On-Orbit Range Set Applications
Marcus J. Holzinger
Aerospace Engineering Department
Texas A&M University
Daniel J. Scheeres
Aerospace Engineering Sciences
University of Colorado at Boulder
Abstract
History and methodology of ∆v range set computation is briefly reviewed, followed by a short sum-
mary of the ∆v optimal spacecraft servicing problem literature. Service vehicle placement is approached
from a ∆v range set viewpoint, providing a framework under which the analysis becomes quite geometric
and intuitive. The optimal servicing tour design problem is shown to be a specific instantiation of the
metric- Traveling Salesman Problem (TSP), which in general is an NP-hard problem. The ∆v-TSP is
argued to be quite similar to the Euclidean-TSP, for which approximate optimal solutions may be found
in polynomial time. Applications of range sets are demonstrated using analytical and simulation results.
1 INTRODUCTION
Range sets, traditionally used in aircraft settings, define a set of reachable points given a fuel constraint and
starting position(s). These sets are excellent mission planning and scenario evaluation tools, as they allow
complex, nonlinear differential systems to be expressed as intuitive, end-user friendly geometric problems.
Several spacecraft applications can benefit directly from the computation of time-independent, characteristic
velocity (∆v)-constrained range sets. In particular, mission design activities and on-orbit service planning
may be performed in a geometric, intuitive framework given orbit range sets. This paper is concerned with
the application of the ∆v metric and range sets to the service vehicle placement and on-orbit service tour
planning problems.
Range set computation for small, linearized regions about reference orbits has been studied intermittently
for more than 40 years by both Breakwell [1] and Marec [2]. More recently, Xue et al. have examined free-
time, single-impulse range sets [3] and Li et al. have examined reachable domain sets using two types of thrust
[4]. It has also been shown that spacecraft ∆v may be viewed as a distance metric [5], and that using ∆v
as a generalized independent parameter (GIP), a Generalied Metric Range Set (GMRS) representing time-
independent, ∆v-constrained orbit range sets may be computed [6]. Using Gauss’ Variational Equations
(GVEs) with ∆v as the independent parameter, time-independent range sets for arbitrary orbit transfers
under J
2
perturbations have been validated and the optimal control policy derived [7]. With the solution to
the on-orbit range set computation under J
2
perturbations in hand, specific applications may be explored.
The problem of designing optimal refueling strategies (also known as ’refueling tours,’ or simply ’tours’)
is a particular case of the Traveling Salesman Problem (TSP) that has been extensively studied for nearly
200 years, first considered by W. R. Hamilton and T. Kirkman, then formally defined in the 1930s by K.
Menger [8]. Interestingly, because minimum-∆v cost has been proven to be a properly defined distance
metric (namely the triangle inequality property) [5], algorithms applying to the metric-TSP subproblem
may be used [9]. Additionally, there are tour optimization algorithms specifically designed for the Euclidean
distance metric that can approximate the optimal path with a cost multiple of 1 +1c (for c >0 arbitrarily
large) in polynomial time [10], rather than the exponential time of the general problem [9].
1
Supposing that these same algorithms may be applied to ∆v metric-TSP (here dubbed ’∆v-TSP’) sce-
narios, realistic computation of ∆v-TSP solutions appears to be entirely feasible. Euclidean-TSP solutions
have been found for in excess of 85, 000 nodes, far exceeding the current number of active on-orbit objects
(1,300), as well as all trackable objects with diameters greater than 10cm (19,000) [11].
There are a number of past efforts examining the on-orbit optimal servicing tour problem. Alfriend et al.
examined the servicing of Geosynchronous objects [12, 13] in which the analysis is restricted to spacecraft with
circular orbits, identical periods, and small differential inclinations. Optimal servicing strategies between
spacecraft in a circular orbit constellation have also been considered with minimum fuel use, equalized fuel
use, and combinations thereof [14]. Further treatment of the same problem has also utilized network graph
theory formalisms to aid analysis and propose new fuel expenditure and transfer strategies [15].
