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Orbital resonances have been exploited in different contexts, with the latest interplanetary application being the ESA/NASA mission Solar Orbiter, which uses repeated flybys of Venus to change the ecliptic inclination with low fuel consumption. The b-plane formalism is a useful framework to represent close approaches at the boundaries of the sphere of influence of the flyby planet. In the presented work, this representation is exploited to prune the design of perturbed resonant interplanetary trajectories in a reverse cascade, replacing the patched conics approximation with a continuity link between flybys and interplanetary legs. The design strategy splits the flyby time and state variables in a two-layer optimization problem. Its core numerically integrates the perturbed orbital motion with the Picard-Chebyshev integration method. The analytical pruning provided by the b-plane formalism is also used as starting guess to ensure the fast convergence of both the numerical integration and the trajectory design algorithm. The proposed semi-analytical strategy allows to take advantage of complex gravitational perturbing effects optimizing artificial maneuvers in a computationally efficient way. The method is applied to the design of a Solar Orbiter-like quasi-ballistic first resonant phase with Venus.
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Acta Astronautica 194 (2022) 216–228
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Research paper
B-plane and Picard–Chebyshev integration method: Surfing complex orbital
perturbations in interplanetary multi-flyby trajectories
Alessandro Masat ,1, Camilla Colombo2
Politecnico di Milano, 20156, Milano, Italy
ARTICLE INFO
Keywords:
b-plane
Picard–Chebyshev
Flyby design
Perturbation surf
ABSTRACT
Orbital resonances have been exploited in different contexts, with the latest interplanetary application being
the ESA/NASA mission Solar Orbiter, which uses repeated flybys of Venus to change the ecliptic inclination
with low fuel consumption. The b-plane formalism is a useful framework to represent close approaches at the
boundaries of the sphere of influence of the flyby planet. In the presented work, this representation is exploited
to prune the design of perturbed resonant interplanetary trajectories in a reverse cascade, replacing the patched
conics approximation with a continuity link between flybys and interplanetary legs. The design strategy splits
the flyby time and state variables in a two-layer optimization problem. Its core numerically integrates the
perturbed orbital motion with the Picard–Chebyshev integration method. The analytical pruning provided
by the b-plane formalism is also used as starting guess to ensure the fast convergence of both the numerical
integration and the trajectory design algorithm. The proposed semi-analytical strategy allows to take advantage
of complex gravitational perturbing effects optimizing artificial maneuvers in a computationally efficient way.
The method is applied to the design of a Solar Orbiter-like quasi-ballistic first resonant phase with Venus.
1. Introduction
Orbital resonances have been exploited in several ways for mission
design purposes and in many different contexts, such as the Earth–
Moon case (for example in the works of Topputo et al. [1] to actually
reach the moon with low fuel consumption, of Ceriotti et al. [2] for
increasing the observed regions by polar orbits, and Short et al. [3] as
the actual scientific orbit of the Transiting Exoplanet Survey Satellite
mission) or the exploration of Jupiter’s and Saturn’s moon systems (for
example the works of Lantoine et al. [4], Campagnola et al. [5,6] and
Vaquero et al. [7]). The planned introduction of the Lunar Gateway
in 2024 has drawn the attention of more recent works on the cis-lunar
space. The 9:2 resonant Near Rectilinear Halo Orbits are extensively an-
alyzed by Zimovan et al. [8], both as possible candidates for the hosting
the Gateway and in terms of the transfer possibilities toward other cis-
lunar orbits by McGuire et al. with and without the aid of low-thrust
propulsion [9]. Singh et al. [10] investigate eclipse-aware low-thrust
transfer strategies to such orbits, proposing a method whose concept
resembles the one of this work, leveraging the perturbation effects
through the use of high-fidelity analogues of the invariant manifolds
of the Circular Restricted Three Body Problem. Other applications also
regard pure interplanetary orbits, for instance the ESA/NASA mission
Corresponding author.
E-mail address: alessandro.masat@polimi.it (A. Masat).
1Ph.D. Candidate, Department of Aerospace Science and Technology.
2Associate Professor, Department of Aerospace Science and Technology.
Solar Orbiter [11] as the latest example: resonant trajectories with
Venus are exploited to raise the orbital inclination up to almost 30
degrees [12] over the ecliptic, to better observe the near-polar regions
of the Sun.
In this last case, the use of resonant close encounters allows to
save a considerable amount of fuel because of the repeated sequential
flyby maneuvers. Nonetheless, such a phenomenon remains difficult to
accurately model and understand, especially at the boundaries of the
planet’s sphere of influence where none of the two dynamics, planetary
or interplanetary, has a dominant role. This effect is amplified for shal-
low encounters, where either the small relative velocity with respect to
the flyby planet or the high miss distance worsen the patched conics
approximation. However, accurate predictions are required for steep
close approaches too: a small deviation from the nominal condition may
be amplified by several orders of magnitude during the flyby, requiring
trajectory correction maneuvers.
The b-plane formalism presents an analytic theory for the charac-
terization of flybys, based on a manipulation of Öpik’s variables [13]
originally proposed by Carusi et al. [14] further developed by Valsecchi
et al. [1517]. Fixed values of the post-encounter semi-major axis are
represented as circles in the b-plane, which can therefore be targeted a
priori as the link with the orbital period is well known [15].
https://doi.org/10.1016/j.actaastro.2022.01.045
Received 28 August 2021; Received in revised form 9 December 2021; Accepted 31 January 2022
Acta Astronautica 194 (2022) 216–228
217
A. Masat and C. Colombo
On the other side, the Picard–Chebyshev method is a semi-analytical
technique to globally integrate the evolution of a generic dynamical
system accounting for a generic perturbation source. Picard iterations
are performed to update the coefficients of a Chebyshev polynomial
interpolation of an initial solution guess. A derivation of the method
can be found in the work of Fukushima [18]. Bai and Junkins [19,
20] proposed a modified version of the method, making it suitable
to GPU (Graphics Processing Unit) computing platforms, condensing
the algorithm steps in a series of matrix operations. Nonetheless, the
modified Picard–Chebyshev method has been continuously developed
in the past few years, both in its formulation and implementation side
and outlining possible applications for Earth orbits where it contributed
to increase the efficiency of the numerical analyses. Junkins et al. [21]
analyzed the performances of the method comparing the efficiency
against the Runge–Kutta–Nystrom 12(10) integrator, proposing also a
second order version. Later, Koblick and Shankar [22] extended the
analysis to the propagation of accurate orbits testing difference force
models with NASA’s Java Astrodynamics toolkit. Woollands et al. [23
25] applied the method as numerical integrator for the solution of the
Lambert two-point boundary value problem, assessing also the benefits
of adopting the Kuustanheimo–Stiefel formulation of the dynamics
and proposing a solution for the multi-revolution trajectory design.
Swenson et al. [26] applied the modified Picard–Chebyshev method
on the circular restricted three-body problem, using the differential
correction approach. Singh et al. [27] used the method as the numerical
integration scheme for their feasibility study on quasi-frozen, near polar
and low altitude lunar orbits, including the N-bodies and the spherical
harmonics perturbations. The fixed point nature of the method was
exploited by Koblick et al. [28] to design low-thrust trajectories as
an optimal control problem, discretizing the control impulses and also
included the Earth’s oblateness J2perturbation. Macomber et al. [29]
introduced the concepts of cold, warm, ho starts of the method, ad-
dressing possible efficiency improvements by means of better initial
conditions, and variable-precision force models taking advantage of the
fixed-point nature of the algorithm. Woollands et al. [30]extended the
optimal low-thrust design to a high-fidelity model for the non-spherical
Earth, considering an arbitrary number of spherical harmonics in the
perturbing acceleration. Woollands and Junkins [31] developed the
Adaptive Picard–Chebyshev method, including an integral error feed-
back that accelerates the convergence of the Picard iterations and an
empirical low to determine segment length and polynomial degree of
the method, based on previous stability analyses. Atallah et al. [32]
compared the method with other sequential integration techniques on
different Earth-based orbital cases.
In this paper a proof of concept is proposed, where the early basic
formulation of the modified Picard–Chebyshev integration method [20]
is combined with the b-plane flyby prediction capabilities and applied
to the design of multi-flyby trajectories in reverse cascade. The exit
requirements of the current flyby are computed to meet the entrance
condition of the next one. They are consequently back-integrated to
obtain a new entrance condition to be targeted, within a dynamic
programming-like backward recursion logic. The proposed method ex-
tends the unperturbed design algorithm [33] previously developed
by the authors of this work, that exploits the b-plane formalism to
design a series of two body resonant orbits in the patched conics case.
