Angles-Only Initial Relative Orbit Determination Performance Analysis using Cylindrical Coordinates

Initial orbit determination (IOD), as a basis for initial orbit association and accurate orbit determination (OD), has a crucial role in the process of obtaining space debris orbit information. Among the traditional methods, the Gooding method has better convergence and stability. In this study, the Gooding method is enhanced to solve the issues discovered. A novel initial orbit determination (IOD) method is developed using the proposed improvement measures of the single-parameter initial value determination (SIVD) method, the fitted-curve noise suppression method, the restricted corrective value solution method, the removal of trivial solutions, etc. The experimental results verify the effectiveness of the improved method. The success rate of the initial orbit determination reached 99%, and the accuracy of the solved orbit parameters was significantly improved, especially the semi-major axis (SMA) error of less than 50 km, accounting for 88% of the total. It can be seen that the method meets the demand of space-based space debris cataloging for initial orbit association and can serve in the field of space situational awareness, which has important practical significance and application potential.

An investigation has been initiated to examine the concept of determining an initial orbit using angles-only observations over a short arc from a single space based sensor. The methods of Laplace and Gauss for angles-only initial orbit determination were the methods of choice for this study. To apply these methods, one has to solve a system of non-linear equations which have multiple solutions. Typical solution schemes recommend using an iterative solution technique with a first guess to start. However, since only one solution will be found using such a technique, the success of this approach will depend entirely upon how good (or bad) the first guess is. We have found that this technique is not satisfactory for performing initial orbit determination over short arcs of data. In this paper we will discuss past methods used in solving these non-linear equations and then we will present our approach which involves finding all the necessary solutions and determining which solution is correct.

The Padé code has been developed to treat hydrodynamic turbulence in protoplanetary disks. It solves the compressible equations of motion in cylindrical coordinates. Derivatives are computed using nondiffusive and conservative fourth-order Padé differencing, which has higher resolving power compared to both dissipative shock-capturing schemes used in most astrophysics codes, as well as nondiffusive central finite-difference schemes of the same order. The fourth-order Runge–Kutta method is used for time stepping. A previously reported error-corrected Fargo approach is used to reduce the time step constraint imposed by rapid Keplerian advection. Artificial bulk viscosity is used when shock capturing is required. Tests for correctness and scaling with respect to the number of processors are presented. Finally, efforts to improve efficiency and accuracy are suggested.

In this paper, optical target monitoring orbit determination using the Gooding algorithm based on the 400 km altitude space platform was researched. The measurement error was set to 3″ and 6″ respectively for 800 km, 1500 km altitude low Earth orbits and 36 000 km altitude geosynchronous orbit to determine the initial orbit and precise orbit. Simulation results show that the initial orbit determination accuracy of the 4~15 min arc is about 10 km, and the 1~2 min arc is about 100 km when the measurement data error is between 3″ and 6″. The error of 15 min initial determination arc is 100 m. As the arc is less than 10 min, the improved error accuracy of the orbit is in the order of km scale. As the measurement data error is 3″, the initial orbit determination accuracy of the 15~20 min arc is about tens of kilometers, and that of the 8~10 min arc is 100 km. The improved orbit has an error of km. When the measurement data error is 6″, the accuracy of the initial orbit determination of the 20 min arc is at the magnitude of 10 km, the accuracy of the initial orbit determination of the 8~15 min arc is at the magnitude of 100 km, and the error accuracy of the improved orbit is at the magnitude of 10 km.

This paper extends Hodges–Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. Copyright © 2007 John Wiley & Sons, Ltd.

T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.

Space-based initial orbit determination (IOD) is crucial for space situational awareness. However, classical algorithms often fail to converge or become trapped in trivial solutions when applied to very short arc scenarios with angles-only measurements. To overcome this challenge, this paper proposes a simple and fast algorithm based on the online sequential extreme learning machine (OSELM). First, the dynamic model and angles-only measurement model of the space target were constructed. Then, an OSELM-based orbit determination algorithm is designed, including the generation of the training dataset, data preprocessing, and network structure design, which focuses on establishing a nonlinear mapping model from angles-only measurements to the orbital state of space targets using the OSELM. Finally, numerical simulations for low Earth orbit (LEO) scenarios demonstrate the algorithm's high accuracy in estimating the target initial state, along with strong noise resistance and generalization capability.

