Artificial Equilibrium Points Research Articles

Utilizing a control technique for orbital maintenance near [formula omitted] point and Lyapunov exponents

In present article, we have explored the orbit maintenance strategy around an artificial equilibrium point in the Sun–Jupiter system. Two control strategies, proportional control and proportional derivative control laws are discussed for orbit maintenance. The maximum Lyapunov exponent is used to check the chaotic behavior of planetary orbit. We anticipate that the recently established Lyapunov exponent criteria may be used to study motion of planets in generalized stellar binary systems. This research is being guided by the conclusions of recent observations as well as those that are expected from existing and upcoming planet search missions.

Artificial equilibrium points and their linear stability analysis in the solar sail problem with triaxial second primary

Motion of a solar sail or a spacecraft equipped with a solar sail is one of the interesting problem of celestial mechanics. This paper deals with stationary solution of solar sail problem with radiating first primary and triaxial second primary under the frame of circular restricted three body problem (RTBP). First, equations of motion is derived with triaxiality and sail lightness number and then artificial equilibrium points (AEPs) are determined. It is noticed that impact of triaxiality parameters (σ1andσ2) of second primary is less than that of sail lightness number (β) due to which a considerable deviations in the positions of AEPs are occurred. Further, linear stability test is performed for all AEPs and it is found that similar to equilibrium points in classical RTBP, collinear AEPs are unstable for all mass parameter μ∈(0,1/2] and triangular AEPs are stable in the range 0<μ<μc=0.03758902. Moreover, stability range decreases with the values of β and σ1 but increases with σ2. After this we analyze the periodic orbits of the solar sail in the vicinity of AEPs and it is noticed that nature of motion is unaffected but time period is influenced from the sail number β and triaxiality parameters σ1 and σ2. The findings are useful to describe the more generalized solar sail problem with other perturbations such as oblateness, albedo, drag forces etc.

Formation flying for magnetic sails around artificial equilibrium points

This paper investigates the problem of magnetic sail-based spacecraft formation control around the artificial equilibrium points (AEPs), which can eliminate the requirement of the propellant. The thrusts are achieved by utilizing the interaction between the solar wind and the artificial magnetosphere generated by superconducting current coil onboard. The circular restricted three-body problem (CRTBP) of magnetic sail is discussed including the allowed regions and linear stability of AEPs, the locations of collinear AEPs and the possibility of existence of periodic orbits around the collinear AEPs. Next, the dynamical models of magnetic sail formation around the collinear AEPs are established. A novel fast fixed-time nonsingular terminal sliding mode controller (FFNTSM) based on fixed-time disturbance observer (FTDO) is developed to account for external disturbances. Several numerical simulations are conducted to substantiate that spacecraft formation can be precisely controlled by the proposed propellantless propulsion method in the presence of external disturbances.

Sliding mode control for attitude maneuvers of Helianthus solar sail

The recent successes of demonstration missions based on solar sail technology have paved the way for designing new mission scenarios with propellantless propulsion systems. In this context, the Italian Space Agency has been supporting a preliminary study of Helianthus, an innovative concept of deep-space mission, which is designed to use a solar sail to generate an artificial equilibrium point in the Sun–Earth gravitational field. Exploiting such a vantage point, located along the Sun–Earth line, Helianthus will be able to perform scientific observations of the solar corona and send to Earth an early warning information in case of solar flares. The attitude control system required to maintain the spacecraft at the unstable artificial equilibrium point is actuated by electrochromic devices, which change their optical properties in response to an electric input, thus generating suitable control torques. The aim of this work is to illustrate a possible control strategy that guarantees the spacecraft station keeping with feasible attitude maneuvers. A nonlinear sliding mode controller is used for the robustness properties it guarantees, and its performance is illustrated in some test maneuvers.

