Symplectic orbit propagation based on Deprit’s radial intermediary

Abstract

The constantly challenging requirements for orbit prediction have opened the need for better onboard propagation tools. Runge-Kutta (RK) integrators have been widely used for this purpose; however RK integrators are not symplectic, which means that RK integrators may lead to incorrect global behavior and degraded accuracy. Emanating from Deprit's radial intermediary, obtained by the elimination of the parallax transformation, we present the development of symplectic integrators of different orders for spacecraft orbit propagation. Through a set of numerical simulations, it is shown that these integrators are more accurate and substantially faster than Runge-Kutta-based methods. Moreover, it is also shown that the proposed integrators are more accurate than analytic propagation algorithms based on Deprit's radial intermediary solution, and even other previously-developed symplectic integrators.

Full text

Astrodynamics Vol. 2, No. 4, 375–386, 2018 https://doi.org/10.1007/s42064-018-0033-x
Symplectic orbit propagation based on Deprit’s radial intermediary
Leonel Palacios (B), Pini Gurfil
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
ABSTRACT
The constantly challenging requirements for orbit prediction have opened the need for better
onboard propagation tools. Runge–Kutta (RK) integrators have been widely used for this
purpose; however RK integrators are not symplectic, which means that RK integrators may
lead to incorrect global behavior and degraded accuracy. Emanating from Deprit’s radial
intermediary, obtained by the elimination of the parallax transformation, we present the
development of symplectic integrators of different orders for spacecraft orbit propagation.
Through a set of numerical simulations, it is shown that these integrators are more accurate
and substantially faster than Runge–Kutta-based methods. Moreover, it is also shown that
the proposed integrators are more accurate than analytic propagation algorithms based
on Deprit’s radial intermediary solution, and even other previously-developed symplectic
integrators.
KEYWORDS
symplectic integration
spacecraft orbit propagation
Deprit’s radial intermediary
symplecticity
Hamiltonian dynamics
Research Article
Received: 05 November 2017
Accepted: 26 May 2018
©2018 Tsinghua University
Press
1 Introduction
The need to propagate a satellite orbit, possibly for long
time spans, without onboard position measurements,
is a topic of special interest, because in some small
satellites there are no GPS receivers due to insufficient
power. Even if a GPS receiver is available, it is
susceptible to malfunctions or degraded accuracy due
to geometric dilution of precision. In these cases, orbit
prediction starts from a given epoch, and propagated by
integrating orbital models. To that end, Runge–Kutta
integrators are widely used. However, these integrators
are not symplectic, meaning that the resulting solutions
exhibit continuously growing energy errors and loss of
physical fidelity, compromising the global behavior of
the dynamical system in consideration, and leading to
accuracy degradation. A relatively high computational
cost is also associated with these types of integrators,
especially when complex dynamical models are used
and/or long time spans are required.
Analytic alternatives to numerical propagation, such
as the use of intermediary orbits in the main problem
of artificial satellite theory, have been presented
in the literature [1,2]. One of the main features of
using intermediaries is that the phase space dimension of
Blmmoreno@princeton.edu
the dynamical system is reduced, guaranteeing that
the residual between the original problem and the
intermediary is free from first-order secular effects.
Additionally, the use of intermediary solutions implies
that no differential equations have to be integrated
onboard. Relevant work on intermediaries has been
developed by Cid and Lahulla [3,4] using polar-
nodal variables and a contact transformation. This
operation removes the argument of latitude from the
main problem Hamiltonian, resulting in an integrable
problem whose solution is expressed in terms of elliptic
integrals. Then, a more general class, called natural
intermediaries, was introduced by Deprit [5], rendering
the main problem Hamiltonian integrable after a contact
transformation that turns such a problem into the
intermediary, while allowing the inclusion of additional
first-order, short-periodic effects. Specifically, Deprit’s
radial intermediary (DRI) provides analytic solutions
free of elliptic integrals, as opposed to the Cid-
Lahulla intermediary, leading to faster computational
evaluation. Given that fast computations are required
in onboard propagators, Gurfil and Lara [6] proposed a
method that reorganizes the DRI solution into a more
convenient way to improve computational time. They
also presented a comparison against direct numerical

