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Venus transfer design by combining invariant manifolds and low-thrust arcs
Abstract
The design of interplanetary trajectories based on patched circular
restricted three body models is gradually becoming a valuable
alternative to the classical patched conic approach. The main advantage
offered by such a model is the possibility to exploit the manifold
dynamics to move naturally far from or toward a body. Generally,
propulsive maneuvers are required to match these structures. Low-thrust
arcs offer the possibility to have a significant propellant mass
reduction when moving from manifold to manifold. The aim of this paper
is to present a methodology to design low-thrust trajectories between
two planetary orbits connecting the manifolds of two circular three body
systems. The approach is based on a grid search on the main parameters
governing the solution to identify those trajectories moving within the
manifold images on given Poincarè sections. The value of the
Jacoby constant of the target libration point periodic orbit is chosen
as stop condition for the thrusting phases. Ballistic arcs follow up to
the proper Poincarè section intersection. A grid search for an
Earth to Venus transfer is presented as test case.
... In recent years, design methods for fuel-optimal trajectories using a patching approach have been investigated [1,2]. In particular, instead of a patched conic (two-body) approach, which connects the trajec- tories designed in two-body problems, a patched three-body approach, which connects sets of trajectories designed in the circular restricted three-body problem (CR3BP), has been used to design energy-efficient interplanetary transfer trajectories [2]. ...
... In recent years, design methods for fuel-optimal trajectories using a patching approach have been investigated [1,2]. In particular, instead of a patched conic (two-body) approach, which connects the trajec- tories designed in two-body problems, a patched three-body approach, which connects sets of trajectories designed in the circular restricted three-body problem (CR3BP), has been used to design energy-efficient interplanetary transfer trajectories [2]. One of the biggest advantages to considering a three-body problem is that the region reachable by the spacecraft can be used as a gateway for interplanetary transfer. ...
... The invariant manifold of periodic orbits around the Lagrange points has been used as a candidate for sets of trajectories [6][7][8][9]. This set-oriented approach has also been applied to the design of low-thrust and energy-efficient trajectories [2,5,10]. Dellnitz et al. presented an Earth-Venus transfer trajectory by connecting low-thrust trajectories [5]. ...
A method by which to incorporate the Electric Delta-V Earth Gravity Assist (EDVEGA) scheme into a patched three-body approach to design an interplanetary transfer trajectory is presented in this paper. The EDVEGA scheme is a promising technique to reduce fuel consumption using Earth gravity assist and electric propulsion. In a patched three-body approach, the dimension of the problem that connects some trajectories is reduced by the use of special attainable sets. However, because of the singularity associated with Earth's center, it is difficult to connect an EDVEGA trajectory with a trajectory designed in the three-body problem. Therefore, this paper proposes a design method for an Earth–Mars transfer trajectory that combines the EDVEGA scheme with a patched three-body approach using Levi-Civita regularization and a special contour plot. Through comparison with conventional invariant manifold techniques, we demonstrate that fuel consumption is reduced by the proposed method.
... Researchers have carried out extensive investigations on this problem. According to the inter-planetary super highway theory [14] and invariant manifold theory [15,16], the low-energy escaping orbit is positively located in the family of the transit orbits. The family of the transit orbits connects the Earth-Moon L 2 point and the Moon's gravitational vicinity. ...
... Nevertheless, it is found that the Sun's gravitational influence on the dynamic behaviors of trajectories and minimum energy requirements is ignored, which is not suitable for the trajectories in the Earth-Moon system. Since the fact that a probe can rely on the Sun's gravitational effect to raise its perigee and reach the gravitational vicinity of the Moon has been proved [14,30], the Sun's gravitational effect on the escaping orbits must be taken into consideration in the design of the low-energy trajectory. ...
Due to the limits of size and weight of probes, analysis and design of the low-energy escaping orbit play a significant role in saving the energy in space exploration. In this paper, we research the mechanism of the evolution of the probe orbital energy and develop a design method for the low-energy escaping orbits in the Earth–Moon system. A dynamic model accounting for the Sun–Earth–Moon-probe elliptical four-body problem is presented by considering Moon’s eccentricity and Sun’s direct gravitational influence. Considering the influences from phase of Earth, Moon and Sun, the equations of the probe orbital energy and its variation are derived and theoretical analysis is implemented based on corresponding energy expressions. Then, the Poincaré mapping technique is utilized to search two types of low-energy families of escaping orbits and the results of numerical simulation confirm the theoretical predictions. The dynamic model proposed in this paper is more accurate and has practical value comparing with the circular restricted three-body problem, and the escaping strategy can save over 25 % of energy relative to the hyperbolic escaping.