The application of range set computation to both service vehicle placement and optimal servicing tour
scenario analysis is examined in this paper. The Background Theory section briefly reviews ∆v-metrics,
optimal control policy, and the computation of ∆v-range sets. The utility of ∆v range set computation
to both the sevice vehicle placement problem and the optimal service tour problem is then demonstrated
in the Applications section. Specific contributions of this paper are a) demonstration of the utility of ∆v
range sets to the problem of placing service vehicles on-orbit, and b) the observation that the minimum-∆v
servicing tour problem is a specific instantiation of not just the standard TSP, but the metric-TSP, and
nearly identical to the Euclidean-TSP, for which well-developed algorithms may be applied.
2 BACKGROUND THEORY
Prior to discussing applications, there are several specific theoretical results serving as the foundation of
∆v range set theory that must be briefly introduced. First, the result demonstrating that minimum ∆v
costs between initial and final states is a metric is stated. Second, the Generalized Independent Parameter
(GIP) HJB PDE is given and discussed. Lastly, the combination of both ∆v metrics and the GIP HJB
PDE is shown to provide a means by which optimal ∆v range sets may be computed in the presence of J
2
perturbations. Before discussing these items, several assumptions (initially outlined in [7]) are first made.
[A1] The user is unconcerned with the duration of an optimal maneuver from an initial orbit element set
œ
0
to a final orbit element set œ
f
. The problem is considered a ‘free-time’ optimal control problem.
[A2] The general accelerations a
r
, a
θ
, and a
h
in the rotating Hill frame are the result of corresponding
control accelerations u
r
, u
θ
, and u
h
.
[A3] For the first part of this analysis dynamics in the ascending node Ω are ignored (but are discussed in
§2.4).
[A4] Similar to the ascending node Ω, the argument of periapsis (ω) dynamics are ignored for the first
portion of this analysis (but are discussed in §2.4).
[A5] The control inputs u are impulsive. Because of mapping ambiguities between time and ∆v, and because
under control accelerations the true anomaly f changes, it is expedient to consider impulsive control
inputs.
[A6] When integrating in the ∆v independent parameter space, it is very convenient to use the true anomaly
f as a control variable. This is perfectly sensible, as a spacecraft operator can use f as a timing/phasing
control variable. The parameter f is considered a control parameter with f ∈[0, 2π).
2.1 Minimum ∆v Cost as a Metric
The following Theorem from [5] describing how performance indices for minimized Optimal Control Problems
(OCPSs) is introduced, followed by a short discussion on how ∆v is also a distance metric.
Theorem 2.1. Optimal Control Problem Distance
The function
d
OCP
(a, b)= inf
u∈U
∫
t
b
t
a
L(x(τ), u(τ), τ )dτ
s.t.
˙
x(t)=f(x(t), u(t), t)
h(x(t),t)≤0
g(x
a
, t
a
, x
b
, t
b
)=0
(1)
is a distance metric defined over the arguments a and b, where x ∈R
n
, u ∈R
m
, t ∈[t
0
, t
f
], L∶R
n
×R
m
×R →
R, L(⋅, ⋅, ⋅) ≥ 0 ∀x, u, t is the trajectory cost, f ∶ R
n
×R
m
×R → R
n
describes the system dynamics, and
g ∶ R
n
×R ×R
n
×R → R
r
is a function defining boundary conditions. The arguments a = (x
a
, t
a
) and
b = (x
b
, t
b
), each defined on the cartesian product of the state space and time coordinates (R
n
×R) must
be members of the set of boundary values that satisfy the boundary condition equation g(x
a
, t
a
, x
b
, t
b
) = 0,
defined as
(a, b)∈G ≡{(α, β) g(x
α
, t
α
, x
β
, t
β
)=0}
The boundary values a and b must also be members one another’s reachability sets, defined as
b ∈R(U, h, a)≡{b =(x
b
, t
b
)(x
b
, t
b
) is reachable given U, h, and a} (2)
and
a ∈R(U, h, b)≡{a =(x
a
, t
a
)(x
a
, t
a
) is reachable given U, h, and b} (3)
The set R(U, h, a) is the reachability set relative to a, and is a function of the allowable control set U and
trajectory inequality constraints h (see [16] for a brief optimal controls reachability introduction). Using this
definition, given (a, b)∈G, it is required that b ∈R(U, h, a)or a ∈R(U, h, b). A shorthand notation a, b ∈G, R
is used to represent both of these cases.
