Continuous Low-thrust Propulsion Research Articles

NetSat—Challenges and lessons learned of a formation of 4 nano-satellites

The objective of the NetSat mission was to develop underlying technologies for small satellite formation flying and to demonstrate it at the level of four nano-satellites in three-dimensional configurations. Thus, base technologies, such as orbit determination, attitude determination and control, inter-satellite communication, and multi-satellite operations were studied and implemented on a 3U CubeSat platform. Technological challenges concerned in particular the precision 3-axis attitude control system in combination with an electric continuous low-thrust propulsion system. Different control approaches for efficient transitions between various formation topologies were analyzed, taking the limitations of the hardware platform (propellant mass, thrust level, power budget, attitude control capabilities) as well as external orbit perturbations into account and allowing for distributed control of multi-satellite formations. In a research program of over 15 years, the essential technologies for formations of small satellites were realized subsequently and then tested in orbit on multiple missions. On this basis, NetSat was launched on 28. September 2020 on a Soyuz rocket into a circular orbit of an altitude of 575km. During the operations phase several new operations concepts developed in the project, namely the Compass IoT network spanning all ground and satellite systems with more than 50 nodes, or a flexible operations frontend with auto-operations capabilities could be tested and allowed for operating such a multi-satellite network. Additionally, the implemented inter-satellite link among the satellites could be established and tested extensively, marking the first time that four 3U CubeSats were able to successfully demonstrate inter-satellite communication in orbit.

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.

Circular orbit transfer employing recursive nonlinear state transformations and nonlinear feedback

A recursive application of approximate feedback linearization is applied to low-Earth circular orbit rendezvous utilizing continuous low thrust propulsion. A family of solutions is established up to the $$\rho $$ -th degree utilizing the null space that appears at each step in the computations. A convergence analysis is presented to guarantee that the recursive process converges as $$\rho \rightarrow \infty $$ . During the recursive computations, patterns were detected as part of the solution process that illuminated a family of exact nonlinear control solutions obtained by judiciously selecting the null space coefficients. Examples of various analytic solutions are presented. The relationship between the $$\rho $$ -th degree exact solution obtained through the recursive approximations and a known exact solution is illustrated where it is shown that different exact solutions can have different performance in terms of fuel usage leading to the possibility of optimization considerations in selecting the desired exact solution.

Motion Planning on a Class of 6-D Lie Groups via a Covering Map

This paper presents an approach to motion planning for left-invariant kinematic systems defined on the six-dimensional frame bundles of symmetric spaces of constant cross-sectional curvature. A covering map is used to convert the original differential equation into two coupled equations each evolving on a three-dimensional Lie group. These lower dimensional systems lend themselves to a minimal global representation that avoids singularities associated with the use of exponential coordinates. Open-loop and closed-loop kinematic control problems are addressed to demonstrate the use of this mapping for analytical and numerical motion planning methods. The approach is applied to a spacecraft docking problem using two different types of actuation: 1) a fully actuated continuous low-thrust propulsion system; and 2) an underactuated single impulsive thruster and reaction wheel system.

Pursuer’s Control Strategy for Orbital Pursuit-Evasion-Defense Game with Continuous Low Thrust Propulsion

This paper studies the orbital pursuit-evasion-defense problem with the continuous low thrust propulsion. A control strategy for the pursuer is proposed based on the fuzzy comprehensive evaluation and the differential game. First, the system is described by the Lawden’s equations, and simplified by introducing the relative state variables and the zero effort miss (ZEM) variables. Then, the objective function of the pursuer is designed based on the fuzzy comprehensive evaluation, and the analytical necessary conditions for the optimal control strategy are presented. Finally, a hybrid method combining the multi-objective genetic algorithm and the multiple shooting method is proposed to obtain the solution of the orbital pursuit-evasion-defense problem. The simulation results show that the proposed control strategy can handle the orbital pursuit-evasion-defense problem effectively.

