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An electric solar wind sail (E-sail) is a propellantless propulsion system that generates thrust in the interplanetary space by extracting momentum from the stream of solar wind ions by means of a number of long, electrically charged tethers. In its usual configuration, the E-sail consists of a series of radially deployed tethers that belong to a fundamental plane, typically referred to as “nominal plane”. Specifically, the tether assembly is unfolded and maintained stretched via centrifugal force by rotating the spacecraft around a spin axis, which must be suitably oriented during the flight to govern, although within certain limits, the direction of the thrust vector induced by the E-sail. In this article, different E-sail geometric configurations are analyzed, in which the addition of azimuthal conducting tethers to the radial ones gives the E-sail a spider web shape, thus allowing its moment of inertia to be reduced significantly compared with a conventional configuration of equal propulsive performance. More precisely, this article presents an analytical model that enables a quick description of both the thrust vector and the inertia characteristics of a web-shaped E-sail.

In this paper, a spacecraft formation control framework is presented aimed at guaranteeing a long-term bounded relative motion in the asteroid's uncertain dynamical environment. The mean-orbital-elements-based invariant boundedness conditions via an approximate gravitational field are adopted to develop a nominal steering control strategy, while a control correction resulting from a Lyapunov redesign method is devised to account for uncertainties, i.e., asteroid's gravity uncertainty and solar perturbations. Contrary to traditional formation flying missions where the relative orbit has to keep some prescribed strict geometry, the proposed (loose) formation control law with only bounded requirement allows more degrees of freedom in formation design, substantially reducing the fuel consumption necessary for formation maintenance.

Orbits that are frozen in an averaged model, including the effect of a disturbing body laying on the equatorial plane of the primary body and the influence of the oblateness of the primary body, have been applied to probes orbiting the Moon. In this scenario, the main disturbing body is represented by the Earth, which is characterized by a certain obliquity with respect to the equatorial plane of the Moon. As a consequence of this, and of the perturbing effects that are not included in the averaged model, such solutions are not perfectly frozen. However, the orbit eccentricity, inclination, and argument of pericenter present limited variations and can be set to guarantee the fulfillment of requirements useful for lunar telecommunication missions and navigation services. Taking advantage of this, a practical case of a Moon-based mission was investigated to propose useful solutions for potential near-future applications.

This study analyzes the optimal transfer trajectory of a spacecraft propelled by a spin-stabilized electric solar wind sail (E-sail) with a single conducting tether and a spin axis with a fixed direction in an inertial (heliocentric) reference frame. The approach proposed in this study is useful for rapidly analyzing the optimal transfer trajectories of the current generation of small spacecraft designed to obtain in-situevidence of the E-sail propulsion concept. In this context, starting with the recently proposed thrust model for a single-tether E-sail, this study discusses the optimal control law and performance in a typical two-dimensional interplanetary transfer by considering the (binary) state of the onboard electron emitter as the single control parameter. The resulting spacecraft heliocentric trajectory is a succession of Keplerian arcs alternated with propelled arcs, that is, the phases in which the electron emitter is switched on. In particular, numerical simulations demonstrated that a single-tether E-sail with an inertially fixed spin axis can perform a classical mission scenario as a circle-to-circle two-dimensional transfer by suitably varying a single control parameter.

The dynamics of a spacecraft propelled by a continuous radial thrust resembles that of a nonlinear oscillator. This is analyzed in this work with a novel method that combines the definition of a suitable homotopy with a classical perturbation approach, in which the low thrust is assumed to be a perturbation of the nominal Keplerian motion. The homotopy perturbation method provides the analytical (approximate) solution of the dynamical equations in polar form to estimate the corresponding spacecraft propelled trajectory with a short computational time. The accuracy of the analytical results was tested in an orbital-targeting mission scenario.

