HAL Id: hal-03185532https://hal.inria.fr/hal-03185532v2
Preprint submitted on 14 Apr 2021 (v2), last revised 24 Nov 2021 (v3)
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Control of a Solar Sail: An augmented version ofβEffective Controllability Test for Fast Oscillating
Control Systems. Application to Solar SailingβAlesia Herasimenka, Jean-Baptiste Caillau, Lamberto DellβElce, Jean-Baptiste
Pomet
To cite this version:Alesia Herasimenka, Jean-Baptiste Caillau, Lamberto DellβElce, Jean-Baptiste Pomet. Control ofa Solar Sail: An augmented version of βEffective Controllability Test for Fast Oscillating ControlSystems. Application to Solar Sailingβ. 2021. οΏ½hal-03185532v2οΏ½
Control of a solar sailAn augmented version of βEffective Controllability Test
for Fast Oscillating Control Systems. Application to Solar Sailingβ
Alesia Herasimenkaβ Jean-Baptiste CaillauUniversitΓ© CΓ΄te dβAzur, CNRS, Inria, LJAD, France
Lamberto DellβElce Jean-Baptiste PometInria, UniversitΓ© CΓ΄te dβAzur, CNRS, LJAD, France
April 13, 2021
Abstract
Geometric tools for the assessment of local controllability often require that the control sethas the origin within its interior. This study gets rid of this assumption, and investigates thecontrollability of fast-oscillating dynamical systems subject to positivity constraints on the controlvariable, i.e., the convex exterior approximation of the control set is a cone with vertex at the origin.A constructive methodology is offered to determine whether the averaged state of the system withcontrols in the exterior convex cone can be locally moved to an arbitrary direction of the tangentmanifold. The controllability of a solar sail in orbit about a planet is analysed to illustrate thecontribution. It is shown that, given an initial orbit, a minimum cone angle exists which allows thesail to move slow orbital elements to any arbitrary direction.This is an extended version of a document by the same authors untitled β Effective ControllabilityTest for Fast Oscillating Control Systems. Application to Solar Sailingβ, preprint hal-03185532v1.This work was partially supported by ESA.
1 Introduction
A classical approach to study controllability of a general control system, affine with respect to thecontrol, i.e. of the form Β€π₯ = π0(π₯) + π’1π1(π₯) + Β· Β· Β· + π’πππ(π₯), is to evaluate the rank of the collection ofvector fields obtained by iterated Lie brackets the initial vector fields π0, . . . , ππ. The so called Liealgebra rank condition (LARC) requires that this rank be equal to the dimension of the state space,one also says that the family {π0, . . . , ππ} is βbracket generatingβ in this case. It is always necessary,at least in the real analytic case, but sufficiency requires additional conditions. A well known suchadditional condition (classical, stated e.g. as [1, theorem 5, chap. 4]) requires that the drift π0 berecurrent, a property that is true if all solutions of Β€π₯ = π0(π₯) are periodic, but is more general. TheLARC plus this recurrence property imply controllability if there are no constraints on the control inRπ; when the control π’ = (π’1, . . . , π’π) is constrained to a subset π of Rπ, the origin has to be in π forthe condition to be relevant, but [1, theorem 5, chap. 4] asserts controllability under the condition thatπ not only contains the origin but is a neighborhood of the origin. Here, we are interested in systemswhere the origin is on the boundary of π; this is motivated by solar sail control, see Fig. 1 where π isthe set in blue.
βEmail adress: [email protected].
Figure 1: Example of orbital control with solar sails. Equations are of the form of System (1), thecontrol π’ = (π’1, π’2, π’3) is homogeneous to a force, and the solar sail only allows forces contained in theset π figured in blue in the picture (for some characteristics of the sail). The minimal convex conecontaining the control set π is depicted in red. Neither π nor this cone are neighborhoods of the origin.
