Nonlinear Orbit Control with Longitude Tracking

Nonlinear Orbit Control with Longitude Tracking

Mirko Leomanni, Gianni Bianchini, Andrea Garulli, Antonio Giannitrapani

Abstract— The growing level of autonomy of unmanned space the coordination of multiple spacecraft does indeed require to missions has attracted a significant amount of research in track the (relative) true longitude, the techniques developed the aerospace field towards feedback orbit control. Existing to this purpose often rely on linearization assumptions [15], Lyapunov-based controllers can be used to to transfer a space- craft between two elliptic orbits of given size and orientation, [16], [17], [18]. Therefore, their applicability is limited to but do not consider the stabilization of the spacecraft phase spacecraft separated by a very short distance. angle along the orbit, which is a key requirement for application A unified approach has been presented in [19]. By using to formation flying missions. This paper presents a control law backstepping and forwarding techniques (see, e.g., [20]), the based on the orbital element parametrization, which is able to authors derived a passivity-based controller able to track a track a given true longitude (i.e. a reference phase angle), in addition to the parameters describing the reference orbit shape given true longitude, in addition to five modified equinoctial and orientation. A numerical simulation of an orbital rendez- orbital elements describing the orbit shape and orientation. vous demonstrates the effectiveness of the proposed approach. Nevertheless, the obtained results are limited to the case of perfectly circular reference orbits, which leaves out many scenarios of theoretical and practical interest. It is known, I.INTRODUCTION for instance, that low altitude orbits cannot have zero eccen- A fundamental research topic in astrodynamics deals with tricity, due to the asymmetry of the Earth’s gravity field. transferring a spacecraft between two elliptic orbits. Histor- In this paper, a nonlinear control law is proposed which ically, this problem has been tackled by using optimization asymptotically stabilizes the six modified equinoctial ele- techniques [1], [2], [3], [4], or feedback stabilization methods ments, including the true longitude, of any closed orbit. The [5], [6], [7], [8]. In the former approach, no closed-form solution is arrived at by using a design procedure inspired solution is available in general and a two-point boundary by backstepping and damping control techniques. The simple value problem is solved numerically to get the optimal open- structure of the controller makes it suitable for a number of loop thrusting profile. The related computations are lengthy, space missions involving orbit reconfiguration and formation thus making this approach not suitable for applications flying maneuvers. A numerical simulation of a rendez-vous requiring on-line computation of the control signals, such as maneuver is performed to illustrate the proposed approach, formation flying or rendez-vous. Existing Lyapunov-based and to validate the obtained theoretical results. stabilization methods, on the other hand, provide simple The paper is organized as follows. In Section II, a brief feedback controllers, but usually do not consider the transfer introduction to the orbital element parametrization is given time and the injection point on the final orbit. and the considered orbit control problem is formulated. Most of the literature available on the orbit stabilization Section III is devoted to the design of the controller, which problem describes the trajectory of an orbiting body either in is demonstrated by the numerical simulation in Section IV. terms of cartesian position and velocity or by an equivalent Some concluding remarks are outlined in Section V. set of variables introduced by Kepler, known as the orbital elements. The latter parametrization is useful because it cap- Notation tures the constants of the orbital motion. References [5], [7], The notation is fairly standard. Rn is the real n−space; [8] developed nonlinear orbital element feedback schemes for a real vector or matrix v, vT denotes its transpose. To based on the Jurdjevic-Quinn conditions [9]. Similar results save space, cos(·), sin(·) are abbreviated with c(·) and s(·), have been derived in [6], by using the cartesian coordinate respectively. Moreover, representation of the orbital elements. Such techniques have c(φ) −s(φ) proven to be effective in low-thrust applications [10], [11], R(φ)= s(φ) c(φ) [12], but do not address the stabilization of the spacecraft   phase angle along the orbit. This angle is often referred to is the counter-clockwise rotation operator by an angle φ in as the true longitude. R2. The continuous time index is denoted as t ∈ R+. Motivated by the increasing number of distributed space missions, the orbit control problem is also widely discussed II. PROBLEM FORMULATION in the formation flying literature. The interested reader is Classical orbital elements are commonly used as a referred to [13], [14] for a survey on recent results. While parametrization of the position r ∈ R3 and velocity r˙ ∈ R3 of an orbiting body, since they provide a clear physical The authors are with the Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Universit`a di Siena, Siena, Italy. Email: insight of the body motion. The semi-major axis a > 0 {leomanni,giannibi,garulli,giannitrapani}@dii.unisi.it. and eccentricity e ∈ [0, 1] define the shape of the orbit. The inclination i ∈ [0, π], longitude of the ascending node Ω ∈ [0, 2π] and argument of perigee ω ∈ [0, 2π] define the ψ˙ = f(ψ)+ g(ψ)u, (2) orientation of the orbital plane with respect to a given inertial, right-handed reference frame centered at the central body where the vector fields f(ψ) and g(ψ) are given by (e.g., the Earth). The true anomaly ν(t) ∈ [0, 2π] defines the T instantaneous angle at which the spacecraft is located relative µ 2 f(ψ)= 3 (1 + ζX ) 00000 , (3) to the perigee position, as illustrated in Fig. 1. ψ2 h q i Z Inclined orbit ψ2 η 0 0 µ 1+ζX Line of nodes 3  ψ2 2 q  µ 1+ζX 0 0  q ψ2 qX ψ2 s(ψ ) − ψ2 ψ4η  Equatorial orbit  µ 1+ζX µ 1 µ 1+ζX  g(ψ)=   . r ν  q ψ2 qY −q ψ2 c(ψ ) qψ2 ψ3η   µ 1+ζX µ 1 µ 1+ζX  Perigee  2  Y  q q ψ2q 1+h c  ω  0 0 µ 2(1+ζX ) (ψ1)   2  Ω 2  q ψ 1+h s  i  0 0 µ 2(1+ζX ) (ψ1)    L  q (4) In (3)-(4),

