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Passivity-based distributed acquisition and station-keeping control of a satellite constellation in areostationary orbit Emmanuel Sin, He Yin and Murat Arcak Abstract— We present a distributed control law to assemble a cluster of satellites into an equally-spaced, planar constellation in a desired circular orbit about a planet. We assume each satellite only uses local information, transmitted through com- munication links with neighboring satellites. The same control law is used to maintain relative angular positions in the presence of disturbance forces. The stability of the constellation in the desired orbit is proved using a compositional approach. We first show the existence and uniqueness of an equilibrium of the interconnected system. We then certify each satellite and communication link is equilibrium-independent passive with respective storage functions. By leveraging the skew symmetric Fig. 1. Depiction of constellation. Each satellite may share state informa- coupling structure of the constellation and the equilibrium- tion with its neighbors via communication links independent passivity property of each subsystem, we show that the equilibrium of the interconnected system is stable with a Lyapunov function composed of the individual subsystem distributed control strategy is appealing for satellite con- storage functions. We further prove that the angular velocity of each satellite converges to the desired value necessary to stellations in situations where centralized control is difficult maintain circular, areostationary orbit. Finally, we present or impossible. For example, as thousands of satellites are simulation results to demonstrate the efficacy of the proposed employed in constellations, the resulting uplink/downlink control law in acquisition and station-keeping of an equally- demands on a network of Earth-based ground stations may spaced satellite constellation in areostationary orbit despite the become unmanageable. A distributed strategy is also critical presence of unmodeled disturbance forces. for a constellation orbiting a planet without ground stations. I. INTRODUCTION Passivity-based methods are well suited for distributed A satellite constellation is a group of satellites that are control of large-scale, interconnected systems [2]–[3]. We coordinated to achieve objectives that may not be possible model our constellation as an interconnected system where with a single satellite. Constellations have been applied we assume each satellite has a communication link with to serve as telecommunications or broadcasting networks, neighboring satellites, sharing relative angular position in- provide global imagery and weather services, and enable formation. An internal feedback control law is designed for global positioning and navigation capabilities. The control the satellites and we certify that each satellite and commu- of such constellations can be divided into two different nication link is equilibrium independent passive with respect problems: acquisition and station-keeping. Acquisition refers to proposed storage functions. A constellation coordination to the process of forming the constellation once the satellites control law is introduced to interconnect the subsystems have been deployed by the delivery vehicle. For example, we in a skew-symmetric coupling structure. The equilibrium- may spread out a cluster of satellites in a desired orbital independent passivity property of each subsystem and the plane to form an equally-spaced constellation. Once the skew-symmetry of their interconnection enables us to prove desired constellation is acquired, station-keeping refers to the stability of the constellation at equilibrium. arXiv:2005.12214v1 [eess.SY] 25 May 2020 the process of maintaining relative positions and velocities in the presence of disturbances. The acquisition of a small A. Preliminaries spacecraft constellation in low Earth orbit, using a centralized We use a compositional approach to certify the stability approach, is studied in [1]. A centralized approach may be of a large system consisting of interconnected, dissipative used if, for example, a large number of ground stations are subsystems. We briefly state results that extend the works in available to measure and control the satellites. [4], [5] and [6], which are used in a later section to prove In this paper, we shift our focus to a distributed ap- stability of the constellation under a closed-loop acquisition proach of acquiring and station-keeping a constellation. A and station-keeping control law. Consider the system S The first two authors contributed equally to this work. described by E. Sin and H. Yin are Graduate Students of the Department of Mechanical Engineering at the University of California, Berkeley femansin,he yin x˙(t) = f (t;x(t);u(t)) ; y(t) = h(t;x(t);u(t)) ; (1) [email protected]. M. Arcak is a Professor of the Department of