In this paper, the closed-form along-track boundedness condition for spacecraft relative motion around a slowly rotating asteroid is presented. As opposed to the previously proposed criteria where three necessary constraints must be satisfied, the current approach only requires to enforce one constraint to suppress the along-track drift, thus allowing two additional degrees of freedom in spacecraft formation design. In particular, to adapt the derived results applicable for formation with a relatively large size, the boundedness condition via first-order analysis is then extended to obtain a second-order result. In the latter case, the value of mean semimajor axis difference is found to be the root of a quadratic (algebraic) equation. Moreover, an optimal single-impulse maneuver is analyzed, by means of which the bounded relative motion can be guaranteed in the presence of initialization errors. Illustrative examples are provided to validate the analytical results.

A diffractive sail is a solar sail whose exposed surface is covered by an advanced diffractive metamaterial film with engineered optical properties. This study examines the optimal performance of a diffractive solar sail with a Sun-facing attitude in a typical orbit-to-orbit heliocentric transfer. A Sun-facing attitude, which can be passively maintained through the suitable design of the sail shape, is obtained when the sail nominal plane is perpendicular to the Sun–spacecraft line. Unlike an ideal reflective sail, a Sun-facing diffractive sail generates a large transverse thrust component that can be effectively exploited to change the orbital angular momentum. Using a recent thrust model, this study determines the optimal control law of a Sun-facing ideal diffractive sail and simulates the minimum transfer times for a set of interplanetary mission scenarios. It also quantifies the performance difference between Sun-facing diffractive sail and reflective sail. A case study presents the results of a potential mission to the asteroid 16 Psyche.

The aim of this correspondence is to analyze the performance of a solar sail-based spacecraft in a two-impulse orbit transfer between circular and coplanar heliocentric orbits of assigned radii. In particular, assuming a transfer trajectory with a sweep angle and a flight time equal to those of a classical Hohmann orbit, the sail propulsive acceleration is used to minimize the total velocity variation required by the two tangential impulses. In this sense, the article quantifies the optimal performance of a solar sail augmented Hohmann transfer as a function of the sail characteristics.

This study made use of a shape-based method to analyze the orbital dynamics of a spacecraft subject to a continuous propulsive acceleration acting along the circumferential direction. Under the assumption of a logarithmic spiral trajectory, an exact solution to the equations of motion exists, which allows the spacecraft state variables and flight time to be expressed as a function of the angular coordinate. There is also a case characterized by specific initial conditions in which the time evolution of the state variables may be analytically determined. In this context, the presented solution is used to analyze circle-to-circle trajectories, where the combination of two impulsive maneuvers and a logarithmic spiral path are used to accomplish the transfer. The determined results are then applied to the achievement of the Earth—Mars and the Earth—Venus transfers using actual data from a recent thruster developed by NASA.

The aim of this paper is to analyze optimal trajectories of a solar sail-based spacecraft in missions towards the outer Solar System region. The paper proposes a simplified approach able to estimate the minimum flight time required to reach a given (sufficiently high) heliocentric distance. In particular, the effect of a set of solar photonic assists on the overall mission performance is analyzed with a simplified numerical approach. A comparison with results taken from the existing literature show the soundness of the proposed approach.

In this study, new analytical solutions to the equations of motion of a propelled spacecraft are investigated using a shape-based approach. There is an assumption that the spacecraft travels a two-dimensional spiral trajectory in which the orbital radius is proportional to an assigned power of the spacecraft angular coordinate. The exact solution to the equations of motion is obtained as a function of time in the case of a purely radial thrust, and the propulsive acceleration magnitude necessary for the spacecraft to track the prescribed spiral trajectory is found in a closed form. The analytical results are then specialized to the case of a generalized sail, that is, a propulsion system capable of providing an outward radial propulsive acceleration, the magnitude of which depends on a given power of the Sun-spacecraft distance. In particular, the conditions for an outward radial thrust and the required sail performance are quantified and thoroughly discussed. It is worth noting that these propulsion systems provide a purely radial thrust when their orientation is Sun-facing. This is an important advantage from an engineering point of view because, depending on the particular propulsion system, a Sun-facing attitude can be stable or obtainable in a passive way. A case study is finally presented, where the generalized sail is assumed to start the spiral trajectory from the Earth’s heliocentric orbit. The main outcome is that the required sail performance is in principle achievable on the basis of many results available in the literature.