ζX ψ3 X = R(ψ1) ζY −ψ    4  Fig. 1. Orbital elements. qX = ψ3 +(2+ ζX ) c(ψ1)

qY = ψ4 +(2+ ζX ) s(ψ1) It is well known that ω is indeterminate for circular orbits η = ψ5 s(ψ1) − ψ6 c(ψ1) (i.e., when e = 0) and Ω is indeterminate for equatorial 2 2 2 orbits (i.e., when i =0). These singularities can be avoided h = ψ5 + ψ6 , by adopting a different parameterization of the orbit using and µ is the gravitational parameter of the central body. the modified equinoctial elements ψ = [ψ ... ψ ]T , defined 1 6 Notice that ζ does not affect the system dynamics. as [21] Y The control objective is to track the reference trajectory ψ1 = L = Ω+ ω + ν specified by the orbital elements 2 ψ2 = p = a(1 − e ) r r r r r r r T ψ = e = e · c(Ω + ω) ψ (t) = [ψ1(t), ψ2, ψ3, ψ4, ψ5, ψ6] , 3 X (1) ψ4 = eY = e · s(Ω + ω) which are the solution to equation (2) with u =0, i.e., ψ5 = hX = tan(i/2) c(Ω) ψ = h = tan(i/2) s(Ω). 6 Y ψ˙r = f(ψr), (5) In this parameterization, L is the true longitude shown in Fig. 1, p is the orbit semi-parameter, eX , eY are the corresponding to the given initial condition components of the eccentricity vector, and , are the hX hY r r r r r r r T components of the inclination vector. Notice that any closed ψ (0) = [L (0), p , eX , eY , hX , hY ] ∈ Ψ. Keplerian orbit is such that ψ = p > 0. Moreover, the 2 Let ψ˜ = ψ − ψr denote the tracking error. Then, the error escape to parabolic orbits (i.e., e = 1) is not possible dynamics evolves according to the time-varying system with continuous feedback [22], which is the case considered in this paper. Therefore, in the following we restrict our ˜˙ ˜ ˜ r ˜ r (6) attention to the case e < 1. Hence, the state vector ψ must ψ = f(ψ; ψ )+ g(ψ + ψ )u, belong to the set where f˜(ψ˜; ψr) = f(ψ˜ + ψr) − f(ψr). The orbit control R6 2 2 problem considered in this paper can be formulated as Ψ= {ψ ∈ : ψ2 > 0, ψ3 + ψ4 < 1}. follows. The dynamics of the orbital elements ψ in (1), in the Problem 1: Find a continuous state feedback control law presence of forcing inputs, are described by Gauss’s vari- u = u(ψ˜; ψr) such that the error system (6) is globally ational equations. Let us introduce the control input vector asymptotically stable, which in turn guarantees that T u = [uθ ur uh] , where uθ, ur and uh denote the along- track, radial and cross-track components of the accelera- lim ψ˜(t)=0 t tion, respectively. The resulting dynamics can be expressed →∞ as [23]: for any initial condition ψ(0), ψr(0) ∈ Ψ. III. CONTROL SYSTEM DESIGN Note that, by virtue of the properties of the considered In order to derive a solution to Problem 1, we first problem, all Fij and Gij are positive functions of χ and r, except for and . The vector r in (12) introduce a diffeomorphic coordinate transformation x = ψ F33 F43 H(x; ψ ) can be computed as x(ψ˜; ψr) in system (6), defined as follows r ∂x ˜ r x = ψ˜ (7) H(x; ψ )= gh(ψ + ψ ), (13) 1 1 ∂ψ˜ ˜ ψ2 where gh(·) denotes the third