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Acta Astronautica

Acta Astronautica 69 (2011) 809–821

0094-57

doi:10.1

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colin.m1 Te

journal homepage: www.elsevier.com/locate/actaastro

Hybrid solar sail and solar electric propulsion for novel Earthobservation missions$

Matteo Ceriotti n, Colin R. McInnes 1

Advanced Space Concepts Laboratory, Department of Mechanical Engineering, University of Strathclyde, 75 Montrose Street, James Weir Building, G1 1XJ

Glasgow, United Kingdom

a r t i c l e i n f o

Article history:

Received 1 February 2011

Received in revised form

20 May 2011

Accepted 2 June 2011Available online 25 June 2011

Keywords:

Hybrid propulsion

Pole-sitter

Solar sailing

Solar electric propulsion

Earth observation

65/$ - see front matter & 2011 Elsevier Ltd. A

016/j.actaastro.2011.06.007

s paper was presented during the 61st IAC in

esponding author. Tel.: þ44 0 141 548 5726

ail addresses: matteo.ceriotti@strath.ac.uk (M

cinnes@strath.ac.uk (C.R. McInnes).

l.: þ44 0 141 548 2049.

a b s t r a c t

In this paper we propose a pole-sitter spacecraft hybridising solar electric propulsion

(SEP) and solar sailing. The intriguing concept of a hybrid propulsion spacecraft is

attractive: by combining the two forms of propulsion, the drawbacks of the two

systems cancel each other, potentially enabling propellant mass saving, increased

reliability, versatility and lifetime over the two independent systems. This almost

completely unexplored concept will be applied to the continuous monitoring of the

Earth’s polar regions through a pole-sitter, i.e. a spacecraft that is stationary above one

pole of the Earth. The continuous, hemispherical, real-time view of the pole will enable

a wide range of new applications for Earth observation and telecommunications. In this

paper, families of 1-year-periodic, minimum-propellant orbits are found, for different

values of the sail lightness number and distance from the pole. The optimal control

problem is solved using a pseudo-spectral method. The process gives a reference control

to maintain these orbits. In addition, for stability issues, a feedback control is designed

to guarantee station-keeping in the presence of injection errors, sail degradation and

temporary SEP failure. Results show that propellant mass can be saved using a medium-

sized solar sail. Finally, it is shown that the feedback control is able to maintain the

spacecraft on-track with only minimal additional effort from the SEP thruster.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Geostationary spacecraft provide a continuous hemi-spherical view of equatorial and medium-latitude zones,but unfortunately these platforms cannot cover high-latitude regions. Conversely, a pole-sitter is a spacecraftthat is constantly above one of the Earth’s poles, i.e. lyingon the Earth’s polar axis [1]. This type of mission wouldprovide a continuous, hemispherical, real-time view of

ll rights reserved.

Prague.

.

. Ceriotti),

the pole, and will enable a wide range of new applicationsin climate science and telecommunications.

Traditionally, polar observation is performed with onespacecraft or a constellation of spacecraft in highly inclinedlow or medium Earth orbits [2]; however, the observation isnot continuous, but it relies on the passage of one spacecraftabove regions of interest. Therefore, the temporal coverageof the entire polar region can be quite poor, and furthermoredifferent areas are imaged at different times, hence missingthe possibility to have a real-time complete view of the pole.

The hemispheric view of the pole is currently recon-structed through a composite image that is made of severalimages taken at different times. The main science applica-tions for polar image composites are the generation ofatmospheric motion vectors (AMV) and the identificationof storm systems. According to Lazzara et al. [3], these two

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821810

applications among others would benefit from a truepole-sitter spacecraft. For example, both at the North Poleand South Pole, there is an interval of latitudes in whichthe AMV are not available, due to a gap between geosta-tionary and polar orbiting satellites. Citing Ref. [3], ‘‘thepossibility of seeing the deep polar regions with dramati-cally increased temporal resolution, and even the oppor-tunity for focused observations on a specific geographicregion, with potential high spatial and temporal resolution,are capabilities that would improve the short term fore-casting and understanding of atmospheric phenomena thatoccur in these portions of the world’’.

Another significant benefit to the polar regions from apole-sitter platform will be for communications. It is wellknown that line-of-sight telecommunications to conven-tional spacecraft in geostationary orbits is not possible athigh latitudes and polar regions. While the distance of thespacecraft from the Earth could preclude high-bandwidthtelecommunications, a pole-sitter will be able to alwayssee the South Pole, providing a continuous flow of datawith scientific South Pole stations. The same spacecraftcould accomplish both observation and telecommunicationtasks over the polar regions. Furthermore, telecommunica-tion with polar regions will be a key issue in future aschanges to the arctic ice pack opens navigation channels forshipping. Such developments will necessitate real-timepolar imaging for route planning.

Here we propose to hybridise solar electric propulsion(SEP) with solar sailing, in order to get benefit from combin-ing the two propulsion systems on the same spacecraft.

Solar electric propulsion is a mature technology thatprovides a spacecraft with a relatively low thrust (of theorder of a fraction of a Newton [4]). Nevertheless, despitetheir high efficiency, the thrusting time and hence themission duration is always limited by the mass ofpropellant on-board.

