The present status and prospects in the research of orbital dynamics and control near small celestial bodies

Small celestial body exploration is of great significance to deep space activities. The dynamics and control of orbits around small celestial bodies is of top priority in the exploration research. It includes the modeling of dynamics environment and the orbital dynamics mechanism. This paper introduced state-ofthe-art researches, major challenges, and future trends in this field. Three topics are mainly discussed： the gravitational field modeling of irregular-shaped small celestial bodies, natural orbital dynamics and control, and controlled orbital dynamics. Finally, constructive suggestions are made for China’s future space exploration missions.

A survey on orbital dynamics and navigation of asteroid missions

Asteroid exploration is currently one of the most concerned topics among international space agencies. Or- bital dynamics and navigation are obviously crucial for asteroid exploration. This paper aims to give a brief review on the dynamics, control and navigation of asteroid reconnaissance orbits, including the heliocentric transfer orbit and near as- teroid orbit. The developments in optimization techniques of the transfer segment are discussed in detail. We surveyed global researches in this field and made comments on several important progresses. The final section proposed a prospec- tive of future studies with emphasis on the key techniques of these issues in the asteroid exploration missions.

Spacecraft motion analysis about rapid rotating small body

The orbital dynamics equation of a spacecraft around an irregular sphere small body is established based on the small body’s gravitational potential approximated with a tri-axial ellipsoid. According to the Jacobi integral constant, the spacecraft zero-velocity curves in the vicinity of the small body is described and feasible motion region is analyzed. The limited condition and the periapsis radius corresponding to different eccentricity against impact surface are presented. The stability of direct and retrograde equator orbits is analyzed based on the perturbation solutions of mean orbit elements.

A Power Reduction Method for Pilot Channel of LEO Satellite Based on Dynamic Compensation

Since the lower power requirement of code division multiple access(CDMA) than that of other multiple access, the CDMA technology is suitable to be used in low earth orbit(LEO) satellite communication system whose space power is limited due to the small size of satellite. The pilot channel of CDMA technology is very important for earth mobile station(EMS) in LEO system to recover carrier and code, but the power requirement of pilot channel is very higher than that of other channels. In this paper, a power reduction method for pilot channel is proposed. By the new method, the power of pilot channel transmitted from LEO satellite is reduced to a lower level. For improving the signal to noise ratio(SNR) of pilot channel with lower power, coherent integration is employed in EMS at the pre-processing stage. Considering the high dynamic situation of LEO satellite, the long period of time for integration will deteriorate the receiving performance of EMS, therefore, a dynamic compensation module is added to carrier tracking loop against the high dynamic. Meanwhile, the transfer function of the new tracking loop and the condition for steadystate zero error are deduced. Numerical examples are provided to demonstrate effectiveness of the proposed approach.

Pseudospectral method for optimal propellantless rendezvous using geomagnetic Lorentz force

A charged spacecraft is subject to the Lorentz force when it orbits a central body with a magnetic field. The induced Lorentz force provides a new mean of propellantless electromagnetic propulsion for orbital control. Modeling the Earth magnetic field as a tilted dipole that co-rotates with the Earth, this paper develops a nonlinear dynamical model that describes the relative motion of the Lorentz spacecraft about an arbitrary reference orbit. Based on the proposed dynamical model, feasibility of Lorentz-propelled rendezvous with no restrictions on the initial states is investigated. The rendezvous problem is then formulated as an optimal control problem, and solved with the Gauss pseudospectral method（GPM）. Numerical simulations substantiate the validity of proposed model and method, and results show that the propellantless rendezvous is achieved at both fixed and free final time.

Fast and accurate adaptive collocation iteration method for orbit dynamic problems

For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.

Non-equatorial equilibrium points around an asteroid with gravitational orbit-attitude coupling perturbation

A recently proposed orbital dynamics model in the close proximity of an asteroid,which is called“attitude-restricted orbital dynamics”,includes the perturbation caused by the spacecraft’s gravitational orbit-attitude coupling.This orbital model improves the precision of classical point-mass orbital model with only the non-spherical gravity.Equatorial equilibrium points have been investigated in the previous paper.In this paper,the inplane non-equatorial equilibrium points,which are outside the asteroid’s equatorial plane but within its longitudinal principal plane,are further studied for a uniformly-rotating asteroid.These non-equatorial equilibrium points are more diverse than those in the classical point-mass orbital dynamics without gravitational orbit-attitude coupling perturbation(GOACP).Two families of them have been found.The equatorial equilibrium points studied before and the non-equatorial ones studied here give a complete map of equilibrium points in the asteroid’s principal planes.Compared with the classical point-mass orbital dynamics without GOACP,the equatorial equilibrium points have extended the longitude range of equilibrium points around an asteroid,while the non-equatorial ones studied here will extend the latitude range.These equatorial and non-equatorial equilibrium points provide natural hovering positions for the asteroid close-proximity operations.

Out-of-plane equilibrium points and invariant manifolds about an asteroid with gravitational orbit–attitude coupling perturbation

By considering the spacecraft as an extended,rigid body with a prior known attitude instead of a point mass,the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid,because it includes the perturbation caused by the gravitational orbit–attitude coupling of the spacecraft(GOACP).The GOACP is defined as the difference between the gravity acting on a non-spherical,extended body(the real case of a spacecraft)and the gravity acting on a point mass(the approximation of a spacecraft in classical orbital dynamics).Inplane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies,including equatorial and in-plane non-equatorial equilibrium points.In this study,out-of-plane equilibrium points outside the principal planes of the asteroid were examined.Out-ofplane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP.The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP.Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP,respectively,while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges.Additionally,the invariant manifolds of out-of-plane equilibrium points were calculated,and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds.In practice,these equilibrium points can provide natural hovering positions for operations in proximity to asteroids,and their invariant manifolds can be used for transfers to or from the equilibrium points.

Super twisting controller for on-orbit servicing to non-cooperative target

A relative position and attitude coupled controller is proposed for rendezvous and docking between two docking ports located in different spacecraft. It is concerned with servicing to a tumbling non-cooperative target spacecraft in arbitrary orbit subjected to external disturbances.By considering both kinematic and dynamical coupled effects of relative rotation on relative translation, a coupled dynamic model is established to represent the relative motion of docking port on target spacecraft with respect to another on the service spacecraft. The spacecraft control is based on the second order sliding mode algorithm of super twisting（ST）. It is schemed to manipulate the relative position and attitude synchronously. A formal proof of the finite time convergence property of the closed-loop system is derived theoretically by the second method of Lyapunov. Numerical simulations with the designed ST controller are presented to validate the analytic analysis by contrast with the twisting control algorithm. Simulation results demonstrate that the proposed relative position and attitude integrated controller is characterized by high precision, strong robustness and high reliability.