A Closed-Loop Hardware Simulation of Decentralized Satellite

AAS 03-004

A CLOSED-LOOP HARDWARE SIMULATION OF DECENTRALIZED SATELLITE FORMATION CONTROL Takuji Ebinuma∗ and E. Glenn Lightsey† A closed-loop simulation system has been developed to demonstrate autonomous satellite formation control using the Global Positioning System (GPS). The developed system is capable of realistic simulations of GPSbased guidance, navigation, and control in real-time and enables research into orbit control strategies for autonomous formation flying of multiple spacecraft. A sample formation flying scenario employing a decentralized controller architecture has been investigated in this paper. The technical issues surrounding the decentralized architecture implementation, such as data flow and timing requirements, are also investigated. It is expected that the decentralized architecture promotes agility, adaptability, scalability, and affordability of future multiple spacecraft platforms.

INTRODUCTION In recent years, there has been significant interest in the use of formation flying spacecraft for a variety of earth and space science missions [1]. Formation flying may provide smaller and cheaper satellites that, working together, have more capability than larger and more expensive satellites. Several decentralized architectures have been proposed for autonomous establishment and maintenance of satellite formations. In such architectures, each satellite cooperatively maintains the shape of the formation without a central supervisor, and processes only local measurement information. The Global Positioning System (GPS) sensors are ideally suited to provide such local position and velocity measurements to the individual satellite. An investigation of the feasibility of a decentralized approach to satellite formations was originally presented by Carpenter [2]. He extended a decentralized linear-quadratic-Gaussian (LQG) framework proposed by Speyer [3] in a fashion similar to a linearized Kalman filter (LKF) which processed GPS position fix solutions. The new decentralized LQG architecture was demonstrated in a numerical simulation for an ideal scenario that is similar to the TechSat-21 mission [4] that has been proposed by NASA and the U.S. Air Force. Another decentralized architecture was proposed by Park et al. [5] using carrier differential-phase GPS (CDGPS). Busse et al. [6] recently demonstrated the decentralized CDGPS architecture in a hardware-in-the-loop (HWIL) simulation on the Formation Flying Test-Bed (FFTB) at NASA’s Goddard Space Flight Center (GSFC), which features four 16-channel GPS signal generators. Although the early experiments represented a step forward by utilizing GPS signal simulators for a spacecraft formation flying simulation, only open-loop performance was considered. Open-loop ∗ Research Engineer, Center for Space Research, 3925 West Braker Lane, Suite 200, Austin, TX 78759. email: [email protected]. † Assistant Professor, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712. email: [email protected]

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tests can provide some information about the basic performance of a system, but they are limited to scripted trajectories that do not allow for control of the vehicles based on the sensor measurements. Satellite formation operations require precise relative navigation, but this alone is insufficient to allow autonomous on-orbit operations. The guidance algorithm relies on the navigation state and must operate in concert with the navigation system in a real-time mode. In this research, hardware experimentation has been extended to include closed-loop integrated guidance and navigation of spacecraft formations using GPS receivers and an inter-spacecraft communication system. A hardware closed-loop simulation employing the decentralized LQG architecture proposed by Carpenter is presented in this paper. The technical issues surrounding the integration of guidance and navigation systems in a real-time HWIL test are also investigated to ensure that no practical issues (such as timing mismatch) are overlooked.

DECENTRALIZED CONTROL A brief introduction to decentralized control is presented in this section. More complete mathematical definitions are presented by Carpenter [2]. Subsequently, the subscript i denotes time epoch, and the superscript j denotes local information of each satellite in the formation. Consider the solution to the discrete decentralized LQG control problem obtained by minimizing the cost function    N K X  X xTi W i xi + J =E (uji )T V ji uji  (1)   i=1

j=1

over control vectors uji , subject to the measurement y ji given by y ji = H j xi + v ji , and the state xi given by xi+1 = Φi+1 xi +

K X

(2)

Λj uji + wi ,

(3)

j=1

where W i and V ji are weight matrices for the state and the control, respectively, Φi is the state transition matrix, H j and Λj are the measurement and control matrices of spacecraft j, respectively, K is the number of spacecraft in the formation, N is the number of epochs, and v ji and wi are white noise vectors, such that n o n o n o E v ji = 0 , E v ji (v jk )T = Rji δik , E v ji (v li )T = 0 , (4) © ª E {wi } = 0 , E wi (wk )T = Qi δik . Speyer [3] decomposed the state space into a control-dependent partition partition xD i as follows: D xi = xC i + xi .