The aim of this paper is to evaluate the minimum flight time of a solar sail-based spacecraft towards Earth-synchronous (heliocentric) circular displaced orbits. These are special displaced non-Keplerian orbits characterized by a period of one year, which makes them suitable for the observation of Earth’s polar regions. The solar sail is modeled as a flat and purely reflective film with medium-low performance, that is, with a characteristic acceleration less than one millimeter per second squared. Starting from a circular parking orbit of radius equal to one astronomical unit, the optimal steering law is sought by considering the characteristic acceleration that is required for the maintenance of the target Earth-synchronous displaced orbit. The indirect approach used for the calculation of the optimal transfer trajectory allows the minimum flight time to be correlated with several Earth-synchronous displaced orbits, each one being characterized by given values of Earth- spacecraft distance and displacement over the ecliptic. The proposed mathematical model is validated by comparison with results available in the literature, in which a piecewise-constant steering law is used to find the optimal flight time for a transfer towards a one-year Type I non-Keplerian orbit.

This paper deals with the problem of determining an analytical control law capable of maintaining highly elliptical heliosynchronous polar orbits around Venus. The problem is addressed using the Smart Dust concept, a propellantless propulsion system that extracts momentum from the solar radiation pressure using a reflective coating. The modulation of the thrust magnitude is performed by exploiting the property of electrochromic materials of changing their optical characteristics through the application of a suitable electrical voltage. The propulsive acceleration can, therefore, be switched from a minimum to a maximum value (or vice versa) so as to obtain a simple on–off control law. The required Smart Dust performance is described in closed form as a function of the semimajor axis and eccentricity of the working orbit. The soundness of the analytical control law is validated through a numerical integration of the equations of motion, in which the orbital perturbations due to the oblateness of Venus and to the gravitational attraction of the Sun are also included.

This paper analyzes reachable domains of spacecraft using a single impulsive maneuver. In particular, compact expressions are obtained of the envelopes of spacecraft trajectories in closed form, in both cases of either radial or tangential impulse. Suitable bounds are enforced on the magnitude of the velocity variation to obtain an elliptic transfer trajectory after the maneuver with a pericenter radius greater than the primary body’s radius. Three different cases are investigated: (1) the impulse point is fixed and the velocity variation may be varied; (2) a fixed impulse magnitude and free maneuver point; (3) both free impulse point and magnitude. Finally, some mission scenarios are analyzed to show the effectiveness of the proposed analytical method.

In-situmeasurements are necessary for a long-term analysis of the spatial structure of the geomagnetic tail. This type of mission requires the use of a propellantless propulsion system, such as a classical solar sail, to continuously rotate the design orbit apse line such that it remains parallel to the Sun-Earth direction. To reduce the mission costs, this paper suggests the employment of Sun-pointing smart dusts, which are here investigated in terms of propulsive acceleration level necessary to guarantee a mission’s feasibility. A Sun-pointing smart dust can be thought of as a millimeter-scale solar sail, whose geometric configuration allows it to passively maintain an alignment with the Sun-spacecraft line. The smart dust external surface is coated with an electrochromic reflective film in such a way that it may change, within some limits, its propulsive acceleration magnitude. A suitable control law is necessary for the smart dust to enable an artificial precession of its Earth-centred orbit, similar to what happens in the GeoSail mission. This paper analyzes the required control law using an optimal approach. In particular, the proposed mathematical model provides a set of approximate equations that allow a simple and effective tradeoff analysis between the propulsive requirements, in terms of the smart dust acceleration, and the characteristics of the design orbit.

