Comparison of three filters in asteroid-based autonomous navigation

At present,optical autonomous navigation has become a key technology in deep space exploration programs.Recent studies focus on the problem of orbit determination using autonomous navigation,and the choice of filter is one of the main issues.To prepare for a possible exploration mission to Mars,the primary emphasis of this paper is to evaluate the capability of three filters,the extended Kalman filter(EKF),unscented Kalman filter(UKF) and weighted least-squares(WLS) algorithm,which have different initial states during the cruise phase.One initial state is assumed to have high accuracy with the support of ground tracking when autonomous navigation is operating; for the other state,errors are set to be large without this support.In addition,the method of selecting asteroids that can be used for navigation from known lists of asteroids to form a sequence is also presented in this study.The simulation results show that WLS and UKF should be the first choice for optical autonomous navigation during the cruise phase to Mars.     

星载GPS自主定轨的长期稳定性

通过模拟长达100天的星载GPS伪距观测资料,进行卡尔曼滤波定轨仿真计算．重点研究：1)采用简化的动力学模型与简化状态转移矩阵,是否保证滤波的长期稳定性；2)模型误差矩阵Q的选取对滤波定轨精度的影响；3)与事后最小二乘批处理相比较,在简化模型下自主定轨的精度．同时给出了相应的结论．     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .     

姿态卡尔曼滤波对FAST馈源舱 BDS/SINS组合导航位姿精度的提高

500 m球面射电望远镜(The Five-hundred-meter Aperture Spherical radio Telescope, FAST)馈源舱位置和姿态测量精度直接影响望远镜接收机的对准精度.为了提高馈源舱位姿精度,提出姿态卡尔曼滤波算法,采用北斗导航系统的多天线技术解算馈源舱姿态,将它与捷联惯导解算的馈源舱姿态对比,两者的差值作为姿态卡尔曼滤波的量测量.为了降低北斗导航系统解算姿态的复杂度,采用了罗德里格矩阵解算方法.以望远镜跟踪观测模式时,北斗导航系统和捷联惯导解算的实际导航结果作为测试数据,测试结果表明采用姿态卡尔曼滤波的组合导航结果精度优于传统卡尔曼滤波的组合导航结果,尤其在航向角和yf轴的位置精度上.     

Lunar capture trajectories and homoclinic connections through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for many decades. This paper is concerned with a topological study of such transfers, with emphasis to trajectories that allow performing lunar capture and those that exhibit homoclinic connections, in the context of the circular restricted three-body problem. A fundamental theorem stated by Conley locates capture trajectories in the phase space and can be condensed in a sentence: “if a crossing asymptotic orbit exists then near any such there is a capture orbit”. In this work this fundamental theoretical assertion is used together with an original cylindrical isomorphic mapping of the phase space associated with the third body dynamics. For a given energy level, the stable and unstable invariant manifolds of the periodic Lyapunov orbit around the collinear interior Lagrange point are computed and represented in cylindrical coordinates as tubes that emanate from the transformed periodic orbit. These tubes exhibit complex geometrical features. Their intersections correspond to homoclinic orbits and determine the topological separation of long-term lunar capture orbits from short-duration capture trajectories. The isomorphic mapping is proven to allow a deep insight on the chaotic motion that characterizes the dynamics of the circular restricted three-body, and suggests an interesting interpretation, and together corroboration, of Conley’s assertion on the topological location of lunar capture orbits. Moreover, an alternative three-dimensional representation of the phase space is profitably employed to identify convenient lunar periodic orbits that can be entered with modest propellant consumption, starting from the Lyapunov orbit.     