The scope of the study is limited to so called fast-oscillating dynamical systems, of the form1
d πΌd π‘
= Y
πβοΈπ=1
π’ππΉπ (πΌ, π)
d πd π‘
= π(πΌ)(1)
where Y > 0 is a small parameter, πΌ β π denotes the slow component of the state and the angleπ β S1 = R/2πZ the fast component. π is a real analytic manifold of dimension π and the state spaceis naturally π Γ S1. Each πΉπ, 1 β€ π β€ π, can be considered either as a smooth map π Γ S1 β ππ or asa vector field on π Γ S1 whose projection on the second factor of the product is zero, there will be noambiguity. From the smooth map π : π β R, one defines the drift πΉ0 = π π/ππ, it is a vector fieldon π Γ S1 whose projection on the first factor of the product is zero. The control π’ = (π’1, . . . , π’π) isconstrained to belong to some fixed bounded subset π of Rπ. When the control is a function of time, ithas to have values in π for all time. We assume π bounded to ensure that the variable πΌ is slow. Thesolutions of the differential equation associated to πΉ0 are obviously all periodic (the one starting from(πΌ0, π0) has period 2π/π(πΌ0)), so [1, theorem 5, chap. 4] yields controllability of System (1) if the LARCholds and π is a neighborhood of 0. As stated above, we are interested in cases where the LARC doeshold but 0 is on the boundary of π.
The target here is to apply this methodology to orbital control of a spacecraft in orbit around aplanet using solar sails. The possibility of using solar radiation pressure (SRP) as an inexhaustible sourceof propulsion intrigued researchers since decades and triggered research in that direction. The potentialexploitation of these devices in space missions of various nature, e.g., interplanetary transfers, planetescapes, de-orbiting was proposed, see for instance [2]. Most available contributions offer numericalsolutions to optimal transfers and locally-optimal feedback strategies, but a thorough analysis of thecontrollability of solar sails is not available, yet. Orbital control with SRP is challenging becausethe sail cannot generate a force with positive component toward the direction of the Sun (just likeone cannot sail exactly against the wind in marine sailing). When incoming SRP is only partiallyreflected (which is systematically true in real-life missions), the possible directions of the control forceare contained in a convex cone with revolution symmetry with respect to an axis throught the origin,see Fig. 1, and its angle approaches zero as the portion of reflected SRP is decreased. This is a typical
1It would be natural to also have a βsmallβ term as a perturbation on the dynamics of the fast variable. In order tofacilitate the notation, we do not consider this possibility here.
case of systems of the type (1), where the LARC is satisfied, recurrence of the drift is met from thestructure of the system, as noticed above, but the control set π, or the convex cone it generates, is nota neighborhood of the zero control. In [3], we have outlined non-controllability of these sails for somereflectivity coefficients. Here, it is shown that, for a given orbit, a minimum cone angle exists whichmakes the system controllable. In turn, this result may serve as a mission design requirement.
In Section 2, we give a sufficient condition for controllability and introduce an optimisation problemwhose solution is equivalent to checking whether that condition is satisfied. Numerical solution of theproblem is tackled in Section 3. Finally, the application of the methodology to solar sails orbital controlis discussed in Section 4.
2 Controllability of fast-oscillating systems
2.1 A condition for controllability
Consider System (1) and the associated vector fields πΉ0, . . . , πΉπ on πΓS1 defined right after Eq. (1). LetY be a small positive parameter, the drift vector field is πΉ0 and the control vector fields are YπΉ1, . . . , YπΉπ.
Proposition 1 Under the following three conditions:(0) π(.) does not vanish on π,(i) the LARC holds everywhere, i.e. {πΉ0, πΉ1, . . . , πΉπ} is bracket generating,(ii) the control set π contains the origin,(iii) for all πΌ β π,
cone
{πβοΈπ=1
π’ππΉπ (πΌ, π), π’ β π, π β S1}= ππΌπ , (2)
System (1) is controllable in the following sense: for any (πΌ0, π0) and any (πΌ π , π π ) in π Γ S1, there isa time π and an integrable control π’(.) : [0, π] β π that drives initial condition (πΌ0, π0) at time 0 to(πΌ π , π π ) at time π .
Proof . As in [4, chapter 8] or [1, chapter 3], we associate to the vector fields πΉ0, . . . , πΉπ, the family ofvector fields
E = { πΉ1 + π’1πΉ1 + Β· Β· Β· + π’ππΉπ , (π’1, . . . , π’π) β π}
made of all the vector fields obtained by fixing, in (1), the control to a constant value that belongs to π(let us put a bar to emphasize that these controls are constants and not functions of time). We denoteby πE ((πΌ, π)) the accessible set from (πΌ, π) of this family of vector fields in all positive (unspecified)time, i.e. the set of points that can be reached from (πΌ, π) by following successively the flow of a finitenumer of vector fields in E, each for a certain positive time, which is the same as the set of points thatcan be reached, for the control system (1), with piecewise constant controls. We are going to show that,under assumptions (i) to (iii), πE ((πΌ, π)) is the whole manifold π Γ S1 for any (πΌ, π) in π Γ S1. Thisobviously implies the Proposition.
