In this paper the value and possibility of eccentric orbits in formation flying mission are illustrated in detail. A new set of closed-form formulae that can be used in the formation design of eccentric orbits are derived by linearizing a general solution, which describes relative motion of two spacecraft in arbitrary elliptic orbits of 10 <≤ e . The new formulae are shown to be in excellent agreement with the exact solution and are examined carefully. By comparing with the trajectory described by Hill′s solution, the characteristics of relative motion of two close spacecraft in eccentric orbits are demonstrated clearly. Based on those characteristics, five candidate formations and their values in space missions are proposed.
INTRODUCTION
In recent years a significant attention has focused on
spacecraft formation flying. Clusters of spacecraft that would maintain a constant or nearly constant shape and size are of interest for large distributed aperture sensing.1 Another potential application is to form clusters from many small inexpensive spacecraft, each with a particular type of sensor and some computer power. For such missions, it might be unnecessary to maintain extremely precise relative positions, rather it might be sufficient to know the relative position accurately, and to remain in close enough proximity to share information and computing ability among themselves.2
With the desire to minimize the propellant consumed for formation-keeping, the natural orbital motions of spacecraft to maintain the geometry of the array are considered. Thus the need for formation-keeping would be reduced to eliminating the effect of perturbations on the array. The well-known C-W equation 3 that describes the natural orbital motion of a spacecraft near a circular reference orbit is used in the design and control of formations-flying mission and four formations are proposed based on its solution (Hill′s solution).4 However C-W equation can only produce a result valid for circular (or near circular) orbits. The exact applicable conditions of Hill′s solution are presented in Ref. 5,6. Due to the assumptions made in the derivation of C-W equation, spacecraft formation flying mission has traditionally been carried out in near-
circular orbits. However, there are some benefits to expand the topic to eccentric orbits, which allows for longer dwell time over regions of interest and shorter occultation time. Owing to the deficiency of appropriate formulae that could be used for designing formations of eccentric orbits, the potential value of eccentric orbits in formation-flying mission has not been well understood.
Relative motion of spacecraft in eccentric orbits was examined by Robert G.Melton, a state transition matrix was proposed and expanded up to second order in eccentricity.7 However errors are on the order of 10-20% for reference orbits of eccentricity less than 0.2. Tschauner and Hempel solved the problem for motion relative to an eccentric orbit and presented T-H equation.8,9 The T-H equation has been effectively used in the fuel-optimal rendezvous10 and formation flying control11 in eccentric orbits. However, solution of the T-H equation is a little complex in form and can not be conveniently applied in formation design.
In this paper, a set of simplified formulae that are more effective for formation design of eccentric orbits are derived by linearizing the analytical solution of motion relative to an arbitrary eccentric orbit. Specific characteristics about the relative motion of two close spacecraft in eccentric orbits can be inferred from the new set of formulae. By exploiting those characteristics, five candidate formations are proposed at the end of this paper.
EQUATION AND ANALYTICAL SOLUTION
Assume that a reference spacecraft rS (may be virtual) and a formation spacecraft fS both run in elliptic orbits around the Earth with eccentricities of e1 and e2 respectively. The two orbits have the same semimajor axis a. To analyze the geometric relationship between fS and rS , two coordinate systems, i.e., the Earth-centric inertial coordinate system OXYZ and the relative coordinate system oxyz, should be set up first (see Fig.1).
In OXYZ, the origin O is in the center of the Earth. Let plane OXY be the same as the orbital plane of
rS and axis OZ be perpendicular to plane OXY. Axis OX is along the intersection of two orbital planes and directing from Earth-center O to the ascending-cross point of fS arriving at the orbital plane of rS , and OY can be determined by right-hand rule.
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2
In oxyz, the origin o is defined as the mass center of reference spacecraft rS , axis ox as the radial direction in the reference orbit plane. Axis oz called cross track axis is perpendicular to the plane of reference orbit, then the third axis oy called in-track axis can be established by right-hand rule.
Now suppose that all the disturbing forces are neglected, differential equations that describe relative motion between formation spacecraft fS and reference spacecraft rS in the relative coordinate system oxyz can be written as follows
=+
=++−+
=++−−−−
0
02
02
32
32
2
32
132
131
2
zR
z
yR
xyxy
xR
RR
RR
yxyx
µ
µϕϕϕ
µµµϕϕϕ
&&
&&&&&&&
&&&&&&&
(1)
where 1R is radial vector from Earth center to spacecraft rS , 2R is radial vector from Earth center to spacecraft fS , ϕ is true anomaly of rS , µ is gravity constant of the Earth. Dot represents differentiation with respect to time.
