AAS 13-926
MINIMUM-TIME LOW-EARTH ORBIT TO HIGH-EARTH ORBIT
LOW-THRUST TRAJECTORY OPTIMIZATION
Kathryn F. Schubert∗
Anil V. Rao†
The problem of many revolution low-thrust Earth-orbit trajectory optimization is
considered. The objective is to transfer a spacecraft from a parking orbit to a desired terminal orbit in minimum time. The minimum-time orbital transfer problem
is posed as a nonlinear optimal control problem, and the optimal control problem is
solved using a direct transcription variable-order Gaussian quadrature collocation
method. It is found that the thrust-to-mass ratio holds a power relationship to the
transfer time, final mass, and final true longitude. Using these power relationships
obtained through regression on the collected results, it is possible to estimate the
performance of a given thrust-to-mass ratio without solving for the optimal trajectory. In addition, the key structure of the optimal orbital transfers are identified.
The results presented provide insight into the structure of the optimal performance
for a range of small thrust-to-mass ratios and highlight the interesting features of
the optimal solutions. Finally, a discussion of the performance of the variableorder Gaussian quadrature collocation method is provided.
INTRODUCTION
The use of low-thrust propulsion in a variety of space missions has been of great interest to the
space community ever since the success of NASA’s Deep Space 1 (DS1) mission near the beginning
of the 21st century. A key feature of low-thrust propulsion is that the specific impulses are 10 to
20 times greater than the specific impulses of traditional chemical systems. As a result, the use of
low-thrust propulsion can significantly reduce the amount of required propellant which in turn can
reduce the mass of the spacecraft that must be delivered to orbit via a launch vehicle. Low-thrust
trajectory design problems are, however, computationally challenging because the applied thrust
is extremely small, with a typical specific force (thrust-to-mass ratio) of O(10−3 ) to O(10−4 ).
In addition, because forces such as non-spherical mass distributions and third-body perturbations
are often much larger than the thrust, low-thrust orbital transfers typically have an extremely long
duration ranging from months to years. Finally, low-thrust orbital transfers between widely spaced
orbits about the same planet generally require hundreds of orbital revolutions, thus creating further
computational issues due to the structure of the solutions.
A variety of low-thrust trajectory optimization problems have been considered previously. Reference 1 studied the problem of minimum-fuel power-limited transfers between coplanar elliptic orbits
using orbital averaging with polar coordinates and orbital elements. Reference 2 considered a 100revolution low-Earth orbit (LEO) to geo-stationary orbit (GEO) coplanar transfer using direct col∗
†
Ph.D. Student, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611.
Associate Professor, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL
32611. Associate Fellow, AIAA. Corresponding Author.
1
location paired with a Runge-Kutta parallel-shooting scheme that included J2 oblateness and thirdbody perturbations from the moon. Reference 3 performed a study of using collocation methods for
interplanetary low-thrust trajectory optimization and demonstrated the results on an Earth-to-Mars
transfer. Reference 4 showed the effectiveness of a genetic algorithm for designing near-optimal
low-thrust interplanetary trajectories and applies the approach to a minimum-time Earth-to-Mars
and Earth-to-Mercury transfers. Reference 5 studied near-optimal, minimum-time LEO to GEO
and geo-synchronous transfer orbit (GTO) to GEO transfers using a direct optimization approach
and orbital averaging. Moreover, Reference 5 used an acceleration model similar to that of a solar
electric propulsion engine and included the effects of Earth shadowing, second-order oblateness, and
solar cell degradation. Reference 6 provided a numerical optimization study of low-thrust trajectory
optimization using higher-order collocation methods. Reference 7 described a hybrid optimization
method that integrates a multi-objective genetic algorithm with a calculus-of-variations-based lowthrust trajectory optimizer, and generates novel trajectories for both an Earth-to-Mars and Earthto-Mercury transfer. Reference 8 later solved a minimum-fuel low-thrust near-polar Earth-orbit
transfer with over 578 revolutions via discretization and sequential quadratic programming (SQP)
using an acceleration model that incorporated oblateness effects through fourth-order. Reference 9
described a computational implementation of a shape-based method for the design of low-thrust,
gravity assist trajectories. Reference 10 considered a minimum-fuel transfer from a low, elliptic,
and inclined orbit to GEO using a single shooting method combined with a homotopic approach.
Reference 11 developed an anti-aliasing method in order to obtain high-accuracy solutions to lowthrust trajectory optimization problems. Reference 12 presented a direct optimization method that
employed orbital averaging and a parameterized control law technique to solve three common nearoptimal, minimum-time Earth orbit transfers that contained second-order oblateness and shadow
effects. More recently, Reference 13 developed a collocation approach using orbital averaging in
conjunction with hybrid control formulations to solve LEO to GEO and GTO to GEO transfers.