The newly extended version of the strategy proposed here uses the
unperturbed b-plane solution to prune the trajectory design in the
perturbed environment. Starting from the Keplerian initial guesses for
the patched conics interplanetary arcs, a continuity link between the
planetary and interplanetary legs is introduced at the boundaries of
the planet’s sphere of influence. The core of the presented approach
numerically integrates the full dynamics using the Picard–Chebyshev
method, embedded in a multi-layer optimization problem that aims
to minimize an artificial correction at a user-specified point in the
interplanetary cruise. This application also tests the Picard–Chebyshev
integration techniques to interplanetary orbits, where the fixed point
nature of the algorithm introduces further benefits compared to the
sole Earth case. In particular, the numerical propagation scheme is used
to remove the patched conics approximation, and to surf the complex
perturbing accelerations from the N-bodies and general relativity. In
summary, the b-plane formalism is used for both the preliminary design
of the patched-conics initial trajectory guess and for the description of
the optimization variables. The Picard–Chebyshev integration scheme
is then used at the core of the optimization, exploiting the fixed point
nature for increased computational performance when including the
effects of N-bodies and general relativity perturbations.
The case of Solar Orbiter’s resonant close approaches with
Venus [12] is studied, achieving a quasi-ballistic transfer that surfs
the chaotic perturbed environment, requiring a single artificial control
impulse easily achievable by low thrust propulsion technologies.
The current implementation of the design strategy only requires a
generic two-body patched conics solution in the b-plane formalism, not
necessarily resonant. A generic solution from the Lambert problem [34]
with consequent b-plane description of the planetocentric phase would
suffice for the full extension to the design of non-resonant interplane-
tary arcs. In this work the resonant case is analyzed, since its connection
with the b-plane formalism is straightforward [15] and the already
available unperturbed design routine [33].
As the Picard–Chebyshev method can be parallelized, the whole de-
sign strategy is well suited to be used with high performance computing
facilities. In spite of this, the serial execution of the initial formulation
of the modified Picard–Chebyshev method will be shown to be already
efficient, because of the minimized need to read the database for
the ephemerides of the bodies. Further performance improvements
are therefore expected including the latest adaptive version of the
integrator [31].
The first steps toward the development of an efficient tool for the
continuous design of perturbed multi-flyby trajectories are made, with
particular focus onto the resonant ones. Addressing the behavior of the
natural dynamics is fundamental before implementing any real mission
maneuver design strategy.
This article is outlined as follows: first, a review of the b-plane
representation of flybys is given in Section 2, followed by a recap of the
Picard–Chebyshev integration method in Section 3, then the concept
of proposed design algorithm is presented in Section 4. Finally, the
application to a Solar Orbiter-like first resonant phase with Venus is
shown in Section 5.
2. Close encounters in the b-plane
Assuming the planet in a circular orbit around the Sun, an interme-
diate frame needs to be defined for the b-plane flyby representation.
Such a frame was first introduced in the framework of Öpik’s the-
ory [13] by Greenberg et al. [35] and later used by Carusi et al. [14]
for the characterization of close encounters, particularly aiming to
find analytic expressions for post-flyby orbital parameters. Consider
a frame centered on the planet’s center of mass, the (, , )axes
are directed as the heliocentric position, velocity 𝐯and angular mo-
mentum of the planet, respectively, as shown in Fig. 1.𝐔and 𝐔
denote the pre-encounter and post-encounter planetocentric velocities,
respectively.
All the involved quantities are non-dimensional, such that the
planet’s distance from the Sun and the Sun’s gravitational parameter
are both equal to 1. The non-dimensionalization gives in turn 𝐯= 1
and makes the orbital period of the planet equal to 2. The angles ,
and appear in the works of Carusi and Valsecchi [14,15] for other
analyses, whereas are not necessary for the purposes of the presented
design algorithm. identifies the flyby turn angle, and the pre
and post encounter angles between the corresponding planetocentric
velocity 𝐔or 𝐔and the planet’s velocity 𝐯, and identifies the
direction for the rotation of 𝐔into 𝐔caused by the flyby, measured
counter-clockwise from the major circles identified by 𝐔and 𝐯.
Acta Astronautica 194 (2022) 216–228
218
A. Masat and C. Colombo
Fig. 1. Graphical representation of the reference frame of analysis.
Source: Picture re-drawn based on original from [14].
The flyby effect, interplanetary-wise in any patched conics approx-
imation, is modeled as an instantaneous rotation of the planetocentric
velocity vector 𝐔without magnitude change. With the above defined
quantities it is possible to introduce the b-plane reference frame, whose
axes (
, ,
)are defined as by Öpik [13]:
=𝐔
𝐔;
=𝐯×𝐔
𝐔 𝐯;
=
×. (1)
In the following, the definition b-plane will be used to identify the
plane perpendicular to the axis, because
2+2=2(2)
with the impact parameter as in Milani et al. [36].
Recalling [14], from an interplanetary point of view the flyby can
be modeled as an instantaneous rotation of 𝐔into 𝐔. The superscript
shall denote the post-encounter quantities in the following lines.
2.1. B-plane circles
A certain post-encounter semi-major axis is fully determined by
[15]:
cos =1 − 1∕2
2(3)
From the b-plane properties and some spherical geometry analysis,
the b-plane locus of points of a given post-encounter semi-major axis
is a circle centered on the
axis [15]:
2+22sin
cos − cos +2(cos + cos )
cos − cos = 0 (4)
which is equivalent to
2+2− 2 +2=2(5)
with the center’s coordinate and the radius explicitly defined as
=sin
cos − cos =
sin
cos − cos (6)
where, analogously to ,is the angle between 𝐔and 𝐯, and =
𝐔2. As already mentioned, any reachable post-encounter semi-
major axis can be drawn as a circle in the b-plane, and need not be
resonant. The sole exception are flybys that do not modify the value of
, and thus feature , which are defined as the straight horizontal
line [15]:
= cot (7)
2.2. Perturbations in the b-plane
Previous results by the authors of this work [33] led to the semi-
analytical definition of the b-plane circles arising from the effects of a
generic perturbation source. All the perturbing effects can be condensed
in three angular variations:
of the turn angle ,;
of the angle that identifies the direction of the rotation of 𝐔
into 𝐔,;
of the post-encounter angle ,.
Figs. 2(a) and 2(b) compare the resonant circles drawn with the
unperturbed theory (Fig. 2(a) on the left) and the new perturbed
model (Fig. 2(b) on the right) with the simulated resonant samples,
highlighted in yellow, coming from the planetary protection analysis of
the upper stage of the launcher of Solar Orbiter [12,37].3The b-plane
circles, on purpose nearly visible and drawn in light gray, have become
the black bounded belt shaped loci of points, because also almost
perfectly phased resonant returns have been considered extending each
circle over its own neighborhood.
In the case of Figs. 2(a) and 2(b) the angles , and 
remain small in magnitude, nevertheless the difference they make in
the characterization of the b-plane circles is significant. This gives a
further proof to the need of precise models for the flyby phase, which
is required if the desired post-encounter prediction must be accurate.
3. Picard–Chebyshev integration method
Picard iterations [40] are a method that can be used to obtain
an approximation of the solution of initial/boundary value problems.
Denoting the state of dimension with 𝐱, the independent variable with
, the initial/boundary condition with 𝐱0and the dynamics function
with 𝐟(𝐱, ), the problem is defined as:
𝐱
 =𝐟(𝐱, ),𝐱0=𝐱(0)(8)
Starting from an initial approximation 𝐱(0)()of the actual solu-
tion 𝐱()in the interval [0, ]of the initial/boundary value problem
presented in Eq. (8), the th Picard iteration improves the previous
approximation 𝐱(−1)()of 𝐱()with 𝐱()()as in [40]:
𝐱()() = 𝐱(0)() +
0
𝐟𝐱(−1)(),  (9)
The method converges for a good enough initial approximation
𝐱(0)()and for +∞ [40].
In the analytical Picard iteration context, performing more than
one iteration is in general hard. The increasingly complex expressions
for 𝐱()()make it difficult to retrieve closed form solutions after the
first 2–3 steps [19]. At the same time, numerically computing the
integral functions by quadrature might not suffice in accuracy, as only
the first few iterations in general improve the function approximation.
In the attempt to develop parallelizable routines for the integration
of the dynamical motion, the Picard–Chebyshev method was built
combining the Picard iterations with the Chebyshev polynomial ap-
proximation [41]. A possible derivation of the method that follows
the work of Fukushima [18] is reported in Appendix, and can be
summarized in three steps:
1. Select a good enough initial guess 𝐱(0)().
2. Approximate 𝐟(𝐱, )and 𝐱(0)()with their Chebyshev polynomial
expansion.
3More detailed information about this analysis and the related validation
can be found in the work of Colombo et al. [38], Colombo et al. [39] and
Masat [33].