Multiple Spacecraft Formation Flying Control around Artificial Equilibrium Point Using Propellantless Approach

This paper demonstrates a detailed analysis of the feasibility for compact formation system around an L2-type artificial equilibrium point by means of continuous low-thrust propulsion in the hybrid form of solar sail and Coulomb force propulsion. Firstly, in view of non-ideal solar sail, the position of L2-type artificial equilibrium point and numerical periodic orbits around L2 utilized as leader’s nominal trajectory are given. Secondly, considering the external disturbances in the deep space environment, the nonlinear dynamic model of the spacecraft formation system based on the circular restricted three-body problem (CRTBP) is derived, under the assumption that the leader covers the nominal trajectory and each follower adjusts its propulsive acceleration vector (that is, both its sail attitude and electrostatic charge) in order to track a desired relative trajectory. Thirdly, based on a new double power combination function reaching law, a fast integral terminal sliding mode control methodology (MFITSM) is ameliorated to achieve orbital tracking rapidly, which has better robustness against external disturbances and the buffeting effect during spacecraft propulsion simultaneously. To properly allocate control inputs, a novel optimal allocation scheme is designed to calculate the charge product of the spacecrafts and sail attitude angles, which can make the magnitude of the acceleration required from the Coulomb propulsion system minimum and avoid formation geometry instabilities by balancing electrostatic interaction between adjacent spacecraft. Finally, several numerical examples are conducted to validate the superiority of the proposed control algorithm.

Collinear artificial equilibrium point maintenance with a wrinkled solar sail

Propellantless propulsion systems allow innovative mission scenarios to be envisaged, including the generation and the stabilization of artificial equilibrium points in the circular restricted three-body problem. This paper discusses the generation of collinear, L1-type, artificial equilibrium points in the Sun-[Earth+Moon] system, using a (photonic) solar sail. The main contribution of this paper is to investigate the spacecraft dynamics and its control when the sail propulsive acceleration is described with a recent thrust model that accounts for the presence of wrinkles on the membrane surface. In particular, an H∞ static output-feedback control based on reflectivity control devices is proposed to stabilize the spacecraft around the L1-type artificial equilibrium point, and a procedure is described for estimating the required area covered by those devices as a fraction of the whole solar sail surface.

Searching for orbits to observe the poles of celestial bodies

The objective of the present paper is to show a method to find orbits near artificial equilibrium points for a satellite equipped with a continuous thrust that allows it to stay near the poles of a celestial body. The physical system includes the presence of a moon of the celestial body under observation, and the perturbation caused by this moon is counteracted by an algorithm to help the satellite to stay close to its original position, instead of escape from it. The equations of motion are changed under some approximations, and analytical solutions for these equations are obtained and analyzed. Initial conditions are used such that their secular terms are nullified. These solutions are restricted to a short period of time, but we propose a method in which there are periodic updates in the thrust. Thus, the solutions can be extended for the duration of the mission. A numerical simulation is obtained, whose results are required to be in agreement with the analytical solution using these periodic adjustments of the thrust. This agreement means that the motion of the spacecraft remains bounded close to its initial position for longer times. Several systems with different sizes and mass parameters are used to show the results of the research, like Sun-Earth-Moon, Sun-Ida-Dactyl, Sun-Saturn-Titan and Sun-Mars-Phobos systems. The results also indicate the locations of points that require minimum magnitude of the thrust.

Feedback control law of solar sail with variable surface reflectivity at Sun-Earth collinear equilibrium points

A solar sail generates thrust without consuming any propellant, so it constitutes a promising option for mission scenarios requiring a continuous propulsive acceleration, such as the maintenance of a (collinear) L1-type artificial equilibrium point in the Sun-[Earth+Moon] circular restricted three-body problem. The usefulness of a spacecraft placed at such an artificial equilibrium point is in its capabilities of solar observation, as it guarantees a continuous monitoring of solar activity and is able to give an early warning in case of catastrophic solar flares. Because those vantage points are known to be intrinsically unstable, a suitable control system is necessary for station keeping purposes. This work discusses on how to stabilize an L1-type artificial equilibrium point with a solar sail by suitably adjusting its lightness number and thrust vector orientation. A full-state feedback control law is assumed, where the control gains are chosen with a linear-quadratic regulator approach. In particular, the numerical simulation results show that an L1-type artificial equilibrium point can be maintained with small required control torques, by using a set of reflectivity control devices.