It is seen that previous research on low-thrust trajectory optimization for Earth-orbit transfers
has focused primarily on circular orbit-raising transfers and has employed either orbital averaging or low-order collocation methods. In this research the problem of determining high-accuracy
solutions to low-thrust orbital transfer problems with large changes in orbit size, inclination, and
eccentricity is considered using a fourth-order oblate gravity model. The low-thrust trajectory optimization problem is then solved using a high-accuracy variable-order Gaussian quadrature collocation method.14, 15, 16, 17, 18, 19, 20, 21 As a result, solutions to the optimal control problem are obtained
without having to replace the equations of motion with averaged approximations over each orbital
revolution. Using the aforementioned approach, the performance requirements for a minimum-time
low-thrust trajectory optimization from a circular low-Earth orbit to a highly elliptic low-periapsis
high-Earth orbit is studied. Numerical solutions are presented for a variety of specific force values
of O(10−3 ). It is found that the thrust-to-mass ratio holds a power function relationship with the
minimum transfer time, final true longitude, and final mass. It is then possible to utilize the power
function regressions on the results obtained in this study to estimate the transfer time and number
of revolutions that may be required for transfers beyond those for which the results were obtained
without having to solve another trajectory optimization problem. In addition, it is found that as the
initial mass increases, the transfers themselves follow a distinct structure. Finally, in this study the
performance of the variable-order Gaussian quadrature collocation method is discussed, where it
is found that the mesh refinement method places collocation points in the appropriate locations in
order to meet the required accuracy tolerance on the numerical approximation.
2
This paper is organized as follows. First, the low-thrust orbital transfer trajectory optimization
problem is described. Second, the initial guess generation method that is used to solve the trajectory
optimization problem using the MATLAB optimal control software GPOPS − II is described.21
Third, numerical results are presented for a range of thrust-to-mass ratios which show the power
function relationship that the thrust-to-mass ratio has with the minimum transfer time, the final true
longitude, and the final mass. Fourth, the key features of the optimal trajectories are identified.
Finally, the performance of the variable-order Gaussian quadrature collocation method is discussed.
LOW-THRUST EARTH-ORBIT OPTIMAL CONTROL PROBLEM
Consider the problem of transferring a spacecraft from an initial low-Earth orbit (LEO) to a final
high-Earth orbit (HEO) using low-thrust propulsion. The objective is to determine the minimumtime trajectory and control that transfers a spacecraft from a circular low-Earth orbit (LEO) with an
inclination of i0 to a highly elliptic low periapsis high-Earth orbit (HEO) with a terminal eccentricity
ef , inclination if , and argument of periapsis ωf . In the remainder of this section we describe
the low-thrust optimal control problem for the aforementioned LEO to HEO transfer. First, the
dynamics of the spacecraft, modeled as a point mass, are described using modified equinoctial
elements together with a fourth-order oblate gravity model and a propulsion system that has a high
specific impulse and a small thrust-to-mass ratio of O(10−3 ). Second, we describe the boundary
conditions for the orbit transfer in terms of both classical orbital elements and modified equinoctial
elements. Third, we describe the path constraints in effect during the orbital transfer. Finally, we
describe the minimum-time optimal control problem.
Equations of Motion
The state of the spacecraft is comprised of the modified equinoctial elements (p, f, g, h, k, L),22
where p is the semi-parameter, f and g are modified equinoctial elements that describe the eccentricity of the orbit, h and k are modified equinoctial elements that describe the inclination of the
orbit, and L is the true longitude, together with the mass, m. The control is the thrust direction,
u, where u is expressed in radial-transverse-normal components u = (ur , uθ , uh ). The differential
equations of motion of the spacecraft are given as
r
dp
p 2p
=
∆θ = Fp ,
dt
µ q
r df
p
sin L ∆r + 1q (q + 1) cos L + f ∆θ − gq h sin L − k cos L ∆h = Ff ,
=
dt
µ
r dg
p
− cos L ∆r + 1q (q + 1) sin L + g ∆θ + fq h sin L − k cos L ∆h = Fg ,
=
dt
µ
r 2
dh
p s cos L
=
∆h = Fh ,
dt
µ 2q
r 2
p s sin L
dk
=
∆h = Fk ,
dt
µ 2q
2
r
q
p
dL
√
h sin L − k cos L ∆h + µp
=
= FL ,
dt
µ
p
T
dm
= − = Fm ,
dt
Ve
3
(1)
where
q = 1 + f cos L + g sin L,
r = p/q,
α2 = h2 − k 2 ,
s2 = 1 +
√
h2 + k 2 .
(2)
In this research, time is replaced as the independent variable in favor of the true longitude, L,
because L provides a more intuitive understanding of the transfer. For example, each increment of
2π in L is an orbit revolution, and because the spacecraft moves from a LEO to a HEO, more orbital
revolutions will be completed in a given amount of time near the start of the transfer than will be
completed near the terminus of the transfer. Using true longitude as the independent variable, the
differential equation for L is replaced with the following differential equation for t
dt
1
=
= FL−1 ,
dL
FL
(3)
while the remaining six differential equations for (p, f, g, h, k, m) are given as
dp
dL
df
dL
dg
dL
dh
dL
dk
dL
dm
dL
= FL−1 Fp ,
= FL−1 Ff ,
= FL−1 Fg ,
(4)
FL−1 Fh ,
=
= FL−1 Fk ,
= FL−1 Fm .