Acta Astronautica 194 (2022) 216–228
219
A. Masat and C. Colombo
Fig. 2. Visual accuracy improvement of the b-plane circle model. The analytical belts are bounded by the black circles, the yellow dots highlight the numerically detected resonances
on the whole simulated cloud of initial conditions.
3. Perform a Picard iteration to update the coefficients of the
interpolating Chebyshev polynomials.
The Picard iterations halt when the stopping conditions are met, based
on the maximum difference between two consecutive iterations drop-
ping below some user-specified tolerance.
The so defined method allows to easily perform several more Picard
iterations than the analytical case. The involved expressions remains
always of the same type, i.e. the Chebyshev polynomials. The function
approximation becomes an interpolation through nodes that should be
close to the true trajectory, instead of a global function whose value
after the iterations still depend on the initial guess choice. Further-
more, few iterations suffice to drop below a low tolerance if the real
solution 𝐱()differs from the initial guess 𝐱(0)()only because of small
perturbations [40]. Starting from the unperturbed Keplerian solution
for the generic weakly perturbed two body problem, a relatively fast
convergence of the method is ensured [18]. In the context of orbital
simulations, Macomber [42] referred to this type of initial guess as
warm-starting the Picard–Chebyshev iteration method, because the
analytic solution of the dominant dynamics part is used to reduce the
number of iterations required. Differently, the cold start was defined
by simply setting all the trajectory samples as equal to the initial
condition. In general, the closer the initial guess to the true trajectory,
the lower the number of iterations will be. Semi-analytic initial guesses
or results of propagations from simpler models are also an options,
and in the case of three-body-like perturbed trajectories would be a
better choice compared to the Keplerian approximation. Macomber also
introduced the concept of hot start in the case of time spans covering
multiple Earth planetary orbits [42], where the first orbit was used to
compute the difference between the Keplerian guess and the converged
trajectory. The near-periodicity of the spherical harmonics perturbation
was then exploited, including this difference in the starting trajectory,
achieving a further reduction of the iterations required for convergence.
3.1. Matrix form for vectorized and parallel computation
The method is suitable for parallel or vector implementation, in-
deed Fukushi-ma also proposed a vectorized version [43]. More recent
works over this technique by Bai and Junkins developed the modi-
fied Picard–Chebyshev method [20] and a CUDA implementation for
NVIDIA GPUs [19]. For compactness and to better highlight the par-
allelization possibilities, the method is presented following the matrix
formulation by Koblick et al. [44].
For Chebyshev nodes and the integration interval [0, −1], the
independent variable is sampled for = 0,1,,  − 1 up-front as
=2+1(10)
with
= − cos
− 1 , 1=−1 +0
2, 2=−1 0
2(11)
Given the -dimensional sampled states 𝐲(−1)() = 𝐲(−1)
,  =
0,,  as a matrix 𝐲(−1) of dimension ×computed at the Picard
iteration 1, the whole process can be summarized in three sequential
steps to obtain the states at the iteration . The first one collects the
evaluations of the dynamics function 𝐟in the ×force matrix 𝐅[44]:
𝐅()
+1 =2𝐟𝐲(−1)
, , = 0,,  − 1 (12)
Secondly, identifying with 𝐀,𝐂,𝐒the method’s constant matrices
whose definition can be found in [44], the ×matrix 𝐁is obtained
by rows as [44]
𝐁1=𝐒𝐀𝐅 + 2𝐲0,𝐁=𝐀𝐅, = 2,,  (13)
Third and last, the ×matrix of the state guesses 𝐲()for the th
Picard iteration is
𝐲()=𝐂𝐁 (14)
The iteration process stops when the maximum state difference
between two consecutive Picard iterations 𝐲()and 𝐲(−1) drops below
a specified relative or absolute tolerance, upon user’s choice.
Despite the proved theoretical convergence, large integration spans
may lead to numerical instabilities, due to the cumulation of round-off
errors even with large as multiple orbital revolutions take place [18
20]. Fukushima [18] suggests a piece-wise approach as a workaround,
which has been implemented in this work and uses the modified
Picard–Chebyshev method to integrate orbit by orbit in sequence4until
the end of the span.
The core steps of the proposed algorithm follow the presented
scheme [19,20], together with the automatic generation of the Kep-
lerian initial guess spanning one nominal orbital period.
4. Continuous multi-flyby design in the relativistic N-body prob-
lem
Significant trajectory deflections can be achieved using flybys, how-
ever such an amplifying effect requires a high precision measure of
the entrance state to the planet’s sphere of influence. In fact, it is well
known that even small errors on the entrance conditions can lead to
completely undesired exit states, which might be disastrous for the
4The proposed implementation automatically handles either forward or
backward integration.
Acta Astronautica 194 (2022) 216–228
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A. Masat and C. Colombo
forthcoming mission phases. This issue can be mitigated increasing the
precision of the models used to simulate the trajectories, nonetheless
the high computational complexity of some perturbation effects hinders
their practical use for the mission analysis. Among those, other than
their computational burden, complex gravitational fields generated by
the N-body environment build an overall chaotic dynamical system.
This makes it extremely difficult to search for solutions similar to
each other, since such systems are characterized by diverging trajec-
tories even for small differences on the initial conditions. This work
introduces an efficient computational framework to account for such
perturbing effects, taking also advantage of the chaotic force envi-
ronment to minimize the artificial trajectory correction maneuvers.
Being the goal the development of the design technique itself to exploit
chaotic perturbations, without focusing on the particular test-case tra-
jectory, solar radiation pressure effects are neglected. Their modeling
strategy is well known, as well as its optimal exploitation through the
solar sail technology. It is true that even without a sail its magnitude
may be higher than some of the effects of the N-bodies, nonetheless
their inclusion in the dynamical model would not be a conceptual
novelty, and moreover it would not change the computational setup
proposed in this work. On top of the Newtonian gravitational effects,
general relativity contributions are included as well, to highlight that
even perturbations with the most complex physics can be exploited
by the proposed setup. General relativity effects have been previ-
ously implemented by the authors of this work [33,39], based on the
post-Newtonian model of the Einstein–Infeld–Hoffmann equations as
presented by Seidelmann [45]. The same set of equations is used by
the Jet Propulsion Laboratory (JPL) for their simulations generating
ephemerides data [46], which are also used in this work to fetch the
state of the N-bodies at each sampling time .
The b-plane theory is used to prune the optimization of a given
multi-flyby trajectory. Knowledge of desired macro-properties are as-
sumed to be known, such as semi-major axis, eccentricity, inclination
and flyby planets and times, the overall algorithm can be summarized
in two steps:
1. Obtaining the unperturbed patched-conics solution using the
b-plane theory, as explained in Section 4.1, for the interplane-
tary orbits and the planetocentric details of all of the possibly
multiple flybys.
2. Making the solution continuous in time and space, account-
ing for perturbing effects and exploiting them to minimize the
corrections required to enter subsequent flybys.
The presented steps are explained in more detail in the following
sections.
4.1. Patched conics b-plane solution for resonant orbits
Valsecchi et al. [16,17] found an analytical solution for the compu-
tation of the post-encounter orbital parameters for a given b-plane point
at the entrance of the sphere of influence. They successfully identify
fixed values of eccentricity and inclination that conserve the Tisserand
parameter, for each point belonging to a fixed semi-major axis circle.
Although analytical, the relationship is unfortunately given as a full
algorithmic procedure made of highly non-linear equations: this makes
it difficult to build the inverse relation, i.e. to retrieve the b-plane
entrance to the sphere of influence given the full set of post-encounter
orbital parameters, even in a numerical or optimization context as
convexity cannot be in general ensured.
An alternative approach was developed in a previous work [33],
defining an efficient optimization problem that uses the spherical ge-
ometry relations that generate the b-plane circles.
Specifically mentioning to the case of resonances, another optimiza-
tion layer was developed [33]: find a set of intermediate resonant
trajectories to gradually move from an initial interplanetary orbit to
a final one, which is not reachable with a single flyby, for a fixed
Fig. 3. Block-scheme diagram of the unperturbed design algorithm developed in [33].
number of intermediate flybys. A set of intermediate tentative 𝐯
targets is defined, which the algorithm tries to match while preserving
the resonance condition. A block-scheme diagram of the unperturbed
design algorithm is given in Fig. 3.
In the unperturbed and patched conics context, 1–2 s only [33] are
required by a MATLAB®implementation of this approach to design
a set of resonant orbits with Venus, which are already very close
to the actual optimized mission profile from Solar Orbiter’s mission
redbook [12].