376 L. Palacios, P. Gurfil
integration, showing that their algorithm is faster than
Runge–Kutta methods and even more accurate when
using large time steps. Later, Lara [7] has introduced
an improved version that includes second-order secular
and periodic terms of the main problem.
Continuing with the efforts to find alternatives to
generic numerical integrators for onboard propagation,
in the present research we propose the use of geometric
numerical integrators as propagation tools. Contrary
to generic integration methods, this type of integrators
provide high processing speed and stability by
incorporating the underlying geometric properties of
the dynamical system under consideration, leading
to improved qualitative behavior, favorable error pro-
pagation, accurate and faster long-time integration [8].
A comprehensive work has been published in treatises
such as the one published by Blanes and Casas [8],
Hairer and Warner [9], Feng and Qin [10]. Specifically,
symplectic integrators were shown to be structure-
preserving numerical integrators. There have been
advances within the context of such integrators, for
instance, in the work done by Simo et al. [11]
the necessary conditions for the conservation of
symplecticity and energy and momentum were provided.
Other examples of such advances are the works by
Yoshida [12], with the construction of high-order
symplectic methods through composition or Blanes et
al. [13] and Farres et al. [14] with the development of
high-precision symplectic integrators for astronomy.
Symplectic integration techniques have also been
developed to propagate the motion of spacecraft. An
example of such work is the one done by Tsuda
and Scheeres [15], where a numerical method for
deriving a symplectic state transition matrix for an
arbitrary Hamiltonian dynamical system was presented.
This matrix was applied to long-term propagation of
spacecraft, among other applications. Additionally, the
work of Imre and Palmer [16] described a symplectic
numerical method to propagate relative orbits using an
arbitrary number of zonal and tesseral terms in the
geopotential.
In this paper, we develop 4th and 6th order symplectic
integrators for the propagation of satellite orbit based
on Deprit’s radial intermediary. Then, its performance
is compared against Runge–Kutta numerical methods,
the DRI solution algorithm presented in [6], and a
symplectic integrator in Cartesian variables developed
by the Authors in a previous work [17] (in which
the effect of drag on the symplectic and variational
integrators has been extensively investigated, showing
their superior performance over other integrators).
This paper is organized as follows. The main problem
in artificial satellite theory as well as background on
Deprit’s radial intermediary are presented in Section
2. Next, in Section 3, the development of symplectic
integrators is presented. Numerical simulations and
propagator performance analysis can be found in
Section 4.
2 The main problem Hamiltonian
The motion of a spacecraft around an oblate Earth can
be described using the Whittaker polar-nodal chart [18]
with the variables
r, θ, ν, R, Θ, N 
where ris distance to the attraction center, θis the
argument of latitude, νis the right ascension of the
ascending node, R= dr/dtis the radial velocity, Θ is
the modulus of the angular momentum vector, and N
is the polar component of the angular momentum. It is
then possible to map from polar-nodal variables to the
classical orbital elements:
a, e, I, Ω, ω, f 
namely the semimajor axis, eccentricity, inclination,
right ascension of the ascending node, argument of
periapsis and true anomaly, respectively, through the
relations
a=−µr2
r2R2+ Θ2−2µr (1)
e=s1−Θ2
µa (2)
I= cos−1N
Θ(3)
Ω = ν(4)
cos f=Θ2−µr
µre ,sin f=ΘR
µe (5)
ω=θ−f(6)
where µis the gravitational parameter. Then, the J2-
perturbed dynamics of the spacecraft, also known as
the main problem in artificial satellites theory, can be