In contrast, solar sailing [5] is a propellant-less spacecraftpropulsion system: it exploits the solar radiation pressuredue to solar photons impinging on a large, highly reflectingsurface (the sail) to generate thrust. Despite the original ideaof solar sailing being rather old [6], only very recently aspacecraft successfully deployed a solar sail: the JapaneseAerospace Exploration Agency (JAXA) Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) [7].

Due to the interesting potential of enabling missionsthat are not constrained by propellant mass, studies onpotential solar sail missions have been undertaken, whileothers are still ongoing [8,9]. Solar sails seem to be suitablefor potentially long duration missions that require a small,but continuous, thrust. They have been proposed for inter-planetary transfers [10], or to generate artificial equilibriumpoints [11], for example in the proximity of the Lagrangepoints of the Sun–Earth [12] or Sun–Moon [13] system.

The idea to combine solar sail propulsion and conven-tional reaction-mass propulsion on the same spacecraftwas proposed for interstellar travel in 1992 [14], usingnuclear electric propulsion. Ten years later, Leipold andGotz [15] suggested the hybridisation of solar sailing withsolar electric propulsion, and since then, the conceptremained almost completely unexplored. Current researchranges from artificial equilibria in the Sun–Earth system for

Earth observation [16] to interplanetary transfers [17], todisplaced periodic orbits in the Earth–Moon system [18]. Inaddition, IKAROS is exploiting hybrid propulsion [7].

The reason for this interest is due to the fact that in thehybrid system, at the cost of increased spacecraft com-plexity, the two propulsion systems complement eachother, cancelling their reciprocal disadvantages and lim-itations. In principle, SEP can provide the missing accel-eration component towards the Sun, that the sail cannotgenerate. Similarly, the hybrid spacecraft can be seen asan SEP spacecraft, in which an auxiliary solar sail providespart of the acceleration, enabling saving of propellantmass, and lower demand on the electric thruster, possiblywith some intervals in which it could be switched off.Moreover, the reliability of the system is increased: thesolar sail is only providing part of the necessary accelera-tion, and hence it can be smaller than what would berequired for a pure sail spacecraft; also, an SEP-onlydegraded mission could be envisaged, in the case thatthe sail fails to deploy. Certainly, a mechanism to jettisonthe sail shall be designed, and various configurations andoperation modes related to the two thrusting modesshould be analysed. Moreover, the amount of thrustrequired by the SEP-only system would be higher, tocompensate for the missing sail acceleration; SEP-onlyscenarios will be presented later in the paper.

Overall, the hybrid spacecraft can be seen as a way togradually introduce solar sails for space applications, andhence to reduce the advancement degree of difficulty(AD2) [19] in the technology readiness level (TRL) scale.

In this work we propose hybrid propulsion for thepole-sitter spacecraft, thus extending the pure SEP space-craft proposed by Driver [1] and the hybrid spacecraftstationary above the Lagrangian point L1 proposed by Baigand McInnes [16], providing a practical realisation of thesolar sail pole-sitter orbits by Forward [20].

Previous work by the authors focused on the optimisa-tion procedure, including a shape-based approach to findfirst-guess solutions [21], and a comparison between pureSEP and hybrid sails in terms of spacecraft mass [22]. Inthis paper we present the design and control of optimalorbits for a pole-sitter mission. After explaining thedynamics that will be used, we will present the optimalcontrol problem that is solved to obtain minimumpropellant consumption orbits. After a comparisonbetween the hybrid sail and pure SEP performance, afeedback control system will be designed to keep thespacecraft on track when subject to injection error, saildegradation, or temporary SEP failure.

2. Equations of motion

We consider the circular restricted three-body problem(CR3BP), that describes the motion of a negligible mass(the spacecraft) under the gravitational attraction of twomasses (the primaries, Sun and Earth in this case), thatrotate of circular motion around their common centre ofmass. We use a synodic reference frame centred at thecentre of mass, and having the x axis collinear with the twoprimaries, pointing towards the Earth, the z axis is aligned

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 811

with the angular velocity of the primaries x¼oz and the yaxis completes the right-hand system (Fig. 1).

The dynamics of a spacecraft at position r is governed by:

€rþ2x_r�rU ¼ asþaT

where r is the position vector and the effective potential,which takes into account gravitational attraction andcentrifugal acceleration, is

U ¼1

2ðr2

x þr2y Þþ

1�mr1þmr2

The system is normalised such that o¼1, m1þm2¼1,and the unit of distance is the separation of the twoprimaries. With these assumptions, the position alongthe x-axis of m1 is �m, and the position of m2 is 1�m, andm¼3.0404�10�6.

In addition, as and aT are the accelerations providedby the solar sail and the SEP thruster, respectively. Theformer is expressed as [5]:

as ¼1

2b0

m0

m

1�mr2

1

gcosanþhsinath i

cosa

Here n is the component normal to the non-ideal sail andparallel t to it, in the plane of the Sun vector r1. b0 isthe lightness number at the beginning of the mission,b0¼s*A/m0: values of b0, ranging from 0 (no sail, hence pureSEP spacecraft) to 0.05 can be assumed for near- to mid-termtechnology [23]. m0 and m are the spacecraft mass at thebeginning of the mission and at any given time, respectively.Note that, in the hybrid case, the spacecraft mass varies in

Fig. 1. Restricted three-body problem.

