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Attitude control for a full-scale flexible electric solar wind sail spacecraft on heliocentric and displaced non-Keplerian orbits
... The deflection of the main tethers has a significant effect on the central spacecraft attitude [21]. Although it is challenging to develop a highfidelity full-scale dynamic model for the E-sail, many effects describing the nonlinear rigid-flexible coupling motion of the E-sail on the central spacecraft have yielded remarkable results [11,12,22,23]. In these previous studies, the dynamics of the tether with highorder modes were captured using the NPFEM [11] and ANCF [12] in the global inertial coordinate system and the RNCF [22,23] in the non-inertial coordinate system. ...
... Although it is challenging to develop a highfidelity full-scale dynamic model for the E-sail, many effects describing the nonlinear rigid-flexible coupling motion of the E-sail on the central spacecraft have yielded remarkable results [11,12,22,23]. In these previous studies, the dynamics of the tether with highorder modes were captured using the NPFEM [11] and ANCF [12] in the global inertial coordinate system and the RNCF [22,23] in the non-inertial coordinate system. Meanwhile, the attitude motions of the central spacecraft are described by utilizing the floating frame of reference method (FFRM) [22,23] and natural coordinate formulation (NCF) [11,12]. ...
... In these previous studies, the dynamics of the tether with highorder modes were captured using the NPFEM [11] and ANCF [12] in the global inertial coordinate system and the RNCF [22,23] in the non-inertial coordinate system. Meanwhile, the attitude motions of the central spacecraft are described by utilizing the floating frame of reference method (FFRM) [22,23] and natural coordinate formulation (NCF) [11,12]. Specifically, the latter can prevent the singularity of the Euler angles. ...
This study examines the impact of electric solar wind sail (E-sail) parameters on the attitude stability of E-sail’s central spacecraft by using a comprehensive rigid–flexible coupling dynamic model. In this model, the nodal position finite element method is used to model the elastic deformation of the tethers through interconnected two-node tensile elements. The attitude dynamics of the central spacecraft is described using a natural coordinate formulation. The rigid–flexible coupling between the central spacecraft and its flexible tethers is established using Lagrange multipliers. Our research reveals the significant influences of parameters such as tether numbers, tether’s electric potential, and solar wind velocity on attitude stability. Specifically, solar wind fluctuations and the distribution of electric potential on the main tethers considerably affect the attitude stability of the spacecraft. For consistent management, the angular velocities of the spacecraft must remain at target values. Moreover, the attitude stability of a spacecraft has a pronounced dependence on the geometrical configuration of the E-sail, with axisymmetric E-sails proving to be more stable.
... The referenced nodal coordinates formulation (RNCF) was proposed by describing the ANCF in a non-inertia reference frame [17]. Since the rigid motions of the structures can be described by the reference frame, the RNCF model can be used for the attitude control of flexible structures, such as the solar sail [18,19] and large-scale hoop-column antenna [20]. To reduce the dimension of the RNCF model, a general nonlinear order-reduction method was developed for flexible multibody systems by incorporating the modal derivative techniques into the RNCF [21]. ...
... Highly flexible beam structures play a significant role in extensive engineering applications, such as large deployable space structures [1,2], flexible manipulators [3], flexible robots [4], and compliant mechanisms [5]. There are great demands for beam models with high accuracy and efficiency in simulating the mechanical system with large deformations. ...
A new recursive geometrically exact formulation (RGEF) for three-dimensional Euler–Bernoulli beams with large deformations is proposed in this work. The proposed RGEF introduces three major innovations compared to the existing beam formulations. First, each element only has three degrees of freedom, which can significantly reduce the dimension of the equations of motion. Second, the computational complexity is only due to the recursive scheme, meaning the calculation time increases linearly with the number of elements. Third, the inertia force and mass matrix can be explicitly integrated in advance by adopting the velocity approximation approach, which can significantly reduce computational costs. Moreover, the proposed RGEF is singularity-free due to the interpolation of relative rotation vectors. The proposed RGEF can be used for both open-loop and closed-loop multibody systems, and several static and dynamic benchmark examples are presented to demonstrate the effectiveness of the formulation.
... Flexible beams have been widely used in many engineering applications, such as soft robots [1][2][3], deployable antennae [4,5], and flexible spacecraft [6,7]. The accurate and efficient simulation of the dynamics of the beams undergoing large displacements and rotations has been a hot spot for decades. ...
A spacetime variational integration approach for flexible beams is developed based on the absolute nodal coordinate formulation (ANCF), where the discretization is accomplished in a consistent way that regards displacements in both space and time. Specifically, the Hermite interpolation functions are used for spatial discretization, considering that the nodal position and its axial gradient are essential to interpolate displacements for ANCF beam elements. The quadratic polynomials with the use of variables at three consecutive time steps are adopted for temporal discretization. The discrete equations of motion are derived by a discrete variational principle based on the full spacetime discretization, which ultimately renders a variational integrator. Compared to variational integrators for the large rotation vector formulations and geometrically exact beam formulations, the present approach has the advantage of avoiding nonlinearity due to spatial rotational parametrizations as well as the complex expressions for inertia forces, such that computation complexity is significantly reduced. Numerical experiments are performed to validate the capabilities of the present spacetime variational integrator for flexible beams with large displacements, rotations, and deformations. The results reveal its good energy-momentum preserving behavior for conservative systems.