2

(5) xC i

and a data-dependent (6)

A local LKF algorithm operating only on the local mesurements of spacecraft j can be described as xC i

=

Φi xC i−1 +

K X

Λj uji−1

(7)

j=1

¯ Dj x i

=

ˆ Dj Φi x i−1

(8)

=

ˆ j ΦT Φi P i−1 i

(9)

K ji

=

¯ j (H j )T P i

(10)

ˆ Dj x i

=

¯ Dj x i

(11)

ˆj P i

=

¯ j (I − K j H j )T + K j Rj (K j )T , (I − K ji H j )P i i i i i

¯j P i

K ji

+ Qi h i−1 ¯ j (H j )T + Rj HjP i i h i Dj ¯ + K ji y ji − H j (xC + x ) i i

(12)

P ji

where and are the local Kalman gain and local state error covariance matrices, respectively. Even though this decentralized approach does not require a global Kalman filter, it does require the global state error covariance matrix P i to be maintained locally via ¯ i = Φi P ˆ i−1 ΦT + Q P (13) i i  −1 K X ˆ i = P ¯ −1 + P sji (H j )T (Rji )−1 H j  . (14) i j=1

A significant practical issue with the local reconstruction of the globally optimal covariance matrix via Eq. (14) is that the measurement updates may not occur at all local filters uniformly. This issue requires an additional transmission of a semaphore sji ∈ {0, 1} from each satellite j to all the other satellites every time a local measurement update is successfully executed. ˆD ˆ Dj The global estimate x and i is then obtained by a linear combination of the local estimates x i j an additional data vector hi , which contains non-local information, via ˆD x i =

K ³ X

j ˆ i (P ˆ j )−1 x ˆ Dj P i i + hi

´ ,

(15)

j=1

where (16)

=

¯ Dj ; hj0 = 0 F i hji−1 + Gji x i ˆ i (P ¯ i )−1 Φi P

=

ˆ i (P ¯ j )−1 . ˆ i−1 (P ¯ j )−1 Φ−1 − P F iP i i−1 i

(18)

hji

=

Fi Gji

(17)

The main advantage of this decentralized algorithm is that each spacecraft need not reconstruct ˆD the globally optimal state estimate x i via Eq. (15) to compute the globally optimized control. By exchanging an additional vector h i j l T ˆ ˆ j −1 x ˆ Dj αlj , l = 1, 2, . . . , K (19) i = (Λ ) S i+1 Φi P i (P i ) i + hi between the satellites l and j, the globally optimal control uji can be obtained locally via ( ) K h i−1 X uji = − V ji + (Λj )T S i+1 Λj (Λj )T S i+1 xC αjl , i + i

(20)

l=1

where the controller Riccati matrix S i can be found via a backward sweep of Si

=

ΦTi S i+1 Φi −

K n h i o X (Lji )T V ji + (Λj )T S i+1 Λj Lji + W i

(21)

j=1

Lji

=

h i−1 − V ji + (Λj )T S i+1 Λj (Λj )T S i+1 Φi , 3

(22)

and may be stored on-board each satellite as a table-lookup for use in Eq. (20). Note that αlj i has the dimension of the local controller, which is almost always smaller than that of the state. Since the dimension of the controller is the minimum data exchange rate required to execute optimal control, the inter-spacecraft link budget is much cheaper than passing raw measurements for a global Kalman filter. This can be a significant advantage when the number of spacecraft in the formation is large.

