JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 37, No. 2, March–April 2014
Virtual Formation Method for Precise Autonomous Absolute Orbit Control
S. De Florio∗ University of Glasgow, Glasgow, Scotland G12 8QQ, United Kingdom S. D’Amico† DLR, German Aerospace Center, Wessling 82234, Germany and G. Radice‡ University of Glasgow, Glasgow, Scotland G12 8QQ, United Kingdom DOI: 10.2514/1.61575 This paper analyzes the problem of precise autonomous orbit control of a spacecraft in a low Earth orbit. Autonomous onboard orbit control means that the spacecraft maintains its ground track close to a reference trajectory, without operator intervention. The problem is formulated as a specific case of two spacecraft in formation in which one, the reference, is virtual and affected only by the Earth’s gravitational field. A new parametrization, the relative Earth-fixed elements, is introduced to describe the relative motion of the two spacecraft’s subsatellite points on the Earth surface. These relative elements enable the translation of absolute-into-relative orbit-control requirements and the straightforward use of modern control-theory techniques for orbit control. As a demonstration, linear and quadratic optimum regulators are designed and compared, by means of numerical simulations, with an analytical autonomous orbit-control algorithm that has been validated in flight. The differences of the numerical and analytical control methods are identified and discussed.
Nomenclature δe = relative eccentricity vector amplitude δ A = spacecraft reference area, m2 ej = relative eccentricity vector component A −1 δF = relative Earth-fixed elements vector = dynamic model matrix, s δ a = semimajor axis, m h = normalized altitude difference B δi = relative inclination vector amplitude, rad = control matrix δ C = output matrix ij = relative inclination vector component, rad δκ = relative orbital elements vector CD = drag coefficient δ e = eccentricity Lj = phase difference vector component, rad δr = normalized relative position vector ej = eccentricity vector component δ G rj = component of normalized relative position vector = gain matrix δ G = disturbance gain matrix u = relative argument of latitude, rad d δv = normalized relative velocity vector gj = gain δ i = inclination, rad vj = component of normalized relative velocity vector ϵ = normalized relative orbital elements vector, m J2 = geopotential second-order zonal coefficient η m = spacecraft mass, kg = normalized altitude ∕ κ = mean orbital elements vector n = reference spacecraft mean motion, rad s λ R = Earth’s equatorial radius, m = longitude, rad E μ = Earth’s gravitational coefficient, m3∕s2 sj = characteristic root
Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 ξ T = coordinates transformation matrix = thrusters execution error, % ρ ∕ 3 t = time, s = atmospheric density, kg m ϕ = relative perigee, rad u = argument of latitude, rad φ u = constant value of argument of latitude, rad = latitude, rad ∕ Ω = right ascension of the ascending node, rad v = spacecraft velocity, m s Ω_ ∕ Δh = altitude difference, m = secular rotation of the line of nodes, rad s Δ ω = argument of periapsis, rad vj = impulsive maneuver velocity increment component, ω ’ ∕ ∕ E = Earth s rotation rate, rad s m s θ δa = relative semimajor axis = relative ascending node, rad R− = set of negative real numbers R = set of positive real numbers Z = set of integer numbers Received 27 December 2012; revision received 3 June 2013; accepted for publication 3 June 2013; published online 5 February 2014. Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All Subscripts rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright c = closed-loop Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include d = disturbance the code 1533-3884/14 and $10.00 in correspondence with the CCC. *Ph.D. Candidate, Space Advanced Research Team, School of E = Earth Engineering; [email protected]. g = gravity †Research Lead, GSOC/Space Flight Technology, Münchner Straβe 20. N = cross-track ‡Reader (Aerospace Sciences), Space Advanced Research Team, School of R = radial Engineering. R = reference orbit
425 426 DE FLORIO, D’AMICO, AND RADICE
r = reduced II. Virtual Formation Model: Parametrization T = along-track A. Relative Orbital Elements The absolute orbit-control system design can be formulated as the Superscripts control problem of two spacecraft in formation in which one is virtual T = vector transpose and not affected by nongravitational orbit perturbations. The most appropriate parametrization to represent the relative motion of the real spacecraft with respect to the reference is the set of relative orbital elements (ROEs) [26], shown in Eq. (1), where the subscript R refers to the reference orbit: 0 1 0 1 δ − ∕ I. Introduction a a aR aR B δ C B − C B ex C B ex exR C N AUTONOMOUS orbit-control system can fulfil very strict B δ C B − C δκ B ey C B ey eyR C A control requirements on different orbit parameters in real time B δ C B − C (1) B ix C B i iR C and with a significant reduction in flight dynamics ground operations @ δ A @ Ω − Ω A [1]. In the last two decades, several studies on the autonomous orbit iy R sin i δ − control of satellites in a low Earth orbit (LEO) have been conducted u u uR – [2 8]. The TacSat-2 [9], Demeter [10], and autonomous orbit- These parameters are obtained as a nonlinear combination of the keeping (AOK) PRISMA [11] in-flight experiments represent mean orbital elements κ a; ex;ey;i;Ω;u [27–29]. The relative milestones in demonstrating that this technology has now reached a orbit representation of Eq. (1) is based on the relative eccentricity and sufficient level of maturity to be applied routinely to LEO missions. inclination vectors [30] defined in Cartesian and polar notations as The next step is to define a general and rigorous formalization of the autonomous orbit-control problem and explore new control methods. δex cos ϕ δix cos θ This paper analyzes the problem of autonomous absolute orbit δe δe δi δi (2) δe sin ϕ δi sin θ control as a specific case of two spacecraft in formation in which one, y y ’ the reference, is virtual and is affected only by the Earth s The phases of the relative e∕i vectors are termed relative perigee ϕ gravitational field. The control actions are implemented by means of and relative ascending node θ because they characterize the relative in-plane and out-of-plane velocity increments. A new param- geometry and determine the angular locations of the perigee and etrization, the relative Earth-fixed elements (REFEs), analogous to ascending node of the relative orbit. The normalized position δr the relative orbital elements (ROEs) [12–14], is introduced to T T δrRδrT δrN ∕aR and velocity δv δvRδvT δvN ∕ naR vectors describe the relative motion of the real and reference subsatellite of the spacecraft relative to the reference orbit in the RTN orbital points on the Earth surface. The REFEs allow the general formali- frame (R pointing along the orbit’s radius, N pointing along the zation of the absolute orbit-control requirements, which are usually angular momentum vector, and T N × R pointing in the direction expressed through specific Earth-fixed operational parameters such of motion for a circular orbit) can be described through the relative as altitude deviation, phase difference, etc. [15]. A direct mapping orbital elements [26,30] as between REFE and ROE enables the direct translation of absolute δr T into relative orbit-control requirements. By means of this new p δκ formalization, the deviation between the actual and the reference δv T 0 v 10 1 nominal orbit [16] can be defined in an Earth-fixed coordinate system δ analogous to the orbital frame [17]. This approach allows moreover B 1 − cos u − sin u 000CB a C B CB C the straightforward use of modern control-theory techniques for orbit B CB δ C B − 3∕2 u 2 sin u −2 cos u 01∕ tan i 1 CB ex C control. Three types of control algorithms are compared. The first is B CB C B CB δ C the one used in the AOK [3,11] on the PRISMA [18,19] mission. The B 0 0 0 sin u − cos u 0 CB ey C B CB C AOK controller adopts a guidance law for the orbital longitude of the B 0 sin u − cos u 000CB δ C B CB ix C ascending node and implements an analytical feedback control B CB C B − 3∕2 2 cos u 2 sin u 000CB δi C algorithm using along- and anti-along-track velocity increments. The @ A@ y A second and third controllers are the linear and the quadratic optimum 000cosu sin u 0 δu regulators from modern control theory [20,21]. This research finds its
Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 natural place and exploitation in the steadily increasing orbit-control (3) accuracy requirements for current and future Earth observation missions such as TerraSAR-X [16,22] or Sentinel-1 [23], whose nominal frozen repeat cycle orbits have very strict control require- ments within an orbital tube defined in an Earth-fixed frame. Following a thorough theoretical explanation of the concepts just mentioned, the final part of this paper is dedicated to the presentation of numerical simulations in which the different types of control paradigms are compared. As a case study, the spacecraft MANGO of the PRISMA formation is used because this research is based on the experience and flight data acquired during that mission. The control algorithm implementation used for this research is, in fact, grounded on the expertise gained in the realization of the PRISMA mission navigation and control flight software [24]. The simulation results are evaluated from a performance and operational point of view to formulate a first conclusion about the advantages and disadvantages of the control techniques considered here. The DLR, German Aerosapce Center/German Space Operations Center formation- flying simulation platform [25] is used to perform the analyses in a test environment very representative of reality. This test platform includes an orbit propagator and control flight software, and it simulates actuators and navigation errors. Fig. 1 Geometry of the relative Earth-fixed elements. DE FLORIO, D’AMICO, AND RADICE 427
Fig. 2 Earth-fixed and orbital reference frames.