Proof:
The proof for Theorem 2.1 is given in detail in [5]. ◻
A common performance index used to compute ∆v cost is the L
1
norm of the input acceleration u(t)
[17], defined as
∆v =d
∆v
(a, b)=
∫
t
f
t
0
u(τ)dτ (4)
Through a straightforward application of Theorem 2.1, it is clear that using (4) as the performance index
of choice also defines a metric distance in terms of ∆v between any two orbits and epochs a =(œ
0
, t
0
) and
b =(œ
f
, t
f
).
2.2 Generalized Metric Range Sets
∆v range sets have very special properties which are now briefly outlined. The Generalized Independent
Parameter (GIP) HJB PDE derived in [6] is given as
∂
˜
V
∂s
+opt
u ∈U
˜
L(x, u, s)
˜
l(x, u, s)
+
∂
˜
V
∂x
T
˜
f(x, u, s)
˜
l(x, u, s)
=0
(5)
where s is a general independent parameter,
˜
V is the value function, and
˜
l is the slope of the mapping
function from time t to s. As discussed in detail in [6], if the independent parameter is also the performance
index cost (s =P ) and P is a metric, (5) reduces to
∂
˜
V
∂s
+opt
u ∈U
∂
˜
V
∂x
T
˜
f(x, u, s)
˜
L(x, u, s)
=0 (6)
which has the same functional form as traditionally defined minimum time reachability problems [18, 19, 20,
21]. As such, existing numerical toolboxes may be used to propagate (6) and compute the zero-level sets,
representing the boundaries of the range sets. For efforts discussed in this paper, I. M. Mitchell’s toolbox is
used [22].
2.3 ∆v Range Set Computation
This section summarizes an extended, detailed derivation of the ∆v metric range computation given in
[7]. Choosing to make the independent parameter in (5) ∆v, as defined by (4), the independent parameter
mapping function is chosen such that l(x(t), u(t), t)≡u(t). Applying Lemma 2.III from [6], choosing the
state-space coordinates to be constants of motion (e.g. the classical orbit elements œ, where
˙
œ = f(œ)u,
shown in Fig. 1(a)) allows (6) to be well defined during periods in which u(t)= 0. Additionally, since a
time-independent approach is now being considered (through the use of constants of motion as coordinates
and per [A1]), the orbit true anomaly f is also considered a control variable ([A6]). Reparameterizing the
instantaneous control acceleration direction
ˆ
u into Hill-frame azimuth and elevation coordinates (as shown
in Fig. 1(b)), where
ˆ
u is written as
ˆ
u =
u
u
2
=
sin β cos γ
cos β cos γ
sin γ
=
ˆ
u(β, γ)
ˆ
i
x
ˆ
i
y
ˆ
i
z
ˆ
i
e
ˆ
i
p
ˆ
i
h
ˆ
i
r
ˆ
i
θ
Ω
ω
i
f
i
ˆ
i
h
(a) Orbit element definition and coordinate frames
ˆ
i
r
ˆ
i
θ
ˆ
i
h
β
γ
u
r
v
u
r
u
θ
u
h
(b) Control acceleration u decomposition and pa-
rameterization in the rotating Hill frame
Figure 1: Reference frames and control vector decomposition
Lastly, for reasons made explicitly clear in §2.4, neither the ascending node (Ω) nor the argument of
periapsis (ω) are directly considered in the analysis, leaving the semi-major axis a, the eccentricity e, and
inclination i as the remaining state-space components. Thus, after all of the above assumptions and choices
are accounted for the final form of the ∆v HJB PDE to be obtained
∂
˜
V
∂∆v
+ sup
f,β,γ
