Analytical results for solar sail optimal missions with modulated radial thrust

The aim of this paper is to analyze the optimal trajectories of a spacecraft subjected to a modulated radial thrust, whose magnitude is inversely proportional to the square of the distance from the primary body. This case is representative of a Sun-facing solar sail with a passive attitude control system. In this study the sailcraft is assumed to perform a finite number of reorientation maneuvers to set the propelling acceleration to zero and generate suitable coasting arcs along the trajectory. Accordingly, the resulting generalized orbit is a sequence of either propelled or ballistic conic arcs, whose main characteristics (in terms of semimajor axis, eccentricity, and perihelion radius) can be calculated in closed form. As a result, the sailcraft optimal performance can be studied using an analytical approach. In particular, some compact relationships are drawn and discussed that allow one to find the optimal sailcraft characteristics required to reach a prescribed final orbit.     

Optimization of the interplanetary trajectories of spacecraft with a solar electric propulsion power plant of minimal power

The problem of optimizing the interplanetary trajectories of a spacecraft (SC) with a solar electric propulsion system (SEPS) is examined. The problem of investigating the permissible power minimum of the solar electric propulsion power plant required for a successful flight is studied. Permissible ranges of thrust and exhaust velocity are analyzed for the given range of flight time and final mass of the spacecraft. The optimization is performed according to Portnyagin’s maximum principle, and the continuation method is used for reducing the boundary problem of maximal principle to the Cauchy problem and to study the solution/ parameters dependence. Such a combination results in the robust algorithm that reduces the problem of trajectory optimization to the numerical integration of differential equations by the continuation method.     

Symmetries in the Optimal Control of Solar Sail Spacecraft

The theory of optimal control is applied to obtain minimum-time trajectories for solar sail spacecraft for interplanetary missions. We consider the gravitational and solar radiation forces due to the Sun. The spacecraft is modelled as a flat sail of mass m and surface area A and is treated dynamically as a point mass. Coplanar circular orbits are assumed for the planets. We obtain optimal trajectories for several interrelated problem families and develop symmetry properties that can be used to simplify the solution-finding process. For the minimum-time planet rendezvous problem we identify different solution branches resulting in multiple solutions to the associated boundary value problem. We solve the optimal control problem via an indirect method using an efficient cascaded computational scheme. The global optimizer uses a technique called Adaptive Simulated Annealing. Newton and Quasi-Newton Methods perform the terminal fine tuning of the optimization parameters.     

Trajectory optimization by making use of the closed solution of constant thrust-acceleration motion

The problem of fuel optimal rendezvous and transfer maneuvers in a central gravitational field is considered. By using analytical results and a parametrization of the control functions, the original optimal control problem can be solved by a sequence of mathematical programming problems. After introducingg KS-variables and piecewise-constant thrust accelerations, all necessary trajectory integrations are performed in closed form. This optimization procedure leads to a considerable reduction in computing time and allows the solution of a wide class of problems: The propulsion system may be thrust-limited or power-limited, one may consider rendezvous or transfer maneuvers with fixed or free final time. A numerical example for a 3-dimensional maneuver is included.     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .     

Three-body problem, its Lagrangian points and how to exploit them using an alternative transfer to L4 and L5

An alternative transfer strategy to send spacecraft to stable orbits around the Lagrangian equilibrium points L4 and L5 based in trajectories derived from the periodic orbits around L1 is presented in this work. The trajectories derived, called Trajectories G, are described and studied in terms of the initial generation requirements and their energy variations relative to the Earth through the passage by the lunar sphere of influence. Missions for insertion of spacecraft in elliptic orbits around L4 and L5 are analysed considering the restricted three-body problem Earth–Moon-particle and the results are discussed starting from the thrust, time of flight and energy variation relative to the Earth.     

Optimal low-thrust takeoff from an orbit about an oblate planet

Future space missions to the outer planets may depend upon the use of low-thrust propulsion systems. As these planets are decidedly oblate, the question of the effect of that oblateness on a low-thrust trajectory is of some interest. In this paper the problem of optimal energy increase is attacked under the assumption that the coefficients for the second zonal harmonic, i.e.,J ₂, and the nondimensional thrust acceleration are the same order of magnitude. Using a two variable asymptotic expansion technique, a near optimal control program is generated and the first order uniformly valid approximation for the corresponding trajectory is obtained. Tangential thrust is shown to be a good near-optimal thrust program even in the presence of oblateness effects. The optimal control program is found to be oscillatory and quite similar to the optimal control for energy increase in an inverse square gravitational field.This research was supported by the National Aeronautics and Space Administration under Grant NGR-23-005-329.     

