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Content uploaded by Giacomo Borelli
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All content in this area was uploaded by Giacomo Borelli on Nov 07, 2019
Content may be subject to copyright.
Italian Association of Aeronautics and Astronautics
XXV International Congress
9-12 September 2019| Rome, Italy
LOW THRUST MULTI-REVOLUTION TRANSFER DESIGN
USING ARTIFICIAL POTENTIAL GUIDANCE IN THE
PHASE SPACE
Giacomo Borelli1*, Marco Nugnes2*, Camilla Colombo3*
*Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di
Milano, Via La Masa 34, 20156 Milano- Italy
1 giacomo.borelli@mail.polimi.it
2 marco.nugnes@polimi.it
3 camilla.colombo@polimi.it
ABSTRACT
The semi-analytical treatment of spacecraft’s perturbed orbits can be used to interpret and
evaluate efficiently its orbit evolution. The present study proposes a semi-analytical
methodology to design continuous low thrust multi-revolutions manoeuvres in the mean orbital
elements phase space introducing an artificial perturbation formulated as disturbing averaged
potential function. The method proposed aims to be computationally efficient both for an
onboard implementation of the guidance with high autonomy requirements and for preliminary
mission design. The low thrust is modelled in the averaged domain, introducing a pre-defined
steering law, which will be tuned over many revolutions to obtain the artificial potential
perturbation effects. The potentials introduced to perturb the orbit phase space are expressed
in function of constant parameters and the transfer problem is translated into a parameter
optimisation problem to shape the artificial potential function. The methodology developed is
preliminary applied to the orbit planar dynamics reduced to the equatorial plane perturbed by
the Earth’s oblateness, studying the conservative transfers for a test case, an heliotropic control
of a swarm of satellites and a reconfiguration of an elliptic equatorial constellation with equally
spaced oriented apogees.
Keywords: Orbit manoeuvres, semi-analytical methods, phase space, low thrust.
1 INTRODUCTION
Multi-revolutions transfers can be cumbersome to design with the traditional methods used to
deal with continuous low thrust. In particular, the optimal control methods are characterised by
a substantial computational effort needed to compute the low thrust action along the trajectory.
For satellites with limited resources and high autonomy requirements, simpler and
computationally cheaper methods are often preferred. In literature the semi-analytic
formulations of the dynamics over one orbit revolution have been exploited to simplify the low
thrust multi-revolutions transfer problem [1][2][3][4]. For example, in Gao [1], a suboptimal
steering law is employed and tuned using a direct optimisation method both for the minimum
fuel and minimum time cases in the average domain. Blended control laws have been also
investigated, both with direct optimisation methods, Kluever and Oleson [4], or with closed-
loop tuning based on the errors with the target state, Huang et al. [3]. Closed-loop guidance
schemes using Lyapunov based controls have been also explored [2][5] for the robustness to
the external perturbations, uncertainties and errors. The present work develops a methodology
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
2
for planning the low thrust manoeuvre in the orbital elements averaged perturbed phase space,
making an extensive use of semi-analytical methods to reduce the computational effort in the
trajectory computation. The method is useful as cheap and efficient algorithm to be
implemented onboard a highly autonomous and low resources spacecraft or for preliminary
mission design considerations on the low thrust manoeuvre.
2 GUIDANCE MODEL
2.1 Methodology description
The dynamics describing the low thrust action is simplified using the averaging procedure over
one orbital revolution.
(1)
where the rate is obtained applying to the i-th Gauss’ equation the approximation on
the rate of eccentric anomaly , valid for small perturbing accelerations relative
to the primary body spherical gravity [1]. In this equation n is the mean motion, r is the
osculating position of the spacecraft and a the orbit semi-major axis. The orbital elements are
considered constant during the integral operation. The
function represents the
averaged simplification of the Gauss’ equation of the i-th orbital element, obtained from the
knowledge of the acceleration vector dependence on the fast variable, i.e. the eccentric
anomaly, within one revolution with analytical integration or with quadrature methods. In this
work, great care is placed upon the introduction of a steering law, , analytically
averageable. The variables will represent the parameters that define the steering scheme
over one revolution, i.e. thrust angles and burn arcs widths.
