OPTIMAL AUTONOMOUS ORBIT CONTROL OF REMOTE SENSING

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Jun 15, 2007 - a satellite in Low Earth Orbit (LEO) using a standard LQR controller. ... these feedback control laws operate on the orbit elements, Gauss' variational ... AOK control results, thus giving the possibility to assess advantages and ..... Gauss' variational equations of motion adapted for near-circular orbits (see Ref.
AAS 09-162 OPTIMAL AUTONOMOUS ORBIT CONTROL OF REMOTE SENSING SPACECRAFT Sergio De Florio∗* and Simone D’Amico* This paper analyses the problem of the autonomous control of a satellite in Low Earth Orbit (LEO) using an optimum controller. The type of controller considered is a standard Linear Quadratic Regulator (LQR). As the problem can also be treated as a two-satellites-formation control in which there is no cancellation of the common perturbative forces, its formulation is similar to that used in formation flying control problems. The driving orbit control requirement is to keep the satellite orbit within a maximum absolute distance of 250 m (r.m.s.) from a sun-synchronous, phased and frozen reference orbit. The control action is realized by means of in-plane and out of-plane thrusts whose cost is minimized by the optimal solution of the control problem. The PRISMA dual satellite mission flight software development and test environment is used as a first test-bed to validate the control algorithms. The PRISMA flight software and test environment allows a very realistic validation of the proposed control techniques. The TerraSAR-X mission scenario is use as a second test-bed for the validation of the control algorithm as it is a very representative example of LEO satellite for Earth observation with high demanding orbit control accuracy requirements.

INTRODUCTION The continuous enhancement of performances of space-borne remote sensing sensors poses an increasing accuracy demand on the orbit determination and control of satellites for Earth observation. Though there is a consolidated experience in ground-in-the loop orbit control with strict requirements (Ref. 1, 2, 3, 4 and 5) an autonomous on-board orbit control system could achieve even better results in a more efficient and economical way (Ref. 6 and 7). In fact the fulfillment of strict requirements on different orbit parameters can be achieved in real time and with a significant reduction of ground operations. So far autonomous orbit control has been realized (e.g. Ref. 6 and 7) and experimented (e.g. Ref. 7) using a guidance law that implements an analytical solution for an in-plane orbits control by along-track and anti-along-track velocity increments. This is not a mathematical optimal solution to the control problem but has the advantage of being of very simple and straightforward implementation. This paper analyses the problem of the autonomous control of a satellite in Low Earth Orbit (LEO) using a standard LQR controller. The problem is formulated as a two spacecraft formation flying control problem in which one of the spacecraft is virtual and not affected by nongravitational orbit perturbations. The parameterization used for the representation of the relative motion is the set of relative orbital elements which are obtained as a non-linear combination of the absolute orbital elements (Ref. 8, 9 and 10). The driving orbit control requirement is to keep the satellite orbit within a maximum absolute distance of 250 m (r.m.s.) (Ref. 1) from a sun-synchronous, phased and frozen reference orbit. As these feedback control laws operate on the orbit elements, Gauss’ variational equations of motion provide a convenient set of equations relating the effect of a control acceleration vector to the osculating orbit element time derivatives. The control action is realized by means of in-plane and out of-plane thrusts whose cost is minimized by the optimal solution of the control problem. The PRISMA (Ref. 11, 12 and 13) dual satellite mission flight software development and test environment is used as a first test-bed to validate the control algorithms. PRISMA is a micro-satellite mission created by the Swedish National Space Board (SNSB) and Swedish Space Corporation (SSC), which serves as a test platform for autonomous formation flying and rendezvous of spacecraft. PRISMA comprises a fully maneuverable micro-satellite (MANGO) as well as a smaller sub-satellite (TANGO) which will be launched together in a clamped configuration in autumn 2009 and separated in orbit after completion of all checkout operation. The design orbit of the satellites is ∗ Space

Flight Technology Department, German Aerospace Center (DLR), D-82230 Wessling, Germany.

sun-synchronous, near-polar at approximately 700 km altitude with local time of the ascending node at 6.00 or 18.00. One of the PRISMA secondary objectives is the demonstration of autonomous orbit keeping of the MANGO spacecraft which will be performed at the end of the mission, through the Autonomous Orbit Keeping (AOK) experiment (Ref. 6). AOK will demonstrate autonomous control of the orbit’s LAN using GPS based navigation. The PRISMA flight software and test environment allows a very realistic validation of the proposed control techniques. In addition the LRQ control performances can be directly compared with the AOK control results, thus giving the possibility to assess advantages and disadvantages of an optimal control with respect to an analytical closed form solution of the Gauss’ variational equations. TerraSAR-X mission scenario is used as a second test-bed for the validation of the control algorithm as it is a representative example of LEO satellite for Earth observation with high demanding orbit control accuracy requirements. TerraSARX is a German national SAR-satellite system based on a public-private partnership agreement between the German Aerospace center DLR and EADS Astrium GmbH and was launched on 15th June 2007. TerraSARX is the first satellite providing continuously global SAR data in X-Band at a varying geometrical resolution between 1 and 16 meters. TerraSAR-X will provide single or dual polarized data. The design orbit of the satellites is sun-synchronous, frozen, near-polar at approximately 514 km altitude.