As shown in Section 2.2, when accounting for perturbing effects in
the b-plane, a modification of the circles is inevitably introduced, as
already shown in Figs. 2(a) and 2(b) Nevertheless, the 𝐯variation due
to perturbing effects is much smaller than the difference between the
two set of circles, in relative terms [33].
4.2. Integration method adaptation and precision assessment
The SPICE toolkit [46] is used together with JPL’s ephemerides
data to retrieve the states of the bodies at any integration step,
required for the computation of both the Newtonian and the relativistic
perturbations due to the Solar System bodies. This aspect has been
noted to be the most computationally expensive task in the general
integration accounting for N-body effects, for instance making around
60% of the total account in the work by Colombo et al. [38]. In fact,
a binary source must be scanned seeking for the closest saved samples,
which must then be interpolated to fit the actual supplied time, for each
step and for each of the bodies in the integration. Time steps cannot
be foreseen with the standard integration methods, that continuously
adapt the step size and sequentially move forward or backward from a
given state, thus requiring repeated toolkit calls.
The fixed point nature of the modified Picard–Chebyshev method
brings a significant advantage to this regard: consider the restricted
N-body problem equation for a test particle written in barycentric
Cartesian coordinates:
𝐫()=−
=1
(𝐫() − 𝐫())
𝐫() − 𝐫()3(15)
with 𝐫(),
𝐫()and
𝐫()position, velocity and acceleration vectors respec-
tively.
If the time is used as the independent variable to integrate the mo-
tion of the test particle with the modified Picard–Chebyshev method,
it must be sampled a-priori on the Chebyshev nodes, by the definition
of the method itself. Using a dataset for the ephemerides instead of
requiring a custom integration of the full N-body problem makes 𝐫
sole function of the time . In turn, it is possible to sample also the
states of the bodies a-priori, as the sampling times are never going
to change through the whole integration process, and such samples can
be then given as input not only to the dynamics function evaluation, but
become a parameter for all the required Picard iterations. This aspect
can dramatically speed up numerical simulations in the interplanetary
environment, provided that the precision achieved is satisfactory.
This section aims indeed at discussing the accuracy of the method.
Note that the inclusion of general relativity effects does not add any
Acta Astronautica 194 (2022) 216–228
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A. Masat and C. Colombo
Fig. 4. Picard–Chebyshev and Runge–Kutta RK78 integration errors, as relative position difference with respect to JPL’s data for the asteroid 2010RF12, for the pre-flyby 4(a), flyby
4(b) and post-flyby legs 4(c), as well as globally for the whole integration span 4(d). The color scale is the same for all the sub-figures, and reported in 4(d). (For interpretation
of the references to color in this figure legend, the reader is referred to the web version of this article.)
conceptual complexity to the problem. Its dynamics can be simply
summarized as
𝐫() = 𝐟(𝐫(),
𝐫()) = 1,, , thus also the velocity
samples for all the bodies are required to keep the ephemerides data
as an integration parameter. Furthermore the integration is performed
piece-wise orbit-by-orbit, as suggested by Fukushima [18]: new time
nodes are generated, thus new ephemerides are sampled, one orbital
period by one orbital period until the end of the time span is reached,
or only once for the time spent within the sphere of influence in case
of flyby phases.
Figs. 4(a),4(b),4(c),4(d) show the evolution of the relative position
error with respect to JPL’s data for the near-Earth asteroid 2010RF12
from 1st January 1989, 100 years forward in time, integrating in
the Sun-centered J2000 reference frame and varying the number of
Chebyshev nodes per orbit from 15 to 200. Such asteroid was chosen
because it performs a flyby of Earth, so that the hyperbolic phase could
be tested too. The three different legs (pre-flyby in Fig. 4(a), flyby in
Fig. 4(b), post-flyby in 4(c)) are shown on their own, as well as the
overall global view is given in Fig. 4(d). The color scale is the same for
all the sub-figures, and reported in 4(d). For each leg, the initial guess
is the Keplerian solution, elliptical or hyperbolic depending on the
current status, generated from the initial state (entrance to the sphere
of influence in case of planetary flybys). The flyby time is known, so
three legs in total are shown.5It can be clearly seen that the integration
accuracy increases with increasing number of nodes per orbit, depicted
with the same color scale in all the sub-plots, and converges to the
precision of a more traditional simulation strategy plotted with the
black solid line. The latter was performed using the Runge–Kutta RK78
method, the same dynamical model was adopted in both cases.6
5The flyby detection routine is necessary to use the integrator as a whole,
nevertheless for the design purposes of this work all the legs and times are
known beforehand.
6The test case was extensively discussed by Masat [33] for the validation
of the implementation of relativistic effects.
Fig. 5 presents instead the relative relationship between the exe-
cution time of the sequential modified Picard–Chebyshev method and
the number of Chebyshev nodes. A detailed analysis of the absolute
computational performances, including scalability, speedup properties
of a parallelized version and implementation language influence, will
be included in future works upon completion of the whole integra-
tion strategy. To provide a first order of magnitude, a MATLAB®
non-parallel implementation with a MEX®function for the dynam-
ics7requires about 15 s to complete the full integration presented in
Fig. 4(d), on a single core of a local workstation equipped with an
Intel®CoreTM i7-7700 CPU (3.60 GHz).
4.3. B-plane pruned recursive refinement: reverse cascade flyby design
A recursive strategy for multi-flyby design can be built wrapping up
the concepts presented so far. The whole multi-flyby problem is broken
down in a discrete set of orbital arcs, each being covered between
two gravity assist maneuvers. When placed in a backward design, the
proposed algorithm tries to give an optimal solution to the problem:
how should flyby occur, so that flyby + 1 happens according to some
already specified features and accounting for any perturbing effect?
The b-plane design strategy [33] provides a unique entrance (and
thus exit) to the sphere of influence in the patched conics approxi-
mation. All that remains to do, conceptually, is to properly provide
the interface conditions between the two legs, accounting for all the
possible perturbations sources and replacing the zero/infinity link with
a continuity relationship. In the following lines the subscripts  and 
shall denote the specific points of entrance and exit to/from the sphere
of influence for the current flyby.
7The MEX®function was generated with MATLAB®’s Code Generation
Toolbox.
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A. Masat and C. Colombo
Fig. 5. Picard–Chebyshev serial execution runtimes for the asteroid 2010RF12, reported as relative runtime with respect to the maximum runtime obtained for the different number
of Chebyshev nodes.
Consider the entrance conditions to flyby + 1, happening at the
time (+1)
 , as the Sun-centric position 𝐫(+1)
 and velocity 𝐯(+1)
 , already
fulfilling the mission requirements for >(+1)
 together with possible
future maneuvers already defined. Consider also a deep space correc-
tion maneuver happening at the time
 > ()
. The whole entrance
condition (+1)
 ,𝐫(+1)
 ,𝐯(+1)
 is back-integrated in the perturbed en-
vironment with the modified Picard–Chebyshev method to the time
<(+1)
 , obtaining the connection state (
𝐫,
𝐯).
Assume that the unperturbed solution for flyby is expressed in the
b-plane formalism, which can also mean a manipulation of the solution
of the Lambert problem [34] with the related planetocentric phase and
not necessarily from the already mentioned b-plane algorithm [33],
particularly as:
the outgoing time ();
the outgoing b-plane coordinates (,  );
the outgoing planetocentric asymptotic velocity 𝐔.
Based on this, the time spent in the flyby phase ()can be estimated
with the time law for the hyperbolic motion,8forcing the remaining b-
plane coordinate such that the distance from the flyby planet equals
the radius of the sphere of influence. In turn, ()can be used to get
another estimate, that is the actual exit from the sphere of influence
()
 =()+()∕2. The time ()
 is actually the outer optimization
variable of the proposed algorithm. Intuitively, the time estimate aris-
ing from ()and ()might not be the best possible time when to
abandon the sphere of influence starting the phase toward flyby + 1
and performing the minimum cost correction maneuver at
, especially
because of perturbing effects acting on the way. The claim that is made
treats ()
 as a very good starting guess for an outer optimization layer,
using a ‘‘perturbation’’ ()of the exit time as optimization variable and
bounding the search to a relatively small domain. A similar reasoning is
made for the b-plane coordinates (,  )and the outgoing planetocentric
velocity 𝐔, considering the unperturbed solution as initial optimiza-
tion guess and searching over small variations thereof. Theoretical
support comes in this case from the results of the perturbed b-plane
circles: the relatively small difference between the selected points in
the perturbed and unperturbed cases suggests to use the variations
of the b-plane coordinates (,  )as two optimization variables and
to bound them again in a relatively small search space. The set of
optimization variables is completed with 𝐔, a variation of 𝐔bounded
in a small domain as well. Note that the usage of the b-plane interface
between flyby and interplanetary leg combined with the small bounded
variation approach has also a more practical reason: despite working
8Not reported here. See for instance Vallado [34] for more details.
