Formation flying solar-sail gravity tractors in displaced ... - Springer Link

Celest Mech Dyn Astr (2009) 105:159–177 DOI 10.1007/s10569-009-9211-8 ORIGINAL ARTICLE

Formation flying solar-sail gravity tractors in displaced orbit for towing near-Earth asteroids Shengping Gong · Junfeng Li · Hexi BaoYin

Received: 22 September 2008 / Revised: 31 March 2009 / Accepted: 7 April 2009 / Published online: 2 June 2009 © Springer Science+Business Media B.V. 2009

Abstract Several methods of asteroid deflection have been proposed in literature and the gravitational tractor is a new method using gravitational coupling for near-Earth object orbit modification. One weak point of gravitational tractor is that the deflection capability is limited by the mass and propellant of the spacecraft. To enhance the deflection capability, formation flying solar sail gravitational tractor is proposed and its deflection capability is compared with that of a single solar sail gravitational tractor. The results show that the orbital deflection can be greatly increased by increasing the number of the sails. The formation flying solar sail gravitational tractor requires several sails to evolve on a small displaced orbit above the asteroid. Therefore, a proper control should be applied to guarantee that the gravitational tractor is stable and free of collisions. Two control strategies are investigated in this paper: a loose formation flying realized by a simple controller with only thrust modulation and a tight formation realized by the sliding-mode controller and equilibrium shaping method. The merits of the loose and tight formations are the simplicity and robustness of their controllers, respectively. Keywords Asteroid deflection · Solar sail · Formation flying · Gravitational tractor · NEO deflection–displaced orbit near an asteroid

List of symbols µs Gravitational constant of the Sun µa Gravitational constant of the asteroid n a Average angular velocity of the asteroid around the Sun S. Gong · J. Li · H. BaoYin (B) School of Aerospace, Tsinghua University, 10084 Beijing, China e-mail: [email protected] S. Gong e-mail: [email protected] J. Li e-mail: [email protected]

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r0 r ri r˙i

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Vector pointing from the Sun to the asteroid Vector from the asteroid to the sail Vector from the asteroid to the i-th sail Velocity in the rotating frame Velocity in the inertial frame Lightness number of the i-th sail Normal vector of the i-th sail Radius of the displaced orbit Displacement of the displaced orbit Angle between the displaced orbit direction and the Sun-asteroid line Angular velocity of the Earth around the Sun Number of the individual in the formation flying

1 Introduction The methods of NEAs (Near Earth Asteroids) deflection can be classified as HEIMs (High Energy Impulse Methods) and LDLTMs (Long Duration Low Thrust Methods). There are usually two ways of implementing HEIMs: striking at the Asteroid at high relative velocity and a stand-off nuclear blast explosion. There are several kinds of LDLTMs being discussed in literature, such as “rendezvous and push” methods, surface ablation of the object using a laser or solar concentrator, exploitation of solar flux induced perturbations, mass driver, space tug and non-contact gravitational tractor. The typical representatives of HEIMs and LDLTMs are direct impact and “rendezvous and push” methods. Izzo et al. (2005) compares the direct impact method and “rendezvous and push” method. Since the resulting perturbation is due to the reaction principle stated by Newton’s third law, one might think of the impact case as being able to expel at once the entire final spacecraft mass with a relative high velocity and all spacecraft mass is used as reaction mass. The formula tells us that any mass expelled after a time from the deflection start contributes increasingly less to the miss-distance. Therefore, the direct impact method is superior to “rendezvous and push” method in theory (Vasile and Colombo 2008). However, the optimum direction impact is impractical with the consideration of transfer trajectory and the time to impact and the actual achievable miss-distance is different from the optimum value. Ahrens and Harris (1992) present several methods of deflection, including the deflections by nuclear explosion radiation and by surface nuclear explosion. Both methods utilize the energy released by the nuclear explosion to eject the mass of the asteroid, which will disturb the velocity of the asteroid. For the radioactive stand-off explosion, velocity change of 1 cm/s for an asteroid of diameter 100 m, 1 km and 10 km will require the energy about 0.01–0.1 kton, 0.01–0.1 Mton and 0.01–0.1 Gton, respectively. It is mentioned that the method is more effective to deflect a small asteroid because the required escape velocity of the ejection is much larger for larger asteroid. McInnes (2004) considers impacting the asteroid with a solar sail, which can perform a head-to-head impact when solar sail evolves on a retrograde orbit. The relative velocity of impact can reach as high as 60 km/s, the release energy of which is comparable with that of the nuclear explosion. Melsoh (1993) proposes a creative strategy in which solar sail is used to focus sunlight onto the surface of the asteroid to generate thrust as the surface’s layers vaporize. According the formula given in the paper, a 0.5 km solar sail collector operating for a year can deflect an asteroid up to 2.2 km in diameter, which is much more effective compared with other strategies. Joseph (2002) proposes another asteroid hazard mitigation method using Yarkovsky effect. The requirements

