Passivity-Based Distributed Acquisition and Station-Keeping Control of a Satellite Constellation in Areostationary Orbit

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 communication 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 Σ 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 {emansin,he yin x˙(t) = f (t,x(t),u(t)) , y(t) = h(t,x(t),u(t)) , (1) }@berkeley.edu. 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) ∈ R is the state, u(t) ∈ R is the input, and n [email protected]. y(t) ∈ R y is the output. Furthermore, suppose there exists a n nonempty set X ⊂ R x where, for everyx ¯ ∈ X , there exists of a planet lie in the same plane (e.g., equatorial plane), n a uniqueu ¯ ∈ 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, ∀(x,x¯) s.t. x 6= x¯, (2a) ˙ ˙ ¯ ~r = re˙ r + rθeθ (6b) V(t,x¯,x¯) = 0, V(x¯,x¯) = 0, (2b) ¨ ˙ 2 ˙ ¨ ¯ ~r = r¨− rθ er + 2˙rθ + rθ eθ . (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 θ. We use er and eθ as the unit n n n ∀(t,x,x¯,u,u¯) ∈ 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 µ 1 (OSEIP) if, for some ε > 0, it is EID with respect to rï = riθi − 2 + τr,i + (~aperturb,i)r (7a) ri mi > > s(u − u¯,y − y¯) = (u − u¯) (y − y¯) − ε(y − y¯) (y − y¯) . (4) −2˙riθ˙i 1 1 θï = + τθ,i + (~aperturb,i)θ . (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 ω := θ, 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 µ 1 convergence properties of the constellation. v˙i = riωi − 2 + τr,i (8b) ri mi A. Satellite Model −2viωi 1 ω˙i = + τθ,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 θ˙ = ω from the set of equations. gravitational pull from the central body, the thrust applied by i i The θ 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 µ 1 th th ¨ ~ shown in Fig 1. If the i and j subsystems have access ~r = − 3~r + τ +~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 ∈ 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, µ 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, ~τ ∈ 3 R direction of a communication link to point in the direction is thrust, and ~a ∈ 3 represents the specific forces due perturb R of the orbital motion. Hence, the incidence matrix D of the to perturbations. graph is defined as: It is well known that two-body motion in an inertial frame is planar. Since atmospheric drag acts against the direction +1 if ith node is positive end of lth link  th th of motion, a satellite under atmospheric drag remains in Dil = −1 if i node is negative end of l link planar motion. Furthermore, if a satellite and the moons  0 otherwise. In this application, for a constellation with N satellites that communication links [8]-[9]. The links can be expressed as only communicate with neighbors, the incidence matrix D is subsystems Λl for l = 1,...,M :

 1 0 0  ˙ rel θl = el (12a)  ..  rel −1 . 0  N×M yl = hl(θ ) , (12b) D =   ∈ R , (9) l  ..   0 . 1  where el is the input and yl is the output of each communi- 0 0 −1 rel cation link. The subsystem Λl keeps track of a state θl ∈ R where M := N − 1. Note that we assume the 1st and Nth and outputs a signal of interest that is measured through the satellites do not communicate; hence, they do not share a function hl : R 7→ R, that we assume is strictly increasing and communication link. All other satellites have two links each. onto, and lima→∞ hl(a) = ∞. Let us refer to satellite inputs and outputs in compact form > > III. CONTROL STRATEGY as u := u1,...,uN and z := z1,...,zN , respectively. Similarly, we refer to the communication link inputs and out- We now describe an internal feedback control strategy for > > puts collectively as e := e1,...,eM and y := y1,...,yM , each satellite that renders a linear map between the input (to respectively. be designed with a simple state feedback law) and the output variable of interest. Subsequently, we add a constellation coordination term that regulates the relative angular spacing error between neighboring satellites.