... The thrust vectoring of a spinning E-sail-propelled spacecraft is, in fact, a complex task, which requires a precise design of the attitude control system as thoroughly discussed in the two interesting papers by Toivanen and Janhunen [38,39] that analyzed this specific important problem by using an elegant analytical approach. In this regard, the E-sail-related scientific literature contains more recent and useful works [40,41] which investigate the connection between a generic E-sail attitude maneuver and the spacecraft response from the structural viewpoint, taking the intrinsic flexibility of such a large space structure into account [42][43][44]. From the perspective of attitude control system design, a constrained (and generally small) value of the target sail pitch angle could improve the effectiveness of the maneuver for the variation in the inclination of the sail nominal plane at the expense, however, of a worsening of the orbital transfer performance in terms of the required total flight time [45]. The aim of this paper is to investigate the effects of a constraint on the maximum value of the sail pitch angle, on the performance of a spacecraft equipped with an E-sail propulsion system in a typical interplanetary mission scenario [46]. ...
The Electric Solar Wind Sail (E-sail) deflects charged particles from the solar wind through an artificial electric field to generate thrust in interplanetary space. The structure of a spacecraft equipped with a typical E-sail essentially consists in a number of long conducting tethers deployed from a main central body, which contains the classical spacecraft subsystems. During flight, the reference plane that formally contains the conducting tethers, i.e., the sail nominal plane, is inclined with respect to the direction of propagation of the solar wind (approximately coinciding with the Sun–spacecraft direction in a preliminary trajectory analysis) in such a way as to vary both the direction and the module of the thrust vector provided by the propellantless propulsion system. The generation of a sail pitch angle different from zero (i.e., a non-zero angle between the Sun–spacecraft line and the direction perpendicular to the sail nominal plane) allows a transverse component of the thrust vector to be obtained. From the perspective of attitude control system design, a small value of the sail pitch angle could improve the effectiveness of the E-sail attitude maneuver at the expense, however, of a worsening of the orbital transfer performance. The aim of this paper is to investigate the effects of a constraint on the maximum value of the sail pitch angle, on the performance of a spacecraft equipped with an E-sail propulsion system in a typical interplanetary mission scenario. During flight, the E-sail propulsion system is considered to be always on so that the entire transfer can be considered a single propelled arc. A heliocentric orbit-to-orbit transfer without ephemeris constraints is analyzed, while the performance analysis is conducted in a parametric form as a function of both the maximum admissible sail pitch angle and the propulsion system’s characteristic acceleration value.
... Yuan et al. [27] proposed an order reduction method for geometrically nonlinear flexible bodies based on the RNCF and modeled a large truss structure in space. The RNCF has been successfully applied to the dynamic modeling of a full-scale electric sail [28] and a large-scale hoop-column antenna [29,30], showing its effectiveness and applicability. ...
A novel modeling framework combining arbitrary Lagrange-Euler and referenced nodal coordinate formulation (ALE-RNCF) is proposed for deployment dynamics and control of a hub-spoke tethered satellite formation. The ALE-RNCF approach allows for an accurate analysis of the intricate coupling effect between the orbit, attitude, and deployment dynamics, and its strengths lie in overcoming the accuracy loss and low-efficiency issues when dealing with spatial and temporal multiscale problems. Specifically, the orbital and attitude motions are separated with vibrations of the variable-length ALE tethers through the RNCF, which is the main distinguishing feature over the widely-used absolute nodal coordinate formulation. To achieve stable deployment, the control torque is added to the central satellite by employing the proportional-differential algorithm, where the maximum tension of tethers or the spinning angular velocity is selected as the control object. Various cases with different deployment velocities, target tensions, and orbital heights are simulated and corresponding effects on the deployment performance are analyzed. The proposed ALE-RNCF approach provides a comprehensive understanding of the orbit-attitude-structure coupled behavior during the deployment of the hub-spoke tethered satellite formation and contributes to the development of effective control strategies.
... However, when a LFSS is complex, there are large numbers of low-frequency and closefrequency modes, which make it difficult to establish an accurate dynamics model by selecting some modes. Therefore, it is necessary to study the control problem for a full-scale dynamics model, and we have performed attitude control simulation for a full-scale flexible electric solar wind sail spacecraft (Zeng et al., 2023). ...
The challenges in the dynamic modeling and control of large flexible space structures (LFSSs) include the high number of degrees of freedom in the model, the geometric nonlinearities in the flexible components, and the nonlinear coupling effects between overall motions and vibrations of the structure. In this study, we propose a systematic method to solve the mentioned problems in the research of LFSSs. We use the referenced nodal coordinate formulation (RNCF) to build the dynamic model. The modeling method is general and the model can better describe the large deformations of flexible structures compared with the modal-based dynamic model. The key feature of our work is that we use a hybrid control strategy for the attitude and vibration control directly performing on the full-scale dynamic model. Specifically, the control strategy combines a proportional and derivative (PD) control algorithm based on SO(3) for the attitude control and an analytical linear-quadratic (LQ) vibration control method for the vibration control. The simulation results are presented to demonstrate the effectiveness of the proposed hybrid control strategy both in the three-axis attitude maneuver task and on-orbit pointing scenario.
The spinning circular solar sail is a promising spacecraft for long-duration missions. This work reveals its structural dynamic and stability behavior under the periodically time-varying solar radiation pressure and gravitational force. The geometric stiffness generated by the centrifugal force due to spinning and the coupling effect between the deformation and solar radiation pressure are taken into account. The von Kármán plate theory is adopted by neglecting the high-frequency in-plane vibrations and considering the effect of the in-plane internal force on the transverse vibration. The partial differential equation of the spinning solar sail is derived and further spatially discretized into periodically time-varying equations of motion. Effects of Poisson ratio and radius ratio on natural frequencies and mode shapes are analyzed, and curve veering phenomena are then observed. Steady-state periodic responses of the solar sail under the solar radiation pressure with different orbit distances, incident angles, and spinning angular velocities are analyzed. The stability analysis is rigorously performed by the Floquet theory rather than the commonly used approach of conducting the eigenvalue analysis at a series of specific discrete time nodes. Moreover, the stability boundary associated with transverse vibrations is determined, which contributes to the parameter design of the spinning solar sail.
