Nonlinear Dynamics and Output Feedback Control of Multiple Spacecraft in Elliptical Orbits

Proceedings of the American Control Conference Chicago, Illinois June 2000

Nonlinear Dynamics and Output Feedback Control of

Multiple Spacecraft in Elliptical Orbits

Qiguo Yan, Guang Yang, Vikram

Kapila, and Marcio

S.

de Queiroz

Department of Mechanical, Aerospace, and Manufacturing Engineering

Polytechnic University, Brooklyn, NY 11201

[qyanol, gyang0l]@utopia.poly.edu, [vkapila, queirox]Qpoly.edu

Abstract

This paper considers the problem of relative position control for multiple spacecraft formation flying. Specif- ically, the nonlinear dynamics describing the motion of a follower spacecraft relative to a leader spacecraft are developed for the case where the leader spacecraft is in an elliptical orbit. Next, a Lyapunov-based, nonlinear, output feedback control law is designed which guaran- tees global uniform ultimate boundedness of the posi- tion and velocity tracking errors in the presence of un- known, spacecraft masses and disturbance force parame- ters. Simulation results are provided to illustrate the per- formance of the output feedback control design method- ology for formation maintenance in ideal, naturally at- tractive, orbits.

1. Introduction

Recent years have witnessed an intense research activ- ity towards the development of technologies for multiple spacecraft formation flying (MSFF). The identification of MSFF as an enabling technology for space missions in the next century by the U.S. Air Force and NASA has in part contributed to the increased focus in this field [6--8,10, 121. In particular, MSFF is expected to enable the Earth and space science missions, such as dis- tributed aperture radar, enhanced stellar optical inter- ferometer, virtual co-observing and stereo-imaging plat- forms for space science and Earth observing, etc.

The early efforts in MSFF typically focused on dy- namics, ideal formation design, and feedback control is- sues related to the linearized dynamics of relative mo- tion between a leader-follower spacecraft pair, viz., the Hill’s equations [3, 9, 11-13]. Specifically, in early re- search, the linearized spacecraft relative motion dynamic model (also known as the Clohessy-Wiltshire model) was developed for spacecraft rendezvous problem in circular orbits [3,13]. Note that for ideal MSFF, the initial con- ditions for the formation flying spacecraft must be cho- sen to enable the spacecraft t o undergo periodic motions such that a relative spatial pattern persists for several or- bits with minimal propellant expenditure [2]. In previous research, [ll, 121 identified the set of feasible initial con- ditions that annihilate the secular growth in time in the solution of Hill’s equations; thus, yielding periodic rela- tive motion between the leader-follower spacecraft pair. In addition, spatial patterns for formation design have also been proposed based on the Hill’s equations [ll].

This work was supported in part by the Air Force Office

of Scientific Research under grant F49620-93-C-0063, the Air Force Research Lab/VAAD, WPAFB, OH, under IPA: Vis- iting Faculty Grant, and the NASA/New York Space Grant Consortium under grant 32310-5891.

Finally, a number of MSFF control designs have been developed which utilize the Hill’s linearized relative mo- tion equations [9,13].

Unfortunately, a caveat of ideal formation and feed- back control design based on Hill’s equations is that it is predicated on the linearization of nonlinear dynamics of spacecraft relative motion. Since Hill’s equations ne- glect the influence of nonlinear terms (e.g., higher-order terms)

,

non-circular orbits, long formation baseline, long mission duration, etc., on the relative motion dynam- ics, control designs based on Hill’s equations necessitate prohibitive fuel consumption and endanger formation in- tegrity in general orbits with long mission duration. In fact, it can be shown that the previously developed ideal, no-thrust, formation initialization techniques fail to hold

the designed formations for the nonlinear dynamics of

spacecraft relative motion. In addition, since Hill’s equa- tions are typically valid for only short period maneuvers, formation control schemes based on Hill’s equations are unlikely to yield good precision for MSFF in general el- liptical orbits for long duration.

It is clear from the preceding discussion that there exists an urgent need for developing MSFF nonlinear dynamic modeling, formation initialization, and control schemes. Reviewing the current state of MSFF control research, [14] developed a model-based, relative position controller with local asymptotic position tracking errors. More recently,

[4]

proposed a nonlinear, adaptive con- troller which ensures global asymptotic position tracking errors. However, the framework of [4] is based on the as- sumption that the leader spacecraft remains in a circular orbit.

In this paper, to address the aforementioned short- comings in the MSFF schemes, we consider the problem of relative position control for MSFF using the nonlin- ear system dynamics. Specifically, nonlinear dynamics describing the motion of a follower spacecraft relative to a leader spacecraft are developed for the case where the leader spacecraft is in an elliptical orbit. Next, a Lyapunov-based, nonlinear, output feedback, robust con- trol law is designed which guarantees global uniform ul- timate boundedness (GUUB) of the position and veloc- ity tracking errors in the presence of unknown, space- craft masses and exogenous disturbance force parame- ters. This control design methodology is evaluated for the formation maintenance problem for the case of ideal, naturally attractive, spatial formation.