Now define the families E1 and E2 (with E β E1 β E2) as follows:
E1 = E βͺ {βπΉ0} , E2 = { exp(π‘ πΉ0)β π , π β E1, π‘ β R} , E3 = cone (E2) ,
where exp(π‘ πΉ0)β π denotes the pullback of the vector field π by the diffΓ©omorphism exp(π‘ πΉ0) andcone (E2) denotes the family made of all vector fields that are finite combinations of the form
βπ _πππ
with each ππ in E2 and each _π a positive number (conic combination). One has, for all (πΌ, π),2
πE1 ((πΌ, π)) = πE ((πΌ, π))
because on the one hand condition (ii) implies πΉ0 β E, and on the other hand, for any (πΌ β², πβ²),exp(βπ‘ πΉ0) ((πΌ β², πβ²)) = exp
((βπ‘ + 2ππ/π(πΌ β²)) πΉ0
)((πΌ, π)) for all positive integers π, but for fixed π‘ and
πΌ β², βπ‘ + 2ππ/π(πΌ β²) is positive for π large enough. Since πΉ0 and βπΉ0 now belong to E1, we haveexp(π‘ πΉ0) ((πΌ, π)) β πE ((πΌ, π)) for all (πΌ, π) in π Γ S1 and all π‘ in R, hence exp(π‘ πΉ0) is according to [1,
Chapter 3, Definition 5 and the following lemma], a βnormalizerβ of the family E1 and, according toTheorem 9 in the same chapter of the same reference, this implies that2
πE2 ((πΌ, π)) β πE1 ((πΌ, π)) (3)
where the overline denotes topological closure (for the natural topology on π Γ S1). Now, [4, Corollary8.2] or [1, chapter 3, Theorem 8(b)] tell us that2
πE3 ((πΌ, π)) β πE2 ((πΌ, π)) . (4)
These inclusions are of interest because condition (iii) implies that πE3 ((πΌ, π)) is the whole manifold:indeed, (2) (written in terms of the πΌ-directions only, but adding πΉ0 and βπΉ0 yields the whole tangentspace to π Γ S1) obviously implies that πE3 ((πΌ, π)) is, for any (πΌ, π), a neighborhood of (πΌ, π), obtainedfor small times, hence accessible sets are closed and open in the connected manifold). Together with(3)-(4), this implies πE ((πΌ, π)) = π Γ S1, and finally πE ((πΌ, π)) = π Γ S1 from condition (i) and [4,Corollary 8.1]. οΏ½
Remark 1 If all vector fields are real analytic, (iii) implies (i).Indeed, if (i) does not hold, there is at least one point (πΌ, π) and a non zero element β of πβ
(πΌ ,π) (π Γ S1),i.e. a nonzero linear form β on the vector space ππΌπ Γ ππS1, such that γβ, π (πΌ, π)γ = 0 for any vectorfield π obtained as a Lie bracket of any order of πΉ0, . . . , πΉπ. In particular,
(a) γβ, πΉ0(πΌ, π)γ = 0 , (b)β¨β, (ad π
πΉ0πΉπ) (πΌ, π)
β©= 0 for π β N, π = 1, . . . , π .
From (a), β vanishes on the direction of π/ππ and hence can be considered as a nonzero linear form onππΌπ. Define the smooth maps ππ : Rβ R by ππ (π‘) =
β¨β, exp(βπ‘πΉ0)β πΉπ (πΌ, π)
β©for π = 1, . . . , π, where
the star β denotes the pullback by a diffeomorphism. Classical properties of the Lie bracket3 implyd π ππ
dπ‘ π(0) =
β¨β , (ad π
πΉ0πΉπ) (πΌ, π)
β©, and the right-hand side is zero from (b); if all the vector fields are real
analytic, so are the maps ππ , and they must be identically zero if all their derivatives at zero are zero.Seen the particular form of πΉ0, one has (exp(βπ‘πΉ0)β πΉπ) (πΌ, π) = πΉπ (πΌ, π + π‘π(πΌ) ); since π(πΌ) β 0, onefinally deduces that γβ, πΉπ (πΌ, π)γ = 0 for all π in S1; this contradicts point (iii).