Orbital rendezvous performance comparison of differential geometric and ZEM/ZEV feedback guidance algorithms

In this paper, the performance of two distinct classes of feedback guidance algorithms is evaluated for a spacecraft rendezvous problem utilizing a continuous low-thrust propulsion system. They are the DG (Differential Geometric) and ZEM/ZEV (Zero-Effort-Miss/Zero- Effort-Velocity) feedback guidance algorithms. Even though these two guidance algorithms do not attempt to minimize the onboard fuel consumption or ΔV directly, the ΔV requirement is used as a measure of their orbital rendezvous performance for various initial conditions and a wide range of the rendezvous time (within less than one orbital period of the target vehicle). For the DG guidance, the effects of its guidance parameter and terminal time on the closed-loop performance are evaluated by numerical simulations. For the ZEM/ZEV guidance, its nearfuel- optimality is further demonstrated for a rapid, short-range orbital rendezvous, in comparison with the corresponding open-loop optimal solutions. Furthermore, the poor ΔV performance of the ZEM/ZEV guidance for a slow, long-range orbital rendezvous is remedied by simply adding an initial drift phase. The ZEM/ZEV feedback guidance algorithm and its appropriate variants are then shown to be a simple practical solution to a non-impulsive rendezvous problem, in comparison with the DG guidance as well as the open-loop optimal guidance.

Loosely-displaced geostationary orbits with hybrid sail propulsion

To overcome the congestion of geostationary orbit slots, previous work proposed to use vertically-displaced, non-Keplerian geostationary orbits by means of continuous low-thrust propulsion in the form of hybrid solar sail and solar electric propulsion (hybrid sail). This work extends and generalizes that concept by loosening the position constraint and introducing a station-keeping box. Sub-optimal orbits are first found with an inverse method that still satisfy the geostationary position constraint (i.e., no station-keeping box), which will be referred to as ideal displaced geostationary orbits. For these sub-optimal orbits, it is found that the hybrid sail saves propellant mass compared to the pure solar electric propulsion case: for solar sail lightness numbers of up to a value of 0.2 and the most favorable time during the year (i.e., at summer solstice), the hybrid sail saves up to 71.6% propellant mass during a single day compared to the use of pure solar electric propulsion. Subsequently, the sub-optimal orbits are used as a first-guess for a direct optimization algorithm based on Gauss pseudospectral transcription, which loosens the position constraint. This enables a more flexible trajectory around the ideal displaced geostationary orbit and lets the solar sail contribute more efficiently to the required acceleration. It therefore leads to a further propellant savings of up to 73.8%. Finally, the mass budget shows that by using by using far-term solar sail technology, the hybrid propulsion system enables an evident reduction in the required initial mass of the spacecraft for a given payload mass with a relatively long mission duration.

Continuous low thrust trajectory optimization for preliminary design

Mission design using continuous low thrust propulsion is an interesting but challenging task. In this paper, a virtual central gravitational field method is proposed for continuous low thrust trajectory design in preliminary stage. The basic idea of presented method is to form a virtual gravity using continuous thrust, in which the spacecraft flies in a virtual conic orbit to accomplish trajectory maneuver. Therefore, the continuous thrust non-Keplerian trajectory can be parameterized, and solved analytically. Instead of providing some special initial guesses for the more accurate optimizer, a large number of feasible initial guesses can be located quickly using the proposed method. Three examples are presented as the applications of the proposed technique to test its suitability, including the Earth to Mars rendezvous mission; the Earth–Mars–Ceres rendezvous mission; and a near-Earth collision-speed maximal interception mission. In order to verify the efficiency of the proposed method, the solutions generated by the shape-based method are used to compare. It shows that the virtual central gravitational field method can parameterize a large number of non-Keplerian orbits using parameters of virtual gravity, and provide initial guess for more accurate optimizer. Compared with the shape-based method, it requires more fuel cost using the proposed method in rendezvous trajectory design, especially in long flight time case. However, there is no constraint about the direction of thrust, so it is simple and flexible to change the velocity and position. The capability of rapid maneuver in short flight time outperforms the shape-based method discussed in prior papers, and it can be applied to providing initial guesses for continuous thrust rapid maneuver trajectory mission design in preliminary stage.

Extension of frozen orbits and Sun-synchronous orbits around terrestrial planets using continuous low-thrust propulsion