The Electric Solar Wind Sail (E-sail) is an innovative propellantless propulsion system conceived by Pekka Janhunen in 2004 for use in interplanetary space. An E-sail consists of a network of electrically charged tethers maintained at a high voltage level by an electron emitter. The electrostatic field surrounding the E-sail extracts momentum from the incoming solar wind ions, thus giving rise to the generation of a continuous thrust. In a geocentric context, the same physical principle is also exploited by the plasma brake, a promising option for reducing the decay time of satellites in low Earth orbits after the end of their operational life. This paper discusses the scientific advances of both E-sail and plasma brake concepts from their first design to the current state of the art. A general description of the E-sail architecture is first presented with particular emphasis on the proposed tether deployment mechanisms and thermo-structural analyses that have been carried out over the recent years. The working principle of an E-sail is then illustrated and the evolution of the thrust and torque vector models is retraced to emphasize the subsequent refinements that these models have encountered. The dynamic behavior of an E-sail is also analyzed by illustrating the mathematical tools that have been proposed and developed for both orbital dynamics and attitude control. A particular effort is devoted to reviewing the numerous mission scenarios that have been studied to date. In fact, the extensive literature about E-sail-based mission scenarios demonstrates the versatility of such an innovative propulsion system in an interplanetary framework. Credit is given to the very recent studies on environmental uncertainties, which highlight the importance of using suitable control strategies for the compensation of solar wind fluctuations. Finally, the applications of the plasma brake are thoroughly reviewed.
The Electric Solar Wind Sail is an innovative propulsion system that gains thrust from the interaction of the incoming ions from the solar wind with an artificial electric field produced by means of long charged tethers, which are deployed and maintained stretched by rotating the spacecraft around a spin axis. Under the combined interaction between solar wind dynamic pressure and centrifugal force, the tethers reach an equilibrium configuration whose spatial shape must be known for obtaining a reasonable estimate of the propulsive acceleration, a fundamental information for preliminary mission analysis purposes. This problem has been addressed in recent papers, which deal with the analytical expressions of thrust and torque vectors of a spinning and axially-symmetric Electric Solar Wind Sail. The torque acting on the sail induces a perturbation on the orientation of the thrust vector, which is here studied by analyzing the attitude dynamics. Numerical simulations show that the spacecraft motion is characterized by an undamped precession combined with a nutation motion. The effect due to the torque acting on the spacecraft is to align the thrust direction along the Sun-sail line, thus reducing the maneuvering capabilities. This paper proposes an effective control law which is able to remove the torque by suitably adjusting the tether electric voltage. It is shown that the proposed solution maintains the nominal thrust magnitude, and requires a small electric voltage modulation.
This paper investigates the dynamic characteristics of thrust-induced sail plane coning and attitude motion of an electric solar wind sail (E-sail) by considering high-order modes of flexible elastic tethers. The tethers of the E-sail are assumed elastic and discretized into inter-connected 2-noded tensile elements using the nodal position finite element method, while the central spacecraft and the remote units are simplified as lumped masses. The E-sail is assumed in the heliocentric ecliptic orbital plane at a distance of 1 au from the Sun. The influences of the propulsive force models and the initial E-sail orientation on the dynamic characteristics of the sail plane coning and attitude motion of E-sail are analyzed. The current work derives an analytic expression of the coning motion frequency under the assumption of small coning angle. Through parametric analyses, the current work shows that the magnitude of the propulsive force significantly influences the increment magnitude of the E-sail's orbital radius while has little effect on the angles of sail and thrust angles and the E-sail spin rate. The parametric analyses also show that the initial E-sail orientation significantly influences the thrust vector and the variation of sail angle. Finally, the relationships of the sail and thrust angles, as well as the dimensionless acceleration of the E-sail, are given in polynomial expressions by curve-fitting of simulation results of E-sails with the consideration of coning motion. The relationships are compared with the previous results of E-sails without including coning motion. It shows the coning motion has negligible effect on the macro dynamic behaviors of E-sail.
We develop a referenced nodal coordinate formulation (RNCF) to study the dynamics of flexible bodies undergoing large-distance travels and/or high-speed rotations. RNCF is similar to the absolute nodal coordinate formulation (ANCF) but is presented in a noninertia reference coordinate system (RCS). The position vector and rotation matrix of the RCS describe translational and rotational motions of the system, whereas the nodal coordinates and slopes in a structure depict its large deformations, such that the generalized coordinates with multiple scales in length and time are automatically separated. We develop a parameter-irrelevant technique to derive the rotation equations of the system, where the influences of large deformations on the rotatory inertia tensors are embodied. The derived governing equations are simple and elegant, and consistent with the governing equations for rigid bodies, the floating frame of reference method, as well as ANCF. We verify the RNCF approach by three typical examples, including the spin-up maneuver, the high speed motor, and the flexible slider-crank mechanism. The results indicate that to achieve the same accuracy, the computational cost for RNCF is much lower than that for the corresponding ANCF in high-speed rotating systems. Moreover, the electrical solar wind sail spacecraft system is formulated by RNCF, and its propulsive efficiencies with respect to the spin rates of the E-sails are studied by full-scale models with over ten thousand degrees of freedom. RNCF provides an effective way to formulate and study the dynamics of vehicles, trains, ships, aircrafts, and spacecrafts.