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of long lead time and changing the diurnal thermal wave of the asteroid makes the method impractical. For the “rendezvous and push” method, the pusher has to be attached to the surface of the asteroid, where the attraction is weak and the surface characteristics are unknown. Besides, the self-rotation of the asteroid will influence the implementation of the thrust. For the direct impact method, the chance of success is dependent on the composition of the asteroid. Barerock and porous regolith will generate completely different results for the same size asteroids. The nuclear explosion may produce space nuclear pollution and may also generate some small fragmentations that threaten the Earth in another way and creates orbital debris on a solar system scale. The solar collector seems very attractive with the orbital deflection distance and working time considered. However, the large solar collector has to be delivered to the asteroid first and extra station keeping propulsion system is necessary to keep the collector properly orientated. Besides, the method is also very sensitive to the material composition and shape of the asteroid. In a word, all the deflection methods discussed above are dependent on the structure or material composition or shape or self-rotation of the asteroid. Lu and Love (2005) proposes the gravitational tractor concept, since the gravity tractor never makes physical contact with the asteroid, the asteroid’s composition, structure, spin rate have less influences on the mission compared with other methods. Nor would it require new technology, since ion engines have already been used on a number of spacecraft, such as Deep Space 1. However, the spacecraft thruster should be canted to keep the exhaust plume from the asteroid surface, which would reduce the efficiency of the tractor because the net towing force is reduced. To cancel the restriction of exhaust plume, McInnes (2007) considers placing the tractor on a displaced orbit, and all the propellant can be utilized as long as the radius of the displaced orbit is larger than that of the asteroid. The results show that a 20-tonne gravitational tractor hovering for 1 year can deflect a typical asteroid of about 200 m diameter given a lead time of roughly 20 years. Deflecting a larger asteroid would require a heavier spacecraft, or more time spent hovering, or more lead time. For a long time thrusting, the tractor needs plenty of propellant to hover above the asteroid. About 7 ton propellant is required for a year hovering with the specific impulse of 450 s and thrust of 1 N. The mass reduces to about 1 ton if the specific impulse of the propellant is 3,000 s. For deflections of larger asteroids, the mass requirement will exceed present launch capacity for a single tractor, especially for urgent deflection mission. The formation flying gravitational tractor that handles a swarm of several small spacecraft or solar sails can provide a possible solution for large asteroids. Solar sail with specific impulse tending to infinite in theory can hover for a time as long as the life span of the sail. Wie (2007) discusses the orbital deflection of Apophis using a solar sail gravitational tractor. The gravity of the asteroid is cancelled by the solar radiation pressure force and only the attitude control of sail is required, which consumes about several kilograms propellant a year. The weakness of gravitational tractor is the relatively low efficiency compared with the nuclear explosion. Consequently, gravitational tractor is a method that needs more lead time or larger mass but is less dependent on characteristics of the asteroid. In Wie’s presentation on the first solar sailing symposium, an advanced concept, tethered solar sail, that has the similar thought as the space tow proposed by Greschik (2006) and Greschik and Edward (2007) is proposed to increase the deflection capability. The space tow has many advantages over regular solar sail, but it also faces new challenges such as attitude control. SSFFGT (formation flying solar sail gravitational tractor) around the asteroid combining a swarm of solar sails is a good option to deflect large asteroid or increase the orbital deflection. Solar sail formation flying around displaced solar orbit and planetary displaced orbit has been investigated in Gong et al. (2007a, b, c). The dynamics and control of the relative motion around the asteroid are similar, but different from those of the

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Earth-centered displaced orbit because the gravity field is much weaker and the relative distance is much smaller. Collision avoidance is a key issue for the SSFFGT around the asteroid. Equilibrium shaping was an algorithm originally used to automatically assemble collective robotics. Only relative measurements but no mutual communications are needed to guide the individual to avoid collisions. The theory and the applications of the algorithm to satellite swarm are presented in Gazi (2003), Gazi and Passino (2004), and Izzo and Pettazzi (2007), respectively. Broschart and Scheeres (2005) investigates the control of hovering spacecraft near small bodies with the assumption of a uniform gravitational field for the asteroid. However, the gravitational field of the asteroid is dependent on its shape and rotation, which can be determined through near-Earth asteroid rendezvous mission (Miller et al. 2002). Close-orbit dynamics around an irregularly shaped, non-principal axis rotator is investigated (Scheeres et al. 1998) and the corresponding active control for a nonuniformly gravitational field can be designed if the close dynamics is known. Sawai et al. (2002) discusses the control of hovering spacecraft over a uniformly rotating spherical small body and application of this control to a nonuniformly rotating ellipsoid and results show that the stability is similar. In this paper, SSFFGT around the asteroid is investigated and a uniform gravitational field model of the asteroid is adopted for analysis. A static equilibrium is generated if the solar sail is used to cancel the gravity of the asteroid. In this case, the equilibrium is unstable and an active control is necessary. The sail can also evolve on a displaced orbit above the asteroid and this orbit is stabilizable using only the thrust modulation, which means that the magnitude of the thrust is adjusted while the thrust direction is not changed. A simple control strategy that includes the collision avoidance is employed to achieve a loose formation flying in the vicinity of a displaced orbit. For tight formation flying, the equilibrium shaping algorithm and sliding-mode control are employed to control the relative configuration without collisions. The dynamics and control details of the asteroid 2004 MN4 is analyzed in this paper. The Keplerian elements of the asteroid used in the following simulation examples are the ones employed: aa = 0.92239 AU, ea = 0.19104, i a = 3.3312 deg, ωa = 126.365 deg, a = 204.462 deg. The mass of the asteroid is 4.7 × 1010 kg.