Frozen orbits and Sun-synchronous orbits are useful in exploration of terrestrial planets’ surface and atmosphere with a view to future human exploration. This work therefore develops novel orbits around terrestrial planets using continuous low-thrust propulsion to enable one new and unique investigations of the planets. This paper considers the use of continuous acceleration by solar electric propulsion, to achieve artificial frozen orbits and artificial Sun-synchronous orbits around terrestrial planets. These artificial orbits are similar to natural frozen orbits and Sun-synchronous orbits around Mercury, Venus, the Earth, and Mars, and the orbital parameters can be designed arbitrarily with the help of continuous low-thrust control. The control strategies to achieve the artificial orbits take into account $J_{2}$ , $J_{3}$ , and $J_{4}$ perturbations of terrestrial planets. It is proved that the control strategies minimize characteristic velocity. Relevant formulas are derived, and numerical results are presented. For the natural frozen orbits, the arguments of periapsis are about 270° for Mercury, Venus, and Mars, whereas about 90° for the Earth. By exerting both radial and transverse thrusts simulation shows that the control acceleration and characteristic velocity of the artificial frozen orbit around Mercury are the smallest among these plants. The characteristic velocity within one orbital period for Mercury is only 0.0089 m/s. The natural Sun-synchronous orbits exist around the Earth and Mars, but not around Mercury and Venus. By offsetting the perturbation acceleration in norm direction, the control acceleration and characteristic velocity of the artificial Sun-synchronous orbit around Mars are less than those of the others. The characteristic velocity within one orbital period is only 18.0885 m/s for the artificial Sun-synchronous orbit around Mars. The relationships between the control thrusts and the primary orbital parameters of the artificial orbits around other terrestrial planets are always similar to those around the Earth. Finally, one constellation of the artificial frozen orbit and the artificial Sun-synchronous orbit is designed by using the multiobjective evolutionary algorithm based on decomposition (MOEA/D), in which a Gaussian process is used to build a surrogate model in lieu of the expensive problem. Simulation shows that the control scheme effectively extends the orbital parameters’ selection ranges of the two types of artificial orbits around terrestrial planets, compared with the natural frozen orbit and Sun-synchronous orbit. The optimization result of the constellation orbits around Mars shows that the optimization framework is effective.

On the R4BP when third primary is an oblate spheroid

The present paper deals with the restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate body. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and m 2. We have studied the motion of m 4, moving under the influence of the three primaries m i , i=1,2,3, but the motion of the primaries is not being influenced by infinitesimal mass m 4. The aim of this study is to find the locations of equilibrium points and their stability. We obtain three collinear and five non-collinear equilibrium points. The collinear equilibrium points are unstable for all the mass parameter. The non-collinear points are stable for different mass parameter and oblateness factor. Also, we have considered the autonomous coplanar circular restricted four-body problem with the infinitesimal mass as a low-thrust spacecraft. Artificial equilibrium points are created with the use of continuous low-thrust propulsion. The obtained results show that, in absence of thrust there are unstable equilibrium points close to the third primary. Also, using the constant low-thrust, the artificial equilibrium points can be generated, which move from the natural equilibrium points. Further, it is proved that in certain region closed to the third primary, these points are stable. We have drawn the zero velocity surfaces to determine the possible allowed boundary regions. We observed that for increasing values of oblateness coefficient A, the corresponding possible boundary regions increase where the particle can freely move from one side to another side. Further, for different values of Jacobi constant C, we can find the boundary region where the particle can move in possible allowed partitions. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C.

Extension of Highly Elliptical Earth Orbits Using Continuous Low-Thrust Propulsion

Extension of highly elliptical orbits, with free selection of orbit period, using low-thrust propulsion is investigated. These newly proposed orbits, termed Taranis orbits, are enabled by existing low-thrust propulsion technology, offering a radically new set of tools for mission design and facilitating novel Earth Observation science. One example, using general and special perturbation techniques, is the application of continuous low-thrust to alter the "critical inclination" of an orbit from natural values of 63.4 or 116.6 deg, to any inclination required to optimally fulfil mission goals. Continuous acceleration is used to compensate for the drift in argument of perigee caused by Earth’s gravitational field. Pseudospectral optimization techniques are applied to the 90 deg inclination Taranis orbit, generating fuel-optimal control profiles, with a fuel saving of from general perturbation results. This orbit provides an alternative solution for high-latitude imaging from distances equivalent to geostationary orbits and enables continuous visibility of frigid and neighboring temperate regions using three spacecraft, enabling high-quality imaging prohibited using conventional orbits. Finally, order of magnitude mission lifetimes and a Strawman mass budget are developed, where mission lifetimes for spacecraft with initial mass of 1000, 1500, and 2500 kg are limited to 4.3, 6, and 7.4 years, respectively.