The propulsive characteristics of an Electric Solar Wind Sail are usually evaluated using a simplified model in which all the sail tethers are coplanar and form a sort of rigid disk. However, the three-dimensional arrangement of the tethers is fundamental information in the study of the spacecraft performance, and must be accounted for in refined mission analyses. In this paper, a Finite Element approach is chosen to estimate the deflected shape of the tethers, thus allowing important information on the structural response of the sail to be obtained. A parametric code is developed to perform a static analysis of an Electric Solar Wind Sail, whose requirements are given in terms of payload mass and spacecraft characteristic acceleration. In particular, the tether structural response is investigated using three different beam models, which are compared in terms of accuracy and computational efficiency. The analysis is specialized to the noteworthy case of a Sun-facing sail that is placed at a distance of one astronomical unit from the Sun. The numerical results, which concern a set of possible sail configurations, are compared with those taken from analytical models.
An Electric Solar Wind Sail is a propellantless propulsion system that gains thrust from the interaction of solar wind particles with a grid of long and charged tethers, which are deployed by spinning the spacecraft about its symmetry axis. In an ideal arrangement, the tethers are all stretched out and the sail takes the shape of a spoked wheel. Actually, the solar wind dynamic pressure warps the tethers and, therefore, the expressions of thrust and torque vectors are difficult to predict in analytical form. Recent works have shown that the bending of the tethers induces a disturbance torque, which can be counterbalanced through a modulation of the tether electrical voltage. Under the assumption that the Electric Solar Wind Sail behaves like an axially-symmetric rigid body, this paper proves that a modulation of the tether electrical voltage is also a feasible option for actively controlling and maintaining the spacecraft attitude. The proposed control law, which is analytically derived as a function of time and spacecraft attitude, is validated through numerical simulations.
The working principle of the Electric Solar Wind Sail, an innovative propellantless propulsion system proposed in 2004, is based on the electrostatic interaction between a spinning grid of tethers, kept at a high positive potential, and the incoming ions from the solar wind. Similar to the well-known solar sail concept, the E-sail could simplify the feasibility of advanced (deep space) missions which would otherwise require a significative amount of propellant, if enabled by conventional thrusters. However, the intrinsic variability of the solar wind properties makes accurate trajectory tracking a difficult task, since the perturbations of the solar wind dynamic pressure have the same order of magnitude as their mean value. To circumvent such a problem, in a recent study the plasma dynamic pressure was modelled as a random variable with a gamma probability density function and the sail grid voltage was suggested to be varied as a function of the instantaneous value of the solar wind properties. The aim of this paper is to improve those results, by discussing a more accurate statistical model of the solar wind dynamic pressure, which is used in the numerical simulations to estimate the actual impact of the solar wind uncertainties on the spacecraft heliocentric trajectory. In particular, the paper proposes a control law that is able to accurately track a nominal, non-Keplerian orbit.
The electric sail is an innovative propellantless propulsion concept that gains continuous momentum from the solar wind. In this work, the electric sail’s attitude dynamics is established from a multibody perspective. Firstly, the dynamics of a single charged tether is described by a dumbbell model. Comparisons with an elastic multipoint model show that the dumbbell model is simpler and accurate enough to describe the motion of tether when the spin rate meets a specific lower bound. Then, the electric sail’s attitude dynamics model is established through Kane’s method based on the dumbbell model of the tethers. Finally, a case study is performed for the electric sail. It is shown that a 4-km-radius electric sail with 12 tethers of which the voltage is 25 kV could provide the sailcraft with a characteristic acceleration of 0.1 mm/s2. The results also indicate that the satellite bus’s acceleration oscillates with the undiminished out-of-plane swing of the electric sail, in which the period and amplitude are in inverse proportional to the spin rate and its square, respectively. Besides, a dynamical equilibrium point for system’s attitude exists when the electric sail is perpendicular to the solar wind. The presented work offers a basic referential model for the real-time feedback control problems of the electric sail’s attitude.
The Electric Solar Wind Sail is an innovative propulsion system concept that gains propulsive acceleration from the interaction with charged particles released by the Sun. The aim of this paper is to obtain analytical expressions for the thrust and torque vectors of a spinning sail of given shape. Under the only assumption that each tether belongs to a plane containing the spacecraft spin axis, a general analytical relation is found for the thrust and torque vectors as a function of the spacecraft attitude relative to an orbital reference frame. The results are then applied to the noteworthy situation of a Sun-facing sail, that is, when the spacecraft spin axis is aligned with the Sun-spacecraft line, which approximatively coincides with the solar wind direction. In that case, the paper discusses the equilibrium shape of the generic conducting tether as a function of the sail geometry and the spin rate, using both a numerical and an analytical (approximate) approach. As a result, the structural characteristics of the conducting tether are related to the spacecraft geometric parameters.
Displaced solar orbits for spacecraft propelled by electric sails are investigated. Since the propulsive thrust is induced by the sail attitude, the orbital and attitude dynamics of electric-sail-based spacecraft are coupled and required to be investigated together. However, the coupled dynamics and control of electric sails have not been discussed in most published literatures. In this paper, the equilibrium point of the coupled dynamical system in displaced orbit is obtained, and its stability is analyzed through a linearization. The results of stability analysis show that only some of the orbits are marginally stable. For unstable displaced orbits, linear quadratic regulator is employed to control the coupled attitude-orbit system. Numerical simulations show that the proposed strategy can control the coupled system and a small torque can stabilize both the attitude and orbit. In order to generate the control force and torque, the voltage distribution problem is studied in an optimal framework. The numerical results show that the control force and torque of electric sail can be realized by adjusting the voltage distribution of charged tethers.