This paper deals with the optimization of the transfer trajectory of a solar sail-based spacecraft between circular and coplanar heliocentric orbits. The problem is addressed using both a direct and an indirect approach, while an ideal and an optical force model are used to describe the propulsive acceleration of a flat solar sail. In the direct approach, the total flight time is partitioned into arcs of equal duration, within which the sail attitude is assumed to be constant with respect to an orbital reference frame, and a nonlinear programming solver is used to optimize the transfer trajectory. The aim of the paper is to compare the performance of the two (direct and indirect) approaches in term of optimal (minimum) flight time. In this context, the simulation results show that a direct transcription method using a small number of arcs is sufficient to obtain a good estimate of the global minimum flight time obtained through the classical calculus of variation.

The aim of this paper is to analyze a potential mission scenario that could be used to perform an in situ test of the Electric Solar Wind Sail. Such an advanced propulsive system works only outside the Earth’s magnetosphere; so, a translunar mission (where the spacecraft is supposed to be inserted as a secondary “piggyback” payload) is hypothesized, in which at least a part of the spacecraft trajectory is around or beyond the Moon’s orbit. The analysis of possible selenocentric trajectories is conducted with a three-dimensional generalization of the classical patched conic approximation. In particular, two cases are considered: a closed selenocentric orbit, and a Moon flyby that inserts the spacecraft on a post-flyby geocentric orbit with at least some branches outside the Earth’s magnetosphere.

A finite element approach is used to calculate the components of forces and moments acting on a square solar sail at a sun-sail distance equal to one astronomical unit. The model takes into account the deformation effect induced by the solar radiation pressure, where the incidence of the reflected photons changes as a function of the local orientation of the sail surface. Assuming a specular reflection model, the analysis shows that the maximum value of the transversal thrust component takes place when the solar zenith angle is about 36°, which is in accordance with the result available for a classical flat solar sail. Notably, the modulus of the moment due to the solar radiation pressure takes its maximum value approximately at the same (solar zenith) angle.

This paper investigates the optimal transfer trajectories from a circular parking orbit towards the apocenter of a rectilinear ellipse, where the spacecraft reaches a quasi-stationary condition relative to an inertial reference frame. The spacecraft is equipped with a propulsion system that provides a circumferential continuous propulsive acceleration, that is, an acceleration whose direction is perpendicular to the primary body-spacecraft line. The performance index to minimize is the total flight time, and an indirect method is used to analyze the transfer trajectories. In this context, the optimal transfer performance is obtained as a function of the spacecraft propulsive acceleration magnitude through an interpolation procedure of numerical simulations. The results obtained with a continuous thrust propulsion system are also compared with those derived from a multi-impulse transfer. Finally, the paper investigates a heliocentric mission scenario in which the spacecraft minimizes the flight time required to reach a rectilinear ellipse with a given value of the aphelion radius.

This paper analyzes the locally-optimal heliocentric transfer of a spacecraft propelled by an electric solar wind sail, an innovative propellantless propulsion system that generates a propulsive acceleration exploiting the momentum of solar wind particles. The potentialities of such an advanced thruster are investigated in terms of flight times required to achieve a given heliocentric orbit. The problem is addressed using a locally-optimal formulation, by minimizing a scalar performance index that depends on the time derivatives of the osculating orbital elements. The proposed algorithm gives an estimate of the globally-optimal flight time with reduced computational efforts compared to a traditional optimization approach. Also, when the performance index involves a single orbital parameter and the transfer trajectory is two-dimensional, the proposed approach provides an analytical solution to the locally-optimal control problem. The procedure discussed in the paper is used to quantify the near-optimal performance of an electric solar wind sail in some advanced mission scenarios, such as the design of a heliocentric non-Keplerian orbit for solar activity monitoring, the exploration of the Solar System boundaries, and the rendezvous with comets 1P Halley and 67P/Churyumov-Gerasimenko.