2 Deflection capability of SSFFGTs At present, only a 20 m × 20 m solar sail ground validation project has been successfully completed in 2005 (Murphy et al. 2005; Lichodziejewski et al. 2005). Cosmos 1 solar sail spacecraft was attempted on June 21, 2005 and did not achieve its mission goal because of a boost rocket failure. A 40 m × 40 m solar sail is currently being developed by NASA and industries for a possible flight validation experiment via the New Millennium Program (NMP) Space Technology 9 program. Solar Polar Imager (SPI) mission, one of the solar sail roadmap missions envisioned by NASA, requires a 160 m × 160 m solar sail. Consequently, it is expected that a 100 m × 100 m class solar sail will be available within 15 years. In Wie (2007), a 90 m × 90 m weighing 2,500 kg SSGT (soar sail gravitational tractor) is controlled to hover at an altitude of 340 m above Apophis and a 5-year towing using this SSGT spacecraft and a 3-year coasting time will result in an orbital deflection of 30 km. Large orbital deflection will require sails of much larger size and mass, which will be beyond the solar sail technology at present or in near future. Therefore, the deflection capability of the SSGT is greatly restricted by the size of solar sail. With the consideration of the solar sail technology at present and in near future, the concept of SSGT formation flying is proposed and it increases the deflection capability by increasing the number of the SSGTs not the size of a single SSGT.

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Fig. 1 A actual-size illustration of a six-sail formation

Several SSGTs can be placed in the vicinity of the asteroid, either at static equilibrium points or on displaced orbits above asteroids. As shown in Fig. 1, six 100 m × 100m solar sails are uniformly placed on a displaced orbit with a radius of 200 m and the number can be increased by placing the SSGTs on different displaced orbits of different sizes and altitudes. We know that the gravitational tractor uses the gravitational coupling to modify the orbit of an asteroid. The mutual gravity between the asteroid and spacecraft will determine the acceleration exerted on the asteroid. Only a component of the gravity is used to deflect the asteroid for a spacecraft on the displaced orbit and the ratio of efficient component is determined by the ratio of the radius to the displacement of the displaced orbit because only the component of the gravity along the direction perpendicular to the displaced orbit plane is cancelled by the solar radiation pressure force. A 100 m sail will generate a solar thrust of about 0.056 N, which can cancel the gravity exerted on a 2,500 kg spacecraft on a displaced orbit of 197-m radius and 200-m displacement above the Apophis. The deflection ability of SSFFGT is estimated assuming that sails are on the reference displaced orbit. An option of six 100m sails with each weighting 2500 kg evolving on a 35-deg displaced orbit of 230-m radius and 167-m altitude is proposed. The along-track acceleration produced by the SSGTs in a time of 5 years is illustrated in Fig. 2a. The orbital deflection and velocity increment of the asteroid and the propellant required for station keeping are shown in Fig. 2b–d, respectively. It is found that a 5-year towing of Apophis using six 2500-kg SSGTs will result in an orbital deflection of 95.6 km and velocity increment of about 2.453 mm/s, and an additional 3-year coasting time will result in a deflection of 328.14 km. The orbital deflection is about 10 times of that generated by a single SSGT in Wie (2007), where a 5-year towing using a 90m × 90m weighing 2,500 kg SSGT and a 3-year coasting time result in an orbital deflection of 30 km. The distance between the asteroid and the Sun changes with the position of the asteroid, and the distance variation will lead to the solar thrust variation because the Sun angle remains constant. The along-track acceleration of the asteroid is introduced by the along-track component of the solar thrust. Therefore, the oscillation of along-track acceleration of the asteroid

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∆S (km)

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∆mp (kg)

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Fig. 2 The results of six SSFFGTs on a displaced orbit

in Fig. 2a is induced by the oblateness of the asteroid’s orbit. In addition, an additional control force is required to compensate the solar thrust variation to keep the equilibrium position of the sails. A large ratio of the control thrust is used to cancel the solar thrust variation and the results show that about 90 kg propellants are required for a 5-year station keeping of each sail, about 15 kg every year. The along-track acceleration will be constant and extra control force compensating the solar thrust variation will vanish if the orbit of the asteroid is circular. It means that the SSFFGT will be much more efficient for asteroids of small eccentricity. In the next section, two formation control strategies of the SSFFGTs will be presented to validate the feasibility of the SSFFGT concept for towing asteroids.

3 Dynamical equations and simplifications As shown in Fig. 3, the solar pressure force should be always along the z axis to generate an off-axis displaced orbit (McInnes and Simmons 1992). Since only the along-track thrust force provides an effective V of the target asteroid (Wie 2007), the solar thrust utilized to deflect the asteroid can be given approximately by f d = 2P A cos2 γ sin γ , where P is the solar pressure at the distance of the asteroid and A is the area of the sail. For a given area of the sail, the optimal angle maximizing f d is γ ≈ 35 degree. To illustrate the dynamics of the sails, two rotating frames are defined. One of them is defined as follows: the origin point ‘o’ is at the mass center of the asteroid; the x 2 axis, denoted as e1 , is defined by the angular momentum direction of the asteroid around the Sun; the z 2 axis, denoted as e3 , is along the axis perpendicular to the displaced orbit plane; the y2 axis forms a right handed triad with x2 and z 2 axes. The other frame ox yz is obtained by rotating the frame ox2 y2 z 2 along z 2 axis with an angular velocity of ω3 . Therefore, the plane spanned by x and y (dotted line in Fig. 3) is parallel to the displaced orbit plane and rotates along the z axis with an angular velocity of ω3 .

