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Variational and symplectic integrators for satellite relative orbit propagation including drag

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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 integrate 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 atmospheric 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.
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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
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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
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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,R, 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, 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+Θ22μr(1a)
e=1Θ2
μa(1b)
I=cos1N
Θ(1c)
Ω=ν(1d)
cos f=Θ2μr
μre ,sin f=ΘR
μe(1e)
ω=θf,(1f)
where μis the gravitational parameter. Then, the J2-perturbed dynamics of the leader space-
craft can be defined by the Hamiltonian:
HL(r, R,Θ,N)=1
2R2+Θ2
r2μ
r11
2J2
α2
r23sin
2θ1N2
Θ21,
(2)
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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)
dt=HL
(R, N)and d(R, N)
dt=− HL
(r)+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
r3κ
r413sin
2Isin2θBLRvL(7)
˙
Θ=−κsin2Isin 2θ
r3BLΘvL(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.
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)=1
2Ξ2+P2+Ψ2+ωψRΞ+ψωξ(r+ξ)ωψP
ωξΨμ
rFκ
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.
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Variational and symplectic integrators for satellite… Page 5 of 26 31
and
ωξ=−κsin 2Isin θ
r3Θ
ψ=Θ
r2.(13)
Next, from Hamilton’s equations we obtain the relations
d(ξ,, ψ)
dt=HF
(Ξ, P)and d(Ξ, P)
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=ΞR+ωψ(15)
˙=P(r+ξ)ωψ+ψωξ(16)
˙
ψ=Ψωξ(17)
˙
Ξ=Pωψ+(r+ξ)σsin Isin θBFΞvF(18)
˙
P=−Ξωψ+Ψω
ξ+σ sin Icos θBFPvF(19)
˙
Ψ=−Pωξ+ψσ cos IBFΨv
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
Fκ
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 rR3\{0}, and the next ones to the velocity vector v=dr
dtR3. 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+Z2μ
rμJ2α2
2r313z
r2,(21)
or by the Lagrangian
L=1
2v2
x+v2
y+v2
z+μ
r+μJ2α2
2r313z
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
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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
r313
2J2α
r25z2
r21Bvvx(24)
¨y=−μy
r313
2J2α
r25z2
r21Bvvy(25)
¨z=−μz
r313
2J2α
r25z2
r23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=rFrLv=vFvL.(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)
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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=JxH(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 t0.(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 AB):
Φh
2S=Φh/2
1SΦh/2
1S,(33)
where Φh/2
1Sis the map corresponding to the 1st order symplectic Euler method, Φh/2
1Sis 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
2pHqk+1/2,pk(34)
pk+1=pkh
2qHqk+1/2,pk+∇
qHqk+1/2,pk+1(35)
qk+1=qk+1/2+h
2pHqk+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
1SΦh/2
1S,(37)
we obtain the following method:
pk+1/2=pkh
2qHqk,pk+1/2(38)
qk+1=qk+h
2pHqk,pk+1/2+∇pHqk+1,pk+1/2(39)
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31 Page 8 of 26 L. Palacios, P. Gurfil
pk+1=pk+1/2h
2qHqk+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
r3μ
r2κ
r413sin
2θ1N2
1/2
Θ2
1/2
h
2BLRvL(41)
Θ1/2=Θ+h
2κsin 2θ
r31N2
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
22κsin2θ
r3Θ1/2N1/2
Θ1/2+h
22κsin2θ+
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
+13sin
2θ+1N2
1/2
Θ2
1/2
h
2BLRvL(47)
Θ+=Θ1/2+h
2κsin 2θ+
r3
+1N2
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
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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)σsin 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
2ωξΨ1/2h
2ωψΞ1/2+fP,(54)
with fPexpressed as:
fP=P+h
2σ 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
2ωξfΨωψfΞ+fP,(56)
with the elements:
Λ=1
1+h2
4ω2
ξ+ω2
ψ
(57)
I1/2=cos1N1/2
Θ1/2
.(58)
Finally, a drag term is added to these half-indexed momenta:
P1/2=Λh
2ωξfΨωψfΞ+fPh
2BFPvF(59)
Ξ1/2=h
2ωψP1/2+fΞh
2BFΞvF(60)
Ψ1/2=−h
2ωξP1/2+fΨh
2BFΨv
F.(61)
Then, the new explicit positions are obtained using a similar approach as with the half-indexed
momenta:
+=Πh
2ωξ+fψωψ+fξ+f(62)
ξ+=h
2ωψ+++fξ(63)
ψ+=−h
2ωξ+++fψ,(64)
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31 Page 10 of 26 L. Palacios, P. Gurfil
with the elements:
Π=1
1+h2
4ω2
ξ++ω2
ψ+
(65)
fξ=ξ+hΞ1/2hR
1/2+h
2ωψ(66)
f=+hP
1/2+h
2ψωξ(r+ξ)ωψh
2r+ωψ+(67)
fψ=ψ+hΨ1/2h
2ωξ.(68)
Lastly, the new momenta complete the method for the follower spacecraft:
Ξ+=Ξ1/2+h
2P1/2ωψ+(r++ξ+)σsin I1/2sin θ+
h
2BFΞvF(69)
P+=P1/2+h
2Ψ1/2ωξΞ1/2ωψ++σsin I1/2cos θ+
h
2BFPvF(70)
