Initial Orbit Determination Research Articles

Robust Initial Satellite Orbit Determination method using a Modified Kalman Filter

The control segment in satellite navigation systems is responsible for estimating satellite orbit and clock bias which is required for a reliable Position, Navigation and Timing (PNT) user service. Initial orbit determination is a crucial step which accounts for all unknowns/anomalous parameters such as satellite orbit manoeuvres, on board and receiver clock frequency variations and environmental effects. It is vital that the estimates of the orbits and clock are insensitive to these factors. In this paper, an initial orbit determination method is presented using existing robust methodology for estimation of initial satellite position and its propagation using a variant of the Kalman Filter (KF) which allows the initial position determination process to be independent of satellite and receiver anomalies. The derivation of this KF variant is presented. Preliminary results obtained from simulated data are shown. The said method is checked for robustness by comparing results obtained for a given satellite position data set with that from the conventional Kalman Filter. The conventional KF exhibits divergence due to anomalies which are eliminated by the use of the method presented in this paper.

A space surveillance satellite for cataloging high-altitude small debris

This paper proposes a space surveillance satellite whose mission is to detect, catalog and maintain space debris as small as 10 cm in the Geosynchronous Earth Orbit (GEO) region. To fulfill the objective, the satellite is designed to be placed on a zero-inclination orbit of 4163 km in altitude. The payload is a large optical telescope with a 1.5 m Entrance Pupil Diameter (EPD) and a 20° × 20° Field of View (FoV). Based on these configurations, a 60-day simulation experiment is organized to evaluate the debris detecting and cataloging performances of the satellite. The simulation results show that the satellite is capable of detecting small debris higher than 4163 km. About 94.7% of 10 cm debris in the GEO region are detected more than 60 times over the 60-day simulation span by the satellite. In addition, the rules-based cataloging and maintaining capability of the satellite is close to 50.0% with regard to an entire 10 cm reference population of 3267 debris. For larger size debris, the results are more optimistic. This paper also assesses the performances of our Initial Orbit Determination (IOD) and Un-Correlated Track (UCT) association methods for 74 debris in the GEO region. The IOD success rate is 97.4% by applying a range-search based method. The True Positive (TP) association rate is 89.7% and the False Positive (FP) rate is 1.8%. Accordingly, 66 of the 74 space objects are correctly identified that each has at least two detection arcs, making accurate orbit determination possible.

相机阵列在空间目标初轨确定中的应用

相机阵列在空间目标初轨确定中的应用

一种适用于相机阵列的空间目标初轨确定方法

一种适用于相机阵列的空间目标初轨确定方法

Initial orbit determination using Doppler shift of Fraunhofer lines

Techniques for tracking objects in low Earth orbit include line-of-sight angle measurements and range measurements using RADAR or laser reflection. However, active ranging techniques are less effective for higher orbits because of the greater signal loss from distant targets. In this paper, using computer simulation, we show how incorporating measurements of the Doppler shift of Solar Fraunhofer lines can improve initial orbit determination. Results indicate that for Doppler measurement errors of less than 5 m/s, prediction accuracies for some orbital configurations can exceed Gooding’s angles-only method, while using only two observations of the target.

Maneuver-free approach to range-only initial relative orbit determination for spacecraft proximity operations

Range-only relative orbit determination for spacecraft proximity operations suffers from a well-known mirror solution problem in the context of Clohessy-Wiltshire dynamics during coasting flight. One approach proposed in previous work is to perform specific orbital maneuvers so as to avoid ambiguous relative state estimates. Alternatively, if the range-sensor offset from the spacecraft center-of-mass (COM) is considered, the relative orbit may be determined by using range-only measurements. This research developed a maneuver-free analytic solution to the range-only initial relative orbit determination (IROD) problem for close-in proximity operations by utilizing the range-sensor offsetting for enhanced observability to exclude mirror solutions. As a result, the initial relative orbit determination is reduced to a problem of solving linear equations. Based on these equations, the relative state observability is explored and observable conditions with respect to the range-sensor offset are obtained. The uncertainty of the relative orbit estimation is also derived, given as approximate analytic mean and covariance solutions. Overall, it has been strictly theoretically proven the range-only problem of non-periodic coasting close-in operations can be analytically solved. All these theoretical results are verified by a set of numerical simulation examples.