column of g(·) in (4), and the x2 = 1+ r − 1 (8) s ψ2 right hand side of (13) can be expressed in terms of χ and r r ψ by inverting the coordinate transformation (7)-(11). The ψ2 r 0 r ˜ ˜2 ˜ expression of ∂x/∂ψ is reported in Appendix (a). x3 ψ +ψ2 ˜ r ψ3 +ψ3 = ψr R(ψ1 +ψ1) r (9) x4 2 −ψ˜ −ψ The structure of system (12) allows one to tackle the  0 ˜ r  4 4   ψ2+ψ2   control design problem in a two-step procedure. q  ψ˜2  r − r ζ ψ˜2+ψ X Step 1 + 2 − r , 0 ζY " #   In the first step, we assume uh =0 and derive a feedback x5 = ψ˜5 (10) stabilizer for the fourth order subsystem ˜ u x6 = ψ6, (11) χ˙ = F (χ; ψr)+ G(χ; ψr) θ (14) ur where   using the inputs uθ and ur. The stabilization of the full ζr ψr X = R(ψr) 3 . system will be addressed afterwards, using the input uh. ζr 1 −ψr  Y   4  For system (14), let us first introduce the following non- A similar transformation is used in [19] for the case of singular transformation in the input variables r r circular reference orbits (i.e., ζX = ζY = 0). System (6) 1 uθ = (v − F12k1x1) (15) in the new coordinate set has the form G21 r r 1 F (χ; ψ ) G(χ; ψ ) uθ r ur = (w − F x ) , (16) x˙ = + + H(x; ψ ) uh , 43 3 02 1 02 2 ur G42  ×   ×   (12) where v and w are the new input variables and k1 is any T where χ = [x1 ...x4] and given positive constant, to be treated as a design parameter. Hence, system (14) becomes 0 F12 F13 0 00 0 0 x˙ 1 = F12x2 + F13x3 (17) F (χ, ψr) =  χ 0 0 −F33 −F12 x˙ 2 = −F12k1x1 + v (18)  0 F42 F12 +F43 0    x˙ = −F x − F x (19)   3 33 3 12 4 0 0 x˙ 4 = F42x2 + F12x3 + w. (20) G 0 G(χ; ψr) = 21 ,  0 0  Let 1 1  0 G42  x4∗ = F13k1k2− x1 − F33x3 + λ3 , (21)   F12  
being where k2 > 0 is constant and λ3(x1, x2, x3) is any given continuous function such that sgn(λ3)= sgn(x3). Similarly µ r 2 F12 = (x3 +1+ ζX ) to backstepping control design, we use x as a virtual input ψr3 4∗ r 2 for system (17)-(19), and consider the transformed state µ r F13 = (x3 +2+2ζ ) vector ψr3 X 2 T T r z = [z1 z2 z3 z4] = [x1 x2 x3 (x4 − x∗)] . (22) µ r 3 4 F42 = (x2 +2)(x3 +1+ ζ ) ψr3 X In the new coordinates, equations (17)-(20) read r 2 r F33 = F13 ζY z˙1 = F12z2 + F13z3 (23)

r z˙2 = −F12k1z1 + v (24) F43 = F13 ζ X 1 z˙3 = −F13k1k2− z1 − F12z4 − λ3 (25) ψr 1 2 z˙4 = F42z2 + F12z3 + w − x˙ ∗. (26) G21 = r 4 s µ (x3 +1+ ζX ) Consider the Lyapunov function candidate r ψ2 k 1 k k G42 = . V (z)= 1 z2 + z2 + 2 z2 + 2 z2. (27) s µ 1 2 1 2 2 2 3 2 4 The time derivative of (27), along the trajectories of (23)- where d is a given positive scalar continuous function, (26), reads r r ∂V2(x; ψ ) ∂V1(z; ψ ) r = x5 x6 , (39) V˙1(z; ψ ) = (v+F42k2z4)z2 − k2λ3z3 + (w−x˙ ∗)k2z4. ∂x ∂χ 4   (28) and the expression of ∂V (z; ψr)/∂χ is reported in Ap- In order to render V˙ in (28) negative semidefinite, we make 1 1 pendix (b). With this particular choice, (37) is negative the following choice for the inputs v and w semidefinite and vanishes if and only if z2 = z3 = z4 = 0