Equation (3) represents the first-order solution of the Clohessy– intersection points R1 and R tR of the reference orbit projection with Wiltshire equations [31] expressed in terms of relative orbital the circle of latitude φ at times t and tR (Fig. 1). The minus sign in elements. the second member of Eq. (4) is due to the convention that δλδt is positive when δt is negative and vice versa. The time difference δt can B. Relative Earth-Fixed Elements be written as δt −δφn sin iR, where vφ n sin iR is the φ The constraints on the relative position of the subsatellite points component of the velocity of the reference subsatellite point moving ’ (SSPs) of the real and reference spacecraft and on the difference of on the Earth s surface, and n is the reference spacecraft mean motion. their altitudes determine the virtual formation’s geometry to be Hence, Eq. (4) can be written as maintained in the RTN orbital frame. Referring to Fig. 1, the λφη _ jωE − ΩRj cos φ reference frame has the origin in the subsatellite point considered δLλ δλ − cot iR δφ (7) at the altitude of the satellite, the λ axis tangent to the local circle n sin iR of latitude and pointing eastward, the φ axis tangent to the local meridian and pointing northward, and the η axis pointing along the Because the absolute orbit-control requirements are formulated in orbit radius. The relative position of the real and reference SSPs is terms of REFE [Eqs. (4–6)] but the control is realized in terms of ROE defined in the λRφRηR frame of the reference spacecraft by the [Eq. (1)], a direct mapping between the two systems is required. T phase difference vector δL δLλ; δLφ and by δh Δh∕aR, the Referring to Figs. 2b and 2c for the transformation from RTN to T altitude difference normalized to aR: λφη and using Eqs. (5–7), the REFE vector δF δLλδLφδh can be mapped into RTN coordinates with the transformation
δLλ δλδ δλδ δλ − δφ cot iR − cos φjω − Ω_ Rjδt (4) t 0 t E δF TEFδr (8)
0 1 δLφ δφ (5) _ _ jωE−ΩRj jωE−ΩRj 1 B 0 cos φ cos φ cos iR − 1 C T B n n sin iR C EF @ 0 sin iR cos iR A δh δη (6) 10 0 (9) where δλ; δφ; δη λ − λR; φ − φR; η − ηR , iR is the reference π ∈ orbit inclination, φ is the real spacecraft’s latitude at time t, ωE Equation (7) is singular for iR 0 k (with k Z) because, for −5 −1 7.292115 × 10 rad s is the Earth’s rotation rate, Ω_ R is the secular those values of iR, the orbital and equatorial planes are parallel. It is δλ rotation of the reference orbit line of nodes, δt t − tR, and tR is the interesting to remark that the term δt 0 of Eq. (4) can be written RTN δφ time at which the reference spacecraft passes at latitude φ. Figure 1 directly in the coordinate system by imposing Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 δ δ δ depicts the case in which t>tR. The phase difference vector’s rT sin iR rN cos iR 0, and substituting the solution rT −δ ∕ π component δLλ δLλ φ; δλ; δφ; δt is the distance, normalized to rN cos iR sin iR, singular for iR 0 k , in the equation δλ δ − δ aR, of the real and reference ground tracks measured along the λ axis rT cos iR rNpsin iR. Finally, from Eqs. (3) and (8) and 2 at latitude φ. δLλ is also a function of δt because the real and reference substituting cos φ 1 − sin u sin i , vector δF and its time spacecraft pass at latitude φ at different times, and the coordinates derivative d δF ∕dt, evaluated fixing u u , can be written in terms systems λRφRηR and λφη move with the SSPs while the Earth is of relative orbital elements using the transformation matrix δλ T T T T rotating. The quantity δt 0 is the normalized distance, measured EF p, with p defined in Eq. (3): λ along the axis, between the intersection points S and R1 of the real and reference orbit projections with the circle of latitude φ at time t δF u; δκ T u δκ (10) (Figs. 1 and 2). Referring to Fig. 2a, as in the coordinate system λ φ η ≈ δφ π − −δφ −δλ −δφ R R R is R1 cot iR ; , and S ; ,it δλ − ≈ δλ − δφ δλ d δF d δκ results in δt 0 R1 S cot iR. The quantity δt is the T u (11) normalized distance measured along the λ axis between the dt dt 0 1 h i 3 su τ 1−cu ciR cu B − uτ 2τsu −2τcu τciR − 1 1 τ C B 2 siR cu siR C B C T u B 3 C (12) B − usiR 2susiR −2cusiR suciR 1 − cu ciR siR C @ 2 A 1 −cu −su 000 428 DE FLORIO, D’AMICO, AND RADICE p _ 2 ~ ~ with τ jωE − ΩRj∕n 1 − sin u sin i , su sin u, cu is the Jacobian evaluated at κR of the vectors sum Ag κ Ad κ , cos u, siR sin iR, and ciR cos iR. Because u t is periodic, the and δκ is the relative orbital elements vector. vectorial function δF u; δκ t is obtained from the function Making the proper modifications to matrix A~ κR for the δF u t ; δκ t by considering only the subset u; δκ t of the normalization in aR and the introduction of δiy [Eq. (1)], and adding function’s domain u t ; δκ t . This is why the term dT∕dt δκ the control term B κ Δv: dT∕du du ∕dt δκ 0 does not appear in Eq. (11) if the variation of ϵ A κ ϵ x B κ Δv i is considered negligible. This procedure is justified by the control _ R d (20) design approach explained at the end of Sec. III.A. Equations (10) A κ A κ A κ ϵ δκ and (11) are valid under the assumptions of near-circular orbits, and where R g R d R [Eqs. (A2) and (A4)], aR is the relative orbital elements vector [Eq. (1)] normalized to the separations between the real and reference spacecraft are small when x compared to the reference orbit radius; see Eq. (3). Table 1 shows the semimajor axis, d results from the direct (dyadic) vectors product A~ κ T Δv Δ Δ Δ T form assumed by the relative Earth-fixed elements when evaluated at d R 1;aR;aR; 1; 1; 1 , and vR; vT ; vN is the the ascending node (u 0). The phase difference vector component vector of impulsive velocity increments in the RTN orbital frame. B κ δLλ at the ascending node is commonly used as operational parameter Matrix [Eq. (A5)] represents the Gauss variational equations of for the maintenance of phased orbits. If the inclination remains equal motion adapted for near-circular nonequatorial orbits [15]. The to its nominal value, the control of the phase difference at the equator elements of matrix B κ are computed with good approximation [29] will be the most effective way to monitor the displacement between using the mean orbital elements. The Gauss equations provide the real and reference ground tracks. It can also be noticed that the relationships between the impulsive velocity increments in the RTN maintenance of the altitude deviation δh requires the control of the orbital frame and the increments of the orbital elements. semimajor axis as well as the eccentricity vector. Equation (20) can be written in the general form ϵ_ AI ϵ B Δv (21) x 0A x 0 III. Virtual Formation Model: Controller Design _ d 0 d The state–space representation approach is used here for the Equation (21) is the representation of a tracking system [20] in which development of the autonomous orbit control with techniques from the atmospheric drag perturbation vector xd is represented as an modern control theory. In general, the linear model used has the form additional state variable, the disturbance input, which is assumed