이 논문의 목적은 이온층의 대표적인 교란현상인 Equatorial Spread F(ESF)가 GPS 관측 값을 이용한 저궤도위성의 상대궤도결정에 미치는 영향을 분석하는 것이다. 궤도결정은 우주기반의 Differential GPS(DGPS)방법을 이용했으며, 궤도결정 성능의 분석을 위해 ESF가 발생하지 않는 시점인 2004년 10월 한 달간에 대해 GRACE 두 위성의 상대궤도를 결정하였다. 궤도결정 정밀도를 검증하기 위해 상대궤도 정보로부터 계산된 기선거리를 K-Band Ranging 관측 값과 비교하였고, 그 결과 기선거리의 평균오차는 약 2~3mm(<TEX>$1{\sigma}$</TEX>)였다. 이 결과를 바탕으로 하여 ESF가 우주기반 DGPS 자료처리에 미치는 영향을 분석하기 위해 ESF가 자주 발생한 2003년과 거의 발생하지 않은 2004년에 대해 2주씩 자료처리를 수행하였다. 그 결과 2003년의 2주간 평균 기선 거리 오차는 약 15mm로 2004년의 2주간 평균 3.5mm보다 상대적으로 켰으며, K-Band Ranging 센서를 통해 그 당시의 ESF로 보이는 이온층의 교란현상이 상대적으로 많이 발생하였음을 확인 하였다. 이 연구를 통해 ESF가 자주 발생할 것으로 예상되는 태양활동 극대기의 경우 우주기반 DGPS 자료처리 시 ESF의 영향을 고려해야 하며, 이를 극복하기 위한 방안이 필요함을 확인하였다. In this paper, relative orbit of Low Earth Orbit satellites is determined using only GPS measurements and the effects of Equatorial Spread-F (ESF), that is one of biggest ionospheric irregularities, are investigated. First, relative orbit determiation process is constructed based on doubly differenced GPS observations. In order to see orbit determination performance, relative orbit of two GRACE satellites is estimated for one month in 2004 when no ESF is observed. The root mean square of the achieved baselines compared with that from K-Band Ranger sensor is about 2~3 mm and average of 95% of ambiguities are resolved. Based on this performance, the relative orbit is estimated for two weeks of two difference years, 2003 when there are lots of ESF occurred, and 2004 when only few ESF occurred. For 2003, the averaged baseline error over two weeks is about 15 mm. That is about 4 times larger than the case of 2004 (3.6 mm). Ionospheric status achieved from K-Band Ranging sensor also shows that more Equatorial Spread-F occurred at 2003 than 2004. Investigation on raw observations and screening process revealed that the ionospheric irregualarities caused by Equatorial Spread-F gave significant effects on GPS signal like signal loss or enhancement ionospheric error, From this study, relative orbit determination using GPS observations should consider the effect of Equatorial Spread-F and adjust orbit determination strategy, especially at the time of solar maximum.

Images are commonly used for spacecraft navigation. When close to the observed body it is often advantageous to perform such image-based navigation with observations of landmarks. This work demonstrates that it is possible to solve the initial orbit determination (IOD) problem using only observations of unknown landmarks if the spacecraft is assumed to be orbiting an ellipsoidal central body of known shape. Further, the solution developed here is noniterative and is based purely on geometric arguments. Example autonomous IOD results using only unknown landmarks are shown for a spacecraft in a circular Mars orbit and an elliptical Earth orbit.