Autonomous Initial Orbit Determination with Optical Observations of Unknown Planetary Landmarks

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.

A combination method using evolutionary algorithms in initial orbit determination for too short arc

With the combination of two evolutionary algorithms EDA and DE, a new method of initial orbit determination for satellites based on ground-based too-short-arc is established. Compared with other algorithms, the proposed method focuses on the most densely populated region in the solution space rather than the individual with best fitness value. Both the global information and local information are well fused in the search of optimum. In the method (a,e,M) are treated as variables of the optimization, and the optimization procedure is carried out as a two-stage hierarchical optimization problem which has three variables for each stage. Kernel density estimation is applied to build the probability distribution model without any assumptions of the specified distribution, accompanied by handling semi-major axis and eccentricity as a pair of dependent variables in the construction of the probability for the correlation between them in the practice. Numerical experiments with real ground-based observations show that the proposed method is applicable to too-short-arc with even 3 s, and the result of bias in several kilometers can be achieved with 5″ error added to angular measurements.

Improved tracklet association for space objects using short-arc optical measurements

Initial orbit determination (IOD) for space objects is challenging, especially in the case where only optical observations, i.e. angles-only observations, are available and the optical observing arcs are very short (i.e. the too-short arc (TSA) problem). One approach to address the TSA problem is to associate several short-arc tracklets to targets across varying time intervals. In order to achieve better association and run-time performance, this study proposes an improvement to the traditional initial value problem (IVP) solution that determines the association by searching for the global minimum of a new loss function defined in a nonsingular canonical space. The improved IVP method was validated using optical data of space objects at different altitudes collected from the Mount Stromlo Observatory and compared with traditional IVP and another popular tracklet association method: the boundary value problem (BVP) approach. Results illustrate that the improved IVP method is superior to IVP and BVP in terms of association performance, and it also achieves good run-time performance. In addition, traditional methods suffer the drawback of incorrectly associating tracklets from different objects in the same constellation. A new approach dubbed the common ellipse method is presented to address this issue. The common ellipse method is tested with 86 Iridium constellation tracklets, and results show that it significantly improves the true negative rate for the tested scenario.

Initial orbit determination of BDS-3 satellites based on new code signals

For the two newly launched satellites (PRN number 27 and 28) of the future global BeiDou navigation satellite system (BDS-3), there is no available broadcast ephemeris data and other initial orbit information, but the initial orbit is the fundamental of the comprehensive analysis of the satellites and their signals. Precise orbit determination (POD) also requires determination of a priori initial value with a certain precision in order to avoid problems such as filter divergence during POD. Compared with the Newton iteration method, which relies on the initial value, this study utilizes the Bancroft algorithm to directly solve the nonlinear equations with the advantage of numerical stability. The initial orbits of these two satellites are calculated based on new code signals, and their results are analyzed and discussed. The experimental results show that, with the exception of very few epochs, when the new code signal is utilized, the median and robust variance factor of the observed residuals computed using pseudo-range observations and the solved initial orbits are less than 4 and 2 m, respectively. It also shows that this solution can be used for rapid initial orbit recovery after maneuvers of the new BeiDou satellites.

Non-iterative angles-only initial relative orbit determination with [formula omitted] perturbations

An approximate solution to the angles-only initial relative orbit determination problem is developed and evaluated in the context of the two-body problem with J2 perturbations. The algorithm is non-iterative and requires the singular value decomposition of a 6 × 6 matrix and the solution of a fifteenth-order polynomial. The performance of the algorithm is investigated for non-circular low-Earth orbits and near-circular geostationary orbits with and without measurement error.

A geometrical approach to association of space-based very short-arc LEO tracks

Conventional methods of association between uncorrelated tracks (UCTs) require knowledge of UCT covariance and their accurate propagations, which are difficult to achieve in some cases. For space-based angles-only very short-arc (SBVSA) LEO tracks, errors of initial tracks determined through the initial orbit determination (IOD) could be very large, for example, the semi-major axes (SMA) could have errors exceeding 100 km. Therefore, the conventional methods may fail if two tracks are apart by tens of orbital periods because of the need to correctly propagate the errors over the time span. In this paper, a geometrical approach to associating the SBVSA LEO tracks is proposed. This approach uses an analytical orbital propagator to propagate two UCTs to the middle of the time interval, and then adjust the SMA of one UCT to minimise position difference between the two UCTs at the middle point. After several iterations, an acceptable along track bias is obtained and a judgement on whether the two UCTs are associated is made. Preliminary simulations show that, the true positive rate is about 90% for LEO near-circular UCT associations. Furthermore, the geometrical method is tested on the non-circular LEO track association problem, the true positive rate is about 80%. A comparison with the Covariance Based Track Association (CBTA) method is also presented in the paper.

