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Dynamical behaviors and relative trajectories of the spacecraft formation flying
Aerospace Science and Technology 6 (2002) 295–301 www.elsevier.com/locate/aescte
Dynamical behaviors and relative trajectories of the spacecraft formation flying H. Baoyin a,∗ , Li Junfeng b , Gao Yunfeng b a Dept. of Mechanical Engineering, Iwate University, Morioka 020-8551, Japan b Dept. of Engineering Mechanics, Tsinghua University, Beijing, 100084, PR China
Received 31 July 2001; received in revised form 15 January 2002; accepted 15 January 2002
Abstract To describe the relative motion of spacecraft formation flying, this paper presents a method based on relative orbital elements, which is suitable to elliptical orbit with arbitrary eccentricity. The long time formation flying conditions are theoretically derived taken into account the relationship between relative motion and relative orbital elements. These conditions include that both the orbital periods of all participating spacecrafts should be the same and other relative orbital elements should be small enough. The expected relative distance of the spacecrafts would determine the magnitudes of such relative orbital elements. Theoretical analysis and numerical simulation results show that the spacecrafts with sufficient small relative orbital elements can keep long time formation flying without any active control when the orbital perturbations are not considered. The results also show that Hill’s equation is only suitable for describing a short time formation flying. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Spacecraft formation flying; Orbital elements; Relative motion; Hill’s equation
1. Introduction The innovative concept of replacing the performance capabilities of a single large spacecraft with a fleet of small satellites constrained to move in a specific formation to achieve the required performance specifications is being considered for numerous space missions [4,16]. A practical implementation of the concept relies on the control of relative distances and orientation between the participating spacecraft. The earlier efforts in spacecraft formation flying typically focused on orbital dynamics, formation design and feedback control issues related to the linearized dynamics of relative motion between a leader-follower spacecraft pair, namely Hill’s equation or C-W equation [2,4,6,8,9, 11,14,16], which was developed earlier for the spacecraft rendezvous-docking problem in circular orbit [2,11]. Unfortunately, the idea of formation and 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 neglect the influence of nonlinear effect, non-circular orbit, long formation baseline, * Corresponding author.
E-mail address: [email protected] (H. Baoyin).
long mission duration, etc, the control design based on Hill’s equations necessitate prohibitive fuel consumption and endanger formation integrity in general orbits with long mission duration. Therefore many researchers have considered the problem of relative position control [1,3,5, 7,10,12,13,15] for spacecraft formation flying by use of the nonlinear system dynamics [7,12,15]. This paper presents a method of describing the relative motion of spacecraft formation flying, which is suitable for elliptical orbit with arbitrary eccentricity, based on relative orbital elements. It has been proven that the orbital periods of each spacecraft should be the same and other relative orbital elements should be small enough in near distance formation. The results of theoretical analysis and numerical simulations show that the spacecrafts with sufficiently closed orbital elements can keep long time formation flying without any active control when the orbital perturbations are not considered.
2. Orbital elements and coordinates The spacecraft orbital dynamics and relative motion are described by the relative orbital element method in this section. The definitions of coordinate frames and orbital
1270-9638/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 7 0 - 9 6 3 8 ( 0 2 ) 0 1 1 5 1 - 3
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H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
Nomenclature OXY Z Ox y z Sxyz λ r Ω i ω a e tp f
M E θ n x, y, z
Earth’s equatorial inertial coordinate system Orbital coordinate system Spacecraft body coordinate system The vernal equinox Spacecraft position vector Right-ascension of the ascending node Inclination Argument of perigee Semi-major axis Eccentricity Time of across ascending node True anomaly
AXx AXx
2
Ax1 X
Mean anomaly Eccentric anomaly ω + f , argument of latitude Mean motion Relative displacements in body coordinate of the leader spacecraft Transformation matrix from coordinate OXY Z to Ox y z Transformation matrix from coordinate (Sxyz)2 to OXY Z Transformation matrix from coordinate OXY Z to (Sxyz)1 .
Definitions of orbital elements Ω, i, ω, a, e and tp are same as in common textbooks, and they can completely define the position of a spacecraft in the inertial space. In the orbital plane, Keplerian equation is given by r=
a(1 − e2 ) , 1 + e cos f
(4)
in other coordinate systems the spacecraft’s position vector r can be expressed as {r}(x ) = [r 0 0]T
and {r}(X) = AXx {r}(x ) ,
(5)
where {r}(x ) and {r}(X) denote the vectors expressed in coordinates Ox y z and OXY Z, respectively. Consequently the transformation matrix AXx can be written as AXx = A(Ω)A(i)A(θ ).
Fig. 1. Coordinate frames and orbital elements.
elements are shown as Fig. 1. The origin of coordinate OXY Z is at the earth’s center, Z-axis points in the direction of the North Pole of earth, X-axis lies in the equatorial plane and points in the direction of the vernal equinox λ, and Y -axis is normal to XZ plane and forms the right handed frame. The origin of coordinate Ox y z is also at the earth’s center, and x -axis points to the mass center of the spacecraft, whose unit vector is defined as r , (1) x = |r| z is normal to the orbital plane and its direction is the same as the direction of the orbital angle velocity vector, and the unit vector of y defined as (2), i.e. Ox y z is a right handed frame: y =
z × x . |z × x |
(2)
The origin of body coordinate Sxyz is at the spacecraft’s mass center and its unit vectors are x = x,
y = y,
z = z .
(3)
(6)
Definitions of matrices A(Ω), A(i) and A(θ ) are given in (A.1a) of Appendix A.
