Dynamics And Control Of Multi-tethered Satellite Formations Near Libration Points

Spacecraft formation flying is the new development trend of space science and technology in 21st century, some new space missions could be implemented and the capability of human in exploring deep space could be enhanced by the formation flying near libration points because of their unique orbits, dynamic characteristic and space environment. Some space missions have been or will be performed near libration points of Sun-Earth system in the formation flying plans that belong to National Aeronautics and Space Administration (NASA) and European Space Agency (ESA). However, formation flying in the vicinity of libration points is challenging with its own unique problems, such as high power consumption to the formation station-keeping, the formation reconfiguration and the inherent instability of dynamics near the collinear libration points. To alleviate these concerns, several propellant-free propulsion techniques have been proposed, such as tethered systems. The variable and controllable baseline can be achieved by deploying or retracting the tethers. The potential range of applications for multi-tethered formations near libration points of Sun-Earth system is the greatest in space interferometry for deep space, which make use of the advantage of tethered satellite system and formation flying near libration points. Simultaneously, some new relevant research issues are proposed. But the researchs are just during primary stage, no one mission has been perfomed with this technology. Therefore, our nation may achieve and even ahead of international level in this research field.There have been some investigations of the dynamics and control of tethered satellite systems over the years. For almost all of these studies, it is assumed that the system consist two satellites and one tether. Some previous studies were accomplished on the dynamics and control of multi-tethered formations, but almost all of them focused on tethered systems in low Earth orbits (LEO), and it was assumed that there was no coupling between the attitude dynamics and the orbit dynamics. Comparatively few studies presented on the dynamics and control of tethered systems near libration points. Additionally, unlike LEO, the libration point orbits (LPO) are inherently unstable Non-Keplerian orbit, thus the existing conclusions with tethered systems in LEO can not be applied to that in LPO. Hence, in order to solve some outstanding problems in this research field and promote the development of dynamics and control of multi-tethered satellite formations near libration points, a new formulation for the analysis of the coupling dynamics is developed, and the controller is designed for the station-keeping in this paper. The main contents are as follows:(1) Based on the Hill's restricted three-body problem(HRTBP), a new formulation for the analysis of the coupling dynamics of a multi-tethered system near collinear libration points of Sun-Earth system is developed in the framework of the Lagrangian formulation. The system consists of a parent satellite and n subsatellites, connected together in a hub-spoke configuration via variable-length tethers. The resulted equations governing the dynamics of the multi-tethered formation can be used for the dynamics analysis and control design of static formations or dynamic formations, station-keeping with constant length or reconfiguration with variable length. Next, some insight can be achieved from equilibrium positions and an approximate analysis of the linearized equations for static formation with non-rotating tethers. The formulation and some results obtained in this paper are validated by comparing with the silimar work of relevant references.(2) For the tethered static formations in which the parent satellite is located very close to libration points, the numerical simulations using the full nonlinear and time-varying dynamic equations are performed to study the coupling dynamics during station-keeping and reconfiguration stage. First, the stability of tethered formations are compared with the untethered case with the same initial locations for each satellite. Then, the relevant parameters are varied to investigate their impact on the dynamic characteristics of system. The result shows that the tethered formation has the better stability than the untethered case and longer tether lengths and large mass ratios make the parent satellite move away from its equilibrium position earlier. However, the change of them has almost no impact on attitude dynamics of tethered formations, and the parent satellite will move away from its initial position for both deployment and retrieval, but the excursions during retrieval are larger than those during deployment at the same time. The attitude motions during deployment are basically stable, whereas they are unstable during retrieval.The excursion of parent satellite from its equilibrium position, the in-plane relative angle and angular velocities increase as the deployment rate increases, but the out-of-plane librations approach to the stable quasi-periodic motion for the faster deployment rate.(3) For the tethered dynamic formations in which the parent satellite is predefined on larger periodic Halo orbits, numerical simulations using the full nonlinear and time-varying dynamic equations are performed to study the coupling dynamics during station-keeping and reconfiguration stage. The results show that spin-stabilizing for the multi-tethered satellite formation on a large halo orbit is dynamically possible. The attitude stability is with very few effects of orbital amplitudes and directions, and the smaller orbit amplitudes of the parent satellite are with the better stability than the larger ones. The Southern Halo orbit is more helpful to make the system remain its planar configuration compared with the Northern Halo orbit. The faster in-plane spin rate is more helpful to make the attitude motions and the orbit motions remain stable.The orbit and attitude motions of the tethered formation with the small mass ratio are more stable than those with big mass ratio, but the change of tether lengths in an appropriate range has almost no impact on attitude dynamics of tethered formations. The parent satellite will move away from its nominal orbits for both deployment and retrieval, whereas the excursions during deployment are larger than those during retrieval at the same time. The attitude motions during deployment are unstable, while basically stable during retrieval. The slower tether length rate is helpful to make the smaller change range of the out-of-plane libration angles for the deployment with different tether length rate.(4) Based on the optimal control theory, a new method named "Approximating Sequence of Riccati Equations" (ASRE) is introduced to design nonlinear optimal tracking controller for the station keeping of periodic libration point orbits. The numerical approach based on interval mixed energy and precise integration method is applied to solve the ASRE precisely and efficiently. The numerical solutions of periodic libration point orbits of Sun-Earth system with high precision are chosen as nominal reference orbits. The simulations illustrating the effectiveness of the nonlinear tracking controller are presented. The control performance of the proposed method in this paper is compared with the LQR (Linear Quadratic Regulator) method with the same parameters. The results show that the advantages of the nonlinear controller, such as high tracking accuracy, low fuel cost and little computional time, in maintaining periodic libration point orbits. Meanwhile, the actual trajectory of satellite tracks the nominal orbit with good accuracy under the influence of initial injection errors, and the control forces is small and the total fuel cost of the control can be quite reasonable. Furthermore, a number of the problem parameters related to the controller are varied in order to demonstrate their effects on control performance, and some advice is given in choosing reasonable value.(5) Based on the above nonlinear optimal control method, the nonlinear tracking control of rotating tethered satellite formations are considered, in which the parent satellite follows larger Halo orbits centered about the second libration point of the Sun-Earth system. The simulations illustrating the effectiveness of the nonlinear tracking controller are performed. The results show that the actual trajectory of the parent satellite tracks the nominal orbit with good accuracy under the influence of initial injection errors and the perturbations of subsatellties, the control forces are reasonable, and the total fuel cost of the control is low.