2. System Model

We begin with the consideration of a MSFF system composed of two spacecraft, i.e., a leader spacecraft and

a foZlower spacecraft. The leader spacecraft, which pro-

vides the basic reference motion trajectory, is considered to be in an ideal, elliptical orbit around the Earth. The follower spacecraft navigates in proximity of the leader spacecraft.

A schematic drawing of the MSFF system is given in Figure 1 where we make the following considerations i)

the inertial coordinate frame { X , Y, Z } is attached to the center of the Earth, ii)

R(t)

E

R3

denotes the position vector from the origin of the inertial coordinate frame to the leader spacecraft,

iii)

a right-hand coordinate frame

{ z l , y l , z l } is attached to the leader spacecraft with the sl-axis perpendicular to the instantaneous vector

R

and contained in the orbital plane of the leader spacecraft, the y/-axis pointing along the direction of the vector

R,

and the z/-axis pointing along the orbital angular mo- mentum of the leader spacecraft, and iw) p ( t ) E

R3

de- notes the position vector from the origin of the moving coordinate frarne (51, y/, z l } to the follower spacecraft.

The nonlinear position dynamics of the leader and follower spacecraft in the inertial coordinate frame

{ X ,

Y,

2 ) are given by [4,13]

mlfi

+

mi

( M

+

ml)

G

y

+

Fdl = 211 (1)

R

IlRll

and

respectively, where ml, m f are the masses, Fdl(t),

F d f ( t ) E

R3

are bounded disturbance force vectors due to solar radiation, aerodynamics, and magnetic fields, and

u , ( t ) , u f ( t ) E

R3

are the actual control input vectors of the leader and follower spacecraft, respectively, M is the Earth’s mass, and

G

is the universal gravity constant. Since M

>>

m l , m f , let ( M

+

m i ) G M M G , a = 1, f . Next, after some simple algebraic manipulations on (1) and

(a),

the dynamic equation describing the position of the follower spacecraft relative to the leader spacecraft in the coordinate frame

{ X ,

Y , Z } can be written as

(3)

where p

f

M G .

To write the dynamics of (3) in terms of the moving coordinate frame (21, yl, z l } , we must obtain an expres- sion for

p ( t )

in the moving coordinate frame (21, yl, 21).

Hence, we consider the homogeneous form of (l), with

(A4

+

m,)G M p, given by

R

R + p - - - , = O .

IlRll (4)

Using a polar coordinate frame fixed at the center of the Earth (see the R-B coordinate frame in Figure l), the motion of the leader spacecraft given by the vector dif- ferential equation (4) can be alternatively characterized by the planar dynamics

where 7-1

2

(1

RI1

and 8 represents the true anomaly of the leader spacecraft. Employing standard orbital mechanics techniques, simple manipulations of (5), (6) yield

where al is the semi-major axis of the elliptical orbit of the leader spacecraft, el is the orbital eccentricity of the leader spacecraft, and n is the average orbital angular velocity defined by n e $ , with T as the orbital period. Finally, differentiating (8), we obtain

-2n2el(1

+

el C O S B ( ~ ) ) ~ sinB(t) (1 - e l 2 ) 3

e ( t )

=

.

(9)

Next, note that the relative position vector p ( t ) ex- pressed in (51, yl, zl} is given by p = zil

+

yjl

+

z&,

where i l , j l , & denote the unit vectors, while the angu- lar velocity of the moving coordinate frame {zl, yl, z l } is given by

8kl.

Hence, the relative acceleration vector

p ( t )

is given by

p

= ( 2 - 2ey - e2a: - &)Z^l

+

(y

+

28k - 82y

+

& ) j ,

+ Z i l . (10)

After substituting the right-hand side of (10) into (3), the nonlinear position dynamics of the follower spacecraft relative to the leader spacecraft can be arranged into the following advantageous form

m f

Note that, in (ll), q ( t )

f

[ z ( t )

z(t)lT

E

R3

is the relative position vector, C ( 8 ) 4 2 8 1 0 0 E

R3x3 is the Coriolis-like matrix and N E

R3

is a nonlin- ear term defined as

ij+

~ ( 8 1 4

+

~ ( q , 8 , 8 ,

R )

+

(

y(t)

I:

:

:]

In addition, in (ll),

R

in the moving coordinate frame {zl,yl, zl} is given by

R

= [O rl 0IT, and Fd(t) E

R3

is the composite, disturbance force given by

We will assume that Fd(t) can be linearly parameterized as

where Y f ( t ) E RSxP is a known regression matrix and

B f

E RP is a vector containing some constant disturbance parameters. Finally, note that r l ( t ) ,

8 ( t ) ,

and e ( t ) in (11) are computed from (7), (8), and (9), respectively.