Earth–Moon Weak Stability Boundaries in the restricted three and four body problem

This work deals with the structure of the lunar Weak Stability Boundaries (WSB) in the framework of the restricted three and four body problem. Geometry and properties of the escape trajectories have been studied by changing the spacecraft orbital parameters around the Moon. Results obtained using the algorithm definition of the WSB have been compared with an analytical approximation based on the value of the Jacobi constant. Planar and three-dimensional cases have been studied in both three and four body models and the effects on the WSB structure, due to the presence of the gravitational force of the Sun and the Moon orbital eccentricity, have been investigated. The study of the dynamical evolution of the spacecraft after lunar capture allowed us to find regions of the WSB corresponding to stable and safe orbits, that is orbits that will not impact onto lunar surface after capture. By using a bicircular four body model, then, it has been possible to study low-energy transfer trajectories and results are given in terms of eccentricity, pericenter altitude and inclination of the capture orbit. Equatorial and polar capture orbits have been compared and differences in terms of energy between these two kinds of orbits are shown. Finally, the knowledge of the WSB geometry permitted us to modify the design of the low-energy capture trajectories in order to reach stable capture, which allows orbit circularization using low-thrust propulsion systems.     

On the construction of low-energy cislunar and translunar transfers based on the libration points

There exist cislunar and translunar libration points near the Moon, which are referred to as the LL ₁ and LL ₂ points, respectively. They can generate the different types of low-energy trajectories transferring from Earth to Moon. The time-dependent analytic model including the gravitational forces from the Sun, Earth, and Moon is employed to investigate the energy-minimal and practical transfer trajectories. However, different from the circular restricted three-body problem, the equivalent gravitational equilibria are defined according to the geometry of the instantaneous Hill boundary due to the gravitational perturbation from the Sun. The relationship between the altitudes of periapsis and eccentricities is achieved from the Poincaré mapping for all the captured lunar trajectories, which presents the statistical feature of the fuel cost and captured orbital elements rather than generating a specified Moon-captured segment. The minimum energy required by the captured trajectory on a lunar circular orbit is deduced in the spatial bi-circular model. The idea is presented that the asymptotical behaviors of invariant manifolds approaching to/traveling from the libration points or halo orbits are destroyed by the solar perturbation. In fact, the energy-minimal cislunar transfer trajectory is acquired by transiting the LL ₁ point, while the energy-minimal translunar transfer trajectory is obtained by transiting the LL ₂ point. Finally, the transfer opportunities for the practical trajectories that have escaped from the Earth and have been captured by the Moon are yielded by the transiting halo orbits near the LL ₁ and LL ₂ points, which can be used to generate the whole of the trajectories.     

Direct and indirect capture of near-Earth asteroids in the Earth–Moon system

Near-Earth asteroids have attracted attention for both scientific and commercial mission applications. Due to the fact that the Earth–Moon \(\hbox {L}_{1}\) and \(\hbox {L}_{2}\) points are candidates for gateway stations for lunar exploration, and an ideal location for space science, capturing asteroids and inserting them into periodic orbits around these points is of significant interest for the future. In this paper, we define a new type of lunar asteroid capture, termed direct capture. In this capture strategy, the candidate asteroid leaves its heliocentric orbit after an initial impulse, with its dynamics modeled using the Sun–Earth–Moon restricted four-body problem until its insertion, with a second impulse, onto the \(\hbox {L}_{2}\) stable manifold in the Earth–Moon circular restricted three-body problem. A Lambert arc in the Sun-asteroid two-body problem is used as an initial guess and a differential corrector used to generate the transfer trajectory from the asteroid’s initial obit to the stable manifold associated with Earth–Moon \(\hbox {L}_{2}\) point. Results show that the direct asteroid capture strategy needs a shorter flight time compared to an indirect asteroid capture, which couples capture in the Sun–Earth circular restricted three-body problem and subsequent transfer to the Earth–Moon circular restricted three-body problem. Finally, the direct and indirect asteroid capture strategies are also applied to consider capture of asteroids at the triangular libration points in the Earth–Moon system.     

