Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

An ergodic algorithm for long-term coverage of elliptical orbits

This paper deals with the coverage analysis problem of elliptical orbits. An algorithm based on ergodic theory, for long-term coverage of elliptical orbits, is proposed. The differential form of the invariant measure is constructed via the perturbation on mean orbital elements resulted from the J2 term of non-spherical shape of the earth. A rigorous proof for this is then given. Different from the case of circular orbits, here the flow and its space of the dynamical system are defined on a physical space, and the real-value function is defined as the characteristic function on station mask. Therefore, the long-term coverage is reduced to a double integral via Birkhoff-Khinchin theorem. The numerical implementation indicates that the ergodic algorithm developed is available for a wide range of eccentricities.

Angles-only relative navigation in highly elliptical orbits

For angles-only relative navigation system only measures line-of-sight information,there are inherent problems in the ability to determine the range between Chaser and Target. Angles-only relative navigation is an attractive alternative for inspecting or rendezvous with noncooperative target,if adequate accuracy can be achieved. Angles-only relative navigation model considering J2 perturbation is presented for tracking and rendezvous with noncooperative target in highly elliptical orbit. Impulsive out-of-plane maneuvers of the Chaser are used to improve the navigation accuracy. The first impulse burns in cross-track directions to change the orbit inclination of the Chaser. The second impulse burns after one orbit period to change the orbit of the Chaser back. The simulation results show that the relative navigation system without maneuvers can't correct the initial state errors,while impulsive out-ofplane maneuvers of the Chaser improves the navigation accuracy. Angles-only relative navigation with chaser vehicle maneuvers to improve observability is effective when the spacecrafts are in highly elliptical orbits.

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.

Simulation for MSS-2 low-perigee elliptical orbit satellites:an example of lithospheric magnetic field modelling

A future constellation of at least four geomagnetic satellites(designated Macao Scientific Satellite-1(MSS-1)and Macao Scientific Satellite-2(MSS-2))was recently proposed,to continue high-quality geomagnetic observations in the post-Swarm period,focusing especially on collecting data that will provide a global,three-dimensional survey of the geomagnetic field.In this paper,we present a simulation of two years of orbits(2020.01.01-2022.01.01)of two satellites(tentatively denoted as MSS-2)that are constellated in elliptical(200×5,300 km)low-perigee orbits.By comparing error variances of Gauss coefficients,we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits,including a near circular reference orbit of 300×350km,and elliptical orbit of 180×5,300 km,220×5,300 km,200×3,000 km and 200×1,500 km.We find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km,data that will be useful in advancing the quality of lithospheric magnetic field modelling;this number of observations reflects the fact that only 4.5%of the flight time of these satellites will be below250 km(just 6.4%of their flight time below 300 km).By combining observations from the MSS-2 satellites’elliptical orbits of 200×5,300km with observations from a circular reference orbit,the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree n=200 and by a factor of 1,300 at n=250.The planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes,thus dramatically improving the spatial resolution of satellite-derived lithospheric field models,(up to 80%at n=150).In addition,lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes,thus improving the model predictions at all spherical harmonic degrees(around 52%-62%at n=150).The upper limit of the expected improvement to the field model at the orbital apogee is not as good as at the perigee.However,data from the MSS-1 orbit can help fill the gap between data from the MSS-2 orbits and from the circular reference orbit for the low-degree part of the model.The feasibility of even lower-altitude flight requires further discussion with satellite engineers.

Adaptive control for autonomous rendezvous of spacecraft on elliptical orbit

A strategy for spacecraft autonomous rendezvous on an elliptical orbit in situation of no orbit information is developed. Lawden equation is used to describe relative motion of two spacecraft. Then an adaptive gain factor is introduced, and an adaptive control law for auton- omous rendezvous on the elliptical orbit is designed using Lyapunov approach. The relative motion is proved to be ultimately bounded under this control law, and the final relative position error can achieve the expected magnitude. Simulation results indicate that the adaptive control law can realize autonomous rendezvous on the elliptical orbit with relative state information only.