This paper analyzes the performance of an Electric Solar Wind Sail for generating and maintaining an elliptic, heliocentric, displaced non-Keplerian orbit. In this sense, this paper extends and completes recent studies regarding the performances of an Electric Solar Wind Sail that covers a circular, heliocentric, displaced orbit of given characteristics. The paper presents the general equations that describe the elliptic orbit maintenance in terms of both spacecraft attitude and performance requirements, when a refined thrust model (recently proposed for the preliminary mission design) is taken into account. In particular, the paper also discusses some practical applications on particular mission scenarios in which an analytic solution of the governing equations has been found.
The electric solar wind sail is a propulsion system that uses long centrifugally spanned and electrically charged tethers to extract solar wind momentum for spacecraft thrust. The sail solar zenith angle and phase can be controlled by modulating the voltage of each tether separately to produce net torque for attitude control and thrust vectoring. In this paper, we cover the basics of the electric sail control based on our set of dynamical models including single tether simulation and rigid body and fully dynamical simulations for the entire sail. Short descriptions of these models are given. We also consider real solar wind conditions based on available solar wind data and address the effects of the solar wind density and velocity variations on the sail dynamics, control, and navigation.
Displaced orbits for spacecraft propelled by electric sails are investigated as an alternative to the use of solar sails. The orbital dynamics of electric sails based spacecraft are studied within a spherical coordinate system, which permits finding the solutions of displaced electric sail orbits and optimize transfer trajectory. Transfer trajectories from Earth's orbit to displaced orbit are also studied in an optimal framework, by using genetic algorithm and Gauss pseudospectral method. The initial guesses for the state and control histories used in the Gauss pseudospectral method are interpolated from the best solution of a genetic algorithm. Numerical simulations show that the electric sail is able to perform the transfer from Earth's orbit to displaced orbit in acceptable time, and the hybrid optimization method has the capability to search the feasible and optimal solution without any initial value guess.
The electric solar wind sail is a propulsion system that uses long centrifugally spanned and electrically charged tethers to extract the solar wind momentum for spacecraft thrust. The sail angle with respect to the sun direction can be controlled by modulating the voltage of each tether separately to produce net torque for attitude control and thrust vectoring. A solution for the voltage modulation that maintains any realistic sail angle under constant solar wind is obtained. Together with the adiabatic invariance of the angular momentum, the tether spin rate and coning angle is solved as functions of temporal changes in the solar wind dynamic pressure, the tether length, or the sail angle. The obtained modulation also gives an estimate for the fraction of sail performance (electron gun power) to be reserved for sail control. We also show that orbiting around the Sun with a xed sail angle leads to a gradual increase (decrease) in the sail spin rate when spiraling outward (inward). This eect arises from the fact that the modulation of the electric sail force can only partially cancel the Coriolis eect, and the remaining component lays in the spin plane having a cumulative eect on the spin rate.
An electric sail uses the solar wind dynamic pressure to produce a small but continuous thrust by interacting with an electric field generated around a number of charged tethers. Because of the weakness of the solar wind dynamic pressure, quantifiable in about 2 nPa at Earth's distance from the sun, the required tether length is of the order of some kilometers. Equipping a 100-kg spacecraft with 100 of such tethers, each one being of 10-km lengths, is sufficient to obtain a spacecraft acceleration of about 1 mm/s(2). These values render the electric sail a potentially competitive propulsion means for future mission applications. The aim of this paper is to provide a preliminary analysis of the electric sail performance and to investigate the capabilities of this propulsion system in performing interplanetary missions. To this end, the minimum-time rendezvous/transfer problem between circular and coplanar orbits is considered, and an optimal steering law is found using an indirect approach. The main differences between electric sail and solar sail performances are also emphasized.
The electric solar wind sail (E-sail) is a new type of propellantless
propulsion system for Solar System transportation, which uses the
natural solar wind for producing spacecraft propulsion. This paper
discusses a mass breakdown and a performance model for an E-sail
spacecraft that hosts a scientific payload of prescribed mass. In
particular, the model is able to estimate the total spacecraft mass and
its propulsive acceleration as a function of various design parameters
such as the tethers number and their length. A number of subsystem
masses are calculated assuming existing or near-term E-sail technology.
In light of the obtained performance estimates, an E-sail represents a
promising propulsion system for a variety of transportation needs in the
Solar System.
Electric sail for extracting momentum from the solar wind was described. Another way to use the solar wind momentum is to use a magnetic sail, which requires a superconducting one-dimensional loop wire of length smaller than the linear extent of the electric wire mesh. It was found that the electric sail does not need any magnetic fields or superconductors and its power requirements are modest. Accurate maneuvering is difficult but the method is expected to be well suited for missions whose purpose is just to travel fast out of the solar system across the heliosphere to interstellar space.
An electric sail is capable of guaranteeing the fulfilment of a class of trajectories that would be otherwise unfeasible through
conventional propulsion systems. In particular, the aim of this paper is to analyze the electric sail capabilities of generating
a class of displaced non-Keplerian orbits, useful for the observation of the Sun’s polar regions. These orbits are characterized
through their physical parameters (orbital period and solar distance) and the spacecraft propulsion capabilities. A comparison
with a solar sail is made to highlight which of the two systems is more convenient for a given mission scenario. The optimal
(minimum time) transfer trajectories towards the displaced orbits are found with an indirect approach.