v = −F42k2z4 − λ2 (29) and uh = 0. Then, according to the previous analysis, limt χ(t)=0. Using this fact and observing that x5 and w =x ˙ 4∗ − λ4, (30) →∞ x6 are constant for uh =0, it can be verified from (13), (34) ˙ where λ2(z) and λ4(z) are continuous functions such that and (38)-(39) that the largest invariant set in which V2 =0 sgn(λ2) = sgn(z2) and sgn(λ4) = sgn(z4), respectively. is the trivial one x = 0. Hence, the proposed control law With this particular choice, (28) boils down to renders the equilibrium point of the full system (34) globally asymptotically stable with the Lyapunov function (36). V˙1(z) = −λ2z2 − k2λ3z3 − k2λ4z4 (31) , Summarizing, the feedback control law defined by uθ and given by (32)-(33), combined with as in (38), is a which is indeed negative semidefinite. Clearly, (31) vanishes ur uh solution to Problem 1. in the set {z : z2 = z3 = z4 =0}. Notice from (23) that z1 is constant in this set. Moreover, it follows from (24)-(25) that Remark 1: The proposed method for the solution of Prob- lem 1 actually provides the parameterization of a class z˙ 6= 0 and z˙ 6= 0 for z 6= 0. Hence, the largest invariant 2 3 1 of stabilizing controllers via the tunable design parameters set in which V˙ = 0 is z = 0. Since χ = 0 for z = 0, one 1 , , , , and . It can be foreseen that such can conclude that the equilibrium point χ = 0 is globally k1 k2 λ2 λ3 λ4 d asymptotically stabilized by the proposed control law. parameters can be exploited to enforce further control spec- ifications or to optimize suitable performance indices. One The final expression of the control inputs u and u is θ r such specification concerns the magnitude of the controls obtained from (15)-(16), (22) and (29)-(30) as in (32), (33) and (38). For circular reference orbits, i.e., for r 1 λ2 F33 = F43 =0, it turns out that the magnitude of the control uθ(χ; ψ )=− [F12k1x1 + F42k2(x4 −x4∗)]− (32) G21 G21 inputs can be made arbitrarily small locally by scaling the r 1 λ4 tuning parameters of the controller. This is not guaranteed, ur(χ; ψ )=− (F43 x3 − x˙ 4∗) − , (33) G42 G42 however, for eccentric reference orbits, since in this case the term in (33) requires the exact cancellation of with x given by (21). (F43x3 − x˙ 4∗) 4∗ a part of the system dynamics (without further assumptions Step 2 on λ2, λ3 and λ4). A systematic method for exploiting the design parameters for performance is out of the scope of the The stabilization of the full system can be tackled by present paper and is the subject of current investigation. introducing the previously derived control inputs uθ, ur in (12) and rewriting the resulting dynamics as IV. NUMERICAL SIMULATION r Fcl(χ; ψ ) r Consider the problem of transferring a spacecraft from x˙ = + H(x; ψ ) uh, (34) 02 1 a near-circular equatorial orbit to an higher altitude elliptic  ×  where orbit, with a prescribed longitude (i.e. phase) along the final orbit. This problem may occur, for instance, in formation r r r r uθ(χ; ψ ) flying applications, in which an actively controlled chaser Fcl(χ; ψ )= F (χ; ψ )+ G(χ; ψ ) r . (35) ur(χ; ψ )   spacecraft is required to intercept a passive target spacecraft. Given the structure of (34), it turns out that the origin of The orbital elements of the initial and the reference orbit are the full system can be globally stabilized via a damping reported in Table I. The reference eccentricity and inclination controller, as explained next. Consider the Lyapunov function vectors correspond to a target eccentricity of 0.5 and a target inclination of 35 deg. 1 1 V (x; ψr)= V (z; ψr)+ x2 + x2, (36) 2 1 2 5 2 6 TABLE I where z is defined by (22). The time derivative of (36), along ORBITALELEMENTSOFTHEINITIALANDTHEREFERENCEORBIT the trajectories of (34), can be written as r Orbital element Initial orbit Reference orbit r ∂V2(x; ψ ) r r V˙2(x; ψ )= V˙1(z)+ H(x; ψ )uh, (37) True longitude L(0) = 2.44 rad L (0)= 3.92 rad ∂x Semi-parameter p(0) = 6778 km pr = 19800 km e . −4 er . where V˙ (z) is given by (31). X (0) = 9 4 · 10 X = −0 13 1 Eccentricity vector −4 r eY (0) = 3.4 · 10 eY = 0.48 Let r r hX (0) = 0 h = 0.30 ∂V2(x; ψ ) Inclination vector X u (x; ψr)= −d H(x; ψr), (38) h (0) = 0 hr = 0.08 h ∂x Y Y )