1r

n m

1r

αθ

tSail

1θ

Fig. 2. (a) Definition of the cone and centre-line angles (plane of the figure is per

and clock angles.

general, due to the SEP propellant consumption, and so doesthe acceleration from the sail. snffi1:53� 10�3 kg=m2 isthe critical sail loading for the Sun.

The sail acceleration is controlled through the space-craft attitude: the vector n can be described using thecone angle a (angle between n and r1, see Fig. 2a) andthe clock angle d (angle measured around r1, starting fromthe vertical plane, of the component of n perpendicular tor1, see Fig. 2b). In the hybrid spacecraft, thin film solarcells (TFSC) cover an area ATF¼0.05A on the sail, and areused to power the SEP thruster (see Fig. 3 for a potentialconfiguration). The area ratio is a conservative estimationbased on previous studies [16]. It is assumed that theactual direction of the sail acceleration m is related to nthrough the coefficients g and h [16], that can be com-puted as a function of the hemispherical reflectance of thesail, ~rs ¼ 0:9, and of the thin film ~rTF ¼ 0:4 [15]

g ¼ 1þ ~rsþATF

Að~rTF�~rsÞ

h¼ 1�~rs�ATF

Að~rTF�~rsÞ ð1Þ

The SEP thruster is assumed to be variable-thrust andmounted on a gimbal, such that the thrust vector Tis completely controllable (refer again to Fig. 3). Thepropellant mass flow _m is related to the thrust throughNewton’s law and the conservation of mass

_m ¼ T=ve ð2Þ

where the exhaust gas velocity is ve¼ Ispg0, Isp is thespecific impulse in seconds and g0¼9.81 m/s2.

α 1rδ

1 1 1ˆˆ ˆ≡r

n

θ

m

1r

1

1

ˆˆ

ˆˆ≡ z r

z r

×

×

×θϕ

pendicular to the sail, containing the Sun vector r1) and (b) solar sail cone

Fig. 3. Configuration of the hybrid sail/SEP spacecraft, with steerable

thrust.

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821812

3. Pole-sitter mission

A pole-sitter spacecraft, during the operational phase ofits mission, is constantly aligned with the polar axis of theEarth. If we neglect precession of the equinoxes and nutation(which are long-term, and thus irrelevant in this analysis),the polar axis of the Earth does not change its directionwhile the Earth is orbiting the Sun. Therefore, in the synodicreference frame, the same axis rotates with a motion ofapparent precession. Its angular velocity is the opposite ofthat of the primaries, or �x. Therefore the polar axis spans afull conical surface every year (see Fig. 4). The cone half angleis the tilt of the axis relative to the ecliptic, i.e. deq¼23.51.

Consequently, the pole-sitter shall follow Earth’s polaraxis, and describe a 1-year-periodic orbit which can bedescribed by

rðtÞ ¼

dðtÞsindeqcosotþð1�mÞ�dðtÞsindeqsinot

dðtÞcosdeq

264

375 ð3Þ

where d(t) is the distance from the centre of the Earth, andis a continuous function of time. The North Pole case isconsidered here, the South Pole case being symmetric to it.

In this work, we search for optimal periodic pole-sitterorbits, that minimise the SEP propellant consumptionover a period (one year), while maintaining the pole-sittercondition (3) at each time during the mission.

3.1. Optimal control problem

As seen from Eq. (3), the pole-sitter spacecraft remainsabove the pole even if the distance from the Earth varieswith time. Therefore, in the general case, the function d(t),i.e. the shape of the orbit, is not known a priori, and it isfound by solving an optimal control problem, together withthe control history that guarantees minimal fuel consump-tion. The problem is the one of finding the state time historyx(t) and control time history u(t), between initial and finaltime, t0 and tf, minimising the cost function:

J¼jðx0,xf ,t0,tf Þþ

Z tf

t0

Lðxðt0Þ,uðt0Þ,t0Þdt0

subject to the dynamical constraints:

_xðtÞ ¼ fðxðtÞ,uðtÞ,tÞ

Fig. 4. Apparent precession of Earth’s polar axis due to rotation of

reference frame.

state and control bounds:

xlrxðtÞrxu

ulruðtÞruu

initial and final state bounds:

x0,lrxðt0Þrx0,u

u0,lruðt0Þru0,u

xf ,lrxðtf Þrxf ,u

uf ,lruðtf Þruf ,u

and general non-linear path constraints:

cðx,u,tÞr0

For this specific case, the time span in normalised unitsis t0¼0, tf¼2p. The state vector is composed of theposition, velocity and mass x¼ r v m

� �Tof the space-

craft, which leads to the following dynamical constraints

_x ¼

v

�rU�2x� vþasþT=m

�T=ve

264

375

The controls (sail normal n and thrust vector T) areexpressed in Cartesian components in the reference framein Fig. 2b, instead of angles and magnitude, to avoid theambiguity related to the use of angular variables:

u¼ nr1ny1

nj1Tr1

Ty1Tj1

h iT

Since the sail normal n is a unit vector, a path constraintshall be added to guarantee the uniqueness of the solution:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2r1þn2

y1þn2

j1

q¼ 1 ð4Þ

The pole-sitter constraint in Eq. (3) is enforced throughthe path constraints:

atan2ð�r2,y,r2,xÞ�ot¼ 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

2,xþr22,y

q�r2,ztanðdeqÞ ¼ 0

We assume that the spacecraft is injected in the pole-sitter orbit at the winter solstice, at some point in the x–zplane. To guarantee that the orbit is periodic andsymmetric with respect to the same plane, the initialvelocity can only have a positive y component. Additionallyit is enforced that the position and velocity at time tf matchthose (partially to be determined) at time t0.

