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).