... Low-energy trajectories are special solutions of the restricted 3-body problem, whose applications were extensively investigated in the past decades [20][21][22][23][24][25]. Their use in the design of an Earth-Venus transfer, though attractive because it leads to savings in the total delta-V (propellant mass) required [26,27], has a major drawback in the longer transfer time. The combined use of high-and low-energy trajectory, developed here, allows preserving the saving in propellant mass while limiting the increase in the transit time [28]. ...
Exploration of Venus is recently driven by the interest of the scientific community in understanding the evolution of Earth-size planets, and is leading the implementation of missions that can benefit from new design techniques and technology. In this work, we investigate the possibility to implement a microsatellite exploration mission to Venus, taking advantage of (i) weak capture, and (ii) nonlinear orbit control. This research considers the case of a microsatellite, equipped with a high-thrust and a low-thrust propulsion system, and placed in a highly elliptical Earth orbit, not specifically designed for the Earth-Venus mission of interest. In particular, to minimize the propellant mass, phase (i) of the mission was designed to inject the microsatellite into a low-energy capture around Venus, at the end of the interplanetary arc. The low-energy capture is designed in the dynamical framework of the circular restricted 3-body problem associated with the Sun-Venus system. Modeling the problem with the use of the Hamiltonian formalism, capture trajectories can be characterized based on their state while transiting in the equilibrium region about the collinear libration point L1. Low-energy capture orbits are identified that require the minimum velocity change to be established. These results are obtained using the General Mission Analysis Tool, which implements planetary ephemeris. After completing the ballistic capture, phase (ii) of the mission starts, and it is aimed at driving the microsatellite toward the operational orbit about Venus. The transfer maneuver is based on the use of low-thrust propulsion and nonlinear orbit control. Convergence toward the desired operational orbit is investigated and is proven analytically using the Lyapunov stability theory, in conjunction with the LaSalle invariance principle, under certain conditions related to the orbit perturbing accelerations and the low-thrust magnitude. The numerical results prove that the mission profile at hand, combining low-energy capture and low-thrust nonlinear orbit control, represents a viable and effective strategy for microsatellite missions to Venus.
... Similar transfers have been used in the trajectory design for the GRAIL [60], SMART-1 [61], CAPSTONE and KPLO missions. The use of the invariant manifolds to achieve ballistic or nearly propellant-free transfers has featured heavily in preliminary mission design studies to reduce overall capture ∆v [18,19,35,48,62,63,64,65,66,67,68,69]. It has performed a particularly key role in the investigation of asteroids and other large bodies, since their masses make traditional methods of orbital insertion difficult [70,71,72,73,74]; ...
With renewed interest in space exploration, the question of designing efficient transfers between celestial bodies is as relevant as ever. Combined with the rise of scientific computing in the latter half of the 20th century, significant attention has been given to using modern computing methods to design more efficient orbital transfers to reduce costs and improve overall mission lifetime.
Preliminary mission design is often performed in simplified, time-independent models of motion. In these, classical dynamical systems theory identifies dynamical structures which can be used to create low-energy transfers between points in space, or used to create orbits that exist as a delicate balance of gravity to achieve mission objectives. However, in more realistic, time-dependent models such structures are not guaranteed to exist. Attention has thus been given to techniques well-developed in fluid dynamics to identify similar structures in astrodynamics systems, but numerical and computational difficulties have frustrated these efforts.
This PhD is separated into two parts. The first studies the use of time-independent dynamical structures in space mission design in combination with high-performance computing techniques. An intensive optimisation procedure is used to construct transfers that use the invariant manifolds to retrieve asteroids into two of the equilibrium points of the Sun-Earth system. This PhD improves on the state of the art in this field by improving the methods used to find and construct the retrieval transfers. As a result, 27 more asteroids that are considered ‘easily retrievable’ are found, and the velocity required to compute the transfers is generally reduced for those already considered easily retrievable. Moreover, it is revealed that these transfers exist across a range of transfer times, allowing greater flexibility for mission designers.
The second part of this thesis uses techniques from fluid mechanics to find analogous structures to those used in the asteroid retrieval study directly in time-dependent models of motion, rather than needing to use simplified models. This thesis makes two improvements to the current body of research: the first is the presentation of an improved numerical method to compute Lagrangian Coherent Structures (LCS) in three-dimensional dynamical systems called DA-LCS. This numerical method uses a direct computer implementation of an algebra of polynomials to compute more accurate and less numerically noisy quantities that signal LCS in general dynamical systems, and greatly outperforms standard approaches in astrodynamics systems. Since the relevant quantities are computed as polynomial expansions, the method also allows the computation of all relevant quantities completely automatically.
This numerical method is then applied to a series of test cases from astrodynamics, where three-dimensional LCS is constructed and shown to perform the role of generalised unstable manifolds in astrodynamics systems by separating qualitatively different behaviour. The effect of orbit parameterisation and integration time is also elaborated in an effort to provide the space community with in-depth knowledge of how to use LCS in astrodynamics in future studies.