3. Theorems of spacecraft’s relative motion In this section a pair of spacecrafts are taken as leader (subscript 1) and follower spacecraft (subscript 2), respectively. Here orbital elements of spacecraft are used to describe motion in body coordinate of the leader spacecraft. Thus the relative displacement in the coordinates (Sxyz)1 can be written as (7) { r}(1) = Ax1 X AXx2 {r 2 }(2 ) − {r 1 }(1 ) where Ax X = ATXx = A(−θ1 )A(−i1)A(−Ω1 ). 1
(8)
1
The complete form of matrix Ax X AXx is listed in (A.1b). 1 2 Near distance formation flying is considered in this paper, it is to say that the orbital elements of leader and follower spacecrafts are very close, i.e. Ω = Ω2 − Ω1 , i = i2 −
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
i1 , and θ = θ2 − θ1 are all small. In this way, through simplifying matrix of (A.1b), Eq. (7) can be simplified as r2 − r1 x = r2 sin θ + r2 Ω cos i1 cos θ . (9) y r2 i sin θ2 − r2 Ω sin i1 cos θ2 z (1) From Eq. (9), several important results on the relative motion of formation flying can be achieved.
297
Now going back to the Hill’s equation. For the formation flying, if the leader spacecraft is moving in a circular orbit, the relative dynamics are given by x¨ − 2ny˙ − n2 x + n2 [g(x, y, z, r)(y + r) − r] = Fx , (16) y¨ + 2nx˙ − n2 y + 2g(x, y, z, r)x = Fy , z¨ + n2 g(x, y, z, r)z = Fz ,
Theorem 1. In the near distance long time formation flying, the orbital periods of the participating spacecraft should be the same.
where Fx , Fy , Fz are components of external force, perturbations and controls along the x, y, z axes, respectively, and
2y (x 2 + y 2 + z2 ) −3/2 + g(x, y, z, r) = 1 + . r r2
Proof. Compared with the distance from earth center to the formation, the distance from the leader spacecraft to the follower d is very small in the formation flying. Thus by Eq. (9), it can be obtained that
If x, y, z r, Hill’s equation can be derived from Eq. (16) x¨ − 2ny˙ − 3n2 x = Fx , (17) y¨ + 2nx˙ = Fy , z¨ + n2 z = Fz .
|sin θ + Ω cos i cos θ |
d . r2
(10)
This means that θ should not be allowed to grow continuously along with the time. Otherwise d would be so large that the spacecraft could not be closely formatted. On the other hand, according to the relationship among M, n, tp and E, there is (E − sin E) = M = (n2 − n1 )t + n2 tp2 − n1 tp1 = (n2 − n1 )t + c
(11)
From the solution it is clear that if y˙0 = −2nx0 the relative motion is not periodic. Therefore the initial condition for a periodic motion is
(12)
y˙0 = −2nx0.
where c = n2 tp2 − n1 tp1 is a constant, thus | E| |(n2 − n1 )t + c| − 2.
In addition, with the relations θ = ω + f and f = E + Θ, there is | θ| |(n2 − n1 )t + c| − 2 − |Θ| − | ω|,
(13)
where 0 Θ π/2 . Eq. (13) shows that, if n2 = n1 , θ would continuously grow with the time. This result conflicts with condition (10), it is to say that condition n1 = n2 (or a1 = a2 ) has to be satisfied if a long time near distance formation is expected. Theorem 2. If the orbital periods of the participating spacecraft are not the same, the order of maximum relative distance is as same as the distance from earth center to the formation. Proof. According to Eq. (9), there is |y| = |r2 sin θ + r2 Ω cos i1 cos θ | = |r2 | · |sin θ + Ω cos i1 cos θ| = |r2 | · |sin( θ + ϕ)| · 1 + ( Ω cos i1 )2 .
(14)
And, if n1 = n2 , θ will grow endlessly with time, thus |y|max = |r2 | · 1 + ( Ω cos i1 )2 , (15) x and z are similar.
When the external forces are not considered, the solution is given by
x = 2 x0 + y˙n0 − 2ny˙0 + 3x0 cos(nt) + x˙n0 sin(nt), y = y − 2x˙0 + 2 2y˙0 + 3x sin(nt) 0 0 n n (18)
x˙0 + 2 cos(nt) − 3( y ˙ + 2nx )t, 0 0 n z = z˙n0 sin(nt) + z0 cos(nt).
(19)
This is not true according to Theorem 3. And according to Theorem 2, the relative distance cannot endlessly grow with the time. Theorem 3. When the eccentricities of two spacecrafts are small, the relative velocity of the formation flying is approximately zero, viz. y˙ ≈ 0.
(20)
Proof. When e1 and e2 are small, Eqs. (9) can be approximately simplified and expressed by mean anomaly M that (see (2) of Appendix A) x − e cos M1 ω − n tp + Ω cos i1 + 2 e sin M1 a (21) = y − Ω sin i1 cos(ω2 + M2 ) + i sin(ω2 + M2 ) z (1) and, getting relative velocity e sin M + θ + Ω cos i vx 1 1 µ e cos M1 vy = . (22) a Ω sin i1 cos θ2 − i sin θ2 v z
(OXY Z)
On the other hand, there is the relative motion relation d˜ d (r 2 − r 1 ) = (r 2 − r 1 ) + n × (r 2 − r 1 ), dt dt
(23)
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Table 1 Corresponding orbital elements of Figs. 2(a)–4, 5(b)
a1 (km) e1 Ω1 (deg) ω1 (deg) i1 (deg) tp1 (s) da (km) de dΩ (rad) dω (rad) di (rad) dtp (s)
Fig. 2(a)
Fig. 2(b)
Fig. 2(c)
Fig. 2(d)
Fig. 3
Fig. 4
Fig. 5(b)
7000 0.0 0.0 0.0 0.0 0.0 100 0.0 0.0 0.0 0.0 0.0
7000 0.0 0.0 0.0 0.0 0.0 100 0.05 0.0 0.0 0.0 0.0
7000 0.05 0.0 0.0 0.0 0.0 100 0.01 0.0 0.0 0.0 0.0
7000 0.0 0.0 0.0 0.0 0.0 0.0 0.001 0.0 0.0 0.0 0.0
7000 0.001 0.0 0.0 80.0 0.0 0.0 0.001 0.001 0.001 0.001 0.001
17000 0.2 0.0 0.0 80.0 0.0 0.0 0.0 0.0 0.0 0.001 1.0
7000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 −5.39 × 10−4 0.0 0.5
˜
where dtd denotes relative derivative in leader’s body coordinates. Thus x˙ vx2 − vx1 0 −n 0 x y˙ = vy2 − vy1 + n 0 0 y . (24) z˙ 0 0 0 z vz2 − vz1 (OXY Z) Comparing Eqs. (21) and (22) with Eq. (24), it can be found that y˙ ≈ −nx + nx ≈ 0.