The analytical solution of equation (1) can be expressed as follows
+=
−+−+
++++−
−=
−−+−+
++++−
=
)sin(sin
)]sin(2cos1)sin(
2cos1[
)]cos(2cos1)cos(
2cos1[
2
2
12
f
rfrf
rfrf
iRz
iiRy
RiiRx
ρθ
ρρϕθρρϕθ
ρρϕθρρϕθ
(2)
Where θ is true anomaly of fS , i is relative inclination of two orbit planes, fρ , rρ are defined as angles of perigee measured along the flying direction from OX axis to the perigees of fS and rS respectively.
Actually two independent variables ϕ ,θ and seven parameters i, e1, e2, fρ , rρ , tpr, tpf (the times of perigee passage of Sr and Sf respectively) are included in solution (2). If seven parameters are randomly chosen, the relative trajectory of two spacecraft in eccentric orbits is very complex and irregular. However the
relative trajectory could become very simple if those parameters are carefully designed.
SIMPLIFICATION IN CLOSE PROXIMITY
For many formation-flying missions, it is required to maintain spacecraft clusters in close proximity, which can be expressed in the following four constraints for two spacecraft in eccentric orbits: (a) The difference between eccentricities of two orbits
e∆ is very small. (b) The difference between times of perigee passage t∆
is very small. (c) The difference between angles of perigee rf ρρ − is
very small. (d) The relative inclination of two orbit planes i is very
small. Assume that i, e∆ , t∆ and rf ρρ − are less than 0.01
and are considered as the first order terms, the second and higher order terms are neglected during the simplification.
True anomaly can be considered as a function of two independent variables: the time of perigee passage tp and orbit eccentricity e. Under constraints (a) and (b), the true anomaly ϕ can be expressed by θ according to Taylor first-order expansion
tt
ee
∆∂∂
+∆∂∂
+=θθ
θϕ (3)
In order to obtain the partial differentiation of true anomaly with respect to time and eccentricity, eccentric anomaly is introduced. From Kepler Equation, we have
21sin
cos1sin
eEeE
eE
−=
−=
∂∂ θ (4)
Een
tE
cos1 −=
∂∂ (5)
Where n denotes mean angular velocity. With relationship of eccentric anomaly and true
anomaly, the partial differentiation of true anomaly with respect to eccentricity can be obtained
21)cos2(sin
ee
e −+
=∂∂ θθθ (6)
The partial differentiation of true anomaly with respect to time can be inferred from Momentum Reservation
2
3)cos1( θ
µθ ept
+=∂∂ (7)
Where p is semilatus rectum Substitute (6)(7) into (3), we can obtain
tep
eee
∆++∆−+
−= 2
32)cos1(
1)cos2(sin
θµθθ
θϕ (8)
where 12 eee −=∆
X
Y
Z
rS
fS
fρ
rρ x
yzθ
ϕ
perigee of rS
perigee of fS
Fig.1 the Earth-centric coordinate system OXYZ and the relative coordinate system oxyz
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3
)cos1
sin1cos1sin1
arcsin
cos1sin1
cos1sin1
(arcsin1
02
0222
02
022
01
0211
01
021
θθ
θθ
ϕϕ
ϕϕ
eee
ee
eee
ee
nt
+
−+
+
−−
+
−−
+
−=∆
0ϕ and 0θ are initial true anomaly of rS and fS respectively. According to Taylor first-order expansion, 1R can be expressed by 2R as well.