Ψ+=Ψ1/2+h
2P1/2ωξ+ψ+σcos I1/2h
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+=Xhμx1/2
r3
1/213
2J2
α2
r2
1/25z2
1/2
r2
1/21hBXv(75)
Y+=Yhμy1/2
r3
1/213
2J2
α2
r2
1/25z2
1/2
r2
1/21hBYv(76)
Z+=Zhμz1/2
r3
1/213
2J2
α2
r2
1/25z2
1/2
r2
1/23hBZv, (77)
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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+1tk. 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)=
K1
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})=
K1
k=0D1L(qk,qk+1,h)δqk+D2L(qk,qk+1,h)δqk+1
=
K1
k=0D2L(qk1,qk,h)+D1L(qk,qk+1,h)δqk
+D1L(q0,q1,h)δq0+D2L(qN1,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(qn1,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
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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+1qk
h+h
2
Lqk+1,qk+1qk
h.(83)
In particular, this approximation leads to the second-order variational integrator Φh
2V,dened
as
pk=1
2D2Lqk,v
k+1/2+1
2D2Lqk+1,v
k+1/2h
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+1qk)/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
2fxhBXv(86)
y+=y+hY +h2
2fyhBYv(87)
z+=z+hZ +h2
2fzhBZv(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
r313
2J2
α2
r25z2
r21 (92)
fy=−μy
r313
2J2
α2
r25z2
r21 (93)
123
Variational and symplectic integrators for satellite… Page 13 of 26 31
fz=−μz
r313
2J2
α2
r25z2
r23.(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
2SΦβh
2SΦγh
2S(95)
and the constants
γ=1
221/3and β=12γ. (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
2SΦw2h
2SΦw1h
2SΦw0h
2SΦw1h
2SΦw2h
2SΦw3h
2S,(97)
with the constants
w1=−1.17767998417887 (98)
w2=0.235573213359357 (99)
w3=0.784513610477560 (100)
w0=12(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=AAref ,(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 30000
Follower 7000 0.04 30100.3
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(LaraandGurl2012, 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=6378.1363 km,J2=1.0826266 ×103.
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 ×104km/s
and 3.1677 ×104km/s respectively, while CRK4 with 3.7976 ×103km/s, CRK4r with
3.7951 ×103km/s and SY6e with 1.0193 ×103km/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 ×1005 and 3.3837 ×1005 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 ×105km/s, and is followed by VA6 and SY6 sharing the same value
of 9.444 ×105km/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 ×1013. 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 ×1030.1014 1.2707
CRK4r 1.6083 3.7951 ×1030.1014 0.4747
SY4e 1.2459 1.0193 ×1030.1015 0.8866
VA 4 0 .1523 3.7292 ×1040.1015 0.6460
SY4 0.1087 3.1677 ×1040.1015 0.7014
SY6e 0.0720 2.1531 ×1050.1015 1.8766
VA 6 4 .7425 ×1005 9.4441 ×1050.1015 0.8255
SY6 3.3837 ×1005 9.4441 ×1050.1015 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 ×103and 4.3231 ×103km, respectively. In the
case of the velocity error, CRK4 and CRK4r present the largest values with 3.7974 ×103
and 3.795 ×103km/s, respectively, followed by SY6e with 5.7597 ×104km/s. Again,
VA6 and SY6 produce the lowest errors with the same maximum value of 9.1609 ×105
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 ×1031.2355
CRK4r 1.7803 3.7950 ×1030.4700
SY6e 13.172 5.7597 ×1041.8671
VA 6 4.2116 ×1039.1609 ×1050.7973
SY6 4.3231 ×1039.1609 ×1050.8940
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
rLδrTrLδrTvL
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
hLδrThLhT
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=R1/2Rh
2Θ2
1/2
r3μ
r2κ
r413sin
2θ1N2
1/2
Θ2
1/2
+h
2BLRvL=0 (109)
f2=Θ1/2Θh
2κsin 2θ
r31N2
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+rhR
1/2=0 (115)
g2=θ+θh
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+νh
22κsin2θ
r3Θ1/2N1/2
Θ1/2
h
22κsin2θ+
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+=1hκsin 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)
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... 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). ...
... = 0 (17) and the equations of motion are expressed as ...
... the constant h being the selected time step [8,17]. The operation in Eq. (37) leads to ...
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Preliminaries of Differential Manifolds.- Symplectic Algebra and Geometry Preliminaries.- Hamiltonian Mechanics and Symplectic Geometry.- Symplectic Difference Schemes for Hamiltonian Systems.- The Generating Function Method.- The Calculus of Generating Function and Formal Energy.- Symplectic Runge-Kutta Methods.- Composition Scheme.- Formal Power Series and B-Series.- Volume-Preserving Methods for Source-Free Systems.- Free Systems.- Contact Algorithms for Contact Dynamic Systems.- Poisson Bracket and Lie-Poisson Schemes.- KAM Theorem of Symplectic Algorithms.- Lee-Variational Integrator.- Structure Preserving Schemes for Birkhoff Systems.- Multisymplectic and Variational Integrators.