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z ( z 2) Displaced Orbit

ω3 n

x x 2 f

Va

Sun

Sail

γ

r0

y

Asteroid

y2

In this paper, only the gravitational forces of the Sun and the asteroid are considered to model the dynamics of the SSGT. To maximize the along-track force, the SSGTs are desired df to evolve on a 35-deg displaced orbit. The angular velocity of frame ox 2 y2 z 2 is e1 and the dt angular velocity of the frame ox yz relative to frame ox2 y2 z 2 is ω3 e3 . Therefore, the angular velocity of the frame ox yz can be written as df (1) e 1 + ω 3 e3 dt The angular acceleration can be obtained by calculating the derivative of the angular velocity, given by ω=

d2 f df d2 f df d2 f e1 + ω3 e˙ 3 = e1 + e1 − ω 3 e 1 × e3 = ω 3 e2 2 2 dt dt dt dt 2 dt The derivatives of the angles can be obtained by some algebraic manipulations. ω˙ =

(2)

df (1 + ea cos f )2 = na 3 dt 2 2 (1 − ea )

(3)

−2n a2 (1 + ea cos f )4 ea sin f d2 f = dt 2 (1 − ea2 )3

(4)

The equations of motion of the i-th sail in an inertial frame can be written as d 2 (r0 + ri ) µa µs βi µs =− ri + (r0 + ri ) − (ni · s)2 ni + fci + f pi 3 3 2 dt |r0 + ri | |ri | |r0 + ri |2

(5)

where s is the unit vector of the solar light direction; fci is the control force exerted on the i-th spacecraft; f pi is the unknown perturbation acceleration with a upper bound of f pm . The transformations of the velocity and acceleration between the inertial frame and the rotating frame ox yz are given by dri = r˙i + ω × ri dt d 2 ri = rï + 2ω × r˙i + ω × (ω × ri ) + ω˙ × ri d 2t

(6) (7)

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Therefore, the equations of motion of the sail in the rotating frame can be written in following form: (¨r0 + rï ) + 2ω × (˙r0 + r˙i ) + ω × [ω × (r0 + ri )] + ω˙ × (r0 + ri ) µa βi µs µs ri + =− (r0 + ri ) − (ni · s)2 ni + fci + f pi |r0 + ri |3 |ri |3 |r0 + ri |2

(8)

The motion of the asteroid is induced by the gravity of the Sun. Therefore, the constraint of the following equation is always satisfied. r¨0 + 2ω × r˙0 + ω × (ω × r0 ) + ω˙ × r0 = −

µs r0 |r0 |3

(9)

Then, equations of motion can be rearranged by substituting Eq. 9 into Eq. 8 rï + 2ω × r˙i + ω × (ω × ri ) + ω˙ × ri µa µs µs βi µs = r0 − ri + (r0 + ri ) − (ni · s)2 ni + fci + f pi (10) |r0 |3 |r0 + ri |3 |ri |3 |r0 + ri |2 This is the complete dynamics of the sail in the rotating frame.

4 Simplifications of the equations of motion The displaced orbit above the asteroid can be determined by the radius ρ0 , displacement z 0 , the Sun angle γ , and angular velocity ω3 (McInnes and Simmons 1992). The angular velocity of the displaced orbit is chosen so that a component of the gravity of the asteroid provides the centripetal force of the sail on the displaced orbit. Then, the angular velocity of the displaced orbit is determined by the orbit size, given by µa ω3 = (11) 3 ρ02 + z 02 2 The control acceleration along the z axis can be written as f c0 =

µa ρ02 + z 02

23 z 0

(12)

The magnitudes of the terms induced by the eccentricity of the asteroid orbit are shown in df d2 f 2 /ω E and Fig. 4 and the values of /ω E (maximum values over one orbit) are dependt dt 2 dent on the eccentricity and semi-major axis of the asteroid orbit. The results show that the ω3 detervalues are very small when the orbit of the asteroid is quasi-circular. The value of ωE mined by the displaced orbit of ρ0 = 197 m and z 0 = 200 is about 2,300. Therefore, the angular velocity of the asteroid rotation around the Sun is very small compared with that of the spacecraft rotation on the displaced orbit. The gravitational force induced by the Sun can be analyzed similarly, and ri is small compared with r0 . It is known for Fig. 4 that the angular velocity of the asteroid and Earth around the Sun is close to each other. Therefore, the pertinent terms of the righ-hand of Eq. 10 can be linearized as

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Fig. 4 The angular velocity and acceleration induced by the eccentricity of the asteroid orbit

µs µs µs µs ri T = 3 ≈ ω2E ri r − + r r r − I r + o (r ) 0 0 i 0 i 0 |r0 | |r0 |3 |r0 + ri |3 |r0 |3 |r0 |5