System Dynamics In order to demonstrate the application of the decentralized algorithm to a spacecraft formation, an example is presented. Each satellite is assumed to be orbiting the Earth in a near-circular orbit and to remain in the vicinity of the formation origin. For an intuitive interpretation of the relative motion of the satellites with respect to the formation origin, it is suitable to express the differential position vector r and velocity vector v in a reference frame aligned with the local radial (x), alongtrack (y), and cross-track (z) directions. The corresponding unit vectors of this local reference frame are defined by rR rR × vR ex = , ey = , ez = ey × ex , (23) |r R | |r R × v R | where r R is the inertial position of the formation origin, and the inertial velocity v R = r˙ R + ω ⊕ × r R

(24)

is obtained by correcting the derivative of the WGS84 position for the angular velocity ω ⊕ of the Earth’s rotation. The motion of each satellite in the local reference frame is then described by the ClohessyWiltshire (CW) equations. For the time interval ∆t, the state transition matrix is given by   4 − 3C 0 0 S/n 2(1 − C)/n 0 6(S − n∆t) 1 0 2(C − 1)/n (4S − 3n∆t)/n 0     0 0 C 0 0 S/n j   , Φ = (25) 3nS 0 0 C 2S 0     6n(C − 1) 0 0 −2S 4C − 3 0  0 0 −nS 0 0 C where n is the instantaneous frame rotation rate n = (eTy )/|r R | ,

(26)

S = sin n∆t , C = cos n∆t .

(27)

and It is also assumed that each satellite has small thrusters that can apply an impulsive velocity vector £ uj = ujx

ujy

ujz

¤T

.

(28)

Then the state equation defined in Eq. (3) gives   1       r1 r 0 0 0  v1   v 1  I  0 0  1   1  2  2       Φ 0 ... 0 w r          r 0 0 0  2  0 Φ2   2        w2  v     v  0 1 I  2     = . +  .  . (29)      +   ui +   ui + · · · + 0 uK ..  ..   ..   ..   ..   ..  i  ..   ..  .  .   .  . . .          wK i 0 ΦK i+1  r K   r K  0 0 0 0 0 I v K i+1 vK i 

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Measurement Models In this research, it is assumed that the navigation system onboard each satellite in the formation consists of a GPS receiver and proximity sensors. The GPS receivers are one of the primary sensors for Earth orbiting spacecraft navigation systems. Most GPS receivers are capable of providing a position solution r j in the WGS84 frame. The differential position measurement y jGPS in the reference frame is then given by y jGPS = (ex , ey , ez )T (r j − r R ) .

(30)

There is much work currently going into the development of various proximity sensors for the formation flying missions. One of the examples is the VISNAV sensor [7], which is a vision-based system processing measurements from an optical sensor to estimate the sensor location (and orientation) with respect to the known light sources (beacons). The navigation system can also be extended to support pseudolite augmentation systems [8] for GPS-based relative navigation. Although there are diverse measurement types, this paper assumes that the local proximity sensors are capable of providing the relative positions of all the other satellites in the formation given by T l j y lj REL = (ex , ey , ez ) (r − r ) .

(31)

The measurement vector in Eq. (2) then becomes h y ji = (y jGPS )T

T (y 1j REL )

···

(j−1)j

(y REL )T

(j+1)j

(y REL )T

···

T (y Kj REL )

iT i

.

(32)

HARDWARE SETUP The decentralized controller performance is investigated in closed-loop HWIL simulations using two actual space-capable GPS receivers. Figure 1 shows the hardware setup, including: a GPS signal simulator, two GPS receivers, flight computers, and an external controller. The core component of the facility is the Spirent STR4760 GPS signal simulator, which is capable of simultaneously simulating L1 signals for two vehicles on up to 16 channels each. Each receiver is connected to one of the RF outputs of the simulator via a coaxial cable and provides its position fix solution to the flight computer which runs the decentralized controller. In addition to the STR4760 system, the closed-loop simulation requires the STR4762 remote control option that allows the simulations to be controlled by an external computer in real-time. The external controller runs two satellite trajectory propagators simultaneously and provides position, velocity, and acceleration vectors of the two satellites to the STR4760 system every 100 ms. The propagators are also capable of accepting control vectors from the flight computers for real-time trajectory updates. In this research, the external controller is also modified to provide numerically simulated relative measurements defined in Eq. (31) to the flight computers.

