Relative Orbit Determination Research Articles

Relative Orbit Determination Algorithm of Space Targets with Passive Observation

Relative Orbit Determination Algorithm of Space Targets with Passive Observation

Fast angles-only relative navigation using polynomial dynamics

A fast and efficient method for initial relative orbit determination from bearing-angle measurements is introduced. The range ambiguity problem for angles-only relative navigation is addressed by modeling nonlinear effects with a novel second-order mapping from relative orbit elements (ROE) to relative position coordinates. This model is used to form a system of polynomial constraint equations linking the line-of-sight measurements to the ROE. An efficient method for solving this system is developed around the insight that the ROE scale with the ratio of the inter-spacecraft separation to the orbit radius and are therefore small for most applications of interest. The method uses a truncated expansion of the quadratic formula to recursively eliminate unknowns, reduce the dimension of the system, and ultimately acquire an approximate solution. Strategies for improving robustness, efficiency, and accuracy are developed and the method is applied to general second-order systems as well as to a broad range of intial relative orbit determination scenarios. Modifications to the constraint equations and a solution algorithm are introduced to address the challenge of bias in the bearing-angle measurements.

Absolute and relative POD of LEO satellites in formation flying: Undifferenced and uncombined approach

Absolute or relative precise orbit determination (POD) is an essential prerequisite for many low earth orbit (LEO) missions. The POD of LEO satellites typically relays on processing the onboard global navigation satellite system (GNSS) measurements. The absolute POD is usually based on an ionosphere-free (IF) combination, and currently, integer ambiguity resolution (IAR) can be achieved only when external GNSS satellite phase bias (SPB) products are used. The use of these products is not flexible in multi-frequency/multi-constellation scenarios and is difficult to achieve in real-time missions. For relative POD, the double-differenced (DD) with IAR is the most general method. However, the differencing process amplifies observation noise and loses the opportunity to impose dynamic constraints on some eliminated parameters. In this contribution, based on the use of undifferenced and uncombined (UDUC) observations, a new model for both absolute and relative POD is proposed. In this model, the ambiguities of common-view satellites are constructed into DD form, thus IAR can be achieved without any external SPB products. Working with the UDUC observations, multi-frequency scenarios can be easily applied, and residuals can be separated for each frequency. In addition, with precise GNSS satellite clock/orbit products, both the absolute and relative orbits can be derived, which supports absolute and relative LEO POD. Based on onboard GPS observations of T-A and T-B satellites in formation flying, the performance of the UDUC POD model with DD ambiguity was evaluated. With the UDUC algorithm and IAR, the proposed model presented a consistency of 2.8–3.8 cm in 3D with the reference orbits, and the orbit difference was reduced by 16.3% and 10.6% for T-A and T-B compared with the IF-based POD, respectively. In addition, the relative orbit of the two satellites derived from the proposed model showed a consistency of 1.1–1.5 mm, which proved the feasibility of the UDUC POD model with DD ambiguity for formation flying missions.

Deep-neural-network-based angles-only relative orbit determination for space non-cooperative target