(13)

where I is a unit matrix. ωE A small parameter ε = is employed to reduce the terms. ω3 ω=

df e1 + ω3 e3 ≈ ω3 e3 + ω3 εe1 ≈ ω3 e3 dt

(14)

df d2 f e1 − ω3 e2 ≈ ε 2 ω32 e1 − εω32 e2 ≈ 0 2 dt dt µs µs ri ri 2 2 2 ≈ ω ≈ ε ≈0 r − + r r + o ω r + o (r ) 0 0 i 3 i E i |r0 | |r0 | |r0 |3 |r0 + ri |3 ω˙ =

(15) (16)

The equations of motion can be simplified by discarding the small terms and they are consistent with the results of two body problem after simplification, which can be written as rï + 2ω3 × r˙i + ω3 × (ω3 × ri ) = −

µa βi µs ri + (ni · s)2 ni + fci + f pi 3 |ri | |r0 + ri |2

(17)

This is the simplified dynamical equations of the sail in the vicinity of the asteroid. The dynamics on the displaced orbit is dominated by the gravity of the asteroid. The influences of the asteroid’s orbit eccentricity on the sail’s orbit are negligible.

5 Loose formation flying gravitational tractors 5.1 Stability analysis For the stability analysis, the simplified dynamical equations of motion are considered. Different from the regular spacecraft, the solar sail cannot be controlled to evolve on the displaced orbit perpendicular to the heliocentric velocity of the asteroid because the sunlight is almost

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perpendicular to the sail normal in this case. According to the results given in McInnes and Simmons (1992), the normal of the sail must always point in the z axis to generate this kind of displaced orbit. To maximize the along-track component of the solar thrust, a 35-deg displaced orbit is adopted for the sails in this paper. Then, the equations of motion in scalar form can be given by ⎧ µs i ⎪ x¨ − 2ω3 y˙i − ω32 xi = − 3 xi + f cx ⎪ ⎪ i r ⎪ i ⎪ ⎨ µs i yï + 2ω3 x˙i − ω32 yi = − 3 yi + f cy (18) ⎪ r ⎪ i ⎪ µ ⎪ s ⎪ ⎩ z¨ i = ais (t) − 3 z i + f czi ri βi µs βi µs cos2 γ ≈ cos2 γ . |r0 + ri |2 |r0 |2 When no active control is applied, the stability of the displaced orbit can be analyzed if the orbit of the asteroid is circular. Since any point on the displaced orbit is equivalent in terms of stability and control analysis, an equilibrium point, x = ρ0 , y = 0, z = z 0 , is selected for analysis. In this case, the variational equations can be obtained by perturbing the equilibrium solution as x → ρ0 + δx, y → δy, z → z 0 + δz.

⎧ µ µ µs ⎪ s s ⎪ δ x¨ − 2ω3 δ y˙ − ω2 δx = − ⎪ + 5 ρ02 δx + 5 ρ0 z 0 δz ⎪ 3 3 ⎪ r r r0 ⎪ 0 0 ⎪ ⎨ µs 2 δ y¨ + 2ω3 δ x˙ − ω3 δy = − 3 δy ⎪

r0 ⎪ ⎪ ⎪ ⎪ µ µ µs s s 2 ⎪ ⎪ ⎩ δ z¨ = − 3 + 5 z 0 δz + 5 ρ0 z 0 δx r0 r0 r0 where ais (t) =

With the consideration of Eq. 11, the characteristic polynomial of the system can be written as

F(λ) = λ4 + 2ω32 λ2 + (1 − 9 cos2 ϑ)ω34 λ2 = 0 (19) z0 . ri0 The sufficient conditions of the stability are that all the eigenvalues of the system are pure imaginaries. The necessary condition is obtained as √ ρ0 ≥ 2 2z 0 (20) where cos ϑ =

When the eccentricity of the asteroid orbit is small, the solar radiation pressure force can be written in the form of series of the eccentricity βi µs βi µs aa2 βi µs (1 + ea cos f )2 cos2 γ = 2 cos2 γ = 2 cos2 γ 2 2 aa |r0 | aa (1 − ea2 )2 |r0 | βi µs 2ea cos f + ea2 (2 + cos2 f ) − ea4 = 2 cos2 γ 1 + aa (1 − ea2 )2 s = ai0 [1 + 2ea cos f + o(ea )]

ais (t) =

s + fε = ai0

(21)

The force can be regarded as the summation of a constant force and a small periodic perturbation force. The equations of motion can be decoupled by transferring the coordinates

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into another orthogonal coordinate frame because all the eigenvalues of the system are pure imaginaries. ⎡ ⎤ ⎡ ⎤ ξ xi ⎣ ς ⎦ = C T ⎣ yi ⎦ (22) η zi where Cis a unit orthogonal matrix whose columns are the corresponding eigenvectors of the system. The perturbation force can be transferred into the new frame as f ε = C T f ε and the decoupled equations of motion in the new frame can be written as ⎧ 2 ⎪ ξ¨ + ω1i ξ = c1,3 f ε ⎪ ⎨ 2 ς¨ + ω2i ς = c2,3 f ε ⎪ ⎪ ⎩ 2 η¨ + ω3i η = c3,3 f ε

