Content uploaded by Pini Gurfil

Author content

All content in this area was uploaded by Pini Gurfil on Apr 17, 2018

Content may be subject to copyright.

Celest Mech Dyn Astr (2018) 130:31

https://doi.org/10.1007/s10569-018-9826-8

ORIGINAL ARTICLE

Variational and symplectic integrators for satellite

relative orbit propagation including drag

Leonel Palacios1·Pini Gurﬁl2

Received: 9 October 2017 / Revised: 9 February 2018 / Accepted: 13 March 2018

© Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract Orbit propagation algorithms for satellite relative motion relying on Runge–

Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior

and degraded accuracy. Thus, attempts have been made to apply symplectic methods to inte-

grate satellite relative motion. However, so far all these symplectic propagation schemes

have not taken into account the effect of atmospheric drag. In this paper, drag-generalized

symplectic and variational algorithms for satellite relative orbit propagation are developed in

different reference frames, and numerical simulations with and without the effect of atmo-

spheric drag are presented. It is also shown that high-order versions of the newly-developed

variational and symplectic propagators are more accurate and are signiﬁcantly faster than

Runge–Kutta-based integrators, even in the presence of atmospheric drag.

Keywords Symplectic integration ·Variational integration ·Geometric numerical

integration ·Satellite relative motion ·Hamiltonian dynamics

1 Introduction

Generic integration methods, such as the Runge–Kutta variety, are widely used in engi-

neering, embedded in simulation software, or utilized as onboard propagators, but they are

non-symplectic. The resulting solutions exhibit continuously growing energy errors, and a

concomitant loss of physical ﬁdelity. This compromises the global behavior of the dynam-

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.

BLeonel Palacios

lmmoreno@princeton.edu

Pini Gurﬁl

pgurﬁl@technion.ac.il

1Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544-5263, USA

2Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, 3200003 Haifa, Israel

123

31 Page 2 of 26 L. Palacios, P. Gurﬁl

ical system in consideration and leads to accuracy degradation. Moreover, when complex

dynamical models are used and/or long-time spans are required, a number of these integra-

tion methods are usually associated with high computational cost.

Contrary to generic integration methods, geometric methods, which include variational

and symplectic integrators, provide low computational cost and stability by incorporating

the underlying geometric properties of the dynamical system under consideration, leading to

improved qualitative behavior, favorable error propagation, and faster long-time integration.

All these features are relevant for fast and accurate satellite orbit propagation, in particular

for satellites with limited computational power (e.g., CubeSats). In the present research,

we develop and compare the efﬁciency of geometric numerical integration vis-à-vis generic

numerical methods. In particular, we are interested in the use of symplectic and variational

numerical integration for satellite relative orbit propagation.

Extensive treatises have been written to present the foundations of geometric integrators.

Examples of such include Blanes and Casas (2016), Hairer et al. (2006), Leimkuhler and

Reich (2005) and Feng and Qin (2010). Speciﬁcally, these works show that the symplectic

and variational approaches provide a systematic way of constructing structure-preserving

numerical integrators. In the context of a variational integrators, Lew et al. (2004), Marsden

and West (2001) and Simo et al. (1992) provide the necessary conditions for the conservation

of symplecticity, energy and momentum. Symplectic methods have also seen recent advances

with the construction of high-order symplectic methods through composition (Yoshida 1990),

and with the development of high precision symplectic integrators for astronomy (Farres et al.

2013;Blanesetal.2013).

The use of variational and symplectic integrators has been ubiquitous. For instance,

Kharevych et al. (2006), developed general-purpose variational integrators for Lagrangian

systems, and then applied them to nonlinear elasticity problems, and for improving the per-

formance of computer animations. Additionally, there have been studies with variational

and symplectic integrators including dissipation effects. Kane et al. (2000) developed vari-

ational algorithms for dissipative systems and presented detailed comparisons with explicit

and implicit Newark integrators, showing that the Newark algorithms are indeed variational

and that both types of integrators accurately capture the energy decay. Another example

with dissipative effects is the work by Modin and Söderlind (2011), wherein three geometric

integration schemes for Hamiltonian systems were introduced.

Lie group variational integrators have also been developed for a variety of mechanical

systems under the assumption of rigid-body motion. For instance, Nordkvist and Sanyal

(2010) designed a method to propagate the motion of underwater vehicles. Following a similar

approach, Lee et al. (2005,2007,2011) presented integrators for the rigid-body motion of

pendulums and spacecraft. Fahnestock et al. (2006) also presented Lie group variational

integrators for the motion of spacecraft, but, in contrast to Lee et al. (2005,2007,2011),

they used polyhedral-body representation to improve the calculation of the gravitational

potential.

A symplectic approach for integrating spacecraft relative dynamics was introduced by

Imre and Palmer (2007), who developed a symplectic numerical method to propagate relative

orbits using an arbitrary number of zonal and tesseral terms in the geopotential, without drag.

Tsuda and Scheeres (2009) developed a numerical method for deriving a symplectic state

transition matrix for an arbitrary Hamiltonian dynamical system. The state transition matrix

was then applied to long-term propagation of spacecraft ﬂying in formation, among other

applications.

The approach presented herein focuses on the development of explicit symplectic and

variational integrators for the propagation of satellite relative motion in different reference

123

Variational and symplectic integrators for satellite… Page 3 of 26 31

frames, including atmospheric drag. As opposed to previous studies, a drag model is incor-

porated into the geometric integrator map, in addition to J2. The performance of fourth- and

sixth-order versions is then examined in simulated scenarios, wherein the effect of atmo-

spheric drag is taken into consideration.

2 Relative motion under J2and drag

Among the several formulations of spacecraft relative motion, two are of interest in this

work. The ﬁrst is described in terms of relative coordinates in the Local-Vertical-Local-

Horizontal (LVLH) reference frame. The other is obtained from the absolute motion in

Cartesian coordinates, rotated into the LVLH frame.

2.1 Relative motion from relative coordinates

Consider two spacecraft, a leader and a follower. The motion of the leader spacecraft can

be described in terms of the Whittaker polar-nodal chart (Lara and Gurﬁl 2012) with the

variables

r,θ,ν,R,Θ, N,

where ris the distance to the attraction center, θis the argument of latitude, νis the right

ascension of the ascending node, R=dr/dtis the radial velocity, Θis the modulus of the

angular momentum vector, and Nis the polar component of the angular momentum. The

mapping from the 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, can be obtained through the relations

a=− μr2

r2R2+Θ2−2μr(1a)

e=1−Θ2

μa(1b)

I=cos−1N

Θ(1c)

Ω=ν(1d)

cos f=Θ2−μr

μre ,sin f=ΘR

μe(1e)

ω=θ−f,(1f)

where μis the gravitational parameter. Then, the J2-perturbed dynamics of the leader space-

craft can be deﬁned by the Hamiltonian:

HL(r,θ,ν, R,Θ,N)=1

2R2+Θ2

r2−μ

r1−1

2J2

α2

r23sin

2θ1−N2

Θ2−1,

(2)