The objective is to maximise the final mass of thespacecraft, i.e. minimise the fuel consumption, after oneperiod:

J¼�mf ¼�mðtf Þ ð5Þ

To solve this optimal control problem, the tool PSOPTwas used [24]. PSOPT implements a direct pseudo-spectralmethod to solve the optimal control problem. By discretis-ing the time interval into a finite number of nodes, theinfinite dimensional optimal control problem is transformedinto a finite dimension non-linear programming (NLP)problem. Pseudo-spectral methods use Legendre or Cheby-shev polynomials to interpolate the time dependent vari-ables at the nodes. The advantage of using pseudo-spectralmethods is that the derivatives of the state functions at thenodes are computed by matrix multiplication only, and that

Table 1Optimal pole-sitter orbits for three different values of lightness number.

b0 dmin (AU) dmax (AU) mf (kg) Tmax (N)

0.0 0.015675 0.020332 843.430417 0.180648

0.05 0.013116 0.023422 901.896219 0.141085

0.1 0.011896 0.028363 925.192867 0.134256

Fig. 6. Earth as seen from the pole-sitter spacecraft at 0.018 AU using an

instrument with 30’ field of view.

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 813

the integral in the cost function is approximated using wellknown Gauss quadrature rules. We refer to [25,26] for adetailed description of pseudo-spectral methods.

The optimisation process is initialised with a first-guesssolution. The generation of this sub-optimal trajectory, andcorresponding control history, is the subject of previouswork [21]. Here we provide a brief outline of the procedure.The first guess is generated using a shape-based approach,in which a specific orbit for the spacecraft and initial massm0 are assigned, and then the controls that enable that orbitare obtained from the equations of motion, with an iterativeprocess. The orbit is discretised into a finite number ofpoints in time. At each point, the sail cone and clock anglesare computed numerically, minimising the magnitude of theSEP acceleration. Once as is known, aT can be computed bydifferencing. Assuming that the thrust remains constantfrom one point to the next along the orbit, Eq. (2) can beintegrated to find the mass change. With this new value ofmass, the procedure iterates on to the next point on the orbit.

3.2. Solutions

Optimal orbits are sought for three different spacecraft: apure SEP (no solar sail, b0¼0) and two different hybridspacecraft, with increasing size of the sail (b0¼0.05, 0.1).The initial mass is m0¼1000 kg and Isp¼3000 s based oncurrent ion engine technology (existing NSTAR/DS1 [27]).

Each case leads to a different optimal orbit (Fig. 5). Thepure SEP spacecraft optimal orbit is symmetric with respectto summer and winter, while if a solar sail is added, thespacecraft goes farther from the Earth in summer and closerin winter. All the orbits are at about 0.018 AU (or about 2.7millions of km) from Earth’s centre, as can be noted inTable 1. Fig. 6 shows the view of the Earth from this distance.Note that the Lagrangian point L1 of the Earth–Sun system isabout 1.5 million km from the Earth. Fig. 5 shows the threeorbits; in the same plot, the cone described by Earth’s polaraxis is also superimposed. Table 1 also highlights thepropellant mass saving that a hybrid solar sail offers withrespect to pure SEP. Also, the presence of the sail allows us toreduce the maximum thrust needed, thus allowing the use ofa smaller engine.

Fig. 5. Optimal hybrid pole-sitter orbits for three different values of b0.

However, it shall be noted that these results assume aspacecraft with the same initial mass, both in the case ofSEP only or hybrid propulsion. If a sail is present, then thedry mass of the spacecraft shall be higher or the payloadmass reduced. A complete investigation of the massbudget of the hybrid spacecraft, including a discussionon the benefit of the hybrid propulsion in terms of masssaving, is presented in [22].

It was stated that the thrust vector is assumed comple-tely controllable, from a trajectory design point of view.However this cannot be achieved in practice. Although agimbal system can be used, the thruster cannot ejectpropellant in the direction of the solar sail, in the case ofthe hybrid system. Also it is difficult to engineer a thrusterthat could rotate 1801. Therefore, we analysed the steeringangle needed for the thruster, with respect to the mainspacecraft bus and the sail. Fig. 7 shows the evolutionduring one orbit of the angle between the sail normal andthe thrust vector: this is the angle represented in Fig. 3. Twoimportant considerations can be made: the first is that theangle variation is limited to about 21, for all the three casespresented. This implies very little movement of the thruster,and possibly a lightweight gimbal mechanism. The second isthat the angle is around 561, and therefore the thruster doesnot thrust towards the sail, during the whole orbit.

Additional families of orbits can be found by consider-ing different mission requirements. For example, a camerawith a fixed focal length may require that the variation ofthe distance from the Earth (and hence the excursionalong the z axis) shall be limited. However, the value of

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821814

the distance itself is not assigned, as the camera could bemounted with a suitable focal length.