Symplectic orbit propagation based on Deprit’s radial intermediary 377
defined by the main problem Hamiltonian:
H(r, θ, R, Θ, N )
=1
2R2+Θ2
r2−µ
r1−1
2J2
α2
r23 sin2θ1−N2
Θ2−1
(7)
where the constans αand J2are the mean equatorial
radius and the second order zonal harmonic coefficient,
respectively. Next, from Hamilton equations [19], we
can obtain the relations:
d (r, θ, ν)
dt=∂H
∂(R, Θ, N )and d (R, Θ, N)
dt=−∂H
∂(r, θ, ν)
(8)
which lead to the final form of the equations of motion,
˙r=R(9)
˙
θ=Θ
r2+2λsin2θcos2I
Θr3(10)
˙ν=−2λcos Isin2θ
Θr3(11)
˙
R=−µ
r2+Θ2
r3−λ
r41−3 sin2Isin2θ(12)
˙
Θ = −λsin2Isin 2θ
r3(13)
˙
N= 0 (14)
with the constant λdefined as
λ=3µJ2α2
2
In inertial Earth-centered Cartesian coordinates,
defined by the unit vectors (ix,iy,iz), the motion of
a spacecraft may also be described with the Cartesian
variables x, y, z, ˙x, ˙y, ˙z
where the first three terms correspond to the
components of the position vector of the spacecraft
r=xix+yiy+ziz, the next ones to the velocity
vector v= ˙xix+ ˙yiy+ ˙zizand ˙
ζ= dζ/dt. With this
in consideration, the Cartesian J2-perturbed absolute
motion of a spacecraft around the Earth is described by
the Hamiltonian [20]
H(x, y, z, ˙x, ˙y, ˙z) = 1
2˙x2+ ˙y2+ ˙z2−
µ
r−µJ2α2
2r31−3z
r2(15)
or by the Lagrangian
L(x, y, z, ˙x, ˙y, ˙z) = 1
2˙x2+ ˙y2+ ˙z2+
µ
r+µJ2α2
2r31−3z
r2(16)
Next, from the the Euler-Lagrange equations [19], we
obtain the relations
d
dt
∂L
∂( ˙x, ˙y, ˙z)−∂L
∂(x, y, z)= 0 (17)
and the equations of motion are expressed as
¨x=−µx
r31−3
2J2α
r25z2
r2−1 (18)
¨y=−µy
r31−3
2J2α
r25z2
r2−1 (19)
¨z=−µz
r31−3
2J2α
r25z2
r2−3 (20)
The relations between the Cartesian coordinates and
the polar-nodal variables can be found as follows [6]. Let
the angular momentum vector be h=r×v. Define the
unit vectors n1and n2as
n1=


h×iz
kh×izkif h×iz6=0
ixif h×iz=0
and n2=ˆ
h×n1
(21)
Then
r=krk(cos θ=ˆ
r·n1
sin θ=ˆ
r·n2(cos ν=ix·n1
sin ν=iy·n1
(22)
and
R=r·v
rΘ = khkN=h·iz(23)
The inverse transformation can be found using the
relations:
Rν=


cos ν−sin ν0
sin νcos ν0
0 0 1



RI=


1 0 0
0 cos I−sin I
0 sin Icos I



Rθ=


cos θ−sin θ0
sin θcos θ0
0 0 1


(24)
and:
r=RνRIRθhr0 0 iT
v=h×r
r2+Rr
r(25)

378 L. Palacios, P. Gurfil
where
cos I=N
Θsin I=s1−N
Θ2
h=RνRIh00ΘiT(26)
It is well known that the main problem presented in
this section is not integrable except for the equatorial
case [21,22], and several approximate solutions to the
dynamics obtained from the Hamiltonian in Eq. (7)
have been proposed. In this work, we are particularly
interested in the solution obtained by using Deprit’s
radial intermediary [5]. Deprit’s radial intermediary is
obtained after reducing the main problem Hamiltonian
in Eq. (7) by applying the elimination of the parallax,
which is a canonical transformation formulated as
(r, θ, ν, R, Θ, N )7→ r0, θ0, ν0, R0,Θ0, N 0
which removes short-periodic terms of the original
Hamiltonian without reducing the number of degrees
of freedom. The main problem Hamiltonian is written
using Deprit’s radial intermediary as
Hr0, R0,Θ0, N 0=1
2 R02+Θ02
r02!−µ
r0+
Θ02
2r02
α2
p02J21−3
2sin2I0(27)
with
p0=Θ02
µand sin2I0= 1 −N02
Θ02(28)
and the first-order contact transformation required to
obtain the primed variables is presented in Appendix
A. Then, from Hamilton’s equations, we obtain the
relations
d (r0, θ0, ν0)
dt=∂H
∂(R0,Θ0, N 0)and
dR0,Θ0, N 0
dt=−∂H
∂(r0, θ0, ν0)(29)
and the equations of motion
˙r0=R0(30)
˙
θ0=Θ0
r02−κ
2r02Θ03+3κN02
r02Θ05(31)
˙ν0=−3κN 0
2r02Θ04(32)
˙
R0=Θ02
r03−µ
r02+κ
2r03Θ02−3κN02
2r03Θ04(33)
˙
Θ0= 0 (34)
˙
N0= 0 (35)
with the constant κdefined as
κ=µ2J2α2(36)
With Deprit’s intermediary, it is also possible to
obtain closed-form solutions in terms of trigonometric
functions accurate up to the first order of J2. Such
solutions not only account for secular and long-period
terms, but also for short-periodic effects.
3 Symplectic numerical integration
Several types of geometric integrator have been
developed and presented in literature, however the most
known could be the so-called symplectic propagation
algorithms, which are obtained from Hamiltonian
systems [8]. In this work, we pay special attention to this
type of integrators, because they have shown favorable
performance for different mechanical systems, including
the motion of spacecraft.
3.1 Basic construction of symplectic
integrators
The basic method Φh
2used in this work is known as
the St¨ormer–Verlet method [23]. This method has
second-order accuracy and it can be obtained using the
composition operation
Φh
2= Φh/2
1◦Φh/2
1∗(37)
where Φh/2
1is the map corresponding to the first-order
symplectic Euler method
qn+1 =qn+h∇pH(qn+1, pn) (38)
pn+1 =pn−h∇qH(qn+1, pn) (39)
and Φh/2
1∗is its adjoint method,
qn+1 =qn+h∇pH(qn, pn+1) (40)
pn+1 =pn−h∇qH(qn, pn+1) (41)
the constant hbeing the selected time step [8,17]. The
operation in Eq. (37) leads to
qn+1/2=qn+h
2∇pHqn+1/2, pn(42)