123

31 Page 4 of 26 L. Palacios, P. Gurﬁl

where αis the mean equatorial radius. Next, from Hamilton’s equations (Schaub and Junkins

2009, p. 279) we obtain the relations

d(r,θ,ν)

dt=∂HL

∂(R,Θ, N)and d(R,Θ, N)

dt=− ∂HL

∂(r,θ,ν)+QL,(3)

where QLrepresents the non-conservative forces acting on the leader, so that the equations

of motion, including the effects of drag,1take the form

˙r=R(4)

˙

θ=Θ

r2+2κsin2θcos2I

Θr3(5)

˙ν=−2κcos Isin2θ

Θr3(6)

˙

R=−μ

r2+Θ2

r3−κ

r41−3sin

2Isin2θ−BLRvL(7)

˙

Θ=−κsin2Isin 2θ

r3−BLΘvL(8)

˙

N=0.(9)

In these last equations, the constant κ, the ballistic coefﬁcient BLand the magnitude of

the velocity are deﬁned as

κ=3μJ2α2

2BL=1

2ρCD

AL

mL

and vL=R2+Θ2

r21/2

,

with ρas the atmospheric density, CDis the drag coefﬁcient and AL/mLis the area to mass

ratio of the leader spacecraft. The follower spacecraft’s dynamics may be described in terms

of relative coordinates in the LVLH reference frame (Alfriend et al. 2009), with the variables

ξ, , ψ, Ξ, P,Ψ.

The ﬁrst three variables correspond to the relative position components of the follower

with respect to the leader, and the others to the relative momenta. Taking the angular velocity

of the rotating LVLH frame as (Kechichian 1998)

ω=ωξ0ωψT,(10)

(note that ω=0) then the motion of the J2-perturbed dynamics of the follower spacecraft

can be determined by the Hamiltonian

HF(ξ,, ψ, Ξ, P,Ψ)=1

2Ξ2+P2+Ψ2+ωψ−RΞ+ψωξ−(r+ξ)ωψP

−ωξΨ−μ

rF−κ

3r3

F+κZ2

r5

F

,

(11)

where the orbital position of the follower rFand its third position component resolved in the

inertial reference frame, Z, are deﬁned as

rF=(r+ξ)2+2+ψ2and Z=(r+ξ)sin Isin θ+sin Icos θ+ψcos I

(12)

1Throughout this study, we exclude the effect of the atmospheric velocity on the overall drag force.

123

Variational and symplectic integrators for satellite… Page 5 of 26 31

and

ωξ=−κsin 2Isin θ

r3Θ,ω

ψ=Θ

r2.(13)

Next, from Hamilton’s equations we obtain the relations

d(ξ,, ψ)

dt=∂HF

∂(Ξ, P,Ψ)and d(Ξ, P,Ψ)

dt=− ∂HF

∂(ξ,, ψ)+QF,(14)

where QFdenotes the non-conservative forces acting on the follower. The equations of

motion, including the effect of the atmospheric drag, are then

˙

ξ=Ξ−R+ωψ(15)

˙=P−(r+ξ)ωψ+ψωξ(16)

˙

ψ=Ψ−ωξ(17)

˙

Ξ=Pωψ+(r+ξ)σ−sin Isin θ−BFΞvF(18)

˙

P=−Ξωψ+Ψω

ξ+σ −sin Icos θ−BFPvF(19)

˙

Ψ=−Pωξ+ψσ −cos I−BFΨv

F,(20)

with the ballistic coefﬁcient BFand the velocity vFexpressed as:

BF=1

2ρCD

AF

mF

and vF=Ξ2+P2+Ψ2

and

σ=−μ

r3

F−κ

r5

F+5κZ2

r7

F

,=2κZ

r5

F

.

2.2 Relative motion from absolute Cartesian elements

The absolute motion of a spacecraft around the Earth is usually derived in an Earth-centered

inertial reference frame using the Cartesian variables

x,y,z,v

x,v

y,v

z,

where the ﬁrst three terms correspond to the components of the position vector of the space-

craft r∈R3\{0}, and the next ones to the velocity vector v=dr

dt∈R3. Depending on the

context, the velocity variables may be replaced by the momenta:

X,Y,Z.

With this in consideration, the Cartesian J2-perturbed absolute motion of a spacecraft is

described by the Hamiltonian

H=1

2X2+Y2+Z2−μ

r−μJ2α2

2r31−3z

r2,(21)

or by the Lagrangian

L=1

2v2

x+v2

y+v2

z+μ

r+μJ2α2

2r31−3z

r2,(22)

where the constants μ,J2and αare deﬁned as before. Also, under these circumstances, the

momenta are equal to the velocities, X=vx,Y=vyand Z=vz.Itiscommontoﬁndthe

123

31 Page 6 of 26 L. Palacios, P. Gurﬁl

equations of motion in terms of position and velocity. Therefore, from Lagrange’s equations,

we ﬁnd the relations

d

dt

∂L

∂(vx,v

y,v

z)−∂L

∂(x,y,z)=Q,(23)

where Qstands for the non-conservative forces acting on the spacecraft. Next, the equations

of motion, including the effects of atmospheric drag, are expressed as

¨x=−μx

r31−3

2J2α

r25z2

r2−1−Bvvx(24)

¨y=−μy

r31−3

2J2α

r25z2

r2−1−Bvvy(25)

¨z=−μz

r31−3

2J2α

r25z2

r2−3−Bvvz,(26)

with the ballistic coefﬁcient Band the magnitude of the velocity of the spacecraft vdeﬁned

as before,

B=1

2ρCD

A

m,v=v2

x+v2

y+v2

z.(27)

After obtaining the absolute motion of the leader and the follower with these equations,

the relative position and velocity vectors, expressed in the Earth-centered inertial frame, are

obtained,

δr=rF−rL,δv=vF−vL.(28)

Then, the relative motion in the LVLH reference frame is obtained through a standard

transformation (Alfriend et al. 2009), included in “Appendix A” for convenience.

3 Geometric numerical integration

Several types of geometric numerical integrators have been presented in the literature; two

types, symplectic and variational integrators, are of special interest in this work. The main

reason for this choice is the favorable performance obtained from previous applications of

such integrators to a variety of mechanical systems, including the motion of spacecraft. In the

next section, we brieﬂy present the basic theory behind these methods [further details may

be found in the works by Hairer et al. (2006), Feng and Qin (2010) and Blanes and Casas

(2016)], and then provide dedicated derivations for satellite relative orbit propagation.