in a backward time recursion, a perturbed trajectory that minimizes
the maneuver cost at
may in general excessively differ from the
mission requirements. The b-plane intrinsically constrains the interface
to be an actual flyby, furthermore the small and bounded search space
should ensure a perturbed trajectory not too different from the desired
profile for <()
 . Given the initial values ()
,  , , 𝐔and the generic
variations (), ,  , 𝐔, the initial conditions 𝐫()
,𝐯()
at the
time ()
=()
 +()for the forward Picard–Chebyshev integration from
()
to
are uniquely defined through the following steps:
1. the flyby planet’s state 𝐫()
,𝐯()
can be retrieved by reading the
ephem-erides database for the time ()
;
2. from (+,  + )the third b-plane coordinate is fixed by
requiring the distance from the planet to equal the radius of the
sphere of influence;
3. the b-plane coordinates (+, ,  + )can be converted into
the planetocentric cartesian coordinates 𝐫, because the axes of
the b-plane reference frame are uniquely defined as in Eq. (1)
and the planetocentric velocity vector is 𝐔+𝐔;
4. The Sun-centric coordinates 𝐫()
,𝐯()
are retrieved by the sim-
ple summations 𝐫()
 =𝐫 +𝐫()
and 𝐯()
 = (𝐔+𝐔) +
𝐯()
.
The initial value problem identified by 𝐫()
,𝐯()
at the time ()
=
()
 +()is solved numerically forward in time with the modified
Picard–Chebyshev method, to the connection maneuver at an arbitrary
time
. Using a concise notation, the initial value problem to be numer-
ically integrated will be identified by the dynamics functions 𝐫()and
𝐯(), with ()

, for position and velocity respectively, and setting
the initial conditions:
0=()
,𝐫(0) = 𝐫()
,𝐯(0) = 𝐯()
 (16)
In general, the forward-integrated state 𝐫(
),𝐯(
)at the correc-
tion maneuver time
will differ from the back-integrated state that
leads to flyby + 1 by
𝐫=𝐫(
) −
𝐫𝟎and
𝐯=𝐯(
) −
𝐯𝟎(17)
The physics of the correction maneuver performed at the maneuver
time
embeds the mandatory constraint of the position where it is
to happen, theoretically defined as
𝐫=𝟎. Note that the dynamical
motion is numerically integrated, thus leaving the maneuver position
as a pure equality constraint might severely affect the computational
performance of the optimization: a full Picard–Chebyshev integration
would be required to evaluate the constraint function, since the opti-
mization variables are nothing but the b-plane form of the initial state
𝐫()
,𝐯()
and their simulation to the connection time
would always
Acta Astronautica 194 (2022) 216–228
223
A. Masat and C. Colombo
Fig. 6. Block-scheme diagram of features and steps embedded in the solution of the optimization problem of Eq. (18).
be needed. At the opposite side, the actual maneuver to be designed
may not have any physical sense if omitted, as the continuity require-
ment may be lost. Nonetheless, in a numerical context an absolutely
negligible value of
𝐫suffices to satisfy the physical meaning of the
correction maneuver. These observations led to the choice of explicitly
implementing the position constraint with a penalty method [47], that
is penalizing the objective function (the correction
𝐯in this case)
adding a large term direct function of the position difference
𝐫.
Therefore, defining =
𝐯, omitting the explicit dependencies on
the optimization variables for conciseness and denoting the components
of 𝐔with ′(1,2,3)
minimize
, ,𝐔
, ()
+
, ()
subject to  ,
,
′(1,2,3) ′(1,2,3)

(18)
with =
𝐫and the weighting factor of the penalty method
sufficiently large. The choice of is in general arbitrary, it will be
discussed in Section 5for the presented test case.
A block-scheme diagram summarizing all the presented features and
steps of the algorithmic optimization problem in Eq. (18) is given in
Fig. 6. The initial unperturbed solution expressed in the b-plane formal-
ism is converted to a Cartesian state and used to prune the optimization
process. Subsequently, the fixed-point nature of the Picard–Chebyshev
nature requires to sample an initial trajectory guess on fixed time
nodes: as already mentioned, this feature is also exploited to perform
the sampling of the N-bodies ephemerides only once, not only for the
Picard–Chebyshev integration but also for the whole optimization run.
Finally, the ‘‘closed loop’’ that can be seen in Fig. 6 is entered, where
each objective function evaluation involves the Picard–Chebyshev for-
ward integration of some coordinates, generated from the optimization
variables expressed as b-plane variations. The exit conditions strongly
depend on the chosen implementation, although they can follow any
already existing scheme (for instance, relative state and objective func-
tion variations smaller than some user-defined tolerance in this work,
as it is discussed in Section 5.2).
Finally, assuming the position constraint to be fulfilled in the opti-
mization problem of Eq. (18) whose result gives
, ()
=
, ()
+

, ()
, for optimizing the flyby time it is enough to use its defini-
tion ()
=()
 +(), with ()
minimize
()
, ()
 +()
subject to ()
(19)
the new optimization variable:
Table 1
Optimization levels for the full design of the arc to + 1.
+(Eq. (18))Optimize
←←
,, (Eq. (19))
(Eq. (19))Optimize
←←←←←←
()to + 1, maneuver at
to + 1, maneuver at
Optimize
←←←←←←
to + 1, optimal
Note that no choice has been made yet about the optimization algo-
rithms, which might be sensitive to the search space size and the func-
tion relative steepness within the different regions. Moreover, it should
also be tailored on the available computational resources, i.e. preferring
parallelizable routines over dominantly sequential algorithms for high
performance computing facilities.
The optimization problem of Eq. (18) is a sub-problem of the opti-
mization problem of Eq. (19). This somehow enhances the flexibility of
the approach, i.e. the former might be used for search space exploration
purposes without the need of a finely refined solution in terms of
starting time ()
. Note that the optimization problem of Eq. (18) is
explicitly dependent on the maneuvering point
, which in fact can
and for practical applications should be optimized as well. In this work
it shall remain a problem parameter, as more focus is put toward
exploring the effect of small variations of the departure time ()
.
Completing the description, another more optimization level can be
easily defined to find the best
similarly to what done for ()
in
the optimization problem of Eq. (19), and in the presented formalism
it shall straightforwardly include the innermost level defined by the
optimization problem of Eq. (18). A summary of the relationship among
the different optimization levels is given in Table 1.
Some observations regarding the expected computational perfor-
mances of the optimization can be made based on the analysis of the
Picard–Chebyshev integration, already presented in Figs. 4(a),4(b),
4(c),4(d) for the accuracy behavior and Fig. 5 for the computational
time variation with increasing number of Chebyshev nodes. For se-
quential executions, the higher the value of
the higher the runtime
will be, if the number of nodes per period is kept constant. Note also
that the optimization problem of Eq. (18) is going to benefit from
the minimal ephemerides overhead as a whole: the boundary times
are fixed, thus the ephemerides dataset can be scanned only once and
the related values can be considered as parameters not only within
the Picard–Chebyshev integration, but also for all the iterations of
the optimization algorithm. Finally, the penalty approach [47] used
to define the optimization problem of Eq. (18) allows for massively
parallel and brute force strategies to be implemented as well, because
all the remaining constraints are of boundary type.
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A. Masat and C. Colombo
Table 2
Maneuvering point, apocenter of unperturbed initial pruning solution.
𝐫(+1)
 [km]𝐯(+1)
 [km/s]
67030683.03 85738232.37 2563856.42 30.54 4.05 1.79
5. Application: Solar Orbiter’s first resonant phase with venus
The presented design procedure was tested on a phase of the ongo-
ing mission Solar Orbiter [11], particularly taking the initial data from
the trajectory profile with launch in January 20179available in the
mission redbook [12]. For this concept validation phase, the algorithm
has been entirely implemented in MATLAB®. A small computational
acceleration is introduced compiling the Picard–Chebyshev iterations
into a MEX®function with MATLAB®Coder. The optimization prob-
lem of Eq. (18) is solved with the fmincon.m function of MATLAB®’s
Optimization Toolbox, using both the Interior-point and Sequential
Quadratic Programming methods [47], based on the dimension of the
search space. The selected local optimization algorithms and tolerances
sufficed for the test case of this work to converge, proving the method-
ology concept of efficiently designing trajectories that take advantage
of chaotic perturbations. The use of global search approaches and/or
different tolerance setups could for sure increase the robustness of the
approach, at the price of possibly increasing the total computational
load. In the case of practical use by mission analysts, the optimization
algorithm selection should also be tailored on the hardware availability,
possibly exploiting the full potential of supercomputing facilities.