Remark 2 Another point of view would be to introduce the averaged system as defined in [5]. It hasstate πΌ, its control at each time is an integrable function π’ : S1 β π (since π is bounded, π’ is then inany πΏ π, 1 β€ π β€ β; we write π’ β πΏ2(S1,π)), and its dynamics read
d πΌ
dπ‘= F (πΌ) π’(Β·) (5)
where the equation is to be understood as
d πΌ (π‘)dπ‘
= F (πΌ (π‘)) π’(π‘) (Β·)
and where F (πΌ) is the linear map πΏ2(S1,π) β ππΌπ
F (πΌ) π’(Β·) = 1
2π
β«S1
πβοΈπ=1
π’π (π)πΉπ (πΌ, π) dπ (6)
(readβ«S1
asβ« 2π
0). Conditions (i) and (iii) then imply controllability of System (5) in the sense that, for
each πΌ0 in π, there is an open dense subset A of π such that, for all πΌ π β A, there is a time π and acontrol π’ : [0, π] β πΏ2(S1,π) such that the solution of Eq. (5) starting at πΌ0 at time 0 arrives at πΌ πat time π . Here, the results from [1] are not needed, π can be taken small as πΌ π gets close to πΌ0, andA = π if π is convex. The relation with controllability of System (1) in time β π/Y occurs for small Yonly and under some conditions, according to the convergence result in [5] (established there only if πis an Euclidean ball centered at the origin, but extendable mutatis mutandis).
2 In the terminology of [4, Section 8.2], βπΉ0 is compatible with E, the vector fields in E2 are compatible with E1, andthe vector fields in E3 are compatible with E2.
3For any two vector fields π and π, one has ddπ‘ (exp(βπ‘π )β π) (πΌ, π) = (exp(βπ‘π )β [π, π]) (πΌ, π).
Remark 3 (Localisation) Assume that (i) holds everywhere (it is the case of our satellite application).If condition (iii) is only known to hold at one point πΌ in π, it remains true in a neighbourhood π
of πΌ. Although the results from [1] that we invoked cannot be localized in general, this very specialconfiguration allows one to have a local result on π Γ S1 in that case because the augmented family ofvector fields in the sense of [1, chapter 3, section 2] allows to go in all directions and is augmented bytransporting only along solutions of the drift, that does not move the slow variable πΌ.
Condition (i) can be checked via a finite number of differentiations, and (ii) by inspection. One goalof this paper is to give a verifiable check, relying on convex optimisation, of the property (iii) at a pointπΌ, and to give consequences on solar sailing.
2.2 Formulation as an optimisation problem
Assume condition (iii) does not hold at some point πΌ in π. Then the set generating the convex cone in(iii) is contained in some half-space, and there exists a nonzero ππΌ in πβ
πΌπ such that, for all π’ in π and
all π in S1, β¨ππΌ ,
πβοΈπ=1
π’ππΉπ (πΌ, π)β©β€ 0.
Actually this property even holds true for all π’ in
πΎ := cone(π). (7)
This implies the weaker property that, for any square integrable control function π’ : S1 β πΎ,
γππΌ , F (πΌ) π’γ β€ 0 (8)
with F defined in Eq. (6). Let π0, . . . , ππ in ππΌπ be the vertices of an π-simplex containing 0 in itsinterior. The negation of Eq. (8) is that, for all π = 0, . . . , π, there exists a control π’ in πΏ2(S1,Rπ)valued in πΎ such that
2π F (πΌ) π’ = ππ .
This condition together with the cone constraint of Eq. (7) define feasibility conditions for the optimalcontrol problem over ππΌπ (remember that πΌ is fixed) with quadratic costβ«
S1|π’(π) |2 dπ β min . (9)
This is indeed a control problem associated with the following dynamics with state πΏπΌ β ππΌπ and timeπ, where πΌ appears as a fixed parameter:
d πΏπΌ
dπ=
πβοΈπ=1
π’π πΉπ (πΌ, π),
with control constraint π’ β πΎ and boundary conditions πΏπΌ (0) = 0 and πΏπΌ (2π) = ππ . Note that we havechosen a quadratic cost in order to preserve the convexity of the problem.