... Demeyer and Gurfil [34] developed a systematic technique to design transfers from the Earth to the prescribed distant retrograde orbits in the Sun-Earth planar CRTBP by using invariant manifold theory. Finocchietti et al. [35] presented a methodology to design patched three-body interplanetary transfers by combining invariant manifolds with low-thrust arcs. The low-energy transfers between low Earth-parking orbits and halo orbits of the Earth-Moon system are investigated by Zanzottera et al. [36]. ...
Single-impulse and low-thrust low-energy transfers from a Lyapunov orbit around (Formula presented.) point of the Sun–Earth system to the periodic orbits around the equilibrium points (Formula presented.) of the Earth–Moon system are investigated. In our computation, the series expansions of invariant manifolds are used to generate the departure state, the series expansions of periodic orbits around the triangular libration points, and Lissajous orbits around the collinear libration points are used to generate the target orbits. In order to construct the first guesses for single-impulse and low-thrust low-energy transfers, we establish the corresponding optimization problems under some suitable assumptions and solve them by means of an improved cooperative evolutionary algorithm (ICEA). For computing optimal transfers, the nonlinear programming problems are established by direct description and solved by a local optimization method with the initial-guess transfers computed by ICEA. Numerical results show that the low-thrust transfers outperform the corresponding single-impulse transfers in terms of propellant mass.
... 16,18,[21][22][23] Dynamical systems theory has also been suggested as a design tool for interplanetary trajectory design. 24 29,30 As alternatives to the high energy arcs, low-thrust arcs have also been investigated for transfers between the two systems 22,[31][32][33][34] The past investigations on the system-to-system transfer design strategies have successfully contributed numerous design techniques and insight into the fourbody regime. However, interplanetary trajectory design techniques from the Earth-Moon libration point orbits warrants further examination. ...
This investigation is focused specifically on transfers from Earth-Moon L-1/L-2 libration point orbits to Mars. Initially, the analysis is based on the circular restricted three-body problem to utilize the framework of the invariant manifolds. Various departure scenarios are compared, including arcs that leverage manifolds associated with the Sun-Earth L-2 orbits as well as non-manifold trajectories. For the manifold options, ballistic transfers from Earth-Moon L-2 libration point orbits to Sun-Earth L-1/L-2 halo orbits are first computed. This autonomous procedure applies to both departure and arrival between the Earth-Moon and Sun-Earth systems. Departure times in the lunar cycle, amplitudes and types of libration point orbits, manifold selection, and the orientation/location of the surface of section all contribute to produce a variety of options. As the destination planet, the ephemeris position for Mars is employed throughout the analysis. The complete transfer is transitioned to the ephemeris model after the initial design phase. Results for multiple departure/arrival scenarios are compared.
In the circular restricted three-body problem (CR3BP), transit orbit is a class of orbit which can pass through the bottleneck region of the zero velocity curve and escapes from the vicinity of the primary or the secondary. This kind of orbit plays a very important role in the design of space exploration missions. A kind of low-energy interplanetary transfer, which is called Interplanetary Superhighway (IPS), can be realized by utilizing transit orbits. To use the transit orbit in actual mission design, a key issue is to find an algorithm which can separate the states corresponding to transit orbits from the states corresponding to other types of orbits rapidly. In fact, the distribution of transit orbit in the phase space has been investigated by numerical method, and a Fourier series approximation method has been introduced to describe the boundary of transit orbits. However, the Fourier series approximation method needs several hundred sets of Fourier series. The coefficients of these Fourier series are neither easy to be computed nor convenient to be stored, which makes the method can hardly be used in actual mission design. In this paper, the support vector machine (SVM) is used to classify the trajectories in the CR3BP. Using the Gaussian kernel, the 6-dimensional states in the CR3BP are mapped into an infinite-dimensional space, and the bound of the transit orbits is described by a hyperplane. A training data generation method is introduced, which reduces the size of training data by generating the states near the hyperplane. The numerical results show that the proposed algorithm gives the good correct rate of classification, and its computing speed is much faster than that of the Fourier series approximation method.
Space missions which reach destinations such as the moon, asteroids, or the satellites of Jupiter are complex and challenging to design, requiring new and unusual kinds of orbits to meet their goals, orbits that cannot be found by classical approaches to the problem. In addition, libration point orbits are seeing greater use.
This book guides the reader through the trajectory design of both libration point missions and a new class of low energy trajectories which have recently been discovered and make possible missions which classical trajectories (conic sections) could not. Low energy trajectories are achieved by making use of gravity as much as possible, using the natural dynamics arising from the presence of a third body (or more bodies). The term "low-energy" is used to refer to the low fuel and therefore low energy required to control the trajectory from a given starting condition to a targeted final condition.