Solar Orbiter’s first resonant phase with Venus is reproduced ac-
counting for perturbing effects from the bodies and general rela-
tivity. Following the notation from the mission redbook [12], the two
gravity assist maneuvers are identified with V2 and V3, with V standing
for the flyby planet (Venus) and the numbers 2 and 3 representing the
second and third close approach with Venus from the mission launch,
respectively. The interplanetary leg between the two flybys is identified
with V2–V3. The goal is to design flyby V2 so that V3 can lead to a
desired post-encounter trajectory almost ballistically, i.e. minimizing
the correction maneuver required in the phase between V2 and V3.
The maneuver is designed with maneuvering time
at the apocenter
of the first nominal orbit after V2, nevertheless as already mentioned
even this aspect can and should be optimized.
5.1. Boundary conditions, b-plane pruning and method parameters
Generally, the required boundary condition is the state vector that
allows a specified entrance to flyby + 1. It may come from a previous
step of the presented flyby design algorithm, as the output of the
back-integration of 𝐫(+1)
 ,𝐯(+1)
 , or simply being given, if no close
approach is to happen after flyby + 1. Considering the Solar Orbiter-
like mission, flyby V3 may be entered as the interplanetary state written
in the ecliptic J2000 reference frame reported in Table 2, at the time
(+1)
 = 8119.84 MJD2000.
Solar Orbiter’s first resonant phase with Venus is in a 3/4 resonance,
which means that in the unperturbed and patched conics case flyby V2
is to happen 3 Venus’ periods before flyby V3, and in the meantime
Solar Orbiter would have traveled for 4 of its orbital periods. The
output of the b-plane preliminary unperturbed design [33] enforcing
the 3/4 resonance has produced the pruning quantities reported in
Table 3, together with the exit time from flyby set as ()
 = 7446.52
MJD2000.
9Later discarded, the actual mission left Earth on February 2020.
Table 3
Retrieved optimal b-plane coordinates (, )and planetocentric velocity 𝐔.
[km][km][km/s][km/s][km/s]
8057.07 5497.19 3.08 17.78 3.66
Table 4
Maneuvering point, apocenter of unperturbed initial pruning solution.
r[km]
v[km/s]
133524954.60 32036518.08 4418791.75 5.09 20.43 1.65
The maneuvering time is set as a parameter, particularly at the
nominal10 apocenter of the first interplanetary resonant orbit, at the
time
= 7570.92 MJD2000, with the correspondent state (
𝐫,
𝐯)reported
in Table 4.
The maximum values where to bound ,  ,  ′(1,2,3)
 have
been set as 1% of the impact parameter [36]=2+2and of 𝐔
for the b-plane coordinates and the velocity components respectively.
The boundary value for the exit time variation  is set to 1% of
Venus’ orbital period. Trivially, the optimization starts with all the
variables ,  , 𝐔, ()set equal to zero. The cost functions and
are in all the cases computed as the relative values 𝐯
𝐯and
𝐫
𝐫with respect to the known maneuvering point, to remove the
possible dimension sensitivity.
Specifically for the modified Picard–Chebyshev method, 160 nodes
per period are used and the iterations are stopped when the maximum
of the relative difference between two consecutive state updates drops
below 10-14. The first arc to be designed, i.e. the one defining the
optimal exit and the maneuver, spans less than one orbital period, thus
proportional nodes to the defined 160 per period based on its total
time length are set, according to the fixed nodes per period logic. The
optimal time found is then used for a single run of the optimization
problem of Eq. (18) with 200 Chebyshev nodes, assessing the influence
of the number of nodes in the design precision, comparing both the
node cases against a relativistic simulation.
5.2. Optimization implementation
Despite the narrow region where the optimization variables are set
to vary, it has been observed that even the smallest variations have a
relevant impact in the convergence of the algorithm, especially if the
position constraint is made strict. For this reason and to keep a robust
approach in the concept validation phase, the optimization problem
of Eq. (18) is solved several times in a continuation procedure, using
the result of the previous step as the new starting guess. Particularly:
the search space dimension is reduced by 10 times for each op-
timization problem, up to an absolute minimum of 108starting
from the already introduced ±1% for each variable;
within the optimization solver, the initial minimum relative step
size between two iterations is of 106, reduced by a factor 10 each
time up to 1015;
the penalty factor is initially set to 105to improve the con-
vergence also for the contribution, although the position con-
straint is then made stricter by raising the value of by a factor
10 each time, up to 109;
the Interior-point algorithm in fmincon.m is selected for the
first half optimization problems, whereas Sequential Quadratic
Programming is used in the last ones because of the smaller search
space;
10 It is set as if the orbital parameters were exactly equal to the desired
trajectory after the maneuver, assuming the difference between pre and post
maneuver orbits to be small.
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A. Masat and C. Colombo
Table 5
Optimization results, in terms of position difference residual 𝐫and correction effort
𝐯at the maneuvering time.
𝐫[m]𝐯[m/s]
0.52 0.52 1.19 1.28 1.57 0.22
Table 6
Optimization results, in terms of initial position residual and vmagnitude.
𝐯[m/s]𝐯
v[]𝐫[m]𝐫
r[]
2.04 9.66 × 10−5 1.39 1.02 × 10−11
Table 7
Optimization results, in terms of initial optimal state 𝐫()∗
 ,𝐯()∗
 .
𝐫()∗
 [km]𝐯()∗
 [km/s]
64960957.28 85998225.22 2682290.24 31.00 3.45 1.7
MATLAB®’s globalsearch algorithm solves the current opti-
mization problem if the previous step has returned the starting
guess without improvements, searching for a global minimizer.11
The optimization problem of Eq. (19) is solved with a grid search
approach. The time span is always sampled with the initial supplied
value plus 40 evenly spaced values of (), reducing  by a factor
10 for 5 times, from the initial grid size equal to ±1% of Venus’ orbital
period. The best value from the previous search is used as starting
point for the new one. This approach resembles the algorithm used in
MATLAB®’s patternsearch.m function, implemented manually in
this work to keep a low number of trial ()in this concept validation
phase.
5.3. Results
Solving the optimization problem of Eq. (18) with the above de-
scribed implementation took about 2–3 min on a single core of a local
workstation equipped with an Intel®CoreTM i7-7700 CPU (3.60 GHz).
The 200 nodes algorithm12 converged to the residual 𝐫and impulsive
action 𝐯for the required maneuver presented in Table 5.
One should note both how small the correction effort is, despite
the execution point
is yet to be optimized, and the fulfillment of
the position constraint. The presented maneuver is modeled as a single
impulse, nevertheless given its magnitude it can be easily achieved by
the current low thrust propulsion technologies, as shown in Table 6
Most of the computational time was required to fulfill the position
constraints. If the presented algorithm was to be used with a wider but
still good position tolerance, it is likely to run significantly faster even
before its parallel implementation.
The best starting time obtained was ()∗
 = 7446.52 MJD2000,
slightly higher than the initial guess ()
. The interplanetary optimal
starting state the initial condition is given in Table 7.
The b-plane coordinates (,  )and the planetocentric velocity 𝐔′∗
retrieved from 𝐫()∗
 ,𝐯()∗
 and Venus’ position at ()∗
 are presented in
Table 8, proving the optimal pruning brought by the b-plane prediction.
11 For performance reasons a maximum of half of the iterations can run the
global search, in any case the presented test case at most two were experienced
out of all the ten steps.
12 The difference with the 𝐯resulting from the 160 nodes run is negligible,
the position constraint is slightly worse fulfilled but in the same order of
magnitude.
Table 8
Retrieved optimal b-plane coordinates (, )and planetocentric velocity 𝐔.
[km][km]
[km/s]
[km/s]
[km/s]
8057.22 5700.49 3.25 17.76 3.67
Fig. 7. Visual representation of the b-plane pruning strategy. The pruning point
corresponds to the blue dot, whereas the optimized point is depicted in dark orange.
The value of looks slightly (2%) out of the initial bounds despite
the constraint, which may have two different explanations. First, the
optimization variables are updated concurrently: variations on 𝐔also
change the orientation of the b-plane axes, which in turn result on dif-
ferent b-plane coordinates for a given fixed position in space. Secondly,
the domain reduction sequential procedure may find a minimum close
to the initial boundaries, centering there the next narrower search. The
difference is in any case rather small in magnitude, as it can be also
seen in the real-scale difference shown in Fig. 7: with the pruning point
identified by the blue dot, whereas the optimized one is plotted in dark
orange.