Remark 4 Given πΌπ such that there is some curve in π connecting πΌ0 to πΌπ with tangent vectorππ at πΌ0, Problem (9) can serve as a proxy to find an admissible curve for the averaged dynamics ofSystem (5); this proxy being all the more accurate as πΌπ is close to πΌ0 in the manifold of slow variables.
We show in the next section that this problem is accurately approximated by a convex optimisationafter approximating πΎ by a polyhedral cone and truncating the Fourier series of the control. Eventually,by proving feasibility of these control problems for every vertex ππ , one has an effective check of localcontrollability around πΌ of the original problem.
3 Discretisation of the semi-infinite optimisation problem
The discretisation of Problem (9) is achieved in two steps. First, πΎ is approximated by the polyhedralcone πΎπ β πΎ generated by πΊ1, . . . , πΊπ chosen in ππΎ: admissible controls are given by a conicalcombination of the form
π’(π) =πβοΈπ=1
πΎ π (π)πΊ π , πΎ π (π) β₯ 0, π β S1, π = 1, . . . , π.
Second, an π-dimensional basis of trigonometric polynomials, Ξ¦(π) =(1, πππ , π2ππ , . . . , π (πβ1)ππ ), is used
to model functions πΎ π asπΎ π (π) = (Ξ¦(π) |π π)π»
where π π β Cπ are complex-valued vectors (serving as design variables of the finite-dimensional problem),and where (Β·|Β·)π» is the Hermitian product on Cπ . Positivity constraints on the functions πΎ π define asemi-infinite optimisation problem; these constraints are enforced by leveraging on the formalism ofsquared functional systems outlined in [6] which allows to recast continuous positivity constraints intolinear matrix inequalities (LMI). Specifically, given a trigonometric polynomial π(π) = (Ξ¦(π) |π)π» andthe linear operator Ξβ : CπΓπ β Cπ associated to Ξ¦(π) (more details are provided in Appendix A), itholds
(βπ β S1) : π(π) β₯ 0 ββ (β π οΏ½ 0) : Ξβ(π ) = π.
For an admissible control π’ valued in πΎπ, one hasβ«S1
πβοΈπ=1
π’π (π)πΉπ (π) dπ =
πβοΈπ=1
(πΏ ππ π + οΏ½ΜοΏ½ ππ π
)with πΏ π (πΌ) in CπΓπ defined by
πΏ π (πΌ) =1
2
πβοΈπ=1
β«S1πΊπ ππΉπ (πΌ, π)Ξ¦π» (π) dπ,
where Ξ¦π» (π) denotes the Hermitian transpose and where πΊ π = (πΊπ π)π=1,...,π. We note that thecomponents of πΏ π (πΌ) are Fourier coefficients of the function
βππ=1πΊπ ππΉπ (πΌ, π). The discrete Fourier
transform (DFT) can be used to approximate πΏ π (πΌ). Since vector fields πΉπ are smooth, truncation ofthe series is justified by the fast decrease of the coefficients. Finally, for a control π’ valued in πΎπ withcoefficients πΎ π that are truncated Fourier series of order π β 1, the πΏ2 norm over S1 is easily expressedin terms the coefficients π π using orthogonality of the family of complex exponentials:
1
2
β«S1
|π’(π) |2 dπ =1
2
πβοΈπ ,π=1
πβ1βοΈπ=0
πΊππ πΊ π
(ππππ ππ + ππππ ππ
)=
πβοΈπ, π=1
πΊππ πΊ π (π π |ππ)π» .
As a result, for every vertex ππ , the finite-dimensional convex programming approximation ofProblem (9) is
minπ π βCπ , ππ βCπΓπ
πβοΈπ ,π=1
πΊππ πΊπ (π π |ππ)π» subject to
πβοΈπ=1
(πΏ ππ π + οΏ½ΜοΏ½ ππ π
)= ππ
π π οΏ½ 0, Ξβ (π π
)= π π , π = 1, . . . , π.