3.1 Symplecticity in Hamiltonian systems

The most known geometric numerical integrator is the symplectic propagation algorithm,

obtained from Hamiltonian systems (Blanes and Casas 2016). Let us consider, in gen-

eral, a system with ddegrees of freedom and phase space variables x=(q,p)T=

(q1,...,qd,p1,..., pd)T,where(qi,pi)are the usual canonical conjugate coordinates and

momenta, respectively. Given a Hamiltonian function H(q,p), the equations of motion are

˙qi=∂H

∂pi

,and ˙pi=−∂H

∂qi

for i=1,...,d.(29)

123

Variational and symplectic integrators for satellite… Page 7 of 26 31

This system of equations can be rewritten in a more compact notation as

˙x=J∇xH(x),(30)

where Jis the orthoskew matrix

J=0dId

−Id0d.(31)

The element Idis the d×didentity matrix, 0dis the corresponding zero matrix and

∇x=∂

∂x1,..., ∂

∂xd. One of the most distinctive features of such systems is that the ﬂow

φt:xn→ xn+1is a symplectic transformation, or in other words, its Jacobian matrix

φ

t(x)≡∂(qn+1,pn+1)

∂(qn,pn)satisﬁes

φ

t(x)TJφ

t(x)=φ

t(x)Jφ

t(x)T=Jfor t≥0.(32)

When carrying out simulations, symplectic integrators respect the previous property. This

has two important consequences. First, symplecticity results in stable energy, rather than

energy damping or growth, ensuring globally correct behavior. Second, discrete momenta

are preserved. Thus, simulations using these integrators usually have a greater physical ﬁdelity

and a lower computational cost, even when constraints, dissipation and external forces are

contemplated (Blanes and Casas 2016).

3.2 Symplectic integration

We start with the introduction of the basic symplectic method Φh

2S. The basic method pre-

sented here, also known as the Störmer–Verlet method (Hairer et al. 2003), has second-order

accuracy, and is obtained using the composition operation (denoted as A◦B):

Φh

2S=Φh/2

1S◦Φh/2

1S∗,(33)

where Φh/2

1Sis the map corresponding to the 1st order symplectic Euler method, Φh/2

1S∗is its

adjoint method, his the time step and the Sdenotes “symplectic”, which is used to distinguish

the symplectic method from other variants. This leads to the method

qk+1/2=qk+h

2∇pHqk+1/2,pk(34)

pk+1=pk−h

2∇qHqk+1/2,pk+∇

qHqk+1/2,pk+1(35)

qk+1=qk+1/2+h

2∇pHqk+1/2,pk+1,(36)

where the subindex kis the current iteration number. On the other hand, when using the

composition operation

Φh

2S=Φh/2

1S∗◦Φh/2

1S,(37)

we obtain the following method:

pk+1/2=pk−h

2∇qHqk,pk+1/2(38)

qk+1=qk+h

2∇pHqk,pk+1/2+∇pHqk+1,pk+1/2(39)

123

31 Page 8 of 26 L. Palacios, P. Gurﬁl

pk+1=pk+1/2−h

2∇qHqk+1,pk+1/2.(40)

Next, we apply this basic method for developing a symplectic integrator for the relative

motion of satellites.

3.2.1 Semi-implicit symplectic relative propagation in the LVLH reference frame

Starting with the motion of the leader, consider the Hamiltonian in Eq. (2). Using the basic

method in Eqs. (38)–(40) with the compact notation Ak+1=A+,Ak+1/2=A1/2and

Ak=A, the half-indexed momenta are obtained as

R1/2=R+h

2Θ2

1/2

r3−μ

r2−κ

r41−3sin

2θ1−N2

1/2

Θ2

1/2

−h

2BLRvL(41)

Θ1/2=Θ+h

2−κsin 2θ

r31−N2

1/2

Θ2

1/2−h

2BLΘvL(42)

N1/2=N.(43)

Then, the new canonical coordinates are

r+=r+hR

1/2(44)

θ+=θ+h

2Θ1/2

r2+2κsin2θ

r3Θ1/2N2

1/2

Θ2

1/2

+h

2Θ1/2

r2

++2κsin2θ+

r3

+Θ1/2N2

1/2

Θ2

1/2 (45)

ν+=ν+h

2−2κsin2θ

r3Θ1/2N1/2

Θ1/2+h

2−2κsin2θ+

r3

+Θ1/2N1/2

Θ1/2.(46)

Finally, the new momenta complete the method for the leader spacecraft:

R+=R1/2+h

2Θ2

1/2

r3

+−μ

r2

+−κ

r4

+1−3sin

2θ+1−N2

1/2

Θ2

1/2

−h

2BLRvL(47)

Θ+=Θ1/2+h

2−κsin 2θ+

r3

+1−N2

1/2

Θ2

1/2−h

2BLΘvL(48)

N+=N1/2.(49)

The present algorithm is implicit and an iterative method to solve nonlinear equations is

required to propagate it in time. For the upcoming simulations, the Newton–Raphson method

(Engeln-Mulges 1996) is selected, and the elements of the Jacobian matrix are presented in

“Appendix B”. Next, the propagation algorithm of the follower spacecraft is obtained also

from the basic method presented in Eqs. (38)–(40). First, the half-indexed momenta for Ξ1/2

123

Variational and symplectic integrators for satellite… Page 9 of 26 31

and Ψ1/2are obtained in terms of P1/2as:

Ξ1/2=h

2ωψP1/2+fΞ(50)

Ψ1/2=−h

2ωξP1/2+fΨ,(51)

with fΞand fΨdeﬁned as:

fΞ=Ξ+h

2(r+ξ)σ−sin I1/2sin θ(52)

fΨ=Ψ+h

2ψσ −cos I1/2.(53)

Next, the half-indexed momenta for P1/2is expressed in terms of Ξ1/2and Ψ1/2as:

P1/2=h

2ωξΨ1/2−h

2ωψΞ1/2+fP,(54)

with fPexpressed as:

fP=P+h

2σ −sin I1/2cos θ.(55)

The substitution of Eqs. (50)and(51) into Eq. (54), renders the half-indexed momenta

P1/2explicit as expressed below:

P1/2=Λh

2ωξfΨ−ωψfΞ+fP,(56)

with the elements:

Λ=1

1+h2

4ω2

ξ+ω2

ψ

(57)

I1/2=cos−1N1/2

Θ1/2

.(58)

Finally, a drag term is added to these half-indexed momenta:

P1/2=Λh

2ωξfΨ−ωψfΞ+fP−h

2BFPvF(59)

Ξ1/2=h

2ωψP1/2+fΞ−h

2BFΞvF(60)

Ψ1/2=−h

2ωξP1/2+fΨ−h

2BFΨv

F.(61)

Then, the new explicit positions are obtained using a similar approach as with the half-indexed

momenta:

+=Πh

2ωξ+fψ−ωψ+fξ+f(62)

ξ+=h

2ωψ+++fξ(63)

ψ+=−h

2ωξ+++fψ,(64)

123

31 Page 10 of 26 L. Palacios, P. Gurﬁl

with the elements:

Π=1

1+h2

4ω2

ξ++ω2

ψ+

(65)

fξ=ξ+hΞ1/2−hR

1/2+h

2ωψ(66)

f=+hP

1/2+h

2ψωξ−(r+ξ)ωψ−h

2r+ωψ+(67)

fψ=ψ+hΨ1/2−h

2ωξ.(68)

Lastly, the new momenta complete the method for the follower spacecraft:

Ξ+=Ξ1/2+h

2P1/2ωψ+(r++ξ+)σ−sin I1/2sin θ+

−h

2BFΞvF(69)

P+=P1/2+h

2Ψ1/2ωξ−Ξ1/2ωψ++σ−sin I1/2cos θ+

−h

2BFPvF(70)

Ψ+=Ψ1/2+h

2−P1/2ωξ+ψ+σ−cos I1/2−h

2BFΨv

F.(71)

The propagator corresponding to the follower motion is explicit, and it does not require

the use of the Newton–Raphson method to advance in time, making the whole algorithm a

semi-implicit symplectic propagator.