Low energy trajectory technology allows space agencies and aerospace companies to envision missions in the near future involving long duration observations and/or constellations of spacecraft using little fuel. A proper understanding of low energy trajectory technology begins with a study of the restricted three-body problem, a classic problem of astrodynamics, which we approach from a rigorous and geometric point of view.
We develop systematic and computationally efficient methods for the design of both libration point orbits and low energy trajectories based on fundamental ideas of invariant manifold theory.
Furthermore, we develop the computational techniques needed to design trajectories for a spacecraft in the field of N bodies by patching together solutions of the 3 body problem. These computational methods are key for the development of some NASA and ESA mission trajectories, such as low energy libration point orbit missions (e.g., the Genesis Discovery Mission and Terrestrial Planet Finder, to mention only two), low energy lunar missions and low energy tours of outer planet moon systems, such as the a mission to tour and explore in detail the icy moons of Jupiter.
This book can serve as a valuable resource for graduate students and advanced undergraduates in aerospace engineering, as well as a manual for practitioners who work on libration point and deep space missions in industry and at government laboratories. The book contains a wealth of background material, but also brings the reader up to a portion of the research frontier. Furthermore, the book will also have appeal to students of applied mathematics, especially those with an interest in dynamical systems and real world applications.
In this paper, a technique for the analysis and the design of low-energy interplanetary transfers, exploiting the invariant manifolds of the restricted three-body problem, is presented. This approach decomposes the full four-body problem describing the dynamics of an interplanetary transfer between two planets, in two three-body problems each one having the Sun and one of the planets as primaries; then the transit orbits associated to the invariant manifolds of the Lyapunov orbits are generated for each Sun-planet system and linked by means of a Lambert's arc defined in an intermediate heliocentric two-body system. The search for optimal transit orbits is performed by means of a dynamical Poincaré section of the manifolds. A merit function, defined on the Poincaré section, is used to optimally generate a transfer trajectory given the two sections of the manifolds. Due to the high multimodality of the resulting optimization problem, an evolutionary algorithm is used to find a first guess solution which is then refined, in a further step, using a gradient method. In this way all the parameters influencing the transfer are optimized by blending together dynamical system theory and optimization techniques. The proposed patched conic-manifold method exploits the gravitational attractions of the two planets in order to change the two-body energy level of the spacecraft and to perform a ballistic capture and a ballistic repulsion. The effectiveness of this approach is demonstrated by a set of solutions found for transfers from Earth to Venus and to Mars.
A method to incorporate low-thrust propulsion into the invariant manifolds technique is presented in this paper. The low-thrust propulsion is introduced by means of special attainable sets that are used in conjunction with invariant manifolds to define a first-guess solution. This is later optimized in a more refined model where an optimal control formalism is used. Planar low-energy low-thrust transfers to the moon, as well as spatial low-thrust stable-manifold transfers to halo orbits in the Earth moon system, are presented. These solutions are not achievable via patched-conics methods or standard invariant manifolds techniques. The aim of the work is to demonstrate the usefulness of the proposed method in delivering efficient solutions, which are compared with known examples.
The Genesis spacecraft will collect solar wind samples from a halo orbit about the Sun-Earth L1 point for two years, returning those samples to Earth in 2003 for analysis and examination. The solar wind will imbed itself into a set of ultra-pure material collectors that will be deployed throughout the collection phase of the mission. Analysis of the samples collected by the mission will contribute to our understanding of the origins of the solar system.
The T6 ion engine is a 22-cm diameter, 4.5-kW Kaufman-type ion thruster produced by QinetiQ, Ltd., and is baselined for the European Space Agency BepiColombo mission to Mercury and is being qualified under ESA sponsorship for the extended range AlphaBus communications satellite platform. The heritage of the T6 includes the T5 ion thruster now successfully operating on the ESA GOCE spacecraft. As a part of the T6 development program, an engineering model thruster was subjected to a suite of performance tests and plume diagnostics at the Jet Propulsion Laboratory. The engine was mounted on a thrust stand and operated over its nominal throttle range of 2.5 to 4.5 kW. In addition to the typical electrical and flow measurements, an EB mass analyzer, scanning Faraday probe, thrust vector probe, and several near-field probes were utilized. Thrust, beam divergence, double ion content, and thrust vector movement were all measured at four separate throttle points. The engine performance agreed well with published data on this thruster. At full power the T6 produced 143 mN of thrust at a specific impulse of 4120 seconds and an efficiency of 64%; optimization of the neutralizer for lower flow rates increased the specific impulse to 4300 seconds and the efficiency to nearly 66%. Measured beam divergence was less than, and double ion content was greater than, the ring-cusp-design NSTAR thruster that has flown on NASA missions. The measured thrust vector offset depended slightly on throttle level and was found to increase with time as the thruster approached thermal equilibrium.
