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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

2R2+Θ2

r2−µ

r1−1

2J2

α2

r23 sin2θ1−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

r41−3 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

2r31−3z

r2(15)

or by the Lagrangian

L(x, y, z, ˙x, ˙y, ˙z) = 1

2˙x2+ ˙y2+ ˙z2+

µ

r+µJ2α2

2r31−3z

r2(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

r31−3

2J2α

r25z2

r2−1 (18)

¨y=−µy

r31−3

2J2α

r25z2

r2−1 (19)

¨z=−µz

r31−3

2J2α

r25z2

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

Θ2

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

Hr0, R0,Θ0, N 0=1

2 R02+Θ02

r02!−µ

r0+

Θ02

2r02

α2

p02J21−3

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

dR0,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∇pHqn+1/2, pn(42)

Symplectic orbit propagation based on Deprit’s radial intermediary 379

pn+1 =pn−h

2∇qHqn+1/2, pn+

∇qHqn+1/2, pn+1(43)

qn+1 =qn+1/2+h

2∇pHqn+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∇qHqn, pn+1/2(46)

qn+1 =qn+h

2∇pHqn, pn+1/2+

∇pHqn+1, pn+1/2(47)

pn+1 =pn+1/2−h

2∇qHqn+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

2Θ2

r3−µ

r2+κ

2r3Θ21−3N2

Θ2 (58)

Θ1/2= Θ (59)

N1/2=N(60)

The new positions are

r+=r+hR1/2(61)

θ+=θ+h

2Θ

r2−κ

2r2Θ3+3κN2

r2Θ5+

h

2Θ

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

2Θ2

r3

+

−µ

r2

+

+κ

2r3

+Θ21−3N2

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=p1−3

2sin2I−1

2sin2Icos 2θ(A5)

∆θ=3

2−7

4sin2I+2−3 sin2Iϕsin 2θ−

5−6 sin2I+1−2 sin2Icos 2θσ(A6)

∆ν= cos I(3 + cos2θ)σ−3

2+ 2ϕsin 2θ(A7)

∆R=Θ

p(1 + ϕ)2sin2Isin 2θ(A8)

∆Θ = −Θ sin2I3

2+ 2ϕcos 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

˜

Θ1 + 2−12 cos2Iε(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.