Angles-only relative orbit determination for space situational awareness is subjected to a well-known range observability problem, that is, the range between two spacecraft is weakly observable or unobservable. However, in previous studies, the solutions to this problem using nonlinear relative motion dynamics have been limited by the computational complexity and long arc of observation. To this end, this study develops a simple and fast algorithm to address the angles-only relative orbit determination problem by exploiting a deep neural network (DNN), which has a significantly strong capability of capturing nonlinearity. Emphasis is placed on the construction of a nonlinear mapping model from the line-of-sight angles to the relative orbit state by training the designed DNN. First, a training dataset generator including nonlinear relative motion dynamics and a line-of-sight measurement model is established to generate the training data for the DNN. Second, the DNN frame, including the network structure, data processing, and network training algorithm for angles-only relative orbit determination, is designed. Subsequently, a digital simulation system comprising error models, reference missions and trajectories, and computation models for error estimation is established. Thus, the anti-noise and generalization performance of the nonlinear mapping model on GEO-type orbits are verified using digital simulations. The results indicate that the proposed algorithm is effective, whereby the estimation accuracy for the relative position is generally better than that for the relative velocity. In the case of a co-elliptical orbit, the estimated errors of distance and velocity in each direction are less than 9.7% and 48.2%, respectively, whereas the maximum average errors are approximately 1.1% and 4.2%, respectively, where only three sets of angle measurements with an interval of 600 s are available. However, in the case of a non-coplanar orbit, the interval between angles can be decreased to 50 s, when the corresponding estimated error of distance in each direction is less than 9.9%, and the maximum average error is approximately 1.1%. Additionally, the sensitivity of the proposed algorithm to the arc length, number, and interval of the measurements is analyzed.

Range-Based Relative Navigation for a Swarm of Centimeter-Scale Femto-Spacecraft

In-orbit relative navigation between a networked swarm of centimeter-scale femto-spacecraft would add considerable value to a range of space mission concepts and applications, such as for multipoint sensing and distributed sparse aperture interferometry. For a swarm of networked femto-spacecraft, relative position determination would be possible by inferring coarse range estimates from the received signal strength indication associated with the communication link between swarm members. This is particularly advantageous for highly resource-constrained devices. In this paper, algorithms for swarm relative positioning using interspacecraft range estimates are presented that can be applied to centralized, decentralized, and distributed network configurations. Relative navigation filters for initial relative orbit determination and state estimation are presented for femto-spacecraft swarm deployment and dispersal scenarios. The algorithms presented could also find use in terrestrial applications, in static and dynamic wireless sensor networks.

Relative Kinematic Orbit Determination for GRACE-FO Satellite by Jointing GPS and LRI

As the first in-orbit formation satellites equipped with a Laser Ranging Interferometer (LRI) instrument, Gravity Recovery and Climate Experiment Follow-on (GRACE-FO) satellites are designed to evaluate the effective ability of the new LRI ranging system applied to satellite-to-satellite tracking. To evaluate the application of LRI in GRACE-FO, a relative kinematic orbit determination scheme for formation satellites integrating Kalman filters and GPS/LRI is proposed. The observation equation is constructed by combining LRI and spaceborne GPS data, and the intersatellite baselines of GRACE-FO formation satellites are calculated with Kalman filters. The combination of GPS and LRI techniques can limit the influence of GPS observation errors and improve the stability of orbit determination of the GRACE-FO satellites formation. The linearization of the GPS/LRI observation model and the process of the GPS/LRI relative kinematic orbit determination are provided. Relative kinematic orbit determination is verified by actual GPS/LRI data of GRACE-FO-A and GRACE-FO-B satellites. The quality of relative kinematic orbit determination is evaluated by reference orbit check and K-Band Ranging (KBR) check. The result of the reference orbit check indicates that the accuracy of GRACE-FO relative kinematic orbit determination along X, Y, and Z (components of the baseline vector) directions is better than 2.9 cm. Compared with the relative kinematic orbit determination by GPS only, GPS/LRI improves the accuracy of the relative kinematic orbit determination by approximately 1cm along with X, Y and Z directions, and by about 1.8 cm in 3D directions. The overall accuracy of relative kinematic orbit determination is improved by 25.9%. The result of the KBR check indicates that the accuracy of the intersatellite baseline determination is about +/−10.7 mm.