(25)
Theorem 3 shows that initial condition (20) should be hold in the near distance formation, it conflicts with Hill’s equation’s periodic condition (19). Theorem 4. In small eccentricity orbit, the relative distance is directly proportional to both the semi-major axis and the relative orbital elements.
Table 2 Initial values of Fig. 5(a) and Fig. 5(b). Here n = −0.001078 (1/s) Fig. 5(a) Fig. 5(b)
y0 (km)
y˙0 (km/s)
x0 (km)
x˙0 (km/s)
−7.54864 −7.54864
0.0 2 × (−4.3845 × 10−6 )
−0.0041 −0.0041
0.0 0.0
It is obvious that 2 y x + r1 2 + = 1. r2 r2
(27)
Hence the relative trajectory is a moving circle placed in a circular band when both r1 and r2 are variable. If r1 is a constant and b r2 a, the relative trajectory is placed in a band between two circles with radius b and a. When θ is small, it could not form a large circle band with radius r2 . If leader spacecraft is in a circular orbit, i.e. e1 = 0, and θ is small, using Eq. (A.5), Eq. (27) can be approximately rewritten as
Proof. According to Eq. (21), it is straightforward.
(y − a( ω + M))2 x2 + = 1. (2ae2)2 (ae2)2
4. Simulations
This is an ellipse, of which semi-major axis and semi-minor axis are keeping a fixed radio of 2 : 1. In addition, if e2 = 0, then
In this section various dynamical behaviors and formation patterns of two spacecraft formation flying will be discussed. Here subscript 1 denotes the leader spacecraft’s orbital elements, subscript 2 does the follower’s, and d∗ = ∗1 − ∗2 denotes the difference of corresponding orbital elements of leader and follower. Simulations used the precise mathematical model given in Eq. (7). Each figures corresponding orbital elements are listed in Table 1 and initial values for Hill equation are placed in Table 2. Simulation 1. Two spacecrafts are on the same orbital plane, i.e. Ω = 0, i = 0. According to (A.1) of Appendix A, there is x r2 cos θ − r1 . (26) = y r2 sin θ z (1) 0
x = 0,
y = a θ.
(28)
(29)
The precise solution is θ θ cos . 2 2 When θ is small, the above solution can be simplified as x = a(1 − cos θ ),
y = 2a sin
θ θ θ 2 ≈ 0, y = 2a sin cos = a θ. 2 2 2 Fig. 2(a), (b) and (c) show that the formation flying is not succeeded, since a1 = a2 . In Fig. 2(a), the relative trajectory goes back to initial position rounding about after
x =a
1 45.51 circles (N = n2n−n = a13 /( a23 − a13 ) ≈ 45.51, 1 where a1 = 7000 Km and a2 = 7100 Km). Figs. 2(b) and 2(c) are circular bands. Fig. 2(d) shows a formation flying case, since where a1 = a2 .
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
(a)
(b)
(c)
(d)
299
Fig. 2. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xy plane. (c) The relative trajectory mapped into xy plane. (d) The relative trajectory mapped into xy plane.
(a)
(b)
Fig. 3. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xz plane.
Simulation 2. Here the leader spacecraft is considered on the elliptical orbits with both small and large eccentricities. Fig. 3 shows a formation flying case. It shows that the relative trajectory is an approximate ellipse. Its mapping in xy plane (Fig. 3(a)) is a small ellipse whose center is different from the coordinate origin. Fig. 3(b) shows that
it is a rotated ellipse in xz plane. In fact, the above two conclusions can also be obtained from Eq. (21), which are
x ea
2
y − ( ω − n tp + Ω cos i1 ) + 2 ea
2 =1
(30)
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H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
(a)
(b)
Fig. 4. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xz plane.
(a)
(b)
Fig. 5. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xy plane.
and
sin i1 z − Ω e x 2 x 2 + = 1. (31) ea ia Fig. 4 is a very interesting formation with a large eccentricity orbit, where the follower spacecraft is always following after its leader. The mappings of relative trajectory in each coordinates plane are no longer ellipses. Simulation 3. Comparing with Hill’s equation. For convenience, a simple formation is considered here. Two spacecrafts are moving on a same circular orbit, and its orbital elements are given in Fig. 5(a) in Table 1. In fact, it is clear that the relative trajectory is a point in the coordinate planes, i.e. the relative distance is invariable, where initial condition is holding as y˙0 = −2nx0 (but actually formatted case), the curves are periodically divergent, shown as Fig. 5(a). When the initial condition is given as y˙0 = −2nx0 , that is a not formatted case in Fig. 5(b), but Hill’s equation give a periodic curve. Here the curves in Figs. 5(a) and 5(b) are derived from Hill’s equation. The results show that in the unformatted case the relative motion of Hill’s equation is a rounding trajectory, but in the formatted case it is a divergent trajectory.