In 1991, the Japanese Hiten mission used a low energy transfer with a
ballistic capture at the Moon which required less ΔV than a standard
Hohmann transfer to the Moon. In this paper, we apply the same dynamical
systems techniques used to produce the “Petit Grand Tour” of Jovian moons
to reproduce a Hiten-like mission. We decouple the Sun-Earth-Moon-
Spacecraft 4-body problem into two 3-body problems. Using the invariant
manifold theory of the Lagrange points of the 3-body systems, we are able to
construct low energy transfer trajectories from the Earth and ballistic capture
trajectories at the Moon. The techniques used in the design and construction
of this trajectory may be applied in many situations.
The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem.
This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.
Application of low thrust propulsion to interconnect ballistic trajectories on invariant manifolds associated with multiple
circular restricted three body systems has been investigated. Sun-planet three body models have been coupled to compute the
two ballistic trajectories, where electric propulsion is used to interconnect these trajectories as no direct intersection
in the Poincarè sections exists. The ability of a low thrust to provide the energy change required to transit the spacecraft
between two systems has been assessed for some Earth to Mars transfers. The approach followed consists in a planetary escape
on the unstable manifold starting from a periodic orbit around one of the two collinear libration points near the secondary
body. Following the planetary escape and the subsequent coasting phase, the electric thruster is activated and executes an
ad-hoc thrusting phase. The complete transfer design, composed of the three discussed phases, and possible applications to
Earth–Mars missions is developed where the results are outlined in this paper.
A third-order analytical solution for halo-type periodic motion about the collinear points of the circular-restricted problem is presented. The three-dimensional equations of motion are obtained by a Lagrangian formulation. The solution is constructed using the method of successive approximations in conjunction with a technique similar to the Lindstedt-Poincar method. The theory is applied to the Sun-Earth system.
A lagrangian formulation for the three-dimensional motion of a satellite in the vicinity of the collinear points of the circular-restricted problem is reconsidered. It is shown that the influence of the primaries can be expressed in the form of two third-body disturbing functions. By use of this approach, the equations for the Lagrangian and for the motion itself are readily developed into highly compact expressions. All orders of the non-linear developments are shown to be easily obtainable using well-known recursive relationships. The resulting forms for these equations are well suited for use in the initial phase of canonical or non-canonical investigations.
In this paper novel Earth–Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun–Earth
problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed
in the Sun–Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory
is optimized in the full n-body problem which takes into account planets’ orbital inclinations and eccentricities. Accurate, efficient solutions with
reasonable flight times are presented and compared with known results.
KeywordsRestricted three-body problem–Invariant manifolds–Low-thrust transfer–
n-body models–Ballistic capture–Lunar-gravity assist
The low-thrust version of the low energy transfers to the Moon exploiting the structure of the invariant manifolds associated
to the Lagrange point orbits is presented in this paper. A method to systematically produce low-energy, low-thrust transfers
executing ballistic lunar capture is discussed. The coupled restricted three-body problems approximation is used to deliver
appropriate first guess for the subsequent optimization of the transfer trajectory within a complete four-body model using
direct transcription and multiple shooting strategy. It is shown that less propellant than standard low energy transfers to
the Moon is required. This paper follows previous works by the same authors aimed at integrating together knowledge coming
from dynamical system theory and optimal control problems for the design of efficient low-energy, low-thrust transfers.
Of the three collinear libration points of the Sun–Earth Circular Restricted Three-Body Problem (CR3BP), L3 is that located opposite to the Earth with respect to the Sun and approximately at the same heliocentric distance. Whereas
several space missions have been launched to the other two collinear equilibrium points, i.e., L1 and L2, taking advantage of their dynamical and geometrical characteristics, the region around L3 is so far unexploited. This is essentially due to the severe communication limitations caused by the distant and permanent
opposition to the Earth, and by the gravitational perturbations mainly induced by Jupiter and the close passages of Venus,
whose effects are more important than those due to the Earth. However, the adoption of a suitable periodic orbit around L3 to ensure the necessary communication links with the Earth, or the connection with one or more relay satellites located at
L4 or L5, and the simultaneous design of an appropriate station keeping-strategy, would make it possible to perform valuable fundamental
physics and astrophysics investigations from this location. Such an opportunity leads to the need of studying the ways to
transfer a spacecraft (s/c) from the Earth’s vicinity to L3. In this contribution, we investigate several trajectory design methods to accomplish such a transfer, i.e., various types
of two-burn impulsive trajectories in a Sun-s/c two-body model, a patched conics strategy exploiting the gravity assist of
the nearby planets, an approach based on traveling on invariant manifolds of periodic orbits in the Sun–Earth CR3BP, and finally
a low-thrust transfer. We examine advantages and drawbacks, and we estimate the propellant budget and time of flight requirements
of each.