A. Internal Feedback Control For each subsystem, we propose the following thrust control laws in the radial and tangential directions: 2 µ τr,i = mi −riωi + 2 − kv(vi − vd) − kr(ri − rd) (10a) ri ri τθ,i = mi 2viωi − kω (ωi − ωd) + ui , (10b) kc where rd, vd, and ωd are the desired radius, radial velocity, and angular velocity for every satellite to maintain an areostationary orbit. The term ui is a constellation coordination Fig. 2. Interconnected system control law to be designed. The controller gains kr, kv, kω , kc > 0 are discussed and chosen in the subsequent stability We construct an interconnection between the satellites analysis and simulation results. Σ1,...,ΣN and the communication links Λ1,...,ΛM as shown If we substitute the thrust control laws (10a)-(10b) into the in Fig 2 and define the following input-output mappings: equations of motion (8a)-(8c), the dynamics of each satellite, Σ for i = 1,...,N, take the form of    ˙ rel i ω1 − ω2 θ1 ω − ω θ˙ rel >  2 3   2  rel r˙i = vi (11a) e := D z =   ≡   =: θ˙ (13a)  .   .  v˙ = −k (v − v ) − k (r − r ) (11b)  .   .  i v i d r i d ˙ rel ωN−1 − ωN θM kω 1 ω˙i = − (ωi − ωd) + ui (11c)  h (θ rel)  ri kc 1 1  .  rel zi = ωi , (11d) u := −Dy = −D .  = −Dh(θ ) . (13b) rel hM(θM ) where the output variable zi of interest is the angular velocity of the satellite. Note that we have transformed the radial Note that the input applied to the ith satellite, dynamics (11a) - (11b) to be independent of the ω state. M rel B. Constellation Coordination Control ui = − ∑ Dilhl(θl ) , (14) l=1 The subsystems are dynamically decoupled, however, we may coordinate their relative motion through a constellation is based only on local information since Dil = 0 when the coordination control law where we use feedback of local ith subsystem does not have access to information on the lth information from spatially neighboring subsystems. We as- communication link. Hence, we have a distributed control sume that this local information is shared via inter-satellite architecture where local controllers act on local information. e e IV. STABILITY ANALYSIS define ri = ri −r¯i, and vi = vi −v¯i = vi, then (11a) and (11b) We first show the existence and uniqueness of an equilib- can be rewritten as e e rium point whose stability will be subsequently analyzed. At r˙i 0 1 ri e = e . (19) equilibrium, the right-hand sides of (11a), (11b), (11c) for all v˙i −kr −kv vi i = 1,...,N, and (12a) for all l = 1,...,M must equal zero. It can be verified that the equilibrium point (r¯ ,v¯ ) of (11a)– The equilibrium states of the radial dynamics (11a)–(11b) i i (11b) is exponentially stable if and only if k > 0 and k > 0. (r ,v ) = (r ,v ) = (r , ) r v may be found by inspection to be ¯i ¯i d d d 0 . We now proceed to prove stability of the tangential com- e For the right-hand side of (12a) to vanish, l must equal zero ponent of the subsystems under the influence of both the l = ,...M for 1 . In other words, internal feedback law (10b) and the constellation coordina- e¯ = D>ω¯ = 0. (15) tion law (13b). In the internal feedback law (10b), we utilize a positive parameter kc to scale down the magnitude of the > By definition of D given in (9), we have D 1 = 0. Since constellation coordination control input ui. More specifically, > nullity(D ) = 1, the span of 1 constitutes the entire null we assume that kc is a time-varying parameter: > space of D . Therefore, ω¯ = ω01 is the unique solution to kc(t) ≥ kc > 0, k˙c(t) ≤ 0, ∀ t ≥ 0, (20) (15), where ω0 is the common angular velocity of all N ¯ satellites. That is, all satellites must have the same angular that decreases and converges to a positive limit kc. velocity. Finally, the right-hand side of (11c) must vanish: We propose the following storage function¯ for the ith k 1 subsystem: − ω (ω − ω ) + u¯ = 0, for i = 1,...,N. (16) r 0 d k i i c kc(t) 2 Si(t,ωi,ω¯i) = (ωi − ω¯i) . (21) > > N 2 From (13b) and the fact that 1 D = 0 , we have ∑i=1 ui = > > rel 1 2 1 u = −1 Dh(θ ) = 0. Adding (16) from i = 1 to i = N We can verify that Si(t,ωi,ω¯i) ≥ kc(ωi − ω¯i) > 0, for all 2 ¯ yields the following equation: (ωi, ω¯i) such that ωi 6= ω¯i, and that Si(t,ω¯i,ω¯i) = 0. N If we take the derivative of the storage function we get kω −(ω0 − ωd) = 0, ˙ ∑ r