Angles-only initial relative orbit determination algorithm for non-cooperative spacecraft proximity operations

This research furthers the development of a closed-form solution to the angles-only initial relative orbit determination problem for non-cooperative target close-in proximity operations when the camera offset from the vehicle center-of-mass allows for range observability. In previous work, the solution to this problem had been shown to be non-global optimal in the sense of least square and had only been discussed in the context of Clohessy-Wiltshire. In this paper, the emphasis is placed on developing a more compact and improved solution to the problem by using state augmentation least square method in the context of the Clohessy-Wiltshire and Tschauner-Hempel dynamics, derivation of corresponding error covariance, and performance analysis for typical rendezvous missions. A two-body Monte Carlo simulation system is used to evaluate the performance of the solution. The sensitivity of the solution accuracy to camera offset, observation period, and the number of observations are presented and discussed.

Initial Orbit Determination Using Very Short Arc Data Based on Double-Station Observation

A method of initial orbit determination (IOD) of a space object, which uses very short arc (VSA) data of double-station observations, is presented. Closed-form solutions of the initial orbit are derived, and these solutions utilize radial measurements (the range, velocity, and acceleration) from two radar stations. The performance of the method based on the geometric dilution of precision (GDOP) metric is analyzed in detail. The analytical approximation expressions of the GDOP metric are also derived, and these expressions can serve as guidance for choosing an observation arc in order to obtain better IOD performance. Comparison of the analytical results with Monte Carlo simulations shows good consistency. Theoretical analysis and simulation results show that several seconds of VSA data can produce an IOD accuracy of several kilometers in position and several meters per second in velocity, provided that highly accurate measurements of phase-derived velocity and acceleration are available.

Estimation of Distribution Algorithm for Initial Orbit Determination of Too-short-arc Based on Kernel Density Estimation

A new approach of the initial orbit determination for too-short-arc with angular measurements is implemented by building the probabilistic model in the solution space with the estimation of distribution algorithm. Without any assumption about distribution, the non-parametric kernel density estimation is employed in the model building. The method, unlike other evolutionary algorithms, such as the genetic algorithm and particle swarm optimization, considers the fitness as well as the characteristics of the solution space. Numerical experiments with real observations indicate that without any constraints, the proposed technique has a good performance for the observations of usual accuracy.

Joint Probabilistic Data Association and Smoothing Applied to Multiple Space Object Tracking

A foundational aspect of space domain awareness is the ability to identify and track space objects, including space object discovery and custody. This paper demonstrates the power of combining an efficient multiple hypothesis joint probabilistic data association (MH-JPDA) algorithm with a fixed-interval smoother to simultaneously track multiple space objects. For newly discovered objects, statistical initial orbit determination (SIOD) is possible with a single short optical tracklet, but results in large initial uncertainties. Combining these uncertainties with closely spaced objects can result in highly ambiguous data associations, which can lead to poor state estimates and even filter divergence. This paper invokes MH-JPDA to probabilistically update multiple tracks with multiple simultaneous observations in a sequential filter, while avoiding assigning one-to-one associations. Once sufficient information has been collected, the space objects become uniquely distinguishable among each other. Subsequently, the smoother is applied to achieve improved association of the prior observations. MH-JPDA allows for immediate track formation (using SIOD) and sequential processing of incoming observations, providing statistically rigorous real-time state estimates, whereas smoothing produces a more-refined, higher-confidence overall track estimate at user-defined intervals. This paper demonstrates this approach within the Constrained Admissible Region, Multiple Hypothesis Filter (CAR-MHF) software by tracking a simulated break-up scenario.