Fig. 8 shows the difference between the designed trajectory with
respect to a relativistic simulation of the same case, both featuring the
optimized maneuver at
. It can be observed that the two trajectories ba-
sically coincide even if using the lower number of Chebyshev nodes, for
a relative difference that remains in the order of 108and as expected
from what already seen in Figs. 4(a),4(b),4(c),4(d). Again as expected
the higher number of nodes brought a more accurate solution, with
the difference from the relativistic simulation reduced by more than 10
times. Even if small, the error inevitably cumulates and gets amplified
if multiple gravity assists are present, thus a higher number of nodes
should be kept the more precise the design needs to be. The periodic
‘‘hills’’ visible in Fig. 8 happen far from the domain boundaries, and
are located where the Chebyshev nodes become more sparse.13 In this
particular case, they correspond to the neighborhood of the pericenter
of the pre-maneuver arc. The periodic error increase likely due to
the faster orbital dynamics nearby the pericenter, not followed by the
Chebyshev nodes thickness, as the domain boundaries are located at the
correction maneuver (at the apocenter). This effect could be mitigated
by adapting the Picard–Chebyshev integration intervals so that the
node distribution becomes denser nearby the pericenter, for instance
splitting the optimization horizon into two sub-intervals, the first from
the flyby exit to the pericenter, the second from the pericenter to the
connection maneuver point, and finally following the same concept for
the fixed post-maneuver arc. If the full trajectory was required with
13 Because of the definition of Chebyshev nodes in Eqs. (10) and (11).
Acta Astronautica 194 (2022) 216–228
226
A. Masat and C. Colombo
Fig. 8. Design difference with respect to relativistic simulation between V2 and V3 for the two node cases, and without maneuver.
Fig. 9. Solar Orbiter’s continuous first resonant phase with Venus.
as high precision as possible this should be considered, anyway, even
with the tested setup, the long-term error evolution remains low, and
already allows for a precise design at the event points (flybys and cor-
rection maneuver) compared against standard simulation techniques.
The ‘‘noise’’ over those hills is explained by interpolation of the Picard–
Chebyshev solution over the standard simulation time steps, necessary
to visualize the presented difference measurement. The impact of the
correction maneuver, despite small, can also be assessed: at the time
of the close approach V3 the position would differ of thousands of
kilometers from the desired condition, definitely destroying the correct
occurrence of the flyby.
Finally, Figs. 9(a) and 9(b) show the continuous trajectory that
embeds the planetocentric phases for both the flybys V2 and V3,
together with the pre-V2 and post-V3 solutions and all generated with
the Picard–Chebyshev approach. As expected they are all very similar
to the original mission profile (Fig. 9(a)), and zooming over the flyby
regions the new continuity feature can be recognized (Fig. 9(b)).
6. Conclusion
The complexity of the multi-flyby design problem in the continuous
environment was successfully broken down into a backward recursive
approach that designs each of the flybys in cascade, considering the
next encounter as the target condition for how to perform the current
one. Given the results of an unperturbed patched conics analysis, the
b-plane was proven to be a powerful formalism to enforce a continu-
ity condition with, and particularly well suited for pruning purposes,
making the dimension of the optimization search space minimal.
A first possible development direction is the inclusion of tighter
mission constraints, such as a minimum pericenter distance as Solar
Orbiter [11] needs. This aspect might be tackled with the proposed
strategy before the design of any maneuver, seeking for quasi-ballistic
solutions that surf the effects of orbital perturbations, even if chaotic,
aiming to minimize the required artificial corrections.
Despite it might be already satisfactory, the computational perfor-
mance of the method is for sure what can be improved the most by
future works. First of all, the newest adaptive version of the modified
Picard–Chebyshev method can be adopted and the sequential execu-
tion can be accelerated by a complete implementation in a compiled
programming language, instead of the MATLAB®platform as proposed
in this work. Furthermore, although the multi-step solution of the
optimization problem of Eq. (18) proposed in the presented application
is extremely robust, it could be better tailored by prior analysis of the
search space or made more lightweight already scanning with finer
tolerances. Most importantly, the whole approach can and is built to
be parallelized, both for the modified Picard–Chebyshev method and
the solution of the optimization problem.
In conclusion, a systematic framework about how to surf a complex
perturbation environment such as the relativistic N-body problem is
proposed. Provided the model to be sufficiently accurate, the available
technology might become the new bottleneck for practical purposes:
some uncertainty is inevitably introduced by the execution of the
control maneuvers, as well as the connected orbital determination mea-
surements. Future works might also deepen this aspect using models of
real life equipment, studying in turn what consequences non-precise
Acta Astronautica 194 (2022) 216–228
227
A. Masat and C. Colombo
measurements or thruster firings might have on the high-fidelity de-
signed trajectory, together with the possible required mission planning
actions.
Looking to possible applications in other environments, the pre-
sented approach may be used in the design of moon tour missions
toward the giant planets, which feature the available fuel as a major
constraint. Quasi-ballistic solutions are always sought for, to swing
by the numerous bodies multiple times maximizing the exploration
outcome. For instance the currently being planned JUICE [48] could
benefit from this design strategy, since moving from the interplanetary
to the jovian system would not require major changes at all.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Funding sources
The research leading to these results has received funding from
the European Research Council (ERC) under the European Union’s
Horizon2020 research and innovation programme as part of project
COMPASS (Grant agreement No 679086), www.compass.polimi.it.
Appendix
The expansions of 𝐟(𝐲, )and 𝐲()in terms of the Chebyshev polyno-
mials of the first kind ()[41] are:
𝐟(𝐲, ) =
+∞
=0
𝐅()
𝐲() =
+∞
=0
𝐘()
(20)
where () = cos(arccos ). Denoting with 0and initial and final
integration times respectively and introducing =with 0
and for generality:
=2− (+)
,−1 1(21)
Fukushima [18] presented the method for scalar functions, whereas
in this work 𝐅and 𝐘are vectors of the same dimension of the
state 𝐲, representing the coefficients of the th term in the Chebyshev
expansion.
Truncating the expansion to the order for the state and to order
− 1 for the dynamics function, and highlighting the th Picard
iteration yields [18]
𝐟()(𝐲(), ) =
()−1
=0
𝐅()
()
𝐲()() =
()
=0
𝐘()
()
(22)
Note that is not necessarily fixed for all the iterations. The
physical time is discretized by selecting the zeros of the polynomial
as the points of evaluation of 𝐟(𝐲, )and 𝐲(), to obtain [18,41]:
()
=+
2
2cos(2− 1)
2(), = 1,, ()(23)
Applying the Picard iteration brings the approximation 𝐲()
:
𝐲()
=𝐲(−1)()
=
(−1)
=0
()
 𝐘(−1)
(24)
where, since (cos ) = cos [41]:
()
 = cos(2− 1)
2()(25)
𝐟can now be evaluated at the points 𝐲()
:
𝐟()
=𝐟𝐲()
, ()
(26)
After a re-arrangement of the terms and exploiting the orthogonality
of the Chebyshev polynomials [41], the following recursions for the
coefficients 𝐘()
are obtained:
𝐘()
=
 ()
()
=1
()
 𝐠()
, = 1,, ()
𝐘()
0=𝐲0
()
=1
𝐘()
(
0)
(27)
with
()
 = sin(2− 1)
2()
𝐠()
=𝐟()
()
1
(28)
Note that the expression for 𝐘()
0in Eq. (27) simply means 𝐲()(0) =
𝐲0[18].
The recursion can be summarized as:
𝐘(−1)
Eq. (24)
𝐲()
Eq. (26)
𝐟()
Eq. (27)
𝐘()
(29)
In the above lines the two different subscripts and are used to
distinguish the quantities where the function evaluation is performed
and the ones that build the actual Chebyshev coefficients.
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... This work takes inspiration from the results obtained in [12,13], whose aim was to optimize a multi-flyby trajectory arc while taking advantage of the natural dynamics of the N-body relativistic environment. Despite the conceptual complexity, the combined b-plane and Picard-Chebyshev (PC) approach led to an overall computationally feasible algorithm, for a Matlab ® sequential implementation converging to an optimal trajectory in a few minutes only. ...
... Differently from the previous developments of the method, this work introduces an augmentation of the dynamical system being integrated to enable stable massive parallelism for the short-term propagation of large sets of initial conditions. The results obtained in [12,13], as all the simulated trajectories analyzed to find the optimal one, are re-run as a direct application of the proposed approach, discussing the parallelization options in deep detail. Within a single run, the common time nodes of all the trajectory arcs allow to rework the iterative refinement process of PC. ...
... The test case follows what already computed in [12,13]. The first resonant phase with Venus of a Solar Orbiter-like mission was reproduced, designing a continuous trajectory even during flyby injections and exits. ...