DEVIATION OF THE REAL ORBIT FROM THE REFERENCE Reference Orbit Table 1 shows the MANGO and TerraSAR-X satellites nominal mean orbital elements. The mean orbital elements of a satellite in LEO deviate from their nominal values under the action of perturbing forces. Orbital elements values are dictated by specific mission requirements. The orbit of MANGO (Ref. 6) is only sunsynchronous while that of TerraSAR-X (Ref. 1) is also phased and frozen by design for the fulfillment of Synthetic Aperture Radar (SAR) payload requirements. Table 1 MANGO and TerraSAR-X satellites nominal orbital elements Mean orbital elements

MANGO

TerraSAR-X

Semi-major axis (a) Eccentricity vector x-component (ex ) Eccentricity vector y-component (ey ) Inclination (i) Right Ascension of Ascending Node (Ω) Argument of latitude (u)

7087297 [m] 0.00067 [-] 0.0013 [-] 98.1877 [deg] 189.8914 [deg] 0.0 [deg]

6892944 [m] 0.00057 [-] - 0.0012 [-] 97.4401 [deg] 104.2749 [deg] 68.0 [deg]

A reference orbit is a target orbit representing the desired mean nominal motion of the satellite over a long time interval. Basically, the reference orbit is a trade-off. First of all it shall be as realistic as possible because the spacecraft actual orbit is desired to be as close as possible to the target. On the other hand the reference orbit shall be as simple as possible, because the desired orbit has to be completely periodic relative to an Earth-fixed coordinate system. Thus the reference orbit model should at least consider the non-spherical terms of the Earth gravity potential which cause the sun synchronous secular motion of Ω. Other secular, long-periodic and short periodic terms can be included in the reference orbit depending on the flight control accuracy requirements. Deviations of real from reference orbital elements leading to a violation of flight control requirements have to be corrected by orbit maneuvers.

Orbit Parameterization The set of orbital elements chosen for the absolute state representation and shown in Eq. (1) are the semimajor axis, the components of the eccentricity vector, the orbital plane inclination and the mean argument of latitude sum of the argument of perigee and of the mean anomaly.

    κ=   

a ex ey i Ω u





      =      

a e cos ω e sin ω i Ω ω+M

       

(1)

The choice of this parameterization of the state is dictated by the fact that it does not lead to singular equations as the eccentricity value tends to zero. Nevertheless this set of orbital elements leads to singular equations as the inclination angle tends to zero but this case is out of interest when dealing with remote sensing satellites orbits. As this absolute orbit control problem can be formulated as a two spacecraft formation flying control problem in which one of the spacecraft is virtual and not affected by non-gravitational orbit perturbations, the most appropriate parameterization to represent the relative motion of the real satellite with respect to the reference is a set of relative orbital elements, shown in Eq. (2), which are obtained as a non-linear combination of the absolute orbital elements defined in Eq. (1) (Ref. 12 and 14).     ²=   

δa δex δey δix δiy δu





      =      

(a − ar )/a ex − ex r ey − ey r i − ir (Ω − Ωr ) sin i u − ur

       

(2)

where the subscript r refers to the reference orbit. It is noteworthy the fact that in an ideal two body problem, the orbital element u is the only not invariant element because it represents the time variable. The relative orbit representation defined in Eq. (2) is based on the relative eccentricity and inclination vectors defined in cartesian and polar notations as µ δe =

δex δey



µ = δe

cos φ sin φ



µ δi =

δix δiy



µ = δi

cos θ sin θ

¶ (3)

The phases of the relative e/i vectors are termed relative perigee φ and relative ascending node θ because they characterize the geometry of the relative orbit and determine the angular locations of the perigee and ascending node of the relative orbit (Ref. 12 and 14). The position of the satellite relative to the reference orbit in the RTN orbital frame (R pointing along the orbit radius, N pointing along the angular momentum vector and T = N × R pointing in the direction of motion for a circular orbit) can be described in means of orbital elements as                  

δrR /a δrT /a δrN /a δvR /v δvT /v δvN /v





                 =                  

δa − δe cos(u − φ) 3 δiy − uδa + 2δe sin(u − φ) δu + tan i 2 δi sin(u − θ) δe sin(u − φ) 3 − δu + 2δe sin(u − φ) 2 δi cos(u − θ)

                    

(4)

with

p φ = arctan(δey /δex ) where δe = (δex )2 + (δey )2 p θ = arctan(δiy /δix ) where δi = (δix )2 + (δiy )2 . Eq. (4) represents the first order solution of the Clohessy-Wiltshire equations. Uncontrolled motion Figures 1 and 2 show respectively the evolution of MANGO’s real orbit absolute and relative elements with respect to their respective reference orbits.