(23)

(24)

where ωij (j = 1, 2, 3) are the imaginary parts of the eignvalues, and ci, j is the element of the i-th row and j-th column of matrix C. With assumption that the eccentricity is small, the perturbation force induced by the oblateness can be written in the form of a periodic function: s s f ε = ai0 2ea cos f ≈ ai0 2ea cos (n a t + f 0 )

(25)

The solution including the perturbation force is obtained by constant variation method. ⎧ a s e sin (n a t + f 0 ) ⎪ ⎪ ξ(t) = Aξ sin(ωi1 t + φ1 ) + c1,3 i0 ⎪ ⎪ 2 ⎪ ⎪ ωi1 − n a 2 ⎪ ⎪ s ⎨ a e sin (n a t + f 0 ) ς(t) = Aς sin(ωi2 t + φ2 ) + c2,3 i0 (26) 2 ⎪ ωi2 − n a 2 ⎪ ⎪ s e sin ⎪ ⎪ ai0 (n a t + f 0 ) ⎪ 3 ⎪ ⎪ 2 ⎩ η(t) = Aη sin(ωi t + φ3 ) + c3,3 3 ωi − n a 2 where Aξ and φi are constants determined by the initial condition. The perturbed solution in the rotating frame can be obtained by transferring back the solution given by Eq. 26. ⎤ ⎡ ⎡ ⎤ ξ (t) xi (t) ⎣ yi (t) ⎦ = C ⎣ ς (t) ⎦ (27) z i (t) η (t) It can be found that the perturbed solution is also stable with small periodic perturbations, and a long- period solution induced by the oblateness is added to the unperturbed solution. Natural formation flying in the vicinity of the asteroid exists without the consideration of orbital deflection if the size of the displaced orbit satisfies Eq. 20. To balance the solar thrust of a 100m sail and the component force along the z axis of the gravity, the displacement of the displaced orbit should be very small with the constraint of Eq. 20, which is not practical for deflection mission because collision avoidance should be considered because of the small distances between the sails. Therefore, active control should be employed to stabilize the formation and avoid collisions between sails. A loose formation flying can be

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realized by distributing the sails uniformly on the displaced orbit and controlling them with a thrust modulation. The formation controller includes the stability controller and collision avoidance controller, where the stability controller stabilizes the sails at each instant and the collision avoidance controller will generate an extra control force if certain sail is threatened by collision harzard. 5.2 Stability controller Only the control force along the z axis is employed to stabilize the sail on a displaced orbit. In the rotating frame, the controlled equations of motion can be written as ⎧ µs ⎪ xï − 2ω3 y˙i − ω32 xi = − 3 xi ⎪ ⎪ ri ⎪ ⎪ ⎨ µs 2 yï + 2ω3 x˙i − ω3 yi = − 3 yi (28) ri ⎪ ⎪ ⎪ µ ⎪ s ⎪ ⎩ zï = ais (t) − 3 z i + f czi ri The input feedback linearization method is employed to stabilize each sail. To impose the altitude a constant value, the control law can be designed as µs f zi = 3 z i − ais (t) − 2λz z˙ i − λ2z (z i − z 0 ) (29) ri where λz is the damped coefficient of the feedback linearization. Substitution of Eq. 29 into Eq. 28 and choosing λz = ω3 generate the controlled equations of motion. ⎧ µs ⎪ xï − 2ω3 y˙i − ω32 xi = − 3 xi ⎪ ⎪ ri ⎪ ⎨ µ s (30) yï + 2ω3 x˙i − ω32 yi = − 3 yi ⎪ ⎪ ⎪ r ⎪ i ⎩ zï = −2ω3 z˙ i − ω32 (z i − z 0 ) The equations can be linearized in the vicinity of the equilibrium solution to obtain the variational equations, whose dimensions can be reduced by decoupling the equations. The coordinate transformation that removes the constant induced by dimension reduction will introduce a new coordinate frame, and the decoupled equations with new coordinates can be given by δ xï + ω32 (4 − 3 sin2 ϑ)δxi − 3ω32 sin ϑ cos ϑδz i = 0 (31) δ z¨ i + 2ω3 z˙ i + ω32 δz i = 0 The eigenvalues of the system are calculated as λ1,2 =−ω3 and λ3,4 = ± 4 − 3 sin2 ϑω3 i. The two negative eigenvalues are related to the altitude of the solar sail, which is determined by the control law. In our case, the altitude is designed to converge to the desired altitude exponentially. The other two pure imaginaries are related to the oscillation frequencies in the displaced orbit plane. 5.3 Collision avoidance strategy Six sails are desired to be static at the six equilibria uniformly distributed on the displaced orbit. Because only the altitude is accurately imposed by the stability controller, the sail