The electric solar wind sail (E-sail) is a space propulsion concept that uses the natural solar wind dynamic pressure for producing spacecraft thrust. In its baseline form, the E-sail consists of a number of long, thin, conducting, and centrifugally stretched tethers, which are kept in a high positive potential by an onboard electron gun. The concept gains its efficiency from the fact that the effective sail area, i.e., the potential structure of the tethers, can be millions of times larger than the physical area of the thin tethers wires, which offsets the fact that the dynamic pressure of the solar wind is very weak. Indeed, according to the most recent published estimates, an E-sail of 1 N thrust and 100 kg mass could be built in the rather near future, providing a revolutionary level of propulsive performance (specific acceleration) for travel in the solar system. Here we give a review of the ongoing technical development work of the E-sail, covering tether construction, overall mechanical design alternatives, guidance and navigation strategies, and dynamical and orbital simulations.
A displaced non-Keplerian orbit is a trajectory whose orbital plane does not contain the center of mass of the primary body, so that its orbital maintenance requires the application of a suitable continuous thrust. Although the latter could be provided, in principle, by a low-thrust electric propulsion system, innovative propellantless propulsive technologies are well suited to such a mission scenario, due to their ability to generate thrust without requiring any propellant, thus significantly extending mission lifetime. This chapter focuses on the possibility of maintaining a displaced non-Keplerian orbit by means of both solar sails and electric solar wind sails (or E-sails). In fact, these advanced propulsion systems are both capable of generating a propulsive acceleration without consuming any propellant, by exploiting the solar radiation pressure (in case of solar sails) or the solar wind dynamic pressure (E-sails). This analysis uses recent models to provide a mathematical description of the propulsive acceleration generated by both propulsion systems, and different scenarios involving non-Keplerian orbits are analyzed. Particular focus is given to Type II displaced orbits, non-Keplerian orbits lying on the ecliptic plane, and heliostationary positions. Performance and attitude requirements are provided for each scenario. A linear stability analysis is also performed, in order to identify the combination of orbital parameters that characterize stable non-Keplerian orbits. The results suggest the feasibility of the mission scenarios discussed, but for most of them performance requirements are very demanding. A possible exception is non-Keplerian orbits lying on the ecliptic, which represent a very promising near-term scenario.
This paper investigates the attitude control and stability analysis of an electric solar wind sail (E-sail) by considering elastic deflection of tethers while assuming main spacecraft and remote units as point masses. The attitude and orbital motion of the E-sail is analyzed by a high-order high-fidelity E-sail model derived from the nodal position finite element method, where the attitude angles are implicitly described via the nodal coordinates. To overcome the difficulty in handling the stability analysis of high-order model under the Lyapunov framework, the E-sails attitude dynamics is approximated explicitly by a reduced order analytical model with only three attitude angles. A sliding mode control law is proposed for the E-sail attitude control based on the reduced order analytical E-sail model and its stability is proved by the Lyapunov theory. Finally, two schemes are derived to map the control torque to either the control thrust at remote units or the voltages of main tethers respectively, which are applied to the high-fidelity E-sail model for attitude control. Numerical simulation demonstrates that the proposed control law performs similarly with the high-fidelity and reduced order analytical E-sail models if proper control gains are selected. It shows that the control law developed from the reduced order analytical E-sail model can stably control the attitude of a real E-sail. The investigation also indicates that the high-order flexible E-sail model provides an effective virtual testbed to evaluate the E-sail attitude control strategy derived from the reduced order attitude dynamics.
An electric solar wind sail (electric sail or E-sail) is a new type of propulsion system that does not require propellant. This paper proposes a nonlinear model predictive control law based on deep learning to control the attitude of a barbell electric sail. A barbell E-sail is composed of two tip satellites connected via long conductive tethers to a central insulated confluence point. The two tethers are insulated from each other at the confluence point such that the voltages of the two tethers can be independently controlled. The attitude dynamics of the system is modelled using a nonsingular description of tether orientation. To reduce online computation loads, the proposed control scheme has a two-stage design, namely, an offline stage and an online stage. In the offline stage, a large amount of data is generated using the nonlinear model predictive control law and then taken as a dataset to train a deep neural network. In the online stage, feedback control of the system attitude is achieved with extremely low computational cost by performing real-time mapping from the system state to the control output using the trained deep neural network. Finally, the efficacy and performance of the proposed deep learning-based control law are demonstrated via numerical case studies.
This paper investigates the spin rate bounds, the configuration stability subject to the solar wind fluctuations, and sail angle control of an electric solar wind sail (E-sail) by a high-fidelity tether dynamic model. This model describes the elastic deformation of tethers with inter-connected 2-noded tensile elements discretized by the nodal position finite element method. The E-sail is assumed to be an axisymmetric system spinning in the plane normal to the heliocentric ecliptic plane. The upper and lower spin rate bounds are revisited to reveal the physics that dictates these bounds and analytic expressions are provided to ensure the proper operation of E-sail. Then, the influences of the solar wind fluctuations on the configuration stability of the E-sail are investigated by parametric analysis with different E-sail configurations, sail angles, and spin rates. Finally, an alternative sail angle control strategy for the E-sail is proposed by applying control force at the remote units with a simple PD control. Numerical analysis demonstrates that the sail angle of E-sail can be controlled quickly by the control law at the remote units with a high-precision.