In order to find optimal orbits that have reducedexcursion along the z axis, but still minimise the propel-lant consumption by settling at the correct altitude, we

0 50 100 150 200 250 300 35055

55.2

55.4

55.6

55.8

56

56.2

56.4

56.6

56.8

57

t, d

Ang

le, d

eg

Pure SEPβ = 0.05β = 0.1

Fig. 7. Angle between the thrust vector T and the solar sail normal n,

during the three orbits in Fig. 5.

Table 2Characteristics of optimal orbits obtained minimising a weighed sum of

propellant mass and velocity in z, for b0¼0.05.

w dmin (AU) dmax (AU) mf (kg)

5�10�15 0.014744 0.019960 899.638660

5�10�14 0.016982 0.017942 894.916402

1�10�13 0.017262 0.017763 894.270074

2�10�13 0.017411 0.017668 893.914791

Fig. 8. Optimal orbits obtained constraining the ma

can devise the following objective function

J¼�mf þw

2p

Z 2p

0v2

z dt

in which the final mass is weighed together with theintegral of the square of the velocity component alongthe z axis. Hence, a velocity vz along the orbit is penalisingthe cost according to the weight w. By varying the weight,families of optimal orbits are found, trading off propellantconsumption with altitude change throughout the orbit.Table 2 shows, for 5 different values of the weight, theminimum and maximum distance from the Earth, and therespective final mass after one year, assuming an initialmass of 1000 kg and b0¼0.05. It can be seen that, as theweight increases, the orbit converges to a ‘‘flat’’ orbitparallel to the x–y plane, at a distance of about 0.017 AU.This is the minimum consumption altitude for a constant-distance orbit [21]. The propellant cost of having a flatorbit with respect to a free one can be quantified as about8 kg in the first year for b0¼0.05 (compare Tables 1 and 2).

Another family of orbits can be found assuming thatthe payload cannot get further from the Earth than adefined distance. In this case, optimal orbits can be found

ximum distance from the Earth, for b0¼0.05.

Table 3Characteristics of optimal orbits with constrained maximum distance,

for b0¼0.05.

dmin (AU) dmax (AU) mf (kg)

0.018 0.013179 898.507259

0.017 0.013196 896.688888

0.016 0.013201 894.203213

0.015 0.013183 890.818185

0.014 0.013105 886.186387

0.013 0.012838 879.749327

0.012 0.012000 870.546037

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 815

using the same objective function as in Eq. (5), but addingthe following inequality constraints:

0rrzrdmaxcosðdeqÞ

This leads to the family of orbits plotted in Fig. 8;Table 3 describes some of their characteristics. Once againlimiting the altitude is counterbalanced by additionalpropellant consumption. Note that values of dmax smallerthan those in the table could increase the resolution ofpossible imagery, however the level of thrust increasesquickly to a level that becomes unrealistic, as was shownin [21]. Therefore, we limit our analysis to this range ofdistances, and refer to the same work for a more detaileddiscussion of the different types of pole-sitter orbits.

4. Spacecraft feedback control

Due to the instability of pole-sitter orbits (which wasverified numerically), a feedback control is necessary tokeep the spacecraft on track, counterbalancing errors andsmall perturbations that are not considered in the dynamicsfor the reference solution, as well as errors in the spacecraftmodel (e.g. degradation and non-idealness of the sail).

In this section, a linear quadratic regulator (LQR) willbe designed with this aim [28]. It is important to under-line that we would like to control the spacecraft usingonly the thruster; the reason of this choice is twofold:first, the thruster offers three independent controlvariables, and the thrust vector can in principle beoriented in any direction and have an arbitrary magnitude(this does not happen with the sail force): this guaranteesthe controllability of the system; second, we assume thatsteering the thruster or varying the thrust magnitude ischeaper and quicker (in terms of angular momentumchange) than changing the attitude of the solar sail. Since

Fig. 9. Simulink block scheme for

the dynamics of the orbit are relatively slow, we neglectthe response time of the actuator when designing thefeedback control. Also, we assume that the spacecraft hasperfect knowledge of its position in space at any time. Thiscan be achieved with a proper tracking system from Earth.

The feedback control is designed as an additionalcontrol component, which sums to the reference (orfeedforward) control. The control system is implementedusing MATLAB/Simulink (Fig. 9).

In this paper, a superscript tilde (�) will denote thereference values, as given by the trajectory optimisation.The reference control is transformed into cone/clockangles, such that the control vector over time becomes:

~uðtÞ ¼ a d aT dT T� �T

where aT and dT are the clock and cone angles of thethrust vector T, respectively, defined in an analogous wayas for n. This is done to reduce the control vector to theminimum number of variables and avoid the additionalpath constraint in Eq. (4), with respect to the Cartesianversion. The reference control is fed into the dynamics ofthe real system, after passing though a saturation block,which enforces the following control bounds

ul ¼ 0 �1 �1 �1 0� �T

uu ¼ p=2 þ1 þ1 þ1 Tmax

h iT

For the feedback control loop, the mass state iscompletely neglected (there is no necessity to follow thereference mass), therefore the reference state vector issimply ~xðtÞ ¼ ½r,v�T .