Direct Uncertainty Estimates from Angles-Only Initial Orbit Determination

Initial orbit determination methods yield only a point solution, with no information about the quality of or confidence in the solution. However, probabilistic information may be inferred directly from the initial orbit determination process, given measurement uncertainty characteristics. A state uncertainty estimate is derived directly from Gauss’s angles-only initial orbit determination method using a novel technique based on sigma points. This method is tested across a range of simulated orbit types and observation geometries. Comparisons with a standard least-squares-based approach are made, and the direct uncertainty estimates are shown to be more realistic than the least-squares estimates. The state and uncertainty estimates from this method are also shown to be suitable to initialize an unscented Kalman filter for processing further measurements.

An initial orbit determination method from relative position increment measurements

Both Global Navigation Satellite Systems and X-ray pulsar-based navigation system use relative position information to provide accurate spacecraft navigation. To supplement that navigation information, an absolute reference position from other types of measurements should be provided; however, that may increase the complexity or decrease the autonomy of the navigation system. To overcome those drawbacks, this paper proposes an initial orbit determination method using only relative position increment measurements and their corresponding epoch. The proposed method can be summarised as three steps: first, determining the orbital plane from a set of relative position vectors; next, determining the shape of the orbit by solving an ellipse fitting problem in two dimensions; finally, estimating the absolute position and all six orbital elements according to Kepler's equation and geometric relations. Monte Carlo simulations were done to analyse the feasibility and features of the proposed method. The results show that the algorithm we used does not need an initial guess and is robust. It can obtain an acceptable result from a 20% portion of an orbit when the signal-to-noise ratio is set at 15 dB with determining the inclination and right ascension of the ascending node quickly and accurately.

Gaussian Mixture Approximation of Angles-Only Initial Orbit Determination Likelihood Function

A method is developed to approximate the angles-only orbit determination likelihood function using a Gaussian mixture to incorporate information about an admissible region. The resulting probability density function can provide the a priori information for a Gaussian mixture orbit determination filter. This admissible region approach can compensate for the effects of measurement uncertainty over a short track of angles-only data better than deterministic approaches can. The new technique starts with a nonlinear batch least-squares solution. The solution enforces soft constraints on an admissible region defined in terms of minimum periapsis and maximum apoapsis. Although this soft-constrained solution lies in or near the admissible region, it does not characterize that region well. It provides a starting point to develop a Gaussian mixture approximation of the batch least-squares likelihood function as modified through multiplication by a finite-support function that is zero outside the admissible region and 1 inside that region. This Gaussian mixture is optimized to fit the resulting probability density in the two-dimensional subspace of position/velocity space that has the most uncertainty. The procedure starts by approximating the finite-support distribution using a single optimized Gaussian mixand. It automatically increases the number of mixands and reoptimizes until the resulting mixture fits the finite-support distribution to a prespecified precision. This optimal fitting procedure allows the Gaussian mixture to use a low number of mixands while fitting the finite-support probability density function well. By approximating the product of a finite-support function and the original likelihood function, the new method gains the capability to transition smoothly between regimes where the admissibility constraints dominate, i.e., high-altitude/short-measurement-arc cases, and those where they are less important, i.e., low-altitude/long-measurement-arc cases.

Nonlinear Kalman Filtering for Improved Angles-Only Navigation Using Relative Orbital Elements

This paper addresses the design and validation of an accurate estimation architecture for autonomous angles-only navigation in orbits of arbitrary eccentricity. The proposed filtering strategy overcomes the major deficiencies of existing approaches in the literature, which mainly focus on applications in near-circular orbits and generally suffer from poor dynamical observability due to linearizing the filter dynamics and measurement models. Consequently, traditional angles-only navigation solutions require conducting known orbital maneuvers to reconcile the ambiguous range. In contrast, the algorithms developed in this work enable accurate maneuver-free reconstruction of the relative orbital motion. This is done through the full exploitation of nonlinearities in the measurement model using the unscented Kalman filter to improve dynamical observability and filter performance. The filter estimates mean relative orbit elements, adopting a state transition matrix subject to secular and long-period perturbation effects to decouple observable from unobservable parameters. The complete state is then reconciled with the angle measurements in the measurement model through a nonlinear transformation that includes the conversion from mean to osculating orbital elements. The resulting linear dynamics model is supplemented by either first-order Gauss–Markov processes (that is, differential empirical accelerations) or by a covariance-matching approach to online adaptive process noise tuning to increase performance at minimal computational complexity. Finally, the estimation architecture is completed by a novel deterministic algorithm for batch initial relative orbit determination to accurately initialize the sequential filter.