5. Conclusions Hill’s equation is not relevant to the formation flying when the leader spacecraft is in an elliptical orbit. Furthermore, even if in a circular orbit it can only be used in short time formation flying. The proposed method is available to arbitrary elliptical orbits and longtime mission. The orbital periods of the participating spacecrafts in longtime formation flying should be equal; otherwise the relative distance would increase up to distances larger than the semi-major axis. Moreover, the relative orbital elements should be small, and have the same order as d/a, where d is the expecting formation distance and a is the semi-major axis. When the orbital perturbations are not considered, the spacecrafts with sufficiently close orbital elements can keep long time formation flying without any active control. Appendix A (1) Definitions of transformation matrices cos Ω − sin Ω 0 A(Ω) = sin Ω cos Ω 0 , 0 0 1
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
1 0 0 A(i) = 0 cos i − sin i and 0 sin i cos i cos θ − sin θ 0 A(θ ) = sin θ cos θ 0 . 0 0 1
(A.1a)
Ax X AXx = ATXx AXx = 1 2 2 1 Cθ1 (Cθ2 C Ω + Ci2 Sθ2 S Ω) + Sθ1 (Si1 Si2 Sθ2 + Ci1 (Ci2 C ΩSθ2 − Cθ2 S Ω)) − Cθ2 (C ΩSθ1 + Ci1 Cθ1 S Ω) + Sθ2 (Ci1 Ci2 Cθ1 C Ω + Cθ1 Si1 Si2 − Ci2 Sθ1 S Ω) − Ci2 C ΩSi1 Sθ2 + Ci1 Si2 Sθ2 + Cθ2 Si1 S Ω − Cθ1 C ΩSθ2 + Cθ2 (Si1 Si2 Sθ1 + Ci2 Cθ1 S Ω) + Ci1 Sθ1 (Ci2 Cθ2 C Ω + Sθ2 S Ω) Cθ1 Cθ2 Si1 Si2 + Sθ1 (C ΩSθ2 − Ci2 Cθ2 S Ω) + Ci1 Cθ1 (Ci2 Cθ2 C Ω + Sθ2 S Ω) − Ci2 Cθ2 C ΩSi1 + Ci1 Cθ2 Si2 − Si1 Sθ2 S Ω Ci2 Si1 Sθ1 − Si2 (Ci1 C ΩSθ1 + Cθ1 S Ω) Ci2 Cθ1 Si1 + Si2 (−Ci1 Cθ1 C Ω + Sθ1 S Ω) , Ci1 Ci2 + C ΩSi1 Si2 3×3
(A.1b)
where Ω = Ω1 − Ω2 , C and S represent cosine and sine functions respectively. (2) True anomaly f and mean anomaly M have a relation 5 e2 sin M + e2 sin 2M f = M + 2e − 4 4 13 + e3 sin 3M + · · · , (A.2) 12 so, when e is small, we can truncate the high order terms of the right hand of Eq. (A.2), and can get e2 f ≈ M + 2e − sin M = M + 2e sin M. (A.3) 4 Substituting Eq. (A.3) into Eq. (4), and considering a1 = a2 = a, it yields a1 (1 − e12 ) ≈ a(1 − e1 cos M1 ), 1 + e1 cos f1 a2 (1 − e22 ) r2 = ≈ a(1 − e2 cos M2 ). 1 + e2 cos f2
r1 =
(A.4)
Substituting Eq. (A.4) into Eq. (9), we get Eqs. (21) and (22). (3) When e1 = 0, θ 1, e2 1, and if formatting condition a1 = a2 = a is satisfied, according to (26), there is
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a(1−e2 ) 1+e2 cos f2 − a r2 cos θ − r1 x = = a(1−e2 ) θ r2 sin θ y 1+e2 cos f2 0 z (1) 0 −ae2 cos f2 a(1 − e2 cos f2 ) − a a θ = a( ω + f ) = 0 0 −ae2 cos M2 (A.5) = a( ω + M + 2e2 sin M2 ) . 0 References [1] R.W. Beard, F.Y. Hadaegh, Finite thrust control for satellite formation flying with state constraints, in: Proceeding of the American Control Conference, San Diego, CA, 1999, pp. 4383–4387. [2] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satellite rendezvous, J. Aerospace Sci. 27 (9) (1960) 653–658. [3] M.S. de Queiroz, V. Kapla, Q. Yan, Adaptive nonlinear control of satellite formation flying, in: Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, OR, 1999, pp. 1596– 1604. [4] V. Kapila, A. Sparks, J.M. Buffington, Q. Yan, Spacecraft formation flying: dynamics and control, in: Proceeding of the American Control Conference, San Diego, CA, 1999, pp. 4137–4141. [5] J. Lawton, R.W. Beard, F.Y. Hadaegh, An adaptive control approach to satellite formation flying with relative distance constraints, in: Proceeding of the American Control Conference, San Diego, CA, 1999, pp. 1545–1549. [6] C.L. Leonard, W.M. Hollister, E.V. Bergmann, Orbital formation keeping with differential drag, J. Guidance Control Dynam. 12 (1) (1989) 108–113. [7] M. Mesbahi, F.Y. Hadaegh, Formation flying control of multiple spacecraft via graphs, matrix inequalities, and switching, J. Guidance Control Dynam. 24 (2) (2001) 369–377. [8] D.C. Redding, N.J. Adams et al., Linear-quadratic stationkeeping for the STS orbit, J. Guidance Control Dynam. 12 (1) (1989) 248–255. [9] C. Sabol, R. Burns, C. McLaughlin, Formation flying design and evolution, in: Proceeding of the AAS/AIAA Space Flight Mechanics Meeting, Breckenridge, CO, 1999, pp. 99–121. [10] C.J. Schumacher, R. Kumar, Adaptive control of UAVs in closecoupled formation flight, in: Proceeding of the American Control Conference, Chicago, IL, 2000, pp. 849–853. [11] R.H. Vassar, R.B. Sherwood, Formationkeeping for a pair of satellite in a circular orbit, J. Guidance Control Dynam. 8 (2) (1985) 235–242. [12] P.K.C. Wang, F.Y. Hadaegh, Coordination and control of multiple microspacecraft moving in formation, J. Astronautical Sci. 44 (3) (1996) 315–355. [13] G.Q. Xing, S.A. Parvez, Implementation of autonomous GPS guidance and control for the spacecraft formation flying, in: Proceeding of the American Control Conference, San Diego, CA, 1999, pp. 4163–4167. [14] Q. Yan, V. Kapila, A.G. Sparks, Pulse-based periodic control for spacecraft formation flying, in: Proceeding of the American Control Conference, Chicago, IL, 2000, pp. 374–378. [15] Q. Yan, G. Yang, V. Kapila, M.S. de Queiroz, Nonlinear dynamics and output feedback control of multiple spacecraft in elliptical orbits, in: Proceeding of the American Control Conference, Chicago, IL, 2000, pp. 839–843. [16] R.K. Yedavalli, A.G. Sparks, Satellite formation flying control design based on hybrid control system stability analysis, in: Proceeding of the American Control Conference, Chicago, IL, 2000, pp. 2210–2214.