Next, the spacecraft acceleration, ∆ = (∆r , ∆θ , ∆h ), is modeled as
∆ = ∆g + ∆T ,
(5)
where ∆g is the gravitational acceleration due to the oblateness of the Earth and ∆T is the thrust
specific force. The acceleration due to Earth oblateness is expressed in rotating radial coordinates
as
∆g = QT
(6)
r δg,
where Qr = ir iθ ih is the transformation from rotating radial coordinates to Earth centered
inertial coordinates and whose columns are defined as
ir =
r
krk
, ih =
r×v
||r × v||
, iθ = ih × ir .
(7)
Furthermore, the vector δg is defined as
δg = δgn in − δgr ir ,
(8)
where in is the local North direction and is defined as
in =
en − (eT
n ir )ir
||en − (eT
n ir )ir ||
4
(9)
and en = (0, 0, 1). The oblate earth perturbations are then expressed as
δgr
δgn
k
4
µ X
Re
(k + 1)
= − 2
Pk (s)Jk ,
r
r
k=2
4 µ cos(φ) X Re k ′
= −
Pk (s)Jk ,
r2
r
k=2
(10)
(11)
where Re is the equatorial radius of the earth, Pk (s) (s ∈ [−1, +1]) is the k th -degree Legendre
′
polynomial, Pk is the derivative of Pk with respect to s, s = sin(φ), and Jk represents the zonal
harmonic coefficients for k = (2, 3, 4). Next, the acceleration due to thrust is given as
∆T =
T
u.
m
(12)
The physical constants used in this study are given in Table 1.
Table 1: Physical Constants.
Quantity
Value
Ve
7.355 × 104 m · s−1
µ
3.9860047 × 1014 m3 · s−2
Re
6378140 m
J2
1082.639 × 10−6
−2.565 × 10−6
J3
−1.608 × 10−6
J4
Boundary Conditions
The spacecraft starts in a circular low-Earth orbit (LEO) at time t0 = 0. The initial orbit is
specified in classical orbital elements23 as
a(t0 ) = a0 ,
Ω(t0 ) = Ω0 ,
e(t0 ) = e0 ,
ω(t0 ) = ω0 ,
i(t0 )
ν(t0 )
= i0 ,
(13)
= ν0 ,
where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the longitude of the
ascending node, ω is the argument of periapsis, and ν is the true anomaly, and it is noted that ω0 and
ν0 are arbitrarily set to zero. Equation (13) can be expressed equivalently in terms of the modified
equinoctial elements as
p(t0 )
= a0 (1 − e20 ),
h(t0 )
= tan(i0 /2) sin Ω0 ,
f (t0 ) = e0 cos(ω0 + Ω0 ),
k(t0 )
= tan(i0 /2) cos Ω0 ,
g(t0 ) = e0 sin(ω0 + Ω0 ),
L(t0 ) = Ω0 + ω0 + ν0 .
5
(14)
The spacecraft terminates in a highly elliptic low periapsis high-Earth orbit (HEO) with a final
inclination that is different from the initial inclination and a specified argument of periapsis. The
terminal orbit is specified in classical orbital elements as
a(tf ) = af ,
Ω(tf ) = free,
e(tf ) = ef ,
ω(tf ) = ωf ,
i(tf )
ν(tf )
= if ,
(15)
= free.
Equation (15) can be expressed equivalently in terms of the modified equinoctial elements as
p(tf ) = af (1 − e2f ),
(f 2 (tf ) + g 2 (tf ))1/2 = ef ,
(h2 (tf ) + k 2 (tf ))1/2 = tan(if /2),
(16)
f (tf )h(tf ) + g(tf )k(tf ) = ef tan(if /2) cos ωf ,
g(tf )h(tf ) − k(tf )f (tf ) ≤ ef tan(if /2) sin ωf .
The last two terminal conditions ensure that ω stays between the third and fourth quadrants.
Path Constraints
During the transfer the thrust direction must be a vector of unit length. Thus, the following
equality path constraint is enforced throughout the orbital transfer:
||u||2 = 1.
(17)
In terms of the radial, transverse, and normal components, the path constraint of Eq. (17) is given as
u2r + u2θ + u2h = 1.
(18)
Optimal Control Problem
In this study the objective is to determine the trajectory (p(t), f (t), g(t), h(t), k(t), L(t), m(t))
and the control (ur (t), uθ (t), uh (t), that transfer the spacecraft from the initial conditions defined
in Eq. (14) to the terminal conditions defined in Eq. (16) while satisfying the dynamic constraints
of Eqs. (3) and (4), the path constraint of Eq. (18) and minimizing the time required to complete the
transfer. Consequently, the objective function for this problem is given as
J = tf .
