Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Asteroid 522 Helga is chaotic and stable

Celestial Mechanics and Dynamical Astronomy -     

Lunar capture trajectories and homoclinic connections through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for many decades. This paper is concerned with a topological study of such transfers, with emphasis to trajectories that allow performing lunar capture and those that exhibit homoclinic connections, in the context of the circular restricted three-body problem. A fundamental theorem stated by Conley locates capture trajectories in the phase space and can be condensed in a sentence: “if a crossing asymptotic orbit exists then near any such there is a capture orbit”. In this work this fundamental theoretical assertion is used together with an original cylindrical isomorphic mapping of the phase space associated with the third body dynamics. For a given energy level, the stable and unstable invariant manifolds of the periodic Lyapunov orbit around the collinear interior Lagrange point are computed and represented in cylindrical coordinates as tubes that emanate from the transformed periodic orbit. These tubes exhibit complex geometrical features. Their intersections correspond to homoclinic orbits and determine the topological separation of long-term lunar capture orbits from short-duration capture trajectories. The isomorphic mapping is proven to allow a deep insight on the chaotic motion that characterizes the dynamics of the circular restricted three-body, and suggests an interesting interpretation, and together corroboration, of Conley’s assertion on the topological location of lunar capture orbits. Moreover, an alternative three-dimensional representation of the phase space is profitably employed to identify convenient lunar periodic orbits that can be entered with modest propellant consumption, starting from the Lyapunov orbit.     

Lissajous trajectories for lunar global positioning and communication systems

This work proposes a Lunar Global Positioning System (LGPS) and a Lunar Global Communication System (LGCS) using two constellations of satellites on Lissajous trajectories around the collinear L ₁ and L ₂ libration points in the Earth–Moon system. This solution is compared against a Walker constellation around the Moon similar to the one used for the Global Positioning System (GPS) on the Earth to evaluate the main differences between the two cases and the advantages of adopting the Lissajous constellations. The problem is first studied using the Circular Restricted Three Body Problem to find out its main features. The study is then repeated with higher fidelity using a four-body model and higher-order reference trajectories to simulate the Earth-Moon-spacecraft dynamics more accurately. The LGPS performance is evaluated for both on-ground and in-flight users, and a visibility study for the LGCS is used to check that communication between opposite sides of the Moon is possible. The total ΔV required for the transfer trajectories from the Earth to the constellations and the trajectory control is calculated. Finally, the estimated propellant consumption and the total number of satellites for the Walker constellation and the Lissajous constellations is used as a performance index to compare the two proposed solutions.     

GCM simulation of balloon trajectories on Titan

Possible trajectories of passive balloons in Titan's troposphere are simulated with the instantaneous wind field predicted by a GCM (general circulation model). In most areas the basic motion of a balloon is a predominantly eastward or westward drift, depending on altitude, latitude and season of the balloon release point. Some meridional oscillation is always superposed on this basic motion, resulting in a wavy trajectory, with a maximum extent (of 40^°) at high latitudes of the winter hemisphere. As a general rule, the meridional oscillation can be maximised if a balloon is deployed in altitudes and latitudes where the mean zonal wind is eastward and smaller than the phase speed of Saturn's gravitational tide on Titan. A balloon's groundtrack does not repeat as it makes successive circuits around Titan, but rather makes a spiral or braided pattern. The summer pole is rather difficult to access for a balloon not directly introduced there because of small meridional oscillation, while the winter pole can be readily accessed and left several times. A preferred zonal drift direction can be achieved by choosing a proper altitude and hemisphere, but choosing a preferred meridional direction is not possible.     

The chaotic rotation of Hyperion

A plot of spin rate versus orientation when Hyperion is at the pericenter of its orbit (surface of section) reveals a large chaotic zone surrounding the synchronous spin-orbit state of Hyperion, if the satellite is assumed to be rotating about a principal axis which is normal to its orbit plane. This means that Hyperion's rotation in this zone exhibits large, essentially random variations on a short time scale. The chaotic zone is so large that it surrounds the ½ and 2 states, and libration in the 3/2 state is not possible. Stability analysis shows that for libration in the synchronous and ½ states, the orientation of the spin axis normal to the orbit plane is unstable, whereas rotation in the 2 state is attitude stable. Rotation in the chaotic zone is also attitude unstable. A small deviation of the principal axis from the orbit normal leads to motion through all angles in both the chaotic zone and the attitude unstable libration regions. Measures of the exponential rate of separation of nearby trajectories in phase space (Lyapunov characteristic exponents) for these three-dimensional motions indicate the the tumbling is chaotic and not just a regular motion through large angles. As tidal dissipation drives Hyperion's spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. Capture from the chaotic state into the synchronous or ½ state is impossible since they are also attitude unstable. The 3/2 state does not exist. Capture into the stable 2 state is possible, but improbable. It is expected that Hyperion will be found tumbling chaotically.     