Dynamical behaviors and relative trajectories of the spacecraft formation flying H. Baoyin a,∗ , Li Junfeng b , Gao Yunfeng b a Dept. of Mechanical Engineering, Iwate University, Morioka 020-8551, Japan b Dept. of Engineering Mechanics, Tsinghua University, Beijing, 100084, PR China
Received 31 July 2001; received in revised form 15 January 2002; accepted 15 January 2002
Abstract To describe the relative motion of spacecraft formation flying, this paper presents a method based on relative orbital elements, which is suitable to elliptical orbit with arbitrary eccentricity. The long time formation flying conditions are theoretically derived taken into account the relationship between relative motion and relative orbital elements. These conditions include that both the orbital periods of all participating spacecrafts should be the same and other relative orbital elements should be small enough. The expected relative distance of the spacecrafts would determine the magnitudes of such relative orbital elements. Theoretical analysis and numerical simulation results show that the spacecrafts with sufficient small relative orbital elements can keep long time formation flying without any active control when the orbital perturbations are not considered. The results also show that Hill’s equation is only suitable for describing a short time formation flying. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Spacecraft formation flying; Orbital elements; Relative motion; Hill’s equation
1. Introduction The innovative concept of replacing the performance capabilities of a single large spacecraft with a fleet of small satellites constrained to move in a specific formation to achieve the required performance specifications is being considered for numerous space missions [4,16]. A practical implementation of the concept relies on the control of relative distances and orientation between the participating spacecraft. The earlier efforts in spacecraft formation flying typically focused on orbital dynamics, formation design and feedback control issues related to the linearized dynamics of relative motion between a leader-follower spacecraft pair, namely Hill’s equation or C-W equation [2,4,6,8,9, 11,14,16], which was developed earlier for the spacecraft rendezvous-docking problem in circular orbit [2,11]. Unfortunately, the idea of formation and 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 neglect the influence of nonlinear effect, non-circular orbit, long formation baseline, * Corresponding author.
E-mail address: [email protected] (H. Baoyin).
long mission duration, etc, the control design based on Hill’s equations necessitate prohibitive fuel consumption and endanger formation integrity in general orbits with long mission duration. Therefore many researchers have considered the problem of relative position control [1,3,5, 7,10,12,13,15] for spacecraft formation flying by use of the nonlinear system dynamics [7,12,15]. This paper presents a method of describing the relative motion of spacecraft formation flying, which is suitable for elliptical orbit with arbitrary eccentricity, based on relative orbital elements. It has been proven that the orbital periods of each spacecraft should be the same and other relative orbital elements should be small enough in near distance formation. The results of theoretical analysis and numerical simulations show that the spacecrafts with sufficiently closed orbital elements can keep long time formation flying without any active control when the orbital perturbations are not considered.
2. Orbital elements and coordinates The spacecraft orbital dynamics and relative motion are described by the relative orbital element method in this section. The definitions of coordinate frames and orbital
1270-9638/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 7 0 - 9 6 3 8 ( 0 2 ) 0 1 1 5 1 - 3
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H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
Nomenclature OXY Z Ox y z Sxyz λ r Ω i ω a e tp f
M E θ n x, y, z
Earth’s equatorial inertial coordinate system Orbital coordinate system Spacecraft body coordinate system The vernal equinox Spacecraft position vector Right-ascension of the ascending node Inclination Argument of perigee Semi-major axis Eccentricity Time of across ascending node True anomaly
AXx AXx
2
Ax1 X
Mean anomaly Eccentric anomaly ω + f , argument of latitude Mean motion Relative displacements in body coordinate of the leader spacecraft Transformation matrix from coordinate OXY Z to Ox y z Transformation matrix from coordinate (Sxyz)2 to OXY Z Transformation matrix from coordinate OXY Z to (Sxyz)1 .