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may oscillate in the vicinity of its corresponding reference point in the displaced orbit plane. Therefore, the neighboring sails may collide with each other with perturbations considered. A collision avoidance strategy used to achieve safe multi-sail formation flying is necessary. The avoidance controller is triggered in the case of the distance between two sails being within a specified value. The control strategy applies an extra control force in the z axis to drive one of them out of the plane. Therefore, only one of them needs the avoidance control and the force is given in the following form: ri − r j < L w L − ri − r j i fa = (32) ri − r j ≥ L 0 where L is the minimal distance allowed between two sails and w is the feedback coefficient. Then, the total control force exerted on the i-th solar sail is a summation of two components that can be given by µs f zi = 3 z i − ais (t) − 2λz z˙ i − λ2z (z i − z 0 ) + f ai (33) ri There are several parameters should be chosen properly to guarantee the stability of the system. The minimal distance L is chosen based on the radius of the displaced orbit. If L is too small, the strategy may be not quick enough to avoid the collision. If L is too large, the sails may jump frequently from the plane like elastic balls and in worse case the system may become unstable. The control parameter coefficient w is also very important for this strategy. If it is too small, the control force may be not large enough to avoid the collision. On the contrary, the sail will jump very high above the plane when the avoidance strategy is triggered. Luckily, the simulations show that large ranges of the parameters that guarantee the stability of the system exist. 5.4 Numerical example The displaced orbit size are ρ0 = 206m and z 0 = 146m, and the offset angle is chosen to be γ = 35◦ . The control parameters for the feedback linearization λz and collision avoidance w are 2 × 10−4 and 1 × 10−5 , respectively. The minimal distance for the collision avoidance is L = 100m. The initial position of each sail is randomly selected in the circle of radius 10 m around the reference point. The controlled trajectories of the sail are shown in Fig. 5. We can see from the results that one sail jumps once from the plane to avoid the collision. Other sails vibrate in the vicinity of their reference orbits without any collision warning. The merit of the loose formation is that the implementation is very simple because thrust direction is fixed and only thrust modulation is required. If there are no perturbations or the perturbations are well understood, the loose formation is effective. As the unknown perturbations increase, plenty of propellant is required for the avoidance controller. And the gravitational environment is uncertain because of the spinning motion of an irregularly shape asteroid, therefore, a robust formation strategy that is insensitive to the perturbations is necessary. In the next section, a more robust formation that considered the unknown perturbation is discussed.

6 Tight formation flying tractor The loose formation is simple but not robust enough. A tight formation that controls sails exactly on the desired relative configuration is investigated in this section. The complete

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145 200 -200

-200

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145 200

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Fig. 5 The controlled formation of the solar sails

dynamics with unknown perturbations is employed to design the control law and the equilibrium shaping is used to navigate the sails to achieve the target configuration. First, the equilibrium shaping used by Izzo and Pettazzi (2007) for satellite swarm is introduced simply. 6.1 Equilibrium shaping Equilibrium shaping is a method used to assemble swarm robotics initially. The theoretical results have been discussed by Gazi and Passino (2004), and each individual follows the designed velocity field that is defined by different velocity contributions such as gather behavior and avoidance behavior. Another dock behavior is added to the velocity field in Izzo’s work, which takes effect when the individual approaches the target position. We use the same model as Izzo’s and the underlying kinematical field of the swarm vehicles is designed. The velocity field is defined as a sum of three different behaviors: gather, avoid and dock. The gather behavior attracts N different individuals towards the N targets globally. The analytical expression of this behavior contribution to the i-th individual desired velocity may be written in the following form: viG =

N

c j ψG ξ j − ri (ξ j − ri )

(34)

j=1

where c j is a constant; ξ j (j = 1. . .N ) is the target position; ψG is a function mapping from positive real to positive real. The linear gather can be chosen as viG =

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The dock behavior takes effect only if the agent is in the neighborhood of the target. The velocity of the vehicle should decelerate when approaching the target or it will rush beyond the target position and turn around. k D determines the radius of the sphere of influence of the dock behavior. The expression used for this behavior is: viD =

N

d j ψ D ξ j − ri , k D (ξ j − ri )

(36)

j=1

where d j is a constant. The specific expression employed in the paper is given by viD

=

N

dje

−

|ξ j −ri |2 kD

(ξ j − ri )

(37)

j=1

This avoid behavior will introduce a repulsive contribution to the velocity field when two different individuals are in the vicinity of each other. The expression that describes the desired velocity for this kind of behavior is given below: viA =

N

bψ A ri − r j , k A (ri − r j )

(38)

j=1

where b is a constant; ψ A is a mapping from positive real to positive real that is negligible whenever the mutual distance is considered to be not dangerous according to the value k A . The specific expression can be chosen as viA =

N

be

−

|ri −r j | kA

(ri − r j ).

(39)

j=1

Therefore, the desired velocity field of all the individuals can be written as r˙d = vG + v D + v A = g(r, ξ , λ)

(40)

T where rd = r1T · · · rTN is a vector including the position vectors of all the vehicles; T

T ξ = ξ 1T · · · ξ TN is a vector of the target positions; λ = c j d j b is a vector of the unknown parameters. Two conditions have to be satisfied to guarantee that the vehicles will converge to the target configuration when they follow the desired velocity field. The first condition is that the vector of the target positions is an equilibrium solution that can be written as g(rd = ξ , ξ , λ) = 0

(41)

The second condition is that the equilibrium solution is stable, which means that the real ∂g(rd = ξ , ξ , λ) are negative. Both the conditions can be satisfied parts of eigenvalues of ∂rd by choosing the unknown parameters λ. The symmetry of the target configuration can be utilized to reduce the constraint equations, which has been discussed in Izzo and Pettazzi (2007).