KeywordsTwo-body impulsive maneuvers-Patched conics-Gravity assist-Circular restricted three-body problem-Libration points-Periodic orbits-Invariant manifolds-Low-thrust
There has recently been considerable interest in sending a spacecraft to orbit Europa, the smallest of the four Galilean moons of Jupiter. The trajectory design involved in effecting a capture by Europa presents formidable challenges to traditional conic analysis since the regimes of motion involved depend heavily on three-body dynamics. New three-body perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a three-body approach provide low-fuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft "leap-frogs" between moons, orbiting each for a desired duration in a temporary capture orbit. We call this concept the "Petit Grand Tour." For this application, we apply dynamical systems techniques developed in a previous paper to design a Europa capture orbit. We show how it is possible, using a gravitational boost from Ganymede, to go from a jovicentric orbit beyond the orbit of Ganymede to a ballistic capture orbit around Europa. The main new technical result is the employment of dynamical channels in the phase space - tubes in the energy surface which naturally link the vicinity of Ganymede to the vicinity of Europa. The transfer Delta-V necessary to jump from one moon to another is less than half that required by a
standard Hohmann transfer.
This paper concerns heteroclinic connections and resonance transitions in the planar circular restricted 3-body problem, with applications to the dynamics of comets and asteroids and the design of space missions such as the Genesis Discovery Mission and low energy Earth to Moon transfers. The existence of a heteroclinic connection between pairs of equal energy periodic orbits around two of the libration points is shown numerically. This is applied to resonance transition and the construction
of orbits with prescribed itineraries. Invariant manifold structures are relevant for transport between the interior and exterior Hill’s regions, and other resonant phenomena throughout the solar system.
The Genesis spacecraft will collect solar wind samples from a halo orbit about the Sun-Earth L1 point for two years, returning those samples to Earth in 2003 for analysis and examination. The solar wind will imbed itself into a set of ultra-pure material collectors that will be deployed throughout the collection phase of the mission. Analysis of the samples collected by the mission will contribute to our understanding of the origins of the solar system.
Introduction The subject of spacecraft trajectory optimization has a long and interesting history. The problem can be simply stated as the determination of a trajectory for a spacecraft that satisfies specified initial and terminal conditions, that is conducting a required mission, while minimizing some quantity of importance. The most common objective is to minimize the propellant required or equivalently to maximize the fraction of the spacecraft that is not devoted to propellant. of course, as is common in the optimization of continuous dynamical systems, it is usually necessary to provide some practical upper bound for the final time or the optimizer will trade time for propellant. There are also spacecraft trajectory problems where minimizing flight time is the important thing, or problems, for example those using continuous thrust, where minimizing flight time and minimizing propellant use are synonymous. Except in very special (integrable) cases, which reduce naturally to parameter optimization problems, the problem is a continuous optimization problem of an especially complicated kind.
The design of a suitable space trajectory often results in a standard optimal control problem where the control effort has to be minimized for a given transfer time and a given mission target.
The common crucial point of the whole optimization procedure is the detection of an enough accurate and easy to compute initial guess to refine by means of a local optimizer. The detection of such a suitable initial guess is not that hard in the classical keplerian model where analytic approximations are available, but it is
fundamental in more refined models, as the Circular Restricted Three Body Problem (CR3BP), offering non conventional transfer solutions.
The aim of this contribution is to provide an efficient tool for finding a set of suitable initial guesses for non conventional transfers, in order to offer a set of flexible and feasible options to be refined with more sophisticated optimizers.
The approach consists in creating \emph{pseudo-trajectories}, that follow the natural dynamics but involving local discontinuities in order to enlarge the set of candidate trajectories. The effectiveness of the method is addressed considering a heteroclinic connection in the Sun-Earth model -- a path that was already exploited for real mission applications.
Despite the fact the Mercury is a small, rocky, burned planet, so close to the Sun that it is hard to be seen from Earth except during twilight, it has played a very important role in the history of astronomy and fundamental physics and plays a crucial role in the understanding of the formation and evolution of our Solar System.
The fundamental mathematical principles of celestial mechanics and space navigation are presented in an introduction for students of aerospace engineering and science. Chapters are devoted to hypergeometric functions and elliptic integrals, basic topics in analytical dynamics, the two-body problem, two-body orbits and the initial-value problem, solving the Kepler equation, the two-body orbital boundary-value problem, and solving Lambert's problem. Also examined are non-Keplerian motion, patched-conic orbits and perturbation methods, variation of parameters, two-body orbital transfer, numerical integration of differential equations, the celestial position fix, and space navigation. Diagrams, graphs, and sample problems are provided.
In Part I it is shown that the preference for near-commensurability of mean motions demonstrated in a previous paper extends
to the small satellites of Jupiter, including the retrograde ones, and to the retrograde satellite of Saturn, implying that
stability of near-commensurable configurations is the reason for such a preference. In Part II it is proved that if, at a
certain epoch, a system of n gravitating point-masses has each radius vector from the (assumed stationary) centre of mass of the system perpendicular
to every velocity vector (hereinafter called a “mirror configuration”), then the behaviour of each of the point-masses under
the internal gravitational forces of the system after the epoch will be a mirror image of its behaviour prior to the epoch.