(19)
SOLUTION OF OPTIMAL CONTROL PROBLEM
The aforementioned minimum-time low-thrust optimal control problem was solved using the
optimal control software GPOPS − II.21 GPOPS − II is a MATLAB software that implements
the Legendre-Gauss-Radau collocation method19, 18, 24 together with a variable-order ph-adaptive
mesh refinement method.20 GPOPS − II transcribes the optimal control problem to a large sparse
nonlinear programming problem (NLP). In this study the NLP created by GPOPS − II was solved
using the open-source NLP solver IPOPT 25 with all first- and second-derivatives approximated
using a sparse finite-difference method based on the sparsity structure of the NLP as described in
References 24 and 21.
6
An important aspect of solving the optimal control problem effectively using GPOPS − II was
to develop a systematic method for generating an initial guess of the solution. Because in this
research we are interested in solving a problem whose solution will result in a large number of
orbital revolutions, the initial guess must itself consist of a number of orbital revolutions that are
somewhat close to the number of orbital revolutions of the optimal solution. For this study the
following initial guess procedure was devised that consisted of solving a series of optimal control
sub-problems. The goal of each sub-problem was to determine the state and control that transfer the
spacecraft from an initial orbit to a terminal orbit that is as close in proximity to the desired terminal
semi-parameter pd , eccentricity ed , and inclination id (for simplicity, the sub-problem did not take
into account the desired terminal argument of periapsis). Each sub-problem is evaluated over at
most one revolution using the terminal state of the previous sub-problem as the initial state of the
current sub-problem. The continuous-time optimal control sub-problem is then stated as follows.
Minimize the cost functional
J=
p(tf ) − pd f 2 (tf ) + g 2 (tf ) − e2d h2 (tf ) + k 2 (tf ) − (tan( i2d ))2
+
+
(1 + pd )2
(1 + e2d )2
(1 + (tan( i2d ))2 )2
(20)
subject to the dynamic constraints Eqs. (3) and (4), the path constraint Eq. (18), and the boundary
conditions
p(r) (t0 )
= p(r−1) (tf ),
p(r) (tf )
= free,
f (r) (t0 )
= f (r−1) (tf ),
f (r) (tf )
= free,
g (r) (t
= free,
g (r) (t
0)
(r)
h (t0 )
k (r) (t0 )
m(r) (t0 )
L(r) (t0 )
=
=
=
=
=
g (r−1) (t
f ),
(r−1)
h
(tf ),
k (r−1) (tf ),
m(r−1) (tf ),
L(r−1) (tf ),
f)
(r)
h (tf )
k (r) (tf )
m(r) (tf )
L(r) (tf )
= free,
(21)
= free,
= free,
≤ L(r−1) (tf ) + 2π rad
for r = 1, . . . , R where R represents the total number of revolutions. The initial conditions for the
first revolution when r = 1 are simply the initial conditions stated in Eq. (14). Once the desired
terminal conditions as stated in Eq. (16) are obtained within a user specified tolerance, the subproblem solutions are then combined to form the initial guess. The collocation distribution of each
sub-problem is maintained and used as the initial mesh.
RESULTS
The aforementioned LEO to HEO Earth-orbit transfer problem was solved using the following
initial and terminal orbits, respectively:26
a(t0 ) = 6656000 m,
Ω(t0 ) = 180 deg,
e(t0 ) = 0,
ω(t0 ) = 0 deg,
i(t0 )
ν(t0 )
= 28.5 deg,
= 0 deg,
a(tf ) = 26565000 m,
Ω(tf ) = free,
e(tf ) = 0.7355,
ω(tf ) = −90 deg,
i(tf )
= 63.4 deg,
ν(t0 )
7
(22)
= free.
(23)
The problem was solved for thrust values of T = (320, 420, 520) mN and initial mass values of
m0 = (120, 170, 220, 270, 320) kg. All computations were performed on a 2.8 GHz Quad-Core
Intel Xeon Mac Pro running Mac OS-X 10.7.5 with 32 GB of RAM. The results presented below
are divided into three parts. First, relationships are identified between the specific force and transfer
time, specific force and number of revolutions, and specific force and final mass. Second, the key
features of the optimal trajectories are described and analyzed. Finally, the performance of the
variable-order collocation method is discussed.
Specific Force vs. Transfer Time, Number of Revolutions, and Final Mass
A key feature of the results is the ability to estimate the transfer time, the total number of revolutions, and the final mass as a function of specific force. Figures 1–3 show the specific force as
a function of the final time, tf , final true longitude, L(tf ), and final mass, m(tf ), respectively for
T = (320, 420, 520) mN and m0 = (120, 170, 220, 270, 320) kg. It is seen that T /m0 decreases as
tf , L(tf ), and m(tf ) increase in a manner similar to that of a power function y = ax−b (where a
and b are positive constants). The observation that T /m0 holds a power function relationship with
tf , L(tf ), and m(tf ) provides a way to estimate the final values of any of these quantities by performing a power regression on the data in Figures 1, 2, and 3. The power fit regression coefficients
for T /m0 vs. tf , L(tf ), and m(tf ) are shown in Table 2. From these power function regressions
(also shown in Figures 1–3), it is possible to estimate tf , L(tf ), or m(tf ) at points beyond those
for which the results were obtained. For example, the minimum transfer time, tf , can be estimated
using the power fit regressions as follows. First, given a desired final mass, m(tf ), the specific
force, T /m0 , can be estimated for a given value of T using Figure 3. Next, given the value of T /m0
estimated from the power regression of Figure 3, the minimum transfer time, tf , can be estimated
using Figure 1. Following this procedure, given a desired payload of 260 kg and T = 520 mN the
estimated minimum transfer time is 41.49 days. Furthermore, Figures 1 and 2 show that the transfer
time and number of revolutions become impractically large as T /m0 becomes small. For instance,
a thrust-to-mass ratio of 1.9677 × 10−4 would require a transfer time of 365 days and 1912 orbital
revolutions, both of which may be inordinately large and unusable in a practical setting.