)]sinsin(cos1[ 22
22211 e
eEEt
tEEEeaR ∆
∂∂
+∆∂
∂−−≈
With result of (4)(5), the above formula can become
θθ sin1
cos221
etaneeaRR
−
∆+∆+≈ (9)
Substitute (8)(9) into (2), and neglect the second and higher order terms, the relative trajectory can be simplified as
++
−=
+−−
+−
+∆−
++∆
=
−
∆−∆−=
θρθ
θρρθ
θθθ
θθ
cos1)sin(
sin)1(
cos1)1)((
1)cos1(
cos1)cos2(sin
1sincos
2
2
2
2
eieaz
eea
eetan
eeeay
etaneeax
f
rf
(10) In stead of orbital elements, formula (10) can also be expressed in terms of initial relative position and velocity as
+−+
++
+−=
++
++
+−+−+
++
−−=
−+−=
)11(cos1
)sin()cos1(cos1
cos)cos(
cos1cos1
cos1)cos2()cos2)(cos(
cos1cos2
)sin(
)sin()cos(
00
0
000
00
00
0
000
0
0
000
θθθθ
θθθθθ
θθ
θθθθθ
θθθ
θθ
θθθ
θθ
eez
ee
zz
eey
eeex
eexy
xxx
&&
&&
&&
Formula (11) is the closed-form T-H solutions.10 Both Formula (10) and (11) are closed periodic
solutions and have no singularity at e=0. It is clear that formula (11) is reduced to Hill′s solution when e=0. All the previously published formations for passive aperture forming with circular orbits can be inferred from formula (10). In fact, formula (10) not only extend and generalize Hill′s solution but also is more accurate for the same applicable conditions. For practical formation flying mission, the reference orbit can not be perfect circular. Errors will be induced with neglecting the small eccentricity (even to e=0.005) of the reference orbit.12 Fuel penalty is needed to maintain the formation designed by Hill′s solution.12
Formula (10) shows that the in-plane relative motion is closely related to three variables: e∆ , t∆ and rf ρρ − . It is noted that formula (10) will be invalid when all the three variables are zero, i.e., two spacecraft run in the
same eccentric orbits with only little relative inclination. In fact, Hill′s solution also can not produce valid result for two spacecraft in the same circular orbits but little relative inclination. In that case, the mode of the relative motion changes, i.e., multiple frequency terms become dominant.5,6 Since practical flights can not exactly make the three variables zero, formula (10) can describe the practical relative motion of two close spacecraft in eccentric orbits.
Although the simplified formula (10) is verified to be equal to the closed-form T-H solution, it provides more direct geometric meanings for relative motion than T-H solution. By carefully examining effects of the three variables e∆ , t∆ and rf ρρ − in formula (10), characteristics of relative motion of two close spacecraft in eccentric orbits can be demonstrated clearly.
RELATIVE MOTION ANALYSIS
In this section numerical examples are designed to demonstrate some conclusions. We suppose the semimajor axis of two eccentric orbits is 7500km, the relative inclination is 0.01. If not indicated, eccentricity of the reference orbit is 0.4, rf ρρ − and e∆ are 0.01, and t∆ is 4 seconds. Case 1: when rf ρρ − is not zero, while e∆ and t∆ are zeros
It is inferred from formula (10) that the trajectory of out-of-plane motion is an ellipse. Center of the ellipse is not in origin. Direction of the ellipse′s major axis depends on the value of fρ or rρ . When 0=fρ or
πρ =f , the major and minor axis of out-of-plane ellipse are along in-track and cross-track direction as showed in Fig.2. When 2/πρ ±=f , the minor axis diminishes to zero, the out-of-plane trajectory becomes a line (Fig.4). In other cases, the out-of-plane trajectory is a slantwise ellipse (Fig.3). Case 2: when t∆ is not zero, while e∆ is zero
It can be inferred from formula (10) that: (a) If rf ρρ − is zero, the trajectory of in-plane motion
is a circle with center in )1
,0(2e
tan
−
∆− and radius of
21 e
tean
−
∆ (Fig.5).
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Fig.2 0=fρ Fig.3 4πρ =f Fig.4
2πρ =f
Fig.5 rf ρρ = Fig.6 rf ρρ < Fig.7 rf ρρ >
(b) If rf ρρ − is not zero, symmetry of in-plane motion
in Fig.5 is destroyed. At the same time, it is possible to make the in-plane motion centered in origin if
tne rf ∆=−− )(1 2 ρρ is satisfied. In this case, the in-track motion is composed of a symmetric term t∆ and an asymmetric term rf ρρ − . The sign of
rf ρρ − changes the asymmetry of in-plane motion as showed in Fig.6 and Fig.7.
(c) For out-of-plane motion, the in-track motion is composed of the symmetric term rf ρρ − and the asymmetric term t∆ . Compared with the case in Fig.2, the out-of-plane motion in Fig.8 and Fig.9 becomes asymmetric due to effect of t∆ . Also it is possible to make the asymmetric trajectory centered in origin if tne rf ∆=−− )(1 2 ρρ is satisfied. For the same reason, the sign of t∆ changes the asymmetry of out-of-plane motion, as showed in Fig.8 and Fig.9.