GPS Receivers The GPS receivers (Figure 2) used in this study are Zarlink’s GPS Orion receivers [9], which have been modified for space applications. The original receiver provides C/A code tracking on 12 channels at the L1 frequency. To support user specific software adaptations for the GPS receiver, the GPS Architect development kit was made available by Mitel Semiconductor [10]. For use on low Earth orbit satellites and other space applications, numerous software modifications and enhancements have been made to the original firmware of the Orion receiver [11]. These modifications include the fixes related to the implicit assumption of a low speed vehicle in the Doppler prediction and the time tagging error of the raw measurements. Aside from these fixes, an open-loop Doppler and visibility prediction algorithm has been added to the receiver code to ensure robust tracking and rapid signal acquisition under the conditions of a high-dynamic space vehicle [12].

5

R e la tiv e M e a s u r e m e n ts

E x te r n a l C o n tr o lle r

C o n tro l V e c to r

M o tio n D a ta

F lig h t C o m p u te r s G P S R e c e iv e r s

G P S S ig n a l S im u la to r

G P S S ig n a ls

In te r s a te llite C o m m u n ic a tio n

Figure 1: Closed-Loop GPS Test Facility

Figure 2: GPS Orion Receivers

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Also, an active alignment of measurement epochs and navigation solutions to the integer second of GPS system time has been implemented, which ensures synchronized measurements among multiple independent receivers (typically better than 100 ns in the absence of multipath). To improve the overall navigation performance of the Orion receiver, integrated carrier phase measurements have been made available using a 3rd order phase lock loop (PLL) assisted by a 2nd order frequency lock loop (FLL) [13]. The loop provides accurate tracking and stable acquisition over a wide range of dynamic conditions. Raw measurement accuracies obtained in signal simulator tests are better than 1 m for C/A code pseudorange, 1 mm for L1 carrier phase, and 10 cm/s for L1 Doppler measurements in the absence of environmental error sources such as multipath [14].

Decentralized Controller Implementation As an example, the sequence of events of the decentralized controller on satellite 1 in a twosatellite formation gives the following result: 2

1

¯ } at the epoch i = 0, initialize the controller ¯ } and {r 2 , v 2 , P 1. Given {r 10 , v 10 , P 0 0 0 0      1 0 0 r0 v 10  0 0 D1 1 C     ˆ0 =  x0 =  0 , h0 = 0 , r 20  , x v 20 0 0 # # " " ¯1 0 ¯1 0 1 P P 0 0 ˆ ˆ P0 = , P0 = , ¯2 ¯2 0 P 0 P 0 0 and transmit the local control vector u10 if available. 2. Receive the local control vector u20 from satellite 2, then propagate the control dependent part of the state state vector via     0 0 · 1 ¸     Φ 0 I C   1 0 2 xC 2 x0 +   u0 +   u0 , 1 = 0 0 0 Φ 0 I and the data dependent part via ·

¯ D1 x 1

Φ1 = 0

¸ 0 ˆ D1 . x Φ2 0

Also propagate the local and global covariance matrices via · 1 ¯1 = Φ P 1 0

¸ · 0 ˆ 1 Φ1 P 0 Φ2 0

0 Φ2

· 1 Φ ¯ P1 = 0

· ¸ 0 ˆ Φ1 P0 0 Φ2

0 Φ2

3. Given the local measurement

"

y 11

y 1GP S = 12 y REL

¸T + Q1 , ¸T + Q1 .

# ,

update the data dependent part of the local state vector via £ 1 ¤ 1 1 C ¯ D1 ˆ D1 ¯ D1 x 1 =x 1 + K 1 y 1 − H (x1 + x 1 ) , 7

and the local state error covariance matrix via ˆ 1 = (I − K 1 H 1 )P ¯ 1 (I − K 1 H 1 )T + K 1 R1 (K 1 )T , P 1 1 1 1 1 1 1 where

·

I H = −I

0 0

1

0 I

0 0

¸ .

If the local measurement update is successfully executed, transmit a semaphore s11 = 1 to satellite 2. Otherwise, s11 = 0. 4. After the reception of the semaphore s21 from the satellite 2, update the global state error covariance matrix via i−1 h ˆ1 = P ¯ −1 + s1 (H 1 )T (R1 )−1 H 1 + s2 (H 2 )T (R2 )−1 H 2 , P 1

1

1

where

· H2 =

1

0 I

0 0

I −I

1

0 0

¸ .