This paper analyzes the performance of an electric solar wind sail for generating and maintaining a heliocentric circular displaced orbit. Previous research on this subject was based on a simplified mathematical model of the spacecraft thrust. However, recent studies have proposed a more accurate algorithm for evaluating both the modulus and the direction of the propulsive thrust as a function of some important parameters related to the spacecraft attitude. Therefore, a reappraisal of the problem is motivated by the need to revise past results, taking into account new information available on the propulsion system. Within this context, this paper focuses on circular displaced orbits that are characterized in terms of orbital period, heliocentric distance and elevation angle. The attitude configuration and the value of the spacecraft characteristic acceleration required for orbital maintenance are calculated. An in-depth analysis of the linear stability of displaced orbits is given. It is shown that displaced orbits are unstable when the elevation angle exceeds about 20°.

Despite the existence of many studies about the structural analysis of a square solar sail, the need for obtaining reliable numerical results still poses a number of practical issues to be solved. The aim of this paper is to propose a new method that improves the existing analysis techniques. In this sense, the solar sail is modeled using distributed sail-boom connections, and its structural behavior in free flight is studied, using the inertia relief method, at different incidence angles of the incoming solar radiation. The proposed approach is able to circumvent the onset of numerical convergence problems by means of suitable strategies. A nonlinear analysis is carried out starting from an initial geometrical configuration in which the whole solar sail is perturbed using a linear combination of the first global buckling modes, obtained with a static eigenvalue analysis. Key points of the procedure are the application of a correct sail pre-stress, a clever choice of the type of elements to be used in the finite element analysis and the use of a suitable mesh refinement. The performance of the new approach have been successfully tested on square solar sails with side length varying from relatively small to medium-to-large sizes, in the range of 10–100 m. A detailed analysis is presented for a reference 20 m × 20 m square solar sail, where the paper shows that the suggested procedure is able to guarantee accurate results without the need of additional stabilization technique. In particular, the vibration global mode shapes and frequencies of the solar sail are correctly described even in presence of unsymmetrical loading conditions. In other terms, the numerical analysis is completed without any convergence problem and any disturbing local modes.

The capability to “detect and avoid” potential collisions is one of the main technical challenges restricting widespread operations of unmanned aircraft into non-segregated airspaces. In fact, to operate into prescribed environments, an unmanned aircraft needs an onboard technology to replace the capability of the human pilot to “see and avoid” collision hazards. Such a technology is a “sense and avoid” system. This article focuses on the “avoid function” of such a system and proposes a suitable solution. The approach to the problem is to schematize a generic obstacle through a moving ellipsoid that represents the region of space the unmanned aircraft must not violate. The obtained solution enables situations of potential conflict to be detected and avoided through a set of possible actions such as speed changes in magnitude and/or direction. Thousands of test cases have been considered to validate this solution. Simulations show that the proposed algorithm is able to detect and avoid situations of potential conflict in the three-dimensional space and in real-time, even without the assistance of a human operator. As such, it can be considered as a fundamental step for the development of a prototype of “sense and avoid” system for promoting the integration of unmanned aircraft into non-segregated airspaces.

The problem to define a methodology for the analysis of aircraft performances, in the phase of conceptual design, is addressed. The proposed approach is based on a numerical optimization procedure where a scalar objective function, the take-off weight, is minimized. Deterministic and stochastic approaches as well as hybridizations between these two search techniques are considered. More precisely, we consider two-stage strategies where the optimum localization is performed by a genetic algorithm, while a gradient-based method is used to terminate the optimization process. Also, another type of hybridization strategy is investigated where a partially converged gradient-based method is incorporated in the genetic algorithm as a new operator. A detailed discussion is made and various different solutions are critically compared.The problem to define a methodology for the analysis of aircraft performances, in the phase of conceptual design, is addressed. The proposed approach is based on a numerical optimization procedure where a scalar objective function, the take-off weight, is minimized. Deterministic and stochastic approaches as well as hybridizations between these two search techniques are considered. More precisely, we consider two-stage strategies where the optimum localization is performed by a genetic algorithm, while a gradient-based method is used to terminate the optimization process. Also, another type of hybridization strategy is investigated where a partially converged gradient-based method is incorporated in the genetic algorithm as a new operator. A detailed discussion is made and various different solutions are critically compared.