Case 3: when e∆ is not zero, while t∆ and rf ρρ − are zeros
Compared with the elliptic in-plane motion described in Hill′s solution with center in origin and 2:1 ratio of two major axes, Fig.10 and Fig.11 show that the in-plane motion of eccentric orbits is asymmetrical. The larger the orbit eccentricity is, the greater extent
asymmetry of in-plane motion becomes. The sign of e∆ changes the asymmetric trajectory of in-plane
motion as showed in Fig.10 and Fig.11. The out-of-plane motion is closely related to the
value of fρ or rρ . Fig.12 shows the figure-8-shaped out-of-plane motion when 0=fρ or πρ =f . In comparison, the same case for circular reference orbits only result in one-dimensional out-of-plane motion. The figure-8-shaped out-of-plane motion could be used to provide higher uv-plane coverage per orbit for aperture filling observation.12 Fig.13 and Fig.14 show
the out-of-plane motion when 2π
ρ ±=f . Compared
with the same case described by Hill′s solution, orbit eccentricity also makes the out-of-plane motion asymmetric.
FORMATION DESIGN
In our development, the seven orbit parameters are treated as design variables, and values are chosen to obtain appropriate formations. 6.1 Out-of-plane linear formation
Fig.2 shows that spacecraft in eccentric orbits can form an elliptic out-of-plane motion. Furthermore, if
ie
erf sin1 2−
=− ρρ is satisfied, a circular out-of- plane
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American Institute of Aeronautics and Astronautics
motion is made. In this case, several formation spacecraft with little difference in relative inclination can form a linear formation aligned with reference spacecraft at any time, showed in Fig.15.
Fig.15 out-of-plane linear formation 6.2 Out-of-plane circular motion symmetric about cross track direction
In Comparison with the circular out-of-plane motion described by Hill′s solution, Fig.2 have three important differences: (a) Center of circle can not be in origin. (b) Only one spacecraft can be placed on the circle. (c) The velocity of formation spacecraft passing the projective circle is not even.
For a specific reference spacecraft in eccentric orbit, two formation spacecraft can form circular out-of-plane motions as showed in Fig.16, one centered before it, and the other centered after it. At any time, the distances from two formation spacecraft to the reference spacecraft maintain equal. It is seen from the equal time separation mark “ o” in Fig.16, the two formation spacecraft speed up when close to the
reference spacecraft, and slow down when away from it.
Fig.16 out-of-plane circular motion
6.3 In-plane circular motion symmetric about origin
Fig.17 in-plane circular motion The in-plane circular motion showed in Fig.5 is a
peculiar formation that can not be formed from Hill′s solution. When two spacecraft run in the same circular orbit with only difference in initial position, the in-plane relative motion is a fixed point, however, when the same condition happens in eccentric orbit, the fixed
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6
point enlarges into a circle. The larger the eccentricity, the larger the radius of circle. It is noted that the in-plane circle never encompass the origin, because the former will always be the former if two spacecraft running in the same orbit.
If two formation spacecraft run in tandem about the reference spacecraft in the same eccentric orbit, and their separations in time of perigee passage equal, then the three spacecraft make a straight line with equal separation at any time when viewed from the reference spacecraft shown in Fig.17. Opposite to Fig.16, the two formation spacecraft slow down when close to the reference spacecraft and speed up when away from it. 6.4 Asymmetric encircle formation
Hill′s solution shows that times that the formation spacecraft appears before and after (or left to and right to) the reference spacecraft equal. However for space missions that need the formation spacecraft stay longer in some direction, Hill′s solution can not provide a satisfied design because of its symmetry in trajectory. In that case, if eccentric orbit is exploited, asymmetric motion can be formed, such as Fig.6, Fig.7, Fig.10, Fig.11, Fig.13 and Fig.14. In order to illustrate the uneven motion of formation spacecraft, trajectories in those Figures are time-evenly marked by “ o” . Fig.6, Fig.7, Fig.8 and Fig.9 show a longer stay time of formation spacecraft before and after the reference spacecraft. Fig.10 and Fig.11 show a longer stay time on the right and left hand of the reference spacecraft. Fig.13 and Fig.14 show a longer stay time above and below the reference spacecraft. Furthermore, the larger the orbit eccentricity, the more highly uneven the distribution of time on trajectory. For example, the orbit eccentricity of the reference spacecraft is 0.4 in Fig.10, the ratio of passing time of the right hand to that of left hand is 3:1. When the orbit eccentricity increases to 0.8, the ratio rises to 7:1.
In conclusion, by carefully choosing the seven orbit parameters the formation spacecraft can stayed longer in any of the six directions in space, etc., before, behind, right to, left to, above and below the reference spacecraft. 6.5 Out-of-plane circle-like formation
When eei
+∆
=12sin is satisfied in Fig.13 (Fig.14), the
lower (upper) part of trajectory approaches a half circle;
when eei
−∆
=12sin is satisfied in Fig.13 (Fig.14), the
upper (lower) part of trajectory approaches a half circle. Although the variation of the projection from a half circle becomes larger with increase of orbit eccentricity,
the maximum variation will be less than 5% of the radius when orbit eccentricity increases to 0.4.