Preprint
Full-text available
The orbital propagation of large sets of initial conditions under high accuracy requirements is currently a bottleneck in the development of space missions, e.g. for planetary protection compliance analyses. The proposed approach can include any force source in the dynamical model through efficient Picard-Chebyshev (PC) numerical simulations. A two-level augmentation of the integration scheme is proposed, to run an arbitrary number of simulations within the same algorithm call, fully exploiting high performance and GPU (Graphics Processing Units) computing facilities. The performances obtained with implementation in C and NVIDIA CUDA programming languages are shown, on a test case taken from the optimization of a Solar Orbiter-like first resonant phase with Venus.
... This work takes inspiration from the results obtained in [12,13], whose aim was to optimize a multi-flyby trajectory arc while taking advantage of the natural dynamics of the N-body relativistic environment. Despite the conceptual complexity, the combined b-plane and Picard-Chebyshev (PC) approach led to an overall computationally feasible algorithm, for a Matlab ® sequential implementation converging to an optimal trajectory in a few minutes only. ...
... Differently from the previous developments of the method, this work introduces an augmentation of the dynamical system being integrated to enable stable massive parallelism for the short-term propagation of large sets of initial conditions. The results obtained in [12,13], as all the simulated trajectories analyzed to find the optimal one, are re-run as a direct application of the proposed approach, discussing the parallelization options in deep detail. Within a single run, the common time nodes of all the trajectory arcs allow to rework the iterative refinement process of PC. ...
... The test case follows what already computed in [12,13]. The first resonant phase with Venus of a Solar Orbiter-like mission was reproduced, designing a continuous trajectory even during flyby injections and exits. ...
Article
Full-text available
The orbital propagation of large sets of initial conditions under high accuracy requirements is currently a bottleneck in the development of space missions, e.g. for planetary protection compliance analyses. The proposed approach can include any force source in the dynamical model through efficient Picard-Chebyshev (PC) numerical simulations. A two-level augmentation of the integration scheme is proposed, to run an arbitrary number of simulations within the same algorithm call, fully exploiting high performance and GPU (Graphics Processing Units) computing facilities. The performances obtained with implementation in C and NVIDIA® CUDA® programming languages are shown, on a test case taken from the optimization of a Solar Orbiter-like first resonant phase with Venus.
... This work takes inspiration from the results obtained in [12,13], whose aim was to optimize a multi-flyby trajectory arc while taking advantage of the natural dynamics of the N-body relativistic environment. Despite the conceptual complexity, the combined b-plane and Picard-Chebyshev (PC) approach led to an overall computationally feasible algorithm, for a Matlab ® sequential implementation converging to an optimal trajectory in a few minutes only. ...
... Differently from the previous developments of the method, this work introduces an augmentation of the dynamical system being integrated to enable stable massive parallelism for the short-term propagation of large sets of initial conditions. The results obtained in [12,13], as all the simulated trajectories analyzed to find the optimal one, are re-run as a direct application of the proposed approach, discussing the parallelization options in deep detail. Within a single run, the common time nodes of all the trajectory arcs allow to rework the iterative refinement process of PC. ...
... The test case follows what already computed in [12,13]. The first resonant phase with Venus of a Solar Orbiter-like mission was reproduced, designing a continuous trajectory even during flyby injections and exits. ...
Preprint
Full-text available
The orbital propagation of large sets of initial conditions under high accuracy requirements is currently a bottleneck in the development of space missions, e.g. for planetary protection compliance analyses. The proposed approach can include any force source in the dynamical model through efficient Picard-Chebyshev (PC) numerical simulations. A two-level augmentation of the integration scheme is proposed, to run an arbitrary number of simulations within the same algorithm call, fully exploiting high performance and GPU (Graphics Processing Units) computing facilities. The performances obtained with implementation in C and NVIDIA ® CUDA ® programming languages are shown, on a test case taken from the optimization of a Solar Orbiter-like first resonant phase with Venus.
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Designing long-duration lunar orbiter missions is challenging due to the Moon’s highly nonlinear gravity field and the third-body perturbations induced by the Earth, Sun and other large bodies. The absence of a Lunar atmosphere has offered the possibility for mission designers to search for extremely low-altitude, quasi-stable lunar orbits. In addition to the reduced amount of propellant required for station-keeping maneuvers, these orbits present great opportunities for unique scientific studies such as high resolution imaging and characterization of the polar ice deposits in deep craters. Prior to the GRAIL mission, mission planning for Lunar orbiters had suffered from inaccuracies, mainly due to the lack of an accurate Lunar gravity model, which resulted in severe deviations with respect to the spacecraft’s nominal orbit. We study station-keeping feasibility for spacecraft in near-polar and extremely low-altitude, quasi-frozen orbits around the Moon, that are perturbed by a high-fidelity lunar gravity model and third-body effects from the Earth and Sun. For several candidate orbits, we compare the trade-space between mission duration and ∆V budget, considering impulsive maneuvers applied once every ‘N ∈ {2,6,10,14,18}’ orbits at periselene or aposelene. Additionally, we investigate the propulsive cost for different orbit insertion dates, the location of impulsive corrections for arresting argument of periselene (ω) drift, and controlling periselene altitude.
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We present the results of a comprehensive study in which the precision and efficiency of six numerical integration techniques, both implicit and explicit, are compared for solving the gravitationally perturbed two-body problem in astrodynamics. Solution of the perturbed two-body problem is fundamental for applications in space situational awareness, such as tracking orbit debris and maintaining a catalogue of over twenty thousand pieces of orbit debris greater than the size of a softball, as well as for prediction and prevention of future satellite collisions. The integrators used in the study are a 5th/4th and 8th/7th order Dormand-Prince, an 8th order Gauss-Jackson, a 12th/10th order Runga-Kutta-Nystrom, Variable-step Gauss Legendre Propagator and the Adaptive-Picard-Chebyshev methods. Four orbit test cases are considered, low Earth orbit, Sun-synchronous orbit, geosynchronous orbit, and a Molniya orbit. A set of tests are done using a high fidelity spherical-harmonic gravity (70 × 70) model with and without an exponential cannonball drag model. We present three metrics for quantifying the solution precision achieved by each integration method. These are conservation of the Hamiltonian for conservative systems, round-trip-closure, and the method of manufactured solutions. The efficiency of each integrator is determined by the number of function evaluations required for convergence to a solution with a prescribed accuracy. The present results show the region of applicability of the selected methods as well as their associated computational cost. Comparison results are concisely presented in several figures and are intended to provide the reader with useful information for selecting the best integrator for their purposes and problem specific requirements in astrodynamics.
Conference Paper
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This paper captures trajectory analysis of a representative low thrust, high power Solar Electric Propulsion (SEP) vehicle to move a mass around cis-lunar space in the range of 20 to 40 kW power to the Electric Propulsion (EP) system. These cis-lunar transfers depart from a selected Near Rectilinear Halo Orbit (NRHO) and target other cis-lunar orbits. The NRHO cannot be characterized in the classical two-body dynamics more familiar in the human spaceflight community, and the use of low thrust orbit transfers provides unique analysis challenges. Among the target orbit destinations documented in this paper are transfers between a Southern and Northern NRHO, transfers between the NRHO and a Distant Retrograde Orbit (DRO) and a transfer between the NRHO and two different Earth Moon Lagrange Point 2 (EML2) Halo orbits. Because many different NRHOs and EML2 halo orbits exist, simplifying assumptions rely on previous analysis of orbits that meet current abort and communication requirements for human mission planning. Investigation is done into the sensitivities of these low thrust transfers to EP system power. Additionally, the impact of the Thrust to Weight ratio of these low thrust SEP systems and the ability to transit between these unique orbits are investigated.
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An adaptive self-tuning Picard–Chebyshev numerical integration method is presented for solving initial and boundary value problems by considering high-fidelity perturbed two-body dynamics. The current adaptation technique is self-tuning and adjusts the size of the time interval segments and the number of nodes per segment automatically to achieve near-maximum efficiency. The technique also uses recent insights on local force approximations and adaptive force models that take advantage of the fixed-point nature of the Picard iteration. In addition to developing the adaptive method, an integral quasi-linearization “error feedback” term is introduced that accelerates convergence to a machine precision solution by about a of two. The integral quasi linearization can be implemented for both first- and second-order systems of ordinary differential equations. A discussion is presented regarding the subtle but significant distinction between integral quasi linearization for first-order systems, second-order systems that can be rearranged and integrated in first-order form, and second-order systems that are integrated using a kinematically consistent Picard–Chebyshev iteration in cascade form. The enhanced performance of the current algorithm is demonstrated by solving an important problem in astrodynamics: the perturbed two-body problem for near-Earth orbits. The adaptive algorithm has proven to be more efficient than an eighth-order Gauss–Jackson and a 12th/10th-order Runge–Kutta while maintaining machine precision over several weeks of propagation.