Evidence-based robust design of deflection actions for near Earth objects

This paper presents a novel approach to the robust design of deflection actions for near Earth objects (NEO). In particular, the case of deflection by means of solar-pumped laser ablation is studied here in detail. The basic idea behind laser ablation is that of inducing a sublimation of the NEO surface, which produces a low thrust thereby slowly deviating the asteroid from its initial Earth threatening trajectory. This work investigates the integrated design of the space-based laser system and the deflection action generated by laser ablation under uncertainty. The integrated design is formulated as a multi-objective optimisation problem in which the deviation is maximised and the total system mass is minimised. Both the model for the estimation of the thrust produced by surface laser ablation and the spacecraft system model are assumed to be affected by epistemic uncertainties (partial or complete lack of knowledge). Evidence Theory is used to quantify these uncertainties and introduce them in the optimisation process. The propagation of the trajectory of the NEO under the laser-ablation action is performed with a novel approach based on an approximated analytical solution of Gauss’ variational equations. An example of design of the deflection of asteroid Apophis with a swarm of spacecraft is presented.     

On target for Venus – set oriented computation of energy efficient low thrust trajectories

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.     

Navigating to the Moon along low-energy transfers

This paper presents a navigation strategy to fly to the Moon along a Weak Stability Boundary transfer trajectory. A particular strategy is devised to ensure capture into an uncontrolled relatively stable orbit at the Moon. Both uncertainty in the orbit determination process and in the control of the thrust vector are included in the navigation analysis. The orbit determination process is based on the definition of an optimal filtering technique that is able to meet accuracy requirements at an acceptable computational cost. Three sequential filtering techniques are analysed: an extended Kalman filter, an unscented Kalman filter and a Kalman filter based on high order expansions. The analysis shows that only the unscented Kalman filter meets the accuracy requirements at an acceptable computational cost. This paper demonstrates lunar weak capture for all trajectories within a capture corridor defined by all the trajectories in the neighbourhood of the nominal one, in state space. A minimum Δv strategy is presented to extend the lifetime of the spacecraft around the Moon. The orbit determination and navigation strategies are applied to the case of the European Student Moon Orbiter.     

Artificial equilibrium points for a generalized sail in the elliptic restricted three-body problem

Different types of propulsion systems with continuous and purely radial thrust, whose modulus depends on the distance from a massive body, may be conveniently described within a single mathematical model by means of the concept of generalized sail. This paper discusses the existence and stability of artificial equilibrium points maintained by a generalized sail within an elliptic restricted three-body problem. Similar to the classical case in the absence of thrust, a generalized sail guarantees the existence of equilibrium points belonging only to the orbital plane of the two primaries. The geometrical loci of existing artificial equilibrium points are shown to coincide with those obtained for the circular three body problem when a non-uniformly rotating and pulsating coordinate system is chosen to describe the spacecraft motion. However, the generalized sail has to provide a periodically variable acceleration to maintain a given artificial equilibrium point. A linear stability analysis of the artificial equilibrium points is provided by means of the Floquet theory.     

Ground-Based Observational Support for Spacecraft Exploration of the Outer Planets

This report presents both a retrospective of ground-based support for spacecraft missions to the outer solar system and a perspective of support for future missions. Past support is reviewed in a series of case studies involving the author. The most basic support is essential, providing the mission with information without which the planned science would not have been accomplished. Another is critical, without which science would have been returned, but missing a key element in its understanding. Some observations are enabling by accomplishing one aspect of an experiment which would otherwise not have been possible. Other observations provide a perspective of the planet as a whole which is not available to instruments with narrow fields of view and limited spatial coverage, sometimes motivating a re-prioritizing of experiment objectives. Ground-based support is also capable of providing spectral coverage not present in the complement of spacecraft instruments. Earth-based observations also have the capability of filling in gaps of spacecraft coverage of atmospheric phenomena, as well as providing surveillance of longer-term behavior than the coverage available to the mission. Future missions benefiting from ground-based support would include the Juno mission to Jupiter in the next decade, a flagship-class mission to the Jupiter or to the Saturn systems currently under consideration, and possible intermediate-class missions which might be proposed in NASA’s New Frontiers category. One of the principal benefits of future 30 m-class giant telescopes would be to improve the spatial resolution of maps of temperature and composition which are derived from observations of thermal emission at mid-infrared and longer wavelengths. In many situations, this spatial resolution is competitive with those of the relevant instruments on the spacecraft themselves.     

Optimal station keeping by electric propulsion with thrust operation constraints

The orbit of a geostationary satellite has to be corrected from time to time in order to compensate for the effects of various perturbations. This is usually done by means of a system of thrusters mounted on the satellite. In this paper a method is developed to find the optimal thrusting strategy for the case of an electric propulsion system under given limitations on thrust magnitude and operation times. Optimization techniques are applied to minimize a cost function which is a weighted combination of fuel consumption and station keeping errors.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

The Existence and Stability of Families of Displacement Two-Body Orbits

The two-body problem is considered with an additional thrust induced acceleration. Stationary solutions are obtained to this problem in a rotating frame of reference which generates families of displaced circular orbits when viewed from an internal frame of reference. The existence and stability of these orbits is considered along with applications such as in-situ observations of Saturn's ring system and spacecraft proximity operations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Earth–Mars halo to halo low thrust manifold transfers

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.     