For circular reference orbit, a circular out-of plane motion can be formed by one spacecraft in near-circular orbit. While for eccentric reference orbit, such an out-of-plane motion can only be approximately made by two formation spacecraft with opposite relative motion
in Fig.13 and Fig.14. When eei
−∆
=12sin is satisfied, an
inside circle-like out-of-plane formation is formed
shown in Fig.18. When eei
+∆
=12sin is satisfied, an
outside circle-like out-of-plane formation is formed shown in Fig.19.
Actually at most 4 spacecraft can be placed on such a circle-like formation in Fig.18 and Fig.19. At any time, spacecraft 1 is symmetric with spacecraft 2 about cross-track direction, with spacecraft 3 about in-track direction and with spacecraft 4 about the origin. Compared with the circular out-of-plane motion described by Hill′s solution, it will take shorter time for two spacecraft to form such a circular formation. For example, if eccentricity of the reference orbit is 0.4, only ¼ of orbit period is needed to form an inside circle formation, and only ¾ of orbit period is needed to form an outside circle formation. In addition, formations in Fig.18 and Fig.19 encircle the reference spacecraft twice during one orbit period.
Fig.18 inside circle-like formation
Fig.19 outside circle-like formation
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American Institute of Aeronautics and Astronautics
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CONCLUSIONS
The goal of this paper is to explore the value of eccentric orbits in formation flying missions. A new set of formulae that could be used for formation design of eccentric orbits is presented. Rather than linearizing the original nonlinear differential equations, the new formulae are derived by simplifying the accurate relative motion of two random eccentric orbits under constraint of close proximity. The simplified formulae are expressed in terms of orbit parameters and are verified similar to the close-form T-H solutions, but have a more direct geometric meaning than T-H solutions. Thus they are more fitful for formation design. The new formulae are examined carefully and basic characteristics of relative motion of two close spacecraft in eccentric orbits are demonstrated clearly. By comparing with the trajectory described by Hill′s solution, it is concluded that the trajectory of relative motion is distributed unevenly in both space and time due to effect of orbit eccentricity. By fully exploiting the characteristics and carefully designing orbit parameters, five candidate formations in eccentric orbits are proposed.
REFERENCES
1Edmund M.C.Kong, David W.Miller, Raymond J.Sedwick. “ Exploiting Orbital Dynamics for Aperture Synthesis Using Distributed Satellite Systems: Applications to a Visible Earth Imager System” . The Journal of the Astronautical Sciences. Vol.47. Nos.1 and 2. January-June 1999.
2David F.Chichka,“ Satellite Clusters with Constant Apparent Distribution” , Journal of Guidance, Control, and Dynamics, Vol.24, No.1, January-February 2001.
3W. H. Clohessy and R. S. Wiltshire, the Martin Company, “ Termianl Guidance System for Satellite Rendezvous” , Journal of The Aerospace Sciences, September, 1960
4C.Sabol, R.Burns, C.McLaughlin.“ Formation Flying Design and Evolution” . AAS/AIAA Spaceflight Mechanics Meeting, Breckenridge, CO, 7-10 February 1999, AAS paper 99-126.
5Zhang Zhen Min, Xu Bin, Zhang Heng,“ Analysis of Spacecraft Relative Motion and Applicability of Hill′s Solution” , Chinese Space Science and Technology, 2002, 22 (2).
6Zhang Heng, Xu Bin, Zhang Zhen Min. “ An Analytical Approach to Analyzing Spacecraft Formation-Flying Motion” . AIAA Guidance, Navigation and Control Conference, Monterey, California, 5-8 Aug. 2002, AIAA-2002-4847.
7Robert G.Melton, “ Relative Motion of Satellites in Elliptical Orbits” , AAS 97-734.
9J. Tschauner and P.Hempel,“ Rendezvous with a Target in an Elliptical Orbits,” AIAA Journal, Vol.8, 1970, pp. 1878-1879.
10Mayer Humi, “ Fuel-Optimal Rendezvous in a General Central Force Field” , Journal of Guidance, Control and Dynamics, Vol.16, No.1, January-February 1993.
11Michael Tillerson and Jonathan P.How, “ Formation Flying Control in Eccentric orbits” , AIAA-2001-4092.
12Inalhan, G., Tillerson, M., and How, J.P., “ Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits” , Journal of Guidance, Control and Dynamics, Vol.25, No.1, January-February 2002.
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