Figure 1 Real orbit elements evolution of MANGO satellite in free motion

The analysis was performed by numerical orbit propagation over 30 days and using orbit propagation model and satellite physical parameters given in Table 2. The noisy pattern of the plots in Fig. 2 is due to the short periods gravitational perturbations not modeled by the 20x20 gravitational harmonics reference orbit. Fig. 3 shows the δe and δi vectors evolution of MANGO and TerraSAR-X in free motion. At the altitudes of MANGO (about 700 km) and TerraSAR-X satellites (about 514 km) the evolution of δe and δi, i.e. the evolution of the relative motion of the controlled satellite with respect to the ideal reference satellite is mainly driven by J2 . As the reference satellite’s trajectory is not affected by non gravitational perturbation the atmospheric drag plays also a fundamental role in the evolution of δe as well as the moon and sun third body gravitational perturbation in varying δi. A clear understanding of the evolution of δe and δi in uncontrolled motion is fundamental in deciding the orbit control strategy. At each instant of time the place along the orbit of a correction maneuver of δe and δi can be determined with Eq. (5) that is derived from an analytical solution of the Gauss equations (Ref. 14). µ uem = arctan

∆δey ∆δex



µ uim = arctan

∆δiy ∆δix

¶ (5)

where uem and uim are respectively the orbital places for a maneuver finalized to bring vectors δe and δi to the origin and ∆δe , ∆δi are the corresponding correction maneuvers. From Eq. (5) follows also that for

polar orbits (i = 90◦ ) at the orbits nodes ( u = 0◦ and u = 180◦ ) only ∆δex or ∆δix corrections are possible while at the highest latitudes ( u = 90◦ and u = 270◦ ) only ∆δey or ∆δiy corrections are possible if only along-track and cross-track maneuvers are used. These considerations are relevant for near-polar orbits.

Figure 2 Relative orbital elements evolution of MANGO satellite in free motion with respect to its reference orbit.

Figure 3 δe and δi vectors evolution of MANGO and TerraSAR-X satellites in free motion with respect to their respective reference orbits.

Table 2 Propagation parameters Physical Properties

MANGO

TerraSAR-X

Mass Drag area Drag coefficient SRP effective area SRP coefficient

154.4 [kg] 1.3 [m2 ] 2.5 [-] 2.5 [m2 ] 1.3 [-]

1341.17 [kg] 3.2[m2 ] 2.3[-] 10 [m2 ] 1.3 [-]

Propagation

Real orbit

Reference orbit

Earth gravity field Atmospheric density Sun and Moon ephemerides Solar Radiation Pressure

GRACE GGM01S 90x90 Harris-Priester Analytical formulas (Ref. 15) Canon-Ball model with conical Earth’s shadow IERS First order effects Shampine and Gordon DE

GRACE GGM01S 20x20 -

Solid Earth, polar and ocean tides Relativity effects Numerical integration method

RK4R: Runge-Kutta 4th-order with Richardson extrapolation

OPTIMAL ORBIT CONTROL LQR Controller The type of controller considered is a standard linear quadratic regulator. The process is modeled by a linear dynamic model in its state-space representation: x˙ = Ax + Bu

(6)

where x is the orbit error vector defined in Eq. (2), u is the control input vector, typically a velocity increment ∆v, A is a 6 × 6 matrix and B is a 6 × 3 matrix. A linear control law u = −Kx(t)

(7)

is sought where K is a suitable gain matrix such that a specified cost function Λ is minimized. The cost function Λ of Eq. (8) is expressed as the integral of a quadratic form in the state plus a second quadratic form in the control. Z Λ=

T

£ T ¤ x (τ )Q(τ )x(τ ) + uT (τ )R(τ )u(τ ) dτ

(8)

t

where Q is the 6 × 6 state weighting symmetric matrix and R is the 3 × 3 control weighting symmetric matrix (Ref. 16). Taking the control window as infinite ( T = ∞ ) in Eq. (8), the optimum gain matrix is found to be K = −R−1 BT M where M is the solution of the algebraic Riccati Eq. (10)

(9)

0 = MA + AT M − MBR−1 BT M + Q

(10)

Linear Dynamic Model Gauss’ variational equations of motion adapted for near-circular orbits (see Ref. 17) provide a convenient set of equations relating the effect of a control acceleration vector u to the osculating orbital element time derivatives. After some manipulation, the Gauss equations can be used to represent the control problem in the form of Eq. (6) using mean elements. The Gauss’ variational equations for near-circular orbit are presented in Eq. (11).  da  dt        dex      dt             dey       dt     κ˙ o =   = Γ(κo ) a =    di          dt          dΩ      dt        du dt