Artificial equilibrium points and bi-impulsive maneuvers to observe 243 Ida

In the restricted three-body problem, the traditional Lagrange points L1 and L2 are the only equilibrium points near the asteroid 243 Ida. The thrust generated by a solar sail over a spacecraft enables the existence of new artificial equilibrium points, which depend on the position of the spacecraft with respect to the asteroid and the attitude of the solar sail. Such equilibrium points generate new spots to observe the body from above or below the plane of motion. Such points are very good observational locations due to their stationary condition. This work provides a preliminary analysis to observe Ida through the use of artificial equilibrium points as spots combined with transfer maneuvers between them. Such combination can be used to observe the asteroid from more different points of view in comparison to fixed ones. The analyses are made for a spacecraft equipped with a solar sail and capable of performing bi-impulsive maneuvers. The solar radiation pressure is used both to maintain the equilibrium condition and to reduce the costs of the transfers and/or to create transfers with longer duration. This is a new aspect of the present research, because it combines the continuous thrust with initial and final small impulses, which are feasible for most of the spacecraft, because the magnitudes of the impulses are very low. These combined maneuvers may reduce the transfer times of the maneuvers in most of the cases, compared with the maneuvers based only on continuous thrust. Several options involved in these transfers are shown, like to minimize the fuel spent (Δv) as a function of the transfer time or to extend the duration of the travel between the points. Extended transfer times can be useful when observations are required during the transfers.

Artificial Collinear Lagrangian Point Maintenance With Electric Solar Wind Sail

This article discusses the maintenance of an $L_1$ -type artificial equilibrium point in the Sun–[Earth+Moon] circular restricted three-body problem by means of an electric solar wind sail. The reference configuration instability is compensated for with a feedback control law that adjusts the grid voltage as a function of the distance from the natural $L_1$ point. Two different control strategies are analyzed assuming the solar wind fluctuations to be modeled through a statistical approach.

Harmonic Based Model Predictive Control for Set-Point Tracking

This paper presents a novel model predictive control (MPC) formulation for set-point tracking. Stabilizing predictive controllers based on terminal ingredients may exhibit stability and feasibility issues in the event of a reference change for small to moderate prediction horizons. In the MPC for tracking formulation, these issues are solved by the addition of an artificial equilibrium point as a new decision variable, providing a significantly enlarged domain of attraction and guaranteeing recursive feasibility for any reference change. However, it may suffer from performance issues if the prediction horizon is not large enough. This paper presents an extension of this formulation where a harmonic artificial reference is used in place of the equilibrium point. The proposed formulation achieves even greater domains of attraction and can significantly outperform other MPC formulations when the prediction horizon is small. We prove the asymptotic stability and recursive feasibility of the proposed controller, as well as provide guidelines for the design of its main ingredients. Finally, we highlight its advantages with a case study of a ball and plate system.

Solar sail cooperative formation flying around [formula omitted]-type artificial equilibrium points

The aim of this paper is to propose a distributed control architecture for a solar sail-based formation, flying around an L2-type artificial equilibrium point in the Sun-[Earth + Moon] circular restricted three-body problem. Two typical cases, depending on whether the formation structure is leaderless or includes a virtual leader, are investigated. In particular, the virtual leader case is further analyzed according to whether the state information of the virtual leader is available to all of the sails or to a part of the formation structure only. The protocols of the consensus-based algorithms are formulated on a general directed (unidirectional) communication topology, by exploring each available local neighbor-to-neighbor information interaction in a cooperative manner. In that case, a synchronized formation tracking may be achieved while increasing the reliability of the formation system. Illustrative examples show the effectiveness of the proposed approach in a typical mission scenario.

Families of halo orbits in the elliptic restricted three-body problem for a solar sail with reflectivity control devices

Solar sail halo orbits designed in the Sun-Earth circular restricted three-body problem (CR3BP) provide inefficient reference orbits for station-keeping since the disturbance due to the eccentricity of the Earth’s orbit has to be compensated for. This paper presents a strategy to compute families of halo orbits around the collinear artificial equilibrium points in the Sun-Earth elliptic restricted three-body problem (ER3BP) for a solar sail with reflectivity control devices (RCDs). In this non-autonomous model, periodic halo orbits only exist when their periods are equal to integer multiples of one year. Here multi-revolution halo orbits with periods equal to integer multiples of one year are constructed in the CR3BP and then used as seeds to numerically continue the halo orbits in the ER3BP. The linear stability of the orbits is analyzed which shows that the in-plane motion is unstable while the out-of-plane motion is neutrally stable and a bifurcation is identified. Finally, station-keeping is performed which shows that a reference orbit designed in the ER3BP is significantly more efficient than that designed in the CR3BP, while the addition of RCDs improve station-keeping performance and robustness to uncertainty in the sail lightness number.