The key aspect of this work is to use the low thrust averaged dynamics to follow a target
artificial potential function evolution, which is then used to perturb the long-term orbit phase
space in the desired fashion. It is well known that the perturbed long-term orbit dynamics can
be expressed in the Lagrange’s form from the averaged disturbing potential function
,. Such
formulation highlights the property of the disturbing potential function of being an integral of
motion, where the iso-surfaces of the potential function in the orbital elements phase space
represent the long-term evolution of the orbit in time. In this work, an additional artificial
potential function is used to act on the perturbed orbital element phase space ,
inducing different evolutions by modifying the orbit phase space topology.
(2)
In Eq. (2), the functions represent the Lagrange’s expression of the perturbed dynamics for
the i-th element [6]. To obtain an artificial perturbation on the orbit phase space , its
dynamic effects will be introduced with a low thrust steering scheme. The mapping of the time
evolution of the low thrust steering scheme dynamics to the one defined by the artificial
potential function is reduced to the condition in Eq. (3), equating the averaged dynamics of the
artificial potential function with the averaged steering scheme dynamics over one revolution.
(3)
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
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Provided that the artificial potential is known, and the average of the low thrust dynamics can
be obtained analytically, the condition in Eq. (3) becomes an algebraic problem to be solved
each orbit revolution by finding the proper steer parameters. The methodology proposed
includes the search of an artificial perturbation which allows the fulfilment of an orbit transfer.
For this purpose, a candidate nonlinear function
is introduced, dependent on the control
parameters vector . The manoeuvre design is then translated into a parametric optimisation
problem which shapes the artificial potential to drive the orbit to the desired condition while
minimising a trajectory performance index. As performance index, the cumulated
along the
manoeuvre is considered, integrating its averaged rate over one revolution as follows.
(4)
where e is the orbit eccentricity. The general formulation of the optimisation problem
approached with a single shooting, shown in Figure 1, is as follows:
min{x} J (5)
with: and J = ΔV subject to:
where the low thrust orbit dynamics propagated inside the cost function and constraint function
evaluation are driven by the artificial potential introduced
by imposing the solution of
the algebraic nonlinear problem of Eq. (3). The parametric optimisation problem is solved with
a multistart approach, using single and multiple shooting methods. The numerical routine
employed is the interior-point algorithm embedded in the MATLAB® built-in function
fmincon.m. Multiple runs of the optimisation problem are performed to improve the optimality
and convergence of the gradient based search, notoriously sensitive to the initial guess. In the
multiple shooting approach, the artificial potential function is defined piecewise to improve
convergence property and optimality of the phase space trajectory. The time discretisation of
the phases is done dynamically in function of the parameters, according to Eq. (6) where
stands for the time at the end of the j-th phase, which are then included among the optimisation
variables. In the multi-phase approach, the multiple shooting node guesses of orbital elements,
, are also included in the optimisation variables vector which becomes as Eq. (7), and
continuation equality constraints are imposed at nodes for the orbital elements.
(6)
(7)
Figure 1: Scheme of the parametric optimisation used for the transfer problem.
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
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Once the parametric optimisation is solved, the phase space trajectory to reach the target
condition corresponds to the iso-surface of the potential function
, which represents the
analytic description of the desired orbit long-term dynamics. The phase space trajectory can
then be implemented by solving the low thrust mapping condition of Eq. (3) at each revolution,
resulting in a cheap and highly autonomous onboard guidance. The methodology is thought to
be advantageous for highly perturbed environments where the natural effects can be exploited
to reduce the manoeuvre cost. In the present work, the methodology is only applied to a
simplified model in a low perturbed environment with the aim of assessing its behaviour.
2.2 Simplified conservative model
The simplified model studied describes the orbit planar dynamics constrained onto the Earth’s
equatorial plane and subjected only to conservative natural and artificial perturbations. The
system is formulated in rotating equatorial reference frame at a constant frequency with
respect to the geocentric inertial equatorial frame. In the orbit model, only conservative
perturbation effects are accounted, both describing the the natural perturbations and the
artificial control perturbations with low thrust. Therefore, the orbit can be described with two
scalar parameters, the eccentricity e and the angle, which describes the orientation of the orbit
pericentre with respect to the rotating frame. The orientation angle is defined in Eq. (8), and it
is function of the longitude of perigee , defined for equatorial orbits [6], and the frequency
of rotation .