kc(t) i=1 i S˙ (t,ω ,ω¯ ) = k (t)(ω − ω¯ )ω˙ + (ω − ω¯ )2 i i i c i i i 2 i i which requires that ω0 = ωd, and therefore ω¯ = ωd1. Sub- kω 1 stituting this value for ω0 back into (16), we get = kc(t)(ωi − ω¯i) − (ωi − ωd) + ui + ri kc(t) M ˙ ¯ rel kc(t) 2 u¯i = − ∑ Dilhl(θl ) = 0 for i = 1,...,N, (17) (ωi − ω¯i) (22) l=1 2 ˙ kc(t)kω kc(t) 2 which amounts to = (ui − u¯i)(ωi − ω¯i) − − (ωi − ω¯i) (23) ri 2 ¯ rel h1(θ1 ) = 0, M rel where we have used ω¯i = ωd,u ¯i = −∑ Dilhl(θ¯ ) = 0. ¯ rel ¯ rel l=1 l −hl−1(θl−1) + hl(θl ) = 0, l = 2,...,M, (18) We note that ri(t) > 0, ∀i = 1,...,N is always satisfied (i.e., ¯ rel the radius is always positive). Hence, the storage function S , −hM(θM ) = 0. i described by (21), certifies that the tangential component of ¯ rel A solution θl for l = 1,...,M exists and is unique since the Σi subsystems (11c)–(11d), is OSEIP, as defined in (4). hl is onto and strictly increasing. In summary, there exists For the links Λl, we propose a unique equilibrium point for a desired constellation given rel rel Z θl by (r¯i,v¯i,ω¯i) = (rd,0,ωd), i = 1,...,N and θ¯ , l = 1,...,M rel ¯ rel ¯ rel l Tl(θl ,θl ) = hl(z) − hl(θl ) dz. (24) pµ 3 ¯rel that satisfy (18). Furthermore, we note that ωd = /rd for θl a circular orbit at a given altitude. Since hl is strictly increasing, we can verify that We use a compositional approach to analyze the stabil- rel ¯ rel rel ¯ rel ¯ rel ¯ rel Tl(θl ,θl ) > 0 for all θl 6= θl and Tl(θl ,θl ) = 0. ity properties of the closed-loop constellation under our If we take the derivative of the storage function we get proposed internal feedback and coordination control laws. ˙ rel ¯ rel ˙ rel rel ¯ rel First, we show the stability of an equilibrium point for the Tl(θl ,θl ) = θl hl(θl ) − hl(θl ) radial component of each individual Σ subsystem (11a)– i = (e − e¯ )(y − y¯ ) (25) (11b). Second, we propose storage functions for each of l l l l N N the interconnected subsystems, comprised of the tangential where we have usede ¯l = ∑i=1 Dilz¯i = ∑i=1 Dilω¯i = 0 and ¯ rel component of the Σi subsystems (11c)–(11d), i = 1,...,N and y¯l = hl(θl ). We note that the storage function Tl certifies the Λl subsystems (12), l = 1,...,M, and certify that they are that each communication link Λl is EIP as defined in (3). EID as defined in (2). We then use the storage functions to Now that we have shown that each of the subsystems compose a Lyapunov function for the interconnected system. is equilibrium-independent passive, we note that the in- For the radial component of the Σi subsystem (11a) - (11b), terconnected system as shown in Fig 2 may be brought we choose kr, kv so that the closed-loop system is stable. We into the canonical form of Fig 3 where the upper block Finally, use our constellation coordination control law (13b) andu ¯ = −Dy¯, then k˙ (t) = −k (t)k (ω − ω¯ )>R−1(ω − ω¯ ) + c (ω − ω¯ )>(ω − ω¯ ). c ω 2 (30) Note that the expression above is negative semi-definite. As a ¯ rel ¯ rel result, (r¯i,v¯i,ω¯i,θl ) = (rd,0,ωd,θl ), for all Σi, i = 1,...N and all Λl, l = 1,...M is a stable equilibrium point of the ¯ rel interconnected system shown in Fig 2, where θl satisfies equations (18). Due to the time-varying parameters r and kc, the interconnected constellation is a non-autonomous system for which the Lasalle-Krasovskii Invariance Principle is not applicable. Although we may not conclude asymptotic stability of an equilibrium, we may prove the weaker result [10] that ωi, i = Fig. 3. Interconnected system in canonical form 1,...,N converges to the desired ωd value. Physically, this signifies that the constellation will maintain a circular orbit. has the subsystems along its diagonal and the lower block As shown in [10], x(t) is bounded by using (27) and contains a skew symmetric matrix. As shown in [4], since the dynamics are locally Lipschitz in x and bounded in t, the equilibrium-independent passive subsystems are coupled implying thatx ˙(t) is also bounded for all t ≥ 0. Hence, x(t) through a skew symmetric interconnection matrix, an equi- is uniformly continuous for t ≥ 0. Define a negative semi- librium point of the interconnected system, if it exists, is definite function stable and the sum of the individual subsystems provides a > −1 W(x) = −kckω (ω − ω¯ ) R (ω − ω¯ ). (31) Lyapunov function. ¯ Let us sum the storage functions for all the Σi subsystems As a result, W(·) is uniformly continuous on the bounded and Λl subsystems: domain of x(t). From (30) we can verify that N M V˙ (t,x(t),x¯) ≤ W(x(t)) rel ¯ rel V(t,x,x¯) = ∑ Si(t,ωi,ω¯i) + ∑ Tl(θl ,θl ) (26) i=1 l=1 Integrate it over [0,T], then where we use x := (ω,θ rel),x ¯ := (ω¯ ,θ¯ rel). The time-varying Z T V(T,x(T),x¯) −V(0,x(0),x¯) ≤ W(x(t))dt , Lyapunov function (26) can be lower and upper bounded: 0 V (x,x¯) ≤ V (t,x,x¯) ≤ V¯ (x,x¯) , (27) which implies ¯ Z ∞ where − W(x(t))dt ≤ V(0,x(0),x¯) < ∞. N M 0 kc 2 rel ¯ rel V(x,x¯) = ∑ ¯ (ωi − ω¯i) + ∑ Tl(θl ,θl ) (28) Using Barbalat’s Lemma, since W (·) is uniformly continuous ¯ i=1 2 l=1 R ∞ and 0 W(x(t))dt exists, W (x(t)) → 0 as t → ∞, which N k¯ M implies that x(t) approaches E = {x : W (x) = 0}. In other V¯ (x,x¯) = c (ω − ω¯ )2 + T (θ rel,θ¯ rel) (29) ∑ i i ∑ l l l words, ω (t) → ω¯ = ω . i=1 2 l=1 i i d and k¯c := kc(0) ≥ kc(t) ≥ kc ∀t ≥ 0. We note that V(x,x¯) and V. EXAMPLE ¯ ¯ V¯ (x,x¯) are positive definite and radially unbounded. Consider a cluster of N = 10 satellites that have been batch If we take the time derivative of (26), we get: deployed into a nearly-circular, equatorial, prograde orbit N M around the planet Mars at a desired altitude of approximately ˙ ˙ ˙ V(t,x,x¯) = ∑ Si + ∑ Tl 17032 km above the Martian surface. Assuming the equato- i=1 l=1 rial radius of Mars is 3396.2 km, each satellite in this orbit N p k (t)k k˙ (t) has desired equilibrium states of (r ,v ,ω ) = (r ,0, µ/r3) = (u − u¯ )(ω − ω¯ ) − c ω − c (ω − ω¯ )2 d d d d d ∑ i i i i i i where r = 20428.2km. This specific orbit, from the class of i=1 ri 2 d M areosynchronous (i.e., Martian synchronous) orbits, is known + ∑ {(el − e¯l)(yl − y¯l)}. as an areostationary orbit. Similar to satellites in geosta- l=1 tionary orbit about Earth, the position of an areostationary > satellite appears fixed in the sky relative to an observer on If we define R := blkdiag(r1,...,rN), and usee ¯ = D ω¯ then the surface of Mars. By equally spacing the 10 satellites ˙ > −1 kc(t) > within this orbit, the resulting constellation may serve as = −kc(t)k (ω − ω¯ ) R (ω − ω¯ ) + (ω − ω¯ ) (ω − ω¯ ) ω 2 a telecommunication network or navigation system for the + (ω − ω¯ )>(u − u¯) + (ω − ω¯ )>D(y − y¯) exploration of Mars. After deployment we assume the following In the acquisition phase, we consider a generous acquisi- initial conditions for all i = 1,...,N satellites: tion time of t f = 355 Martian days (Sols), or approximately −8 −1 ri = (20428.0 ± 0.1)km, vi = (0 ± 1) × 10 ms , ωi = 1 Earth year. Although the constellation may be acquired −5 −1 −3 (7.0879 ± 0.0100) × 10 rads , θi = (0 ± 5) × 10 rad. in less time, it may not be necessary. In various design Note that the initial conditions prescribe nearly circular proposals for manned missions to explore Mars [12], plans orbits. The angular position θi is measured with respect to include an initial uncrewed cargo mission so that supplies a reference horizontal line in the orbital plane. and infrastructure are in place before the crewed missions We assume each m = 100kg satellite is equipped with arrive. We assume that a satellite constellation to serve as a throtteable, continuous-thrust propulsion system with a a telecommunications network would be launched in this maximum thrust of τmax = 100mN in each of the radial and initial mission. Given that subsequent crewed missions would tangential directions of motion. In this example, we do not require approximately two years to arrive, due to launch consider motion normal to the orbital plane. Solar electric window constraints, 1 Earth year would provide sufficient propulsion systems, which use electricity generated by solar time to deploy and test the satellite constellation before use panels to accelerate propellant at high exhaust speeds, are by a crewed mission. capable of throtteable, continuous-thrust. Although electric VI. RESULTS propulsion systems have high specific impulse (i.e., they are fuel efficient), they have much weaker thrust compared to We implement the thrust controls laws described by (10) traditional chemical rockets. The NASA Evolutionary Xenon where the formation control law ui