Definitions of orbital elements Ω, i, ω, a, e and tp are same as in common textbooks, and they can completely define the position of a spacecraft in the inertial space. In the orbital plane, Keplerian equation is given by r=
a(1 − e2 ) , 1 + e cos f
(4)
in other coordinate systems the spacecraft’s position vector r can be expressed as {r}(x ) = [r 0 0]T
and {r}(X) = AXx {r}(x ) ,
(5)
where {r}(x ) and {r}(X) denote the vectors expressed in coordinates Ox y z and OXY Z, respectively. Consequently the transformation matrix AXx can be written as AXx = A(Ω)A(i)A(θ ).
Fig. 1. Coordinate frames and orbital elements.
elements are shown as Fig. 1. The origin of coordinate OXY Z is at the earth’s center, Z-axis points in the direction of the North Pole of earth, X-axis lies in the equatorial plane and points in the direction of the vernal equinox λ, and Y -axis is normal to XZ plane and forms the right handed frame. The origin of coordinate Ox y z is also at the earth’s center, and x -axis points to the mass center of the spacecraft, whose unit vector is defined as r , (1) x = |r| z is normal to the orbital plane and its direction is the same as the direction of the orbital angle velocity vector, and the unit vector of y defined as (2), i.e. Ox y z is a right handed frame: y =
z × x . |z × x |
(2)
The origin of body coordinate Sxyz is at the spacecraft’s mass center and its unit vectors are x = x,
y = y,
z = z .
(3)
(6)
Definitions of matrices A(Ω), A(i) and A(θ ) are given in (A.1a) of Appendix A.
3. Theorems of spacecraft’s relative motion In this section a pair of spacecrafts are taken as leader (subscript 1) and follower spacecraft (subscript 2), respectively. Here orbital elements of spacecraft are used to describe motion in body coordinate of the leader spacecraft. Thus the relative displacement in the coordinates (Sxyz)1 can be written as (7) { r}(1) = Ax1 X AXx2 {r 2 }(2 ) − {r 1 }(1 ) where Ax X = ATXx = A(−θ1 )A(−i1)A(−Ω1 ). 1
(8)
1
The complete form of matrix Ax X AXx is listed in (A.1b). 1 2 Near distance formation flying is considered in this paper, it is to say that the orbital elements of leader and follower spacecrafts are very close, i.e. Ω = Ω2 − Ω1 , i = i2 −
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
i1 , and θ = θ2 − θ1 are all small. In this way, through simplifying matrix of (A.1b), Eq. (7) can be simplified as r2 − r1 x = r2 sin θ + r2 Ω cos i1 cos θ . (9) y r2 i sin θ2 − r2 Ω sin i1 cos θ2 z (1) From Eq. (9), several important results on the relative motion of formation flying can be achieved.
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Now going back to the Hill’s equation. For the formation flying, if the leader spacecraft is moving in a circular orbit, the relative dynamics are given by x¨ − 2ny˙ − n2 x + n2 [g(x, y, z, r)(y + r) − r] = Fx , (16) y¨ + 2nx˙ − n2 y + 2g(x, y, z, r)x = Fy , z¨ + n2 g(x, y, z, r)z = Fz ,
Theorem 1. In the near distance long time formation flying, the orbital periods of the participating spacecraft should be the same.
where Fx , Fy , Fz are components of external force, perturbations and controls along the x, y, z axes, respectively, and
2y (x 2 + y 2 + z2 ) −3/2 + g(x, y, z, r) = 1 + . r r2
Proof. Compared with the distance from earth center to the formation, the distance from the leader spacecraft to the follower d is very small in the formation flying. Thus by Eq. (9), it can be obtained that
If x, y, z r, Hill’s equation can be derived from Eq. (16) x¨ − 2ny˙ − 3n2 x = Fx , (17) y¨ + 2nx˙ = Fy , z¨ + n2 z = Fz .
|sin θ + Ω cos i cos θ |
d . r2
(10)
This means that θ should not be allowed to grow continuously along with the time. Otherwise d would be so large that the spacecraft could not be closely formatted. On the other hand, according to the relationship among M, n, tp and E, there is (E − sin E) = M = (n2 − n1 )t + n2 tp2 − n1 tp1 = (n2 − n1 )t + c
(11)
From the solution it is clear that if y˙0 = −2nx0 the relative motion is not periodic. Therefore the initial condition for a periodic motion is
(12)
y˙0 = −2nx0.
where c = n2 tp2 − n1 tp1 is a constant, thus | E| |(n2 − n1 )t + c| − 2.
In addition, with the relations θ = ω + f and f = E + Θ, there is | θ| |(n2 − n1 )t + c| − 2 − |Θ| − | ω|,
(13)
where 0 Θ π/2 . Eq. (13) shows that, if n2 = n1 , θ would continuously grow with the time. This result conflicts with condition (10), it is to say that condition n1 = n2 (or a1 = a2 ) has to be satisfied if a long time near distance formation is expected. Theorem 2. If the orbital periods of the participating spacecraft are not the same, the order of maximum relative distance is as same as the distance from earth center to the formation. Proof. According to Eq. (9), there is |y| = |r2 sin θ + r2 Ω cos i1 cos θ | = |r2 | · |sin θ + Ω cos i1 cos θ| = |r2 | · |sin( θ + ϕ)| · 1 + ( Ω cos i1 )2 .
(14)
And, if n1 = n2 , θ will grow endlessly with time, thus |y|max = |r2 | · 1 + ( Ω cos i1 )2 , (15) x and z are similar.