∂
˜
V
∂œ
T
f(œ)
ˆ
u(β, γ)=0 (7)
where here œ
T
= [
a e i
]
T
. The optimal control policy (f
∗
, β
∗
, γ
∗
) for the ∆v range computation
problem in terms of the states a, e, and i and adjoints p
a
, p
e
, and p
i
is
(f
∗
, β
∗
, γ
∗
)=
(0, 0, 0) if p
a
≥0, p
a
≥
−ae
1−e
2
p
e
,
and −2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
≤p
i
≤2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
(π, 0, 0) if p
a
<0, p
a
≤
−ae
1−e
2
p
e
and 2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
≤p
i
≤−2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
(0, π, 0) if p
a
≥0, p
a
<
−ae
1−e
2
p
e
and 2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
≤p
i
≤−2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
(π, π, 0) if p
a
<0, p
a
>
−ae
1−e
2
p
e
and −2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
≤p
i
≤2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
(π, 0, −
π
2
) if p
i
>2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
, p
i
>−2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
p
i
>−2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
, and p
i
>2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
(π, 0,
π
2
) if p
i
>−2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
, p
i
>2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
p
i
>2(ap
a
+(1 −e)p
e
)(p
2
a
+p
2
e
)
1
2
, and p
i
>−2
1−e
1+e
(−ap
a
+(1 +e)p
e
)(p
2
a
+p
2
e
)
1
2
(8)
Table 1 summarizes each of the 6 optimal maneuvers indicated by the optimal control policy. Also, after
Table 1: Optimal Control Policy Equivalence
Case f
∗
, β
∗
, γ
∗
Description
H
1
0, 0, 0 Raise apoapsis
H
2
π, 0, 0 Raise periapsis
H
3
0, π, 0 Lower apoapsis
H
4
π, π, 0 Lower periapsis
H
5
π, 0, −
π
2
Increase i at apoapsis
H
6
π, 0,
π
2
Decrease i at apoapsis
significant simplification the resulting state dynamics are found
da
d∆v
=
a
1
2
µ
1
2
(1 −e
2
)
1
2
[a(1 +e cos f
∗
)]cos β
∗
cos γ
∗
(9)
de
d∆v
=
a
1
2
µ
1
2
(1 −e
2
)
1
2
2(1 −e
2
)(e +cos f
∗
)
1 +e cos f
∗
cos β
∗
cos γ
∗
(10)
di
d∆v
=
a
1
2
µ
1
2
(1 −e
2
)
1
2
(1 −e
2
)cos(ω
∗
+f
∗
)
1 +e cos f
∗
sin γ
∗
(11)
Along optimal trajectories, the necessary condition for optimality on the adjoint variables p
a
, p
e
, and p
i
associated with the states a, e, and i, is that dpd∆v =−∂H∂x, yielding
dp
a
d∆v
=
a
1
2
µ
1
2
(1 −e
2
)
1
2
−3(1 +e cos f
∗
)cos β
∗
cos γ
∗
p
a
−
(1 −e
2
)(e +cos f
∗
)
a(1 +e cos f
∗
)
cos β
∗
cos γ
∗
p
e
−
1 −e
2
a(e +cos f
∗
)
cos(ω
∗
+f
∗
)cos f
∗
sin γ
∗
p
i
(12)
dp
e
d∆v
=
a
1
2
µ
1
2
(1 −e
2
)
1
2
−
2a (e +cos f
∗
)
(1 −e
2
)
cos β
∗
cos γ
∗
p
a
+
2e(e +cos f
∗
)
(1 +e cos f
∗
)
cos β
∗
cos γ
∗
p
e
+
1 +e cos f
∗
cos f
∗
(e +cos f
∗
)
2
cos(ω
∗
+f
∗
)sin γ
∗
p
i
(13)
dp
i
d∆v
=0
(14)
Taken together, (12), (13), and (14) define the dynamics of the adjoint variables p
a
, p
e
, and p
i
in the ∆v
integration space along time-independent, ∆v-optimal trajectories. Interestingly, the optimal control policy
(8) is piecewise defined, and regions of optimality in the adjoint space can be used to view the optimal policy
using a graph topology, such as shown in Fig. 2.