According to the methodology presented, a low thrust steering scheme should be introduced to
act on the orbit long-term dynamics. The selected steering scheme uses a sequence of burn and
coasting arcs to obtain a null effect on the semimajor axis and the desired effect on the
magnitude and orientation of the eccentricity vector on the equatorial plane. In this work the
simplifications of constant thrust acceleration magnitude and direction within the burn arcs is
considered, which allows the analytical simplifications in the average procedure of Eq. (1).
Using symmetric arcs, the conservative condition can be fulfilled, while studies on the
suboptimality of the action on the e and discriminated the decision of location and thrust
angles within the burn arcs [1]. The proposed conservative steering features a pair of symmetric
arcs centred at periapsis/apoapsis and one at E = {90°, 270°} locations, with an “inertial”
control acceleration directed respectively perpendicular and parallel to the semi-major axis, as
shown in Figure 2b. It is worth noticing that the possibility of other steering schemes to act on
the planar conservative dynamics is possible, but in this work the sensitivity of the guidance
(8)
(a)
(b)
Figure 2: (a) Rotating reference frame. (b) Conservative inertial low thrust steering scheme.
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
5
with respect to different steering scheme definition is not presented. The averaged dynamics of
the steering law, computed analytically applying Eq. (1), is reported in Eq. (9).
(9)
where and are the half arcs widths shown in Figure 2b, while and are the constant
magnitudes of the thrust accelerations involved in the steering scheme.
The class of artificial potentials to perturb the phase space are introduced in a simple
manner considering the following nonlinear parametrised function:
(10)
where the eccentricity function is preliminary considered only as a linear term. The simplicity
of the nonlinear function has been favoured at this early stage of research. The control
parameters , and univocally define a function topology of the phase space, together
with the natural perturbing effects. The natural perturbations on the orbital elements accounted
in this work on the orbit are the effect due the Earth’s oblateness and the frame rotation
contribution. The total perturbing potential, reported in Eq. (11) results in the dynamics
expressed in Eq. (12) applying Eq. (2).
(11)
(12)
Using the steering law and potential dynamics introduced, the low thrust mapping
condition of Eq. (3) reduces to:
(13)
The latter algebraic problem is solved determining the and half arcs widths at each orbit
evolution along the propagation of the orbit. The constant acceleration magnitude is taken equal
to , where and are free to assume positive and negative values
according to the guiding potential rates required.
3 CONSERVATIVE TRANSFERS RESULTS
3.1 Test case
The methodology is applied to a transfer test case in the planar equatorial conservative
phase space, with initial and target conditions reported in Table 1. The rotating reference frame
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
6
frequency is considered equal to the apparent mean motion of the Sun on the ecliptic . The
rotating frame will then keep approximately a constant orientation with respect to the Sun
direction, neglecting the obliquity of the ecliptic. Initial conditions are taken considering the
Sun at the spring equinox, and the frame first axis aligned in such direction. Figure 3 shows the
phase space trajectory computed with the single and multiple shooting methods of shaping
artificial potential perturbation, together with the total potential function isolines computed
from the single shooting solution. It can be noticed, from Figure 3, how the multiple shooting
algorithm switches between different isolines, while the single shooting phase space trajectory
follows a single isoline from initial to target condition defined by the
integral of motion.