Dynamics of Solar Radiation Pressure–Assisted Maneuvers Between Lissajous Orbits

The analysis of the phase space geometry around the artificial equilibrium points of the circular restricted three-body problem, including solar radiation pressure, is used to investigate “propellant-free” transfers between Lissajous orbits in the Sun-Earth system. It is assumed that the required transfer maneuvers are achieved by means of instantaneous changes in the spacecraft’s lightness parameter and/or in the incidence angle of the sail with respect to the Sun-line direction. Using a semi-analytic solution of the linearized equations of motion around the libration points , this paper develops a systematic methodology for general solar radiation pressure assisted maneuvers.

Determination of thrusts to generate artificial equilibrium points in binary systems with applications to a planar solar sail

In this work, the artificial equilibrium points for a general two-body system are derived, visualized, and summarized as functions of the direction of the thrust, for several directions. The results for the Sun-Earth system are also presented. Planar solar sail applications are also considered.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

A concept of hazardous NEO detection and impact warning system

In 2013, the well-known Chelyabinsk meteor entered the Earth's atmosphere over Chelyabinsk, Russia. It is estimated that the meteor exploded at altitude near 30 km, which damaged thousands of buildings and injured a thousand of residents. The estimated size of the meteor is approximately 20 m. Because the meteor approached to Earth from Sun direction, no ground-based observatories could not detect until the impact.Considering such situations, the paper proposes a concept to detect Chelyabinsk-class small Near-Earth Objects. The concept addresses a “last-minute” warning system of NEO impact, in the same manner of “Tsunami” warning.To achieve the mission objective, two locations are assumed for the space telescope installation point i.e., Sun-Earth Lagrange point 1, SEL1 and Artificial Equilibrium Point, AEP. SEL1 is one of the natural equilibrium points, on the other hand, AEP is artificially equilibrated point by Sun and Earth gravity, centrifugal force and low-thrust acceleration. The magnitude of the acceleration to keep AEP is sufficiently small near 1 au radius orbit around the Sun i.e., the order of μm/s2 which can be achieved by solar sail. Through some cases of numerical simulations considering the size of NEOs and detector capability, this paper will show the feasibility of the proposed concept.

Solar-Sail Orbital Motion About Asteroids and Binary Asteroid Systems

Solar radiation pressure is a major orbital perturbation for missions to small bodies like asteroids and binary asteroid systems. This Paper studies the utilization of solar radiation pressure on a solar sail for asteroid mission applications, specifically to generate artificial equilibrium points and displaced periodic orbits in these systems. For the single-asteroid case, contours of artificial equilibrium points for constant sail acceleration magnitudes are found in the Hill + solar radiation pressure problem as well as periodic orbits around these artificial equilibrium points. The binary system is modeled by first adding a fourth-body perturbation to the Hill + solar radiation pressure dynamics, demonstrating the effect of the smaller asteroid as an oscillatory motion superimposed on the unperturbed orbit. Truly periodic orbits are obtained in the bicircular + solar radiation pressure problem, showing the existence of so-called pole-sitter-like orbits above the binary system’s orbital plane. Higher-fidelity dynamical effects are investigated for these pole-sitter-like orbits for binary system 1999 KW4, showing feasibility of the orbits around aphelion. All results are generated for near-term sail technology and for a simple, fixed sail attitude relative to the sun. They therefore enable operable and unique vantage points from where to monitor the asteroid(s) over extended periods of time.

Artificial Equilibrium Points near Irregular-Shaped Asteroids with Continuous Thrust

Artificial equilibrium points near irregular-shaped asteroids are generated by using continuous thrust to enlarge the feasible regions of body-fixed hovering missions. The dynamics of artificial equilibrium points and feasible sets with constraints of the control magnitude and direction are investigated. The linearized motion near artificial equilibrium points is derived with the generalized effective potential for analysis of topological classification, bifurcations, and stability. Collision and annihilation of equilibrium points and Hopf bifurcation are found during numerical continuation. A parametric study is conducted to investigate the influence of the control magnitude on the sets of artificial equilibrium points. The linearly stable regions are obtained by a grid searching. Moreover, a method based on the polyhedral model of asteroids is proposed to solve the feasible artificial equilibrium points with constraints of the control direction and observation. The observable area at feasible artificial equilibrium points is also obtained by the proposed method. In numerical simulations, 433 Eros is selected as a representative irregular-shaped asteroid to validate the effectiveness of the methods.

Artificial Equilibrium Points in the Low-Thrust Restricted Three-Body Problem When Both the Primaries Are Oblate Spheroids

This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A1(0<A1<1) and A2(0<A2<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.