Symplectic orbit propagation based on Deprit’s radial intermediary 379
pn+1 =pn−h
2∇qHqn+1/2, pn+
∇qHqn+1/2, pn+1(43)
qn+1 =qn+1/2+h
2∇pHqn+1/2, pn+1(44)
where qcorresponds to position, pto momenta and the
subindex nstands for the current number of iteration.
It is also possible to use the composition operation
Φh
2= Φh/2
1∗◦Φh/2
1(45)
to obtain the alternative method
pn+1/2=pn−h
2∇qHqn, pn+1/2(46)
qn+1 =qn+h
2∇pHqn, pn+1/2+
∇pHqn+1, pn+1/2(47)
pn+1 =pn+1/2−h
2∇qHqn+1, pn+1/2(48)
Arbitrarily high-order versions of basic symplectic
methods can be obtained using specific sequences of
compositions, while preserving the desirable geometric
properties of the basic method. One of the commonly
used approaches to increase the order of the basic
method is known as triple jump composition or the
Suzuki–Yoshida technique [12,24]. If, for instance, a
fourth-order method is to be obtained with Φh
2as the
basic method, then the operation
Φh
4= Φγh
2◦Φβh
2◦Φγh
2(49)
may be used along with the constants γand βdefined
as
γ=1
2−21/3and β= 1 −2γ(50)
It is important to mention that, in order to build
a fourth-order method using γand βas defined in
Eq. (50), it is necessary that the basic method be at
least second order. However, this composition method
can be applied, in general, to a method Φh
2kof order
2kas
Φh
2k+2 = Φγh
2k◦Φβh
2k◦Φγh
2k(51)
with
γ=1
2−21/(2k+1) (52)
and βdefined as in Eq. (50). Following a similar
line of thought, a sixth-order method [12] can also
be obtained through composition operations departing
from the basic method Φh
2as
Φh
6= Φw3h
2◦Φw2h
2◦Φw1h
2◦Φw0h
2◦Φw1h
2◦Φw2h
2◦Φw3h
2(53)
using the constants
w1=−1.17767998417887 (54)
w2= 0.235573213359357 (55)
w3= 0.784513610477560 (56)
w0= 1 −2 (w1+w2+w3) (57)
3.2 Symplectic integrator for Deprit’s radial
intermediary
In this section, we derive a symplectic integrator derived
from Deprit’s radial intermediary in Eq. (27) and the
basic symplectic method Φh
2presented in Eqs. (46–
48). Dropping the primes in the polar nodal variables
and using the notation simplification qn+1/2=q1/2and
qkn+1 =q+we have
R1/2=R+h
2Θ2
r3−µ
r2+κ
2r3Θ21−3N2
Θ2 (58)
Θ1/2= Θ (59)
N1/2=N(60)
The new positions are
r+=r+hR1/2(61)
θ+=θ+h
2Θ
r2−κ
2r2Θ3+3κN2
r2Θ5+
h
2Θ
r2
+
−κ
2r2
+Θ3+3κN2
r2
+Θ5(62)
ν+=ν+h
2−3
2
κN
Θ4r2+h
2−3
2
κN
Θ4r2
+(63)
and the new momenta
R+=R1/2+h
2Θ2
r3
+
−µ
r2
+
+κ
2r3
+Θ21−3N2
Θ2 (64)
Θ+= Θ1/2(65)
N+=N1/2(66)
It is important to remember that all variables in this
symplectic integrator are primed. This integrator is of
second order, fully explicit, and requires the contact
transformation from primed to original variables, as
presented in Appendix A.