2 n

0

0 0 0 0 0 n

    sin u     an        cos u     −   an +       0           0     2 − n 

2 cos u an 2 sin u an 0 0 0

 0

     0     aR     0   aT   ·   cos u     aN an    sin u   an sin i    sin u  an tan i

       

(11)

where aR , aT and aN are the perturbing accelerations respectively along the R, T and N axes. The real orbit can be modeled using Gauss’ equations as shown in Eq. (12) ˜ o ) + Γ(κo )ac κ˙ o = Γ(κo )(ag + ap ) + Γ(κo )ac = A(κ

(12)

where κo is the osculating orbital elements vector, ag , ap and ac are in order the Earth gravity field, the perturbations and control accelerations vectors. Mean orbital elements are the most appropriate in the representation of the secular deviation of the real from the reference orbit. Using Brouwer’s analytical transformation (Ref. 18) κ = ξ(κo )

(13)

Eq. (12) can be written in function of mean elements µ ˜ κ˙ = A(κ) +

∂ξ ∂κo

¶ Γ(κo )ac

(14)

As matrix ∂ξ/∂κo is approximately a 6 × 6 identity matrix with the off-diagonal terms being of order J2 or smaller (see Ref. 9), it is reasonable in the problem here studied to approximate Eq. (14) as ˜ κ˙ ' A(κ) + Γ(κ)ac

(15)

In the same way the reference orbit can be modeled by Eq (16) this time without the control acceleration and non-gravitational perturbations terms. ˜ r) κ˙r = Γ(κr )ag = A(κ

(16)

h i ˜ ˜ r ) + Γ(κ)ac (κ˙ − κ˙ r ) = A(κ) − A(κ

(17)

Subtracting Eq. (16) from Eq. (12)

˜ and linearizing A(κ) around κr ¯ ¯ ˜ ∂ A(κ) ¯ ˜ ˜ r) + A(κ) ' A(κ ¯ ∂κ ¯

(κ − κr )

(18)

κr

Eq. (19), is thus obtained from Eq. (17) ¯ ¯ ˜ ∂ A(κ) ¯ (κ˙ − κ˙ r ) = ¯ ∂κ ¯

(κ − κr ) + Γ(κ)ac

(19)

κr

Eq. (19) is in the form of Eq. (6) and in the parameterization defined in Eq. (1). With the appropriate variable transformation Eq. (19) can be written in the parameterization defined in Eq. (2) thus obtaining Eq. (20). x˙ = A(κr )x + B(κ)δvc

(20)

where B is obtained from Eq. (4) and δvc is the velocity increment vector. State-space Matrices A and B ˜ of Eq. (19) represents the orbit model of the satellite in free motion (cfr. Eq. (12)). A simple Vector A analytical orbit model expressed by Equations (21) and (22) has been used for the realization of the LQR controller. This model consists of the averaged equations of motion of a satellite perturbed by the gravity field zonal term J2 and the atmospheric drag perturbation (neglecting the atmospheric rotational speed). ˜ g (κ) and A ˜ d (κ) represent the different For details related to the model see Ref. 19 and 20. Vectors A contributions respectively of the gravity field and the atmospheric drag.           ˜ Ag (κ) = α(a, e)          



0

                  ¤ 

£ ¤ − 5(cos i)2 − 1 ey £ ¤ 5(cos i)2 − 1 ex 0 −2 cos i √ £ n + 5(cos i)2 − 1 + 1 − e2 3(cos i)2 − 1 α(a, e)

(21)



v3 an2

   µ ¶   cos f  ex 1 + v  e    µ ¶  ˜ d (κ) = −β  ey 1 + cos f v A  e     0     0   0

                       

(22)

with α(a, e) = (3/4)nJ2 (RE /p)2 where RE = 6378140 m is the equatorial radius of the terrestrial spheroid and p = a(1 − e2 ) is the parameter of the orbit β = (A/m)CD ρ where A is the satellite drag area, m the satellite mass, CD the drag coefficient and ρ the atmospheric density v is the magnitude of the satellite velocity written in terms of orbital elements as v = an (orbit assumed circular) f = u − ω is the true anomaly where sin f = (ex sin u − ey cos u)/e and cos f = (ex cos u + ey sin u)/e ˜ g and A ˜ d and transMatrix A of Eq. (19) is thus obtained computing the Jacobian matrices of vectors A forming in relative orbital elements as in Eq. (23). A(κ) = A(κ)g + A(κ)d = Tg

˜ ˜ ∂ A(κ) ∂ A(κ) d g + Td ∂κ ∂κ

(23)

where Tg and Td are the transformation matrices. The terms of matrices Ag and Ad of Eq. (23) and matrix B of Eq. (20) are given in the Appendix. Weighting Matrices Q and R In the performance function defined in Equations (8) the quadratic form xT Qx represents a penalty on the deviation of the real from the reference orbit and the weighting matrix Q specifies the importance of the various components of the state vector relative to each other. The term uT Ru is instead included in an attempt to limit the magnitude of the control signal u and to prevent saturation of the actuator. Overall the gains matrices choice is a trade-off between control action cost (i.e., small gains to limit propellant consumption and avoid thruster saturation phenomena) and control accuracy (i.e. large gains to limit the excursion of the state from its reference value). To choose the weighting matrices the maximum size technique has been adopted. The aim of this method is to confine the individual states and control actions within prescribed maximum limits given respectively by ²imax and uimax where the subscript max indicates the maximum value of the associated quantity. The terms of Q and R will be thus chosen with the rule imposed by Equations (24) and (25).