Micro/nano satellites have become the protagonists in most formation flying missions currently, there is an urgent demand to find an approach to quickly calculate multiple-impulsive scheme on board for relative orbital transfer, while considering the optimality of total impulsive consumption. This paper presents an optimal control approach based on multiple impulses for in-plane satellite formation reconfiguration in near circular orbit. A new type of relative orbit motion equation is established based on the initial small deviation in the cylindrical coordinates system in this paper, which is suitable to calculate the solution of relative orbital maneuver by means of multiple impulses. The formation reconfiguration problem is investigated on account of the relative orbit motion, and the orbital motion equations with initial deviations are presented. The formation reconfiguration problem in orbital plane is discussed and analyzed selectively. An optimal three-impulsive approach for relative orbital transfer in plane is obtained by analysis solution based on graphical and numerical way, which is achieved by the combination method from four-impulsive solution. And the method is proved to be effectivity and optimality.

I have constructed and demonstrated a new fast-running algorithm to perform refined orbit determination for any spacecraft in GEO (geosynchronous Earth orbit), without user requirement for an a priori orbit estimate. We now have a refined method for GEO initial orbit determination (IOD). This is enabled by the use of equinoctial orbit elements, by a one-dimensional search in the equinoctial orbit element mean argument of orbit longitude, by the use of sensor locations to reduce the one-dimensional search, and by convergence boundaries of the nonlinear least squares (LS) algorithm. I shall refer herein to the new algorithm as LS_GEO_IOD. Algorithm LS_GEO_IOD uses any desired perturbative acceleration model for numerical trajectory integration, and its success appears to be independent of input measurement type.

With the recent flurry of research on satellite formation flying, a need has become apparent for a set of linearized equations of relative motion that capture the effect of the J2 geopotential disturbance force. Typically, Hill's linearized equations of relative motion have been used for this analysis, but they fail to capture the effect of the J2 disturbance force on a satellite cluster. In this paper, a new set of constant coefficient, linearized differential equations of motion is derived. These equations are similar in form to Hill's equations, but they capture the effects of the J2 disturbance force. A numerical simulator is employed to check the fidelity of the equations. It is shown that with the appropriate initial conditions, the new lin earized equations of motion have periodic errors (on the order of centimeters) that do not grow in time. The new linearized equations of motion also allow for insight into the effects of the J2 disturbance on a satellite cluster. This includes 'tumbling', the period of the relative orbit, and satellite separation due to differential J2 effects. Overall, a new high fidelity set of linearized equations are produced that are well suited to model satellite relative motion in the presence of the J2 disturbance force.

Initial orbit determination is highly enabling for operational awareness and safety in space. Reliable orbit determination in non-Keplerian systems, such as multi-body orbital mechanics, is challenging due to chaotic and unpredictable motion of these systems over long time scales. Typical approaches to non-Keplerian orbit determination use explicit integration with numerical shooting methods to propagate candidate trajectories. These methods often break down for measurement gaps on the order of days or even hours. This work presents a novel algorithm for non-Keplerian orbit determination using sparse grid collocation methods. The approach provides many benefits, including a large region of convergence without a close initial guess, as well as increased stability and performance over long observing time scales. A further advancement is made for estimating periodic orbits, in which a repeating orbit and its natural period are determined simultaneously from angles-only measurements. Orbit determination using collocation may significantly reduce sensor tasking requirements and enhance capabilities in complex orbital environments such as cislunar space.

The paper presents a method for solving the problem of initial orbit determination of an unknown resident space object based on a limited number of angular measurements. The angular measurements were obtained from a matched detection of the unknown object with an estimation of the velocity magnitude in the frame plane within two pointing regions of the optical system. Based on the estimated velocity magnitude, two position radius vectors were calculated, which were subsequently used to determine the velocity vector and orbital parameters of the unknown space object. The proposed method allowed the unambiguous computation of the orbital parameters of the Indefix/Ariane 42P satellite (NORAD: 27422) with an angular precision of up to 0.7 deg. in the inclination, longitude of the ascending node, and argument of latitude determination and an error in the determination of the semi-major axis of less than 27.5 km.

Linear Equations of Relative Motion