3.2.2 Fully explicit symplectic relative propagation from the absolute Cartesian motion

Now, we present a symplectic integrator derived from Cartesian variables. Using the Hamil-

tonian in Eq. (21), the second-order symplectic method Φh

2Spresented in Eqs. (34)–(36)

yields the half-indexed positions

x1/2=x+h

2X(72)

y1/2=y+h

2Y(73)

z1/2=z+h

2Z.(74)

Next, the new momenta are deﬁned as

X+=X−hμx1/2

r3

1/21−3

2J2

α2

r2

1/25z2

1/2

r2

1/2−1−hBXv(75)

Y+=Y−hμy1/2

r3

1/21−3

2J2

α2

r2

1/25z2

1/2

r2

1/2−1−hBYv(76)

Z+=Z−hμz1/2

r3

1/21−3

2J2

α2

r2

1/25z2

1/2

r2

1/2−3−hBZv, (77)

123

Variational and symplectic integrators for satellite… Page 11 of 26 31

with r1/2=x2

1/2+y2

1/2+z2

1/2and v=√X2+Y2+Z2. Finally, the new positions

complete the method:

x+=x1/2+h

2X+(78)

y+=y1/2+h

2Y+(79)

z+=z1/2+h

2Z+.(80)

This algorithm is fully explicit. It is used to propagate the motion of the leader and the

follower. Then, for every time step, the relative motion in the LVLH reference frame is

obtained from Eq. (28). The coordinate transformation is presented in “Appendix A”.

3.3 Variational integration

The main idea of a variational integrator is to discretize Hamilton’s principle for a given

problem (Blanes and Casas 2016;Haireretal.2006). The phase space of the continuous

Lagrangian is replaced by discrete variables and a discrete Lagrangian,

L(qk,qk+1,h)≈tk+1

tk

L(q(t), ˙q(t))dt,

where q(t)is the exact solution of the Euler–Lagrange equations joining qkand qk+1in the

given time interval. It is worth noticing that the discrete Lagrangian Lis a function of two

positions qkand qk+1and the time step h=tk+1−tk. The next step is to consider discrete

curve points {qk}K

k=0and evaluate the discrete action along this sequence by adding up Lon

each adjacent pair:

S({qk}K

k=0)=

K−1

k=0

L(qk,qk+1,h).

Then, variations of this action are computed (just as in the continuous case) with boundaries

q0and qNheld ﬁxed,

δS({qk})=

K−1

k=0D1L(qk,qk+1,h)δqk+D2L(qk,qk+1,h)δqk+1

=

K−1

k=0D2L(qk−1,qk,h)+D1L(qk,qk+1,h)δqk

+D1L(q0,q1,h)δq0+D2L(qN−1,qN,h)δqN,

where the subindexes 1 and 2 denote partial derivation with respect to the ﬁrst and second

argument of the discrete Lagrangian L, respectively. Now, we require that the variations of

the action be zero for any choice of δqk, and the discrete Euler–Lagrange equations of motion

are obtained,

D2L(qn−1,qn,h)+D1L(qn,qn+1,h)=0.

These equations, which provides the map (qk,qk+1)→ (qk+1,qk+2), are referred to

as the variational integrator. Depending on how the action integral is approximated, it is

indeed possible to rewrite a variational integrator into a position-momentum form, rather than

123

31 Page 12 of 26 L. Palacios, P. Gurﬁl

involving two positions. If this is done, then the position and momentum may be obtained

using the relations

pk=−D1L(qk,qk+1,h)(81)

pk+1=D2L(qk,qk+1,h). (82)

In general, any integrator which is a solution of the discrete Euler-Lagrange equations for

some discrete Lagrangian, is called a variational integrator. In our basic method, a discrete

Lagrangian is obtained by approximating the continuous one using a trapezoidal rule as

L(qk,qk+1,h)=h

2

Lqk,qk+1−qk

h+h

2

Lqk+1,qk+1−qk

h.(83)

In particular, this approximation leads to the second-order variational integrator Φh

2V,deﬁned

as

pk=1

2D2Lqk,v

k+1/2+1

2D2Lqk+1,v

k+1/2−h

2D1Lqk,v

k+1/2(84)

pk+1=1

2D2Lqk,v

k+1/2+1

2D2Lqk+1,v

k+1/2

+h

2D1Lqk+1,v

k+1/2,(85)

where, in this case, in particular, vk+1/2=(qk+1−qk)/his deﬁned only for brevity. Next,

this basic method is used to obtain a variational integrator for satellite relative motion.

3.3.1 Fully explicit variational relative propagation from absolute Cartesian motion

A variational integrator is obtained for the relative motion of satellites with the Lagrangian

presented in Eq. (22) deﬁned using Cartesian variables. The method Φh

2Vpresented in

Eqs. (84)–(85) yields the new positions

x+=x+hX +h2

2fx−hBXv(86)

y+=y+hY +h2

2fy−hBYv(87)

z+=z+hZ +h2

2fz−hBZv(88)

and the new momenta completes the method,

X+=X+h

2fx+h

2fx+−hBXv(89)

Y+=Y+h

2fy+h

2fy+−hBYv(90)

Z+=Z+h

2fz+h

2fz+−hBZv, (91)

where

fx=−μx

r31−3

2J2

α2

r25z2

r2−1 (92)

fy=−μy

r31−3

2J2

α2

r25z2

r2−1 (93)

123

Variational and symplectic integrators for satellite… Page 13 of 26 31

fz=−μz

r31−3

2J2

α2

r25z2

r2−3.(94)

Notice that fx+,fy+and fz+are obtained using Eqs. (92)–(94) with the new positions.

This method is fully explicit and as with the symplectic integrator presented in Sect. 3.2.2,

it is used to propagate the motion of the leader and the follower. Then, again, for every time

step, the relative motion in the LVLH reference frame is obtained from Eqs. (28)andthe

transformation algorithm in “Appendix A”.