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6.2 Gravitational environments and sliding-mode control The equilibrium shaping method is utilized to design the velocity field of the sails when they are in the vicinity of the asteroid. Sliding mode control is used to control the sails to follow the velocity generated by the equilibrium shaping method. The full equations of motion for the i-th sail are written in following form: rï = fgi + f pi + fci

(42)

µa µs µs where fgi = r0 − ri − [2ω × r˙i + ω × (ω × ri ) + ω˙ × ri ]. (r0 + ri ) − 3 3 |r0 | |r0 + ri | |ri |3 The equations of motion for six sails can be obtained by combining the equations for each one, given by r¨ = fg + f p + fc

(43)

where r = [ r1T · · · rTN ]T , fg= [ fg1T · · · fgN T ]T , fc = [ fc1T · · · fcN T ]T , f p = [ f p1T · · · f pN T ]T . The sliding manifold is chosen to be in the following form: σ = [σ1 · · · σ N ] = r˙d − r˙ = 0

(44)

where r˙d is the velocity field scheduled by the Equilibrium Shaping technique, which is in fact a method to build such a sliding manifold. Whenever the system is on the sliding manifold it will stay on it if and only if the following relation is satisfied at each instant: σ˙ = 0

(45)

Substitution of Eq. 43 into Eq. 45 generates the relation given by σ˙ = r¨d − fg + f p + fc = 0

(46)

It can then define the equivalent control fe as a feedback that keeps the state of the system on the manifold for all the time instants: fe = r¨d − fg

(47)

The equivalent control guarantees that the system will never leave the manifold after intersected with it. Another control contribution, labeled as fn , should be applied to guarantee that the system will converge to the manifold when σ = 0. A Lyapunov method can be used to obtain the value of the control contribution. The Lyapunov function is defined as 1 σ ·σ (48) 2 The control feedback must be derived to impose the time derivative of V to be negative definite along the trajectories of the system. The condition on the total time derivative of the Lyapunov function can be given by ˙ = σ · σ˙ = −σ · fn + f p < 0 (49) V V =

The additional feedback law can be derived in order to drive the motion of the system towards the sliding manifold. fn = f ε sign (σ ) where the sign function is defined componentwise and f ε should satisfy f ε > antee the negativity of V˙ .

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(50) f pm

to guar-


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z (m)

150 Final position

100

50 Asteroid 300

0 -300

200 100

-200 -100

0

0

x (m)

-100

100

y (m)

-200

200 300

-300

Fig. 6 Controlled formation configuration

Therefore, the total control force can be given by fc = fe + fn = r¨d − fg + f ε sign (σ )

(51)

6.3 Numerical example Six sails are controlled uniformly distributed on a displaced orbit to form a hexagon. The displaced orbit size is chosen to be ρ0 = 206 m, z 0 = 146m. The parameters for the equilibrium shaping are k A = 4 × 105 , k D = 3 × 104 , ci = 0.01, di = 0.03, and b = 0.0171. The initial positions of the sails are randomly selected in the vicinity of the displaced orbit. Figure 6 shows that they will converge to the finial configurations regardless of the initial positions. The results illustrate that only the control forces along the z axis are required to keep the formation when the sails are in final configuration, which is very similar with that of the loose formation. An important merit of sliding mode control is that it provides robustness to uncertainties. The additional forces can be designed to guarantee that the system with bounded perturbations is asymptotically stable (Jovan et al. 2002). The weak point is that the signal function will introduce the chattering of the system. To reduce the chattering, the signal function can be substituted by saturation function. The details of the sliding mode control are not discussed in this paper. The tight formation will keep the desired relative configuration with various perturbation considered. However, the controller is more complex than that of the loose formation. A controller able to adjust the magnitude and direction of the thrust is required. For both stability controller and equilibrium shaping method, the control force will converge to the same value, which is the difference between gravity of the asteroid and solar radiation pressure force. Therefore, the propellant costs for both methods are comparative.

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7 Conclusions and discusses The deflection capability of the SSFFGT increases linearly with the number of sails in the formation. Besides the control strategies, there are no extra difficulties to realize the SSFFGT compared with a single SSGT. Two formation flying control strategies are investigated in this paper. A very simple control strategy with only the thrust modulation along the z axis is employed to stabilize the system and avoid collisions. Another tight formation strategy with each sail controlled accurately is also discussed. The merits of the loose and tight formations are the simple implementation of the control force and the robustness to perturbations, respectively. The two SSFFGTs generate similar results in orbital deflections and propellant consumptions. The control force required to keep the relative configuration is almost perpendicular to the displaced orbit plane and it can be realized by small vanes that are attached to the corners of the sail or pulsed plasma thruster. The attitude control is not considered here and the attitude of the sail is assumed to keep fixed with respect to the rotating frame. The control methods proposed by Wie (2004a, b) or passive control discussed in Gong et al. (2007a, b, c) can be used to realize the desired attitude. Acknowledgments Research for this paper was supported by the National Natural Science Foundation of China (Grants No.10602027, No. 10832004 and No. 10672084). The authors are thankful to Bong Wie, Iowa State University, for several useful comments and fruitful discussions.

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