(10)
4 Controllability of a non-ideal solar sail
4.1 Orbital dynamics
The equations of motion of a solar sail in orbit about a planet are now introduced. Consider a referenceframe with origin at the center of the planet, π₯1 toward the Sun-planet direction, π₯2 toward an arbitrarydirection orthogonal to π₯1, and π₯3 completes the right-hand frame. Slow variables consist of Eulerangles denoted πΌ1, πΌ2, πΌ3 orienting the orbital plane and perigee via a 1 β 2 β 1 rotation as shown inFig. 2. Then, πΌ4 and πΌ5 are semi-major axis and eccentricity of the orbit, respectively. These coordinatesdefine on an open set of R5 a standard local chart of the five-dimensional configuration manifold π
[7]. The fast variable, π β S1, is the mean anomaly of the satellite. The motion of the sail is governedby Eq. (1). Vector fields πΉπ (πΌ, π) are detailed in Appendix B, and π = 3. These fields are deduced byassuming that:
(i) Solar eclipses are neglected
(ii) SRP is the only perturbation
(iii) Orbit semi-major axis, πΌ4, is much smaller than the Sun-planet distance (so that radiation pressurehas reasonably constant magnitude)
(iv) The period of the heliocentric orbit of the planet is much larger than the orbital period of the sail(so that motion of the reference frame is neglected).
We note that removing the first assumption may be problematic, since eclipses would introducediscontinuities (or very sharp variations) in the vector fields, which could jeopardise the convergence ofDFT coefficients. Other assumptions are only introduced to facilitate the presentation of the resultsand are not critical for the methodology.
Figure 2: Orbital orientation using Euler angles πΌ1, πΌ2, πΌ3. Here, β and π denote the angular momentumand eccentricity vectors of the orbit.
4.2 Solar sail models
Solar sails are satellites that leverage on SRP to modify their orbit. Interaction between photonsand sailβs surface results in a thrust applied to the satellites. Its magnitude and direction dependon several variables, namely distance from Sun, orientation of the sail, cross-sectional area, opticalproperties (reflectivity and absorptivity coefficients of the surface) [8]. A realistic sail model combinesboth absorptive and reflective forces. Here, a simplified model is used by assuming that the sail isflat with surface π΄, and that only a portion π of the incoming radiation is reflected in a specularway (π β [0, 1] is referred to as reflectivity coefficient in the reminder). Hence, denoting π the unitvector orthogonal to the sail, πΏ the angle between π and π₯1 (recall that π₯1 is the direction of the Sun),π‘ = sinβ1 πΏ π₯1 Γ (π Γ π₯1) a unit vector orthogonal to π₯1 in the plane generated by π and π₯1, the force permass unit, π, of the sail is given by
πΉ (π) = π΄π
πcos πΏ [(1 + π cos 2πΏ) π₯1 + π sin 2πΏ π‘]
where π is the SRP magnitude and is a function of the Sun-sail distance. By virtue of assumption (iii),π is assumed to be constant, and the small parameter Y is set to Y = π΄π/π. Control set is thus given by
π =
{πΉ (π)Y
, β π β R3, |π| = 1, (π|π₯1) β₯ 0
}Fig. 3 shows control sets for different reflectivity coefficients of the sail. When π = 0 the sail is perfectlyabsorptive. In this particular case, Lie algebra of the system is not full rank. Conversely, π = 1represents a perfectly-reflective sail, which is the ideal case. This set is symmetric with respect to π₯1,and πΎ = cone(π) is a circular cone with angle obtained by solving
tanπΌ = minπΏβ[0, π/2]
(πΉ (π) | π‘)(πΉ (π) | π₯1)
= minπΏβ[0, π/2]
π sin 2πΏ
(1 + π cos 2πΏ)
which yields
πΌ = tanβ1
(πβοΈ
1 β π2
)(11)
Fig. 1 and 4 show the minimal convex cone of angle πΌ including the control set.