Precise Relative Orbit Determination for Chinese TH-2 Satellite Formation Using Onboard GPS and BDS2 Observations

TH-2 is China’s first short-range satellite formation system used to realize interferometric synthetic aperture radar (InSAR) technology. In order to achieve the mission goal of InSAR processing, the relative orbit must be determined with high accuracy. In this study, the precise relative orbit determination (PROD) for TH-2 based on global positioning system (GPS), second-generation BeiDou navagation satellite system (BDS2), and GPS + BDS2 observations was performed. First, the performance of onboard GPS and BDS2 measurements were assessed by analyzing the available data, code multipath errors and noise levels of carrier phase observations. The differences between the National University of Defense Technology (NDT) and the Xi’an Research Institute of Surveying and Mapping (CHS) baseline solutions exhibited an RMS of 1.48 mm outside maneuver periods. The GPS-based orbit was used as a reference orbit to evaluate the BDS2-based orbit and the GPS + BDS2-based orbit. It is the first time BDS2 has been applied to the PROD of low Earth orbit (LEO) satellite formation. The results showed that the root mean square (RMS) of difference between the PROD results using GPS and BDS2 measurements in 3D components was 2.89 mm in the Asia-Pacific region. We assigned different weights to geostationary Earth orbit (GEO) satellites to illustrate the impact of GEO satellites on PROD, and the accuracy of PROD was improved to 7.08 mm with the GEO weighting strategy. Finally, relative orbits were derived from the combined GPS and BDS2 data. When BDS2 was added on the basis of GPS, the average number of visible navigation satellites from TH-2A and TH-2B improved from 7.5 to 9.5. The RMS of the difference between the GPS + BDS2-based orbit and the GPS-based orbit was about 1.2 mm in 3D. The overlap comparison results showed that the combined orbit consistencies were below 1 mm in the radial (R), along-track (T), and cross-track (N) directions. Furthermore, when BDS2 co-worked with GPS, the average of the ambiguity dilution of precision (ADOP) reduced from 0.160 cycle to 0.153 cycle, which was about a 4.4% reduction. The experimental results indicate that millimeter-level PROD results for TH-2 satellite formation can be obtained by using onboard GPS and BDS2 observations, and multi-GNSS can further improve the accuracy and reliability of PROD.

Precise absolute and relative orbit determination for distributed InSAR satellite system

Precise absolute and relative orbit determination for distributed InSAR satellite system

Fully Decentralized Cooperative Navigation for Spacecraft Constellations

This article proposes a method to decentralize the navigation burden and improve the fault tolerance for a spacecraft constellation. The constellation body reference system is introduced, which is the perifocal frame of one satellite in the constellation. The structure of the proposed navigation method is constructed to enable each spacecraft to estimate its own orbit in this body reference system. This step is essentially the relative orbit determination (OD) based on inter-satellite range measurements. Thereafter, the approach to transfer an orbit from the constellation body reference system to an inertial reference system is developed. The essential requirements on absolute measurements to realize the coordinate transfer are presented. By dividing the absolute OD into relative OD and coordinate transfer, each navigation subsystem operated in a spacecraft can be independent with others, and the absolute measurements collected by any spacecraft can contribute to the absolute OD of the whole constellation. The proposed method applies to constellations in any geometric configuration. A Walker constellation is taken as an example for numerical simulations. The results show that the proposed method has a lower computation burden compared to an integrated navigation system. With the same type of absolute measurements, the proposed method has higher accuracy and convergence velocity than conventional decentralized algorithms. When a spacecraft occurs with fault, the orbit results of other spacecraft are not affected using the proposed method, which is beyond the ability of conventional methods.

Neural-aided GNC reconfiguration algorithm for distributed space system: development and PIL test