21 5. Update the data vector h1 via Eq. (16) and obtain α11 1 and α1 via Eq. (19) using     0 0 I  0 1 2    Λ = 0 , and Λ = 0 , 0 I

respectively. Then transmit α21 1 to satellite 2. 12 6. From the locally available α11 1 and α1 received from satellite 2, the globally optimal control vector of satellite 1 can be obtained via £ ¤−1 © 1 T ª 11 12 u11 = − V 11 + (Λ1 )T S 1 Λ1 (Λ ) S 1 xC , 1 + α1 + α1

where the controller Riccati matrix S 1 is read from a pre-computed memory source. Then transmit the control vector to satellite 2. 7. Update the current epoch i, and go to 2. In the current implementation, the receipt of the output string from the GPS receiver is used as a trigger for the control process. Although the output string trigger provides enough synchronization for this experiment, more precise alignment between the navigation and control systems can be achieved by utilizing the 1 pulse-per-second (1PPS) signal from the GPS Orion receiver, which is aligned with a UTC or GPS system integer second [15].

CLOSED-LOOP SIMULATION RESULTS This section presents preliminary closed-loop simulation results of the decentralized control algorithm for satellite formations. Figure 3 depicts the nominal trajectories of the two satellites (♦) selected for this experiment. This example mission could be of interest for synthetic aperture imaging or optical interferometry. Alfriend et al. [16] describes this type of satellite formation in more detail. The orbits of the satellites in the formation differ slightly from the circular orbit of the formation center, so that they suborbit around the center in an elliptical epicycle. Since the primary goal of this research is not to design a definitive control low for formation flying missions, but to develop and test the capability of demonstrating technologies for such missions, the simulation scenario in this paper is a rather simplified example. The propagator model consists of the

8

Radial [km]

1 0 −1 −2 −2 0 Cross−Track [km]

0 2 2

Along−Track [km]

Figure 3: Example Formation Flying Mission J2 gravitational perturbation and atmospheric drag, and the physical properties of the two satellites are identical. The commanded maneuvers are perfectly executed, and the GPS measurements are bias-free (no broadcast ephemeris, ionosperic delay, and multipath errors). Table 1 summarizes the simulator and controller parameters used in this research. More extensive design and analysis of the distributed control system can be considered as future work. Table 1: Simulator and Controller Parameters Start date and time 6-Nov-2001, 16:30:00 GPS YUMA almanac Week 1138 Initial hub orbit radius, eccentricity, and inclination 400 km, 0, 56 deg Formation horizontal radius 2 km Number of Satellites 2 Measurement rate 1/60 Hz Maneuver rate 1/60 Hz GPS position fix measurement noise (x, y, z) 3, 1, 1 m (1σ) Proximity sensor measurement noise (x, y, z) 0.1, 0.1, 0.1 m (1σ) Initial position uncertainties (x, y, z) 100, 300, 300 m (1σ) Initial velocity uncertainties (x, y, z) 0.1, 0.3, 0.3 m/s (1σ) Process noise intensity 9.81e-6 m/sec3/2 Cost function position, velocity weight 4e-2, 4e-8 Cost function control weight 1e3 Figures 4 and 5 show the resulting formation trajectories and tracking errors, respectively. It is verified that the decentralized controller can be successfully employed in a real-time HWIL test to maintain the shape of the formation. Figure 6 depicts the local state errors of satellite 1, while Figure 7 shows the reconstructed global state errors. The absolute state accuracy strongly depends on the GPS position fix solutions and less on the relative state measurements. A local positioning accuracy of 0.8 m is achievable in both the along-track and cross-track directions, while the accuracy of the radial component is slightly worse (about 2.5 m) because of the less favorable vertical dilution of precision (DOP). Although the proximity sensor measurements do not affect the global absolute state accuracy very much, they improve the relative position state as shown in Figure 8. The reconstructed global absolute states of the two satellites provide the relative positioning accuracy of 0.08 m, which is similar to the simulated proximity sensor measurement noise defined in Table 1. This indicates that the decentralized controller properly processed the relative measurements.