Fd

( t )

=

y.

(t)e.f

i (13)

The dynamic model of ( l l ) , (12) has the following property which will be exploited in the subsequent con- trol design. The left-hand side of the dynamic equation (1 1) can be linearly parameterized as

mfq

+

m f C ( e ) q

+

mfN(q,

e , e ,

R )

+

+

Fd

ml

= y ( i i , 4 , 4 , e ,

8,

R,

" 1 , cl)$, (14)

where

Y

E

R3x(2+P)

is the regression matrix and (b E R(2+P) is the system's constant parameter vector defined as

3. Output Feedback Control Law Design

In this section, we develop an output feedback control law that tracks a prespecified spacecraft relative position trajectory. In particular, we consider that a desired po- sition trajectory qd(t) E

R3

for the follower spacecraft relative to the leader spacecraft is given. Furthermore, we assume that

& ( t )

and its first two time derivatives are bounded functions of time. The closeness of the con- trol objective is quantified by the position tracking error

e ( t ) E

R3

defined as

e ( t )

f

qd(t) (16)

The above control objective is to be met with the as- sumption that the relative velocity 4(t) cannot be mea- sured, and under the constraint that the satellite masses and disturbance-related parameters are not known pre- cisely (i.e.,

4

defined in (15) is unknown, with the excep- tion of lower and upper bounds on the parameters).

To facilitate the output feedback control design and stability analysis, we define the variable

q

f

e + w + e , (17)

where w(t) E

R3

is a pseudo-velocity tracking error signal characterized by the output of the dynamic filter [l]

p

= -

(IC

+

1 ) p

+

( k 2

+

1) e, (18)

w = -ke + p , (19)

where p ( t ) E

R3

is an auxiliary filter variable, and k is a positive, scalar control gain given as

k = l + k , , (20)

with k , being an additional, positive, scalar control gain yet to be specified.

To determine the dynamics for w(t), differentiate (19)

with respect to time and use (17)-(19) to obtain

w

= -w -

kq

+ e . ₍₂₁₎

Finally, note that the dynamics of e ( t ) can be produced by simply rearranging (17), i.e.,

e = -e + q - w. ₍₂₂₎

To develop the open-loop dynamics for the tracking er- ror signal q ( t ) , differentiate (17) with respect to time and pre-multiply the resulting expression by

m f

to produce

mf7j = m f

(&

-

Q)

+

mfw

+

m f e . (23)

After utilizing (11), (21), and (22) to substitute for

m f q ( t ) , G ( t ) , and e @ ) , respectively, in (23), we obtain

mf7j = rrifud

+

m f c ( e ) q

+

m f N ( q ,

8,

e,

R ) "f

+-U/ m1

+

Fd -

u f

- 2 r r t . f ~ - k,rri,fq, (24) where (20) has been used. Next, it follows from (16) and

(22) that

q

= e - q + w + q d . (25)

NOW, using (25) in (24) and rearranging terms yield mf7j =

Y

(.) (b - " , f - k,nlfq - m f C ( 8 ) q , ₍₂₆₎ where Y(.)(b = m f q d

+

m f C ( e ) ( e

+

w

+

qd)

+

r n f N ( q ,

e , 8 ,

R )

(27) "f -2mfW

+

&UI --- Fd,

with

Y

(.)

E J R 3 X ( 2 + p ) being the regression matrix com- posed of known functions and (b being defined in (15).

Based on the form of the open-loop dynamics of (26),

the control input

u f ( t )

is designed as follows

uf

=

Y ( . ) d

- kw + e , (28)

where

4

E R2+P denotes a constunt, best-guess estimate of the unknown parameter vector I$ and k is the control

gain (20). Using (28), the closed-loop dynamics for q ( t )

becomes

where

4

E R2+P represents the constant, parameter esti- mation error

i

fi

4 - d .

Note that the follower spacecraft control input u f ( t )

depends on the leader spacecraft control input u l ( t ) , which is in turn designed t o make the leader spacecraft track some desired trajectory. If ul(t) is properly de- signed t o ensure that

R ( t )

+

& ( t )

as

!

--$ 00 where

&(t)

denotes some bounded, desired trajectory for the leader spacecraft, then it is reasonable to assume that

ul(t),

R(t)

E

L,.