380 L. Palacios, P. Gurfil
3.3 Cartesian symplectic integrator
The symplectic integrator presented in this section,
obtained from Cartesian variables, was developed by the
Authors in Ref. [17]. Using the Hamiltonian in Eq. (15),
the second-order symplectic method Φh
2presented in
Eqs. (42–44) yields the half-indexed positions
x1/2=x+h
2X(67)
y1/2=y+h
2Y(68)
z1/2=z+h
2Z(69)
Next, the new momenta are defined as
X+=X+hfx(70)
Y+=Y+hfy(71)
Z+=Z+hfz(72)
with fx,fyand fzdefined as
fx=−µx1/2
r3
1/2"1−3
2J2
α2
r2
1/2 5z2
1/2
r2
1/2
−1!# (73)
fy=−µy1/2
r3
1/2"1−3
2J2
α2
r2
1/2 5z2
1/2
r2
1/2
−1!# (74)
fz=−µz1/2
r3
1/2"1−3
2J2
α2
r2
1/2 5z2
1/2
r2
1/2
−3!# (75)
and r1/2=qx2
1/2+y2
1/2+z2
1/2. Finally, the new
positions complete the method:
x+=x1/2+h
2X+(76)
y+=y1/2+h
2Y+(77)
z+=z1/2+h
2Z+(78)
This algorithm is fully explicit and its performance
will be examined in a set of simulated scenarios in
the next section, along with the symplectic integrator
presented in Section 3.2.
4 Numerical simulations
In this section, several performance parameters are
obtained to test the efficiency of the integration methods
presented in this work. These are provided in terms of
Hamiltonian, position and velocity errors defined as
∆H=kH(t)−H0k
kH0kand ∆A=kA−Aref k(79)
where the generic variable Astands for either the
position or velocity vector. The performance parameters,
obtained by the symplectic integrators through different
time steps, are compared to those obtained by
two generic integrators, the DRI solution algorithm
presented in Ref. [6], and the symplectic integrator
in Cartesian variables developed by the Authors in
Ref. [17]. The generic integrators selected for the
comparison are the 4th order, fixed-step, classical
Runge–Kutta integrator (CRK4) and the 4th order,
fixed-step, Runge–Kutta due to Dormand and Prince
(DP4). Additionally, reference “ground truth” is
obtained using an 8th order, variable step, Runge–Kutta
integrator due to Dormand and Prince [25]. These last
three integrators will propagate the motion obtained
from the full J2equations in inertial Earth–centered
Cartesian coordinates presented in Eqs. (18–20).
All the simulations and figure plots presented in this
section were carried out on a PC with a processor
Intel(R) Core(TM) i5-3570 with 3.40 GHz, 4.00 GB of
RAM and MATLAB 2017a. In the following sections,
the simulation time corresponds to 100 orbits and the
time step is 50 seconds (unless otherwise told). To
facilitate reading, the acronyms presented in Table 1 are
used. Additionally, the implementation algorithm for
the symplectic integrator in Section 3.2 is summarized
in Table 2. The spacecraft initial osculating orbital
elements and initial Cartesian absolute state can be
found in Table 3 and Table 4, respectively, and the
parameter values used in the simulations are
µ= 398 600.4415 km3/s2, α = 6378.1363 km,
J2= 1.0826266 ×10−3
4.1 4th Order symplectic integrator
The simulations starts with the 4th order symplectic
algorithms. Figure 1 shows that SY4, SYC4 and DGL
preserve the Hamiltonian, with SY4 and DGL having
almost the same maximum Hamiltonian error values of
5.9043×10−7and 5.886×10−7, respectively, while SYC4
yields a maximum ∆Hof 5.51753 ×10−8. In the same
figure, it is also noticed that DP4 and CRK4 present

