around the collinear libration point L2. 72 Page 6 of 9 J. Astrophys. Astr. (2018) 39:72

Table 2. Time period of orbit with regards to lightness num- orbit in xy-plane around the collinear libration point β ber ( ). L2. Figure 2(c) shows the projection of Lissajous orbit in yz-plane around the collinear libration point L . β 2 Lightness number ( ) Time Period Figure 2(d) shows the projection of Lissajous orbit (Dimensionless Unit) in xz-plane around the collinear libration point L2. β = 0.00 2.97279 We numerically computed the time period of orbit β = 0.01 2.97696 around L2 and obtained with the effect of solar sail β = 0.02 2.98114 (β = 0.01), the time period is 2.97696 which is β = 0.03 2.98535 greater than the classical case (β = 0). For the clas- β = 0.04 2.98957 sical case (β = 0), the time period is 2.97279. As β = 0.05 2.99381 increasing value of lightness number β, the orbital period of orbit increases (as shown in Table 2). We obtained the time series plots of three Lissajous positions states around L2 in 40 days shown in Fig. 3(a)–(d), 5. Numerical simulation results while Fig. 4(a)–(d) depicts the time series plots of three Lissajous velocities states around L2 in 40 days. In this section, a numerical simulation was performed Figure 4(d) shows that velocity changes as time increa- about the collinear libration point L2. Lindstedt– ses. Furthermore, we have used the linear quadratic Poincaré linear order approximation was used to obtain regulator (LQR) method to stabilize the trajectory. For the initial guess, and using differential correction the computation of gain matrix and stabilized trajec- scheme, we found the initial condition for Lissajous tories, we took the weight matrices P = I6×6 and orbit; the time of simulation was 40 days. R = 10 I3×3. We obtained stabilized 3-D trajec- We computed Lissajous orbits around collinear tory and projection of stablized trajectory in the xy- μ = . Libration L2 point with mass ratio 0 000954 and xz-planes using LQR as shown in Figs. 5, 6, and lightness number β = 0.01 along with 5600 km and 7 respectively. All the numerical simulations were amplitude. Figure 2(a) shows the three-dimensional done using MATLAB/Simulink to design a linear Lissajous orbit around the collinear libration point state-feedback controller and LQR for the nonlinear L2. Figure 2(b) shows the projection of Lissajous dynamical system.

(a)(b) x 104 6000 1.5

4000 1

2000 0.5

0 0 x(km) y(km)

−2000 −0.5

−4000 −1

−6000 −1.5 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Time (days) Time (days) (c) (d) x 104 x 104 5 5 x y z

0 0 z(km) Positions

−5 −5 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Time (days) Time (days) Figure 3. (a) Time series plot of Lissajous position state x(t) in time of 40 days. (b) Time series plot of Lissajous position state y(t) in time of 40 days. (c) Time series plot of Lissajous position state z(t) in time of 40 days. (d) Combinations of Lissajous positions in time of 40 days. J. Astrophys. Astr. (2018) 39:72 Page 7 of 9 72

(a)b)( x 104 x 104 1.5 3

1 2

0.5 1

0 0 x dot y dot

−0.5 −1

−1 −2

−1.5 −3 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Time (days) Time (days) (c) (d) x 104 x 104 x dot 8 8 y dot z dot 6 6

4 4

2 2

0 0 z dot Velocities −2 −2

−4 −4

−6 −6

−8 −8 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Time (days) Time (days) Figure 4. (a) Time series plot of Lissajous velocity state x˙(t) in time of 40 days. (b) Time series plot of Lissajous velocity state y˙(t) in time of 40 days. (c) Time series plot of Lissajous velocity state z˙(t) in time of 40 days. (d) Combinations of Lissajous velocities in time of 40 days.

0.1

−0.1

z(km) −0.2

−0.3

−0.4 2 8 0 6 4 x 10−3 −2 2 0 −4 −2 x 10 −4 −4 y(km) x(km) Figure 5. Stabilized 3-D trajectory using LQR.

x 10−3 0.5

−0.5

−1

−1.5 y(km)

−2

−2.5

−3

−3.5 −4 −2 0 2 4 6 8 x(km) −4 x 10 Figure 6. Projection of stabilized trajectory in the xy-plane using LQR. 72 Page 8 of 9 J. Astrophys. Astr. (2018) 39:72

0.05

−0.05

−0.1

−0.15

z(km) −0.2

−0.25

−0.3

−0.35

−0.4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 x(km) x 10−3 Figure 7. Projection of stabilized trajectory in the xz-plane using LQR.

6. Conclusions Farquhar, R. W., Kamel, A. A. 1973, Celestial Mechanics 7(4), 458 In this study, we took the model radial solar sail Farrés, A., Jorba, À. 2010a, Acta Astronautica 67(7), circular-restricted three-body problem (RSCRTBP) in 979 the Sun–Jupiter system. We have investigated the Lis- Farrés, A., Jorba, À. 2010b, Celestial Mechanics and Dynam- sajous orbit around L point for the lightness number ical Astronomy 107(1), 233 2 Garner, C., Diedrich, B., Leipold, M. 1999, A summary fo β = 0.01 in the Sun–Jupiter system. Simulation results solar sail technology developments and proposed demon- have been demonstrated as the time series plots of three stration missions Lissajous positions and velocities state about the libra- Goldstein, B. E., Buffington, A., Cummings, A. C., Fisher, tion point L2. On considering the lightness number, the R. R., Jackson, B. V., Liewer, P. C., Mewaldt, R. A., time period of orbit for solar sail increases compared to Neugebauer, M. 1998, in: Missions to the Sun II, Vol- the classical case β = 0. LQR was used to stabilized ume 3442, p. 65. International Society for Optics and trajectory using a technique called state-feedback con- Photonics trol mechanism. We have also found the stabilized 3-D Grøtte, M. E., Holzinger, M. J. 2017, Advances in Space trajectory and projections on the xy- and xz-planes for Research 59, 1112. https://doi.org/10.1016/j.asr.2016.11. the model. We have chosen different weight matrices 020 Q and R to acquire gain matrix, and with the help of Hu, X., Gong, S., Li, J. 2014, Advances in Space Research gain matrix, we conclude that the collinear points are 54, 72. https://doi.org/10.1016/j.asr.2014.03.008 Li, M., Peng, H., Zhong, W.2016, Nonlinear Dynamics 83(4), linearly stable. The LQR based controller is useful for 2241 fuel-efficient station-keeping mission around the libra- Macdonald, M., McInnes, C. 2011, Advances in Space tion point orbit. Research 48(11), 1702 Macdonald, M., Hughes, G., McInnes, C., Lyngvi, A., Acknowledgements Falkner, P.,Atzei, A. 2006, J. Spacecraft and Rockets 43(5), 960 McInnes, C. R. 1993, J. Spacecraft and Rockets 30, 782. We are thankful to the Science and Engineering https://doi.org/10.2514/3.26393 Research Board, Government of India, for providing Richardson, D. L. 1980, Celestial Mechanics and Dynamical financial support through SERB research project No.- Astronomy 22(3), 241 EMR/2016/001145. Shahid, K., Kumar, K. 2010, J. Spacecraft and Rockets 47(4), 614 Simo, J., McInnes, C. R. 2009, Communications in References Nonlinear Science and Numerical Simulation 14(12), 4191 Biggs, J. D., McInnes, C. R. 2010, J. Guidance, Control, Sood, R. 2012, Solar sail applications for mission design in Dynamics 33(3), 1017 sun-planet systems from the perspective of the circular J. Astrophys. Astr. (2018) 39:72 Page 9 of 9 72

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