Qii =

ki kij , kij ∈ R for i = 1, 2, . . . , 6 and j = 1, 2, . . . , 6 , Qij = ²2imax 2²imax ²jmax

(24)

Rii =

hi hij , Rij = , hij ∈ R for i = 1, 2, 3 and j = 1, 2, 3 u2imax 2uimax ujmax

(25)

The choice of diagonal Q and R matrices is usually a good starting point of a trial-and-error design procedure aimed at obtaining desired properties of the controller. Control Strategy An optimal control strategy is dictated in the case here considered by the following main drivers: 1. Minimizing the total number of orbital maneuvers required for the fulfillment of the control accuracy requirements. 2. Maneuvering in the appropriate orbit places in order to maximize the efficiency of the maneuver performed. 3. Giving to the LQR controller the right information through matrices Q and R in order to obtain an actual optimal control. From these considerations stem the following guidelines for the orbital control: 1. The LQR of Equations (6) and (7) will be implemented in the discrete domain with the control action u constituted by a velocity increment dv . 2. The value of vector ² components will be computed at the orbits nodes only. 3. The LQR will operate a control action every n (with n ≥ 1) number of orbits at u = 0◦ + k 180◦ and/or u = 45◦ + k 180◦ and/or u = 90◦ + k 180◦ (with k ≥ 0). 4. As ix has a periodic variation and iy has a secular variation Q5i will be imposed much larger than Q4i . 5. The terms B61 and B63 of control matrix B will be imposed to be null if the controller is activated in u 6= 90◦ in order to avoid the controller controlling δu with dvs in radial and cross track directions. 6. Parameter dVRmax is imposed very small in order to keep maneuvers in the radial direction negligible with respect to maneuvers in the two other orbital frame directions. 7. The only off-diagonal terms introduced in matrices Q and R are Q16 = Q61 = −1/(2δamax δumax ) which gives to the controller the information that δa and δu are coupled with opposite sign through the Kepler equations and thus making it controlling δu by means of along-track and anti-along-track dvs. It has to be stressed that the correct calibration of matrices Q and R is crucial for the realization of an effective optimal controller.

NUMERICAL SIMULATIONS The numerical simulations whose results are shown in this section have in first place the purpose of showing the absolute control performances of the LQR controller in two mission scenarios and its potentialities as an alternative to traditional analytical-based control methods. The orbit propagation model and satellite physical parameters used for the simulations are given in Table 2. Table 3 collects the navigation accuracy and the parameters used for the accuracy modeling of sensors and actuators. The propulsion system is characterized by a Minimum Impulse Value (MIV) and a Minimum Impulse Bit (MIB). Consequently, the thrusters can only realize ∆v’s which are larger than MIV and integer multiples of MIB. Furthermore, the performance error of the thruster system is quantified by Equation (26)

¯ ¯ ¯ ∆Vr − ∆Vp ¯ ¯ ¯ · 100 η=¯ ¯ ∆Vp

(26)

where ∆vp is the planned velocity increment computed by the controller and ∆vr is the actual velocity increment realized by the propulsion system. Finally the attitude control error, which causes thrusters misalignment, is treated as Gaussian noise with zero bias and a 0.2 [◦ ] standard deviation in three axes. In all the simulation presented the control strategy criteria explained in the preceding section has been adopted in the choice of terms Qij and Rij . Table 3 Navigation, sensors and actuators accuracy modeling Absolute Navigation Accuracy

MANGO

TerraSAR-X

Mean σ

0 [m] 2 [m]

0 [m] 2 [m]

Attitude Control Accuracy

MANGO

TerraSAR-X

Mean σ

0 [◦ ] 0.2 [◦ ]

0 [◦ ] 0.2 [◦ ]

Propulsion System Accuracy

MANGO

TerraSAR-X

MIV MIB η

−4

8 · 10−5 [m/s] 4 · 10−5 [m/s] 5

7 · 10 [m/s] 7 · 10−5 [m/s] 5

MANGO Autonomous Orbits Control Fig. 4 is representative of some first results of the application of the LQR in the autonomous control of MANGO’s orbit.

Figure 4 LQR control - Deviation of real from reference orbit of MANGO satellite in RTN frame and LAN deviation with respect to the reference.