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full N -body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for   N 10 000  , some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding time-scale of the instability. For trajectories of   N ≈100 000  systems this time-scale could be arbitrarily large – and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well-known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For N up to a few hundred, the instability time-scales of N -body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger N (up to a few thousand) the values of the these time-scales were found to saturate, increasing significantly more slowly with N . We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time variations of the time-dependent Liapunov exponents. The results presented here go some way towards resolving the long-standing apparent paradoxes concerning the local instability of trajectories. This now appears to be a manifestation of mechanisms driving evolution in gravitational systems and their interactions – and may thus be a useful diagnostic of such processes.     

Detection of Ordered and Chaotic Motion Using the Dynamical Spectra

Two simple and efficient numerical methods to explore the phase space structure are presented, based on the properties of the "dynamical spectra". 1) We calculate a "spectral distance" D of the dynamical spectra for two different initial deviation vectors. D → 0 in the case of chaotic orbits, while D → const ≠ 0 in the case of ordered orbits. This method is by orders of magnitude faster than the method of the Lyapunov Characteristic Number (LCN). 2) We define a sensitive indicator called ROTOR (ROtational TOri Recongnizer) for 2D maps. The ROTOR remains zero in time on a rotational torus, while it tends to infinity at a rate ∝ N = number of iterations, in any case other than a rotational torus. We use this method to locate the last KAM torus of an island of stability, as well as the most important cantori causing stickiness near it. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simulation of Martian chaotic terrain and outflow channels

Outflow channeling and associated chaotic terrain were created under temperature and pressure conditions suggested for a diluvian period on Mars 3.5 to 0.5 by ago. Pressures under which both features were formed ranged from 130 to 34 mbar at a constant ambient temperature of 266°K. Analogs of the collapse structures and channels evolved in a high-altitude/low-temperature chamber are found on the Martian surface. Similarities exist not only in their overall morphology but in the finer details of the megastructures themselves. The critical factor that allowed channelized flow to occur was the sudden release of liquid water derived from melting of subsurface ground ice and ice layers under the low atmospheric pressure and temperature conditions within the chamber. Experimentation may indicate the existence of substantially thick water ice layers beneath the Martian regolith prior to the outflow channeling episode.     

Optimal switching strategy for radially accelerated trajectories

This paper studies the problem of a spacecraft subject to an outward radial thrust, with constant modulus, that may be switched on or off at suitable time intervals. The problem is to find the optimal strategy to guarantee the possibility of transferring the spacecraft from an initial to a final position in a given time interval using the least amount of thrust level. The problem is solved in an optimal framework, using an indirect approach. A number of different mission scenarios are studied in detail: escape missions, flyby missions and rendezvous missions. In the latter case the spacecraft uses a hybrid system comprising an high thrust propulsion system for the final impulsive maneuver. The optimal switching strategy allows one to substantially decrease the thrust level when compared to the continuous case (without thrust modulation).     

Logarithmic spiral trajectories generated by Solar sails

Analytic solutions to continuous thrust-propelled trajectories are available in a few cases only. An interesting case is offered by the logarithmic spiral, that is, a trajectory characterized by a constant flight path angle and a fixed thrust vector direction in an orbital reference frame. The logarithmic spiral is important from a practical point of view, because it may be passively maintained by a Solar sail-based spacecraft. The aim of this paper is to provide a systematic study concerning the possibility of inserting a Solar sail-based spacecraft into a heliocentric logarithmic spiral trajectory without using any impulsive maneuver. The required conditions to be met by the sail in terms of attitude angle, propulsive performance, parking orbit characteristics, and initial position are thoroughly investigated. The closed-form variations of the osculating orbital parameters are analyzed, and the obtained analytical results are used for investigating the phasing maneuver of a Solar sail along an elliptic heliocentric orbit. In this mission scenario, the phasing orbit is composed of two symmetric logarithmic spiral trajectories connected with a coasting arc.     