0 1 2-1
0
1 = 0
= 0.25
= 0.5
= 0.75
= 1
Figure 3: Control sets for different reflectivity coefficients π
Figure 4: Convexification of the control set
4.3 Simulation and results
Problem (10) is solved by means of CVX, a package for specifying and solving convex programs [9], [10].Table 1 lists initial conditions and parameters used for the simulations. Because of the symmetries ofthe problem, the results do not depend on πΌ1 (first Euler angle) or πΌ4 (semi-major axis), so we havenot included them in the table. Figure 5 shows the magnitude of Fourier coefficients of vector fields.Polynomials are truncated at order 10. At this order, the magnitude of coefficients is reduced of a
Table 1: Simulation parametersInitial conditions
πΌ2 20 degπΌ3 30 degπΌ5 0.5
Constants for Figs. 7 and 8DFT order, π 10Number of generators, π 10Direction of displacement π5Cone angle, πΌ 80 deg
0 5 10 1510
-4
10-3
10-2
10-1
100
Figure 5: Convergence of coefficients of the DFT. The norm is evaluated asβοΈβπ
οΏ½οΏ½οΏ½β«S1πΉπeπππdπ
οΏ½οΏ½οΏ½2/βοΈβπ
οΏ½οΏ½οΏ½β«S1πΉπdπ
οΏ½οΏ½οΏ½2.factor 103 with respect to zeroth-order terms. The possibility to truncate polynomials at low-order isconvenient when multiple instances of Problem (10) need to be solved.
A major takeoff of the proposed methodology is the assessment of a minimum cone angle required tohave local controllability of the system. To this purpose, Problem (10) is solved for various πΌ between0 and 90 deg, and for all πΒ±π = Β±π/ππΌπ , π = 1, . . . , π (these vertices do not define a simplex, but theresulting computation is obviously sufficient to test local controllability). The minimum cone anglenecessary for local controllability is the smallest angle such that Problem (10) is feasible for all vertices.When it is the case, we define Zβ(πΒ±π) to be the inverse of the value function,
Zβ(πΒ±π) =2
βπ’β22,
and set Zβ(πΒ±π) = 0 when the problem is not feasible. For the orbit at hand, feasibility occurs forπΌ = 19.4 deg as depicted in Fig. 6. This angle may serve as a minimal requirement for the design of thesail. Specifically, the reflectivity coefficient associated to this cone angle can be evaluated by invertingEq. (11), namely
π =tanπΌ
β1 + tan2 πΌ
Β·
In the example at hand, π ' 0.3 is the minimum reflectivity that satisfies the controllability criterion(indeed, the precise value of the minimum π depends on orbital conditions). In addition, opticalproperties degrade in time [11], so that this result may be also used to investigate degradation of thecontrollability of a sail during its lifetime.
Figures 7 and 8 show controls and trajectory for π5 = π/ππΌ5 (i.e., increase of orbital eccentricity)with πΌ = 80 deg. Periodic control obtained as solution of Problem (10) is applied for several orbits.The displacement of the averaged state is clearly toward the desired direction, namely all slow variables
0 30 60 900
10
20
Figure 6: Grey lines show the resulting displacement Zβ toward all positive and negative base vectors,i.e., πΒ±π = Β±π/ππΌπ , π = 1, . . . , π. Black line shows the minimum of these curves. The minimal angle πΌensuring local controllability is highlighted in red. One can notice that some curves do not strictlyincrease, but are constant instead. It means that the control is inside the cone, and increasing πΌ doesnot change the result.
0 100 200 3000
(a) Black line shows control in Sun direction, the redone combines the two other components. When theycoincide, the control is on the coneβs boundary.
(b) The polyhedral approximation πΎπ of πΎ.
Figure 7: Control force solution of Problem (10).
but πΌ5 exhibit periodic variations, while πΌ5 has a positive secular drift. The structure of the control arcsis such that control is on the surface of the cone in the middle of the maneuver whereas it is at theinterior at the beginning and end. We note that no initial guess is required to solve Problem (10). Assuch, a priori knowledge of this structure is not necessary.
5 Conclusions
A methodology to verify local controllability of a system with conical constraints on the control set wasproposed. A convex optimisation problem needs to be solved to this purpose. Controllability of solarsails is investigated as case study, and it is shown that a minimum cone angle πΌ exists that satisfies theproposed criterion. This angle yields a minimum requirement for the surface reflectivity of the sail.
6 Acknowledgments
The authors thank Ariadna FarrΓ©s for her help on the solar sail application.