to satisfy the model x_ d A0xd, and I is the 6 × 6 identity matrix. If the ϵ Aϵ BΔv x _ d (13) feedback controller is designed to compute the control inputs Δvj always in the same place of the orbit (u u ), xd can be assumed as y Cϵ (14) A ≡ constant [Eq. (A3)] (i.e., 0 0). This design approach stems from the consideration that the implementation of an orbit-control strategy Δv −Gy − G x d d (15) implies the specification not only of the magnitude of the corrective maneuvers but also the in-orbit location that maximizes their ϵ A B where is the state vector; and are the dynamic model and efficiency. The choice of the less-expensive maneuver’s in-orbit x control input matrix, respectively; d is the modelled state location depends on the operational parameter that has to be Δv G perturbation; is the impulsive velocity increment vector; and controlled. An out-of-plane maneuver Δv to change the orbit’s G N and d are the output and disturbance gain matrices, respectively. inclination, for example, according to the Gauss equations [Eq. (A5)] y The output vector is composed of relative Earth-fixed elements is most effective if placed at the node (u 0) while at the orbit’s introduced by Eq. (10). highest latitude (u π∕2)ifΩ has to be changed. On the other hand, while the semimajor axis can be changed with an along-track A. Linear Dynamic Model maneuver ΔvT with the same effectiveness anywhere along the orbit, The state–space model of the orbital motion of the real and the ΔvT most effective [11] to control δLλ (Table 1) has to be reference spacecraft is given by computed at the equator (ascending or descending node) for reasons of symmetry. κ_ A~ κ A~ κ (16) g d B. Reduced Model ~ κ_R Ag κR (17) In case no eccentricity or inclination are to be actively controlled, the model can be reduced to Eq. (22) by considering only the states δ δ δ Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 where κ a; ex;ey;i;Ω;u is the mean orbital elements vector. a, iy, and u. The elements aij agij adij are given by ~ ~ Vector functions Ag and Ad [Eqs. (A1) and (A3)] describe the Eqs. (A2) and A4). The use of this model allows the design of a linear κ δ δ behavior of the mean orbital elements under the influence of the J2 controller for the relative Earth-fixed elements Lλ and Lφ. The gravitational perturbation and atmospheric drag [29,32]. The mean passive control of δh can be achieved by a proper in-orbit placement orbital elements κR of the reference spacecraft are affected only by of the along-track maneuvers as explained in the next section: the Earth’s gravitational field as they define the nominal trajectory 0 1 0 1 δ [15,16]. The linearization around κR of the difference between a11 00 a 2 B C B C Eqs. (16) and (17) yields A 3 a nJ2 B C ϵ B δ C r 2 2 @ a51 00A r aR@ iy A 4 RE 1 − e d δκ a 00 δu A~ κR δκ A~ κR (18) 61 dt d 0 1 0 p 1 02 0 μa B C B C where 1 B C A B C Br @ 0 0 sin u A xdr − CDρ@ 0 A n m ∂ A~ κ A~ κ −20− sin u∕ tan i 0 A~ κ g d R ∂κ (19) κR (22)
Table 1 Relative Earth-fixed elements (specific case)
iR u δLλ δLφ δh _ iR 0 δiy∕ sin iR jωE − ΩRj∕n δu − 2δey δu − 2δey sin iR δa − δex DE FLORIO, D’AMICO, AND RADICE 429
– ∕ ω − Ω_ ∕ In this case, Eqs. (10 12) have the following form: where the terms c12 1 sin iR, c13 j E Rj n and c21 ∕ ω − Ω_ ∕ – ar21 sin iR ar31 j E Rj n are obtained from Eqs. (23 25) δF u; ϵr aR Tr u ϵr (23) imposing u 0. Δ − δ δ ∕ The control input is given by vT g1 Lλ g2d Lλ dt aR. The objective here is to find the gains g1 and g2 that place the poles of δF A A − B GC d T A ϵ the closed-loop dynamic matrix c r r at the locations aR r u r r (24) A dt desired. The characteristic polynomial of c is
I − A 2 − 0 h i 1 js cj s s b1c21g2 ar11 s b1 ar21 c12 ar31 c13 g1 τ 1−cu ciR − 3 uτ 1 cu τ 2 B 2 cu siR C s s a^ g s a^ g (29) B C 1 2 2 1 Tr u B − 3 − C (25) @ 2 usiR 1 cu ciR siR A One of the three poles of Ac is placed in the origin regardless of the 100 value of the gains. This is due to the fact that the part of the system depending on δiy is not controllable by ΔvT. Indeed the controllability test matrix of Eq. (30) has rank 2 < 3. However, δLλ C. Linear Control can be controlled by means of variations of δu, which compensate the The linear control law for the system Eq. (21) has the general form variations of δiy (Table 1): [20] of Eq. (15). G and Gd are the gain matrices, and y Cϵ is the 0 1 ’ C b a b a2 b system s output. The terms of matrix will be computed from B 1 r11 1 r11 1 C Eq. (12) because the goal of the proposed absolute orbit controller Q 2 B C ctr BAB A B @ 0 ar b1 ar ar b1 A (30) is the maintenance of one or more relative Earth-fixed elements r r r r 21 11 21
within their control windows. For the closed-loop system to be 0 ar31 b1 ar11 ar31 b1 asymptotically stable, the characteristic roots [20] of the closed-loop p A A − BGC 2 dynamics matrix c must have negative real parts. The closed-loop poles s −a^ a^ p− 4a^ ∕2 of Eq.p (29) are G 1 1 2 This can be accomplished by a suitable choice of the gain matrix if real (or complex-conjugate) if ja^1j > 2 a^2 (or ja^1j < 2 a^2) and the system is controllable. Once the gains and thus the poles of matrix a^ b c g > 0.Ifa^ > 0, the poles are placed on the left of A G 2 1 21 1 1 c have been set, matrix d is obtained substituting Eq. (15) in the imaginary axis of the complex plane, and the closed-loop system Eq. (20), imposing y Cϵ and the steady-state condition ϵ_ 0: is stable. These stability conditions impose constraints on the value of the gains as resumed by Table 2. If the poles chosen are G CA−1B −1CA−1I − δ ∕ d c c (26) complex-conjugate, the contribution of the term g2 d Lλ dt aR to the control action ΔvT will be negligible because aRd δLλ ∕dt has −1 The linear control system is designed by means of pole placement. an order of magnitude of 30∕86; 400 m · s [11], and the value of g2 The choice of which relative Earth-fixed element has to be controlled is limited in the range indicated in the second row of Table 2. This is determined by the mission requirements, whereas the best place means that to also control d δLλ ∕dt, the poles have to be on the and direction of the orbital maneuvers is also dictated by the Gauss negative real axis because, in this way, a suitably large value of g2 can equations [Eq. (A5)]. First, an in-plane orbit-control system will be be obtained. The gains are related to the poles by the following considered with the single control input (ΔvT ) and two outputs (δLλ equation: and d δLλ ∕dt), computed at the orbit’s ascending node. This is the basic control required for the maintenance of a repeat-track orbit [15]. s s s1 s2 ar g 1 2 ;g 11 (31) Second, an in-plane/out-of-plane controller with two control inputs 1 2 b1 ar21 c12 ar31 c13 b1c21 (ΔvT and ΔvN) and three outputs (δLλ, d δLλ ∕dt, and δiy)is designed. In this case, the in-orbit phasing δLφ can be restrained in a − with s1, s2 ∈ R . The gain values chosen as a first guess are g1 control window as well. The design of these regulators will be based Δ ∕ δ Δ ∕ δ ∕ sgn c21 vTδ aR Lλ and g2 sgn c21 vT δ aRd Lλ on the reduced model [Eq. (22)]. L Δ ΔMAX