Origin of the moon by tidal capture and some geophysical consequences

Of the many proposed modes of origin of the Moon, some violate physical laws; many are in conflict with observations; all are improbable. Perhaps the least improbable - based on recent tidal theory calculations and on the interpretation of lunar rock data - is capture of the Moon as it passed near the Earth in adirect (prograde) orbit, shortly after the formation of Moon and Earth, about 4.5 billion years ago. (Capture of the Moon from an initiallyretrograde orbit which had been proposed some years ago, leads to physically unacceptable consequences.) The effects of capture on the Earth would have been cataclysmic, leading to intensive heating of its interior, to volcanism, and to the immediate formation of an atmosphere and hydrosphere. Thus capture of a Moon may have given rise to the unique properties of the Earth (in the Solar System) and to the early evolution of life, about 3.5 billion years ago.Presented at the NATO Advanced Study Institute on Lunar Studies in Patras, Greece, September, 1971.     

Design and analysis of Weak Stability Boundary trajectories to Moon

An algorithm is developed to find Weak Stability Boundary transfer trajectories to Moon in high fidelity force model using forward propagation. The trajectory starts from an Earth Parking Orbit (circular or elliptical). The algorithm varies the control parameters at Earth Parking Orbit and on the way to Moon to arrive at a ballistic capture trajectory at Moon. Forward propagation helps to satisfy launch vehicle’s maximum payload constraints. Using this algorithm, a number of test cases are evaluated and detailed analysis of capture orbits is presented.     

Sequential filter design for precision orbit determination and physical constant refinement

Earth-based spacecraft tracking data have historically been processed with classical least squares filtering techniques both for navigation purposes and for physical constant determination. The small, stochastic non-gravitational forces acting on the spacecraft are described to motivate the use of sequential estimation as an alternative to the least squares fitting procedures. The stochastic forces are investigated both in terms of their effect on the tracking data and their influence on estimation accuracy. A flexiible sequential filter design which leaves the existing trajectory, variational equations, data observable and partial computations undisturbed is described. A detailed filter design is presented that meets the precision demands and flexibility requirements of deep space navigation and scientific problems, one which provides a high degree of numerical integrity and numerical analysis capability, facilitates the efficient computation of multiple solutions, and makes few demands on the supporting computational structure.This paper presents the results of one phase of research carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by NASA.     

A Research on the Pulsar Timing Based on Kalman Filtering

The pulsar is a high-speed rotationing neutron star with a stable rotational period, being not disturbed and destroyed artificially, and can be taken as the reference quantity of the absolute time. In this article a kind of pulsar time service method based on the Kalman filtering algorithm is proposed, and the simulation analysis of the clock error control based on the Kalman filtering and of the effect of the pulsar catalogue error and the measuring accuracy of the pulsar time of arrival (TOA) on the accuracy of time service is made by taking a certain solar synchronous orbit as an example. The result shows that by utilizing this method the clock error of the satellite-borne clock can be effectively eliminated and its time-dependent increase is restrained, thereby solving the problem that the accuracy of the spacecraft-borne low cost clock can not meet the needs.     

Comparing analytical and numerical approaches to meteoroid orbit determination using Hayabusa telemetry

Fireball networks establish the trajectories of meteoritic material passing through Earth's atmosphere, from which they can derive pre‐entry orbits. Triangulated atmospheric trajectory data require different orbit determination methods to those applied to observational data beyond the Earth's sphere of influence, such as telescopic observations of asteroids. Currently, the vast majority of fireball networks determine and publish orbital data using an analytical approach, with little flexibility to include orbital perturbations. Here, we present a novel numerical technique for determining meteoroid orbits from fireball network data and compare it to previously established methods. The re‐entry of the Hayabusa spacecraft, with its known pre‐Earth orbit, provides a unique opportunity to perform this comparison as it was observed by fireball network cameras. As initial sightings of the Hayabusa spacecraft and capsule were made at different altitudes, we are able to quantify the atmosphere's influence on the determined pre‐Earth orbit. Considering these trajectories independently, we found the orbits determined by the novel numerical approach to align closer to JAXA's telemetry in both cases. Using simulations, we determine the atmospheric perturbation to become significant at ~90 km—higher than the first observations of typical meteorite dropping events. Using further simulations, we find the most substantial differences between techniques to occur at both low entry velocities and Moon passing trajectories. These regions of comparative divergence demonstrate the need for perturbation inclusion within the chosen orbit determination algorithm.     