Electrical Engineer- nx nu ing & Computer Sciences at the University of California, Berkeley, where x(t) 2 R is the state, u(t) 2 R is the input, and n [email protected]. y(t) 2 R y is the output. Furthermore, suppose there exists a n nonempty set X ⊂ R x where, for everyx ¯ 2 X , there exists of a planet lie in the same plane (e.g., equatorial plane), n a uniqueu ¯ 2 R u satisfying f (t;x¯;u¯) = 0 then the gravitational perturbations from the moons may be Definition 1. The system (1) is equilibrium independent approximated as planar. Hence, for certain examples, we may dissipative (EID) with supply rate s(·;·) if there exist con- use a polar coordinate system to represent the satellite orbital n tinuously differentiable functions V : R×R x ×X 7! R and kinematics in the plane: n V : R x × X 7! R satisfying the conditions ¯ ~r = rer (6a) V(t;x;x¯) ≥ V(x;x¯) > 0; 8(x;x¯) s.t. x 6= x¯; (2a) ˙ ˙ ¯ ~r = re˙ r + rqeq (6b) V(t;x¯;x¯) = 0; V(x¯;x¯) = 0; (2b) ¨ ˙ 2 ˙ ¨ ¯ ~r = r¨− rq er + 2˙rq + rq eq : (6c) > V˙ (t;x;x¯) := ∇tV(t;x;x¯) + ∇xV(t;x;x¯) f (t;x;u) We denote the magnitude of the radial position with r and ≤ s(u − u¯;y − y¯) ; (2c) the angular position with q. We use er and eq as the unit n n n 8(t;x;x¯;u;u¯) 2 R×R x ×X ×R u ×R u , wherey ¯ = h(t;x¯;u¯). vectors in the radial and tangential directions of the orbital A system is equilibrium-independent passive (EIP) if it is plane, respectively. EID with respect to the supply rate If we include the specific forces from the right-hand side of (5), we get the following model representing the ith satellite’s s(u − u;y − y) = (u − u)>(y − y) ¯ ¯ ¯ ¯ (3) motion in the radial and tangential directions, respectively: and it is output strictly equilibrium-independent passive ˙ 2 m 1 (OSEIP) if, for some e > 0, it is EID with respect to r¨i = riqi − 2 + tr;i + (~aperturb;i)r (7a) ri mi > > s(u − u¯;y − y¯) = (u − u¯) (y − y¯) − e(y − y¯) (y − y¯) : (4) −2˙riq˙i 1 1 q¨i = + tq;i + (~aperturb;i)q : (7b) II. SYSTEM DYNAMICS ri miri ri Instead of creating a monolithic model of the constellation, Finally, if we implement a change of variables so that v := r˙ ˙ we decompose it into subsystems and consider the intercon- and w := q, we get the following set of first-order differential nections between them. By characterizing the input-output equations to describe each satellite of the constellation properties of each individual subsystem and the interconnec- r˙i = vi (8a) tions that exist between them, we may certify stability and 2 m 1 convergence properties of the constellation. v˙i = riwi − 2 + tr;i (8b) ri mi A. Satellite Model −2viwi 1 w˙i = + tq;i : (8c) In our constellation, we refer to the constituent satellites ri miri as subsystems. Each satellite is under the influence of the Note that we exclude q˙ = w from the set of equations. gravitational pull from the central body, the thrust applied by i i The q state does not appear in the equations of motion (8), the satellite, and natural perturbing forces (e.g., atmospheric hence, it is not needed in our state feedback controller design. drag, gravity from moons, solar radiation pressure). To model Furthermore, we omit the terms representing specific forces the motion of a satellite orbiting a planet, we start with due to perturbations. Through an example simulation we will the central-force problem (or restricted two-body problem) show that our state feedback controller based on the model where we assume that the barycenter of the system is co- described by (8) is robust to unmodeled disturbances that are located with the center of a spherically, symmetric central present in the simulation model, described by (7). body (i.e., the mass of the satellite is negligible). The satellite’s motion can be described by the following second- B. Interconnections order ordinary differential equation known as the fundamen- We assume that only neighboring satellites may com- tal orbital differential equation (FODE) with specific force municate with each other. The topology of this particular perturbations [7]: information exchange is illustrated by the undirected graph m 1 th th ¨ ~ shown in Fig 1. If the i and j subsystems have access ~r = − 3~r + t +~aperturb ; (5) th th k~rk2 m to relative state information, then the i and j nodes of the graph are connected by a link l = 1;:::;M. Although the where ~r 2 3 is the position vector pointing from the center R communication is assumed to be bidirectional, we assign an of the planet to the satellite, m is the gravitational parameter orientation to the graph by considering one of the nodes of of the central body (i.e., gravitational constant multiplied by a link to be the positive end. As a convention, we set the the mass of the planet), m is the mass of the satellite, ~t 2 3 R direction of a communication link to point in the direction is thrust, and ~a 2 3 represents the specific forces due perturb R of the orbital motion.

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