δ dδL ∕ ∈ dt MAX, where vTδL , vTdδL , aR LλMAX , and aRd Lλ dt MAX R are limits imposed by design, and sgn c21 is the sign of c21. The 1. In-Plane Control with One Input and Two Outputs dynamics of the closed-loop system can be verified and adjusted by In this case, the design of the feedback system is finalized to computing the poles with Eq. (31), adjusting their placement, and Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 control δLλ and d δLλ ∕dt, computed at the ascending node iterating the process. The control input is (Table 1), by means of along-track velocity increments. This means that the orbit controller is designed to work only once per orbit at Δ −GCϵ − δ δ δ vT r g2c21 a g1c12 iy g1c13 u aR (32) most. The system components are 0 1 0 1 0 1 The following subsystem of Eq. (27) is considered for the a 00 δ r11 a b1 G B C B C B C determination of the disturbance gain matrix d [Eq. (15)]: A B C ϵ B δ C B B C r @ ar21 00A r aR@ iy A r @ 0 A d δa ar 00 δu 0 aR a δa aR b Δv x y aRδa (33) 31 0 1 dt r11 1 T d1 xd1 B C ∕ G G x B C for which Eq. (26) yields g0 1 b1, and then g1 g2 d g0 00 dr @ 0 A (27) 0 Table 2 Gain constraints ∕ with ar , ar , ar , b1 2 n, and xd1 given by Eq. (22), and g1, Pole type c21 i > 0 and g1 > 0 c21 i < 0 and g1 < 0 ∈ 11 21 31 y Cϵ p p g2 R. The output r is a 2 a^ a 2 a^ Real g > r11 2 g < r11 2 0 1 2 b c 2 b c δa 1 21 p p 1 21 δLλ ar ar 2 a^2 ar 2 a^2 ar 0 c12 c13 @ δ A 11 11 11 11 aR d δLλ aR iy (28) Complex < g2 < < g2 < c 00 b1c21 b1c21 b1c21 b1c21 dt 21 δu 430 DE FLORIO, D’AMICO, AND RADICE
D. Linear Quadratic Optimum Regulator Control G 1 00 (34) d b1 The linear quadratic optimum regulator (LQR) [20] is best suited for a multiple-input/multiple-output system like that here considered. ∕ Here, instead of seeking a gain matrix to achieve specified closed- The term xd1 b1 for a small satellite in a low Earth orbit has an order of magnitude of 10−8 m∕s and is negligible in the computation of loop locations of the poles, a gain is sought to minimize a specified Λ Δv . This is not surprising because G represents the instantaneous cost function expressed as the integral of a quadratic form in the T d ε Δv effect of the drag and not its integration over time. state plus a second quadratic from in the control : The maneuvers have to be computed at the ascending node but can Z T T T T be executed with the same effectiveness in any place along the orbit as Λ ϵ τ C QyCϵ τ Δv τ RΔv τ dτ (40) they change δu by means of semimajor axis increments [first row of t Eq. (A5)]. Nevertheless, the along-track maneuvers can be located Q CT τ Q τ C × [26] to be the most effective on the control of the relative eccentricity where y is the 6 6 state weighting symmetric matrix, Q is the weighting matrix of the output y Cϵ, and R is the vector components δex and δey. Solving the second and third of the y Gauss equations ϵ_ B κ Δv in u and imposing that the effect of the 3 × 3 control weighting symmetric matrix. The optimum steady-state [T ∞ in Eq. (37)] gain matrix G for system Eq. (21) is velocity increment ΔvT is decreasing the magnitude of δex and δey, the eccentricity vector can be passively controlled with a proper in- G −R−1BT M orbit location [33] of the along-track maneuver: (41) M δe where is the steady-state solution to the Riccati equation u arctan y kπ;k 0if δe Δv < 0; M δe x T x −M _ MA AT M − MBR −1BT M Q 0 (42) k 1if δexΔvT > 0 (35) The disturbance gain matrix Gd for the case of Eq. (21), in which xd is 2. In-Plane/Out-of-Plane Control with Two Inputs and Three Outputs constant, is given by In this case, the design of the system is finalized to control the G −R−1BT AT −1MI (43) relative Earth-fixed elements δLλ and δLφ at the ascending node by d c means of along-track and cross-track velocity increments. At the A A − BG M ascending node, δLλ and δLφ are related to each other by the equation where c is the closed-loop dynamics matrix, is given I δLλ k δi k δLφ, where k and k are the values displayed in by Eq. (42), and is the identity matrix. As already remarked in 1 y 2 1 2 G x Table 1. The only chance of controlling δLλ and δLφ at the same time Sec. III.C.1, the term d d can be neglected. In the performance function defined by Eqs. (40), the quadratic form yT Qy represents a δLφ is thus selecting δiy as one of the controlled outputs. The velocity increment Δv to control δi has to be placed at the orbit’s highest penalty on the deviation of the real from the reference orbit, and the N y Q latitude (u π∕2) to maximize its effectiveness [fifth row of weighting matrix specifies the importance of the various Eq. (A5)]. The execution of Δv will be placed with the rule of components of the state vector relative to each other. The term T ΔvT RΔv Eq. (35). The system components are A and ϵ from Eq. (27) and is instead included to limit the magnitude of the control r r Δv 0 1 signal and to prevent saturation of the actuator. Overall, the gain b 0 matrices choice is a tradeoff between control action cost (i.e., small 1 g g 0 B @ 0 b A G 1 2 (36) gains to limit propellant consumption and avoid thruster saturation r 2 00g phenomena) and control accuracy (i.e., large gains to limit the 00 N excursion of the state from its reference value). The choice of the ∕ ∕ weighting matrices is done here with the maximum size technique. where b1 2 n, b2 sin u n are given by the first and fifth rows of The aim of this method is to confine the individual states and control matrix B [Eq. (A5)], and the term of Br relating ΔvN and δu has been y Cϵ actions within prescribed maximum limits given respectively by omitted by design. The output r is Δ Q R yiMAX and viMAX . The terms of y and will be thus chosen 0 1 0 10 1 imposing the following equations: δ δ Lλ 0 c12 c13 a @ d δLλ A @ A@ δ A aR dt aR c21 00 iy (37) ki kij δ δ Qyii ;Qyij ; iy 010 u y2 2y y iMAX iMAX jMAX (44) Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 kij ∈ R for i 1; 2; 3 and j 1; 2; 3 where c12, c13, and c21 are the same as in Eq. (28). The system is controllable as the rank of the controllability matrix is 3. The A A − B GC characteristic polynomial of c r r is h h R i ;R ij ; I − A 3 − 2 ii Δv2 ij 2Δv Δv js cj s b1g2c21 b2gN ar11 s iMAX iMAX jMAX (45) − ∈ b1g1 ar21 c12 ar31 c13 b2gN b1g2c21 ar11 s hij R for i R; T; N and j R; T; N 3 2 ar31 c13b1b2g1gN s a^1 g2;gN s The choice of diagonal Qy and R matrices is usually a good starting a^2 g1;g2;gN s a^2 g1;gN (38) point in a trial-and-error design procedure aimed at obtaining the desired properties of the controller. The proper control gains can be found by splitting the problem into two distinct subproblems. The first-guess values of g1 and g2 are the same found solving the problem of the previous section. The cross-track IV. Numerical Simulations δ ∕Δ maneuver gain is found with the relation gN aR iyMAX vNMAX , Two types of numerical simulations were run to validate and δ Δ ∈ where iyMAX , vNMAX R are imposed by design. The placement of compare the control methods introduced in the previous sections. A the closed-loop poles and the dynamic response of the system can then first set of simulations was based on an orbit propagation model, be verified by finding the roots of Eq. (38). The control input is which included the gravitational field and a constant atmospheric drag as the orbital perturbations. This is also the perturbation model Δ − δ − δ δ vT g2c21 a g1 c12 iy c13 u aR on which the analytical algorithm of the AOK controller [3,11] is based. By means of these ideal simulations, the performance of the Δ − δ vN gN iyaR (39) different controllers could be compared under theoretical design DE FLORIO, D’AMICO, AND RADICE 431