for all i = 1,...,N Thruster [11] is an example of a solar electric propulsion satellites is given by (14) and the interconnection between system with a maximum thrust of 236 mN. We expect that the satellites is described by the incidence matrix D in (9). state-of-the-art will continue to develop, allowing for even In this example, the measurement output from each of the rel higher thrust magnitudes in the future, but we maintain a communication links, hl(θl ), l = 1,...,M, in (12b) is of conservative thrust limit for this example. the form: In addition to the gravitational pull of Mars, we introduce rel rel rel hl(θ ) = θ − θ , (34) perturbations due to the gravity of Mars’ two moons. Since l l d rel 2π the inclinations of Phobos and Deimos with respect to Mars’ where θd = N represents the desired, equal angular spacing equator are 1.093◦ and 0.930◦, respectively, we approximate between neighboring satellites. The model (7) is used for their orbits as equatorial in this example. Note that since simulation where the specific force perturbations due to Phobos and Deimos have orbital eccentricities of 0.0151 and Phobos and Deimos are included using (32). 0.0003, respectively, their orbits are nearly circular. We use To regulate the radial distance, radial velocity, and angular the values of 9234.42 km and 23455.50 km for the radial dis- velocity of each satellite about the areostationary orbit, we −5 −4 4 tance of each moon’s orbit at its respective periapsis. Finally, use the gains kr = 1 × 10 , kv = 1 × 10 , and kw = 1 × 10 . 13 3 −2 we use values of µ = 4.282837 × 10 m s , µPhobos = In the acquisition phase (0 ≤ t ≤ t f ), we use a time-varying 5 3 −2 5 3 −2 7.161 × 10 m s , and µDeimos = 1.041 × 10 m s for constellation coordination gain the standard gravitational parameter of Mars, Phobos, and c kc(t) = (k¯c − kc)exp(− t) + kc , (35) Deimos, respectively. We find the specific force perturbation ¯ t f ¯ acting on each satellite by each moon, ~a (where p = p,i where k¯c > kc > 0 and c > 0 . We can simply calculate the {Phobos,Deimos}), by computing ¯ time derivative of kc as c (~ap,i)r µp cosθi sinθi k˙ (t) = − (k¯ − k )exp(−c t ) < 0, ∀ t ≥ 0 . (36) = − ~rp,i , (32) c c c t (~a ) 3 -sinθ cosθ t f ¯ f p,i θ k~rp,ik2 i i th Note that the constellation coordination gain function, (35), where ~rp,i, the expression for the relative position of the i satisfies the condition in (20) used for the stability analysis. satellite with respect to the moon p in the Mars-centered 11 9 For this example, we choose k¯c = 1 × 10 , kc = 1 × 10 , inertial coordinate system, is rel ¯ c = 30. Since the relative angle θl is far from the desired rel r cosθ − r cosθ relative angle θd at the beginning of the acquisition phase, ~r = i i p p . (33a) p,i r sin − r sin the magnitude of control input ui derived with (34) is large. i θi m θp rel We initially need a large kc to scale it down. As θl rel The radial and tangential components of the acceleration are converges to θd , the magnitude of ui decreases and we found by rotating ~rp,i by the appropriate rotation matrix. require less scaling. Therefore, the constantly decreasing The mission objectives are (1) spread out the initial cluster parameter kc allows the thrust commands τr,i and τθ,i in of satellites into an equally-spaced constellation, and (2) (10) to stay within a reasonable range during the acquisition regulate the satellites’ deviations from the desired areosta- phase. After acquisition, we enter the station-keeping phase tionary orbit as well as their relative angular positions with where we use a constant value of kc. respect to the desired spacings, in the presence of unmodeled The simulated states of each satellite¯ are shown in the perturbations. We call these distinct phases of the mission as first three subplots of Fig 4. Despite the perturbed ini- acquisition and station-keeping. tial conditions and the specific force perturbations due to Fig. 4. Absolute radial positions, radial and angular velocities, and relative Fig. 6. Orbital position of satellites during different stages of the 303.06 angular spacing between neighboring satellites Sols acquisition phase