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1

0.5

0.5

0 Sat 2

Sat 1 −0.5 −1

1

0 −1 Along−Track [km]

Radial [km]

Radial [km]

1

0 Sat 1Sat 2 −0.5 −1

−2

−2

−1 0 1 Cross−Track [km]

−2

−1 −0.5

Radial [km]

Cross−Track [km]

−1.5

Sat 1

0

Sat 2

0.5 1

1 Sat 2

0 −1 −2

Sat 1 −1

1.5

−2 −1

0

0

1 1

0 −1 Along−Track [km]

−2

1

Cross−Track [km]

Along−Track [km]

Figure 4: Satellite Formation Flying Result

Sat 1 Tracking Error

Sat 2 Tracking Error 500 Radial [m]

Radial [m]

500

0

−500

0

20

40

60

−500

80

0

0

20

40

60

40

60

80

0

20

40

60

80

0

20

40 60 Time [min]

80

500 Cross−Track [m]

Cross−Track [m]

20

0

−500

80

500

0

−500

0

500 Along−Track [m]

Along−Track [m]

500

−500

0

0

20

40 60 Time [min]

0

−500

80

Figure 5: Tracking Errors

10

Sat 1 Local Velocity 0.2

5

0.1

Radial [m/s]

Radial [m]

Sat 1 Local Position 10

0 −5 −10

0

20

40

60

Along−Track [m/s]

Along−Track [m]

0 −5 0

20

40

60

20

40

60

80

0

20

40

60

80

0

20

40 60 Time [min]

80

0.1 0 −0.1 −0.2

80

0.2 Cross−Track [m/s]

10 Cross−Track [m]

0

0.2

5

5 0 −5 −10

−0.1 −0.2

80

10

−10

0

0

20

40 60 Time [min]

0.1 0 −0.1 −0.2

80

Figure 6: Local State Errors of Satellite 1 Sat 1 Global Velocity 0.2

5

0.1

Radial [m/s]

Radial [m]

Sat 1 Global Position 10

0 −5 −10

0

20

40

60

Along−Track [m/s]

Along−Track [m]

0 −5 0

20

40

60

20

40

60

80

0

20

40

60

80

0

20

40 60 Time [min]

80

0.1 0 −0.1 −0.2

80

0.2 Cross−Track [m/s]

10 Cross−Track [m]

0

0.2

5

5 0 −5 −10

−0.1 −0.2

80

10

−10

0

0

20

40 60 Time [min]

0.1 0 −0.1 −0.2

80

Figure 7: Reconstructed Global State Errors of Satellite 1

11

Global Relative Position

Global Relative Velocity 0.2 Radial [m/s]

Radial [m]

0.5

0

−0.5

0

20

40

60

Along−Track [m]

Along−Track [m/s] 0

20

40

60

0

20

40

60

80

0

20

40

60

80

0

20

40 60 Time [min]

80

0.1 0 −0.1 −0.2

80

0.2 Cross−Track [m/s]

0.5 Cross−Track [m]

−0.1

0.2

0

0

−0.5

0

−0.2

80

0.5

−0.5

0.1

0

20

40 60 Time [min]

0.1 0 −0.1 −0.2

80

Figure 8: Reconstructed Global Relative State Errors

CONCLUSIONS A unique closed-loop simulation facility has been developed for design and analysis of GPS-based navigation and control architectures for formation flying satellites. The preliminary results from the decentralized controller implementation show that the designed system is capable of supporting hardware and software technology development for future distributed spacecraft operations. Although the current simulation setup is capable of simulating only two satellite formations due to local hardware limitations, the system is currently being extended to allow multiple spacecraft formation flying simulations by connecting remotely located GPS simulation systems together through the internet. The inter-spacecraft communication system will then be modeled as an IP network. The developed closed-loop capability has been transferred to the formation flying testbed at NASA Goddard Space Flight Center, which is capable of simulating the GPS signals for up to four spacecraft.

ACKNOWLEDGEMENTS The authors are grateful to Oliver Montenbruck at German Space Operation Center (GSOC) for his effort to modify the Orion GPS receiver software to work efficiently in the low Earth orbit dynamics environment. This research was completed under NASA Goddard Space Flight Center (GSFC) contract NAG5-11278. The authors would like to thank Russell Carpenter of NASA GSFC for his support.

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