conditions. A second set of simulations was run to compare the Table 5 Actuators behavior of the controllers in a realistic orbit environment. Tables 3–6 accuracy display the spacecraft physical parameters, the orbit propagation Accuracy Value model, and the sensor, actuators, and navigation models used for the Propulsion system simulations. The propulsion system is characterized by a minimum − MIV 7 · 10 4 m∕s impulse value (MIV) and a minimum impulse bit (MIB). −5 ∕ Δ MIB 7 · 10 m s Consequently, the thrusters can only deliver values of v, which ξ 5% are larger than MIV and integer multiples of MIB. Furthermore, the Attitude control execution error of the thrusters is quantified by the relation Mean 0 deg ξ Δ − Δ ∕Δ Δ j Vreal Vcmd Vcmdj · 100, where vcmd is the velocity Δ increment commanded by the onboard controller, and vreal is the actual velocity increment executed by the propulsion system. Finally, Table 6 Absolute navigation accuracy the attitude-control error, which causes thruster misalignment, is treated as Gaussian noise with zero bias and a 0.2 deg standard Navigation accuracy Position Velocity deviation in the spacecraft’s three body axes. The values of the Mean, m 3.04 4.35 · 10−3 navigation accuracy refer to the magnitude of the absolute position σ, m 1.6 2.7 · 10−3 and velocity vectors in the RTN frame and are typical of an onboard GPS-based navigation system [34]. Further details about the navigation accuracy in the simulations are given in Appendix B. The initial state used for the orbit propagation is the same shown in node for the analytical controller, whereas it occurs following the rule of Eq. (35) for the linear regulators. Figure 4 shows how the phase Table 7, and the run time is one month. The reference orbit has been δ generated with the only inclusion of the GRACE GGM01S 30 × 30 difference vector component Lλ is maintained in its control window gravitational field in the forces model. The precise orbit determina- by means of along-track maneuvers, which change the value of the ’ δ tion (POD) ephemerides of the spacecraft TANGO of the PRISMA orbit s semimajor axis. The guided time evolution of a determines δ mission [18], the formation’s target satellite, which flies in free that of u and thus the timing of the real with respect to the virtual motion, have been exploited to calibrate the atmospheric density spacecraft in passing at the ascending node [Eq. (4)]. The drift of the δ model used for the simulations to have a high degree of realism. phase difference vector component Lφ is actually used to δ δ Figure 3 shows the time evolution of the difference between compensate the drift of iy in the control of Lλ (Table 1). The linear TANGO’s real (POD) and reference orbital (RO) elements. Figure 3 and LQR regulators are able to control the eccentricity vector (and includes also the difference between TANGO’s orbit and an orbit thus δh) with the maneuvers’ location rule of Eq. (35), whereas the propagated with the calibrated model. analytical controller has no eccentricity vector control capability as it executes the maneuvers only at the orbit’s ascending node. Figure 5 A. In-Plane Orbit Control shows the orbital maneuvers commanded by the onboard controllers 1. Ideal Simulations Scenario and executed by the spacecraft thrusters. Table 9 collects the control performance and the maneuvers budget. The goal of controlling δLλ In these simulations, a constant atmospheric drag is the only by means of along- and anti-along-track velocity increments is achieved perturbation force included in the orbit propagation model and no with very similar performances by the three controllers. The main thruster, attitude, and navigation errors are included. The gravita- difference between the AOK analytical controller and the numerical tional field model used for the orbit propagation is the GRACE regulators is that the AOK’s maneuvers computation is based on a long- GGM01S 40 × 40, whereas that used for the RO generation is the × term prediction of δa highly dependent on the correct estimation of the GRACE GGM01S 30 30. Figure 4 shows the REFE of the ∕ spacecraft, computed at the ascending node (u 0), in case the orbit- semimajor axis decay rate da dt. On the other hand, the linear control system is designed with the analytical algorithms of the AOK regulators compute the orbital maneuvers with a pure feedback logic experiment [3,11] (AOK), Eq. (32) (linear), and Eq. (41) (LQR). based on the values of the control gains. This fundamental difference Table 8 collects the parameters used for the design of the linear between the two control strategies is demonstrated by examining Fig. 5. regulators [Eqs. (31), (44), and (45)]. The maneuver duty cycle The linear and LQR regulators command groups of equal-sized ≈ −4 ∕ imposed in all of the simulations was two orbital periods (i.e., the consecutive maneuvers ( 8 · 10 m s) at nonregular time intervals, controllers could compute and command maneuvers once every third whereas the AOK control system commands larger maneuvers −3 orbit). The execution of the maneuvers takes place at the ascending (≈2.5 · 10 m∕s) at a deterministic maneuvers cycle of two days. The AOK algorithm has an optimal control performance, in terms of control Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 accuracy and Δv budget, if it has an accurate knowledge of the Table 3 PRISMA spacecraft physical semimajor axis decay rate, as in the case of this simulation. The constant properties value of the atmospheric density used for the orbit propagation was, in Spacecraft physical property MANGO TANGO fact, given as input to the AOK software. Mass, kg 155.12 38.45 Drag area, m2 0.5625 0.2183 Table 7 Initial state Drag coefficient 2.5 2.5 SRP effective area 0.5625 0.2183 Parameter Value ECI state X −3; 967; 394.8566 m Table 4 Propagation parameters Y −289; 822.105 m Z 5,883,191.2151 m Propagation Model − ∕ vx 6; 126.365 m s Orbit vy 1; 487.7675 m∕s Earth gravitational field GRACE GGM01S 40 × 40 vz −4; 071.5062 m∕s Atmospheric density Harris–Priester Mean orbital elements Sun and Moon ephemerides Analytical formulas [17] a 7,130,522.2961 m Solid Earth, polar, and ocean tides IERS ex −0.004058 Relativity effects First-order effects ey 0.002774 Numerical integration method Dormand–Prince I 98.28 deg RO Ω 351.74 deg Earth gravitational field GRACE GGM01S 30 × 30 U 123.38 deg 432 DE FLORIO, D’AMICO, AND RADICE
Fig. 3 Evolution in time of TANGO’s orbital elements (POD) with respect to the reference and the propagated actual orbit.
