Optimal Multiple-Impulse Circular Orbit Phasing Sergey P. Trofimov1 Keldysh Institute of Applied Mathematics, Moscow 125047, Russia and Mikhail Yu. Ovchinnikov2 Keldysh Institute of Applied Mathematics, Moscow 125047, Russia
Nomenclature
b
=
shape coefficient of the primer vector hodograph
G r
=
Newtonian gravity gradient matrix
g r
=
Newtonian gravitational acceleration
N
=
number of full revolutions for two-impulse maneuvers
n0
=
mean motion for the circular reference orbit
p
=
primer vector
pr
=
radial component of the primer vector
p
=
circumferential component of the primer vector
r
=
spacecraft’s position vector
r0
=
radius of the circular reference orbit
T0
=
orbital period for the circular reference orbit
t0
=
start time of the phasing maneuver
tf
=
end time of the phasing maneuver
W
=
matrix of linear system (13) given by Eq. (18)
r
=
radial deviation from the circular reference orbit
x
=
spacecraft’s phase vector in the rotating reference frame
z
=
out-of-plane deviation from the circular reference orbit
1
Researcher, Spaceflight Mechanics and Control Department, 4 Miusskaya Pl.;
[email protected].
2
Head of Attitude Control Systems & Orientation Division, Spaceflight Mechanics and Control Department, 4 Mi-
usskaya Pl.;
[email protected]. IAA Member (M2). Senior Member AIAA.
=
circumferential (phase) deviation from the circular reference orbit
t
=
phasing maneuver duration
V
=
vector of the impulse magnitudes
vi
=
i-th impulse vector
vi
=
magnitude of the i-th impulse
=
phase change required
=
phase angle
=
Earth’s gravitational constant
=
scale coefficient of the primer vector hodograph
i
=
dimensionless time of applying the i-th impulse
Φ
=
state transition matrix for the linearized equations of motion
=
phase coefficient of the primer vector hodograph
I.
Introduction
T
HE problem of orbit phasing is among the most studied ones in astrodynamics. It can be formulated as a problem
of changing the spacecraft’s phase—some angle describing the position of spacecraft along its orbit—faster or slower than it naturally changes. The phasing maneuver allows a spacecraft to move from one point of the orbit to another in a specified time and is therefore required for rendezvous and station-keeping operations. There are a lot of different formulations of the phasing problem, depending on the orbit shape, the thrust type, and the cost function (see, e.g., [1–4]). A wide variety of optimal transfer scenarios have been discovered and investigated. However, most of the results are related to locally optimal maneuvers. Global optimality can rarely be proven analytically whereas stochastic or metaheuristic optimization techniques do not reveal the qualitative structure of the global solution. In this Note, the fundamental problem of circular orbit phasing by performing a multiple-impulse maneuver is considered. Being a special case of the problem of multiple-impulse rendezvous between close near-circular orbits, it can be treated in the same way as proposed in the well-known papers [5, 6] of Prussing. However, it is shown below that the peculiar properties of the considered problem allow a simpler procedure of finding the globally optimal solution; it appears to be comprised of two- and four-impulse maneuvers (sometimes with a final coast arc) and “switches” between them as the phasing maneuver duration grows.
Two-impulse maneuvers do not require any optimization since they are unambiguously calculated as least-delta-v solutions to the multiple-revolution Lambert problem. More difficult are the cases of three- and four-impulse maneuvers. Since the spacecraft does not deviate much from the circular reference orbit, it is reasonable to use the linearized model of orbital dynamics given by the well-known Hill-Clohessy-Wiltshire equations. Locally optimal four-impulse solutions are obtained in linear approximation using the efficient numerical algorithm independently developed in [7] and [8]. Nonexistence of locally optimal three-impulse solutions is, to the authors’ knowledge, first proven here based on the results from [6]. In fact, the globally optimal solution is constructed in the circular orbit phasing problem using the existing methods of calculating a rendezvous maneuver with different number of impulses. Despite nothing methodologically new being introduced in the Note, we believe that the globally optimal solution to such a basic astrodynamics problem as circular orbit phasing is worth being presented explicitly.
II.