2 0.4 0.2 (m/s θ

u 0 0 0.5 1 1.5 2 2.5 ×105

) 4 2 2 (m/s r 0 u 0 0.5 1 1.5 2 2.5 ×105

) 0.1 2 0

Initial orbit (m/s -0.1 h Controlled orbit u 0 0.5 1 1.5 2 2.5 Reference orbit Time (s) ×105

Fig. 2. Orbital transfer trajectory. Fig. 4. Control input signals.

0 synchronization of the phase of the orbiting body along a reference orbit. The stability of the closed loop system -0.2 has been proved by standard Lyapunov arguments, and the -0.4 applicability of the proposed controller has been illustrated -0.6 by numerical simulation of a spacecraft rendez-vous maneu- 1

x ver. Several performance-related aspects still remain to be -0.8 investigated. Future work should be tailored to the evaluation -1 of the controller performance with respect to metrics such as the settling time, the fuel consumption and the magnitude of -1.2 the control inputs. -1.4 APPENDIX -1.6 0 0.5 1 1.5 2 2.5 The following expressions are used in the paper. 5 (a) The Jacobian of the coordinate transformation (7)-(11), Time (s) ×10 with respect to the original coordinates ψ˜, in (13) is Fig. 3. True longitude tracking error. given by 1 0 0 001×2   1 1×2 0 2 r r 0 0 0 A numerical simulation relying on the dynamic model (6)  ψ2 √ψ2/ψ2    has been performed to demonstrate the application of the  r r r r  ∂x ψ2 ψ2 ψ2 ψ2 = ζ 2 (1 + ζ ) c(ψ1) s(ψ1) 01×2 . proposed approach to the considered rendez-vous problem,  ψ2 Y ψ X ψ2 ψ2  ∂ψ˜  − − 2  and to validate the obtained theoretical results. To ensure a  r r r r   ψ2 −ψ2 ψ2 ψ2  ζ ζ s(ψ1) c(ψ1) 01×2  q ψ2 X 2 2 r Y q ψ2 q ψ2  fast convergence towards the reference orbit, the following  ψ2 √ψ2 /ψ2 −    tuning parameters have been empirically selected for the  02×1 02×1 02×1 02×1 I2×2  nonlinear controller: k = 10 5, k = 10 4, λ = 10 4 z , 1 − 2 − 2 − 2 (b) The gradient ∂V /∂χ of the Lyapunov function V in λ =7 · 10 4 z , λ =7 · 10 4 z , and d = 10 4. 1 1 3 − 3 4 − 4 − (39) is given by The resulting trajectory is depicted in Fig. 4, in terms of cartesian coordinates. As expected, the asymptotic tracking F13 k2 ∂λ3 k1z1 − k1z4 − z4 of the reference trajectory is achieved. In Fig. 3, it can be F12 F12 ∂z1 seen that the true longitude tracking error is steered to zero r T  k2 ∂λ3  ∂V1(z; ψ ) z2 − z4 after a short initial transient, so that the correct phase is =  F12 ∂z2  , ∂χ  k ∂λ  acquired along the reference orbit. The control input signals  2 3   k2z3 + γk2z4 − z4  are reported for completeness in Fig. 4.  F12 ∂z3   k z   2 4  V. CONCLUSIONS where   This paper has studied the trajectory tracking problem for 1 √τF13 k1 √τ r the class of reference trajectories consisting of unperturbed γ = τ − z1 + F33 z3 −1 −τζ z3 F √F12 k2 √F12 Y closed orbits about a central body. A feedback control law 12 h T  T i has been derived for this problem, which allows for the and [z1 z2 z3 z4] = [x1 x2 x3 (x4 − x4∗)] . REFERENCES [12] J. T. Betts, “Optimal low–thrust orbit transfers with eclipsing,” Optimal Control Applications and Methods, vol. 36, no. 2, pp. 218–240, 2015. [1] T. N. Edelbaum, “Optimum power-limited orbit transfer in strong [13] D. P. Scharf, F. Y. Hadaegh, and S. R. Ploen, “A survey of spacecraft gravity fields,” AIAA Journal, vol. 3, no. 5, pp. 921–925, 1965. formation flying guidance and control. part ii: control,” in American [2] C. R. Hargraves and S. W. Paris, “Direct trajectory optimization Control Conference, vol. 4, 2004, pp. 2976–2985. using nonlinear