Fig. 4 Relative Earth-fixed elements (in-plane control).
2. Realistic Simulations Scenario Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 strict control of the relative eccentricity vector because the solar These simulations were run using the parameters in Tables 3–6. radiation pressure perturbing force is this time included in the orbit’s The regulators have been designed with the same parameters of perturbation forces, and the orbit is not at frozen eccentricity. The δ AOK controller has, in this case, the additional disadvantage of Table 8, with the exception of the imposition of LλMAX 15 m for all of the regulators. Figure 6 shows the relative Earth-fixed elements inaccuracies in the onboard estimation of the atmospheric drag [3,11] of the spacecraft. A reference orbit acquisition based on the control of and for this reason has a performance slightly worse than the linear δLλ [11] was also demonstrated. Table 10 collects the control regulators. The reliance of the AOK analytical controller on a correct performance and the maneuvers budget during the steady-state phase onboard estimation of the atmospheric drag can be noticed by following the reference orbit acquisition. The degradation of the comparing Figs. 5 and 7 (top). The loss of accuracy in the onboard control performance with respect to the ideal case (Table 9) is mainly estimation of da∕dt [11] entails the loss of determinism in the caused by the inclusion of the onboard navigation error in the maneuver cycle. Table 11 offers an overview of the different pole simulation model. The placement of the maneuvers with the rule of placements in open and closed loop. The LQR is always able to find Eq. (35), not optimized [33] from time to time, is not sufficient for a an optimal placement for all of the poles, while the linear in-plane regulator can place two poles only. Nevertheless, poles s5 and s6, δ δ ∕ Table 8 Regulator design parameters from which the control of Lλ and d Lλ dt by means of along-track velocity increments depends upon, are placed in a similar way by AOK Linear LQR both methods. δLλ ,m 5 10 10 δ MAX∕ ∕ —— ∕ ∕ d Lλ dt MAX, m s 100 86; 400 200 86; 400 B. In-Plane/Out-of-Plane Orbit Control Δ ∕ —— —— −6 vRMAX , m s 10 Δ ∕ —— −3 −3 Figures 8 and 9 show the ROE and the REFE, respectively, of the vTMAX , m s 10 10 −6 spacecraft in case of in-plane/out-of-plane orbit control realized with ΔvN , m∕s —— —— 10 MAX the linear controller of Eq. (39) and in a realistic simulation scenario DE FLORIO, D’AMICO, AND RADICE 433
Fig. 5 Commanded and executed orbital maneuvers.
Fig. 6 Relative Earth-fixed elements (in-plane control).
– δ (Tables 3 6). The gains were computed imposing the limits LλMAX and eight (12 h) for the out-of-plane maneuvers placed at
Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 δ ∕ ∕ ∕ δ π∕ 10 m, d Lλ dt MAX 100 86400 m s, and iyMAX 40 m on the u 2. −3 −2 δ δ outputs and ΔvT 1 · 10 and ΔvN 1.5 · 10 m∕s on the Unlike the case of the previous section, all the three states a, iy, MAX MAX δ δ δ φ and u are controlled as all the poles are placed on the left of the inputs. The value of iyMAX was chosen to keep L in a control window of 1500 m. The duty cycle imposed for the in-plane imaginary axis (Table 11). As expected, the out-of-plane velocity δ maneuver at the ascending node was four orbital periods (6 h) increments allow the control of Lφ (Fig. 9). The same considerations
Table 9 Control performance and maneuvers budget with in-plane control
δLλ, m Minimum Maximum Mean σ RMS ΔvT , m∕s Minimum Maximum Total AOK −7.4 9.2 0.18 3.6 3.6 AOK 0.0018 0.0025 0.0373 Linear −0.6 9.9 5.3 1.7 5.6 Linear 0.0007 0.0011 0.0379 LQR −1.2 9.1 4.8 1.8 5.1 LQR 0.0007 0.001 0.0374
Table 10 Control performance and maneuvers budget with in-plane control δ σ Δ ∕ Lλ, m Minimum Maximum Mean RMS vT , m s Minimum Maximum Total AOK −37.2 23.3 −9.8 13.5 16.7 AOK −0.0078 0.0107 0.0942 Linear −15.4 24.0 6.5 7.8 10.2 Linear −0.0044 0.0112 0.0946 LQR −13.3 24.5 3.9 7.5 8.4 LQR −0.0049 0.0114 0.1228 434 DE FLORIO, D’AMICO, AND RADICE
Table 11 Poles
Open loop Linear in-plane LQR in-plane In-plane/out-of-plane s —— − −10 − −10 —— 1 0 8.77 · 10 8.78 · 10 j s —— − −10 −10 —— 2 0 8.77 · 10 8.78 · 10 j s − −12 —— − −12 − −7 —— 3 2.38 · 10 3.16 · 10 6.11 · 10 j s − −7 − −12 −7 − −1 4 5.71 · 10 0 3.16 · 10 6.11 · 10 j 3.56 · 10 s − −5 − −1 − −1 − −1 5 1.94 · 10 4.35j 1.61 · 10 2.87 · 10 1.61 · 10 s − −5 − − −5 − −4 − −4 6 1.94 · 10 4.35j 7.72 · 10 1.93 · 10 1.15 · 10
of the previous section regarding the in-plane control are valid here. V. Conclusions δ Δ In Fig. 8, one can also observe that ix is not influenced at all by vN The problem of the autonomous absolute orbit control can be π∕ as the out-of-plane maneuvers are executed only when u 2 formalized as a specific case of two spacecraft in formation in which [Eq. (A5)]. Figure 10 shows the orbital maneuvers commanded by one, the reference, is virtual and affected only by the Earth’s the onboard controller and executed by the spacecraft thrusters. A gravitational field. A new parametrization, the relative Earth-fixed regular maneuver cycle of the in-plane and out-of-plane maneuvers elements, has been introduced for the translation of absolute orbit- cannot be detected. Table 12 collects the control performance and the control requirements in terms of relative orbit control. Methods maneuvers budget. The in-plane control accuracy and cost is very developed for the formation keeping can thus be used for the orbit similar to that of the simple linear controller of the previous section control of a single satellite. This approach allows also the straight- (Table 10). The out-of-plane Δv is 0.51 m∕s, rather expensive forward use of modern control-theory numerical techniques for orbit compared to the in-plane maneuvers budget. These simulation results control. Indeed, a bridge between the worlds of control theory and confirm that the in-plane and out-of-plane control problems can be orbit control is built by this formalization. As a demonstration, a treated separately in the design of the regulator. linear and a quadratic optimum regulator have been designed and
Fig. 7 Commanded and executed orbital maneuvers. Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575
Fig. 8 Relative orbital elements (in-plane and out-of-plane control). DE FLORIO, D’AMICO, AND RADICE 435
Fig. 9 Relative Earth-fixed elements (in-plane and out-of-plane control).
Fig. 10 Commanded and executed orbital maneuvers.
Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 tested. These two numerical control methods have been compared, by very good control accuracies. With the onboard availability of means of numerical simulations, with an analytical algorithm that an accurate orbit model, this type of analytical controller has an has been validated in flight with the autonomous orbit-keeping optimal control performance in terms of accuracy and costs as well as experiment on the PRISMA mission. a deterministic maneuver cycle, the duration of which depends on the The main difference between these methods is that the maneuvers’ size of the control window. On the other hand, the numerical computation by the analytical controller is based on a long-term regulators have a simpler flight software implementation and an orbit prediction, whereas the linear regulators compute the control enhanced flexibility given by the possibility of varying the type actions with a pure feedback logic based on the values of the control of onboard controller simply by uploading to the spacecraft gains. The accuracy of the orbit model therefore plays a critical different gain configurations. The type of control of these feedback role in the implementation of the analytical controller. For the systems is, in fact, entirely determined by the type and value of implementation of the numerical feedback regulators, the critical the gains. issue is not the prediction accuracy of the model but its reliability in The step forward is the design of a predictive control system with defining the stability conditions of the closed-loop system in the the same formal approach explained in this paper. This kind of determination of the gains. The PRISMA analytical controller has numerical regulator can combine the advantages of the analytical and demonstrated in-flight to be robust, cost-effective, and capable of numerical orbit-control systems.
Table 12 Control performance and maneuvers budget with in-plane and out-of-plane control
δL, m Minimum Maximum Mean σ RMS Δv, m∕s Minimum Maximum Total
δLλ −17.5 26.3 2.4 6.7 7.1 ΔvT −0.0021 0.0032 0.094 δLφ −153.9 1269.2 403.4 293.1 498.6 ΔvN −0.0044 0.0312 0.51 436 DE FLORIO, D’AMICO, AND RADICE
Appendix A: Linearized Orbit Model
A1 Gravity Field
0 1 0 1 0 0 B 2 C B C B − 5 cos i − 1 ey C B 0 C B C B C B 2 − C B C 3 R 2 nJ B 5 cos i 1 ex C B 0 C A~ E 2 B C B C (A1) g − 2 2 B C B C 4 a 1 e B 0 C B 0 C B C B C @ −2 cos i A @ 0 A p 5 cos2 i − 1 3 cos2 i − 1 1 − e2 n
0 1 B 000000C B C B a a a a 00C 0 1 B g21 g22 g23 g24 C 0 ::: 0 B C B C 2 B ag ag ag ag 00C B C A 3 RE nJ2 B 31 32 33 34 C − . . . . g B C B . . . C 4 a 1 − e2 2 B 000000C @ A B C B C 3n∕2a aR ::: 0 B a a a a 00C @ g51 g52 g53 g54 A
ag61 ag62 ag63 ag64 00 2 2 7aR 4 5 cos i − 1 4e a 5 cos2 i − 1 e a − e e a − 5 cos2 i − 1 y 1 g21 2a y g22 1 − e2 x y g23 1 − e2 2 7aR 4e a 10e sin i cos ia − 5 cos2 i − 1 e a 5 cos2 i − 1 x 1 g24 y g31 2a x g32 1 − e2 2 4 5 cos i − 1 7aR cos i a e e a −10e sin i cos ia g33 2 x y g34 x g51 1 − e a sin iR 8e cos i 8e cos i 2 sin i a − x a − y a g52 2 g53 2 g54 1 − e sin iR 1 − e sin iR sin iR p 7aR − 2 − 2 − − 2 ag61 5 cos i 1 3 cos i 1 1 e 2a p e a x 4 5 cos2 i − 1 3 3 cos2 i − 1 1 − e2 g62 1 − e2 p p e a y 4 5 cos2 i − 1 3 3 cos2 i − 1 1 − e2 a −2 5 3 1 − e2 sin i cos i (A2) g63 1 − e2 g64 Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575
Fig. B1 Accuracy of the onboard estimated position in RTN (realistic simulations scenario). DE FLORIO, D’AMICO, AND RADICE 437
Fig. B2 Accuracy of the onboard estimated orbital elements (realistic simulations scenario).
Table B1 Navigation error (position and velocity in RTN)
R,m T,m N, m Position (three-dimensional), m vR, m∕s vT, m∕s vN, m∕s Velocity (three-dimensional), m∕s Mean 0.86 −0.30 −1.13 3.04 3 · 10−5 −3.5 · 10−5 −2.3 · 10−5 4.35 · 10−3 σ 2.32 1.51 1.42 1.60 3 · 10−3 3.7 · 10−3 2 · 10−3 2.7 · 10−3 RMS 2.47 1.54 1.81 3.43 3 · 10−3 3.7 · 10−3 2 · 10−3 5.1 · 10−3
A2 Atmospheric Drag Table B2 Navigation error (orbital elements) a,m aex,m aey ,m i, deg Ω,m u, deg 0 p 1 − −6 − −4 −5 μa Mean 2.78 0.18 0.01 3 · 10 3.8 · 10 2.7 · 10 p σ 3.94 2.99 3.71 2.2 · 10−5 6.1 · 10−3 5.6 · 10−3 B μ∕ C − − − B ex cos u p a C RMS 4.82 2.99 3.71 2.2 · 10 5 6.1 · 10 3 5.6 · 10 3 A B μ∕ C A~ − ρB ey sin u a C d CD B C (A3) m B 0 C @ A 0 A3 Control Matrix 0 0 1 0 1 02 0 a 00000 B d11 C B C B C B sin u 2 cos u 0 C B a a a C B − C d21 d22 000d26 1 B cos u 2 sin u 0 C B C B B C; Δt 1s Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 B C nΔt B 0 0 cos u C A B ad 0 ad 00ad C A ρB 31 33 36 C @ A d CD B C 0 0 sin u m B C B 000000C −20− sin u∕ tan i B C B C @ 000000A (A5) 000000 Appendix B: Navigation Accuracy r μ Figures B1 and B2 and Tables B1 and B2 show the accuracy of the 1 RTN ad − onboard estimated absolute position in the orbital frame and of 11 2aR a the orbital elements in the realistic simulations scenario. 1 ad21 n ex cos u aR 2 r r References μ μ “ a − a sin u [1] Wertz, J. R., and Gurevich, G., Applications of Autonomous On-Board d22 a d26 a Orbit Control,” 11th AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 2001-238, Feb. 2001. 1 a n e sin u aR [2] Bonaventure, F., Baudry, V., Sandre, T., and Gicquel, A.-H., d31 y “ 2 r Autonomous Orbit Control for Routine Station Keeping on a LEO μ Mission,” Proceedings of the 23rd International Symposium on Space a − Flight Dynamics, Pasadena, CA, Oct.–Nov. 2012. d33 a r [3] De Florio, S., and D’Amico, S., “The Precise Autonomous Orbit μ Keeping Experiment on the Prisma Mission,” Journal of the − Astronautical Sciences, Vol. 56, No. 4, 2008, pp. 477–494. ad36 cos u (A4) a doi:10.1007/BF03256562 438 DE FLORIO, D’AMICO, AND RADICE