The dynamics and control strategies of a full-scale flexible electric solar wind sail (E-sail) spacecraft are studied in this work. The configuration of the E-sail is an umbrella shape consisting of 100 main tethers of 20 km long, with a remote unit attached at the free end of each main tether; such remote units are also connected by an auxiliary tether with 100 separated segments; the electric voltages of each main or each segment of the auxiliary tether can be independently prescribed to control the orbit, orientation, and vibrations of the E-sail. Two computational effective formulations, the reference nodal coordinate formulation (RNCF) and the nonlinear floating frame of reference method (FFRM) in a large deformed configuration, are developed to describe the dynamics of the E-sail spacecraft. The RNCF is accurate and comprehensive, but requires over 10000 degrees of freedoms to describe the highly flexible and nonlinear dynamics of the E-sail. The FFRM requires only tens of degrees of freedom, and can be adopted for controller designs. The time scales of the orbit motions, the orientation motions and the structural vibrations of the E-sail are quite different. The modes in FFRM are categorized into three groups, such that the influences of the electric voltages on each category of modes are distinctively different, and the controllable properties of each category of modes are explored for controller designs. Although it is designed from the simplified FFRM approach, the resultant controller is implemented to the full-scale RNCF approach to verify its effectiveness, which validates that the E-sail spacecraft can be adopted for solar system exploration missions.
This paper investigates the modelling of rigid-flexible coupling effect on the attitude dynamics and spin control of an electric solar wind sail (E-sail) by developing a rigid-flexible coupling dynamic model. The model considers the attitude dynamics of the central spacecraft, the elastic deformation of the tethers and the rigid-flexible coupling between the spacecraft and the tether. The attitude and translation dynamics of the central spacecraft is described by the natural coordinate formulation, while the tether deformation is described by the high-fidelity nodal position finite element method. The latter enables a natural coupling between the motion of the flexible tethers and the rigid-body dynamics of the central spacecraft at the anchor points where the tethers connected to the spacecraft by Lagrange multipliers. Based on the model, the influence of the rigid-flexible coupling, E-sail orientation and geometrical configuration on the dynamic characteristics of the E-sail is investigated by a parametric analysis. It is found that the deformation motion of flexible tethers will cause the offset of centres of mass and thrust of E-sail, which generates disturbance torques on the central spacecraft. Through the nonlinear rigid-flexible coupling, the disturbance causes the tension fluctuations and the undesired fluctuations of the E-sail's attitude and spin rate. The parametric analysis indicates that the E-sail is more stable if the spin plane passes the centre of mass of the central spacecraft. Finally, the controllability of E-sail spin rate is investigated by applying simple feedback torque controls at the central spacecraft or at the central spacecraft and the remote units simultaneously. The analysis demonstrates the spin rate cannot be controlled by the central spacecraft along due to the rigid-flexible coupling and must be controlled at the remote units with finite control input.
The paper focuses on the attitude dynamics and control of a barbell electric sail (E-sail). The barbell E-sail is composed of two tip satellites connected via long conductive tethers to a central insulated confluence point. The two tethers are insulated from each other at the confluence point such that the voltages of the two tethers can be independently controlled. A nonsingular formulation based on the dumbbell assumption is proposed to describe the attitude dynamics of the barbell E-sail, and exploited to develop a nonlinear model predictive controller for attitude adjustment and maintenance of the barbell E-sail by regulating the tether voltages. The nonlinear model predictive control law explicitly accounts for the nonlinear attitude dynamics and mission-related constraints of the barbell E-sail. Finally, the efficiency and performance of the proposed control law are demonstrated via numerical case studies.
This paper studies the flight dynamics and control strategy for electric solar wind sails based on the nodal position finite element method, where the coupling effects between tether dynamics and the electrical field are considered. A modified throttling control strategy is proposed to control the attitude of electric sails by modulating individual tether voltage synchronously with the spinning motion of the sails. The effects of four critical physical parameters (tether numbers, tether length, sail spin rate, and mass of remote units) are investigated. The results show that the effect of the relative velocity of the solar wind has a significant effect on the spin rate of the sail in attitude maneuvering, which in turn affects the attitude dynamics and orbit motion of the sail. Numerical results show that the proposed control strategy work successfully stabilizes the spin rate of sail when the new type sail is adopted.
The electric solar wind sail is a propulsion system that extracts the solar wind momentum for the thrust force of a spacecraft by using an interaction between solar wind protons and the electric potential structure around charged long thin conducting tethers. The system enables a spacecraft to generate a thrust force without consuming
reaction mass. This paper investigates the capability of the electric solar wind sail as a propulsion system for deep space exploration missions. The shape of the conducting tether that is determined by the equilibrium of the solar wind force and centrifugal force is numerically calculated for formulating an advanced solar wind force
model. The conducting tethers deviate from the ideal sail spin plane, and the maximum value of the thrust direction varies from 13∘ to 19∘. To estimate the spacecraft thrust vector, which is calculated as the sum of solar wind force vectors exerted on each tether, best-fit polynomial equations are proposed. We performed numerical simulations for a two-dimensional orbital transfer mission to investigate the capability of the electric solar wind sail. Results of numerical simulations show that the electric solar wind sail enables spacecraft to perform Earth-Venus, Earth-Mars, and Earth-Itokawa transfer missions. Additionally, this paper performs three-dimensional
simulations for an Earth–Ryugu transfer mission. The electric solar wind sail achieves a more complicated orbital transfer in a reasonable mission time.
This paper studies the dynamic characteristics of an electric solar wind sail (E-sail). A high-fidelity multiphysics model is developed by the nodal position finite element method to investigate the coupling effects of orbital and self-spinning motions of the E-sail, and the interaction between the axial/transverse elastic motions of tether and the Coulomb force. Furthermore, parametric study is conducted to better understand these coupling effects. The simulation results show that the coupling effects have a significant impact on the dynamic behavior of E-sail and the induced thrust. Furthermore, the analysis indicates a strong dependence of the thrust on sail and coning angles of E-sail even in the case of small sail angle. Finally, the influence of the initial self-spin rate and the sail angle on the dynamic behavior of a flexible E-sail is investigated. It shows that a high spin rate is needed to hold the geometrical configuration of the E-sail, and the difference in the orbital maneuvering is distinct when the E-sail inclines to the incident solar wind. It implies that a suitable control strategy should be employed to accomplish the thrust vectoring for the orbit maneuvering. The analysis provides an effective and robust way to design the E-sail in the mission planning phase.