Table 2: Power fit regression coefficients for T /m0 vs. tf , L(tf ), and m(tf ).
T /m0 vs. tf
T /m0 vs. L(tf )
T /m0 vs. m(tf )
for T = 320 mN
T /m0 vs. m(tf )
for T = 420 mN
T /m0 vs. m(tf )
for T = 520 mN
a
0.0825
0.8559
0.2872
0.3798
0.4689
b
1.023
1.109
0.9970
0.9986
0.9982
Key Features of the Optimal Trajectories
Figure 4 shows the components of the state and Figure 5 shows the classical orbital elements a, e,
and i for T = 420 mN and m0 = (120, 170, 220, 270, 320) kg. Figure 6 shows the classical orbital
element ω for (T, m0 ) = (420 mN, 320 kg) only. It is seen from Figure 4a that p overshoots the
desired terminal value, whereas Figure 5a shows that a increases more gradually to meet the required
terminal value. Next, Figure 5b shows that e increases slowly as it approaches 0.2, then begins a
more rapid increase to the terminal value. Therefore, a majority of the eccentricity change occurs
near the end of the transfer. The noticeable increase in de/dt is also evident by examining Figures 4b
and 4c which show that the modified equinoctial elements f and g, associated with e, change more
8
5.0
Power Fit
320 mN - 120 kg
420 mN - 120 kg
520 mN - 120 kg
320 mN - 170 kg
420 mN - 170 kg
520 mN - 170 kg
320 mN - 220 kg
420 mN - 220 kg
520 mN - 220 kg
320 mN - 270 kg
420 mN - 270 kg
520 mN - 270 kg
320 mN - 320 kg
420 mN - 320 kg
520 mN - 320 kg
4.5
4.0
T /m0 (× 10−3 )
3.5
3.0
2.5
2.0
1.5
1.0
0.5
20
30
40
50
60
tf (days)
70
80
90
100
Figure 1: Specific force, T /m0 , vs. final time, tf .
rapidly in the second half of the transfer. Figure 5c shows that i gradually increases throughout
the transfer. As the orbit transitions from circular to slightly elliptic, ω oscillates between −180
and 180 deg, as seen in Figure 6. Near e = 0.01, the behavior of ω changes such that it gradually
approaches the desired terminal value.
The control components for the case (T, m0 ) = (420 mN, 220 kg) are shown in Figure 7. At
approximately 22 days, it is seen that all of the control components undergo a distinct change in
behavior. From Figure 5b, 22 days corresponds to e = 0.2, which recalling from above is the point
in the transfer where de/dt increases notably. In particular, the tangential control component, uθ ,
shown in Figure 7b, becomes more influential starting at e = 0.2 as it plays an important role in
increasing the eccentricity.
Another important feature of the optimal trajectories is that regardless of the initial mass, the state
and control components maintain the key characteristics previously described, such as the overshoot
in p, the slow then rapid rate of changes in f and g, and the distinct change in control components
near e = 0.2. Combined with the ability to estimate tf , L(tf ), and m(tf ), knowledge of the overall
structures associated with the optimal states and control components provides valuable insight into
generating initial guesses for problems with specific forces different from those examined in this
study.
Performance of Collocation Method
Figure 8 shows a three dimensional view of the optimal trajectory for (T, m0 ) = (320 mN, 320 kg).
The solution shown in Figure 8 contains approximately 422 orbital revolutions and a minimum
9
5.0
Power Fit
320 mN - 120 kg
420 mN - 120 kg
520 mN - 120 kg
320 mN - 170 kg
420 mN - 170 kg
520 mN - 170 kg
320 mN - 220 kg
420 mN - 220 kg
520 mN - 220 kg
320 mN - 270 kg
420 mN - 270 kg
520 mN - 270 kg
320 mN - 320 kg
420 mN - 320 kg
520 mN - 320 kg
4.5
4.0
T /m0 (× 10−3 )
3.5
3.0
2.5
2.0
1.5
1.0
0.5
100
150
200
250
300
L(tf ) (revs)
350
400
450
Figure 2: Specific force, T /m0 , vs. final number of revolutions, L(tf ).