It is further shown that, if a mirror configuration of the system exists at two separate epochs, then the orbit of each point-mass
is periodic. The authors argue that such periodic orbits are the more stable (under the action of external forces) the shorter
the interval of time between mirror configurations, and they show that the frequent occurrence of mirror configurations between
any two point-masses requires that the mean motions of the two point-masses be nearly commensurable. In Part III, various
near-commensurable pairs of orbits in the solar system are shown to behave according to the above arguments. The relationship
of the present work to the more restricted “symmetry theorem” of Griffin is discussed.
To date, ozone has only been identified in the atmospheres of Earth and Mars. This study
reports the first detection of ozone in the atmosphere of Venus by the SPICAV ultraviolet
instrument onboard the Venus Express spacecraft. Venusian ozone is characterized by a
vertically confined and horizontally variable layer residing in the thermosphere at a mean
altitude of 100 km, with local concentrations of the order of 107-108 molecules.cm-3. The
observed ozone concentrations are consistent with values expected for a chlorine-catalyzed
destruction scheme, indicating that the key chemical reactions operating in Earth’s upper
stratosphere may also operate on Venus.
Two-impulse trajectories as well as mixed invariant-manifold and low-thrust efficient transfers to the Moon are discussed. Exterior trajectories executing ballistic lunar capture are formalized through the definition of special attainable sets. The coupled restricted three-body problems approximation is used to design appropriate first guesses for the subsequent optimization. The introduction of the Moon-perturbed Sun-Earth restricted three-body problem allows to formalize the idea of ballistic escape from the Earth and to take explicitly advantage of lunar fly-by. Then, accurate first guess solutions are optimized, through a direct method approach and multiple shooting technique.
Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based
on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure
and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy
efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on
the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design
can be suitably augmented and compute approximations to trajectories for a mission to Venus.
In the circular restricted three-body problem (CR3BP) the weak stability boundary (WSB) is defined as a boundary set in the phase space between stable and unstable motion relative to the second primary. At a given energy level, the boundaries of such region are provided by the stable manifolds of the central objects of the L1 and L2 libration points, i.e., the two planar Lyapunov orbits. Besides, the unstable manifolds of libration point orbits (LPOs) around L1 and L2 have been identified as responsible for the weak or temporary capture around the second primary of the system. These two issues suggest the existence of natural dynamical channels between the Earth's vicinity and the Sun–Earth libration points L1 and L2. Furthermore, it has been shown that the Sun–Earth L2 central unstable manifolds can be linked, through an heteroclinic connection, to the central stable manifolds of the L2 point in the Earth–Moon three-body problem. This concept has been applied to the design of low energy transfers (LETs) from the Earth to the Moon. In this contribution we consider all the above three issues, i.e., weak stability boundaries, temporary capture and low energy transfers, and we discuss the role played by the invariant manifolds of LPOs in each of them. The study is made in the planar approximation.
Mariner 10, the first dual-planet, gravity-assist mission, was launched by an Atlas/Centaur Mariner launch vehicle from the National Aeronautics and Space Administration—Kennedy Space Center in Cape Canaveral, Florida on 3 November 1973. Shortly after liftoff, a series of earth and Moon observations were made. These were followed by the initial trajectory correction maneuver and a period of interplanetary cruise operations. An additional trajectory correction maneuver was made several weeks prior to the encounter with Venus to refine the flyby on 5 February 1974 to 5000 km (3000 miles) above the surface of the planet.Extensive scientific observations of Venus took place over a period of about one week. Several thousand TV images were transmitted to Earth, many of which showed spectacular ultraviolet cloud formations and motions.The post-Venus trajectory required only a modest correction to place the spacecraft on a flight path that passed within the planned 1000 km (620 miles) of the surface of Mercury on 19 March 1974. Extensive TV imaging, together with other scientific observations, provided the first in-depth information concerning Mercury.The Mariner 10 mission is described, including engineering highlights of the flight and the key scientific results. The post-Mercury operation plan is discussed, the initial results of the second encounter with Mercury are given, and the possibilities of a third encounter are presented.
Venus Express is the first European mission to planet Venus. The mission aims at a comprehensive investigation of Venus atmosphere and plasma environment and will address some important aspects of the surface physics from orbit. In particular, Venus Express will focus on the structure, composition, and dynamics of the Venus atmosphere, escape processes and interaction of the atmosphere with the solar wind and so to provide answers to the many questions that still remain unanswered in these fields. Venus Express will enable a breakthrough in Venus science after a long period of silence since the period of intense exploration in the 1970s and the 1980s.