Lawden’s Primer Vector Theory and Four-Impulse Solutions
Let us first briefly revisit the fundamentals of Lawden’s primer vector theory. We consider a spacecraft moving in a circular orbit in the Newtonian gravitational field
g r
r3
r
(1)
though the theory is valid for any motion in an arbitrary field. According to Lawden [9], the necessary local optimality conditions for impulsive maneuvers are conveniently expressed in terms of the primer vector, the adjoint vector to the spacecraft’s velocity. The primer vector evolution is governed by the equation
p G r p
(2)
where G r g r . In order for an impulsive maneuver to be locally optimal, the primer vector must be a continuously differentiable function, with its magnitude being equal to 1 when impulses are applied and strictly less than 1 at other moments. Moreover, the optimal impulse direction coincides with the primer vector direction at the moment of applying this impulse. The initial values of p and p are set so that the boundary conditions for r and r are satisfied. Once it is achieved, the locally optimal solution is usually considered to be found. It is worth noting that the important extension has been made for linear systems: the number of impulses cannot exceed the number of boundary conditions (i.e., the dimension of phase space) and, moreover, the necessary optimality conditions are also sufficient [10, 11].
The orbit phasing problem can be treated as a rendezvous between close coplanar near-circular orbits. The motion equations linearized about the circular reference orbit have the same form in both Cartesian and cylindrical coordinates and are called the Hill-Clohessy-Wiltshire (HCW) equations. Cylindrical coordinates are more appropriate to describe the phasing maneuver since no restrictions are imposed on the phase angle . The corresponding HCW equations are
r 3n02 r 2n0 r0
(3)
r0 2n0 r
(4)
z n02 z
(5)
The origin of the uniformly rotating frame moves along the circular reference orbit; hence the spacecraft’s coordinates are calculated as follows: r r0 r , n0t , z z . Equation (5) is redundant for a coplanar rendezvous and no longer used below. In linear approximation, the gravity gradient matrix G r in Eq. (2) is evaluated along the circular reference orbit. The equations for the radial and circumferential primer vector components appear to be of the same form as (3) and (4):
pr 3n02 pr 2n0 p
(6)
p 2n0 pr
(7)
The solution to these equations can be obtained independently of the real spacecraft trajectory:
pr t A cos n0t B sin n0t 2C
(8)
p t 2B cos n0t 2 A sin n0t 3Ct D
(9)
The unknown constants A, B, C, D should be determined based on Lawden’s local optimality conditions. The primer vector hodograph—a time-parameterized curve in the pr , p plane—is generally cycloid-like but can also take one of the following “degenerate” shapes: an ellipse, a straight line, or an isolated point. The last two options are, however, not available for locally optimal four-impulse maneuvers (direct consequence of the optimality conditions). As for the cycloid and the ellipse, they have to be inscribed in the unit circle (Figs. 1 and 2) and are described by the expressions
pr cos 2b
(10)
p 2sin 3b
(11)
where n0t . Due to axial symmetry, the points 1 , 2 , 3 , 4 where the hodograph intersects or is tangent to the unit circle satisfy the relations 1 4 , 2 3 . Note that for locally optimal four-impulse maneuvers without coast arcs, 0 (t0 ) 1 and f (t f ) 4 , which results in
n0 t
Fig. 1 Cycloidal primer vector hodograph.
t0 t f 2
(12)
Fig. 2 Elliptical primer vector hodograph.
Defining the phasing maneuver duration t t f t0 , we thus define 4 n0 t 2 . Then, one can calculate all the other primer vector hodograph parameters using a fairly simple algorithm that has been independently developed in [7] and [8]. It gives us the optimal impulse application times and directions. Finally, the impulse magnitudes are obtained by solving the linear system [8]
W V
1
Φ 4 x f Φ 4 x0
(13)
where
V v1 v2 v3 v4
T
(14)
x r r0 r r0
(15)
x0 x (t0 ) , x f x (t f )
(16)
T
and Φ is the HCW state transition matrix:
4 3cos 6 sin Φ 3n0 sin 6n0 1 cos
0 sin n0 1 2 1 cos n0 0 cos 0 2sin
2 1 cos n0 4sin 3 n0 2sin 4 cos 3
(17)
The system matrix
f1 4 n0 f n W 2 4 0 f 3 4 f 4 4
f1 3 n0 f 2 3 n0 f 3 3 f 4 3
f1 3 n0 f 2 3 n0 f 3 3 f 4 3
f1 4 n0 f 2 4 n0 f 3 4 f 4 4
(18)
is formed of values of the functions
f1 3cos 4 sin 2b 3 cos 3 sin
(19)
f 2 2 1 sin 2 cos 3 sin b 4 cos 12 sin 9 2
(20)
f3 1 3sin 2 2b cos 3 sin
(21)
f 4 6 1 cos sin b 4sin 12 cos 9
(22)
All the V components should be positive; otherwise, there exists no locally optimal four-impulse phasing maneuver of a given duration. However, if coast arcs are permitted, there may be a faster four-impulse phasing maneuver. To illustrate the theory numerically, consider a sample problem of GEO satellite station-keeping. Suppose that the spacecraft has been drifted 10 km east (west) from its orbital position. The delta-v budget required for a locally optimal four-impulse correction maneuver is evaluated for the T0 , 4T0 range of maneuver duration values. The corresponding graph, shown in Fig. 3a, demonstrates that not any maneuver duration value allows an optimal four-impulse maneuver to exist. It is nevertheless natural to permit maneuvers with a final coast arc by “filling in” all the blank intervals with horizontal lines (Fig. 3b). This is what Shen and Tsiotras [12] called the sliding rule.