The control shall be designed such as to follow a time-varying reference in a non-linear system. Therefore, inprinciple, the gain matrix shall be time-variant and designedusing adaptive control theory. However, if we assume that

the feedback control loop.

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821816

the dynamics of the reference trajectory is slow enough, thenwe can approximate the time varying problem as a sequenceof time-invariant problems, and use classic linear feedbackcontrol theory for computing the gain matrix. However, theoptimal control problem shall be solved at each instant oftime, and the gain matrix updated. The disadvantage of sucha control method is that the gain matrix shall be computedin real-time, as a function of the current spacecraft state,with computational load on the spacecraft, and cannot becomputed offline.

We consider a generic instant of time t: from thetrajectory design, it is known that at that time the space-craft shall be at reference state x¼ ~xðtÞ and with referencecontrol u¼ ~uðtÞ. However, in general, the spacecraft willbe at a state x with control u. If we assume that the realstate and control are not distant from the reference stateand control, the dynamics of the CR3BP with control canbe linearised around the reference state and control, inthe following way

A6�6ðx,uÞ ¼@f

@x

����x,u

; B6�3ðx,uÞ ¼@f

@½aT ,dT ,T�

����x,u

where the matrix B is computed differentiating only withrespect to the controls that will be used (those related tothe thruster). The derivatives can be computed analyti-cally offline starting from the expression of f, and thenevaluated at a specific point. However their completeexpression is omitted here for conciseness.

Hence the dynamics of the system in the vicinity of x,ucan be expressed as

d _x ¼AdxþBdu ð6Þ

where dx¼ x�x and du¼ ½aT ,dT ,T�T�½aT ,dT ,T�T representthe error with respect to the reference condition (seeFig. 10).

In order for the control to be valid, the spacecraft shallbe in the vicinity of the reference at any time (i.e. linearapproximation).

Within the linear dynamics, the problem is that offollowing a linearised time-varying reference trajectory,subject to the (linearised) dynamics given by Eq. (6). It canbe proven that if the reference follows linear dynamics, thenthe optimal gain matrix is the same as in the case of a

Reference

( ) ( ),t tx uLinearised dynamics of the reference

O

Linearisation around ( )tx

( )t=x x

( )tδx

Fig. 10. Linearisation around the reference.

system pursuing dx¼0, except that this time the transient ison the error with respect to the linearised dynamics of thereference, rather than on dx. However, in this case, since thelinearisation is done around the reference, and it is updatedat every instant of time, it results that the reference statecorresponds to dx¼0 at any time. Therefore, pursuing nullerror is equivalent to dx¼0. This can also be explainedconsidering that the feedback control is responsible fortackling small excursions off the reference path, while thereference control is responsible for ensuring that the space-craft follows the reference states.

The optimal control problem to solve is the one ofminimising the quadratic cost function:

JðduÞ ¼Z 1

0ðdxT QdxþduT Rduþ2dxT NduÞdt ð7Þ

which aims at minimising the state error and the controlsover an infinite amount of time, subject to the dynamicsin Eq. (6). The matrices Q, R and N are weights thatquantify the relative cost of each state and control in thecost function. For this problem, considering the normal-isation of the variables, by trial and error they were set to

Q ¼ 1000UI6�6

R¼ diagð½1 1 1033�Þ

N¼ 06�3

The large number in R can be explained by the factthat the unit of force is small in the non-dimensionalsystem (1 N is of the order of 10�18).

Minimising Eq. (7) under the assumption of a controlproportional to the state error, as du¼�Kdx, leads to thewell known algebraic Riccati equation [28], which can besolved to compute the gain matrix K3�6 for A, B.

This control is added to the reference, to find the totalcontrol to be applied:

u¼ ~uþ 0 0 du� �T

to force the spacecraft to follow the reference states.

4.1. Test cases

We consider a North Pole observation mission in whichthe maximum distance of the payload from the Earth isconstrained to dmax¼0.018 AU. The spacecraft mass at injec-tion into the orbit is m0¼1000 kg and the lightness numberis b0¼0.05, corresponding to a sail size of about 180 m�180 m. The specific impulse of the thruster is 3200 s.

The optimal orbit for this spacecraft, which will beused as a reference, is represented in dotted black in thefollowing figures (see for example Fig. 12c), and itsreference control is in Fig. 11, split into angular compo-nents and thrust magnitude. The maximum thrust avail-able from the SEP system is set to Tmax¼0.15 N. It wasnoted that the performances are very similar on thedifferent orbits proposed in the preceding sections.

We present three different scenarios in which thefeedback control plays a fundamental role to guaranteethe success of the mission.

−1

0

1

2

α, δ

, rad

−2

0

2

4

α T, δ

T, ra

d

0 50 100 150 200 250 300 3500

0.05

0.1

T, N

Time, d

α

α

δ

δ

Fig. 11. Reference control for selected orbit.

−1

0

1

2x 105

Pos

ition

, km

rrr

−40

−20

0

20

Vel

ocity

, m/s

vvv

0 50 100 150 200 250 300 350−2

−1

0

1

2

Mas

s, k

gTime, d

−1

0

1

2

3

α T, δ

T, ra

d

0 50 100 150 200 250 300 3500

0.05

0.1

T, N

Time, d

F/BRef

F/BRef

δ

α

0.995

1

1.005

−5

0

5

x 10−3

0.014

0.016

0.018

0.02

x, AU

y, AU

z, A

U

Initial positionRefNo F/BWith F/B

Fig. 12. Initial error of Drx¼100,000 km. (a) State error, (b) controls

(dashed line is the reference) and (c) trajectory (reference, without

control and with control).