When the external forces are not considered, the solution is given by
x = 2 x0 + y˙n0 − 2ny˙0 + 3x0 cos(nt) + x˙n0 sin(nt), y = y − 2x˙0 + 2 2y˙0 + 3x sin(nt) 0 0 n n (18)
x˙0 + 2 cos(nt) − 3( y ˙ + 2nx )t, 0 0 n z = z˙n0 sin(nt) + z0 cos(nt).
(19)
This is not true according to Theorem 3. And according to Theorem 2, the relative distance cannot endlessly grow with the time. Theorem 3. When the eccentricities of two spacecrafts are small, the relative velocity of the formation flying is approximately zero, viz. y˙ ≈ 0.
(20)
Proof. When e1 and e2 are small, Eqs. (9) can be approximately simplified and expressed by mean anomaly M that (see (2) of Appendix A) x − e cos M1 ω − n tp + Ω cos i1 + 2 e sin M1 a (21) = y − Ω sin i1 cos(ω2 + M2 ) + i sin(ω2 + M2 ) z (1) and, getting relative velocity e sin M + θ + Ω cos i vx 1 1 µ e cos M1 vy = . (22) a Ω sin i1 cos θ2 − i sin θ2 v z
(OXY Z)
On the other hand, there is the relative motion relation d˜ d (r 2 − r 1 ) = (r 2 − r 1 ) + n × (r 2 − r 1 ), dt dt
(23)
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Table 1 Corresponding orbital elements of Figs. 2(a)–4, 5(b)
a1 (km) e1 Ω1 (deg) ω1 (deg) i1 (deg) tp1 (s) da (km) de dΩ (rad) dω (rad) di (rad) dtp (s)
Fig. 2(a)
Fig. 2(b)
Fig. 2(c)
Fig. 2(d)
Fig. 3
Fig. 4
Fig. 5(b)
7000 0.0 0.0 0.0 0.0 0.0 100 0.0 0.0 0.0 0.0 0.0
7000 0.0 0.0 0.0 0.0 0.0 100 0.05 0.0 0.0 0.0 0.0
7000 0.05 0.0 0.0 0.0 0.0 100 0.01 0.0 0.0 0.0 0.0
7000 0.0 0.0 0.0 0.0 0.0 0.0 0.001 0.0 0.0 0.0 0.0
7000 0.001 0.0 0.0 80.0 0.0 0.0 0.001 0.001 0.001 0.001 0.001
17000 0.2 0.0 0.0 80.0 0.0 0.0 0.0 0.0 0.0 0.001 1.0
7000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 −5.39 × 10−4 0.0 0.5
˜
where dtd denotes relative derivative in leader’s body coordinates. Thus x˙ vx2 − vx1 0 −n 0 x y˙ = vy2 − vy1 + n 0 0 y . (24) z˙ 0 0 0 z vz2 − vz1 (OXY Z) Comparing Eqs. (21) and (22) with Eq. (24), it can be found that y˙ ≈ −nx + nx ≈ 0.
(25)
Theorem 3 shows that initial condition (20) should be hold in the near distance formation, it conflicts with Hill’s equation’s periodic condition (19). Theorem 4. In small eccentricity orbit, the relative distance is directly proportional to both the semi-major axis and the relative orbital elements.
Table 2 Initial values of Fig. 5(a) and Fig. 5(b). Here n = −0.001078 (1/s) Fig. 5(a) Fig. 5(b)
y0 (km)
y˙0 (km/s)
x0 (km)
x˙0 (km/s)
−7.54864 −7.54864
0.0 2 × (−4.3845 × 10−6 )
−0.0041 −0.0041
0.0 0.0
It is obvious that 2 y x + r1 2 + = 1. r2 r2
(27)
Hence the relative trajectory is a moving circle placed in a circular band when both r1 and r2 are variable. If r1 is a constant and b r2 a, the relative trajectory is placed in a band between two circles with radius b and a. When θ is small, it could not form a large circle band with radius r2 . If leader spacecraft is in a circular orbit, i.e. e1 = 0, and θ is small, using Eq. (A.5), Eq. (27) can be approximately rewritten as
Proof. According to Eq. (21), it is straightforward.
(y − a( ω + M))2 x2 + = 1. (2ae2)2 (ae2)2
4. Simulations
This is an ellipse, of which semi-major axis and semi-minor axis are keeping a fixed radio of 2 : 1. In addition, if e2 = 0, then
In this section various dynamical behaviors and formation patterns of two spacecraft formation flying will be discussed. Here subscript 1 denotes the leader spacecraft’s orbital elements, subscript 2 does the follower’s, and d∗ = ∗1 − ∗2 denotes the difference of corresponding orbital elements of leader and follower. Simulations used the precise mathematical model given in Eq. (7). Each figures corresponding orbital elements are listed in Table 1 and initial values for Hill equation are placed in Table 2. Simulation 1. Two spacecrafts are on the same orbital plane, i.e. Ω = 0, i = 0. According to (A.1) of Appendix A, there is x r2 cos θ − r1 . (26) = y r2 sin θ z (1) 0
x = 0,
y = a θ.
(28)
(29)
The precise solution is θ θ cos . 2 2 When θ is small, the above solution can be simplified as x = a(1 − cos θ ),
y = 2a sin
θ θ θ 2 ≈ 0, y = 2a sin cos = a θ. 2 2 2 Fig. 2(a), (b) and (c) show that the formation flying is not succeeded, since a1 = a2 . In Fig. 2(a), the relative trajectory goes back to initial position rounding about after
x =a
1 45.51 circles (N = n2n−n = a13 /( a23 − a13 ) ≈ 45.51, 1 where a1 = 7000 Km and a2 = 7100 Km). Figs. 2(b) and 2(c) are circular bands. Fig. 2(d) shows a formation flying case, since where a1 = a2 .