III.
Multiple-Revolution Lambert Problem and Two-Impulse Solutions
To obtain the globally optimal solution, we also have to consider three- and two-impulse maneuvers. (Recall that, according to Neustadt [10], no more than four impulses are needed for coplanar maneuvers.) The case of three-impulse maneuvers, which is usually the most difficult to analyze, appears to be extremely simple: for the circular orbit phasing
problem, there are no locally optimal three-impulse solutions. Indeed, as Prussing discovered in [6], for optimal threeimpulse rendezvous between two coplanar circular orbits, the phase change f 0 is constrained to be proportional to the difference r between orbits’ radii. It is evidently not possible when phasing the orbit ( r 0 ) since it implies zero phase change and no phasing maneuver occurred.
a) without final coasting
b) with final coasting
Fig. 3 Delta-v budget for locally optimal four-impulse phasing maneuvers. It remains to examine two-impulse maneuvers. They can be obtained in the nonlinear model from the very beginning since the linear approximation gives no noticeable gain in computing time and difficulty of implementation. If the maneuver duration is fixed, any optimization procedures can be avoided. Given the values of t0 and t f , the impulses v1 , v2 are applied at the two positions r (t0 ) , r (t f ) of the circular reference orbit separated by the angle
n0 t ; the impulse magnitudes and directions are calculated by numerically solving the multiple-revolution Lambert problem. Since the transfer trajectory should be close enough to the reference orbit, the number of revolutions N equals to the integer part of n0 t 2 t T0 2 . Among the two feasible solutions (usually referred to as short-path and long-path) with N full revolutions, we simply choose the least-delta-v one. The two-impulse solution to the above mentioned sample problem of GEO satellite station-keeping is depicted in Fig. 4 closely resembling Fig. 12 in paper [12] by Shen and Tsiotras. The dashed line denotes the no-coasting maneuver budget curve whereas the solid bold line is obtained by applying the sliding rule. To solve the multiple-revolution Lambert problem, the extremely robust algorithm of Gooding [13] based on the work of Lancaster and Blanchard [14] is used.
IV.
Combining Four-Impulse and Two-Impulse Solutions into Globally Optimal Solution
Now it is time to combine the four-impulse and two-impulse solutions (see Figs. 3b and 4) into the globally optimal multiple-impulse solution. The rule of combination is quite obvious: for a given phasing maneuver duration, the solution with less total delta-v is selected. The result is shown in Fig. 5. Several conclusions follow from this graph. First, for the fast phasing ( t T0 ), two-impulse maneuvers only fit. Second, as the phasing maneuver duration grows, the most efficient maneuvering scheme switches between the two-impulse and four-impulse modes. As usual, the nonincreasing delta-v budget curve describes a trade-off between the maneuver duration and fuel consumption.
Fig. 4
Delta-v budget for two-impulse phasing.
V.
Fig. 5
Globally optimal 2/4-impulse solution.
Conclusions
The globally optimal solution for the circular orbit phasing problem is constructed based on two-impulse and fourimpulse locally optimal maneuvers. The resulting trade-off curve can be exploited to satisfactorily evaluate the minimum delta-v cost of phasing. Provided the maneuver duration is moderate (as compared to the reference orbit period), the impulse magnitudes and directions calculated are precise enough and can be used as an initial guess in some direct optimization procedure to allow for nonlinear effects of real orbital dynamics.
Acknowledgments The work was supported by the Russian Science Foundation (RSF) under Grant 14-11-00621.
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