Symplectic orbit propagation based on Deprit’s radial intermediary 381
Table 1 Acronyms used in the simulations
Acronym Meaning
DP8 8th order, adaptive step, Runge–Kutta method
due to Dormand and Prince
DP4 4th order, fixed step, Runge–Kutta method due
to Dormand and Prince
CRK4 Classical 4th order Runge–Kutta method
SY4, SY6 4th & 6th order explicit symplectic integrator
as in Eqs. (58–66)
SYC4, SYC6 4th & 6th order explicit symplectic integrator
as in Eqs. (67–78)
DGL Deprit-Gurfil-Lara analytic propagator as in
Appendix B
Table 2 Algorithm 1: Propagation of the symplectic integrator
1: input: initial and final time, time step, initial state
(r0, θ0, ν0, R0,Θ, N ) of the spacecraft
2: Transform initial state from original to primed variables
using Appendix A
3: for every time step
4: if desired order is 4 then
5: Use composition in Eq. (49) with constants from Eq. (50)
6: Propagate the state of the spacecraft with Eqs. (58–66)
7: Transform the state from primed to original space using
Appendix A
8: else if desired order is 6 then
9: Use composition in Eq. (53) with constants from
Eq. (54–57)
10: Repeat steps 6 and 7
11: end if
12: end for
Table 3 Summary of initial osculating orbit elements
a(km) e I Ωω θ
7000 0.005 55◦0◦10◦25◦
Table 4 Summary of initial Cartesian absolute positions and
velocities
x(km) y(km) z(km) ˙x(km/s) ˙y(km/s) ˙z(km/s)
6313.5040 1688.6292 2411.6125 −3.1956 3.9440 5.6327
continuous error growth reaching, to maximum values
of 1.6134 ×10−7and 7.9941 ×10−6, respectively.
Figure 2 shows that DGL and CRK4 produce the
largest position errors, with maximum final values of
0.5796 km and 0.1664 km, respectively. SYC4 yields a
maximum position error value of 0.0489 km. The lowest
errors are provided by SY4 and DP4, showing values of
Fig. 1 Hamiltonian error.
Fig. 2 Position and velocity errors.
0.0098 km and 0.0035 km, respectively. Similar results
are found in terms of velocity errors. DGL presents
the largest error with a value of 1.243 ×10−3km/s,
followed by CRK4 with 1.6793 ×10−4km/s, SYC4 with
7.3752 ×10−5km/s, SY4 with 8.2043 ×10−6km/s
and finally DP4 with 3.4963 ×10−6km/s. Despite
the apparent advantage presented by DP4 in terms of
position and velocity errors, it is the slowest of the
methods with a computational time of 0.1706 seconds.
On the other hand, DGL, SY4 and SYC4 required the
shortest computational times with 0.0247, 0.0183 and
0.0152 seconds, which is ∼85.50%, ∼89.27% and
∼91.07 faster than DP4, respectively. Compared to
CRK4, whose computational time is 0.1136 seconds,
these last propagators are ∼78.23%, ∼83.89% and
∼86.59% faster.
The error trends presented in the previous simulation