3.4 Higher order symplectic and variational integrators

Higher order symplectic and variational methods can be obtained using composition opera-

tions. For example, a fourth-order method may be obtained with Φh

2S(or with Φh

2V)asthe

basic method by the operation

Φh

4S=Φγh

2S◦Φβh

2S◦Φγh

2S(95)

and the constants

γ=1

2−21/3and β=1−2γ. (96)

This approach for constructing methods of arbitrarily high order is known as the triple-

jump composition or the Suzuki–Yoshida technique (Suzuki 1990; Yoshida 1990). It is

important to mention that, in order to build higher order methods using γand βas deﬁned

in Eq. (96), it is necessary that the basic method (in this example Φh

2S) be at least of second

order. Following this logic, a sixth-order method can also be obtained through composition

operations (Yoshida 1990) emanating, for example, from the basic method Φh

2S,

Φh

6S=Φw3h

2S◦Φw2h

2S◦Φw1h

2S◦Φw0h

2S◦Φw1h

2S◦Φw2h

2S◦Φw3h

2S,(97)

with the constants

w1=−1.17767998417887 (98)

w2=0.235573213359357 (99)

w3=0.784513610477560 (100)

w0=1−2(w1+w2+w3).(101)

These methods are used for obtaining high-order versions of the previous variational and

symplectic integrators for satellite relative motion.

4 Numerical simulations

In this section, several performance parameters are used to test the efﬁciency of the integration

methods presented in this work. The performance parameters obtained by the variational and

symplectic integrators are compared to those obtained from the full J2equations of motion

in inertial Earth-centered Cartesian coordinates, propagated using a fourth-order, ﬁxed-step,

classical Runge–Kutta integrator. The reference “ground truth” is obtained using an 8th-

order, variable step, Runge–Kutta integrator due to Dormand and Prince (Hairer et al. 1993).

The performance parameters are formulated in terms of the Hamiltonian, position errors and

velocity errors, for different time steps. These errors are deﬁned as

123

31 Page 14 of 26 L. Palacios, P. Gurﬁl

Tab l e 1 Acronyms used in the simulations

Acronym Method Order Nature Time step Equationsc

DP8aRK Dormand and Prince 8 Explicit Adaptive NA

CRK4aClassic RK 4 Explicit Fixed NA

CRK4rbClassic RK 4 Explicit Fixed NA

SY4e Symplectic 4 Semi-implicit Fixed (41)–(49)and(59)–(71)

SY6e Symplectic 6 Semi-implicit Fixed (41)–(49)and(59)–(71)

VA4 Variational 4 Explicit Fixed (86)–(91)

VA6 Variational 6 Explicit Fixed (86)–(91)

SY4 Symplectic 4 Explicit Fixed (72)–(80)

SY6 Symplectic 6 Explicit Fixed (72)–(80)

RK Runge–Kutta, NA Not applicable

aMethod applied to Eqs. (24)–(26)

bMethod applied to Eqs. (4)–(9)and(15)–(20)

cEquations corresponding to the acronym

Tab l e 2 Algorithm 1:

propagating the variational

integrator

1: input: initial and ﬁnal time, time step, initial state

x,y,z,v

x,v

y,v

zof the leader and follower

2: for every time step

3: if desired order is 4 then

4: Use composition in Eq. (95) with constants from Eq. (96)

5: Propagate the state of the leader and follower with

Eqs. (86)–(91)

6: Obtain the relative motion with Eq. (28)

7: Transform onto the LVLH reference frame using “Appendix A”

8: else if desired order is 6 then

9: Use composition in Eq. (97) with constants from

Eqs. (98)–(101)

10: Repeat steps 5 to 7

13: end if

14: end for

ΔH=H(t)−H0

H0and ΔA=A−Aref ,(102)

where the generic variable Astands for either the position or velocity vector. All the simula-

tions in this section were carried out on a PC with an Intel(R) Core(TM) i5-3570 3.40 GHz

processor, 4.00 GB of RAM and MATLAB 2017a. In the following sections, the simulation

duration is 100 orbits, and the time step is 50 s (unless otherwise told). For the sake of con-

ciseness, the acronyms presented in Table 1are used. Additionally, the implementation of

the algorithm of the variational integrator in Sect. 3.3.1 is summarized in Table 2.Therest

of the integrators are propagated in a similar way.

123

Variational and symplectic integrators for satellite… Page 15 of 26 31

Tab l e 3 Summary of initial

osculating orbit elements Spacecraft a(km) eIΩω θ

Leader 7000 0.04 30◦0◦0◦0◦

Follower 7000 0.04 30◦1◦0◦−0.3◦

Tab l e 4 Summary of initial Cartesian absolute positions and velocities

Spacecraft x(km) y(km) z(km) ˙x(km/s) ˙y(km/s) ˙z(km/s)

Leader 6714.6010 0 0 0 6.8073 3.9330

Follower 6714.0215 86.7153 −17.5827 −0.0791 6.8069 3.9329

Fig. 1 Relative motion in the

LVLH frame

-100

80 1

0

z (km)

75

y (km)

0

x (km)

100

70 -1

65 -2

4.1 J2-perturbed relative motion

A quasi-periodic relative orbit is ﬁrst obtained without dissipation using the algorithm devel-

opedby(LaraandGurﬁl2012, Sect. 4.2.4). The leader and follower initial osculating orbit

elements and initial Cartesian absolute state can be found in Tables 3and 4, respectively.

These initial conditions are used to propagate the relative motion observed in Fig. 1.The

parameter values used in the simulations are

μ=398600.4415 km3/s2,α=6378.1363 km,J2=1.0826266 ×10−3.

Starting ﬁrst with the fourth-order variational and symplectic integrators, Fig. 2shows

the time evolution of the Hamiltonian error. In Fig. 2a, it is noticed that all the integrators

in consideration preserve the Hamiltonian for the current integration time and time step.

However, when longer time spans and time steps are considered (for example, 200 orbits and

h=500 s), it is then noticed that CRK4 breaks, while CRK4r remains bounded, as observed

in Fig. 2b. In the case of SY4e, VA4 and SY6, these too remain bounded as noticed in Fig. 2c.

There is a notable difference when the Hamiltonian error is calculated individually for the

leader and the follower using Eq. (21), as observed in Fig. 3. It can be observed that VA4 and

SY4 preserve such Hamiltonians (with minimum differences between them), while CRK4

does not.

Figure 4presents the position and velocity errors. Because its algorithm requires fewer

steps, CRK4r is the faster integrator in the current test case with 0.4681 s; however, it presents

one of the highest position errors with a maximum value of 1.6083 km. CRK4 and SY4e

also present high position errors with maximum values of 1.6374 and 1.2459 km for each

one of them. Contrary to these, VA4 and SY4 present the lowest errors, with maximum

errorvaluesof0.15238 km for VA4 and 0.10874 km for SY4. In terms of velocity errors,

123

31 Page 16 of 26 L. Palacios, P. Gurﬁl

Fig. 2 Hamiltonian errors (J2

only). a100 orbits and h=50 s.

b200 orbits and h=500 s. c200

orbits and h=500 s

020406080100

Time (Orbits)

10-5

100

Hamiltonian Error

(Normalized)

CRK4 CRK4r SY4 VA4 SY4e

0 50 100 150 200

Time (Orbits)

10-5

100

Hamiltonian Error

(Normalized)

CRK4 CRK4r

0 50 100 150 200

Time (Orbits)

10-5

100

Hamiltonian Error

(Normalized)

SY4 VA4 SY4e

(a)

(b)

(c)

VA4 and SY4 also present the lowest errors with maximum values of 3.7292 ×10−4km/s

and 3.1677 ×10−4km/s respectively, while CRK4 with 3.7976 ×10−3km/s, CRK4r with

3.7951 ×10−3km/s and SY6e with 1.0193 ×10−3km/s yield the highest errors.