H
3
H
2
H
1
H
4
H
5
H
6
Raise
Apoapsis
Lower
Apoapsis
Raise
Periapsis
Lower
Periapsis
Raise
Inclination
Lower
Inclination
Hohman
Transfer
Bi-Elliptic
Transfer
Bi-Elliptic
Transfer w/
Plane Change
Figure 2: Optimal basis maneuvers visualized as nodes on a bi-directional graph. Classical maneuver com-
binations are emphasized.
Interestingly, the optimal control policy (8) found using a first-principles optimal controls approach can
exactly reproduce several classical orbital maneuvers, namely the Hohman, Bi-Elliptic, and Bi-Elliptic with
plane change transfers. The following section discusses the applicability of the approach described here to
J
2
-perturbed dynamics.
2.4 Applicability to Orbit Range with J
2
Perturbations
The the analysis thus far derived the required equations to compute optimal orbit range sets in a, e, and
i using the classical GVE equations of motion. If J
2
perturbations are also considered, the time dynamics
of the mean semi-major axis ¯a, mean eccentricity ¯e, and mean inclination
¯
i have the same form as the un-
perturbed orbit elements a, e, and i, while in general (with the exception of particular critical regions) the
mean ascending node
¯
Ω and mean argument of periapsis ¯ω drift at different rates over time. This property,
that (¯a, ¯e,
¯
i)still behave as constants of motion while the orientation coordinates
¯
Ω and ¯ω drift over time, can
be exploited to realize significant ∆v savings in the general Two-Point Boundary Value Problem (TPBVP).
Using these properties, trajectories in regions where J
2
-induced drift rates in
¯
Ω and ¯ω are non-zero (and
different from one another) can place themselves in intermediate orbits and allow J
2
perturbations to perturb
them until a desired instantaneous value of Ω and ω is reached. Because in [A1] it is assumed that the user
is unconcerned with the total time of the orbit transfer, this effectively allows the user to transfer from any
initial instantaneous orbit element set œ
0
to any final instantaneous orbit element set œ
f
incurring only
the ∆v cost associated with transitioning from (a
0
, e
0
, i
0
) to (a
f
, e
f
, i
f
). Thus, the orbit element range set
computation method developed in this paper provides the general minimum-fuel, free-time optimal transfer
set from an initial orbit to any final orbit under J
2
perturbations.
3 APPLICATIONS
Servicing vehicle placement and optimal servicing tour applications are now discussed. Fig. 3 plots range
sets for three initial orbits with ∆v =1000ms to aid the reader in range set visualization. This particular
plot corresponds with example 2 in [7].
Figure 3: Range set for three initial orbits (LEO, GTO, and GEO) with a final ∆v constraint of 1,000 ms.
Projections of the range set in the a-e, a-i, and e-i planes are also shown along with markers indicating
initial orbits to aid in visualization.
Note that the Low Earth Orbit (LEO) range set starting at a = 6700km, e = 0 is not very large. This
is entirely expected, as gravitational dynamics are much stronger at smaller semi-major axes. Conversely,
both the Geostationary Transfer Orbit (GTO) and Geostationary Orbit (GEO) range sets are much larger.
Also, the relative ease of inclination change at higher radii is particularly emphasized in the GTO and GEO
range sets.
3.1 Servicing Vehicle Placement
An intuitive application of ∆v range sets are service vehicle placement exercises. Whether client spacecraft
are known a-priori or planners desire to place a service vehicle such that the greatest number of potential
clients are reachable, ∆v range sets may be used to visually evaluate candidate placement orbits. A notional
illustration of this approach is shown in Fig. 4.
œ
1
œ
2
œ
n
œ
A
S/C
œ
B
S/C
œ
C
S/C
...
!"#$%&'%(&)*+&
!"#$%&'%(&),+&!"#$%&'%(&)-+&
Figure 4: Notional illustration of how ∆v range set computation can be used to aid service vehicle placement
mission planning activities.