382 L. Palacios, P. Gurfil
are maintained though different time steps. For the
next simulations, the same initial conditions and final
simulation time are used, but the time steps are 0.1, 1,
20, 40, 60, 80, 100, 120, 140, 160, 180 and 200 seconds.
The maximum Hamiltonian error is observed in Fig. 3,
where it can be noticed that both SY4 and DGL are
the only integrators preserving the Hamiltonian, while
SYC4, CRK4 and DP4 show a continuous error growth
rate through the different time steps.
Figure 4 shows that DGL presents the lowest position
and velocity errors, with almost constant values, closely
followed by SY4, DP4 and SYC4. However, the
error values for CRK4 are notably large, with final
position and velocity errors above 120 km and 0.1 km/s,
respectively. The computational time advantage is also
retained by SYC4, SY4 and DGL, as indicated in Fig. 5,
SYC4 being the fastest and DP4 the slowest.
Fig. 3 Maximum Hamiltonian error vs time step.
Fig. 4 Maximum position and velocity errors vs time step.
Fig. 5 Computational time vs time step.
4.2 6th Order symplectic integrator
Next, the performance of the 6th order symplectic
integrators is tested using the previous scenario
conditions. SY6 presents no significant improvement
with respect to Hamiltonian error with a maximum
error value of 5.9043 ×10−7, but SYC6 does improve
substantially with a value of 1.4279 ×10−11 as observed
in Fig. 6.
Both integrators also present better error values in
terms of position and velocity as in Fig. 7. For instance,
the SY6 maximum position and velocity error values
are 7.4631 ×10−3km and 6.6537 ×10−6km/s, while
for SYC6 they are 3.1188 ×10−5km and 4.0833 ×
10−8km/s, respectively. Although SY6 error values
are not visible in Fig. 7 in comparison with Fig. 2,
they represent an improvement of 2.4 m in position
and 0.0016 m/s in velocity error. Nevertheless, SYC6
is the integrator with the highest precision for the
current selection of final time and time step. Even
though the composition procedure to increase the order
of the symplectic integrators requires additional steps,
this process does not have a significant impact on the
computational time. SY6 presents a computational time
of 0.0219 seconds, that is ∼78.71% faster than CRK4
and ∼85.95% faster than DP4 (similar to those values
obtained by DGL). On the other hand, SYC6 requires
Fig. 6 Hamiltonian error.

Symplectic orbit propagation based on Deprit’s radial intermediary 383
Fig. 7 Position and velocity errors.
0.0187 seconds, which is ∼81.80% faster than CRK4
and ∼87.99% faster than DP4.
SY6 preserves the maximum Hamiltonian error
through different time steps, as observed in Fig. 8, and
although SYC6 produces a growing Hamiltonian error
for larger time step, the resulting values are the lowest
among the integrators in consideration. In terms of
position errors, SY6 present improvements producing
now the smallest error for all time steps, followed closely
by SYC6 and DGL, as indicated in the magnified view,
within Fig. 9. A similar trend can be observed in terms
of velocity errors in the same figure. Next, in Fig. 10 it is
noticed that SYC6 is slightly faster than SY6 and DGL,
and this tendency is sustained during the whole set of
time steps. The same figure shows that SY6, SYC6 and
DGL have a clear gap in terms of computational time
Fig. 8 Maximum Hamiltonian error vs time step.
Fig. 9 Maximum position and velocity errors vs time step.
Fig. 10 Computational time vs time step.
with respect to CRK4 and DP4. A summary of the
results obtained in this section is presented in Table 5.
Table 5 Summary of results (only maximum values and h=
50 sec.)
Method Pos. error Vel. error H error Comp.
(km) (km/s) (Norm.) time (s)
CRK4 0.1664 1.6793×10−04 7.9941×10−06 0.1136
DP4 3.5064×10−03 3.4963×10−06 1.6134×10−07 0.1706
DGL 0.57965 1.243×10−03 5.8860×10−07 0.0247
SY4 9.8683×10−03 8.2043×10−06 5.9043×10−07 0.0183
SYC4 0.048952 7.3752×10−05 5.5175×10−08 0.0152
SY6 7.4631×10−03 6.6537×10−06 5.9043×10−07 0.0219
SYC6 3.1188×10−05 4.0833×10−08 1.4279×10−11 0.0187