This paper presents a neural-aided Guidance, Navigation & Control algorithm for reconfiguration of distributed space systems. The guidance algorithm is based on Artificial Potential Fields (APF) in the Relative Orbital Elements (ROE) space. Since the relative orbit determination measurements are typically referred to the Cartesian metrics (e.g. range or range rate), a linear mapping between the set of ROE and the Cartesian coordinates expressed in the Local-Vertical-Local-Horizontal (LVLH) reference frame is derived. The navigation and control algorithms rely on the relative dynamics expressed in the same ROE set of coordinates. To cope with uncertainties and nonlinearities of the system, a Radial Basis Function Neural Network (RBFNN) is employed to reconstruct the perturbed dynamics. The Artificial Neural Network (ANN) is coupled with an adaptive Extended Kalman Filter for state estimation. A feedback control is designed to track the desired state, whose stability is analyzed using Lyapunov theory. The guidance, navigation and control algorithms are tested in a high-fidelity numerical orbit propagator. Moreover, the algorithm is tested in relevant Processor-In-the-Loop (PIL) simulations using a TI C2000-Delfino MCU F28379D. The results demonstrate the effectiveness of the algorithm for relative reconfiguration maneuvers involving relative distances ~102 m with limited fuel consumption and constrained available thrust (⩽1mN). In particular along-track maneuvers, relative plane change and formation enlargement are analysed in the paper, showing the comparison between the proposed algorithm and the legacy one without the neural network. The benefit of implementing a neural network is particularly highlighted when the nonlinearities or unmodelled terms in the on-board dynamics become prominent.

Rotation based analytic range-only initial relative orbit solution for natural periodic motion

Range-only Initial relative orbit determination (IROD) for natural-periodic relative motion (periodically coasting flight) during close-in proximity suffers from a mirror-solution (also named orbital ambiguity) problem if linear relative dynamics is used to propagate the orbit. Approaches proposed in previous work were to perform specific orbital maneuvers, measure by double range-sensor or utilize prior information so as to avoid mirror solutions. Alternatively, if the range-sensor offset from the vehicle center-of-mass (COM) is included, the range-only IROD problem may be uniquely solved. This research develops a novel solution to the range-only IROD problem for natural periodic motion in the context of Clohessy–Wiltshire dynamics, where the range-sensor offset is included to provide state observability to exclude the mirror solutions and certain attitude maneuver is introduced to construct analytic solution. Therefore, the range-only IROD is reduced to a problem of solving linear equations. The uniquely analytic solution is achieved based on these equations. Three rotation schemes for attitude maneuver are analyzed among which the y-mirror rotation scheme is proven to be effective for de-ambiguity. The proposed algorithm is verified and tested by a set of Monte Carlo simulations. The sensitivity of the solution accuracy to different orbits, range-sensor offset, ranging accuracy, rotating accuracy and measurement span is presented and discussed.

Analytic Initial Relative Orbit Solution for Angles-Only Space Rendezvous Using Hybrid Dynamics Method

A closed-form solution to the angles-only initial relative orbit determination (IROD) problem for space rendezvous with non-cooperated target is developed, where a method of hybrid dynamics with the concept of virtual formation is introduced to analytically solve the problem. Emphasis is placed on developing the solution based on hybrid dynamics (i.e., Clohessy-Wiltshire equations and two-body dynamics), obtaining formation geometries that produce relative orbit state observability, and deriving the approximate analytic error covariance for the IROD solution. A standard Monte Carlo simulation system based on two-body dynamics is used to verify the feasibility and evaluate the performance proposed algorithms. The sensitivity of the solution accuracy to the formation geometry, observation numbers is presented and discussed.

A numerical approach to the problem of angles-only initial relative orbit determination in low earth orbit

A practical and effective numerical method is presented, aiming at solving the problem of initial relative orbit determination using solely line-of-sight measurements. The proposed approach exploits the small discrepancies which can be observed between a linear and a more advanced relative motion model. The method consists in systematically performing a series of least-squares adjustments at varying intersatellite distances in the vicinity of a family of collinear solutions coming from the linear theory. The solution presenting the smallest fitting residuals is then selected. The investigations specifically focus on the rendezvous in low Earth near-circular orbit with a noncooperative target. The objective is to determine the relative state of the formation using only bearing observations when the spacecraft are separated by a few dozen kilometers without any a priori additional information. The method is validated with flight data coming from the ARGON (2012) and AVANTI (2016) experiments. Both cases demonstrate that an observation time span of a few maneuver-free orbits is enough to compute a solution which can compete with Two-Line Elements in terms of accuracy.