New Challenges to Trajectory Design by the Use of Electric Propulsion and Other New Means of Wandering in the Solar System

The design of spacecraft trajectories is a crucial part of a space mission design. Often the mission goal is tightly related to the spacecraft trajectory. A geostationary orbit is indeed mandatory for a stationary equatorial position. Visiting a solar system planet implies that a proper trajectory is used to bring the spacecraft from Earth to the vicinity of the planet. The first planetary missions were based on conventional trajectories obtained with chemical engine rockets. The manoeuvres could be considered 'impulsive' and clear limitations to the possible missions were set by the energy required to reach certain orbits. The gravity-assist trajectories opened a new way of wandering through the solar system, by exploiting the gravitational field of some planets. The advent of other propulsion techniques, as electric or ion propulsion and solar sail, opened a new dimension to the planetary trajectory, while at the same time posing new challenges. These 'low thrust' propulsion techniques cannot be considered 'impulsive' anymore and require for their study mathematical techniques which are substantially different from before. The optimisation of such trajectories is also a new field of flight dynamics, which involves complex treatments especially in multi-revolution cases as in a lunar transfer trajectory. One advantage of these trajectories is that they allow to explore regions of space where different bodies gravitationally compete with each other. We can exploit therefore these gravitational perturbations to save fuel or reduce time of flight. The SMART-1 spacecraft, first European mission to the Moon, will test for the first time all these techniques. The paper is a summary report on various activities conducted by the project team in these areas.     

Scientific exploration of near‐Earth objects via the Orion Crew Exploration Vehicle

Abstract— A study in late 2006 was sponsored by the Advanced Projects Office within NASA's Constellation Program to examine the feasibility of sending the Orion Crew Exploration Vehicle (CEV) to a near‐Earth object (NEO). The ideal mission profile would involve two or three astronauts on a 90 to 180 day flight, which would include a 7 to 14 day stay for proximity operations at the target NEO. This mission would be the first human expedition to an interplanetary body beyond the Earth‐Moon system and would prove useful for testing technologies required for human missions to Mars and other solar system destinations. Piloted missions to NEOs using the CEV would undoubtedly provide a great deal of technical and engineering data on spacecraft operations for future human space exploration while conducting in‐depth scientific investigations of these primitive objects. The main scientific advantage of sending piloted missions to NEOs would be the flexibility of the crew to perform tasks and to adapt to situations in real time. A crewed vehicle would be able to test several different sample collection techniques and target specific areas of interest via extra‐vehicular activities (EVAs) more efficiently than robotic spacecraft. Such capabilities greatly enhance the scientific return from these missions to NEOs, destinations vital to understanding the evolution and thermal histories of primitive bodies during the formation of the early solar system. Data collected from these missions would help constrain the suite of materials possibly delivered to the early Earth, and would identify potential source regions from which NEOs originate. In addition, the resulting scientific investigations would refine designs for future extraterrestrial resource extraction and utilization, and assist in the development of hazard mitigation techniques for planetary defense.     

Use of Programs for Two-Parameter Multiple Impulse Correction of Altitude and Inclination of Circular Orbits

The paper describes the method for the development of programs for two-parameter multiple impulse correction of altitude and inclination of circular orbits. The considered corrections feature the simultaneous adjustment of altitude and inclination of the orbit under limited thrust of the propulsion system of the spacecraft. Low thrust-to-weight ratio of the spacecraft leads to the need for a correction program consisting of several burns of the propulsion system. Thrust vector orientation, burn time, and its operation duration are determined as propulsion system parameters.     

Optimal Control of a Spacecraft During Descent in the Martian Atmosphere

The way for optimal controlling a spacecraft under its motion in the Martian atmosphere is examined. The minimum final velocity is taken as an optimality criterion. A procedure for calculating the spacecraft trajectories is developed based on the formalism of the Pontryagin maximum principle. The high efficiency of two-parameter control of the spacecraft is shown. The results can be used for exploring Mars and other planets.     

On the loss of telemetry data in full-sky surveys from space

In this paper we discuss the issue of loosing telemetry (TM) data due to different reasons (e.g. spacecraft–ground transmissions) while performing a full-sky survey with space-borne instrumentation. This is a particularly important issue considering the current and future space missions (like Planck from ESA and MAP from NASA) operating from an orbit far from Earth with short periods of visibility from ground stations. We consider, as a working case, the Low Frequency Instrument (LFI) on-board the Planck satellite albeit the approach developed here can be easily applied to any kind of experiment that makes use of an observing (scanning) strategy which assumes repeated pointings of the same region of the sky on different time scales. The issue is addressed by means of a Monte Carlo approach. Our analysis clearly shows that, under quite general conditions, it is better to cover the sky more times with a lower fraction of TM retained than less times with a higher guaranteed TM fraction. In the case of Planck, an extension of mission time to allow a third sky coverage with 95% of the total TM guaranteed provides a significant reduction of the probability to loose scientific information with respect to an increase of the total guaranteed TM to 98% with the two nominal sky coverages.