In terms of computational efﬁciency, SY4e, VA4 and SY4 required computational times of

0.8866, 0.6667 and 0.7239 s, respectively; that is ∼32.18, ∼49.89 and ∼48.2% faster than

CRK4. It is also worth noticing that SY4e, despite requiring the Newton–Raphson method

in every iteration, is also ∼88.99% faster than the reference DP8, whose computational time

is 7.7426 s. However, the high values in position errors obtained with all these integrators in

the current test case requires exploring higher order variational and symplectic integrators.

Next, the sixth-order variational and symplectic integrators are tested. As illustrated in

Fig. 5, there is no signiﬁcant change in the values of the Hamiltonian errors, but there are

signiﬁcant changes in position errors. SY6e presents a lower position error than before, with

a maximum value of 0.0720 km. However, it is observed that the VA6 and SY6 position

errors are the lowest, with values of 4.7425 ×10−05 and 3.3837 ×10−05 km, respectively.

123

Variational and symplectic integrators for satellite… Page 17 of 26 31

Fig. 3 Leader and follower

Hamiltonian error (J2only)

0 20406080100

Time (Orbits)

10

-10

10-5

Leader Hamiltonian

Error (Normalized)

CRK4 VA4 SY4

020 40 60 80 100

Time (Orbits)

10

-10

10-5

Follower Hamiltonian

Error (Normalized)

CRK4 VA4 SY4

Fig. 4 Position and velocity

errors (J2only)

0 20 40 60 80 100

Time (Orbits)

10-5

10 0

Position Error (km)

CRK4r CRK4 SY4e VA4 SY4

0 20 40 60 80 100

Time (Orbits)

10

-10

10-5

10 0

Velocity Error (km/s)

CRK4r CRK4 SY4e VA4 SY4

In terms of velocity errors, SY6e presents the lowest velocity error with a maximum

value of 2.1531 ×10−5km/s, and is followed by VA6 and SY6 sharing the same value

of 9.444 ×10−5km/s. Despite having increased the order of the variational and symplectic

integrators, there is not much impact on the computational time of VA6 and SY6 with values of

0.8341 and 0.9105 s, respectively. These values mean that VA6 is ∼33.65% faster than CRK4

123

31 Page 18 of 26 L. Palacios, P. Gurﬁl

Fig. 5 Hamiltonian, position and

velocity errors (J2only)

0 20 40 60 80 100

Time (Orbits)

10-5

100

Hamiltonian Error

(Normalized)

CRK4 CRK4r SY6 VA6 SY6e

0 20 40 60 80 100

0 20 40 60 80 100

Time (Orbits)

10

-10

10-5

10 0

Position Error (km)

CRK4r CRK4 SY6e VA6 SY6

Time (Orbits)

10

-10

10-5

100

Velocity Error (km/s)

CRK4r CRK4 SY6e VA6 SY6

while SY6 is ∼27.58% faster. This is not the case for SY6e, which presents a computational

time of 1.8766 s (CRK4 is ∼32.99% faster), although it is ∼75.53% faster than the reference

DP8. When the Hamiltonian error is computed individually for the leader and the follower, its

behavior is quite different from the one calculated with relative motion, as showed in Fig. 6.

It is seen that VA6 and SY6 do preserve the Hamiltonian while CRK4 does not.

The integration error trend observed so far is maintained when using different time steps.

For the next simulations, the time steps are 0.1, 1, 20, 40, 60, 80, 100, 120, 140, 160, 180 and

200 s. The maximum Hamiltonian and position errors are depicted in Fig. 7. It is seen that

the Hamiltonian is consistently preserved by VA6, SY6 and SY6e for the current selection of

ﬁnal time and time step. Regarding the position errors, VA6 and SY6 yield the lowest values.

In the case of the velocity errors, shown in Fig. 8, SY6e is the one exhibiting the lowest

values, followed by VA6 and SY6. However, for time steps larger than 150 s, these three

are almost the same. VA6 and SY6 also require the lowest computational time, as shown in

Fig. 8. To conclude this set of simulations, a summary of the results obtained in this section

with a time step of 50 s is presented in Table 5.

123

Variational and symplectic integrators for satellite… Page 19 of 26 31

Fig. 6 Leader and follower

Hamiltonian error (J2only)

0 20406080100

Time (Orbits)

10-15

10-10

10-5

Leader Hamiltonian

Error (Normalized)

CRK4 VA6 SY6

0 20406080100

Time (Orbits)

10-15

10-10

10-5

Follower Hamiltonian

Error (Normalized)

CRK4 VA6 SY6

Fig. 7 Maximum Hamiltonian

and position error versus time

step (J2only)

0 50 100 150 200

Step Size (sec)

0.1

0.2

0.3

0.4

Max. Hamiltonian Error

(Normalized)

CRK4 CRK4r SY6e VA6 SY6

120 140 160 180 200

Step Size (sec)

0.1

0.101

Max. Ham. Error

0 50 100 150 200

Step Size (sec)

10-10

10-5

100

105

Max. Position Error (km)

CRK4 CRK4r SY6e VA6 SY6

4.2 Relative motion perturbed by J2and drag

In the next set of simulations, the quasi-periodic relative orbit used in the previous section is

used, but the effects of drag are added with CD=2.2, (A/m)L=0.01, (A/m)F=0.007

and ρ=1.1371 ×10−13. Due to the advantages obtained in the previous section by using

sixth-order variational and symplectic integrators, its corresponding fourth-order versions

123

31 Page 20 of 26 L. Palacios, P. Gurﬁl

Fig. 8 Computational time

versus time step (J2only)

0 50 100 150 200

Step Size (sec)

10-6

10-4

10-2

Max. Velocity Error (km/s)

CRK4 CRK4r SY6e VA6 SY6

0 50 100 150 200

Step Size (sec)

10 -1

100

101

102

Comp. Time (sec)

CRK4 CRK4r SY6e VA6 SY6

Tab l e 5 Summary of results (only J2, maximum values, h=50 s)

Method Pos. error (km) Vel. error (km/s) H error (norm.) Comp. time (s)