transfer time of tf = 73.43 days. GPOPS − II required 27 mesh refinement iterations to compute
this solution, and the final mesh contained 3986 mesh intervals and 17974 Legendre-Gauss-Radau
collocation points. The entire mesh refinement history is shown in Table 3 where, as might be
expected, the number of collocation points increases with each mesh refinement iteration and the
error estimate tends to decrease as the mesh refinement progresses. An interesting feature of the
mesh refinement method is the placement of the collocation points. Specifically, Figure 9 shows
the number of collocation points placed in a band that is ±15% of the orbital period around both
the periapses and apoapses of the transfer trajectory as a function of the orbital revolution alongside the eccentricity. It is seen in Figure 9 that the number of collocation points near the periapses
increases as the eccentricity increases. This last result is consistent with the fact that the vehicle is
traveling the fastest near periapsis and is traveling the slowest near apoapsis. In order to visualize
the collocation point placement, Figures. 10 and 11 show the thrust direction for the first and last
orbit revolutions, respectively. It is seen during the first orbital revolution, shown in Figure 10, that
the optimal thrust direction increases the size of the orbit and the inclination as much as possible.
In addition, the thrust direction during the first revolution rotates the orbit slightly. Next, in the
last orbital revolution, shown in Figure 11, a large number of collocation points are utilized near
periapsis. Results obtained for m0 = 320 kg are shown in Table 4, where it is seen that as thrust
decreases, the number of collocation points increases.
10
5.0
320 mN Power Fit
420 mN Power Fit
520 mN Power Fit
320 mN - 120 kg
420 mN - 120 kg
520 mN - 120 kg
320 mN - 170 kg
420 mN - 170 kg
520 mN - 170 kg
320 mN - 220 kg
420 mN - 220 kg
520 mN - 220 kg
320 mN - 270 kg
420 mN - 270 kg
520 mN - 270 kg
320 mN - 320 kg
420 mN - 320 kg
520 mN - 320 kg
4.5
4.0
T /m0 (× 10−3 )
3.5
3.0
2.5
2.0
1.5
1.0
0.5
100
120
140
160
180
200
220
m(tf ) (kg)
240
260
Figure 3: Specific force, T /m0 , vs. final mass, m(tf ).
11
280
300
0.8
18
0.6
14
0.4
f
p (km ×103 )
22
10
6
0
0.2
120 kg
170 kg
220 kg
270 kg
320 kg
10
20
30
t (days)
40
50
60
0
120 kg
170 kg
220 kg
270 kg
320 kg
10
(a) p(t) vs. t.
0.7
20
30
t (days)
40
50
60
(b) f (t) vs. t.
-0.1
120 kg
170 kg
220 kg
270 kg
320 kg
-0.2
0.5
h
g
-0.3
0.3
-0.4
120 kg
170 kg
220 kg
270 kg
320 kg
-0.5
0.1
0
10
20
30
t (days)
40
50
-0.6
0
60
10
(c) g(t) vs. t.
20
30
t (days)
40
50
60
(d) h(t) vs. t.
0.6
350
250
k
L (revs)
0.4
150
0.2
120 kg
170 kg
220 kg
270 kg
320 kg
0
0
10
20
30
t (days)
40
50
120 kg
170 kg
220 kg
270 kg
320 kg
50
0
0
60
(e) k(t) vs. t.
10
20
30
t (days)
(f) L(t) vs. t.
Figure 4: Modified equinoctial elements for T = 420 mN.
12
40
50
60
30
0.8
0.7
25
a (km ×103 )
0.6
0.5
e
20
15
0.4
0.3
0.2
120 kg
170 kg
220 kg
270 kg
320 kg
10
5
0
10
20
30
t (days)
40
50
120 kg
170 kg
220 kg
270 kg
320 kg
0.1
60
20
10
0
(a) a vs. t.
30
t (days)
40
50
60
(b) e vs. t.
70
i (deg)
60
50
40
120 kg
170 kg
220 kg
270 kg
320 kg
30
20
0
20
10
30
t (days)
40
50
60
(c) i vs. t.
Figure 5: Classical orbital elements a, e, and i vs. t for T = 420 mN.
180
ω (deg)
90
0
-90
-180
0
10
20
30
40
50
t (days)
Figure 6: Classical orbital element ω vs. t for (T, m0 ) = (420 mN, 320 kg).
13
60
1.0
ur
0.5
0
-0.5
-1.0
0
5
10
15
20
25
30
35
40
25
30
35
40
25
30
35
40
t (days)
(a) ur vs. t.
1.0
uθ
0.5
0
-0.5
-1.0
0
5
10
15
20
t (days)
(b) uθ vs. t.
1.0
uh
0.5
0
-0.5
-1.0
0
5
10
15
20
t (days)
(c) uh vs. t.
Figure 7: Control variables for (T, m0 ) = (420 mN, 220 kg).
14
40
z (km × 103 )
30
20
10
0
-10
-20
-20
-10
0
0
20
10
x (km × 103 )
y (km ×
103 )
Figure 8: Optimal trajectory for (T, m0 ) = (320 mN, 320 kg).
15
Table 3: Mesh refinement history for (T, m0 ) = (320 mN, 320 kg).