In the notional scenario shown in Fig. 4 there are several client orbits of interest (œ
1
, œ
2
, . . . , œ
n
). There
are three candidate service vehicle placement orbits considered: œ
A
S/C
, œ
B
S/C
, and œ
C
S/C
. With an arbitrary
∆v constraint, there are three corresponding range sets (Range set ‘A’, ‘B’, and ‘C’). Upon inspection it is
clear that Range Set ‘C’ only includes œ
1
and Range Set ‘B’ only includes œ
2
, . . . , œ
n
. However, Range set
‘A’ envelopes all client orbits of interest. This is perhaps the most intuitive and easily accessible application
of ∆v range sets; a casual inspection of range sets and potential clients during mission planning activities
requires no special knowledge of optimal controls or even astrodynamics. The complex, multi-dimensional
optimal control problem involved in computing fuel-optimal trajectories from an initial orbit to any possible
final orbit is neatly visualized as a ∆v range surface. Objects within or on the edge of the volume defined
by these sets are reachable, while those outside are not.
3.2 Servicing Tour Optimization
It has been long realized that choosing a fuel-optimal sequence of client spacecraft to service is an instantiation
of the Traveling Salesman Problem (TSP) [12, 13]. Because ∆v is a metric (see §2.1), computing time-
independent, ∆v-optimal tours is a more specific subproblem called the metric-TSP [9]. Specific algorithms
have been designed for the Euclidean metric that can identify nearly optimal sequences with a cost factor of
1 +1c, c >0, c arbitrarily large [10]. Using these algorithms, Euclidean distance TSPs have been solved for
in excess of 85, 000 nodes, far exceeding the 1,300 active spacecraft on orbit [11].
Fig. 5 demonstrates how evaluating tour optimization problems using time-independent, ∆v-optimal
trajectories is notionaly accomplished. On the left-hand side, the various client orbits are plotted along with
their optimal transfer paths in orbit-element space. Note that in orbit element space, differences in orbits do
not satisfy the triangle inequality. On the right-hand side, the client orbits (nodes) and associated transfer
differences are transformed onto the ∆v metric space, where the triangle-inequality does apply. It is on this
space that the metric-TSP is solved.
It must be emphasized that computing the ∆v range set for each of the n nodes requires approximately
O(nk
3
) operations, whereas computing all inter-orbit optimal transfers requires only O(n
2
). Unless n is
a
e
œ
1
œ
2
œ
n
œ
1
œ
2
œ
n
d
∆v
(1,n)
d
∆v
(2,n)
d
∆v
(1, 2)
Figure 5: The optimal connecting trajectories in orbit element space (left) may be represented using a non-
directional graph (right) where the edges between nodes (orbits) are weighted according to ∆v metric costs
associated with such a transfer.
arbitrarily high, to compute the individual distances between client spacecraft (nodes), it is likely much more
computationally optimal to simply utilize classical orbit transfer results. Still, constructing the probulem in
the ∆v metric space provides insight into the problem.
4 Conclusions & Future Work
The application of ∆v range set computation to service vehicle placement and optimal tour design is described
and motivated. Existing literature is cited to demonstrate that ∆v costs from minimum ∆v connecting
trajectories are metrics. Combined with the Generalized Independent Parameter (GIP) Hamilton-Jacobi-
Bellman (HJB) PDE, when ∆v is the independent paramter, resulting solutions are instances of Metric Range
Sets. The time-independent, ∆v optimal control policy is introduced and shown to reproduce classical orbit
transfer maneuvers, including the Hohman, Bi-Elliptic, and Bi-Elliptic with plane change transfers. It is
argued that the inclusion of J
2
perturbations allows the range set computation to ignore explicit inclusion
of the argument of periapsis (ω) and ascending node (Ω), as they secularly drift with time and may be
controlled through maneuver timing. Finally, both service vehicle placement and ∆v-optimal tour selection
are both introduced as applications for ∆v-optimal range sets.
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