[4] Garulli, A., Giannitrapani, A., Leomanni, M., and Scortecci, F., Mission,” Proceedings of the 35th Annual AAS Guidance and Control “Autonomous Low-Earth-Orbit Station-Keeping with Electric Propul- Conference, AAS Paper 2012-072, Feb. 2012. sion,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, [19] D’Amico, S., Bodin, P., Delpech, M., and Noteborn, R., “PRISMA,” Nov.–Dec. 2011, pp. 1683–1693. Distributed Space Missions for Earth System Monitoring, Vol.31, Space doi:10.2514/1.52985 Technology Library, Springer, New York, 2013, pp. 599–637. [5] Julien, E., and Lamy, A., “Out-of-Plane Autonomous Orbit Control,” [20] Friedland, B., Control System Design: An Introduction to State Space Journal of Aerospace Engineering, Sciences and Applications, Vol. 4, Methods, McGraw–Hill, New York, 1986, pp. 14–111, 190–221, 337– No. 3, 2012, pp. 28–40. 377, Chaps. 2, 3, 5, 9. doi:10.7446/jaesa.0403.03 [21] Kirk, D. E., Optimal Control Theory: An Introduction, Prentice–Hall, [6] Königsmann, H. J., Collins, J. T., Dawson, S., and Wertz, J. R., Englewood Cliffs, NJ, 1970, pp. 1–47, 184–310. “Autonomous Orbit Maintenance System,” Acta Astronautica, Vol. 39, [22] Kahle, R., Kazeminejad, B., Kirschner, M., Yoon, Y., Kiehlin, R., and Nos. 9–12, 1996, pp. 977–985. D’Amico, S., “First In-Orbit Experience of TerraSAR-X Flight doi:10.1016/S0094-5765(97)00084-2 Dynamics Operations,” Proceedings of the 20th International [7] Galski, R. L., and Orlando, V., “Autonomous Orbit Control Procedure, Symposium on Space Flight Dynamics, Annapolis, MD, Sept. 2007. Using a Simplified GPS Navigator and a New Longitude Phase Drift [23] Martin, M. A., Shurmer, I., and Marc, X., “Sentinel-1: Operational Prediction Method, Applied to the CBERS Satellite,” Journal of Approach to the Orbit Control Strategy,” Proceedings of the 23rd Aerospace Engineering, Sciences and Applications, Vol.4, No. 3, July– International Symposium on Space Flight Dynamics, Pasadena, CA, Sept. 2012, pp. 54–61. Oct.–Nov. 2012. doi:10.7446/jaesa.0403.05 [24] De Florio, S., and D’Amico, S., “Max-Path Unit Tests of the PRISMA [8] Orlando, V., and Kuga, H. K., “A Survey of Autonomous Orbit Control Flight Software v. 6.0.0,” DLR, German Aerospace Center, Rept. Investigations at INPE,” Proceedings of the 17th International PRISMA-DLR-TN-17, Cologne, Germany, 2007. Symposium on Space Flight Dynamics, Moscow, June 2003. [25] D’Amico, S., De Florio, S., and Yamamoto, T., “Offline and Hardware- [9] Plam, Y., Allen, R. V., Wertz, J. R., and Bauer, T., “Autonomous Orbit in-the-Loop Validation of the GPS-Based Real-Time Navigation Control Experience on TacSat-2 Using Microcosm’s Orbit Control Kit System for the PRISMA Formation Flying Mission,” Proceedings of the (OCK),” 31st Annual AAS Guidance and Control Conference, AAS 3rd International Symposium on Formation Flying, Missions and Paper 2008-008, Feb. 2008. Technology, ESA Paper SP-654, Paris, April 2008. [10] Lamy, A., Julien, E., and Flamenbaum, D., “Four Year Experience of [26] D’Amico, S., Ardaens, J.-S., and Larsson, R., “Spaceborne Operational Implementation of Autonomous Orbit Control: Lessons Autonomous Formation Flying Experiment on the PRISMA Mission,” Learned, Feedback and Perspectives,” Proceedings of the 21st Journal of Guidance, Control and Dynamics, Vol. 35, No. 3, 2012, International Symposium on Space Flight Dynamics, Toulouse, France, pp. 834–850. 2009. [27] Brouwer, D., “Solution of the Problem of Artificial Satellite Theory [11] De Florio, S., D’Amico, S., and Radice, G., “Flight Results of the Without Drag,” Astronomical Journal, Vol. 64, No. 1274, 1959, Precise Autonomous Orbit Keeping Experiment on the Prisma pp. 378–397. Mission,” Journal of Spacecraft and Rockets, Vol. 50, No. 3, 2013, doi:10.1086/107958 pp. 662–674. [28] Walter, H. G., “Conversion of Osculating Orbital Elements into Mean doi:10.2514/1.A32347 Elements,” Astronomical Journal, Vol. 72, No. 8, Oct. 1967, pp. 994– [12] Gurfil, P., and Kholshevnikov, K. V., “Manifolds and Metrics in the 997. Relative Spacecraft Motion Problem,” Journal of Guidance, Control, doi:10.1086/110374 and Dynamics, Vol. 29, No. 4, 2006, pp. 1004–1010. [29] Schaub, H., and Junkins, J. L., “Analytical Mechanics of Space doi:10.2514/1.15531 Systems,” AIAA Education Series, AIAA, Reston, VA, 2003, pp. 525, [13] Alfriend, K.T., Vadali, S. R., Gurfil, P., How, J. P., and Breger, L. S., 649–651, 693–698. Spacecraft Formation Flying: Dynamics, Control and Navigation, [30] D’Amico, S., and Montenbruck, O., “Proximity Operations of Elsevier Astrodynamics Series, Elsevier, Oxford, England, U.K., 2010, Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector Chap. 6. Separation,” Journal of Guidance, Control, and Dynamics, Vol. 29, [14] Schaub, H., “Relative Orbit Geometry Through Classical Orbit Element No. 3, May–June 2006, pp. 554–563. Differences,” Journal of Guidance, Control, and Dynamics, Vol. 27, [31] Clohessy, W. H., and Wiltshire, R. S., “Terminal Guidance System for No. 5, Sept.–Oct. 2004, pp. 839–848. Satellite Rendezvous,” Journal of Aerospace Sciences, Vol. 27, No. 9, doi:10.2514/1.12595 1960, pp. 653–658. [15] Micheau, P., “Orbit Control Techniques for LEO Satellites,” Spaceflight [32] Liu, J., “Satellite Motion About an Oblate Earth,” AIAA Journal, Vol.12, Dynamics, Vol.1, Cepadues-Editions, Toulouse, France, 1995, pp. 739– No. 11, 1974, pp. 1511–1516. 902. doi:10.2514/3.49537 [16] D’Amico, S., Arbinger, C., Kirschner, M., and Campagnola, S., [33] D’Amico, S., Kirschner, M., and Arbinger, C., “Precise Orbit Control of “Generation of an Optimum Target Trajectory for the TerraSAR-X LEO Repeat Observation Satellites with Ground-in-the-Loop,” DLR, Repeat Observation Satellite,” Proceedings of the 18th International German Aerospace Center, TN-04-05, Oberpfaffenhofen, Germany, Symposium on Space Flight Dynamics, Munich, Oct. 2004. 2004. Downloaded by STANFORD UNIVERSITY on August 2, 2015 | http://arc.aiaa.org DOI: 10.2514/1.61575 [17] Montenbruck, O., and Gill, E., Satellite Orbits: Model, Methods, [34] Ardaens, J.-S., D’Amico, S., and Montenbruck, O., “Final and Applications, Springer–Verlag, Heidelberg, Germany, 2000, Commissioning of the Prisma GPS Navigation System,” Journal of Chaps. 3, 5. Aerospace Engineering, Sciences and Applications, Vol.4, No. 3, July– [18] Bodin, P., Noteborn, R., Larsson, R., Karlsson, T., D’Amico, S., Sept. 2012, pp. 104–118. Ardaens, J.-S., Delpech, M., and Berges, J.-C., “The Prisma Formation doi:10.7446/jaesa.0403.10 Flying Demonstrator: Overview and Conclusions from the Nominal