An accurate and robust geometrically exact beam formulation (GEBF) is developed to simulate the dynamics of a beam with large deformations and large rotations. The undeformed configuration of the centroid line of the beam can be either straight or curved, and cross sections of the beam can be either uniform or nonuniform with arbitrary shapes. The beam is described by the position of the centroid line and a local frame of a cross section, and a rotation vector is used to characterize the rotation of the cross section. The elastic potential energy of the beam is derived using continuum mechanics with the small-strain assumption and linear constitutive relation, and a factor naturally arises in the elastic potential energy, which can resolve a drawback of the traditional GEBF. Shape functions of the position vector and rotation vector are carefully chosen, and numerical incompatibility due to independent discretization of the position vector and rotation vector is resolved, which can avoid the shear locking problem. Numerical singularity of the rotation vector with its norm equal to zero is eliminated by Taylor polynomials. A rescaling strategy is adopted to resolve the singularity problem with its norm equal to 2mp, where m is a nonzero integer. The current formulation can be used to handle linear and nonlinear dynamics of beams under arbitrary concentrated and distributed loads. Several benchmark problems are simulated using the current formulation to validate its accuracy, adaptiveness, and robustness.
The shape of a rotating electric solar wind sail under the centrifugal force and solar wind dynamic pressure is modeled to address the sail attitude maintenance and thrust vectoring. The sail rig assumes centrifugally stretched main tethers that extend radially outward from the spacecraft in the sail spin plane. Furthermore, the tips of the main tethers host remote units that are connected by auxiliary tethers at the sail rim. Here, we derive the equation of main tether shape and present both a numerical solution and an analytical approximation for the shape as parametrized both by the ratio of the electric sail force to the centrifugal force and the sail orientation with respect to the solar wind direction. The resulting shape is such that near the spacecraft, the roots of the main tethers form a cone, whereas towards the rim, this coning is flattened by the centrifugal force, and the sail is coplanar with the sail spin plane. Our approximation for the sail shape is parametrized only by the tether root coning angle and the main tether length. Using the approximate shape, we obtain the torque and thrust of the electric sail force applied to the sail. As a result, the amplitude of the tether voltage modulation required for the maintenance of the sail attitude is given as a torque-free solution. The amplitude is smaller than that previously obtained for a rigid single tether resembling a spherical pendulum. This implies that less thrusting margin is required for the maintenance of the sail attitude. For a given voltage modulation, the thrust vectoring is then considered in terms of the radial and transverse thrust components.
An electric sail is a new propulsion system uses the solar wind dynamic pressure as a thrust force. We analyzed the probability of the orbital control not changing sail plane angle but switching an electron gun connected with tethers. First, locally optimal switching laws are derived from Lagrange variational equations analytically. By theoretical calculations and numerical simulations, these switching laws are effective for changing some orbital elements. Moreover, mission applicability is studied. Assuming minimum time transfer problem between circular and coplanar orbits, optimal control laws are studied with direct approach. We conclude that the switching orbital control method is effective for exploration of other planets.
We produced a 1 km continuous piece of multifilament electric solar wind sail tether of μm-diameter aluminum wires using a custom made automatic tether factory. The tether comprising 90 704 bonds between 25 and 50 μm diameter wires is reeled onto a metal reel. The total mass of 1 km tether is 10 g. We reached a production rate of 70 m∕24 h and a quality level of 1[per thousand] loose bonds and 2[per thousand] rebonded ones. We thus demonstrated that production of long electric solar wind sail tethers is possible and practical.
We present a novel ultrasonic wire-to-wire bonding method for bonding two micrometer-thick metal
wires together. A special jig and an industrial wire bonder perform the bonding. This wire-to-wire bonding
is the core unit process to produce space tether for the Electric Solar Wind Sail. The proposed method
was validated experimentally with 38 bonds where a 25 um and a 50 um by diameter Al wires that were
first bonded together after which the bond was pull tested. The measured average pull force was
(74 ± 15) mN whereas the lowest pull force value was 40 mN. The results show that wire-to-wire bonds
of sufficient strength can be produced for the Electric Solar Wind Sail tether application. Tether manufacturing
was demonstrated with a separate test where a 1.4 m long tether was produced featuring more
than 100 wire-to-wire bonds.
In the first part of this work on high-performance solar sails (SSs),
heliocentric-orbiting SS dynamics are investigated within a corotating
reference frame (with arbitrary angular velocity) within which
stationary solutions out of the ecliptic plane are found. The period of
these 'halo-type' sun-centered orbits may be chosen to minimize sail
performance requirements. Complex trajectories are obtainable by the
joining of several halo orbits. In the second part, attention is given
to the equations of motion of a geocentric-orbiting SS in a corotating
reference frame corresponding to earth-centered halo-type orbits when
viewed from an inertial reference frame; from the joining of these,
also, complex trajectories may be formed.
Solar sail halo orbits. Part i - heliocentric case
- Mclnnes
Flight dynamics and control strategy of electric solar wind sails
- Li
- H D Curtis
H.D. Curtis, Orbital Mechanics for Engineering Students, third ed., Butterworth-Heinemann, Boston, 2014, pp. 405-457.
- S Boyd
S. Boyd, et al., Linear Matrix Inequalities in System and Control Theory, Society for
Industrial and Applied Mathematics, Philadelphia, 1994.