Randomization-based algorithms for servicing spacecraft motion planning on elliptical orbit

Randomization-based motion planning algorithms are presented to solve problems of servicing spacecraft maneuvering in proximity to servicing targets on an elliptical orbit.The feasible trajectories of position and attitude for spacecraft are obtained by these algorithms under a variety of constraints.The state transition matrix is applied to computation of relative motion on elliptical orbits without performing numerical integration.The pseudo body coordinate system is built for identifying the planners on three coordinate axes with different functions.Finally,motion planning algorithm for translation and attitude taking account of the dependent variable (i.e.time) is used to obtain feasible trajectories.As the simulation examples indicate,the effectiveness of these methods is verified for relative motion while getting close to large structures,and the paper concludes with a detailed analysis of the results.

Elliptical formation control based on relative orbit elements

A new set of relative orbit elements（ROEs）is used to derive a new elliptical formation flying model.In-plane and out-of-plane motions can be completely decoupled,which benefts elliptical formation design.The inverse transformation of the state transition matrix is derived to study the relative orbit control strategy.Impulsive feedback control laws are developed for both in-plane and out-of-plane relative motions.Control of in-plane and out-of-plane relative motions can be completely decoupled using the ROE-based feedback control law.A tangential impulsive control method is proposed to study the relationship of fuel consumption and maneuvering positions.An optimal analytical along-track impulsive control strategy is then derived.Different typical orbit maneuvers,including formation establishment,reconfguration,long-distance maneuvers,and formation keeping,are taken as examples to demonstrate the performance of the proposed control laws.The effects of relative measurement errors are also considered to validate the high accuracy of the proposed control method.

Switching LPV control for electromagnetic formation flying on highly elliptical orbit

The electromagnetic force generated by the interaction of electromagnetic coils can be used to replace the conventional propellant consumption mode in close relative motion control,thereby promoting the application of formation flight technology for long-term and continuous space missions.Herein,a hysteresis-switching logic-based switching linear parameter varying(LPV)controller synthesis technique with guaranteed performance for electromagnetic formation flying on a highly elliptical orbit is proposed.First,considering that the relative dynamics model of an elliptical orbit is characterized by time-varying uncertainty,the LPV model is described.By introducing switching LPV controllers among different scheduled parameter subsets,conservativeness can be reduced.Second,the system modeling error,the uncertainty caused by a simplified electromagnetic coil model,and external disturbance are considered to derive switching LPV controller synthesis conditions based on the guaranteed H_(∞)performance.Derivation analysis shows that the proposed switching LPV controller not only ensures the robustness of the system against uncertainties,but also realizes the control input constraints.Finally,numerical simulations and comparative analyses are performed to demonstrate the effectiveness and advantages of the proposed control method.

Tangent-impulse transfer from elliptic orbit to an excess velocity vector

The two-body orbital transfer problem from an elliptic parking orbit to an excess veloc-ity vector with the tangent impulse is studied. The direction of the impulse is constrained to be aligned with the velocity vector, then speed changes are enough to nullify the relative velocity. First, if one tangent impulse is used, the transfer orbit is obtained by solving a single-variable function about the true anomaly of the initial orbit. For the initial circular orbit, the closed-form solution is derived. For the initial elliptic orbit, the discontinuous point is solved, then the initial true anomaly is obtained by a numerical iterative approach; moreover, an alternative method is proposed to avoid the singularity. There is only one solution for one-tangent-impulse escape trajectory. Then, based on the one-tangent-impulse solution, the minimum-energy multi-tangent-impulse escape trajectory is obtained by a numerical optimization algorithm, e.g., the genetic method. Finally, several examples are provided to validate the proposed method. The numerical results show that the minimum-energy multi-tangent-impulse escape trajectory is the same as the one-tangent-impulse trajectory.