CRK4 1.6374 3.7976 ×10−30.1014 1.2707

CRK4r 1.6083 3.7951 ×10−30.1014 0.4747

SY4e 1.2459 1.0193 ×10−30.1015 0.8866

VA 4 0 .1523 3.7292 ×10−40.1015 0.6460

SY4 0.1087 3.1677 ×10−40.1015 0.7014

SY6e 0.0720 2.1531 ×10−50.1015 1.8766

VA 6 4 .7425 ×10−05 9.4441 ×10−50.1015 0.8255

SY6 3.3837 ×10−05 9.4441 ×10−50.1015 0.9363

will not be included here. The addition of drag considerably changes the error behavior, as

illustrated in Fig. 9, where the position and velocity error are displayed. It is observed that

SY6e has the largest position error with a maximum value of 13.172 km. It is followed by

CRK4 and CRK4r with maximum values of 1.825 and 1.7803 km. VA6 and SY6 display

the lower errors with values of 4.2116 ×10−3and 4.3231 ×10−3km, respectively. In the

case of the velocity error, CRK4 and CRK4r present the largest values with 3.7974 ×10−3

and 3.795 ×10−3km/s, respectively, followed by SY6e with 5.7597 ×10−4km/s. Again,

VA6 and SY6 produce the lowest errors with the same maximum value of 9.1609 ×10−5

km/s. These last integrators are still faster than CRK4 (1.2355) with computational times of

0.7973 and 0.8940 s, respectively, that is ∼35.45 and ∼27.63% faster than CRK4. On the

other hand, SY6e is now the slowest method with a computational time of 1.8671 s.

123

Variational and symplectic integrators for satellite… Page 21 of 26 31

Fig. 9 Position and velocity

errors (J2+drag)

0 20 40 60 80 100

Time (Orbits)

10-5

100

Position Error (km)

CRK4r CRK4 SY6e VA6 SY6

0 20 40 60 80 100

Time (Orbits)

10-10

10-5

Velocity Error (km/s)

CRK4r CRK4 SY6e VA6 SY6

Tab l e 6 Summary of results

(J2+drag, maximum values,

h=50 sec.)

Method Pos. error (km) Vel. error (km/s) Comp. time (s)

CRK4 1.8250 3.7974 ×10−31.2355

CRK4r 1.7803 3.7950 ×10−30.4700

SY6e 13.172 5.7597 ×10−41.8671

VA 6 4.2116 ×10−39.1609 ×10−50.7973

SY6 4.3231 ×10−39.1609 ×10−50.8940

As in the previous section, the error trend observed in the last ﬁgures is maintained when

using different time steps. In the following simulations, the range of time steps previously

used is used here again. With regard to the maximum position and velocity errors, Fig. 10

shows that VA6 and SY6 keep providing the lowest values throughout the different time

steps, followed by SY6e. Additionally, the computational time advantage over CRK4 is also

maintained, as indicated in Fig. 11, in which it can also be observed that SY6e requires the

largest computation times. A summary of the results can be found in Table 6.

5 Conclusions

Explicit and semi-implicit variational and symplectic integrators were developed for satellite

relative orbit propagation in different reference frames. The proposed algorithms preserve

symplecticity and its corresponding fourth- and sixth-order versions yield more accurate

propagation than the Runge–Kutta integrators used in this paper. Moreover, the explicit

versions of the geometric algorithms are considerably faster than these generic integrators.

123

31 Page 22 of 26 L. Palacios, P. Gurﬁl

Fig. 10 Maximum position and

velocity errors versus time step

(J2+drag)

0 50 100 150 200

Step Size (sec)

10-5

100

Max. Position Error (km)

CRK4 CRK4r SY6e VA6 SY6

0 50 100 150 200

Step Size (sec)

10-5

Max. Velocity Error (km/s)

CRK4 CRK4r SY6e VA6 SY6

Fig. 11 Computational time

versus time step (J2+drag)

050100150200

Step Size (sec)

10-1

100

101

102

Comp. Time (sec)

CRK4 CRK4r SY6e VA6 SY6

Both features, accuracy and computational speed were indeed veriﬁed in terms of Hamil-

tonian, position and velocity errors in a set of simulated scenarios including the effects of J2

and drag using a range of time steps. Even in the presence of drag, which introduces physical

dissipation into the equations of motion, the high-order explicit variational and symplectic

integrators consistently present the lowest integration errors and the fastest computational

times in comparison with the Runge–Kutta variants.

Appendix A: transformation from earth-centered inertial to LVLH

The relative position between two satellites in elliptic orbits can be expressed in the LVLH

reference frame using the relations (Alfriend et al. 2009)

123

Variational and symplectic integrators for satellite… Page 23 of 26 31

ξ=δrTrL

rL

(103)

=δrT(hL×rL)

hL×rL(104)

ψ=δrThL

hL

,(105)

where hL=rL×vL. Next, the relative velocity expressed also in the LVLH reference frame

is obtained by differentiating the previous equations:

˙

ξ=δvTrL+δrTvL

rL−δrTrLδrTvL

r3

L

(106)

˙=δvT(hL×rL)+δrT˙

hL×rL+hL×vL

hL×rL

−δrT(hL×rL)(

hL×rL)T˙

hL×rL+hL×vL

hL×rL3(107)

˙

ψ=δvThL+δrT˙

hL

hL−δrThLhT

L˙

hL

h3

L

,(108)

where ˙

hL=rL×˙

vL.

Appendix B: elements of the Jacobian matrix

The exact calculation of the elements of the Jacobian matrix are used to improve the speed

of the Newton–Raphson method in the semi-implicit symplectic integrator presented in

Sect. 3.2.1. First, the system of equations of interest is re-arranged in the form f(x)=0

and then the expression is differentiated with respect to the unknown elements. For the

half-indexed momenta, Eqs. (41)and(42) are re-arranged as

f1=R1/2−R−h

2Θ2

1/2

r3−μ

r2−κ

r41−3sin

2θ1−N2

1/2

Θ2

1/2

+h

2BLRvL=0 (109)

f2=Θ1/2−Θ−h

2−κsin 2θ

r31−N2

1/2

Θ2

1/2+h

2BLΘvL=0,(110)

and then each of these equations is differentiated with respect to the unknown elements R1/2

and Θ1/2,

∂f1

∂R1/2=1 (111)

∂f1

∂Θ1/2=−hΘ1/2

r3+3κsin2θ

r4

N2

1/2

Θ3

1/2(112)

∂f2

∂R1/2=0 (113)

123

31 Page 24 of 26 L. Palacios, P. Gurﬁl

∂f2

∂Θ1/2=1+hκsin 2θ

r3

N2

1/2

Θ3

1/2

.(114)

Then, for the new positions, Eqs. (44)–(46) are re-arranged as

g1=r+−r−hR

1/2=0 (115)

g2=θ+−θ−h

2Θ1/2

r2+2κsin2θ

r3Θ1/2N2

1/2

Θ2

1/2

−h

2Θ1/2

r2

++2κsin2θ+

r3

+Θ1/2N2

1/2

Θ2

1/2=0 (116)

g3=ν+−ν−h

2−2κsin2θ

r3Θ1/2N1/2

Θ1/2

−h

2−2κsin2θ+

r3

+Θ1/2N1/2

Θ1/2=0.(117)