384 L. Palacios, P. Gurfil
5 Conclusions
Explicit symplectic integrators of different orders were
obtained based on DRI. These were applied to propagate
the orbital motion of spacecraft including the effect of J2,
yielding more accurate results and substantially faster
processing speeds in comparison with Runge–Kutta
integrators. Furthermore, the developed integrators
produce more accurate results than those obtained with
DRI-based analytic propagators.
A set of simulated scenarios was used to verify the results
in terms of Hamiltonian, position and velocity errors
using a set of different time steps. Throughout these
simulations, the symplectic integrators consistently
sustained the lowest errors and computational times
with respect to the selected contenders.
Appendix A. First order contact trans-
formation for Deprit’s radial intermediary
The transformation from prime polar-nodal variables to
original ones, and vice versa, requires the computation
of the first order corrections [6,7]:
∆ξ=δξ (A1)
where ξis any of the polar nodal variables and the
variable δis defined as
δ=−1
2J2
α2
p2(A2)
These corrections, as well as δ, must be expressed in
prime variables for the direct transformation
∆ξd=ξ−ξ0(A3)
and in original variables for the inverse transformation
∆ξi=ξ0−ξ(A4)
Then, the first order corrections are the same for both
the direct and inverse transformations, and they are
expressed as
∆r=p1−3
2sin2I−1
2sin2Icos 2θ(A5)
∆θ=3
2−7
4sin2I+2−3 sin2Iϕsin 2θ−
5−6 sin2I+1−2 sin2Icos 2θσ(A6)
∆ν= cos I(3 + cos2θ)σ−3
2+ 2ϕsin 2θ(A7)
∆R=Θ
p(1 + ϕ)2sin2Isin 2θ(A8)
∆Θ = −Θ sin2I3
2+ 2ϕcos 2θ+σsin 2θ(A9)
∆N= 0 (A10)
where σ=pR/Θ and ϕ=p/r −1. It is important to
remember that the right-hand side terms in Eqs. (A5–
A10) must be expressed in prime variables for the direct
transformation, and in original variables in the case of
the inverse one.
Appendix B. Analytical propagation of
Deprit’s radial intermediary
The variables used in DRI are primed variables.
Therefore, all the variables presented in this
Appendix should be understood as primed variables.
Departing from primed polar nodal initial conditions
(r0, θ0, ν0, R0), with Θ and Nas constants, the analytic
propagation of the spacecraft is obtained using the
algorithm developed by Gurfil and Lara in Ref. [6].
First, from initial conditions, auxiliary constants are
defined as
cos I=N
Θ(B1)
ε=−1
4J2
α2
p2(B2)
˜
Θ = Θp1−(2 −6 cos2I)ε(B3)
τ=Θ
˜
Θ1 + 2−12 cos2Iε(B4)
χ= 6εN
˜
Θ(B5)
˜p=˜
Θ2
µ(B6)
a=−µ
R2
0+˜
Θ2/r2
0−2µ/r0
(B7)
e=r1−˜p
a(B8)
as well as the initial values of the anomalies
f0from ecos f0=˜p
r0
−1, e sin f0=R0s˜p
µ(B9)

Symplectic orbit propagation based on Deprit’s radial intermediary 385
and
u0= 2 arctan r1−e
1 + etan f0
2!(B10)
`0=u0−esin u0(B11)
Then, for a given time step, the motion is propagated
using the sequence
`=`0+rµ
a3t(B12)
ufrom `=u−esin u(B13)
f= 2 arctan r1 + e
1−etan u
2!(B14)
r=a(1 −ecos u) (B15)
θ=θ0+τ(f−f0) (B16)
ν=ν0+χ(f−f0) (B17)
R=µ
˜
Θesin f(B18)
together with the constant values of Θ and N.
Acknowledgements
This work was supported by the European Commission
Horizon 2020 Program in the framework of the Sensor
Swarm Sensor Network Project under grant agreement
687351.
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Leonel M. Palacios
is a postdoctoral
research associate at the High Contrast
Imaging Laboratory of Princeton
University. He received his Ph.D.
degree in aerospace engineering from
the University of Glasgow, the United
Kingdom, in 2016. In May 2016, he joined
the Asher Space Research Institute of
the Technion–Israel Institute of Technology as a postdoctoral
fellow working for the Satellite Swarm Sensor Network
(S3NET), a project aimed for the development of the full
potential of “swarms” of satellites through the optimized
and enhanced use of their on-board resources. His areas of
expertise include astrodynamics and dynamics and control of
distributed space systems. E-mail: lmmoreno@princeton.edu.
Pini Gurfil
is a full professor of
aerospace engineering at the Technion–
Israel Institute of Technology, and
director of the Asher Space Research
Institute. He received his Ph.D. in
aerospace engineering from the Technion
in March 2000. From 2000 to 2003, he was
with the Department of Mechanical and
Aerospace Engineering, Princeton University. In September
2003, he joined the Faculty of Aerospace Engineering at the
Technion. Dr. Gurfil is the founder and director of the
Distributed Space Systems Laboratory, a research laboratory
aimed at development and validation of spacecraft formation
flying algorithms and technologies. He has been conducting
research in astrodynamics, distributed space systems, and
satellite dynamics and control.