Mesh Iteration
Number of Collocation Points
Error Estimate
1
6993
1.2704 × 10−3
2
9364
3.8726 × 10−4
3
11498
1.6538 × 10−4
4
13369
1.3257 × 10−4
5
14530
9.4626 × 10−5
6
15118
3.2589 × 10−4
7
16183
4.9533 × 10−5
8
16674
3.5061 × 10−5
9
17051
3.2682 × 10−5
10
17328
5.6966 × 10−5
11
17551
4.2430 × 10−5
12
17604
1.6136 × 10−5
13
17642
2.4355 × 10−5
14
17671
1.7279 × 10−5
15
17724
1.5806 × 10−5
16
17761
2.8130 × 10−5
17
17827
2.3848 × 10−5
18
17864
1.3401 × 10−5
19
17890
1.0876 × 10−5
20
17900
1.5433 × 10−5
21
17916
1.2159 × 10−5
22
17932
1.0399 × 10−5
23
17933
1.1282 × 10−5
24
17939
1.0712 × 10−5
25
17947
1.4341 × 10−5
26
17672
1.1912 × 10−5
27
17974
9.9086 × 10−6
16
replacements
1.0
60
Number of collocation points near apoapsis
Number of collocation points near periapsis
Eccentricity
50
40
0.6
Eccentricity
Number of collocation points
0.8
30
0.4
20
0.2
10
0
50
0
100
150
250
200
300
350
400
0
450
L (revs)
Figure 9: Collocation points near periapsis and apoapsis for (T, m0 ) = (320 mN, 320 kg).
Table 4: Transfer time, number of orbital revolutions, and final mass for m0 = 320 kg.
Thrust (mN)
Number of
Collocation Points
Error
Estimate
520
8763
420
320
10588
17974
Final
Time (days)
Number of
Orbital Revolutions
Final
Mass (kg)
9.9191 × 10−6
46.65
287.40
291.63
9.9656 × 10
−6
56.96
332.69
291.90
9.9086 × 10
−6
73.43
422.10
292.40
17
z (km ×103 )
5
0
-5
-5
-5
0
0
5
5
x (km ×103 )
y (km ×103 )
Figure 10: Optimal thrust direction of the initial revolution for (T, m0 ) = (320 mN, 320 kg).
40
z (km ×103 )
30
20
10
0
-30
-20
-20
-10
0
0
10
20
x (km ×103 )
y (km ×103 )
Figure 11: Optimal thrust direction of the final revolution for (T, m0 ) = (320 mN, 320 kg).
18
DISCUSSION
The results of this study highlight several interesting aspects of using low-thrust trajectory optimization for a low-Earth orbit (LEO) to high-Earth orbit (HEO) maneuver with an inclination
change. First, it is seen that the thrust-to-mass ratio holds a power function relationship with the
transfer time, final mass, and final true longitude. These power function relationships provide a simple mechanism for estimating the overall performance of orbital transfers of this type at values of the
parameters beyond those for which the results have been collected without needing to actually solve
the problem at these other parameter values. Second, it is seen that the transfers themselves follow a well-defined structure as the initial mass changes. In particular, as the initial mass increases,
the minimum transfer time increases in a manner such that the overall structure of the solution remains similar. Third, it is interesting to see that the Gaussian quadrature collocation method used to
solve the problem places the collocation points in the regions near the periapses of the transfer orbit
(where the vehicle is traveling the fastest) while placing considerably fewer collocation points near
the apoapsis of the transfer orbit (where the vehicle is traveling the slowest). In addition, it is seen
that the number of collocation points required to meet a desired accuracy tolerance increases as the
thrust decreases. This last result is consistent with the fact that a smaller thrust will require a longer
transfer time and a larger number of orbital revolutions.
CONCLUSIONS
The problem of minimum-time low-thrust LEO to HEO transfers with varying specific forces of
O(10−3 ) have been considered. The low-thrust Earth-orbit transfer problem was described in detail
and posed as a nonlinear optimal control problem. The optimal control problem was then solved
using a variable-order Gaussian quadrature collocation method. Initial guesses were generated by
solving a series of optimal control sub-problems until the desired terminal conditions were reached
within a specified tolerance. The results of this research identified power fit regressions such that
for a given specific force, the transfer time, number of revolutions, and final mass can be estimated.
Furthermore, the results highlighted key features of the optimal trajectories and examined the performance of the collocation method.
ACKNOWLEDGMENTS
The authors gratefully acknowledge support for this research from the National Aeronautics and
Space Administration under Grant NNX12AF98G.
REFERENCES
[1] Haissig, C. M., Mease, K. D., and Vinh, N. X., “Minimum-Fuel, Power-Limited Transfers Between
Coplanar Elliptic Orbits,” Acta Astronautica, Vol. 29, No. 1, January 1993, pp. 1–15.
[2] Scheel, W. A. and Conway, B. A., “Optimization of Very-Low-Thrust, Many-Revolution Spacecraft
Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 6, November-December 1994,
pp. 1185–1192.
[3] Tang, S. and Conway, B. A., “Optimization of Low-Thrust Interplanetary Trajectories Using Collocation and Nonlinear Programming,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 3,
May-June 1995, pp. 599–604.