After differentiating each of these equations with respect to the unknown elements r+,θ+

and ν+, we obtain the expressions:

∂g1

∂r+=1 (118)

∂g1

∂θ+=0 (119)

∂g1

∂ν+=0 (120)

∂g2

∂r+=hΘ1/2

r3

++3κsin2θ+

Θ1/2r4

+N1/2

Θ1/22(121)

∂g2

∂θ+=1−hκsin 2θ+

Θ1/2r3

+N1/2

Θ1/22

(122)

∂g2

∂ν+=0 (123)

∂g3

∂r+=−h3κsin2θ+N1/2

r4

+Θ2

1/2

(124)

∂g3

∂θ+=h2κsin θ+cos θ+N1/2

r3

+Θ2

1/2

(125)

∂g3

∂ν+=1.(126)

References

Alfriend, K., Vadali, S.R., Gurﬁl, P., How, J., Breger, L.: Spacecraft Formation Flying: Dynamics, Control,

and Navigation. Butterworth-Heinemann, Oxford (2009)

Blanes, S., Casas, F.: A Concise Introduction to Geometric Numerical Integration, 1st edn. CRC Press, Baco

Raton (2016)

123

Variational and symplectic integrators for satellite… Page 25 of 26 31

Blanes, S., Casas, F., Farres, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting

methods for numerical integration in dynamical astronomy.Appl. Numer. Math. 68, 58–72 (2013). https://

doi.org/10.1016/j.apnum.2013.01.003

Engeln-Mulges, G.: Numerical Algorithms with C, 1st edn. Springer, New York (1996)

Fahnestock, E.G., Lee, T., Leok, M., McClamroch, N.H., Scheeres, D.J.: (2006) Polyhedral potential and

variational integrator computation of the full two body problem. In: AIAA/AAS Astrodynamics Specialist

Conference. https://doi.org/10.2514/6.2006-6289,http://arxiv.org/abs/math/0608695

Farres, A., Laskar, J., Blanes, S., Casas, F., Makazaga, J., Murua, A.: High precision symplectic integrators for

the solar system. Celest. Mech. Dyn. Astron. 116(2), 141–174 (2013). https://doi.org/10.1007/s10569-

013-9479- 6

Feng, K., Qin, M.: Symplectic Geometric Algorithms for Hamiltonian Systems, 1st edn. (2010). https://doi.

org/10.1007/978- 3-642- 01777-3,https:// books.google.com/books?id=L8wGeoxvuDsC&pgis=1

Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn.

Springer, Berlin (1993)

Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Stormer–Verlet

method. Acta Numer. 12(2003), 399–450 (2003). https://doi.org/10.1017/S0962492902000144,http://

www.journals.cambridge.org/abstract_S0962492902000144

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Algorithms for Ordinary Differential

Equations, 2nd edn. Springer, New York (2006)

Imre, E., Palmer, P.L.: High-precision, symplectic numerical, relative orbit propagation. J. Guid. Control Dyn.

30(4), 965–973 (2007). https://doi.org/10.2514/1.26846

Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for

conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295–1325

(2000). https://doi.org/10.1002/1097-0207(20001210)49:10<1295::AID-NME993> 3.0.CO;2-W

Kechichian, J.A.: Motion in general elliptic orbit with respect to a dragging and presessing coordinate frame.

J. Astronaut. Sci. 46(1), 25–45 (1998)

Kharevych, L., Yang, W., Tong, Y., Kanso, E., Marsden, J.E., Schröder, P., et al.: Geometric, variational

integrators for computer animation. In: Eurographics/ACM SIGGRAPH Symposium on Computer Ani-

mation, vol. 24, pp. 43–51. (2006). https://doi.org/10.1145/1073204.1073300,http:// portal.acm.org/

citation.cfm?id=1218064.1218071

Lara, M., Gurﬁl, P.: Integrable approximation of J2-perturbed relative orbits. Celest. Mech. Dyn. Astronaut.

114(3), 229–254 (2012). https://doi.org/10.1007/s10569-012- 9437-8

Lee, D., Springmann, J., Spangelo, S., Cutler, J.: Satellite dynamics simulator development using lie group

variational integrator. In: AIAA Modeling and Simulation Technologies Conference, August, pp. 1–20.

(2011). https://doi.org/10.2514/6.2011-6430

Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem in orbital

mechanics. Celest. Mech. Dyn. Astron. 98(2), 121–144 (2007). https://doi.org/10.1007/s10569-007-

9073-x

Lee, T.L.T., McClamroch, N., Leok, M.: A lie group variational integrator for the attitude dynamics of a

rigid body with applications to the 3D pendulum. In: Proceedings of 2005 IEEE Conference on Control

Applications, 2005 CCA, pp. 962–967. (2005). https://doi.org/10.1109/CCA.2005.1507254

Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and

Computational Mathematics. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/

CBO9780511614118

Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Method Eng. 60(1),

153–212 (2004). https://doi.org/10.1002/nme.958

Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. (January 2003):1–

158. (2001). https://doi.org/10.1017/S096249290100006X,http://www3.nd.edu/~izaguirr/ papers/acta_

numerica.pdf

Modin, K., Söderlind, G.: Geometric integration of Hamiltonian systems perturbed by Rayleigh damping. BIT

Numer. Math. 51(4), 977–1007 (2011). https://doi.org/10.1007/s10543-011-0345- 1

Nordkvist, N., Sanyal, A.K.: A lie group variational integrator for rigid body motion in SE(3) with applications

to underwater vehicle dynamics. Proc. IEEE Conf. Decis. Control 3, 5414–5419 (2010). https://doi.org/

10.1109/CDC.2010.5717622

Schaub, H., Junkins, J.L.: Analytical Mechanics of Space Systems, 2nd edn. American Institute of Aeronautics

and Astronautics Inc, Reston (2009)

Simo, J.C., Tarnow, N., Wong, K.K.: Exact energy-momentum conserving algorithms and symplectic schemes

for nonlinear dynamics. Comput. Methods Appl. Mech. Eng. 100(1), 63–116 (1992). https://doi.org/10.

1016/0045-7825(92)90115-Z

123

31 Page 26 of 26 L. Palacios, P. Gurﬁl

Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and

Monte Carlo simulations. Phys. Lett. A 146(6), 319–323 (1990)

Tsuda, Y., Scheeres, D.J.: Computation and applications of an orbital dynamics symplectic state transition

matrix. Adv. Astronaut. Sci. 134(4), 899–918 (2009). https://doi.org/10.2514/1.42358

Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990).

https://doi.org/10.1016/0375-9601(90)90092-3

123

- A preview of this full-text is provided by Springer Nature.
- Learn more

Preview content only

Content available from Celestial Mechanics and Dynamical Astronomy

This content is subject to copyright. Terms and conditions apply.