1
1.0005
1.001
1.0015
1.002 I4
I5
0 2 4 6 8 10-4
-2
0
2
410
-3
I1
I2
I3
Figure 8: For verification, controls resulting from the optimisation problem are injected into realdynamical equations. Plots of trajectories of slow variables correspond to the desired movement(increase of eccentricity, πΌ5). Moreover, this trajectory is stable over multiple orbits.
A. Positive polynomials
Consider the basis of trigonometric polynomials Ξ¦ = (1, πππ , π2ππ , . . . , π (πβ1)ππ). Its correspondingsquared functional system is S2(π) = Ξ¦(π)Ξ¦π» (π) where Sπ» (π) denotes conjugate transpose of S(π).Let Ξπ» : Cπ β CπΓπ be a linear operator mapping coefficients of polynomials in Ξ¦(π) to the squaredbase, so that application of Ξπ» on Ξ¦(π) yields
Ξπ» (Ξ¦(π)) = Ξ¦(π)Ξ¦π» (π)
and define its adjoint operator Ξβπ»
: CπΓπ β Cπ as
(π |Ξπ» (π))π» β‘ (Ξβπ» (π ) |π)π» , π β CπΓπ , π β Cπ .
Theory of squared functional system postulated by Nesterov [6] proves that polynomial (Ξ¦(π) |π)π»is non-negative for all π β S1 if and only if there is a Hermitian positive semidefinite matrix π such thatπ = Ξβ
π»(π ), namely
(βπ β S1) : (Ξ¦(π) |π)π» β₯ 0 ββ (βπ οΏ½ 0) : π = Ξβπ» (π ).
In fact in this case it holds
(Ξ¦(π) |π)π» = (Ξ¦(π) |Ξβπ» (π ))π» = (Ξπ» (Ξ¦(π)) |π )π» ,
= (Ξ¦(π)Ξ¦π» (π) |π )π» = Ξ¦π» (π)πΞ¦(π) β₯ 0.
For trigonometric polynomials Ξβ is given by
Ξβ(π ) =
(π |π0)...
(π |ππβ1)
where ππ π = 0, . . . , π β 1 are Toeplitz matrices such that
π0 = πΌ, π(π,π)π
=
{2 if π β π = π
0 otherwise π = 1, . . . , π β 1
B. Equations of motion of solar sails
Slow component of the state vector consists of three Euler angles, which position the orbital plane andperigee in space, πΌ1, πΌ2, πΌ3, and of the semi-major axis and eccentricity, πΌ4 and πΌ5, respectively. Meananomaly is the fast variable, π. Keplerβs equation is used to relate π to the eccentric anomaly, π, andthen to the true anomaly, \, as
π = π β πΌ5 sinπ, tan\
2=
βοΈ1 + πΌ51 β πΌ5
tanπ
2
Vector fields of the equations of motion are given by
πΉπ =
3βοΈπ=1
π π π πΉ(πΏπ πΏπ» )π
where, π π π are components of the rotation matrix from the reference to the local-vertical local-horizontalframes,
π = π 1(πΌ3 + \)π 2(πΌ2)π 1(πΌ1)
(π π (π₯) denoting a rotation of angle π₯ about the π-th axis), and vector fields πΉ (πΏπ πΏπ» )π
can be deducedfrom Gauss variational equations (GVE) expressed with classical orbital element [12] by replacingthe right ascension of the ascending node, inclination, and argument of perigee with πΌ1, πΌ2, and πΌ3,respectively. Rescaling time such that the planetary constant equals 1, these fields are
πΉ(πΏπ πΏπ» )1 =
βοΈπΌ4
(1 β πΌ25
)
00
βcos \πΌ5
2πΌ4 πΌ5
1 β πΌ25sin \
sin \
πΉ(πΏπ πΏπ» )2 =
βοΈπΌ4
(1 β πΌ25
)
00
2 + πΌ5 cos \1 + πΌ5 cos \
sin \
πΌ5
2πΌ4 πΌ5
1 β πΌ25(1 + πΌ5 cos \)
πΌ5 cos2 \ + 2 cos \ + πΌ51 + πΌ5 cos \
πΉ
(πΏπ πΏπ» )3 =
βοΈπΌ4
(1 β πΌ25
)1 + πΌ5 cos \
sin (πΌ3 + \)sin πΌ2
cos (πΌ3 + \)βsin (πΌ3 + \) cos πΌ2
sin πΌ200
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