programming and collocation,” Journal of Guidance, [14] E. N. Hartley, “A tutorial on model predictive control for spacecraft Control, and Dynamics, vol. 10, no. 4, pp. 338–342, 1987. rendezvous,” in European Control Conference, 2015, pp. 1355–1361. [3] F. Fahroo and I. M. Ross, “Direct trajectory optimization by a [15] H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, “Spacecraft Chebyshev pseudospectral method,” Journal of Guidance, Control, and formation flying control using mean orbit elements,” Journal of the Dynamics, vol. 25, no. 1, pp. 160–166, 2002. Astronautical Sciences, vol. 48, no. 1, pp. 69–87, 2000. [4] B. Bonnard, J.-B. Caillau, and E. Tr´elat, “Geometric optimal control of [16] S. D’Amico and O. Montenbruck, “Proximity operations of formation- elliptic Keplerian orbits,” Discrete and Continuous Dynamical Systems flying spacecraft using an eccentricity/inclination vector separation,” series S, pp. 929–956, 2005. Journal of Guidance, Control, and Dynamics, vol. 29, no. 3, pp. 554– [5] M. R. Ilgen, “Low thrust OTV guidance using Lyapunov optimal 563, 2006. feedback control techniques,” Advances in the Astronautical Sciences, [17] L. Breger and J. P. How, “Gauss’s variational equation-based dynamics vol. 85 Part 2, pp. 1527–1545, 1993. and control for formation flying spacecraft,” Journal of Guidance, [6] D. E. Chang, D. F. Chichka, and J. E. Marsden, “Lyapunov-based Control, and Dynamics, vol. 30, no. 2, pp. 437–448, 2007. transfer between elliptic Keplerian orbits,” Discrete and Continuous [18] M. Leomanni, E. Rogers, and S. B. Gabriel, “Explicit model predic- Dynamical Systems Series B, vol. 2, no. 1, pp. 57–68, 2002. tive control approach for low-thrust spacecraft proximity operations,” [7] A. E. Petropoulos, “Low-thrust orbit transfers using candidate Lya- Journal of Guidance, Control, and Dynamics, vol. 37, no. 6, pp. 1780– punov functions with a mechanism for coasting,” in AAS/AIAA Astro- 1790, 2014. dynamics Specialist Conference, 2004. [19] C. M. Kellett and L. Praly, “Nonlinear control tools for low thrust [8] P. Gurfil, “Nonlinear feedback control of low-thrust orbital transfer orbital transfer,” in Proceedings of the 6th IFAC Symposium on in a central gravitational field,” Acta Astronautica, vol. 60, no. 8, pp. Nonlinear Control Systems, 2004. 631–648, 2007. [20] R. Sepulchre, M. Jankovic, and P. V. Kokotovic, Constructive nonlin- [9] V. Jurdjevic and J. P. Quinn, “Controllability and stability,” Journal of ear control. Springer Science & Business Media, 2012. differential equations, vol. 28, no. 3, pp. 381–389, 1978. [21] M. J. H. Walker, B. Ireland, and J. Owens, “A set of modified [10] C. Kluever, “Geostationary orbit transfers using solar electric propul- equinoctial orbit elements,” Celestial mechanics, vol. 36, no. 4, pp. sion with specific impulse modulation,” Journal of Spacecraft and 409–419, 1985. Rockets, vol. 41, no. 3, pp. 461–466, 2004. [22] P. Gurfil, “Control-theoretic analysis of low-thrust orbital transfer [11] A. Garulli, A. Giannitrapani, M. Leomanni, and F. Scortecci, “Au- using orbital elements,” Journal of Guidance, Control, and Dynamics, tonomous low-Earth-orbit station-keeping with electric propulsion,” vol. 26, no. 6, pp. 979–983, 2003. Journal of Guidance Control and Dynamics, vol. 34, no. 6, pp. 1683– [23] R. H. Battin, An introduction to the mathematics and methods of 1693, 2011. astrodynamics. AIAA, 1999.