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
(a)
(b)
(c)
(d)
299
Fig. 2. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xy plane. (c) The relative trajectory mapped into xy plane. (d) The relative trajectory mapped into xy plane.
(a)
(b)
Fig. 3. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xz plane.
Simulation 2. Here the leader spacecraft is considered on the elliptical orbits with both small and large eccentricities. Fig. 3 shows a formation flying case. It shows that the relative trajectory is an approximate ellipse. Its mapping in xy plane (Fig. 3(a)) is a small ellipse whose center is different from the coordinate origin. Fig. 3(b) shows that
it is a rotated ellipse in xz plane. In fact, the above two conclusions can also be obtained from Eq. (21), which are
x ea
2
y − ( ω − n tp + Ω cos i1 ) + 2 ea
2 =1
(30)
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H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
(a)
(b)
Fig. 4. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xz plane.
(a)
(b)
Fig. 5. (a) The relative trajectory mapped into xy plane. (b) The relative trajectory mapped into xy plane.
and
sin i1 z − Ω e x 2 x 2 + = 1. (31) ea ia Fig. 4 is a very interesting formation with a large eccentricity orbit, where the follower spacecraft is always following after its leader. The mappings of relative trajectory in each coordinates plane are no longer ellipses. Simulation 3. Comparing with Hill’s equation. For convenience, a simple formation is considered here. Two spacecrafts are moving on a same circular orbit, and its orbital elements are given in Fig. 5(a) in Table 1. In fact, it is clear that the relative trajectory is a point in the coordinate planes, i.e. the relative distance is invariable, where initial condition is holding as y˙0 = −2nx0 (but actually formatted case), the curves are periodically divergent, shown as Fig. 5(a). When the initial condition is given as y˙0 = −2nx0 , that is a not formatted case in Fig. 5(b), but Hill’s equation give a periodic curve. Here the curves in Figs. 5(a) and 5(b) are derived from Hill’s equation. The results show that in the unformatted case the relative motion of Hill’s equation is a rounding trajectory, but in the formatted case it is a divergent trajectory.
5. Conclusions Hill’s equation is not relevant to the formation flying when the leader spacecraft is in an elliptical orbit. Furthermore, even if in a circular orbit it can only be used in short time formation flying. The proposed method is available to arbitrary elliptical orbits and longtime mission. The orbital periods of the participating spacecrafts in longtime formation flying should be equal; otherwise the relative distance would increase up to distances larger than the semi-major axis. Moreover, the relative orbital elements should be small, and have the same order as d/a, where d is the expecting formation distance and a is the semi-major axis. When the orbital perturbations are not considered, the spacecrafts with sufficiently close orbital elements can keep long time formation flying without any active control. Appendix A (1) Definitions of transformation matrices cos Ω − sin Ω 0 A(Ω) = sin Ω cos Ω 0 , 0 0 1
H. Baoyin et al. / Aerospace Science and Technology 6 (2002) 295–301
1 0 0 A(i) = 0 cos i − sin i and 0 sin i cos i cos θ − sin θ 0 A(θ ) = sin θ cos θ 0 . 0 0 1
(A.1a)
Ax X AXx = ATXx AXx = 1 2 2 1 Cθ1 (Cθ2 C Ω + Ci2 Sθ2 S Ω) + Sθ1 (Si1 Si2 Sθ2 + Ci1 (Ci2 C ΩSθ2 − Cθ2 S Ω)) − Cθ2 (C ΩSθ1 + Ci1 Cθ1 S Ω) + Sθ2 (Ci1 Ci2 Cθ1 C Ω + Cθ1 Si1 Si2 − Ci2 Sθ1 S Ω) − Ci2 C ΩSi1 Sθ2 + Ci1 Si2 Sθ2 + Cθ2 Si1 S Ω − Cθ1 C ΩSθ2 + Cθ2 (Si1 Si2 Sθ1 + Ci2 Cθ1 S Ω) + Ci1 Sθ1 (Ci2 Cθ2 C Ω + Sθ2 S Ω) Cθ1 Cθ2 Si1 Si2 + Sθ1 (C ΩSθ2 − Ci2 Cθ2 S Ω) + Ci1 Cθ1 (Ci2 Cθ2 C Ω + Sθ2 S Ω) − Ci2 Cθ2 C ΩSi1 + Ci1 Cθ2 Si2 − Si1 Sθ2 S Ω Ci2 Si1 Sθ1 − Si2 (Ci1 C ΩSθ1 + Cθ1 S Ω) Ci2 Cθ1 Si1 + Si2 (−Ci1 Cθ1 C Ω + Sθ1 S Ω) , Ci1 Ci2 + C ΩSi1 Si2 3×3
(A.1b)
where Ω = Ω1 − Ω2 , C and S represent cosine and sine functions respectively. (2) True anomaly f and mean anomaly M have a relation 5 e2 sin M + e2 sin 2M f = M + 2e − 4 4 13 + e3 sin 3M + · · · , (A.2) 12 so, when e is small, we can truncate the high order terms of the right hand of Eq. (A.2), and can get e2 f ≈ M + 2e − sin M = M + 2e sin M. (A.3) 4 Substituting Eq. (A.3) into Eq. (4), and considering a1 = a2 = a, it yields a1 (1 − e12 ) ≈ a(1 − e1 cos M1 ), 1 + e1 cos f1 a2 (1 − e22 ) r2 = ≈ a(1 − e2 cos M2 ). 1 + e2 cos f2
r1 =
(A.4)
Substituting Eq. (A.4) into Eq. (9), we get Eqs. (21) and (22). (3) When e1 = 0, θ 1, e2 1, and if formatting condition a1 = a2 = a is satisfied, according to (26), there is
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