Grey shaded area corresponds to critical eccentricity values that results in a perigee altitude
below 400 km above the Earth’s surface. The piecewise trajectory computed with multiple
shooting has been discretised in four phases, and the upper boundary of manoeuvre time is set
to half a year for both shooting methods. From Table 1 the improvement of the multiple
shooting approach to the optimality of the phase space trajectory in term of performance index
is evident. As expected, the capability of defining a piecewise phase space isoline with multiple
shooting overcomes the limitation of the finite class of disturbing functions topology defined
which may not be able to drive the system for the whole phase space trajectory efficiently. The
CPU time for 5 and 10 runs respectively of the multiple and single shooting parametric
optimisation problems are reported in Table 1, where two Intel(R) Core(TM) i7-5500U CPU@
2.40 GHz are used in parallel for the computations. The transfer problem is also solved
considering various upper limits for the total manoeuvre time, and the solutions in terms of
and manoeuvre time used are shown in Figure 4.
Figure 4: Test case transfer problem solutions for different upper limit of manoeuvre times.
Initial Conditions
Target Conditions
Performance
Figure 3: Phase space trajectories of test case transfer.
Table 1: Test case transfer
conditions and performance.
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
7
The solutions are influenced by the manoeuvre time limit and show an incremental
improvement in the minimisation of the performance index for the solutions obtained with
upper time limit below 0.8 years, reported in Figure 4. This characteristic is associated with a
greater exploitation of the natural regression on the element to minimise the low thrust control
action using the whole manoeuvre time available. The plateau region corresponds to the
situation in which the natural regression of will drive the orbit orientation passed the desired
condition for longer manoeuvre times.
3.2 Heliotropic swarm control
The conservative phase space shaping method for low thrust manoeuvres is here applied to the
control of multiple spacecraft starting from different conditions in the phase space to the
same eccentric heliotropic orbit on the equatorial plane. Stable heliotropic orbits maintain the
apogee orientation fixed towards to the Sun direction, guaranteeing a useful coverage of the
Earth’s surface at the same local hour being advantageous for telecommunications and Earth
observations missions [7] [8]. The same approximation done in the transfer test case for the Sun
direction on the equator is used. The target heliotropic orbit is defined fixing the eccentricity
and retrieving the semi-major axis from the frozen condition with respect to the effect in Eq.
(14), obtained setting to zero the natural rate of in Eq. (12). The transfer problem is solved
with the multiple shooting method using four phases and upper manoeuvre time limit of 0.5
years. The phase space trajectories for a grid of initial conditions are shown in Figure 5, where
the colormap represents the required of each trajectory.
3.3 Elliptic constellation configuration
At last, the conservative guidance is applied to a reconfiguration of an elliptic equatorial
constellation with equally spaced apogees from the same injection orbit used to provide
enhanced coverage for equatorial countries, [9]. In this case the frequency of rotation of the
reference frame is determined from the target orbits eccentricity and semimajor axis, imposing
the frozen condition with respect to of Eq. (15). In such a way, regardless of the time needed
to perform the transfer for each satellite, the perigee equispaced condition is guaranteed for a
simultaneous control of the whole constellation. Initial and target conditions for the
constellation are reported in Table 3, while the phase space trajectories computed with a four
phases multiple shooting approach are shown in Figure 6, with an upper limit of manoeuvre
time set equal to 0.5 years.
Figure 5: Phase space trajectories of the
heliotropic control of multiple spacecraft.
Figure 6: Phase space trajectories of the
constellation reconfiguration solutions.
Low thrust manoeuvres in the phase space Borelli G., Nugnes M., Colombo C.
8
4 CONLCUSION
A methodology for designing orbit transfer perturbing the averaged perturbed orbit phase space
was defined in this work, using semi-analytical techniques. The method can be the basis for an
onboard autonomous guidance implementation or can be used for mission preliminary studies,
thanks to its computational efficiency. The applications to a simplified orbit model of
conservative planar orbit transfer on the equatorial plane are studied in this paper. Possible
future developments will address the extension to the 3D non-conservative orbit dynamics,
together with the applications in higher perturbed environments to better exploit the natural
perturbing effects reducing the low thrust manoeuvre cost.
5 ACKNOWLEDGEMENTS
The research leading to these results has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme as part
of project COMPASS (Grant agreement No 679086), www. compass.polimi.it.
REFERENCES
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(14)
(15)
Target Heliotropic Orbit
Elliptic Constellation Conditions (N=12 satellites)
Table 2: Target heliotropic orbit.
Table 3: Elliptic constellation initial and target orbits.
