Initial Relative Orbit Determination Research Articles

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.

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.

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.

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.

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.

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

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

Initial Relative Orbit Determination Analytical Covariance and Performance Analysis for Proximity Operations

This research furthers the development of a closed-form solution to the angles-only initial relative orbit determination problem for close-in proximity operations when the camera offset from the vehicle center of mass allows for range observability. Emphasis is placed on developing closed-form error covariance equations for the initial relative orbit state solution and verification of the analytic covariance equations through systematic nonlinear Monte Carlo simulation of typical rendezvous missions. Closed-form analytic estimates of the relative state error covariance based on angle measurement errors, attitude knowledge errors, and camera center-of-mass offset uncertainties for three or more observations are obtained. A two-body Monte Carlo simulation system is used to evaluate the performance of the closed-form relative state estimation algorithms and associated closed-form covariance equations. The sensitivity of the solution accuracy to spacecraft trajectories, camera offset, camera accuracy, attitude knowledge, and the time interval between measurements is presented and discussed.

Bearings-Only Initial Relative Orbit Determination

A method for performing bearings-only initial relative orbit determination of a nearby space object in the absence of any information regarding the space object’s geometry and relative orbit is presented. To resolve the range ambiguity characteristic of a single optical sensor system, a second optical sensor is included at a known baseline distance on the observing spacecraft. To formulate an initial estimate of the space object’s relative orbit and its associated uncertainty, the angle measurements from both sensors are used to bound a region for all possible relative positions of the space object. A parameterized probability distribution in relative position that reflects uniform relative range uncertainty across the bounded region is constructed at two unique times. Linkage of the positional distributions is performed using a second-order relative Lambert solver to formulate a full-state probability density function in relative position and velocity, which can be further refined through processing subsequent measurement data in a Bayesian framework.

Relative navigation to non-cooperative targets in LEO: achievable accuracy from radar tracking measurements

Any future space debris removal or on-orbit servicing mission faces the problem of the initial relative orbit determination of the servicing satellite to the non-cooperative target. In this work, we analyse the relative navigation accuracy that can be achieved in low Earth orbit, by using ground-based orbit determination from radar tracking measurements for the target, and classical GPS-based orbit determination for the servicing satellite. The analysis is based on the radar tracking measurements obtained from a 10 × 10 × 34 cm small object at an altitude of 635 km. The results show that the relative orbit can be determined with accuracy down to 2 m (RMS) in the semi-major axis, and down to 20 m (RMS) in both the radial and normal separations. From the results, we derive requirements on radar-tracking campaigns.