Fig. 5 in contrast shows the results for a classical autonomous LAN control method (AOK experiment, Ref. 6) in the same scenario. In comparing these two cases it has to be clear that the LQR controls all the relative orbital elements, while the AOK controller is required to control only the LAN, an Earth fixed reference parameter, with an accuracy of 20 m (r.m.s) and by means of along-track and anti-along-track velocity increments. Tables 4 and 5 collects controller parameters and control performances in the cases depicted in the figures. The LQR controller is activated one time per orbit in the orbit places indicated in Table 4.

Figure 5 AOK control - Relative orbital elements of MANGO satellite and LAN deviation with respect to the reference.

Table 4 MANGO controller parameters Case 1 ◦

Case 2 u = 80

AOK ◦

Maneuver places

u=0

δamax δexmax δeymax δixmax δiymax δumax

10/aM AN 1000/aM AN 50/aM AN 200/aM AN 20/aM AN 20/aM AN

As case 1

-

dVRmax dVT max dVN max

1 dVmin 1 dVmin 3 dVmin

As case 1

-

In this case the constraints on δex , δey and δix have been relaxed and the control of δa and δu has been coupled. At the altitude of MANGO a number of expensive out-of-plane maneuvers are required for the attainment of a fine control of δrN . Without a substantial relaxation of the accuracy control requirements of δrN , the LQR controller proves in this case to be very expensive compared to the AOK controller. It is straightforward to note that the control of δiy is more efficient at higher latitudes (compare cases 1 and 2 of Tables 4 and 5). It can be concluded that if the control requirements concern only an Earth fixed

reference parameter as the LAN, a traditional control method using along-track and anti-along-track velocity increments gives better performances and lower dv budgets than an LQR used in the configuration here considered. Nevertheless the LQR controller is more robust than a traditional autonomous orbit control in the sense that it is less sensitive to the dynamic model used. The LQR controller can be used as a competitive alternative control method for 3 orbital axes if used in a different configuration (matrices Q, R and B), but this point requires further investigations. Table 5 MANGO control performances Case 1

Case 2

AOK

R mean [m] T mean [m] N mean [m] LAN mean [m]

9 113 -15 17

-46 -66 -0.85 11

22.6 -7213 -487 -2.628

R std. dev. [m] T std. dev. [m] N std. dev. [m] LAN std. dev. [m]

25 54 37 40

60 98 20 36

12.5 4678 317 18

dVR budget[m/s] dVT budget [m/s] dVN budget [m/s]

0 0.19 1.84

0 0.18 1.44

0.1397 -

dVT OT budget [m/s]

2.03

1.62

0.1397

TerraSAR-X Autonomous Orbits Control Fig. 6 shows the evolution of the relative orbital elements of TerraSAR-X with respect to the reference orbit during two month of simulated autonomous orbital control by the LQR controller activated once per day.

Figure 6 LQR control - Evolution of relative orbital elements with respect to the reference orbit for TerraSAR-X during two months simulation.

Fig. 7 shows the relative motion of TerraSAR-X real orbit with respect to the reference in the RT N frame and the Earth fixed parameter LAN.

Figure 7 LQR control - Relative motion of TerraSAR-X real orbit with respect to the reference (0,0) during two months simulation.

Tables 6 and 7 collect controller parameters and control performances in the cases depicted in the figures and for the actual ground base orbit control. The main orbit control (Ref. 1 and 4) requirement for TerraSARX is to maintain the satellite orbit in a tube defined around the reference orbit with a radius of 250 m. The LQR controller is activated one or two times each 16 orbits (once a day) in the orbit places indicated in Table 6. Relative orbital parameters δa and δu have been coupled in order to control δu by means of along-track and anti-along-track maneuvers. Table 6 TerraSAR-X controller parameters Case 1

Case 2

Case 3

Ground based

Maneuver places

u = 0◦ , 90◦

u = 80◦

u = 45◦

Optimal place

δamax δexmax δeymax δixmax δiymax δumax

5/aM AN 200/aM AN 200/aM AN 800/aM AN 20/aM AN 3000/aM AN

As case 1

As case 1

-

dVRmax dVT max dVN max

1 dVmin 30 dVmin 5 dVmin

1 dVmin 55 dVmin 5 dVmin

1 dVmin 55 dVmin 5 dVmin

-

The dv budget of the ground based control, given in Table 7, is based on values averaged on the period from 20 June 2007 to 27 January 2009 and excluding the orbital maneuvers actuated during periods in which

the satellite was in safe-mode. The best control performance in terms of accuracy and cost is achieved by activating the controller two times in one orbit, once per day, at the ascending node and at the highest northern latitude (case 1 of Table 6). From the analysis of these first simulation results it can be concluded that the LQR controller allows an accurate in-plane and out-of-plane control but is more expensive than the optimized ground based control. Table 7 TerraSAR-X control performances Case 1

Case 2

Case 3

Ground based

R mean [m] T mean [m] N mean [m]

