Control for Satellites Formation Flying
Yunjun Xu
1
; Norman Fitz-Coy
2
; Rick Lind
3
; and Andrew Tatsch
4
Abstract: In this paper, a controller is designed for a satellite formation flying system around the Earth based on an uncertainty model
derived from a nonlinear relative position equation. In this model, nonzero eccentricity and varying semimajor axis are included as
parametric uncertainties. J
2
perturbation, atmospheric drag, and actuation and sensor noise are bounded by functional uncertainties. The
controller design based on the nominal mission an 800 km altitude circular reference orbit is capable of achieving desired perfor-
mance, is robust to uncertainties, and satisfies fuel consumption requirements even in a challenge nonnominal mission a 0.1 eccentricity
and 7,978 km semimajor axis elliptic reference orbit with the same control gain. In this nonnominal mission, the designed controller
is able to keep formation with almost the same level of the V budget 43.86 m/s/year as used in the nominal mission 39.65 m/s/year.
For comparison, linear quadratic regulator LQR and sliding mode controllers SMC are developed and extensively tuned to get the
same V consumption as that of the designed controller for the nominal mission. However, as shown in the simulation, the designed
linear robust controller LQR and nonlinear robust controller SMC have a serious V consumption penalty 1.72 km/s/year for SMC
or are unstable for LQR in the nonnominal mission.
DOI: 10.1061/ASCE0893-1321200720:110
CE Database subject headings: Satellites; Dynamics; Models; Control systems.
Introduction
Formation flying systems FFS have been investigated and pro-
posed for military or nonmilitary services. Due to their potential
advantages over the conventional large size monolithic satellite,
such as the low cost, flexibility, improved observation efficiency,
increased reliability, and enhanced survivability, many missions
are considering the use of satellite or spacecraft FFS to achieve
goals that are difficult for the conventional large-size single sys-
tem. An ongoing FFS mission list includes the New Millennium
Program DS1, DS2, EO-1, EO-3, ST-5, and ST-6, micro-
arcsecond X-ray imaging mission MAXIMLuquette and San-
ner 2003, terrestrial planet finder TPFLuquette and Sanner
2003, Stellar Imager Luquette and Sanner 2003, SPHERES
Miller et al. 2003, ACE Dahl et al. 1999, TerraSAR-X/
TanDEM-X synthetic aperture radar SAR D’ Amico et al.
2005, Gemini Gill et al. 2001, and GRACE Kirschner et al.
2004. The typical distance among space satellites in a formation
ranges from 30 m to 200 km, with the control precision in a range
from 1 m to 50 km.
In relative dynamic models for FFS, the Clohessy-Wiltshire
CW equation is most widely used Redding and Adams 1989;
Kapila et al. 1999; Robertson et al. 1999; Morton and Weininger
1999; Aorpimai et al. 1999; Vassar and Sherwood 1985; Sedwick
et al. 1999; Yan et al. 2000; Yedavalli and Sparks 2000. Based on
this linear model, Redding and Adams 1989 and Kapila et al.
1999 designed LQ controllers for a pulse-based thruster,
whereas Robertson et al. 1999 implemented a PD controller
with a thruster mapping. Sedwick et al. 1999 derived an analyti-
cal expression for an exact cancellation of differential J
2
for a
satellite formation in a polar, circular orbit. Yan et al. 2000
designed a pulse-based control for the formation satellites’ peri-
odic motion which guarantees the global stability. Yedavalli and
Sparks 2000 proposed a hybrid control scheme to achieve a
balance between discrete control efforts and continuous dynamics
performance. Ulybyshev 2003 designed a discrete LQR control-
ler for an 800 km altitude Earth orbit formation mission with fuel
consumption on the order of V =50 m/s/year for formation sat-
ellites station keeping within a 20–40 km control precision.
All the above-mentioned controllers are designed using the
CW equation. However, the CW equation is a linear approxima-
tion of the real system, and does not contain atmospheric drag,
high order harmonic terms, and uncertainties. Furthermore, the
CW equation is only applicable in the near circular reference orbit
case. Among these effects, higher order harmonic terms espe-
cially J
2
, which changes the orbit period, drifts the perigee, and
changes the nodal precession rate Alfriend et al. 2000 and ec-
centricity are the dominant ones which affect the drifting of the
CW equation. Therefore, mitigating the relative distance drift ef-
fects coming from J
2
and elliptic reference orbits have been in-
vestigated by many researchers.
Schweighart and Sedwick 2001 studied the effect of J
2
on
the orbital elements, and a linearized J
2
form is added to the right
of the CW equation. Yeh et al. 2000 designed a Lyapunov-type
nonlinear controller for the J
2
perturbation rejection based on the
CW equation; the V budget is on the order of 100 m/s for only
1
Assistant Professor, School of Aerospace and Mechanical
Engineering, Univ. of Oklahoma, Norman, OK 73019. E-mail:
yjxu@ou.edu
2
Associate Professor, Mechanical and Aerospace Engineering, Univ.
of Florida, Gainesville, FL 32611. E-mail: nfc@ufl.edu
3
Assistant Professor, Mechanical and Aerospace Engineering, Univ. of
Florida, Gainesville, FL 32611. E-mail: ricklind@ufl.edu
4
Research Assistant, Mechanical and Aerospace Engineering, Univ. of
Florida, Gainesville, FL 32611. E-mail: atatsch@ufl.edu
Note. Discussion open until June 1, 2007. Separate discussions must
be submitted for individual papers. To extend the closing date by one
month, a written request must be filed with the ASCE Managing Editor.
The manuscript for this paper was submitted for review and possible
publication on April 15, 2005; approved on November 18, 2005. This
paper is part of the Journal of Aerospace Engineering, Vol. 20, No. 1,
January 1, 2007. ©ASCE, ISSN 0893-1321/2007/1-10–21/$25.00.
10 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / JANUARY 2007