[4] Rauwolf, G. A. and Coverstone-Carroll, V. L., “Near-Optimal Low-Thrust Orbit Transfers Generated by
a Genetic Algorithm,” Journal of Spacecraft and Rockets, Vol. 33, No. 6, November-December 1996,
pp. 859–862.
[5] Kluever, C. A. and Oleson, S. R., “Direct Approach for Computing Near-Optimal Low-Thrust EarthOrbit Transfers,” Journal of Spacecraft and Rockets, Vol. 35, No. 4, July-August 1998, pp. 509–515.
19
[6] Herman, A. L. and Spencer, D. B., “Optimal, Low-Thrust Earth-Orbit Transfers Using Higher-Order
Collocation Methods,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 1, January-February
2002, pp. 40–47.
[7] Coverstone-Carroll, V., Hartmann, J. W., and Mason, W. J., “Optimal Multi-Objective Low-Thrust
Spacecraft Trajectories,” Computer Methods in Applied Mechanics and Engineering, Vol. 186, No. 2-4,
June 2000, pp. 387–402.
[8] Betts, J. T., “Very low-thrust trajectory optimization using a direct SQP method,” Journal of Computational and Applied Mathematics, Vol. 120, No. 1-2, August 2000, pp. 27–40.
[9] Petropoulos, A. and Longuski, J. M., “Shaped-Based Algorithm for the Automated Design of LowThrust, Gravity Assist Trajectories,” Journal of Spacecraft and Rockets, Vol. 41, No. 5, SeptemberOctober 2004, pp. 787–796.
[10] Haberkorn, T., Martinon, P., and Gergaud, J., “Low-Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 6, 2004.
[11] Ross, I. M., Gong, Q., and Sekhavat, P., “Low-Thrust, High-Accuracy Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4, July-August 2007, pp. 921–933.
[12] Yang, G., “Direct Optimization of Low-thrust Many-revolution Earth-orbit Transfers,” Chinese Journal
of Aeronautics, Vol. 22, No. 4, August 2009, pp. 426–433.
[13] Falck, R. D. and Dankanich, J. D., “Optimization of Low-Thrust Spiral Trajectories by Collocation,”
No. 2012-4423, AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, August 2012.
[14] Benson, D. A., Huntington, G. T., Thorvaldsen, T. P., and Rao, A. V., “Direct Trajectory Optimization
and Costate Estimation via an Orthogonal Collocation Method,” Journal of Guidance, Control, and
Dynamics, Vol. 29, No. 6, November-December 2006, pp. 1435–1440.
[15] Huntington, G. T., Benson, D. A., and Rao, A. V., “Optimal Configuration of Tetrahedral Spacecraft
Formations,” Journal of the Astronautical Sciences, Vol. 55, No. 2, April-June 2007, pp. 141–169.
[16] Rao, G. T. H. A. V., “Optimal Reconfiguration of Tetrahedral Spacecraft Formations Using the Gauss
Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 3, May-June 2008,
pp. 141–169.
[17] Rao, A. V., Benson, D. A., Darby, C. L., Francolin, C., Patterson, M. A., Sanders, I., and Huntington, G. T., “Algorithm 902: GPOPS, A Matlab Software for Solving Multiple-Phase Optimal Control
Problems Using the Gauss Pseudospectral Method,” ACM Transactions on Mathematical Software,
April-June 2010.
[18] Garg, D., Patterson, M. A., Hager, W. W., Rao, A. V., Benson, D. A., and Huntington, G. T., “A Unified
Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral Methods,”
Automatica, Vol. 46, No. 11, November 2010, pp. 1843–1851.
[19] Garg, D., Patterson, M. A., Darby, C. L., Francolin, C., Huntington, G. T., Hager, W. W., and Rao, A. V.,
“Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal
Control Problems via a Radau Pseudospectral Method,” Computational Optimization and Applications,
Vol. 49, No. 2, June 2011, pp. 335–358.
[20] Rao, A. V., Patterson, M. A., and Hager, W. W., “A ph Collocation Scheme for Optimal Control,”
Automatica, January 2013.
[21] Patterson, M. A. and Rao, A. V., “GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal
Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear
Programming,” ACM Transactions on Mathematical Software, Vol. 39, No. 3, July 2013.
[22] Walker, M. J. H., Owens, J., and Ireland, B., “A Set of Modified Equinoctial Orbit Elements,” Celestial
Mechanics, Vol. 36, No. 4, 1985, pp. 409–419.
[23] Prussing, J. E. and Conway, B. A., Orbital Mechanics, Oxford University Press, 2nd ed., 2013.
[24] Patterson, M. A. and Rao, A. V., “Exploiting Sparsity in Direct Collocation Pseudospectral Methods for
Solving Optimal Control Problems,” Journal of Spacecraft and Rockets, Vol. 49, No. 2, March-April
2011, pp. 364–377.
[25] Biegler, L. T. and Zavala, V. M., “Large-Scale Nonlinear Programming Using IPOPT: An Integrating
Framework for Enterprise-Wide Optimization,” Computers and Chemical Engineering, Vol. 33, No. 3,
March 2008, pp. 575–582.
[26] Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
20