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 817

4.1.1. Injection error

In this first scenario, we hypothesise an injection errorof the spacecraft into the pole-sitter orbit starting point.In other words, we consider a perturbed initial state(in position or velocity) at the winter solstice, and startpropagating from that state. The aim is to assess thecapability of the feedback control to bring the spacecraftto the reference orbit.

It was found that the control is able to recover thespacecraft within 20,000 km and 20 m/s from the referencein less than 1 year if the initial displacement is less than100,000 km in any direction (x, y or z), and less than 100 m/sin velocity (being more sensitive to variations of vy).

Figs. 12 and 13 show, respectively, the cases of aninjection error in position and velocity. In each figure, plot(a) shows the state error dx with respect to the reference.This vector shall be driven to zero by the feedback controlloop. Even if the mass state is not followed, it is plotted toshow the gain or loss of mass with respect to referenceconditions. Plot (b) shows the control history includingthe feedback. For sake of comparison, the referencecontrol is also plotted in the same graph, as a dashedline. Finally, plot (c) represents the reference trajectory,the actual trajectory of the spacecraft and the trajectory ofthe spacecraft if no feedback control is used.

The history of the error in the state vector with respectto the reference (a) shows the initial displacement of onecomponent, then a transient period in which all thecomponents are far from the reference, and finally settle-ment in its neighbourhood.

Comparing thrust magnitude history (b) with the refer-ence, it can be seen that an initial saturation of the thrust isnecessary to bring the spacecraft back to the reference state.Also, oscillations around the reference thrust can be seen,even in the vicinity of the reference state, as is common inmany feedback control loops.

If the tests are repeated with increasing values of initialdisplacement (either position or velocity), the control satu-rates for a longer and longer time at the beginning of the

−4

−2

0

2x 105

Pos

ition

, km

rrr

−100

−50

0

50

100

Vel

ocity

, m/s

vvv

0 50 100 150 200 250 300 350−6

−4

−2

0

Mas

s, k

g

Time, d

−1

0

1

2

3

α T, δ

T, ra

d

0 50 100 150 200 250 300 3500

0.05

0.1

T, N

Time, d

F/BRef

F/BRef

δ

α

0.9951

1.005

−5

0

5

x 10−3

0

5

10

15

x 10−3

x, AUy, AU

z, A

U

Initial positionRefNo F/BWith F/B

Fig. 13. Initial error of Dvz¼–100 m/s. (a) State error, (b) controls

(dashed line is the reference) and (c) trajectory (reference, without

control and with control).

−10

−5

0

5x 104

Pos

ition

, km

rrr

−10

0

10

20

Vel

ocity

, m/s

vvv

0 50 100 150 200 250 300 350−8

−6

−4

−2

0

Mas

s, k

gTime, d

−1

0

1

2

3

α T, δ

T, ra

d

0 50 100 150 200 250 300 3500

0.05

0.1

T, N

Time, d

F/BRef

F/BRef

δ

α

0.995

1

1.005

−6−4

−20

24

6

x 10−3

0.01

0.012

0.014

0.016

x, AU

y, AU

z, A

U

Initial positionRefNo F/BWith F/B

Fig. 14. Sail degradation: reflectance reduced to rs¼0.8. (a) State error,

(b) controls (dashed line is the reference) and (c) trajectory (reference,

without control and with control).

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821818

orbit, and eventually even the saturated thrust is notenough to bring the spacecraft on-track and the trajectorydiverges indefinitely from the reference.

It is worth noting that an error in the injection doesnot necessarily translate into additional propellant con-sumption to recover the reference trajectory. However, it

2x 105

r

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 819

always implies a transient trajectory that is not fulfillingthe pole-sitter constraint, i.e. the spacecraft is not abovethe pole.

Finally, the trajectory plot (c) clearly shows that, if nocontrol were used, the spacecraft would not be able totrack the reference, but instead would fall into an orbitaround the Earth, or escape from it.

−2

−1

0

1

Pos

ition

, km

rr

−100

−50

0

50

100

Vel

ocity

, m/s

vvv

0 50 100 150 200 250 300 350−5

0

5

Mas

s, k

g

Time, d

1

2

3

, δT,

rad

F/BRef

α

4.1.2. Sail degradation

The sail optical degradation is a continuous process,whose modelling requires the integration of additionaldifferential equations. However, this is beyond the scopeof this paper. In the current framework, we assess theimpact of a guessed one-step-reduced reflectance on thefeedback control loop. We consider the worst-case sce-nario in which the actual sail reflectance is reduced tors¼0.8 from the beginning of the mission. This scenariocan also model the presence of unexpected wrinkles afterdeployment.

As it can be seen from Fig. 14, the thruster is able tocompensate for the degradation of the sail, keeping thespacecraft in the vicinity of the reference. However,differently from the case of error in the injection, therequired thrust is constantly above the reference, andthis translates into additional propellant consumption.For the considered degradation, the propellant needed isabout 7.5 kg in the first year. This value can be lowered ifa new reference optimal trajectory, or at least control law,could be redesigned taking into account the degradedsolar sail.