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.

Robust Orbit Determination with Flash Lidar Around Small Bodies

This work explores the limits of using flash lidar measurements around an asteroid for relative orbit determination by comparing different filtering methods and testing the robustness of these methods to model errors. Previous work showed that flash lidar can provide accurate orbit determination with an onboard shape model and is less computationally expensive than using the state-of-the-art optical navigation methods. Here, the performance of three filters is compared for a circular terminator orbit and an eccentric orbit, both around the asteroid Itokawa. An iterative least-squares filter is presented that counters the filter saturation seen with other Kalman filtering methods. Estimating an off-nominal pointing bias is added to the estimation state, and the bias and pointing jitter are accurately resolved with the iterative least-squares filter. Using a low-fidelity onboard shape model tests the filter robustness and increases computational efficiency, and the filters do not diverge. The majority of the state errors are captured with a sequential consider covariance analysis. The results of these studies add confidence to pursuing the use of flash lidar measurements for autonomous navigation in proximity to small bodies and in complex dynamical environments.

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.

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.

Relative positioning of formation-flying spacecraft using single-receiver GPS carrier phase ambiguity fixing

In recent years, differential carrier phase-based relative positioning (or “baseline determination”) with precision at the millimeter and submillimeter levels has been demonstrated for the GRACE, TanDEM-X and Swarm missions in offline processing. Specific features of such missions have included the use of spacecraft of similar shapes placed in almost identical orbits as well as the use of consistent geodetic-class GPS receivers. These elements have proven to be advantageous for the computation of baseline solutions with such precision levels. Particularly, they have allowed to fully leverage the use of differential GPS techniques, including the estimation and use of carrier phase integer ambiguities. Similarly, the aforementioned spacecraft and orbit characteristics have made it possible to tightly constrain the relative dynamics of formations in the generation of reduced-dynamic solutions. Other than the former examples, prospective formation-flying mission proposals, such as SAOCOM-CS and PICOSAR, may comprise spacecraft with very different characteristics, including dissimilar GPS/GNSS receivers. Such cases may no longer provide favorable conditions for relative orbit determination strategies. As an alternative, absolute orbit solutions may be computed individually for each spacecraft and used for the generation of precise baseline products. This study aims at the assessment of the potential of single-receiver ambiguity fixing for the generation of precise baseline solutions. Results using flight data from the GRACE, TanDEM-X and Swarm missions exhibit baseline accuracy better than 5 mm (3D RMS) for a one-month test period in June 2016. As such, the presented solutions may be considered for prospective formation-flying remote sensing missions with baseline precision requirements at the subcentimeter level. Likewise, the method is considered of particular interest for future multi-spacecraft formations and swarms that require efficient determination of a large number of individual baselines.

Angles-only relative orbit determination in low earth orbit

The paper provides an overview of the angles-only relative orbit determination activities conducted to support the Autonomous Vision Approach Navigation and Target Identification (AVANTI) experiment. This in-orbit endeavor was carried out by the German Space Operations Center (DLR/GSOC) in autumn 2016 to demonstrate the capability to perform spaceborne autonomous close-proximity operations using solely line-of-sight measurements. The images collected onboard have been reprocessed by an independent on-ground facility for precise relative orbit determination, which served as ultimate instance to monitor the formation safety and to characterize the onboard navigation and control performances. During two months, several rendezvous have been executed, generating a valuable collection of images taken at distances ranging from 50 km to only 50 m. Despite challenging experimental conditions characterized by a poor visibility and strong orbit perturbations, angles-only relative positioning products could be continuously derived throughout the whole experiment timeline, promising accuracy at the meter level during the close approaches. The results presented in the paper are complemented with former angles-only experience gained with the PRISMA satellites to better highlight the specificities induced by different orbits and satellite designs.

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.