-126 1230 -111

-219 2087 45

-208 2270 -32

R std. dev. [m] T std. dev. [m] N std. dev. [m]

86 573 99

188 921 115

114 1022 157

dVR budget [m/s] dVT budget [m/s] dVN budget [m/s]

0 0.52 0.15

0 0.49 0.14

0 0.49 0.16

0.10 (mean value) 0.30 (mean value)

dVT OT budget [m/s]

0.67

0.63

0.65

0.40 (mean value)

CONCLUSION The problem of the autonomous control of a satellite in low Earth orbit using an LQR controller has been analyzed. The problem has been formulated as a two spacecraft formation flying control problem in which one of the spacecraft is virtual and not affected by non-gravitational orbit perturbations. The parameterization used for the representation of the relative motion is the set of relative orbital elements which are obtained as a non-linear combination of the absolute orbital elements. A linearized analytical orbit model consisting of the averaged equations of motion of a satellite perturbed by the gravity field zonal term J2 and the atmospheric drag perturbation has been used to represent the two spacecraft formation dynamic system controlled by inplane and out-of-plane velocity increments. The driving orbit control requirement was to keep the satellite orbit within a maximum absolute distance of 250 m (r.m.s.) from a sun-synchronous, phased and frozen reference orbit. The PRISMA mission flight software development and test environment is used as a first testbed to validate the control algorithms as it allows a very realistic validation of the proposed control techniques and a direct comparison with the foreseen results of a planned autonomous orbit keeping experiment using traditional sub-optimal control techniques. The TerraSAR-X mission scenario has been used as a second testbed for the validation of the control algorithm as it is a very representative example of LEO satellite for Earth observation with high demanding orbit control accuracy requirements. From the first simulations results it can be assessed that the LQR controller implemented and calibrated in the way described in this paper is very competitive in terms of control accuracy with respect to a ground-based optimized orbital control but more expensive in terms of dv budgets and number of maneuvers performed. The frequency of the maneuver performed and their overall cost is proportional to the required control accuracy. The minimization of the dv budgets maintaining similar control accuracy has to be the object of further investigations.

REFERENCES [1] S. D’Amico, M. K. M, and C. Arbinger, “Precise Orbit Control of LEO Repeat Observation Satellites with Ground-In-The-Loop Case of study: TerraSAR-X,” No. 04-05, 2004. [2] S. D’Amico, C. Arbinger, M. Kirschner, and S. Campagnola, “Generation of an Optimum Target Trajectory for the TerraSAR-X Repeat Observation Satellite,” 18th International Symposium on Space Flight Dynamics, Munich, Germany, 11-15 October 2004. [3] C. Arbinger, S. D’Amico, and M. Eineder, “Precise Ground-In-the-Loop Orbit Control for Low Earth Observation Satellites,” 18th International Symposium on Space Flight Dynamics, Munich, Germany, 11-15 October 2004. [4] R. Kahle, B. Kazeminejad, M. Kirschner, Y. Yoon, R. Kiehling, and S. D’Amico, “First In-Orbit Experience of TerraSAR-X Flight Dynamics Operations,” ISSFD, Annapolis, USA, 2007. [5] O. Montenbruck, R. Kahle, S. D’Amico, and J.-S. Ardaens, “Navigation and Control of the TanDEM-X Formation,” Journal of the Astronautical Sciences, Tapley Symposium Special Issue, Vol. Version 2.0, 19 June 2008 2008. [6] S. D. Florio, S. D’Amico, and M. G. Fernandez, “The Precise Autonomous Orbit Keeping Experiment on the Prisma Formation Flying Mission,” Galveston, USA, 18th AAS/AIAA Space Flight Mechanics Meeting, 27-31 January 2008. [7] A. Lamy, M. C. Charmeau, D. Laurichesse, M. Grondin, and R. Bertrand, “Experiment of Autonomous Orbit Control on the Demeter Satellite,” 18th International Symposium on Space Flight Dynamics, Munich, Germany, 11-15 October 2004. [8] L. Berger and J. P. How, “Gauss’s Variational Equation-Based Dynamics and Control for Formation Flying Spacecraft,” Journal of Guidance, Control and Dynamics, Vol. 30, March-April 2007. [9] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems. Reston, VA, USA: AIAA Education Series, 2003. [10] J. S. Ardaens and S. D’Amico, “Control of Formation Flying Spacecraft at a Lagrange Point,” No. 00-08, 2008. [11] S. Persson, P. Bodin, E. Gill, J. Har, and J. J¨orgensen, “Prisma An Autonomous Formation Flying Mission,” Sardinia, Italy, ESA Small Satellite Systems and Services Symposium (4S), 25-29 September 2006. [12] S. D’Amico, E. Gill, and O. Montenbruck, “Relative Orbit Control Design for the Prisma Formation Flying Mission,” AIAA Guidance, Navigation, and Control Conference, Keystone, Colorado, 21-24 August 2006. [13] E. Gill, S. D’Amico, and O. Montenbruck, “Autonomous Formation Flying for the PRISMA Mission,” Journal of Spacecraft and Rockets, Vol. 44, May 2007, pp. 671–681. [14] S. D’Amico and O. Montenbruck, “Proximity Operations of Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector Separation,” Journal of Guidance, Control and Dynamics, Vol. 29, May-June 2006, pp. 554–563. [15] O. Montenbruck and E. Gill, Satellite Orbits - Model, Methods and Applications. Heidelberg, Germany: Springer Verlag, 2000. [16] B. Friedland, Control System Design - An Introduction to State-space Methods. McGraw Hill, 1986. [17] P. Micheau, Spaceflight Dynamics, Vol. I. Toulouse, France: Cepadues-Editions, 1995. [18] D. Brouwer, “Solution of the Problem of Artificial Satellite Theory Without Drag,” The Astronomical Journal, Vol. 64, No. 1274, 1959, pp. 378–397. [19] J. Liu, “Satellite Motion about an Oblate Earth,” AIAA Journal, Vol. 12, November 1974, pp. 1511–1516. [20] C. Chao, Applied Orbit Perturbation and Maintenance. Reston, VA, USA: AIAA, 2005.