More specifically,

where

cu,L,,

are some positive bounding constants. Based on these arguments, it is not difficult t o show that the following bounds exist. First, since q ( t ) #

-R(t)

(i.e., the follower spacecraft cannot be located a t the Earth's center), then

where is some positive bounding constant. Next,

(33)

where

z ( t )

E

R9

is defined as

(34)

and Cl,& are some positive bounding constants which

only depend on the desired relative motion trajectory,

(31), (32), and bounds on the unknown parameters.

The combination of the error systems of (21), (22),

and (29) gives a stability result for the position tracking

error as delineated by the following theorem.

Theorem 1. For a sufficiently large

k,,

the output ensures the GUUB of the augmented state vector and

z ( t

fs’

as feedback control law described by (18), (19),

illustrated by

l l q / ~

l l m / I 2 e x P ( - $ t )

+ &

( l - e x p ( - E t ) ) , (35) where X I

e-

1 min { 1 , m f } A2 A 1 =

5

m a x { l , m f } , (36) 2

E is some positive constant, and

p

is another constant satisfying 0

<

,B

<

1. Thus, GUUB of all components of

z ( t ) ,

viz., e ( t ) , w(t), and

v ( t ) ,

follows. Note that (35) can also be used to describe velocity tracking. Specifically, the equality in (22) can be used to form the following inequality

From (34) and (35), we know that the terms on the right- hand side of (37) are GUUB; hence, we can conclude that e ( t ) is also GUUB, despite the controller’s lack of direct velocity measurements.

U

Note that the above stability result ensures GUUB of the position tracking error for arbitrary, sufficiently smooth, desired relative position trajectories q d ( t ) . How- ever, tracking an arbitrary spatial formation geometry for formation maintenance will require prohibitive fuel expenditure. Thus, for spacecraft formation mainte- nance, q d ( t ) must be generated to yield an ideal, nat- urally attractive, no-thrust, spatial formation geometry.

As a result, control effort will only be required to com- pensate for non-ideal conditions, e.g., disturbance forces. In contrast, for formation configuration maneuvers, q d ( t )

may represent a more aggressive trajectory; hence de- manding significant control effort. The efficacy of our nonlinear control design for formation configuration ma- neuvers has recently been illustrated in [4].

1141

I

llell

+

llvll

+

ll4

.

(37)

Proof: See 151 for details of a similar proof.

4. Simulation Results

The problem data for simulating the output feedback control law described in Theorem 1 is adopted from [13, 151 and is given by

A4

= 5.974 x kg, mp = 410 kg, ml = 1550 kg,

G

= 6.673 x

&,

U / = Fd/ = 0

N,

Fd = [1.9106, -1.9106, -1.5171 X

lo-’

N. ₍₃₈₎

The orbital elements and initial position for the leader spacecraft were selected as Tl = 24 hours, el = 1.425 x

lo-’, 00 = 0 rad. The desired formation trajectory is generated online by the evolution of unperturbed, rela- tive motion dynamics

i d

+

c ( e ) G d

+

N(qd,

e,e,

R )

= 0, (39)

with qd(0) = [-100,-50,-0.2]T m and &(O) =

[-26.1805,10.6574, 0.0398IT

E.

Note that this desired

trajectory yields an ideal, no-thrust, periodic q d ( t ) in the moving frame {zl,yl,zl} [15].

In the following simulation, the control gain

k

in (18), (19), and (28) was set to k = 27 while 2 ~ 3 0 % of paramet- ric uncertainty was assumed, i.e.,

where f d i ( t ) , 2 = 1 , 2 , 3 , are the z / , y ~ , z ~ com- ponents of Fd. The actual initial conditions for the relative position and velocity were chosen to be

q(0) = [-97.154, -343.632,3.497 x m, G(0) =

[-135.694, -5.38 x 39.81271

2.

The phase portrait of the actual trajectory q ( t ) of the follower spacecraft relative t o the leader spacecraft is il- lustrated in Figure 2 where

“*”

represents the leader spacecraft at the origin of the moving frame. Figure 3

depicts the position tracking error

e ( t ) .

Note that the maximum, steady-state tracking errors in Figure 4 were smaller than 0.1 m.

T

5. Conclusion

In this paper, we developed the dynamic equation of relative motion for

MSFF

when the leading spacecraft is in an elliptical orbit. This dynamic model has wider application and yields greater precision compared with the linearized Hill’s equation model which is restricted to circular spacecraft orbits. Next, using a Lyapunov- based design and analysis framework, we developed an output feedback controller which was shown to guarantee GUUB of the position and velocity tracking errors in the presence of unknown, spacecraft masses and disturbance force parameters. Simulation results were provided to demonstrate the efficacy of the output feedback con- troller for formation maintenance with ideal, no-thrust, naturally attractive, formation geometry.

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4

Follower Swcecrsfi

Figure 3: Position tracking error

Figure 1: Schematic representation of the

MSFF

system Figure 4: Steady-state position tracking error