be conducted by decreasing (increasing) the altitude of a spacecraft, causing it to speed up (slown down) in the tangential direction to gain (reduce) angular position. Finally, we present Fig 6, where the angular positions of the satellites are depicted at different times during the acquisition phase. The central red body represents Mars whereas the two gray bodies are the moons, Phobos and Deimos. We note that the orbit of the outer moon, Deimos, is very close to that of the areostationary orbit at a distance of approximately 3000 km. Despite the close proximity, the effect of the unmodeled gravitational perturbation is mitigated by the proposed control law. An animation of the acquisition phase is available at https://youtu.be/-2y_IWRPuzU.

VII.CONCLUSION

Fig. 5. Radial and tangential thrust commands to each satellite during We have presented a control strategy to coordinate a acquisition phase large number of satellites to not only acquire but also to maintain an equally-spaced constellation in areostationary orbit. The proposed distributed control law is implemented Phobos and Deimos, each satellite regulates to the desired on each satellite using only local information from neigh- equilibrium point for an areostationary orbit (illustrated by boring satellites. We proved that the closed-loop system, the dotted lines). The fourth subplot of Fig 4 shows that comprised of the satellites and communication links, is stable the angular spacing between each pair of satellites reaches at equilibrium due to the equilibrium-independent passive the desired value of 36◦. All angular spacings reach within property of each subsystem and the skew-symmetric coupling a 0.5◦ tolerance of the desired value in 303.06 Sols (or structure of their interconnections. We further proved that the approximately 311 solar Earth days). angular velocities of each satellite converge to the desired In Fig 5, we plot the radial and tangential thrust inputs value necessary for a circular, areostationary orbit. We then commanded by our feedback laws (10). We observe that the demonstrated the efficacy of the acquisition and station- control histories remain within the maximum thrust value keeping control strategy on a simulation example. of 100 mN throughout the acquisition phase. We also note Regarding the practical implementation of our approach that, although the constellation coordination term appears in to constellation acquisition and station-keeping, we note that the tangential thrust control law, most of the control action although the proposed control strategy is not optimal (with occurs in the radial direction. This behavior signifies that the respect to a minimum-acquisition-time or minimum-fuel 2 ωi term in the radial thrust law (10a) dominates the other objective), it is a simple, distributed, and computationally terms. The controller exhibits the same strategy as traditional inexpensive approach that may be tuned to achieve specific station-keeping methods where orbital phasing maneuvers mission constraints on time or fuel. Given the time and (i.e., adjusting a satellite’s position within an orbit) can maximum thrust constraints of our example mission, our simulation results showed that the commanded thrust profiles are achievable with the current state-of-the-art in electric propulsion. We also note that the proposed strategy exhibits robustness to perturbed initial conditions and unmodeled disturbances. Future work will investigate delay robustness although we do not deem the communication delay between satellites to be significant relative to the slow time scales in which the constellation evolves in our example. If we assume that communication delay is proportional to inter-satellite link distance, the worst delay is when the areostationary constellation is completely acquired and the 10 satellites are equally spaced with a line-of-sight distance of 12625 km between each pair. Considering that the delay between a ground station and a geostationary satellite at an altitude of 36000 km is approximately a quarter of a second, we can deduce that the communication delay between our satellites will be relatively small compared to the time it takes a circular, areostationary orbit to be influenced by low-thrust propulsion or the time we allow for the acquisition phase.

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