APPENDIX: LINEARIZED ORBIT MODEL Gravity Field          3 RE 2  Ag = nJ2 ·  4 p        

0

0

0

0

0

A(2, 1)g

A(2, 2)g

A(2, 3)g

A(2, 4)g

0

A(3, 1)g

A(3, 2)g

A(3, 3)g

A(3, 4)g

0

0

0

0

A(5, 1)g

A(5, 2)g

A(5, 3)g

A(5, 4)g

A(6, 1)g

A(6, 2)g

A(6, 3)g

A(6, 4)g

¤ 7 £ 5(cos i)2 − 1 ey 2a ¤ 4ex £ A(2, 2)g = − 5(cos i)2 − 1 ey (1 − e2 ) " # £ ¤ 4e2y 2 A(2, 3)g = − 5(cos i) − 1 +1 (1 − e2 ) A(2, 4)g = 10ey sin i cos i ¤ 7 £ 5(cos i)2 − 1 ex 2a · ¸ £ ¤ 4e2x A(3, 2)g = 5(cos i)2 − 1 + 1 (1 − e2 ) A(3, 1)g = −

¤ 4ey £ 5(cos i)2 − 1 ex (1 − e2 )

A(3, 4)g = −10ex sin i cos i 7 sin i cos i a 8ex A(5, 2)g = − sin i cos i (1 − e2 ) A(5, 1)g =

A(5, 3)g = −

8ey sin i cos i (1 − e2 )

A(5, 4)g = 2(sin i)2 √ £ ¤ª 3n 7 © 5(cos i)2 − 1 + 1 − e2 3(cos i)2 − 1 − 2a 2a √ £ ¤ª © £ ¤ ex A(6, 2)g = 4 5(cos i)2 − 1 + 3 1 − e2 3(cos i)2 − 1 (1 − e2 ) √ ¤ £ ¤ª 4ey © £ A(6, 3)g = 4 5(cos i)2 − 1 + 3 1 − e2 3(cos i)2 − 1 2 (1 − e ) √ ¡ ¢ A(6, 4)g = −2 sin i cos i 5 + 3 1 − e2 A(6, 1)g = −



  0     0 0     0 0     0 0     0 0 

A(2, 1)g =

A(3, 3)g =

0

Atmospheric Drag



A(1, 2)d

   A(2, 1)d     A(3, 1)d   A Ad = C D ·   0 m     0     0

3 A(1, 2)d = − 2a

A(2, 1)d =

r

0

0

0 0

A(2, 2)d

A(2, 3)d

0

0

A(3, 2)d

A(3, 3)d

0

0

0

0

0

0

0

0

0

0

0

0

0

0

µ a

µ ¶ 1 cos f n 1+ ex 2 e

r · µ ¶ ¸ µ e2x cos f ex A(2, 2)d = − 1+ 1−2 2 + 2 cos u a e e e r · ¸ µ sin u 2ey A(2, 3)d = − − 3 cos f ex a e2 e r A(2, 6)d =

µ ex sin f a e

µ ¶ 1 cos f A(3, 1)d = n 1 + ey 2 e r · ¸ µ cos u 2ex A(3, 2)d = − − 3 cos f ey a e2 e à ! # r " e2y cos f µ ey A(3, 3)d = − 1+ 1−2 2 + 2 sin u a e e e r A(3, 6)d =

µ ey sin f a e

0



  A(2, 6)d     A(3, 6)d      0     0     0

Control Matrix  0     sin u    an    cos u  −  an  B=   0       0     2 − n



2 an

0

2 cos u an

0

2 sin u an

0

0

cos u an

0

sin u an

0

sin u an tan i

                         