−1

0α T

0 50 100 150 200 250 300 3500

0.05

0.1

T, N

Time, d

F/BRef

δ

10

15

x 10−3

, AU

Initial positionRefNo F/BWith F/B

4.1.3. SEP failure

This case involves a sudden, temporary failure of theSEP thruster. Also in this case the feedback control canbring back the spacecraft on-track, provided that theduration of the failure is below a certain limit, whichdepends on the position along the orbit. Table 4 reportsthe maximum failure duration in four different positionsalong the orbit. As expected, the most critical period iswinter, in which the part of total acceleration provided bythe SEP is highest. Here the SEP cannot be switched off forlonger than 20 days, otherwise it would be impossible torecover the mission. In the summer solstice, instead, thefailure time can go up to 35 days. These data refer to thefirst year of the mission. In the following years, the massof the spacecraft is lower (while the thrust remains thesame), and therefore it is expected an increase in thesevalues.

Table 4Maximum SEP failure duration.

Starting epoch Maximum

duration (d)

21 December (t¼0) 20

21 March (t¼p/2) 30

21 June (t¼p) 35

21 September (t¼3p/2) 30

Fig. 15 shows one case of SEP failure. The period of 20days without thrust is visible in plot (a), and it is followedby about 35 days of full thrust to meet the reference orbit.

0.9951

1.005

−5

0

5

x 10−3

5

x, AU

y, AU

z

Fig. 15. SEP failure on 21 March (t¼p/2), lasting for 20 days. (a) State

error, (b) controls (dashed line is the reference) and (c) trajectory

(reference, without control and with control).

M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821820

Note that even a failure of 20 days, as in figure, resultsin an additional mass consumption of only about 1 kg (seeFig. 15c). In fact, the mass used to bring the spacecraft ontrack after the failure is almost completely compensatedfor by the mass saved during the engine failure. However,as before, during the transient the spacecraft is notprecisely above the North Pole.

5. Conclusion

In this work, the hybridisation of solar electric propul-sion (SEP) and solar sailing was proposed for a pole-sittermission. In addition, a feedback control system wasdesigned to keep the spacecraft on-track under unstabledynamics.

Minimum fuel consumption pole-sitter orbits werepresented using hybrid solar electric propulsion (SEP)and solar sail propulsion. A direct method based onpseudo-spectral discretisation was demonstrated to bereliable and suitable for this aim, guaranteeing fast con-vergence to the optimal solution. The combination of thetwo propulsion systems allows propellant saving over thepure SEP spacecraft, potentially enabling longer missions;at the same time, the presence of low-thrust propulsionenables orbits that cannot be achieved with a pure sail.The analysis took into account free-distance orbits, as wellas orbits that constrain their distance from the Earth tomeet possible payload requirements, at the cost of somepropellant mass.

The proposed feedback control proved that is possibleto maintain the spacecraft in the reference condition,despite the instability of the natural dynamics, only byusing the SEP thruster. Moreover, it was shown thatvery little variation on the reference thrust vector areenough to respond to large injection errors, and relativelylong SEP failures. This allows keeping the sail on thereference attitude, and avoiding fast slew manoeuvres ofthe sail.

Acknowledgements

This work was funded by the European ResearchCouncil, as part of project 227571 VISIONSPACE. Theauthors thank Dr. James Biggs for useful discussions andDr. Victor Becerra for providing PSOPT.

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M. Ceriotti, C.R. McInnes / Acta Astronautica 69 (2011) 809–821 821

Dr. Matteo Ceriotti received his M.Sc. summacum laude from Politecnico di Milano (Italy) in2006 with a thesis on planning and schedulingfor planetary exploration. In 2010, he receivedhis Ph.D. on ‘‘Global Optimisation of MultipleGravity Assist Trajectories’’ from the Depart-ment of Aerospace Engineering of the Univer-sity of Glasgow, United Kingdom. Since 2009,Matteo is a research fellow at the AdvancedSpace Concepts Laboratory, University ofStrathclyde, Glasgow (http://www.strath.ac.uk/space), leading the research theme ‘‘Orbital

Dynamics of Large Gossamer Spacecraft’’. The

current research topic is non-Keplerian motion of spacecraft with hybridsolar sail and solar electric propulsion. His main research interests arespace mission analysis and trajectory design, orbital dynamics, trajectoryoptimisation, and spacecraft autonomy. He is AIAA member.

Colin McInnes is a Director of the AdvancedSpace Concepts Laboratory at the Universityof Strathclyde. His work spans highly non-Keplerian orbits, orbital dynamics and mis-sion applications for solar sails, spacecraftcontrol using artificial potential field meth-ods and is reported in over 100 journalpapers. Recent work is exploring newapproaches to spacecraft orbital dynamicsat extremes of spacecraft length-scale tounderpin future space-derived products andservices. McInnes has been the recipient of

awards including the Royal Aeronautical

Society Pardoe Space Award (2000), the Ackroyd Stuart Propulsion Prize(2003) and a Leonov medal by the International Association of SpaceExplorers (2007).