Deep Space 1 (DS1), currently scheduled for launch in July or August 1998, is the first mission of NASA's New Millennium program, chartered to flight validate high-risk, advanced technologies important for future space and Earth science programs. DS1's payload of technologies will be rigorously exercised during the two-year mission. Several features of the project present unique or unusual opportunities and challenges in the design of the mission that are likely to be encountered in future missions. The principal mission-driving technology is solar electric propulsion (SEP); this will be the first mission to rely on SEP as the primary source of propulsion. Another important technology for the mission design is the autonomous on-board navigation system, which requires frequent (at least weekly) intervals of several hours during which it collects visible images of distant asteroids and stars for its use in orbit determination and maneuver planning. The mission design accommodates the needs of these and other technologies for operational use and for acquiring sufficient validation data to assess their viability for future missions. DS1's mission profile includes encounters with an asteroid and a comet.
The Voyager project, which involves the 1977 launch of two advanced three-axis attitude stabilized spacecraft for the exploration of the Jovian and Saturnian systems, as well as interplanetary space, is discussed. The missions include investigation of the gravitational fields, atmospheric dynamics and magnetospheres of Jupiter and Saturn, the atmospheres, surface composition and features of Titan, the Io flux tube, the Great Red Spot of Jupiter, and earth occultation by Saturn's rings. To reduce energy required to reach Saturn, gravity-assist swingbys of Jupiter will be employed; a continuation to Uranus by the second satellite may be implemented by reliance on gravity-assist at Saturn.
Our Solar System is connected by a vast Interplanetary Superhighway System (ISSys) providing low energy transport throughout. The Outer Planets with their satellites and rings are smaller replicas of the Solar System with their own ISSys, also providing low energy transport within their own satellite systems. This low energy transport system is generated by all of the Lagrange points of the planets and satellites within the Solar System. Figures show the tubular passage-ways near L1 of Jupiter and the ISSys of Jupiter schematically. These delicate and resilient dynamics may be used to great effect to produce free temporary captures of a spacecraft by a planet or satellite, low energy interplanetary and inter-satellite transfers, as well as precision impact orbits onto the surface of the satellites. Additional information is contained in the original extended abstract.
In the summer of 1996, we supervised two undergraduate students during a nine-week summer program at the Geometry Center. They worked on a project using dynamical systems techniques to compute and visualize orbits in the circular restricted three-body problem. This project was motivated by recent interest in the space science community to send missions near to the Sun-Earth libration points. A fuller understanding of the geometry of the phase space of the circular restricted three-body problem could provide new possibilities for baseline trajectory design. To this end, the goal of this project was to develop computational and visualization tools to aid in trajectory design. In particular, we wanted to be able to easily and interactively explore the geometry of the halo orbits and their stable and unstable manifolds. This report provides a summary of the mathematics underlying the project and a brief discussion of the results. Keywords: restricted three-body problem, halo orbits, (un)st...
Venus express: the first European mission to Venus, International Astronautic Confer-ence
- J Fabrega
- T Shirmann
- R Schmidt
- D Mccoy
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Assessment of Mission Design Including Utilization of Libration Points and Weak Stability Boundaries
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- F Topputo
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Periodic Orbits, The Carnegie Institution
- F R Moulton
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SMART-1 Mission Analysis: Collection of Notes on the Moon Mission, ESOC, Mission Analysis Section Working Paper N. 417
- J.-L Cano
- J Schoenmaekers
- R Jehn
- M Hechler
Cano, J. Schoenmaekers, R. Jehn, M. Hechler, SMART-1 Mission Analysis: Collection of Notes on the Moon Mission, ESOC, Mission Analysis Section Working Paper N. 417, August 1999.
An approach to the design of low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem, Paper AAS 04-245
- F Topputo
- M Vasile
- A Ercoli
- Finzi
F. Topputo, M. Vasile, A. Ercoli Finzi, An approach to the design of low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem, Paper AAS 04-245, 14th AAS/ AIAA Space Flight Mechanics Conference, Maui, Hawaii, 2004.
Poincarè Map, Floquet Theory and Stability of Periodic Orbits
- W S Koon
W.S. Koon, Poincarè Map, Floquet Theory and Stability of Periodic Orbits, CDS140A, 2006.
Mission Concepts to Outer Planet Satellites Using Non-Conic Low Energy Trajectories , Forum on Innovative Approaches to Outer Planetary Exploration
- M W Lo
- Petit Tour
M.W., Lo, Petit Grand Tour: Mission Concepts to Outer Planet Satellites Using Non-Conic Low Energy Trajectories, Forum on Innovative Approaches to Outer Planetary Exploration, 2001–2020, p. 52.
Shoot the Moon. Spaceflight Mechanics
- Koon
An approach to the design of low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem
- F Topputo
- M Vasile
- A Ercoli Finzi
Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System
- M Lo
- S Ross