Optimal Spacecraft Formation Reconfiguration with Collision Avoidance Using Particle Swarm Optimization

This paper presents an energy-optimal trajectory planning method for spacecraft formation reconfiguration in deep space environment using continuous low-thrust propulsion system. First, we employ the Legendre pseudospectral method (LPM) to transform the optimal reconfiguration problem to a parameter optimization nonlinear programming (NLP) problem. Then, to avoid the computational complexity for calculating the gradient information caused by traditional optimization methods, we use particle swarm optimization (PSO) algorithm to solve the NLP problem. Meanwhile, in order to avoid the collision between any pair of Legendre-Gauss-Lobatto (LGL) points, we insert some test points in the region where collision may happen most likely. What’s more, the collision avoidance constraints are also checked at these test points. Finally, numerical simulation shows that the energy-optimal trajectories for spacecraft reconfiguration could be generated by the method we proposed in a relative short time, so that it could be adopted on-board for practical spacecraft formation problems. DOI: http://dx.doi.org/10.5755/j01.itc.41.2.854

Periodic Orbits with Low-Thrust Propulsion in the Restricted Three-body Problem

Periodic orbits around nonequilibrium points are generated systematically by using continuous low-thrust propulsion in the restricted three-body problem, with a mass ratio varying from 0 to 1/2. A continuous constant acceleration is applied to cancel the gravitational forces of two primary bodies and the centrifugal force at a nonequilibrium point, which is changed into an artificial equilibrium point. The equations of motion are linearized to analytically generate periodic orbits with constant acceleration. Then, periodic orbits around artificial equilibrium points which exist on the line connecting two primary bodies are investigated. The frequencies of these periodic motions are expressed by a parameter that is a function of the mass ratio and the position of the orbits around artificial equilibrium points. By choosing the frequencies of motions that are small-integer resonant, we have found the existence of points at which in-plane and out-of-plane motions are synchronized.

Instability of Fixed, Low-Thrust Drag Compensation

FORCED drag compensation using continuous low-thrust propulsion has been considered for satellites in low Earth orbit. This simple, but nonoptimal, scheme merely requires that the thrust vector is directed opposite to the drag vector and that the magnitude of the two are equal. In principle, the drag force acting on the spacecraft could be determined onboard using accurate accelerometers. However, for small, low-cost spacecraft such sensors may be unavailable.An alternative strategy would be to Ž x the thrust magnitude equal to the expected air drag that would be experienced by the spacecraft. The thrust levelwould be periodically updated based on ground-based orbit determination. In this Engineering Note, it is shown that such a forced circular orbit with a Ž fixed thrust level is exponentially unstable for all physically reasonable atmosphere models.

GEOSAIL: Exploring the Geomagnetic Tail Using a Small Solar Sail

Conventional geomagnetic tail missions require a spacecraft to be injected into a long elliptical orbit to explore the spatial structure of the geomagnetic tail. However, because the elliptical orbit is inertially fixed and the geomagnetic tail is directed along the sun-Earth line, the apse line of the elliptical orbit is precisely aligned with the geomagnetic tail only once every year. To artificially precess the apse line of the elliptical orbit in a sun-synchronous manner, which would keep the spacecraft in the geomagnetic tail during the entire year, would require continuous low-thrust propulsion or periodic impulses from a high-thrust propulsion system. Both of these options require reaction mass that will ultimately limit the mission lifetime. It is demonstrated that sun-synchronous apse-line precession can be achieved using only a small, low-cost solar sail. Because solar sails do not require reaction mass, a geomagnetic tail mission can be configured that provides a continuous science return by permanently stationing a science payload within the geomagnetic tail.

Optimal Low-Thrust Interception of Earth-Crossing Asteroids

The spectacular collision of the Shoemaker–Levy 9 asteroid with Jupiter in July 1994 was a dramatic reminder of the inevitability of such catastrophes in the Earth’s future unless steps are taken to develop methods for Earthapproaching object detection and possible interdiction. In this work, optimal (minimum-time) trajectories are determined for the interception of asteroids that pose a threat of collision with the Earth. An impulsive-thrust escape from the Earth is used initially to reduce  ight time but is followed with continuous low-thrust propulsion using values of thrust and speciŽ c impulse representative of electric motors. The continuous optimization problem is formulatedas a nonlinearprogrammingproblemusing the collocationmethod inwhich thedifferential equations of motion are included as nonlinear constraint equations. The use of low-thrust propulsion after Earth escape is shown to dramatically decrease the mass of the interceptor vehicle at launch.

Optimal in-plane orbital evasive maneuvers using continuous thrust propulsion

Introduction I N recent years, much research has been accomplished on optimal maneuvering in space using impulsive and continuous thrust to address problems such as orbit transfer, stationkeeping, and rendezvous. Very little research, however, has addressed the optimal intercept/avoidance problem, which relates directly to antisatellite, space reconnaissance, and collision-avoidance objectives. The optimal intercept/avoidance problem is becoming more relevant to spacecraft mission planning because of current and emerging capabilities to intercept and avoid interception. Where time permits, any target under intercept might be maneuvered to attempt to prevent a successful intercept. This option motivates this study of optimal evasive-maneuver strategies using a continuous low thrust propulsion system.