Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits.Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris.This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories,considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow.Firstly,a set of mean-element orbital dynamic equations of a chaser(spacecraft)and a target(debris)are derived by using the orbital averaging technique,and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem.Subsequently,the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle,and a practical optimization procedure is adopted to find the specific initial costate variables,wherein the necessary conditions of the optimal solutions are all satisfied.Afterwards,the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics.A simple correction strategy about the initialization of the mean elements,specifically the differential mean true longitude,is suggested,which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements.Finally,numerical examples are presented,and specifically,the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems,demonstrating the effectiveness of the optimization method proposed in this article.

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 orΔV directly,theΔV 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 poorΔV 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.

Optimal maneuver strategy to improve the observability of angles-only rendezvous

This paper proposes an optimal maneuver strategy to improve the observability of angles-only rendezvous from the perspective of relative navigation.A set of dimensionless relative orbital elements(ROEs)is used to parameterize the relative motion,and the objective function of the observability of anglesonly navigation is established.An analytical solution of the optimal maneuver strategy to improve the observability of anglesonly navigation is obtained by means of numerical analysis.A set of dedicated semi-physical simulation system is built to test the performances of the proposed optimal maneuver strategy.Finally,the effectiveness of the method proposed in this paper is verified through the comparative analysis of the objective function of the observability of angles-only navigation and the performances of the angles-only navigation filter under different maneuver schemes.Compared with the cases without orbital maneuver,it is concluded that the tangential filtering accuracy with the optimal orbital maneuver at the terminal time is increased by 35%on average,and the radial and normal filtering accuracy is increased by 30%on average.

Fast calculation method for mission opportunities in orbital interception and rendezvous problems

This paper proposes a fast calculation method to solve all mission opportunities for orbital interception and orbital rendezvous under the impulse-magnitude constraint.Different from the existing search methods,the proposed method does not need to solve Lambert's problem in the whole process.Three cases are considered for either departure time or transfer time being free,or both being free.For fixed departure time,the feasible windows of transfer time are obtained by solving a single-variable nonlinear equation only of terminal true anomaly.Similarly,for fixed interception(or rendezvous)time,the feasible windows of departure time are obtained.For free departure time and free transfer time,all mission opportunities are obtained by using a onedimensional search strategy.The hyperbolic-transfer and the multiple-revolution cases are also analyzed.Numerical results show that the proposed method is superior to the typical pork-chop plot method and the two-dimensional launch window method in computational time.

Optimal four-impulse rendezvous between coplanar elliptical orbits

Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods proposed for fuel optimal orbital rendezvous, Lawden＇s primer vector theory is favored by many researchers with its clear physical concept and simplicity in solu- tion. Prussing has applied the primer vector optimization theory to minimum-fuel, multiple-impulse, time-fixed orbital ren- dezvous in a near circular orbit and achieved great success. Extending Prussing＇s work, this paper will employ the primer vec- tor theory to study trajectory optimization problems of arbitrary eccentricity elliptical orbit rendezvous. Based on linearized equations of relative motion on elliptical reference orbit （referred to as T-H equations）, the primer vector theory is used to deal with time-fixed multiple-impulse optimal rendezvous between two coplanar, coaxial elliptical orbits with arbitrary large ec- centricity. A parameter adjustment method is developed for the prime vector to satisfy the Lawden＇s necessary condition for the optimal solution. Finally, the optimal multiple-impulse rendezvous solution including the time, direction and magnitudes of the impulse is obtained by solving the two-point boundary value problem. The rendezvous error of the linearized equation is also analyzed. The simulation results confirmed the analyzed results that the rendezvous error is small for the small eccentric- ity case and is large for the higher eccentricity. For better rendezvous accuracy of high eccentricity orbits, a combined method of multiplier penalty function with the simplex search method is used for local optimization. The simplex search method is sensitive to the initial values of optimization variables, but the simulation results show that initial values with the primer vector theory, and the local optimization algorithm can improve the rendezvous accuracy effectively with fast convergence, because the optimal results obtained by the primer vector theory are already very close to the actual optimal solution.