Invariant properties of families of moon-to-earth trajectories

Analytical techniques are employed to demonstrate certain invariant properties of families of moon-to-earth trajectories. The analytical expressions which demonstrate these properties have been derived from an earlier analytical solution of the restricted three-body problem which was developed by the method of matched asymptotic expansions. These expressions are given explicitly to orderµ ^1/2 where is the dimensionless mass of the moon. It is also shown that the inclusion of higher order corrections does not affect the nature of the invariant properties but only increases the accuracy of the analytic expressions.The results are compared with the work of Hoelker, Braud, and Herring who first discovered invariant properties of earth-to-moon trajectories by exact numerical integration of the equations of motion. (Similar properties for moon-to-earth trajectories follow from the principle of reflection). In each instance the analytical expressions result in properties which are equivalent, to orderµ ^1/2, with those found by numerical integration. Some quantitative comparisons are presented which show the analytical expressions to be quite accurate for calculating particular geometrical characteristics.

Nomenclature

Orbital Elements near the Moon energy - angular momentum - semi-major axis - eccentricity - inclination - argument of node - argument of pericynthion Orbital Elements near the Earth h _e energy - l _e angular momentum - i inclination - argument of node - argument of perigee - t _f time of flight Other symbols parameters used in matehing - U a function of the energy near the earth - a function of the angular momentum near the earth - r _p perigee radius - perincynthion radius - radius at node near moon - true anomaly of node near moon - initial angle between node near moon and earth-moon line - a function ofU, , andi - earth phase angle - dimensionless mass of the moon - U ₀, U₁ U=U ₀+U ₁ - i ₀, i_1/2, i₁ i=i ₀+µ ^1/2 i _1/2+µ i ₁ - ₀, _1/2, ₁ = ₀+µ ^1/2 i _1/2+µ i ₁ - _p longitude of vertex line - _n latitude of vertex line - R _o ,S _o ,N _o functions ofU ₀ and - a function ofU ₀, and      

Design and analysis of Weak Stability Boundary trajectories to Moon

An algorithm is developed to find Weak Stability Boundary transfer trajectories to Moon in high fidelity force model using forward propagation. The trajectory starts from an Earth Parking Orbit (circular or elliptical). The algorithm varies the control parameters at Earth Parking Orbit and on the way to Moon to arrive at a ballistic capture trajectory at Moon. Forward propagation helps to satisfy launch vehicle’s maximum payload constraints. Using this algorithm, a number of test cases are evaluated and detailed analysis of capture orbits is presented.     

Numerical studies of chaotic Hilda-type orbits

Earlier work indicates a comparatively rapid chaotic evolution of the orbits of some Hilda asteroids that move at the border of the domain occupied by the characteristic parameters of the objects at the 3/2 mean motion resonance. A simple Jupiter–Saturn model of the forces leads to numerical results on some of these cases and allows a search for additional resonances that can contribute to the chaotic evolution. In this context the importance of the secondary resonances that depend on the period of revolution of the argument of perihelion is pointed out. Among the studied additional resonances there are three-body resonances with arguments that depend on the mean longitudes of Jupiter, Saturn, and asteroid, but on slowly circulating angular elements of the asteroid as well, and the frequency of these arguments is close to a rational ratio with respect to the frequency of the libration due to the basic resonance.     

A new method for reconstructing type III trajectories

A local density approximation (LDA) method is developed for reconstructing the trajectories of type III radio bursts through the interplanetary medium. The method uses the measured source directions and the measured frequency drift rates of the type III burst to determine the locations of the radio source in the interplanetary medium at consecutive frequency levels. The technique is used to reconstruct the trajectory of an actual type III burst and the results are compared to the trajectory obtained from the global density law method. The LDA method represents an improvement in that it utilizes more observed data on the type III burst and that it takes full account of the local density variations at the source locations.     

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J Differ Equ 190:539–558, 2003): this result has been obtained by the use of the Poincaré-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre in \mathbbR²{\mathbb{R}^2} , we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (Celest Mech Dyn Astron 77:49–75, 77:49–75, 2000; Celest Mech Dyn Astron 94(4):433–449, 2006). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).     

Automated design of gravity-assist trajectories to Mars and the outer planets

In this paper, a new approach to planetary mission design is described which automates the search for gravity-assist trajectories. This method finds all conic solutions given a range of launch dates, a range of launch energies and a set of target planets. The new design tool is applied to the problems of finding multiple encounter trajectories to the outer planets and Venus gravity-assist trajectories to Mars. The last four-planet grand tour opportunity (until the year 2153) is identified. It requires an Earth launch in 1996 and encounters Jupiter, Uranus, Neptune, and Pluto. Venus gravity-assist trajectories to Mars for the 30 year period 1995–2024 are examined. It is shown that in many cases these trajectories require less launch energy to reach Mars than direct ballistic trajectories.Assistant Professor, School of Aeronautics and AstronauticsGraduate Student, School of Aeronautics and Astronautics