星载GPS卡尔曼滤波定轨算法

利用卡尔曼滤波定轨算法，处理神舟4号星载GPS伪距实测资料，重点在于研究卡尔曼滤波中模型误差方差矩阵的选取准则，GPS信号中断或连续野值对递推滤波的影响，如何自主监控滤波的运行状态，即是正常还是趋于发散，目的在于评价该算法用于星上自主定轨长期平稳运行的可靠性．     

An orbit determination algorithm by means of the satellite-borne GPS data and Kalman filter

Using the Kalman filter algorithm, we have processed on-board GPS data of Shenzhou 4. The research focuses on three problems, namely, the selection criteria for the model error variance matrix of the Kalman filter, the effects of GPS signal interruption or runs of outliers on the recursive filtering, and the method to monitor the filter running status (normal or divergent). The aim is to evaluate the reliability of long-time stationary running of this algorithm used for on-board autonomous orbit determination     

A method of astronomical autonomous orbit and attitude determinations for satellites

A method of realtime autonomous orbit determination for earth satellites using the extended Kalman filtering is proposed. The observed quantities are: the satellite-sun direction vector measured by a sun sensor, the satellite-earth and satellite-moon direction vectors measured by an ultraviolet sensor, and the geocentric distance measured by a radar altimeter. At the same time the satellite attitude to the earth is also determined. Results of our simulation of the autonomous orbit determination show that the precision of the orbit determinations is better than 200 m. The effects of the sampling period, orbital inclination, orbital eccentricity and orbital altitude on the precision of orbit determination are analyzed and compared, and certain principles helpful for improving the precision of orbit determination are suggested.     

An Orbital Prediction Algorithm for LEO Satellites Based on Optical Observations of Short Arcs at Single Station

The forecasting technique of the target tracking based on the short arcs at single station is an important way to guarantee that high-precision photoelectric theodolites can normally track and capture the targets in unconventional environments. We construct the tracking prediction algorithm based on nonlinear filter, which can provide the guiding data for the closed loop tracking under normal circumstances. At the same time we also construct the target prediction algorithm based on the nonlinear transformation, without valid observational data, which can provide a track guidance for the theodolite and ensure that the targets will not be lost. It is demonstrated that the nonlinear filtering is more effective than the EKF (extended Kalman filter) in the tracking prediction algorithm of the short arcs at single station. The results indicate that the nonlinear filter designed in this paper can be used as the guiding algorithm for the optical tracking equipments. And its guiding accuracy is in the same order of magnitude of the theodolite's random measurement accuracy. When the systematic error of the equipments reaches 50″, the accuracy can achieve 20″ for predictions in 60 s. This still satisfies the requirement of the field of view of the tracking equipments.     

Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit

The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J ₂ perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J ₂-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.     

Lunar shape does not record a past eccentric orbit

The Moon has long been known to have an overall shape not consistent with expected past tidal forces. It has recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a past non-synchronous spin-orbit resonance. Here I show that the match between the lunar shape and the proposed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) spin and shape evolution scenarios also completely ignore the physics of the capture into such resonances, which require prior permanent deformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric, the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentricity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin-orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunar eccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impacts in the early history of the Moon would have frequently disrupted this putative resonant rotation, making the rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) unstable. I conclude that the present lunar shape cannot be used to support the hypothesis of an early eccentric lunar orbit.     

Application of the extended Kalman filter in satellite prediction

The extended Kalman filter is used in this paper to process single-station laser ranging data over a few revolutions to improve the satellite orbit. The aim is to provide accurate short-term predictions of the satellite position. The dynamical model includes the perturbations due to the Earth's oblateness, air drag, solar radiation pressure and the gravitational attractions of the Sun and the Moon.The proposed method is tested with simulated and real LAGEOS data. The results show that the above aim is achievable. Moreover, the computing program based on the present method can be realized on mini-computers.